LMFDB - Embedded newform 6525.2.a.cd.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.23642$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} -3.05552 q^{8} +O(q^{10})$	$q+1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} -3.05552 q^{8} +2.30259 q^{11} +5.57932 q^{13} -4.05142 q^{14} -2.83539 q^{16} -1.94031 q^{17} -3.59158 q^{19} +2.84698 q^{22} -1.66263 q^{23} +6.89840 q^{26} +1.54419 q^{28} +1.00000 q^{29} +1.70877 q^{31} +2.60530 q^{32} -2.39904 q^{34} -9.16633 q^{37} -4.44071 q^{38} -8.54038 q^{41} -3.56165 q^{43} -1.08512 q^{44} -2.05571 q^{46} +11.5640 q^{47} +3.73698 q^{49} -2.62931 q^{52} +9.66312 q^{53} +10.0121 q^{56} +1.23642 q^{58} -9.83385 q^{59} +5.42714 q^{61} +2.11276 q^{62} +8.89204 q^{64} +5.20114 q^{67} +0.914391 q^{68} -6.02596 q^{71} +15.5143 q^{73} -11.3334 q^{74} +1.69257 q^{76} -7.54498 q^{77} +12.2465 q^{79} -10.5595 q^{82} +10.7179 q^{83} -4.40370 q^{86} -7.03562 q^{88} -2.53756 q^{89} -18.2820 q^{91} +0.783531 q^{92} +14.2980 q^{94} -5.89998 q^{97} +4.62048 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8	$ 9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8} + 2 q^{11} + q^{13} - 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} + 3 q^{22} + 16 q^{23} + 6 q^{26} - 4 q^{28} + 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} - 10 q^{41} + 3 q^{43} - 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} - 9 q^{52} + 22 q^{53} + 22 q^{56} + 2 q^{58} + 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} + 5 q^{67} + 39 q^{68} - 10 q^{73} - 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} - 8 q^{82} + 46 q^{83} + 28 q^{86} + 2 q^{88} + 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} + 7 q^{97} + 51 q^{98}+O(q^{100}) $ 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8 + 2 * q^11 + q^13 - 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 + 3 * q^22 + 16 * q^23 + 6 * q^26 - 4 * q^28 + 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 - 10 * q^41 + 3 * q^43 - 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 - 9 * q^52 + 22 * q^53 + 22 * q^56 + 2 * q^58 + 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 + 5 * q^67 + 39 * q^68 - 10 * q^73 - 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 - 8 * q^82 + 46 * q^83 + 28 * q^86 + 2 * q^88 + 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 + 7 * q^97 + 51 * q^98

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 $

$2$	1.23642	0.874282	0.437141	−	0.899393i	$-0.355991\pi$
$2$	1.23642	0.874282	0.437141	+	0.899393i	$0.355991\pi$
$3$	0	0
$3$	0	0
$4$	−0.471260	−0.235630
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.27673	−1.23849	−0.619244	−	0.785198i	$-0.712561\pi$
$7$	−3.27673	−1.23849	−0.619244	+	0.785198i	$0.712561\pi$
$8$	−3.05552	−1.08029
$9$	0	0
$10$	0	0
$11$	2.30259	0.694258	0.347129	−	0.937817i	$-0.387157\pi$
$11$	2.30259	0.694258	0.347129	+	0.937817i	$0.387157\pi$
$12$	0	0
$13$	5.57932	1.54743	0.773713	−	0.633536i	$-0.218397\pi$
$13$	5.57932	1.54743	0.773713	+	0.633536i	$0.218397\pi$
$14$	−4.05142	−1.08279
$15$	0	0
$16$	−2.83539	−0.708848
$17$	−1.94031	−0.470594	−0.235297	−	0.971923i	$-0.575606\pi$
$17$	−1.94031	−0.470594	−0.235297	+	0.971923i	$0.575606\pi$
$18$	0	0
$19$	−3.59158	−0.823965	−0.411982	−	0.911192i	$-0.635164\pi$
$19$	−3.59158	−0.823965	−0.411982	+	0.911192i	$0.635164\pi$
$20$	0	0
$21$	0	0
$22$	2.84698	0.606977
$23$	−1.66263	−0.346682	−0.173341	−	0.984862i	$-0.555456\pi$
$23$	−1.66263	−0.346682	−0.173341	+	0.984862i	$0.555456\pi$
$24$	0	0
$25$	0	0
$26$	6.89840	1.35289
$27$	0	0
$28$	1.54419	0.291825
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.70877	0.306903	0.153452	−	0.988156i	$-0.450961\pi$
$31$	1.70877	0.306903	0.153452	+	0.988156i	$0.450961\pi$
$32$	2.60530	0.460556
$33$	0	0
$34$	−2.39904	−0.411432
$35$	0	0
$36$	0	0
$37$	−9.16633	−1.50693	−0.753467	−	0.657485i	$-0.771620\pi$
$37$	−9.16633	−1.50693	−0.753467	+	0.657485i	$0.771620\pi$
$38$	−4.44071	−0.720378
$39$	0	0
$40$	0	0
$41$	−8.54038	−1.33378	−0.666892	−	0.745154i	$-0.732376\pi$
$41$	−8.54038	−1.33378	−0.666892	+	0.745154i	$0.732376\pi$
$42$	0	0
$43$	−3.56165	−0.543146	−0.271573	−	0.962418i	$-0.587544\pi$
$43$	−3.56165	−0.543146	−0.271573	+	0.962418i	$0.587544\pi$
$44$	−1.08512	−0.163588
$45$	0	0
$46$	−2.05571	−0.303098
$47$	11.5640	1.68678	0.843391	−	0.537300i	$-0.180556\pi$
$47$	11.5640	1.68678	0.843391	+	0.537300i	$0.180556\pi$
$48$	0	0
$49$	3.73698	0.533854
$50$	0	0
$51$	0	0
$52$	−2.62931	−0.364620
$53$	9.66312	1.32733	0.663666	−	0.748029i	$-0.269000\pi$
$53$	9.66312	1.32733	0.663666	+	0.748029i	$0.269000\pi$
$54$	0	0
$55$	0	0
$56$	10.0121	1.33793
$57$	0	0
$58$	1.23642	0.162350
$59$	−9.83385	−1.28026	−0.640129	−	0.768268i	$-0.721119\pi$
$59$	−9.83385	−1.28026	−0.640129	+	0.768268i	$0.721119\pi$
$60$	0	0
$61$	5.42714	0.694874	0.347437	−	0.937703i	$-0.387052\pi$
$61$	5.42714	0.694874	0.347437	+	0.937703i	$0.387052\pi$
$62$	2.11276	0.268320
$63$	0	0
$64$	8.89204	1.11150
$65$	0	0
$66$	0	0
$67$	5.20114	0.635421	0.317710	−	0.948188i	$-0.397086\pi$
$67$	5.20114	0.635421	0.317710	+	0.948188i	$0.397086\pi$
$68$	0.914391	0.110886
$69$	0	0
$70$	0	0
$71$	−6.02596	−0.715150	−0.357575	−	0.933884i	$-0.616396\pi$
$71$	−6.02596	−0.715150	−0.357575	+	0.933884i	$0.616396\pi$
$72$	0	0
$73$	15.5143	1.81581	0.907905	−	0.419177i	$-0.137681\pi$
$73$	15.5143	1.81581	0.907905	+	0.419177i	$0.137681\pi$
$74$	−11.3334	−1.31749
$75$	0	0
$76$	1.69257	0.194151
$77$	−7.54498	−0.859830
$78$	0	0
$79$	12.2465	1.37784	0.688922	−	0.724835i	$-0.258084\pi$
$79$	12.2465	1.37784	0.688922	+	0.724835i	$0.258084\pi$
$80$	0	0
$81$	0	0
$82$	−10.5595	−1.16610
$83$	10.7179	1.17644	0.588221	−	0.808700i	$-0.299828\pi$
$83$	10.7179	1.17644	0.588221	+	0.808700i	$0.299828\pi$
$84$	0	0
$85$	0	0
$86$	−4.40370	−0.474863
$87$	0	0
$88$	−7.03562	−0.749999
$89$	−2.53756	−0.268981	−0.134491	−	0.990915i	$-0.542940\pi$
$89$	−2.53756	−0.268981	−0.134491	+	0.990915i	$0.542940\pi$
$90$	0	0
$91$	−18.2820	−1.91647
$92$	0.783531	0.0816887
$93$	0	0
$94$	14.2980	1.47472
$95$	0	0
$96$	0	0
$97$	−5.89998	−0.599053	−0.299526	−	0.954088i	$-0.596829\pi$
$97$	−5.89998	−0.599053	−0.299526	+	0.954088i	$0.596829\pi$
$98$	4.62048	0.466739
$99$	0	0
$100$	0	0
$101$	−4.33789	−0.431636	−0.215818	−	0.976434i	$-0.569242\pi$
$101$	−4.33789	−0.431636	−0.215818	+	0.976434i	$0.569242\pi$
$102$	0	0
$103$	7.62183	0.751001	0.375501	−	0.926822i	$-0.377471\pi$
$103$	7.62183	0.751001	0.375501	+	0.926822i	$0.377471\pi$
$104$	−17.0477	−1.67167
$105$	0	0
$106$	11.9477	1.16046
$107$	4.19703	0.405742	0.202871	−	0.979205i	$-0.434973\pi$
$107$	4.19703	0.405742	0.202871	+	0.979205i	$0.434973\pi$
$108$	0	0
$109$	4.02677	0.385695	0.192847	−	0.981229i	$-0.438228\pi$
$109$	4.02677	0.385695	0.192847	+	0.981229i	$0.438228\pi$
$110$	0	0
$111$	0	0
$112$	9.29082	0.877900
$113$	7.40527	0.696629	0.348314	−	0.937378i	$-0.386754\pi$
$113$	7.40527	0.696629	0.348314	+	0.937378i	$0.386754\pi$
$114$	0	0
$115$	0	0
$116$	−0.471260	−0.0437554
$117$	0	0
$118$	−12.1588	−1.11931
$119$	6.35788	0.582826
$120$	0	0
$121$	−5.69807	−0.518006
$122$	6.71024	0.607516
$123$	0	0
$124$	−0.805274	−0.0723157
$125$	0	0
$126$	0	0
$127$	12.9643	1.15040	0.575199	−	0.818014i	$-0.304925\pi$
$127$	12.9643	1.15040	0.575199	+	0.818014i	$0.304925\pi$
$128$	5.78371	0.511213
$129$	0	0
$130$	0	0
$131$	17.6941	1.54594	0.772971	−	0.634442i	$-0.218770\pi$
$131$	17.6941	1.54594	0.772971	+	0.634442i	$0.218770\pi$
$132$	0	0
$133$	11.7686	1.02047
$134$	6.43081	0.555537
$135$	0	0
$136$	5.92866	0.508378
$137$	−22.5543	−1.92694	−0.963471	−	0.267813i	$-0.913699\pi$
$137$	−22.5543	−1.92694	−0.963471	+	0.267813i	$0.913699\pi$
$138$	0	0
$139$	−3.20075	−0.271484	−0.135742	−	0.990744i	$-0.543342\pi$
$139$	−3.20075	−0.271484	−0.135742	+	0.990744i	$0.543342\pi$
$140$	0	0
$141$	0	0
$142$	−7.45064	−0.625243
$143$	12.8469	1.07431
$144$	0	0
$145$	0	0
$146$	19.1822	1.58753
$147$	0	0
$148$	4.31973	0.355079
$149$	−0.612365	−0.0501669	−0.0250835	−	0.999685i	$-0.507985\pi$
$149$	−0.612365	−0.0501669	−0.0250835	+	0.999685i	$0.507985\pi$
$150$	0	0
$151$	12.4363	1.01205	0.506026	−	0.862518i	$-0.331114\pi$
$151$	12.4363	1.01205	0.506026	+	0.862518i	$0.331114\pi$
$152$	10.9741	0.890121
$153$	0	0
$154$	−9.32878	−0.751734
$155$	0	0
$156$	0	0
$157$	−0.378638	−0.0302186	−0.0151093	−	0.999886i	$-0.504810\pi$
$157$	−0.378638	−0.0302186	−0.0151093	+	0.999886i	$0.504810\pi$
$158$	15.1419	1.20463
$159$	0	0
$160$	0	0
$161$	5.44799	0.429362
$162$	0	0
$163$	2.54446	0.199297	0.0996487	−	0.995023i	$-0.468228\pi$
$163$	2.54446	0.199297	0.0996487	+	0.995023i	$0.468228\pi$
$164$	4.02474	0.314280
$165$	0	0
$166$	13.2519	1.02854
$167$	−10.7799	−0.834175	−0.417088	−	0.908866i	$-0.636949\pi$
$167$	−10.7799	−0.834175	−0.417088	+	0.908866i	$0.636949\pi$
$168$	0	0
$169$	18.1289	1.39453
$170$	0	0
$171$	0	0
$172$	1.67846	0.127982
$173$	23.1221	1.75794	0.878970	−	0.476877i	$-0.158231\pi$
$173$	23.1221	1.75794	0.878970	+	0.476877i	$0.158231\pi$
$174$	0	0
$175$	0	0
$176$	−6.52875	−0.492123
$177$	0	0
$178$	−3.13750	−0.235165
$179$	23.7081	1.77203	0.886014	−	0.463659i	$-0.153464\pi$
$179$	23.7081	1.77203	0.886014	+	0.463659i	$0.153464\pi$
$180$	0	0
$181$	11.6809	0.868233	0.434116	−	0.900857i	$-0.357061\pi$
$181$	11.6809	0.868233	0.434116	+	0.900857i	$0.357061\pi$
$182$	−22.6042	−1.67554
$183$	0	0
$184$	5.08020	0.374517
$185$	0	0
$186$	0	0
$187$	−4.46774	−0.326714
$188$	−5.44965	−0.397457
$189$	0	0
$190$	0	0
$191$	8.68267	0.628256	0.314128	−	0.949381i	$-0.398288\pi$
$191$	8.68267	0.628256	0.314128	+	0.949381i	$0.398288\pi$
$192$	0	0
$193$	−2.73727	−0.197033	−0.0985164	−	0.995135i	$-0.531410\pi$
$193$	−2.73727	−0.197033	−0.0985164	+	0.995135i	$0.531410\pi$
$194$	−7.29487	−0.523741
$195$	0	0
$196$	−1.76109	−0.125792
$197$	26.2457	1.86993	0.934964	−	0.354741i	$-0.115431\pi$
$197$	26.2457	1.86993	0.934964	+	0.354741i	$0.115431\pi$
$198$	0	0
$199$	9.94800	0.705195	0.352597	−	0.935775i	$-0.385299\pi$
$199$	9.94800	0.705195	0.352597	+	0.935775i	$0.385299\pi$
$200$	0	0
$201$	0	0
$202$	−5.36347	−0.377372
$203$	−3.27673	−0.229982
$204$	0	0
$205$	0	0
$206$	9.42380	0.656587
$207$	0	0
$208$	−15.8196	−1.09689
$209$	−8.26994	−0.572044
$210$	0	0
$211$	7.76180	0.534344	0.267172	−	0.963649i	$-0.413911\pi$
$211$	7.76180	0.534344	0.267172	+	0.963649i	$0.413911\pi$
$212$	−4.55384	−0.312759
$213$	0	0
$214$	5.18930	0.354733
$215$	0	0
$216$	0	0
$217$	−5.59917	−0.380096
$218$	4.97879	0.337206
$219$	0	0
$220$	0	0
$221$	−10.8256	−0.728210
$222$	0	0
$223$	−9.63976	−0.645526	−0.322763	−	0.946480i	$-0.604612\pi$
$223$	−9.63976	−0.645526	−0.322763	+	0.946480i	$0.604612\pi$
$224$	−8.53687	−0.570394
$225$	0	0
$226$	9.15603	0.609050
$227$	9.21807	0.611825	0.305913	−	0.952060i	$-0.401038\pi$
$227$	9.21807	0.611825	0.305913	+	0.952060i	$0.401038\pi$
$228$	0	0
$229$	−6.89586	−0.455691	−0.227846	−	0.973697i	$-0.573168\pi$
$229$	−6.89586	−0.455691	−0.227846	+	0.973697i	$0.573168\pi$
$230$	0	0
$231$	0	0
$232$	−3.05552	−0.200605
$233$	−13.6229	−0.892466	−0.446233	−	0.894917i	$-0.647235\pi$
$233$	−13.6229	−0.892466	−0.446233	+	0.894917i	$0.647235\pi$
$234$	0	0
$235$	0	0
$236$	4.63430	0.301667
$237$	0	0
$238$	7.86102	0.509554
$239$	−0.749144	−0.0484581	−0.0242290	−	0.999706i	$-0.507713\pi$
$239$	−0.749144	−0.0484581	−0.0242290	+	0.999706i	$0.507713\pi$
$240$	0	0
$241$	−12.5380	−0.807645	−0.403822	−	0.914837i	$-0.632319\pi$
$241$	−12.5380	−0.807645	−0.403822	+	0.914837i	$0.632319\pi$
$242$	−7.04522	−0.452884
$243$	0	0
$244$	−2.55760	−0.163733
$245$	0	0
$246$	0	0
$247$	−20.0386	−1.27503
$248$	−5.22117	−0.331545
$249$	0	0
$250$	0	0
$251$	11.7441	0.741281	0.370640	−	0.928776i	$-0.379138\pi$
$251$	11.7441	0.741281	0.370640	+	0.928776i	$0.379138\pi$
$252$	0	0
$253$	−3.82836	−0.240687
$254$	16.0294	1.00577
$255$	0	0
$256$	−10.6330	−0.664560
$257$	13.9304	0.868955	0.434477	−	0.900683i	$-0.356933\pi$
$257$	13.9304	0.868955	0.434477	+	0.900683i	$0.356933\pi$
$258$	0	0
$259$	30.0356	1.86632
$260$	0	0
$261$	0	0
$262$	21.8774	1.35159
$263$	25.8059	1.59126	0.795631	−	0.605781i	$-0.207139\pi$
$263$	25.8059	1.59126	0.795631	+	0.605781i	$0.207139\pi$
$264$	0	0
$265$	0	0
$266$	14.5510	0.892180
$267$	0	0
$268$	−2.45109	−0.149724
$269$	−11.4590	−0.698667	−0.349333	−	0.936998i	$-0.613592\pi$
$269$	−11.4590	−0.698667	−0.349333	+	0.936998i	$0.613592\pi$
$270$	0	0
$271$	−21.4783	−1.30472	−0.652358	−	0.757911i	$-0.726220\pi$
$271$	−21.4783	−1.30472	−0.652358	+	0.757911i	$0.726220\pi$
$272$	5.50154	0.333580
$273$	0	0
$274$	−27.8866	−1.68469
$275$	0	0
$276$	0	0
$277$	−7.16907	−0.430747	−0.215374	−	0.976532i	$-0.569097\pi$
$277$	−7.16907	−0.430747	−0.215374	+	0.976532i	$0.569097\pi$
$278$	−3.95748	−0.237354
$279$	0	0
$280$	0	0
$281$	14.0042	0.835420	0.417710	−	0.908580i	$-0.362833\pi$
$281$	14.0042	0.835420	0.417710	+	0.908580i	$0.362833\pi$
$282$	0	0
$283$	−10.3513	−0.615324	−0.307662	−	0.951496i	$-0.599547\pi$
$283$	−10.3513	−0.615324	−0.307662	+	0.951496i	$0.599547\pi$
$284$	2.83980	0.168511
$285$	0	0
$286$	15.8842	0.939253
$287$	27.9846	1.65188
$288$	0	0
$289$	−13.2352	−0.778541
$290$	0	0
$291$	0	0
$292$	−7.31127	−0.427859
$293$	26.6063	1.55435	0.777177	−	0.629282i	$-0.216651\pi$
$293$	26.6063	1.55435	0.777177	+	0.629282i	$0.216651\pi$
$294$	0	0
$295$	0	0
$296$	28.0079	1.62793
$297$	0	0
$298$	−0.757142	−0.0438601
$299$	−9.27634	−0.536465
$300$	0	0
$301$	11.6706	0.672680
$302$	15.3765	0.884819
$303$	0	0
$304$	10.1835	0.584066
$305$	0	0
$306$	0	0
$307$	−29.2808	−1.67114	−0.835571	−	0.549383i	$-0.814863\pi$
$307$	−29.2808	−1.67114	−0.835571	+	0.549383i	$0.814863\pi$
$308$	3.55565	0.202602
$309$	0	0
$310$	0	0
$311$	−5.03104	−0.285284	−0.142642	−	0.989774i	$-0.545560\pi$
$311$	−5.03104	−0.285284	−0.142642	+	0.989774i	$0.545560\pi$
$312$	0	0
$313$	27.5875	1.55934	0.779670	−	0.626191i	$-0.215387\pi$
$313$	27.5875	1.55934	0.779670	+	0.626191i	$0.215387\pi$
$314$	−0.468157	−0.0264196
$315$	0	0
$316$	−5.77131	−0.324662
$317$	−7.70663	−0.432848	−0.216424	−	0.976300i	$-0.569439\pi$
$317$	−7.70663	−0.432848	−0.216424	+	0.976300i	$0.569439\pi$
$318$	0	0
$319$	2.30259	0.128920
$320$	0	0
$321$	0	0
$322$	6.73601	0.375383
$323$	6.96878	0.387753
$324$	0	0
$325$	0	0
$326$	3.14602	0.174242
$327$	0	0
$328$	26.0953	1.44087
$329$	−37.8921	−2.08906
$330$	0	0
$331$	18.9943	1.04402	0.522010	−	0.852940i	$-0.325182\pi$
$331$	18.9943	1.04402	0.522010	+	0.852940i	$0.325182\pi$
$332$	−5.05092	−0.277205
$333$	0	0
$334$	−13.3285	−0.729305
$335$	0	0
$336$	0	0
$337$	20.1983	1.10027	0.550136	−	0.835075i	$-0.314576\pi$
$337$	20.1983	1.10027	0.550136	+	0.835075i	$0.314576\pi$
$338$	22.4149	1.21921
$339$	0	0
$340$	0	0
$341$	3.93459	0.213070
$342$	0	0
$343$	10.6921	0.577317
$344$	10.8827	0.586755
$345$	0	0
$346$	28.5887	1.53694
$347$	23.7995	1.27763	0.638813	−	0.769362i	$-0.279426\pi$
$347$	23.7995	1.27763	0.638813	+	0.769362i	$0.279426\pi$
$348$	0	0
$349$	−22.8554	−1.22342	−0.611710	−	0.791082i	$-0.709518\pi$
$349$	−22.8554	−1.22342	−0.611710	+	0.791082i	$0.709518\pi$
$350$	0	0
$351$	0	0
$352$	5.99894	0.319745
$353$	−17.5338	−0.933231	−0.466615	−	0.884460i	$-0.654527\pi$
$353$	−17.5338	−0.933231	−0.466615	+	0.884460i	$0.654527\pi$
$354$	0	0
$355$	0	0
$356$	1.19585	0.0633801
$357$	0	0
$358$	29.3132	1.54925
$359$	26.6594	1.40703	0.703513	−	0.710682i	$-0.251613\pi$
$359$	26.6594	1.40703	0.703513	+	0.710682i	$0.251613\pi$
$360$	0	0
$361$	−6.10055	−0.321082
$362$	14.4425	0.759081
$363$	0	0
$364$	8.61556	0.451578
$365$	0	0
$366$	0	0
$367$	22.5869	1.17902	0.589512	−	0.807760i	$-0.299320\pi$
$367$	22.5869	1.17902	0.589512	+	0.807760i	$0.299320\pi$
$368$	4.71420	0.245745
$369$	0	0
$370$	0	0
$371$	−31.6634	−1.64388
$372$	0	0
$373$	−11.0978	−0.574623	−0.287311	−	0.957837i	$-0.592761\pi$
$373$	−11.0978	−0.574623	−0.287311	+	0.957837i	$0.592761\pi$
$374$	−5.52402	−0.285640
$375$	0	0
$376$	−35.3340	−1.82221
$377$	5.57932	0.287350
$378$	0	0
$379$	−28.2474	−1.45097	−0.725485	−	0.688238i	$-0.758384\pi$
$379$	−28.2474	−1.45097	−0.725485	+	0.688238i	$0.758384\pi$
$380$	0	0
$381$	0	0
$382$	10.7354	0.549273
$383$	7.24537	0.370221	0.185110	−	0.982718i	$-0.440736\pi$
$383$	7.24537	0.370221	0.185110	+	0.982718i	$0.440736\pi$
$384$	0	0
$385$	0	0
$386$	−3.38442	−0.172262
$387$	0	0
$388$	2.78043	0.141155
$389$	−34.4482	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$389$	−34.4482	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$390$	0	0
$391$	3.22601	0.163147
$392$	−11.4184	−0.576717
$393$	0	0
$394$	32.4508	1.63485
$395$	0	0
$396$	0	0
$397$	13.6412	0.684632	0.342316	−	0.939585i	$-0.388789\pi$
$397$	13.6412	0.684632	0.342316	+	0.939585i	$0.388789\pi$
$398$	12.2999	0.616540
$399$	0	0
$400$	0	0
$401$	−10.3741	−0.518059	−0.259029	−	0.965869i	$-0.583403\pi$
$401$	−10.3741	−0.518059	−0.259029	+	0.965869i	$0.583403\pi$
$402$	0	0
$403$	9.53376	0.474910
$404$	2.04428	0.101707
$405$	0	0
$406$	−4.05142	−0.201069
$407$	−21.1063	−1.04620
$408$	0	0
$409$	38.5394	1.90565	0.952825	−	0.303520i	$-0.0981620\pi$
$409$	38.5394	1.90565	0.952825	+	0.303520i	$0.0981620\pi$
$410$	0	0
$411$	0	0
$412$	−3.59187	−0.176959
$413$	32.2229	1.58558
$414$	0	0
$415$	0	0
$416$	14.5358	0.712677
$417$	0	0
$418$	−10.2251	−0.500128
$419$	−9.87578	−0.482464	−0.241232	−	0.970468i	$-0.577551\pi$
$419$	−9.87578	−0.482464	−0.241232	+	0.970468i	$0.577551\pi$
$420$	0	0
$421$	−10.2857	−0.501292	−0.250646	−	0.968079i	$-0.580643\pi$
$421$	−10.2857	−0.501292	−0.250646	+	0.968079i	$0.580643\pi$
$422$	9.59686	0.467168
$423$	0	0
$424$	−29.5259	−1.43390
$425$	0	0
$426$	0	0
$427$	−17.7833	−0.860594
$428$	−1.97789	−0.0956051
$429$	0	0
$430$	0	0
$431$	15.2705	0.735553	0.367776	−	0.929914i	$-0.380119\pi$
$431$	15.2705	0.735553	0.367776	+	0.929914i	$0.380119\pi$
$432$	0	0
$433$	13.8370	0.664966	0.332483	−	0.943109i	$-0.392114\pi$
$433$	13.8370	0.664966	0.332483	+	0.943109i	$0.392114\pi$
$434$	−6.92294	−0.332312
$435$	0	0
$436$	−1.89766	−0.0908813
$437$	5.97146	0.285654
$438$	0	0
$439$	−6.48836	−0.309673	−0.154836	−	0.987940i	$-0.549485\pi$
$439$	−6.48836	−0.309673	−0.154836	+	0.987940i	$0.549485\pi$
$440$	0	0
$441$	0	0
$442$	−13.3850	−0.636661
$443$	22.1026	1.05012	0.525062	−	0.851064i	$-0.324042\pi$
$443$	22.1026	1.05012	0.525062	+	0.851064i	$0.324042\pi$
$444$	0	0
$445$	0	0
$446$	−11.9188	−0.564372
$447$	0	0
$448$	−29.1368	−1.37659
$449$	−0.247203	−0.0116662	−0.00583311	−	0.999983i	$-0.501857\pi$
$449$	−0.247203	−0.0116662	−0.00583311	+	0.999983i	$0.501857\pi$
$450$	0	0
$451$	−19.6650	−0.925990
$452$	−3.48981	−0.164147
$453$	0	0
$454$	11.3974	0.534908
$455$	0	0
$456$	0	0
$457$	−35.4153	−1.65666	−0.828328	−	0.560243i	$-0.810708\pi$
$457$	−35.4153	−1.65666	−0.828328	+	0.560243i	$0.810708\pi$
$458$	−8.52619	−0.398403
$459$	0	0
$460$	0	0
$461$	−13.5690	−0.631969	−0.315985	−	0.948764i	$-0.602335\pi$
$461$	−13.5690	−0.631969	−0.315985	+	0.948764i	$0.602335\pi$
$462$	0	0
$463$	−7.88217	−0.366316	−0.183158	−	0.983084i	$-0.558632\pi$
$463$	−7.88217	−0.366316	−0.183158	+	0.983084i	$0.558632\pi$
$464$	−2.83539	−0.131630
$465$	0	0
$466$	−16.8437	−0.780268
$467$	20.8411	0.964411	0.482206	−	0.876058i	$-0.339836\pi$
$467$	20.8411	0.964411	0.482206	+	0.876058i	$0.339836\pi$
$468$	0	0
$469$	−17.0428	−0.786961
$470$	0	0
$471$	0	0
$472$	30.0475	1.38305
$473$	−8.20102	−0.377083
$474$	0	0
$475$	0	0
$476$	−2.99622	−0.137331
$477$	0	0
$478$	−0.926258	−0.0423661
$479$	−28.9598	−1.32321	−0.661604	−	0.749853i	$-0.730124\pi$
$479$	−28.9598	−1.32321	−0.661604	+	0.749853i	$0.730124\pi$
$480$	0	0
$481$	−51.1419	−2.33187
$482$	−15.5023	−0.706110
$483$	0	0
$484$	2.68527	0.122058
$485$	0	0
$486$	0	0
$487$	−23.3354	−1.05743	−0.528714	−	0.848800i	$-0.677326\pi$
$487$	−23.3354	−1.05743	−0.528714	+	0.848800i	$0.677326\pi$
$488$	−16.5827	−0.750665
$489$	0	0
$490$	0	0
$491$	25.9951	1.17314	0.586570	−	0.809898i	$-0.300478\pi$
$491$	25.9951	1.17314	0.586570	+	0.809898i	$0.300478\pi$
$492$	0	0
$493$	−1.94031	−0.0873872
$494$	−24.7762	−1.11473
$495$	0	0
$496$	−4.84502	−0.217548
$497$	19.7455	0.885706
$498$	0	0
$499$	−31.9575	−1.43061	−0.715307	−	0.698810i	$-0.753713\pi$
$499$	−31.9575	−1.43061	−0.715307	+	0.698810i	$0.753713\pi$
$500$	0	0
$501$	0	0
$502$	14.5207	0.648089
$503$	4.75172	0.211869	0.105934	−	0.994373i	$-0.466217\pi$
$503$	4.75172	0.211869	0.105934	+	0.994373i	$0.466217\pi$
$504$	0	0
$505$	0	0
$506$	−4.73346	−0.210428
$507$	0	0
$508$	−6.10957	−0.271068
$509$	−10.8427	−0.480595	−0.240297	−	0.970699i	$-0.577245\pi$
$509$	−10.8427	−0.480595	−0.240297	+	0.970699i	$0.577245\pi$
$510$	0	0
$511$	−50.8361	−2.24886
$512$	−24.7143	−1.09223
$513$	0	0
$514$	17.2239	0.759712
$515$	0	0
$516$	0	0
$517$	26.6272	1.17106
$518$	37.1367	1.63169
$519$	0	0
$520$	0	0
$521$	−33.8929	−1.48487	−0.742437	−	0.669916i	$-0.766330\pi$
$521$	−33.8929	−1.48487	−0.742437	+	0.669916i	$0.766330\pi$
$522$	0	0
$523$	−25.7652	−1.12664	−0.563318	−	0.826240i	$-0.690475\pi$
$523$	−25.7652	−1.12664	−0.563318	+	0.826240i	$0.690475\pi$
$524$	−8.33853	−0.364270
$525$	0	0
$526$	31.9070	1.39121
$527$	−3.31554	−0.144427
$528$	0	0
$529$	−20.2357	−0.879812
$530$	0	0
$531$	0	0
$532$	−5.54610	−0.240454
$533$	−47.6496	−2.06393
$534$	0	0
$535$	0	0
$536$	−15.8922	−0.686438
$537$	0	0
$538$	−14.1681	−0.610832
$539$	8.60473	0.370632
$540$	0	0
$541$	4.96888	0.213629	0.106814	−	0.994279i	$-0.465935\pi$
$541$	4.96888	0.213629	0.106814	+	0.994279i	$0.465935\pi$
$542$	−26.5563	−1.14069
$543$	0	0
$544$	−5.05509	−0.216735
$545$	0	0
$546$	0	0
$547$	9.03353	0.386246	0.193123	−	0.981175i	$-0.438138\pi$
$547$	9.03353	0.386246	0.193123	+	0.981175i	$0.438138\pi$
$548$	10.6289	0.454046
$549$	0	0
$550$	0	0
$551$	−3.59158	−0.153006
$552$	0	0
$553$	−40.1287	−1.70644
$554$	−8.86399	−0.376595
$555$	0	0
$556$	1.50839	0.0639699
$557$	22.5265	0.954477	0.477238	−	0.878774i	$-0.341638\pi$
$557$	22.5265	0.954477	0.477238	+	0.878774i	$0.341638\pi$
$558$	0	0
$559$	−19.8716	−0.840479
$560$	0	0
$561$	0	0
$562$	17.3151	0.730393
$563$	14.0601	0.592561	0.296280	−	0.955101i	$-0.404254\pi$
$563$	14.0601	0.592561	0.296280	+	0.955101i	$0.404254\pi$
$564$	0	0
$565$	0	0
$566$	−12.7986	−0.537967
$567$	0	0
$568$	18.4125	0.772570
$569$	5.59236	0.234444	0.117222	−	0.993106i	$-0.462601\pi$
$569$	5.59236	0.234444	0.117222	+	0.993106i	$0.462601\pi$
$570$	0	0
$571$	−20.9909	−0.878443	−0.439221	−	0.898379i	$-0.644746\pi$
$571$	−20.9909	−0.878443	−0.439221	+	0.898379i	$0.644746\pi$
$572$	−6.05424	−0.253140
$573$	0	0
$574$	34.6007	1.44421
$575$	0	0
$576$	0	0
$577$	−25.7139	−1.07048	−0.535242	−	0.844699i	$-0.679779\pi$
$577$	−25.7139	−1.07048	−0.535242	+	0.844699i	$0.679779\pi$
$578$	−16.3643	−0.680665
$579$	0	0
$580$	0	0
$581$	−35.1197	−1.45701
$582$	0	0
$583$	22.2502	0.921510
$584$	−47.4042	−1.96160
$585$	0	0
$586$	32.8966	1.35894
$587$	−9.26970	−0.382602	−0.191301	−	0.981531i	$-0.561271\pi$
$587$	−9.26970	−0.382602	−0.191301	+	0.981531i	$0.561271\pi$
$588$	0	0
$589$	−6.13717	−0.252878
$590$	0	0
$591$	0	0
$592$	25.9901	1.06819
$593$	−31.5912	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$593$	−31.5912	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$594$	0	0
$595$	0	0
$596$	0.288584	0.0118208
$597$	0	0
$598$	−11.4695	−0.469022
$599$	13.1545	0.537480	0.268740	−	0.963213i	$-0.413393\pi$
$599$	13.1545	0.537480	0.268740	+	0.963213i	$0.413393\pi$
$600$	0	0
$601$	34.2150	1.39566	0.697829	−	0.716265i	$-0.254150\pi$
$601$	34.2150	1.39566	0.697829	+	0.716265i	$0.254150\pi$
$602$	14.4297	0.588113
$603$	0	0
$604$	−5.86073	−0.238470
$605$	0	0
$606$	0	0
$607$	3.51939	0.142848	0.0714238	−	0.997446i	$-0.477246\pi$
$607$	3.51939	0.142848	0.0714238	+	0.997446i	$0.477246\pi$
$608$	−9.35714	−0.379482
$609$	0	0
$610$	0	0
$611$	64.5193	2.61017
$612$	0	0
$613$	−5.90592	−0.238538	−0.119269	−	0.992862i	$-0.538055\pi$
$613$	−5.90592	−0.238538	−0.119269	+	0.992862i	$0.538055\pi$
$614$	−36.2034	−1.46105
$615$	0	0
$616$	23.0538	0.928866
$617$	30.3173	1.22053	0.610264	−	0.792198i	$-0.291063\pi$
$617$	30.3173	1.22053	0.610264	+	0.792198i	$0.291063\pi$
$618$	0	0
$619$	−23.1044	−0.928646	−0.464323	−	0.885666i	$-0.653702\pi$
$619$	−23.1044	−0.928646	−0.464323	+	0.885666i	$0.653702\pi$
$620$	0	0
$621$	0	0
$622$	−6.22049	−0.249419
$623$	8.31491	0.333130
$624$	0	0
$625$	0	0
$626$	34.1098	1.36330
$627$	0	0
$628$	0.178437	0.00712042
$629$	17.7855	0.709155
$630$	0	0
$631$	−27.6228	−1.09965	−0.549824	−	0.835280i	$-0.685305\pi$
$631$	−27.6228	−1.09965	−0.549824	+	0.835280i	$0.685305\pi$
$632$	−37.4196	−1.48847
$633$	0	0
$634$	−9.52865	−0.378431
$635$	0	0
$636$	0	0
$637$	20.8498	0.826099
$638$	2.84698	0.112713
$639$	0	0
$640$	0	0
$641$	−10.7286	−0.423756	−0.211878	−	0.977296i	$-0.567958\pi$
$641$	−10.7286	−0.423756	−0.211878	+	0.977296i	$0.567958\pi$
$642$	0	0
$643$	−18.0482	−0.711751	−0.355876	−	0.934533i	$-0.615817\pi$
$643$	−18.0482	−0.711751	−0.355876	+	0.934533i	$0.615817\pi$
$644$	−2.56742	−0.101171
$645$	0	0
$646$	8.61635	0.339006
$647$	26.9395	1.05910	0.529551	−	0.848278i	$-0.322360\pi$
$647$	26.9395	1.05910	0.529551	+	0.848278i	$0.322360\pi$
$648$	0	0
$649$	−22.6433	−0.888829
$650$	0	0
$651$	0	0
$652$	−1.19910	−0.0469605
$653$	13.4857	0.527737	0.263868	−	0.964559i	$-0.415002\pi$
$653$	13.4857	0.527737	0.263868	+	0.964559i	$0.415002\pi$
$654$	0	0
$655$	0	0
$656$	24.2153	0.945450
$657$	0	0
$658$	−46.8507	−1.82643
$659$	27.5560	1.07343	0.536716	−	0.843763i	$-0.319665\pi$
$659$	27.5560	1.07343	0.536716	+	0.843763i	$0.319665\pi$
$660$	0	0
$661$	18.1148	0.704583	0.352292	−	0.935890i	$-0.385403\pi$
$661$	18.1148	0.704583	0.352292	+	0.935890i	$0.385403\pi$
$662$	23.4849	0.912768
$663$	0	0
$664$	−32.7488	−1.27090
$665$	0	0
$666$	0	0
$667$	−1.66263	−0.0643772
$668$	5.08015	0.196557
$669$	0	0
$670$	0	0
$671$	12.4965	0.482422
$672$	0	0
$673$	−45.5751	−1.75679	−0.878396	−	0.477933i	$-0.841386\pi$
$673$	−45.5751	−1.75679	−0.878396	+	0.477933i	$0.841386\pi$
$674$	24.9736	0.961948
$675$	0	0
$676$	−8.54342	−0.328593
$677$	10.2385	0.393497	0.196748	−	0.980454i	$-0.436962\pi$
$677$	10.2385	0.393497	0.196748	+	0.980454i	$0.436962\pi$
$678$	0	0
$679$	19.3327	0.741920
$680$	0	0
$681$	0	0
$682$	4.86482	0.186283
$683$	−41.9562	−1.60541	−0.802705	−	0.596376i	$-0.796607\pi$
$683$	−41.9562	−1.60541	−0.802705	+	0.596376i	$0.796607\pi$
$684$	0	0
$685$	0	0
$686$	13.2199	0.504738
$687$	0	0
$688$	10.0987	0.385008
$689$	53.9137	2.05395
$690$	0	0
$691$	50.6633	1.92732	0.963662	−	0.267126i	$-0.0860739\pi$
$691$	50.6633	1.92732	0.963662	+	0.267126i	$0.0860739\pi$
$692$	−10.8965	−0.414224
$693$	0	0
$694$	29.4263	1.11701
$695$	0	0
$696$	0	0
$697$	16.5710	0.627671
$698$	−28.2589	−1.06961
$699$	0	0
$700$	0	0
$701$	−17.2666	−0.652151	−0.326076	−	0.945344i	$-0.605726\pi$
$701$	−17.2666	−0.652151	−0.326076	+	0.945344i	$0.605726\pi$
$702$	0	0
$703$	32.9216	1.24166
$704$	20.4747	0.771671
$705$	0	0
$706$	−21.6792	−0.815907
$707$	14.2141	0.534577
$708$	0	0
$709$	−18.3307	−0.688425	−0.344213	−	0.938892i	$-0.611854\pi$
$709$	−18.3307	−0.688425	−0.344213	+	0.938892i	$0.611854\pi$
$710$	0	0
$711$	0	0
$712$	7.75357	0.290577
$713$	−2.84104	−0.106398
$714$	0	0
$715$	0	0
$716$	−11.1727	−0.417543
$717$	0	0
$718$	32.9622	1.23014
$719$	10.6931	0.398785	0.199392	−	0.979920i	$-0.436103\pi$
$719$	10.6931	0.398785	0.199392	+	0.979920i	$0.436103\pi$
$720$	0	0
$721$	−24.9747	−0.930107
$722$	−7.54286	−0.280716
$723$	0	0
$724$	−5.50474	−0.204582
$725$	0	0
$726$	0	0
$727$	−5.74209	−0.212962	−0.106481	−	0.994315i	$-0.533958\pi$
$727$	−5.74209	−0.212962	−0.106481	+	0.994315i	$0.533958\pi$
$728$	55.8609	2.07034
$729$	0	0
$730$	0	0
$731$	6.91070	0.255601
$732$	0	0
$733$	6.03079	0.222752	0.111376	−	0.993778i	$-0.464474\pi$
$733$	6.03079	0.222752	0.111376	+	0.993778i	$0.464474\pi$
$734$	27.9269	1.03080
$735$	0	0
$736$	−4.33164	−0.159667
$737$	11.9761	0.441146
$738$	0	0
$739$	23.7939	0.875273	0.437637	−	0.899152i	$-0.355816\pi$
$739$	23.7939	0.875273	0.437637	+	0.899152i	$0.355816\pi$
$740$	0	0
$741$	0	0
$742$	−39.1494	−1.43722
$743$	24.4388	0.896574	0.448287	−	0.893890i	$-0.352034\pi$
$743$	24.4388	0.896574	0.448287	+	0.893890i	$0.352034\pi$
$744$	0	0
$745$	0	0
$746$	−13.7216	−0.502383
$747$	0	0
$748$	2.10547	0.0769836
$749$	−13.7525	−0.502507
$750$	0	0
$751$	38.9811	1.42244	0.711220	−	0.702970i	$-0.248143\pi$
$751$	38.9811	1.42244	0.711220	+	0.702970i	$0.248143\pi$
$752$	−32.7885	−1.19567
$753$	0	0
$754$	6.89840	0.251225
$755$	0	0
$756$	0	0
$757$	−6.78690	−0.246674	−0.123337	−	0.992365i	$-0.539360\pi$
$757$	−6.78690	−0.246674	−0.123337	+	0.992365i	$0.539360\pi$
$758$	−34.9257	−1.26856
$759$	0	0
$760$	0	0
$761$	−8.25388	−0.299203	−0.149601	−	0.988746i	$-0.547799\pi$
$761$	−8.25388	−0.299203	−0.149601	+	0.988746i	$0.547799\pi$
$762$	0	0
$763$	−13.1946	−0.477678
$764$	−4.09180	−0.148036
$765$	0	0
$766$	8.95833	0.323678
$767$	−54.8662	−1.98110
$768$	0	0
$769$	−12.3938	−0.446930	−0.223465	−	0.974712i	$-0.571737\pi$
$769$	−12.3938	−0.446930	−0.223465	+	0.974712i	$0.571737\pi$
$770$	0	0
$771$	0	0
$772$	1.28997	0.0464269
$773$	−38.1311	−1.37148	−0.685740	−	0.727846i	$-0.740521\pi$
$773$	−38.1311	−1.37148	−0.685740	+	0.727846i	$0.740521\pi$
$774$	0	0
$775$	0	0
$776$	18.0275	0.647150
$777$	0	0
$778$	−42.5925	−1.52701
$779$	30.6735	1.09899
$780$	0	0
$781$	−13.8753	−0.496499
$782$	3.98872	0.142636
$783$	0	0
$784$	−10.5958	−0.378421
$785$	0	0
$786$	0	0
$787$	37.5823	1.33967	0.669833	−	0.742512i	$-0.266366\pi$
$787$	37.5823	1.33967	0.669833	+	0.742512i	$0.266366\pi$
$788$	−12.3686	−0.440612
$789$	0	0
$790$	0	0
$791$	−24.2651	−0.862767
$792$	0	0
$793$	30.2798	1.07527
$794$	16.8663	0.598561
$795$	0	0
$796$	−4.68810	−0.166165
$797$	−14.9045	−0.527945	−0.263972	−	0.964530i	$-0.585033\pi$
$797$	−14.9045	−0.527945	−0.263972	+	0.964530i	$0.585033\pi$
$798$	0	0
$799$	−22.4377	−0.793790
$800$	0	0
$801$	0	0
$802$	−12.8268	−0.452930
$803$	35.7231	1.26064
$804$	0	0
$805$	0	0
$806$	11.7878	0.415206
$807$	0	0
$808$	13.2545	0.466292
$809$	−12.5838	−0.442424	−0.221212	−	0.975226i	$-0.571001\pi$
$809$	−12.5838	−0.442424	−0.221212	+	0.975226i	$0.571001\pi$
$810$	0	0
$811$	−27.9894	−0.982841	−0.491420	−	0.870922i	$-0.663522\pi$
$811$	−27.9894	−0.982841	−0.491420	+	0.870922i	$0.663522\pi$
$812$	1.54419	0.0541906
$813$	0	0
$814$	−26.0963	−0.914675
$815$	0	0
$816$	0	0
$817$	12.7919	0.447533
$818$	47.6509	1.66608
$819$	0	0
$820$	0	0
$821$	−1.20609	−0.0420929	−0.0210464	−	0.999778i	$-0.506700\pi$
$821$	−1.20609	−0.0420929	−0.0210464	+	0.999778i	$0.506700\pi$
$822$	0	0
$823$	−25.7664	−0.898162	−0.449081	−	0.893491i	$-0.648249\pi$
$823$	−25.7664	−0.898162	−0.449081	+	0.893491i	$0.648249\pi$
$824$	−23.2887	−0.811299
$825$	0	0
$826$	39.8411	1.38625
$827$	56.1443	1.95233	0.976165	−	0.217029i	$-0.0696368\pi$
$827$	56.1443	1.95233	0.976165	+	0.217029i	$0.0696368\pi$
$828$	0	0
$829$	−14.6534	−0.508935	−0.254467	−	0.967081i	$-0.581900\pi$
$829$	−14.6534	−0.508935	−0.254467	+	0.967081i	$0.581900\pi$
$830$	0	0
$831$	0	0
$832$	49.6116	1.71997
$833$	−7.25089	−0.251229
$834$	0	0
$835$	0	0
$836$	3.89730	0.134791
$837$	0	0
$838$	−12.2106	−0.421809
$839$	8.72873	0.301349	0.150675	−	0.988583i	$-0.451855\pi$
$839$	8.72873	0.301349	0.150675	+	0.988583i	$0.451855\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−12.7174	−0.438271
$843$	0	0
$844$	−3.65783	−0.125908
$845$	0	0
$846$	0	0
$847$	18.6710	0.641545
$848$	−27.3987	−0.940876
$849$	0	0
$850$	0	0
$851$	15.2402	0.522427
$852$	0	0
$853$	−42.3533	−1.45015	−0.725075	−	0.688670i	$-0.758195\pi$
$853$	−42.3533	−1.45015	−0.725075	+	0.688670i	$0.758195\pi$
$854$	−21.9876	−0.752402
$855$	0	0
$856$	−12.8241	−0.438319
$857$	49.8924	1.70429	0.852145	−	0.523305i	$-0.175301\pi$
$857$	49.8924	1.70429	0.852145	+	0.523305i	$0.175301\pi$
$858$	0	0
$859$	−33.4585	−1.14159	−0.570795	−	0.821092i	$-0.693365\pi$
$859$	−33.4585	−1.14159	−0.570795	+	0.821092i	$0.693365\pi$
$860$	0	0
$861$	0	0
$862$	18.8807	0.643081
$863$	36.8016	1.25274	0.626371	−	0.779525i	$-0.284540\pi$
$863$	36.8016	1.25274	0.626371	+	0.779525i	$0.284540\pi$
$864$	0	0
$865$	0	0
$866$	17.1084	0.581368
$867$	0	0
$868$	2.63867	0.0895622
$869$	28.1988	0.956579
$870$	0	0
$871$	29.0189	0.983267
$872$	−12.3039	−0.416662
$873$	0	0
$874$	7.38325	0.249742
$875$	0	0
$876$	0	0
$877$	−19.7336	−0.666358	−0.333179	−	0.942864i	$-0.608121\pi$
$877$	−19.7336	−0.666358	−0.333179	+	0.942864i	$0.608121\pi$
$878$	−8.02235	−0.270741
$879$	0	0
$880$	0	0
$881$	38.2999	1.29036	0.645179	−	0.764032i	$-0.276783\pi$
$881$	38.2999	1.29036	0.645179	+	0.764032i	$0.276783\pi$
$882$	0	0
$883$	32.9689	1.10949	0.554745	−	0.832020i	$-0.312816\pi$
$883$	32.9689	1.10949	0.554745	+	0.832020i	$0.312816\pi$
$884$	5.10169	0.171588
$885$	0	0
$886$	27.3281	0.918105
$887$	−15.0022	−0.503724	−0.251862	−	0.967763i	$-0.581043\pi$
$887$	−15.0022	−0.503724	−0.251862	+	0.967763i	$0.581043\pi$
$888$	0	0
$889$	−42.4806	−1.42475
$890$	0	0
$891$	0	0
$892$	4.54284	0.152105
$893$	−41.5330	−1.38985
$894$	0	0
$895$	0	0
$896$	−18.9517	−0.633131
$897$	0	0
$898$	−0.305647	−0.0101996
$899$	1.70877	0.0569905
$900$	0	0
$901$	−18.7494	−0.624634
$902$	−24.3143	−0.809577
$903$	0	0
$904$	−22.6269	−0.752561
$905$	0	0
$906$	0	0
$907$	−45.4276	−1.50840	−0.754199	−	0.656646i	$-0.771974\pi$
$907$	−45.4276	−1.50840	−0.754199	+	0.656646i	$0.771974\pi$
$908$	−4.34411	−0.144164
$909$	0	0
$910$	0	0
$911$	59.7103	1.97829	0.989146	−	0.146937i	$-0.0469415\pi$
$911$	59.7103	1.97829	0.989146	+	0.146937i	$0.0469415\pi$
$912$	0	0
$913$	24.6790	0.816755
$914$	−43.7882	−1.44839
$915$	0	0
$916$	3.24975	0.107375
$917$	−57.9789	−1.91463
$918$	0	0
$919$	−1.59828	−0.0527224	−0.0263612	−	0.999652i	$-0.508392\pi$
$919$	−1.59828	−0.0527224	−0.0263612	+	0.999652i	$0.508392\pi$
$920$	0	0
$921$	0	0
$922$	−16.7770	−0.552520
$923$	−33.6208	−1.10664
$924$	0	0
$925$	0	0
$926$	−9.74569	−0.320263
$927$	0	0
$928$	2.60530	0.0855231
$929$	35.5116	1.16510	0.582549	−	0.812795i	$-0.302055\pi$
$929$	35.5116	1.16510	0.582549	+	0.812795i	$0.302055\pi$
$930$	0	0
$931$	−13.4216	−0.439877
$932$	6.41994	0.210292
$933$	0	0
$934$	25.7684	0.843168
$935$	0	0
$936$	0	0
$937$	52.7492	1.72324	0.861621	−	0.507552i	$-0.169449\pi$
$937$	52.7492	1.72324	0.861621	+	0.507552i	$0.169449\pi$
$938$	−21.0720	−0.688026
$939$	0	0
$940$	0	0
$941$	−27.2935	−0.889743	−0.444872	−	0.895594i	$-0.646751\pi$
$941$	−27.2935	−0.889743	−0.444872	+	0.895594i	$0.646751\pi$
$942$	0	0
$943$	14.1995	0.462399
$944$	27.8828	0.907508
$945$	0	0
$946$	−10.1399	−0.329677
$947$	21.1187	0.686266	0.343133	−	0.939287i	$-0.388512\pi$
$947$	21.1187	0.686266	0.343133	+	0.939287i	$0.388512\pi$
$948$	0	0
$949$	86.5592	2.80983
$950$	0	0
$951$	0	0
$952$	−19.4266	−0.629621
$953$	3.73049	0.120842	0.0604211	−	0.998173i	$-0.480756\pi$
$953$	3.73049	0.120842	0.0604211	+	0.998173i	$0.480756\pi$
$954$	0	0
$955$	0	0
$956$	0.353042	0.0114182
$957$	0	0
$958$	−35.8066	−1.15686
$959$	73.9043	2.38650
$960$	0	0
$961$	−28.0801	−0.905810
$962$	−63.2330	−2.03871
$963$	0	0
$964$	5.90867	0.190305
$965$	0	0
$966$	0	0
$967$	22.8453	0.734654	0.367327	−	0.930092i	$-0.380273\pi$
$967$	22.8453	0.734654	0.367327	+	0.930092i	$0.380273\pi$
$968$	17.4106	0.559597
$969$	0	0
$970$	0	0
$971$	−42.3917	−1.36042	−0.680208	−	0.733019i	$-0.738110\pi$
$971$	−42.3917	−1.36042	−0.680208	+	0.733019i	$0.738110\pi$
$972$	0	0
$973$	10.4880	0.336230
$974$	−28.8524	−0.924492
$975$	0	0
$976$	−15.3881	−0.492560
$977$	−0.0842590	−0.00269568	−0.00134784	−	0.999999i	$-0.500429\pi$
$977$	−0.0842590	−0.00269568	−0.00134784	+	0.999999i	$0.500429\pi$
$978$	0	0
$979$	−5.84297	−0.186742
$980$	0	0
$981$	0	0
$982$	32.1409	1.02566
$983$	15.0122	0.478813	0.239407	−	0.970919i	$-0.423047\pi$
$983$	15.0122	0.478813	0.239407	+	0.970919i	$0.423047\pi$
$984$	0	0
$985$	0	0
$986$	−2.39904	−0.0764011
$987$	0	0
$988$	9.44339	0.300434
$989$	5.92170	0.188299
$990$	0	0
$991$	3.36536	0.106904	0.0534521	−	0.998570i	$-0.482978\pi$
$991$	3.36536	0.106904	0.0534521	+	0.998570i	$0.482978\pi$
$992$	4.45185	0.141346
$993$	0	0
$994$	24.4137	0.774357
$995$	0	0
$996$	0	0
$997$	−1.94573	−0.0616220	−0.0308110	−	0.999525i	$-0.509809\pi$
$997$	−1.94573	−0.0616220	−0.0308110	+	0.999525i	$0.509809\pi$
$998$	−39.5130	−1.25076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.cd.1.6	yes	9
3.2	odd	2		6525.2.a.cb.1.4	yes	9
5.4	even	2		6525.2.a.ca.1.4	✓	9
15.14	odd	2		6525.2.a.cc.1.6	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.4	✓	9	5.4	even	2
6525.2.a.cb.1.4	yes	9	3.2	odd	2
6525.2.a.cc.1.6	yes	9	15.14	odd	2
6525.2.a.cd.1.6	yes	9	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} -3.05552 q^{8} +O(q^{10})\)	\(q+1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} -3.05552 q^{8} +2.30259 q^{11} +5.57932 q^{13} -4.05142 q^{14} -2.83539 q^{16} -1.94031 q^{17} -3.59158 q^{19} +2.84698 q^{22} -1.66263 q^{23} +6.89840 q^{26} +1.54419 q^{28} +1.00000 q^{29} +1.70877 q^{31} +2.60530 q^{32} -2.39904 q^{34} -9.16633 q^{37} -4.44071 q^{38} -8.54038 q^{41} -3.56165 q^{43} -1.08512 q^{44} -2.05571 q^{46} +11.5640 q^{47} +3.73698 q^{49} -2.62931 q^{52} +9.66312 q^{53} +10.0121 q^{56} +1.23642 q^{58} -9.83385 q^{59} +5.42714 q^{61} +2.11276 q^{62} +8.89204 q^{64} +5.20114 q^{67} +0.914391 q^{68} -6.02596 q^{71} +15.5143 q^{73} -11.3334 q^{74} +1.69257 q^{76} -7.54498 q^{77} +12.2465 q^{79} -10.5595 q^{82} +10.7179 q^{83} -4.40370 q^{86} -7.03562 q^{88} -2.53756 q^{89} -18.2820 q^{91} +0.783531 q^{92} +14.2980 q^{94} -5.89998 q^{97} +4.62048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8	\( 9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8} + 2 q^{11} + q^{13} - 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} + 3 q^{22} + 16 q^{23} + 6 q^{26} - 4 q^{28} + 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} - 10 q^{41} + 3 q^{43} - 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} - 9 q^{52} + 22 q^{53} + 22 q^{56} + 2 q^{58} + 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} + 5 q^{67} + 39 q^{68} - 10 q^{73} - 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} - 8 q^{82} + 46 q^{83} + 28 q^{86} + 2 q^{88} + 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} + 7 q^{97} + 51 q^{98}+O(q^{100}) \) 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8 + 2 * q^11 + q^13 - 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 + 3 * q^22 + 16 * q^23 + 6 * q^26 - 4 * q^28 + 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 - 10 * q^41 + 3 * q^43 - 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 - 9 * q^52 + 22 * q^53 + 22 * q^56 + 2 * q^58 + 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 + 5 * q^67 + 39 * q^68 - 10 * q^73 - 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 - 8 * q^82 + 46 * q^83 + 28 * q^86 + 2 * q^88 + 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 + 7 * q^97 + 51 * q^98

Introduction

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