LMFDB - Embedded newform 8624.2.a.df.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

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:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 539)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.27614$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27614 q^{3} +0.246676 q^{5} +7.73309 q^{9} +O(q^{10})$	$q+3.27614 q^{3} +0.246676 q^{5} +7.73309 q^{9} -1.00000 q^{11} +3.17078 q^{13} +0.808144 q^{15} +6.49256 q^{17} -4.32335 q^{19} -3.15700 q^{23} -4.93915 q^{25} +15.5063 q^{27} +6.48417 q^{29} +1.78122 q^{31} -3.27614 q^{33} +8.38424 q^{37} +10.3879 q^{39} +0.553067 q^{41} -5.69023 q^{43} +1.90756 q^{45} +10.2971 q^{47} +21.2705 q^{51} -10.1744 q^{53} -0.246676 q^{55} -14.1639 q^{57} -6.45660 q^{59} +3.38149 q^{61} +0.782155 q^{65} +3.65484 q^{67} -10.3428 q^{69} -0.345158 q^{71} -2.97942 q^{73} -16.1813 q^{75} +3.77595 q^{79} +27.6014 q^{81} -6.34157 q^{83} +1.60156 q^{85} +21.2430 q^{87} +0.246676 q^{89} +5.83553 q^{93} -1.06646 q^{95} +4.81068 q^{97} -7.73309 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 22 q^{9}+O(q^{10}) $ 10 * q + 22 * q^9	$ 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93} + 64 q^{95} - 22 q^{99}+O(q^{100}) $ 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.27614	1.89148	0.945740	−	0.324924i	$-0.105339\pi$
$3$	3.27614	1.89148	0.945740	+	0.324924i	$0.105339\pi$
$4$	0	0
$5$	0.246676	0.110317	0.0551584	−	0.998478i	$-0.482434\pi$
$5$	0.246676	0.110317	0.0551584	+	0.998478i	$0.482434\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.73309	2.57770
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.17078	0.879417	0.439709	−	0.898140i	$-0.355082\pi$
$13$	3.17078	0.879417	0.439709	+	0.898140i	$0.355082\pi$
$14$	0	0
$15$	0.808144	0.208662
$16$	0	0
$17$	6.49256	1.57468	0.787339	−	0.616520i	$-0.211458\pi$
$17$	6.49256	1.57468	0.787339	+	0.616520i	$0.211458\pi$
$18$	0	0
$19$	−4.32335	−0.991844	−0.495922	−	0.868367i	$-0.665170\pi$
$19$	−4.32335	−0.991844	−0.495922	+	0.868367i	$0.665170\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.15700	−0.658279	−0.329140	−	0.944281i	$-0.606759\pi$
$23$	−3.15700	−0.658279	−0.329140	+	0.944281i	$0.606759\pi$
$24$	0	0
$25$	−4.93915	−0.987830
$26$	0	0
$27$	15.5063	2.98418
$28$	0	0
$29$	6.48417	1.20408	0.602040	−	0.798466i	$-0.294355\pi$
$29$	6.48417	1.20408	0.602040	+	0.798466i	$0.294355\pi$
$30$	0	0
$31$	1.78122	0.319917	0.159958	−	0.987124i	$-0.448864\pi$
$31$	1.78122	0.319917	0.159958	+	0.987124i	$0.448864\pi$
$32$	0	0
$33$	−3.27614	−0.570303
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.38424	1.37836	0.689180	−	0.724590i	$-0.257971\pi$
$37$	8.38424	1.37836	0.689180	+	0.724590i	$0.257971\pi$
$38$	0	0
$39$	10.3879	1.66340
$40$	0	0
$41$	0.553067	0.0863746	0.0431873	−	0.999067i	$-0.486249\pi$
$41$	0.553067	0.0863746	0.0431873	+	0.999067i	$0.486249\pi$
$42$	0	0
$43$	−5.69023	−0.867752	−0.433876	−	0.900973i	$-0.642854\pi$
$43$	−5.69023	−0.867752	−0.433876	+	0.900973i	$0.642854\pi$
$44$	0	0
$45$	1.90756	0.284363
$46$	0	0
$47$	10.2971	1.50199	0.750995	−	0.660308i	$-0.229574\pi$
$47$	10.2971	1.50199	0.750995	+	0.660308i	$0.229574\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	21.2705	2.97847
$52$	0	0
$53$	−10.1744	−1.39756	−0.698780	−	0.715336i	$-0.746273\pi$
$53$	−10.1744	−1.39756	−0.698780	+	0.715336i	$0.746273\pi$
$54$	0	0
$55$	−0.246676	−0.0332617
$56$	0	0
$57$	−14.1639	−1.87605
$58$	0	0
$59$	−6.45660	−0.840577	−0.420289	−	0.907390i	$-0.638071\pi$
$59$	−6.45660	−0.840577	−0.420289	+	0.907390i	$0.638071\pi$
$60$	0	0
$61$	3.38149	0.432956	0.216478	−	0.976287i	$-0.430543\pi$
$61$	3.38149	0.432956	0.216478	+	0.976287i	$0.430543\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.782155	0.0970144
$66$	0	0
$67$	3.65484	0.446510	0.223255	−	0.974760i	$-0.428332\pi$
$67$	3.65484	0.446510	0.223255	+	0.974760i	$0.428332\pi$
$68$	0	0
$69$	−10.3428	−1.24512
$70$	0	0
$71$	−0.345158	−0.0409627	−0.0204813	−	0.999790i	$-0.506520\pi$
$71$	−0.345158	−0.0409627	−0.0204813	+	0.999790i	$0.506520\pi$
$72$	0	0
$73$	−2.97942	−0.348715	−0.174357	−	0.984682i	$-0.555785\pi$
$73$	−2.97942	−0.348715	−0.174357	+	0.984682i	$0.555785\pi$
$74$	0	0
$75$	−16.1813	−1.86846
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.77595	0.424828	0.212414	−	0.977180i	$-0.431868\pi$
$79$	3.77595	0.424828	0.212414	+	0.977180i	$0.431868\pi$
$80$	0	0
$81$	27.6014	3.06682
$82$	0	0
$83$	−6.34157	−0.696078	−0.348039	−	0.937480i	$-0.613152\pi$
$83$	−6.34157	−0.696078	−0.348039	+	0.937480i	$0.613152\pi$
$84$	0	0
$85$	1.60156	0.173713
$86$	0	0
$87$	21.2430	2.27749
$88$	0	0
$89$	0.246676	0.0261476	0.0130738	−	0.999915i	$-0.495838\pi$
$89$	0.246676	0.0261476	0.0130738	+	0.999915i	$0.495838\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.83553	0.605116
$94$	0	0
$95$	−1.06646	−0.109417
$96$	0	0
$97$	4.81068	0.488451	0.244226	−	0.969718i	$-0.421466\pi$
$97$	4.81068	0.488451	0.244226	+	0.969718i	$0.421466\pi$
$98$	0	0
$99$	−7.73309	−0.777205
$100$	0	0
$101$	6.86468	0.683061	0.341531	−	0.939871i	$-0.389055\pi$
$101$	6.86468	0.683061	0.341531	+	0.939871i	$0.389055\pi$
$102$	0	0
$103$	5.07835	0.500385	0.250192	−	0.968196i	$-0.419506\pi$
$103$	5.07835	0.500385	0.250192	+	0.968196i	$0.419506\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.897562	−0.0867706	−0.0433853	−	0.999058i	$-0.513814\pi$
$107$	−0.897562	−0.0867706	−0.0433853	+	0.999058i	$0.513814\pi$
$108$	0	0
$109$	5.71569	0.547464	0.273732	−	0.961806i	$-0.411742\pi$
$109$	5.71569	0.547464	0.273732	+	0.961806i	$0.411742\pi$
$110$	0	0
$111$	27.4679	2.60714
$112$	0	0
$113$	−5.99001	−0.563493	−0.281747	−	0.959489i	$-0.590914\pi$
$113$	−5.99001	−0.563493	−0.281747	+	0.959489i	$0.590914\pi$
$114$	0	0
$115$	−0.778754	−0.0726192
$116$	0	0
$117$	24.5200	2.26687
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.81193	0.163376
$124$	0	0
$125$	−2.45175	−0.219291
$126$	0	0
$127$	−2.31399	−0.205334	−0.102667	−	0.994716i	$-0.532738\pi$
$127$	−2.31399	−0.205334	−0.102667	+	0.994716i	$0.532738\pi$
$128$	0	0
$129$	−18.6420	−1.64133
$130$	0	0
$131$	14.4800	1.26513	0.632564	−	0.774508i	$-0.282003\pi$
$131$	14.4800	1.26513	0.632564	+	0.774508i	$0.282003\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.82502	0.329205
$136$	0	0
$137$	−10.9962	−0.939470	−0.469735	−	0.882807i	$-0.655651\pi$
$137$	−10.9962	−0.939470	−0.469735	+	0.882807i	$0.655651\pi$
$138$	0	0
$139$	1.05087	0.0891334	0.0445667	−	0.999006i	$-0.485809\pi$
$139$	1.05087	0.0891334	0.0445667	+	0.999006i	$0.485809\pi$
$140$	0	0
$141$	33.7348	2.84098
$142$	0	0
$143$	−3.17078	−0.265154
$144$	0	0
$145$	1.59949	0.132830
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.4185	1.50890	0.754452	−	0.656356i	$-0.227903\pi$
$149$	18.4185	1.50890	0.754452	+	0.656356i	$0.227903\pi$
$150$	0	0
$151$	−19.1564	−1.55893	−0.779463	−	0.626448i	$-0.784508\pi$
$151$	−19.1564	−1.55893	−0.779463	+	0.626448i	$0.784508\pi$
$152$	0	0
$153$	50.2076	4.05904
$154$	0	0
$155$	0.439384	0.0352922
$156$	0	0
$157$	−23.0214	−1.83731	−0.918653	−	0.395065i	$-0.870722\pi$
$157$	−23.0214	−1.83731	−0.918653	+	0.395065i	$0.870722\pi$
$158$	0	0
$159$	−33.3327	−2.64346
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.93865	0.386825	0.193412	−	0.981118i	$-0.438044\pi$
$163$	4.93865	0.386825	0.193412	+	0.981118i	$0.438044\pi$
$164$	0	0
$165$	−0.808144	−0.0629139
$166$	0	0
$167$	7.39244	0.572044	0.286022	−	0.958223i	$-0.407667\pi$
$167$	7.39244	0.572044	0.286022	+	0.958223i	$0.407667\pi$
$168$	0	0
$169$	−2.94613	−0.226625
$170$	0	0
$171$	−33.4328	−2.55667
$172$	0	0
$173$	18.2383	1.38663	0.693316	−	0.720634i	$-0.256149\pi$
$173$	18.2383	1.38663	0.693316	+	0.720634i	$0.256149\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.1527	−1.58994
$178$	0	0
$179$	2.37373	0.177421	0.0887103	−	0.996057i	$-0.471725\pi$
$179$	2.37373	0.177421	0.0887103	+	0.996057i	$0.471725\pi$
$180$	0	0
$181$	21.6459	1.60893	0.804463	−	0.594002i	$-0.202453\pi$
$181$	21.6459	1.60893	0.804463	+	0.594002i	$0.202453\pi$
$182$	0	0
$183$	11.0782	0.818928
$184$	0	0
$185$	2.06819	0.152056
$186$	0	0
$187$	−6.49256	−0.474783
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.22154	−0.522532	−0.261266	−	0.965267i	$-0.584140\pi$
$191$	−7.22154	−0.522532	−0.261266	+	0.965267i	$0.584140\pi$
$192$	0	0
$193$	−3.66604	−0.263887	−0.131944	−	0.991257i	$-0.542122\pi$
$193$	−3.66604	−0.263887	−0.131944	+	0.991257i	$0.542122\pi$
$194$	0	0
$195$	2.56245	0.183501
$196$	0	0
$197$	−3.81615	−0.271889	−0.135945	−	0.990716i	$-0.543407\pi$
$197$	−3.81615	−0.271889	−0.135945	+	0.990716i	$0.543407\pi$
$198$	0	0
$199$	−19.3819	−1.37395	−0.686973	−	0.726683i	$-0.741061\pi$
$199$	−19.3819	−1.37395	−0.686973	+	0.726683i	$0.741061\pi$
$200$	0	0
$201$	11.9738	0.844565
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.136428	0.00952856
$206$	0	0
$207$	−24.4133	−1.69684
$208$	0	0
$209$	4.32335	0.299052
$210$	0	0
$211$	−24.1279	−1.66103	−0.830517	−	0.556993i	$-0.811955\pi$
$211$	−24.1279	−1.66103	−0.830517	+	0.556993i	$0.811955\pi$
$212$	0	0
$213$	−1.13078	−0.0774801
$214$	0	0
$215$	−1.40364	−0.0957275
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.76100	−0.659587
$220$	0	0
$221$	20.5865	1.38480
$222$	0	0
$223$	14.6952	0.984061	0.492030	−	0.870578i	$-0.336255\pi$
$223$	14.6952	0.984061	0.492030	+	0.870578i	$0.336255\pi$
$224$	0	0
$225$	−38.1949	−2.54633
$226$	0	0
$227$	9.19531	0.610314	0.305157	−	0.952302i	$-0.401291\pi$
$227$	9.19531	0.610314	0.305157	+	0.952302i	$0.401291\pi$
$228$	0	0
$229$	−17.4339	−1.15207	−0.576033	−	0.817427i	$-0.695400\pi$
$229$	−17.4339	−1.15207	−0.576033	+	0.817427i	$0.695400\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.7505	−1.81799	−0.908997	−	0.416802i	$-0.863151\pi$
$233$	−27.7505	−1.81799	−0.908997	+	0.416802i	$0.863151\pi$
$234$	0	0
$235$	2.54005	0.165695
$236$	0	0
$237$	12.3705	0.803553
$238$	0	0
$239$	22.5326	1.45752	0.728758	−	0.684772i	$-0.240098\pi$
$239$	22.5326	1.45752	0.728758	+	0.684772i	$0.240098\pi$
$240$	0	0
$241$	7.22658	0.465505	0.232752	−	0.972536i	$-0.425227\pi$
$241$	7.22658	0.465505	0.232752	+	0.972536i	$0.425227\pi$
$242$	0	0
$243$	43.9073	2.81665
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.7084	−0.872244
$248$	0	0
$249$	−20.7759	−1.31662
$250$	0	0
$251$	12.8842	0.813245	0.406622	−	0.913596i	$-0.366706\pi$
$251$	12.8842	0.813245	0.406622	+	0.913596i	$0.366706\pi$
$252$	0	0
$253$	3.15700	0.198479
$254$	0	0
$255$	5.24692	0.328575
$256$	0	0
$257$	−31.0932	−1.93954	−0.969771	−	0.244016i	$-0.921535\pi$
$257$	−31.0932	−1.93954	−0.969771	+	0.244016i	$0.921535\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	50.1426	3.10375
$262$	0	0
$263$	7.50215	0.462603	0.231301	−	0.972882i	$-0.425702\pi$
$263$	7.50215	0.462603	0.231301	+	0.972882i	$0.425702\pi$
$264$	0	0
$265$	−2.50978	−0.154174
$266$	0	0
$267$	0.808144	0.0494576
$268$	0	0
$269$	1.93752	0.118133	0.0590663	−	0.998254i	$-0.481188\pi$
$269$	1.93752	0.118133	0.0590663	+	0.998254i	$0.481188\pi$
$270$	0	0
$271$	22.9786	1.39585	0.697926	−	0.716170i	$-0.254107\pi$
$271$	22.9786	1.39585	0.697926	+	0.716170i	$0.254107\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.93915	0.297842
$276$	0	0
$277$	−5.52095	−0.331722	−0.165861	−	0.986149i	$-0.553040\pi$
$277$	−5.52095	−0.331722	−0.165861	+	0.986149i	$0.553040\pi$
$278$	0	0
$279$	13.7743	0.824648
$280$	0	0
$281$	−7.28000	−0.434288	−0.217144	−	0.976140i	$-0.569674\pi$
$281$	−7.28000	−0.434288	−0.217144	+	0.976140i	$0.569674\pi$
$282$	0	0
$283$	−13.2600	−0.788225	−0.394113	−	0.919062i	$-0.628948\pi$
$283$	−13.2600	−0.788225	−0.394113	+	0.919062i	$0.628948\pi$
$284$	0	0
$285$	−3.49389	−0.206960
$286$	0	0
$287$	0	0
$288$	0	0
$289$	25.1534	1.47961
$290$	0	0
$291$	15.7605	0.923895
$292$	0	0
$293$	5.56718	0.325238	0.162619	−	0.986689i	$-0.448006\pi$
$293$	5.56718	0.325238	0.162619	+	0.986689i	$0.448006\pi$
$294$	0	0
$295$	−1.59269	−0.0927297
$296$	0	0
$297$	−15.5063	−0.899765
$298$	0	0
$299$	−10.0102	−0.578902
$300$	0	0
$301$	0	0
$302$	0	0
$303$	22.4897	1.29200
$304$	0	0
$305$	0.834132	0.0477623
$306$	0	0
$307$	−14.2467	−0.813102	−0.406551	−	0.913628i	$-0.633269\pi$
$307$	−14.2467	−0.813102	−0.406551	+	0.913628i	$0.633269\pi$
$308$	0	0
$309$	16.6374	0.946468
$310$	0	0
$311$	−3.56834	−0.202342	−0.101171	−	0.994869i	$-0.532259\pi$
$311$	−3.56834	−0.202342	−0.101171	+	0.994869i	$0.532259\pi$
$312$	0	0
$313$	−9.11394	−0.515150	−0.257575	−	0.966258i	$-0.582924\pi$
$313$	−9.11394	−0.515150	−0.257575	+	0.966258i	$0.582924\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.7542	−0.660181	−0.330090	−	0.943949i	$-0.607079\pi$
$317$	−11.7542	−0.660181	−0.330090	+	0.943949i	$0.607079\pi$
$318$	0	0
$319$	−6.48417	−0.363044
$320$	0	0
$321$	−2.94054	−0.164125
$322$	0	0
$323$	−28.0696	−1.56183
$324$	0	0
$325$	−15.6610	−0.868715
$326$	0	0
$327$	18.7254	1.03552
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.6748	1.41122	0.705609	−	0.708602i	$-0.250674\pi$
$331$	25.6748	1.41122	0.705609	+	0.708602i	$0.250674\pi$
$332$	0	0
$333$	64.8361	3.55299
$334$	0	0
$335$	0.901561	0.0492575
$336$	0	0
$337$	33.5644	1.82837	0.914185	−	0.405298i	$-0.132832\pi$
$337$	33.5644	1.82837	0.914185	+	0.405298i	$0.132832\pi$
$338$	0	0
$339$	−19.6241	−1.06584
$340$	0	0
$341$	−1.78122	−0.0964585
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.55131	−0.137358
$346$	0	0
$347$	26.7567	1.43637	0.718187	−	0.695850i	$-0.244972\pi$
$347$	26.7567	1.43637	0.718187	+	0.695850i	$0.244972\pi$
$348$	0	0
$349$	−19.9411	−1.06742	−0.533711	−	0.845667i	$-0.679203\pi$
$349$	−19.9411	−1.06742	−0.533711	+	0.845667i	$0.679203\pi$
$350$	0	0
$351$	49.1670	2.62434
$352$	0	0
$353$	−10.0436	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$353$	−10.0436	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$354$	0	0
$355$	−0.0851420	−0.00451887
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.0519	0.899964	0.449982	−	0.893038i	$-0.351430\pi$
$359$	17.0519	0.899964	0.449982	+	0.893038i	$0.351430\pi$
$360$	0	0
$361$	−0.308682	−0.0162464
$362$	0	0
$363$	3.27614	0.171953
$364$	0	0
$365$	−0.734951	−0.0384691
$366$	0	0
$367$	−0.115028	−0.00600443	−0.00300221	−	0.999995i	$-0.500956\pi$
$367$	−0.115028	−0.00600443	−0.00300221	+	0.999995i	$0.500956\pi$
$368$	0	0
$369$	4.27692	0.222648
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.5106	−0.906664	−0.453332	−	0.891342i	$-0.649765\pi$
$373$	−17.5106	−0.906664	−0.453332	+	0.891342i	$0.649765\pi$
$374$	0	0
$375$	−8.03226	−0.414784
$376$	0	0
$377$	20.5599	1.05889
$378$	0	0
$379$	−7.72561	−0.396838	−0.198419	−	0.980117i	$-0.563581\pi$
$379$	−7.72561	−0.396838	−0.198419	+	0.980117i	$0.563581\pi$
$380$	0	0
$381$	−7.58096	−0.388384
$382$	0	0
$383$	−10.6044	−0.541860	−0.270930	−	0.962599i	$-0.587331\pi$
$383$	−10.6044	−0.541860	−0.270930	+	0.962599i	$0.587331\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−44.0030	−2.23680
$388$	0	0
$389$	−22.2172	−1.12646	−0.563228	−	0.826302i	$-0.690441\pi$
$389$	−22.2172	−1.12646	−0.563228	+	0.826302i	$0.690441\pi$
$390$	0	0
$391$	−20.4970	−1.03658
$392$	0	0
$393$	47.4386	2.39296
$394$	0	0
$395$	0.931435	0.0468656
$396$	0	0
$397$	−19.8781	−0.997654	−0.498827	−	0.866702i	$-0.666236\pi$
$397$	−19.8781	−0.997654	−0.498827	+	0.866702i	$0.666236\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.7759	−1.23725	−0.618624	−	0.785687i	$-0.712309\pi$
$401$	−24.7759	−1.23725	−0.618624	+	0.785687i	$0.712309\pi$
$402$	0	0
$403$	5.64787	0.281340
$404$	0	0
$405$	6.80859	0.338322
$406$	0	0
$407$	−8.38424	−0.415591
$408$	0	0
$409$	21.7215	1.07406	0.537029	−	0.843564i	$-0.319546\pi$
$409$	21.7215	1.07406	0.537029	+	0.843564i	$0.319546\pi$
$410$	0	0
$411$	−36.0251	−1.77699
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.56431	−0.0767890
$416$	0	0
$417$	3.44279	0.168594
$418$	0	0
$419$	−25.6815	−1.25462	−0.627311	−	0.778769i	$-0.715845\pi$
$419$	−25.6815	−1.25462	−0.627311	+	0.778769i	$0.715845\pi$
$420$	0	0
$421$	28.7580	1.40158	0.700789	−	0.713369i	$-0.252832\pi$
$421$	28.7580	1.40158	0.700789	+	0.713369i	$0.252832\pi$
$422$	0	0
$423$	79.6285	3.87167
$424$	0	0
$425$	−32.0677	−1.55551
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.3879	−0.501534
$430$	0	0
$431$	9.84034	0.473992	0.236996	−	0.971511i	$-0.423837\pi$
$431$	9.84034	0.473992	0.236996	+	0.971511i	$0.423837\pi$
$432$	0	0
$433$	0.424750	0.0204122	0.0102061	−	0.999948i	$-0.496751\pi$
$433$	0.424750	0.0204122	0.0102061	+	0.999948i	$0.496751\pi$
$434$	0	0
$435$	5.24014	0.251245
$436$	0	0
$437$	13.6488	0.652910
$438$	0	0
$439$	−3.98983	−0.190424	−0.0952121	−	0.995457i	$-0.530353\pi$
$439$	−3.98983	−0.190424	−0.0952121	+	0.995457i	$0.530353\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.9380	0.947285	0.473642	−	0.880717i	$-0.342939\pi$
$443$	19.9380	0.947285	0.473642	+	0.880717i	$0.342939\pi$
$444$	0	0
$445$	0.0608489	0.00288451
$446$	0	0
$447$	60.3416	2.85406
$448$	0	0
$449$	9.97100	0.470561	0.235280	−	0.971928i	$-0.424399\pi$
$449$	9.97100	0.470561	0.235280	+	0.971928i	$0.424399\pi$
$450$	0	0
$451$	−0.553067	−0.0260429
$452$	0	0
$453$	−62.7591	−2.94868
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.9248	1.63371	0.816856	−	0.576841i	$-0.195715\pi$
$457$	34.9248	1.63371	0.816856	+	0.576841i	$0.195715\pi$
$458$	0	0
$459$	100.675	4.69912
$460$	0	0
$461$	−26.8042	−1.24839	−0.624197	−	0.781267i	$-0.714574\pi$
$461$	−26.8042	−1.24839	−0.624197	+	0.781267i	$0.714574\pi$
$462$	0	0
$463$	28.1253	1.30709	0.653547	−	0.756886i	$-0.273280\pi$
$463$	28.1253	1.30709	0.653547	+	0.756886i	$0.273280\pi$
$464$	0	0
$465$	1.43948	0.0667544
$466$	0	0
$467$	−27.7857	−1.28577	−0.642884	−	0.765964i	$-0.722262\pi$
$467$	−27.7857	−1.28577	−0.642884	+	0.765964i	$0.722262\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−75.4213	−3.47523
$472$	0	0
$473$	5.69023	0.261637
$474$	0	0
$475$	21.3537	0.979773
$476$	0	0
$477$	−78.6795	−3.60249
$478$	0	0
$479$	−9.27456	−0.423766	−0.211883	−	0.977295i	$-0.567960\pi$
$479$	−9.27456	−0.423766	−0.211883	+	0.977295i	$0.567960\pi$
$480$	0	0
$481$	26.5846	1.21215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.18668	0.0538843
$486$	0	0
$487$	−10.6356	−0.481944	−0.240972	−	0.970532i	$-0.577466\pi$
$487$	−10.6356	−0.481944	−0.240972	+	0.970532i	$0.577466\pi$
$488$	0	0
$489$	16.1797	0.731672
$490$	0	0
$491$	19.9832	0.901829	0.450915	−	0.892567i	$-0.351098\pi$
$491$	19.9832	0.901829	0.450915	+	0.892567i	$0.351098\pi$
$492$	0	0
$493$	42.0989	1.89604
$494$	0	0
$495$	−1.90756	−0.0857387
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.9536	0.490351	0.245175	−	0.969479i	$-0.421154\pi$
$499$	10.9536	0.490351	0.245175	+	0.969479i	$0.421154\pi$
$500$	0	0
$501$	24.2187	1.08201
$502$	0	0
$503$	−25.5069	−1.13729	−0.568647	−	0.822582i	$-0.692533\pi$
$503$	−25.5069	−1.13729	−0.568647	+	0.822582i	$0.692533\pi$
$504$	0	0
$505$	1.69335	0.0753531
$506$	0	0
$507$	−9.65192	−0.428657
$508$	0	0
$509$	−5.16877	−0.229102	−0.114551	−	0.993417i	$-0.536543\pi$
$509$	−5.16877	−0.229102	−0.114551	+	0.993417i	$0.536543\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−67.0389	−2.95984
$514$	0	0
$515$	1.25271	0.0552008
$516$	0	0
$517$	−10.2971	−0.452867
$518$	0	0
$519$	59.7512	2.62279
$520$	0	0
$521$	−15.4837	−0.678352	−0.339176	−	0.940723i	$-0.610148\pi$
$521$	−15.4837	−0.678352	−0.339176	+	0.940723i	$0.610148\pi$
$522$	0	0
$523$	−10.2761	−0.449344	−0.224672	−	0.974434i	$-0.572131\pi$
$523$	−10.2761	−0.449344	−0.224672	+	0.974434i	$0.572131\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.5647	0.503766
$528$	0	0
$529$	−13.0334	−0.566669
$530$	0	0
$531$	−49.9294	−2.16675
$532$	0	0
$533$	1.75366	0.0759593
$534$	0	0
$535$	−0.221407	−0.00957224
$536$	0	0
$537$	7.77666	0.335588
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.5212	1.39820	0.699099	−	0.715025i	$-0.253585\pi$
$541$	32.5212	1.39820	0.699099	+	0.715025i	$0.253585\pi$
$542$	0	0
$543$	70.9150	3.04325
$544$	0	0
$545$	1.40992	0.0603944
$546$	0	0
$547$	32.7759	1.40139	0.700697	−	0.713459i	$-0.252872\pi$
$547$	32.7759	1.40139	0.700697	+	0.713459i	$0.252872\pi$
$548$	0	0
$549$	26.1494	1.11603
$550$	0	0
$551$	−28.0333	−1.19426
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.77567	0.287611
$556$	0	0
$557$	−14.3574	−0.608341	−0.304170	−	0.952618i	$-0.598379\pi$
$557$	−14.3574	−0.608341	−0.304170	+	0.952618i	$0.598379\pi$
$558$	0	0
$559$	−18.0425	−0.763116
$560$	0	0
$561$	−21.2705	−0.898043
$562$	0	0
$563$	−13.3422	−0.562305	−0.281152	−	0.959663i	$-0.590717\pi$
$563$	−13.3422	−0.562305	−0.281152	+	0.959663i	$0.590717\pi$
$564$	0	0
$565$	−1.47759	−0.0621627
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.6491	−0.781809	−0.390904	−	0.920431i	$-0.627838\pi$
$569$	−18.6491	−0.781809	−0.390904	+	0.920431i	$0.627838\pi$
$570$	0	0
$571$	−32.0508	−1.34128	−0.670642	−	0.741781i	$-0.733981\pi$
$571$	−32.0508	−1.34128	−0.670642	+	0.741781i	$0.733981\pi$
$572$	0	0
$573$	−23.6588	−0.988359
$574$	0	0
$575$	15.5929	0.650268
$576$	0	0
$577$	16.4352	0.684206	0.342103	−	0.939662i	$-0.388861\pi$
$577$	16.4352	0.684206	0.342103	+	0.939662i	$0.388861\pi$
$578$	0	0
$579$	−12.0104	−0.499137
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.1744	0.421380
$584$	0	0
$585$	6.04848	0.250074
$586$	0	0
$587$	11.8148	0.487647	0.243824	−	0.969820i	$-0.421598\pi$
$587$	11.8148	0.487647	0.243824	+	0.969820i	$0.421598\pi$
$588$	0	0
$589$	−7.70083	−0.317307
$590$	0	0
$591$	−12.5022	−0.514273
$592$	0	0
$593$	20.7452	0.851905	0.425952	−	0.904746i	$-0.359939\pi$
$593$	20.7452	0.851905	0.425952	+	0.904746i	$0.359939\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−63.4978	−2.59879
$598$	0	0
$599$	−8.71038	−0.355897	−0.177948	−	0.984040i	$-0.556946\pi$
$599$	−8.71038	−0.355897	−0.177948	+	0.984040i	$0.556946\pi$
$600$	0	0
$601$	−46.2310	−1.88580	−0.942902	−	0.333072i	$-0.891915\pi$
$601$	−46.2310	−1.88580	−0.942902	+	0.333072i	$0.891915\pi$
$602$	0	0
$603$	28.2632	1.15097
$604$	0	0
$605$	0.246676	0.0100288
$606$	0	0
$607$	−30.1304	−1.22295	−0.611477	−	0.791262i	$-0.709424\pi$
$607$	−30.1304	−1.22295	−0.611477	+	0.791262i	$0.709424\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.6499	1.32088
$612$	0	0
$613$	41.7643	1.68684	0.843421	−	0.537253i	$-0.180538\pi$
$613$	41.7643	1.68684	0.843421	+	0.537253i	$0.180538\pi$
$614$	0	0
$615$	0.446958	0.0180231
$616$	0	0
$617$	8.94182	0.359984	0.179992	−	0.983668i	$-0.442393\pi$
$617$	8.94182	0.359984	0.179992	+	0.983668i	$0.442393\pi$
$618$	0	0
$619$	−30.9126	−1.24248	−0.621241	−	0.783620i	$-0.713371\pi$
$619$	−30.9126	−1.24248	−0.621241	+	0.783620i	$0.713371\pi$
$620$	0	0
$621$	−48.9532	−1.96442
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.0910	0.963639
$626$	0	0
$627$	14.1639	0.565651
$628$	0	0
$629$	54.4352	2.17047
$630$	0	0
$631$	−30.5300	−1.21538	−0.607691	−	0.794174i	$-0.707904\pi$
$631$	−30.5300	−1.21538	−0.607691	+	0.794174i	$0.707904\pi$
$632$	0	0
$633$	−79.0464	−3.14181
$634$	0	0
$635$	−0.570805	−0.0226517
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.66914	−0.105589
$640$	0	0
$641$	−10.4287	−0.411907	−0.205953	−	0.978562i	$-0.566030\pi$
$641$	−10.4287	−0.411907	−0.205953	+	0.978562i	$0.566030\pi$
$642$	0	0
$643$	−16.4973	−0.650590	−0.325295	−	0.945613i	$-0.605464\pi$
$643$	−16.4973	−0.650590	−0.325295	+	0.945613i	$0.605464\pi$
$644$	0	0
$645$	−4.59852	−0.181067
$646$	0	0
$647$	41.0615	1.61429	0.807147	−	0.590350i	$-0.201010\pi$
$647$	41.0615	1.61429	0.807147	+	0.590350i	$0.201010\pi$
$648$	0	0
$649$	6.45660	0.253444
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.1454	−0.788350	−0.394175	−	0.919035i	$-0.628970\pi$
$653$	−20.1454	−0.788350	−0.394175	+	0.919035i	$0.628970\pi$
$654$	0	0
$655$	3.57187	0.139565
$656$	0	0
$657$	−23.0401	−0.898881
$658$	0	0
$659$	−6.46125	−0.251694	−0.125847	−	0.992050i	$-0.540165\pi$
$659$	−6.46125	−0.251694	−0.125847	+	0.992050i	$0.540165\pi$
$660$	0	0
$661$	−14.3775	−0.559221	−0.279611	−	0.960114i	$-0.590205\pi$
$661$	−14.3775	−0.559221	−0.279611	+	0.960114i	$0.590205\pi$
$662$	0	0
$663$	67.4443	2.61932
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.4705	−0.792620
$668$	0	0
$669$	48.1434	1.86133
$670$	0	0
$671$	−3.38149	−0.130541
$672$	0	0
$673$	27.0790	1.04382	0.521909	−	0.853001i	$-0.325220\pi$
$673$	27.0790	1.04382	0.521909	+	0.853001i	$0.325220\pi$
$674$	0	0
$675$	−76.5878	−2.94786
$676$	0	0
$677$	−15.8329	−0.608506	−0.304253	−	0.952591i	$-0.598407\pi$
$677$	−15.8329	−0.608506	−0.304253	+	0.952591i	$0.598407\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30.1251	1.15440
$682$	0	0
$683$	−37.0041	−1.41592	−0.707962	−	0.706251i	$-0.750385\pi$
$683$	−37.0041	−1.41592	−0.707962	+	0.706251i	$0.750385\pi$
$684$	0	0
$685$	−2.71250	−0.103639
$686$	0	0
$687$	−57.1159	−2.17911
$688$	0	0
$689$	−32.2608	−1.22904
$690$	0	0
$691$	14.5966	0.555283	0.277641	−	0.960685i	$-0.410447\pi$
$691$	14.5966	0.555283	0.277641	+	0.960685i	$0.410447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.259223	0.00983290
$696$	0	0
$697$	3.59082	0.136012
$698$	0	0
$699$	−90.9145	−3.43870
$700$	0	0
$701$	−40.3475	−1.52390	−0.761952	−	0.647633i	$-0.775759\pi$
$701$	−40.3475	−1.52390	−0.761952	+	0.647633i	$0.775759\pi$
$702$	0	0
$703$	−36.2480	−1.36712
$704$	0	0
$705$	8.32155	0.313408
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.763572	0.0286766	0.0143383	−	0.999897i	$-0.495436\pi$
$709$	0.763572	0.0286766	0.0143383	+	0.999897i	$0.495436\pi$
$710$	0	0
$711$	29.1998	1.09508
$712$	0	0
$713$	−5.62331	−0.210594
$714$	0	0
$715$	−0.782155	−0.0292509
$716$	0	0
$717$	73.8201	2.75686
$718$	0	0
$719$	−7.59172	−0.283123	−0.141562	−	0.989929i	$-0.545212\pi$
$719$	−7.59172	−0.283123	−0.141562	+	0.989929i	$0.545212\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.6753	0.880493
$724$	0	0
$725$	−32.0263	−1.18943
$726$	0	0
$727$	2.08478	0.0773204	0.0386602	−	0.999252i	$-0.487691\pi$
$727$	2.08478	0.0773204	0.0386602	+	0.999252i	$0.487691\pi$
$728$	0	0
$729$	61.0421	2.26082
$730$	0	0
$731$	−36.9442	−1.36643
$732$	0	0
$733$	37.7859	1.39566	0.697828	−	0.716266i	$-0.254150\pi$
$733$	37.7859	1.39566	0.697828	+	0.716266i	$0.254150\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.65484	−0.134628
$738$	0	0
$739$	−36.8466	−1.35543	−0.677713	−	0.735327i	$-0.737029\pi$
$739$	−36.8466	−1.35543	−0.677713	+	0.735327i	$0.737029\pi$
$740$	0	0
$741$	−44.9106	−1.64983
$742$	0	0
$743$	44.4863	1.63204	0.816022	−	0.578020i	$-0.196175\pi$
$743$	44.4863	1.63204	0.816022	+	0.578020i	$0.196175\pi$
$744$	0	0
$745$	4.54340	0.166457
$746$	0	0
$747$	−49.0399	−1.79428
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.6146	−0.971179	−0.485590	−	0.874187i	$-0.661395\pi$
$751$	−26.6146	−0.971179	−0.485590	+	0.874187i	$0.661395\pi$
$752$	0	0
$753$	42.2105	1.53824
$754$	0	0
$755$	−4.72542	−0.171976
$756$	0	0
$757$	−6.38731	−0.232151	−0.116075	−	0.993240i	$-0.537031\pi$
$757$	−6.38731	−0.232151	−0.116075	+	0.993240i	$0.537031\pi$
$758$	0	0
$759$	10.3428	0.375418
$760$	0	0
$761$	−40.1614	−1.45585	−0.727925	−	0.685657i	$-0.759515\pi$
$761$	−40.1614	−1.45585	−0.727925	+	0.685657i	$0.759515\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	12.3850	0.447780
$766$	0	0
$767$	−20.4725	−0.739218
$768$	0	0
$769$	0.481136	0.0173502	0.00867510	−	0.999962i	$-0.497239\pi$
$769$	0.481136	0.0173502	0.00867510	+	0.999962i	$0.497239\pi$
$770$	0	0
$771$	−101.866	−3.66861
$772$	0	0
$773$	−24.1864	−0.869924	−0.434962	−	0.900449i	$-0.643238\pi$
$773$	−24.1864	−0.869924	−0.434962	+	0.900449i	$0.643238\pi$
$774$	0	0
$775$	−8.79772	−0.316023
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.39110	−0.0856701
$780$	0	0
$781$	0.345158	0.0123507
$782$	0	0
$783$	100.545	3.59319
$784$	0	0
$785$	−5.67881	−0.202686
$786$	0	0
$787$	37.8595	1.34954	0.674772	−	0.738026i	$-0.264242\pi$
$787$	37.8595	1.34954	0.674772	+	0.738026i	$0.264242\pi$
$788$	0	0
$789$	24.5781	0.875004
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.7220	0.380749
$794$	0	0
$795$	−8.22237	−0.291618
$796$	0	0
$797$	27.0930	0.959683	0.479842	−	0.877355i	$-0.340694\pi$
$797$	27.0930	0.959683	0.479842	+	0.877355i	$0.340694\pi$
$798$	0	0
$799$	66.8547	2.36515
$800$	0	0
$801$	1.90756	0.0674005
$802$	0	0
$803$	2.97942	0.105141
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.34758	0.223445
$808$	0	0
$809$	15.7254	0.552876	0.276438	−	0.961032i	$-0.410846\pi$
$809$	15.7254	0.552876	0.276438	+	0.961032i	$0.410846\pi$
$810$	0	0
$811$	32.3327	1.13536	0.567678	−	0.823251i	$-0.307842\pi$
$811$	32.3327	1.13536	0.567678	+	0.823251i	$0.307842\pi$
$812$	0	0
$813$	75.2811	2.64023
$814$	0	0
$815$	1.21824	0.0426732
$816$	0	0
$817$	24.6008	0.860674
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.55190	0.333364	0.166682	−	0.986011i	$-0.446695\pi$
$821$	9.55190	0.333364	0.166682	+	0.986011i	$0.446695\pi$
$822$	0	0
$823$	−28.6991	−1.00039	−0.500194	−	0.865913i	$-0.666738\pi$
$823$	−28.6991	−1.00039	−0.500194	+	0.865913i	$0.666738\pi$
$824$	0	0
$825$	16.1813	0.563362
$826$	0	0
$827$	9.66064	0.335933	0.167967	−	0.985793i	$-0.446280\pi$
$827$	9.66064	0.335933	0.167967	+	0.985793i	$0.446280\pi$
$828$	0	0
$829$	−16.4010	−0.569629	−0.284815	−	0.958583i	$-0.591932\pi$
$829$	−16.4010	−0.569629	−0.284815	+	0.958583i	$0.591932\pi$
$830$	0	0
$831$	−18.0874	−0.627445
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.82353	0.0631060
$836$	0	0
$837$	27.6201	0.954690
$838$	0	0
$839$	−15.0128	−0.518301	−0.259150	−	0.965837i	$-0.583443\pi$
$839$	−15.0128	−0.518301	−0.259150	+	0.965837i	$0.583443\pi$
$840$	0	0
$841$	13.0444	0.449807
$842$	0	0
$843$	−23.8503	−0.821448
$844$	0	0
$845$	−0.726738	−0.0250005
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−43.4416	−1.49091
$850$	0	0
$851$	−26.4690	−0.907346
$852$	0	0
$853$	−57.4026	−1.96543	−0.982714	−	0.185133i	$-0.940729\pi$
$853$	−57.4026	−1.96543	−0.982714	+	0.185133i	$0.940729\pi$
$854$	0	0
$855$	−8.24706	−0.282044
$856$	0	0
$857$	−32.1817	−1.09931	−0.549654	−	0.835393i	$-0.685240\pi$
$857$	−32.1817	−1.09931	−0.549654	+	0.835393i	$0.685240\pi$
$858$	0	0
$859$	23.9856	0.818380	0.409190	−	0.912449i	$-0.365811\pi$
$859$	23.9856	0.818380	0.409190	+	0.912449i	$0.365811\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.7485	−1.25093	−0.625467	−	0.780251i	$-0.715091\pi$
$863$	−36.7485	−1.25093	−0.625467	+	0.780251i	$0.715091\pi$
$864$	0	0
$865$	4.49894	0.152969
$866$	0	0
$867$	82.4060	2.79865
$868$	0	0
$869$	−3.77595	−0.128090
$870$	0	0
$871$	11.5887	0.392669
$872$	0	0
$873$	37.2015	1.25908
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.3446	−0.855826	−0.427913	−	0.903820i	$-0.640751\pi$
$877$	−25.3446	−0.855826	−0.427913	+	0.903820i	$0.640751\pi$
$878$	0	0
$879$	18.2389	0.615182
$880$	0	0
$881$	12.7302	0.428890	0.214445	−	0.976736i	$-0.431206\pi$
$881$	12.7302	0.428890	0.214445	+	0.976736i	$0.431206\pi$
$882$	0	0
$883$	−10.0508	−0.338235	−0.169117	−	0.985596i	$-0.554092\pi$
$883$	−10.0508	−0.338235	−0.169117	+	0.985596i	$0.554092\pi$
$884$	0	0
$885$	−5.21786	−0.175396
$886$	0	0
$887$	−18.9974	−0.637871	−0.318936	−	0.947776i	$-0.603325\pi$
$887$	−18.9974	−0.637871	−0.318936	+	0.947776i	$0.603325\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−27.6014	−0.924682
$892$	0	0
$893$	−44.5180	−1.48974
$894$	0	0
$895$	0.585541	0.0195725
$896$	0	0
$897$	−32.7947	−1.09498
$898$	0	0
$899$	11.5497	0.385205
$900$	0	0
$901$	−66.0579	−2.20071
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.33951	0.177491
$906$	0	0
$907$	59.1419	1.96378	0.981888	−	0.189464i	$-0.0606749\pi$
$907$	59.1419	1.96378	0.981888	+	0.189464i	$0.0606749\pi$
$908$	0	0
$909$	53.0852	1.76073
$910$	0	0
$911$	11.5307	0.382028	0.191014	−	0.981587i	$-0.438822\pi$
$911$	11.5307	0.382028	0.191014	+	0.981587i	$0.438822\pi$
$912$	0	0
$913$	6.34157	0.209875
$914$	0	0
$915$	2.73273	0.0903414
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.8583	−0.918962	−0.459481	−	0.888188i	$-0.651964\pi$
$919$	−27.8583	−0.918962	−0.459481	+	0.888188i	$0.651964\pi$
$920$	0	0
$921$	−46.6742	−1.53797
$922$	0	0
$923$	−1.09442	−0.0360233
$924$	0	0
$925$	−41.4110	−1.36159
$926$	0	0
$927$	39.2713	1.28984
$928$	0	0
$929$	51.3944	1.68619	0.843097	−	0.537761i	$-0.180730\pi$
$929$	51.3944	1.68619	0.843097	+	0.537761i	$0.180730\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−11.6904	−0.382726
$934$	0	0
$935$	−1.60156	−0.0523765
$936$	0	0
$937$	−8.07437	−0.263778	−0.131889	−	0.991264i	$-0.542104\pi$
$937$	−8.07437	−0.263778	−0.131889	+	0.991264i	$0.542104\pi$
$938$	0	0
$939$	−29.8585	−0.974397
$940$	0	0
$941$	24.3020	0.792223	0.396112	−	0.918202i	$-0.370359\pi$
$941$	24.3020	0.792223	0.396112	+	0.918202i	$0.370359\pi$
$942$	0	0
$943$	−1.74603	−0.0568586
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.5660	−1.31822	−0.659109	−	0.752047i	$-0.729067\pi$
$947$	−40.5660	−1.31822	−0.659109	+	0.752047i	$0.729067\pi$
$948$	0	0
$949$	−9.44710	−0.306666
$950$	0	0
$951$	−38.5083	−1.24872
$952$	0	0
$953$	−4.52347	−0.146529	−0.0732647	−	0.997313i	$-0.523342\pi$
$953$	−4.52347	−0.146529	−0.0732647	+	0.997313i	$0.523342\pi$
$954$	0	0
$955$	−1.78138	−0.0576440
$956$	0	0
$957$	−21.2430	−0.686690
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.8273	−0.897653
$962$	0	0
$963$	−6.94092	−0.223668
$964$	0	0
$965$	−0.904322	−0.0291112
$966$	0	0
$967$	48.3508	1.55486	0.777428	−	0.628972i	$-0.216524\pi$
$967$	48.3508	1.55486	0.777428	+	0.628972i	$0.216524\pi$
$968$	0	0
$969$	−91.9599	−2.95418
$970$	0	0
$971$	12.7515	0.409217	0.204608	−	0.978844i	$-0.434408\pi$
$971$	12.7515	0.409217	0.204608	+	0.978844i	$0.434408\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−51.3076	−1.64316
$976$	0	0
$977$	21.9973	0.703757	0.351879	−	0.936046i	$-0.385543\pi$
$977$	21.9973	0.703757	0.351879	+	0.936046i	$0.385543\pi$
$978$	0	0
$979$	−0.246676	−0.00788379
$980$	0	0
$981$	44.2000	1.41120
$982$	0	0
$983$	−51.4813	−1.64200	−0.820999	−	0.570930i	$-0.806583\pi$
$983$	−51.4813	−1.64200	−0.820999	+	0.570930i	$0.806583\pi$
$984$	0	0
$985$	−0.941350	−0.0299939
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.9640	0.571223
$990$	0	0
$991$	12.8667	0.408724	0.204362	−	0.978895i	$-0.434488\pi$
$991$	12.8667	0.408724	0.204362	+	0.978895i	$0.434488\pi$
$992$	0	0
$993$	84.1144	2.66929
$994$	0	0
$995$	−4.78104	−0.151569
$996$	0	0
$997$	30.0438	0.951496	0.475748	−	0.879582i	$-0.342177\pi$
$997$	30.0438	0.951496	0.475748	+	0.879582i	$0.342177\pi$
$998$	0	0
$999$	130.008	4.11328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.df.1.10		10
4.3	odd	2		539.2.a.l.1.7	✓	10
7.6	odd	2	inner	8624.2.a.df.1.1		10
12.11	even	2		4851.2.a.cg.1.3		10
28.3	even	6		539.2.e.o.177.3		20
28.11	odd	6		539.2.e.o.177.4		20
28.19	even	6		539.2.e.o.67.3		20
28.23	odd	6		539.2.e.o.67.4		20
28.27	even	2		539.2.a.l.1.8	yes	10
44.43	even	2		5929.2.a.bv.1.3		10
84.83	odd	2		4851.2.a.cg.1.4		10
308.307	odd	2		5929.2.a.bv.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.7	✓	10	4.3	odd	2
539.2.a.l.1.8	yes	10	28.27	even	2
539.2.e.o.67.3		20	28.19	even	6
539.2.e.o.67.4		20	28.23	odd	6
539.2.e.o.177.3		20	28.3	even	6
539.2.e.o.177.4		20	28.11	odd	6
4851.2.a.cg.1.3		10	12.11	even	2
4851.2.a.cg.1.4		10	84.83	odd	2
5929.2.a.bv.1.3		10	44.43	even	2
5929.2.a.bv.1.4		10	308.307	odd	2
8624.2.a.df.1.1		10	7.6	odd	2	inner
8624.2.a.df.1.10		10	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27614 q^{3} +0.246676 q^{5} +7.73309 q^{9} +O(q^{10})\)	\(q+3.27614 q^{3} +0.246676 q^{5} +7.73309 q^{9} -1.00000 q^{11} +3.17078 q^{13} +0.808144 q^{15} +6.49256 q^{17} -4.32335 q^{19} -3.15700 q^{23} -4.93915 q^{25} +15.5063 q^{27} +6.48417 q^{29} +1.78122 q^{31} -3.27614 q^{33} +8.38424 q^{37} +10.3879 q^{39} +0.553067 q^{41} -5.69023 q^{43} +1.90756 q^{45} +10.2971 q^{47} +21.2705 q^{51} -10.1744 q^{53} -0.246676 q^{55} -14.1639 q^{57} -6.45660 q^{59} +3.38149 q^{61} +0.782155 q^{65} +3.65484 q^{67} -10.3428 q^{69} -0.345158 q^{71} -2.97942 q^{73} -16.1813 q^{75} +3.77595 q^{79} +27.6014 q^{81} -6.34157 q^{83} +1.60156 q^{85} +21.2430 q^{87} +0.246676 q^{89} +5.83553 q^{93} -1.06646 q^{95} +4.81068 q^{97} -7.73309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 22 q^{9}+O(q^{10}) \) 10 * q + 22 * q^9	\( 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93} + 64 q^{95} - 22 q^{99}+O(q^{100}) \) 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more