LMFDB - Embedded newform 1045.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.47888$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.47888 q^{2} -2.61641 q^{3} +22.0181 q^{4} +5.00000 q^{5} +14.3350 q^{6} -5.28814 q^{7} -76.8037 q^{8} -20.1544 q^{9} +O(q^{10})$	\(q-5.47888 q^{2} -2.61641 q^{3} +22.0181 q^{4} +5.00000 q^{5} +14.3350 q^{6} -5.28814 q^{7} -76.8037 q^{8} -20.1544 q^{9} -27.3944 q^{10} +11.0000 q^{11} -57.6084 q^{12} -39.7591 q^{13} +28.9731 q^{14} -13.0820 q^{15} +244.653 q^{16} +64.4867 q^{17} +110.424 q^{18} +19.0000 q^{19} +110.091 q^{20} +13.8359 q^{21} -60.2677 q^{22} -191.874 q^{23} +200.950 q^{24} +25.0000 q^{25} +217.835 q^{26} +123.375 q^{27} -116.435 q^{28} +222.545 q^{29} +71.6749 q^{30} +188.084 q^{31} -725.995 q^{32} -28.7805 q^{33} -353.315 q^{34} -26.4407 q^{35} -443.763 q^{36} +361.329 q^{37} -104.099 q^{38} +104.026 q^{39} -384.018 q^{40} -140.965 q^{41} -75.8054 q^{42} -348.345 q^{43} +242.199 q^{44} -100.772 q^{45} +1051.25 q^{46} -226.137 q^{47} -640.112 q^{48} -315.036 q^{49} -136.972 q^{50} -168.723 q^{51} -875.421 q^{52} -287.816 q^{53} -675.957 q^{54} +55.0000 q^{55} +406.149 q^{56} -49.7117 q^{57} -1219.30 q^{58} -237.919 q^{59} -288.042 q^{60} -505.931 q^{61} -1030.49 q^{62} +106.579 q^{63} +2020.42 q^{64} -198.796 q^{65} +157.685 q^{66} +1022.68 q^{67} +1419.88 q^{68} +502.019 q^{69} +144.865 q^{70} +1103.96 q^{71} +1547.93 q^{72} +450.787 q^{73} -1979.68 q^{74} -65.4102 q^{75} +418.344 q^{76} -58.1696 q^{77} -569.946 q^{78} -196.574 q^{79} +1223.26 q^{80} +221.370 q^{81} +772.331 q^{82} +80.6312 q^{83} +304.641 q^{84} +322.433 q^{85} +1908.54 q^{86} -582.267 q^{87} -844.840 q^{88} +897.644 q^{89} +552.118 q^{90} +210.252 q^{91} -4224.70 q^{92} -492.104 q^{93} +1238.98 q^{94} +95.0000 q^{95} +1899.50 q^{96} +1423.40 q^{97} +1726.04 q^{98} -221.699 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	−5.47888	−1.93708	−0.968538	−	0.248864i	$-0.919943\pi$
$2$	−5.47888	−1.93708	−0.968538	+	0.248864i	$0.919943\pi$
$3$	−2.61641	−0.503528	−0.251764	−	0.967789i	$-0.581011\pi$
$3$	−2.61641	−0.503528	−0.251764	+	0.967789i	$0.581011\pi$
$4$	22.0181	2.75227
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	14.3350	0.975372
$7$	−5.28814	−0.285533	−0.142766	−	0.989756i	$-0.545600\pi$
$7$	−5.28814	−0.285533	−0.142766	+	0.989756i	$0.545600\pi$
$8$	−76.8037	−3.39427
$9$	−20.1544	−0.746460
$10$	−27.3944	−0.866287
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−57.6084	−1.38584
$13$	−39.7591	−0.848245	−0.424123	−	0.905605i	$-0.639417\pi$
$13$	−39.7591	−0.848245	−0.424123	+	0.905605i	$0.639417\pi$
$14$	28.9731	0.553099
$15$	−13.0820	−0.225184
$16$	244.653	3.82270
$17$	64.4867	0.920018	0.460009	−	0.887914i	$-0.347846\pi$
$17$	64.4867	0.920018	0.460009	+	0.887914i	$0.347846\pi$
$18$	110.424	1.44595
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	110.091	1.23085
$21$	13.8359	0.143774
$22$	−60.2677	−0.584051
$23$	−191.874	−1.73950	−0.869748	−	0.493496i	$-0.835719\pi$
$23$	−191.874	−1.73950	−0.869748	+	0.493496i	$0.835719\pi$
$24$	200.950	1.70911
$25$	25.0000	0.200000
$26$	217.835	1.64312
$27$	123.375	0.879391
$28$	−116.435	−0.785862
$29$	222.545	1.42502	0.712509	−	0.701663i	$-0.247559\pi$
$29$	222.545	1.42502	0.712509	+	0.701663i	$0.247559\pi$
$30$	71.6749	0.436199
$31$	188.084	1.08971	0.544853	−	0.838532i	$-0.316585\pi$
$31$	188.084	1.08971	0.544853	+	0.838532i	$0.316585\pi$
$32$	−725.995	−4.01060
$33$	−28.7805	−0.151819
$34$	−353.315	−1.78215
$35$	−26.4407	−0.127694
$36$	−443.763	−2.05446
$37$	361.329	1.60546	0.802730	−	0.596342i	$-0.203380\pi$
$37$	361.329	1.60546	0.802730	+	0.596342i	$0.203380\pi$
$38$	−104.099	−0.444396
$39$	104.026	0.427115
$40$	−384.018	−1.51797
$41$	−140.965	−0.536953	−0.268476	−	0.963286i	$-0.586520\pi$
$41$	−140.965	−0.536953	−0.268476	+	0.963286i	$0.586520\pi$
$42$	−75.8054	−0.278501
$43$	−348.345	−1.23540	−0.617699	−	0.786415i	$-0.711935\pi$
$43$	−348.345	−1.23540	−0.617699	+	0.786415i	$0.711935\pi$
$44$	242.199	0.829839
$45$	−100.772	−0.333827
$46$	1051.25	3.36954
$47$	−226.137	−0.701818	−0.350909	−	0.936409i	$-0.614127\pi$
$47$	−226.137	−0.701818	−0.350909	+	0.936409i	$0.614127\pi$
$48$	−640.112	−1.92484
$49$	−315.036	−0.918471
$50$	−136.972	−0.387415
$51$	−168.723	−0.463255
$52$	−875.421	−2.33460
$53$	−287.816	−0.745935	−0.372967	−	0.927845i	$-0.621660\pi$
$53$	−287.816	−0.745935	−0.372967	+	0.927845i	$0.621660\pi$
$54$	−675.957	−1.70345
$55$	55.0000	0.134840
$56$	406.149	0.969177
$57$	−49.7117	−0.115517
$58$	−1219.30	−2.76037
$59$	−237.919	−0.524990	−0.262495	−	0.964933i	$-0.584545\pi$
$59$	−237.919	−0.524990	−0.262495	+	0.964933i	$0.584545\pi$
$60$	−288.042	−0.619767
$61$	−505.931	−1.06193	−0.530966	−	0.847393i	$-0.678171\pi$
$61$	−505.931	−1.06193	−0.530966	+	0.847393i	$0.678171\pi$
$62$	−1030.49	−2.11084
$63$	106.579	0.213139
$64$	2020.42	3.94613
$65$	−198.796	−0.379347
$66$	157.685	0.294086
$67$	1022.68	1.86479	0.932394	−	0.361443i	$-0.117716\pi$
$67$	1022.68	1.86479	0.932394	+	0.361443i	$0.117716\pi$
$68$	1419.88	2.53214
$69$	502.019	0.875884
$70$	144.865	0.247353
$71$	1103.96	1.84530	0.922651	−	0.385637i	$-0.126018\pi$
$71$	1103.96	1.84530	0.922651	+	0.385637i	$0.126018\pi$
$72$	1547.93	2.53369
$73$	450.787	0.722749	0.361374	−	0.932421i	$-0.382308\pi$
$73$	450.787	0.722749	0.361374	+	0.932421i	$0.382308\pi$
$74$	−1979.68	−3.10990
$75$	−65.4102	−0.100706
$76$	418.344	0.631413
$77$	−58.1696	−0.0860914
$78$	−569.946	−0.827355
$79$	−196.574	−0.279953	−0.139976	−	0.990155i	$-0.544703\pi$
$79$	−196.574	−0.279953	−0.139976	+	0.990155i	$0.544703\pi$
$80$	1223.26	1.70956
$81$	221.370	0.303662
$82$	772.331	1.04012
$83$	80.6312	0.106632	0.0533158	−	0.998578i	$-0.483021\pi$
$83$	80.6312	0.106632	0.0533158	+	0.998578i	$0.483021\pi$
$84$	304.641	0.395703
$85$	322.433	0.411445
$86$	1908.54	2.39306
$87$	−582.267	−0.717536
$88$	−844.840	−1.02341
$89$	897.644	1.06910	0.534551	−	0.845136i	$-0.320481\pi$
$89$	897.644	1.06910	0.534551	+	0.845136i	$0.320481\pi$
$90$	552.118	0.646649
$91$	210.252	0.242202
$92$	−4224.70	−4.78756
$93$	−492.104	−0.548697
$94$	1238.98	1.35948
$95$	95.0000	0.102598
$96$	1899.50	2.01945
$97$	1423.40	1.48994	0.744972	−	0.667096i	$-0.232463\pi$
$97$	1423.40	1.48994	0.744972	+	0.667096i	$0.232463\pi$
$98$	1726.04	1.77915
$99$	−221.699	−0.225066
$100$	550.453	0.550453
$101$	−123.580	−0.121749	−0.0608747	−	0.998145i	$-0.519389\pi$
$101$	−123.580	−0.121749	−0.0608747	+	0.998145i	$0.519389\pi$
$102$	924.415	0.897360
$103$	−29.1266	−0.0278634	−0.0139317	−	0.999903i	$-0.504435\pi$
$103$	−29.1266	−0.0278634	−0.0139317	+	0.999903i	$0.504435\pi$
$104$	3053.64	2.87918
$105$	69.1796	0.0642975
$106$	1576.91	1.44493
$107$	−783.431	−0.707824	−0.353912	−	0.935279i	$-0.615149\pi$
$107$	−783.431	−0.707824	−0.353912	+	0.935279i	$0.615149\pi$
$108$	2716.49	2.42032
$109$	835.864	0.734507	0.367253	−	0.930121i	$-0.380298\pi$
$109$	835.864	0.734507	0.367253	+	0.930121i	$0.380298\pi$
$110$	−301.338	−0.261195
$111$	−945.382	−0.808394
$112$	−1293.76	−1.09151
$113$	−1877.23	−1.56279	−0.781394	−	0.624038i	$-0.785491\pi$
$113$	−1877.23	−1.56279	−0.781394	+	0.624038i	$0.785491\pi$
$114$	272.365	0.223766
$115$	−959.368	−0.777926
$116$	4900.02	3.92203
$117$	801.322	0.633181
$118$	1303.53	1.01695
$119$	−341.015	−0.262695
$120$	1004.75	0.764338
$121$	121.000	0.0909091
$122$	2771.94	2.05704
$123$	368.822	0.270371
$124$	4141.26	2.99916
$125$	125.000	0.0894427
$126$	−583.936	−0.412866
$127$	−2053.36	−1.43469	−0.717347	−	0.696716i	$-0.754644\pi$
$127$	−2053.36	−1.43469	−0.717347	+	0.696716i	$0.754644\pi$
$128$	−5261.66	−3.63336
$129$	911.412	0.622057
$130$	1089.18	0.734824
$131$	−2398.66	−1.59978	−0.799892	−	0.600144i	$-0.795110\pi$
$131$	−2398.66	−1.59978	−0.799892	+	0.600144i	$0.795110\pi$
$132$	−633.692	−0.417847
$133$	−100.475	−0.0655057
$134$	−5603.17	−3.61224
$135$	616.876	0.393276
$136$	−4952.81	−3.12279
$137$	−1524.08	−0.950445	−0.475222	−	0.879866i	$-0.657632\pi$
$137$	−1524.08	−0.950445	−0.475222	+	0.879866i	$0.657632\pi$
$138$	−2750.50	−1.69666
$139$	720.990	0.439954	0.219977	−	0.975505i	$-0.429402\pi$
$139$	720.990	0.439954	0.219977	+	0.975505i	$0.429402\pi$
$140$	−582.175	−0.351448
$141$	591.666	0.353385
$142$	−6048.48	−3.57449
$143$	−437.350	−0.255756
$144$	−4930.84	−2.85349
$145$	1112.72	0.637287
$146$	−2469.81	−1.40002
$147$	824.261	0.462475
$148$	7955.78	4.41866
$149$	1218.12	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$149$	1218.12	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$150$	358.374	0.195074
$151$	−1733.87	−0.934438	−0.467219	−	0.884142i	$-0.654744\pi$
$151$	−1733.87	−0.934438	−0.467219	+	0.884142i	$0.654744\pi$
$152$	−1459.27	−0.778700
$153$	−1299.69	−0.686757
$154$	318.704	0.166766
$155$	940.420	0.487331
$156$	2290.46	1.17553
$157$	816.459	0.415035	0.207518	−	0.978231i	$-0.433462\pi$
$157$	816.459	0.415035	0.207518	+	0.978231i	$0.433462\pi$
$158$	1077.00	0.542290
$159$	753.043	0.375599
$160$	−3629.98	−1.79359
$161$	1014.65	0.496683
$162$	−1212.86	−0.588217
$163$	2556.28	1.22836	0.614181	−	0.789165i	$-0.289486\pi$
$163$	2556.28	1.22836	0.614181	+	0.789165i	$0.289486\pi$
$164$	−3103.79	−1.47784
$165$	−143.902	−0.0678956
$166$	−441.769	−0.206554
$167$	−4270.26	−1.97870	−0.989350	−	0.145557i	$-0.953503\pi$
$167$	−4270.26	−1.97870	−0.989350	+	0.145557i	$0.953503\pi$
$168$	−1062.65	−0.488007
$169$	−616.214	−0.280480
$170$	−1766.57	−0.797000
$171$	−382.934	−0.171250
$172$	−7669.90	−3.40014
$173$	−78.0383	−0.0342956	−0.0171478	−	0.999853i	$-0.505459\pi$
$173$	−78.0383	−0.0342956	−0.0171478	+	0.999853i	$0.505459\pi$
$174$	3190.17	1.38992
$175$	−132.204	−0.0571066
$176$	2691.18	1.15259
$177$	622.493	0.264347
$178$	−4918.08	−2.07093
$179$	3901.90	1.62928	0.814642	−	0.579964i	$-0.196933\pi$
$179$	3901.90	1.62928	0.814642	+	0.579964i	$0.196933\pi$
$180$	−2218.81	−0.918781
$181$	−609.545	−0.250316	−0.125158	−	0.992137i	$-0.539944\pi$
$181$	−609.545	−0.250316	−0.125158	+	0.992137i	$0.539944\pi$
$182$	−1151.94	−0.469164
$183$	1323.72	0.534712
$184$	14736.6	5.90433
$185$	1806.64	0.717984
$186$	2696.18	1.06287
$187$	709.353	0.277396
$188$	−4979.11	−1.93159
$189$	−652.425	−0.251095
$190$	−520.494	−0.198740
$191$	1102.56	0.417689	0.208844	−	0.977949i	$-0.433030\pi$
$191$	1102.56	0.417689	0.208844	+	0.977949i	$0.433030\pi$
$192$	−5286.23	−1.98698
$193$	−731.225	−0.272719	−0.136359	−	0.990659i	$-0.543540\pi$
$193$	−731.225	−0.272719	−0.136359	+	0.990659i	$0.543540\pi$
$194$	−7798.64	−2.88613
$195$	520.130	0.191012
$196$	−6936.49	−2.52788
$197$	−2667.26	−0.964642	−0.482321	−	0.875995i	$-0.660206\pi$
$197$	−2667.26	−0.964642	−0.482321	+	0.875995i	$0.660206\pi$
$198$	1214.66	0.435970
$199$	4170.49	1.48562	0.742810	−	0.669503i	$-0.233493\pi$
$199$	4170.49	1.48562	0.742810	+	0.669503i	$0.233493\pi$
$200$	−1920.09	−0.678855
$201$	−2675.76	−0.938972
$202$	677.081	0.235838
$203$	−1176.85	−0.406889
$204$	−3714.97	−1.27500
$205$	−704.826	−0.240133
$206$	159.581	0.0539735
$207$	3867.10	1.29846
$208$	−9727.18	−3.24259
$209$	209.000	0.0691714
$210$	−379.027	−0.124549
$211$	447.757	0.146089	0.0730447	−	0.997329i	$-0.476728\pi$
$211$	447.757	0.146089	0.0730447	+	0.997329i	$0.476728\pi$
$212$	−6337.16	−2.05301
$213$	−2888.42	−0.929160
$214$	4292.33	1.37111
$215$	−1741.72	−0.552487
$216$	−9475.66	−2.98489
$217$	−994.615	−0.311147
$218$	−4579.60	−1.42280
$219$	−1179.44	−0.363924
$220$	1211.00	0.371115
$221$	−2563.93	−0.780402
$222$	5179.64	1.56592
$223$	3331.27	1.00035	0.500176	−	0.865924i	$-0.333269\pi$
$223$	3331.27	1.00035	0.500176	+	0.865924i	$0.333269\pi$
$224$	3839.17	1.14516
$225$	−503.860	−0.149292
$226$	10285.1	3.02724
$227$	−3425.96	−1.00171	−0.500857	−	0.865530i	$-0.666982\pi$
$227$	−3425.96	−1.00171	−0.500857	+	0.865530i	$0.666982\pi$
$228$	−1094.56	−0.317934
$229$	−1556.52	−0.449161	−0.224580	−	0.974456i	$-0.572101\pi$
$229$	−1556.52	−0.449161	−0.224580	+	0.974456i	$0.572101\pi$
$230$	5256.26	1.50690
$231$	152.195	0.0433494
$232$	−17092.2	−4.83690
$233$	897.777	0.252426	0.126213	−	0.992003i	$-0.459718\pi$
$233$	897.777	0.252426	0.126213	+	0.992003i	$0.459718\pi$
$234$	−4390.35	−1.22652
$235$	−1130.68	−0.313863
$236$	−5238.53	−1.44491
$237$	514.317	0.140964
$238$	1868.38	0.508861
$239$	−2546.17	−0.689113	−0.344557	−	0.938766i	$-0.611971\pi$
$239$	−2546.17	−0.689113	−0.344557	+	0.938766i	$0.611971\pi$
$240$	−3200.56	−0.860813
$241$	1840.90	0.492044	0.246022	−	0.969264i	$-0.420877\pi$
$241$	1840.90	0.492044	0.246022	+	0.969264i	$0.420877\pi$
$242$	−662.945	−0.176098
$243$	−3910.32	−1.03229
$244$	−11139.7	−2.92272
$245$	−1575.18	−0.410753
$246$	−2020.73	−0.523729
$247$	−755.423	−0.194601
$248$	−14445.5	−3.69876
$249$	−210.964	−0.0536920
$250$	−684.860	−0.173257
$251$	−5123.87	−1.28851	−0.644254	−	0.764812i	$-0.722832\pi$
$251$	−5123.87	−1.28851	−0.644254	+	0.764812i	$0.722832\pi$
$252$	2346.68	0.586615
$253$	−2110.61	−0.524478
$254$	11250.1	2.77911
$255$	−843.617	−0.207174
$256$	12664.7	3.09196
$257$	−7205.25	−1.74884	−0.874419	−	0.485172i	$-0.838757\pi$
$257$	−7205.25	−1.74884	−0.874419	+	0.485172i	$0.838757\pi$
$258$	−4993.51	−1.20497
$259$	−1910.76	−0.458412
$260$	−4377.11	−1.04406
$261$	−4485.26	−1.06372
$262$	13142.0	3.09890
$263$	−505.925	−0.118619	−0.0593093	−	0.998240i	$-0.518890\pi$
$263$	−505.925	−0.118619	−0.0593093	+	0.998240i	$0.518890\pi$
$264$	2210.45	0.515316
$265$	−1439.08	−0.333592
$266$	550.489	0.126890
$267$	−2348.60	−0.538322
$268$	22517.6	5.13239
$269$	−4482.16	−1.01592	−0.507959	−	0.861381i	$-0.669600\pi$
$269$	−4482.16	−1.01592	−0.507959	+	0.861381i	$0.669600\pi$
$270$	−3379.79	−0.761805
$271$	−7155.46	−1.60392	−0.801962	−	0.597375i	$-0.796210\pi$
$271$	−7155.46	−1.60392	−0.801962	+	0.597375i	$0.796210\pi$
$272$	15776.9	3.51696
$273$	−550.104	−0.121955
$274$	8350.25	1.84108
$275$	275.000	0.0603023
$276$	11053.5	2.41067
$277$	2872.05	0.622978	0.311489	−	0.950250i	$-0.399172\pi$
$277$	2872.05	0.622978	0.311489	+	0.950250i	$0.399172\pi$
$278$	−3950.22	−0.852224
$279$	−3790.72	−0.813422
$280$	2030.74	0.433429
$281$	1134.34	0.240815	0.120407	−	0.992725i	$-0.461580\pi$
$281$	1134.34	0.240815	0.120407	+	0.992725i	$0.461580\pi$
$282$	−3241.67	−0.684534
$283$	7559.87	1.58794	0.793971	−	0.607956i	$-0.208010\pi$
$283$	7559.87	1.58794	0.793971	+	0.607956i	$0.208010\pi$
$284$	24307.2	5.07876
$285$	−248.559	−0.0516608
$286$	2396.19	0.495418
$287$	745.444	0.153318
$288$	14632.0	2.99375
$289$	−754.470	−0.153566
$290$	−6096.48	−1.23447
$291$	−3724.20	−0.750228
$292$	9925.49	1.98920
$293$	−2140.73	−0.426835	−0.213418	−	0.976961i	$-0.568459\pi$
$293$	−2140.73	−0.426835	−0.213418	+	0.976961i	$0.568459\pi$
$294$	−4516.03	−0.895851
$295$	−1189.59	−0.234783
$296$	−27751.4	−5.44937
$297$	1357.13	0.265146
$298$	−6673.93	−1.29735
$299$	7628.72	1.47552
$300$	−1440.21	−0.277168
$301$	1842.10	0.352747
$302$	9499.66	1.81008
$303$	323.336	0.0613041
$304$	4648.41	0.876988
$305$	−2529.66	−0.474910
$306$	7120.85	1.33030
$307$	8341.80	1.55079	0.775394	−	0.631478i	$-0.217551\pi$
$307$	8341.80	1.55079	0.775394	+	0.631478i	$0.217551\pi$
$308$	−1280.79	−0.236946
$309$	76.2069	0.0140300
$310$	−5152.45	−0.943998
$311$	−2614.67	−0.476734	−0.238367	−	0.971175i	$-0.576612\pi$
$311$	−2614.67	−0.476734	−0.238367	+	0.971175i	$0.576612\pi$
$312$	−7989.57	−1.44975
$313$	2807.28	0.506954	0.253477	−	0.967341i	$-0.418426\pi$
$313$	2807.28	0.506954	0.253477	+	0.967341i	$0.418426\pi$
$314$	−4473.28	−0.803955
$315$	532.897	0.0953186
$316$	−4328.18	−0.770505
$317$	5036.07	0.892283	0.446142	−	0.894962i	$-0.352798\pi$
$317$	5036.07	0.892283	0.446142	+	0.894962i	$0.352798\pi$
$318$	−4125.83	−0.727563
$319$	2447.99	0.429659
$320$	10102.1	1.76476
$321$	2049.77	0.356409
$322$	−5559.17	−0.962114
$323$	1225.25	0.211067
$324$	4874.15	0.835760
$325$	−993.978	−0.169649
$326$	−14005.5	−2.37943
$327$	−2186.96	−0.369844
$328$	10826.6	1.82257
$329$	1195.84	0.200392
$330$	788.424	0.131519
$331$	−7781.47	−1.29217	−0.646085	−	0.763265i	$-0.723595\pi$
$331$	−7781.47	−1.29217	−0.646085	+	0.763265i	$0.723595\pi$
$332$	1775.35	0.293479
$333$	−7282.37	−1.19841
$334$	23396.3	3.83289
$335$	5113.42	0.833959
$336$	3385.00	0.549604
$337$	−11454.5	−1.85153	−0.925763	−	0.378105i	$-0.876576\pi$
$337$	−11454.5	−1.85153	−0.925763	+	0.378105i	$0.876576\pi$
$338$	3376.16	0.543310
$339$	4911.60	0.786907
$340$	7099.38	1.13241
$341$	2068.92	0.328559
$342$	2098.05	0.331724
$343$	3479.79	0.547786
$344$	26754.2	4.19328
$345$	2510.10	0.391707
$346$	427.563	0.0664333
$347$	−869.433	−0.134506	−0.0672531	−	0.997736i	$-0.521423\pi$
$347$	−869.433	−0.134506	−0.0672531	+	0.997736i	$0.521423\pi$
$348$	−12820.4	−1.97485
$349$	−2974.25	−0.456184	−0.228092	−	0.973640i	$-0.573249\pi$
$349$	−2974.25	−0.456184	−0.228092	+	0.973640i	$0.573249\pi$
$350$	724.327	0.110620
$351$	−4905.28	−0.745939
$352$	−7985.95	−1.20924
$353$	−6426.36	−0.968954	−0.484477	−	0.874804i	$-0.660990\pi$
$353$	−6426.36	−0.968954	−0.484477	+	0.874804i	$0.660990\pi$
$354$	−3410.56	−0.512060
$355$	5519.82	0.825244
$356$	19764.4	2.94245
$357$	892.233	0.132274
$358$	−21378.1	−3.15605
$359$	−12663.8	−1.86175	−0.930877	−	0.365333i	$-0.880955\pi$
$359$	−12663.8	−1.86175	−0.930877	+	0.365333i	$0.880955\pi$
$360$	7739.67	1.13310
$361$	361.000	0.0526316
$362$	3339.63	0.484881
$363$	−316.585	−0.0457752
$364$	4629.35	0.666604
$365$	2253.94	0.323223
$366$	−7252.51	−1.03578
$367$	−2472.95	−0.351736	−0.175868	−	0.984414i	$-0.556273\pi$
$367$	−2472.95	−0.351736	−0.175868	+	0.984414i	$0.556273\pi$
$368$	−46942.4	−6.64958
$369$	2841.07	0.400814
$370$	−9898.38	−1.39079
$371$	1522.01	0.212989
$372$	−10835.2	−1.51016
$373$	5462.62	0.758294	0.379147	−	0.925336i	$-0.376217\pi$
$373$	5462.62	0.758294	0.379147	+	0.925336i	$0.376217\pi$
$374$	−3886.46	−0.537337
$375$	−327.051	−0.0450369
$376$	17368.1	2.38216
$377$	−8848.18	−1.20876
$378$	3574.56	0.486390
$379$	−13313.2	−1.80437	−0.902183	−	0.431354i	$-0.858036\pi$
$379$	−13313.2	−1.80437	−0.902183	+	0.431354i	$0.858036\pi$
$380$	2091.72	0.282377
$381$	5372.42	0.722408
$382$	−6040.80	−0.809095
$383$	−10374.4	−1.38409	−0.692045	−	0.721855i	$-0.743290\pi$
$383$	−10374.4	−1.38409	−0.692045	+	0.721855i	$0.743290\pi$
$384$	13766.6	1.82950
$385$	−290.848	−0.0385012
$386$	4006.30	0.528278
$387$	7020.69	0.922175
$388$	31340.6	4.10072
$389$	14026.3	1.82818	0.914088	−	0.405515i	$-0.132908\pi$
$389$	14026.3	1.82818	0.914088	+	0.405515i	$0.132908\pi$
$390$	−2849.73	−0.370004
$391$	−12373.3	−1.60037
$392$	24195.9	3.11754
$393$	6275.86	0.805536
$394$	14613.6	1.86859
$395$	−982.868	−0.125199
$396$	−4881.39	−0.619442
$397$	8448.30	1.06803	0.534015	−	0.845475i	$-0.320683\pi$
$397$	8448.30	1.06803	0.534015	+	0.845475i	$0.320683\pi$
$398$	−22849.6	−2.87776
$399$	262.883	0.0329839
$400$	6116.32	0.764541
$401$	−1731.10	−0.215579	−0.107789	−	0.994174i	$-0.534377\pi$
$401$	−1731.10	−0.215579	−0.107789	+	0.994174i	$0.534377\pi$
$402$	14660.2	1.81886
$403$	−7478.05	−0.924338
$404$	−2721.00	−0.335087
$405$	1106.85	0.135802
$406$	6447.81	0.788176
$407$	3974.61	0.484065
$408$	12958.6	1.57241
$409$	−5031.45	−0.608286	−0.304143	−	0.952626i	$-0.598370\pi$
$409$	−5031.45	−0.608286	−0.304143	+	0.952626i	$0.598370\pi$
$410$	3861.66	0.465155
$411$	3987.61	0.478575
$412$	−641.312	−0.0766874
$413$	1258.15	0.149902
$414$	−21187.4	−2.51522
$415$	403.156	0.0476871
$416$	28864.9	3.40197
$417$	−1886.40	−0.221529
$418$	−1145.09	−0.133990
$419$	−13018.7	−1.51791	−0.758957	−	0.651141i	$-0.774291\pi$
$419$	−13018.7	−1.51791	−0.758957	+	0.651141i	$0.774291\pi$
$420$	1523.21	0.176964
$421$	−8015.54	−0.927918	−0.463959	−	0.885857i	$-0.653572\pi$
$421$	−8015.54	−0.927918	−0.463959	+	0.885857i	$0.653572\pi$
$422$	−2453.21	−0.282987
$423$	4557.66	0.523879
$424$	22105.3	2.53191
$425$	1612.17	0.184004
$426$	15825.3	1.79985
$427$	2675.44	0.303216
$428$	−17249.7	−1.94812
$429$	1144.29	0.128780
$430$	9542.70	1.07021
$431$	−7743.27	−0.865384	−0.432692	−	0.901542i	$-0.642436\pi$
$431$	−7743.27	−0.865384	−0.432692	+	0.901542i	$0.642436\pi$
$432$	30184.1	3.36165
$433$	−12036.8	−1.33591	−0.667957	−	0.744200i	$-0.732831\pi$
$433$	−12036.8	−1.33591	−0.667957	+	0.744200i	$0.732831\pi$
$434$	5449.38	0.602715
$435$	−2911.34	−0.320892
$436$	18404.2	2.02156
$437$	−3645.60	−0.399068
$438$	6462.02	0.704949
$439$	−5792.60	−0.629762	−0.314881	−	0.949131i	$-0.601965\pi$
$439$	−5792.60	−0.629762	−0.314881	+	0.949131i	$0.601965\pi$
$440$	−4224.20	−0.457684
$441$	6349.36	0.685602
$442$	14047.5	1.51170
$443$	12209.2	1.30943	0.654715	−	0.755876i	$-0.272789\pi$
$443$	12209.2	1.30943	0.654715	+	0.755876i	$0.272789\pi$
$444$	−20815.6	−2.22492
$445$	4488.22	0.478117
$446$	−18251.6	−1.93776
$447$	−3187.09	−0.337236
$448$	−10684.3	−1.12675
$449$	−11137.5	−1.17062	−0.585312	−	0.810808i	$-0.699028\pi$
$449$	−11137.5	−1.17062	−0.585312	+	0.810808i	$0.699028\pi$
$450$	2760.59	0.289190
$451$	−1550.62	−0.161897
$452$	−41333.1	−4.30121
$453$	4536.50	0.470515
$454$	18770.4	1.94040
$455$	1051.26	0.108316
$456$	3818.04	0.392097
$457$	−5050.83	−0.516997	−0.258499	−	0.966012i	$-0.583228\pi$
$457$	−5050.83	−0.516997	−0.258499	+	0.966012i	$0.583228\pi$
$458$	8528.00	0.870059
$459$	7956.05	0.809056
$460$	−21123.5	−2.14106
$461$	−14495.7	−1.46449	−0.732247	−	0.681039i	$-0.761528\pi$
$461$	−14495.7	−1.46449	−0.732247	+	0.681039i	$0.761528\pi$
$462$	−833.859	−0.0839711
$463$	−3289.31	−0.330167	−0.165083	−	0.986280i	$-0.552789\pi$
$463$	−3289.31	−0.330167	−0.165083	+	0.986280i	$0.552789\pi$
$464$	54446.2	5.44742
$465$	−2460.52	−0.245385
$466$	−4918.81	−0.488969
$467$	−11246.7	−1.11442	−0.557212	−	0.830370i	$-0.688129\pi$
$467$	−11246.7	−1.11442	−0.557212	+	0.830370i	$0.688129\pi$
$468$	17643.6	1.74268
$469$	−5408.10	−0.532458
$470$	6194.89	0.607976
$471$	−2136.19	−0.208982
$472$	18273.0	1.78196
$473$	−3831.79	−0.372486
$474$	−2817.88	−0.273058
$475$	475.000	0.0458831
$476$	−7508.51	−0.723008
$477$	5800.76	0.556810
$478$	13950.2	1.33487
$479$	6153.91	0.587013	0.293507	−	0.955957i	$-0.405178\pi$
$479$	6153.91	0.587013	0.293507	+	0.955957i	$0.405178\pi$
$480$	9497.49	0.903123
$481$	−14366.1	−1.36182
$482$	−10086.0	−0.953126
$483$	−2654.75	−0.250094
$484$	2664.19	0.250206
$485$	7117.01	0.666323
$486$	21424.2	1.99963
$487$	−16533.9	−1.53844	−0.769221	−	0.638983i	$-0.779355\pi$
$487$	−16533.9	−1.53844	−0.769221	+	0.638983i	$0.779355\pi$
$488$	38857.4	3.60449
$489$	−6688.26	−0.618514
$490$	8630.21	0.795660
$491$	10206.2	0.938086	0.469043	−	0.883175i	$-0.344599\pi$
$491$	10206.2	0.938086	0.469043	+	0.883175i	$0.344599\pi$
$492$	8120.78	0.744132
$493$	14351.2	1.31104
$494$	4138.87	0.376957
$495$	−1108.49	−0.100653
$496$	46015.3	4.16562
$497$	−5837.92	−0.526894
$498$	1155.85	0.104005
$499$	−15144.1	−1.35861	−0.679303	−	0.733858i	$-0.737718\pi$
$499$	−15144.1	−1.35861	−0.679303	+	0.733858i	$0.737718\pi$
$500$	2752.27	0.246170
$501$	11172.7	0.996330
$502$	28073.0	2.49594
$503$	7980.79	0.707447	0.353723	−	0.935350i	$-0.384915\pi$
$503$	7980.79	0.707447	0.353723	+	0.935350i	$0.384915\pi$
$504$	−8185.69	−0.723452
$505$	−617.901	−0.0544479
$506$	11563.8	1.01595
$507$	1612.27	0.141229
$508$	−45211.1	−3.94866
$509$	−11452.5	−0.997297	−0.498648	−	0.866804i	$-0.666170\pi$
$509$	−11452.5	−0.997297	−0.498648	+	0.866804i	$0.666170\pi$
$510$	4622.07	0.401312
$511$	−2383.83	−0.206368
$512$	−27295.0	−2.35601
$513$	2344.13	0.201746
$514$	39476.7	3.38763
$515$	−145.633	−0.0124609
$516$	20067.6	1.71207
$517$	−2487.51	−0.211606
$518$	10468.8	0.887979
$519$	204.180	0.0172688
$520$	15268.2	1.28761
$521$	6757.83	0.568265	0.284132	−	0.958785i	$-0.408294\pi$
$521$	6757.83	0.568265	0.284132	+	0.958785i	$0.408294\pi$
$522$	24574.2	2.06050
$523$	−8709.54	−0.728187	−0.364093	−	0.931362i	$-0.618621\pi$
$523$	−8709.54	−0.728187	−0.364093	+	0.931362i	$0.618621\pi$
$524$	−52814.0	−4.40303
$525$	345.898	0.0287547
$526$	2771.90	0.229773
$527$	12128.9	1.00255
$528$	−7041.23	−0.580360
$529$	24648.5	2.02585
$530$	7884.54	0.646193
$531$	4795.12	0.391884
$532$	−2212.27	−0.180289
$533$	5604.65	0.455468
$534$	12867.7	1.04277
$535$	−3917.16	−0.316549
$536$	−78545.9	−6.32960
$537$	−10209.0	−0.820389
$538$	24557.2	1.96791
$539$	−3465.39	−0.276929
$540$	13582.4	1.08240
$541$	−12151.9	−0.965717	−0.482859	−	0.875698i	$-0.660402\pi$
$541$	−12151.9	−0.965717	−0.482859	+	0.875698i	$0.660402\pi$
$542$	39203.9	3.10692
$543$	1594.82	0.126041
$544$	−46817.0	−3.68982
$545$	4179.32	0.328481
$546$	3013.95	0.236237
$547$	2620.47	0.204832	0.102416	−	0.994742i	$-0.467343\pi$
$547$	2620.47	0.204832	0.102416	+	0.994742i	$0.467343\pi$
$548$	−33557.4	−2.61588
$549$	10196.7	0.792690
$550$	−1506.69	−0.116810
$551$	4228.35	0.326921
$552$	−38556.9	−2.97299
$553$	1039.51	0.0799357
$554$	−15735.6	−1.20676
$555$	−4726.91	−0.361525
$556$	15874.9	1.21087
$557$	6748.33	0.513350	0.256675	−	0.966498i	$-0.417373\pi$
$557$	6748.33	0.513350	0.256675	+	0.966498i	$0.417373\pi$
$558$	20768.9	1.57566
$559$	13849.9	1.04792
$560$	−6468.80	−0.488137
$561$	−1855.96	−0.139677
$562$	−6214.91	−0.466477
$563$	−10564.6	−0.790844	−0.395422	−	0.918500i	$-0.629402\pi$
$563$	−10564.6	−0.790844	−0.395422	+	0.918500i	$0.629402\pi$
$564$	13027.4	0.972609
$565$	−9386.16	−0.698900
$566$	−41419.6	−3.07596
$567$	−1170.64	−0.0867056
$568$	−84788.4	−6.26346
$569$	−14429.3	−1.06311	−0.531554	−	0.847024i	$-0.678392\pi$
$569$	−14429.3	−1.06311	−0.531554	+	0.847024i	$0.678392\pi$
$570$	1361.82	0.100071
$571$	2518.52	0.184583	0.0922914	−	0.995732i	$-0.470581\pi$
$571$	2518.52	0.184583	0.0922914	+	0.995732i	$0.470581\pi$
$572$	−9629.63	−0.703908
$573$	−2884.75	−0.210318
$574$	−4084.20	−0.296988
$575$	−4796.84	−0.347899
$576$	−40720.3	−2.94563
$577$	16272.3	1.17405	0.587023	−	0.809570i	$-0.300300\pi$
$577$	16272.3	1.17405	0.587023	+	0.809570i	$0.300300\pi$
$578$	4133.65	0.297469
$579$	1913.18	0.137322
$580$	24500.1	1.75398
$581$	−426.389	−0.0304468
$582$	20404.4	1.45325
$583$	−3165.97	−0.224908
$584$	−34622.1	−2.45321
$585$	4006.61	0.283167
$586$	11728.8	0.826813
$587$	16427.8	1.15511	0.577554	−	0.816352i	$-0.304007\pi$
$587$	16427.8	1.15511	0.577554	+	0.816352i	$0.304007\pi$
$588$	18148.7	1.27286
$589$	3573.60	0.249996
$590$	6517.65	0.454792
$591$	6978.64	0.485724
$592$	88400.1	6.13720
$593$	−2287.42	−0.158403	−0.0792014	−	0.996859i	$-0.525237\pi$
$593$	−2287.42	−0.158403	−0.0792014	+	0.996859i	$0.525237\pi$
$594$	−7435.53	−0.513609
$595$	−1705.07	−0.117481
$596$	26820.7	1.84332
$597$	−10911.7	−0.748050
$598$	−41796.9	−2.85819
$599$	−18881.4	−1.28793	−0.643967	−	0.765053i	$-0.722713\pi$
$599$	−18881.4	−1.28793	−0.643967	+	0.765053i	$0.722713\pi$
$600$	5023.74	0.341822
$601$	19215.9	1.30421	0.652107	−	0.758127i	$-0.273885\pi$
$601$	19215.9	1.30421	0.652107	+	0.758127i	$0.273885\pi$
$602$	−10092.6	−0.683297
$603$	−20611.6	−1.39199
$604$	−38176.5	−2.57182
$605$	605.000	0.0406558
$606$	−1771.52	−0.118751
$607$	4696.83	0.314067	0.157033	−	0.987593i	$-0.449807\pi$
$607$	4696.83	0.314067	0.157033	+	0.987593i	$0.449807\pi$
$608$	−13793.9	−0.920094
$609$	3079.11	0.204880
$610$	13859.7	0.919938
$611$	8991.00	0.595314
$612$	−28616.8	−1.89014
$613$	−10769.8	−0.709604	−0.354802	−	0.934941i	$-0.615452\pi$
$613$	−10769.8	−0.709604	−0.354802	+	0.934941i	$0.615452\pi$
$614$	−45703.7	−3.00399
$615$	1844.11	0.120913
$616$	4467.64	0.292218
$617$	−3983.54	−0.259921	−0.129960	−	0.991519i	$-0.541485\pi$
$617$	−3983.54	−0.259921	−0.129960	+	0.991519i	$0.541485\pi$
$618$	−417.529	−0.0271771
$619$	−16962.9	−1.10145	−0.550724	−	0.834687i	$-0.685648\pi$
$619$	−16962.9	−1.10145	−0.550724	+	0.834687i	$0.685648\pi$
$620$	20706.3	1.34127
$621$	−23672.4	−1.52970
$622$	14325.5	0.923471
$623$	−4746.87	−0.305264
$624$	25450.3	1.63273
$625$	625.000	0.0400000
$626$	−15380.7	−0.982009
$627$	−546.829	−0.0348297
$628$	17976.9	1.14229
$629$	23300.9	1.47705
$630$	−2919.68	−0.184639
$631$	8131.87	0.513034	0.256517	−	0.966540i	$-0.417425\pi$
$631$	8131.87	0.513034	0.256517	+	0.966540i	$0.417425\pi$
$632$	15097.6	0.950236
$633$	−1171.51	−0.0735601
$634$	−27592.0	−1.72842
$635$	−10266.8	−0.641614
$636$	16580.6	1.03375
$637$	12525.5	0.779089
$638$	−13412.2	−0.832282
$639$	−22249.7	−1.37744
$640$	−26308.3	−1.62489
$641$	7020.87	0.432617	0.216309	−	0.976325i	$-0.430598\pi$
$641$	7020.87	0.432617	0.216309	+	0.976325i	$0.430598\pi$
$642$	−11230.5	−0.690391
$643$	−7716.34	−0.473255	−0.236627	−	0.971600i	$-0.576042\pi$
$643$	−7716.34	−0.473255	−0.236627	+	0.971600i	$0.576042\pi$
$644$	22340.8	1.36700
$645$	4557.06	0.278192
$646$	−6712.98	−0.408852
$647$	17962.9	1.09149	0.545744	−	0.837952i	$-0.316247\pi$
$647$	17962.9	1.09149	0.545744	+	0.837952i	$0.316247\pi$
$648$	−17002.0	−1.03071
$649$	−2617.11	−0.158290
$650$	5445.88	0.328623
$651$	2602.32	0.156671
$652$	56284.4	3.38078
$653$	8960.44	0.536982	0.268491	−	0.963282i	$-0.413475\pi$
$653$	8960.44	0.536982	0.268491	+	0.963282i	$0.413475\pi$
$654$	11982.1	0.716417
$655$	−11993.3	−0.715445
$656$	−34487.6	−2.05261
$657$	−9085.35	−0.539503
$658$	−6551.89	−0.388175
$659$	−6505.36	−0.384542	−0.192271	−	0.981342i	$-0.561585\pi$
$659$	−6505.36	−0.384542	−0.192271	+	0.981342i	$0.561585\pi$
$660$	−3168.46	−0.186867
$661$	24294.0	1.42954	0.714770	−	0.699360i	$-0.246531\pi$
$661$	24294.0	1.42954	0.714770	+	0.699360i	$0.246531\pi$
$662$	42633.8	2.50303
$663$	6708.29	0.392954
$664$	−6192.77	−0.361937
$665$	−502.374	−0.0292951
$666$	39899.2	2.32142
$667$	−42700.4	−2.47881
$668$	−94023.2	−5.44591
$669$	−8715.96	−0.503705
$670$	−28015.8	−1.61544
$671$	−5565.24	−0.320184
$672$	−10044.8	−0.576618
$673$	5718.46	0.327534	0.163767	−	0.986499i	$-0.447635\pi$
$673$	5718.46	0.327534	0.163767	+	0.986499i	$0.447635\pi$
$674$	62757.6	3.58655
$675$	3084.38	0.175878
$676$	−13567.9	−0.771954
$677$	30343.3	1.72258	0.861291	−	0.508111i	$-0.169656\pi$
$677$	30343.3	1.72258	0.861291	+	0.508111i	$0.169656\pi$
$678$	−26910.1	−1.52430
$679$	−7527.15	−0.425428
$680$	−24764.1	−1.39656
$681$	8963.70	0.504390
$682$	−11335.4	−0.636443
$683$	−9626.03	−0.539282	−0.269641	−	0.962961i	$-0.586905\pi$
$683$	−9626.03	−0.539282	−0.269641	+	0.962961i	$0.586905\pi$
$684$	−8431.49	−0.471325
$685$	−7620.40	−0.425052
$686$	−19065.3	−1.06110
$687$	4072.49	0.226165
$688$	−85223.6	−4.72256
$689$	11443.3	0.632736
$690$	−13752.5	−0.758767
$691$	−12955.4	−0.713235	−0.356617	−	0.934251i	$-0.616070\pi$
$691$	−12955.4	−0.713235	−0.356617	+	0.934251i	$0.616070\pi$
$692$	−1718.26	−0.0943907
$693$	1172.37	0.0642638
$694$	4763.52	0.260549
$695$	3604.95	0.196753
$696$	44720.2	2.43551
$697$	−9090.38	−0.494007
$698$	16295.6	0.883663
$699$	−2348.95	−0.127104
$700$	−2910.88	−0.157172
$701$	−30309.7	−1.63307	−0.816534	−	0.577298i	$-0.804107\pi$
$701$	−30309.7	−1.63307	−0.816534	+	0.577298i	$0.804107\pi$
$702$	26875.5	1.44494
$703$	6865.24	0.368318
$704$	22224.6	1.18980
$705$	2958.33	0.158039
$706$	35209.3	1.87694
$707$	653.509	0.0347634
$708$	13706.1	0.727553
$709$	7555.14	0.400197	0.200098	−	0.979776i	$-0.435874\pi$
$709$	7555.14	0.400197	0.200098	+	0.979776i	$0.435874\pi$
$710$	−30242.4	−1.59856
$711$	3961.83	0.208974
$712$	−68942.3	−3.62882
$713$	−36088.4	−1.89554
$714$	−4888.44	−0.256226
$715$	−2186.75	−0.114377
$716$	85912.6	4.48422
$717$	6661.81	0.346987
$718$	69383.4	3.60636
$719$	26371.8	1.36787	0.683937	−	0.729541i	$-0.260266\pi$
$719$	26371.8	1.36787	0.683937	+	0.729541i	$0.260266\pi$
$720$	−24654.2	−1.27612
$721$	154.025	0.00795590
$722$	−1977.88	−0.101951
$723$	−4816.53	−0.247757
$724$	−13421.0	−0.688936
$725$	5563.62	0.285003
$726$	1734.53	0.0886701
$727$	−30840.0	−1.57330	−0.786652	−	0.617397i	$-0.788187\pi$
$727$	−30840.0	−1.57330	−0.786652	+	0.617397i	$0.788187\pi$
$728$	−16148.1	−0.822100
$729$	4254.00	0.216126
$730$	−12349.0	−0.626108
$731$	−22463.6	−1.13659
$732$	29145.9	1.47167
$733$	−14225.4	−0.716818	−0.358409	−	0.933565i	$-0.616681\pi$
$733$	−14225.4	−0.716818	−0.358409	+	0.933565i	$0.616681\pi$
$734$	13549.0	0.681339
$735$	4121.30	0.206825
$736$	139299.	6.97641
$737$	11249.5	0.562255
$738$	−15565.9	−0.776407
$739$	−7512.30	−0.373944	−0.186972	−	0.982365i	$-0.559867\pi$
$739$	−7512.30	−0.373944	−0.186972	+	0.982365i	$0.559867\pi$
$740$	39778.9	1.97608
$741$	1976.49	0.0979869
$742$	−8338.91	−0.412576
$743$	18453.7	0.911171	0.455586	−	0.890192i	$-0.349430\pi$
$743$	18453.7	0.911171	0.455586	+	0.890192i	$0.349430\pi$
$744$	37795.4	1.86243
$745$	6090.59	0.299520
$746$	−29929.0	−1.46887
$747$	−1625.07	−0.0795962
$748$	15618.6	0.763468
$749$	4142.90	0.202107
$750$	1791.87	0.0872399
$751$	−18837.6	−0.915304	−0.457652	−	0.889131i	$-0.651309\pi$
$751$	−18837.6	−0.915304	−0.457652	+	0.889131i	$0.651309\pi$
$752$	−55325.1	−2.68284
$753$	13406.1	0.648799
$754$	48478.1	2.34147
$755$	−8669.34	−0.417894
$756$	−14365.2	−0.691080
$757$	27964.9	1.34267	0.671335	−	0.741154i	$-0.265721\pi$
$757$	27964.9	1.34267	0.671335	+	0.741154i	$0.265721\pi$
$758$	72941.6	3.49519
$759$	5522.21	0.264089
$760$	−7296.35	−0.348245
$761$	1447.92	0.0689711	0.0344855	−	0.999405i	$-0.489021\pi$
$761$	1447.92	0.0689711	0.0344855	+	0.999405i	$0.489021\pi$
$762$	−29434.8	−1.39936
$763$	−4420.17	−0.209726
$764$	24276.3	1.14959
$765$	−6498.46	−0.307127
$766$	56840.0	2.68109
$767$	9459.45	0.445320
$768$	−33135.9	−1.55689
$769$	1829.23	0.0857787	0.0428893	−	0.999080i	$-0.486344\pi$
$769$	1829.23	0.0857787	0.0428893	+	0.999080i	$0.486344\pi$
$770$	1593.52	0.0745799
$771$	18851.9	0.880588
$772$	−16100.2	−0.750595
$773$	29324.3	1.36445	0.682227	−	0.731141i	$-0.261012\pi$
$773$	29324.3	1.36445	0.682227	+	0.731141i	$0.261012\pi$
$774$	−38465.5	−1.78632
$775$	4702.10	0.217941
$776$	−109322.	−5.05728
$777$	4999.32	0.230823
$778$	−76848.3	−3.54132
$779$	−2678.34	−0.123185
$780$	11452.3	0.525715
$781$	12143.6	0.556379
$782$	67791.8	3.10004
$783$	27456.5	1.25315
$784$	−77074.4	−3.51104
$785$	4082.30	0.185609
$786$	−34384.7	−1.56038
$787$	3537.69	0.160235	0.0801175	−	0.996785i	$-0.474470\pi$
$787$	3537.69	0.160235	0.0801175	+	0.996785i	$0.474470\pi$
$788$	−58728.1	−2.65495
$789$	1323.71	0.0597277
$790$	5385.02	0.242519
$791$	9927.07	0.446227
$792$	17027.3	0.763936
$793$	20115.4	0.900779
$794$	−46287.2	−2.06886
$795$	3765.21	0.167973
$796$	91826.4	4.08882
$797$	14049.0	0.624392	0.312196	−	0.950018i	$-0.398935\pi$
$797$	14049.0	0.624392	0.312196	+	0.950018i	$0.398935\pi$
$798$	−1440.30	−0.0638924
$799$	−14582.8	−0.645686
$800$	−18149.9	−0.802119
$801$	−18091.5	−0.798042
$802$	9484.49	0.417592
$803$	4958.66	0.217917
$804$	−58915.2	−2.58430
$805$	5073.27	0.222124
$806$	40971.3	1.79051
$807$	11727.1	0.511543
$808$	9491.40	0.413251
$809$	10718.3	0.465805	0.232902	−	0.972500i	$-0.425178\pi$
$809$	10718.3	0.465805	0.232902	+	0.972500i	$0.425178\pi$
$810$	−6064.30	−0.263059
$811$	−41621.6	−1.80214	−0.901069	−	0.433677i	$-0.857216\pi$
$811$	−41621.6	−1.80214	−0.901069	+	0.433677i	$0.857216\pi$
$812$	−25912.0	−1.11987
$813$	18721.6	0.807620
$814$	−21776.4	−0.937670
$815$	12781.4	0.549340
$816$	−41278.7	−1.77089
$817$	−6618.55	−0.283420
$818$	27566.7	1.17830
$819$	−4237.50	−0.180794
$820$	−15519.0	−0.660909
$821$	14452.6	0.614372	0.307186	−	0.951649i	$-0.400613\pi$
$821$	14452.6	0.614372	0.307186	+	0.951649i	$0.400613\pi$
$822$	−21847.6	−0.927037
$823$	−22237.5	−0.941860	−0.470930	−	0.882171i	$-0.656082\pi$
$823$	−22237.5	−0.941860	−0.470930	+	0.882171i	$0.656082\pi$
$824$	2237.03	0.0945759
$825$	−719.512	−0.0303639
$826$	−6893.25	−0.290372
$827$	12399.7	0.521380	0.260690	−	0.965423i	$-0.416050\pi$
$827$	12399.7	0.521380	0.260690	+	0.965423i	$0.416050\pi$
$828$	85146.3	3.57372
$829$	4359.93	0.182662	0.0913309	−	0.995821i	$-0.470888\pi$
$829$	4359.93	0.182662	0.0913309	+	0.995821i	$0.470888\pi$
$830$	−2208.84	−0.0923736
$831$	−7514.45	−0.313687
$832$	−80330.0	−3.34728
$833$	−20315.6	−0.845010
$834$	10335.4	0.429119
$835$	−21351.3	−0.884901
$836$	4601.79	0.190378
$837$	23204.9	0.958277
$838$	71328.0	2.94032
$839$	10050.2	0.413552	0.206776	−	0.978388i	$-0.433703\pi$
$839$	10050.2	0.413552	0.206776	+	0.978388i	$0.433703\pi$
$840$	−5313.25	−0.218243
$841$	25137.1	1.03067
$842$	43916.2	1.79745
$843$	−2967.89	−0.121257
$844$	9858.78	0.402077
$845$	−3081.07	−0.125434
$846$	−24970.9	−1.01479
$847$	−639.865	−0.0259575
$848$	−70415.0	−2.85149
$849$	−19779.7	−0.799572
$850$	−8832.87	−0.356429
$851$	−69329.4	−2.79269
$852$	−63597.5	−2.55730
$853$	8399.76	0.337166	0.168583	−	0.985687i	$-0.446081\pi$
$853$	8399.76	0.337166	0.168583	+	0.985687i	$0.446081\pi$
$854$	−14658.4	−0.587353
$855$	−1914.67	−0.0765852
$856$	60170.4	2.40255
$857$	32488.0	1.29495	0.647473	−	0.762088i	$-0.275826\pi$
$857$	32488.0	1.29495	0.647473	+	0.762088i	$0.275826\pi$
$858$	−6269.40	−0.249457
$859$	24174.0	0.960195	0.480097	−	0.877215i	$-0.340601\pi$
$859$	24174.0	0.960195	0.480097	+	0.877215i	$0.340601\pi$
$860$	−38349.5	−1.52059
$861$	−1950.38	−0.0771997
$862$	42424.5	1.67631
$863$	11034.2	0.435237	0.217618	−	0.976034i	$-0.430171\pi$
$863$	11034.2	0.435237	0.217618	+	0.976034i	$0.430171\pi$
$864$	−89569.7	−3.52688
$865$	−390.192	−0.0153375
$866$	65948.0	2.58777
$867$	1974.00	0.0773247
$868$	−21899.6	−0.856359
$869$	−2162.31	−0.0844089
$870$	15950.9	0.621592
$871$	−40661.0	−1.58180
$872$	−64197.4	−2.49312
$873$	−28687.8	−1.11218
$874$	19973.8	0.773025
$875$	−661.018	−0.0255388
$876$	−25969.1	−1.00162
$877$	43568.5	1.67754	0.838770	−	0.544486i	$-0.183275\pi$
$877$	43568.5	1.67754	0.838770	+	0.544486i	$0.183275\pi$
$878$	31736.9	1.21990
$879$	5601.02	0.214923
$880$	13455.9	0.515453
$881$	−22140.5	−0.846690	−0.423345	−	0.905969i	$-0.639144\pi$
$881$	−22140.5	−0.846690	−0.423345	+	0.905969i	$0.639144\pi$
$882$	−34787.4	−1.32806
$883$	−15766.1	−0.600873	−0.300436	−	0.953802i	$-0.597132\pi$
$883$	−15766.1	−0.600873	−0.300436	+	0.953802i	$0.597132\pi$
$884$	−56453.0	−2.14787
$885$	3112.46	0.118220
$886$	−66892.8	−2.53647
$887$	−44875.3	−1.69872	−0.849360	−	0.527814i	$-0.823012\pi$
$887$	−44875.3	−1.69872	−0.849360	+	0.527814i	$0.823012\pi$
$888$	72608.8	2.74391
$889$	10858.4	0.409652
$890$	−24590.4	−0.926149
$891$	2435.07	0.0915577
$892$	73348.4	2.75324
$893$	−4296.60	−0.161008
$894$	17461.7	0.653251
$895$	19509.5	0.728638
$896$	27824.4	1.03744
$897$	−19959.8	−0.742965
$898$	61020.9	2.26759
$899$	41857.1	1.55285
$900$	−11094.1	−0.410891
$901$	−18560.3	−0.686274
$902$	8495.65	0.313608
$903$	−4819.67	−0.177618
$904$	144178.	5.30453
$905$	−3047.73	−0.111945
$906$	−24855.0	−0.911425
$907$	11834.8	0.433260	0.216630	−	0.976254i	$-0.430494\pi$
$907$	11834.8	0.433260	0.216630	+	0.976254i	$0.430494\pi$
$908$	−75433.2	−2.75698
$909$	2490.69	0.0908810
$910$	−5759.72	−0.209816
$911$	−2825.70	−0.102766	−0.0513828	−	0.998679i	$-0.516363\pi$
$911$	−2825.70	−0.102766	−0.0513828	+	0.998679i	$0.516363\pi$
$912$	−12162.1	−0.441588
$913$	886.943	0.0321506
$914$	27672.9	1.00146
$915$	6618.61	0.239130
$916$	−34271.7	−1.23621
$917$	12684.4	0.456791
$918$	−43590.2	−1.56720
$919$	−45198.9	−1.62239	−0.811194	−	0.584777i	$-0.801182\pi$
$919$	−45198.9	−1.62239	−0.811194	+	0.584777i	$0.801182\pi$
$920$	73683.0	2.64050
$921$	−21825.5	−0.780864
$922$	79420.2	2.83684
$923$	−43892.6	−1.56527
$924$	3351.05	0.119309
$925$	9033.21	0.321092
$926$	18021.7	0.639559
$927$	587.029	0.0207989
$928$	−161566.	−5.71517
$929$	−18769.0	−0.662854	−0.331427	−	0.943481i	$-0.607530\pi$
$929$	−18769.0	−0.662854	−0.331427	+	0.943481i	$0.607530\pi$
$930$	13480.9	0.475329
$931$	−5985.68	−0.210712
$932$	19767.4	0.694744
$933$	6841.04	0.240049
$934$	61619.4	2.15872
$935$	3546.77	0.124055
$936$	−61544.4	−2.14919
$937$	−52303.2	−1.82356	−0.911778	−	0.410684i	$-0.865290\pi$
$937$	−52303.2	−1.82356	−0.911778	+	0.410684i	$0.865290\pi$
$938$	29630.3	1.03141
$939$	−7344.97	−0.255265
$940$	−24895.6	−0.863834
$941$	−3800.28	−0.131653	−0.0658265	−	0.997831i	$-0.520968\pi$
$941$	−3800.28	−0.131653	−0.0658265	+	0.997831i	$0.520968\pi$
$942$	11703.9	0.404814
$943$	27047.5	0.934027
$944$	−58207.6	−2.00688
$945$	−3262.13	−0.112293
$946$	20993.9	0.721535
$947$	23789.0	0.816303	0.408152	−	0.912914i	$-0.366173\pi$
$947$	23789.0	0.816303	0.408152	+	0.912914i	$0.366173\pi$
$948$	11324.3	0.387970
$949$	−17922.9	−0.613068
$950$	−2602.47	−0.0888792
$951$	−13176.4	−0.449289
$952$	26191.2	0.891660
$953$	36869.3	1.25322	0.626608	−	0.779335i	$-0.284443\pi$
$953$	36869.3	1.25322	0.626608	+	0.779335i	$0.284443\pi$
$954$	−31781.7	−1.07858
$955$	5512.81	0.186796
$956$	−56061.9	−1.89662
$957$	−6404.94	−0.216345
$958$	−33716.5	−1.13709
$959$	8059.55	0.271383
$960$	−26431.2	−0.888606
$961$	5584.59	0.187459
$962$	78710.1	2.63796
$963$	15789.6	0.528362
$964$	40533.1	1.35423
$965$	−3656.13	−0.121964
$966$	14545.1	0.484451
$967$	9297.05	0.309176	0.154588	−	0.987979i	$-0.450595\pi$
$967$	9297.05	0.309176	0.154588	+	0.987979i	$0.450595\pi$
$968$	−9293.24	−0.308570
$969$	−3205.74	−0.106278
$970$	−38993.2	−1.29072
$971$	7040.23	0.232679	0.116340	−	0.993209i	$-0.462884\pi$
$971$	7040.23	0.232679	0.116340	+	0.993209i	$0.462884\pi$
$972$	−86098.0	−2.84115
$973$	−3812.70	−0.125621
$974$	90587.1	2.98008
$975$	2600.65	0.0854230
$976$	−123778.	−4.05945
$977$	−53709.3	−1.75876	−0.879382	−	0.476117i	$-0.842044\pi$
$977$	−53709.3	−1.75876	−0.879382	+	0.476117i	$0.842044\pi$
$978$	36644.2	1.19811
$979$	9874.08	0.322346
$980$	−34682.5	−1.13050
$981$	−16846.4	−0.548280
$982$	−55918.6	−1.81714
$983$	24413.8	0.792145	0.396072	−	0.918219i	$-0.370373\pi$
$983$	24413.8	0.792145	0.396072	+	0.918219i	$0.370373\pi$
$984$	−28326.9	−0.917712
$985$	−13336.3	−0.431401
$986$	−78628.3	−2.53959
$987$	−3128.81	−0.100903
$988$	−16633.0	−0.535593
$989$	66838.2	2.14897
$990$	6073.30	0.194972
$991$	15654.4	0.501796	0.250898	−	0.968014i	$-0.419274\pi$
$991$	15654.4	0.501796	0.250898	+	0.968014i	$0.419274\pi$
$992$	−136548.	−4.37037
$993$	20359.5	0.650643
$994$	31985.2	1.02063
$995$	20852.4	0.664389
$996$	−4645.03	−0.147775
$997$	−3976.33	−0.126311	−0.0631553	−	0.998004i	$-0.520116\pi$
$997$	−3976.33	−0.126311	−0.0631553	+	0.998004i	$0.520116\pi$
$998$	82972.9	2.63172
$999$	44579.0	1.41183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.1	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-5.47888 q^{2} -2.61641 q^{3} +22.0181 q^{4} +5.00000 q^{5} +14.3350 q^{6} -5.28814 q^{7} -76.8037 q^{8} -20.1544 q^{9} +O(q^{10})\)	\(q-5.47888 q^{2} -2.61641 q^{3} +22.0181 q^{4} +5.00000 q^{5} +14.3350 q^{6} -5.28814 q^{7} -76.8037 q^{8} -20.1544 q^{9} -27.3944 q^{10} +11.0000 q^{11} -57.6084 q^{12} -39.7591 q^{13} +28.9731 q^{14} -13.0820 q^{15} +244.653 q^{16} +64.4867 q^{17} +110.424 q^{18} +19.0000 q^{19} +110.091 q^{20} +13.8359 q^{21} -60.2677 q^{22} -191.874 q^{23} +200.950 q^{24} +25.0000 q^{25} +217.835 q^{26} +123.375 q^{27} -116.435 q^{28} +222.545 q^{29} +71.6749 q^{30} +188.084 q^{31} -725.995 q^{32} -28.7805 q^{33} -353.315 q^{34} -26.4407 q^{35} -443.763 q^{36} +361.329 q^{37} -104.099 q^{38} +104.026 q^{39} -384.018 q^{40} -140.965 q^{41} -75.8054 q^{42} -348.345 q^{43} +242.199 q^{44} -100.772 q^{45} +1051.25 q^{46} -226.137 q^{47} -640.112 q^{48} -315.036 q^{49} -136.972 q^{50} -168.723 q^{51} -875.421 q^{52} -287.816 q^{53} -675.957 q^{54} +55.0000 q^{55} +406.149 q^{56} -49.7117 q^{57} -1219.30 q^{58} -237.919 q^{59} -288.042 q^{60} -505.931 q^{61} -1030.49 q^{62} +106.579 q^{63} +2020.42 q^{64} -198.796 q^{65} +157.685 q^{66} +1022.68 q^{67} +1419.88 q^{68} +502.019 q^{69} +144.865 q^{70} +1103.96 q^{71} +1547.93 q^{72} +450.787 q^{73} -1979.68 q^{74} -65.4102 q^{75} +418.344 q^{76} -58.1696 q^{77} -569.946 q^{78} -196.574 q^{79} +1223.26 q^{80} +221.370 q^{81} +772.331 q^{82} +80.6312 q^{83} +304.641 q^{84} +322.433 q^{85} +1908.54 q^{86} -582.267 q^{87} -844.840 q^{88} +897.644 q^{89} +552.118 q^{90} +210.252 q^{91} -4224.70 q^{92} -492.104 q^{93} +1238.98 q^{94} +95.0000 q^{95} +1899.50 q^{96} +1423.40 q^{97} +1726.04 q^{98} -221.699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more