LMFDB - Embedded newform 1850.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{11} -0.618034 q^{12} -3.85410 q^{13} +1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.61803 q^{18} -4.47214 q^{19} +0.763932 q^{21} +3.61803 q^{22} +3.85410 q^{23} +0.618034 q^{24} +3.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} +6.32624 q^{29} +9.61803 q^{31} -1.00000 q^{32} +2.23607 q^{33} +4.47214 q^{34} -2.61803 q^{36} +1.00000 q^{37} +4.47214 q^{38} +2.38197 q^{39} +7.38197 q^{41} -0.763932 q^{42} +0.763932 q^{43} -3.61803 q^{44} -3.85410 q^{46} -3.23607 q^{47} -0.618034 q^{48} -5.47214 q^{49} +2.76393 q^{51} -3.85410 q^{52} +8.47214 q^{53} -3.47214 q^{54} +1.23607 q^{56} +2.76393 q^{57} -6.32624 q^{58} -9.23607 q^{59} +8.38197 q^{61} -9.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -2.23607 q^{66} +10.0902 q^{67} -4.47214 q^{68} -2.38197 q^{69} -14.9443 q^{71} +2.61803 q^{72} +4.09017 q^{73} -1.00000 q^{74} -4.47214 q^{76} +4.47214 q^{77} -2.38197 q^{78} +11.5623 q^{79} +5.70820 q^{81} -7.38197 q^{82} +5.52786 q^{83} +0.763932 q^{84} -0.763932 q^{86} -3.90983 q^{87} +3.61803 q^{88} -10.4721 q^{89} +4.76393 q^{91} +3.85410 q^{92} -5.94427 q^{93} +3.23607 q^{94} +0.618034 q^{96} -8.47214 q^{97} +5.47214 q^{98} +9.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 - 5 * q^11 + q^12 - q^13 - 2 * q^14 + 2 * q^16 + 3 * q^18 + 6 * q^21 + 5 * q^22 + q^23 - q^24 + q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 + 17 * q^31 - 2 * q^32 - 3 * q^36 + 2 * q^37 + 7 * q^39 + 17 * q^41 - 6 * q^42 + 6 * q^43 - 5 * q^44 - q^46 - 2 * q^47 + q^48 - 2 * q^49 + 10 * q^51 - q^52 + 8 * q^53 + 2 * q^54 - 2 * q^56 + 10 * q^57 + 3 * q^58 - 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^63 + 2 * q^64 + 9 * q^67 - 7 * q^69 - 12 * q^71 + 3 * q^72 - 3 * q^73 - 2 * q^74 - 7 * q^78 + 3 * q^79 - 2 * q^81 - 17 * q^82 + 20 * q^83 + 6 * q^84 - 6 * q^86 - 19 * q^87 + 5 * q^88 - 12 * q^89 + 14 * q^91 + q^92 + 6 * q^93 + 2 * q^94 - q^96 - 8 * q^97 + 2 * q^98 + 10 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−1.00000	−0.353553
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−3.61803	−1.09088	−0.545439	−	0.838150i	$-0.683637\pi$
$11$	−3.61803	−1.09088	−0.545439	+	0.838150i	$0.683637\pi$
$12$	−0.618034	−0.178411
$13$	−3.85410	−1.06894	−0.534468	−	0.845189i	$-0.679488\pi$
$13$	−3.85410	−1.06894	−0.534468	+	0.845189i	$0.679488\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	2.61803	0.617077
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	0.763932	0.166704
$22$	3.61803	0.771367
$23$	3.85410	0.803636	0.401818	−	0.915720i	$-0.368378\pi$
$23$	3.85410	0.803636	0.401818	+	0.915720i	$0.368378\pi$
$24$	0.618034	0.126156
$25$	0	0
$26$	3.85410	0.755852
$27$	3.47214	0.668213
$28$	−1.23607	−0.233595
$29$	6.32624	1.17475	0.587376	−	0.809314i	$-0.300161\pi$
$29$	6.32624	1.17475	0.587376	+	0.809314i	$0.300161\pi$
$30$	0	0
$31$	9.61803	1.72745	0.863725	−	0.503964i	$-0.168125\pi$
$31$	9.61803	1.72745	0.863725	+	0.503964i	$0.168125\pi$
$32$	−1.00000	−0.176777
$33$	2.23607	0.389249
$34$	4.47214	0.766965
$35$	0	0
$36$	−2.61803	−0.436339
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	4.47214	0.725476
$39$	2.38197	0.381420
$40$	0	0
$41$	7.38197	1.15287	0.576435	−	0.817143i	$-0.304444\pi$
$41$	7.38197	1.15287	0.576435	+	0.817143i	$0.304444\pi$
$42$	−0.763932	−0.117877
$43$	0.763932	0.116499	0.0582493	−	0.998302i	$-0.481448\pi$
$43$	0.763932	0.116499	0.0582493	+	0.998302i	$0.481448\pi$
$44$	−3.61803	−0.545439
$45$	0	0
$46$	−3.85410	−0.568256
$47$	−3.23607	−0.472029	−0.236015	−	0.971750i	$-0.575841\pi$
$47$	−3.23607	−0.472029	−0.236015	+	0.971750i	$0.575841\pi$
$48$	−0.618034	−0.0892055
$49$	−5.47214	−0.781734
$50$	0	0
$51$	2.76393	0.387028
$52$	−3.85410	−0.534468
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	−3.47214	−0.472498
$55$	0	0
$56$	1.23607	0.165177
$57$	2.76393	0.366092
$58$	−6.32624	−0.830676
$59$	−9.23607	−1.20243	−0.601217	−	0.799086i	$-0.705317\pi$
$59$	−9.23607	−1.20243	−0.601217	+	0.799086i	$0.705317\pi$
$60$	0	0
$61$	8.38197	1.07320	0.536600	−	0.843836i	$-0.319708\pi$
$61$	8.38197	1.07320	0.536600	+	0.843836i	$0.319708\pi$
$62$	−9.61803	−1.22149
$63$	3.23607	0.407706
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.23607	−0.275241
$67$	10.0902	1.23271	0.616355	−	0.787468i	$-0.288609\pi$
$67$	10.0902	1.23271	0.616355	+	0.787468i	$0.288609\pi$
$68$	−4.47214	−0.542326
$69$	−2.38197	−0.286755
$70$	0	0
$71$	−14.9443	−1.77356	−0.886779	−	0.462193i	$-0.847063\pi$
$71$	−14.9443	−1.77356	−0.886779	+	0.462193i	$0.847063\pi$
$72$	2.61803	0.308538
$73$	4.09017	0.478718	0.239359	−	0.970931i	$-0.423063\pi$
$73$	4.09017	0.478718	0.239359	+	0.970931i	$0.423063\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−4.47214	−0.512989
$77$	4.47214	0.509647
$78$	−2.38197	−0.269705
$79$	11.5623	1.30086	0.650431	−	0.759566i	$-0.274589\pi$
$79$	11.5623	1.30086	0.650431	+	0.759566i	$0.274589\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	−7.38197	−0.815202
$83$	5.52786	0.606762	0.303381	−	0.952869i	$-0.401885\pi$
$83$	5.52786	0.606762	0.303381	+	0.952869i	$0.401885\pi$
$84$	0.763932	0.0833518
$85$	0	0
$86$	−0.763932	−0.0823769
$87$	−3.90983	−0.419178
$88$	3.61803	0.385684
$89$	−10.4721	−1.11004	−0.555022	−	0.831836i	$-0.687290\pi$
$89$	−10.4721	−1.11004	−0.555022	+	0.831836i	$0.687290\pi$
$90$	0	0
$91$	4.76393	0.499396
$92$	3.85410	0.401818
$93$	−5.94427	−0.616392
$94$	3.23607	0.333775
$95$	0	0
$96$	0.618034	0.0630778
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	5.47214	0.552769
$99$	9.47214	0.951985
$100$	0	0
$101$	12.4721	1.24102	0.620512	−	0.784197i	$-0.286925\pi$
$101$	12.4721	1.24102	0.620512	+	0.784197i	$0.286925\pi$
$102$	−2.76393	−0.273670
$103$	16.2705	1.60318	0.801590	−	0.597873i	$-0.203987\pi$
$103$	16.2705	1.60318	0.801590	+	0.597873i	$0.203987\pi$
$104$	3.85410	0.377926
$105$	0	0
$106$	−8.47214	−0.822887
$107$	−8.32624	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$107$	−8.32624	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$108$	3.47214	0.334106
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	0	0
$111$	−0.618034	−0.0586612
$112$	−1.23607	−0.116797
$113$	−10.9443	−1.02955	−0.514775	−	0.857325i	$-0.672125\pi$
$113$	−10.9443	−1.02955	−0.514775	+	0.857325i	$0.672125\pi$
$114$	−2.76393	−0.258866
$115$	0	0
$116$	6.32624	0.587376
$117$	10.0902	0.932837
$118$	9.23607	0.850249
$119$	5.52786	0.506738
$120$	0	0
$121$	2.09017	0.190015
$122$	−8.38197	−0.758868
$123$	−4.56231	−0.411369
$124$	9.61803	0.863725
$125$	0	0
$126$	−3.23607	−0.288292
$127$	−0.472136	−0.0418953	−0.0209476	−	0.999781i	$-0.506668\pi$
$127$	−0.472136	−0.0418953	−0.0209476	+	0.999781i	$0.506668\pi$
$128$	−1.00000	−0.0883883
$129$	−0.472136	−0.0415693
$130$	0	0
$131$	8.65248	0.755970	0.377985	−	0.925812i	$-0.376617\pi$
$131$	8.65248	0.755970	0.377985	+	0.925812i	$0.376617\pi$
$132$	2.23607	0.194625
$133$	5.52786	0.479327
$134$	−10.0902	−0.871658
$135$	0	0
$136$	4.47214	0.383482
$137$	−19.3262	−1.65115	−0.825576	−	0.564291i	$-0.809150\pi$
$137$	−19.3262	−1.65115	−0.825576	+	0.564291i	$0.809150\pi$
$138$	2.38197	0.202766
$139$	−1.85410	−0.157263	−0.0786314	−	0.996904i	$-0.525055\pi$
$139$	−1.85410	−0.157263	−0.0786314	+	0.996904i	$0.525055\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	14.9443	1.25410
$143$	13.9443	1.16608
$144$	−2.61803	−0.218169
$145$	0	0
$146$	−4.09017	−0.338505
$147$	3.38197	0.278940
$148$	1.00000	0.0821995
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	0	0
$151$	17.7082	1.44107	0.720537	−	0.693417i	$-0.243896\pi$
$151$	17.7082	1.44107	0.720537	+	0.693417i	$0.243896\pi$
$152$	4.47214	0.362738
$153$	11.7082	0.946552
$154$	−4.47214	−0.360375
$155$	0	0
$156$	2.38197	0.190710
$157$	7.52786	0.600789	0.300394	−	0.953815i	$-0.402882\pi$
$157$	7.52786	0.600789	0.300394	+	0.953815i	$0.402882\pi$
$158$	−11.5623	−0.919848
$159$	−5.23607	−0.415247
$160$	0	0
$161$	−4.76393	−0.375450
$162$	−5.70820	−0.448479
$163$	12.4721	0.976893	0.488447	−	0.872594i	$-0.337564\pi$
$163$	12.4721	0.976893	0.488447	+	0.872594i	$0.337564\pi$
$164$	7.38197	0.576435
$165$	0	0
$166$	−5.52786	−0.429045
$167$	−7.14590	−0.552966	−0.276483	−	0.961019i	$-0.589169\pi$
$167$	−7.14590	−0.552966	−0.276483	+	0.961019i	$0.589169\pi$
$168$	−0.763932	−0.0589386
$169$	1.85410	0.142623
$170$	0	0
$171$	11.7082	0.895349
$172$	0.763932	0.0582493
$173$	−8.47214	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$173$	−8.47214	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$174$	3.90983	0.296403
$175$	0	0
$176$	−3.61803	−0.272720
$177$	5.70820	0.429055
$178$	10.4721	0.784920
$179$	18.6525	1.39415	0.697076	−	0.716997i	$-0.254484\pi$
$179$	18.6525	1.39415	0.697076	+	0.716997i	$0.254484\pi$
$180$	0	0
$181$	5.52786	0.410883	0.205441	−	0.978669i	$-0.434137\pi$
$181$	5.52786	0.410883	0.205441	+	0.978669i	$0.434137\pi$
$182$	−4.76393	−0.353126
$183$	−5.18034	−0.382942
$184$	−3.85410	−0.284128
$185$	0	0
$186$	5.94427	0.435855
$187$	16.1803	1.18322
$188$	−3.23607	−0.236015
$189$	−4.29180	−0.312182
$190$	0	0
$191$	4.09017	0.295954	0.147977	−	0.988991i	$-0.452724\pi$
$191$	4.09017	0.295954	0.147977	+	0.988991i	$0.452724\pi$
$192$	−0.618034	−0.0446028
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	8.47214	0.608264
$195$	0	0
$196$	−5.47214	−0.390867
$197$	16.4721	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$197$	16.4721	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$198$	−9.47214	−0.673155
$199$	20.9443	1.48470	0.742350	−	0.670012i	$-0.233711\pi$
$199$	20.9443	1.48470	0.742350	+	0.670012i	$0.233711\pi$
$200$	0	0
$201$	−6.23607	−0.439858
$202$	−12.4721	−0.877536
$203$	−7.81966	−0.548833
$204$	2.76393	0.193514
$205$	0	0
$206$	−16.2705	−1.13362
$207$	−10.0902	−0.701315
$208$	−3.85410	−0.267234
$209$	16.1803	1.11922
$210$	0	0
$211$	−22.2705	−1.53317	−0.766583	−	0.642146i	$-0.778044\pi$
$211$	−22.2705	−1.53317	−0.766583	+	0.642146i	$0.778044\pi$
$212$	8.47214	0.581869
$213$	9.23607	0.632845
$214$	8.32624	0.569170
$215$	0	0
$216$	−3.47214	−0.236249
$217$	−11.8885	−0.807047
$218$	14.9443	1.01215
$219$	−2.52786	−0.170817
$220$	0	0
$221$	17.2361	1.15942
$222$	0.618034	0.0414797
$223$	8.18034	0.547796	0.273898	−	0.961759i	$-0.411687\pi$
$223$	8.18034	0.547796	0.273898	+	0.961759i	$0.411687\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	10.9443	0.728002
$227$	17.7082	1.17533	0.587667	−	0.809103i	$-0.300046\pi$
$227$	17.7082	1.17533	0.587667	+	0.809103i	$0.300046\pi$
$228$	2.76393	0.183046
$229$	17.1246	1.13163	0.565813	−	0.824534i	$-0.308562\pi$
$229$	17.1246	1.13163	0.565813	+	0.824534i	$0.308562\pi$
$230$	0	0
$231$	−2.76393	−0.181853
$232$	−6.32624	−0.415338
$233$	−13.5623	−0.888496	−0.444248	−	0.895904i	$-0.646529\pi$
$233$	−13.5623	−0.888496	−0.444248	+	0.895904i	$0.646529\pi$
$234$	−10.0902	−0.659615
$235$	0	0
$236$	−9.23607	−0.601217
$237$	−7.14590	−0.464176
$238$	−5.52786	−0.358318
$239$	3.14590	0.203491	0.101746	−	0.994810i	$-0.467557\pi$
$239$	3.14590	0.203491	0.101746	+	0.994810i	$0.467557\pi$
$240$	0	0
$241$	−10.4721	−0.674570	−0.337285	−	0.941403i	$-0.609509\pi$
$241$	−10.4721	−0.674570	−0.337285	+	0.941403i	$0.609509\pi$
$242$	−2.09017	−0.134361
$243$	−13.9443	−0.894525
$244$	8.38197	0.536600
$245$	0	0
$246$	4.56231	0.290882
$247$	17.2361	1.09670
$248$	−9.61803	−0.610746
$249$	−3.41641	−0.216506
$250$	0	0
$251$	−3.05573	−0.192876	−0.0964379	−	0.995339i	$-0.530745\pi$
$251$	−3.05573	−0.192876	−0.0964379	+	0.995339i	$0.530745\pi$
$252$	3.23607	0.203853
$253$	−13.9443	−0.876669
$254$	0.472136	0.0296244
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.9443	1.18171	0.590856	−	0.806777i	$-0.298790\pi$
$257$	18.9443	1.18171	0.590856	+	0.806777i	$0.298790\pi$
$258$	0.472136	0.0293939
$259$	−1.23607	−0.0768055
$260$	0	0
$261$	−16.5623	−1.02518
$262$	−8.65248	−0.534552
$263$	8.76393	0.540407	0.270204	−	0.962803i	$-0.412909\pi$
$263$	8.76393	0.540407	0.270204	+	0.962803i	$0.412909\pi$
$264$	−2.23607	−0.137620
$265$	0	0
$266$	−5.52786	−0.338935
$267$	6.47214	0.396088
$268$	10.0902	0.616355
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−4.94427	−0.300343	−0.150172	−	0.988660i	$-0.547983\pi$
$271$	−4.94427	−0.300343	−0.150172	+	0.988660i	$0.547983\pi$
$272$	−4.47214	−0.271163
$273$	−2.94427	−0.178195
$274$	19.3262	1.16754
$275$	0	0
$276$	−2.38197	−0.143378
$277$	7.79837	0.468559	0.234279	−	0.972169i	$-0.424727\pi$
$277$	7.79837	0.468559	0.234279	+	0.972169i	$0.424727\pi$
$278$	1.85410	0.111202
$279$	−25.1803	−1.50751
$280$	0	0
$281$	−5.88854	−0.351281	−0.175641	−	0.984454i	$-0.556200\pi$
$281$	−5.88854	−0.351281	−0.175641	+	0.984454i	$0.556200\pi$
$282$	−2.00000	−0.119098
$283$	−11.2361	−0.667915	−0.333957	−	0.942588i	$-0.608384\pi$
$283$	−11.2361	−0.667915	−0.333957	+	0.942588i	$0.608384\pi$
$284$	−14.9443	−0.886779
$285$	0	0
$286$	−13.9443	−0.824542
$287$	−9.12461	−0.538609
$288$	2.61803	0.154269
$289$	3.00000	0.176471
$290$	0	0
$291$	5.23607	0.306944
$292$	4.09017	0.239359
$293$	−18.6525	−1.08969	−0.544845	−	0.838537i	$-0.683411\pi$
$293$	−18.6525	−1.08969	−0.544845	+	0.838537i	$0.683411\pi$
$294$	−3.38197	−0.197240
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−12.5623	−0.728939
$298$	6.18034	0.358017
$299$	−14.8541	−0.859035
$300$	0	0
$301$	−0.944272	−0.0544269
$302$	−17.7082	−1.01899
$303$	−7.70820	−0.442825
$304$	−4.47214	−0.256495
$305$	0	0
$306$	−11.7082	−0.669313
$307$	6.14590	0.350765	0.175382	−	0.984500i	$-0.443884\pi$
$307$	6.14590	0.350765	0.175382	+	0.984500i	$0.443884\pi$
$308$	4.47214	0.254824
$309$	−10.0557	−0.572050
$310$	0	0
$311$	2.03444	0.115363	0.0576813	−	0.998335i	$-0.481629\pi$
$311$	2.03444	0.115363	0.0576813	+	0.998335i	$0.481629\pi$
$312$	−2.38197	−0.134852
$313$	−5.81966	−0.328947	−0.164473	−	0.986382i	$-0.552592\pi$
$313$	−5.81966	−0.328947	−0.164473	+	0.986382i	$0.552592\pi$
$314$	−7.52786	−0.424822
$315$	0	0
$316$	11.5623	0.650431
$317$	−3.05573	−0.171627	−0.0858134	−	0.996311i	$-0.527349\pi$
$317$	−3.05573	−0.171627	−0.0858134	+	0.996311i	$0.527349\pi$
$318$	5.23607	0.293624
$319$	−22.8885	−1.28151
$320$	0	0
$321$	5.14590	0.287216
$322$	4.76393	0.265484
$323$	20.0000	1.11283
$324$	5.70820	0.317122
$325$	0	0
$326$	−12.4721	−0.690768
$327$	9.23607	0.510756
$328$	−7.38197	−0.407601
$329$	4.00000	0.220527
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	5.52786	0.303381
$333$	−2.61803	−0.143467
$334$	7.14590	0.391006
$335$	0	0
$336$	0.763932	0.0416759
$337$	17.0344	0.927925	0.463963	−	0.885855i	$-0.346427\pi$
$337$	17.0344	0.927925	0.463963	+	0.885855i	$0.346427\pi$
$338$	−1.85410	−0.100850
$339$	6.76393	0.367366
$340$	0	0
$341$	−34.7984	−1.88444
$342$	−11.7082	−0.633107
$343$	15.4164	0.832408
$344$	−0.763932	−0.0411885
$345$	0	0
$346$	8.47214	0.455465
$347$	−12.7639	−0.685204	−0.342602	−	0.939481i	$-0.611308\pi$
$347$	−12.7639	−0.685204	−0.342602	+	0.939481i	$0.611308\pi$
$348$	−3.90983	−0.209589
$349$	12.1803	0.651999	0.325999	−	0.945370i	$-0.394299\pi$
$349$	12.1803	0.651999	0.325999	+	0.945370i	$0.394299\pi$
$350$	0	0
$351$	−13.3820	−0.714277
$352$	3.61803	0.192842
$353$	−29.7082	−1.58121	−0.790604	−	0.612328i	$-0.790233\pi$
$353$	−29.7082	−1.58121	−0.790604	+	0.612328i	$0.790233\pi$
$354$	−5.70820	−0.303388
$355$	0	0
$356$	−10.4721	−0.555022
$357$	−3.41641	−0.180815
$358$	−18.6525	−0.985814
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.52786	−0.290538
$363$	−1.29180	−0.0678017
$364$	4.76393	0.249698
$365$	0	0
$366$	5.18034	0.270781
$367$	27.1246	1.41589	0.707947	−	0.706266i	$-0.249622\pi$
$367$	27.1246	1.41589	0.707947	+	0.706266i	$0.249622\pi$
$368$	3.85410	0.200909
$369$	−19.3262	−1.00608
$370$	0	0
$371$	−10.4721	−0.543686
$372$	−5.94427	−0.308196
$373$	−14.2918	−0.740001	−0.370001	−	0.929032i	$-0.620643\pi$
$373$	−14.2918	−0.740001	−0.370001	+	0.929032i	$0.620643\pi$
$374$	−16.1803	−0.836665
$375$	0	0
$376$	3.23607	0.166887
$377$	−24.3820	−1.25574
$378$	4.29180	0.220746
$379$	16.9098	0.868600	0.434300	−	0.900768i	$-0.356996\pi$
$379$	16.9098	0.868600	0.434300	+	0.900768i	$0.356996\pi$
$380$	0	0
$381$	0.291796	0.0149492
$382$	−4.09017	−0.209271
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0.618034	0.0315389
$385$	0	0
$386$	4.00000	0.203595
$387$	−2.00000	−0.101666
$388$	−8.47214	−0.430108
$389$	−0.145898	−0.00739732	−0.00369866	−	0.999993i	$-0.501177\pi$
$389$	−0.145898	−0.00739732	−0.00369866	+	0.999993i	$0.501177\pi$
$390$	0	0
$391$	−17.2361	−0.871665
$392$	5.47214	0.276385
$393$	−5.34752	−0.269747
$394$	−16.4721	−0.829854
$395$	0	0
$396$	9.47214	0.475993
$397$	10.6525	0.534632	0.267316	−	0.963609i	$-0.413863\pi$
$397$	10.6525	0.534632	0.267316	+	0.963609i	$0.413863\pi$
$398$	−20.9443	−1.04984
$399$	−3.41641	−0.171034
$400$	0	0
$401$	−9.23607	−0.461227	−0.230614	−	0.973045i	$-0.574073\pi$
$401$	−9.23607	−0.461227	−0.230614	+	0.973045i	$0.574073\pi$
$402$	6.23607	0.311027
$403$	−37.0689	−1.84653
$404$	12.4721	0.620512
$405$	0	0
$406$	7.81966	0.388083
$407$	−3.61803	−0.179339
$408$	−2.76393	−0.136835
$409$	−3.81966	−0.188870	−0.0944350	−	0.995531i	$-0.530104\pi$
$409$	−3.81966	−0.188870	−0.0944350	+	0.995531i	$0.530104\pi$
$410$	0	0
$411$	11.9443	0.589167
$412$	16.2705	0.801590
$413$	11.4164	0.561765
$414$	10.0902	0.495905
$415$	0	0
$416$	3.85410	0.188963
$417$	1.14590	0.0561149
$418$	−16.1803	−0.791406
$419$	9.56231	0.467149	0.233575	−	0.972339i	$-0.424958\pi$
$419$	9.56231	0.467149	0.233575	+	0.972339i	$0.424958\pi$
$420$	0	0
$421$	−1.96556	−0.0957954	−0.0478977	−	0.998852i	$-0.515252\pi$
$421$	−1.96556	−0.0957954	−0.0478977	+	0.998852i	$0.515252\pi$
$422$	22.2705	1.08411
$423$	8.47214	0.411929
$424$	−8.47214	−0.411443
$425$	0	0
$426$	−9.23607	−0.447489
$427$	−10.3607	−0.501388
$428$	−8.32624	−0.402464
$429$	−8.61803	−0.416083
$430$	0	0
$431$	−32.3607	−1.55876	−0.779380	−	0.626552i	$-0.784466\pi$
$431$	−32.3607	−1.55876	−0.779380	+	0.626552i	$0.784466\pi$
$432$	3.47214	0.167053
$433$	36.3262	1.74573	0.872864	−	0.487964i	$-0.162260\pi$
$433$	36.3262	1.74573	0.872864	+	0.487964i	$0.162260\pi$
$434$	11.8885	0.570668
$435$	0	0
$436$	−14.9443	−0.715701
$437$	−17.2361	−0.824513
$438$	2.52786	0.120786
$439$	−16.7984	−0.801743	−0.400871	−	0.916134i	$-0.631293\pi$
$439$	−16.7984	−0.801743	−0.400871	+	0.916134i	$0.631293\pi$
$440$	0	0
$441$	14.3262	0.682202
$442$	−17.2361	−0.819836
$443$	18.2705	0.868058	0.434029	−	0.900899i	$-0.357092\pi$
$443$	18.2705	0.868058	0.434029	+	0.900899i	$0.357092\pi$
$444$	−0.618034	−0.0293306
$445$	0	0
$446$	−8.18034	−0.387350
$447$	3.81966	0.180664
$448$	−1.23607	−0.0583987
$449$	−19.5279	−0.921577	−0.460788	−	0.887510i	$-0.652433\pi$
$449$	−19.5279	−0.921577	−0.460788	+	0.887510i	$0.652433\pi$
$450$	0	0
$451$	−26.7082	−1.25764
$452$	−10.9443	−0.514775
$453$	−10.9443	−0.514207
$454$	−17.7082	−0.831087
$455$	0	0
$456$	−2.76393	−0.129433
$457$	29.2361	1.36761	0.683803	−	0.729667i	$-0.260325\pi$
$457$	29.2361	1.36761	0.683803	+	0.729667i	$0.260325\pi$
$458$	−17.1246	−0.800181
$459$	−15.5279	−0.724779
$460$	0	0
$461$	−21.0557	−0.980663	−0.490332	−	0.871536i	$-0.663124\pi$
$461$	−21.0557	−0.980663	−0.490332	+	0.871536i	$0.663124\pi$
$462$	2.76393	0.128590
$463$	−15.5623	−0.723242	−0.361621	−	0.932325i	$-0.617777\pi$
$463$	−15.5623	−0.723242	−0.361621	+	0.932325i	$0.617777\pi$
$464$	6.32624	0.293688
$465$	0	0
$466$	13.5623	0.628262
$467$	−20.3607	−0.942180	−0.471090	−	0.882085i	$-0.656139\pi$
$467$	−20.3607	−0.942180	−0.471090	+	0.882085i	$0.656139\pi$
$468$	10.0902	0.466418
$469$	−12.4721	−0.575910
$470$	0	0
$471$	−4.65248	−0.214375
$472$	9.23607	0.425124
$473$	−2.76393	−0.127086
$474$	7.14590	0.328222
$475$	0	0
$476$	5.52786	0.253369
$477$	−22.1803	−1.01557
$478$	−3.14590	−0.143890
$479$	16.4377	0.751057	0.375529	−	0.926811i	$-0.377461\pi$
$479$	16.4377	0.751057	0.375529	+	0.926811i	$0.377461\pi$
$480$	0	0
$481$	−3.85410	−0.175732
$482$	10.4721	0.476993
$483$	2.94427	0.133969
$484$	2.09017	0.0950077
$485$	0	0
$486$	13.9443	0.632525
$487$	25.3050	1.14668	0.573338	−	0.819319i	$-0.305648\pi$
$487$	25.3050	1.14668	0.573338	+	0.819319i	$0.305648\pi$
$488$	−8.38197	−0.379434
$489$	−7.70820	−0.348577
$490$	0	0
$491$	27.4508	1.23884	0.619420	−	0.785060i	$-0.287368\pi$
$491$	27.4508	1.23884	0.619420	+	0.785060i	$0.287368\pi$
$492$	−4.56231	−0.205685
$493$	−28.2918	−1.27420
$494$	−17.2361	−0.775487
$495$	0	0
$496$	9.61803	0.431862
$497$	18.4721	0.828589
$498$	3.41641	0.153093
$499$	23.7082	1.06132	0.530662	−	0.847583i	$-0.321943\pi$
$499$	23.7082	1.06132	0.530662	+	0.847583i	$0.321943\pi$
$500$	0	0
$501$	4.41641	0.197311
$502$	3.05573	0.136384
$503$	−7.90983	−0.352682	−0.176341	−	0.984329i	$-0.556426\pi$
$503$	−7.90983	−0.352682	−0.176341	+	0.984329i	$0.556426\pi$
$504$	−3.23607	−0.144146
$505$	0	0
$506$	13.9443	0.619898
$507$	−1.14590	−0.0508911
$508$	−0.472136	−0.0209476
$509$	−4.29180	−0.190231	−0.0951153	−	0.995466i	$-0.530322\pi$
$509$	−4.29180	−0.190231	−0.0951153	+	0.995466i	$0.530322\pi$
$510$	0	0
$511$	−5.05573	−0.223652
$512$	−1.00000	−0.0441942
$513$	−15.5279	−0.685572
$514$	−18.9443	−0.835596
$515$	0	0
$516$	−0.472136	−0.0207846
$517$	11.7082	0.514926
$518$	1.23607	0.0543097
$519$	5.23607	0.229838
$520$	0	0
$521$	25.4164	1.11351	0.556757	−	0.830676i	$-0.312046\pi$
$521$	25.4164	1.11351	0.556757	+	0.830676i	$0.312046\pi$
$522$	16.5623	0.724912
$523$	−34.1803	−1.49460	−0.747301	−	0.664486i	$-0.768651\pi$
$523$	−34.1803	−1.49460	−0.747301	+	0.664486i	$0.768651\pi$
$524$	8.65248	0.377985
$525$	0	0
$526$	−8.76393	−0.382126
$527$	−43.0132	−1.87368
$528$	2.23607	0.0973124
$529$	−8.14590	−0.354169
$530$	0	0
$531$	24.1803	1.04934
$532$	5.52786	0.239663
$533$	−28.4508	−1.23234
$534$	−6.47214	−0.280077
$535$	0	0
$536$	−10.0902	−0.435829
$537$	−11.5279	−0.497464
$538$	−4.00000	−0.172452
$539$	19.7984	0.852776
$540$	0	0
$541$	5.32624	0.228993	0.114496	−	0.993424i	$-0.463475\pi$
$541$	5.32624	0.228993	0.114496	+	0.993424i	$0.463475\pi$
$542$	4.94427	0.212375
$543$	−3.41641	−0.146612
$544$	4.47214	0.191741
$545$	0	0
$546$	2.94427	0.126003
$547$	−42.0689	−1.79874	−0.899368	−	0.437193i	$-0.855973\pi$
$547$	−42.0689	−1.79874	−0.899368	+	0.437193i	$0.855973\pi$
$548$	−19.3262	−0.825576
$549$	−21.9443	−0.936559
$550$	0	0
$551$	−28.2918	−1.20527
$552$	2.38197	0.101383
$553$	−14.2918	−0.607749
$554$	−7.79837	−0.331321
$555$	0	0
$556$	−1.85410	−0.0786314
$557$	0.562306	0.0238257	0.0119128	−	0.999929i	$-0.496208\pi$
$557$	0.562306	0.0238257	0.0119128	+	0.999929i	$0.496208\pi$
$558$	25.1803	1.06597
$559$	−2.94427	−0.124529
$560$	0	0
$561$	−10.0000	−0.422200
$562$	5.88854	0.248393
$563$	27.8885	1.17536	0.587681	−	0.809093i	$-0.300041\pi$
$563$	27.8885	1.17536	0.587681	+	0.809093i	$0.300041\pi$
$564$	2.00000	0.0842152
$565$	0	0
$566$	11.2361	0.472287
$567$	−7.05573	−0.296313
$568$	14.9443	0.627048
$569$	−21.8885	−0.917615	−0.458808	−	0.888536i	$-0.651723\pi$
$569$	−21.8885	−0.917615	−0.458808	+	0.888536i	$0.651723\pi$
$570$	0	0
$571$	9.56231	0.400170	0.200085	−	0.979779i	$-0.435878\pi$
$571$	9.56231	0.400170	0.200085	+	0.979779i	$0.435878\pi$
$572$	13.9443	0.583039
$573$	−2.52786	−0.105603
$574$	9.12461	0.380854
$575$	0	0
$576$	−2.61803	−0.109085
$577$	20.6525	0.859774	0.429887	−	0.902883i	$-0.358553\pi$
$577$	20.6525	0.859774	0.429887	+	0.902883i	$0.358553\pi$
$578$	−3.00000	−0.124784
$579$	2.47214	0.102738
$580$	0	0
$581$	−6.83282	−0.283473
$582$	−5.23607	−0.217042
$583$	−30.6525	−1.26950
$584$	−4.09017	−0.169252
$585$	0	0
$586$	18.6525	0.770527
$587$	30.9443	1.27721	0.638603	−	0.769536i	$-0.279512\pi$
$587$	30.9443	1.27721	0.638603	+	0.769536i	$0.279512\pi$
$588$	3.38197	0.139470
$589$	−43.0132	−1.77233
$590$	0	0
$591$	−10.1803	−0.418763
$592$	1.00000	0.0410997
$593$	2.56231	0.105221	0.0526106	−	0.998615i	$-0.483246\pi$
$593$	2.56231	0.105221	0.0526106	+	0.998615i	$0.483246\pi$
$594$	12.5623	0.515438
$595$	0	0
$596$	−6.18034	−0.253157
$597$	−12.9443	−0.529774
$598$	14.8541	0.607429
$599$	38.3607	1.56737	0.783687	−	0.621155i	$-0.213336\pi$
$599$	38.3607	1.56737	0.783687	+	0.621155i	$0.213336\pi$
$600$	0	0
$601$	35.6869	1.45570	0.727850	−	0.685737i	$-0.240520\pi$
$601$	35.6869	1.45570	0.727850	+	0.685737i	$0.240520\pi$
$602$	0.944272	0.0384856
$603$	−26.4164	−1.07576
$604$	17.7082	0.720537
$605$	0	0
$606$	7.70820	0.313124
$607$	−5.96556	−0.242135	−0.121067	−	0.992644i	$-0.538632\pi$
$607$	−5.96556	−0.242135	−0.121067	+	0.992644i	$0.538632\pi$
$608$	4.47214	0.181369
$609$	4.83282	0.195836
$610$	0	0
$611$	12.4721	0.504569
$612$	11.7082	0.473276
$613$	36.1803	1.46131	0.730655	−	0.682747i	$-0.239215\pi$
$613$	36.1803	1.46131	0.730655	+	0.682747i	$0.239215\pi$
$614$	−6.14590	−0.248028
$615$	0	0
$616$	−4.47214	−0.180187
$617$	11.0902	0.446473	0.223237	−	0.974764i	$-0.428338\pi$
$617$	11.0902	0.446473	0.223237	+	0.974764i	$0.428338\pi$
$618$	10.0557	0.404501
$619$	18.2705	0.734354	0.367177	−	0.930151i	$-0.380324\pi$
$619$	18.2705	0.734354	0.367177	+	0.930151i	$0.380324\pi$
$620$	0	0
$621$	13.3820	0.537000
$622$	−2.03444	−0.0815737
$623$	12.9443	0.518601
$624$	2.38197	0.0953550
$625$	0	0
$626$	5.81966	0.232600
$627$	−10.0000	−0.399362
$628$	7.52786	0.300394
$629$	−4.47214	−0.178316
$630$	0	0
$631$	26.3951	1.05077	0.525387	−	0.850864i	$-0.323921\pi$
$631$	26.3951	1.05077	0.525387	+	0.850864i	$0.323921\pi$
$632$	−11.5623	−0.459924
$633$	13.7639	0.547067
$634$	3.05573	0.121358
$635$	0	0
$636$	−5.23607	−0.207624
$637$	21.0902	0.835623
$638$	22.8885	0.906166
$639$	39.1246	1.54775
$640$	0	0
$641$	−22.5066	−0.888956	−0.444478	−	0.895790i	$-0.646611\pi$
$641$	−22.5066	−0.888956	−0.444478	+	0.895790i	$0.646611\pi$
$642$	−5.14590	−0.203092
$643$	33.2361	1.31070	0.655351	−	0.755324i	$-0.272521\pi$
$643$	33.2361	1.31070	0.655351	+	0.755324i	$0.272521\pi$
$644$	−4.76393	−0.187725
$645$	0	0
$646$	−20.0000	−0.786889
$647$	−18.9098	−0.743422	−0.371711	−	0.928348i	$-0.621229\pi$
$647$	−18.9098	−0.743422	−0.371711	+	0.928348i	$0.621229\pi$
$648$	−5.70820	−0.224239
$649$	33.4164	1.31171
$650$	0	0
$651$	7.34752	0.287972
$652$	12.4721	0.488447
$653$	4.72949	0.185079	0.0925396	−	0.995709i	$-0.470502\pi$
$653$	4.72949	0.185079	0.0925396	+	0.995709i	$0.470502\pi$
$654$	−9.23607	−0.361159
$655$	0	0
$656$	7.38197	0.288217
$657$	−10.7082	−0.417767
$658$	−4.00000	−0.155936
$659$	−15.4508	−0.601880	−0.300940	−	0.953643i	$-0.597300\pi$
$659$	−15.4508	−0.601880	−0.300940	+	0.953643i	$0.597300\pi$
$660$	0	0
$661$	1.67376	0.0651018	0.0325509	−	0.999470i	$-0.489637\pi$
$661$	1.67376	0.0651018	0.0325509	+	0.999470i	$0.489637\pi$
$662$	−28.0000	−1.08825
$663$	−10.6525	−0.413708
$664$	−5.52786	−0.214523
$665$	0	0
$666$	2.61803	0.101447
$667$	24.3820	0.944073
$668$	−7.14590	−0.276483
$669$	−5.05573	−0.195466
$670$	0	0
$671$	−30.3262	−1.17073
$672$	−0.763932	−0.0294693
$673$	−17.8541	−0.688225	−0.344113	−	0.938928i	$-0.611820\pi$
$673$	−17.8541	−0.688225	−0.344113	+	0.938928i	$0.611820\pi$
$674$	−17.0344	−0.656142
$675$	0	0
$676$	1.85410	0.0713116
$677$	−3.34752	−0.128656	−0.0643279	−	0.997929i	$-0.520490\pi$
$677$	−3.34752	−0.128656	−0.0643279	+	0.997929i	$0.520490\pi$
$678$	−6.76393	−0.259767
$679$	10.4721	0.401884
$680$	0	0
$681$	−10.9443	−0.419385
$682$	34.7984	1.33250
$683$	35.4164	1.35517	0.677586	−	0.735444i	$-0.263026\pi$
$683$	35.4164	1.35517	0.677586	+	0.735444i	$0.263026\pi$
$684$	11.7082	0.447674
$685$	0	0
$686$	−15.4164	−0.588601
$687$	−10.5836	−0.403789
$688$	0.763932	0.0291246
$689$	−32.6525	−1.24396
$690$	0	0
$691$	−35.7771	−1.36102	−0.680512	−	0.732737i	$-0.738243\pi$
$691$	−35.7771	−1.36102	−0.680512	+	0.732737i	$0.738243\pi$
$692$	−8.47214	−0.322062
$693$	−11.7082	−0.444758
$694$	12.7639	0.484512
$695$	0	0
$696$	3.90983	0.148202
$697$	−33.0132	−1.25046
$698$	−12.1803	−0.461033
$699$	8.38197	0.317035
$700$	0	0
$701$	3.97871	0.150274	0.0751370	−	0.997173i	$-0.476061\pi$
$701$	3.97871	0.150274	0.0751370	+	0.997173i	$0.476061\pi$
$702$	13.3820	0.505070
$703$	−4.47214	−0.168670
$704$	−3.61803	−0.136360
$705$	0	0
$706$	29.7082	1.11808
$707$	−15.4164	−0.579794
$708$	5.70820	0.214527
$709$	43.2148	1.62297	0.811483	−	0.584377i	$-0.198661\pi$
$709$	43.2148	1.62297	0.811483	+	0.584377i	$0.198661\pi$
$710$	0	0
$711$	−30.2705	−1.13523
$712$	10.4721	0.392460
$713$	37.0689	1.38824
$714$	3.41641	0.127856
$715$	0	0
$716$	18.6525	0.697076
$717$	−1.94427	−0.0726102
$718$	−4.47214	−0.166899
$719$	0.583592	0.0217643	0.0108822	−	0.999941i	$-0.496536\pi$
$719$	0.583592	0.0217643	0.0108822	+	0.999941i	$0.496536\pi$
$720$	0	0
$721$	−20.1115	−0.748990
$722$	−1.00000	−0.0372161
$723$	6.47214	0.240701
$724$	5.52786	0.205441
$725$	0	0
$726$	1.29180	0.0479430
$727$	23.1459	0.858434	0.429217	−	0.903201i	$-0.358790\pi$
$727$	23.1459	0.858434	0.429217	+	0.903201i	$0.358790\pi$
$728$	−4.76393	−0.176563
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−3.41641	−0.126360
$732$	−5.18034	−0.191471
$733$	−36.4721	−1.34713	−0.673565	−	0.739128i	$-0.735238\pi$
$733$	−36.4721	−1.34713	−0.673565	+	0.739128i	$0.735238\pi$
$734$	−27.1246	−1.00119
$735$	0	0
$736$	−3.85410	−0.142064
$737$	−36.5066	−1.34474
$738$	19.3262	0.711409
$739$	−22.0902	−0.812600	−0.406300	−	0.913740i	$-0.633181\pi$
$739$	−22.0902	−0.812600	−0.406300	+	0.913740i	$0.633181\pi$
$740$	0	0
$741$	−10.6525	−0.391328
$742$	10.4721	0.384444
$743$	−10.0689	−0.369392	−0.184696	−	0.982796i	$-0.559130\pi$
$743$	−10.0689	−0.369392	−0.184696	+	0.982796i	$0.559130\pi$
$744$	5.94427	0.217928
$745$	0	0
$746$	14.2918	0.523260
$747$	−14.4721	−0.529508
$748$	16.1803	0.591612
$749$	10.2918	0.376054
$750$	0	0
$751$	−18.9443	−0.691286	−0.345643	−	0.938366i	$-0.612339\pi$
$751$	−18.9443	−0.691286	−0.345643	+	0.938366i	$0.612339\pi$
$752$	−3.23607	−0.118007
$753$	1.88854	0.0688224
$754$	24.3820	0.887939
$755$	0	0
$756$	−4.29180	−0.156091
$757$	10.8541	0.394499	0.197250	−	0.980353i	$-0.436799\pi$
$757$	10.8541	0.394499	0.197250	+	0.980353i	$0.436799\pi$
$758$	−16.9098	−0.614193
$759$	8.61803	0.312815
$760$	0	0
$761$	−25.8541	−0.937210	−0.468605	−	0.883408i	$-0.655243\pi$
$761$	−25.8541	−0.937210	−0.468605	+	0.883408i	$0.655243\pi$
$762$	−0.291796	−0.0105707
$763$	18.4721	0.668736
$764$	4.09017	0.147977
$765$	0	0
$766$	−17.8885	−0.646339
$767$	35.5967	1.28532
$768$	−0.618034	−0.0223014
$769$	−11.8885	−0.428712	−0.214356	−	0.976756i	$-0.568765\pi$
$769$	−11.8885	−0.428712	−0.214356	+	0.976756i	$0.568765\pi$
$770$	0	0
$771$	−11.7082	−0.421661
$772$	−4.00000	−0.143963
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	8.47214	0.304132
$777$	0.763932	0.0274059
$778$	0.145898	0.00523070
$779$	−33.0132	−1.18282
$780$	0	0
$781$	54.0689	1.93474
$782$	17.2361	0.616361
$783$	21.9656	0.784985
$784$	−5.47214	−0.195433
$785$	0	0
$786$	5.34752	0.190740
$787$	34.4721	1.22880	0.614399	−	0.788995i	$-0.289398\pi$
$787$	34.4721	1.22880	0.614399	+	0.788995i	$0.289398\pi$
$788$	16.4721	0.586796
$789$	−5.41641	−0.192829
$790$	0	0
$791$	13.5279	0.480995
$792$	−9.47214	−0.336578
$793$	−32.3050	−1.14718
$794$	−10.6525	−0.378042
$795$	0	0
$796$	20.9443	0.742350
$797$	−12.7295	−0.450902	−0.225451	−	0.974255i	$-0.572386\pi$
$797$	−12.7295	−0.450902	−0.225451	+	0.974255i	$0.572386\pi$
$798$	3.41641	0.120940
$799$	14.4721	0.511987
$800$	0	0
$801$	27.4164	0.968711
$802$	9.23607	0.326137
$803$	−14.7984	−0.522223
$804$	−6.23607	−0.219929
$805$	0	0
$806$	37.0689	1.30570
$807$	−2.47214	−0.0870233
$808$	−12.4721	−0.438768
$809$	27.1246	0.953651	0.476825	−	0.878998i	$-0.341787\pi$
$809$	27.1246	0.953651	0.476825	+	0.878998i	$0.341787\pi$
$810$	0	0
$811$	53.1033	1.86471	0.932355	−	0.361544i	$-0.117750\pi$
$811$	53.1033	1.86471	0.932355	+	0.361544i	$0.117750\pi$
$812$	−7.81966	−0.274416
$813$	3.05573	0.107169
$814$	3.61803	0.126812
$815$	0	0
$816$	2.76393	0.0967570
$817$	−3.41641	−0.119525
$818$	3.81966	0.133551
$819$	−12.4721	−0.435812
$820$	0	0
$821$	21.4164	0.747438	0.373719	−	0.927542i	$-0.378082\pi$
$821$	21.4164	0.747438	0.373719	+	0.927542i	$0.378082\pi$
$822$	−11.9443	−0.416604
$823$	33.8885	1.18128	0.590640	−	0.806935i	$-0.298875\pi$
$823$	33.8885	1.18128	0.590640	+	0.806935i	$0.298875\pi$
$824$	−16.2705	−0.566810
$825$	0	0
$826$	−11.4164	−0.397228
$827$	−56.0689	−1.94971	−0.974853	−	0.222849i	$-0.928464\pi$
$827$	−56.0689	−1.94971	−0.974853	+	0.222849i	$0.928464\pi$
$828$	−10.0902	−0.350658
$829$	31.2016	1.08368	0.541839	−	0.840483i	$-0.317728\pi$
$829$	31.2016	1.08368	0.541839	+	0.840483i	$0.317728\pi$
$830$	0	0
$831$	−4.81966	−0.167192
$832$	−3.85410	−0.133617
$833$	24.4721	0.847909
$834$	−1.14590	−0.0396792
$835$	0	0
$836$	16.1803	0.559609
$837$	33.3951	1.15430
$838$	−9.56231	−0.330324
$839$	−5.34752	−0.184617	−0.0923085	−	0.995730i	$-0.529425\pi$
$839$	−5.34752	−0.184617	−0.0923085	+	0.995730i	$0.529425\pi$
$840$	0	0
$841$	11.0213	0.380044
$842$	1.96556	0.0677376
$843$	3.63932	0.125345
$844$	−22.2705	−0.766583
$845$	0	0
$846$	−8.47214	−0.291278
$847$	−2.58359	−0.0887733
$848$	8.47214	0.290934
$849$	6.94427	0.238327
$850$	0	0
$851$	3.85410	0.132117
$852$	9.23607	0.316422
$853$	12.7426	0.436300	0.218150	−	0.975915i	$-0.429998\pi$
$853$	12.7426	0.436300	0.218150	+	0.975915i	$0.429998\pi$
$854$	10.3607	0.354535
$855$	0	0
$856$	8.32624	0.284585
$857$	26.9443	0.920399	0.460199	−	0.887816i	$-0.347778\pi$
$857$	26.9443	0.920399	0.460199	+	0.887816i	$0.347778\pi$
$858$	8.61803	0.294215
$859$	26.5836	0.907020	0.453510	−	0.891251i	$-0.350172\pi$
$859$	26.5836	0.907020	0.453510	+	0.891251i	$0.350172\pi$
$860$	0	0
$861$	5.63932	0.192188
$862$	32.3607	1.10221
$863$	7.41641	0.252457	0.126229	−	0.992001i	$-0.459713\pi$
$863$	7.41641	0.252457	0.126229	+	0.992001i	$0.459713\pi$
$864$	−3.47214	−0.118124
$865$	0	0
$866$	−36.3262	−1.23442
$867$	−1.85410	−0.0629686
$868$	−11.8885	−0.403523
$869$	−41.8328	−1.41908
$870$	0	0
$871$	−38.8885	−1.31769
$872$	14.9443	0.506077
$873$	22.1803	0.750691
$874$	17.2361	0.583019
$875$	0	0
$876$	−2.52786	−0.0854086
$877$	−36.8328	−1.24376	−0.621878	−	0.783114i	$-0.713630\pi$
$877$	−36.8328	−1.24376	−0.621878	+	0.783114i	$0.713630\pi$
$878$	16.7984	0.566918
$879$	11.5279	0.388825
$880$	0	0
$881$	−22.7426	−0.766219	−0.383110	−	0.923703i	$-0.625147\pi$
$881$	−22.7426	−0.766219	−0.383110	+	0.923703i	$0.625147\pi$
$882$	−14.3262	−0.482390
$883$	−29.3050	−0.986190	−0.493095	−	0.869975i	$-0.664135\pi$
$883$	−29.3050	−0.986190	−0.493095	+	0.869975i	$0.664135\pi$
$884$	17.2361	0.579712
$885$	0	0
$886$	−18.2705	−0.613810
$887$	7.88854	0.264871	0.132436	−	0.991192i	$-0.457720\pi$
$887$	7.88854	0.264871	0.132436	+	0.991192i	$0.457720\pi$
$888$	0.618034	0.0207399
$889$	0.583592	0.0195731
$890$	0	0
$891$	−20.6525	−0.691884
$892$	8.18034	0.273898
$893$	14.4721	0.484292
$894$	−3.81966	−0.127749
$895$	0	0
$896$	1.23607	0.0412941
$897$	9.18034	0.306523
$898$	19.5279	0.651653
$899$	60.8460	2.02933
$900$	0	0
$901$	−37.8885	−1.26225
$902$	26.7082	0.889286
$903$	0.583592	0.0194207
$904$	10.9443	0.364001
$905$	0	0
$906$	10.9443	0.363599
$907$	20.1115	0.667790	0.333895	−	0.942610i	$-0.391637\pi$
$907$	20.1115	0.667790	0.333895	+	0.942610i	$0.391637\pi$
$908$	17.7082	0.587667
$909$	−32.6525	−1.08301
$910$	0	0
$911$	−21.8885	−0.725200	−0.362600	−	0.931945i	$-0.618111\pi$
$911$	−21.8885	−0.725200	−0.362600	+	0.931945i	$0.618111\pi$
$912$	2.76393	0.0915229
$913$	−20.0000	−0.661903
$914$	−29.2361	−0.967043
$915$	0	0
$916$	17.1246	0.565813
$917$	−10.6950	−0.353182
$918$	15.5279	0.512496
$919$	29.8885	0.985932	0.492966	−	0.870049i	$-0.335913\pi$
$919$	29.8885	0.985932	0.492966	+	0.870049i	$0.335913\pi$
$920$	0	0
$921$	−3.79837	−0.125161
$922$	21.0557	0.693433
$923$	57.5967	1.89582
$924$	−2.76393	−0.0909267
$925$	0	0
$926$	15.5623	0.511409
$927$	−42.5967	−1.39906
$928$	−6.32624	−0.207669
$929$	28.4508	0.933442	0.466721	−	0.884405i	$-0.345435\pi$
$929$	28.4508	0.933442	0.466721	+	0.884405i	$0.345435\pi$
$930$	0	0
$931$	24.4721	0.802042
$932$	−13.5623	−0.444248
$933$	−1.25735	−0.0411639
$934$	20.3607	0.666222
$935$	0	0
$936$	−10.0902	−0.329808
$937$	−57.0476	−1.86366	−0.931832	−	0.362890i	$-0.881790\pi$
$937$	−57.0476	−1.86366	−0.931832	+	0.362890i	$0.881790\pi$
$938$	12.4721	0.407230
$939$	3.59675	0.117375
$940$	0	0
$941$	−3.81966	−0.124517	−0.0622587	−	0.998060i	$-0.519830\pi$
$941$	−3.81966	−0.124517	−0.0622587	+	0.998060i	$0.519830\pi$
$942$	4.65248	0.151586
$943$	28.4508	0.926487
$944$	−9.23607	−0.300608
$945$	0	0
$946$	2.76393	0.0898632
$947$	34.8328	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$947$	34.8328	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$948$	−7.14590	−0.232088
$949$	−15.7639	−0.511719
$950$	0	0
$951$	1.88854	0.0612402
$952$	−5.52786	−0.179159
$953$	44.4508	1.43990	0.719952	−	0.694024i	$-0.244164\pi$
$953$	44.4508	1.43990	0.719952	+	0.694024i	$0.244164\pi$
$954$	22.1803	0.718115
$955$	0	0
$956$	3.14590	0.101746
$957$	14.1459	0.457272
$958$	−16.4377	−0.531078
$959$	23.8885	0.771401
$960$	0	0
$961$	61.5066	1.98408
$962$	3.85410	0.124261
$963$	21.7984	0.702443
$964$	−10.4721	−0.337285
$965$	0	0
$966$	−2.94427	−0.0947304
$967$	−11.7295	−0.377195	−0.188597	−	0.982054i	$-0.560394\pi$
$967$	−11.7295	−0.377195	−0.188597	+	0.982054i	$0.560394\pi$
$968$	−2.09017	−0.0671806
$969$	−12.3607	−0.397082
$970$	0	0
$971$	26.6738	0.856002	0.428001	−	0.903778i	$-0.359218\pi$
$971$	26.6738	0.856002	0.428001	+	0.903778i	$0.359218\pi$
$972$	−13.9443	−0.447263
$973$	2.29180	0.0734716
$974$	−25.3050	−0.810823
$975$	0	0
$976$	8.38197	0.268300
$977$	−52.4721	−1.67873	−0.839366	−	0.543566i	$-0.817074\pi$
$977$	−52.4721	−1.67873	−0.839366	+	0.543566i	$0.817074\pi$
$978$	7.70820	0.246481
$979$	37.8885	1.21092
$980$	0	0
$981$	39.1246	1.24915
$982$	−27.4508	−0.875992
$983$	−39.7771	−1.26869	−0.634346	−	0.773049i	$-0.718731\pi$
$983$	−39.7771	−1.26869	−0.634346	+	0.773049i	$0.718731\pi$
$984$	4.56231	0.145441
$985$	0	0
$986$	28.2918	0.900994
$987$	−2.47214	−0.0786890
$988$	17.2361	0.548352
$989$	2.94427	0.0936224
$990$	0	0
$991$	−54.1033	−1.71865	−0.859324	−	0.511431i	$-0.829116\pi$
$991$	−54.1033	−1.71865	−0.859324	+	0.511431i	$0.829116\pi$
$992$	−9.61803	−0.305373
$993$	−17.3050	−0.549156
$994$	−18.4721	−0.585901
$995$	0	0
$996$	−3.41641	−0.108253
$997$	−53.7771	−1.70314	−0.851569	−	0.524243i	$-0.824348\pi$
$997$	−53.7771	−1.70314	−0.851569	+	0.524243i	$0.824348\pi$
$998$	−23.7082	−0.750470
$999$	3.47214	0.109854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.t.1.1		2
5.2	odd	4		1850.2.b.j.149.2		4
5.3	odd	4		1850.2.b.j.149.3		4
5.4	even	2		74.2.a.b.1.2	✓	2
15.14	odd	2		666.2.a.i.1.2		2
20.19	odd	2		592.2.a.g.1.1		2
35.34	odd	2		3626.2.a.s.1.1		2
40.19	odd	2		2368.2.a.u.1.2		2
40.29	even	2		2368.2.a.y.1.1		2
55.54	odd	2		8954.2.a.j.1.2		2
60.59	even	2		5328.2.a.bc.1.2		2
185.184	even	2		2738.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.b.1.2	✓	2	5.4	even	2
592.2.a.g.1.1		2	20.19	odd	2
666.2.a.i.1.2		2	15.14	odd	2
1850.2.a.t.1.1		2	1.1	even	1	trivial
1850.2.b.j.149.2		4	5.2	odd	4
1850.2.b.j.149.3		4	5.3	odd	4
2368.2.a.u.1.2		2	40.19	odd	2
2368.2.a.y.1.1		2	40.29	even	2
2738.2.a.g.1.2		2	185.184	even	2
3626.2.a.s.1.1		2	35.34	odd	2
5328.2.a.bc.1.2		2	60.59	even	2
8954.2.a.j.1.2		2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{11} -0.618034 q^{12} -3.85410 q^{13} +1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.61803 q^{18} -4.47214 q^{19} +0.763932 q^{21} +3.61803 q^{22} +3.85410 q^{23} +0.618034 q^{24} +3.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} +6.32624 q^{29} +9.61803 q^{31} -1.00000 q^{32} +2.23607 q^{33} +4.47214 q^{34} -2.61803 q^{36} +1.00000 q^{37} +4.47214 q^{38} +2.38197 q^{39} +7.38197 q^{41} -0.763932 q^{42} +0.763932 q^{43} -3.61803 q^{44} -3.85410 q^{46} -3.23607 q^{47} -0.618034 q^{48} -5.47214 q^{49} +2.76393 q^{51} -3.85410 q^{52} +8.47214 q^{53} -3.47214 q^{54} +1.23607 q^{56} +2.76393 q^{57} -6.32624 q^{58} -9.23607 q^{59} +8.38197 q^{61} -9.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -2.23607 q^{66} +10.0902 q^{67} -4.47214 q^{68} -2.38197 q^{69} -14.9443 q^{71} +2.61803 q^{72} +4.09017 q^{73} -1.00000 q^{74} -4.47214 q^{76} +4.47214 q^{77} -2.38197 q^{78} +11.5623 q^{79} +5.70820 q^{81} -7.38197 q^{82} +5.52786 q^{83} +0.763932 q^{84} -0.763932 q^{86} -3.90983 q^{87} +3.61803 q^{88} -10.4721 q^{89} +4.76393 q^{91} +3.85410 q^{92} -5.94427 q^{93} +3.23607 q^{94} +0.618034 q^{96} -8.47214 q^{97} +5.47214 q^{98} +9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 - 5 * q^11 + q^12 - q^13 - 2 * q^14 + 2 * q^16 + 3 * q^18 + 6 * q^21 + 5 * q^22 + q^23 - q^24 + q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 + 17 * q^31 - 2 * q^32 - 3 * q^36 + 2 * q^37 + 7 * q^39 + 17 * q^41 - 6 * q^42 + 6 * q^43 - 5 * q^44 - q^46 - 2 * q^47 + q^48 - 2 * q^49 + 10 * q^51 - q^52 + 8 * q^53 + 2 * q^54 - 2 * q^56 + 10 * q^57 + 3 * q^58 - 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^63 + 2 * q^64 + 9 * q^67 - 7 * q^69 - 12 * q^71 + 3 * q^72 - 3 * q^73 - 2 * q^74 - 7 * q^78 + 3 * q^79 - 2 * q^81 - 17 * q^82 + 20 * q^83 + 6 * q^84 - 6 * q^86 - 19 * q^87 + 5 * q^88 - 12 * q^89 + 14 * q^91 + q^92 + 6 * q^93 + 2 * q^94 - q^96 - 8 * q^97 + 2 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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