LMFDB - Embedded newform 2475.2.c.a.199.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} -1.00000 q^{11} -4.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -2.00000i q^{22} -1.00000i q^{23} +8.00000 q^{26} +4.00000i q^{28} +7.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} +3.00000i q^{37} +8.00000 q^{41} +6.00000i q^{43} +2.00000 q^{44} +2.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +8.00000i q^{52} -6.00000i q^{53} +5.00000 q^{59} +12.0000 q^{61} +14.0000i q^{62} +8.00000 q^{64} -7.00000i q^{67} -4.00000i q^{68} +3.00000 q^{71} -4.00000i q^{73} -6.00000 q^{74} +2.00000i q^{77} +10.0000 q^{79} +16.0000i q^{82} -6.00000i q^{83} -12.0000 q^{86} +15.0000 q^{89} -8.00000 q^{91} +2.00000i q^{92} +16.0000 q^{94} -7.00000i q^{97} +6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 2 q^{11} + 8 q^{14} - 8 q^{16} + 16 q^{26} + 14 q^{31} - 8 q^{34} + 16 q^{41} + 4 q^{44} + 4 q^{46} + 6 q^{49} + 10 q^{59} + 24 q^{61} + 16 q^{64} + 6 q^{71} - 12 q^{74} + 20 q^{79} - 24 q^{86} + 30 q^{89} - 16 q^{91} + 32 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 - 2 * q^11 + 8 * q^14 - 8 * q^16 + 16 * q^26 + 14 * q^31 - 8 * q^34 + 16 * q^41 + 4 * q^44 + 4 * q^46 + 6 * q^49 + 10 * q^59 + 24 * q^61 + 16 * q^64 + 6 * q^71 - 12 * q^74 + 20 * q^79 - 24 * q^86 + 30 * q^89 - 16 * q^91 + 32 * 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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	0	0
$26$	8.00000	1.56893
$27$	0	0
$28$	4.00000i	0.755929i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	− 8.00000i	− 1.41421i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	2.00000	0.294884
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	8.00000i	1.10940i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	14.0000i	1.77800i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	0	0
$82$	16.0000i	1.76690i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	2.00000i	0.208514i
$93$	0	0
$94$	16.0000	1.65027
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	8.00000i	0.755929i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	10.0000i	0.920575i
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	24.0000i	2.17286i
$123$	0	0
$124$	−14.0000	−1.25724
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	7.00000i	0.598050i	0.954245	+	0.299025i	$0.0966615\pi$
$137$	7.00000i	0.598050i	−0.954245	+	0.299025i	$0.903339\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	8.00000	0.662085
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	20.0000i	1.59111i
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−16.0000	−1.24939
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	− 12.0000i	− 0.914991i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	30.0000i	2.24860i
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	− 16.0000i	− 1.18600i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	16.0000i	1.16692i
$189$	0	0
$190$	0	0
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	− 4.00000i	− 0.281439i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−32.0000	−2.22955
$207$	0	0
$208$	16.0000i	1.10940i
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	36.0000	2.46091
$215$	0	0
$216$	0	0
$217$	− 14.0000i	− 0.950382i
$218$	− 20.0000i	− 1.35457i
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	8.00000i	0.518563i
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	−24.0000	−1.53644
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	36.0000i	2.22409i
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	14.0000i	0.855186i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 20.0000i	− 1.19952i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−8.00000	−0.473050
$287$	− 16.0000i	− 0.944450i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	8.00000i	0.468165i
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 20.0000i	− 1.15857i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	4.00000i	0.230174i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	− 4.00000i	− 0.227921i
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−20.0000	−1.12509
$317$	− 13.0000i	− 0.730153i	−0.930978	−	0.365076i	$-0.881043\pi$
$317$	− 13.0000i	− 0.730153i	0.930978	−	0.365076i	$-0.118957\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	− 4.00000i	− 0.222911i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	− 6.00000i	− 0.326357i
$339$	0	0
$340$	0	0
$341$	−7.00000	−0.379071
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	8.00000i	0.426401i
$353$	− 21.0000i	− 1.11772i	−0.829263	−	0.558859i	$-0.811239\pi$
$353$	− 21.0000i	− 1.11772i	0.829263	−	0.558859i	$-0.188761\pi$
$354$	0	0
$355$	0	0
$356$	−30.0000	−1.59000
$357$	0	0
$358$	− 30.0000i	− 1.58555i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	14.0000i	0.735824i
$363$	0	0
$364$	16.0000	0.838628
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	0	0
$382$	− 34.0000i	− 1.73959i
$383$	− 1.00000i	− 0.0510976i	−0.999674	−	0.0255488i	$-0.991867\pi$
$383$	− 1.00000i	− 0.0510976i	0.999674	−	0.0255488i	$-0.00813332\pi$
$384$	0	0
$385$	0	0
$386$	8.00000	0.407189
$387$	0	0
$388$	14.0000i	0.710742i
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	−4.00000	−0.201517
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	− 28.0000i	− 1.39478i
$404$	4.00000	0.199007
$405$	0	0
$406$	0	0
$407$	− 3.00000i	− 0.148704i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	− 32.0000i	− 1.57653i
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	0	0
$416$	−32.0000	−1.56893
$417$	0	0
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	24.0000i	1.16830i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	36.0000i	1.74013i
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	28.0000	1.34404
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	0	0
$442$	16.0000i	0.761042i
$443$	− 11.0000i	− 0.522626i	−0.965254	−	0.261313i	$-0.915845\pi$
$443$	− 11.0000i	− 0.522626i	0.965254	−	0.261313i	$-0.0841554\pi$
$444$	0	0
$445$	0	0
$446$	38.0000	1.79935
$447$	0	0
$448$	− 16.0000i	− 0.755929i
$449$	35.0000	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$449$	35.0000	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	36.0000	1.68956
$455$	0	0
$456$	0	0
$457$	− 12.0000i	− 0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$	− 12.0000i	− 0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$	− 30.0000i	− 1.40181i
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	11.0000i	0.511213i	0.966781	+	0.255607i	$0.0822752\pi$
$463$	11.0000i	0.511213i	−0.966781	+	0.255607i	$0.917725\pi$
$464$	0	0
$465$	0	0
$466$	−48.0000	−2.22356
$467$	27.0000i	1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$	27.0000i	1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	− 60.0000i	− 2.74434i
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	− 16.0000i	− 0.728780i
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	23.0000i	1.04223i	0.853487	+	0.521115i	$0.174484\pi$
$487$	23.0000i	1.04223i	−0.853487	+	0.521115i	$0.825516\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−28.0000	−1.25724
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	46.0000i	2.05308i
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	32.0000i	1.41421i
$513$	0	0
$514$	−4.00000	−0.176432
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−36.0000	−1.57267
$525$	0	0
$526$	−28.0000	−1.22086
$527$	14.0000i	0.609850i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 32.0000i	− 1.38607i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	20.0000i	0.862261i
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	− 56.0000i	− 2.40541i
$543$	0	0
$544$	16.0000	0.685994
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 14.0000i	− 0.598050i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 20.0000i	− 0.850487i
$554$	4.00000	0.169944
$555$	0	0
$556$	20.0000	0.848189
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	36.0000i	1.51857i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	32.0000	1.33565
$575$	0	0
$576$	0	0
$577$	33.0000i	1.37381i	0.726748	+	0.686904i	$0.241031\pi$
$577$	33.0000i	1.37381i	−0.726748	+	0.686904i	$0.758969\pi$
$578$	26.0000i	1.08146i
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	−48.0000	−1.98286
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 12.0000i	− 0.493197i
$593$	44.0000i	1.80686i	0.428732	+	0.903432i	$0.358960\pi$
$593$	44.0000i	1.80686i	−0.428732	+	0.903432i	$0.641040\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	24.0000i	0.978167i
$603$	0	0
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0	0
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	− 30.0000i	− 1.20192i
$624$	0	0
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	14.0000i	0.558661i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	0	0
$634$	26.0000	1.03259
$635$	0	0
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	− 29.0000i	− 1.14365i	−0.820376	−	0.571824i	$-0.806236\pi$
$643$	− 29.0000i	− 1.14365i	0.820376	−	0.571824i	$-0.193764\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	7.00000i	0.275198i	0.990488	+	0.137599i	$0.0439386\pi$
$647$	7.00000i	0.275198i	−0.990488	+	0.137599i	$0.956061\pi$
$648$	0	0
$649$	−5.00000	−0.196267
$650$	0	0
$651$	0	0
$652$	8.00000i	0.313304i
$653$	− 41.0000i	− 1.60445i	−0.597019	−	0.802227i	$-0.703648\pi$
$653$	− 41.0000i	− 1.60445i	0.597019	−	0.802227i	$-0.296352\pi$
$654$	0	0
$655$	0	0
$656$	−32.0000	−1.24939
$657$	0	0
$658$	− 32.0000i	− 1.24749i
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	14.0000i	0.544125i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 24.0000i	− 0.928588i
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	44.0000	1.69482
$675$	0	0
$676$	6.00000	0.230769
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	− 14.0000i	− 0.536088i
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	0	0
$686$	40.0000	1.52721
$687$	0	0
$688$	− 24.0000i	− 0.914991i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	56.0000	2.12573
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	− 60.0000i	− 2.27103i
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	0	0
$704$	−8.00000	−0.301511
$705$	0	0
$706$	42.0000	1.58069
$707$	4.00000i	0.150435i
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.00000i	− 0.262152i
$714$	0	0
$715$	0	0
$716$	30.0000	1.12115
$717$	0	0
$718$	− 40.0000i	− 1.49279i
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	− 38.0000i	− 1.41421i
$723$	0	0
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	3.00000i	0.111264i	0.998451	+	0.0556319i	$0.0177173\pi$
$727$	3.00000i	0.111264i	−0.998451	+	0.0556319i	$0.982283\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	36.0000i	1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$	36.0000i	1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	−8.00000	−0.294884
$737$	7.00000i	0.257848i
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	0	0
$745$	0	0
$746$	−52.0000	−1.90386
$747$	0	0
$748$	4.00000i	0.146254i
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	32.0000i	1.16692i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	10.0000i	0.363216i
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	34.0000	1.23008
$765$	0	0
$766$	2.00000	0.0722629
$767$	− 20.0000i	− 0.722158i
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	0	0
$772$	8.00000i	0.287926i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	−3.00000	−0.107348
$782$	4.00000i	0.143040i
$783$	0	0
$784$	−12.0000	−0.428571
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 4.00000i	− 0.142494i
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	− 48.0000i	− 1.70453i
$794$	4.00000	0.141955
$795$	0	0
$796$	0	0
$797$	− 53.0000i	− 1.87736i	−0.344795	−	0.938678i	$-0.612051\pi$
$797$	− 53.0000i	− 1.87736i	0.344795	−	0.938678i	$-0.387949\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	− 4.00000i	− 0.141245i
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	56.0000	1.97252
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	0	0
$818$	60.0000i	2.09785i
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	− 39.0000i	− 1.35945i	−0.733465	−	0.679727i	$-0.762098\pi$
$823$	− 39.0000i	− 1.35945i	0.733465	−	0.679727i	$-0.237902\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	52.0000i	1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$	52.0000i	1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	0	0
$832$	− 32.0000i	− 1.10940i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	40.0000i	1.38178i
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	44.0000i	1.51634i
$843$	0	0
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	24.0000i	0.824163i
$849$	0	0
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	48.0000	1.64253
$855$	0	0
$856$	0	0
$857$	− 8.00000i	− 0.273275i	−0.990621	−	0.136637i	$-0.956370\pi$
$857$	− 8.00000i	− 0.273275i	0.990621	−	0.136637i	$-0.0436295\pi$
$858$	0	0
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	0	0
$861$	0	0
$862$	36.0000i	1.22616i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	28.0000i	0.950382i
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−28.0000	−0.948744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 12.0000i	− 0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	− 12.0000i	− 0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$	− 80.0000i	− 2.69987i
$879$	0	0
$880$	0	0
$881$	43.0000	1.44871	0.724353	−	0.689429i	$-0.242138\pi$
$881$	43.0000	1.44871	0.724353	+	0.689429i	$0.242138\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	22.0000	0.739104
$887$	22.0000i	0.738688i	0.929293	+	0.369344i	$0.120418\pi$
$887$	22.0000i	0.738688i	−0.929293	+	0.369344i	$0.879582\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	38.0000i	1.27233i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	70.0000i	2.33593i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	− 16.0000i	− 0.532742i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	36.0000i	1.19470i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	6.00000i	0.198571i
$914$	24.0000	0.793849
$915$	0	0
$916$	30.0000	0.991228
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	0	0
$922$	− 24.0000i	− 0.790398i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−22.0000	−0.722965
$927$	0	0
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	− 48.0000i	− 1.57229i
$933$	0	0
$934$	−54.0000	−1.76693
$935$	0	0
$936$	0	0
$937$	8.00000i	0.261349i	0.991425	+	0.130674i	$0.0417142\pi$
$937$	8.00000i	0.261349i	−0.991425	+	0.130674i	$0.958286\pi$
$938$	− 28.0000i	− 0.914232i
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	−20.0000	−0.650945
$945$	0	0
$946$	12.0000	0.390154
$947$	27.0000i	0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$	27.0000i	0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	0	0
$955$	0	0
$956$	60.0000	1.94054
$957$	0	0
$958$	40.0000i	1.29234i
$959$	14.0000	0.452084
$960$	0	0
$961$	18.0000	0.580645
$962$	24.0000i	0.773791i
$963$	0	0
$964$	16.0000	0.515325
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	0	0
$973$	20.0000i	0.641171i
$974$	−46.0000	−1.47394
$975$	0	0
$976$	−48.0000	−1.53644
$977$	27.0000i	0.863807i	0.901920	+	0.431903i	$0.142158\pi$
$977$	27.0000i	0.863807i	−0.901920	+	0.431903i	$0.857842\pi$
$978$	0	0
$979$	−15.0000	−0.479402
$980$	0	0
$981$	0	0
$982$	16.0000i	0.510581i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	− 56.0000i	− 1.77800i
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	− 40.0000i	− 1.26618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.a.199.2		2
3.2	odd	2		275.2.b.a.199.1		2
5.2	odd	4		2475.2.a.a.1.1		1
5.3	odd	4		99.2.a.d.1.1		1
5.4	even	2	inner	2475.2.c.a.199.1		2
12.11	even	2		4400.2.b.h.4049.1		2
15.2	even	4		275.2.a.b.1.1		1
15.8	even	4		11.2.a.a.1.1	✓	1
15.14	odd	2		275.2.b.a.199.2		2
20.3	even	4		1584.2.a.g.1.1		1
35.13	even	4		4851.2.a.t.1.1		1
40.3	even	4		6336.2.a.bu.1.1		1
40.13	odd	4		6336.2.a.br.1.1		1
45.13	odd	12		891.2.e.b.298.1		2
45.23	even	12		891.2.e.k.298.1		2
45.38	even	12		891.2.e.k.595.1		2
45.43	odd	12		891.2.e.b.595.1		2
55.43	even	4		1089.2.a.b.1.1		1
60.23	odd	4		176.2.a.b.1.1		1
60.47	odd	4		4400.2.a.i.1.1		1
60.59	even	2		4400.2.b.h.4049.2		2
105.23	even	12		539.2.e.h.67.1		2
105.38	odd	12		539.2.e.g.177.1		2
105.53	even	12		539.2.e.h.177.1		2
105.68	odd	12		539.2.e.g.67.1		2
105.83	odd	4		539.2.a.a.1.1		1
120.53	even	4		704.2.a.h.1.1		1
120.83	odd	4		704.2.a.c.1.1		1
165.8	odd	20		121.2.c.a.9.1		4
165.32	odd	4		3025.2.a.a.1.1		1
165.38	even	20		121.2.c.e.3.1		4
165.53	even	20		121.2.c.e.81.1		4
165.68	odd	20		121.2.c.a.81.1		4
165.83	odd	20		121.2.c.a.3.1		4
165.98	odd	4		121.2.a.d.1.1		1
165.113	even	20		121.2.c.e.9.1		4
165.128	odd	20		121.2.c.a.27.1		4
165.158	even	20		121.2.c.e.27.1		4
195.38	even	4		1859.2.a.b.1.1		1
240.53	even	4		2816.2.c.j.1409.2		2
240.83	odd	4		2816.2.c.f.1409.2		2
240.173	even	4		2816.2.c.j.1409.1		2
240.203	odd	4		2816.2.c.f.1409.1		2
255.203	even	4		3179.2.a.a.1.1		1
285.113	odd	4		3971.2.a.b.1.1		1
345.68	odd	4		5819.2.a.a.1.1		1
420.83	even	4		8624.2.a.j.1.1		1
435.173	even	4		9251.2.a.d.1.1		1
660.263	even	4		1936.2.a.i.1.1		1
1155.923	even	4		5929.2.a.h.1.1		1
1320.923	even	4		7744.2.a.k.1.1		1
1320.1253	odd	4		7744.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	15.8	even	4
99.2.a.d.1.1		1	5.3	odd	4
121.2.a.d.1.1		1	165.98	odd	4
121.2.c.a.3.1		4	165.83	odd	20
121.2.c.a.9.1		4	165.8	odd	20
121.2.c.a.27.1		4	165.128	odd	20
121.2.c.a.81.1		4	165.68	odd	20
121.2.c.e.3.1		4	165.38	even	20
121.2.c.e.9.1		4	165.113	even	20
121.2.c.e.27.1		4	165.158	even	20
121.2.c.e.81.1		4	165.53	even	20
176.2.a.b.1.1		1	60.23	odd	4
275.2.a.b.1.1		1	15.2	even	4
275.2.b.a.199.1		2	3.2	odd	2
275.2.b.a.199.2		2	15.14	odd	2
539.2.a.a.1.1		1	105.83	odd	4
539.2.e.g.67.1		2	105.68	odd	12
539.2.e.g.177.1		2	105.38	odd	12
539.2.e.h.67.1		2	105.23	even	12
539.2.e.h.177.1		2	105.53	even	12
704.2.a.c.1.1		1	120.83	odd	4
704.2.a.h.1.1		1	120.53	even	4
891.2.e.b.298.1		2	45.13	odd	12
891.2.e.b.595.1		2	45.43	odd	12
891.2.e.k.298.1		2	45.23	even	12
891.2.e.k.595.1		2	45.38	even	12
1089.2.a.b.1.1		1	55.43	even	4
1584.2.a.g.1.1		1	20.3	even	4
1859.2.a.b.1.1		1	195.38	even	4
1936.2.a.i.1.1		1	660.263	even	4
2475.2.a.a.1.1		1	5.2	odd	4
2475.2.c.a.199.1		2	5.4	even	2	inner
2475.2.c.a.199.2		2	1.1	even	1	trivial
2816.2.c.f.1409.1		2	240.203	odd	4
2816.2.c.f.1409.2		2	240.83	odd	4
2816.2.c.j.1409.1		2	240.173	even	4
2816.2.c.j.1409.2		2	240.53	even	4
3025.2.a.a.1.1		1	165.32	odd	4
3179.2.a.a.1.1		1	255.203	even	4
3971.2.a.b.1.1		1	285.113	odd	4
4400.2.a.i.1.1		1	60.47	odd	4
4400.2.b.h.4049.1		2	12.11	even	2
4400.2.b.h.4049.2		2	60.59	even	2
4851.2.a.t.1.1		1	35.13	even	4
5819.2.a.a.1.1		1	345.68	odd	4
5929.2.a.h.1.1		1	1155.923	even	4
6336.2.a.br.1.1		1	40.13	odd	4
6336.2.a.bu.1.1		1	40.3	even	4
7744.2.a.k.1.1		1	1320.923	even	4
7744.2.a.x.1.1		1	1320.1253	odd	4
8624.2.a.j.1.1		1	420.83	even	4
9251.2.a.d.1.1		1	435.173	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} -1.00000 q^{11} -4.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -2.00000i q^{22} -1.00000i q^{23} +8.00000 q^{26} +4.00000i q^{28} +7.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} +3.00000i q^{37} +8.00000 q^{41} +6.00000i q^{43} +2.00000 q^{44} +2.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +8.00000i q^{52} -6.00000i q^{53} +5.00000 q^{59} +12.0000 q^{61} +14.0000i q^{62} +8.00000 q^{64} -7.00000i q^{67} -4.00000i q^{68} +3.00000 q^{71} -4.00000i q^{73} -6.00000 q^{74} +2.00000i q^{77} +10.0000 q^{79} +16.0000i q^{82} -6.00000i q^{83} -12.0000 q^{86} +15.0000 q^{89} -8.00000 q^{91} +2.00000i q^{92} +16.0000 q^{94} -7.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 2 q^{11} + 8 q^{14} - 8 q^{16} + 16 q^{26} + 14 q^{31} - 8 q^{34} + 16 q^{41} + 4 q^{44} + 4 q^{46} + 6 q^{49} + 10 q^{59} + 24 q^{61} + 16 q^{64} + 6 q^{71} - 12 q^{74} + 20 q^{79} - 24 q^{86} + 30 q^{89} - 16 q^{91} + 32 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 - 2 * q^11 + 8 * q^14 - 8 * q^16 + 16 * q^26 + 14 * q^31 - 8 * q^34 + 16 * q^41 + 4 * q^44 + 4 * q^46 + 6 * q^49 + 10 * q^59 + 24 * q^61 + 16 * q^64 + 6 * q^71 - 12 * q^74 + 20 * q^79 - 24 * q^86 + 30 * q^89 - 16 * q^91 + 32 * q^94

Introduction

L-functions

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