LMFDB - Embedded newform 7500.2.d.c.1249.3

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:	$8$
Coefficient field:	8.0.6724000000.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 $ x^8 + 16x^6 + 86x^4 + 181*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$1.70636i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.c.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.0883282i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.0883282i q^{7} -1.00000 q^{9} +2.26981 q^{11} +2.65177i q^{13} -2.08833i q^{17} -1.76095 q^{19} +0.0883282 q^{21} +4.74010i q^{23} +1.00000i q^{27} -3.70636 q^{29} -4.10620 q^{31} -2.26981i q^{33} -7.11909i q^{37} +2.65177 q^{39} -6.58938 q^{41} +1.79469i q^{43} -10.1110i q^{47} +6.99220 q^{49} -2.08833 q^{51} -0.961440i q^{53} +1.76095i q^{57} +8.97801 q^{59} +9.46618 q^{61} -0.0883282i q^{63} -13.9039i q^{67} +4.74010 q^{69} +6.14774 q^{71} -3.15765i q^{73} +0.200488i q^{77} -13.3522 q^{79} +1.00000 q^{81} -15.8195i q^{83} +3.70636i q^{87} +10.2508 q^{89} -0.234226 q^{91} +4.10620i q^{93} +12.8077i q^{97} -2.26981 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * 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
$7$	0.0883282i	0.0333849i	0.999861	+	0.0166925i	$0.00531362\pi$
$7$	0.0883282i	0.0333849i	−0.999861	+	0.0166925i	$0.994686\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.26981	0.684372	0.342186	−	0.939632i	$-0.388833\pi$
$11$	2.26981	0.684372	0.342186	+	0.939632i	$0.388833\pi$
$12$	0	0
$13$	2.65177i	0.735469i	0.929931	+	0.367735i	$0.119867\pi$
$13$	2.65177i	0.735469i	−0.929931	+	0.367735i	$0.880133\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.08833i	− 0.506494i	−0.967402	−	0.253247i	$-0.918501\pi$
$17$	− 2.08833i	− 0.506494i	0.967402	−	0.253247i	$-0.0814985\pi$
$18$	0	0
$19$	−1.76095	−0.403990	−0.201995	−	0.979387i	$-0.564742\pi$
$19$	−1.76095	−0.403990	−0.201995	+	0.979387i	$0.564742\pi$
$20$	0	0
$21$	0.0883282	0.0192748
$22$	0	0
$23$	4.74010i	0.988379i	0.869354	+	0.494190i	$0.164535\pi$
$23$	4.74010i	0.988379i	−0.869354	+	0.494190i	$0.835465\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−3.70636	−0.688254	−0.344127	−	0.938923i	$-0.611825\pi$
$29$	−3.70636	−0.688254	−0.344127	+	0.938923i	$0.611825\pi$
$30$	0	0
$31$	−4.10620	−0.737495	−0.368748	−	0.929530i	$-0.620213\pi$
$31$	−4.10620	−0.737495	−0.368748	+	0.929530i	$0.620213\pi$
$32$	0	0
$33$	− 2.26981i	− 0.395123i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.11909i	− 1.17037i	−0.810900	−	0.585185i	$-0.801022\pi$
$37$	− 7.11909i	− 1.17037i	0.810900	−	0.585185i	$-0.198978\pi$
$38$	0	0
$39$	2.65177	0.424623
$40$	0	0
$41$	−6.58938	−1.02909	−0.514544	−	0.857464i	$-0.672039\pi$
$41$	−6.58938	−1.02909	−0.514544	+	0.857464i	$0.672039\pi$
$42$	0	0
$43$	1.79469i	0.273688i	0.990593	+	0.136844i	$0.0436959\pi$
$43$	1.79469i	0.273688i	−0.990593	+	0.136844i	$0.956304\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.1110i	− 1.47484i	−0.675432	−	0.737422i	$-0.736043\pi$
$47$	− 10.1110i	− 1.47484i	0.675432	−	0.737422i	$-0.263957\pi$
$48$	0	0
$49$	6.99220	0.998885
$50$	0	0
$51$	−2.08833	−0.292424
$52$	0	0
$53$	− 0.961440i	− 0.132064i	−0.997818	−	0.0660320i	$-0.978966\pi$
$53$	− 0.961440i	− 0.132064i	0.997818	−	0.0660320i	$-0.0210339\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.76095i	0.233244i
$58$	0	0
$59$	8.97801	1.16884	0.584419	−	0.811452i	$-0.301323\pi$
$59$	8.97801	1.16884	0.584419	+	0.811452i	$0.301323\pi$
$60$	0	0
$61$	9.46618	1.21202	0.606010	−	0.795457i	$-0.292769\pi$
$61$	9.46618	1.21202	0.606010	+	0.795457i	$0.292769\pi$
$62$	0	0
$63$	− 0.0883282i	− 0.0111283i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.9039i	− 1.69863i	−0.527888	−	0.849314i	$-0.677016\pi$
$67$	− 13.9039i	− 1.69863i	0.527888	−	0.849314i	$-0.322984\pi$
$68$	0	0
$69$	4.74010	0.570641
$70$	0	0
$71$	6.14774	0.729602	0.364801	−	0.931085i	$-0.381137\pi$
$71$	6.14774	0.729602	0.364801	+	0.931085i	$0.381137\pi$
$72$	0	0
$73$	− 3.15765i	− 0.369575i	−0.982779	−	0.184787i	$-0.940840\pi$
$73$	− 3.15765i	− 0.369575i	0.982779	−	0.184787i	$-0.0591596\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.200488i	0.0228477i
$78$	0	0
$79$	−13.3522	−1.50224	−0.751118	−	0.660167i	$-0.770485\pi$
$79$	−13.3522	−1.50224	−0.751118	+	0.660167i	$0.770485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 15.8195i	− 1.73641i	−0.496202	−	0.868207i	$-0.665272\pi$
$83$	− 15.8195i	− 1.73641i	0.496202	−	0.868207i	$-0.334728\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.70636i	0.397364i
$88$	0	0
$89$	10.2508	1.08658	0.543291	−	0.839544i	$-0.317178\pi$
$89$	10.2508	1.08658	0.543291	+	0.839544i	$0.317178\pi$
$90$	0	0
$91$	−0.234226	−0.0245536
$92$	0	0
$93$	4.10620i	0.425793i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.8077i	1.30043i	0.759751	+	0.650214i	$0.225321\pi$
$97$	12.8077i	1.30043i	−0.759751	+	0.650214i	$0.774679\pi$
$98$	0	0
$99$	−2.26981	−0.228124
$100$	0	0
$101$	−2.72537	−0.271185	−0.135592	−	0.990765i	$-0.543294\pi$
$101$	−2.72537	−0.271185	−0.135592	+	0.990765i	$0.543294\pi$
$102$	0	0
$103$	− 0.359976i	− 0.0354695i	−0.999843	−	0.0177348i	$-0.994355\pi$
$103$	− 0.359976i	− 0.0354695i	0.999843	−	0.0177348i	$-0.00564544\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.0286533i	− 0.00277002i	−0.999999	−	0.00138501i	$-0.999559\pi$
$107$	− 0.0286533i	− 0.00277002i	0.999999	−	0.00138501i	$-0.000440863\pi$
$108$	0	0
$109$	18.9217	1.81237	0.906187	−	0.422877i	$-0.138980\pi$
$109$	18.9217	1.81237	0.906187	+	0.422877i	$0.138980\pi$
$110$	0	0
$111$	−7.11909	−0.675714
$112$	0	0
$113$	4.89380i	0.460370i	0.973147	+	0.230185i	$0.0739331\pi$
$113$	4.89380i	0.460370i	−0.973147	+	0.230185i	$0.926067\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.65177i	− 0.245156i
$118$	0	0
$119$	0.184458	0.0169093
$120$	0	0
$121$	−5.84798	−0.531634
$122$	0	0
$123$	6.58938i	0.594144i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.26997i	0.290163i	0.989420	+	0.145081i	$0.0463444\pi$
$127$	3.26997i	0.290163i	−0.989420	+	0.145081i	$0.953656\pi$
$128$	0	0
$129$	1.79469	0.158014
$130$	0	0
$131$	3.59550	0.314141	0.157070	−	0.987587i	$-0.449795\pi$
$131$	3.59550	0.314141	0.157070	+	0.987587i	$0.449795\pi$
$132$	0	0
$133$	− 0.155542i	− 0.0134872i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.0197i	− 1.45409i	−0.686589	−	0.727046i	$-0.740893\pi$
$137$	− 17.0197i	− 1.45409i	0.686589	−	0.727046i	$-0.259107\pi$
$138$	0	0
$139$	18.9860	1.61037	0.805185	−	0.593024i	$-0.202066\pi$
$139$	18.9860	1.61037	0.805185	+	0.593024i	$0.202066\pi$
$140$	0	0
$141$	−10.1110	−0.851502
$142$	0	0
$143$	6.01901i	0.503335i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.99220i	− 0.576707i
$148$	0	0
$149$	−20.3441	−1.66665	−0.833327	−	0.552780i	$-0.813567\pi$
$149$	−20.3441	−1.66665	−0.833327	+	0.552780i	$0.813567\pi$
$150$	0	0
$151$	13.2609	1.07915	0.539577	−	0.841936i	$-0.318584\pi$
$151$	13.2609	1.07915	0.539577	+	0.841936i	$0.318584\pi$
$152$	0	0
$153$	2.08833i	0.168831i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.8066i	− 1.02208i	−0.859558	−	0.511039i	$-0.829261\pi$
$157$	− 12.8066i	− 1.02208i	0.859558	−	0.511039i	$-0.170739\pi$
$158$	0	0
$159$	−0.961440	−0.0762471
$160$	0	0
$161$	−0.418684	−0.0329970
$162$	0	0
$163$	15.0647i	1.17996i	0.807420	+	0.589978i	$0.200863\pi$
$163$	15.0647i	1.17996i	−0.807420	+	0.589978i	$0.799137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.45128i	0.267068i	0.991044	+	0.133534i	$0.0426326\pi$
$167$	3.45128i	0.267068i	−0.991044	+	0.133534i	$0.957367\pi$
$168$	0	0
$169$	5.96810	0.459085
$170$	0	0
$171$	1.76095	0.134663
$172$	0	0
$173$	− 2.41457i	− 0.183576i	−0.995779	−	0.0917880i	$-0.970742\pi$
$173$	− 2.41457i	− 0.183576i	0.995779	−	0.0917880i	$-0.0292582\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.97801i	− 0.674829i
$178$	0	0
$179$	4.06948	0.304167	0.152084	−	0.988368i	$-0.451402\pi$
$179$	4.06948	0.304167	0.152084	+	0.988368i	$0.451402\pi$
$180$	0	0
$181$	−13.3363	−0.991280	−0.495640	−	0.868528i	$-0.665066\pi$
$181$	−13.3363	−0.991280	−0.495640	+	0.868528i	$0.665066\pi$
$182$	0	0
$183$	− 9.46618i	− 0.699760i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.74010i	− 0.346630i
$188$	0	0
$189$	−0.0883282	−0.00642493
$190$	0	0
$191$	25.2529	1.82724	0.913618	−	0.406574i	$-0.133277\pi$
$191$	25.2529	1.82724	0.913618	+	0.406574i	$0.133277\pi$
$192$	0	0
$193$	− 24.6399i	− 1.77362i	−0.462139	−	0.886808i	$-0.652918\pi$
$193$	− 24.6399i	− 1.77362i	0.462139	−	0.886808i	$-0.347082\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.6201i	− 1.32663i	−0.748340	−	0.663315i	$-0.769149\pi$
$197$	− 18.6201i	− 1.32663i	0.748340	−	0.663315i	$-0.230851\pi$
$198$	0	0
$199$	−9.85708	−0.698750	−0.349375	−	0.936983i	$-0.613606\pi$
$199$	−9.85708	−0.698750	−0.349375	+	0.936983i	$0.613606\pi$
$200$	0	0
$201$	−13.9039	−0.980703
$202$	0	0
$203$	− 0.327376i	− 0.0229773i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.74010i	− 0.329460i
$208$	0	0
$209$	−3.99702	−0.276480
$210$	0	0
$211$	−7.89878	−0.543775	−0.271887	−	0.962329i	$-0.587648\pi$
$211$	−7.89878	−0.543775	−0.271887	+	0.962329i	$0.587648\pi$
$212$	0	0
$213$	− 6.14774i	− 0.421236i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.362693i	− 0.0246212i
$218$	0	0
$219$	−3.15765	−0.213374
$220$	0	0
$221$	5.53777	0.372511
$222$	0	0
$223$	− 9.18174i	− 0.614855i	−0.951572	−	0.307427i	$-0.900532\pi$
$223$	− 9.18174i	− 0.614855i	0.951572	−	0.307427i	$-0.0994681\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.5654i	− 0.767626i	−0.923411	−	0.383813i	$-0.874611\pi$
$227$	− 11.5654i	− 0.767626i	0.923411	−	0.383813i	$-0.125389\pi$
$228$	0	0
$229$	−24.9463	−1.64850	−0.824248	−	0.566229i	$-0.808402\pi$
$229$	−24.9463	−1.64850	−0.824248	+	0.566229i	$0.808402\pi$
$230$	0	0
$231$	0.200488	0.0131911
$232$	0	0
$233$	13.0200i	0.852967i	0.904495	+	0.426484i	$0.140248\pi$
$233$	13.0200i	0.852967i	−0.904495	+	0.426484i	$0.859752\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.3522i	0.867317i
$238$	0	0
$239$	26.4086	1.70823	0.854115	−	0.520084i	$-0.174099\pi$
$239$	26.4086	1.70823	0.854115	+	0.520084i	$0.174099\pi$
$240$	0	0
$241$	−6.23591	−0.401690	−0.200845	−	0.979623i	$-0.564369\pi$
$241$	−6.23591	−0.401690	−0.200845	+	0.979623i	$0.564369\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.66964i	− 0.297122i
$248$	0	0
$249$	−15.8195	−1.00252
$250$	0	0
$251$	7.46802	0.471377	0.235689	−	0.971829i	$-0.424265\pi$
$251$	7.46802	0.471377	0.235689	+	0.971829i	$0.424265\pi$
$252$	0	0
$253$	10.7591i	0.676419i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.39880i	− 0.523903i	−0.965081	−	0.261951i	$-0.915634\pi$
$257$	− 8.39880i	− 0.523903i	0.965081	−	0.261951i	$-0.0843660\pi$
$258$	0	0
$259$	0.628816	0.0390727
$260$	0	0
$261$	3.70636	0.229418
$262$	0	0
$263$	− 10.0248i	− 0.618156i	−0.951037	−	0.309078i	$-0.899980\pi$
$263$	− 10.0248i	− 0.618156i	0.951037	−	0.309078i	$-0.100020\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.2508i	− 0.627339i
$268$	0	0
$269$	17.6192	1.07426	0.537130	−	0.843500i	$-0.319509\pi$
$269$	17.6192	1.07426	0.537130	+	0.843500i	$0.319509\pi$
$270$	0	0
$271$	−4.85426	−0.294876	−0.147438	−	0.989071i	$-0.547103\pi$
$271$	−4.85426	−0.294876	−0.147438	+	0.989071i	$0.547103\pi$
$272$	0	0
$273$	0.234226i	0.0141760i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.6120i	− 1.35862i	−0.733851	−	0.679311i	$-0.762279\pi$
$277$	− 22.6120i	− 1.35862i	0.733851	−	0.679311i	$-0.237721\pi$
$278$	0	0
$279$	4.10620	0.245832
$280$	0	0
$281$	29.2542	1.74516	0.872580	−	0.488472i	$-0.162445\pi$
$281$	29.2542	1.74516	0.872580	+	0.488472i	$0.162445\pi$
$282$	0	0
$283$	29.2529i	1.73890i	0.494017	+	0.869452i	$0.335528\pi$
$283$	29.2529i	1.73890i	−0.494017	+	0.869452i	$0.664472\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.582028i	− 0.0343560i
$288$	0	0
$289$	12.6389	0.743464
$290$	0	0
$291$	12.8077	0.750803
$292$	0	0
$293$	− 13.4104i	− 0.783447i	−0.920083	−	0.391723i	$-0.871879\pi$
$293$	− 13.4104i	− 0.783447i	0.920083	−	0.391723i	$-0.128121\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.26981i	0.131708i
$298$	0	0
$299$	−12.5697	−0.726923
$300$	0	0
$301$	−0.158522	−0.00913704
$302$	0	0
$303$	2.72537i	0.156569i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.5444i	− 0.773022i	−0.922285	−	0.386511i	$-0.873680\pi$
$307$	− 13.5444i	− 0.773022i	0.922285	−	0.386511i	$-0.126320\pi$
$308$	0	0
$309$	−0.359976	−0.0204783
$310$	0	0
$311$	−2.03882	−0.115611	−0.0578055	−	0.998328i	$-0.518410\pi$
$311$	−2.03882	−0.115611	−0.0578055	+	0.998328i	$0.518410\pi$
$312$	0	0
$313$	− 18.2786i	− 1.03317i	−0.856237	−	0.516583i	$-0.827204\pi$
$313$	− 18.2786i	− 1.03317i	0.856237	−	0.516583i	$-0.172796\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7978i	1.05579i	0.849309	+	0.527896i	$0.177019\pi$
$317$	18.7978i	1.05579i	−0.849309	+	0.527896i	$0.822981\pi$
$318$	0	0
$319$	−8.41272	−0.471022
$320$	0	0
$321$	−0.0286533	−0.00159927
$322$	0	0
$323$	3.67745i	0.204619i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.9217i	− 1.04637i
$328$	0	0
$329$	0.893088	0.0492375
$330$	0	0
$331$	−8.32012	−0.457315	−0.228657	−	0.973507i	$-0.573434\pi$
$331$	−8.32012	−0.457315	−0.228657	+	0.973507i	$0.573434\pi$
$332$	0	0
$333$	7.11909i	0.390124i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 17.5942i	− 0.958417i	−0.877701	−	0.479209i	$-0.840924\pi$
$337$	− 17.5942i	− 0.958417i	0.877701	−	0.479209i	$-0.159076\pi$
$338$	0	0
$339$	4.89380	0.265795
$340$	0	0
$341$	−9.32028	−0.504721
$342$	0	0
$343$	1.23591i	0.0667326i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.8192i	− 1.65446i	−0.561861	−	0.827231i	$-0.689915\pi$
$347$	− 30.8192i	− 1.65446i	0.561861	−	0.827231i	$-0.310085\pi$
$348$	0	0
$349$	−22.0376	−1.17964	−0.589822	−	0.807533i	$-0.700802\pi$
$349$	−22.0376	−1.17964	−0.589822	+	0.807533i	$0.700802\pi$
$350$	0	0
$351$	−2.65177	−0.141541
$352$	0	0
$353$	6.11611i	0.325527i	0.986665	+	0.162764i	$0.0520408\pi$
$353$	6.11611i	0.325527i	−0.986665	+	0.162764i	$0.947959\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.184458i	− 0.00976256i
$358$	0	0
$359$	−14.8107	−0.781680	−0.390840	−	0.920459i	$-0.627815\pi$
$359$	−14.8107	−0.781680	−0.390840	+	0.920459i	$0.627815\pi$
$360$	0	0
$361$	−15.8990	−0.836792
$362$	0	0
$363$	5.84798i	0.306939i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.3115i	− 0.694855i	−0.937707	−	0.347428i	$-0.887055\pi$
$367$	− 13.3115i	− 0.694855i	0.937707	−	0.347428i	$-0.112945\pi$
$368$	0	0
$369$	6.58938	0.343029
$370$	0	0
$371$	0.0849222	0.00440894
$372$	0	0
$373$	− 17.0229i	− 0.881410i	−0.897652	−	0.440705i	$-0.854728\pi$
$373$	− 17.0229i	− 0.881410i	0.897652	−	0.440705i	$-0.145272\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.82843i	− 0.506190i
$378$	0	0
$379$	2.83465	0.145606	0.0728031	−	0.997346i	$-0.476806\pi$
$379$	2.83465	0.145606	0.0728031	+	0.997346i	$0.476806\pi$
$380$	0	0
$381$	3.26997	0.167526
$382$	0	0
$383$	10.0485i	0.513453i	0.966484	+	0.256726i	$0.0826439\pi$
$383$	10.0485i	0.513453i	−0.966484	+	0.256726i	$0.917356\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.79469i	− 0.0912292i
$388$	0	0
$389$	−8.37128	−0.424441	−0.212220	−	0.977222i	$-0.568070\pi$
$389$	−8.37128	−0.424441	−0.212220	+	0.977222i	$0.568070\pi$
$390$	0	0
$391$	9.89889	0.500608
$392$	0	0
$393$	− 3.59550i	− 0.181369i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.38397i	0.420779i	0.977618	+	0.210390i	$0.0674733\pi$
$397$	8.38397i	0.420779i	−0.977618	+	0.210390i	$0.932527\pi$
$398$	0	0
$399$	−0.155542	−0.00778682
$400$	0	0
$401$	2.14450	0.107091	0.0535455	−	0.998565i	$-0.482948\pi$
$401$	2.14450	0.107091	0.0535455	+	0.998565i	$0.482948\pi$
$402$	0	0
$403$	− 10.8887i	− 0.542405i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 16.1589i	− 0.800969i
$408$	0	0
$409$	3.58754	0.177392	0.0886962	−	0.996059i	$-0.471730\pi$
$409$	3.58754	0.177392	0.0886962	+	0.996059i	$0.471730\pi$
$410$	0	0
$411$	−17.0197	−0.839521
$412$	0	0
$413$	0.793011i	0.0390215i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 18.9860i	− 0.929747i
$418$	0	0
$419$	9.20715	0.449799	0.224899	−	0.974382i	$-0.427795\pi$
$419$	9.20715	0.449799	0.224899	+	0.974382i	$0.427795\pi$
$420$	0	0
$421$	−40.9085	−1.99376	−0.996880	−	0.0789366i	$-0.974848\pi$
$421$	−40.9085	−1.99376	−0.996880	+	0.0789366i	$0.974848\pi$
$422$	0	0
$423$	10.1110i	0.491615i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.836130i	0.0404632i
$428$	0	0
$429$	6.01901	0.290601
$430$	0	0
$431$	2.26272	0.108991	0.0544956	−	0.998514i	$-0.482645\pi$
$431$	2.26272	0.108991	0.0544956	+	0.998514i	$0.482645\pi$
$432$	0	0
$433$	− 4.57519i	− 0.219870i	−0.993939	−	0.109935i	$-0.964936\pi$
$433$	− 4.57519i	− 0.219870i	0.993939	−	0.109935i	$-0.0350642\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.34709i	− 0.399295i
$438$	0	0
$439$	1.75299	0.0836655	0.0418328	−	0.999125i	$-0.486680\pi$
$439$	1.75299	0.0836655	0.0418328	+	0.999125i	$0.486680\pi$
$440$	0	0
$441$	−6.99220	−0.332962
$442$	0	0
$443$	20.8364i	0.989967i	0.868902	+	0.494983i	$0.164826\pi$
$443$	20.8364i	0.989967i	−0.868902	+	0.494983i	$0.835174\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.3441i	0.962243i
$448$	0	0
$449$	−25.1952	−1.18904	−0.594518	−	0.804082i	$-0.702657\pi$
$449$	−25.1952	−1.18904	−0.594518	+	0.804082i	$0.702657\pi$
$450$	0	0
$451$	−14.9566	−0.704280
$452$	0	0
$453$	− 13.2609i	− 0.623050i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 39.5166i	− 1.84851i	−0.381775	−	0.924255i	$-0.624687\pi$
$457$	− 39.5166i	− 1.84851i	0.381775	−	0.924255i	$-0.375313\pi$
$458$	0	0
$459$	2.08833	0.0974748
$460$	0	0
$461$	14.5860	0.679337	0.339668	−	0.940545i	$-0.389685\pi$
$461$	14.5860	0.679337	0.339668	+	0.940545i	$0.389685\pi$
$462$	0	0
$463$	9.87311i	0.458842i	0.973327	+	0.229421i	$0.0736833\pi$
$463$	9.87311i	0.458842i	−0.973327	+	0.229421i	$0.926317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.1418i	0.978325i	0.872193	+	0.489162i	$0.162697\pi$
$467$	21.1418i	0.978325i	−0.872193	+	0.489162i	$0.837303\pi$
$468$	0	0
$469$	1.22810	0.0567085
$470$	0	0
$471$	−12.8066	−0.590097
$472$	0	0
$473$	4.07360i	0.187304i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.961440i	0.0440213i
$478$	0	0
$479$	−41.4475	−1.89378	−0.946892	−	0.321551i	$-0.895796\pi$
$479$	−41.4475	−1.89378	−0.946892	+	0.321551i	$0.895796\pi$
$480$	0	0
$481$	18.8782	0.860772
$482$	0	0
$483$	0.418684i	0.0190508i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.03970i	0.183056i	0.995802	+	0.0915281i	$0.0291751\pi$
$487$	4.03970i	0.183056i	−0.995802	+	0.0915281i	$0.970825\pi$
$488$	0	0
$489$	15.0647	0.681247
$490$	0	0
$491$	28.9752	1.30763	0.653816	−	0.756654i	$-0.273167\pi$
$491$	28.9752	1.30763	0.653816	+	0.756654i	$0.273167\pi$
$492$	0	0
$493$	7.74010i	0.348597i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.543019i	0.0243577i
$498$	0	0
$499$	−23.6824	−1.06017	−0.530086	−	0.847944i	$-0.677840\pi$
$499$	−23.6824	−1.06017	−0.530086	+	0.847944i	$0.677840\pi$
$500$	0	0
$501$	3.45128	0.154192
$502$	0	0
$503$	13.3422i	0.594898i	0.954738	+	0.297449i	$0.0961358\pi$
$503$	13.3422i	0.594898i	−0.954738	+	0.297449i	$0.903864\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.96810i	− 0.265053i
$508$	0	0
$509$	10.3709	0.459683	0.229841	−	0.973228i	$-0.426179\pi$
$509$	10.3709	0.459683	0.229841	+	0.973228i	$0.426179\pi$
$510$	0	0
$511$	0.278909	0.0123382
$512$	0	0
$513$	− 1.76095i	− 0.0777479i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 22.9501i	− 1.00934i
$518$	0	0
$519$	−2.41457	−0.105988
$520$	0	0
$521$	29.5976	1.29669	0.648347	−	0.761345i	$-0.275460\pi$
$521$	29.5976	1.29669	0.648347	+	0.761345i	$0.275460\pi$
$522$	0	0
$523$	2.13978i	0.0935658i	0.998905	+	0.0467829i	$0.0148969\pi$
$523$	2.13978i	0.0935658i	−0.998905	+	0.0467829i	$0.985103\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.57509i	0.373537i
$528$	0	0
$529$	0.531447	0.0231064
$530$	0	0
$531$	−8.97801	−0.389612
$532$	0	0
$533$	− 17.4735i	− 0.756863i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.06948i	− 0.175611i
$538$	0	0
$539$	15.8709	0.683610
$540$	0	0
$541$	−8.61207	−0.370262	−0.185131	−	0.982714i	$-0.559271\pi$
$541$	−8.61207	−0.370262	−0.185131	+	0.982714i	$0.559271\pi$
$542$	0	0
$543$	13.3363i	0.572316i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.3085i	− 0.996601i	−0.867004	−	0.498300i	$-0.833958\pi$
$547$	− 23.3085i	− 0.996601i	0.867004	−	0.498300i	$-0.166042\pi$
$548$	0	0
$549$	−9.46618	−0.404007
$550$	0	0
$551$	6.52673	0.278048
$552$	0	0
$553$	− 1.17937i	− 0.0501520i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.2790i	1.57956i	0.613389	+	0.789781i	$0.289806\pi$
$557$	37.2790i	1.57956i	−0.613389	+	0.789781i	$0.710194\pi$
$558$	0	0
$559$	−4.75911	−0.201289
$560$	0	0
$561$	−4.74010	−0.200127
$562$	0	0
$563$	− 10.7486i	− 0.453000i	−0.974011	−	0.226500i	$-0.927272\pi$
$563$	− 10.7486i	− 0.453000i	0.974011	−	0.226500i	$-0.0727283\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0883282i	0.00370943i
$568$	0	0
$569$	−7.54687	−0.316381	−0.158191	−	0.987409i	$-0.550566\pi$
$569$	−7.54687	−0.316381	−0.158191	+	0.987409i	$0.550566\pi$
$570$	0	0
$571$	29.6221	1.23965	0.619824	−	0.784741i	$-0.287204\pi$
$571$	29.6221	1.23965	0.619824	+	0.784741i	$0.287204\pi$
$572$	0	0
$573$	− 25.2529i	− 1.05496i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15.5747i	− 0.648381i	−0.945992	−	0.324191i	$-0.894908\pi$
$577$	− 15.5747i	− 0.648381i	0.945992	−	0.324191i	$-0.105092\pi$
$578$	0	0
$579$	−24.6399	−1.02400
$580$	0	0
$581$	1.39731	0.0579700
$582$	0	0
$583$	− 2.18228i	− 0.0903809i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 39.0913i	− 1.61347i	−0.590913	−	0.806735i	$-0.701232\pi$
$587$	− 39.0913i	− 1.61347i	0.590913	−	0.806735i	$-0.298768\pi$
$588$	0	0
$589$	7.23082	0.297941
$590$	0	0
$591$	−18.6201	−0.765930
$592$	0	0
$593$	− 18.6722i	− 0.766775i	−0.923588	−	0.383387i	$-0.874757\pi$
$593$	− 18.6722i	− 0.766775i	0.923588	−	0.383387i	$-0.125243\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.85708i	0.403424i
$598$	0	0
$599$	14.0186	0.572783	0.286392	−	0.958113i	$-0.407544\pi$
$599$	14.0186	0.572783	0.286392	+	0.958113i	$0.407544\pi$
$600$	0	0
$601$	−9.89791	−0.403744	−0.201872	−	0.979412i	$-0.564703\pi$
$601$	−9.89791	−0.403744	−0.201872	+	0.979412i	$0.564703\pi$
$602$	0	0
$603$	13.9039i	0.566209i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.2953i	0.904939i	0.891780	+	0.452469i	$0.149457\pi$
$607$	22.2953i	0.904939i	−0.891780	+	0.452469i	$0.850543\pi$
$608$	0	0
$609$	−0.327376	−0.0132660
$610$	0	0
$611$	26.8121	1.08470
$612$	0	0
$613$	− 16.4288i	− 0.663551i	−0.943358	−	0.331776i	$-0.892352\pi$
$613$	− 16.4288i	− 0.663551i	0.943358	−	0.331776i	$-0.107648\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.78516i	0.152385i	0.997093	+	0.0761924i	$0.0242763\pi$
$617$	3.78516i	0.152385i	−0.997093	+	0.0761924i	$0.975724\pi$
$618$	0	0
$619$	0.422090	0.0169652	0.00848262	−	0.999964i	$-0.497300\pi$
$619$	0.422090	0.0169652	0.00848262	+	0.999964i	$0.497300\pi$
$620$	0	0
$621$	−4.74010	−0.190214
$622$	0	0
$623$	0.905434i	0.0362755i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.99702i	0.159626i
$628$	0	0
$629$	−14.8670	−0.592786
$630$	0	0
$631$	−34.9223	−1.39023	−0.695117	−	0.718897i	$-0.744647\pi$
$631$	−34.9223	−1.39023	−0.695117	+	0.718897i	$0.744647\pi$
$632$	0	0
$633$	7.89878i	0.313949i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.5417i	0.734650i
$638$	0	0
$639$	−6.14774	−0.243201
$640$	0	0
$641$	3.65274	0.144275	0.0721373	−	0.997395i	$-0.477018\pi$
$641$	3.65274	0.144275	0.0721373	+	0.997395i	$0.477018\pi$
$642$	0	0
$643$	34.3967i	1.35647i	0.734844	+	0.678236i	$0.237255\pi$
$643$	34.3967i	1.35647i	−0.734844	+	0.678236i	$0.762745\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 31.4429i	− 1.23615i	−0.786120	−	0.618074i	$-0.787913\pi$
$647$	− 31.4429i	− 1.23615i	0.786120	−	0.618074i	$-0.212087\pi$
$648$	0	0
$649$	20.3783	0.799920
$650$	0	0
$651$	−0.362693	−0.0142151
$652$	0	0
$653$	− 34.8800i	− 1.36496i	−0.730904	−	0.682481i	$-0.760901\pi$
$653$	− 34.8800i	− 1.36496i	0.730904	−	0.682481i	$-0.239099\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.15765i	0.123192i
$658$	0	0
$659$	43.2173	1.68351	0.841754	−	0.539862i	$-0.181523\pi$
$659$	43.2173	1.68351	0.841754	+	0.539862i	$0.181523\pi$
$660$	0	0
$661$	−10.0354	−0.390332	−0.195166	−	0.980770i	$-0.562525\pi$
$661$	−10.0354	−0.390332	−0.195166	+	0.980770i	$0.562525\pi$
$662$	0	0
$663$	− 5.53777i	− 0.215069i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 17.5685i	− 0.680256i
$668$	0	0
$669$	−9.18174	−0.354987
$670$	0	0
$671$	21.4864	0.829473
$672$	0	0
$673$	29.8864i	1.15204i	0.817437	+	0.576019i	$0.195394\pi$
$673$	29.8864i	1.15204i	−0.817437	+	0.576019i	$0.804606\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.6279i	0.869662i	0.900512	+	0.434831i	$0.143192\pi$
$677$	22.6279i	0.869662i	−0.900512	+	0.434831i	$0.856808\pi$
$678$	0	0
$679$	−1.13128	−0.0434147
$680$	0	0
$681$	−11.5654	−0.443189
$682$	0	0
$683$	25.6607i	0.981880i	0.871194	+	0.490940i	$0.163346\pi$
$683$	25.6607i	0.981880i	−0.871194	+	0.490940i	$0.836654\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	24.9463i	0.951760i
$688$	0	0
$689$	2.54952	0.0971290
$690$	0	0
$691$	−34.2631	−1.30343	−0.651716	−	0.758463i	$-0.725950\pi$
$691$	−34.2631	−1.30343	−0.651716	+	0.758463i	$0.725950\pi$
$692$	0	0
$693$	− 0.200488i	− 0.00761590i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.7608i	0.521227i
$698$	0	0
$699$	13.0200	0.492461
$700$	0	0
$701$	42.0813	1.58939	0.794695	−	0.607009i	$-0.207631\pi$
$701$	42.0813	1.58939	0.794695	+	0.607009i	$0.207631\pi$
$702$	0	0
$703$	12.5364i	0.472818i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.240727i	− 0.00905347i
$708$	0	0
$709$	20.7728	0.780139	0.390069	−	0.920785i	$-0.372451\pi$
$709$	20.7728	0.780139	0.390069	+	0.920785i	$0.372451\pi$
$710$	0	0
$711$	13.3522	0.500746
$712$	0	0
$713$	− 19.4638i	− 0.728925i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 26.4086i	− 0.986247i
$718$	0	0
$719$	−25.7878	−0.961721	−0.480860	−	0.876797i	$-0.659676\pi$
$719$	−25.7878	−0.961721	−0.480860	+	0.876797i	$0.659676\pi$
$720$	0	0
$721$	0.0317960	0.00118415
$722$	0	0
$723$	6.23591i	0.231916i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.26839i	0.343746i	0.985119	+	0.171873i	$0.0549818\pi$
$727$	9.26839i	0.343746i	−0.985119	+	0.171873i	$0.945018\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	3.74790	0.138621
$732$	0	0
$733$	− 21.0388i	− 0.777086i	−0.921431	−	0.388543i	$-0.872978\pi$
$733$	− 21.0388i	− 0.777086i	0.921431	−	0.388543i	$-0.127022\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31.5591i	− 1.16249i
$738$	0	0
$739$	8.85805	0.325849	0.162924	−	0.986639i	$-0.447907\pi$
$739$	8.85805	0.325849	0.162924	+	0.986639i	$0.447907\pi$
$740$	0	0
$741$	−4.66964	−0.171544
$742$	0	0
$743$	13.9773i	0.512778i	0.966574	+	0.256389i	$0.0825328\pi$
$743$	13.9773i	0.512778i	−0.966574	+	0.256389i	$0.917467\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.8195i	0.578805i
$748$	0	0
$749$	0.00253090	9.24769e−5 0
$750$	0	0
$751$	−32.8762	−1.19967	−0.599834	−	0.800124i	$-0.704767\pi$
$751$	−32.8762	−1.19967	−0.599834	+	0.800124i	$0.704767\pi$
$752$	0	0
$753$	− 7.46802i	− 0.272150i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.4556i	1.14327i	0.820507	+	0.571636i	$0.193691\pi$
$757$	31.4556i	1.14327i	−0.820507	+	0.571636i	$0.806309\pi$
$758$	0	0
$759$	10.7591	0.390531
$760$	0	0
$761$	37.8792	1.37312	0.686559	−	0.727074i	$-0.259120\pi$
$761$	37.8792	1.37312	0.686559	+	0.727074i	$0.259120\pi$
$762$	0	0
$763$	1.67132i	0.0605059i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.8076i	0.859644i
$768$	0	0
$769$	−14.1944	−0.511862	−0.255931	−	0.966695i	$-0.582382\pi$
$769$	−14.1944	−0.511862	−0.255931	+	0.966695i	$0.582382\pi$
$770$	0	0
$771$	−8.39880	−0.302475
$772$	0	0
$773$	− 36.3783i	− 1.30844i	−0.756306	−	0.654218i	$-0.772998\pi$
$773$	− 36.3783i	− 1.30844i	0.756306	−	0.654218i	$-0.227002\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.628816i	− 0.0225586i
$778$	0	0
$779$	11.6036	0.415742
$780$	0	0
$781$	13.9542	0.499320
$782$	0	0
$783$	− 3.70636i	− 0.132455i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.4472i	1.22791i	0.789341	+	0.613954i	$0.210422\pi$
$787$	34.4472i	1.22791i	−0.789341	+	0.613954i	$0.789578\pi$
$788$	0	0
$789$	−10.0248	−0.356892
$790$	0	0
$791$	−0.432260	−0.0153694
$792$	0	0
$793$	25.1021i	0.891403i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36.9018i	− 1.30713i	−0.756872	−	0.653564i	$-0.773273\pi$
$797$	− 36.9018i	− 1.30713i	0.756872	−	0.653564i	$-0.226727\pi$
$798$	0	0
$799$	−21.1151	−0.747000
$800$	0	0
$801$	−10.2508	−0.362194
$802$	0	0
$803$	− 7.16725i	− 0.252927i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 17.6192i	− 0.620224i
$808$	0	0
$809$	−36.1037	−1.26934	−0.634670	−	0.772783i	$-0.718864\pi$
$809$	−36.1037	−1.26934	−0.634670	+	0.772783i	$0.718864\pi$
$810$	0	0
$811$	13.0666	0.458830	0.229415	−	0.973329i	$-0.426319\pi$
$811$	13.0666	0.458830	0.229415	+	0.973329i	$0.426319\pi$
$812$	0	0
$813$	4.85426i	0.170246i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.16036i	− 0.110567i
$818$	0	0
$819$	0.234226	0.00818453
$820$	0	0
$821$	−25.8859	−0.903424	−0.451712	−	0.892164i	$-0.649187\pi$
$821$	−25.8859	−0.903424	−0.451712	+	0.892164i	$0.649187\pi$
$822$	0	0
$823$	39.1757i	1.36558i	0.730615	+	0.682789i	$0.239233\pi$
$823$	39.1757i	1.36558i	−0.730615	+	0.682789i	$0.760767\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.4245i	1.57956i	0.613388	+	0.789782i	$0.289806\pi$
$827$	45.4245i	1.57956i	−0.613388	+	0.789782i	$0.710194\pi$
$828$	0	0
$829$	−16.1544	−0.561065	−0.280533	−	0.959845i	$-0.590511\pi$
$829$	−16.1544	−0.561065	−0.280533	+	0.959845i	$0.590511\pi$
$830$	0	0
$831$	−22.6120	−0.784401
$832$	0	0
$833$	− 14.6020i	− 0.505929i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.10620i	− 0.141931i
$838$	0	0
$839$	−10.4813	−0.361856	−0.180928	−	0.983496i	$-0.557910\pi$
$839$	−10.4813	−0.361856	−0.180928	+	0.983496i	$0.557910\pi$
$840$	0	0
$841$	−15.2629	−0.526306
$842$	0	0
$843$	− 29.2542i	− 1.00757i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.516541i	− 0.0177486i
$848$	0	0
$849$	29.2529	1.00396
$850$	0	0
$851$	33.7452	1.15677
$852$	0	0
$853$	− 38.2211i	− 1.30867i	−0.756207	−	0.654333i	$-0.772950\pi$
$853$	− 38.2211i	− 1.30867i	0.756207	−	0.654333i	$-0.227050\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.3637i	− 0.866408i	−0.901296	−	0.433204i	$-0.857383\pi$
$857$	− 25.3637i	− 0.866408i	0.901296	−	0.433204i	$-0.142617\pi$
$858$	0	0
$859$	−35.7717	−1.22051	−0.610257	−	0.792204i	$-0.708934\pi$
$859$	−35.7717	−1.22051	−0.610257	+	0.792204i	$0.708934\pi$
$860$	0	0
$861$	−0.582028	−0.0198355
$862$	0	0
$863$	44.2905i	1.50767i	0.657066	+	0.753833i	$0.271797\pi$
$863$	44.2905i	1.50767i	−0.657066	+	0.753833i	$0.728203\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 12.6389i	− 0.429239i
$868$	0	0
$869$	−30.3069	−1.02809
$870$	0	0
$871$	36.8699	1.24929
$872$	0	0
$873$	− 12.8077i	− 0.433476i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 9.80714i	− 0.331164i	−0.986196	−	0.165582i	$-0.947050\pi$
$877$	− 9.80714i	− 0.331164i	0.986196	−	0.165582i	$-0.0529502\pi$
$878$	0	0
$879$	−13.4104	−0.452323
$880$	0	0
$881$	−0.982291	−0.0330942	−0.0165471	−	0.999863i	$-0.505267\pi$
$881$	−0.982291	−0.0330942	−0.0165471	+	0.999863i	$0.505267\pi$
$882$	0	0
$883$	− 16.1321i	− 0.542890i	−0.962454	−	0.271445i	$-0.912499\pi$
$883$	− 16.1321i	− 0.542890i	0.962454	−	0.271445i	$-0.0875014\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 50.4066i	− 1.69249i	−0.532796	−	0.846244i	$-0.678859\pi$
$887$	− 50.4066i	− 1.69249i	0.532796	−	0.846244i	$-0.321141\pi$
$888$	0	0
$889$	−0.288830	−0.00968706
$890$	0	0
$891$	2.26981	0.0760414
$892$	0	0
$893$	17.8050i	0.595822i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.5697i	0.419689i
$898$	0	0
$899$	15.2191	0.507584
$900$	0	0
$901$	−2.00780	−0.0668896
$902$	0	0
$903$	0.158522i	0.00527527i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.1281i	0.867570i	0.901016	+	0.433785i	$0.142822\pi$
$907$	26.1281i	0.867570i	−0.901016	+	0.433785i	$0.857178\pi$
$908$	0	0
$909$	2.72537	0.0903949
$910$	0	0
$911$	−31.4435	−1.04177	−0.520886	−	0.853627i	$-0.674398\pi$
$911$	−31.4435	−1.04177	−0.520886	+	0.853627i	$0.674398\pi$
$912$	0	0
$913$	− 35.9072i	− 1.18835i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.317584i	0.0104876i
$918$	0	0
$919$	49.1184	1.62027	0.810133	−	0.586246i	$-0.199395\pi$
$919$	49.1184	1.62027	0.810133	+	0.586246i	$0.199395\pi$
$920$	0	0
$921$	−13.5444	−0.446304
$922$	0	0
$923$	16.3024i	0.536600i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.359976i	0.0118232i
$928$	0	0
$929$	−16.0095	−0.525256	−0.262628	−	0.964897i	$-0.584589\pi$
$929$	−16.0095	−0.525256	−0.262628	+	0.964897i	$0.584589\pi$
$930$	0	0
$931$	−12.3129	−0.403540
$932$	0	0
$933$	2.03882i	0.0667481i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.2670i	0.988779i	0.869240	+	0.494390i	$0.164608\pi$
$937$	30.2670i	0.988779i	−0.869240	+	0.494390i	$0.835392\pi$
$938$	0	0
$939$	−18.2786	−0.596499
$940$	0	0
$941$	−12.3112	−0.401333	−0.200666	−	0.979660i	$-0.564311\pi$
$941$	−12.3112	−0.401333	−0.200666	+	0.979660i	$0.564311\pi$
$942$	0	0
$943$	− 31.2343i	− 1.01713i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.5844i	1.31882i	0.751785	+	0.659409i	$0.229193\pi$
$947$	40.5844i	1.31882i	−0.751785	+	0.659409i	$0.770807\pi$
$948$	0	0
$949$	8.37336	0.271811
$950$	0	0
$951$	18.7978	0.609562
$952$	0	0
$953$	− 8.03153i	− 0.260167i	−0.991503	−	0.130083i	$-0.958475\pi$
$953$	− 8.03153i	− 0.260167i	0.991503	−	0.130083i	$-0.0415245\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.41272i	0.271945i
$958$	0	0
$959$	1.50332	0.0485447
$960$	0	0
$961$	−14.1391	−0.456101
$962$	0	0
$963$	0.0286533i	0 0.000923340i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 26.9707i	− 0.867320i	−0.901076	−	0.433660i	$-0.857222\pi$
$967$	− 26.9707i	− 0.867320i	0.901076	−	0.433660i	$-0.142778\pi$
$968$	0	0
$969$	3.67745	0.118137
$970$	0	0
$971$	29.7540	0.954850	0.477425	−	0.878673i	$-0.341570\pi$
$971$	29.7540	0.954850	0.477425	+	0.878673i	$0.341570\pi$
$972$	0	0
$973$	1.67700i	0.0537620i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.42199i	0.0454935i	0.999741	+	0.0227467i	$0.00724114\pi$
$977$	1.42199i	0.0454935i	−0.999741	+	0.0227467i	$0.992759\pi$
$978$	0	0
$979$	23.2673	0.743627
$980$	0	0
$981$	−18.9217	−0.604125
$982$	0	0
$983$	2.02962i	0.0647348i	0.999476	+	0.0323674i	$0.0103047\pi$
$983$	2.02962i	0.0647348i	−0.999476	+	0.0323674i	$0.989695\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.893088i	− 0.0284273i
$988$	0	0
$989$	−8.50701	−0.270507
$990$	0	0
$991$	20.2314	0.642672	0.321336	−	0.946965i	$-0.395868\pi$
$991$	20.2314	0.642672	0.321336	+	0.946965i	$0.395868\pi$
$992$	0	0
$993$	8.32012i	0.264031i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.3001i	0.579571i	0.957092	+	0.289786i	$0.0935840\pi$
$997$	18.3001i	0.579571i	−0.957092	+	0.289786i	$0.906416\pi$
$998$	0	0
$999$	7.11909	0.225238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.c.1249.3		8
5.2	odd	4		7500.2.a.e.1.2		4
5.3	odd	4		7500.2.a.f.1.3		4
5.4	even	2	inner	7500.2.d.c.1249.6		8
25.2	odd	20		300.2.m.b.121.2	✓	8
25.9	even	10		1500.2.o.b.349.1		16
25.11	even	5		1500.2.o.b.649.2		16
25.12	odd	20		300.2.m.b.181.2	yes	8
25.13	odd	20		1500.2.m.a.901.2		8
25.14	even	10		1500.2.o.b.649.3		16
25.16	even	5		1500.2.o.b.349.4		16
25.23	odd	20		1500.2.m.a.601.2		8
75.2	even	20		900.2.n.b.721.1		8
75.62	even	20		900.2.n.b.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.121.2	✓	8	25.2	odd	20
300.2.m.b.181.2	yes	8	25.12	odd	20
900.2.n.b.181.1		8	75.62	even	20
900.2.n.b.721.1		8	75.2	even	20
1500.2.m.a.601.2		8	25.23	odd	20
1500.2.m.a.901.2		8	25.13	odd	20
1500.2.o.b.349.1		16	25.9	even	10
1500.2.o.b.349.4		16	25.16	even	5
1500.2.o.b.649.2		16	25.11	even	5
1500.2.o.b.649.3		16	25.14	even	10
7500.2.a.e.1.2		4	5.2	odd	4
7500.2.a.f.1.3		4	5.3	odd	4
7500.2.d.c.1249.3		8	1.1	even	1	trivial
7500.2.d.c.1249.6		8	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.0883282i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.0883282i q^{7} -1.00000 q^{9} +2.26981 q^{11} +2.65177i q^{13} -2.08833i q^{17} -1.76095 q^{19} +0.0883282 q^{21} +4.74010i q^{23} +1.00000i q^{27} -3.70636 q^{29} -4.10620 q^{31} -2.26981i q^{33} -7.11909i q^{37} +2.65177 q^{39} -6.58938 q^{41} +1.79469i q^{43} -10.1110i q^{47} +6.99220 q^{49} -2.08833 q^{51} -0.961440i q^{53} +1.76095i q^{57} +8.97801 q^{59} +9.46618 q^{61} -0.0883282i q^{63} -13.9039i q^{67} +4.74010 q^{69} +6.14774 q^{71} -3.15765i q^{73} +0.200488i q^{77} -13.3522 q^{79} +1.00000 q^{81} -15.8195i q^{83} +3.70636i q^{87} +10.2508 q^{89} -0.234226 q^{91} +4.10620i q^{93} +12.8077i q^{97} -2.26981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

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