LMFDB - Embedded newform 3150.2.g.a.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
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 1050)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.a.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} +6.00000i q^{22} +3.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +10.0000i q^{37} -4.00000i q^{38} -9.00000 q^{41} -1.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{49} +1.00000i q^{52} -9.00000i q^{53} +1.00000 q^{56} -3.00000i q^{58} +9.00000 q^{59} +11.0000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +9.00000i q^{82} -9.00000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -14.0000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 12 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{26} + 6 q^{29} + 10 q^{31} + 6 q^{34} - 18 q^{41} + 12 q^{44} + 6 q^{46} - 2 q^{49} + 2 q^{56} + 18 q^{59} + 22 q^{61} - 2 q^{64} + 24 q^{71} + 20 q^{74} - 8 q^{76} + 20 q^{79} - 2 q^{86} - 12 q^{89} - 2 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 - 12 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 - 2 * q^26 + 6 * q^29 + 10 * q^31 + 6 * q^34 - 18 * q^41 + 12 * q^44 + 6 * q^46 - 2 * q^49 + 2 * q^56 + 18 * q^59 + 22 * q^61 - 2 * q^64 + 24 * q^71 + 20 * q^74 - 8 * q^76 + 20 * q^79 - 2 * q^86 - 12 * q^89 - 2 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	6.00000i	1.27920i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000i	0.188982i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 3.00000i	− 0.393919i
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	− 5.00000i	− 0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	6.00000i	0.683763i
$78$	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
$82$	9.00000i	0.993884i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	− 3.00000i	− 0.312772i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	17.0000i	1.67506i	0.546392	+	0.837530i	$0.316001\pi$
$103$	17.0000i	1.67506i	−0.546392	+	0.837530i	$0.683999\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	− 9.00000i	− 0.828517i
$119$	3.00000	0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	− 11.0000i	− 0.995893i
$123$	0	0
$124$	−5.00000	−0.449013
$125$	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	$0.853673\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	4.00000	0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	− 12.0000i	− 1.00702i
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	6.00000	0.483494
$155$	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
$161$	3.00000	0.236433
$162$	0	0
$163$	5.00000i	0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$	5.00000i	0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−9.00000	−0.698535
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	1.00000i	0.0762493i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	6.00000i	0.449719i
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	− 18.0000i	− 1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	− 12.0000i	− 0.844317i
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	17.0000	1.18445
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	9.00000i	0.618123i
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.339422i
$218$	8.00000i	0.541828i
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	0	0
$236$	−9.00000	−0.585850
$237$	0	0
$238$	− 3.00000i	− 0.194461i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 25.0000i	− 1.60706i
$243$	0	0
$244$	−11.0000	−0.704203
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	5.00000i	0.317500i
$249$	0	0
$250$	0	0
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	10.0000	0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	− 4.00000i	− 0.244339i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 16.0000i	− 0.959616i
$279$	0	0
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	6.00000	0.354787
$287$	9.00000i	0.531253i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	10.0000i	0.585206i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	9.00000i	0.521356i
$299$	3.00000	0.173494
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	− 2.00000i	− 0.115087i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	− 6.00000i	− 0.341882i
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 27.0000i	− 1.51647i	−0.651981	−	0.758236i	$-0.726062\pi$
$317$	− 27.0000i	− 1.51647i	0.651981	−	0.758236i	$-0.273938\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	− 3.00000i	− 0.167183i
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	5.00000	0.276924
$327$	0	0
$328$	− 9.00000i	− 0.496942i
$329$	0	0
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	9.00000i	0.493939i
$333$	0	0
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	− 12.0000i	− 0.652714i
$339$	0	0
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	1.00000	0.0539164
$345$	0	0
$346$	12.0000	0.645124
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	0	0
$352$	6.00000i	0.319801i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 18.0000i	− 0.951330i
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	37.0000i	1.93138i	0.259690	+	0.965692i	$0.416380\pi$
$367$	37.0000i	1.93138i	−0.259690	+	0.965692i	$0.583620\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	−9.00000	−0.467257
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	0	0
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	0	0
$382$	− 15.0000i	− 0.767467i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	−22.0000	−1.11977
$387$	0	0
$388$	14.0000i	0.710742i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	−15.0000	−0.755689
$395$	0	0
$396$	0	0
$397$	− 29.0000i	− 1.45547i	−0.685859	−	0.727734i	$-0.740573\pi$
$397$	− 29.0000i	− 1.45547i	0.685859	−	0.727734i	$-0.259427\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	− 5.00000i	− 0.249068i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	−3.00000	−0.148888
$407$	− 60.0000i	− 2.97409i
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	0	0
$412$	− 17.0000i	− 0.837530i
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	24.0000i	1.17388i
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	25.0000i	1.21698i
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	− 11.0000i	− 0.532327i
$428$	− 18.0000i	− 0.870063i
$429$	0	0
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	38.0000i	1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$	38.0000i	1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	8.00000	0.383131
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−41.0000	−1.95682	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−41.0000	−1.95682	−0.978412	+	0.206666i	$0.933739\pi$
$440$	0	0
$441$	0	0
$442$	− 3.00000i	− 0.142695i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	−19.0000	−0.899676
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	54.0000	2.54276
$452$	− 6.00000i	− 0.282216i
$453$	0	0
$454$	3.00000	0.140797
$455$	0	0
$456$	0	0
$457$	25.0000i	1.16945i	0.811231	+	0.584725i	$0.198798\pi$
$457$	25.0000i	1.16945i	−0.811231	+	0.584725i	$0.801202\pi$
$458$	− 22.0000i	− 1.02799i
$459$	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
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	− 3.00000i	− 0.138823i	−0.997588	−	0.0694117i	$-0.977888\pi$
$467$	− 3.00000i	− 0.138823i	0.997588	−	0.0694117i	$-0.0221122\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	9.00000i	0.414259i
$473$	6.00000i	0.275880i
$474$	0	0
$475$	0	0
$476$	−3.00000	−0.137505
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	11.0000i	0.497947i
$489$	0	0
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	9.00000i	0.405340i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	5.00000	0.224507
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	0	0
$501$	0	0
$502$	− 27.0000i	− 1.20507i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	−18.0000	−0.800198
$507$	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
$511$	−10.0000	−0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	15.0000	0.661622
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 10.0000i	− 0.439375i
$519$	0	0
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	15.0000i	0.653410i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	9.00000i	0.389833i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000i	0.776035i
$539$	6.00000	0.258438
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	− 20.0000i	− 0.859074i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	− 17.0000i	− 0.726868i	−0.931620	−	0.363434i	$-0.881604\pi$
$547$	− 17.0000i	− 0.726868i	0.931620	−	0.363434i	$-0.118396\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	− 10.0000i	− 0.425243i
$554$	10.0000	0.424859
$555$	0	0
$556$	−16.0000	−0.678551
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	0	0
$562$	− 24.0000i	− 1.01238i
$563$	− 21.0000i	− 0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$	− 21.0000i	− 0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	12.0000i	0.503509i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	9.00000	0.375653
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	54.0000i	2.23645i
$584$	10.0000	0.413803
$585$	0	0
$586$	0	0
$587$	33.0000i	1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$	33.0000i	1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	10.0000i	0.410997i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	9.00000	0.368654
$597$	0	0
$598$	− 3.00000i	− 0.122679i
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	1.00000i	0.0407570i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−6.00000	−0.241747
$617$	− 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
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	−30.0000	−1.19618
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	10.0000i	0.397779i
$633$	0	0
$634$	−27.0000	−1.07231
$635$	0	0
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	18.0000i	0.712627i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	12.0000	0.472134
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	0	0
$649$	−54.0000	−2.11969
$650$	0	0
$651$	0	0
$652$	− 5.00000i	− 0.195815i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	19.0000i	0.738456i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	− 18.0000i	− 0.696441i
$669$	0	0
$670$	0	0
$671$	−66.0000	−2.54790
$672$	0	0
$673$	− 37.0000i	− 1.42625i	−0.701039	−	0.713123i	$-0.747280\pi$
$673$	− 37.0000i	− 1.42625i	0.701039	−	0.713123i	$-0.252720\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	30.0000i	1.14876i
$683$	− 18.0000i	− 0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$	− 18.0000i	− 0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	− 1.00000i	− 0.0381246i
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−34.0000	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	0	0
$697$	− 27.0000i	− 1.02270i
$698$	17.0000i	0.643459i
$699$	0	0
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	0	0
$703$	40.0000i	1.50863i
$704$	6.00000	0.226134
$705$	0	0
$706$	30.0000	1.12906
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	15.0000i	0.561754i
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	0	0
$718$	3.00000i	0.111959i
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	17.0000	0.633113
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	7.00000i	0.259616i	0.991539	+	0.129808i	$0.0414360\pi$
$727$	7.00000i	0.259616i	−0.991539	+	0.129808i	$0.958564\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	29.0000i	1.07114i	0.844491	+	0.535570i	$0.179903\pi$
$733$	29.0000i	1.07114i	−0.844491	+	0.535570i	$0.820097\pi$
$734$	37.0000	1.36569
$735$	0	0
$736$	3.00000	0.110581
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	0	0
$742$	9.00000i	0.330400i
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	0	0
$745$	0	0
$746$	8.00000	0.292901
$747$	0	0
$748$	18.0000i	0.658145i
$749$	18.0000	0.657706
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	−3.00000	−0.109254
$755$	0	0
$756$	0	0
$757$	− 14.0000i	− 0.508839i	−0.967094	−	0.254419i	$-0.918116\pi$
$757$	− 14.0000i	− 0.508839i	0.967094	−	0.254419i	$-0.0818843\pi$
$758$	35.0000i	1.27126i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−15.0000	−0.542681
$765$	0	0
$766$	24.0000	0.867155
$767$	− 9.00000i	− 0.324971i
$768$	0	0
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	0	0
$772$	22.0000i	0.791797i
$773$	48.0000i	1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$773$	48.0000i	1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−36.0000	−1.28983
$780$	0	0
$781$	−72.0000	−2.57636
$782$	9.00000i	0.321839i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 26.0000i	− 0.926800i	−0.886149	−	0.463400i	$-0.846629\pi$
$787$	− 26.0000i	− 0.926800i	0.886149	−	0.463400i	$-0.153371\pi$
$788$	15.0000i	0.534353i
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	− 11.0000i	− 0.390621i
$794$	−29.0000	−1.02917
$795$	0	0
$796$	8.00000	0.283552
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	60.0000i	2.11735i
$804$	0	0
$805$	0	0
$806$	−5.00000	−0.176117
$807$	0	0
$808$	12.0000i	0.422159i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	3.00000i	0.105279i
$813$	0	0
$814$	−60.0000	−2.10300
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	38.0000i	1.32864i
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	− 34.0000i	− 1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$	− 34.0000i	− 1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	−9.00000	−0.313150
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	− 3.00000i	− 0.103944i
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	0	0
$838$	21.0000i	0.725433i
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	34.0000i	1.17172i
$843$	0	0
$844$	25.0000	0.860535
$845$	0	0
$846$	0	0
$847$	− 25.0000i	− 0.859010i
$848$	− 9.00000i	− 0.309061i
$849$	0	0
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	− 19.0000i	− 0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$	− 19.0000i	− 0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$	−11.0000	−0.376412
$855$	0	0
$856$	−18.0000	−0.615227
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	− 15.0000i	− 0.510902i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	0	0
$865$	0	0
$866$	38.0000	1.29129
$867$	0	0
$868$	5.00000i	0.169711i
$869$	−60.0000	−2.03536
$870$	0	0
$871$	4.00000	0.135535
$872$	− 8.00000i	− 0.270914i
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	41.0000i	1.38368i
$879$	0	0
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	5.00000i	0.168263i	0.996455	+	0.0841317i	$0.0268116\pi$
$883$	5.00000i	0.168263i	−0.996455	+	0.0841317i	$0.973188\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−24.0000	−0.806296
$887$	54.0000i	1.81314i	0.422053	+	0.906571i	$0.361310\pi$
$887$	54.0000i	1.81314i	−0.422053	+	0.906571i	$0.638690\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	0	0
$892$	19.0000i	0.636167i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	− 24.0000i	− 0.800890i
$899$	15.0000	0.500278
$900$	0	0
$901$	27.0000	0.899500
$902$	− 54.0000i	− 1.79800i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 53.0000i	− 1.75984i	−0.475125	−	0.879918i	$-0.657597\pi$
$907$	− 53.0000i	− 1.75984i	0.475125	−	0.879918i	$-0.342403\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	0	0
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	54.0000i	1.78714i
$914$	25.0000	0.826927
$915$	0	0
$916$	−22.0000	−0.726900
$917$	0	0
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	0	0
$922$	12.0000i	0.395199i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	− 3.00000i	− 0.0984798i
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	12.0000i	0.393073i
$933$	0	0
$934$	−3.00000	−0.0981630
$935$	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	$-0.0104006\pi$
$938$	− 4.00000i	− 0.130605i
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	− 27.0000i	− 0.879241i
$944$	9.00000	0.292925
$945$	0	0
$946$	6.00000	0.195077
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	3.00000i	0.0972306i
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 10.0000i	− 0.322413i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	11.0000	0.352101
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	42.0000i	1.34027i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	9.00000	0.286618
$987$	0	0
$988$	4.00000i	0.127257i
$989$	3.00000	0.0953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 5.00000i	− 0.158750i
$993$	0	0
$994$	−12.0000	−0.380617
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 19.0000i	− 0.601434i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.a.2899.1		2
3.2	odd	2		1050.2.g.e.799.2		2
5.2	odd	4		3150.2.a.bg.1.1		1
5.3	odd	4		3150.2.a.a.1.1		1
5.4	even	2	inner	3150.2.g.a.2899.2		2
15.2	even	4		1050.2.a.j.1.1	✓	1
15.8	even	4		1050.2.a.l.1.1	yes	1
15.14	odd	2		1050.2.g.e.799.1		2
60.23	odd	4		8400.2.a.ci.1.1		1
60.47	odd	4		8400.2.a.a.1.1		1
105.62	odd	4		7350.2.a.r.1.1		1
105.83	odd	4		7350.2.a.cz.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.j.1.1	✓	1	15.2	even	4
1050.2.a.l.1.1	yes	1	15.8	even	4
1050.2.g.e.799.1		2	15.14	odd	2
1050.2.g.e.799.2		2	3.2	odd	2
3150.2.a.a.1.1		1	5.3	odd	4
3150.2.a.bg.1.1		1	5.2	odd	4
3150.2.g.a.2899.1		2	1.1	even	1	trivial
3150.2.g.a.2899.2		2	5.4	even	2	inner
7350.2.a.r.1.1		1	105.62	odd	4
7350.2.a.cz.1.1		1	105.83	odd	4
8400.2.a.a.1.1		1	60.47	odd	4
8400.2.a.ci.1.1		1	60.23	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} +6.00000i q^{22} +3.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +10.0000i q^{37} -4.00000i q^{38} -9.00000 q^{41} -1.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{49} +1.00000i q^{52} -9.00000i q^{53} +1.00000 q^{56} -3.00000i q^{58} +9.00000 q^{59} +11.0000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +9.00000i q^{82} -9.00000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -14.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 12 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{26} + 6 q^{29} + 10 q^{31} + 6 q^{34} - 18 q^{41} + 12 q^{44} + 6 q^{46} - 2 q^{49} + 2 q^{56} + 18 q^{59} + 22 q^{61} - 2 q^{64} + 24 q^{71} + 20 q^{74} - 8 q^{76} + 20 q^{79} - 2 q^{86} - 12 q^{89} - 2 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 - 12 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 - 2 * q^26 + 6 * q^29 + 10 * q^31 + 6 * q^34 - 18 * q^41 + 12 * q^44 + 6 * q^46 - 2 * q^49 + 2 * q^56 + 18 * q^59 + 22 * q^61 - 2 * q^64 + 24 * q^71 + 20 * q^74 - 8 * q^76 + 20 * q^79 - 2 * q^86 - 12 * q^89 - 2 * q^91

Introduction

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Modular forms

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