LMFDB - Embedded newform 7500.2.d.c.1249.8

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			1249.8
Root			$2.70636i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.c.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.32440i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +4.32440i q^{7} -1.00000 q^{9} +0.584296 q^{11} -0.966262i q^{13} -2.32440i q^{17} +5.37899 q^{19} -4.32440 q^{21} +1.35813i q^{23} -1.00000i q^{27} +0.706362 q^{29} +8.48817 q^{31} +0.584296i q^{33} -6.11909i q^{37} +0.966262 q^{39} +11.0615 q^{41} +7.03076i q^{43} +9.06932i q^{47} -11.7004 q^{49} +2.32440 q^{51} -8.90571i q^{53} +5.37899i q^{57} +7.29250 q^{59} -4.81370 q^{61} -4.32440i q^{63} -0.376006i q^{67} -1.35813 q^{69} +10.5605 q^{71} -0.213375i q^{73} +2.52673i q^{77} +7.02594 q^{79} +1.00000 q^{81} -16.1113i q^{83} +0.706362i q^{87} +14.0197 q^{89} +4.17850 q^{91} +8.48817i q^{93} +15.7520i q^{97} -0.584296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.32440i	1.63447i	0.576306	+	0.817234i	$0.304494\pi$
$7$	4.32440i	1.63447i	−0.576306	+	0.817234i	$0.695506\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.584296	0.176172	0.0880859	−	0.996113i	$-0.471925\pi$
$11$	0.584296	0.176172	0.0880859	+	0.996113i	$0.471925\pi$
$12$	0	0
$13$	− 0.966262i	− 0.267993i	−0.990982	−	0.133996i	$-0.957219\pi$
$13$	− 0.966262i	− 0.267993i	0.990982	−	0.133996i	$-0.0427811\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.32440i	− 0.563749i	−0.959451	−	0.281874i	$-0.909044\pi$
$17$	− 2.32440i	− 0.563749i	0.959451	−	0.281874i	$-0.0909562\pi$
$18$	0	0
$19$	5.37899	1.23402	0.617012	−	0.786954i	$-0.288343\pi$
$19$	5.37899	1.23402	0.617012	+	0.786954i	$0.288343\pi$
$20$	0	0
$21$	−4.32440	−0.943661
$22$	0	0
$23$	1.35813i	0.283191i	0.989925	+	0.141595i	$0.0452232\pi$
$23$	1.35813i	0.283191i	−0.989925	+	0.141595i	$0.954777\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	0.706362	0.131168	0.0655841	−	0.997847i	$-0.479109\pi$
$29$	0.706362	0.131168	0.0655841	+	0.997847i	$0.479109\pi$
$30$	0	0
$31$	8.48817	1.52452	0.762260	−	0.647271i	$-0.224090\pi$
$31$	8.48817	1.52452	0.762260	+	0.647271i	$0.224090\pi$
$32$	0	0
$33$	0.584296i	0.101713i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.11909i	− 1.00597i	−0.864295	−	0.502986i	$-0.832235\pi$
$37$	− 6.11909i	− 1.00597i	0.864295	−	0.502986i	$-0.167765\pi$
$38$	0	0
$39$	0.966262	0.154726
$40$	0	0
$41$	11.0615	1.72752	0.863759	−	0.503905i	$-0.168104\pi$
$41$	11.0615	1.72752	0.863759	+	0.503905i	$0.168104\pi$
$42$	0	0
$43$	7.03076i	1.07218i	0.844161	+	0.536090i	$0.180099\pi$
$43$	7.03076i	1.07218i	−0.844161	+	0.536090i	$0.819901\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.06932i	1.32290i	0.749991	+	0.661448i	$0.230058\pi$
$47$	9.06932i	1.32290i	−0.749991	+	0.661448i	$0.769942\pi$
$48$	0	0
$49$	−11.7004	−1.67149
$50$	0	0
$51$	2.32440	0.325481
$52$	0	0
$53$	− 8.90571i	− 1.22329i	−0.791131	−	0.611647i	$-0.790507\pi$
$53$	− 8.90571i	− 1.22329i	0.791131	−	0.611647i	$-0.209493\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.37899i	0.712464i
$58$	0	0
$59$	7.29250	0.949403	0.474701	−	0.880147i	$-0.342556\pi$
$59$	7.29250	0.949403	0.474701	+	0.880147i	$0.342556\pi$
$60$	0	0
$61$	−4.81370	−0.616331	−0.308166	−	0.951333i	$-0.599715\pi$
$61$	−4.81370	−0.616331	−0.308166	+	0.951333i	$0.599715\pi$
$62$	0	0
$63$	− 4.32440i	− 0.544823i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.376006i	− 0.0459364i	−0.999736	−	0.0229682i	$-0.992688\pi$
$67$	− 0.376006i	− 0.0459364i	0.999736	−	0.0229682i	$-0.00731165\pi$
$68$	0	0
$69$	−1.35813	−0.163500
$70$	0	0
$71$	10.5605	1.25330	0.626648	−	0.779302i	$-0.284426\pi$
$71$	10.5605	1.25330	0.626648	+	0.779302i	$0.284426\pi$
$72$	0	0
$73$	− 0.213375i	− 0.0249736i	−0.999922	−	0.0124868i	$-0.996025\pi$
$73$	− 0.213375i	− 0.0249736i	0.999922	−	0.0124868i	$-0.00397478\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.52673i	0.287947i
$78$	0	0
$79$	7.02594	0.790480	0.395240	−	0.918578i	$-0.370661\pi$
$79$	7.02594	0.790480	0.395240	+	0.918578i	$0.370661\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.1113i	− 1.76844i	−0.467068	−	0.884222i	$-0.654690\pi$
$83$	− 16.1113i	− 1.76844i	0.467068	−	0.884222i	$-0.345310\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.706362i	0.0757300i
$88$	0	0
$89$	14.0197	1.48609	0.743043	−	0.669243i	$-0.233382\pi$
$89$	14.0197	1.48609	0.743043	+	0.669243i	$0.233382\pi$
$90$	0	0
$91$	4.17850	0.438026
$92$	0	0
$93$	8.48817i	0.880182i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.7520i	1.59937i	0.600417	+	0.799687i	$0.295001\pi$
$97$	15.7520i	1.59937i	−0.600417	+	0.799687i	$0.704999\pi$
$98$	0	0
$99$	−0.584296	−0.0587239
$100$	0	0
$101$	7.14178	0.710634	0.355317	−	0.934746i	$-0.384373\pi$
$101$	7.14178	0.710634	0.355317	+	0.934746i	$0.384373\pi$
$102$	0	0
$103$	− 1.32553i	− 0.130609i	−0.997865	−	0.0653044i	$-0.979198\pi$
$103$	− 1.32553i	− 0.130609i	0.997865	−	0.0653044i	$-0.0208018\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.6796i	1.70915i	0.519331	+	0.854573i	$0.326181\pi$
$107$	17.6796i	1.70915i	−0.519331	+	0.854573i	$0.673819\pi$
$108$	0	0
$109$	−3.53978	−0.339049	−0.169524	−	0.985526i	$-0.554223\pi$
$109$	−3.53978	−0.339049	−0.169524	+	0.985526i	$0.554223\pi$
$110$	0	0
$111$	6.11909	0.580798
$112$	0	0
$113$	− 17.4882i	− 1.64515i	−0.568658	−	0.822574i	$-0.692537\pi$
$113$	− 17.4882i	− 1.64515i	0.568658	−	0.822574i	$-0.307463\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.966262i	0.0893309i
$118$	0	0
$119$	10.0516	0.921430
$120$	0	0
$121$	−10.6586	−0.968964
$122$	0	0
$123$	11.0615i	0.997383i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.5060i	1.55341i	0.629865	+	0.776705i	$0.283110\pi$
$127$	17.5060i	1.55341i	−0.629865	+	0.776705i	$0.716890\pi$
$128$	0	0
$129$	−7.03076	−0.619024
$130$	0	0
$131$	−18.8660	−1.64833	−0.824166	−	0.566349i	$-0.808355\pi$
$131$	−18.8660	−1.64833	−0.824166	+	0.566349i	$0.808355\pi$
$132$	0	0
$133$	23.2609i	2.01697i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.2508i	1.13209i	0.824374	+	0.566046i	$0.191528\pi$
$137$	13.2508i	1.13209i	−0.824374	+	0.566046i	$0.808472\pi$
$138$	0	0
$139$	16.9026	1.43366	0.716829	−	0.697249i	$-0.245593\pi$
$139$	16.9026	1.43366	0.716829	+	0.697249i	$0.245593\pi$
$140$	0	0
$141$	−9.06932	−0.763774
$142$	0	0
$143$	− 0.564582i	− 0.0472128i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.7004i	− 0.965033i
$148$	0	0
$149$	−12.1625	−0.996388	−0.498194	−	0.867066i	$-0.666003\pi$
$149$	−12.1625	−0.996388	−0.498194	+	0.867066i	$0.666003\pi$
$150$	0	0
$151$	−9.84446	−0.801131	−0.400565	−	0.916268i	$-0.631186\pi$
$151$	−9.84446	−0.801131	−0.400565	+	0.916268i	$0.631186\pi$
$152$	0	0
$153$	2.32440i	0.187916i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.4804i	− 1.47489i	−0.675405	−	0.737447i	$-0.736031\pi$
$157$	− 18.4804i	− 1.47489i	0.675405	−	0.737447i	$-0.263969\pi$
$158$	0	0
$159$	8.90571	0.706269
$160$	0	0
$161$	−5.87311	−0.462866
$162$	0	0
$163$	14.5368i	1.13861i	0.822127	+	0.569305i	$0.192788\pi$
$163$	14.5368i	1.13861i	−0.822127	+	0.569305i	$0.807212\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.49299i	− 0.347678i	−0.984774	−	0.173839i	$-0.944383\pi$
$167$	− 4.49299i	− 0.347678i	0.984774	−	0.173839i	$-0.0556172\pi$
$168$	0	0
$169$	12.0663	0.928180
$170$	0	0
$171$	−5.37899	−0.411341
$172$	0	0
$173$	− 1.99816i	− 0.151917i	−0.997111	−	0.0759586i	$-0.975798\pi$
$173$	− 1.99816i	− 0.151917i	0.997111	−	0.0759586i	$-0.0242017\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.29250i	0.548138i
$178$	0	0
$179$	−13.9793	−1.04486	−0.522431	−	0.852681i	$-0.674975\pi$
$179$	−13.9793	−1.04486	−0.522431	+	0.852681i	$0.674975\pi$
$180$	0	0
$181$	13.5379	1.00627	0.503133	−	0.864209i	$-0.332180\pi$
$181$	13.5379	1.00627	0.503133	+	0.864209i	$0.332180\pi$
$182$	0	0
$183$	− 4.81370i	− 0.355839i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.35813i	− 0.0993166i
$188$	0	0
$189$	4.32440	0.314554
$190$	0	0
$191$	2.54547	0.184184	0.0920920	−	0.995751i	$-0.470645\pi$
$191$	2.54547	0.184184	0.0920920	+	0.995751i	$0.470645\pi$
$192$	0	0
$193$	− 5.60541i	− 0.403486i	−0.979438	−	0.201743i	$-0.935339\pi$
$193$	− 5.60541i	− 0.403486i	0.979438	−	0.201743i	$-0.0646606\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.85621i	− 0.559732i	−0.960039	−	0.279866i	$-0.909710\pi$
$197$	− 7.85621i	− 0.559732i	0.960039	−	0.279866i	$-0.0902900\pi$
$198$	0	0
$199$	−16.9970	−1.20489	−0.602443	−	0.798162i	$-0.705806\pi$
$199$	−16.9970	−1.20489	−0.602443	+	0.798162i	$0.705806\pi$
$200$	0	0
$201$	0.376006	0.0265214
$202$	0	0
$203$	3.05459i	0.214390i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.35813i	− 0.0943969i
$208$	0	0
$209$	3.14292	0.217400
$210$	0	0
$211$	−12.9553	−0.891881	−0.445940	−	0.895063i	$-0.647131\pi$
$211$	−12.9553	−0.891881	−0.445940	+	0.895063i	$0.647131\pi$
$212$	0	0
$213$	10.5605i	0.723591i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	36.7062i	2.49178i
$218$	0	0
$219$	0.213375	0.0144185
$220$	0	0
$221$	−2.24597	−0.151081
$222$	0	0
$223$	− 18.9801i	− 1.27100i	−0.772100	−	0.635501i	$-0.780793\pi$
$223$	− 18.9801i	− 1.27100i	0.772100	−	0.635501i	$-0.219207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 0.385109i	− 0.0255606i	−0.999918	−	0.0127803i	$-0.995932\pi$
$227$	− 0.385109i	− 0.0255606i	0.999918	−	0.0127803i	$-0.00406820\pi$
$228$	0	0
$229$	−10.2685	−0.678562	−0.339281	−	0.940685i	$-0.610184\pi$
$229$	−10.2685	−0.678562	−0.339281	+	0.940685i	$0.610184\pi$
$230$	0	0
$231$	−2.52673	−0.166246
$232$	0	0
$233$	21.6380i	1.41755i	0.705433	+	0.708777i	$0.250753\pi$
$233$	21.6380i	1.41755i	−0.705433	+	0.708777i	$0.749247\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.02594i	0.456384i
$238$	0	0
$239$	7.71601	0.499107	0.249553	−	0.968361i	$-0.419716\pi$
$239$	7.71601	0.499107	0.249553	+	0.968361i	$0.419716\pi$
$240$	0	0
$241$	−25.3264	−1.63142	−0.815708	−	0.578463i	$-0.803653\pi$
$241$	−25.3264	−1.63142	−0.815708	+	0.578463i	$0.803653\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.19751i	− 0.330710i
$248$	0	0
$249$	16.1113	1.02101
$250$	0	0
$251$	−2.39913	−0.151432	−0.0757160	−	0.997129i	$-0.524124\pi$
$251$	−2.39913	−0.151432	−0.0757160	+	0.997129i	$0.524124\pi$
$252$	0	0
$253$	0.793552i	0.0498902i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.3086i	− 0.892548i	−0.894896	−	0.446274i	$-0.852751\pi$
$257$	− 14.3086i	− 0.892548i	0.894896	−	0.446274i	$-0.147249\pi$
$258$	0	0
$259$	26.4614	1.64423
$260$	0	0
$261$	−0.706362	−0.0437227
$262$	0	0
$263$	− 13.0805i	− 0.806580i	−0.915072	−	0.403290i	$-0.867867\pi$
$263$	− 13.0805i	− 0.806580i	0.915072	−	0.403290i	$-0.132133\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.0197i	0.857993i
$268$	0	0
$269$	−30.6749	−1.87028	−0.935141	−	0.354277i	$-0.884727\pi$
$269$	−30.6749	−1.87028	−0.935141	+	0.354277i	$0.884727\pi$
$270$	0	0
$271$	14.2362	0.864789	0.432395	−	0.901684i	$-0.357669\pi$
$271$	14.2362	0.864789	0.432395	+	0.901684i	$0.357669\pi$
$272$	0	0
$273$	4.17850i	0.252894i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.1307i	1.20953i	0.796402	+	0.604767i	$0.206734\pi$
$277$	20.1307i	1.20953i	−0.796402	+	0.604767i	$0.793266\pi$
$278$	0	0
$279$	−8.48817	−0.508173
$280$	0	0
$281$	−9.81651	−0.585604	−0.292802	−	0.956173i	$-0.594588\pi$
$281$	−9.81651	−0.585604	−0.292802	+	0.956173i	$0.594588\pi$
$282$	0	0
$283$	− 6.54547i	− 0.389088i	−0.980894	−	0.194544i	$-0.937677\pi$
$283$	− 6.54547i	− 0.389088i	0.980894	−	0.194544i	$-0.0623227\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	47.8344i	2.82357i
$288$	0	0
$289$	11.5972	0.682187
$290$	0	0
$291$	−15.7520	−0.923399
$292$	0	0
$293$	− 0.869428i	− 0.0507925i	−0.999677	−	0.0253963i	$-0.991915\pi$
$293$	− 0.869428i	− 0.0507925i	0.999677	−	0.0253963i	$-0.00808475\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 0.584296i	− 0.0339043i
$298$	0	0
$299$	1.31231	0.0758930
$300$	0	0
$301$	−30.4038	−1.75244
$302$	0	0
$303$	7.14178i	0.410285i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	21.7261i	1.23997i	0.784613	+	0.619986i	$0.212862\pi$
$307$	21.7261i	1.23997i	−0.784613	+	0.619986i	$0.787138\pi$
$308$	0	0
$309$	1.32553	0.0754070
$310$	0	0
$311$	18.9831	1.07643	0.538216	−	0.842807i	$-0.319098\pi$
$311$	18.9831	1.07643	0.538216	+	0.842807i	$0.319098\pi$
$312$	0	0
$313$	6.08210i	0.343781i	0.985116	+	0.171890i	$0.0549875\pi$
$313$	6.08210i	0.343781i	−0.985116	+	0.171890i	$0.945012\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.97817i	0.111105i	0.998456	+	0.0555526i	$0.0176921\pi$
$317$	1.97817i	0.111105i	−0.998456	+	0.0555526i	$0.982308\pi$
$318$	0	0
$319$	0.412724	0.0231081
$320$	0	0
$321$	−17.6796	−0.986776
$322$	0	0
$323$	− 12.5029i	− 0.695680i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.53978i	− 0.195750i
$328$	0	0
$329$	−39.2193	−2.16223
$330$	0	0
$331$	−13.1307	−0.721730	−0.360865	−	0.932618i	$-0.617519\pi$
$331$	−13.1307	−0.721730	−0.360865	+	0.932618i	$0.617519\pi$
$332$	0	0
$333$	6.11909i	0.335324i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.4960i	0.626225i	0.949716	+	0.313113i	$0.101372\pi$
$337$	11.4960i	0.626225i	−0.949716	+	0.313113i	$0.898628\pi$
$338$	0	0
$339$	17.4882	0.949827
$340$	0	0
$341$	4.95960	0.268577
$342$	0	0
$343$	− 20.3264i	− 1.09752i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.7775i	1.59854i	0.600971	+	0.799271i	$0.294781\pi$
$347$	29.7775i	1.59854i	−0.600971	+	0.799271i	$0.705219\pi$
$348$	0	0
$349$	−10.0870	−0.539946	−0.269973	−	0.962868i	$-0.587015\pi$
$349$	−10.0870	−0.539946	−0.269973	+	0.962868i	$0.587015\pi$
$350$	0	0
$351$	−0.966262	−0.0515752
$352$	0	0
$353$	14.2620i	0.759090i	0.925173	+	0.379545i	$0.123919\pi$
$353$	14.2620i	0.759090i	−0.925173	+	0.379545i	$0.876081\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.0516i	0.531988i
$358$	0	0
$359$	6.60909	0.348815	0.174407	−	0.984674i	$-0.444199\pi$
$359$	6.60909	0.348815	0.174407	+	0.984674i	$0.444199\pi$
$360$	0	0
$361$	9.93349	0.522815
$362$	0	0
$363$	− 10.6586i	− 0.559431i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.54259i	0.498119i	0.968488	+	0.249060i	$0.0801215\pi$
$367$	9.54259i	0.498119i	−0.968488	+	0.249060i	$0.919878\pi$
$368$	0	0
$369$	−11.0615	−0.575840
$370$	0	0
$371$	38.5118	1.99943
$372$	0	0
$373$	1.30338i	0.0674866i	0.999431	+	0.0337433i	$0.0107429\pi$
$373$	1.30338i	0.0674866i	−0.999431	+	0.0337433i	$0.989257\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.682531i	− 0.0351521i
$378$	0	0
$379$	−24.2855	−1.24746	−0.623731	−	0.781639i	$-0.714384\pi$
$379$	−24.2855	−1.24746	−0.623731	+	0.781639i	$0.714384\pi$
$380$	0	0
$381$	−17.5060	−0.896861
$382$	0	0
$383$	− 12.1319i	− 0.619910i	−0.950751	−	0.309955i	$-0.899686\pi$
$383$	− 12.1319i	− 0.619910i	0.950751	−	0.309955i	$-0.100314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.03076i	− 0.357394i
$388$	0	0
$389$	29.2598	1.48353	0.741766	−	0.670659i	$-0.233988\pi$
$389$	29.2598	1.48353	0.741766	+	0.670659i	$0.233988\pi$
$390$	0	0
$391$	3.15684	0.159648
$392$	0	0
$393$	− 18.8660i	− 0.951664i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.29380i	0.315877i	0.987449	+	0.157938i	$0.0504847\pi$
$397$	6.29380i	0.315877i	−0.987449	+	0.157938i	$0.949515\pi$
$398$	0	0
$399$	−23.2609	−1.16450
$400$	0	0
$401$	30.3064	1.51343	0.756714	−	0.653747i	$-0.226804\pi$
$401$	30.3064	1.51343	0.756714	+	0.653747i	$0.226804\pi$
$402$	0	0
$403$	− 8.20179i	− 0.408560i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.57536i	− 0.177224i
$408$	0	0
$409$	−18.4761	−0.913583	−0.456792	−	0.889574i	$-0.651002\pi$
$409$	−18.4761	−0.913583	−0.456792	+	0.889574i	$0.651002\pi$
$410$	0	0
$411$	−13.2508	−0.653614
$412$	0	0
$413$	31.5357i	1.55177i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.9026i	0.827722i
$418$	0	0
$419$	22.4453	1.09653	0.548263	−	0.836306i	$-0.315289\pi$
$419$	22.4453	1.09653	0.548263	+	0.836306i	$0.315289\pi$
$420$	0	0
$421$	−6.25053	−0.304632	−0.152316	−	0.988332i	$-0.548673\pi$
$421$	−6.25053	−0.304632	−0.152316	+	0.988332i	$0.548673\pi$
$422$	0	0
$423$	− 9.06932i	− 0.440965i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.8163i	− 1.00737i
$428$	0	0
$429$	0.564582	0.0272583
$430$	0	0
$431$	34.5914	1.66621	0.833104	−	0.553116i	$-0.186561\pi$
$431$	34.5914	1.66621	0.833104	+	0.553116i	$0.186561\pi$
$432$	0	0
$433$	3.93139i	0.188930i	0.995528	+	0.0944652i	$0.0301141\pi$
$433$	3.93139i	0.188930i	−0.995528	+	0.0944652i	$0.969886\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.30539i	0.349464i
$438$	0	0
$439$	−4.98906	−0.238115	−0.119057	−	0.992887i	$-0.537987\pi$
$439$	−4.98906	−0.238115	−0.119057	+	0.992887i	$0.537987\pi$
$440$	0	0
$441$	11.7004	0.557162
$442$	0	0
$443$	− 9.92754i	− 0.471672i	−0.971793	−	0.235836i	$-0.924217\pi$
$443$	− 9.92754i	− 0.471672i	0.971793	−	0.235836i	$-0.0757828\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.1625i	− 0.575265i
$448$	0	0
$449$	−9.87365	−0.465966	−0.232983	−	0.972481i	$-0.574849\pi$
$449$	−9.87365	−0.465966	−0.232983	+	0.972481i	$0.574849\pi$
$450$	0	0
$451$	6.46320	0.304340
$452$	0	0
$453$	− 9.84446i	− 0.462533i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.6424i	0.591387i	0.955283	+	0.295693i	$0.0955507\pi$
$457$	12.6424i	0.591387i	−0.955283	+	0.295693i	$0.904449\pi$
$458$	0	0
$459$	−2.32440	−0.108494
$460$	0	0
$461$	39.7747	1.85249	0.926246	−	0.376919i	$-0.123016\pi$
$461$	39.7747	1.85249	0.926246	+	0.376919i	$0.123016\pi$
$462$	0	0
$463$	− 4.41868i	− 0.205354i	−0.994715	−	0.102677i	$-0.967259\pi$
$463$	− 4.41868i	− 0.205354i	0.994715	−	0.102677i	$-0.0327408\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.2746i	− 0.521728i	−0.965376	−	0.260864i	$-0.915993\pi$
$467$	− 11.2746i	− 0.521728i	0.965376	−	0.260864i	$-0.0840073\pi$
$468$	0	0
$469$	1.62600	0.0750816
$470$	0	0
$471$	18.4804	0.851530
$472$	0	0
$473$	4.10804i	0.188888i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.90571i	0.407765i
$478$	0	0
$479$	5.55895	0.253995	0.126997	−	0.991903i	$-0.459466\pi$
$479$	5.55895	0.253995	0.126997	+	0.991903i	$0.459466\pi$
$480$	0	0
$481$	−5.91264	−0.269593
$482$	0	0
$483$	− 5.87311i	− 0.267236i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16.6341i	− 0.753761i	−0.926262	−	0.376881i	$-0.876997\pi$
$487$	− 16.6341i	− 0.753761i	0.926262	−	0.376881i	$-0.123003\pi$
$488$	0	0
$489$	−14.5368	−0.657377
$490$	0	0
$491$	1.05925	0.0478032	0.0239016	−	0.999714i	$-0.492391\pi$
$491$	1.05925	0.0478032	0.0239016	+	0.999714i	$0.492391\pi$
$492$	0	0
$493$	− 1.64187i	− 0.0739459i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.6676i	2.04847i
$498$	0	0
$499$	−24.9700	−1.11781	−0.558906	−	0.829231i	$-0.688779\pi$
$499$	−24.9700	−1.11781	−0.558906	+	0.829231i	$0.688779\pi$
$500$	0	0
$501$	4.49299	0.200732
$502$	0	0
$503$	− 12.5464i	− 0.559415i	−0.960085	−	0.279708i	$-0.909762\pi$
$503$	− 12.5464i	− 0.559415i	0.960085	−	0.279708i	$-0.0902375\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0663i	0.535885i
$508$	0	0
$509$	15.4275	0.683810	0.341905	−	0.939735i	$-0.388928\pi$
$509$	15.4275	0.683810	0.341905	+	0.939735i	$0.388928\pi$
$510$	0	0
$511$	0.922717	0.0408186
$512$	0	0
$513$	− 5.37899i	− 0.237488i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.29916i	0.233057i
$518$	0	0
$519$	1.99816	0.0877094
$520$	0	0
$521$	−19.3403	−0.847312	−0.423656	−	0.905823i	$-0.639253\pi$
$521$	−19.3403	−0.847312	−0.423656	+	0.905823i	$0.639253\pi$
$522$	0	0
$523$	− 6.95039i	− 0.303920i	−0.988387	−	0.151960i	$-0.951442\pi$
$523$	− 6.95039i	− 0.303920i	0.988387	−	0.151960i	$-0.0485584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 19.7299i	− 0.859446i
$528$	0	0
$529$	21.1555	0.919803
$530$	0	0
$531$	−7.29250	−0.316468
$532$	0	0
$533$	− 10.6883i	− 0.462962i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 13.9793i	− 0.603252i
$538$	0	0
$539$	−6.83649	−0.294469
$540$	0	0
$541$	5.66780	0.243678	0.121839	−	0.992550i	$-0.461121\pi$
$541$	5.66780	0.243678	0.121839	+	0.992550i	$0.461121\pi$
$542$	0	0
$543$	13.5379i	0.580968i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.3997i	0.530172i	0.964225	+	0.265086i	$0.0854003\pi$
$547$	12.3997i	0.530172i	−0.964225	+	0.265086i	$0.914600\pi$
$548$	0	0
$549$	4.81370	0.205444
$550$	0	0
$551$	3.79951	0.161865
$552$	0	0
$553$	30.3829i	1.29201i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.8970i	1.05492i	0.849579	+	0.527461i	$0.176856\pi$
$557$	24.8970i	1.05492i	−0.849579	+	0.527461i	$0.823144\pi$
$558$	0	0
$559$	6.79355	0.287337
$560$	0	0
$561$	1.35813	0.0573405
$562$	0	0
$563$	25.0285i	1.05482i	0.849609	+	0.527412i	$0.176838\pi$
$563$	25.0285i	1.05482i	−0.849609	+	0.527412i	$0.823162\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.32440i	0.181608i
$568$	0	0
$569$	−2.09245	−0.0877199	−0.0438600	−	0.999038i	$-0.513966\pi$
$569$	−2.09245	−0.0877199	−0.0438600	+	0.999038i	$0.513966\pi$
$570$	0	0
$571$	−11.5320	−0.482598	−0.241299	−	0.970451i	$-0.577573\pi$
$571$	−11.5320	−0.482598	−0.241299	+	0.970451i	$0.577573\pi$
$572$	0	0
$573$	2.54547i	0.106339i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5.84516i	− 0.243337i	−0.992571	−	0.121669i	$-0.961175\pi$
$577$	− 5.84516i	− 0.243337i	0.992571	−	0.121669i	$-0.0388245\pi$
$578$	0	0
$579$	5.60541	0.232953
$580$	0	0
$581$	69.6716	2.89046
$582$	0	0
$583$	− 5.20357i	− 0.215510i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.20276i	− 0.379838i	−0.981800	−	0.189919i	$-0.939177\pi$
$587$	− 9.20276i	− 0.379838i	0.981800	−	0.189919i	$-0.0608226\pi$
$588$	0	0
$589$	45.6577	1.88129
$590$	0	0
$591$	7.85621	0.323161
$592$	0	0
$593$	14.9033i	0.612004i	0.952031	+	0.306002i	$0.0989914\pi$
$593$	14.9033i	0.612004i	−0.952031	+	0.306002i	$0.901009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 16.9970i	− 0.695642i
$598$	0	0
$599$	7.52244	0.307359	0.153679	−	0.988121i	$-0.450888\pi$
$599$	7.52244	0.307359	0.153679	+	0.988121i	$0.450888\pi$
$600$	0	0
$601$	18.6618	0.761232	0.380616	−	0.924733i	$-0.375712\pi$
$601$	18.6618	0.761232	0.380616	+	0.924733i	$0.375712\pi$
$602$	0	0
$603$	0.376006i	0.0153121i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.810016i	0.0328776i	0.999865	+	0.0164388i	$0.00523286\pi$
$607$	0.810016i	0.0328776i	−0.999865	+	0.0164388i	$0.994767\pi$
$608$	0	0
$609$	−3.05459	−0.123778
$610$	0	0
$611$	8.76333	0.354527
$612$	0	0
$613$	− 4.99106i	− 0.201587i	−0.994907	−	0.100794i	$-0.967862\pi$
$613$	− 4.99106i	− 0.201587i	0.994907	−	0.100794i	$-0.0321381\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 42.6100i	− 1.71541i	−0.514140	−	0.857706i	$-0.671889\pi$
$617$	− 42.6100i	− 1.71541i	0.514140	−	0.857706i	$-0.328111\pi$
$618$	0	0
$619$	−36.9631	−1.48567	−0.742836	−	0.669473i	$-0.766520\pi$
$619$	−36.9631	−1.48567	−0.742836	+	0.669473i	$0.766520\pi$
$620$	0	0
$621$	1.35813	0.0545001
$622$	0	0
$623$	60.6268i	2.42896i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.14292i	0.125516i
$628$	0	0
$629$	−14.2232	−0.567115
$630$	0	0
$631$	−33.2368	−1.32313	−0.661567	−	0.749886i	$-0.730108\pi$
$631$	−33.2368	−1.32313	−0.661567	+	0.749886i	$0.730108\pi$
$632$	0	0
$633$	− 12.9553i	− 0.514928i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11.3056i	0.447946i
$638$	0	0
$639$	−10.5605	−0.417766
$640$	0	0
$641$	−27.2363	−1.07577	−0.537885	−	0.843018i	$-0.680777\pi$
$641$	−27.2363	−1.07577	−0.537885	+	0.843018i	$0.680777\pi$
$642$	0	0
$643$	12.8557i	0.506978i	0.967338	+	0.253489i	$0.0815782\pi$
$643$	12.8557i	0.506978i	−0.967338	+	0.253489i	$0.918422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.8408i	1.25179i	0.779907	+	0.625895i	$0.215266\pi$
$647$	31.8408i	1.25179i	−0.779907	+	0.625895i	$0.784734\pi$
$648$	0	0
$649$	4.26098	0.167258
$650$	0	0
$651$	−36.7062	−1.43863
$652$	0	0
$653$	14.5019i	0.567504i	0.958898	+	0.283752i	$0.0915793\pi$
$653$	14.5019i	0.567504i	−0.958898	+	0.283752i	$0.908421\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.213375i	0.00832454i
$658$	0	0
$659$	−33.2386	−1.29479	−0.647396	−	0.762154i	$-0.724142\pi$
$659$	−33.2386	−1.29479	−0.647396	+	0.762154i	$0.724142\pi$
$660$	0	0
$661$	−31.8531	−1.23894	−0.619472	−	0.785019i	$-0.712653\pi$
$661$	−31.8531	−1.23894	−0.619472	+	0.785019i	$0.712653\pi$
$662$	0	0
$663$	− 2.24597i	− 0.0872264i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.959335i	0.0371456i
$668$	0	0
$669$	18.9801	0.733814
$670$	0	0
$671$	−2.81262	−0.108580
$672$	0	0
$673$	26.1914i	1.00960i	0.863235	+	0.504802i	$0.168434\pi$
$673$	26.1914i	1.00960i	−0.863235	+	0.504802i	$0.831566\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.8442i	− 0.570509i	−0.958452	−	0.285254i	$-0.907922\pi$
$677$	− 14.8442i	− 0.570509i	0.958452	−	0.285254i	$-0.0920781\pi$
$678$	0	0
$679$	−68.1179	−2.61413
$680$	0	0
$681$	0.385109	0.0147574
$682$	0	0
$683$	5.62626i	0.215283i	0.994190	+	0.107641i	$0.0343299\pi$
$683$	5.62626i	0.215283i	−0.994190	+	0.107641i	$0.965670\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10.2685i	− 0.391768i
$688$	0	0
$689$	−8.60525	−0.327834
$690$	0	0
$691$	34.4090	1.30898	0.654491	−	0.756070i	$-0.272883\pi$
$691$	34.4090	1.30898	0.654491	+	0.756070i	$0.272883\pi$
$692$	0	0
$693$	− 2.52673i	− 0.0959824i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 25.7113i	− 0.973887i
$698$	0	0
$699$	−21.6380	−0.818425
$700$	0	0
$701$	8.86294	0.334749	0.167374	−	0.985893i	$-0.446471\pi$
$701$	8.86294	0.334749	0.167374	+	0.985893i	$0.446471\pi$
$702$	0	0
$703$	− 32.9145i	− 1.24139i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.8839i	1.16151i
$708$	0	0
$709$	−1.53673	−0.0577132	−0.0288566	−	0.999584i	$-0.509187\pi$
$709$	−1.53673	−0.0577132	−0.0288566	+	0.999584i	$0.509187\pi$
$710$	0	0
$711$	−7.02594	−0.263493
$712$	0	0
$713$	11.5281i	0.431730i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.71601i	0.288160i
$718$	0	0
$719$	19.1353	0.713625	0.356813	−	0.934176i	$-0.383863\pi$
$719$	19.1353	0.713625	0.356813	+	0.934176i	$0.383863\pi$
$720$	0	0
$721$	5.73214	0.213476
$722$	0	0
$723$	− 25.3264i	− 0.941899i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.8094i	1.73606i	0.496507	+	0.868032i	$0.334616\pi$
$727$	46.8094i	1.73606i	−0.496507	+	0.868032i	$0.665384\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.3423	0.604441
$732$	0	0
$733$	− 51.0044i	− 1.88389i	−0.335768	−	0.941945i	$-0.608996\pi$
$733$	− 51.0044i	− 1.88389i	0.335768	−	0.941945i	$-0.391004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.219699i	− 0.00809270i
$738$	0	0
$739$	−13.2056	−0.485775	−0.242887	−	0.970054i	$-0.578095\pi$
$739$	−13.2056	−0.485775	−0.242887	+	0.970054i	$0.578095\pi$
$740$	0	0
$741$	5.19751	0.190935
$742$	0	0
$743$	− 10.6063i	− 0.389107i	−0.980892	−	0.194553i	$-0.937674\pi$
$743$	− 10.6063i	− 0.389107i	0.980892	−	0.194553i	$-0.0623258\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.1113i	0.589481i
$748$	0	0
$749$	−76.4534	−2.79355
$750$	0	0
$751$	28.2581	1.03115	0.515577	−	0.856843i	$-0.327578\pi$
$751$	28.2581	1.03115	0.515577	+	0.856843i	$0.327578\pi$
$752$	0	0
$753$	− 2.39913i	− 0.0874293i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.7474i	1.00849i	0.863559	+	0.504247i	$0.168230\pi$
$757$	27.7474i	1.00849i	−0.863559	+	0.504247i	$0.831770\pi$
$758$	0	0
$759$	−0.793552	−0.0288041
$760$	0	0
$761$	−16.1152	−0.584177	−0.292088	−	0.956391i	$-0.594350\pi$
$761$	−16.1152	−0.584177	−0.292088	+	0.956391i	$0.594350\pi$
$762$	0	0
$763$	− 15.3074i	− 0.554165i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.04646i	− 0.254433i
$768$	0	0
$769$	34.7435	1.25288	0.626441	−	0.779468i	$-0.284511\pi$
$769$	34.7435	1.25288	0.626441	+	0.779468i	$0.284511\pi$
$770$	0	0
$771$	14.3086	0.515313
$772$	0	0
$773$	− 23.4684i	− 0.844101i	−0.906572	−	0.422051i	$-0.861310\pi$
$773$	− 23.4684i	− 0.844101i	0.906572	−	0.422051i	$-0.138690\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	26.4614i	0.949296i
$778$	0	0
$779$	59.4997	2.13180
$780$	0	0
$781$	6.17043	0.220795
$782$	0	0
$783$	− 0.706362i	− 0.0252433i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 25.6217i	− 0.913316i	−0.889642	−	0.456658i	$-0.849046\pi$
$787$	− 25.6217i	− 0.913316i	0.889642	−	0.456658i	$-0.150954\pi$
$788$	0	0
$789$	13.0805	0.465679
$790$	0	0
$791$	75.6258	2.68894
$792$	0	0
$793$	4.65129i	0.165172i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.92304i	− 0.0681177i	−0.999420	−	0.0340588i	$-0.989157\pi$
$797$	− 1.92304i	− 0.0681177i	0.999420	−	0.0340588i	$-0.0108434\pi$
$798$	0	0
$799$	21.0807	0.745781
$800$	0	0
$801$	−14.0197	−0.495362
$802$	0	0
$803$	− 0.124674i	− 0.00439965i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 30.6749i	− 1.07981i
$808$	0	0
$809$	45.4087	1.59648	0.798242	−	0.602336i	$-0.205763\pi$
$809$	45.4087	1.59648	0.798242	+	0.602336i	$0.205763\pi$
$810$	0	0
$811$	−23.9207	−0.839970	−0.419985	−	0.907531i	$-0.637965\pi$
$811$	−23.9207	−0.839970	−0.419985	+	0.907531i	$0.637965\pi$
$812$	0	0
$813$	14.2362i	0.499286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	37.8184i	1.32310i
$818$	0	0
$819$	−4.17850	−0.146009
$820$	0	0
$821$	−31.5862	−1.10237	−0.551184	−	0.834384i	$-0.685824\pi$
$821$	−31.5862	−1.10237	−0.551184	+	0.834384i	$0.685824\pi$
$822$	0	0
$823$	− 8.53252i	− 0.297425i	−0.988880	−	0.148713i	$-0.952487\pi$
$823$	− 8.53252i	− 0.297425i	0.988880	−	0.148713i	$-0.0475129\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17.7933i	0.618733i	0.950943	+	0.309366i	$0.100117\pi$
$827$	17.7933i	0.618733i	−0.950943	+	0.309366i	$0.899883\pi$
$828$	0	0
$829$	46.0216	1.59840	0.799199	−	0.601067i	$-0.205257\pi$
$829$	46.0216	1.59840	0.799199	+	0.601067i	$0.205257\pi$
$830$	0	0
$831$	−20.1307	−0.698325
$832$	0	0
$833$	27.1964i	0.942298i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.48817i	− 0.293394i
$838$	0	0
$839$	−1.01208	−0.0349410	−0.0174705	−	0.999847i	$-0.505561\pi$
$839$	−1.01208	−0.0349410	−0.0174705	+	0.999847i	$0.505561\pi$
$840$	0	0
$841$	−28.5011	−0.982795
$842$	0	0
$843$	− 9.81651i	− 0.338099i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 46.0920i	− 1.58374i
$848$	0	0
$849$	6.54547	0.224640
$850$	0	0
$851$	8.31054	0.284882
$852$	0	0
$853$	9.81334i	0.336002i	0.985787	+	0.168001i	$0.0537312\pi$
$853$	9.81334i	0.336002i	−0.985787	+	0.168001i	$0.946269\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 57.3424i	− 1.95878i	−0.201979	−	0.979390i	$-0.564737\pi$
$857$	− 57.3424i	− 1.95878i	0.201979	−	0.979390i	$-0.435263\pi$
$858$	0	0
$859$	−10.7349	−0.366271	−0.183135	−	0.983088i	$-0.558625\pi$
$859$	−10.7349	−0.366271	−0.183135	+	0.983088i	$0.558625\pi$
$860$	0	0
$861$	−47.8344	−1.63019
$862$	0	0
$863$	− 9.63250i	− 0.327894i	−0.986469	−	0.163947i	$-0.947577\pi$
$863$	− 9.63250i	− 0.327894i	0.986469	−	0.163947i	$-0.0524226\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11.5972i	0.393861i
$868$	0	0
$869$	4.10522	0.139260
$870$	0	0
$871$	−0.363320	−0.0123106
$872$	0	0
$873$	− 15.7520i	− 0.533125i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 0.703817i	− 0.0237662i	−0.999929	−	0.0118831i	$-0.996217\pi$
$877$	− 0.703817i	− 0.0237662i	0.999929	−	0.0118831i	$-0.00378260\pi$
$878$	0	0
$879$	0.869428	0.0293251
$880$	0	0
$881$	9.92656	0.334434	0.167217	−	0.985920i	$-0.446522\pi$
$881$	9.92656	0.334434	0.167217	+	0.985920i	$0.446522\pi$
$882$	0	0
$883$	− 16.8403i	− 0.566723i	−0.959013	−	0.283361i	$-0.908550\pi$
$883$	− 16.8403i	− 0.566723i	0.959013	−	0.283361i	$-0.0914496\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.3918i	1.55768i	0.627221	+	0.778842i	$0.284192\pi$
$887$	46.3918i	1.55768i	−0.627221	+	0.778842i	$0.715808\pi$
$888$	0	0
$889$	−75.7030	−2.53900
$890$	0	0
$891$	0.584296	0.0195746
$892$	0	0
$893$	48.7837i	1.63249i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.31231i	0.0438169i
$898$	0	0
$899$	5.99572	0.199968
$900$	0	0
$901$	−20.7004	−0.689630
$902$	0	0
$903$	− 30.4038i	− 1.01177i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.5101i	0.946662i	0.880885	+	0.473331i	$0.156949\pi$
$907$	28.5101i	0.946662i	−0.880885	+	0.473331i	$0.843051\pi$
$908$	0	0
$909$	−7.14178	−0.236878
$910$	0	0
$911$	−6.50073	−0.215379	−0.107689	−	0.994185i	$-0.534345\pi$
$911$	−6.50073	−0.215379	−0.107689	+	0.994185i	$0.534345\pi$
$912$	0	0
$913$	− 9.41375i	− 0.311550i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 81.5841i	− 2.69414i
$918$	0	0
$919$	−16.8266	−0.555058	−0.277529	−	0.960717i	$-0.589515\pi$
$919$	−16.8266	−0.555058	−0.277529	+	0.960717i	$0.589515\pi$
$920$	0	0
$921$	−21.7261	−0.715899
$922$	0	0
$923$	− 10.2042i	− 0.335874i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.32553i	0.0435363i
$928$	0	0
$929$	31.6407	1.03810	0.519049	−	0.854744i	$-0.326286\pi$
$929$	31.6407	1.03810	0.519049	+	0.854744i	$0.326286\pi$
$930$	0	0
$931$	−62.9363	−2.06265
$932$	0	0
$933$	18.9831i	0.621479i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.35105i	− 0.0768053i	−0.999262	−	0.0384026i	$-0.987773\pi$
$937$	− 2.35105i	− 0.0768053i	0.999262	−	0.0384026i	$-0.0122269\pi$
$938$	0	0
$939$	−6.08210	−0.198482
$940$	0	0
$941$	4.29802	0.140111	0.0700557	−	0.997543i	$-0.477682\pi$
$941$	4.29802	0.140111	0.0700557	+	0.997543i	$0.477682\pi$
$942$	0	0
$943$	15.0230i	0.489217i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.8418i	0.904737i	0.891831	+	0.452369i	$0.149421\pi$
$947$	27.8418i	0.904737i	−0.891831	+	0.452369i	$0.850579\pi$
$948$	0	0
$949$	−0.206176	−0.00669275
$950$	0	0
$951$	−1.97817	−0.0641467
$952$	0	0
$953$	44.6209i	1.44541i	0.691154	+	0.722707i	$0.257102\pi$
$953$	44.6209i	1.44541i	−0.691154	+	0.722707i	$0.742898\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.412724i	0.0133415i
$958$	0	0
$959$	−57.3017	−1.85037
$960$	0	0
$961$	41.0490	1.32416
$962$	0	0
$963$	− 17.6796i	− 0.569716i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 55.1855i	− 1.77465i	−0.461148	−	0.887323i	$-0.652562\pi$
$967$	− 55.1855i	− 1.77465i	0.461148	−	0.887323i	$-0.347438\pi$
$968$	0	0
$969$	12.5029	0.401651
$970$	0	0
$971$	−6.19165	−0.198699	−0.0993497	−	0.995053i	$-0.531676\pi$
$971$	−6.19165	−0.198699	−0.0993497	+	0.995053i	$0.531676\pi$
$972$	0	0
$973$	73.0934i	2.34327i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.1646i	0.773095i	0.922269	+	0.386548i	$0.126333\pi$
$977$	24.1646i	0.773095i	−0.922269	+	0.386548i	$0.873667\pi$
$978$	0	0
$979$	8.19166	0.261806
$980$	0	0
$981$	3.53978	0.113016
$982$	0	0
$983$	9.52304i	0.303738i	0.988401	+	0.151869i	$0.0485292\pi$
$983$	9.52304i	0.303738i	−0.988401	+	0.151869i	$0.951471\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 39.2193i	− 1.24836i
$988$	0	0
$989$	−9.54872	−0.303631
$990$	0	0
$991$	−10.4117	−0.330740	−0.165370	−	0.986232i	$-0.552882\pi$
$991$	−10.4117	−0.330740	−0.165370	+	0.986232i	$0.552882\pi$
$992$	0	0
$993$	− 13.1307i	− 0.416691i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 57.7687i	− 1.82955i	−0.403959	−	0.914777i	$-0.632366\pi$
$997$	− 57.7687i	− 1.82955i	0.403959	−	0.914777i	$-0.367634\pi$
$998$	0	0
$999$	−6.11909	−0.193599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.c.1249.8		8
5.2	odd	4		7500.2.a.f.1.1		4
5.3	odd	4		7500.2.a.e.1.4		4
5.4	even	2	inner	7500.2.d.c.1249.1		8
25.2	odd	20		1500.2.m.a.601.1		8
25.9	even	10		1500.2.o.b.349.3		16
25.11	even	5		1500.2.o.b.649.4		16
25.12	odd	20		1500.2.m.a.901.1		8
25.13	odd	20		300.2.m.b.181.1	yes	8
25.14	even	10		1500.2.o.b.649.1		16
25.16	even	5		1500.2.o.b.349.2		16
25.23	odd	20		300.2.m.b.121.1	✓	8
75.23	even	20		900.2.n.b.721.2		8
75.38	even	20		900.2.n.b.181.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.121.1	✓	8	25.23	odd	20
300.2.m.b.181.1	yes	8	25.13	odd	20
900.2.n.b.181.2		8	75.38	even	20
900.2.n.b.721.2		8	75.23	even	20
1500.2.m.a.601.1		8	25.2	odd	20
1500.2.m.a.901.1		8	25.12	odd	20
1500.2.o.b.349.2		16	25.16	even	5
1500.2.o.b.349.3		16	25.9	even	10
1500.2.o.b.649.1		16	25.14	even	10
1500.2.o.b.649.4		16	25.11	even	5
7500.2.a.e.1.4		4	5.3	odd	4
7500.2.a.f.1.1		4	5.2	odd	4
7500.2.d.c.1249.1		8	5.4	even	2	inner
7500.2.d.c.1249.8		8	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.32440i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +4.32440i q^{7} -1.00000 q^{9} +0.584296 q^{11} -0.966262i q^{13} -2.32440i q^{17} +5.37899 q^{19} -4.32440 q^{21} +1.35813i q^{23} -1.00000i q^{27} +0.706362 q^{29} +8.48817 q^{31} +0.584296i q^{33} -6.11909i q^{37} +0.966262 q^{39} +11.0615 q^{41} +7.03076i q^{43} +9.06932i q^{47} -11.7004 q^{49} +2.32440 q^{51} -8.90571i q^{53} +5.37899i q^{57} +7.29250 q^{59} -4.81370 q^{61} -4.32440i q^{63} -0.376006i q^{67} -1.35813 q^{69} +10.5605 q^{71} -0.213375i q^{73} +2.52673i q^{77} +7.02594 q^{79} +1.00000 q^{81} -16.1113i q^{83} +0.706362i q^{87} +14.0197 q^{89} +4.17850 q^{91} +8.48817i q^{93} +15.7520i q^{97} -0.584296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 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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Database

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