LMFDB - Embedded newform 1425.2.c.m.799.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,0,-2,0,0,-4,0,-16,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.m.799.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} -2.23607i q^{7} -2.23607i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.61803i q^{12} -6.47214i q^{13} -1.38197 q^{14} +1.85410 q^{16} +1.23607i q^{17} +0.618034i q^{18} +1.00000 q^{19} +2.23607 q^{21} +2.47214i q^{22} -3.23607i q^{23} +2.23607 q^{24} -4.00000 q^{26} -1.00000i q^{27} -3.61803i q^{28} -7.47214 q^{29} -10.4721 q^{31} -5.61803i q^{32} -4.00000i q^{33} +0.763932 q^{34} -1.61803 q^{36} +2.76393i q^{37} -0.618034i q^{38} +6.47214 q^{39} -5.00000 q^{41} -1.38197i q^{42} -4.00000i q^{43} -6.47214 q^{44} -2.00000 q^{46} +0.472136i q^{47} +1.85410i q^{48} +2.00000 q^{49} -1.23607 q^{51} -10.4721i q^{52} +5.00000i q^{53} -0.618034 q^{54} -5.00000 q^{56} +1.00000i q^{57} +4.61803i q^{58} +14.7082 q^{59} +7.94427 q^{61} +6.47214i q^{62} +2.23607i q^{63} +0.236068 q^{64} -2.47214 q^{66} -10.9443i q^{67} +2.00000i q^{68} +3.23607 q^{69} +9.18034 q^{71} +2.23607i q^{72} -3.47214i q^{73} +1.70820 q^{74} +1.61803 q^{76} +8.94427i q^{77} -4.00000i q^{78} +2.29180 q^{79} +1.00000 q^{81} +3.09017i q^{82} +6.94427i q^{83} +3.61803 q^{84} -2.47214 q^{86} -7.47214i q^{87} +8.94427i q^{88} +9.94427 q^{89} -14.4721 q^{91} -5.23607i q^{92} -10.4721i q^{93} +0.291796 q^{94} +5.61803 q^{96} -1.70820i q^{97} -1.23607i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} - 6 q^{16} + 4 q^{19} - 16 q^{26} - 12 q^{29} - 24 q^{31} + 12 q^{34} - 2 q^{36} + 8 q^{39} - 20 q^{41} - 8 q^{44} - 8 q^{46} + 8 q^{49} + 4 q^{51}+ \cdots + 16 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 - 6 * q^16 + 4 * q^19 - 16 * q^26 - 12 * q^29 - 24 * q^31 + 12 * q^34 - 2 * q^36 + 8 * q^39 - 20 * q^41 - 8 * q^44 - 8 * q^46 + 8 * q^49 + 4 * q^51 + 2 * q^54 - 20 * q^56 + 32 * q^59 - 4 * q^61 - 8 * q^64 + 8 * q^66 + 4 * q^69 - 8 * q^71 - 20 * q^74 + 2 * q^76 + 36 * q^79 + 4 * q^81 + 10 * q^84 + 8 * q^86 + 4 * q^89 - 40 * q^91 + 28 * q^94 + 18 * q^96 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 0.618034i	− 0.437016i	−0.975835	−	0.218508i	$-0.929881\pi$
$2$	− 0.618034i	− 0.437016i	0.975835	−	0.218508i	$-0.0701190\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	− 2.23607i	− 0.845154i	−0.906327	−	0.422577i	$-0.861126\pi$
$7$	− 2.23607i	− 0.845154i	0.906327	−	0.422577i	$-0.138874\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.61803i	0.467086i
$13$	− 6.47214i	− 1.79505i	−0.440966	−	0.897524i	$-0.645364\pi$
$13$	− 6.47214i	− 1.79505i	0.440966	−	0.897524i	$-0.354636\pi$
$14$	−1.38197	−0.369346
$15$	0	0
$16$	1.85410	0.463525
$17$	1.23607i	0.299791i	0.988702	+	0.149895i	$0.0478936\pi$
$17$	1.23607i	0.299791i	−0.988702	+	0.149895i	$0.952106\pi$
$18$	0.618034i	0.145672i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.23607	0.487950
$22$	2.47214i	0.527061i
$23$	− 3.23607i	− 0.674767i	−0.941367	−	0.337383i	$-0.890458\pi$
$23$	− 3.23607i	− 0.674767i	0.941367	−	0.337383i	$-0.109542\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−4.00000	−0.784465
$27$	− 1.00000i	− 0.192450i
$28$	− 3.61803i	− 0.683744i
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	0	0
$31$	−10.4721	−1.88085	−0.940426	−	0.340000i	$-0.889573\pi$
$31$	−10.4721	−1.88085	−0.940426	+	0.340000i	$0.889573\pi$
$32$	− 5.61803i	− 0.993137i
$33$	− 4.00000i	− 0.696311i
$34$	0.763932	0.131013
$35$	0	0
$36$	−1.61803	−0.269672
$37$	2.76393i	0.454388i	0.973850	+	0.227194i	$0.0729551\pi$
$37$	2.76393i	0.454388i	−0.973850	+	0.227194i	$0.927045\pi$
$38$	− 0.618034i	− 0.100258i
$39$	6.47214	1.03637
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	− 1.38197i	− 0.213242i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−6.47214	−0.975711
$45$	0	0
$46$	−2.00000	−0.294884
$47$	0.472136i	0.0688681i	0.999407	+	0.0344341i	$0.0109629\pi$
$47$	0.472136i	0.0688681i	−0.999407	+	0.0344341i	$0.989037\pi$
$48$	1.85410i	0.267617i
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.23607	−0.173084
$52$	− 10.4721i	− 1.45222i
$53$	5.00000i	0.686803i	0.939189	+	0.343401i	$0.111579\pi$
$53$	5.00000i	0.686803i	−0.939189	+	0.343401i	$0.888421\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	−5.00000	−0.668153
$57$	1.00000i	0.132453i
$58$	4.61803i	0.606378i
$59$	14.7082	1.91485	0.957423	−	0.288690i	$-0.0932198\pi$
$59$	14.7082	1.91485	0.957423	+	0.288690i	$0.0932198\pi$
$60$	0	0
$61$	7.94427	1.01716	0.508580	−	0.861015i	$-0.330171\pi$
$61$	7.94427	1.01716	0.508580	+	0.861015i	$0.330171\pi$
$62$	6.47214i	0.821962i
$63$	2.23607i	0.281718i
$64$	0.236068	0.0295085
$65$	0	0
$66$	−2.47214	−0.304299
$67$	− 10.9443i	− 1.33706i	−0.743687	−	0.668528i	$-0.766925\pi$
$67$	− 10.9443i	− 1.33706i	0.743687	−	0.668528i	$-0.233075\pi$
$68$	2.00000i	0.242536i
$69$	3.23607	0.389577
$70$	0	0
$71$	9.18034	1.08951	0.544753	−	0.838597i	$-0.316623\pi$
$71$	9.18034	1.08951	0.544753	+	0.838597i	$0.316623\pi$
$72$	2.23607i	0.263523i
$73$	− 3.47214i	− 0.406383i	−0.979139	−	0.203191i	$-0.934869\pi$
$73$	− 3.47214i	− 0.406383i	0.979139	−	0.203191i	$-0.0651314\pi$
$74$	1.70820	0.198575
$75$	0	0
$76$	1.61803	0.185601
$77$	8.94427i	1.01929i
$78$	− 4.00000i	− 0.452911i
$79$	2.29180	0.257847	0.128924	−	0.991655i	$-0.458848\pi$
$79$	2.29180	0.257847	0.128924	+	0.991655i	$0.458848\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.09017i	0.341252i
$83$	6.94427i	0.762233i	0.924527	+	0.381116i	$0.124460\pi$
$83$	6.94427i	0.762233i	−0.924527	+	0.381116i	$0.875540\pi$
$84$	3.61803	0.394760
$85$	0	0
$86$	−2.47214	−0.266577
$87$	− 7.47214i	− 0.801097i
$88$	8.94427i	0.953463i
$89$	9.94427	1.05409	0.527045	−	0.849837i	$-0.323300\pi$
$89$	9.94427	1.05409	0.527045	+	0.849837i	$0.323300\pi$
$90$	0	0
$91$	−14.4721	−1.51709
$92$	− 5.23607i	− 0.545898i
$93$	− 10.4721i	− 1.08591i
$94$	0.291796	0.0300965
$95$	0	0
$96$	5.61803	0.573388
$97$	− 1.70820i	− 0.173442i	−0.996233	−	0.0867209i	$-0.972361\pi$
$97$	− 1.70820i	− 0.173442i	0.996233	−	0.0867209i	$-0.0276388\pi$
$98$	− 1.23607i	− 0.124862i
$99$	4.00000	0.402015
$100$	0	0
$101$	−14.1803	−1.41100	−0.705498	−	0.708712i	$-0.749277\pi$
$101$	−14.1803	−1.41100	−0.705498	+	0.708712i	$0.749277\pi$
$102$	0.763932i	0.0756405i
$103$	− 16.9443i	− 1.66957i	−0.550577	−	0.834784i	$-0.685592\pi$
$103$	− 16.9443i	− 1.66957i	0.550577	−	0.834784i	$-0.314408\pi$
$104$	−14.4721	−1.41911
$105$	0	0
$106$	3.09017	0.300144
$107$	− 16.2361i	− 1.56960i	−0.619749	−	0.784800i	$-0.712766\pi$
$107$	− 16.2361i	− 1.56960i	0.619749	−	0.784800i	$-0.287234\pi$
$108$	− 1.61803i	− 0.155695i
$109$	7.70820	0.738312	0.369156	−	0.929367i	$-0.379647\pi$
$109$	7.70820	0.738312	0.369156	+	0.929367i	$0.379647\pi$
$110$	0	0
$111$	−2.76393	−0.262341
$112$	− 4.14590i	− 0.391751i
$113$	12.4721i	1.17328i	0.809848	+	0.586640i	$0.199550\pi$
$113$	12.4721i	1.17328i	−0.809848	+	0.586640i	$0.800450\pi$
$114$	0.618034	0.0578842
$115$	0	0
$116$	−12.0902	−1.12254
$117$	6.47214i	0.598349i
$118$	− 9.09017i	− 0.836818i
$119$	2.76393	0.253369
$120$	0	0
$121$	5.00000	0.454545
$122$	− 4.90983i	− 0.444515i
$123$	− 5.00000i	− 0.450835i
$124$	−16.9443	−1.52164
$125$	0	0
$126$	1.38197	0.123115
$127$	10.7639i	0.955145i	0.878592	+	0.477572i	$0.158483\pi$
$127$	10.7639i	0.955145i	−0.878592	+	0.477572i	$0.841517\pi$
$128$	− 11.3820i	− 1.00603i
$129$	4.00000	0.352180
$130$	0	0
$131$	−6.94427	−0.606724	−0.303362	−	0.952875i	$-0.598109\pi$
$131$	−6.94427	−0.606724	−0.303362	+	0.952875i	$0.598109\pi$
$132$	− 6.47214i	− 0.563327i
$133$	− 2.23607i	− 0.193892i
$134$	−6.76393	−0.584315
$135$	0	0
$136$	2.76393	0.237005
$137$	4.29180i	0.366673i	0.983050	+	0.183336i	$0.0586898\pi$
$137$	4.29180i	0.366673i	−0.983050	+	0.183336i	$0.941310\pi$
$138$	− 2.00000i	− 0.170251i
$139$	15.7639	1.33708	0.668540	−	0.743677i	$-0.266920\pi$
$139$	15.7639	1.33708	0.668540	+	0.743677i	$0.266920\pi$
$140$	0	0
$141$	−0.472136	−0.0397610
$142$	− 5.67376i	− 0.476132i
$143$	25.8885i	2.16491i
$144$	−1.85410	−0.154508
$145$	0	0
$146$	−2.14590	−0.177596
$147$	2.00000i	0.164957i
$148$	4.47214i	0.367607i
$149$	19.4164	1.59065	0.795327	−	0.606181i	$-0.207299\pi$
$149$	19.4164	1.59065	0.795327	+	0.606181i	$0.207299\pi$
$150$	0	0
$151$	11.7082	0.952800	0.476400	−	0.879229i	$-0.341941\pi$
$151$	11.7082	0.952800	0.476400	+	0.879229i	$0.341941\pi$
$152$	− 2.23607i	− 0.181369i
$153$	− 1.23607i	− 0.0999302i
$154$	5.52786	0.445448
$155$	0	0
$156$	10.4721	0.838442
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	− 1.41641i	− 0.112683i
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−7.23607	−0.570282
$162$	− 0.618034i	− 0.0485573i
$163$	11.1803i	0.875712i	0.899045	+	0.437856i	$0.144262\pi$
$163$	11.1803i	0.875712i	−0.899045	+	0.437856i	$0.855738\pi$
$164$	−8.09017	−0.631736
$165$	0	0
$166$	4.29180	0.333108
$167$	− 24.5967i	− 1.90335i	−0.307102	−	0.951677i	$-0.599359\pi$
$167$	− 24.5967i	− 1.90335i	0.307102	−	0.951677i	$-0.400641\pi$
$168$	− 5.00000i	− 0.385758i
$169$	−28.8885	−2.22220
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 6.47214i	− 0.493496i
$173$	13.4721i	1.02427i	0.858906	+	0.512134i	$0.171145\pi$
$173$	13.4721i	1.02427i	−0.858906	+	0.512134i	$0.828855\pi$
$174$	−4.61803	−0.350092
$175$	0	0
$176$	−7.41641	−0.559033
$177$	14.7082i	1.10554i
$178$	− 6.14590i	− 0.460655i
$179$	−22.1246	−1.65367	−0.826836	−	0.562444i	$-0.809861\pi$
$179$	−22.1246	−1.65367	−0.826836	+	0.562444i	$0.809861\pi$
$180$	0	0
$181$	10.4721	0.778388	0.389194	−	0.921156i	$-0.372754\pi$
$181$	10.4721	0.778388	0.389194	+	0.921156i	$0.372754\pi$
$182$	8.94427i	0.662994i
$183$	7.94427i	0.587257i
$184$	−7.23607	−0.533450
$185$	0	0
$186$	−6.47214	−0.474560
$187$	− 4.94427i	− 0.361561i
$188$	0.763932i	0.0557155i
$189$	−2.23607	−0.162650
$190$	0	0
$191$	7.41641	0.536632	0.268316	−	0.963331i	$-0.413533\pi$
$191$	7.41641	0.536632	0.268316	+	0.963331i	$0.413533\pi$
$192$	0.236068i	0.0170367i
$193$	3.23607i	0.232937i	0.993194	+	0.116469i	$0.0371574\pi$
$193$	3.23607i	0.232937i	−0.993194	+	0.116469i	$0.962843\pi$
$194$	−1.05573	−0.0757969
$195$	0	0
$196$	3.23607	0.231148
$197$	− 22.6525i	− 1.61392i	−0.590605	−	0.806961i	$-0.701111\pi$
$197$	− 22.6525i	− 1.61392i	0.590605	−	0.806961i	$-0.298889\pi$
$198$	− 2.47214i	− 0.175687i
$199$	−1.76393	−0.125042	−0.0625209	−	0.998044i	$-0.519914\pi$
$199$	−1.76393	−0.125042	−0.0625209	+	0.998044i	$0.519914\pi$
$200$	0	0
$201$	10.9443	0.771949
$202$	8.76393i	0.616628i
$203$	16.7082i	1.17269i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−10.4721	−0.729628
$207$	3.23607i	0.224922i
$208$	− 12.0000i	− 0.832050i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	5.81966	0.400642	0.200321	−	0.979730i	$-0.435802\pi$
$211$	5.81966	0.400642	0.200321	+	0.979730i	$0.435802\pi$
$212$	8.09017i	0.555635i
$213$	9.18034i	0.629027i
$214$	−10.0344	−0.685940
$215$	0	0
$216$	−2.23607	−0.152145
$217$	23.4164i	1.58961i
$218$	− 4.76393i	− 0.322654i
$219$	3.47214	0.234625
$220$	0	0
$221$	8.00000	0.538138
$222$	1.70820i	0.114647i
$223$	− 9.23607i	− 0.618493i	−0.950982	−	0.309246i	$-0.899923\pi$
$223$	− 9.23607i	− 0.618493i	0.950982	−	0.309246i	$-0.100077\pi$
$224$	−12.5623	−0.839354
$225$	0	0
$226$	7.70820	0.512742
$227$	28.1246i	1.86670i	0.358973	+	0.933348i	$0.383127\pi$
$227$	28.1246i	1.86670i	−0.358973	+	0.933348i	$0.616873\pi$
$228$	1.61803i	0.107157i
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	−8.94427	−0.588490
$232$	16.7082i	1.09695i
$233$	− 2.94427i	− 0.192886i	−0.995339	−	0.0964428i	$-0.969254\pi$
$233$	− 2.94427i	− 0.192886i	0.995339	−	0.0964428i	$-0.0307465\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	23.7984	1.54914
$237$	2.29180i	0.148868i
$238$	− 1.70820i	− 0.110726i
$239$	−0.291796	−0.0188747	−0.00943736	−	0.999955i	$-0.503004\pi$
$239$	−0.291796	−0.0188747	−0.00943736	+	0.999955i	$0.503004\pi$
$240$	0	0
$241$	−16.1803	−1.04227	−0.521134	−	0.853475i	$-0.674491\pi$
$241$	−16.1803	−1.04227	−0.521134	+	0.853475i	$0.674491\pi$
$242$	− 3.09017i	− 0.198644i
$243$	1.00000i	0.0641500i
$244$	12.8541	0.822900
$245$	0	0
$246$	−3.09017	−0.197022
$247$	− 6.47214i	− 0.411812i
$248$	23.4164i	1.48694i
$249$	−6.94427	−0.440075
$250$	0	0
$251$	3.70820	0.234060	0.117030	−	0.993128i	$-0.462663\pi$
$251$	3.70820	0.234060	0.117030	+	0.993128i	$0.462663\pi$
$252$	3.61803i	0.227915i
$253$	12.9443i	0.813799i
$254$	6.65248	0.417413
$255$	0	0
$256$	−6.56231	−0.410144
$257$	− 13.0000i	− 0.810918i	−0.914113	−	0.405459i	$-0.867112\pi$
$257$	− 13.0000i	− 0.810918i	0.914113	−	0.405459i	$-0.132888\pi$
$258$	− 2.47214i	− 0.153908i
$259$	6.18034	0.384028
$260$	0	0
$261$	7.47214	0.462514
$262$	4.29180i	0.265148i
$263$	10.2918i	0.634619i	0.948322	+	0.317310i	$0.102779\pi$
$263$	10.2918i	0.634619i	−0.948322	+	0.317310i	$0.897221\pi$
$264$	−8.94427	−0.550482
$265$	0	0
$266$	−1.38197	−0.0847338
$267$	9.94427i	0.608580i
$268$	− 17.7082i	− 1.08170i
$269$	−20.4721	−1.24821	−0.624104	−	0.781341i	$-0.714536\pi$
$269$	−20.4721	−1.24821	−0.624104	+	0.781341i	$0.714536\pi$
$270$	0	0
$271$	3.18034	0.193192	0.0965959	−	0.995324i	$-0.469205\pi$
$271$	3.18034	0.193192	0.0965959	+	0.995324i	$0.469205\pi$
$272$	2.29180i	0.138961i
$273$	− 14.4721i	− 0.875894i
$274$	2.65248	0.160242
$275$	0	0
$276$	5.23607	0.315174
$277$	5.58359i	0.335486i	0.985831	+	0.167743i	$0.0536478\pi$
$277$	5.58359i	0.335486i	−0.985831	+	0.167743i	$0.946352\pi$
$278$	− 9.74265i	− 0.584325i
$279$	10.4721	0.626950
$280$	0	0
$281$	7.88854	0.470591	0.235296	−	0.971924i	$-0.424394\pi$
$281$	7.88854	0.470591	0.235296	+	0.971924i	$0.424394\pi$
$282$	0.291796i	0.0173762i
$283$	− 9.88854i	− 0.587813i	−0.955834	−	0.293906i	$-0.905045\pi$
$283$	− 9.88854i	− 0.587813i	0.955834	−	0.293906i	$-0.0949554\pi$
$284$	14.8541	0.881429
$285$	0	0
$286$	16.0000	0.946100
$287$	11.1803i	0.659955i
$288$	5.61803i	0.331046i
$289$	15.4721	0.910126
$290$	0	0
$291$	1.70820	0.100137
$292$	− 5.61803i	− 0.328771i
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	1.23607	0.0720889
$295$	0	0
$296$	6.18034	0.359225
$297$	4.00000i	0.232104i
$298$	− 12.0000i	− 0.695141i
$299$	−20.9443	−1.21124
$300$	0	0
$301$	−8.94427	−0.515539
$302$	− 7.23607i	− 0.416389i
$303$	− 14.1803i	− 0.814639i
$304$	1.85410	0.106340
$305$	0	0
$306$	−0.763932	−0.0436711
$307$	− 3.05573i	− 0.174400i	−0.996191	−	0.0871998i	$-0.972208\pi$
$307$	− 3.05573i	− 0.174400i	0.996191	−	0.0871998i	$-0.0277919\pi$
$308$	14.4721i	0.824626i
$309$	16.9443	0.963926
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	− 14.4721i	− 0.819323i
$313$	30.9443i	1.74907i	0.484959	+	0.874537i	$0.338834\pi$
$313$	30.9443i	1.74907i	−0.484959	+	0.874537i	$0.661166\pi$
$314$	4.32624	0.244144
$315$	0	0
$316$	3.70820	0.208603
$317$	− 7.94427i	− 0.446195i	−0.974796	−	0.223097i	$-0.928383\pi$
$317$	− 7.94427i	− 0.446195i	0.974796	−	0.223097i	$-0.0716168\pi$
$318$	3.09017i	0.173288i
$319$	29.8885	1.67344
$320$	0	0
$321$	16.2361	0.906209
$322$	4.47214i	0.249222i
$323$	1.23607i	0.0687767i
$324$	1.61803	0.0898908
$325$	0	0
$326$	6.90983	0.382700
$327$	7.70820i	0.426265i
$328$	11.1803i	0.617331i
$329$	1.05573	0.0582042
$330$	0	0
$331$	−26.3607	−1.44891	−0.724457	−	0.689320i	$-0.757909\pi$
$331$	−26.3607	−1.44891	−0.724457	+	0.689320i	$0.757909\pi$
$332$	11.2361i	0.616659i
$333$	− 2.76393i	− 0.151463i
$334$	−15.2016	−0.831796
$335$	0	0
$336$	4.14590	0.226177
$337$	− 8.47214i	− 0.461507i	−0.973012	−	0.230753i	$-0.925881\pi$
$337$	− 8.47214i	− 0.461507i	0.973012	−	0.230753i	$-0.0741190\pi$
$338$	17.8541i	0.971135i
$339$	−12.4721	−0.677393
$340$	0	0
$341$	41.8885	2.26839
$342$	0.618034i	0.0334195i
$343$	− 20.1246i	− 1.08663i
$344$	−8.94427	−0.482243
$345$	0	0
$346$	8.32624	0.447621
$347$	− 1.70820i	− 0.0917012i	−0.998948	−	0.0458506i	$-0.985400\pi$
$347$	− 1.70820i	− 0.0917012i	0.998948	−	0.0458506i	$-0.0145998\pi$
$348$	− 12.0902i	− 0.648101i
$349$	17.8328	0.954569	0.477284	−	0.878749i	$-0.341621\pi$
$349$	17.8328	0.954569	0.477284	+	0.878749i	$0.341621\pi$
$350$	0	0
$351$	−6.47214	−0.345457
$352$	22.4721i	1.19777i
$353$	26.4721i	1.40897i	0.709719	+	0.704485i	$0.248822\pi$
$353$	26.4721i	1.40897i	−0.709719	+	0.704485i	$0.751178\pi$
$354$	9.09017	0.483137
$355$	0	0
$356$	16.0902	0.852777
$357$	2.76393i	0.146283i
$358$	13.6738i	0.722681i
$359$	25.2361	1.33191	0.665954	−	0.745992i	$-0.268025\pi$
$359$	25.2361	1.33191	0.665954	+	0.745992i	$0.268025\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 6.47214i	− 0.340168i
$363$	5.00000i	0.262432i
$364$	−23.4164	−1.22735
$365$	0	0
$366$	4.90983	0.256641
$367$	33.3050i	1.73850i	0.494369	+	0.869252i	$0.335399\pi$
$367$	33.3050i	1.73850i	−0.494369	+	0.869252i	$0.664601\pi$
$368$	− 6.00000i	− 0.312772i
$369$	5.00000	0.260290
$370$	0	0
$371$	11.1803	0.580454
$372$	− 16.9443i	− 0.878520i
$373$	− 12.1803i	− 0.630674i	−0.948980	−	0.315337i	$-0.897882\pi$
$373$	− 12.1803i	− 0.630674i	0.948980	−	0.315337i	$-0.102118\pi$
$374$	−3.05573	−0.158008
$375$	0	0
$376$	1.05573	0.0544450
$377$	48.3607i	2.49070i
$378$	1.38197i	0.0710807i
$379$	−6.76393	−0.347440	−0.173720	−	0.984795i	$-0.555579\pi$
$379$	−6.76393	−0.347440	−0.173720	+	0.984795i	$0.555579\pi$
$380$	0	0
$381$	−10.7639	−0.551453
$382$	− 4.58359i	− 0.234517i
$383$	− 16.1246i	− 0.823929i	−0.911200	−	0.411965i	$-0.864843\pi$
$383$	− 16.1246i	− 0.823929i	0.911200	−	0.411965i	$-0.135157\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	2.00000	0.101797
$387$	4.00000i	0.203331i
$388$	− 2.76393i	− 0.140317i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	− 4.47214i	− 0.225877i
$393$	− 6.94427i	− 0.350292i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	6.47214	0.325237
$397$	22.3607i	1.12225i	0.827731	+	0.561125i	$0.189631\pi$
$397$	22.3607i	1.12225i	−0.827731	+	0.561125i	$0.810369\pi$
$398$	1.09017i	0.0546453i
$399$	2.23607	0.111943
$400$	0	0
$401$	−20.8328	−1.04034	−0.520171	−	0.854062i	$-0.674132\pi$
$401$	−20.8328	−1.04034	−0.520171	+	0.854062i	$0.674132\pi$
$402$	− 6.76393i	− 0.337354i
$403$	67.7771i	3.37622i
$404$	−22.9443	−1.14152
$405$	0	0
$406$	10.3262	0.512483
$407$	− 11.0557i	− 0.548012i
$408$	2.76393i	0.136835i
$409$	−8.76393	−0.433349	−0.216674	−	0.976244i	$-0.569521\pi$
$409$	−8.76393	−0.433349	−0.216674	+	0.976244i	$0.569521\pi$
$410$	0	0
$411$	−4.29180	−0.211699
$412$	− 27.4164i	− 1.35071i
$413$	− 32.8885i	− 1.61834i
$414$	2.00000	0.0982946
$415$	0	0
$416$	−36.3607	−1.78273
$417$	15.7639i	0.771963i
$418$	2.47214i	0.120916i
$419$	−39.3050	−1.92017	−0.960086	−	0.279704i	$-0.909764\pi$
$419$	−39.3050	−1.92017	−0.960086	+	0.279704i	$0.909764\pi$
$420$	0	0
$421$	11.5279	0.561834	0.280917	−	0.959732i	$-0.409361\pi$
$421$	11.5279	0.561834	0.280917	+	0.959732i	$0.409361\pi$
$422$	− 3.59675i	− 0.175087i
$423$	− 0.472136i	− 0.0229560i
$424$	11.1803	0.542965
$425$	0	0
$426$	5.67376	0.274895
$427$	− 17.7639i	− 0.859657i
$428$	− 26.2705i	− 1.26983i
$429$	−25.8885	−1.24991
$430$	0	0
$431$	−5.18034	−0.249528	−0.124764	−	0.992186i	$-0.539817\pi$
$431$	−5.18034	−0.249528	−0.124764	+	0.992186i	$0.539817\pi$
$432$	− 1.85410i	− 0.0892055i
$433$	19.5279i	0.938449i	0.883079	+	0.469225i	$0.155467\pi$
$433$	19.5279i	0.938449i	−0.883079	+	0.469225i	$0.844533\pi$
$434$	14.4721	0.694685
$435$	0	0
$436$	12.4721	0.597307
$437$	− 3.23607i	− 0.154802i
$438$	− 2.14590i	− 0.102535i
$439$	16.7639	0.800099	0.400049	−	0.916494i	$-0.368993\pi$
$439$	16.7639	0.800099	0.400049	+	0.916494i	$0.368993\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	− 4.94427i	− 0.235175i
$443$	− 13.4164i	− 0.637433i	−0.947850	−	0.318716i	$-0.896748\pi$
$443$	− 13.4164i	− 0.637433i	0.947850	−	0.318716i	$-0.103252\pi$
$444$	−4.47214	−0.212238
$445$	0	0
$446$	−5.70820	−0.270291
$447$	19.4164i	0.918365i
$448$	− 0.527864i	− 0.0249392i
$449$	25.4721	1.20210	0.601052	−	0.799210i	$-0.294748\pi$
$449$	25.4721	1.20210	0.601052	+	0.799210i	$0.294748\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	20.1803i	0.949203i
$453$	11.7082i	0.550099i
$454$	17.3820	0.815776
$455$	0	0
$456$	2.23607	0.104713
$457$	8.88854i	0.415789i	0.978151	+	0.207894i	$0.0666610\pi$
$457$	8.88854i	0.415789i	−0.978151	+	0.207894i	$0.933339\pi$
$458$	− 2.76393i	− 0.129150i
$459$	1.23607	0.0576947
$460$	0	0
$461$	−1.81966	−0.0847500	−0.0423750	−	0.999102i	$-0.513492\pi$
$461$	−1.81966	−0.0847500	−0.0423750	+	0.999102i	$0.513492\pi$
$462$	5.52786i	0.257180i
$463$	− 8.58359i	− 0.398913i	−0.979907	−	0.199457i	$-0.936082\pi$
$463$	− 8.58359i	− 0.398913i	0.979907	−	0.199457i	$-0.0639177\pi$
$464$	−13.8541	−0.643161
$465$	0	0
$466$	−1.81966	−0.0842941
$467$	− 26.1803i	− 1.21148i	−0.795662	−	0.605741i	$-0.792877\pi$
$467$	− 26.1803i	− 1.21148i	0.795662	−	0.605741i	$-0.207123\pi$
$468$	10.4721i	0.484075i
$469$	−24.4721	−1.13002
$470$	0	0
$471$	−7.00000	−0.322543
$472$	− 32.8885i	− 1.51382i
$473$	16.0000i	0.735681i
$474$	1.41641	0.0650578
$475$	0	0
$476$	4.47214	0.204980
$477$	− 5.00000i	− 0.228934i
$478$	0.180340i	0.00824855i
$479$	38.7639	1.77117	0.885585	−	0.464478i	$-0.153758\pi$
$479$	38.7639	1.77117	0.885585	+	0.464478i	$0.153758\pi$
$480$	0	0
$481$	17.8885	0.815647
$482$	10.0000i	0.455488i
$483$	− 7.23607i	− 0.329252i
$484$	8.09017	0.367735
$485$	0	0
$486$	0.618034	0.0280346
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	− 17.7639i	− 0.804135i
$489$	−11.1803	−0.505592
$490$	0	0
$491$	−8.29180	−0.374204	−0.187102	−	0.982341i	$-0.559909\pi$
$491$	−8.29180	−0.374204	−0.187102	+	0.982341i	$0.559909\pi$
$492$	− 8.09017i	− 0.364733i
$493$	− 9.23607i	− 0.415972i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−19.4164	−0.871822
$497$	− 20.5279i	− 0.920801i
$498$	4.29180i	0.192320i
$499$	31.1803	1.39582	0.697912	−	0.716184i	$-0.254113\pi$
$499$	31.1803	1.39582	0.697912	+	0.716184i	$0.254113\pi$
$500$	0	0
$501$	24.5967	1.09890
$502$	− 2.29180i	− 0.102288i
$503$	9.52786i	0.424826i	0.977180	+	0.212413i	$0.0681323\pi$
$503$	9.52786i	0.424826i	−0.977180	+	0.212413i	$0.931868\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	8.00000	0.355643
$507$	− 28.8885i	− 1.28299i
$508$	17.4164i	0.772728i
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−7.76393	−0.343456
$512$	− 18.7082i	− 0.826794i
$513$	− 1.00000i	− 0.0441511i
$514$	−8.03444	−0.354384
$515$	0	0
$516$	6.47214	0.284920
$517$	− 1.88854i	− 0.0830581i
$518$	− 3.81966i	− 0.167826i
$519$	−13.4721	−0.591361
$520$	0	0
$521$	−35.8328	−1.56986	−0.784932	−	0.619582i	$-0.787302\pi$
$521$	−35.8328	−1.56986	−0.784932	+	0.619582i	$0.787302\pi$
$522$	− 4.61803i	− 0.202126i
$523$	− 41.1246i	− 1.79825i	−0.437688	−	0.899127i	$-0.644203\pi$
$523$	− 41.1246i	− 1.79825i	0.437688	−	0.899127i	$-0.355797\pi$
$524$	−11.2361	−0.490850
$525$	0	0
$526$	6.36068	0.277339
$527$	− 12.9443i	− 0.563861i
$528$	− 7.41641i	− 0.322758i
$529$	12.5279	0.544690
$530$	0	0
$531$	−14.7082	−0.638282
$532$	− 3.61803i	− 0.156862i
$533$	32.3607i	1.40170i
$534$	6.14590	0.265959
$535$	0	0
$536$	−24.4721	−1.05704
$537$	− 22.1246i	− 0.954747i
$538$	12.6525i	0.545487i
$539$	−8.00000	−0.344584
$540$	0	0
$541$	24.8328	1.06765	0.533823	−	0.845596i	$-0.320755\pi$
$541$	24.8328	1.06765	0.533823	+	0.845596i	$0.320755\pi$
$542$	− 1.96556i	− 0.0844280i
$543$	10.4721i	0.449402i
$544$	6.94427	0.297733
$545$	0	0
$546$	−8.94427	−0.382780
$547$	− 7.70820i	− 0.329579i	−0.986329	−	0.164790i	$-0.947306\pi$
$547$	− 7.70820i	− 0.329579i	0.986329	−	0.164790i	$-0.0526945\pi$
$548$	6.94427i	0.296645i
$549$	−7.94427	−0.339053
$550$	0	0
$551$	−7.47214	−0.318324
$552$	− 7.23607i	− 0.307988i
$553$	− 5.12461i	− 0.217921i
$554$	3.45085	0.146613
$555$	0	0
$556$	25.5066	1.08172
$557$	− 33.8885i	− 1.43590i	−0.696093	−	0.717952i	$-0.745080\pi$
$557$	− 33.8885i	− 1.43590i	0.696093	−	0.717952i	$-0.254920\pi$
$558$	− 6.47214i	− 0.273987i
$559$	−25.8885	−1.09497
$560$	0	0
$561$	4.94427	0.208747
$562$	− 4.87539i	− 0.205656i
$563$	− 23.6525i	− 0.996833i	−0.866938	−	0.498417i	$-0.833915\pi$
$563$	− 23.6525i	− 0.996833i	0.866938	−	0.498417i	$-0.166085\pi$
$564$	−0.763932	−0.0321673
$565$	0	0
$566$	−6.11146	−0.256884
$567$	− 2.23607i	− 0.0939060i
$568$	− 20.5279i	− 0.861330i
$569$	20.8885	0.875693	0.437847	−	0.899050i	$-0.355741\pi$
$569$	20.8885	0.875693	0.437847	+	0.899050i	$0.355741\pi$
$570$	0	0
$571$	−15.6525	−0.655036	−0.327518	−	0.944845i	$-0.606212\pi$
$571$	−15.6525	−0.655036	−0.327518	+	0.944845i	$0.606212\pi$
$572$	41.8885i	1.75145i
$573$	7.41641i	0.309825i
$574$	6.90983	0.288411
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	− 1.41641i	− 0.0589658i	−0.999565	−	0.0294829i	$-0.990614\pi$
$577$	− 1.41641i	− 0.0589658i	0.999565	−	0.0294829i	$-0.00938606\pi$
$578$	− 9.56231i	− 0.397739i
$579$	−3.23607	−0.134486
$580$	0	0
$581$	15.5279	0.644204
$582$	− 1.05573i	− 0.0437613i
$583$	− 20.0000i	− 0.828315i
$584$	−7.76393	−0.321274
$585$	0	0
$586$	−1.23607	−0.0510615
$587$	12.6525i	0.522224i	0.965309	+	0.261112i	$0.0840891\pi$
$587$	12.6525i	0.522224i	−0.965309	+	0.261112i	$0.915911\pi$
$588$	3.23607i	0.133453i
$589$	−10.4721	−0.431497
$590$	0	0
$591$	22.6525	0.931798
$592$	5.12461i	0.210620i
$593$	13.8885i	0.570334i	0.958478	+	0.285167i	$0.0920491\pi$
$593$	13.8885i	0.570334i	−0.958478	+	0.285167i	$0.907951\pi$
$594$	2.47214	0.101433
$595$	0	0
$596$	31.4164	1.28687
$597$	− 1.76393i	− 0.0721929i
$598$	12.9443i	0.529331i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−35.7771	−1.45938	−0.729689	−	0.683779i	$-0.760335\pi$
$601$	−35.7771	−1.45938	−0.729689	+	0.683779i	$0.760335\pi$
$602$	5.52786i	0.225299i
$603$	10.9443i	0.445685i
$604$	18.9443	0.770831
$605$	0	0
$606$	−8.76393	−0.356010
$607$	− 19.1246i	− 0.776244i	−0.921608	−	0.388122i	$-0.873124\pi$
$607$	− 19.1246i	− 0.776244i	0.921608	−	0.388122i	$-0.126876\pi$
$608$	− 5.61803i	− 0.227841i
$609$	−16.7082	−0.677051
$610$	0	0
$611$	3.05573	0.123622
$612$	− 2.00000i	− 0.0808452i
$613$	− 23.0000i	− 0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$	− 23.0000i	− 0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$	−1.88854	−0.0762154
$615$	0	0
$616$	20.0000	0.805823
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	− 10.4721i	− 0.421251i
$619$	−30.7082	−1.23427	−0.617133	−	0.786858i	$-0.711706\pi$
$619$	−30.7082	−1.23427	−0.617133	+	0.786858i	$0.711706\pi$
$620$	0	0
$621$	−3.23607	−0.129859
$622$	− 2.47214i	− 0.0991236i
$623$	− 22.2361i	− 0.890869i
$624$	12.0000	0.480384
$625$	0	0
$626$	19.1246	0.764373
$627$	− 4.00000i	− 0.159745i
$628$	11.3262i	0.451966i
$629$	−3.41641	−0.136221
$630$	0	0
$631$	−43.4164	−1.72838	−0.864190	−	0.503166i	$-0.832169\pi$
$631$	−43.4164	−1.72838	−0.864190	+	0.503166i	$0.832169\pi$
$632$	− 5.12461i	− 0.203846i
$633$	5.81966i	0.231311i
$634$	−4.90983	−0.194994
$635$	0	0
$636$	−8.09017	−0.320796
$637$	− 12.9443i	− 0.512871i
$638$	− 18.4721i	− 0.731319i
$639$	−9.18034	−0.363169
$640$	0	0
$641$	6.94427	0.274282	0.137141	−	0.990552i	$-0.456209\pi$
$641$	6.94427	0.274282	0.137141	+	0.990552i	$0.456209\pi$
$642$	− 10.0344i	− 0.396028i
$643$	17.1803i	0.677526i	0.940872	+	0.338763i	$0.110009\pi$
$643$	17.1803i	0.677526i	−0.940872	+	0.338763i	$0.889991\pi$
$644$	−11.7082	−0.461368
$645$	0	0
$646$	0.763932	0.0300565
$647$	17.3050i	0.680328i	0.940366	+	0.340164i	$0.110483\pi$
$647$	17.3050i	0.680328i	−0.940366	+	0.340164i	$0.889517\pi$
$648$	− 2.23607i	− 0.0878410i
$649$	−58.8328	−2.30939
$650$	0	0
$651$	−23.4164	−0.917761
$652$	18.0902i	0.708466i
$653$	− 25.8885i	− 1.01310i	−0.862211	−	0.506549i	$-0.830921\pi$
$653$	− 25.8885i	− 1.01310i	0.862211	−	0.506549i	$-0.169079\pi$
$654$	4.76393	0.186284
$655$	0	0
$656$	−9.27051	−0.361953
$657$	3.47214i	0.135461i
$658$	− 0.652476i	− 0.0254362i
$659$	−24.9443	−0.971691	−0.485845	−	0.874045i	$-0.661488\pi$
$659$	−24.9443	−0.971691	−0.485845	+	0.874045i	$0.661488\pi$
$660$	0	0
$661$	−23.1246	−0.899443	−0.449722	−	0.893169i	$-0.648477\pi$
$661$	−23.1246	−0.899443	−0.449722	+	0.893169i	$0.648477\pi$
$662$	16.2918i	0.633199i
$663$	8.00000i	0.310694i
$664$	15.5279	0.602598
$665$	0	0
$666$	−1.70820	−0.0661916
$667$	24.1803i	0.936266i
$668$	− 39.7984i	− 1.53985i
$669$	9.23607	0.357087
$670$	0	0
$671$	−31.7771	−1.22674
$672$	− 12.5623i	− 0.484601i
$673$	34.2492i	1.32021i	0.751173	+	0.660105i	$0.229488\pi$
$673$	34.2492i	1.32021i	−0.751173	+	0.660105i	$0.770512\pi$
$674$	−5.23607	−0.201686
$675$	0	0
$676$	−46.7426	−1.79779
$677$	5.83282i	0.224173i	0.993698	+	0.112087i	$0.0357534\pi$
$677$	5.83282i	0.224173i	−0.993698	+	0.112087i	$0.964247\pi$
$678$	7.70820i	0.296032i
$679$	−3.81966	−0.146585
$680$	0	0
$681$	−28.1246	−1.07774
$682$	− 25.8885i	− 0.991324i
$683$	− 8.34752i	− 0.319409i	−0.987165	−	0.159705i	$-0.948946\pi$
$683$	− 8.34752i	− 0.319409i	0.987165	−	0.159705i	$-0.0510542\pi$
$684$	−1.61803	−0.0618671
$685$	0	0
$686$	−12.4377	−0.474873
$687$	4.47214i	0.170623i
$688$	− 7.41641i	− 0.282748i
$689$	32.3607	1.23284
$690$	0	0
$691$	−7.05573	−0.268413	−0.134206	−	0.990953i	$-0.542848\pi$
$691$	−7.05573	−0.268413	−0.134206	+	0.990953i	$0.542848\pi$
$692$	21.7984i	0.828650i
$693$	− 8.94427i	− 0.339765i
$694$	−1.05573	−0.0400749
$695$	0	0
$696$	−16.7082	−0.633323
$697$	− 6.18034i	− 0.234097i
$698$	− 11.0213i	− 0.417162i
$699$	2.94427	0.111363
$700$	0	0
$701$	17.3475	0.655207	0.327603	−	0.944815i	$-0.393759\pi$
$701$	17.3475	0.655207	0.327603	+	0.944815i	$0.393759\pi$
$702$	4.00000i	0.150970i
$703$	2.76393i	0.104244i
$704$	−0.944272	−0.0355886
$705$	0	0
$706$	16.3607	0.615742
$707$	31.7082i	1.19251i
$708$	23.7984i	0.894398i
$709$	40.4164	1.51787	0.758935	−	0.651166i	$-0.225720\pi$
$709$	40.4164	1.51787	0.758935	+	0.651166i	$0.225720\pi$
$710$	0	0
$711$	−2.29180	−0.0859491
$712$	− 22.2361i	− 0.833332i
$713$	33.8885i	1.26914i
$714$	1.70820	0.0639279
$715$	0	0
$716$	−35.7984	−1.33785
$717$	− 0.291796i	− 0.0108973i
$718$	− 15.5967i	− 0.582065i
$719$	37.5279	1.39955	0.699777	−	0.714362i	$-0.253283\pi$
$719$	37.5279	1.39955	0.699777	+	0.714362i	$0.253283\pi$
$720$	0	0
$721$	−37.8885	−1.41104
$722$	− 0.618034i	− 0.0230008i
$723$	− 16.1803i	− 0.601753i
$724$	16.9443	0.629729
$725$	0	0
$726$	3.09017	0.114687
$727$	− 15.7639i	− 0.584652i	−0.956319	−	0.292326i	$-0.905571\pi$
$727$	− 15.7639i	− 0.584652i	0.956319	−	0.292326i	$-0.0944292\pi$
$728$	32.3607i	1.19937i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.94427	0.182871
$732$	12.8541i	0.475101i
$733$	12.4164i	0.458610i	0.973355	+	0.229305i	$0.0736454\pi$
$733$	12.4164i	0.458610i	−0.973355	+	0.229305i	$0.926355\pi$
$734$	20.5836	0.759754
$735$	0	0
$736$	−18.1803	−0.670136
$737$	43.7771i	1.61255i
$738$	− 3.09017i	− 0.113751i
$739$	4.59675	0.169094	0.0845470	−	0.996419i	$-0.473056\pi$
$739$	4.59675	0.169094	0.0845470	+	0.996419i	$0.473056\pi$
$740$	0	0
$741$	6.47214	0.237760
$742$	− 6.90983i	− 0.253668i
$743$	− 42.1246i	− 1.54540i	−0.634770	−	0.772701i	$-0.718905\pi$
$743$	− 42.1246i	− 1.54540i	0.634770	−	0.772701i	$-0.281095\pi$
$744$	−23.4164	−0.858487
$745$	0	0
$746$	−7.52786	−0.275615
$747$	− 6.94427i	− 0.254078i
$748$	− 8.00000i	− 0.292509i
$749$	−36.3050	−1.32655
$750$	0	0
$751$	−20.4721	−0.747039	−0.373519	−	0.927622i	$-0.621849\pi$
$751$	−20.4721	−0.747039	−0.373519	+	0.927622i	$0.621849\pi$
$752$	0.875388i	0.0319221i
$753$	3.70820i	0.135134i
$754$	29.8885	1.08848
$755$	0	0
$756$	−3.61803	−0.131587
$757$	23.0000i	0.835949i	0.908459	+	0.417975i	$0.137260\pi$
$757$	23.0000i	0.835949i	−0.908459	+	0.417975i	$0.862740\pi$
$758$	4.18034i	0.151837i
$759$	−12.9443	−0.469847
$760$	0	0
$761$	−25.1246	−0.910766	−0.455383	−	0.890296i	$-0.650498\pi$
$761$	−25.1246	−0.910766	−0.455383	+	0.890296i	$0.650498\pi$
$762$	6.65248i	0.240994i
$763$	− 17.2361i	− 0.623988i
$764$	12.0000	0.434145
$765$	0	0
$766$	−9.96556	−0.360070
$767$	− 95.1935i	− 3.43724i
$768$	− 6.56231i	− 0.236797i
$769$	5.58359	0.201349	0.100675	−	0.994919i	$-0.467900\pi$
$769$	5.58359	0.201349	0.100675	+	0.994919i	$0.467900\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	5.23607i	0.188450i
$773$	3.94427i	0.141866i	0.997481	+	0.0709328i	$0.0225976\pi$
$773$	3.94427i	0.141866i	−0.997481	+	0.0709328i	$0.977402\pi$
$774$	2.47214	0.0888591
$775$	0	0
$776$	−3.81966	−0.137118
$777$	6.18034i	0.221718i
$778$	4.94427i	0.177261i
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−36.7214	−1.31399
$782$	− 2.47214i	− 0.0884034i
$783$	7.47214i	0.267032i
$784$	3.70820	0.132436
$785$	0	0
$786$	−4.29180	−0.153083
$787$	− 19.3475i	− 0.689665i	−0.938664	−	0.344832i	$-0.887936\pi$
$787$	− 19.3475i	− 0.689665i	0.938664	−	0.344832i	$-0.112064\pi$
$788$	− 36.6525i	− 1.30569i
$789$	−10.2918	−0.366398
$790$	0	0
$791$	27.8885	0.991602
$792$	− 8.94427i	− 0.317821i
$793$	− 51.4164i	− 1.82585i
$794$	13.8197	0.490441
$795$	0	0
$796$	−2.85410	−0.101161
$797$	− 31.3607i	− 1.11085i	−0.831566	−	0.555426i	$-0.812555\pi$
$797$	− 31.3607i	− 1.11085i	0.831566	−	0.555426i	$-0.187445\pi$
$798$	− 1.38197i	− 0.0489211i
$799$	−0.583592	−0.0206460
$800$	0	0
$801$	−9.94427	−0.351364
$802$	12.8754i	0.454646i
$803$	13.8885i	0.490116i
$804$	17.7082	0.624520
$805$	0	0
$806$	41.8885	1.47546
$807$	− 20.4721i	− 0.720653i
$808$	31.7082i	1.11549i
$809$	16.8328	0.591810	0.295905	−	0.955217i	$-0.404379\pi$
$809$	16.8328	0.591810	0.295905	+	0.955217i	$0.404379\pi$
$810$	0	0
$811$	46.7214	1.64061	0.820304	−	0.571927i	$-0.193804\pi$
$811$	46.7214	1.64061	0.820304	+	0.571927i	$0.193804\pi$
$812$	27.0344i	0.948723i
$813$	3.18034i	0.111539i
$814$	−6.83282	−0.239490
$815$	0	0
$816$	−2.29180	−0.0802289
$817$	− 4.00000i	− 0.139942i
$818$	5.41641i	0.189380i
$819$	14.4721	0.505697
$820$	0	0
$821$	5.12461	0.178850	0.0894251	−	0.995994i	$-0.471497\pi$
$821$	5.12461	0.178850	0.0894251	+	0.995994i	$0.471497\pi$
$822$	2.65248i	0.0925157i
$823$	33.1803i	1.15659i	0.815826	+	0.578297i	$0.196282\pi$
$823$	33.1803i	1.15659i	−0.815826	+	0.578297i	$0.803718\pi$
$824$	−37.8885	−1.31991
$825$	0	0
$826$	−20.3262	−0.707240
$827$	− 36.9443i	− 1.28468i	−0.766421	−	0.642339i	$-0.777964\pi$
$827$	− 36.9443i	− 1.28468i	0.766421	−	0.642339i	$-0.222036\pi$
$828$	5.23607i	0.181966i
$829$	−55.6656	−1.93335	−0.966674	−	0.256012i	$-0.917591\pi$
$829$	−55.6656	−1.93335	−0.966674	+	0.256012i	$0.917591\pi$
$830$	0	0
$831$	−5.58359	−0.193693
$832$	− 1.52786i	− 0.0529692i
$833$	2.47214i	0.0856544i
$834$	9.74265	0.337360
$835$	0	0
$836$	−6.47214	−0.223844
$837$	10.4721i	0.361970i
$838$	24.2918i	0.839146i
$839$	38.2361	1.32006	0.660028	−	0.751241i	$-0.270544\pi$
$839$	38.2361	1.32006	0.660028	+	0.751241i	$0.270544\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	− 7.12461i	− 0.245530i
$843$	7.88854i	0.271696i
$844$	9.41641	0.324126
$845$	0	0
$846$	−0.291796	−0.0100322
$847$	− 11.1803i	− 0.384161i
$848$	9.27051i	0.318351i
$849$	9.88854	0.339374
$850$	0	0
$851$	8.94427	0.306606
$852$	14.8541i	0.508893i
$853$	36.8885i	1.26304i	0.775360	+	0.631520i	$0.217569\pi$
$853$	36.8885i	1.26304i	−0.775360	+	0.631520i	$0.782431\pi$
$854$	−10.9787	−0.375684
$855$	0	0
$856$	−36.3050	−1.24088
$857$	32.7771i	1.11964i	0.828613	+	0.559822i	$0.189130\pi$
$857$	32.7771i	1.11964i	−0.828613	+	0.559822i	$0.810870\pi$
$858$	16.0000i	0.546231i
$859$	21.2918	0.726467	0.363233	−	0.931698i	$-0.381673\pi$
$859$	21.2918	0.726467	0.363233	+	0.931698i	$0.381673\pi$
$860$	0	0
$861$	−11.1803	−0.381025
$862$	3.20163i	0.109048i
$863$	− 34.8197i	− 1.18528i	−0.805469	−	0.592638i	$-0.798087\pi$
$863$	− 34.8197i	− 1.18528i	0.805469	−	0.592638i	$-0.201913\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	12.0689	0.410117
$867$	15.4721i	0.525461i
$868$	37.8885i	1.28602i
$869$	−9.16718	−0.310975
$870$	0	0
$871$	−70.8328	−2.40008
$872$	− 17.2361i	− 0.583687i
$873$	1.70820i	0.0578139i
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	5.61803	0.189816
$877$	− 44.9443i	− 1.51766i	−0.651289	−	0.758830i	$-0.725771\pi$
$877$	− 44.9443i	− 1.51766i	0.651289	−	0.758830i	$-0.274229\pi$
$878$	− 10.3607i	− 0.349656i
$879$	2.00000	0.0674583
$880$	0	0
$881$	−0.944272	−0.0318133	−0.0159067	−	0.999873i	$-0.505063\pi$
$881$	−0.944272	−0.0318133	−0.0159067	+	0.999873i	$0.505063\pi$
$882$	1.23607i	0.0416206i
$883$	− 45.6525i	− 1.53633i	−0.640253	−	0.768164i	$-0.721171\pi$
$883$	− 45.6525i	− 1.53633i	0.640253	−	0.768164i	$-0.278829\pi$
$884$	12.9443	0.435363
$885$	0	0
$886$	−8.29180	−0.278568
$887$	− 18.8328i	− 0.632344i	−0.948702	−	0.316172i	$-0.897602\pi$
$887$	− 18.8328i	− 0.632344i	0.948702	−	0.316172i	$-0.102398\pi$
$888$	6.18034i	0.207399i
$889$	24.0689	0.807244
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 14.9443i	− 0.500371i
$893$	0.472136i	0.0157994i
$894$	12.0000	0.401340
$895$	0	0
$896$	−25.4508	−0.850253
$897$	− 20.9443i	− 0.699309i
$898$	− 15.7426i	− 0.525339i
$899$	78.2492	2.60976
$900$	0	0
$901$	−6.18034	−0.205897
$902$	− 12.3607i	− 0.411566i
$903$	− 8.94427i	− 0.297647i
$904$	27.8885	0.927559
$905$	0	0
$906$	7.23607	0.240402
$907$	− 29.8197i	− 0.990146i	−0.868852	−	0.495073i	$-0.835141\pi$
$907$	− 29.8197i	− 0.990146i	0.868852	−	0.495073i	$-0.164859\pi$
$908$	45.5066i	1.51019i
$909$	14.1803	0.470332
$910$	0	0
$911$	11.7639	0.389756	0.194878	−	0.980827i	$-0.437569\pi$
$911$	11.7639	0.389756	0.194878	+	0.980827i	$0.437569\pi$
$912$	1.85410i	0.0613955i
$913$	− 27.7771i	− 0.919287i
$914$	5.49342	0.181706
$915$	0	0
$916$	7.23607	0.239086
$917$	15.5279i	0.512775i
$918$	− 0.763932i	− 0.0252135i
$919$	−41.2918	−1.36209	−0.681045	−	0.732241i	$-0.738474\pi$
$919$	−41.2918	−1.36209	−0.681045	+	0.732241i	$0.738474\pi$
$920$	0	0
$921$	3.05573	0.100690
$922$	1.12461i	0.0370371i
$923$	− 59.4164i	− 1.95571i
$924$	−14.4721	−0.476098
$925$	0	0
$926$	−5.30495	−0.174332
$927$	16.9443i	0.556523i
$928$	41.9787i	1.37802i
$929$	−47.5967	−1.56160	−0.780799	−	0.624782i	$-0.785188\pi$
$929$	−47.5967	−1.56160	−0.780799	+	0.624782i	$0.785188\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	− 4.76393i	− 0.156048i
$933$	4.00000i	0.130954i
$934$	−16.1803	−0.529437
$935$	0	0
$936$	14.4721	0.473037
$937$	39.9443i	1.30492i	0.757822	+	0.652461i	$0.226263\pi$
$937$	39.9443i	1.30492i	−0.757822	+	0.652461i	$0.773737\pi$
$938$	15.1246i	0.493836i
$939$	−30.9443	−1.00983
$940$	0	0
$941$	36.4721	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$941$	36.4721	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$942$	4.32624i	0.140956i
$943$	16.1803i	0.526904i
$944$	27.2705	0.887579
$945$	0	0
$946$	9.88854	0.321504
$947$	− 36.9443i	− 1.20053i	−0.799802	−	0.600264i	$-0.795062\pi$
$947$	− 36.9443i	− 1.20053i	0.799802	−	0.600264i	$-0.204938\pi$
$948$	3.70820i	0.120437i
$949$	−22.4721	−0.729476
$950$	0	0
$951$	7.94427	0.257611
$952$	− 6.18034i	− 0.200306i
$953$	− 53.2492i	− 1.72491i	−0.506132	−	0.862456i	$-0.668925\pi$
$953$	− 53.2492i	− 1.72491i	0.506132	−	0.862456i	$-0.331075\pi$
$954$	−3.09017	−0.100048
$955$	0	0
$956$	−0.472136	−0.0152700
$957$	29.8885i	0.966159i
$958$	− 23.9574i	− 0.774029i
$959$	9.59675	0.309895
$960$	0	0
$961$	78.6656	2.53760
$962$	− 11.0557i	− 0.356451i
$963$	16.2361i	0.523200i
$964$	−26.1803	−0.843212
$965$	0	0
$966$	−4.47214	−0.143889
$967$	22.5967i	0.726662i	0.931660	+	0.363331i	$0.118361\pi$
$967$	22.5967i	0.726662i	−0.931660	+	0.363331i	$0.881639\pi$
$968$	− 11.1803i	− 0.359350i
$969$	−1.23607	−0.0397082
$970$	0	0
$971$	21.0689	0.676133	0.338066	−	0.941122i	$-0.390227\pi$
$971$	21.0689	0.676133	0.338066	+	0.941122i	$0.390227\pi$
$972$	1.61803i	0.0518985i
$973$	− 35.2492i	− 1.13004i
$974$	14.8328	0.475274
$975$	0	0
$976$	14.7295	0.471479
$977$	5.05573i	0.161747i	0.996724	+	0.0808735i	$0.0257710\pi$
$977$	5.05573i	0.161747i	−0.996724	+	0.0808735i	$0.974229\pi$
$978$	6.90983i	0.220952i
$979$	−39.7771	−1.27128
$980$	0	0
$981$	−7.70820	−0.246104
$982$	5.12461i	0.163533i
$983$	− 57.3050i	− 1.82774i	−0.406002	−	0.913872i	$-0.633078\pi$
$983$	− 57.3050i	− 1.82774i	0.406002	−	0.913872i	$-0.366922\pi$
$984$	−11.1803	−0.356416
$985$	0	0
$986$	−5.70820	−0.181786
$987$	1.05573i	0.0336042i
$988$	− 10.4721i	− 0.333163i
$989$	−12.9443	−0.411604
$990$	0	0
$991$	−37.0557	−1.17711	−0.588557	−	0.808456i	$-0.700304\pi$
$991$	−37.0557	−1.17711	−0.588557	+	0.808456i	$0.700304\pi$
$992$	58.8328i	1.86794i
$993$	− 26.3607i	− 0.836531i
$994$	−12.6869	−0.402405
$995$	0	0
$996$	−11.2361	−0.356028
$997$	12.4721i	0.394997i	0.980303	+	0.197498i	$0.0632817\pi$
$997$	12.4721i	0.394997i	−0.980303	+	0.197498i	$0.936718\pi$
$998$	− 19.2705i	− 0.609997i
$999$	2.76393	0.0874469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.m.799.2		4
5.2	odd	4		1425.2.a.n.1.2	✓	2
5.3	odd	4		1425.2.a.q.1.1	yes	2
5.4	even	2	inner	1425.2.c.m.799.3		4
15.2	even	4		4275.2.a.v.1.1		2
15.8	even	4		4275.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.n.1.2	✓	2	5.2	odd	4
1425.2.a.q.1.1	yes	2	5.3	odd	4
1425.2.c.m.799.2		4	1.1	even	1	trivial
1425.2.c.m.799.3		4	5.4	even	2	inner
4275.2.a.s.1.2		2	15.8	even	4
4275.2.a.v.1.1		2	15.2	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} -2.23607i q^{7} -2.23607i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.61803i q^{12} -6.47214i q^{13} -1.38197 q^{14} +1.85410 q^{16} +1.23607i q^{17} +0.618034i q^{18} +1.00000 q^{19} +2.23607 q^{21} +2.47214i q^{22} -3.23607i q^{23} +2.23607 q^{24} -4.00000 q^{26} -1.00000i q^{27} -3.61803i q^{28} -7.47214 q^{29} -10.4721 q^{31} -5.61803i q^{32} -4.00000i q^{33} +0.763932 q^{34} -1.61803 q^{36} +2.76393i q^{37} -0.618034i q^{38} +6.47214 q^{39} -5.00000 q^{41} -1.38197i q^{42} -4.00000i q^{43} -6.47214 q^{44} -2.00000 q^{46} +0.472136i q^{47} +1.85410i q^{48} +2.00000 q^{49} -1.23607 q^{51} -10.4721i q^{52} +5.00000i q^{53} -0.618034 q^{54} -5.00000 q^{56} +1.00000i q^{57} +4.61803i q^{58} +14.7082 q^{59} +7.94427 q^{61} +6.47214i q^{62} +2.23607i q^{63} +0.236068 q^{64} -2.47214 q^{66} -10.9443i q^{67} +2.00000i q^{68} +3.23607 q^{69} +9.18034 q^{71} +2.23607i q^{72} -3.47214i q^{73} +1.70820 q^{74} +1.61803 q^{76} +8.94427i q^{77} -4.00000i q^{78} +2.29180 q^{79} +1.00000 q^{81} +3.09017i q^{82} +6.94427i q^{83} +3.61803 q^{84} -2.47214 q^{86} -7.47214i q^{87} +8.94427i q^{88} +9.94427 q^{89} -14.4721 q^{91} -5.23607i q^{92} -10.4721i q^{93} +0.291796 q^{94} +5.61803 q^{96} -1.70820i q^{97} -1.23607i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} - 6 q^{16} + 4 q^{19} - 16 q^{26} - 12 q^{29} - 24 q^{31} + 12 q^{34} - 2 q^{36} + 8 q^{39} - 20 q^{41} - 8 q^{44} - 8 q^{46} + 8 q^{49} + 4 q^{51}+ \cdots + 16 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 - 6 * q^16 + 4 * q^19 - 16 * q^26 - 12 * q^29 - 24 * q^31 + 12 * q^34 - 2 * q^36 + 8 * q^39 - 20 * q^41 - 8 * q^44 - 8 * q^46 + 8 * q^49 + 4 * q^51 + 2 * q^54 - 20 * q^56 + 32 * q^59 - 4 * q^61 - 8 * q^64 + 8 * q^66 + 4 * q^69 - 8 * q^71 - 20 * q^74 + 2 * q^76 + 36 * q^79 + 4 * q^81 + 10 * q^84 + 8 * q^86 + 4 * q^89 - 40 * q^91 + 28 * q^94 + 18 * q^96 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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