LMFDB - Embedded newform 7500.2.d.g.1249.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.18
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.04684i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.04684i q^{7} -1.00000 q^{9} +6.28891 q^{11} -1.00629i q^{13} +4.69165i q^{17} +5.97554 q^{19} +1.04684 q^{21} +8.05101i q^{23} -1.00000i q^{27} -6.91519 q^{29} -9.52414 q^{31} +6.28891i q^{33} -7.69791i q^{37} +1.00629 q^{39} +1.56932 q^{41} +9.94897i q^{43} +4.84659i q^{47} +5.90413 q^{49} -4.69165 q^{51} +2.82635i q^{53} +5.97554i q^{57} -4.08209 q^{59} +3.16053 q^{61} +1.04684i q^{63} -3.17036i q^{67} -8.05101 q^{69} +6.50896 q^{71} -0.367806i q^{73} -6.58345i q^{77} +3.32009 q^{79} +1.00000 q^{81} +13.8579i q^{83} -6.91519i q^{87} +2.50645 q^{89} -1.05342 q^{91} -9.52414i q^{93} -0.182689i q^{97} -6.28891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.04684i	− 0.395667i	−0.980236	−	0.197833i	$-0.936610\pi$
$7$	− 1.04684i	− 0.395667i	0.980236	−	0.197833i	$-0.0633905\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.28891	1.89618	0.948088	−	0.318008i	$-0.103014\pi$
$11$	6.28891	1.89618	0.948088	+	0.318008i	$0.103014\pi$
$12$	0	0
$13$	− 1.00629i	− 0.279096i	−0.990215	−	0.139548i	$-0.955435\pi$
$13$	− 1.00629i	− 0.279096i	0.990215	−	0.139548i	$-0.0445649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.69165i	1.13789i	0.822375	+	0.568946i	$0.192649\pi$
$17$	4.69165i	1.13789i	−0.822375	+	0.568946i	$0.807351\pi$
$18$	0	0
$19$	5.97554	1.37088	0.685441	−	0.728128i	$-0.259609\pi$
$19$	5.97554	1.37088	0.685441	+	0.728128i	$0.259609\pi$
$20$	0	0
$21$	1.04684	0.228438
$22$	0	0
$23$	8.05101i	1.67875i	0.543551	+	0.839376i	$0.317080\pi$
$23$	8.05101i	1.67875i	−0.543551	+	0.839376i	$0.682920\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−6.91519	−1.28412	−0.642059	−	0.766655i	$-0.721920\pi$
$29$	−6.91519	−1.28412	−0.642059	+	0.766655i	$0.721920\pi$
$30$	0	0
$31$	−9.52414	−1.71059	−0.855293	−	0.518145i	$-0.826623\pi$
$31$	−9.52414	−1.71059	−0.855293	+	0.518145i	$0.826623\pi$
$32$	0	0
$33$	6.28891i	1.09476i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.69791i	− 1.26553i	−0.774344	−	0.632765i	$-0.781920\pi$
$37$	− 7.69791i	− 1.26553i	0.774344	−	0.632765i	$-0.218080\pi$
$38$	0	0
$39$	1.00629	0.161136
$40$	0	0
$41$	1.56932	0.245086	0.122543	−	0.992463i	$-0.460895\pi$
$41$	1.56932	0.245086	0.122543	+	0.992463i	$0.460895\pi$
$42$	0	0
$43$	9.94897i	1.51720i	0.651555	+	0.758602i	$0.274117\pi$
$43$	9.94897i	1.51720i	−0.651555	+	0.758602i	$0.725883\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.84659i	0.706948i	0.935444	+	0.353474i	$0.115000\pi$
$47$	4.84659i	0.706948i	−0.935444	+	0.353474i	$0.885000\pi$
$48$	0	0
$49$	5.90413	0.843448
$50$	0	0
$51$	−4.69165	−0.656962
$52$	0	0
$53$	2.82635i	0.388229i	0.980979	+	0.194114i	$0.0621833\pi$
$53$	2.82635i	0.388229i	−0.980979	+	0.194114i	$0.937817\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.97554i	0.791479i
$58$	0	0
$59$	−4.08209	−0.531443	−0.265721	−	0.964050i	$-0.585610\pi$
$59$	−4.08209	−0.531443	−0.265721	+	0.964050i	$0.585610\pi$
$60$	0	0
$61$	3.16053	0.404665	0.202332	−	0.979317i	$-0.435148\pi$
$61$	3.16053	0.404665	0.202332	+	0.979317i	$0.435148\pi$
$62$	0	0
$63$	1.04684i	0.131889i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.17036i	− 0.387321i	−0.981069	−	0.193660i	$-0.937964\pi$
$67$	− 3.17036i	− 0.387321i	0.981069	−	0.193660i	$-0.0620360\pi$
$68$	0	0
$69$	−8.05101	−0.969228
$70$	0	0
$71$	6.50896	0.772472	0.386236	−	0.922400i	$-0.373775\pi$
$71$	6.50896	0.772472	0.386236	+	0.922400i	$0.373775\pi$
$72$	0	0
$73$	− 0.367806i	− 0.0430485i	−0.999768	−	0.0215242i	$-0.993148\pi$
$73$	− 0.367806i	− 0.0430485i	0.999768	−	0.0215242i	$-0.00685190\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.58345i	− 0.750254i
$78$	0	0
$79$	3.32009	0.373539	0.186770	−	0.982404i	$-0.440198\pi$
$79$	3.32009	0.373539	0.186770	+	0.982404i	$0.440198\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8579i	1.52110i	0.649281	+	0.760549i	$0.275070\pi$
$83$	13.8579i	1.52110i	−0.649281	+	0.760549i	$0.724930\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.91519i	− 0.741386i
$88$	0	0
$89$	2.50645	0.265683	0.132842	−	0.991137i	$-0.457590\pi$
$89$	2.50645	0.265683	0.132842	+	0.991137i	$0.457590\pi$
$90$	0	0
$91$	−1.05342	−0.110429
$92$	0	0
$93$	− 9.52414i	− 0.987607i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.182689i	− 0.0185493i	−0.999957	−	0.00927465i	$-0.997048\pi$
$97$	− 0.182689i	− 0.0185493i	0.999957	−	0.00927465i	$-0.00295226\pi$
$98$	0	0
$99$	−6.28891	−0.632059
$100$	0	0
$101$	−5.96970	−0.594008	−0.297004	−	0.954876i	$-0.595987\pi$
$101$	−5.96970	−0.594008	−0.297004	+	0.954876i	$0.595987\pi$
$102$	0	0
$103$	− 8.95864i	− 0.882721i	−0.897330	−	0.441360i	$-0.854496\pi$
$103$	− 8.95864i	− 0.882721i	0.897330	−	0.441360i	$-0.145504\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.69495i	0.260530i	0.991479	+	0.130265i	$0.0415829\pi$
$107$	2.69495i	0.260530i	−0.991479	+	0.130265i	$0.958417\pi$
$108$	0	0
$109$	−10.2999	−0.986549	−0.493275	−	0.869874i	$-0.664200\pi$
$109$	−10.2999	−0.986549	−0.493275	+	0.869874i	$0.664200\pi$
$110$	0	0
$111$	7.69791	0.730654
$112$	0	0
$113$	− 3.38070i	− 0.318029i	−0.987276	−	0.159015i	$-0.949168\pi$
$113$	− 3.38070i	− 0.318029i	0.987276	−	0.159015i	$-0.0508317\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00629i	0.0930319i
$118$	0	0
$119$	4.91139	0.450226
$120$	0	0
$121$	28.5503	2.59548
$122$	0	0
$123$	1.56932i	0.141501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.97421i	0.530126i	0.964231	+	0.265063i	$0.0853927\pi$
$127$	5.97421i	0.530126i	−0.964231	+	0.265063i	$0.914607\pi$
$128$	0	0
$129$	−9.94897	−0.875958
$130$	0	0
$131$	6.20733	0.542337	0.271168	−	0.962532i	$-0.412590\pi$
$131$	6.20733	0.542337	0.271168	+	0.962532i	$0.412590\pi$
$132$	0	0
$133$	− 6.25541i	− 0.542413i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.9777i	− 1.10876i	−0.832263	−	0.554381i	$-0.812955\pi$
$137$	− 12.9777i	− 1.10876i	0.832263	−	0.554381i	$-0.187045\pi$
$138$	0	0
$139$	−3.99948	−0.339231	−0.169616	−	0.985510i	$-0.554253\pi$
$139$	−3.99948	−0.339231	−0.169616	+	0.985510i	$0.554253\pi$
$140$	0	0
$141$	−4.84659	−0.408156
$142$	0	0
$143$	− 6.32849i	− 0.529215i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.90413i	0.486965i
$148$	0	0
$149$	−5.68762	−0.465948	−0.232974	−	0.972483i	$-0.574846\pi$
$149$	−5.68762	−0.465948	−0.232974	+	0.972483i	$0.574846\pi$
$150$	0	0
$151$	−5.51150	−0.448519	−0.224260	−	0.974529i	$-0.571996\pi$
$151$	−5.51150	−0.448519	−0.224260	+	0.974529i	$0.571996\pi$
$152$	0	0
$153$	− 4.69165i	− 0.379297i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.0217i	− 1.35848i	−0.733915	−	0.679242i	$-0.762309\pi$
$157$	− 17.0217i	− 1.35848i	0.733915	−	0.679242i	$-0.237691\pi$
$158$	0	0
$159$	−2.82635	−0.224144
$160$	0	0
$161$	8.42809	0.664227
$162$	0	0
$163$	12.2958i	0.963078i	0.876425	+	0.481539i	$0.159922\pi$
$163$	12.2958i	0.963078i	−0.876425	+	0.481539i	$0.840078\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.42858i	0.420076i	0.977693	+	0.210038i	$0.0673589\pi$
$167$	5.42858i	0.420076i	−0.977693	+	0.210038i	$0.932641\pi$
$168$	0	0
$169$	11.9874	0.922106
$170$	0	0
$171$	−5.97554	−0.456961
$172$	0	0
$173$	10.7456i	0.816973i	0.912764	+	0.408486i	$0.133943\pi$
$173$	10.7456i	0.816973i	−0.912764	+	0.408486i	$0.866057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.08209i	− 0.306829i
$178$	0	0
$179$	19.3062	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$179$	19.3062	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$180$	0	0
$181$	16.7512	1.24511	0.622555	−	0.782576i	$-0.286095\pi$
$181$	16.7512	1.24511	0.622555	+	0.782576i	$0.286095\pi$
$182$	0	0
$183$	3.16053i	0.233633i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	29.5053i	2.15764i
$188$	0	0
$189$	−1.04684	−0.0761461
$190$	0	0
$191$	−11.9454	−0.864336	−0.432168	−	0.901793i	$-0.642251\pi$
$191$	−11.9454	−0.864336	−0.432168	+	0.901793i	$0.642251\pi$
$192$	0	0
$193$	26.4298i	1.90246i	0.308485	+	0.951229i	$0.400178\pi$
$193$	26.4298i	1.90246i	−0.308485	+	0.951229i	$0.599822\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0718i	0.788833i	0.918932	+	0.394417i	$0.129053\pi$
$197$	11.0718i	0.788833i	−0.918932	+	0.394417i	$0.870947\pi$
$198$	0	0
$199$	16.5548	1.17354	0.586768	−	0.809755i	$-0.300400\pi$
$199$	16.5548	1.17354	0.586768	+	0.809755i	$0.300400\pi$
$200$	0	0
$201$	3.17036	0.223620
$202$	0	0
$203$	7.23907i	0.508083i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.05101i	− 0.559584i
$208$	0	0
$209$	37.5796	2.59944
$210$	0	0
$211$	−4.10714	−0.282747	−0.141373	−	0.989956i	$-0.545152\pi$
$211$	−4.10714	−0.282747	−0.141373	+	0.989956i	$0.545152\pi$
$212$	0	0
$213$	6.50896i	0.445987i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.97021i	0.676822i
$218$	0	0
$219$	0.367806	0.0248540
$220$	0	0
$221$	4.72118	0.317581
$222$	0	0
$223$	− 6.26147i	− 0.419299i	−0.977777	−	0.209650i	$-0.932768\pi$
$223$	− 6.26147i	− 0.419299i	0.977777	−	0.209650i	$-0.0672323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.4817i	0.695697i	0.937551	+	0.347849i	$0.113088\pi$
$227$	10.4817i	0.695697i	−0.937551	+	0.347849i	$0.886912\pi$
$228$	0	0
$229$	−7.56679	−0.500028	−0.250014	−	0.968242i	$-0.580435\pi$
$229$	−7.56679	−0.500028	−0.250014	+	0.968242i	$0.580435\pi$
$230$	0	0
$231$	6.58345	0.433159
$232$	0	0
$233$	− 15.9832i	− 1.04709i	−0.851997	−	0.523547i	$-0.824608\pi$
$233$	− 15.9832i	− 1.04709i	0.851997	−	0.523547i	$-0.175392\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.32009i	0.215663i
$238$	0	0
$239$	2.91998	0.188877	0.0944387	−	0.995531i	$-0.469894\pi$
$239$	2.91998	0.188877	0.0944387	+	0.995531i	$0.469894\pi$
$240$	0	0
$241$	−23.0477	−1.48463	−0.742315	−	0.670051i	$-0.766272\pi$
$241$	−23.0477	−1.48463	−0.742315	+	0.670051i	$0.766272\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.01315i	− 0.382607i
$248$	0	0
$249$	−13.8579	−0.878206
$250$	0	0
$251$	−9.79130	−0.618021	−0.309011	−	0.951059i	$-0.599998\pi$
$251$	−9.79130	−0.618021	−0.309011	+	0.951059i	$0.599998\pi$
$252$	0	0
$253$	50.6321i	3.18321i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.3147i	1.39195i	0.718066	+	0.695975i	$0.245027\pi$
$257$	22.3147i	1.39195i	−0.718066	+	0.695975i	$0.754973\pi$
$258$	0	0
$259$	−8.05845	−0.500728
$260$	0	0
$261$	6.91519	0.428039
$262$	0	0
$263$	0.729154i	0.0449615i	0.999747	+	0.0224808i	$0.00715645\pi$
$263$	0.729154i	0.0449615i	−0.999747	+	0.0224808i	$0.992844\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.50645i	0.153392i
$268$	0	0
$269$	1.85922	0.113359	0.0566794	−	0.998392i	$-0.481949\pi$
$269$	1.85922	0.113359	0.0566794	+	0.998392i	$0.481949\pi$
$270$	0	0
$271$	19.5534	1.18778	0.593891	−	0.804545i	$-0.297591\pi$
$271$	19.5534	1.18778	0.593891	+	0.804545i	$0.297591\pi$
$272$	0	0
$273$	− 1.05342i	− 0.0637561i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.9612i	1.01910i	0.860441	+	0.509551i	$0.170188\pi$
$277$	16.9612i	1.01910i	−0.860441	+	0.509551i	$0.829812\pi$
$278$	0	0
$279$	9.52414	0.570195
$280$	0	0
$281$	−26.0879	−1.55627	−0.778135	−	0.628096i	$-0.783834\pi$
$281$	−26.0879	−1.55627	−0.778135	+	0.628096i	$0.783834\pi$
$282$	0	0
$283$	25.6093i	1.52232i	0.648565	+	0.761159i	$0.275369\pi$
$283$	25.6093i	1.52232i	−0.648565	+	0.761159i	$0.724631\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.64282i	− 0.0969724i
$288$	0	0
$289$	−5.01158	−0.294799
$290$	0	0
$291$	0.182689	0.0107094
$292$	0	0
$293$	27.7845i	1.62319i	0.584220	+	0.811595i	$0.301400\pi$
$293$	27.7845i	1.62319i	−0.584220	+	0.811595i	$0.698600\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 6.28891i	− 0.364919i
$298$	0	0
$299$	8.10169	0.468533
$300$	0	0
$301$	10.4149	0.600307
$302$	0	0
$303$	− 5.96970i	− 0.342951i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.1422i	− 0.921283i	−0.887586	−	0.460642i	$-0.847619\pi$
$307$	− 16.1422i	− 0.921283i	0.887586	−	0.460642i	$-0.152381\pi$
$308$	0	0
$309$	8.95864	0.509639
$310$	0	0
$311$	28.7381	1.62959	0.814795	−	0.579749i	$-0.196850\pi$
$311$	28.7381	1.62959	0.814795	+	0.579749i	$0.196850\pi$
$312$	0	0
$313$	26.4864i	1.49710i	0.663079	+	0.748550i	$0.269249\pi$
$313$	26.4864i	1.49710i	−0.663079	+	0.748550i	$0.730751\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.2980i	− 1.47705i	−0.674229	−	0.738523i	$-0.735524\pi$
$317$	− 26.2980i	− 1.47705i	0.674229	−	0.738523i	$-0.264476\pi$
$318$	0	0
$319$	−43.4890	−2.43491
$320$	0	0
$321$	−2.69495	−0.150417
$322$	0	0
$323$	28.0351i	1.55992i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.2999i	− 0.569585i
$328$	0	0
$329$	5.07358	0.279716
$330$	0	0
$331$	−31.2634	−1.71839	−0.859195	−	0.511649i	$-0.829035\pi$
$331$	−31.2634	−1.71839	−0.859195	+	0.511649i	$0.829035\pi$
$332$	0	0
$333$	7.69791i	0.421843i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.90526i	− 0.485101i	−0.970139	−	0.242550i	$-0.922016\pi$
$337$	− 8.90526i	− 0.485101i	0.970139	−	0.242550i	$-0.0779839\pi$
$338$	0	0
$339$	3.38070	0.183614
$340$	0	0
$341$	−59.8964	−3.24357
$342$	0	0
$343$	− 13.5085i	− 0.729391i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 21.5530i	− 1.15702i	−0.815674	−	0.578512i	$-0.803634\pi$
$347$	− 21.5530i	− 1.15702i	0.815674	−	0.578512i	$-0.196366\pi$
$348$	0	0
$349$	−17.3958	−0.931178	−0.465589	−	0.885001i	$-0.654158\pi$
$349$	−17.3958	−0.931178	−0.465589	+	0.885001i	$0.654158\pi$
$350$	0	0
$351$	−1.00629	−0.0537120
$352$	0	0
$353$	17.8613i	0.950661i	0.879807	+	0.475331i	$0.157672\pi$
$353$	17.8613i	0.950661i	−0.879807	+	0.475331i	$0.842328\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.91139i	0.259938i
$358$	0	0
$359$	−3.65870	−0.193099	−0.0965493	−	0.995328i	$-0.530781\pi$
$359$	−3.65870	−0.193099	−0.0965493	+	0.995328i	$0.530781\pi$
$360$	0	0
$361$	16.7071	0.879319
$362$	0	0
$363$	28.5503i	1.49850i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 26.0423i	− 1.35940i	−0.733492	−	0.679698i	$-0.762111\pi$
$367$	− 26.0423i	− 1.35940i	0.733492	−	0.679698i	$-0.237889\pi$
$368$	0	0
$369$	−1.56932	−0.0816954
$370$	0	0
$371$	2.95872	0.153609
$372$	0	0
$373$	− 29.2609i	− 1.51507i	−0.652792	−	0.757537i	$-0.726403\pi$
$373$	− 29.2609i	− 1.51507i	0.652792	−	0.757537i	$-0.273597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.95871i	0.358392i
$378$	0	0
$379$	10.2669	0.527374	0.263687	−	0.964608i	$-0.415061\pi$
$379$	10.2669	0.527374	0.263687	+	0.964608i	$0.415061\pi$
$380$	0	0
$381$	−5.97421	−0.306068
$382$	0	0
$383$	− 25.6479i	− 1.31055i	−0.755392	−	0.655273i	$-0.772554\pi$
$383$	− 25.6479i	− 1.31055i	0.755392	−	0.655273i	$-0.227446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.94897i	− 0.505735i
$388$	0	0
$389$	9.58813	0.486137	0.243069	−	0.970009i	$-0.421846\pi$
$389$	9.58813	0.486137	0.243069	+	0.970009i	$0.421846\pi$
$390$	0	0
$391$	−37.7725	−1.91024
$392$	0	0
$393$	6.20733i	0.313118i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.3309i	0.618869i	0.950921	+	0.309435i	$0.100140\pi$
$397$	12.3309i	0.618869i	−0.950921	+	0.309435i	$0.899860\pi$
$398$	0	0
$399$	6.25541	0.313162
$400$	0	0
$401$	18.9779	0.947709	0.473855	−	0.880603i	$-0.342862\pi$
$401$	18.9779	0.947709	0.473855	+	0.880603i	$0.342862\pi$
$402$	0	0
$403$	9.58408i	0.477417i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 48.4115i	− 2.39967i
$408$	0	0
$409$	−19.5102	−0.964719	−0.482359	−	0.875973i	$-0.660220\pi$
$409$	−19.5102	−0.964719	−0.482359	+	0.875973i	$0.660220\pi$
$410$	0	0
$411$	12.9777	0.640144
$412$	0	0
$413$	4.27328i	0.210274i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.99948i	− 0.195855i
$418$	0	0
$419$	−1.84915	−0.0903370	−0.0451685	−	0.998979i	$-0.514382\pi$
$419$	−1.84915	−0.0903370	−0.0451685	+	0.998979i	$0.514382\pi$
$420$	0	0
$421$	−6.67752	−0.325443	−0.162721	−	0.986672i	$-0.552027\pi$
$421$	−6.67752	−0.325443	−0.162721	+	0.986672i	$0.552027\pi$
$422$	0	0
$423$	− 4.84659i	− 0.235649i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.30856i	− 0.160112i
$428$	0	0
$429$	6.32849	0.305542
$430$	0	0
$431$	29.6459	1.42799	0.713996	−	0.700150i	$-0.246884\pi$
$431$	29.6459	1.42799	0.713996	+	0.700150i	$0.246884\pi$
$432$	0	0
$433$	28.0178i	1.34645i	0.739438	+	0.673225i	$0.235091\pi$
$433$	28.0178i	1.34645i	−0.739438	+	0.673225i	$0.764909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.1091i	2.30137i
$438$	0	0
$439$	−10.2248	−0.488004	−0.244002	−	0.969775i	$-0.578460\pi$
$439$	−10.2248	−0.488004	−0.244002	+	0.969775i	$0.578460\pi$
$440$	0	0
$441$	−5.90413	−0.281149
$442$	0	0
$443$	26.5115i	1.25960i	0.776757	+	0.629800i	$0.216863\pi$
$443$	26.5115i	1.25960i	−0.776757	+	0.629800i	$0.783137\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.68762i	− 0.269015i
$448$	0	0
$449$	6.39692	0.301889	0.150945	−	0.988542i	$-0.451768\pi$
$449$	6.39692	0.301889	0.150945	+	0.988542i	$0.451768\pi$
$450$	0	0
$451$	9.86929	0.464727
$452$	0	0
$453$	− 5.51150i	− 0.258953i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.9784i	0.700661i	0.936626	+	0.350330i	$0.113931\pi$
$457$	14.9784i	0.700661i	−0.936626	+	0.350330i	$0.886069\pi$
$458$	0	0
$459$	4.69165	0.218987
$460$	0	0
$461$	−12.1832	−0.567429	−0.283715	−	0.958909i	$-0.591567\pi$
$461$	−12.1832	−0.567429	−0.283715	+	0.958909i	$0.591567\pi$
$462$	0	0
$463$	8.31956i	0.386643i	0.981135	+	0.193321i	$0.0619260\pi$
$463$	8.31956i	0.386643i	−0.981135	+	0.193321i	$0.938074\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.30463i	0.384293i	0.981366	+	0.192146i	$0.0615448\pi$
$467$	8.30463i	0.384293i	−0.981366	+	0.192146i	$0.938455\pi$
$468$	0	0
$469$	−3.31884	−0.153250
$470$	0	0
$471$	17.0217	0.784321
$472$	0	0
$473$	62.5681i	2.87689i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.82635i	− 0.129410i
$478$	0	0
$479$	3.05639	0.139650	0.0698249	−	0.997559i	$-0.477756\pi$
$479$	3.05639	0.139650	0.0698249	+	0.997559i	$0.477756\pi$
$480$	0	0
$481$	−7.74636	−0.353204
$482$	0	0
$483$	8.42809i	0.383491i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.7187i	0.531026i	0.964107	+	0.265513i	$0.0855412\pi$
$487$	11.7187i	0.531026i	−0.964107	+	0.265513i	$0.914459\pi$
$488$	0	0
$489$	−12.2958	−0.556033
$490$	0	0
$491$	16.9523	0.765048	0.382524	−	0.923946i	$-0.375055\pi$
$491$	16.9523	0.765048	0.382524	+	0.923946i	$0.375055\pi$
$492$	0	0
$493$	− 32.4436i	− 1.46119i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.81381i	− 0.305641i
$498$	0	0
$499$	−14.0674	−0.629742	−0.314871	−	0.949135i	$-0.601961\pi$
$499$	−14.0674	−0.629742	−0.314871	+	0.949135i	$0.601961\pi$
$500$	0	0
$501$	−5.42858	−0.242531
$502$	0	0
$503$	5.60755i	0.250028i	0.992155	+	0.125014i	$0.0398976\pi$
$503$	5.60755i	0.250028i	−0.992155	+	0.125014i	$0.960102\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.9874i	0.532378i
$508$	0	0
$509$	11.4650	0.508176	0.254088	−	0.967181i	$-0.418225\pi$
$509$	11.4650	0.508176	0.254088	+	0.967181i	$0.418225\pi$
$510$	0	0
$511$	−0.385033	−0.0170328
$512$	0	0
$513$	− 5.97554i	− 0.263826i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	30.4797i	1.34050i
$518$	0	0
$519$	−10.7456	−0.471680
$520$	0	0
$521$	17.9012	0.784267	0.392133	−	0.919908i	$-0.371737\pi$
$521$	17.9012	0.784267	0.392133	+	0.919908i	$0.371737\pi$
$522$	0	0
$523$	1.34677i	0.0588899i	0.999566	+	0.0294450i	$0.00937398\pi$
$523$	1.34677i	0.0588899i	−0.999566	+	0.0294450i	$0.990626\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 44.6839i	− 1.94646i
$528$	0	0
$529$	−41.8188	−1.81821
$530$	0	0
$531$	4.08209	0.177148
$532$	0	0
$533$	− 1.57919i	− 0.0684025i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.3062i	0.833126i
$538$	0	0
$539$	37.1305	1.59933
$540$	0	0
$541$	11.2977	0.485725	0.242862	−	0.970061i	$-0.421914\pi$
$541$	11.2977	0.485725	0.242862	+	0.970061i	$0.421914\pi$
$542$	0	0
$543$	16.7512i	0.718864i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 25.8147i	− 1.10376i	−0.833924	−	0.551879i	$-0.813911\pi$
$547$	− 25.8147i	− 1.10376i	0.833924	−	0.551879i	$-0.186089\pi$
$548$	0	0
$549$	−3.16053	−0.134888
$550$	0	0
$551$	−41.3220	−1.76037
$552$	0	0
$553$	− 3.47559i	− 0.147797i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.3575i	1.49814i	0.662489	+	0.749072i	$0.269500\pi$
$557$	35.3575i	1.49814i	−0.662489	+	0.749072i	$0.730500\pi$
$558$	0	0
$559$	10.0116	0.423445
$560$	0	0
$561$	−29.5053	−1.24572
$562$	0	0
$563$	− 4.55427i	− 0.191940i	−0.995384	−	0.0959698i	$-0.969405\pi$
$563$	− 4.55427i	− 0.191940i	0.995384	−	0.0959698i	$-0.0305952\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.04684i	− 0.0439630i
$568$	0	0
$569$	37.1409	1.55703	0.778514	−	0.627627i	$-0.215974\pi$
$569$	37.1409	1.55703	0.778514	+	0.627627i	$0.215974\pi$
$570$	0	0
$571$	−4.30015	−0.179956	−0.0899778	−	0.995944i	$-0.528680\pi$
$571$	−4.30015	−0.179956	−0.0899778	+	0.995944i	$0.528680\pi$
$572$	0	0
$573$	− 11.9454i	− 0.499025i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 0.467179i	− 0.0194489i	−0.999953	−	0.00972445i	$-0.996905\pi$
$577$	− 0.467179i	− 0.0194489i	0.999953	−	0.00972445i	$-0.00309544\pi$
$578$	0	0
$579$	−26.4298	−1.09838
$580$	0	0
$581$	14.5069	0.601848
$582$	0	0
$583$	17.7746i	0.736150i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 32.9548i	− 1.36019i	−0.733124	−	0.680095i	$-0.761938\pi$
$587$	− 32.9548i	− 1.36019i	0.733124	−	0.680095i	$-0.238062\pi$
$588$	0	0
$589$	−56.9119	−2.34501
$590$	0	0
$591$	−11.0718	−0.455433
$592$	0	0
$593$	− 15.5285i	− 0.637680i	−0.947809	−	0.318840i	$-0.896707\pi$
$593$	− 15.5285i	− 0.637680i	0.947809	−	0.318840i	$-0.103293\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.5548i	0.677542i
$598$	0	0
$599$	36.0116	1.47140	0.735698	−	0.677310i	$-0.236854\pi$
$599$	36.0116	1.47140	0.735698	+	0.677310i	$0.236854\pi$
$600$	0	0
$601$	−24.1503	−0.985112	−0.492556	−	0.870281i	$-0.663937\pi$
$601$	−24.1503	−0.985112	−0.492556	+	0.870281i	$0.663937\pi$
$602$	0	0
$603$	3.17036i	0.129107i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	29.8098i	1.20994i	0.796248	+	0.604970i	$0.206815\pi$
$607$	29.8098i	1.20994i	−0.796248	+	0.604970i	$0.793185\pi$
$608$	0	0
$609$	−7.23907	−0.293342
$610$	0	0
$611$	4.87709	0.197306
$612$	0	0
$613$	− 32.1330i	− 1.29784i	−0.760857	−	0.648919i	$-0.775221\pi$
$613$	− 32.1330i	− 1.29784i	0.760857	−	0.648919i	$-0.224779\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.2509i	0.694493i	0.937774	+	0.347247i	$0.112883\pi$
$617$	17.2509i	0.694493i	−0.937774	+	0.347247i	$0.887117\pi$
$618$	0	0
$619$	−7.06017	−0.283772	−0.141886	−	0.989883i	$-0.545317\pi$
$619$	−7.06017	−0.283772	−0.141886	+	0.989883i	$0.545317\pi$
$620$	0	0
$621$	8.05101	0.323076
$622$	0	0
$623$	− 2.62384i	− 0.105122i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	37.5796i	1.50078i
$628$	0	0
$629$	36.1159	1.44004
$630$	0	0
$631$	−9.78513	−0.389540	−0.194770	−	0.980849i	$-0.562396\pi$
$631$	−9.78513	−0.389540	−0.194770	+	0.980849i	$0.562396\pi$
$632$	0	0
$633$	− 4.10714i	− 0.163244i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.94129i	− 0.235403i
$638$	0	0
$639$	−6.50896	−0.257491
$640$	0	0
$641$	15.6701	0.618930	0.309465	−	0.950911i	$-0.399850\pi$
$641$	15.6701	0.618930	0.309465	+	0.950911i	$0.399850\pi$
$642$	0	0
$643$	0.766747i	0.0302375i	0.999886	+	0.0151188i	$0.00481264\pi$
$643$	0.766747i	0.0302375i	−0.999886	+	0.0151188i	$0.995187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.1728i	0.517878i	0.965894	+	0.258939i	$0.0833728\pi$
$647$	13.1728i	0.517878i	−0.965894	+	0.258939i	$0.916627\pi$
$648$	0	0
$649$	−25.6719	−1.00771
$650$	0	0
$651$	−9.97021	−0.390763
$652$	0	0
$653$	13.0921i	0.512335i	0.966632	+	0.256167i	$0.0824598\pi$
$653$	13.0921i	0.512335i	−0.966632	+	0.256167i	$0.917540\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.367806i	0.0143495i
$658$	0	0
$659$	9.46731	0.368794	0.184397	−	0.982852i	$-0.440967\pi$
$659$	9.46731	0.368794	0.184397	+	0.982852i	$0.440967\pi$
$660$	0	0
$661$	5.55884	0.216214	0.108107	−	0.994139i	$-0.465521\pi$
$661$	5.55884	0.216214	0.108107	+	0.994139i	$0.465521\pi$
$662$	0	0
$663$	4.72118i	0.183355i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 55.6743i	− 2.15572i
$668$	0	0
$669$	6.26147	0.242083
$670$	0	0
$671$	19.8763	0.767316
$672$	0	0
$673$	− 0.304023i	− 0.0117192i	−0.999983	−	0.00585962i	$-0.998135\pi$
$673$	− 0.304023i	− 0.0117192i	0.999983	−	0.00585962i	$-0.00186519\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.5278i	1.51918i	0.650404	+	0.759589i	$0.274600\pi$
$677$	39.5278i	1.51918i	−0.650404	+	0.759589i	$0.725400\pi$
$678$	0	0
$679$	−0.191246	−0.00733934
$680$	0	0
$681$	−10.4817	−0.401661
$682$	0	0
$683$	− 31.5480i	− 1.20715i	−0.797305	−	0.603576i	$-0.793742\pi$
$683$	− 31.5480i	− 1.20715i	0.797305	−	0.603576i	$-0.206258\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.56679i	− 0.288691i
$688$	0	0
$689$	2.84414	0.108353
$690$	0	0
$691$	9.38385	0.356978	0.178489	−	0.983942i	$-0.442879\pi$
$691$	9.38385	0.356978	0.178489	+	0.983942i	$0.442879\pi$
$692$	0	0
$693$	6.58345i	0.250085i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.36269i	0.278882i
$698$	0	0
$699$	15.9832	0.604540
$700$	0	0
$701$	−20.4347	−0.771809	−0.385904	−	0.922539i	$-0.626111\pi$
$701$	−20.4347	−0.771809	−0.385904	+	0.922539i	$0.626111\pi$
$702$	0	0
$703$	− 45.9992i	− 1.73489i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.24930i	0.235029i
$708$	0	0
$709$	5.91798	0.222254	0.111127	−	0.993806i	$-0.464554\pi$
$709$	5.91798	0.222254	0.111127	+	0.993806i	$0.464554\pi$
$710$	0	0
$711$	−3.32009	−0.124513
$712$	0	0
$713$	− 76.6790i	− 2.87165i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.91998i	0.109048i
$718$	0	0
$719$	8.00127	0.298397	0.149199	−	0.988807i	$-0.452331\pi$
$719$	8.00127	0.298397	0.149199	+	0.988807i	$0.452331\pi$
$720$	0	0
$721$	−9.37822	−0.349263
$722$	0	0
$723$	− 23.0477i	− 0.857152i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 40.2787i	− 1.49385i	−0.664907	−	0.746926i	$-0.731529\pi$
$727$	− 40.2787i	− 1.49385i	0.664907	−	0.746926i	$-0.268471\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−46.6771	−1.72641
$732$	0	0
$733$	− 10.5973i	− 0.391420i	−0.980662	−	0.195710i	$-0.937299\pi$
$733$	− 10.5973i	− 0.391420i	0.980662	−	0.195710i	$-0.0627011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.9381i	− 0.734428i
$738$	0	0
$739$	45.5094	1.67409	0.837044	−	0.547135i	$-0.184282\pi$
$739$	45.5094	1.67409	0.837044	+	0.547135i	$0.184282\pi$
$740$	0	0
$741$	6.01315	0.220898
$742$	0	0
$743$	8.64344i	0.317097i	0.987351	+	0.158549i	$0.0506814\pi$
$743$	8.64344i	0.317097i	−0.987351	+	0.158549i	$0.949319\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 13.8579i	− 0.507032i
$748$	0	0
$749$	2.82117	0.103083
$750$	0	0
$751$	−54.4382	−1.98648	−0.993239	−	0.116086i	$-0.962965\pi$
$751$	−54.4382	−1.98648	−0.993239	+	0.116086i	$0.962965\pi$
$752$	0	0
$753$	− 9.79130i	− 0.356815i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.1378i	1.24076i	0.784302	+	0.620380i	$0.213022\pi$
$757$	34.1378i	1.24076i	−0.784302	+	0.620380i	$0.786978\pi$
$758$	0	0
$759$	−50.6321	−1.83783
$760$	0	0
$761$	3.12475	0.113272	0.0566361	−	0.998395i	$-0.481963\pi$
$761$	3.12475	0.113272	0.0566361	+	0.998395i	$0.481963\pi$
$762$	0	0
$763$	10.7823i	0.390345i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.10778i	0.148323i
$768$	0	0
$769$	53.3949	1.92547	0.962735	−	0.270446i	$-0.0871712\pi$
$769$	53.3949	1.92547	0.962735	+	0.270446i	$0.0871712\pi$
$770$	0	0
$771$	−22.3147	−0.803642
$772$	0	0
$773$	− 0.797218i	− 0.0286739i	−0.999897	−	0.0143370i	$-0.995436\pi$
$773$	− 0.797218i	− 0.0286739i	0.999897	−	0.0143370i	$-0.00456375\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.05845i	− 0.289095i
$778$	0	0
$779$	9.37751	0.335984
$780$	0	0
$781$	40.9342	1.46474
$782$	0	0
$783$	6.91519i	0.247129i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 41.7571i	− 1.48848i	−0.667913	−	0.744239i	$-0.732812\pi$
$787$	− 41.7571i	− 1.48848i	0.667913	−	0.744239i	$-0.267188\pi$
$788$	0	0
$789$	−0.729154	−0.0259586
$790$	0	0
$791$	−3.53904	−0.125834
$792$	0	0
$793$	− 3.18042i	− 0.112940i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.6513i	− 0.979458i	−0.871875	−	0.489729i	$-0.837096\pi$
$797$	− 27.6513i	− 0.979458i	0.871875	−	0.489729i	$-0.162904\pi$
$798$	0	0
$799$	−22.7385	−0.804430
$800$	0	0
$801$	−2.50645	−0.0885611
$802$	0	0
$803$	− 2.31310i	− 0.0816275i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.85922i	0.0654477i
$808$	0	0
$809$	−34.9805	−1.22985	−0.614924	−	0.788586i	$-0.710813\pi$
$809$	−34.9805	−1.22985	−0.614924	+	0.788586i	$0.710813\pi$
$810$	0	0
$811$	9.15152	0.321353	0.160677	−	0.987007i	$-0.448632\pi$
$811$	9.15152	0.321353	0.160677	+	0.987007i	$0.448632\pi$
$812$	0	0
$813$	19.5534i	0.685767i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	59.4504i	2.07991i
$818$	0	0
$819$	1.05342	0.0368096
$820$	0	0
$821$	−3.23251	−0.112815	−0.0564077	−	0.998408i	$-0.517965\pi$
$821$	−3.23251	−0.112815	−0.0564077	+	0.998408i	$0.517965\pi$
$822$	0	0
$823$	45.2046i	1.57573i	0.615846	+	0.787867i	$0.288814\pi$
$823$	45.2046i	1.57573i	−0.615846	+	0.787867i	$0.711186\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.6568i	− 0.822629i	−0.911493	−	0.411314i	$-0.865070\pi$
$827$	− 23.6568i	− 0.822629i	0.911493	−	0.411314i	$-0.134930\pi$
$828$	0	0
$829$	−12.7644	−0.443326	−0.221663	−	0.975123i	$-0.571148\pi$
$829$	−12.7644	−0.443326	−0.221663	+	0.975123i	$0.571148\pi$
$830$	0	0
$831$	−16.9612	−0.588379
$832$	0	0
$833$	27.7001i	0.959753i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.52414i	0.329202i
$838$	0	0
$839$	52.0352	1.79646	0.898228	−	0.439530i	$-0.144855\pi$
$839$	52.0352	1.79646	0.898228	+	0.439530i	$0.144855\pi$
$840$	0	0
$841$	18.8198	0.648959
$842$	0	0
$843$	− 26.0879i	− 0.898513i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 29.8875i	− 1.02695i
$848$	0	0
$849$	−25.6093	−0.878911
$850$	0	0
$851$	61.9760	2.12451
$852$	0	0
$853$	− 35.6260i	− 1.21981i	−0.792474	−	0.609905i	$-0.791208\pi$
$853$	− 35.6260i	− 1.21981i	0.792474	−	0.609905i	$-0.208792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.6745i	− 0.842866i	−0.906860	−	0.421433i	$-0.861527\pi$
$857$	− 24.6745i	− 0.842866i	0.906860	−	0.421433i	$-0.138473\pi$
$858$	0	0
$859$	−21.9223	−0.747978	−0.373989	−	0.927433i	$-0.622010\pi$
$859$	−21.9223	−0.747978	−0.373989	+	0.927433i	$0.622010\pi$
$860$	0	0
$861$	1.64282	0.0559871
$862$	0	0
$863$	− 27.5761i	− 0.938702i	−0.883012	−	0.469351i	$-0.844488\pi$
$863$	− 27.5761i	− 0.938702i	0.883012	−	0.469351i	$-0.155512\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.01158i	− 0.170202i
$868$	0	0
$869$	20.8797	0.708296
$870$	0	0
$871$	−3.19031	−0.108100
$872$	0	0
$873$	0.182689i	0.00618310i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.6879i	− 0.698579i	−0.937015	−	0.349290i	$-0.886423\pi$
$877$	− 20.6879i	− 0.698579i	0.937015	−	0.349290i	$-0.113577\pi$
$878$	0	0
$879$	−27.7845	−0.937149
$880$	0	0
$881$	−37.8861	−1.27641	−0.638207	−	0.769865i	$-0.720324\pi$
$881$	−37.8861	−1.27641	−0.638207	+	0.769865i	$0.720324\pi$
$882$	0	0
$883$	39.3444i	1.32404i	0.749484	+	0.662022i	$0.230302\pi$
$883$	39.3444i	1.32404i	−0.749484	+	0.662022i	$0.769698\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.2764i	0.815121i	0.913178	+	0.407560i	$0.133620\pi$
$887$	24.2764i	0.815121i	−0.913178	+	0.407560i	$0.866380\pi$
$888$	0	0
$889$	6.25402	0.209753
$890$	0	0
$891$	6.28891	0.210686
$892$	0	0
$893$	28.9610i	0.969142i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.10169i	0.270507i
$898$	0	0
$899$	65.8612	2.19659
$900$	0	0
$901$	−13.2602	−0.441762
$902$	0	0
$903$	10.4149i	0.346587i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 13.1961i	− 0.438169i	−0.975706	−	0.219085i	$-0.929693\pi$
$907$	− 13.1961i	− 0.438169i	0.975706	−	0.219085i	$-0.0703071\pi$
$908$	0	0
$909$	5.96970	0.198003
$910$	0	0
$911$	−27.6036	−0.914547	−0.457274	−	0.889326i	$-0.651174\pi$
$911$	−27.6036	−0.914547	−0.457274	+	0.889326i	$0.651174\pi$
$912$	0	0
$913$	87.1508i	2.88427i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.49805i	− 0.214585i
$918$	0	0
$919$	−12.1492	−0.400764	−0.200382	−	0.979718i	$-0.564218\pi$
$919$	−12.1492	−0.400764	−0.200382	+	0.979718i	$0.564218\pi$
$920$	0	0
$921$	16.1422	0.531903
$922$	0	0
$923$	− 6.54993i	− 0.215593i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.95864i	0.294240i
$928$	0	0
$929$	−35.2346	−1.15601	−0.578005	−	0.816033i	$-0.696169\pi$
$929$	−35.2346	−1.15601	−0.578005	+	0.816033i	$0.696169\pi$
$930$	0	0
$931$	35.2804	1.15627
$932$	0	0
$933$	28.7381i	0.940844i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.81965i	0.222788i	0.993776	+	0.111394i	$0.0355316\pi$
$937$	6.81965i	0.222788i	−0.993776	+	0.111394i	$0.964468\pi$
$938$	0	0
$939$	−26.4864	−0.864351
$940$	0	0
$941$	31.2034	1.01720	0.508601	−	0.861002i	$-0.330163\pi$
$941$	31.2034	1.01720	0.508601	+	0.861002i	$0.330163\pi$
$942$	0	0
$943$	12.6346i	0.411439i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.3989i	0.922842i	0.887181	+	0.461421i	$0.152660\pi$
$947$	28.3989i	0.922842i	−0.887181	+	0.461421i	$0.847340\pi$
$948$	0	0
$949$	−0.370121	−0.0120146
$950$	0	0
$951$	26.2980	0.852772
$952$	0	0
$953$	11.7015i	0.379049i	0.981876	+	0.189525i	$0.0606947\pi$
$953$	11.7015i	0.379049i	−0.981876	+	0.189525i	$0.939305\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 43.4890i	− 1.40580i
$958$	0	0
$959$	−13.5855	−0.438700
$960$	0	0
$961$	59.7092	1.92610
$962$	0	0
$963$	− 2.69495i	− 0.0868435i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.9318i	0.544490i	0.962228	+	0.272245i	$0.0877661\pi$
$967$	16.9318i	0.544490i	−0.962228	+	0.272245i	$0.912234\pi$
$968$	0	0
$969$	−28.0351	−0.900618
$970$	0	0
$971$	−12.3229	−0.395460	−0.197730	−	0.980257i	$-0.563357\pi$
$971$	−12.3229	−0.395460	−0.197730	+	0.980257i	$0.563357\pi$
$972$	0	0
$973$	4.18679i	0.134222i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.15644i	− 0.132976i	−0.997787	−	0.0664882i	$-0.978821\pi$
$977$	− 4.15644i	− 0.132976i	0.997787	−	0.0664882i	$-0.0211795\pi$
$978$	0	0
$979$	15.7628	0.503782
$980$	0	0
$981$	10.2999	0.328850
$982$	0	0
$983$	− 13.6896i	− 0.436632i	−0.975878	−	0.218316i	$-0.929944\pi$
$983$	− 13.6896i	− 0.436632i	0.975878	−	0.218316i	$-0.0700563\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.07358i	0.161494i
$988$	0	0
$989$	−80.0993	−2.54701
$990$	0	0
$991$	−7.65679	−0.243226	−0.121613	−	0.992578i	$-0.538807\pi$
$991$	−7.65679	−0.243226	−0.121613	+	0.992578i	$0.538807\pi$
$992$	0	0
$993$	− 31.2634i	− 0.992113i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.5223i	1.63173i	0.578244	+	0.815864i	$0.303738\pi$
$997$	51.5223i	1.63173i	−0.578244	+	0.815864i	$0.696262\pi$
$998$	0	0
$999$	−7.69791	−0.243551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.18		24
5.2	odd	4		7500.2.a.n.1.7		12
5.3	odd	4		7500.2.a.m.1.6		12
5.4	even	2	inner	7500.2.d.g.1249.7		24
25.3	odd	20		1500.2.m.d.1201.3		24
25.4	even	10		1500.2.o.c.49.4		24
25.6	even	5		1500.2.o.c.949.4		24
25.8	odd	20		1500.2.m.d.301.3		24
25.17	odd	20		1500.2.m.c.301.4		24
25.19	even	10		300.2.o.a.289.3	yes	24
25.21	even	5		300.2.o.a.109.3	✓	24
25.22	odd	20		1500.2.m.c.1201.4		24
75.44	odd	10		900.2.w.c.289.2		24
75.71	odd	10		900.2.w.c.109.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.3	✓	24	25.21	even	5
300.2.o.a.289.3	yes	24	25.19	even	10
900.2.w.c.109.2		24	75.71	odd	10
900.2.w.c.289.2		24	75.44	odd	10
1500.2.m.c.301.4		24	25.17	odd	20
1500.2.m.c.1201.4		24	25.22	odd	20
1500.2.m.d.301.3		24	25.8	odd	20
1500.2.m.d.1201.3		24	25.3	odd	20
1500.2.o.c.49.4		24	25.4	even	10
1500.2.o.c.949.4		24	25.6	even	5
7500.2.a.m.1.6		12	5.3	odd	4
7500.2.a.n.1.7		12	5.2	odd	4
7500.2.d.g.1249.7		24	5.4	even	2	inner
7500.2.d.g.1249.18		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.04684i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.04684i q^{7} -1.00000 q^{9} +6.28891 q^{11} -1.00629i q^{13} +4.69165i q^{17} +5.97554 q^{19} +1.04684 q^{21} +8.05101i q^{23} -1.00000i q^{27} -6.91519 q^{29} -9.52414 q^{31} +6.28891i q^{33} -7.69791i q^{37} +1.00629 q^{39} +1.56932 q^{41} +9.94897i q^{43} +4.84659i q^{47} +5.90413 q^{49} -4.69165 q^{51} +2.82635i q^{53} +5.97554i q^{57} -4.08209 q^{59} +3.16053 q^{61} +1.04684i q^{63} -3.17036i q^{67} -8.05101 q^{69} +6.50896 q^{71} -0.367806i q^{73} -6.58345i q^{77} +3.32009 q^{79} +1.00000 q^{81} +13.8579i q^{83} -6.91519i q^{87} +2.50645 q^{89} -1.05342 q^{91} -9.52414i q^{93} -0.182689i q^{97} -6.28891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

L-functions

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