LMFDB - Embedded newform 38.8.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.8706309684$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9097x^{2} - 110520x + 10368000 $ x^4 - 9097x^2 - 110520x + 10368000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-48.0684$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +51.0684 q^{3} +64.0000 q^{4} +338.533 q^{5} +408.547 q^{6} -143.677 q^{7} +512.000 q^{8} +420.979 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +51.0684 q^{3} +64.0000 q^{4} +338.533 q^{5} +408.547 q^{6} -143.677 q^{7} +512.000 q^{8} +420.979 q^{9} +2708.27 q^{10} -2238.40 q^{11} +3268.38 q^{12} +3817.19 q^{13} -1149.41 q^{14} +17288.3 q^{15} +4096.00 q^{16} -19910.7 q^{17} +3367.83 q^{18} +6859.00 q^{19} +21666.1 q^{20} -7337.34 q^{21} -17907.2 q^{22} +97702.1 q^{23} +26147.0 q^{24} +36479.8 q^{25} +30537.6 q^{26} -90187.8 q^{27} -9195.31 q^{28} +35917.5 q^{29} +138307. q^{30} -135693. q^{31} +32768.0 q^{32} -114311. q^{33} -159286. q^{34} -48639.4 q^{35} +26942.7 q^{36} -112724. q^{37} +54872.0 q^{38} +194938. q^{39} +173329. q^{40} -334966. q^{41} -58698.7 q^{42} -789155. q^{43} -143257. q^{44} +142516. q^{45} +781617. q^{46} -345282. q^{47} +209176. q^{48} -802900. q^{49} +291839. q^{50} -1.01681e6 q^{51} +244300. q^{52} -282722. q^{53} -721503. q^{54} -757772. q^{55} -73562.5 q^{56} +350278. q^{57} +287340. q^{58} +1.08671e6 q^{59} +1.10645e6 q^{60} +1.60239e6 q^{61} -1.08555e6 q^{62} -60484.9 q^{63} +262144. q^{64} +1.29225e6 q^{65} -914490. q^{66} +825216. q^{67} -1.27429e6 q^{68} +4.98949e6 q^{69} -389115. q^{70} +1.45443e6 q^{71} +215541. q^{72} +6.35893e6 q^{73} -901795. q^{74} +1.86297e6 q^{75} +438976. q^{76} +321605. q^{77} +1.55950e6 q^{78} -3.41609e6 q^{79} +1.38663e6 q^{80} -5.52643e6 q^{81} -2.67973e6 q^{82} +2.76971e6 q^{83} -469590. q^{84} -6.74044e6 q^{85} -6.31324e6 q^{86} +1.83425e6 q^{87} -1.14606e6 q^{88} -1.95057e6 q^{89} +1.14012e6 q^{90} -548442. q^{91} +6.25293e6 q^{92} -6.92963e6 q^{93} -2.76225e6 q^{94} +2.32200e6 q^{95} +1.67341e6 q^{96} -1.49299e7 q^{97} -6.42320e6 q^{98} -942318. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	51.0684	1.09201	0.546006	−	0.837781i	$-0.316147\pi$
$3$	51.0684	1.09201	0.546006	+	0.837781i	$0.316147\pi$
$4$	64.0000	0.500000
$5$	338.533	1.21117	0.605587	−	0.795779i	$-0.292938\pi$
$5$	338.533	1.21117	0.605587	+	0.795779i	$0.292938\pi$
$6$	408.547	0.772170
$7$	−143.677	−0.158323	−0.0791614	−	0.996862i	$-0.525224\pi$
$7$	−143.677	−0.158323	−0.0791614	+	0.996862i	$0.525224\pi$
$8$	512.000	0.353553
$9$	420.979	0.192492
$10$	2708.27	0.856429
$11$	−2238.40	−0.507064	−0.253532	−	0.967327i	$-0.581592\pi$
$11$	−2238.40	−0.507064	−0.253532	+	0.967327i	$0.581592\pi$
$12$	3268.38	0.546006
$13$	3817.19	0.481884	0.240942	−	0.970540i	$-0.422544\pi$
$13$	3817.19	0.481884	0.240942	+	0.970540i	$0.422544\pi$
$14$	−1149.41	−0.111951
$15$	17288.3	1.32262
$16$	4096.00	0.250000
$17$	−19910.7	−0.982915	−0.491457	−	0.870902i	$-0.663536\pi$
$17$	−19910.7	−0.982915	−0.491457	+	0.870902i	$0.663536\pi$
$18$	3367.83	0.136112
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	21666.1	0.605587
$21$	−7337.34	−0.172890
$22$	−17907.2	−0.358548
$23$	97702.1	1.67439	0.837195	−	0.546905i	$-0.184194\pi$
$23$	97702.1	1.67439	0.837195	+	0.546905i	$0.184194\pi$
$24$	26147.0	0.386085
$25$	36479.8	0.466942
$26$	30537.6	0.340744
$27$	−90187.8	−0.881809
$28$	−9195.31	−0.0791614
$29$	35917.5	0.273472	0.136736	−	0.990608i	$-0.456339\pi$
$29$	35917.5	0.273472	0.136736	+	0.990608i	$0.456339\pi$
$30$	138307.	0.935232
$31$	−135693.	−0.818073	−0.409037	−	0.912518i	$-0.634135\pi$
$31$	−135693.	−0.818073	−0.409037	+	0.912518i	$0.634135\pi$
$32$	32768.0	0.176777
$33$	−114311.	−0.553720
$34$	−159286.	−0.695026
$35$	−48639.4	−0.191756
$36$	26942.7	0.0962458
$37$	−112724.	−0.365857	−0.182929	−	0.983126i	$-0.558558\pi$
$37$	−112724.	−0.365857	−0.182929	+	0.983126i	$0.558558\pi$
$38$	54872.0	0.162221
$39$	194938.	0.526224
$40$	173329.	0.428215
$41$	−334966.	−0.759027	−0.379513	−	0.925186i	$-0.623909\pi$
$41$	−334966.	−0.759027	−0.379513	+	0.925186i	$0.623909\pi$
$42$	−58698.7	−0.122252
$43$	−789155.	−1.51364	−0.756820	−	0.653623i	$-0.773248\pi$
$43$	−789155.	−1.51364	−0.756820	+	0.653623i	$0.773248\pi$
$44$	−143257.	−0.253532
$45$	142516.	0.233141
$46$	781617.	1.18397
$47$	−345282.	−0.485100	−0.242550	−	0.970139i	$-0.577984\pi$
$47$	−345282.	−0.485100	−0.242550	+	0.970139i	$0.577984\pi$
$48$	209176.	0.273003
$49$	−802900.	−0.974934
$50$	291839.	0.330178
$51$	−1.01681e6	−1.07336
$52$	244300.	0.240942
$53$	−282722.	−0.260852	−0.130426	−	0.991458i	$-0.541634\pi$
$53$	−282722.	−0.260852	−0.130426	+	0.991458i	$0.541634\pi$
$54$	−721503.	−0.623533
$55$	−757772.	−0.614142
$56$	−73562.5	−0.0559755
$57$	350278.	0.250525
$58$	287340.	0.193374
$59$	1.08671e6	0.688860	0.344430	−	0.938812i	$-0.388072\pi$
$59$	1.08671e6	0.688860	0.344430	+	0.938812i	$0.388072\pi$
$60$	1.10645e6	0.661309
$61$	1.60239e6	0.903886	0.451943	−	0.892047i	$-0.350731\pi$
$61$	1.60239e6	0.903886	0.451943	+	0.892047i	$0.350731\pi$
$62$	−1.08555e6	−0.578465
$63$	−60484.9	−0.0304758
$64$	262144.	0.125000
$65$	1.29225e6	0.583645
$66$	−914490.	−0.391539
$67$	825216.	0.335202	0.167601	−	0.985855i	$-0.446398\pi$
$67$	825216.	0.335202	0.167601	+	0.985855i	$0.446398\pi$
$68$	−1.27429e6	−0.491457
$69$	4.98949e6	1.82845
$70$	−389115.	−0.135592
$71$	1.45443e6	0.482269	0.241134	−	0.970492i	$-0.422481\pi$
$71$	1.45443e6	0.482269	0.241134	+	0.970492i	$0.422481\pi$
$72$	215541.	0.0680561
$73$	6.35893e6	1.91317	0.956587	−	0.291449i	$-0.0941372\pi$
$73$	6.35893e6	1.91317	0.956587	+	0.291449i	$0.0941372\pi$
$74$	−901795.	−0.258700
$75$	1.86297e6	0.509906
$76$	438976.	0.114708
$77$	321605.	0.0802797
$78$	1.55950e6	0.372096
$79$	−3.41609e6	−0.779533	−0.389766	−	0.920914i	$-0.627444\pi$
$79$	−3.41609e6	−0.779533	−0.389766	+	0.920914i	$0.627444\pi$
$80$	1.38663e6	0.302793
$81$	−5.52643e6	−1.15544
$82$	−2.67973e6	−0.536713
$83$	2.76971e6	0.531694	0.265847	−	0.964015i	$-0.414348\pi$
$83$	2.76971e6	0.531694	0.265847	+	0.964015i	$0.414348\pi$
$84$	−469590.	−0.0864452
$85$	−6.74044e6	−1.19048
$86$	−6.31324e6	−1.07031
$87$	1.83425e6	0.298635
$88$	−1.14606e6	−0.179274
$89$	−1.95057e6	−0.293290	−0.146645	−	0.989189i	$-0.546848\pi$
$89$	−1.95057e6	−0.293290	−0.146645	+	0.989189i	$0.546848\pi$
$90$	1.14012e6	0.164855
$91$	−548442.	−0.0762932
$92$	6.25293e6	0.837195
$93$	−6.92963e6	−0.893346
$94$	−2.76225e6	−0.343017
$95$	2.32200e6	0.277862
$96$	1.67341e6	0.193042
$97$	−1.49299e7	−1.66095	−0.830473	−	0.557059i	$-0.811930\pi$
$97$	−1.49299e7	−1.66095	−0.830473	+	0.557059i	$0.811930\pi$
$98$	−6.42320e6	−0.689382
$99$	−942318.	−0.0976055
$100$	2.33471e6	0.233471
$101$	−3.70396e6	−0.357719	−0.178859	−	0.983875i	$-0.557241\pi$
$101$	−3.70396e6	−0.357719	−0.178859	+	0.983875i	$0.557241\pi$
$102$	−8.13447e6	−0.758977
$103$	1.30203e7	1.17406	0.587030	−	0.809565i	$-0.300297\pi$
$103$	1.30203e7	1.17406	0.587030	+	0.809565i	$0.300297\pi$
$104$	1.95440e6	0.170372
$105$	−2.48393e6	−0.209400
$106$	−2.26177e6	−0.184450
$107$	1.20392e7	0.950065	0.475032	−	0.879968i	$-0.342436\pi$
$107$	1.20392e7	0.950065	0.475032	+	0.879968i	$0.342436\pi$
$108$	−5.77202e6	−0.440905
$109$	1.00573e7	0.743852	0.371926	−	0.928262i	$-0.378697\pi$
$109$	1.00573e7	0.743852	0.371926	+	0.928262i	$0.378697\pi$
$110$	−6.06217e6	−0.434264
$111$	−5.75665e6	−0.399521
$112$	−588500.	−0.0395807
$113$	695305.	0.0453316	0.0226658	−	0.999743i	$-0.492785\pi$
$113$	695305.	0.0453316	0.0226658	+	0.999743i	$0.492785\pi$
$114$	2.80222e6	0.177148
$115$	3.30754e7	2.02798
$116$	2.29872e6	0.136736
$117$	1.60696e6	0.0927587
$118$	8.69367e6	0.487098
$119$	2.86071e6	0.155618
$120$	8.85163e6	0.467616
$121$	−1.44768e7	−0.742886
$122$	1.28191e7	0.639144
$123$	−1.71062e7	−0.828867
$124$	−8.68436e6	−0.409037
$125$	−1.40983e7	−0.645626
$126$	−483879.	−0.0215497
$127$	−2.58936e6	−0.112170	−0.0560852	−	0.998426i	$-0.517862\pi$
$127$	−2.58936e6	−0.112170	−0.0560852	+	0.998426i	$0.517862\pi$
$128$	2.09715e6	0.0883883
$129$	−4.03009e7	−1.65291
$130$	1.03380e7	0.412700
$131$	4.28721e7	1.66619	0.833097	−	0.553127i	$-0.186566\pi$
$131$	4.28721e7	1.66619	0.833097	+	0.553127i	$0.186566\pi$
$132$	−7.31592e6	−0.276860
$133$	−985479.	−0.0363217
$134$	6.60173e6	0.237023
$135$	−3.05316e7	−1.06802
$136$	−1.01943e7	−0.347513
$137$	−4.30326e7	−1.42980	−0.714900	−	0.699227i	$-0.753528\pi$
$137$	−4.30326e7	−1.42980	−0.714900	+	0.699227i	$0.753528\pi$
$138$	3.99159e7	1.29291
$139$	5.90693e7	1.86556	0.932781	−	0.360443i	$-0.117375\pi$
$139$	5.90693e7	1.86556	0.932781	+	0.360443i	$0.117375\pi$
$140$	−3.11292e6	−0.0958782
$141$	−1.76330e7	−0.529735
$142$	1.16355e7	0.341016
$143$	−8.54439e6	−0.244346
$144$	1.72433e6	0.0481229
$145$	1.21593e7	0.331222
$146$	5.08715e7	1.35282
$147$	−4.10028e7	−1.06464
$148$	−7.21436e6	−0.182929
$149$	6.40098e7	1.58524	0.792619	−	0.609717i	$-0.208717\pi$
$149$	6.40098e7	1.58524	0.792619	+	0.609717i	$0.208717\pi$
$150$	1.49037e7	0.360558
$151$	7.98100e7	1.88642	0.943210	−	0.332198i	$-0.107790\pi$
$151$	7.98100e7	1.88642	0.943210	+	0.332198i	$0.107790\pi$
$152$	3.51181e6	0.0811107
$153$	−8.38200e6	−0.189203
$154$	2.57284e6	0.0567663
$155$	−4.59367e7	−0.990829
$156$	1.24760e7	0.263112
$157$	5.08701e7	1.04909	0.524547	−	0.851382i	$-0.324235\pi$
$157$	5.08701e7	1.04909	0.524547	+	0.851382i	$0.324235\pi$
$158$	−2.73287e7	−0.551213
$159$	−1.44381e7	−0.284853
$160$	1.10931e7	0.214107
$161$	−1.40375e7	−0.265094
$162$	−4.42114e7	−0.817018
$163$	1.46234e7	0.264479	0.132240	−	0.991218i	$-0.457783\pi$
$163$	1.46234e7	0.264479	0.132240	+	0.991218i	$0.457783\pi$
$164$	−2.14378e7	−0.379513
$165$	−3.86982e7	−0.670651
$166$	2.21577e7	0.375964
$167$	−9.36180e7	−1.55543	−0.777717	−	0.628615i	$-0.783622\pi$
$167$	−9.36180e7	−1.55543	−0.777717	+	0.628615i	$0.783622\pi$
$168$	−3.75672e6	−0.0611260
$169$	−4.81775e7	−0.767788
$170$	−5.39236e7	−0.841797
$171$	2.88750e6	0.0441606
$172$	−5.05059e7	−0.756820
$173$	−1.10488e8	−1.62239	−0.811194	−	0.584778i	$-0.801182\pi$
$173$	−1.10488e8	−1.62239	−0.811194	+	0.584778i	$0.801182\pi$
$174$	1.46740e7	0.211167
$175$	−5.24130e6	−0.0739275
$176$	−9.16847e6	−0.126766
$177$	5.54965e7	0.752244
$178$	−1.56046e7	−0.207387
$179$	−2.53596e7	−0.330489	−0.165245	−	0.986253i	$-0.552841\pi$
$179$	−2.53596e7	−0.330489	−0.165245	+	0.986253i	$0.552841\pi$
$180$	9.12099e6	0.116570
$181$	6.18364e7	0.775121	0.387560	−	0.921844i	$-0.373318\pi$
$181$	6.18364e7	0.775121	0.387560	+	0.921844i	$0.373318\pi$
$182$	−4.38754e6	−0.0539474
$183$	8.18314e7	0.987055
$184$	5.00235e7	0.591986
$185$	−3.81609e7	−0.443117
$186$	−5.54371e7	−0.631691
$187$	4.45681e7	0.498400
$188$	−2.20980e7	−0.242550
$189$	1.29579e7	0.139610
$190$	1.85760e7	0.196478
$191$	−5.55055e7	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$191$	−5.55055e7	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$192$	1.33873e7	0.136502
$193$	8.64074e7	0.865168	0.432584	−	0.901594i	$-0.357602\pi$
$193$	8.64074e7	0.865168	0.432584	+	0.901594i	$0.357602\pi$
$194$	−1.19439e8	−1.17447
$195$	6.59930e7	0.637348
$196$	−5.13856e7	−0.487467
$197$	−5.25048e7	−0.489291	−0.244646	−	0.969613i	$-0.578672\pi$
$197$	−5.25048e7	−0.489291	−0.244646	+	0.969613i	$0.578672\pi$
$198$	−7.53855e6	−0.0690175
$199$	1.16838e8	1.05099	0.525494	−	0.850798i	$-0.323881\pi$
$199$	1.16838e8	1.05099	0.525494	+	0.850798i	$0.323881\pi$
$200$	1.86777e7	0.165089
$201$	4.21425e7	0.366044
$202$	−2.96317e7	−0.252945
$203$	−5.16051e6	−0.0432969
$204$	−6.50757e7	−0.536678
$205$	−1.13397e8	−0.919314
$206$	1.04162e8	0.830186
$207$	4.11306e7	0.322306
$208$	1.56352e7	0.120471
$209$	−1.53532e7	−0.116328
$210$	−1.98715e7	−0.148068
$211$	1.46668e8	1.07484	0.537422	−	0.843313i	$-0.319398\pi$
$211$	1.46668e8	1.07484	0.537422	+	0.843313i	$0.319398\pi$
$212$	−1.80942e7	−0.130426
$213$	7.42755e7	0.526644
$214$	9.63133e7	0.671797
$215$	−2.67155e8	−1.83328
$216$	−4.61762e7	−0.311767
$217$	1.94960e7	0.129520
$218$	8.04580e7	0.525983
$219$	3.24740e8	2.08921
$220$	−4.84974e7	−0.307071
$221$	−7.60031e7	−0.473651
$222$	−4.60532e7	−0.282504
$223$	−4.95094e7	−0.298965	−0.149482	−	0.988764i	$-0.547761\pi$
$223$	−4.95094e7	−0.298965	−0.149482	+	0.988764i	$0.547761\pi$
$224$	−4.70800e6	−0.0279878
$225$	1.53573e7	0.0898824
$226$	5.56244e6	0.0320543
$227$	2.68314e8	1.52249	0.761243	−	0.648466i	$-0.224589\pi$
$227$	2.68314e8	1.52249	0.761243	+	0.648466i	$0.224589\pi$
$228$	2.24178e7	0.125262
$229$	2.31832e8	1.27570	0.637850	−	0.770161i	$-0.279824\pi$
$229$	2.31832e8	1.27570	0.637850	+	0.770161i	$0.279824\pi$
$230$	2.64603e8	1.43400
$231$	1.64239e7	0.0876665
$232$	1.83898e7	0.0966870
$233$	1.79360e7	0.0928922	0.0464461	−	0.998921i	$-0.485210\pi$
$233$	1.79360e7	0.0928922	0.0464461	+	0.998921i	$0.485210\pi$
$234$	1.28557e7	0.0655903
$235$	−1.16889e8	−0.587540
$236$	6.95494e7	0.344430
$237$	−1.74454e8	−0.851259
$238$	2.28857e7	0.110038
$239$	1.04930e8	0.497170	0.248585	−	0.968610i	$-0.420034\pi$
$239$	1.04930e8	0.497170	0.248585	+	0.968610i	$0.420034\pi$
$240$	7.08131e7	0.330654
$241$	−2.84374e8	−1.30867	−0.654334	−	0.756205i	$-0.727051\pi$
$241$	−2.84374e8	−1.30867	−0.654334	+	0.756205i	$0.727051\pi$
$242$	−1.15814e8	−0.525300
$243$	−8.49849e7	−0.379944
$244$	1.02553e8	0.451943
$245$	−2.71808e8	−1.18081
$246$	−1.36849e8	−0.586098
$247$	2.61821e7	0.110552
$248$	−6.94749e7	−0.289233
$249$	1.41445e8	0.580616
$250$	−1.12786e8	−0.456527
$251$	1.29621e8	0.517390	0.258695	−	0.965959i	$-0.416708\pi$
$251$	1.29621e8	0.517390	0.258695	+	0.965959i	$0.416708\pi$
$252$	−3.87104e6	−0.0152379
$253$	−2.18696e8	−0.849022
$254$	−2.07148e7	−0.0793165
$255$	−3.44224e8	−1.30002
$256$	1.67772e7	0.0625000
$257$	4.68302e8	1.72092	0.860458	−	0.509521i	$-0.170177\pi$
$257$	4.68302e8	1.72092	0.860458	+	0.509521i	$0.170177\pi$
$258$	−3.22407e8	−1.16879
$259$	1.61959e7	0.0579235
$260$	8.27039e7	0.291823
$261$	1.51205e7	0.0526411
$262$	3.42977e8	1.17818
$263$	−4.36231e8	−1.47867	−0.739334	−	0.673338i	$-0.764860\pi$
$263$	−4.36231e8	−1.47867	−0.739334	+	0.673338i	$0.764860\pi$
$264$	−5.85274e7	−0.195770
$265$	−9.57107e7	−0.315937
$266$	−7.88383e6	−0.0256833
$267$	−9.96126e7	−0.320276
$268$	5.28139e7	0.167601
$269$	−1.57156e8	−0.492263	−0.246132	−	0.969236i	$-0.579160\pi$
$269$	−1.57156e8	−0.492263	−0.246132	+	0.969236i	$0.579160\pi$
$270$	−2.44253e8	−0.755207
$271$	3.04557e8	0.929558	0.464779	−	0.885427i	$-0.346134\pi$
$271$	3.04557e8	0.929558	0.464779	+	0.885427i	$0.346134\pi$
$272$	−8.15543e7	−0.245729
$273$	−2.80080e7	−0.0833132
$274$	−3.44261e8	−1.01102
$275$	−8.16563e7	−0.236769
$276$	3.19327e8	0.914227
$277$	−2.88662e8	−0.816038	−0.408019	−	0.912973i	$-0.633780\pi$
$277$	−2.88662e8	−0.816038	−0.408019	+	0.912973i	$0.633780\pi$
$278$	4.72554e8	1.31915
$279$	−5.71240e7	−0.157472
$280$	−2.49034e7	−0.0677961
$281$	−4.74373e7	−0.127540	−0.0637702	−	0.997965i	$-0.520312\pi$
$281$	−4.74373e7	−0.127540	−0.0637702	+	0.997965i	$0.520312\pi$
$282$	−1.41064e8	−0.374579
$283$	−5.34448e8	−1.40169	−0.700847	−	0.713311i	$-0.747195\pi$
$283$	−5.34448e8	−1.40169	−0.700847	+	0.713311i	$0.747195\pi$
$284$	9.30836e7	0.241134
$285$	1.18581e8	0.303429
$286$	−6.83551e7	−0.172779
$287$	4.81268e7	0.120171
$288$	1.37946e7	0.0340280
$289$	−1.39017e7	−0.0338786
$290$	9.72742e7	0.234209
$291$	−7.62445e8	−1.81377
$292$	4.06972e8	0.956587
$293$	−5.89495e8	−1.36913	−0.684564	−	0.728953i	$-0.740007\pi$
$293$	−5.89495e8	−1.36913	−0.684564	+	0.728953i	$0.740007\pi$
$294$	−3.28022e8	−0.752814
$295$	3.67887e8	0.834330
$296$	−5.77149e7	−0.129350
$297$	2.01876e8	0.447133
$298$	5.12078e8	1.12093
$299$	3.72948e8	0.806862
$300$	1.19230e8	0.254953
$301$	1.13383e8	0.239644
$302$	6.38480e8	1.33390
$303$	−1.89155e8	−0.390633
$304$	2.80945e7	0.0573539
$305$	5.42462e8	1.09476
$306$	−6.70560e7	−0.133787
$307$	8.87031e8	1.74966	0.874832	−	0.484427i	$-0.160972\pi$
$307$	8.87031e8	1.74966	0.874832	+	0.484427i	$0.160972\pi$
$308$	2.05827e7	0.0401399
$309$	6.64925e8	1.28209
$310$	−3.67493e8	−0.700622
$311$	−1.05587e9	−1.99044	−0.995222	−	0.0976359i	$-0.968872\pi$
$311$	−1.05587e9	−1.99044	−0.995222	+	0.0976359i	$0.968872\pi$
$312$	9.98082e7	0.186048
$313$	5.41700e8	0.998513	0.499257	−	0.866454i	$-0.333606\pi$
$313$	5.41700e8	0.998513	0.499257	+	0.866454i	$0.333606\pi$
$314$	4.06961e8	0.741821
$315$	−2.04762e7	−0.0369115
$316$	−2.18630e8	−0.389766
$317$	2.77536e8	0.489341	0.244670	−	0.969606i	$-0.421320\pi$
$317$	2.77536e8	0.489341	0.244670	+	0.969606i	$0.421320\pi$
$318$	−1.15505e8	−0.201422
$319$	−8.03976e7	−0.138668
$320$	8.87445e7	0.151397
$321$	6.14821e8	1.03748
$322$	−1.12300e8	−0.187450
$323$	−1.36568e8	−0.225496
$324$	−3.53691e8	−0.577719
$325$	1.39251e8	0.225012
$326$	1.16987e8	0.187015
$327$	5.13608e8	0.812296
$328$	−1.71503e8	−0.268357
$329$	4.96090e7	0.0768023
$330$	−3.09585e8	−0.474222
$331$	−6.09280e8	−0.923461	−0.461731	−	0.887020i	$-0.652771\pi$
$331$	−6.09280e8	−0.923461	−0.461731	+	0.887020i	$0.652771\pi$
$332$	1.77262e8	0.265847
$333$	−4.74546e7	−0.0704245
$334$	−7.48944e8	−1.09986
$335$	2.79363e8	0.405987
$336$	−3.00537e7	−0.0432226
$337$	−5.98158e7	−0.0851356	−0.0425678	−	0.999094i	$-0.513554\pi$
$337$	−5.98158e7	−0.0851356	−0.0425678	+	0.999094i	$0.513554\pi$
$338$	−3.85420e8	−0.542908
$339$	3.55081e7	0.0495027
$340$	−4.31388e8	−0.595240
$341$	3.03735e8	0.414815
$342$	2.31000e7	0.0312263
$343$	2.33682e8	0.312677
$344$	−4.04047e8	−0.535153
$345$	1.68911e9	2.21458
$346$	−8.83906e8	−1.14720
$347$	4.10517e8	0.527445	0.263723	−	0.964599i	$-0.415050\pi$
$347$	4.10517e8	0.527445	0.263723	+	0.964599i	$0.415050\pi$
$348$	1.17392e8	0.149317
$349$	−1.24340e9	−1.56575	−0.782877	−	0.622177i	$-0.786249\pi$
$349$	−1.24340e9	−1.56575	−0.782877	+	0.622177i	$0.786249\pi$
$350$	−4.19304e7	−0.0522746
$351$	−3.44264e8	−0.424930
$352$	−7.33477e7	−0.0896370
$353$	−1.15769e9	−1.40081	−0.700407	−	0.713743i	$-0.746998\pi$
$353$	−1.15769e9	−1.40081	−0.700407	+	0.713743i	$0.746998\pi$
$354$	4.43972e8	0.531917
$355$	4.92374e8	0.584111
$356$	−1.24837e8	−0.146645
$357$	1.46092e8	0.169937
$358$	−2.02877e8	−0.233691
$359$	−1.88893e8	−0.215470	−0.107735	−	0.994180i	$-0.534360\pi$
$359$	−1.88893e8	−0.215470	−0.107735	+	0.994180i	$0.534360\pi$
$360$	7.29680e7	0.0824277
$361$	4.70459e7	0.0526316
$362$	4.94691e8	0.548093
$363$	−7.39304e8	−0.811241
$364$	−3.51003e7	−0.0381466
$365$	2.15271e9	2.31718
$366$	6.54651e8	0.697953
$367$	−2.05858e8	−0.217389	−0.108694	−	0.994075i	$-0.534667\pi$
$367$	−2.05858e8	−0.217389	−0.108694	+	0.994075i	$0.534667\pi$
$368$	4.00188e8	0.418597
$369$	−1.41014e8	−0.146106
$370$	−3.05288e8	−0.313331
$371$	4.06205e7	0.0412988
$372$	−4.43496e8	−0.446673
$373$	−3.70971e8	−0.370134	−0.185067	−	0.982726i	$-0.559250\pi$
$373$	−3.70971e8	−0.370134	−0.185067	+	0.982726i	$0.559250\pi$
$374$	3.56545e8	0.352422
$375$	−7.19976e8	−0.705032
$376$	−1.76784e8	−0.171509
$377$	1.37104e8	0.131782
$378$	1.03663e8	0.0987195
$379$	−1.75116e8	−0.165230	−0.0826150	−	0.996582i	$-0.526327\pi$
$379$	−1.75116e8	−0.165230	−0.0826150	+	0.996582i	$0.526327\pi$
$380$	1.48608e8	0.138931
$381$	−1.32234e8	−0.122492
$382$	−4.44044e8	−0.407572
$383$	6.36056e8	0.578495	0.289248	−	0.957254i	$-0.406595\pi$
$383$	6.36056e8	0.578495	0.289248	+	0.957254i	$0.406595\pi$
$384$	1.07098e8	0.0965212
$385$	1.08874e8	0.0972327
$386$	6.91259e8	0.611766
$387$	−3.32218e8	−0.291363
$388$	−9.55512e8	−0.830473
$389$	−5.13237e8	−0.442074	−0.221037	−	0.975265i	$-0.570944\pi$
$389$	−5.13237e8	−0.442074	−0.221037	+	0.975265i	$0.570944\pi$
$390$	5.27944e8	0.450673
$391$	−1.94532e9	−1.64578
$392$	−4.11085e8	−0.344691
$393$	2.18941e9	1.81950
$394$	−4.20039e8	−0.345981
$395$	−1.15646e9	−0.944149
$396$	−6.03084e7	−0.0488028
$397$	9.24179e8	0.741292	0.370646	−	0.928774i	$-0.379136\pi$
$397$	9.24179e8	0.741292	0.370646	+	0.928774i	$0.379136\pi$
$398$	9.34702e8	0.743160
$399$	−5.03268e7	−0.0396638
$400$	1.49421e8	0.116735
$401$	−2.48504e9	−1.92454	−0.962272	−	0.272091i	$-0.912285\pi$
$401$	−2.48504e9	−1.92454	−0.962272	+	0.272091i	$0.912285\pi$
$402$	3.37140e8	0.258832
$403$	−5.17967e8	−0.394216
$404$	−2.37053e8	−0.178859
$405$	−1.87088e9	−1.39944
$406$	−4.12841e7	−0.0306155
$407$	2.52322e8	0.185513
$408$	−5.20606e8	−0.379488
$409$	−2.08902e8	−0.150977	−0.0754887	−	0.997147i	$-0.524052\pi$
$409$	−2.08902e8	−0.150977	−0.0754887	+	0.997147i	$0.524052\pi$
$410$	−9.07177e8	−0.650053
$411$	−2.19760e9	−1.56136
$412$	8.33299e8	0.587030
$413$	−1.56135e8	−0.109062
$414$	3.29045e8	0.227905
$415$	9.37640e8	0.643973
$416$	1.25082e8	0.0851859
$417$	3.01657e9	2.03722
$418$	−1.22825e8	−0.0822566
$419$	3.34065e7	0.0221861	0.0110931	−	0.999938i	$-0.496469\pi$
$419$	3.34065e7	0.0221861	0.0110931	+	0.999938i	$0.496469\pi$
$420$	−1.58972e8	−0.104700
$421$	7.66238e8	0.500468	0.250234	−	0.968185i	$-0.419492\pi$
$421$	7.66238e8	0.500468	0.250234	+	0.968185i	$0.419492\pi$
$422$	1.17334e9	0.760030
$423$	−1.45356e8	−0.0933777
$424$	−1.44754e8	−0.0922250
$425$	−7.26340e8	−0.458964
$426$	5.94204e8	0.372393
$427$	−2.30226e8	−0.143106
$428$	7.70507e8	0.475032
$429$	−4.36348e8	−0.266829
$430$	−2.13724e9	−1.29633
$431$	2.21182e9	1.33070	0.665350	−	0.746532i	$-0.268282\pi$
$431$	2.21182e9	1.33070	0.665350	+	0.746532i	$0.268282\pi$
$432$	−3.69409e8	−0.220452
$433$	−2.56377e9	−1.51765	−0.758825	−	0.651295i	$-0.774226\pi$
$433$	−2.56377e9	−1.51765	−0.758825	+	0.651295i	$0.774226\pi$
$434$	1.55968e8	0.0915842
$435$	6.20954e8	0.361699
$436$	6.43664e8	0.371926
$437$	6.70139e8	0.384131
$438$	2.59792e9	1.47729
$439$	−3.39319e8	−0.191418	−0.0957088	−	0.995409i	$-0.530512\pi$
$439$	−3.39319e8	−0.191418	−0.0957088	+	0.995409i	$0.530512\pi$
$440$	−3.87979e8	−0.217132
$441$	−3.38004e8	−0.187667
$442$	−6.08025e8	−0.334922
$443$	−1.65782e9	−0.905993	−0.452996	−	0.891512i	$-0.649645\pi$
$443$	−1.65782e9	−0.905993	−0.452996	+	0.891512i	$0.649645\pi$
$444$	−3.68426e8	−0.199760
$445$	−6.60334e8	−0.355225
$446$	−3.96075e8	−0.211400
$447$	3.26888e9	1.73110
$448$	−3.76640e7	−0.0197903
$449$	3.56920e9	1.86084	0.930419	−	0.366498i	$-0.119443\pi$
$449$	3.56920e9	1.86084	0.930419	+	0.366498i	$0.119443\pi$
$450$	1.22858e8	0.0635565
$451$	7.49786e8	0.384875
$452$	4.44995e7	0.0226658
$453$	4.07577e9	2.05999
$454$	2.14652e9	1.07656
$455$	−1.85666e8	−0.0924043
$456$	1.79342e8	0.0885739
$457$	−8.20267e8	−0.402021	−0.201010	−	0.979589i	$-0.564422\pi$
$457$	−8.20267e8	−0.402021	−0.201010	+	0.979589i	$0.564422\pi$
$458$	1.85465e9	0.902056
$459$	1.79570e9	0.866743
$460$	2.11683e9	1.01399
$461$	−3.74605e9	−1.78082	−0.890410	−	0.455159i	$-0.849582\pi$
$461$	−3.74605e9	−1.78082	−0.890410	+	0.455159i	$0.849582\pi$
$462$	1.31391e8	0.0619895
$463$	1.62426e9	0.760540	0.380270	−	0.924875i	$-0.375831\pi$
$463$	1.62426e9	0.760540	0.380270	+	0.924875i	$0.375831\pi$
$464$	1.47118e8	0.0683680
$465$	−2.34591e9	−1.08200
$466$	1.43488e8	0.0656847
$467$	−2.70276e9	−1.22800	−0.613999	−	0.789307i	$-0.710440\pi$
$467$	−2.70276e9	−1.22800	−0.613999	+	0.789307i	$0.710440\pi$
$468$	1.02845e8	0.0463793
$469$	−1.18564e8	−0.0530700
$470$	−9.35115e8	−0.415454
$471$	2.59785e9	1.14562
$472$	5.56395e8	0.243549
$473$	1.76644e9	0.767512
$474$	−1.39563e9	−0.601931
$475$	2.50215e8	0.107124
$476$	1.83085e8	0.0778089
$477$	−1.19020e8	−0.0502118
$478$	8.39436e8	0.351552
$479$	−1.16089e9	−0.482631	−0.241316	−	0.970447i	$-0.577579\pi$
$479$	−1.16089e9	−0.482631	−0.241316	+	0.970447i	$0.577579\pi$
$480$	5.66505e8	0.233808
$481$	−4.30291e8	−0.176301
$482$	−2.27499e9	−0.925368
$483$	−7.16873e8	−0.289486
$484$	−9.26512e8	−0.371443
$485$	−5.05426e9	−2.01169
$486$	−6.79879e8	−0.268661
$487$	2.66603e9	1.04596	0.522978	−	0.852346i	$-0.324821\pi$
$487$	2.66603e9	1.04596	0.522978	+	0.852346i	$0.324821\pi$
$488$	8.20423e8	0.319572
$489$	7.46793e8	0.288815
$490$	−2.17447e9	−0.834962
$491$	−7.10777e8	−0.270987	−0.135493	−	0.990778i	$-0.543262\pi$
$491$	−7.10777e8	−0.270987	−0.135493	+	0.990778i	$0.543262\pi$
$492$	−1.09479e9	−0.414434
$493$	−7.15143e8	−0.268800
$494$	2.09457e8	0.0781719
$495$	−3.19006e8	−0.118217
$496$	−5.55799e8	−0.204518
$497$	−2.08968e8	−0.0763541
$498$	1.13156e9	0.410558
$499$	1.23876e9	0.446309	0.223154	−	0.974783i	$-0.428365\pi$
$499$	1.23876e9	0.446309	0.223154	+	0.974783i	$0.428365\pi$
$500$	−9.02290e8	−0.322813
$501$	−4.78092e9	−1.69855
$502$	1.03697e9	0.365850
$503$	6.01551e8	0.210758	0.105379	−	0.994432i	$-0.466394\pi$
$503$	6.01551e8	0.210758	0.105379	+	0.994432i	$0.466394\pi$
$504$	−3.09683e7	−0.0107748
$505$	−1.25391e9	−0.433259
$506$	−1.74957e9	−0.600349
$507$	−2.46035e9	−0.838434
$508$	−1.65719e8	−0.0560852
$509$	−7.54337e8	−0.253544	−0.126772	−	0.991932i	$-0.540462\pi$
$509$	−7.54337e8	−0.253544	−0.126772	+	0.991932i	$0.540462\pi$
$510$	−2.75379e9	−0.919253
$511$	−9.13631e8	−0.302899
$512$	1.34218e8	0.0441942
$513$	−6.18598e8	−0.202301
$514$	3.74641e9	1.21687
$515$	4.40780e9	1.42199
$516$	−2.57926e9	−0.826457
$517$	7.72877e8	0.245976
$518$	1.29567e8	0.0409581
$519$	−5.64245e9	−1.77167
$520$	6.61631e8	0.206350
$521$	−4.75312e9	−1.47247	−0.736236	−	0.676725i	$-0.763399\pi$
$521$	−4.75312e9	−1.47247	−0.736236	+	0.676725i	$0.763399\pi$
$522$	1.20964e8	0.0372229
$523$	2.09748e9	0.641125	0.320563	−	0.947227i	$-0.396128\pi$
$523$	2.09748e9	0.641125	0.320563	+	0.947227i	$0.396128\pi$
$524$	2.74382e9	0.833097
$525$	−2.67665e8	−0.0807298
$526$	−3.48984e9	−1.04558
$527$	2.70175e9	0.804096
$528$	−4.68219e8	−0.138430
$529$	6.14088e9	1.80358
$530$	−7.65686e8	−0.223401
$531$	4.57482e8	0.132600
$532$	−6.30706e7	−0.0181609
$533$	−1.27863e9	−0.365763
$534$	−7.96901e8	−0.226470
$535$	4.07566e9	1.15069
$536$	4.22511e8	0.118512
$537$	−1.29508e9	−0.360898
$538$	−1.25725e9	−0.348083
$539$	1.79721e9	0.494354
$540$	−1.95402e9	−0.534012
$541$	−5.19805e9	−1.41140	−0.705700	−	0.708510i	$-0.749368\pi$
$541$	−5.19805e9	−1.41140	−0.705700	+	0.708510i	$0.749368\pi$
$542$	2.43646e9	0.657297
$543$	3.15789e9	0.846442
$544$	−6.52435e8	−0.173756
$545$	3.40472e9	0.900934
$546$	−2.24064e8	−0.0589113
$547$	2.13091e9	0.556684	0.278342	−	0.960482i	$-0.410215\pi$
$547$	2.13091e9	0.556684	0.278342	+	0.960482i	$0.410215\pi$
$548$	−2.75408e9	−0.714900
$549$	6.74573e8	0.173990
$550$	−6.53250e8	−0.167421
$551$	2.46358e8	0.0627388
$552$	2.55462e9	0.646456
$553$	4.90812e8	0.123418
$554$	−2.30930e9	−0.577026
$555$	−1.94882e9	−0.483889
$556$	3.78043e9	0.932781
$557$	1.67168e8	0.0409884	0.0204942	−	0.999790i	$-0.493476\pi$
$557$	1.67168e8	0.0409884	0.0204942	+	0.999790i	$0.493476\pi$
$558$	−4.56992e8	−0.111350
$559$	−3.01236e9	−0.729399
$560$	−1.99227e8	−0.0479391
$561$	2.27602e9	0.544259
$562$	−3.79498e8	−0.0901847
$563$	7.02257e9	1.65850	0.829252	−	0.558875i	$-0.188767\pi$
$563$	7.02257e9	1.65850	0.829252	+	0.558875i	$0.188767\pi$
$564$	−1.12851e9	−0.264868
$565$	2.35384e8	0.0549044
$566$	−4.27559e9	−0.991148
$567$	7.94019e8	0.182932
$568$	7.44669e8	0.170508
$569$	2.82902e9	0.643789	0.321894	−	0.946776i	$-0.395680\pi$
$569$	2.82902e9	0.643789	0.321894	+	0.946776i	$0.395680\pi$
$570$	9.48646e8	0.214557
$571$	−5.85928e9	−1.31710	−0.658549	−	0.752538i	$-0.728829\pi$
$571$	−5.85928e9	−1.31710	−0.658549	+	0.752538i	$0.728829\pi$
$572$	−5.46841e8	−0.122173
$573$	−2.83458e9	−0.629429
$574$	3.85015e8	0.0849739
$575$	3.56416e9	0.781843
$576$	1.10357e8	0.0240615
$577$	5.86369e9	1.27074	0.635368	−	0.772210i	$-0.280848\pi$
$577$	5.86369e9	1.27074	0.635368	+	0.772210i	$0.280848\pi$
$578$	−1.11214e8	−0.0239558
$579$	4.41268e9	0.944774
$580$	7.78193e8	0.165611
$581$	−3.97943e8	−0.0841792
$582$	−6.09956e9	−1.28253
$583$	6.32843e8	0.132268
$584$	3.25577e9	0.676409
$585$	5.44010e8	0.112347
$586$	−4.71596e9	−0.968119
$587$	−2.61552e9	−0.533734	−0.266867	−	0.963733i	$-0.585988\pi$
$587$	−2.61552e9	−0.533734	−0.266867	+	0.963733i	$0.585988\pi$
$588$	−2.62418e9	−0.532320
$589$	−9.30720e8	−0.187679
$590$	2.94310e9	0.589960
$591$	−2.68134e9	−0.534312
$592$	−4.61719e8	−0.0914643
$593$	−2.82844e9	−0.557001	−0.278500	−	0.960436i	$-0.589837\pi$
$593$	−2.82844e9	−0.557001	−0.278500	+	0.960436i	$0.589837\pi$
$594$	1.61501e9	0.316171
$595$	9.68445e8	0.188480
$596$	4.09663e9	0.792619
$597$	5.96671e9	1.14769
$598$	2.98358e9	0.570537
$599$	1.27951e9	0.243248	0.121624	−	0.992576i	$-0.461190\pi$
$599$	1.27951e9	0.243248	0.121624	+	0.992576i	$0.461190\pi$
$600$	9.53839e8	0.180279
$601$	7.46347e9	1.40243	0.701214	−	0.712951i	$-0.252642\pi$
$601$	7.46347e9	1.40243	0.701214	+	0.712951i	$0.252642\pi$
$602$	9.07066e8	0.169454
$603$	3.47399e8	0.0645235
$604$	5.10784e9	0.943210
$605$	−4.90086e9	−0.899765
$606$	−1.51324e9	−0.276219
$607$	−3.57702e9	−0.649175	−0.324587	−	0.945856i	$-0.605225\pi$
$607$	−3.57702e9	−0.649175	−0.324587	+	0.945856i	$0.605225\pi$
$608$	2.24756e8	0.0405554
$609$	−2.63539e8	−0.0472807
$610$	4.33970e9	0.774114
$611$	−1.31801e9	−0.233762
$612$	−5.36448e8	−0.0946015
$613$	7.37202e9	1.29263	0.646316	−	0.763070i	$-0.276309\pi$
$613$	7.37202e9	1.29263	0.646316	+	0.763070i	$0.276309\pi$
$614$	7.09625e9	1.23720
$615$	−5.79101e9	−1.00390
$616$	1.64662e8	0.0283832
$617$	6.75777e9	1.15826	0.579129	−	0.815236i	$-0.303393\pi$
$617$	6.75777e9	1.15826	0.579129	+	0.815236i	$0.303393\pi$
$618$	5.31940e9	0.906574
$619$	3.00161e7	0.00508670	0.00254335	−	0.999997i	$-0.499190\pi$
$619$	3.00161e7	0.00508670	0.00254335	+	0.999997i	$0.499190\pi$
$620$	−2.93995e9	−0.495414
$621$	−8.81154e9	−1.47649
$622$	−8.44698e9	−1.40746
$623$	2.80252e8	0.0464345
$624$	7.98466e8	0.131556
$625$	−7.62272e9	−1.24891
$626$	4.33360e9	0.706056
$627$	−7.84061e8	−0.127032
$628$	3.25569e9	0.524547
$629$	2.24442e9	0.359606
$630$	−1.63809e8	−0.0261004
$631$	−9.20882e9	−1.45915	−0.729577	−	0.683899i	$-0.760283\pi$
$631$	−9.20882e9	−1.45915	−0.729577	+	0.683899i	$0.760283\pi$
$632$	−1.74904e9	−0.275606
$633$	7.49008e9	1.17374
$634$	2.22028e9	0.346016
$635$	−8.76583e8	−0.135858
$636$	−9.24041e8	−0.142427
$637$	−3.06483e9	−0.469805
$638$	−6.43180e8	−0.0980529
$639$	6.12286e8	0.0928327
$640$	7.09956e8	0.107054
$641$	−5.16699e9	−0.774880	−0.387440	−	0.921895i	$-0.626641\pi$
$641$	−5.16699e9	−0.774880	−0.387440	+	0.921895i	$0.626641\pi$
$642$	4.91857e9	0.733611
$643$	−7.56140e9	−1.12167	−0.560833	−	0.827929i	$-0.689519\pi$
$643$	−7.56140e9	−1.12167	−0.560833	+	0.827929i	$0.689519\pi$
$644$	−8.98401e8	−0.132547
$645$	−1.36432e10	−2.00197
$646$	−1.09254e9	−0.159450
$647$	−9.89841e9	−1.43681	−0.718407	−	0.695623i	$-0.755128\pi$
$647$	−9.89841e9	−1.43681	−0.718407	+	0.695623i	$0.755128\pi$
$648$	−2.82953e9	−0.408509
$649$	−2.43248e9	−0.349296
$650$	1.11400e9	0.159107
$651$	9.95627e8	0.141437
$652$	9.35897e8	0.132240
$653$	−1.11917e10	−1.57289	−0.786446	−	0.617659i	$-0.788081\pi$
$653$	−1.11917e10	−1.57289	−0.786446	+	0.617659i	$0.788081\pi$
$654$	4.10886e9	0.574380
$655$	1.45136e10	2.01805
$656$	−1.37202e9	−0.189757
$657$	2.67698e9	0.368270
$658$	3.96872e8	0.0543075
$659$	−5.84630e9	−0.795760	−0.397880	−	0.917438i	$-0.630254\pi$
$659$	−5.84630e9	−0.795760	−0.397880	+	0.917438i	$0.630254\pi$
$660$	−2.47668e9	−0.335326
$661$	−9.70737e8	−0.130736	−0.0653682	−	0.997861i	$-0.520822\pi$
$661$	−9.70737e8	−0.130736	−0.0653682	+	0.997861i	$0.520822\pi$
$662$	−4.87424e9	−0.652986
$663$	−3.88136e9	−0.517233
$664$	1.41809e9	0.187982
$665$	−3.33617e8	−0.0439919
$666$	−3.79637e8	−0.0497976
$667$	3.50921e9	0.457899
$668$	−5.99155e9	−0.777717
$669$	−2.52836e9	−0.326474
$670$	2.23491e9	0.287076
$671$	−3.58678e9	−0.458328
$672$	−2.40430e8	−0.0305630
$673$	1.20547e10	1.52442	0.762208	−	0.647332i	$-0.224115\pi$
$673$	1.20547e10	1.52442	0.762208	+	0.647332i	$0.224115\pi$
$674$	−4.78526e8	−0.0602000
$675$	−3.29004e9	−0.411754
$676$	−3.08336e9	−0.383894
$677$	−7.29226e9	−0.903237	−0.451619	−	0.892211i	$-0.649153\pi$
$677$	−7.29226e9	−0.903237	−0.451619	+	0.892211i	$0.649153\pi$
$678$	2.84065e8	0.0350037
$679$	2.14508e9	0.262965
$680$	−3.45111e9	−0.420898
$681$	1.37024e10	1.66257
$682$	2.42988e9	0.293319
$683$	1.41645e10	1.70110	0.850548	−	0.525897i	$-0.176270\pi$
$683$	1.41645e10	1.70110	0.850548	+	0.525897i	$0.176270\pi$
$684$	1.84800e8	0.0220803
$685$	−1.45680e10	−1.73174
$686$	1.86946e9	0.221096
$687$	1.18393e10	1.39308
$688$	−3.23238e9	−0.378410
$689$	−1.07920e9	−0.125700
$690$	1.35129e10	1.56594
$691$	6.15519e9	0.709689	0.354845	−	0.934925i	$-0.384534\pi$
$691$	6.15519e9	0.709689	0.354845	+	0.934925i	$0.384534\pi$
$692$	−7.07124e9	−0.811194
$693$	1.35389e8	0.0154532
$694$	3.28413e9	0.372960
$695$	1.99969e10	2.25952
$696$	9.39135e8	0.105583
$697$	6.66942e9	0.746059
$698$	−9.94724e9	−1.10716
$699$	9.15961e8	0.101439
$700$	−3.35443e8	−0.0369638
$701$	−3.60428e9	−0.395189	−0.197595	−	0.980284i	$-0.563313\pi$
$701$	−3.60428e9	−0.395189	−0.197595	+	0.980284i	$0.563313\pi$
$702$	−2.75412e9	−0.300471
$703$	−7.73176e8	−0.0839334
$704$	−5.86782e8	−0.0633830
$705$	−5.96935e9	−0.641601
$706$	−9.26152e9	−0.990526
$707$	5.32173e8	0.0566350
$708$	3.55177e9	0.376122
$709$	−7.50645e8	−0.0790993	−0.0395497	−	0.999218i	$-0.512592\pi$
$709$	−7.50645e8	−0.0790993	−0.0395497	+	0.999218i	$0.512592\pi$
$710$	3.93899e9	0.413029
$711$	−1.43810e9	−0.150054
$712$	−9.98694e8	−0.103694
$713$	−1.32575e10	−1.36977
$714$	1.16873e9	0.120163
$715$	−2.89256e9	−0.295945
$716$	−1.62302e9	−0.165245
$717$	5.35858e9	0.542916
$718$	−1.51115e9	−0.152360
$719$	−1.21392e10	−1.21798	−0.608989	−	0.793179i	$-0.708425\pi$
$719$	−1.21392e10	−1.21798	−0.608989	+	0.793179i	$0.708425\pi$
$720$	5.83744e8	0.0582852
$721$	−1.87071e9	−0.185881
$722$	3.76367e8	0.0372161
$723$	−1.45225e10	−1.42908
$724$	3.95753e9	0.387560
$725$	1.31026e9	0.127696
$726$	−5.91444e9	−0.573634
$727$	−6.81778e9	−0.658071	−0.329035	−	0.944318i	$-0.606724\pi$
$727$	−6.81778e9	−0.658071	−0.329035	+	0.944318i	$0.606724\pi$
$728$	−2.80802e8	−0.0269737
$729$	7.74625e9	0.740535
$730$	1.72217e10	1.63850
$731$	1.57127e10	1.48778
$732$	5.23721e9	0.493527
$733$	5.46670e9	0.512697	0.256349	−	0.966584i	$-0.417480\pi$
$733$	5.46670e9	0.512697	0.256349	+	0.966584i	$0.417480\pi$
$734$	−1.64687e9	−0.153717
$735$	−1.38808e10	−1.28946
$736$	3.20150e9	0.295993
$737$	−1.84716e9	−0.169969
$738$	−1.12811e9	−0.103313
$739$	−7.76760e9	−0.707997	−0.353998	−	0.935246i	$-0.615178\pi$
$739$	−7.76760e9	−0.707997	−0.353998	+	0.935246i	$0.615178\pi$
$740$	−2.44230e9	−0.221558
$741$	1.33708e9	0.120724
$742$	3.24964e8	0.0292026
$743$	−6.32109e8	−0.0565368	−0.0282684	−	0.999600i	$-0.508999\pi$
$743$	−6.32109e8	−0.0565368	−0.0282684	+	0.999600i	$0.508999\pi$
$744$	−3.54797e9	−0.315846
$745$	2.16695e10	1.92000
$746$	−2.96777e9	−0.261724
$747$	1.16599e9	0.102347
$748$	2.85236e9	0.249200
$749$	−1.72975e9	−0.150417
$750$	−5.75981e9	−0.498533
$751$	−8.06627e9	−0.694917	−0.347458	−	0.937695i	$-0.612955\pi$
$751$	−8.06627e9	−0.694917	−0.347458	+	0.937695i	$0.612955\pi$
$752$	−1.41427e9	−0.121275
$753$	6.61954e9	0.564996
$754$	1.09683e9	0.0931838
$755$	2.70184e10	2.28478
$756$	8.29305e8	0.0698052
$757$	1.82516e10	1.52920	0.764602	−	0.644503i	$-0.222936\pi$
$757$	1.82516e10	1.52920	0.764602	+	0.644503i	$0.222936\pi$
$758$	−1.40093e9	−0.116835
$759$	−1.11684e10	−0.927143
$760$	1.18886e9	0.0982392
$761$	1.39681e10	1.14893	0.574463	−	0.818531i	$-0.305211\pi$
$761$	1.39681e10	1.14893	0.574463	+	0.818531i	$0.305211\pi$
$762$	−1.05787e9	−0.0866146
$763$	−1.44499e9	−0.117769
$764$	−3.55235e9	−0.288197
$765$	−2.83759e9	−0.229158
$766$	5.08845e9	0.409058
$767$	4.14818e9	0.331951
$768$	8.56785e8	0.0682508
$769$	−1.28542e10	−1.01930	−0.509650	−	0.860382i	$-0.670225\pi$
$769$	−1.28542e10	−1.01930	−0.509650	+	0.860382i	$0.670225\pi$
$770$	8.70993e8	0.0687539
$771$	2.39154e10	1.87926
$772$	5.53007e9	0.432584
$773$	−1.22702e10	−0.955486	−0.477743	−	0.878500i	$-0.658545\pi$
$773$	−1.22702e10	−0.955486	−0.477743	+	0.878500i	$0.658545\pi$
$774$	−2.65774e9	−0.206025
$775$	−4.95006e9	−0.381993
$776$	−7.64410e9	−0.587233
$777$	8.27096e8	0.0632532
$778$	−4.10590e9	−0.312593
$779$	−2.29753e9	−0.174133
$780$	4.22355e9	0.318674
$781$	−3.25559e9	−0.244541
$782$	−1.55626e10	−1.16374
$783$	−3.23932e9	−0.241150
$784$	−3.28868e9	−0.243733
$785$	1.72212e10	1.27063
$786$	1.75153e10	1.28658
$787$	1.40869e10	1.03016	0.515079	−	0.857143i	$-0.327763\pi$
$787$	1.40869e10	1.03016	0.515079	+	0.857143i	$0.327763\pi$
$788$	−3.36031e9	−0.244646
$789$	−2.22776e10	−1.61473
$790$	−9.25168e9	−0.667614
$791$	−9.98992e7	−0.00717702
$792$	−4.82467e8	−0.0345088
$793$	6.11663e9	0.435568
$794$	7.39343e9	0.524172
$795$	−4.88779e9	−0.345007
$796$	7.47762e9	0.525494
$797$	2.68842e10	1.88102	0.940510	−	0.339766i	$-0.110348\pi$
$797$	2.68842e10	1.88102	0.940510	+	0.339766i	$0.110348\pi$
$798$	−4.02614e8	−0.0280465
$799$	6.87481e9	0.476812
$800$	1.19537e9	0.0825444
$801$	−8.21151e8	−0.0564559
$802$	−1.98803e10	−1.36086
$803$	−1.42338e10	−0.970100
$804$	2.69712e9	0.183022
$805$	−4.75217e9	−0.321075
$806$	−4.14374e9	−0.278753
$807$	−8.02569e9	−0.537558
$808$	−1.89643e9	−0.126473
$809$	1.57674e10	1.04699	0.523493	−	0.852030i	$-0.324629\pi$
$809$	1.57674e10	1.04699	0.523493	+	0.852030i	$0.324629\pi$
$810$	−1.49670e10	−0.989551
$811$	−2.41951e9	−0.159277	−0.0796387	−	0.996824i	$-0.525377\pi$
$811$	−2.41951e9	−0.159277	−0.0796387	+	0.996824i	$0.525377\pi$
$812$	−3.30273e8	−0.0216484
$813$	1.55532e10	1.01509
$814$	2.01857e9	0.131177
$815$	4.95051e9	0.320330
$816$	−4.16485e9	−0.268339
$817$	−5.41282e9	−0.347253
$818$	−1.67122e9	−0.106757
$819$	−2.30883e8	−0.0146858
$820$	−7.25742e9	−0.459657
$821$	1.94539e10	1.22689	0.613445	−	0.789737i	$-0.289783\pi$
$821$	1.94539e10	1.22689	0.613445	+	0.789737i	$0.289783\pi$
$822$	−1.75808e10	−1.10405
$823$	−9.90466e9	−0.619356	−0.309678	−	0.950842i	$-0.600221\pi$
$823$	−9.90466e9	−0.619356	−0.309678	+	0.950842i	$0.600221\pi$
$824$	6.66639e9	0.415093
$825$	−4.17005e9	−0.258555
$826$	−1.24908e9	−0.0771187
$827$	−1.41557e10	−0.870285	−0.435143	−	0.900362i	$-0.643302\pi$
$827$	−1.41557e10	−0.870285	−0.435143	+	0.900362i	$0.643302\pi$
$828$	2.63236e9	0.161153
$829$	6.77554e8	0.0413050	0.0206525	−	0.999787i	$-0.493426\pi$
$829$	6.77554e8	0.0413050	0.0206525	+	0.999787i	$0.493426\pi$
$830$	7.50112e9	0.455358
$831$	−1.47415e10	−0.891124
$832$	1.00065e9	0.0602355
$833$	1.59863e10	0.958277
$834$	2.41326e10	1.44053
$835$	−3.16928e10	−1.88390
$836$	−9.82602e8	−0.0581642
$837$	1.22379e10	0.721385
$838$	2.67252e8	0.0156880
$839$	−2.55064e10	−1.49102	−0.745509	−	0.666496i	$-0.767793\pi$
$839$	−2.55064e10	−1.49102	−0.745509	+	0.666496i	$0.767793\pi$
$840$	−1.27177e9	−0.0740342
$841$	−1.59598e10	−0.925213
$842$	6.12991e9	0.353884
$843$	−2.42255e9	−0.139276
$844$	9.38673e9	0.537422
$845$	−1.63097e10	−0.929924
$846$	−1.16285e9	−0.0660280
$847$	2.07997e9	0.117616
$848$	−1.15803e9	−0.0652129
$849$	−2.72934e10	−1.53067
$850$	−5.81072e9	−0.324537
$851$	−1.10134e10	−0.612587
$852$	4.75363e9	0.263322
$853$	1.10942e10	0.612034	0.306017	−	0.952026i	$-0.401003\pi$
$853$	1.10942e10	0.612034	0.306017	+	0.952026i	$0.401003\pi$
$854$	−1.84181e9	−0.101191
$855$	9.77514e8	0.0534862
$856$	6.16405e9	0.335899
$857$	1.65157e10	0.896321	0.448160	−	0.893953i	$-0.352079\pi$
$857$	1.65157e10	0.896321	0.448160	+	0.893953i	$0.352079\pi$
$858$	−3.49079e9	−0.188676
$859$	−2.33583e10	−1.25738	−0.628688	−	0.777657i	$-0.716408\pi$
$859$	−2.33583e10	−1.25738	−0.628688	+	0.777657i	$0.716408\pi$
$860$	−1.70979e10	−0.916641
$861$	2.45776e9	0.131229
$862$	1.76946e10	0.940947
$863$	2.90901e9	0.154066	0.0770330	−	0.997029i	$-0.475455\pi$
$863$	2.90901e9	0.154066	0.0770330	+	0.997029i	$0.475455\pi$
$864$	−2.95527e9	−0.155883
$865$	−3.74039e10	−1.96499
$866$	−2.05102e10	−1.07314
$867$	−7.09937e8	−0.0369959
$868$	1.24774e9	0.0647598
$869$	7.64656e9	0.395273
$870$	4.96763e9	0.255760
$871$	3.15001e9	0.161528
$872$	5.14931e9	0.262991
$873$	−6.28517e9	−0.319718
$874$	5.36111e9	0.271622
$875$	2.02559e9	0.102217
$876$	2.07834e10	1.04460
$877$	−2.41950e8	−0.0121123	−0.00605616	−	0.999982i	$-0.501928\pi$
$877$	−2.41950e8	−0.0121123	−0.00605616	+	0.999982i	$0.501928\pi$
$878$	−2.71455e9	−0.135353
$879$	−3.01046e10	−1.49510
$880$	−3.10383e9	−0.153536
$881$	−6.21423e8	−0.0306176	−0.0153088	−	0.999883i	$-0.504873\pi$
$881$	−6.21423e8	−0.0306176	−0.0153088	+	0.999883i	$0.504873\pi$
$882$	−2.70403e9	−0.132700
$883$	−3.24481e10	−1.58609	−0.793044	−	0.609164i	$-0.791505\pi$
$883$	−3.24481e10	−1.58609	−0.793044	+	0.609164i	$0.791505\pi$
$884$	−4.86420e9	−0.236825
$885$	1.87874e10	0.911098
$886$	−1.32626e10	−0.640634
$887$	1.39588e10	0.671609	0.335805	−	0.941932i	$-0.390992\pi$
$887$	1.39588e10	0.671609	0.335805	+	0.941932i	$0.390992\pi$
$888$	−2.94740e9	−0.141252
$889$	3.72030e8	0.0177591
$890$	−5.28267e9	−0.251182
$891$	1.23703e10	0.585881
$892$	−3.16860e9	−0.149482
$893$	−2.36829e9	−0.111290
$894$	2.61510e10	1.22407
$895$	−8.58508e9	−0.400280
$896$	−3.01312e8	−0.0139939
$897$	1.90458e10	0.881103
$898$	2.85536e10	1.31581
$899$	−4.87376e9	−0.223720
$900$	9.82864e8	0.0449412
$901$	5.62920e9	0.256395
$902$	5.99829e9	0.272148
$903$	5.79030e9	0.261694
$904$	3.55996e8	0.0160271
$905$	2.09337e10	0.938806
$906$	3.26062e10	1.45664
$907$	4.01035e10	1.78467	0.892334	−	0.451377i	$-0.149067\pi$
$907$	4.01035e10	1.78467	0.892334	+	0.451377i	$0.149067\pi$
$908$	1.71721e10	0.761243
$909$	−1.55929e9	−0.0688578
$910$	−1.48533e9	−0.0653397
$911$	2.42547e10	1.06287	0.531436	−	0.847099i	$-0.321653\pi$
$911$	2.42547e10	1.06287	0.531436	+	0.847099i	$0.321653\pi$
$912$	1.43474e9	0.0626312
$913$	−6.19971e9	−0.269603
$914$	−6.56213e9	−0.284272
$915$	2.77027e10	1.19549
$916$	1.48372e10	0.637850
$917$	−6.15973e9	−0.263796
$918$	1.43656e10	0.612880
$919$	1.89321e10	0.804626	0.402313	−	0.915502i	$-0.368206\pi$
$919$	1.89321e10	0.804626	0.402313	+	0.915502i	$0.368206\pi$
$920$	1.69346e10	0.716998
$921$	4.52992e10	1.91065
$922$	−2.99684e10	−1.25923
$923$	5.55185e9	0.232398
$924$	1.05113e9	0.0438332
$925$	−4.11216e9	−0.170834
$926$	1.29941e10	0.537783
$927$	5.48128e9	0.225997
$928$	1.17694e9	0.0483435
$929$	−2.42440e10	−0.992086	−0.496043	−	0.868298i	$-0.665214\pi$
$929$	−2.42440e10	−0.992086	−0.496043	+	0.868298i	$0.665214\pi$
$930$	−1.87673e10	−0.765088
$931$	−5.50709e9	−0.223665
$932$	1.14790e9	0.0464461
$933$	−5.39217e10	−2.17359
$934$	−2.16220e10	−0.868326
$935$	1.50878e10	0.603649
$936$	8.22764e8	0.0327951
$937$	1.89408e9	0.0752159	0.0376080	−	0.999293i	$-0.488026\pi$
$937$	1.89408e9	0.0752159	0.0376080	+	0.999293i	$0.488026\pi$
$938$	−9.48515e8	−0.0375262
$939$	2.76638e10	1.09039
$940$	−7.48092e9	−0.293770
$941$	−2.02166e10	−0.790941	−0.395471	−	0.918479i	$-0.629419\pi$
$941$	−2.02166e10	−0.790941	−0.395471	+	0.918479i	$0.629419\pi$
$942$	2.07828e10	0.810078
$943$	−3.27269e10	−1.27091
$944$	4.45116e9	0.172215
$945$	4.38668e9	0.169093
$946$	1.41315e10	0.542713
$947$	−2.28622e10	−0.874768	−0.437384	−	0.899275i	$-0.644095\pi$
$947$	−2.28622e10	−0.874768	−0.437384	+	0.899275i	$0.644095\pi$
$948$	−1.11651e10	−0.425630
$949$	2.42733e10	0.921928
$950$	2.00172e9	0.0757480
$951$	1.41733e10	0.534366
$952$	1.46468e9	0.0550192
$953$	4.18466e9	0.156616	0.0783079	−	0.996929i	$-0.475048\pi$
$953$	4.18466e9	0.156616	0.0783079	+	0.996929i	$0.475048\pi$
$954$	−9.52160e8	−0.0355051
$955$	−1.87905e10	−0.698113
$956$	6.71549e9	0.248585
$957$	−4.10577e9	−0.151427
$958$	−9.28709e9	−0.341272
$959$	6.18278e9	0.226370
$960$	4.53204e9	0.165327
$961$	−9.09997e9	−0.330756
$962$	−3.44233e9	−0.124663
$963$	5.06824e9	0.182880
$964$	−1.81999e10	−0.654334
$965$	2.92518e10	1.04787
$966$	−5.73499e9	−0.204697
$967$	8.31822e8	0.0295827	0.0147913	−	0.999891i	$-0.495292\pi$
$967$	8.31822e8	0.0295827	0.0147913	+	0.999891i	$0.495292\pi$
$968$	−7.41210e9	−0.262650
$969$	−6.97429e9	−0.246245
$970$	−4.04341e10	−1.42248
$971$	−3.06782e10	−1.07538	−0.537690	−	0.843143i	$-0.680703\pi$
$971$	−3.06782e10	−1.07538	−0.537690	+	0.843143i	$0.680703\pi$
$972$	−5.43904e9	−0.189972
$973$	−8.48688e9	−0.295361
$974$	2.13282e10	0.739602
$975$	7.11130e9	0.245716
$976$	6.56338e9	0.225971
$977$	−2.63977e10	−0.905599	−0.452799	−	0.891612i	$-0.649575\pi$
$977$	−2.63977e10	−0.905599	−0.452799	+	0.891612i	$0.649575\pi$
$978$	5.97434e9	0.204223
$979$	4.36616e9	0.148717
$980$	−1.73957e10	−0.590407
$981$	4.23390e9	0.143185
$982$	−5.68622e9	−0.191617
$983$	3.22583e10	1.08319	0.541595	−	0.840640i	$-0.317821\pi$
$983$	3.22583e10	1.08319	0.541595	+	0.840640i	$0.317821\pi$
$984$	−8.75836e9	−0.293049
$985$	−1.77746e10	−0.592617
$986$	−5.72115e9	−0.190070
$987$	2.53345e9	0.0838691
$988$	1.67566e9	0.0552759
$989$	−7.71021e10	−2.53442
$990$	−2.55205e9	−0.0835922
$991$	5.69036e10	1.85730	0.928650	−	0.370956i	$-0.120970\pi$
$991$	5.69036e10	1.85730	0.928650	+	0.370956i	$0.120970\pi$
$992$	−4.44639e9	−0.144616
$993$	−3.11149e10	−1.00843
$994$	−1.67174e9	−0.0539905
$995$	3.95535e10	1.27293
$996$	9.05247e9	0.290308
$997$	−5.83019e8	−0.0186316	−0.00931579	−	0.999957i	$-0.502965\pi$
$997$	−5.83019e8	−0.0186316	−0.00931579	+	0.999957i	$0.502965\pi$
$998$	9.91008e9	0.315588
$999$	1.01664e10	0.322616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.e.1.3	✓	4
3.2	odd	2		342.8.a.o.1.1		4
4.3	odd	2		304.8.a.e.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.e.1.3	✓	4	1.1	even	1	trivial
304.8.a.e.1.2		4	4.3	odd	2
342.8.a.o.1.1		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +51.0684 q^{3} +64.0000 q^{4} +338.533 q^{5} +408.547 q^{6} -143.677 q^{7} +512.000 q^{8} +420.979 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +51.0684 q^{3} +64.0000 q^{4} +338.533 q^{5} +408.547 q^{6} -143.677 q^{7} +512.000 q^{8} +420.979 q^{9} +2708.27 q^{10} -2238.40 q^{11} +3268.38 q^{12} +3817.19 q^{13} -1149.41 q^{14} +17288.3 q^{15} +4096.00 q^{16} -19910.7 q^{17} +3367.83 q^{18} +6859.00 q^{19} +21666.1 q^{20} -7337.34 q^{21} -17907.2 q^{22} +97702.1 q^{23} +26147.0 q^{24} +36479.8 q^{25} +30537.6 q^{26} -90187.8 q^{27} -9195.31 q^{28} +35917.5 q^{29} +138307. q^{30} -135693. q^{31} +32768.0 q^{32} -114311. q^{33} -159286. q^{34} -48639.4 q^{35} +26942.7 q^{36} -112724. q^{37} +54872.0 q^{38} +194938. q^{39} +173329. q^{40} -334966. q^{41} -58698.7 q^{42} -789155. q^{43} -143257. q^{44} +142516. q^{45} +781617. q^{46} -345282. q^{47} +209176. q^{48} -802900. q^{49} +291839. q^{50} -1.01681e6 q^{51} +244300. q^{52} -282722. q^{53} -721503. q^{54} -757772. q^{55} -73562.5 q^{56} +350278. q^{57} +287340. q^{58} +1.08671e6 q^{59} +1.10645e6 q^{60} +1.60239e6 q^{61} -1.08555e6 q^{62} -60484.9 q^{63} +262144. q^{64} +1.29225e6 q^{65} -914490. q^{66} +825216. q^{67} -1.27429e6 q^{68} +4.98949e6 q^{69} -389115. q^{70} +1.45443e6 q^{71} +215541. q^{72} +6.35893e6 q^{73} -901795. q^{74} +1.86297e6 q^{75} +438976. q^{76} +321605. q^{77} +1.55950e6 q^{78} -3.41609e6 q^{79} +1.38663e6 q^{80} -5.52643e6 q^{81} -2.67973e6 q^{82} +2.76971e6 q^{83} -469590. q^{84} -6.74044e6 q^{85} -6.31324e6 q^{86} +1.83425e6 q^{87} -1.14606e6 q^{88} -1.95057e6 q^{89} +1.14012e6 q^{90} -548442. q^{91} +6.25293e6 q^{92} -6.92963e6 q^{93} -2.76225e6 q^{94} +2.32200e6 q^{95} +1.67341e6 q^{96} -1.49299e7 q^{97} -6.42320e6 q^{98} -942318. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more