LMFDB - Embedded newform 2475.2.c.g.199.1

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} +1.00000 q^{11} -2.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +6.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +2.00000 q^{34} +6.00000i q^{37} -6.00000i q^{38} +10.0000 q^{41} +6.00000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} -2.00000i q^{52} +6.00000 q^{56} +6.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -4.00000i q^{62} -7.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -2.00000i q^{73} +6.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +10.0000 q^{79} -10.0000i q^{82} -12.0000i q^{83} +6.00000 q^{86} -3.00000i q^{88} +4.00000 q^{91} -4.00000i q^{92} -8.00000 q^{94} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} + 2 q^{11} + 4 q^{14} - 2 q^{16} + 12 q^{19} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 20 q^{41} + 2 q^{44} - 8 q^{46} + 6 q^{49} + 12 q^{56} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 12 q^{74} + 12 q^{76} + 20 q^{79} + 12 q^{86} + 8 q^{91} - 16 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^11 + 4 * q^14 - 2 * q^16 + 12 * q^19 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^34 + 20 * q^41 + 2 * q^44 - 8 * q^46 + 6 * q^49 + 12 * q^56 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 12 * q^74 + 12 * q^76 + 20 * q^79 + 12 * q^86 + 8 * q^91 - 16 * 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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	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
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.00000i	0.377964i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 6.00000i	− 0.973329i
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 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
$52$	− 2.00000i	− 0.277350i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	6.00000i	0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	6.00000	0.688247
$77$	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
$82$	− 10.0000i	− 1.10432i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	6.00000	0.646997
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 4.00000i	− 0.368230i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.00000i	0.543214i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	10.0000i	0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$	10.0000i	0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	12.0000i	1.04053i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	4.00000i	0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$	4.00000i	0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$	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
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	6.00000i	0.493197i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 18.0000i	− 1.45999i
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 10.0000i	− 0.795557i
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	6.00000i	0.457496i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	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
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	0	0
$202$	− 14.0000i	− 0.985037i
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	2.00000i	0.138675i
$209$	6.00000	0.415029
$210$	0	0
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	0	0
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	− 2.00000i	− 0.135457i
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−12.0000	−0.798228
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	18.0000i	1.18176i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	4.00000i	0.259281i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	− 12.0000i	− 0.763542i
$248$	− 12.0000i	− 0.762001i
$249$	0	0
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	10.0000	0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	32.0000i	1.97320i	0.163144	+	0.986602i	$0.447836\pi$
$263$	32.0000i	1.97320i	−0.163144	+	0.986602i	$0.552164\pi$
$264$	0	0
$265$	0	0
$266$	12.0000	0.735767
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	4.00000	0.241649
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	10.0000i	0.599760i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	0	0
$285$	0	0
$286$	−2.00000	−0.118262
$287$	20.0000i	1.18056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	18.0000	1.04623
$297$	0	0
$298$	2.00000i	0.115857i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−12.0000	−0.691669
$302$	− 14.0000i	− 0.805609i
$303$	0	0
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	2.00000i	0.113961i
$309$	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
$313$	− 34.0000i	− 1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	− 34.0000i	− 1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	10.0000	0.562544
$317$	− 4.00000i	− 0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$317$	− 4.00000i	− 0.224662i	0.993671	−	0.112331i	$-0.0358318\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	0	0
$322$	− 8.00000i	− 0.445823i
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	−20.0000	−1.10770
$327$	0	0
$328$	− 30.0000i	− 1.65647i
$329$	16.0000	0.882109
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	20.0000i	1.07990i
$344$	18.0000	0.970495
$345$	0	0
$346$	6.00000	0.322562
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	− 5.00000i	− 0.266501i
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	24.0000i	1.26844i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−24.0000	−1.23771
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	16.0000i	0.818631i
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	12.0000i	0.601506i
$399$	0	0
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	14.0000	0.696526
$405$	0	0
$406$	−12.0000	−0.595550
$407$	6.00000i	0.297409i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	− 6.00000i	− 0.293470i
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	6.00000i	0.292075i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 12.0000i	− 0.580721i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	− 4.00000i	− 0.190261i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	− 14.0000i	− 0.661438i
$449$	−32.0000	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	− 12.0000i	− 0.564433i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	6.00000i	0.280362i
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	− 12.0000i	− 0.552345i
$473$	6.00000i	0.275880i
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	0	0
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	26.0000i	1.18427i
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	18.0000i	0.814822i
$489$	0	0
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	8.00000i	0.357057i
$503$	40.0000i	1.78351i	0.452517	+	0.891756i	$0.350526\pi$
$503$	40.0000i	1.78351i	−0.452517	+	0.891756i	$0.649474\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	10.0000i	0.443678i
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000i	0.486136i
$513$	0	0
$514$	8.00000	0.352865
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	38.0000i	1.66162i	0.556553	+	0.830812i	$0.312124\pi$
$523$	38.0000i	1.66162i	−0.556553	+	0.830812i	$0.687876\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	32.0000	1.39527
$527$	8.00000i	0.348485i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	12.0000i	0.520266i
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	−24.0000	−1.03664
$537$	0	0
$538$	16.0000i	0.689809i
$539$	3.00000	0.129219
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	0	0
$544$	10.0000	0.428746
$545$	0	0
$546$	0	0
$547$	− 38.0000i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$	− 38.0000i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$	4.00000i	0.170872i
$549$	0	0
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	20.0000i	0.850487i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−10.0000	−0.424094
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	28.0000i	1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$	28.0000i	1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	20.0000	0.834784
$575$	0	0
$576$	0	0
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	18.0000	0.743573
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	− 6.00000i	− 0.246598i
$593$	26.0000i	1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$	26.0000i	1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	8.00000i	0.327144i
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	12.0000i	0.489083i
$603$	0	0
$604$	14.0000	0.569652
$605$	0	0
$606$	0	0
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	− 30.0000i	− 1.21666i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	− 46.0000i	− 1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$	− 46.0000i	− 1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	6.00000	0.241747
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	12.0000i	0.481156i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−34.0000	−1.35891
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	− 30.0000i	− 1.19334i
$633$	0	0
$634$	−4.00000	−0.158860
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	12.0000	0.472134
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	16.0000i	0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$	16.0000i	0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$	0	0
$655$	0	0
$656$	−10.0000	−0.390434
$657$	0	0
$658$	− 16.0000i	− 0.623745i
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	−36.0000	−1.39707
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	− 4.00000i	− 0.153168i
$683$	32.0000i	1.22445i	0.790685	+	0.612223i	$0.209725\pi$
$683$	32.0000i	1.22445i	−0.790685	+	0.612223i	$0.790275\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	− 6.00000i	− 0.228748i
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	0	0
$697$	20.0000i	0.757554i
$698$	18.0000i	0.681310i
$699$	0	0
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	36.0000i	1.35777i
$704$	−7.00000	−0.263822
$705$	0	0
$706$	−24.0000	−0.903252
$707$	28.0000i	1.05305i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	− 12.0000i	− 0.444750i
$729$	0	0
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−20.0000	−0.737210
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	34.0000	1.24483
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	4.00000i	0.145287i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	20.0000	0.722629
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	− 36.0000i	− 1.29066i
$779$	60.0000	2.14972
$780$	0	0
$781$	0	0
$782$	− 8.00000i	− 0.286079i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	2.00000i	0.0712470i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	12.0000i	0.426132i
$794$	14.0000	0.496841
$795$	0	0
$796$	−12.0000	−0.425329
$797$	28.0000i	0.991811i	0.868377	+	0.495905i	$0.165164\pi$
$797$	28.0000i	0.991811i	−0.868377	+	0.495905i	$0.834836\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	28.0000i	0.988714i
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	− 42.0000i	− 1.47755i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	34.0000	1.19390	0.596951	−	0.802278i	$-0.296379\pi$
$811$	34.0000	1.19390	0.596951	+	0.802278i	$0.296379\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	36.0000i	1.25948i
$818$	6.00000i	0.209785i
$819$	0	0
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	− 48.0000i	− 1.67317i	−0.547833	−	0.836587i	$-0.684547\pi$
$823$	− 48.0000i	− 1.67317i	0.547833	−	0.836587i	$-0.315453\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	8.00000	0.278356
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	14.0000i	0.485363i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	32.0000i	1.10542i
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 34.0000i	− 1.17172i
$843$	0	0
$844$	−6.00000	−0.206529
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	50.0000i	1.71197i	0.517003	+	0.855984i	$0.327048\pi$
$853$	50.0000i	1.71197i	−0.517003	+	0.855984i	$0.672952\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	36.0000	1.23045
$857$	− 14.0000i	− 0.478231i	−0.970991	−	0.239115i	$-0.923143\pi$
$857$	− 14.0000i	− 0.478231i	0.970991	−	0.239115i	$-0.0768574\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	8.00000i	0.271538i
$869$	10.0000	0.339227
$870$	0	0
$871$	−16.0000	−0.542139
$872$	− 6.00000i	− 0.203186i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	10.0000i	0.337484i
$879$	0	0
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	− 32.0000i	− 1.07689i	−0.842662	−	0.538443i	$-0.819013\pi$
$883$	− 32.0000i	− 1.07689i	0.842662	−	0.538443i	$-0.180987\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	16.0000i	0.537227i	0.963248	+	0.268614i	$0.0865655\pi$
$887$	16.0000i	0.537227i	−0.963248	+	0.268614i	$0.913434\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	− 8.00000i	− 0.267860i
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	6.00000	0.200446
$897$	0	0
$898$	32.0000i	1.06785i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	− 10.0000i	− 0.332964i
$903$	0	0
$904$	−36.0000	−1.19734
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	18.0000	0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	24.0000i	0.792550i
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	30.0000i	0.984798i
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	6.00000	0.195077
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	− 26.0000i	− 0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	− 26.0000i	− 0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	32.0000i	1.03387i
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 12.0000i	− 0.386896i
$963$	0	0
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	− 3.00000i	− 0.0964237i
$969$	0	0
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	4.00000	0.128168
$975$	0	0
$976$	6.00000	0.192055
$977$	− 36.0000i	− 1.15174i	−0.817541	−	0.575871i	$-0.804663\pi$
$977$	− 36.0000i	− 1.15174i	0.817541	−	0.575871i	$-0.195337\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 28.0000i	− 0.893516i
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	− 12.0000i	− 0.381771i
$989$	24.0000	0.763156
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	− 20.0000i	− 0.635001i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50.0000i	− 1.58352i	−0.610835	−	0.791758i	$-0.709166\pi$
$997$	− 50.0000i	− 1.58352i	0.610835	−	0.791758i	$-0.290834\pi$
$998$	− 16.0000i	− 0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.g.199.1		2
3.2	odd	2		2475.2.c.b.199.2		2
5.2	odd	4		99.2.a.c.1.1	yes	1
5.3	odd	4		2475.2.a.c.1.1		1
5.4	even	2	inner	2475.2.c.g.199.2		2
15.2	even	4		99.2.a.a.1.1	✓	1
15.8	even	4		2475.2.a.j.1.1		1
15.14	odd	2		2475.2.c.b.199.1		2
20.7	even	4		1584.2.a.r.1.1		1
35.27	even	4		4851.2.a.o.1.1		1
40.27	even	4		6336.2.a.f.1.1		1
40.37	odd	4		6336.2.a.b.1.1		1
45.2	even	12		891.2.e.j.595.1		2
45.7	odd	12		891.2.e.c.595.1		2
45.22	odd	12		891.2.e.c.298.1		2
45.32	even	12		891.2.e.j.298.1		2
55.32	even	4		1089.2.a.d.1.1		1
60.47	odd	4		1584.2.a.b.1.1		1
105.62	odd	4		4851.2.a.g.1.1		1
120.77	even	4		6336.2.a.cl.1.1		1
120.107	odd	4		6336.2.a.cm.1.1		1
165.32	odd	4		1089.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.a.a.1.1	✓	1	15.2	even	4
99.2.a.c.1.1	yes	1	5.2	odd	4
891.2.e.c.298.1		2	45.22	odd	12
891.2.e.c.595.1		2	45.7	odd	12
891.2.e.j.298.1		2	45.32	even	12
891.2.e.j.595.1		2	45.2	even	12
1089.2.a.d.1.1		1	55.32	even	4
1089.2.a.h.1.1		1	165.32	odd	4
1584.2.a.b.1.1		1	60.47	odd	4
1584.2.a.r.1.1		1	20.7	even	4
2475.2.a.c.1.1		1	5.3	odd	4
2475.2.a.j.1.1		1	15.8	even	4
2475.2.c.b.199.1		2	15.14	odd	2
2475.2.c.b.199.2		2	3.2	odd	2
2475.2.c.g.199.1		2	1.1	even	1	trivial
2475.2.c.g.199.2		2	5.4	even	2	inner
4851.2.a.g.1.1		1	105.62	odd	4
4851.2.a.o.1.1		1	35.27	even	4
6336.2.a.b.1.1		1	40.37	odd	4
6336.2.a.f.1.1		1	40.27	even	4
6336.2.a.cl.1.1		1	120.77	even	4
6336.2.a.cm.1.1		1	120.107	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} +1.00000 q^{11} -2.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +6.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +2.00000 q^{34} +6.00000i q^{37} -6.00000i q^{38} +10.0000 q^{41} +6.00000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} -2.00000i q^{52} +6.00000 q^{56} +6.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -4.00000i q^{62} -7.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -2.00000i q^{73} +6.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +10.0000 q^{79} -10.0000i q^{82} -12.0000i q^{83} +6.00000 q^{86} -3.00000i q^{88} +4.00000 q^{91} -4.00000i q^{92} -8.00000 q^{94} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} + 2 q^{11} + 4 q^{14} - 2 q^{16} + 12 q^{19} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 20 q^{41} + 2 q^{44} - 8 q^{46} + 6 q^{49} + 12 q^{56} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 12 q^{74} + 12 q^{76} + 20 q^{79} + 12 q^{86} + 8 q^{91} - 16 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^11 + 4 * q^14 - 2 * q^16 + 12 * q^19 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^34 + 20 * q^41 + 2 * q^44 - 8 * q^46 + 6 * q^49 + 12 * q^56 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 12 * q^74 + 12 * q^76 + 20 * q^79 + 12 * q^86 + 8 * q^91 - 16 * q^94

Introduction

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