LMFDB - Embedded newform 4025.2.a.bd.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68001 q^{2} +3.08395 q^{3} +5.18247 q^{4} +8.26504 q^{6} -1.00000 q^{7} +8.52906 q^{8} +6.51078 q^{9} +O(q^{10})$	\(q+2.68001 q^{2} +3.08395 q^{3} +5.18247 q^{4} +8.26504 q^{6} -1.00000 q^{7} +8.52906 q^{8} +6.51078 q^{9} -1.03638 q^{11} +15.9825 q^{12} -4.03133 q^{13} -2.68001 q^{14} +12.4931 q^{16} +4.11805 q^{17} +17.4490 q^{18} -6.23865 q^{19} -3.08395 q^{21} -2.77751 q^{22} -1.00000 q^{23} +26.3032 q^{24} -10.8040 q^{26} +10.8271 q^{27} -5.18247 q^{28} -8.14422 q^{29} +10.0647 q^{31} +16.4234 q^{32} -3.19615 q^{33} +11.0364 q^{34} +33.7419 q^{36} +4.74088 q^{37} -16.7197 q^{38} -12.4325 q^{39} -4.45005 q^{41} -8.26504 q^{42} -4.71008 q^{43} -5.37101 q^{44} -2.68001 q^{46} +4.57220 q^{47} +38.5280 q^{48} +1.00000 q^{49} +12.6999 q^{51} -20.8923 q^{52} -6.63018 q^{53} +29.0167 q^{54} -8.52906 q^{56} -19.2397 q^{57} -21.8266 q^{58} +5.25847 q^{59} -0.922345 q^{61} +26.9735 q^{62} -6.51078 q^{63} +19.0289 q^{64} -8.56573 q^{66} -9.91482 q^{67} +21.3417 q^{68} -3.08395 q^{69} -3.55231 q^{71} +55.5308 q^{72} +10.3525 q^{73} +12.7056 q^{74} -32.3316 q^{76} +1.03638 q^{77} -33.3191 q^{78} +4.18977 q^{79} +13.8579 q^{81} -11.9262 q^{82} -13.7686 q^{83} -15.9825 q^{84} -12.6231 q^{86} -25.1164 q^{87} -8.83935 q^{88} +2.94290 q^{89} +4.03133 q^{91} -5.18247 q^{92} +31.0391 q^{93} +12.2536 q^{94} +50.6491 q^{96} +0.694430 q^{97} +2.68001 q^{98} -6.74764 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9}+O(q^{10}) $ 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9	$ 21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9} + 7 q^{11} + 22 q^{12} + 3 q^{13} + 2 q^{14} + 56 q^{16} - 7 q^{17} + 24 q^{19} - q^{21} - 4 q^{22} - 21 q^{23} + 24 q^{24} - 2 q^{26} + 19 q^{27} - 30 q^{28} + 11 q^{29} + 46 q^{31} + 6 q^{32} + 3 q^{33} + 28 q^{34} + 58 q^{36} - 24 q^{37} + 4 q^{38} + 31 q^{39} + 14 q^{41} - 6 q^{42} - 18 q^{43} + 12 q^{44} + 2 q^{46} + 25 q^{47} + 36 q^{48} + 21 q^{49} + 17 q^{51} + 8 q^{52} - 22 q^{53} - 6 q^{54} + 6 q^{56} - 40 q^{57} - 6 q^{58} + 10 q^{59} + 38 q^{61} + 54 q^{62} - 30 q^{63} + 100 q^{64} + 38 q^{66} - 12 q^{67} - 18 q^{68} - q^{69} + 56 q^{71} - 42 q^{72} + 40 q^{73} - 20 q^{74} + 60 q^{76} - 7 q^{77} - 38 q^{78} + 49 q^{79} + 57 q^{81} - 16 q^{82} + 2 q^{83} - 22 q^{84} + 16 q^{86} + 23 q^{87} - 12 q^{88} + 28 q^{89} - 3 q^{91} - 30 q^{92} + 30 q^{93} + 66 q^{94} + 46 q^{96} + q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) $ 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9 + 7 * q^11 + 22 * q^12 + 3 * q^13 + 2 * q^14 + 56 * q^16 - 7 * q^17 + 24 * q^19 - q^21 - 4 * q^22 - 21 * q^23 + 24 * q^24 - 2 * q^26 + 19 * q^27 - 30 * q^28 + 11 * q^29 + 46 * q^31 + 6 * q^32 + 3 * q^33 + 28 * q^34 + 58 * q^36 - 24 * q^37 + 4 * q^38 + 31 * q^39 + 14 * q^41 - 6 * q^42 - 18 * q^43 + 12 * q^44 + 2 * q^46 + 25 * q^47 + 36 * q^48 + 21 * q^49 + 17 * q^51 + 8 * q^52 - 22 * q^53 - 6 * q^54 + 6 * q^56 - 40 * q^57 - 6 * q^58 + 10 * q^59 + 38 * q^61 + 54 * q^62 - 30 * q^63 + 100 * q^64 + 38 * q^66 - 12 * q^67 - 18 * q^68 - q^69 + 56 * q^71 - 42 * q^72 + 40 * q^73 - 20 * q^74 + 60 * q^76 - 7 * q^77 - 38 * q^78 + 49 * q^79 + 57 * q^81 - 16 * q^82 + 2 * q^83 - 22 * q^84 + 16 * q^86 + 23 * q^87 - 12 * q^88 + 28 * q^89 - 3 * q^91 - 30 * q^92 + 30 * q^93 + 66 * q^94 + 46 * q^96 + q^97 - 2 * q^98 - 2 * 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 $

$2$	2.68001	1.89506	0.947528	−	0.319674i	$-0.103573\pi$
$2$	2.68001	1.89506	0.947528	+	0.319674i	$0.103573\pi$
$3$	3.08395	1.78052	0.890261	−	0.455451i	$-0.150522\pi$
$3$	3.08395	1.78052	0.890261	+	0.455451i	$0.150522\pi$
$4$	5.18247	2.59123
$5$	0	0
$5$	0	0
$6$	8.26504	3.37419
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.52906	3.01548
$9$	6.51078	2.17026
$10$	0	0
$11$	−1.03638	−0.312481	−0.156240	−	0.987719i	$-0.549937\pi$
$11$	−1.03638	−0.312481	−0.156240	+	0.987719i	$0.549937\pi$
$12$	15.9825	4.61375
$13$	−4.03133	−1.11809	−0.559046	−	0.829137i	$-0.688832\pi$
$13$	−4.03133	−1.11809	−0.559046	+	0.829137i	$0.688832\pi$
$14$	−2.68001	−0.716264
$15$	0	0
$16$	12.4931	3.12326
$17$	4.11805	0.998773	0.499387	−	0.866379i	$-0.333559\pi$
$17$	4.11805	0.998773	0.499387	+	0.866379i	$0.333559\pi$
$18$	17.4490	4.11276
$19$	−6.23865	−1.43125	−0.715623	−	0.698487i	$-0.753857\pi$
$19$	−6.23865	−1.43125	−0.715623	+	0.698487i	$0.753857\pi$
$20$	0	0
$21$	−3.08395	−0.672974
$22$	−2.77751	−0.592168
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	26.3032	5.36913
$25$	0	0
$26$	−10.8040	−2.11884
$27$	10.8271	2.08367
$28$	−5.18247	−0.979395
$29$	−8.14422	−1.51234	−0.756172	−	0.654373i	$-0.772933\pi$
$29$	−8.14422	−1.51234	−0.756172	+	0.654373i	$0.772933\pi$
$30$	0	0
$31$	10.0647	1.80767	0.903837	−	0.427876i	$-0.140738\pi$
$31$	10.0647	1.80767	0.903837	+	0.427876i	$0.140738\pi$
$32$	16.4234	2.90328
$33$	−3.19615	−0.556379
$34$	11.0364	1.89273
$35$	0	0
$36$	33.7419	5.62365
$37$	4.74088	0.779395	0.389698	−	0.920943i	$-0.372580\pi$
$37$	4.74088	0.779395	0.389698	+	0.920943i	$0.372580\pi$
$38$	−16.7197	−2.71229
$39$	−12.4325	−1.99079
$40$	0	0
$41$	−4.45005	−0.694982	−0.347491	−	0.937683i	$-0.612966\pi$
$41$	−4.45005	−0.694982	−0.347491	+	0.937683i	$0.612966\pi$
$42$	−8.26504	−1.27532
$43$	−4.71008	−0.718280	−0.359140	−	0.933284i	$-0.616930\pi$
$43$	−4.71008	−0.718280	−0.359140	+	0.933284i	$0.616930\pi$
$44$	−5.37101	−0.809711
$45$	0	0
$46$	−2.68001	−0.395146
$47$	4.57220	0.666924	0.333462	−	0.942763i	$-0.391783\pi$
$47$	4.57220	0.666924	0.333462	+	0.942763i	$0.391783\pi$
$48$	38.5280	5.56104
$49$	1.00000	0.142857
$50$	0	0
$51$	12.6999	1.77834
$52$	−20.8923	−2.89724
$53$	−6.63018	−0.910725	−0.455363	−	0.890306i	$-0.650490\pi$
$53$	−6.63018	−0.910725	−0.455363	+	0.890306i	$0.650490\pi$
$54$	29.0167	3.94867
$55$	0	0
$56$	−8.52906	−1.13974
$57$	−19.2397	−2.54836
$58$	−21.8266	−2.86598
$59$	5.25847	0.684595	0.342297	−	0.939592i	$-0.388795\pi$
$59$	5.25847	0.684595	0.342297	+	0.939592i	$0.388795\pi$
$60$	0	0
$61$	−0.922345	−0.118094	−0.0590471	−	0.998255i	$-0.518806\pi$
$61$	−0.922345	−0.118094	−0.0590471	+	0.998255i	$0.518806\pi$
$62$	26.9735	3.42564
$63$	−6.51078	−0.820281
$64$	19.0289	2.37861
$65$	0	0
$66$	−8.56573	−1.05437
$67$	−9.91482	−1.21129	−0.605644	−	0.795735i	$-0.707085\pi$
$67$	−9.91482	−1.21129	−0.605644	+	0.795735i	$0.707085\pi$
$68$	21.3417	2.58806
$69$	−3.08395	−0.371265
$70$	0	0
$71$	−3.55231	−0.421581	−0.210791	−	0.977531i	$-0.567604\pi$
$71$	−3.55231	−0.421581	−0.210791	+	0.977531i	$0.567604\pi$
$72$	55.5308	6.54437
$73$	10.3525	1.21167	0.605833	−	0.795592i	$-0.292840\pi$
$73$	10.3525	1.21167	0.605833	+	0.795592i	$0.292840\pi$
$74$	12.7056	1.47700
$75$	0	0
$76$	−32.3316	−3.70869
$77$	1.03638	0.118107
$78$	−33.3191	−3.77265
$79$	4.18977	0.471385	0.235693	−	0.971828i	$-0.424264\pi$
$79$	4.18977	0.471385	0.235693	+	0.971828i	$0.424264\pi$
$80$	0	0
$81$	13.8579	1.53976
$82$	−11.9262	−1.31703
$83$	−13.7686	−1.51129	−0.755647	−	0.654979i	$-0.772678\pi$
$83$	−13.7686	−1.51129	−0.755647	+	0.654979i	$0.772678\pi$
$84$	−15.9825	−1.74383
$85$	0	0
$86$	−12.6231	−1.36118
$87$	−25.1164	−2.69276
$88$	−8.83935	−0.942278
$89$	2.94290	0.311946	0.155973	−	0.987761i	$-0.450149\pi$
$89$	2.94290	0.311946	0.155973	+	0.987761i	$0.450149\pi$
$90$	0	0
$91$	4.03133	0.422599
$92$	−5.18247	−0.540310
$93$	31.0391	3.21860
$94$	12.2536	1.26386
$95$	0	0
$96$	50.6491	5.16935
$97$	0.694430	0.0705087	0.0352544	−	0.999378i	$-0.488776\pi$
$97$	0.694430	0.0705087	0.0352544	+	0.999378i	$0.488776\pi$
$98$	2.68001	0.270722
$99$	−6.74764	−0.678164
$100$	0	0
$101$	9.42415	0.937738	0.468869	−	0.883268i	$-0.344662\pi$
$101$	9.42415	0.937738	0.468869	+	0.883268i	$0.344662\pi$
$102$	34.0358	3.37005
$103$	10.0654	0.991770	0.495885	−	0.868388i	$-0.334844\pi$
$103$	10.0654	0.991770	0.495885	+	0.868388i	$0.334844\pi$
$104$	−34.3835	−3.37158
$105$	0	0
$106$	−17.7690	−1.72587
$107$	−5.34125	−0.516358	−0.258179	−	0.966097i	$-0.583122\pi$
$107$	−5.34125	−0.516358	−0.258179	+	0.966097i	$0.583122\pi$
$108$	56.1110	5.39928
$109$	−4.52282	−0.433207	−0.216604	−	0.976260i	$-0.569498\pi$
$109$	−4.52282	−0.433207	−0.216604	+	0.976260i	$0.569498\pi$
$110$	0	0
$111$	14.6206	1.38773
$112$	−12.4931	−1.18048
$113$	−9.22779	−0.868078	−0.434039	−	0.900894i	$-0.642912\pi$
$113$	−9.22779	−0.868078	−0.434039	+	0.900894i	$0.642912\pi$
$114$	−51.5627	−4.82929
$115$	0	0
$116$	−42.2072	−3.91884
$117$	−26.2471	−2.42655
$118$	14.0928	1.29735
$119$	−4.11805	−0.377501
$120$	0	0
$121$	−9.92591	−0.902356
$122$	−2.47190	−0.223795
$123$	−13.7238	−1.23743
$124$	52.1600	4.68411
$125$	0	0
$126$	−17.4490	−1.55448
$127$	−7.22753	−0.641339	−0.320670	−	0.947191i	$-0.603908\pi$
$127$	−7.22753	−0.641339	−0.320670	+	0.947191i	$0.603908\pi$
$128$	18.1508	1.60432
$129$	−14.5257	−1.27891
$130$	0	0
$131$	−12.5103	−1.09303	−0.546515	−	0.837450i	$-0.684046\pi$
$131$	−12.5103	−1.09303	−0.546515	+	0.837450i	$0.684046\pi$
$132$	−16.5640	−1.44171
$133$	6.23865	0.540960
$134$	−26.5719	−2.29546
$135$	0	0
$136$	35.1231	3.01178
$137$	8.00974	0.684318	0.342159	−	0.939642i	$-0.388842\pi$
$137$	8.00974	0.684318	0.342159	+	0.939642i	$0.388842\pi$
$138$	−8.26504	−0.703567
$139$	−15.4003	−1.30624	−0.653118	−	0.757256i	$-0.726539\pi$
$139$	−15.4003	−1.30624	−0.653118	+	0.757256i	$0.726539\pi$
$140$	0	0
$141$	14.1005	1.18747
$142$	−9.52023	−0.798920
$143$	4.17800	0.349382
$144$	81.3395	6.77829
$145$	0	0
$146$	27.7448	2.29617
$147$	3.08395	0.254360
$148$	24.5694	2.01960
$149$	−16.6464	−1.36373	−0.681864	−	0.731479i	$-0.738830\pi$
$149$	−16.6464	−1.36373	−0.681864	+	0.731479i	$0.738830\pi$
$150$	0	0
$151$	9.04004	0.735668	0.367834	−	0.929891i	$-0.380099\pi$
$151$	9.04004	0.735668	0.367834	+	0.929891i	$0.380099\pi$
$152$	−53.2099	−4.31589
$153$	26.8117	2.16760
$154$	2.77751	0.223818
$155$	0	0
$156$	−64.4308	−5.15859
$157$	13.2316	1.05600	0.527999	−	0.849245i	$-0.322943\pi$
$157$	13.2316	1.05600	0.527999	+	0.849245i	$0.322943\pi$
$158$	11.2286	0.893301
$159$	−20.4472	−1.62157
$160$	0	0
$161$	1.00000	0.0788110
$162$	37.1393	2.91794
$163$	5.89731	0.461913	0.230957	−	0.972964i	$-0.425814\pi$
$163$	5.89731	0.461913	0.230957	+	0.972964i	$0.425814\pi$
$164$	−23.0623	−1.80086
$165$	0	0
$166$	−36.8999	−2.86399
$167$	−7.11397	−0.550495	−0.275248	−	0.961373i	$-0.588760\pi$
$167$	−7.11397	−0.550495	−0.275248	+	0.961373i	$0.588760\pi$
$168$	−26.3032	−2.02934
$169$	3.25166	0.250128
$170$	0	0
$171$	−40.6185	−3.10617
$172$	−24.4098	−1.86123
$173$	25.6483	1.95001	0.975004	−	0.222188i	$-0.0713200\pi$
$173$	25.6483	1.95001	0.975004	+	0.222188i	$0.0713200\pi$
$174$	−67.3123	−5.10293
$175$	0	0
$176$	−12.9476	−0.975959
$177$	16.2169	1.21894
$178$	7.88700	0.591156
$179$	7.09135	0.530032	0.265016	−	0.964244i	$-0.414623\pi$
$179$	7.09135	0.530032	0.265016	+	0.964244i	$0.414623\pi$
$180$	0	0
$181$	7.09002	0.526997	0.263499	−	0.964660i	$-0.415124\pi$
$181$	7.09002	0.526997	0.263499	+	0.964660i	$0.415124\pi$
$182$	10.8040	0.800848
$183$	−2.84447	−0.210269
$184$	−8.52906	−0.628771
$185$	0	0
$186$	83.1852	6.09943
$187$	−4.26787	−0.312097
$188$	23.6953	1.72816
$189$	−10.8271	−0.787554
$190$	0	0
$191$	19.5891	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$191$	19.5891	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$192$	58.6842	4.23517
$193$	−12.0977	−0.870811	−0.435405	−	0.900235i	$-0.643395\pi$
$193$	−12.0977	−0.870811	−0.435405	+	0.900235i	$0.643395\pi$
$194$	1.86108	0.133618
$195$	0	0
$196$	5.18247	0.370176
$197$	2.29867	0.163773	0.0818866	−	0.996642i	$-0.473905\pi$
$197$	2.29867	0.163773	0.0818866	+	0.996642i	$0.473905\pi$
$198$	−18.0838	−1.28516
$199$	16.1216	1.14283	0.571416	−	0.820661i	$-0.306394\pi$
$199$	16.1216	1.14283	0.571416	+	0.820661i	$0.306394\pi$
$200$	0	0
$201$	−30.5769	−2.15673
$202$	25.2568	1.77707
$203$	8.14422	0.571612
$204$	65.8167	4.60809
$205$	0	0
$206$	26.9753	1.87946
$207$	−6.51078	−0.452530
$208$	−50.3637	−3.49209
$209$	6.46562	0.447236
$210$	0	0
$211$	3.31907	0.228494	0.114247	−	0.993452i	$-0.463554\pi$
$211$	3.31907	0.228494	0.114247	+	0.993452i	$0.463554\pi$
$212$	−34.3607	−2.35990
$213$	−10.9552	−0.750635
$214$	−14.3146	−0.978526
$215$	0	0
$216$	92.3448	6.28327
$217$	−10.0647	−0.683237
$218$	−12.1212	−0.820952
$219$	31.9266	2.15740
$220$	0	0
$221$	−16.6012	−1.11672
$222$	39.1835	2.62983
$223$	−7.85449	−0.525975	−0.262988	−	0.964799i	$-0.584708\pi$
$223$	−7.85449	−0.525975	−0.262988	+	0.964799i	$0.584708\pi$
$224$	−16.4234	−1.09734
$225$	0	0
$226$	−24.7306	−1.64506
$227$	23.7425	1.57585	0.787923	−	0.615774i	$-0.211156\pi$
$227$	23.7425	1.57585	0.787923	+	0.615774i	$0.211156\pi$
$228$	−99.7093	−6.60341
$229$	−1.09061	−0.0720696	−0.0360348	−	0.999351i	$-0.511473\pi$
$229$	−1.09061	−0.0720696	−0.0360348	+	0.999351i	$0.511473\pi$
$230$	0	0
$231$	3.19615	0.210291
$232$	−69.4626	−4.56044
$233$	19.7955	1.29684	0.648422	−	0.761281i	$-0.275429\pi$
$233$	19.7955	1.29684	0.648422	+	0.761281i	$0.275429\pi$
$234$	−70.3426	−4.59844
$235$	0	0
$236$	27.2519	1.77395
$237$	12.9210	0.839312
$238$	−11.0364	−0.715385
$239$	17.1731	1.11083	0.555417	−	0.831572i	$-0.312559\pi$
$239$	17.1731	1.11083	0.555417	+	0.831572i	$0.312559\pi$
$240$	0	0
$241$	−26.4098	−1.70121	−0.850604	−	0.525807i	$-0.823763\pi$
$241$	−26.4098	−1.70121	−0.850604	+	0.525807i	$0.823763\pi$
$242$	−26.6016	−1.71001
$243$	10.2558	0.657912
$244$	−4.78003	−0.306010
$245$	0	0
$246$	−36.7799	−2.34500
$247$	25.1501	1.60026
$248$	85.8425	5.45100
$249$	−42.4616	−2.69089
$250$	0	0
$251$	−15.0483	−0.949837	−0.474919	−	0.880030i	$-0.657522\pi$
$251$	−15.0483	−0.949837	−0.474919	+	0.880030i	$0.657522\pi$
$252$	−33.7419	−2.12554
$253$	1.03638	0.0651567
$254$	−19.3699	−1.21537
$255$	0	0
$256$	10.5866	0.661665
$257$	−18.6447	−1.16303	−0.581513	−	0.813537i	$-0.697539\pi$
$257$	−18.6447	−1.16303	−0.581513	+	0.813537i	$0.697539\pi$
$258$	−38.9290	−2.42361
$259$	−4.74088	−0.294584
$260$	0	0
$261$	−53.0252	−3.28218
$262$	−33.5277	−2.07135
$263$	−2.26355	−0.139576	−0.0697882	−	0.997562i	$-0.522232\pi$
$263$	−2.26355	−0.139576	−0.0697882	+	0.997562i	$0.522232\pi$
$264$	−27.2602	−1.67775
$265$	0	0
$266$	16.7197	1.02515
$267$	9.07576	0.555427
$268$	−51.3833	−3.13873
$269$	2.42234	0.147692	0.0738462	−	0.997270i	$-0.476473\pi$
$269$	2.42234	0.147692	0.0738462	+	0.997270i	$0.476473\pi$
$270$	0	0
$271$	−1.07451	−0.0652718	−0.0326359	−	0.999467i	$-0.510390\pi$
$271$	−1.07451	−0.0652718	−0.0326359	+	0.999467i	$0.510390\pi$
$272$	51.4470	3.11943
$273$	12.4325	0.752446
$274$	21.4662	1.29682
$275$	0	0
$276$	−15.9825	−0.962034
$277$	−18.3660	−1.10351	−0.551753	−	0.834008i	$-0.686041\pi$
$277$	−18.3660	−1.10351	−0.551753	+	0.834008i	$0.686041\pi$
$278$	−41.2730	−2.47539
$279$	65.5290	3.92312
$280$	0	0
$281$	23.7585	1.41731	0.708656	−	0.705554i	$-0.249302\pi$
$281$	23.7585	1.41731	0.708656	+	0.705554i	$0.249302\pi$
$282$	37.7894	2.25033
$283$	24.6469	1.46511	0.732553	−	0.680710i	$-0.238329\pi$
$283$	24.6469	1.46511	0.732553	+	0.680710i	$0.238329\pi$
$284$	−18.4097	−1.09242
$285$	0	0
$286$	11.1971	0.662098
$287$	4.45005	0.262678
$288$	106.929	6.30087
$289$	−0.0416824	−0.00245191
$290$	0	0
$291$	2.14159	0.125542
$292$	53.6514	3.13971
$293$	4.83385	0.282396	0.141198	−	0.989981i	$-0.454905\pi$
$293$	4.83385	0.282396	0.141198	+	0.989981i	$0.454905\pi$
$294$	8.26504	0.482027
$295$	0	0
$296$	40.4352	2.35025
$297$	−11.2210	−0.651107
$298$	−44.6126	−2.58434
$299$	4.03133	0.233138
$300$	0	0
$301$	4.71008	0.271484
$302$	24.2274	1.39413
$303$	29.0637	1.66966
$304$	−77.9398	−4.47016
$305$	0	0
$306$	71.8557	4.10772
$307$	−9.95089	−0.567927	−0.283964	−	0.958835i	$-0.591650\pi$
$307$	−9.95089	−0.567927	−0.283964	+	0.958835i	$0.591650\pi$
$308$	5.37101	0.306042
$309$	31.0411	1.76587
$310$	0	0
$311$	17.1140	0.970444	0.485222	−	0.874391i	$-0.338739\pi$
$311$	17.1140	0.970444	0.485222	+	0.874391i	$0.338739\pi$
$312$	−106.037	−6.00317
$313$	14.7785	0.835331	0.417666	−	0.908601i	$-0.362848\pi$
$313$	14.7785	0.835331	0.417666	+	0.908601i	$0.362848\pi$
$314$	35.4609	2.00117
$315$	0	0
$316$	21.7133	1.22147
$317$	12.6932	0.712919	0.356460	−	0.934311i	$-0.383984\pi$
$317$	12.6932	0.712919	0.356460	+	0.934311i	$0.383984\pi$
$318$	−54.7987	−3.07296
$319$	8.44051	0.472578
$320$	0	0
$321$	−16.4722	−0.919386
$322$	2.68001	0.149351
$323$	−25.6911	−1.42949
$324$	71.8180	3.98989
$325$	0	0
$326$	15.8049	0.875351
$327$	−13.9482	−0.771335
$328$	−37.9548	−2.09570
$329$	−4.57220	−0.252074
$330$	0	0
$331$	27.3143	1.50133	0.750665	−	0.660683i	$-0.229733\pi$
$331$	27.3143	1.50133	0.750665	+	0.660683i	$0.229733\pi$
$332$	−71.3551	−3.91612
$333$	30.8668	1.69149
$334$	−19.0655	−1.04322
$335$	0	0
$336$	−38.5280	−2.10188
$337$	−13.6129	−0.741541	−0.370771	−	0.928725i	$-0.620906\pi$
$337$	−13.6129	−0.741541	−0.370771	+	0.928725i	$0.620906\pi$
$338$	8.71449	0.474006
$339$	−28.4581	−1.54563
$340$	0	0
$341$	−10.4309	−0.564863
$342$	−108.858	−5.88637
$343$	−1.00000	−0.0539949
$344$	−40.1725	−2.16596
$345$	0	0
$346$	68.7379	3.69537
$347$	21.5633	1.15758	0.578788	−	0.815478i	$-0.303526\pi$
$347$	21.5633	1.15758	0.578788	+	0.815478i	$0.303526\pi$
$348$	−130.165	−6.97758
$349$	25.6897	1.37514	0.687570	−	0.726118i	$-0.258677\pi$
$349$	25.6897	1.37514	0.687570	+	0.726118i	$0.258677\pi$
$350$	0	0
$351$	−43.6476	−2.32973
$352$	−17.0209	−0.907218
$353$	−10.8774	−0.578946	−0.289473	−	0.957186i	$-0.593480\pi$
$353$	−10.8774	−0.578946	−0.289473	+	0.957186i	$0.593480\pi$
$354$	43.4615	2.30995
$355$	0	0
$356$	15.2515	0.808326
$357$	−12.6999	−0.672149
$358$	19.0049	1.00444
$359$	20.6570	1.09024	0.545118	−	0.838359i	$-0.316485\pi$
$359$	20.6570	1.09024	0.545118	+	0.838359i	$0.316485\pi$
$360$	0	0
$361$	19.9208	1.04846
$362$	19.0014	0.998689
$363$	−30.6111	−1.60666
$364$	20.8923	1.09505
$365$	0	0
$366$	−7.62322	−0.398472
$367$	−29.7267	−1.55172	−0.775860	−	0.630905i	$-0.782684\pi$
$367$	−29.7267	−1.55172	−0.775860	+	0.630905i	$0.782684\pi$
$368$	−12.4931	−0.651245
$369$	−28.9733	−1.50829
$370$	0	0
$371$	6.63018	0.344222
$372$	160.859	8.34016
$373$	−30.3763	−1.57283	−0.786413	−	0.617701i	$-0.788064\pi$
$373$	−30.3763	−1.57283	−0.786413	+	0.617701i	$0.788064\pi$
$374$	−11.4379	−0.591442
$375$	0	0
$376$	38.9966	2.01110
$377$	32.8321	1.69094
$378$	−29.0167	−1.49246
$379$	−1.07829	−0.0553881	−0.0276940	−	0.999616i	$-0.508816\pi$
$379$	−1.07829	−0.0553881	−0.0276940	+	0.999616i	$0.508816\pi$
$380$	0	0
$381$	−22.2894	−1.14192
$382$	52.4989	2.68608
$383$	7.58719	0.387688	0.193844	−	0.981032i	$-0.437905\pi$
$383$	7.58719	0.387688	0.193844	+	0.981032i	$0.437905\pi$
$384$	55.9763	2.85653
$385$	0	0
$386$	−32.4220	−1.65023
$387$	−30.6663	−1.55885
$388$	3.59886	0.182705
$389$	19.8445	1.00616	0.503079	−	0.864241i	$-0.332201\pi$
$389$	19.8445	1.00616	0.503079	+	0.864241i	$0.332201\pi$
$390$	0	0
$391$	−4.11805	−0.208259
$392$	8.52906	0.430783
$393$	−38.5812	−1.94616
$394$	6.16046	0.310359
$395$	0	0
$396$	−34.9695	−1.75728
$397$	12.9874	0.651818	0.325909	−	0.945401i	$-0.394330\pi$
$397$	12.9874	0.651818	0.325909	+	0.945401i	$0.394330\pi$
$398$	43.2062	2.16573
$399$	19.2397	0.963191
$400$	0	0
$401$	10.2632	0.512519	0.256259	−	0.966608i	$-0.417510\pi$
$401$	10.2632	0.512519	0.256259	+	0.966608i	$0.417510\pi$
$402$	−81.9464	−4.08712
$403$	−40.5742	−2.02114
$404$	48.8404	2.42990
$405$	0	0
$406$	21.8266	1.08324
$407$	−4.91335	−0.243546
$408$	108.318	5.36254
$409$	23.4909	1.16155	0.580775	−	0.814064i	$-0.302750\pi$
$409$	23.4909	1.16155	0.580775	+	0.814064i	$0.302750\pi$
$410$	0	0
$411$	24.7017	1.21844
$412$	52.1635	2.56991
$413$	−5.25847	−0.258753
$414$	−17.4490	−0.857570
$415$	0	0
$416$	−66.2083	−3.24613
$417$	−47.4938	−2.32578
$418$	17.3279	0.847538
$419$	−10.2429	−0.500398	−0.250199	−	0.968194i	$-0.580496\pi$
$419$	−10.2429	−0.500398	−0.250199	+	0.968194i	$0.580496\pi$
$420$	0	0
$421$	23.9872	1.16907	0.584533	−	0.811370i	$-0.301278\pi$
$421$	23.9872	1.16907	0.584533	+	0.811370i	$0.301278\pi$
$422$	8.89514	0.433009
$423$	29.7686	1.44740
$424$	−56.5492	−2.74627
$425$	0	0
$426$	−29.3600	−1.42249
$427$	0.922345	0.0446354
$428$	−27.6808	−1.33800
$429$	12.8848	0.622082
$430$	0	0
$431$	16.6824	0.803564	0.401782	−	0.915735i	$-0.368391\pi$
$431$	16.6824	0.803564	0.401782	+	0.915735i	$0.368391\pi$
$432$	135.263	6.50785
$433$	−5.02449	−0.241462	−0.120731	−	0.992685i	$-0.538524\pi$
$433$	−5.02449	−0.241462	−0.120731	+	0.992685i	$0.538524\pi$
$434$	−26.9735	−1.29477
$435$	0	0
$436$	−23.4394	−1.12254
$437$	6.23865	0.298435
$438$	85.5636	4.08839
$439$	−19.9912	−0.954126	−0.477063	−	0.878869i	$-0.658299\pi$
$439$	−19.9912	−0.954126	−0.477063	+	0.878869i	$0.658299\pi$
$440$	0	0
$441$	6.51078	0.310037
$442$	−44.4915	−2.11625
$443$	−6.40997	−0.304547	−0.152273	−	0.988338i	$-0.548659\pi$
$443$	−6.40997	−0.304547	−0.152273	+	0.988338i	$0.548659\pi$
$444$	75.7711	3.59594
$445$	0	0
$446$	−21.0501	−0.996753
$447$	−51.3368	−2.42815
$448$	−19.0289	−0.899030
$449$	−20.1421	−0.950565	−0.475282	−	0.879833i	$-0.657654\pi$
$449$	−20.1421	−0.950565	−0.475282	+	0.879833i	$0.657654\pi$
$450$	0	0
$451$	4.61195	0.217168
$452$	−47.8228	−2.24939
$453$	27.8791	1.30987
$454$	63.6302	2.98632
$455$	0	0
$456$	−164.097	−7.68454
$457$	−12.9762	−0.607001	−0.303500	−	0.952831i	$-0.598155\pi$
$457$	−12.9762	−0.607001	−0.303500	+	0.952831i	$0.598155\pi$
$458$	−2.92285	−0.136576
$459$	44.5864	2.08112
$460$	0	0
$461$	−23.4048	−1.09007	−0.545036	−	0.838413i	$-0.683484\pi$
$461$	−23.4048	−1.09007	−0.545036	+	0.838413i	$0.683484\pi$
$462$	8.56573	0.398514
$463$	−15.1716	−0.705083	−0.352541	−	0.935796i	$-0.614682\pi$
$463$	−15.1716	−0.705083	−0.352541	+	0.935796i	$0.614682\pi$
$464$	−101.746	−4.72345
$465$	0	0
$466$	53.0521	2.45759
$467$	6.97074	0.322568	0.161284	−	0.986908i	$-0.448437\pi$
$467$	6.97074	0.322568	0.161284	+	0.986908i	$0.448437\pi$
$468$	−136.025	−6.28775
$469$	9.91482	0.457824
$470$	0	0
$471$	40.8057	1.88023
$472$	44.8498	2.06438
$473$	4.88143	0.224449
$474$	34.6286	1.59054
$475$	0	0
$476$	−21.3417	−0.978193
$477$	−43.1676	−1.97651
$478$	46.0240	2.10509
$479$	3.49742	0.159801	0.0799006	−	0.996803i	$-0.474540\pi$
$479$	3.49742	0.159801	0.0799006	+	0.996803i	$0.474540\pi$
$480$	0	0
$481$	−19.1121	−0.871435
$482$	−70.7787	−3.22388
$483$	3.08395	0.140325
$484$	−51.4408	−2.33822
$485$	0	0
$486$	27.4858	1.24678
$487$	−1.57825	−0.0715174	−0.0357587	−	0.999360i	$-0.511385\pi$
$487$	−1.57825	−0.0715174	−0.0357587	+	0.999360i	$0.511385\pi$
$488$	−7.86674	−0.356110
$489$	18.1870	0.822447
$490$	0	0
$491$	−34.0732	−1.53770	−0.768852	−	0.639427i	$-0.779172\pi$
$491$	−34.0732	−1.53770	−0.768852	+	0.639427i	$0.779172\pi$
$492$	−71.1230	−3.20647
$493$	−33.5383	−1.51049
$494$	67.4026	3.03259
$495$	0	0
$496$	125.739	5.64584
$497$	3.55231	0.159343
$498$	−113.798	−5.09939
$499$	−21.5769	−0.965913	−0.482956	−	0.875644i	$-0.660437\pi$
$499$	−21.5769	−0.965913	−0.482956	+	0.875644i	$0.660437\pi$
$500$	0	0
$501$	−21.9391	−0.980169
$502$	−40.3295	−1.79999
$503$	−4.91991	−0.219368	−0.109684	−	0.993967i	$-0.534984\pi$
$503$	−4.91991	−0.219368	−0.109684	+	0.993967i	$0.534984\pi$
$504$	−55.5308	−2.47354
$505$	0	0
$506$	2.77751	0.123476
$507$	10.0280	0.445358
$508$	−37.4564	−1.66186
$509$	14.1012	0.625027	0.312513	−	0.949913i	$-0.398829\pi$
$509$	14.1012	0.625027	0.312513	+	0.949913i	$0.398829\pi$
$510$	0	0
$511$	−10.3525	−0.457967
$512$	−7.92928	−0.350428
$513$	−67.5464	−2.98224
$514$	−49.9681	−2.20400
$515$	0	0
$516$	−75.2788	−3.31396
$517$	−4.73854	−0.208401
$518$	−12.7056	−0.558252
$519$	79.0983	3.47203
$520$	0	0
$521$	29.8650	1.30841	0.654204	−	0.756318i	$-0.273004\pi$
$521$	29.8650	1.30841	0.654204	+	0.756318i	$0.273004\pi$
$522$	−142.108	−6.21991
$523$	22.7170	0.993345	0.496672	−	0.867938i	$-0.334555\pi$
$523$	22.7170	0.993345	0.496672	+	0.867938i	$0.334555\pi$
$524$	−64.8342	−2.83230
$525$	0	0
$526$	−6.06634	−0.264505
$527$	41.4469	1.80546
$528$	−39.9297	−1.73772
$529$	1.00000	0.0434783
$530$	0	0
$531$	34.2367	1.48575
$532$	32.3316	1.40175
$533$	17.9397	0.777053
$534$	24.3231	1.05257
$535$	0	0
$536$	−84.5641	−3.65261
$537$	21.8694	0.943734
$538$	6.49189	0.279885
$539$	−1.03638	−0.0446401
$540$	0	0
$541$	−30.9193	−1.32932	−0.664661	−	0.747145i	$-0.731424\pi$
$541$	−30.9193	−1.32932	−0.664661	+	0.747145i	$0.731424\pi$
$542$	−2.87970	−0.123694
$543$	21.8653	0.938330
$544$	67.6325	2.89972
$545$	0	0
$546$	33.3191	1.42593
$547$	−19.4587	−0.831992	−0.415996	−	0.909366i	$-0.636567\pi$
$547$	−19.4587	−0.831992	−0.415996	+	0.909366i	$0.636567\pi$
$548$	41.5102	1.77323
$549$	−6.00518	−0.256295
$550$	0	0
$551$	50.8090	2.16453
$552$	−26.3032	−1.11954
$553$	−4.18977	−0.178167
$554$	−49.2211	−2.09120
$555$	0	0
$556$	−79.8115	−3.38476
$557$	−7.00008	−0.296603	−0.148302	−	0.988942i	$-0.547381\pi$
$557$	−7.00008	−0.296603	−0.148302	+	0.988942i	$0.547381\pi$
$558$	175.619	7.43453
$559$	18.9879	0.803102
$560$	0	0
$561$	−13.1619	−0.555696
$562$	63.6731	2.68589
$563$	5.32058	0.224236	0.112118	−	0.993695i	$-0.464237\pi$
$563$	5.32058	0.224236	0.112118	+	0.993695i	$0.464237\pi$
$564$	73.0752	3.07702
$565$	0	0
$566$	66.0540	2.77646
$567$	−13.8579	−0.581976
$568$	−30.2979	−1.27127
$569$	−40.4405	−1.69535	−0.847676	−	0.530514i	$-0.821999\pi$
$569$	−40.4405	−1.69535	−0.847676	+	0.530514i	$0.821999\pi$
$570$	0	0
$571$	15.3192	0.641090	0.320545	−	0.947233i	$-0.396134\pi$
$571$	15.3192	0.641090	0.320545	+	0.947233i	$0.396134\pi$
$572$	21.6523	0.905330
$573$	60.4118	2.52374
$574$	11.9262	0.497790
$575$	0	0
$576$	123.893	5.16220
$577$	5.49935	0.228941	0.114470	−	0.993427i	$-0.463483\pi$
$577$	5.49935	0.228941	0.114470	+	0.993427i	$0.463483\pi$
$578$	−0.111709	−0.00464650
$579$	−37.3087	−1.55050
$580$	0	0
$581$	13.7686	0.571216
$582$	5.73949	0.237910
$583$	6.87139	0.284584
$584$	88.2969	3.65375
$585$	0	0
$586$	12.9548	0.535157
$587$	−32.7728	−1.35268	−0.676338	−	0.736591i	$-0.736434\pi$
$587$	−32.7728	−1.35268	−0.676338	+	0.736591i	$0.736434\pi$
$588$	15.9825	0.659107
$589$	−62.7902	−2.58723
$590$	0	0
$591$	7.08898	0.291602
$592$	59.2280	2.43426
$593$	−31.2505	−1.28330	−0.641652	−	0.766996i	$-0.721751\pi$
$593$	−31.2505	−1.28330	−0.641652	+	0.766996i	$0.721751\pi$
$594$	−30.0723	−1.23388
$595$	0	0
$596$	−86.2695	−3.53374
$597$	49.7184	2.03484
$598$	10.8040	0.441810
$599$	25.6017	1.04606	0.523028	−	0.852316i	$-0.324802\pi$
$599$	25.6017	1.04606	0.523028	+	0.852316i	$0.324802\pi$
$600$	0	0
$601$	33.3933	1.36214	0.681071	−	0.732217i	$-0.261514\pi$
$601$	33.3933	1.36214	0.681071	+	0.732217i	$0.261514\pi$
$602$	12.6231	0.514478
$603$	−64.5532	−2.62881
$604$	46.8497	1.90629
$605$	0	0
$606$	77.8910	3.16410
$607$	39.6763	1.61041	0.805205	−	0.592996i	$-0.202055\pi$
$607$	39.6763	1.61041	0.805205	+	0.592996i	$0.202055\pi$
$608$	−102.460	−4.15530
$609$	25.1164	1.01777
$610$	0	0
$611$	−18.4321	−0.745682
$612$	138.951	5.61675
$613$	29.2538	1.18155	0.590776	−	0.806836i	$-0.298822\pi$
$613$	29.2538	1.18155	0.590776	+	0.806836i	$0.298822\pi$
$614$	−26.6685	−1.07625
$615$	0	0
$616$	8.83935	0.356148
$617$	−15.0304	−0.605101	−0.302550	−	0.953133i	$-0.597838\pi$
$617$	−15.0304	−0.605101	−0.302550	+	0.953133i	$0.597838\pi$
$618$	83.1907	3.34642
$619$	18.7893	0.755207	0.377604	−	0.925967i	$-0.376748\pi$
$619$	18.7893	0.755207	0.377604	+	0.925967i	$0.376748\pi$
$620$	0	0
$621$	−10.8271	−0.434476
$622$	45.8656	1.83904
$623$	−2.94290	−0.117905
$624$	−155.319	−6.21775
$625$	0	0
$626$	39.6066	1.58300
$627$	19.9397	0.796314
$628$	68.5724	2.73634
$629$	19.5232	0.778439
$630$	0	0
$631$	−18.7280	−0.745548	−0.372774	−	0.927922i	$-0.621593\pi$
$631$	−18.7280	−0.745548	−0.372774	+	0.927922i	$0.621593\pi$
$632$	35.7348	1.42145
$633$	10.2358	0.406838
$634$	34.0179	1.35102
$635$	0	0
$636$	−105.967	−4.20186
$637$	−4.03133	−0.159727
$638$	22.6207	0.895562
$639$	−23.1283	−0.914941
$640$	0	0
$641$	24.4090	0.964098	0.482049	−	0.876144i	$-0.339893\pi$
$641$	24.4090	0.964098	0.482049	+	0.876144i	$0.339893\pi$
$642$	−44.1456	−1.74229
$643$	19.3762	0.764121	0.382060	−	0.924137i	$-0.375215\pi$
$643$	19.3762	0.764121	0.382060	+	0.924137i	$0.375215\pi$
$644$	5.18247	0.204218
$645$	0	0
$646$	−68.8524	−2.70896
$647$	−16.9974	−0.668238	−0.334119	−	0.942531i	$-0.608439\pi$
$647$	−16.9974	−0.668238	−0.334119	+	0.942531i	$0.608439\pi$
$648$	118.195	4.64312
$649$	−5.44978	−0.213923
$650$	0	0
$651$	−31.0391	−1.21652
$652$	30.5626	1.19693
$653$	−1.42934	−0.0559344	−0.0279672	−	0.999609i	$-0.508903\pi$
$653$	−1.42934	−0.0559344	−0.0279672	+	0.999609i	$0.508903\pi$
$654$	−37.3813	−1.46172
$655$	0	0
$656$	−55.5948	−2.17061
$657$	67.4026	2.62963
$658$	−12.2536	−0.477694
$659$	−25.6190	−0.997976	−0.498988	−	0.866609i	$-0.666295\pi$
$659$	−25.6190	−0.997976	−0.498988	+	0.866609i	$0.666295\pi$
$660$	0	0
$661$	−33.2265	−1.29236	−0.646181	−	0.763184i	$-0.723635\pi$
$661$	−33.2265	−1.29236	−0.646181	+	0.763184i	$0.723635\pi$
$662$	73.2027	2.84510
$663$	−51.1974	−1.98834
$664$	−117.433	−4.55728
$665$	0	0
$666$	82.7234	3.20547
$667$	8.14422	0.315346
$668$	−36.8679	−1.42646
$669$	−24.2229	−0.936511
$670$	0	0
$671$	0.955901	0.0369021
$672$	−50.6491	−1.95383
$673$	−33.5144	−1.29188	−0.645942	−	0.763386i	$-0.723535\pi$
$673$	−33.5144	−1.29188	−0.645942	+	0.763386i	$0.723535\pi$
$674$	−36.4827	−1.40526
$675$	0	0
$676$	16.8516	0.648140
$677$	20.9404	0.804807	0.402403	−	0.915462i	$-0.368175\pi$
$677$	20.9404	0.804807	0.402403	+	0.915462i	$0.368175\pi$
$678$	−76.2681	−2.92906
$679$	−0.694430	−0.0266498
$680$	0	0
$681$	73.2208	2.80583
$682$	−27.9549	−1.07045
$683$	−15.3782	−0.588430	−0.294215	−	0.955739i	$-0.595058\pi$
$683$	−15.3782	−0.588430	−0.294215	+	0.955739i	$0.595058\pi$
$684$	−210.504	−8.04882
$685$	0	0
$686$	−2.68001	−0.102323
$687$	−3.36340	−0.128322
$688$	−58.8432	−2.24338
$689$	26.7285	1.01827
$690$	0	0
$691$	21.2733	0.809273	0.404637	−	0.914478i	$-0.367398\pi$
$691$	21.2733	0.809273	0.404637	+	0.914478i	$0.367398\pi$
$692$	132.922	5.05293
$693$	6.74764	0.256322
$694$	57.7898	2.19367
$695$	0	0
$696$	−214.219	−8.11996
$697$	−18.3255	−0.694129
$698$	68.8489	2.60597
$699$	61.0483	2.30906
$700$	0	0
$701$	8.49681	0.320920	0.160460	−	0.987042i	$-0.448702\pi$
$701$	8.49681	0.320920	0.160460	+	0.987042i	$0.448702\pi$
$702$	−116.976	−4.41498
$703$	−29.5767	−1.11551
$704$	−19.7212	−0.743270
$705$	0	0
$706$	−29.1516	−1.09714
$707$	−9.42415	−0.354432
$708$	84.0435	3.15855
$709$	−20.2759	−0.761477	−0.380739	−	0.924683i	$-0.624330\pi$
$709$	−20.2759	−0.761477	−0.380739	+	0.924683i	$0.624330\pi$
$710$	0	0
$711$	27.2786	1.02303
$712$	25.1001	0.940667
$713$	−10.0647	−0.376926
$714$	−34.0358	−1.27376
$715$	0	0
$716$	36.7507	1.37344
$717$	52.9610	1.97786
$718$	55.3611	2.06606
$719$	31.2522	1.16551	0.582754	−	0.812648i	$-0.301975\pi$
$719$	31.2522	1.16551	0.582754	+	0.812648i	$0.301975\pi$
$720$	0	0
$721$	−10.0654	−0.374854
$722$	53.3880	1.98690
$723$	−81.4468	−3.02904
$724$	36.7438	1.36557
$725$	0	0
$726$	−82.0381	−3.04472
$727$	32.3133	1.19844	0.599218	−	0.800586i	$-0.295478\pi$
$727$	32.3133	1.19844	0.599218	+	0.800586i	$0.295478\pi$
$728$	34.3835	1.27434
$729$	−9.94509	−0.368337
$730$	0	0
$731$	−19.3963	−0.717399
$732$	−14.7414	−0.544857
$733$	30.4247	1.12376	0.561881	−	0.827218i	$-0.310078\pi$
$733$	30.4247	1.12376	0.561881	+	0.827218i	$0.310078\pi$
$734$	−79.6679	−2.94060
$735$	0	0
$736$	−16.4234	−0.605376
$737$	10.2755	0.378504
$738$	−77.6488	−2.85829
$739$	39.6163	1.45731	0.728654	−	0.684882i	$-0.240146\pi$
$739$	39.6163	1.45731	0.728654	+	0.684882i	$0.240146\pi$
$740$	0	0
$741$	77.5618	2.84930
$742$	17.7690	0.652319
$743$	14.2833	0.524003	0.262001	−	0.965068i	$-0.415618\pi$
$743$	14.2833	0.524003	0.262001	+	0.965068i	$0.415618\pi$
$744$	264.734	9.70563
$745$	0	0
$746$	−81.4089	−2.98059
$747$	−89.6440	−3.27990
$748$	−22.1181	−0.808717
$749$	5.34125	0.195165
$750$	0	0
$751$	32.3361	1.17996	0.589980	−	0.807418i	$-0.299136\pi$
$751$	32.3361	1.17996	0.589980	+	0.807418i	$0.299136\pi$
$752$	57.1208	2.08298
$753$	−46.4081	−1.69121
$754$	87.9904	3.20442
$755$	0	0
$756$	−56.1110	−2.04074
$757$	−17.4509	−0.634263	−0.317131	−	0.948382i	$-0.602720\pi$
$757$	−17.4509	−0.634263	−0.317131	+	0.948382i	$0.602720\pi$
$758$	−2.88983	−0.104963
$759$	3.19615	0.116013
$760$	0	0
$761$	10.2059	0.369962	0.184981	−	0.982742i	$-0.440778\pi$
$761$	10.2059	0.369962	0.184981	+	0.982742i	$0.440778\pi$
$762$	−59.7358	−2.16400
$763$	4.52282	0.163737
$764$	101.520	3.67285
$765$	0	0
$766$	20.3338	0.734689
$767$	−21.1987	−0.765439
$768$	32.6487	1.17811
$769$	28.8172	1.03917	0.519587	−	0.854418i	$-0.326086\pi$
$769$	28.8172	1.03917	0.519587	+	0.854418i	$0.326086\pi$
$770$	0	0
$771$	−57.4995	−2.07080
$772$	−62.6959	−2.25648
$773$	−3.93807	−0.141642	−0.0708212	−	0.997489i	$-0.522562\pi$
$773$	−3.93807	−0.141642	−0.0708212	+	0.997489i	$0.522562\pi$
$774$	−82.1860	−2.95411
$775$	0	0
$776$	5.92284	0.212617
$777$	−14.6206	−0.524513
$778$	53.1836	1.90672
$779$	27.7623	0.994689
$780$	0	0
$781$	3.68154	0.131736
$782$	−11.0364	−0.394662
$783$	−88.1781	−3.15123
$784$	12.4931	0.446181
$785$	0	0
$786$	−103.398	−3.68809
$787$	−30.9868	−1.10456	−0.552281	−	0.833658i	$-0.686242\pi$
$787$	−30.9868	−1.10456	−0.552281	+	0.833658i	$0.686242\pi$
$788$	11.9128	0.424375
$789$	−6.98068	−0.248519
$790$	0	0
$791$	9.22779	0.328103
$792$	−57.5511	−2.04499
$793$	3.71828	0.132040
$794$	34.8063	1.23523
$795$	0	0
$796$	83.5499	2.96135
$797$	−32.8018	−1.16190	−0.580949	−	0.813940i	$-0.697319\pi$
$797$	−32.8018	−1.16190	−0.580949	+	0.813940i	$0.697319\pi$
$798$	51.5627	1.82530
$799$	18.8285	0.666106
$800$	0	0
$801$	19.1605	0.677004
$802$	27.5055	0.971252
$803$	−10.7291	−0.378622
$804$	−158.464	−5.58858
$805$	0	0
$806$	−108.739	−3.83018
$807$	7.47038	0.262970
$808$	80.3792	2.82773
$809$	3.96801	0.139508	0.0697539	−	0.997564i	$-0.477779\pi$
$809$	3.96801	0.139508	0.0697539	+	0.997564i	$0.477779\pi$
$810$	0	0
$811$	−50.1963	−1.76263	−0.881315	−	0.472528i	$-0.843341\pi$
$811$	−50.1963	−1.76263	−0.881315	+	0.472528i	$0.843341\pi$
$812$	42.2072	1.48118
$813$	−3.31374	−0.116218
$814$	−13.1679	−0.461533
$815$	0	0
$816$	158.660	5.55422
$817$	29.3845	1.02803
$818$	62.9559	2.20120
$819$	26.2471	0.917149
$820$	0	0
$821$	26.7036	0.931963	0.465982	−	0.884794i	$-0.345701\pi$
$821$	26.7036	0.931963	0.465982	+	0.884794i	$0.345701\pi$
$822$	66.2008	2.30902
$823$	52.5717	1.83253	0.916267	−	0.400567i	$-0.131187\pi$
$823$	52.5717	1.83253	0.916267	+	0.400567i	$0.131187\pi$
$824$	85.8482	2.99066
$825$	0	0
$826$	−14.0928	−0.490350
$827$	−12.2541	−0.426116	−0.213058	−	0.977040i	$-0.568342\pi$
$827$	−12.2541	−0.426116	−0.213058	+	0.977040i	$0.568342\pi$
$828$	−33.7419	−1.17261
$829$	−31.4723	−1.09308	−0.546540	−	0.837433i	$-0.684055\pi$
$829$	−31.4723	−1.09308	−0.546540	+	0.837433i	$0.684055\pi$
$830$	0	0
$831$	−56.6399	−1.96482
$832$	−76.7118	−2.65950
$833$	4.11805	0.142682
$834$	−127.284	−4.40748
$835$	0	0
$836$	33.5079	1.15889
$837$	108.971	3.76660
$838$	−27.4511	−0.948283
$839$	−48.5824	−1.67725	−0.838625	−	0.544710i	$-0.816640\pi$
$839$	−48.5824	−1.67725	−0.838625	+	0.544710i	$0.816640\pi$
$840$	0	0
$841$	37.3283	1.28718
$842$	64.2861	2.21545
$843$	73.2701	2.52356
$844$	17.2010	0.592081
$845$	0	0
$846$	79.7802	2.74290
$847$	9.92591	0.341058
$848$	−82.8312	−2.84443
$849$	76.0099	2.60865
$850$	0	0
$851$	−4.74088	−0.162515
$852$	−56.7748	−1.94507
$853$	32.3500	1.10764	0.553821	−	0.832636i	$-0.313169\pi$
$853$	32.3500	1.10764	0.553821	+	0.832636i	$0.313169\pi$
$854$	2.47190	0.0845866
$855$	0	0
$856$	−45.5558	−1.55707
$857$	43.0385	1.47017	0.735084	−	0.677976i	$-0.237143\pi$
$857$	43.0385	1.47017	0.735084	+	0.677976i	$0.237143\pi$
$858$	34.5313	1.17888
$859$	21.0942	0.719726	0.359863	−	0.933005i	$-0.382824\pi$
$859$	21.0942	0.719726	0.359863	+	0.933005i	$0.382824\pi$
$860$	0	0
$861$	13.7238	0.467705
$862$	44.7091	1.52280
$863$	8.34066	0.283919	0.141960	−	0.989872i	$-0.454660\pi$
$863$	8.34066	0.283919	0.141960	+	0.989872i	$0.454660\pi$
$864$	177.818	6.04948
$865$	0	0
$866$	−13.4657	−0.457583
$867$	−0.128547	−0.00436568
$868$	−52.1600	−1.77043
$869$	−4.34219	−0.147299
$870$	0	0
$871$	39.9700	1.35433
$872$	−38.5754	−1.30633
$873$	4.52128	0.153022
$874$	16.7197	0.565551
$875$	0	0
$876$	165.458	5.59032
$877$	−11.4469	−0.386534	−0.193267	−	0.981146i	$-0.561908\pi$
$877$	−11.4469	−0.386534	−0.193267	+	0.981146i	$0.561908\pi$
$878$	−53.5765	−1.80812
$879$	14.9074	0.502813
$880$	0	0
$881$	−26.5224	−0.893562	−0.446781	−	0.894643i	$-0.647430\pi$
$881$	−26.5224	−0.893562	−0.446781	+	0.894643i	$0.647430\pi$
$882$	17.4490	0.587537
$883$	−2.39513	−0.0806025	−0.0403012	−	0.999188i	$-0.512832\pi$
$883$	−2.39513	−0.0806025	−0.0403012	+	0.999188i	$0.512832\pi$
$884$	−86.0354	−2.89368
$885$	0	0
$886$	−17.1788	−0.577133
$887$	−40.1901	−1.34945	−0.674726	−	0.738068i	$-0.735738\pi$
$887$	−40.1901	−1.34945	−0.674726	+	0.738068i	$0.735738\pi$
$888$	124.700	4.18467
$889$	7.22753	0.242403
$890$	0	0
$891$	−14.3620	−0.481146
$892$	−40.7057	−1.36293
$893$	−28.5244	−0.954532
$894$	−137.583	−4.60147
$895$	0	0
$896$	−18.1508	−0.606376
$897$	12.4325	0.415108
$898$	−53.9811	−1.80137
$899$	−81.9692	−2.73383
$900$	0	0
$901$	−27.3034	−0.909608
$902$	12.3601	0.411546
$903$	14.5257	0.483384
$904$	−78.7044	−2.61767
$905$	0	0
$906$	74.7163	2.48228
$907$	−59.7687	−1.98459	−0.992294	−	0.123907i	$-0.960457\pi$
$907$	−59.7687	−1.98459	−0.992294	+	0.123907i	$0.960457\pi$
$908$	123.045	4.08339
$909$	61.3585	2.03513
$910$	0	0
$911$	−8.52185	−0.282341	−0.141171	−	0.989985i	$-0.545087\pi$
$911$	−8.52185	−0.282341	−0.141171	+	0.989985i	$0.545087\pi$
$912$	−240.363	−7.95921
$913$	14.2695	0.472250
$914$	−34.7764	−1.15030
$915$	0	0
$916$	−5.65206	−0.186749
$917$	12.5103	0.413126
$918$	119.492	3.94383
$919$	7.32586	0.241658	0.120829	−	0.992673i	$-0.461445\pi$
$919$	7.32586	0.241658	0.120829	+	0.992673i	$0.461445\pi$
$920$	0	0
$921$	−30.6881	−1.01121
$922$	−62.7253	−2.06575
$923$	14.3205	0.471366
$924$	16.5640	0.544914
$925$	0	0
$926$	−40.6600	−1.33617
$927$	65.5334	2.15240
$928$	−133.756	−4.39076
$929$	31.8080	1.04359	0.521793	−	0.853072i	$-0.325263\pi$
$929$	31.8080	1.04359	0.521793	+	0.853072i	$0.325263\pi$
$930$	0	0
$931$	−6.23865	−0.204464
$932$	102.589	3.36043
$933$	52.7787	1.72790
$934$	18.6817	0.611283
$935$	0	0
$936$	−223.863	−7.31720
$937$	−16.9706	−0.554405	−0.277203	−	0.960811i	$-0.589407\pi$
$937$	−16.9706	−0.554405	−0.277203	+	0.960811i	$0.589407\pi$
$938$	26.5719	0.867602
$939$	45.5763	1.48733
$940$	0	0
$941$	45.6926	1.48954	0.744769	−	0.667323i	$-0.232560\pi$
$941$	45.6926	1.48954	0.744769	+	0.667323i	$0.232560\pi$
$942$	109.360	3.56313
$943$	4.45005	0.144914
$944$	65.6944	2.13817
$945$	0	0
$946$	13.0823	0.425342
$947$	−18.3647	−0.596771	−0.298386	−	0.954445i	$-0.596448\pi$
$947$	−18.3647	−0.596771	−0.298386	+	0.954445i	$0.596448\pi$
$948$	66.9629	2.17485
$949$	−41.7343	−1.35475
$950$	0	0
$951$	39.1452	1.26937
$952$	−35.1231	−1.13835
$953$	−32.2793	−1.04563	−0.522815	−	0.852446i	$-0.675118\pi$
$953$	−32.2793	−1.04563	−0.522815	+	0.852446i	$0.675118\pi$
$954$	−115.690	−3.74559
$955$	0	0
$956$	88.9989	2.87843
$957$	26.0302	0.841436
$958$	9.37313	0.302832
$959$	−8.00974	−0.258648
$960$	0	0
$961$	70.2983	2.26769
$962$	−51.2206	−1.65142
$963$	−34.7757	−1.12063
$964$	−136.868	−4.40823
$965$	0	0
$966$	8.26504	0.265923
$967$	17.5358	0.563913	0.281956	−	0.959427i	$-0.409017\pi$
$967$	17.5358	0.563913	0.281956	+	0.959427i	$0.409017\pi$
$968$	−84.6587	−2.72103
$969$	−79.2301	−2.54524
$970$	0	0
$971$	39.8737	1.27961	0.639804	−	0.768538i	$-0.279016\pi$
$971$	39.8737	1.27961	0.639804	+	0.768538i	$0.279016\pi$
$972$	53.1506	1.70481
$973$	15.4003	0.493711
$974$	−4.22974	−0.135529
$975$	0	0
$976$	−11.5229	−0.368839
$977$	−4.11183	−0.131549	−0.0657745	−	0.997835i	$-0.520952\pi$
$977$	−4.11183	−0.131549	−0.0657745	+	0.997835i	$0.520952\pi$
$978$	48.7415	1.55858
$979$	−3.04996	−0.0974772
$980$	0	0
$981$	−29.4471	−0.940172
$982$	−91.3167	−2.91403
$983$	2.63477	0.0840360	0.0420180	−	0.999117i	$-0.486621\pi$
$983$	2.63477	0.0840360	0.0420180	+	0.999117i	$0.486621\pi$
$984$	−117.051	−3.73144
$985$	0	0
$986$	−89.8831	−2.86246
$987$	−14.1005	−0.448823
$988$	130.340	4.14666
$989$	4.71008	0.149772
$990$	0	0
$991$	−1.03819	−0.0329790	−0.0164895	−	0.999864i	$-0.505249\pi$
$991$	−1.03819	−0.0329790	−0.0164895	+	0.999864i	$0.505249\pi$
$992$	165.297	5.24818
$993$	84.2361	2.67315
$994$	9.52023	0.301963
$995$	0	0
$996$	−220.056	−6.97274
$997$	−28.6033	−0.905875	−0.452938	−	0.891542i	$-0.649624\pi$
$997$	−28.6033	−0.905875	−0.452938	+	0.891542i	$0.649624\pi$
$998$	−57.8263	−1.83046
$999$	51.3298	1.62400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.bd.1.20		21
5.2	odd	4		805.2.c.c.484.39	yes	42
5.3	odd	4		805.2.c.c.484.4	✓	42
5.4	even	2		4025.2.a.be.1.2		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.c.c.484.4	✓	42	5.3	odd	4
805.2.c.c.484.39	yes	42	5.2	odd	4
4025.2.a.bd.1.20		21	1.1	even	1	trivial
4025.2.a.be.1.2		21	5.4	even	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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68001 q^{2} +3.08395 q^{3} +5.18247 q^{4} +8.26504 q^{6} -1.00000 q^{7} +8.52906 q^{8} +6.51078 q^{9} +O(q^{10})\)	\(q+2.68001 q^{2} +3.08395 q^{3} +5.18247 q^{4} +8.26504 q^{6} -1.00000 q^{7} +8.52906 q^{8} +6.51078 q^{9} -1.03638 q^{11} +15.9825 q^{12} -4.03133 q^{13} -2.68001 q^{14} +12.4931 q^{16} +4.11805 q^{17} +17.4490 q^{18} -6.23865 q^{19} -3.08395 q^{21} -2.77751 q^{22} -1.00000 q^{23} +26.3032 q^{24} -10.8040 q^{26} +10.8271 q^{27} -5.18247 q^{28} -8.14422 q^{29} +10.0647 q^{31} +16.4234 q^{32} -3.19615 q^{33} +11.0364 q^{34} +33.7419 q^{36} +4.74088 q^{37} -16.7197 q^{38} -12.4325 q^{39} -4.45005 q^{41} -8.26504 q^{42} -4.71008 q^{43} -5.37101 q^{44} -2.68001 q^{46} +4.57220 q^{47} +38.5280 q^{48} +1.00000 q^{49} +12.6999 q^{51} -20.8923 q^{52} -6.63018 q^{53} +29.0167 q^{54} -8.52906 q^{56} -19.2397 q^{57} -21.8266 q^{58} +5.25847 q^{59} -0.922345 q^{61} +26.9735 q^{62} -6.51078 q^{63} +19.0289 q^{64} -8.56573 q^{66} -9.91482 q^{67} +21.3417 q^{68} -3.08395 q^{69} -3.55231 q^{71} +55.5308 q^{72} +10.3525 q^{73} +12.7056 q^{74} -32.3316 q^{76} +1.03638 q^{77} -33.3191 q^{78} +4.18977 q^{79} +13.8579 q^{81} -11.9262 q^{82} -13.7686 q^{83} -15.9825 q^{84} -12.6231 q^{86} -25.1164 q^{87} -8.83935 q^{88} +2.94290 q^{89} +4.03133 q^{91} -5.18247 q^{92} +31.0391 q^{93} +12.2536 q^{94} +50.6491 q^{96} +0.694430 q^{97} +2.68001 q^{98} -6.74764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9}+O(q^{10}) \) 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9	\( 21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9} + 7 q^{11} + 22 q^{12} + 3 q^{13} + 2 q^{14} + 56 q^{16} - 7 q^{17} + 24 q^{19} - q^{21} - 4 q^{22} - 21 q^{23} + 24 q^{24} - 2 q^{26} + 19 q^{27} - 30 q^{28} + 11 q^{29} + 46 q^{31} + 6 q^{32} + 3 q^{33} + 28 q^{34} + 58 q^{36} - 24 q^{37} + 4 q^{38} + 31 q^{39} + 14 q^{41} - 6 q^{42} - 18 q^{43} + 12 q^{44} + 2 q^{46} + 25 q^{47} + 36 q^{48} + 21 q^{49} + 17 q^{51} + 8 q^{52} - 22 q^{53} - 6 q^{54} + 6 q^{56} - 40 q^{57} - 6 q^{58} + 10 q^{59} + 38 q^{61} + 54 q^{62} - 30 q^{63} + 100 q^{64} + 38 q^{66} - 12 q^{67} - 18 q^{68} - q^{69} + 56 q^{71} - 42 q^{72} + 40 q^{73} - 20 q^{74} + 60 q^{76} - 7 q^{77} - 38 q^{78} + 49 q^{79} + 57 q^{81} - 16 q^{82} + 2 q^{83} - 22 q^{84} + 16 q^{86} + 23 q^{87} - 12 q^{88} + 28 q^{89} - 3 q^{91} - 30 q^{92} + 30 q^{93} + 66 q^{94} + 46 q^{96} + q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9 + 7 * q^11 + 22 * q^12 + 3 * q^13 + 2 * q^14 + 56 * q^16 - 7 * q^17 + 24 * q^19 - q^21 - 4 * q^22 - 21 * q^23 + 24 * q^24 - 2 * q^26 + 19 * q^27 - 30 * q^28 + 11 * q^29 + 46 * q^31 + 6 * q^32 + 3 * q^33 + 28 * q^34 + 58 * q^36 - 24 * q^37 + 4 * q^38 + 31 * q^39 + 14 * q^41 - 6 * q^42 - 18 * q^43 + 12 * q^44 + 2 * q^46 + 25 * q^47 + 36 * q^48 + 21 * q^49 + 17 * q^51 + 8 * q^52 - 22 * q^53 - 6 * q^54 + 6 * q^56 - 40 * q^57 - 6 * q^58 + 10 * q^59 + 38 * q^61 + 54 * q^62 - 30 * q^63 + 100 * q^64 + 38 * q^66 - 12 * q^67 - 18 * q^68 - q^69 + 56 * q^71 - 42 * q^72 + 40 * q^73 - 20 * q^74 + 60 * q^76 - 7 * q^77 - 38 * q^78 + 49 * q^79 + 57 * q^81 - 16 * q^82 + 2 * q^83 - 22 * q^84 + 16 * q^86 + 23 * q^87 - 12 * q^88 + 28 * q^89 - 3 * q^91 - 30 * q^92 + 30 * q^93 + 66 * q^94 + 46 * q^96 + q^97 - 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more