LMFDB - Embedded newform 2475.2.c.q.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 825)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$-1.75233 + 1.75233i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.q.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36333i q^{2} +0.141336 q^{4} -2.50466i q^{7} +2.91934i q^{8} +O(q^{10})$	$q+1.36333i q^{2} +0.141336 q^{4} -2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{11} +1.14134i q^{13} +3.41468 q^{14} -3.69735 q^{16} +7.64600i q^{17} -1.77801 q^{19} -1.36333i q^{22} -1.41468i q^{23} -1.55602 q^{26} -0.354000i q^{28} -0.726656 q^{29} +2.85866 q^{31} +0.797984i q^{32} -10.4240 q^{34} +8.42401i q^{37} -2.42401i q^{38} -0.636672 q^{41} -12.6974i q^{43} -0.141336 q^{44} +1.92867 q^{46} +6.14134i q^{47} +0.726656 q^{49} +0.161312i q^{52} +12.0187i q^{53} +7.31198 q^{56} -0.990671i q^{58} -3.41468 q^{59} +4.59465 q^{61} +3.89730i q^{62} -8.48262 q^{64} +9.32131i q^{67} +1.08066i q^{68} -5.85866 q^{71} +7.55602i q^{73} -11.4847 q^{74} -0.251297 q^{76} +2.50466i q^{77} -6.91934 q^{79} -0.867993i q^{82} +6.17997i q^{83} +17.3107 q^{86} -2.91934i q^{88} +3.45331 q^{89} +2.85866 q^{91} -0.199945i q^{92} -8.37266 q^{94} +19.4626i q^{97} +0.990671i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4}+O(q^{10}) $ 6 * q - 16 * q^4	$ 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) $ 6 * q - 16 * q^4 - 6 * q^11 + 12 * q^14 + 20 * q^16 + 2 * q^19 + 16 * q^26 + 4 * q^29 + 34 * q^31 - 12 * q^34 - 8 * q^41 + 16 * q^44 - 60 * q^46 - 4 * q^49 + 44 * q^56 - 12 * q^59 - 6 * q^61 - 68 * q^64 - 52 * q^71 + 28 * q^74 - 48 * q^76 - 12 * q^79 - 56 * q^86 + 4 * q^89 + 34 * q^91 - 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.36333i	0.964019i	0.876166	+	0.482009i	$0.160093\pi$
$2$	1.36333i	0.964019i	−0.876166	+	0.482009i	$0.839907\pi$
$3$	0	0
$3$	0	0
$4$	0.141336	0.0706681
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.50466i	− 0.946674i	−0.880881	−	0.473337i	$-0.843049\pi$
$7$	− 2.50466i	− 0.946674i	0.880881	−	0.473337i	$-0.156951\pi$
$8$	2.91934i	1.03214i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.14134i	0.316550i	0.987395	+	0.158275i	$0.0505932\pi$
$13$	1.14134i	0.316550i	−0.987395	+	0.158275i	$0.949407\pi$
$14$	3.41468	0.912612
$15$	0	0
$16$	−3.69735	−0.924338
$17$	7.64600i	1.85443i	0.374533	+	0.927214i	$0.377803\pi$
$17$	7.64600i	1.85443i	−0.374533	+	0.927214i	$0.622197\pi$
$18$	0	0
$19$	−1.77801	−0.407903	−0.203951	−	0.978981i	$-0.565378\pi$
$19$	−1.77801	−0.407903	−0.203951	+	0.978981i	$0.565378\pi$
$20$	0	0
$21$	0	0
$22$	− 1.36333i	− 0.290663i
$23$	− 1.41468i	− 0.294981i	−0.989063	−	0.147491i	$-0.952880\pi$
$23$	− 1.41468i	− 0.294981i	0.989063	−	0.147491i	$-0.0471196\pi$
$24$	0	0
$25$	0	0
$26$	−1.55602	−0.305160
$27$	0	0
$28$	− 0.354000i	− 0.0668996i
$29$	−0.726656	−0.134937	−0.0674684	−	0.997721i	$-0.521492\pi$
$29$	−0.726656	−0.134937	−0.0674684	+	0.997721i	$0.521492\pi$
$30$	0	0
$31$	2.85866	0.513431	0.256716	−	0.966487i	$-0.417360\pi$
$31$	2.85866	0.513431	0.256716	+	0.966487i	$0.417360\pi$
$32$	0.797984i	0.141065i
$33$	0	0
$34$	−10.4240	−1.78770
$35$	0	0
$36$	0	0
$37$	8.42401i	1.38490i	0.721467	+	0.692449i	$0.243468\pi$
$37$	8.42401i	1.38490i	−0.721467	+	0.692449i	$0.756532\pi$
$38$	− 2.42401i	− 0.393226i
$39$	0	0
$40$	0	0
$41$	−0.636672	−0.0994314	−0.0497157	−	0.998763i	$-0.515832\pi$
$41$	−0.636672	−0.0994314	−0.0497157	+	0.998763i	$0.515832\pi$
$42$	0	0
$43$	− 12.6974i	− 1.93633i	−0.250317	−	0.968164i	$-0.580535\pi$
$43$	− 12.6974i	− 1.93633i	0.250317	−	0.968164i	$-0.419465\pi$
$44$	−0.141336	−0.0213072
$45$	0	0
$46$	1.92867	0.284367
$47$	6.14134i	0.895806i	0.894082	+	0.447903i	$0.147829\pi$
$47$	6.14134i	0.895806i	−0.894082	+	0.447903i	$0.852171\pi$
$48$	0	0
$49$	0.726656	0.103808
$50$	0	0
$51$	0	0
$52$	0.161312i	0.0223700i
$53$	12.0187i	1.65089i	0.564483	+	0.825445i	$0.309076\pi$
$53$	12.0187i	1.65089i	−0.564483	+	0.825445i	$0.690924\pi$
$54$	0	0
$55$	0	0
$56$	7.31198	0.977104
$57$	0	0
$58$	− 0.990671i	− 0.130082i
$59$	−3.41468	−0.444553	−0.222277	−	0.974984i	$-0.571349\pi$
$59$	−3.41468	−0.444553	−0.222277	+	0.974984i	$0.571349\pi$
$60$	0	0
$61$	4.59465	0.588285	0.294142	−	0.955762i	$-0.404966\pi$
$61$	4.59465	0.588285	0.294142	+	0.955762i	$0.404966\pi$
$62$	3.89730i	0.494957i
$63$	0	0
$64$	−8.48262	−1.06033
$65$	0	0
$66$	0	0
$67$	9.32131i	1.13878i	0.822068	+	0.569389i	$0.192820\pi$
$67$	9.32131i	1.13878i	−0.822068	+	0.569389i	$0.807180\pi$
$68$	1.08066i	0.131049i
$69$	0	0
$70$	0	0
$71$	−5.85866	−0.695295	−0.347648	−	0.937625i	$-0.613019\pi$
$71$	−5.85866	−0.695295	−0.347648	+	0.937625i	$0.613019\pi$
$72$	0	0
$73$	7.55602i	0.884365i	0.896925	+	0.442182i	$0.145796\pi$
$73$	7.55602i	0.884365i	−0.896925	+	0.442182i	$0.854204\pi$
$74$	−11.4847	−1.33507
$75$	0	0
$76$	−0.251297	−0.0288257
$77$	2.50466i	0.285433i
$78$	0	0
$79$	−6.91934	−0.778487	−0.389244	−	0.921135i	$-0.627264\pi$
$79$	−6.91934	−0.778487	−0.389244	+	0.921135i	$0.627264\pi$
$80$	0	0
$81$	0	0
$82$	− 0.867993i	− 0.0958537i
$83$	6.17997i	0.678340i	0.940725	+	0.339170i	$0.110146\pi$
$83$	6.17997i	0.678340i	−0.940725	+	0.339170i	$0.889854\pi$
$84$	0	0
$85$	0	0
$86$	17.3107	1.86666
$87$	0	0
$88$	− 2.91934i	− 0.311203i
$89$	3.45331	0.366050	0.183025	−	0.983108i	$-0.441411\pi$
$89$	3.45331	0.366050	0.183025	+	0.983108i	$0.441411\pi$
$90$	0	0
$91$	2.85866	0.299669
$92$	− 0.199945i	− 0.0208457i
$93$	0	0
$94$	−8.37266	−0.863574
$95$	0	0
$96$	0	0
$97$	19.4626i	1.97613i	0.154032	+	0.988066i	$0.450774\pi$
$97$	19.4626i	1.97613i	−0.154032	+	0.988066i	$0.549226\pi$
$98$	0.990671i	0.100073i
$99$	0	0
$100$	0	0
$101$	8.37266	0.833111	0.416555	−	0.909110i	$-0.363237\pi$
$101$	8.37266	0.833111	0.416555	+	0.909110i	$0.363237\pi$
$102$	0	0
$103$	2.54669i	0.250933i	0.992098	+	0.125466i	$0.0400427\pi$
$103$	2.54669i	0.250933i	−0.992098	+	0.125466i	$0.959957\pi$
$104$	−3.33195	−0.326725
$105$	0	0
$106$	−16.3854	−1.59149
$107$	− 14.5653i	− 1.40808i	−0.710158	−	0.704042i	$-0.751376\pi$
$107$	− 14.5653i	− 1.40808i	0.710158	−	0.704042i	$-0.248624\pi$
$108$	0	0
$109$	−2.41468	−0.231284	−0.115642	−	0.993291i	$-0.536893\pi$
$109$	−2.41468	−0.231284	−0.115642	+	0.993291i	$0.536893\pi$
$110$	0	0
$111$	0	0
$112$	9.26063i	0.875047i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−0.102703	−0.00953572
$117$	0	0
$118$	− 4.65533i	− 0.428558i
$119$	19.1507	1.75554
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.26401i	0.567117i
$123$	0	0
$124$	0.404032	0.0362832
$125$	0	0
$126$	0	0
$127$	15.2020i	1.34896i	0.738293	+	0.674480i	$0.235632\pi$
$127$	15.2020i	1.34896i	−0.738293	+	0.674480i	$0.764368\pi$
$128$	− 9.96862i	− 0.881110i
$129$	0	0
$130$	0	0
$131$	−18.7453	−1.63779	−0.818893	−	0.573946i	$-0.805412\pi$
$131$	−18.7453	−1.63779	−0.818893	+	0.573946i	$0.805412\pi$
$132$	0	0
$133$	4.45331i	0.386151i
$134$	−12.7080	−1.09780
$135$	0	0
$136$	−22.3213	−1.91404
$137$	8.74531i	0.747163i	0.927597	+	0.373581i	$0.121870\pi$
$137$	8.74531i	0.747163i	−0.927597	+	0.373581i	$0.878130\pi$
$138$	0	0
$139$	22.3013	1.89157	0.945787	−	0.324787i	$-0.105293\pi$
$139$	22.3013	1.89157	0.945787	+	0.324787i	$0.105293\pi$
$140$	0	0
$141$	0	0
$142$	− 7.98728i	− 0.670278i
$143$	− 1.14134i	− 0.0954433i
$144$	0	0
$145$	0	0
$146$	−10.3013	−0.852544
$147$	0	0
$148$	1.19062i	0.0978681i
$149$	−15.9287	−1.30493	−0.652464	−	0.757820i	$-0.726265\pi$
$149$	−15.9287	−1.30493	−0.652464	+	0.757820i	$0.726265\pi$
$150$	0	0
$151$	−9.88665	−0.804564	−0.402282	−	0.915516i	$-0.631783\pi$
$151$	−9.88665	−0.804564	−0.402282	+	0.915516i	$0.631783\pi$
$152$	− 5.19062i	− 0.421015i
$153$	0	0
$154$	−3.41468	−0.275163
$155$	0	0
$156$	0	0
$157$	0.132007i	0.0105353i	0.999986	+	0.00526767i	$0.00167676\pi$
$157$	0.132007i	0.0105353i	−0.999986	+	0.00526767i	$0.998323\pi$
$158$	− 9.43334i	− 0.750476i
$159$	0	0
$160$	0	0
$161$	−3.54330	−0.279251
$162$	0	0
$163$	− 4.31198i	− 0.337740i	−0.985638	−	0.168870i	$-0.945988\pi$
$163$	− 4.31198i	− 0.337740i	0.985638	−	0.168870i	$-0.0540118\pi$
$164$	−0.0899847	−0.00702663
$165$	0	0
$166$	−8.42533	−0.653932
$167$	− 11.2920i	− 0.873801i	−0.899510	−	0.436901i	$-0.856076\pi$
$167$	− 11.2920i	− 0.873801i	0.899510	−	0.436901i	$-0.143924\pi$
$168$	0	0
$169$	11.6974	0.899796
$170$	0	0
$171$	0	0
$172$	− 1.79459i	− 0.136837i
$173$	− 21.6460i	− 1.64571i	−0.568248	−	0.822857i	$-0.692379\pi$
$173$	− 21.6460i	− 1.64571i	0.568248	−	0.822857i	$-0.307621\pi$
$174$	0	0
$175$	0	0
$176$	3.69735	0.278698
$177$	0	0
$178$	4.70800i	0.352879i
$179$	16.1413	1.20646	0.603230	−	0.797567i	$-0.293880\pi$
$179$	16.1413	1.20646	0.603230	+	0.797567i	$0.293880\pi$
$180$	0	0
$181$	20.7546	1.54268	0.771340	−	0.636423i	$-0.219587\pi$
$181$	20.7546	1.54268	0.771340	+	0.636423i	$0.219587\pi$
$182$	3.89730i	0.288887i
$183$	0	0
$184$	4.12994	0.304463
$185$	0	0
$186$	0	0
$187$	− 7.64600i	− 0.559131i
$188$	0.867993i	0.0633049i
$189$	0	0
$190$	0	0
$191$	−3.31198	−0.239646	−0.119823	−	0.992795i	$-0.538233\pi$
$191$	−3.31198	−0.239646	−0.119823	+	0.992795i	$0.538233\pi$
$192$	0	0
$193$	− 2.13201i	− 0.153465i	−0.997052	−	0.0767326i	$-0.975551\pi$
$193$	− 2.13201i	− 0.153465i	0.997052	−	0.0767326i	$-0.0244488\pi$
$194$	−26.5340	−1.90503
$195$	0	0
$196$	0.102703	0.00733591
$197$	17.9287i	1.27737i	0.769470	+	0.638683i	$0.220520\pi$
$197$	17.9287i	1.27737i	−0.769470	+	0.638683i	$0.779480\pi$
$198$	0	0
$199$	−15.1600	−1.07466	−0.537332	−	0.843371i	$-0.680568\pi$
$199$	−15.1600	−1.07466	−0.537332	+	0.843371i	$0.680568\pi$
$200$	0	0
$201$	0	0
$202$	11.4147i	0.803134i
$203$	1.82003i	0.127741i
$204$	0	0
$205$	0	0
$206$	−3.47197	−0.241904
$207$	0	0
$208$	− 4.21992i	− 0.292599i
$209$	1.77801	0.122987
$210$	0	0
$211$	−13.3213	−0.917076	−0.458538	−	0.888675i	$-0.651627\pi$
$211$	−13.3213	−0.917076	−0.458538	+	0.888675i	$0.651627\pi$
$212$	1.69867i	0.116665i
$213$	0	0
$214$	19.8573	1.35742
$215$	0	0
$216$	0	0
$217$	− 7.15999i	− 0.486052i
$218$	− 3.29200i	− 0.222962i
$219$	0	0
$220$	0	0
$221$	−8.72666	−0.587018
$222$	0	0
$223$	− 12.2534i	− 0.820546i	−0.911963	−	0.410273i	$-0.865433\pi$
$223$	− 12.2534i	− 0.820546i	0.911963	−	0.410273i	$-0.134567\pi$
$224$	1.99868	0.133543
$225$	0	0
$226$	8.17997	0.544123
$227$	8.74531i	0.580447i	0.956959	+	0.290223i	$0.0937296\pi$
$227$	8.74531i	0.580447i	−0.956959	+	0.290223i	$0.906270\pi$
$228$	0	0
$229$	11.4626	0.757473	0.378736	−	0.925505i	$-0.376359\pi$
$229$	11.4626	0.757473	0.378736	+	0.925505i	$0.376359\pi$
$230$	0	0
$231$	0	0
$232$	− 2.12136i	− 0.139274i
$233$	− 7.48469i	− 0.490338i	−0.969480	−	0.245169i	$-0.921157\pi$
$233$	− 7.48469i	− 0.490338i	0.969480	−	0.245169i	$-0.0788435\pi$
$234$	0	0
$235$	0	0
$236$	−0.482618	−0.0314157
$237$	0	0
$238$	26.1086i	1.69237i
$239$	12.9066	0.834860	0.417430	−	0.908709i	$-0.362931\pi$
$239$	12.9066	0.834860	0.417430	+	0.908709i	$0.362931\pi$
$240$	0	0
$241$	−16.8773	−1.08716	−0.543582	−	0.839356i	$-0.682932\pi$
$241$	−16.8773	−1.08716	−0.543582	+	0.839356i	$0.682932\pi$
$242$	1.36333i	0.0876381i
$243$	0	0
$244$	0.649390	0.0415729
$245$	0	0
$246$	0	0
$247$	− 2.02930i	− 0.129122i
$248$	8.34542i	0.529935i
$249$	0	0
$250$	0	0
$251$	−2.28267	−0.144081	−0.0720405	−	0.997402i	$-0.522951\pi$
$251$	−2.28267	−0.144081	−0.0720405	+	0.997402i	$0.522951\pi$
$252$	0	0
$253$	1.41468i	0.0889401i
$254$	−20.7253	−1.30042
$255$	0	0
$256$	−3.37473	−0.210920
$257$	− 26.8667i	− 1.67590i	−0.545749	−	0.837949i	$-0.683755\pi$
$257$	− 26.8667i	− 1.67590i	0.545749	−	0.837949i	$-0.316245\pi$
$258$	0	0
$259$	21.0993	1.31105
$260$	0	0
$261$	0	0
$262$	− 25.5560i	− 1.57886i
$263$	− 12.0187i	− 0.741102i	−0.928812	−	0.370551i	$-0.879169\pi$
$263$	− 12.0187i	− 0.741102i	0.928812	−	0.370551i	$-0.120831\pi$
$264$	0	0
$265$	0	0
$266$	−6.07133	−0.372257
$267$	0	0
$268$	1.31744i	0.0804753i
$269$	−14.3854	−0.877092	−0.438546	−	0.898709i	$-0.644506\pi$
$269$	−14.3854	−0.877092	−0.438546	+	0.898709i	$0.644506\pi$
$270$	0	0
$271$	28.6553	1.74069	0.870344	−	0.492445i	$-0.163897\pi$
$271$	28.6553	1.74069	0.870344	+	0.492445i	$0.163897\pi$
$272$	− 28.2700i	− 1.71412i
$273$	0	0
$274$	−11.9227	−0.720279
$275$	0	0
$276$	0	0
$277$	29.1600i	1.75205i	0.482262	+	0.876027i	$0.339815\pi$
$277$	29.1600i	1.75205i	−0.482262	+	0.876027i	$0.660185\pi$
$278$	30.4040i	1.82351i
$279$	0	0
$280$	0	0
$281$	5.36333	0.319949	0.159975	−	0.987121i	$-0.448859\pi$
$281$	5.36333	0.319949	0.159975	+	0.987121i	$0.448859\pi$
$282$	0	0
$283$	− 10.0420i	− 0.596936i	−0.954420	−	0.298468i	$-0.903524\pi$
$283$	− 10.0420i	− 0.596936i	0.954420	−	0.298468i	$-0.0964757\pi$
$284$	−0.828041	−0.0491352
$285$	0	0
$286$	1.55602	0.0920091
$287$	1.59465i	0.0941292i
$288$	0	0
$289$	−41.4613	−2.43890
$290$	0	0
$291$	0	0
$292$	1.06794i	0.0624963i
$293$	− 20.3540i	− 1.18909i	−0.804061	−	0.594547i	$-0.797332\pi$
$293$	− 20.3540i	− 1.18909i	0.804061	−	0.594547i	$-0.202668\pi$
$294$	0	0
$295$	0	0
$296$	−24.5926	−1.42941
$297$	0	0
$298$	− 21.7160i	− 1.25797i
$299$	1.61462	0.0933762
$300$	0	0
$301$	−31.8026	−1.83307
$302$	− 13.4787i	− 0.775615i
$303$	0	0
$304$	6.57392	0.377040
$305$	0	0
$306$	0	0
$307$	− 25.6226i	− 1.46236i	−0.682184	−	0.731181i	$-0.738970\pi$
$307$	− 25.6226i	− 1.46236i	0.682184	−	0.731181i	$-0.261030\pi$
$308$	0.354000i	0.0201710i
$309$	0	0
$310$	0	0
$311$	−11.9414	−0.677134	−0.338567	−	0.940942i	$-0.609942\pi$
$311$	−11.9414	−0.677134	−0.338567	+	0.940942i	$0.609942\pi$
$312$	0	0
$313$	17.1986i	0.972124i	0.873924	+	0.486062i	$0.161567\pi$
$313$	17.1986i	0.972124i	−0.873924	+	0.486062i	$0.838433\pi$
$314$	−0.179969	−0.0101563
$315$	0	0
$316$	−0.977953	−0.0550142
$317$	− 22.7453i	− 1.27750i	−0.769413	−	0.638752i	$-0.779451\pi$
$317$	− 22.7453i	− 1.27750i	0.769413	−	0.638752i	$-0.220549\pi$
$318$	0	0
$319$	0.726656	0.0406850
$320$	0	0
$321$	0	0
$322$	− 4.83068i	− 0.269203i
$323$	− 13.5946i	− 0.756427i
$324$	0	0
$325$	0	0
$326$	5.87864	0.325588
$327$	0	0
$328$	− 1.85866i	− 0.102628i
$329$	15.3820	0.848036
$330$	0	0
$331$	25.2733	1.38915	0.694574	−	0.719421i	$-0.255593\pi$
$331$	25.2733	1.38915	0.694574	+	0.719421i	$0.255593\pi$
$332$	0.873453i	0.0479370i
$333$	0	0
$334$	15.3947	0.842361
$335$	0	0
$336$	0	0
$337$	17.6880i	0.963528i	0.876301	+	0.481764i	$0.160004\pi$
$337$	17.6880i	0.963528i	−0.876301	+	0.481764i	$0.839996\pi$
$338$	15.9473i	0.867420i
$339$	0	0
$340$	0	0
$341$	−2.85866	−0.154805
$342$	0	0
$343$	− 19.3527i	− 1.04495i
$344$	37.0679	1.99857
$345$	0	0
$346$	29.5106	1.58650
$347$	20.1027i	1.07917i	0.841931	+	0.539585i	$0.181419\pi$
$347$	20.1027i	1.07917i	−0.841931	+	0.539585i	$0.818581\pi$
$348$	0	0
$349$	10.9907	0.588317	0.294159	−	0.955757i	$-0.404961\pi$
$349$	10.9907	0.588317	0.294159	+	0.955757i	$0.404961\pi$
$350$	0	0
$351$	0	0
$352$	− 0.797984i	− 0.0425327i
$353$	− 16.7267i	− 0.890270i	−0.895463	−	0.445135i	$-0.853156\pi$
$353$	− 16.7267i	− 0.890270i	0.895463	−	0.445135i	$-0.146844\pi$
$354$	0	0
$355$	0	0
$356$	0.488078	0.0258681
$357$	0	0
$358$	22.0059i	1.16305i
$359$	−13.5933	−0.717429	−0.358714	−	0.933447i	$-0.616785\pi$
$359$	−13.5933	−0.717429	−0.358714	+	0.933447i	$0.616785\pi$
$360$	0	0
$361$	−15.8387	−0.833615
$362$	28.2954i	1.48717i
$363$	0	0
$364$	0.404032	0.0211771
$365$	0	0
$366$	0	0
$367$	− 28.9987i	− 1.51372i	−0.653578	−	0.756859i	$-0.726733\pi$
$367$	− 28.9987i	− 1.51372i	0.653578	−	0.756859i	$-0.273267\pi$
$368$	5.23057i	0.272662i
$369$	0	0
$370$	0	0
$371$	30.1027	1.56285
$372$	0	0
$373$	− 17.8867i	− 0.926136i	−0.886323	−	0.463068i	$-0.846749\pi$
$373$	− 17.8867i	− 0.926136i	0.886323	−	0.463068i	$-0.153251\pi$
$374$	10.4240	0.539013
$375$	0	0
$376$	−17.9287	−0.924601
$377$	− 0.829359i	− 0.0427142i
$378$	0	0
$379$	−23.7839	−1.22170	−0.610850	−	0.791747i	$-0.709172\pi$
$379$	−23.7839	−1.22170	−0.610850	+	0.791747i	$0.709172\pi$
$380$	0	0
$381$	0	0
$382$	− 4.51531i	− 0.231023i
$383$	31.7360i	1.62163i	0.585300	+	0.810817i	$0.300977\pi$
$383$	31.7360i	1.62163i	−0.585300	+	0.810817i	$0.699023\pi$
$384$	0	0
$385$	0	0
$386$	2.90663	0.147943
$387$	0	0
$388$	2.75077i	0.139649i
$389$	25.7546	1.30581	0.652906	−	0.757439i	$-0.273550\pi$
$389$	25.7546	1.30581	0.652906	+	0.757439i	$0.273550\pi$
$390$	0	0
$391$	10.8166	0.547021
$392$	2.12136i	0.107145i
$393$	0	0
$394$	−24.4427	−1.23140
$395$	0	0
$396$	0	0
$397$	3.03863i	0.152505i	0.997089	+	0.0762523i	$0.0242954\pi$
$397$	3.03863i	0.152505i	−0.997089	+	0.0762523i	$0.975705\pi$
$398$	− 20.6680i	− 1.03600i
$399$	0	0
$400$	0	0
$401$	18.1800	0.907864	0.453932	−	0.891036i	$-0.350021\pi$
$401$	18.1800	0.907864	0.453932	+	0.891036i	$0.350021\pi$
$402$	0	0
$403$	3.26270i	0.162526i
$404$	1.18336	0.0588743
$405$	0	0
$406$	−2.48130	−0.123145
$407$	− 8.42401i	− 0.417563i
$408$	0	0
$409$	20.5946	1.01834	0.509170	−	0.860666i	$-0.329952\pi$
$409$	20.5946	1.01834	0.509170	+	0.860666i	$0.329952\pi$
$410$	0	0
$411$	0	0
$412$	0.359939i	0.0177329i
$413$	8.55263i	0.420847i
$414$	0	0
$415$	0	0
$416$	−0.910768	−0.0446541
$417$	0	0
$418$	2.42401i	0.118562i
$419$	32.7253	1.59874	0.799369	−	0.600841i	$-0.205167\pi$
$419$	32.7253	1.59874	0.799369	+	0.600841i	$0.205167\pi$
$420$	0	0
$421$	32.5640	1.58707	0.793537	−	0.608522i	$-0.208237\pi$
$421$	32.5640	1.58707	0.793537	+	0.608522i	$0.208237\pi$
$422$	− 18.1613i	− 0.884079i
$423$	0	0
$424$	−35.0866	−1.70396
$425$	0	0
$426$	0	0
$427$	− 11.5081i	− 0.556914i
$428$	− 2.05861i	− 0.0995066i
$429$	0	0
$430$	0	0
$431$	−1.82003	−0.0876678	−0.0438339	−	0.999039i	$-0.513957\pi$
$431$	−1.82003	−0.0876678	−0.0438339	+	0.999039i	$0.513957\pi$
$432$	0	0
$433$	− 29.8280i	− 1.43344i	−0.697359	−	0.716722i	$-0.745642\pi$
$433$	− 29.8280i	− 1.43344i	0.697359	−	0.716722i	$-0.254358\pi$
$434$	9.76142	0.468563
$435$	0	0
$436$	−0.341281	−0.0163444
$437$	2.51531i	0.120324i
$438$	0	0
$439$	−18.2220	−0.869688	−0.434844	−	0.900506i	$-0.643197\pi$
$439$	−18.2220	−0.869688	−0.434844	+	0.900506i	$0.643197\pi$
$440$	0	0
$441$	0	0
$442$	− 11.8973i	− 0.565897i
$443$	22.2627i	1.05773i	0.848705	+	0.528866i	$0.177383\pi$
$443$	22.2627i	1.05773i	−0.848705	+	0.528866i	$0.822617\pi$
$444$	0	0
$445$	0	0
$446$	16.7054	0.791022
$447$	0	0
$448$	21.2461i	1.00378i
$449$	25.8760	1.22116	0.610582	−	0.791953i	$-0.290936\pi$
$449$	25.8760	1.22116	0.610582	+	0.791953i	$0.290936\pi$
$450$	0	0
$451$	0.636672	0.0299797
$452$	− 0.848017i	− 0.0398874i
$453$	0	0
$454$	−11.9227	−0.559562
$455$	0	0
$456$	0	0
$457$	10.4626i	0.489422i	0.969596	+	0.244711i	$0.0786930\pi$
$457$	10.4626i	0.489422i	−0.969596	+	0.244711i	$0.921307\pi$
$458$	15.6273i	0.730218i
$459$	0	0
$460$	0	0
$461$	−11.8387	−0.551383	−0.275691	−	0.961246i	$-0.588907\pi$
$461$	−11.8387	−0.551383	−0.275691	+	0.961246i	$0.588907\pi$
$462$	0	0
$463$	27.7546i	1.28987i	0.764238	+	0.644934i	$0.223115\pi$
$463$	27.7546i	1.28987i	−0.764238	+	0.644934i	$0.776885\pi$
$464$	2.68670	0.124727
$465$	0	0
$466$	10.2041	0.472695
$467$	20.8294i	0.963868i	0.876208	+	0.481934i	$0.160065\pi$
$467$	20.8294i	0.963868i	−0.876208	+	0.481934i	$0.839935\pi$
$468$	0	0
$469$	23.3467	1.07805
$470$	0	0
$471$	0	0
$472$	− 9.96862i	− 0.458843i
$473$	12.6974i	0.583825i
$474$	0	0
$475$	0	0
$476$	2.70668	0.124061
$477$	0	0
$478$	17.5960i	0.804821i
$479$	2.64939	0.121054	0.0605269	−	0.998167i	$-0.480722\pi$
$479$	2.64939	0.121054	0.0605269	+	0.998167i	$0.480722\pi$
$480$	0	0
$481$	−9.61462	−0.438389
$482$	− 23.0093i	− 1.04805i
$483$	0	0
$484$	0.141336	0.00642437
$485$	0	0
$486$	0	0
$487$	− 40.5360i	− 1.83686i	−0.395580	−	0.918432i	$-0.629456\pi$
$487$	− 40.5360i	− 1.83686i	0.395580	−	0.918432i	$-0.370544\pi$
$488$	13.4134i	0.607194i
$489$	0	0
$490$	0	0
$491$	16.1214	0.727547	0.363773	−	0.931487i	$-0.381488\pi$
$491$	16.1214	0.727547	0.363773	+	0.931487i	$0.381488\pi$
$492$	0	0
$493$	− 5.55602i	− 0.250230i
$494$	2.76661	0.124476
$495$	0	0
$496$	−10.5695	−0.474584
$497$	14.6740i	0.658218i
$498$	0	0
$499$	28.0666	1.25643	0.628217	−	0.778038i	$-0.283785\pi$
$499$	28.0666	1.25643	0.628217	+	0.778038i	$0.283785\pi$
$500$	0	0
$501$	0	0
$502$	− 3.11203i	− 0.138897i
$503$	− 0.906626i	− 0.0404245i	−0.999796	−	0.0202122i	$-0.993566\pi$
$503$	− 0.906626i	− 0.0404245i	0.999796	−	0.0202122i	$-0.00643419\pi$
$504$	0	0
$505$	0	0
$506$	−1.92867	−0.0857400
$507$	0	0
$508$	2.14859i	0.0953284i
$509$	−14.3013	−0.633895	−0.316948	−	0.948443i	$-0.602658\pi$
$509$	−14.3013	−0.633895	−0.316948	+	0.948443i	$0.602658\pi$
$510$	0	0
$511$	18.9253	0.837205
$512$	− 24.5381i	− 1.08444i
$513$	0	0
$514$	36.6281	1.61560
$515$	0	0
$516$	0	0
$517$	− 6.14134i	− 0.270096i
$518$	28.7653i	1.26387i
$519$	0	0
$520$	0	0
$521$	−20.1214	−0.881533	−0.440766	−	0.897622i	$-0.645293\pi$
$521$	−20.1214	−0.881533	−0.440766	+	0.897622i	$0.645293\pi$
$522$	0	0
$523$	− 12.8459i	− 0.561714i	−0.959750	−	0.280857i	$-0.909381\pi$
$523$	− 12.8459i	− 0.561714i	0.959750	−	0.280857i	$-0.0906187\pi$
$524$	−2.64939	−0.115739
$525$	0	0
$526$	16.3854	0.714436
$527$	21.8573i	0.952121i
$528$	0	0
$529$	20.9987	0.912986
$530$	0	0
$531$	0	0
$532$	0.629414i	0.0272886i
$533$	− 0.726656i	− 0.0314750i
$534$	0	0
$535$	0	0
$536$	−27.2121	−1.17538
$537$	0	0
$538$	− 19.6120i	− 0.845533i
$539$	−0.726656	−0.0312993
$540$	0	0
$541$	17.6040	0.756854	0.378427	−	0.925631i	$-0.376465\pi$
$541$	17.6040	0.756854	0.378427	+	0.925631i	$0.376465\pi$
$542$	39.0666i	1.67805i
$543$	0	0
$544$	−6.10138	−0.261595
$545$	0	0
$546$	0	0
$547$	− 19.9873i	− 0.854594i	−0.904111	−	0.427297i	$-0.859466\pi$
$547$	− 19.9873i	− 0.854594i	0.904111	−	0.427297i	$-0.140534\pi$
$548$	1.23603i	0.0528005i
$549$	0	0
$550$	0	0
$551$	1.29200	0.0550411
$552$	0	0
$553$	17.3306i	0.736974i
$554$	−39.7546	−1.68901
$555$	0	0
$556$	3.15198	0.133674
$557$	− 41.5933i	− 1.76237i	−0.472775	−	0.881183i	$-0.656748\pi$
$557$	− 41.5933i	− 1.76237i	0.472775	−	0.881183i	$-0.343252\pi$
$558$	0	0
$559$	14.4919	0.612944
$560$	0	0
$561$	0	0
$562$	7.31198i	0.308437i
$563$	− 2.28267i	− 0.0962032i	−0.998842	−	0.0481016i	$-0.984683\pi$
$563$	− 2.28267i	− 0.0962032i	0.998842	−	0.0481016i	$-0.0153171\pi$
$564$	0	0
$565$	0	0
$566$	13.6906	0.575458
$567$	0	0
$568$	− 17.1035i	− 0.717645i
$569$	−2.37266	−0.0994670	−0.0497335	−	0.998763i	$-0.515837\pi$
$569$	−2.37266	−0.0994670	−0.0497335	+	0.998763i	$0.515837\pi$
$570$	0	0
$571$	−31.9053	−1.33520	−0.667598	−	0.744522i	$-0.732677\pi$
$571$	−31.9053	−1.33520	−0.667598	+	0.744522i	$0.732677\pi$
$572$	− 0.161312i	− 0.00674479i
$573$	0	0
$574$	−2.17403	−0.0907423
$575$	0	0
$576$	0	0
$577$	− 3.28267i	− 0.136659i	−0.997663	−	0.0683297i	$-0.978233\pi$
$577$	− 3.28267i	− 0.136659i	0.997663	−	0.0683297i	$-0.0217670\pi$
$578$	− 56.5254i	− 2.35115i
$579$	0	0
$580$	0	0
$581$	15.4787	0.642167
$582$	0	0
$583$	− 12.0187i	− 0.497762i
$584$	−22.0586	−0.912792
$585$	0	0
$586$	27.7492	1.14631
$587$	26.3400i	1.08717i	0.839355	+	0.543583i	$0.182933\pi$
$587$	26.3400i	1.08717i	−0.839355	+	0.543583i	$0.817067\pi$
$588$	0	0
$589$	−5.08273	−0.209430
$590$	0	0
$591$	0	0
$592$	− 31.1465i	− 1.28011i
$593$	32.9694i	1.35389i	0.736034	+	0.676945i	$0.236697\pi$
$593$	32.9694i	1.35389i	−0.736034	+	0.676945i	$0.763303\pi$
$594$	0	0
$595$	0	0
$596$	−2.25130	−0.0922167
$597$	0	0
$598$	2.20126i	0.0900164i
$599$	24.5454	1.00290	0.501448	−	0.865188i	$-0.332801\pi$
$599$	24.5454	1.00290	0.501448	+	0.865188i	$0.332801\pi$
$600$	0	0
$601$	−9.70668	−0.395944	−0.197972	−	0.980208i	$-0.563435\pi$
$601$	−9.70668	−0.395944	−0.197972	+	0.980208i	$0.563435\pi$
$602$	− 43.3574i	− 1.76712i
$603$	0	0
$604$	−1.39734	−0.0568570
$605$	0	0
$606$	0	0
$607$	− 1.73599i	− 0.0704615i	−0.999379	−	0.0352307i	$-0.988783\pi$
$607$	− 1.73599i	− 0.0704615i	0.999379	−	0.0352307i	$-0.0112166\pi$
$608$	− 1.41882i	− 0.0575408i
$609$	0	0
$610$	0	0
$611$	−7.00933	−0.283567
$612$	0	0
$613$	12.0187i	0.485429i	0.970098	+	0.242715i	$0.0780378\pi$
$613$	12.0187i	0.485429i	−0.970098	+	0.242715i	$0.921962\pi$
$614$	34.9321	1.40974
$615$	0	0
$616$	−7.31198	−0.294608
$617$	42.1587i	1.69724i	0.528999	+	0.848622i	$0.322567\pi$
$617$	42.1587i	1.69724i	−0.528999	+	0.848622i	$0.677433\pi$
$618$	0	0
$619$	−9.03863	−0.363293	−0.181647	−	0.983364i	$-0.558143\pi$
$619$	−9.03863	−0.363293	−0.181647	+	0.983364i	$0.558143\pi$
$620$	0	0
$621$	0	0
$622$	− 16.2800i	− 0.652770i
$623$	− 8.64939i	− 0.346530i
$624$	0	0
$625$	0	0
$626$	−23.4474	−0.937146
$627$	0	0
$628$	0.0186574i	0 0.000744512i
$629$	−64.4100	−2.56819
$630$	0	0
$631$	10.9614	0.436365	0.218183	−	0.975908i	$-0.429987\pi$
$631$	10.9614	0.436365	0.218183	+	0.975908i	$0.429987\pi$
$632$	− 20.1999i	− 0.803511i
$633$	0	0
$634$	31.0093	1.23154
$635$	0	0
$636$	0	0
$637$	0.829359i	0.0328604i
$638$	0.990671i	0.0392211i
$639$	0	0
$640$	0	0
$641$	44.1587	1.74416	0.872081	−	0.489361i	$-0.162770\pi$
$641$	44.1587	1.74416	0.872081	+	0.489361i	$0.162770\pi$
$642$	0	0
$643$	22.5840i	0.890626i	0.895375	+	0.445313i	$0.146908\pi$
$643$	22.5840i	0.890626i	−0.895375	+	0.445313i	$0.853092\pi$
$644$	−0.500796	−0.0197341
$645$	0	0
$646$	18.5340	0.729209
$647$	− 13.3561i	− 0.525081i	−0.964921	−	0.262541i	$-0.915440\pi$
$647$	− 13.3561i	− 0.525081i	0.964921	−	0.262541i	$-0.0845604\pi$
$648$	0	0
$649$	3.41468	0.134038
$650$	0	0
$651$	0	0
$652$	− 0.609438i	− 0.0238674i
$653$	− 37.5933i	− 1.47114i	−0.677448	−	0.735570i	$-0.736914\pi$
$653$	− 37.5933i	− 1.47114i	0.677448	−	0.735570i	$-0.263086\pi$
$654$	0	0
$655$	0	0
$656$	2.35400	0.0919082
$657$	0	0
$658$	20.9707i	0.817523i
$659$	7.00933	0.273045	0.136522	−	0.990637i	$-0.456407\pi$
$659$	7.00933	0.273045	0.136522	+	0.990637i	$0.456407\pi$
$660$	0	0
$661$	26.6226	1.03550	0.517750	−	0.855532i	$-0.326770\pi$
$661$	26.6226	1.03550	0.517750	+	0.855532i	$0.326770\pi$
$662$	34.4559i	1.33917i
$663$	0	0
$664$	−18.0415	−0.700144
$665$	0	0
$666$	0	0
$667$	1.02799i	0.0398038i
$668$	− 1.59597i	− 0.0617498i
$669$	0	0
$670$	0	0
$671$	−4.59465	−0.177374
$672$	0	0
$673$	35.6960i	1.37598i	0.725720	+	0.687990i	$0.241507\pi$
$673$	35.6960i	1.37598i	−0.725720	+	0.687990i	$0.758493\pi$
$674$	−24.1146	−0.928859
$675$	0	0
$676$	1.65326	0.0635869
$677$	33.2920i	1.27952i	0.768577	+	0.639758i	$0.220965\pi$
$677$	33.2920i	1.27952i	−0.768577	+	0.639758i	$0.779035\pi$
$678$	0	0
$679$	48.7474	1.87075
$680$	0	0
$681$	0	0
$682$	− 3.89730i	− 0.149235i
$683$	36.1787i	1.38434i	0.721736	+	0.692169i	$0.243345\pi$
$683$	36.1787i	1.38434i	−0.721736	+	0.692169i	$0.756655\pi$
$684$	0	0
$685$	0	0
$686$	26.3841	1.00735
$687$	0	0
$688$	46.9466i	1.78982i
$689$	−13.7173	−0.522589
$690$	0	0
$691$	35.3293	1.34399	0.671995	−	0.740555i	$-0.265438\pi$
$691$	35.3293	1.34399	0.671995	+	0.740555i	$0.265438\pi$
$692$	− 3.05936i	− 0.116299i
$693$	0	0
$694$	−27.4066	−1.04034
$695$	0	0
$696$	0	0
$697$	− 4.86799i	− 0.184388i
$698$	14.9839i	0.567149i
$699$	0	0
$700$	0	0
$701$	49.4006	1.86584	0.932918	−	0.360088i	$-0.117253\pi$
$701$	49.4006	1.86584	0.932918	+	0.360088i	$0.117253\pi$
$702$	0	0
$703$	− 14.9780i	− 0.564904i
$704$	8.48262	0.319701
$705$	0	0
$706$	22.8039	0.858237
$707$	− 20.9707i	− 0.788684i
$708$	0	0
$709$	11.1027	0.416971	0.208485	−	0.978025i	$-0.433147\pi$
$709$	11.1027	0.416971	0.208485	+	0.978025i	$0.433147\pi$
$710$	0	0
$711$	0	0
$712$	10.0814i	0.377817i
$713$	− 4.04409i	− 0.151452i
$714$	0	0
$715$	0	0
$716$	2.28135	0.0852582
$717$	0	0
$718$	− 18.5322i	− 0.691614i
$719$	−43.9787	−1.64013	−0.820064	−	0.572271i	$-0.806062\pi$
$719$	−43.9787	−1.64013	−0.820064	+	0.572271i	$0.806062\pi$
$720$	0	0
$721$	6.37860	0.237551
$722$	− 21.5933i	− 0.803621i
$723$	0	0
$724$	2.93338	0.109018
$725$	0	0
$726$	0	0
$727$	44.1880i	1.63884i	0.573193	+	0.819421i	$0.305705\pi$
$727$	44.1880i	1.63884i	−0.573193	+	0.819421i	$0.694295\pi$
$728$	8.34542i	0.309302i
$729$	0	0
$730$	0	0
$731$	97.0840	3.59078
$732$	0	0
$733$	− 47.3879i	− 1.75031i	−0.483840	−	0.875156i	$-0.660758\pi$
$733$	− 47.3879i	− 1.75031i	0.483840	−	0.875156i	$-0.339242\pi$
$734$	39.5347	1.45925
$735$	0	0
$736$	1.12889	0.0416115
$737$	− 9.32131i	− 0.343355i
$738$	0	0
$739$	13.0407	0.479710	0.239855	−	0.970809i	$-0.422900\pi$
$739$	13.0407	0.479710	0.239855	+	0.970809i	$0.422900\pi$
$740$	0	0
$741$	0	0
$742$	41.0399i	1.50662i
$743$	14.1800i	0.520213i	0.965580	+	0.260106i	$0.0837576\pi$
$743$	14.1800i	0.520213i	−0.965580	+	0.260106i	$0.916242\pi$
$744$	0	0
$745$	0	0
$746$	24.3854	0.892812
$747$	0	0
$748$	− 1.08066i	− 0.0395127i
$749$	−36.4813	−1.33300
$750$	0	0
$751$	19.1120	0.697408	0.348704	−	0.937233i	$-0.386622\pi$
$751$	19.1120	0.697408	0.348704	+	0.937233i	$0.386622\pi$
$752$	− 22.7067i	− 0.828027i
$753$	0	0
$754$	1.13069	0.0411773
$755$	0	0
$756$	0	0
$757$	− 24.7347i	− 0.898997i	−0.893281	−	0.449498i	$-0.851603\pi$
$757$	− 24.7347i	− 0.898997i	0.893281	−	0.449498i	$-0.148397\pi$
$758$	− 32.4253i	− 1.17774i
$759$	0	0
$760$	0	0
$761$	9.82003	0.355976	0.177988	−	0.984033i	$-0.443041\pi$
$761$	9.82003	0.355976	0.177988	+	0.984033i	$0.443041\pi$
$762$	0	0
$763$	6.04796i	0.218951i
$764$	−0.468102	−0.0169353
$765$	0	0
$766$	−43.2666	−1.56328
$767$	− 3.89730i	− 0.140723i
$768$	0	0
$769$	15.3026	0.551828	0.275914	−	0.961182i	$-0.411020\pi$
$769$	15.3026	0.551828	0.275914	+	0.961182i	$0.411020\pi$
$770$	0	0
$771$	0	0
$772$	− 0.301330i	− 0.0108451i
$773$	− 5.36927i	− 0.193119i	−0.995327	−	0.0965596i	$-0.969216\pi$
$773$	− 5.36927i	− 0.193119i	0.995327	−	0.0965596i	$-0.0307838\pi$
$774$	0	0
$775$	0	0
$776$	−56.8181	−2.03965
$777$	0	0
$778$	35.1120i	1.25883i
$779$	1.13201	0.0405584
$780$	0	0
$781$	5.85866	0.209639
$782$	14.7466i	0.527338i
$783$	0	0
$784$	−2.68670	−0.0959537
$785$	0	0
$786$	0	0
$787$	− 15.4767i	− 0.551684i	−0.961203	−	0.275842i	$-0.911043\pi$
$787$	− 15.4767i	− 0.551684i	0.961203	−	0.275842i	$-0.0889567\pi$
$788$	2.53397i	0.0902689i
$789$	0	0
$790$	0	0
$791$	−15.0280	−0.534334
$792$	0	0
$793$	5.24404i	0.186221i
$794$	−4.14265	−0.147017
$795$	0	0
$796$	−2.14265	−0.0759444
$797$	− 43.6774i	− 1.54713i	−0.633716	−	0.773566i	$-0.718471\pi$
$797$	− 43.6774i	− 1.54713i	0.633716	−	0.773566i	$-0.281529\pi$
$798$	0	0
$799$	−46.9567	−1.66121
$800$	0	0
$801$	0	0
$802$	24.7853i	0.875198i
$803$	− 7.55602i	− 0.266646i
$804$	0	0
$805$	0	0
$806$	−4.44813	−0.156679
$807$	0	0
$808$	24.4427i	0.859890i
$809$	−26.9966	−0.949150	−0.474575	−	0.880215i	$-0.657398\pi$
$809$	−26.9966	−0.949150	−0.474575	+	0.880215i	$0.657398\pi$
$810$	0	0
$811$	−14.2220	−0.499402	−0.249701	−	0.968323i	$-0.580332\pi$
$811$	−14.2220	−0.499402	−0.249701	+	0.968323i	$0.580332\pi$
$812$	0.257236i	0.00902722i
$813$	0	0
$814$	11.4847	0.402538
$815$	0	0
$816$	0	0
$817$	22.5760i	0.789834i
$818$	28.0773i	0.981699i
$819$	0	0
$820$	0	0
$821$	−23.7801	−0.829930	−0.414965	−	0.909837i	$-0.636206\pi$
$821$	−23.7801	−0.829930	−0.414965	+	0.909837i	$0.636206\pi$
$822$	0	0
$823$	− 1.84934i	− 0.0644638i	−0.999480	−	0.0322319i	$-0.989738\pi$
$823$	− 1.84934i	− 0.0644638i	0.999480	−	0.0322319i	$-0.0102615\pi$
$824$	−7.43466	−0.258998
$825$	0	0
$826$	−11.6600	−0.405705
$827$	15.6987i	0.545896i	0.962029	+	0.272948i	$0.0879987\pi$
$827$	15.6987i	0.545896i	−0.962029	+	0.272948i	$0.912001\pi$
$828$	0	0
$829$	4.34128	0.150779	0.0753895	−	0.997154i	$-0.475980\pi$
$829$	4.34128	0.150779	0.0753895	+	0.997154i	$0.475980\pi$
$830$	0	0
$831$	0	0
$832$	− 9.68152i	− 0.335646i
$833$	5.55602i	0.192505i
$834$	0	0
$835$	0	0
$836$	0.251297	0.00869128
$837$	0	0
$838$	44.6154i	1.54121i
$839$	28.3013	0.977070	0.488535	−	0.872544i	$-0.337531\pi$
$839$	28.3013	0.977070	0.488535	+	0.872544i	$0.337531\pi$
$840$	0	0
$841$	−28.4720	−0.981792
$842$	44.3955i	1.52997i
$843$	0	0
$844$	−1.88278	−0.0648080
$845$	0	0
$846$	0	0
$847$	− 2.50466i	− 0.0860613i
$848$	− 44.4372i	− 1.52598i
$849$	0	0
$850$	0	0
$851$	11.9173	0.408519
$852$	0	0
$853$	− 35.7653i	− 1.22458i	−0.790633	−	0.612290i	$-0.790248\pi$
$853$	− 35.7653i	− 1.22458i	0.790633	−	0.612290i	$-0.209752\pi$
$854$	15.6893	0.536875
$855$	0	0
$856$	42.5213	1.45335
$857$	− 24.7513i	− 0.845487i	−0.906249	−	0.422743i	$-0.861067\pi$
$857$	− 24.7513i	− 0.845487i	0.906249	−	0.422743i	$-0.138933\pi$
$858$	0	0
$859$	10.8039	0.368625	0.184313	−	0.982868i	$-0.440994\pi$
$859$	10.8039	0.368625	0.184313	+	0.982868i	$0.440994\pi$
$860$	0	0
$861$	0	0
$862$	− 2.48130i	− 0.0845134i
$863$	16.6027i	0.565161i	0.959244	+	0.282581i	$0.0911904\pi$
$863$	16.6027i	0.565161i	−0.959244	+	0.282581i	$0.908810\pi$
$864$	0	0
$865$	0	0
$866$	40.6654	1.38187
$867$	0	0
$868$	− 1.01197i	− 0.0343484i
$869$	6.91934	0.234723
$870$	0	0
$871$	−10.6387	−0.360480
$872$	− 7.04928i	− 0.238719i
$873$	0	0
$874$	−3.42920	−0.115994
$875$	0	0
$876$	0	0
$877$	11.1600i	0.376846i	0.982088	+	0.188423i	$0.0603376\pi$
$877$	11.1600i	0.376846i	−0.982088	+	0.188423i	$0.939662\pi$
$878$	− 24.8426i	− 0.838396i
$879$	0	0
$880$	0	0
$881$	−54.5254	−1.83701	−0.918504	−	0.395413i	$-0.870602\pi$
$881$	−54.5254	−1.83701	−0.918504	+	0.395413i	$0.870602\pi$
$882$	0	0
$883$	12.7746i	0.429900i	0.976625	+	0.214950i	$0.0689589\pi$
$883$	12.7746i	0.429900i	−0.976625	+	0.214950i	$0.931041\pi$
$884$	−1.23339	−0.0414835
$885$	0	0
$886$	−30.3514	−1.01967
$887$	− 47.1053i	− 1.58164i	−0.612049	−	0.790820i	$-0.709655\pi$
$887$	− 47.1053i	− 1.58164i	0.612049	−	0.790820i	$-0.290345\pi$
$888$	0	0
$889$	38.0759	1.27703
$890$	0	0
$891$	0	0
$892$	− 1.73184i	− 0.0579864i
$893$	− 10.9193i	− 0.365402i
$894$	0	0
$895$	0	0
$896$	−24.9681	−0.834124
$897$	0	0
$898$	35.2775i	1.17722i
$899$	−2.07727	−0.0692807
$900$	0	0
$901$	−91.8947	−3.06146
$902$	0.867993i	0.0289010i
$903$	0	0
$904$	17.5161	0.582576
$905$	0	0
$906$	0	0
$907$	24.7826i	0.822894i	0.911434	+	0.411447i	$0.134976\pi$
$907$	24.7826i	0.822894i	−0.911434	+	0.411447i	$0.865024\pi$
$908$	1.23603i	0.0410191i
$909$	0	0
$910$	0	0
$911$	6.08273	0.201530	0.100765	−	0.994910i	$-0.467871\pi$
$911$	6.08273	0.201530	0.100765	+	0.994910i	$0.467871\pi$
$912$	0	0
$913$	− 6.17997i	− 0.204527i
$914$	−14.2640	−0.471812
$915$	0	0
$916$	1.62009	0.0535291
$917$	46.9507i	1.55045i
$918$	0	0
$919$	−17.9953	−0.593610	−0.296805	−	0.954938i	$-0.595921\pi$
$919$	−17.9953	−0.593610	−0.296805	+	0.954938i	$0.595921\pi$
$920$	0	0
$921$	0	0
$922$	− 16.1400i	− 0.531543i
$923$	− 6.68670i	− 0.220096i
$924$	0	0
$925$	0	0
$926$	−37.8387	−1.24346
$927$	0	0
$928$	− 0.579860i	− 0.0190348i
$929$	−16.7453	−0.549396	−0.274698	−	0.961531i	$-0.588578\pi$
$929$	−16.7453	−0.549396	−0.274698	+	0.961531i	$0.588578\pi$
$930$	0	0
$931$	−1.29200	−0.0423436
$932$	− 1.05786i	− 0.0346513i
$933$	0	0
$934$	−28.3973	−0.929187
$935$	0	0
$936$	0	0
$937$	31.5853i	1.03185i	0.856635	+	0.515924i	$0.172551\pi$
$937$	31.5853i	1.03185i	−0.856635	+	0.515924i	$0.827449\pi$
$938$	31.8293i	1.03926i
$939$	0	0
$940$	0	0
$941$	14.8421	0.483838	0.241919	−	0.970296i	$-0.422223\pi$
$941$	14.8421	0.483838	0.241919	+	0.970296i	$0.422223\pi$
$942$	0	0
$943$	0.900687i	0.0293304i
$944$	12.6253	0.410918
$945$	0	0
$946$	−17.3107	−0.562818
$947$	2.68802i	0.0873490i	0.999046	+	0.0436745i	$0.0139065\pi$
$947$	2.68802i	0.0873490i	−0.999046	+	0.0436745i	$0.986094\pi$
$948$	0	0
$949$	−8.62395	−0.279945
$950$	0	0
$951$	0	0
$952$	55.9074i	1.81197i
$953$	− 14.2700i	− 0.462249i	−0.972924	−	0.231125i	$-0.925760\pi$
$953$	− 14.2700i	− 0.462249i	0.972924	−	0.231125i	$-0.0742405\pi$
$954$	0	0
$955$	0	0
$956$	1.82417	0.0589980
$957$	0	0
$958$	3.61199i	0.116698i
$959$	21.9041	0.707320
$960$	0	0
$961$	−22.8280	−0.736388
$962$	− 13.1079i	− 0.422615i
$963$	0	0
$964$	−2.38538	−0.0768278
$965$	0	0
$966$	0	0
$967$	− 10.0586i	− 0.323463i	−0.986835	−	0.161732i	$-0.948292\pi$
$967$	− 10.0586i	− 0.323463i	0.986835	−	0.161732i	$-0.0517079\pi$
$968$	2.91934i	0.0938313i
$969$	0	0
$970$	0	0
$971$	13.4520	0.431695	0.215848	−	0.976427i	$-0.430749\pi$
$971$	13.4520	0.431695	0.215848	+	0.976427i	$0.430749\pi$
$972$	0	0
$973$	− 55.8573i	− 1.79070i
$974$	55.2639	1.77077
$975$	0	0
$976$	−16.9880	−0.543774
$977$	− 32.8294i	− 1.05030i	−0.851008	−	0.525152i	$-0.824008\pi$
$977$	− 32.8294i	− 1.05030i	0.851008	−	0.525152i	$-0.175992\pi$
$978$	0	0
$979$	−3.45331	−0.110368
$980$	0	0
$981$	0	0
$982$	21.9787i	0.701369i
$983$	− 43.7746i	− 1.39619i	−0.716003	−	0.698097i	$-0.754031\pi$
$983$	− 43.7746i	− 1.39619i	0.716003	−	0.698097i	$-0.245969\pi$
$984$	0	0
$985$	0	0
$986$	7.57467	0.241227
$987$	0	0
$988$	− 0.286814i	− 0.00912477i
$989$	−17.9627	−0.571180
$990$	0	0
$991$	42.1507	1.33896	0.669480	−	0.742830i	$-0.266517\pi$
$991$	42.1507	1.33896	0.669480	+	0.742830i	$0.266517\pi$
$992$	2.28117i	0.0724271i
$993$	0	0
$994$	−20.0055	−0.634535
$995$	0	0
$996$	0	0
$997$	17.4347i	0.552161i	0.961135	+	0.276081i	$0.0890357\pi$
$997$	17.4347i	0.552161i	−0.961135	+	0.276081i	$0.910964\pi$
$998$	38.2640i	1.21123i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.q.199.4		6
3.2	odd	2		825.2.c.f.199.3		6
5.2	odd	4		2475.2.a.z.1.2		3
5.3	odd	4		2475.2.a.bd.1.2		3
5.4	even	2	inner	2475.2.c.q.199.3		6
15.2	even	4		825.2.a.m.1.2	yes	3
15.8	even	4		825.2.a.i.1.2	✓	3
15.14	odd	2		825.2.c.f.199.4		6
165.32	odd	4		9075.2.a.cd.1.2		3
165.98	odd	4		9075.2.a.cj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.2	✓	3	15.8	even	4
825.2.a.m.1.2	yes	3	15.2	even	4
825.2.c.f.199.3		6	3.2	odd	2
825.2.c.f.199.4		6	15.14	odd	2
2475.2.a.z.1.2		3	5.2	odd	4
2475.2.a.bd.1.2		3	5.3	odd	4
2475.2.c.q.199.3		6	5.4	even	2	inner
2475.2.c.q.199.4		6	1.1	even	1	trivial
9075.2.a.cd.1.2		3	165.32	odd	4
9075.2.a.cj.1.2		3	165.98	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.36333i q^{2} +0.141336 q^{4} -2.50466i q^{7} +2.91934i q^{8} +O(q^{10})\)	\(q+1.36333i q^{2} +0.141336 q^{4} -2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{11} +1.14134i q^{13} +3.41468 q^{14} -3.69735 q^{16} +7.64600i q^{17} -1.77801 q^{19} -1.36333i q^{22} -1.41468i q^{23} -1.55602 q^{26} -0.354000i q^{28} -0.726656 q^{29} +2.85866 q^{31} +0.797984i q^{32} -10.4240 q^{34} +8.42401i q^{37} -2.42401i q^{38} -0.636672 q^{41} -12.6974i q^{43} -0.141336 q^{44} +1.92867 q^{46} +6.14134i q^{47} +0.726656 q^{49} +0.161312i q^{52} +12.0187i q^{53} +7.31198 q^{56} -0.990671i q^{58} -3.41468 q^{59} +4.59465 q^{61} +3.89730i q^{62} -8.48262 q^{64} +9.32131i q^{67} +1.08066i q^{68} -5.85866 q^{71} +7.55602i q^{73} -11.4847 q^{74} -0.251297 q^{76} +2.50466i q^{77} -6.91934 q^{79} -0.867993i q^{82} +6.17997i q^{83} +17.3107 q^{86} -2.91934i q^{88} +3.45331 q^{89} +2.85866 q^{91} -0.199945i q^{92} -8.37266 q^{94} +19.4626i q^{97} +0.990671i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4}+O(q^{10}) \) 6 * q - 16 * q^4	\( 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) \) 6 * q - 16 * q^4 - 6 * q^11 + 12 * q^14 + 20 * q^16 + 2 * q^19 + 16 * q^26 + 4 * q^29 + 34 * q^31 - 12 * q^34 - 8 * q^41 + 16 * q^44 - 60 * q^46 - 4 * q^49 + 44 * q^56 - 12 * q^59 - 6 * q^61 - 68 * q^64 - 52 * q^71 + 28 * q^74 - 48 * q^76 - 12 * q^79 - 56 * q^86 + 4 * q^89 + 34 * q^91 - 4 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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