LMFDB - Embedded newform 5832.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5832,2,Mod(1,5832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5832, 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("5832.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5832 = 2^{3} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5832.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:	$46.5687544588$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 6 x^{10} + 84 x^{9} - 45 x^{8} - 378 x^{7} + 373 x^{6} + 591 x^{5} - 723 x^{4} + \cdots - 8 $ x^12 - 6x^11 - 6x^10 + 84x^9 - 45x^8 - 378x^7 + 373x^6 + 591x^5 - 723x^4 - 159x^3 + 288x^2 - 36*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.720285$ of defining polynomial
Character	$\chi$	$=$	5832.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.00902835 q^{5} -2.05241 q^{7} +O(q^{10})$	$q+0.00902835 q^{5} -2.05241 q^{7} -3.70321 q^{11} -1.63930 q^{13} +6.77052 q^{17} +2.50560 q^{19} +0.691342 q^{23} -4.99992 q^{25} -3.31491 q^{29} +7.95841 q^{31} -0.0185299 q^{35} +5.85989 q^{37} -4.21631 q^{41} +6.96290 q^{43} +9.64787 q^{47} -2.78762 q^{49} -6.78317 q^{53} -0.0334339 q^{55} +2.85946 q^{59} -11.9259 q^{61} -0.0148002 q^{65} -1.30309 q^{67} +2.23080 q^{71} -14.1023 q^{73} +7.60050 q^{77} -14.3738 q^{79} +15.1077 q^{83} +0.0611266 q^{85} -4.29757 q^{89} +3.36451 q^{91} +0.0226214 q^{95} -6.91857 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{5} - 6 q^{7}+O(q^{10}) $ 12 * q - 3 * q^5 - 6 * q^7	$ 12 q - 3 q^{5} - 6 q^{7} + 9 q^{11} - 9 q^{13} - 6 q^{17} - 9 q^{19} + 3 q^{25} - 9 q^{29} - 12 q^{31} + 18 q^{35} - 15 q^{37} - 12 q^{41} - 18 q^{43} - 6 q^{49} - 9 q^{53} - 27 q^{55} + 27 q^{59} - 27 q^{61} - 12 q^{65} - 30 q^{67} - 36 q^{73} - 12 q^{77} - 24 q^{79} + 42 q^{83} - 36 q^{85} - 9 q^{89} - 39 q^{91} + 6 q^{95} - 42 q^{97}+O(q^{100}) $ 12 * q - 3 * q^5 - 6 * q^7 + 9 * q^11 - 9 * q^13 - 6 * q^17 - 9 * q^19 + 3 * q^25 - 9 * q^29 - 12 * q^31 + 18 * q^35 - 15 * q^37 - 12 * q^41 - 18 * q^43 - 6 * q^49 - 9 * q^53 - 27 * q^55 + 27 * q^59 - 27 * q^61 - 12 * q^65 - 30 * q^67 - 36 * q^73 - 12 * q^77 - 24 * q^79 + 42 * q^83 - 36 * q^85 - 9 * q^89 - 39 * q^91 + 6 * q^95 - 42 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.00902835	0.00403760	0.00201880	−	0.999998i	$-0.499357\pi$
$5$	0.00902835	0.00403760	0.00201880	+	0.999998i	$0.499357\pi$
$6$	0	0
$7$	−2.05241	−0.775737	−0.387869	−	0.921715i	$-0.626789\pi$
$7$	−2.05241	−0.775737	−0.387869	+	0.921715i	$0.626789\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.70321	−1.11656	−0.558280	−	0.829653i	$-0.688539\pi$
$11$	−3.70321	−1.11656	−0.558280	+	0.829653i	$0.688539\pi$
$12$	0	0
$13$	−1.63930	−0.454660	−0.227330	−	0.973818i	$-0.573000\pi$
$13$	−1.63930	−0.454660	−0.227330	+	0.973818i	$0.573000\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.77052	1.64209	0.821046	−	0.570863i	$-0.193391\pi$
$17$	6.77052	1.64209	0.821046	+	0.570863i	$0.193391\pi$
$18$	0	0
$19$	2.50560	0.574824	0.287412	−	0.957807i	$-0.407205\pi$
$19$	2.50560	0.574824	0.287412	+	0.957807i	$0.407205\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.691342	0.144155	0.0720774	−	0.997399i	$-0.477037\pi$
$23$	0.691342	0.144155	0.0720774	+	0.997399i	$0.477037\pi$
$24$	0	0
$25$	−4.99992	−0.999984
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.31491	−0.615563	−0.307782	−	0.951457i	$-0.599587\pi$
$29$	−3.31491	−0.615563	−0.307782	+	0.951457i	$0.599587\pi$
$30$	0	0
$31$	7.95841	1.42937	0.714686	−	0.699445i	$-0.246570\pi$
$31$	7.95841	1.42937	0.714686	+	0.699445i	$0.246570\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.0185299	−0.00313212
$36$	0	0
$37$	5.85989	0.963360	0.481680	−	0.876347i	$-0.340027\pi$
$37$	5.85989	0.963360	0.481680	+	0.876347i	$0.340027\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.21631	−0.658478	−0.329239	−	0.944247i	$-0.606792\pi$
$41$	−4.21631	−0.658478	−0.329239	+	0.944247i	$0.606792\pi$
$42$	0	0
$43$	6.96290	1.06183	0.530916	−	0.847425i	$-0.321848\pi$
$43$	6.96290	1.06183	0.530916	+	0.847425i	$0.321848\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.64787	1.40729	0.703643	−	0.710553i	$-0.251555\pi$
$47$	9.64787	1.40729	0.703643	+	0.710553i	$0.251555\pi$
$48$	0	0
$49$	−2.78762	−0.398232
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.78317	−0.931741	−0.465870	−	0.884853i	$-0.654259\pi$
$53$	−6.78317	−0.931741	−0.465870	+	0.884853i	$0.654259\pi$
$54$	0	0
$55$	−0.0334339	−0.00450823
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.85946	0.372271	0.186135	−	0.982524i	$-0.440404\pi$
$59$	2.85946	0.372271	0.186135	+	0.982524i	$0.440404\pi$
$60$	0	0
$61$	−11.9259	−1.52696	−0.763480	−	0.645832i	$-0.776511\pi$
$61$	−11.9259	−1.52696	−0.763480	+	0.645832i	$0.776511\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.0148002	−0.00183574
$66$	0	0
$67$	−1.30309	−0.159197	−0.0795987	−	0.996827i	$-0.525364\pi$
$67$	−1.30309	−0.159197	−0.0795987	+	0.996827i	$0.525364\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.23080	0.264748	0.132374	−	0.991200i	$-0.457740\pi$
$71$	2.23080	0.264748	0.132374	+	0.991200i	$0.457740\pi$
$72$	0	0
$73$	−14.1023	−1.65055	−0.825273	−	0.564734i	$-0.808979\pi$
$73$	−14.1023	−1.65055	−0.825273	+	0.564734i	$0.808979\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.60050	0.866157
$78$	0	0
$79$	−14.3738	−1.61718	−0.808591	−	0.588372i	$-0.799769\pi$
$79$	−14.3738	−1.61718	−0.808591	+	0.588372i	$0.799769\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.1077	1.65828	0.829141	−	0.559040i	$-0.188830\pi$
$83$	15.1077	1.65828	0.829141	+	0.559040i	$0.188830\pi$
$84$	0	0
$85$	0.0611266	0.00663011
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.29757	−0.455542	−0.227771	−	0.973715i	$-0.573144\pi$
$89$	−4.29757	−0.455542	−0.227771	+	0.973715i	$0.573144\pi$
$90$	0	0
$91$	3.36451	0.352697
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.0226214	0.00232091
$96$	0	0
$97$	−6.91857	−0.702474	−0.351237	−	0.936287i	$-0.614239\pi$
$97$	−6.91857	−0.702474	−0.351237	+	0.936287i	$0.614239\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.27168	−0.425048	−0.212524	−	0.977156i	$-0.568168\pi$
$101$	−4.27168	−0.425048	−0.212524	+	0.977156i	$0.568168\pi$
$102$	0	0
$103$	−19.3257	−1.90422	−0.952110	−	0.305754i	$-0.901091\pi$
$103$	−19.3257	−1.90422	−0.952110	+	0.305754i	$0.901091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.19236	−0.695312	−0.347656	−	0.937622i	$-0.613022\pi$
$107$	−7.19236	−0.695312	−0.347656	+	0.937622i	$0.613022\pi$
$108$	0	0
$109$	15.6780	1.50168	0.750838	−	0.660487i	$-0.229650\pi$
$109$	15.6780	1.50168	0.750838	+	0.660487i	$0.229650\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.4634	−1.73689	−0.868445	−	0.495786i	$-0.834880\pi$
$113$	−18.4634	−1.73689	−0.868445	+	0.495786i	$0.834880\pi$
$114$	0	0
$115$	0.00624168	0.000582040 0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−13.8959	−1.27383
$120$	0	0
$121$	2.71377	0.246706
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.0902828	−0.00807514
$126$	0	0
$127$	−2.22850	−0.197747	−0.0988735	−	0.995100i	$-0.531524\pi$
$127$	−2.22850	−0.197747	−0.0988735	+	0.995100i	$0.531524\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.7044	1.37210	0.686048	−	0.727556i	$-0.259344\pi$
$131$	15.7044	1.37210	0.686048	+	0.727556i	$0.259344\pi$
$132$	0	0
$133$	−5.14251	−0.445912
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.793305	0.0677766	0.0338883	−	0.999426i	$-0.489211\pi$
$137$	0.793305	0.0677766	0.0338883	+	0.999426i	$0.489211\pi$
$138$	0	0
$139$	0.104070	0.00882708	0.00441354	−	0.999990i	$-0.498595\pi$
$139$	0.104070	0.00882708	0.00441354	+	0.999990i	$0.498595\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.07068	0.507655
$144$	0	0
$145$	−0.0299282	−0.00248540
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.3841	−1.50608	−0.753042	−	0.657972i	$-0.771414\pi$
$149$	−18.3841	−1.50608	−0.753042	+	0.657972i	$0.771414\pi$
$150$	0	0
$151$	−23.1091	−1.88059	−0.940296	−	0.340356i	$-0.889452\pi$
$151$	−23.1091	−1.88059	−0.940296	+	0.340356i	$0.889452\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.0718513	0.00577124
$156$	0	0
$157$	14.4917	1.15656	0.578281	−	0.815837i	$-0.303724\pi$
$157$	14.4917	1.15656	0.578281	+	0.815837i	$0.303724\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.41892	−0.111826
$162$	0	0
$163$	−9.90188	−0.775575	−0.387788	−	0.921749i	$-0.626761\pi$
$163$	−9.90188	−0.775575	−0.387788	+	0.921749i	$0.626761\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.3513	1.26530	0.632651	−	0.774437i	$-0.281967\pi$
$167$	16.3513	1.26530	0.632651	+	0.774437i	$0.281967\pi$
$168$	0	0
$169$	−10.3127	−0.793284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.70655	−0.357832	−0.178916	−	0.983864i	$-0.557259\pi$
$173$	−4.70655	−0.357832	−0.178916	+	0.983864i	$0.557259\pi$
$174$	0	0
$175$	10.2619	0.775725
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.8038	−1.10649	−0.553245	−	0.833019i	$-0.686611\pi$
$179$	−14.8038	−1.10649	−0.553245	+	0.833019i	$0.686611\pi$
$180$	0	0
$181$	−16.6643	−1.23865	−0.619324	−	0.785136i	$-0.712593\pi$
$181$	−16.6643	−1.23865	−0.619324	+	0.785136i	$0.712593\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.0529052	0.00388967
$186$	0	0
$187$	−25.0726	−1.83349
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.0070	−0.796436	−0.398218	−	0.917291i	$-0.630371\pi$
$191$	−11.0070	−0.796436	−0.398218	+	0.917291i	$0.630371\pi$
$192$	0	0
$193$	6.16966	0.444102	0.222051	−	0.975035i	$-0.428725\pi$
$193$	6.16966	0.444102	0.222051	+	0.975035i	$0.428725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.1722	−0.795984	−0.397992	−	0.917389i	$-0.630293\pi$
$197$	−11.1722	−0.795984	−0.397992	+	0.917389i	$0.630293\pi$
$198$	0	0
$199$	−8.72271	−0.618337	−0.309168	−	0.951007i	$-0.600051\pi$
$199$	−8.72271	−0.618337	−0.309168	+	0.951007i	$0.600051\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.80355	0.477515
$204$	0	0
$205$	−0.0380664	−0.00265867
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.27876	−0.641826
$210$	0	0
$211$	−1.64912	−0.113530	−0.0567652	−	0.998388i	$-0.518079\pi$
$211$	−1.64912	−0.113530	−0.0567652	+	0.998388i	$0.518079\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.0628635	0.00428725
$216$	0	0
$217$	−16.3339	−1.10882
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.0989	−0.746594
$222$	0	0
$223$	−12.1068	−0.810731	−0.405365	−	0.914155i	$-0.632856\pi$
$223$	−12.1068	−0.810731	−0.405365	+	0.914155i	$0.632856\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.75076	0.248947	0.124473	−	0.992223i	$-0.460276\pi$
$227$	3.75076	0.248947	0.124473	+	0.992223i	$0.460276\pi$
$228$	0	0
$229$	−8.75584	−0.578602	−0.289301	−	0.957238i	$-0.593423\pi$
$229$	−8.75584	−0.578602	−0.289301	+	0.957238i	$0.593423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.1182	1.31799	0.658994	−	0.752148i	$-0.270982\pi$
$233$	20.1182	1.31799	0.658994	+	0.752148i	$0.270982\pi$
$234$	0	0
$235$	0.0871044	0.00568206
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0726	−0.651542	−0.325771	−	0.945449i	$-0.605624\pi$
$239$	−10.0726	−0.651542	−0.325771	+	0.945449i	$0.605624\pi$
$240$	0	0
$241$	−1.73261	−0.111607	−0.0558037	−	0.998442i	$-0.517772\pi$
$241$	−1.73261	−0.111607	−0.0558037	+	0.998442i	$0.517772\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.0251676	−0.00160790
$246$	0	0
$247$	−4.10743	−0.261350
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.5483	1.48635	0.743177	−	0.669095i	$-0.233318\pi$
$251$	23.5483	1.48635	0.743177	+	0.669095i	$0.233318\pi$
$252$	0	0
$253$	−2.56019	−0.160958
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0246	1.24910	0.624550	−	0.780985i	$-0.285282\pi$
$257$	20.0246	1.24910	0.624550	+	0.780985i	$0.285282\pi$
$258$	0	0
$259$	−12.0269	−0.747315
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.87540	−0.485618	−0.242809	−	0.970074i	$-0.578069\pi$
$263$	−7.87540	−0.485618	−0.242809	+	0.970074i	$0.578069\pi$
$264$	0	0
$265$	−0.0612409	−0.00376200
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.2541	1.11297	0.556485	−	0.830858i	$-0.312150\pi$
$269$	18.2541	1.11297	0.556485	+	0.830858i	$0.312150\pi$
$270$	0	0
$271$	4.59097	0.278881	0.139441	−	0.990230i	$-0.455470\pi$
$271$	4.59097	0.278881	0.139441	+	0.990230i	$0.455470\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.5158	1.11654
$276$	0	0
$277$	−23.3619	−1.40368	−0.701840	−	0.712335i	$-0.747638\pi$
$277$	−23.3619	−1.40368	−0.701840	+	0.712335i	$0.747638\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.0699	1.85347	0.926737	−	0.375710i	$-0.122601\pi$
$281$	31.0699	1.85347	0.926737	+	0.375710i	$0.122601\pi$
$282$	0	0
$283$	3.77798	0.224577	0.112289	−	0.993676i	$-0.464182\pi$
$283$	3.77798	0.224577	0.112289	+	0.993676i	$0.464182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.65360	0.510806
$288$	0	0
$289$	28.8399	1.69646
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.4008	−1.65919	−0.829596	−	0.558364i	$-0.811429\pi$
$293$	−28.4008	−1.65919	−0.829596	+	0.558364i	$0.811429\pi$
$294$	0	0
$295$	0.0258163	0.00150308
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.13332	−0.0655415
$300$	0	0
$301$	−14.2907	−0.823702
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.107672	−0.00616525
$306$	0	0
$307$	2.27379	0.129772	0.0648860	−	0.997893i	$-0.479332\pi$
$307$	2.27379	0.129772	0.0648860	+	0.997893i	$0.479332\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.42610	0.307686	0.153843	−	0.988095i	$-0.450835\pi$
$311$	5.42610	0.307686	0.153843	+	0.988095i	$0.450835\pi$
$312$	0	0
$313$	−12.0006	−0.678316	−0.339158	−	0.940729i	$-0.610142\pi$
$313$	−12.0006	−0.678316	−0.339158	+	0.940729i	$0.610142\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.4919	1.60026	0.800131	−	0.599825i	$-0.204763\pi$
$317$	28.4919	1.60026	0.800131	+	0.599825i	$0.204763\pi$
$318$	0	0
$319$	12.2758	0.687313
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.9642	0.943914
$324$	0	0
$325$	8.19637	0.454653
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.8014	−1.09168
$330$	0	0
$331$	−4.15436	−0.228344	−0.114172	−	0.993461i	$-0.536422\pi$
$331$	−4.15436	−0.228344	−0.114172	+	0.993461i	$0.536422\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.0117647	−0.000642776 0
$336$	0	0
$337$	−18.1284	−0.987516	−0.493758	−	0.869599i	$-0.664377\pi$
$337$	−18.1284	−0.987516	−0.493758	+	0.869599i	$0.664377\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−29.4717	−1.59598
$342$	0	0
$343$	20.0882	1.08466
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.60879	−0.140047	−0.0700235	−	0.997545i	$-0.522307\pi$
$347$	−2.60879	−0.140047	−0.0700235	+	0.997545i	$0.522307\pi$
$348$	0	0
$349$	14.9714	0.801399	0.400699	−	0.916210i	$-0.368767\pi$
$349$	14.9714	0.801399	0.400699	+	0.916210i	$0.368767\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.41952	0.182003	0.0910014	−	0.995851i	$-0.470993\pi$
$353$	3.41952	0.182003	0.0910014	+	0.995851i	$0.470993\pi$
$354$	0	0
$355$	0.0201405	0.00106895
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.7892	−1.04443	−0.522217	−	0.852813i	$-0.674895\pi$
$359$	−19.7892	−1.04443	−0.522217	+	0.852813i	$0.674895\pi$
$360$	0	0
$361$	−12.7220	−0.669577
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.127320	−0.00666425
$366$	0	0
$367$	−18.5242	−0.966958	−0.483479	−	0.875356i	$-0.660627\pi$
$367$	−18.5242	−0.966958	−0.483479	+	0.875356i	$0.660627\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.9218	0.722786
$372$	0	0
$373$	10.2262	0.529495	0.264747	−	0.964318i	$-0.414711\pi$
$373$	10.2262	0.529495	0.264747	+	0.964318i	$0.414711\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.43413	0.279872
$378$	0	0
$379$	−10.0913	−0.518353	−0.259177	−	0.965830i	$-0.583451\pi$
$379$	−10.0913	−0.518353	−0.259177	+	0.965830i	$0.583451\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.8051	1.57406	0.787032	−	0.616912i	$-0.211616\pi$
$383$	30.8051	1.57406	0.787032	+	0.616912i	$0.211616\pi$
$384$	0	0
$385$	0.0686200	0.00349720
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.38973	−0.0704619	−0.0352310	−	0.999379i	$-0.511217\pi$
$389$	−1.38973	−0.0704619	−0.0352310	+	0.999379i	$0.511217\pi$
$390$	0	0
$391$	4.68074	0.236715
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.129772	−0.00652954
$396$	0	0
$397$	24.2792	1.21854	0.609269	−	0.792964i	$-0.291463\pi$
$397$	24.2792	1.21854	0.609269	+	0.792964i	$0.291463\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.2930	−0.563947	−0.281974	−	0.959422i	$-0.590989\pi$
$401$	−11.2930	−0.563947	−0.281974	+	0.959422i	$0.590989\pi$
$402$	0	0
$403$	−13.0462	−0.649879
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.7004	−1.07565
$408$	0	0
$409$	3.46097	0.171134	0.0855670	−	0.996332i	$-0.472730\pi$
$409$	3.46097	0.171134	0.0855670	+	0.996332i	$0.472730\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.86879	−0.288784
$414$	0	0
$415$	0.136397	0.00669548
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.4479	0.705826	0.352913	−	0.935656i	$-0.385191\pi$
$419$	14.4479	0.705826	0.352913	+	0.935656i	$0.385191\pi$
$420$	0	0
$421$	−23.7355	−1.15680	−0.578400	−	0.815753i	$-0.696323\pi$
$421$	−23.7355	−1.15680	−0.578400	+	0.815753i	$0.696323\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−33.8520	−1.64206
$426$	0	0
$427$	24.4769	1.18452
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.4234	−0.887422	−0.443711	−	0.896170i	$-0.646338\pi$
$431$	−18.4234	−0.887422	−0.443711	+	0.896170i	$0.646338\pi$
$432$	0	0
$433$	−15.6349	−0.751364	−0.375682	−	0.926749i	$-0.622592\pi$
$433$	−15.6349	−0.751364	−0.375682	+	0.926749i	$0.622592\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.73223	0.0828637
$438$	0	0
$439$	6.94419	0.331428	0.165714	−	0.986174i	$-0.447007\pi$
$439$	6.94419	0.331428	0.165714	+	0.986174i	$0.447007\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.9190	1.23145	0.615724	−	0.787962i	$-0.288864\pi$
$443$	25.9190	1.23145	0.615724	+	0.787962i	$0.288864\pi$
$444$	0	0
$445$	−0.0388000	−0.00183930
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.44849	0.351516	0.175758	−	0.984433i	$-0.443762\pi$
$449$	7.44849	0.351516	0.175758	+	0.984433i	$0.443762\pi$
$450$	0	0
$451$	15.6139	0.735230
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.0303760	0.00142405
$456$	0	0
$457$	−27.6678	−1.29425	−0.647124	−	0.762385i	$-0.724028\pi$
$457$	−27.6678	−1.29425	−0.647124	+	0.762385i	$0.724028\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.95762	−0.370624	−0.185312	−	0.982680i	$-0.559329\pi$
$461$	−7.95762	−0.370624	−0.185312	+	0.982680i	$0.559329\pi$
$462$	0	0
$463$	−16.7627	−0.779028	−0.389514	−	0.921021i	$-0.627357\pi$
$463$	−16.7627	−0.779028	−0.389514	+	0.921021i	$0.627357\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.73775	−0.0804133	−0.0402067	−	0.999191i	$-0.512802\pi$
$467$	−1.73775	−0.0804133	−0.0402067	+	0.999191i	$0.512802\pi$
$468$	0	0
$469$	2.67447	0.123495
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−25.7851	−1.18560
$474$	0	0
$475$	−12.5278	−0.574815
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.8545	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$479$	−31.8545	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$480$	0	0
$481$	−9.60613	−0.438002
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.0624633	−0.00283631
$486$	0	0
$487$	5.12802	0.232373	0.116186	−	0.993227i	$-0.462933\pi$
$487$	5.12802	0.232373	0.116186	+	0.993227i	$0.462933\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.58090	−0.206733	−0.103367	−	0.994643i	$-0.532961\pi$
$491$	−4.58090	−0.206733	−0.103367	+	0.994643i	$0.532961\pi$
$492$	0	0
$493$	−22.4436	−1.01081
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.57852	−0.205375
$498$	0	0
$499$	−11.9165	−0.533456	−0.266728	−	0.963772i	$-0.585943\pi$
$499$	−11.9165	−0.533456	−0.266728	+	0.963772i	$0.585943\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.4966	−1.31519	−0.657593	−	0.753373i	$-0.728425\pi$
$503$	−29.4966	−1.31519	−0.657593	+	0.753373i	$0.728425\pi$
$504$	0	0
$505$	−0.0385662	−0.00171617
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.8478	−1.14568	−0.572842	−	0.819666i	$-0.694159\pi$
$509$	−25.8478	−1.14568	−0.572842	+	0.819666i	$0.694159\pi$
$510$	0	0
$511$	28.9436	1.28039
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.174480	−0.00768849
$516$	0	0
$517$	−35.7281	−1.57132
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.29862	0.363569	0.181785	−	0.983338i	$-0.441813\pi$
$521$	8.29862	0.363569	0.181785	+	0.983338i	$0.441813\pi$
$522$	0	0
$523$	−3.37217	−0.147455	−0.0737273	−	0.997278i	$-0.523489\pi$
$523$	−3.37217	−0.147455	−0.0737273	+	0.997278i	$0.523489\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	53.8825	2.34716
$528$	0	0
$529$	−22.5220	−0.979219
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.91181	0.299384
$534$	0	0
$535$	−0.0649352	−0.00280739
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.3232	0.444650
$540$	0	0
$541$	23.7788	1.02233	0.511166	−	0.859482i	$-0.329214\pi$
$541$	23.7788	1.02233	0.511166	+	0.859482i	$0.329214\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.141546	0.00606317
$546$	0	0
$547$	−6.64044	−0.283925	−0.141962	−	0.989872i	$-0.545341\pi$
$547$	−6.64044	−0.283925	−0.141962	+	0.989872i	$0.545341\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.30584	−0.353841
$552$	0	0
$553$	29.5009	1.25451
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.8390	−1.26432	−0.632160	−	0.774838i	$-0.717831\pi$
$557$	−29.8390	−1.26432	−0.632160	+	0.774838i	$0.717831\pi$
$558$	0	0
$559$	−11.4143	−0.482773
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.9367	1.26168	0.630842	−	0.775912i	$-0.282710\pi$
$563$	29.9367	1.26168	0.630842	+	0.775912i	$0.282710\pi$
$564$	0	0
$565$	−0.166694	−0.00701287
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.58683	0.108446	0.0542228	−	0.998529i	$-0.482732\pi$
$569$	2.58683	0.108446	0.0542228	+	0.998529i	$0.482732\pi$
$570$	0	0
$571$	−12.3688	−0.517619	−0.258810	−	0.965928i	$-0.583330\pi$
$571$	−12.3688	−0.517619	−0.258810	+	0.965928i	$0.583330\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.45666	−0.144153
$576$	0	0
$577$	3.83280	0.159562	0.0797808	−	0.996812i	$-0.474578\pi$
$577$	3.83280	0.159562	0.0797808	+	0.996812i	$0.474578\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−31.0071	−1.28639
$582$	0	0
$583$	25.1195	1.04034
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.1778	−0.626454	−0.313227	−	0.949678i	$-0.601410\pi$
$587$	−15.1778	−0.626454	−0.313227	+	0.949678i	$0.601410\pi$
$588$	0	0
$589$	19.9406	0.821638
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.889100	−0.0365110	−0.0182555	−	0.999833i	$-0.505811\pi$
$593$	−0.889100	−0.0365110	−0.0182555	+	0.999833i	$0.505811\pi$
$594$	0	0
$595$	−0.125457	−0.00514322
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−32.6146	−1.33260	−0.666298	−	0.745686i	$-0.732122\pi$
$599$	−32.6146	−1.33260	−0.666298	+	0.745686i	$0.732122\pi$
$600$	0	0
$601$	45.8667	1.87094	0.935470	−	0.353406i	$-0.114977\pi$
$601$	45.8667	1.87094	0.935470	+	0.353406i	$0.114977\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.0245009	0.000996102 0
$606$	0	0
$607$	−4.69428	−0.190535	−0.0952675	−	0.995452i	$-0.530371\pi$
$607$	−4.69428	−0.190535	−0.0952675	+	0.995452i	$0.530371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.8158	−0.639837
$612$	0	0
$613$	−42.0014	−1.69642	−0.848210	−	0.529661i	$-0.822319\pi$
$613$	−42.0014	−1.69642	−0.848210	+	0.529661i	$0.822319\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.6653	−0.992988	−0.496494	−	0.868040i	$-0.665380\pi$
$617$	−24.6653	−0.992988	−0.496494	+	0.868040i	$0.665380\pi$
$618$	0	0
$619$	26.9289	1.08237	0.541183	−	0.840905i	$-0.317977\pi$
$619$	26.9289	1.08237	0.541183	+	0.840905i	$0.317977\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.82037	0.353381
$624$	0	0
$625$	24.9988	0.999951
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39.6745	1.58193
$630$	0	0
$631$	−2.15127	−0.0856408	−0.0428204	−	0.999083i	$-0.513634\pi$
$631$	−2.15127	−0.0856408	−0.0428204	+	0.999083i	$0.513634\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.0201196	−0.000798424 0
$636$	0	0
$637$	4.56975	0.181060
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.606056	0.0239378	0.0119689	−	0.999928i	$-0.496190\pi$
$641$	0.606056	0.0239378	0.0119689	+	0.999928i	$0.496190\pi$
$642$	0	0
$643$	−37.7832	−1.49002	−0.745012	−	0.667051i	$-0.767556\pi$
$643$	−37.7832	−1.49002	−0.745012	+	0.667051i	$0.767556\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.1063	1.02634	0.513172	−	0.858286i	$-0.328470\pi$
$647$	26.1063	1.02634	0.513172	+	0.858286i	$0.328470\pi$
$648$	0	0
$649$	−10.5892	−0.415662
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.6770	0.613487	0.306744	−	0.951792i	$-0.400761\pi$
$653$	15.6770	0.613487	0.306744	+	0.951792i	$0.400761\pi$
$654$	0	0
$655$	0.141784	0.00553998
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.0457	0.508188	0.254094	−	0.967179i	$-0.418223\pi$
$659$	13.0457	0.508188	0.254094	+	0.967179i	$0.418223\pi$
$660$	0	0
$661$	30.4995	1.18629	0.593146	−	0.805095i	$-0.297886\pi$
$661$	30.4995	1.18629	0.593146	+	0.805095i	$0.297886\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.0464284	−0.00180042
$666$	0	0
$667$	−2.29174	−0.0887364
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.1642	1.70494
$672$	0	0
$673$	−8.48374	−0.327024	−0.163512	−	0.986541i	$-0.552282\pi$
$673$	−8.48374	−0.327024	−0.163512	+	0.986541i	$0.552282\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.290563	−0.0111673	−0.00558363	−	0.999984i	$-0.501777\pi$
$677$	−0.290563	−0.0111673	−0.00558363	+	0.999984i	$0.501777\pi$
$678$	0	0
$679$	14.1997	0.544935
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−37.2329	−1.42468	−0.712339	−	0.701836i	$-0.752364\pi$
$683$	−37.2329	−1.42468	−0.712339	+	0.701836i	$0.752364\pi$
$684$	0	0
$685$	0.00716224	0.000273655 0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.1197	0.423625
$690$	0	0
$691$	−23.4010	−0.890216	−0.445108	−	0.895477i	$-0.646835\pi$
$691$	−23.4010	−0.890216	−0.445108	+	0.895477i	$0.646835\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.000939579 0	3.56403e−5 0
$696$	0	0
$697$	−28.5466	−1.08128
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.89418	0.184851	0.0924253	−	0.995720i	$-0.470538\pi$
$701$	4.89418	0.184851	0.0924253	+	0.995720i	$0.470538\pi$
$702$	0	0
$703$	14.6825	0.553763
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.76723	0.329726
$708$	0	0
$709$	35.8407	1.34603	0.673013	−	0.739631i	$-0.265000\pi$
$709$	35.8407	1.34603	0.673013	+	0.739631i	$0.265000\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.50199	0.206051
$714$	0	0
$715$	0.0548082	0.00204971
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.4749	1.47217	0.736083	−	0.676892i	$-0.236674\pi$
$719$	39.4749	1.47217	0.736083	+	0.676892i	$0.236674\pi$
$720$	0	0
$721$	39.6643	1.47718
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.5743	0.615553
$726$	0	0
$727$	3.30430	0.122550	0.0612749	−	0.998121i	$-0.480483\pi$
$727$	3.30430	0.122550	0.0612749	+	0.998121i	$0.480483\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	47.1424	1.74362
$732$	0	0
$733$	23.1301	0.854329	0.427165	−	0.904174i	$-0.359512\pi$
$733$	23.1301	0.854329	0.427165	+	0.904174i	$0.359512\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.82560	0.177753
$738$	0	0
$739$	−2.30737	−0.0848779	−0.0424390	−	0.999099i	$-0.513513\pi$
$739$	−2.30737	−0.0848779	−0.0424390	+	0.999099i	$0.513513\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.6679	−0.464739	−0.232369	−	0.972628i	$-0.574648\pi$
$743$	−12.6679	−0.464739	−0.232369	+	0.972628i	$0.574648\pi$
$744$	0	0
$745$	−0.165978	−0.00608097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.7617	0.539379
$750$	0	0
$751$	31.9105	1.16443	0.582216	−	0.813034i	$-0.302186\pi$
$751$	31.9105	1.16443	0.582216	+	0.813034i	$0.302186\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.208637	−0.00759309
$756$	0	0
$757$	−33.2146	−1.20721	−0.603603	−	0.797285i	$-0.706269\pi$
$757$	−33.2146	−1.20721	−0.603603	+	0.797285i	$0.706269\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.2322	−1.09592	−0.547958	−	0.836506i	$-0.684595\pi$
$761$	−30.2322	−1.09592	−0.547958	+	0.836506i	$0.684595\pi$
$762$	0	0
$763$	−32.1776	−1.16491
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.68752	−0.169257
$768$	0	0
$769$	20.5641	0.741560	0.370780	−	0.928721i	$-0.379090\pi$
$769$	20.5641	0.741560	0.370780	+	0.928721i	$0.379090\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.22103	−0.0798847	−0.0399424	−	0.999202i	$-0.512717\pi$
$773$	−2.22103	−0.0798847	−0.0399424	+	0.999202i	$0.512717\pi$
$774$	0	0
$775$	−39.7914	−1.42935
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.5644	−0.378509
$780$	0	0
$781$	−8.26113	−0.295607
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.130836	0.00466974
$786$	0	0
$787$	5.66777	0.202034	0.101017	−	0.994885i	$-0.467790\pi$
$787$	5.66777	0.202034	0.101017	+	0.994885i	$0.467790\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37.8944	1.34737
$792$	0	0
$793$	19.5502	0.694248
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.0092	−1.38178	−0.690889	−	0.722961i	$-0.742781\pi$
$797$	−39.0092	−1.38178	−0.690889	+	0.722961i	$0.742781\pi$
$798$	0	0
$799$	65.3210	2.31089
$800$	0	0
$801$	0	0
$802$	0	0
$803$	52.2237	1.84293
$804$	0	0
$805$	−0.0128105	−0.000451510 0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.9853	−1.44097	−0.720484	−	0.693472i	$-0.756080\pi$
$809$	−40.9853	−1.44097	−0.720484	+	0.693472i	$0.756080\pi$
$810$	0	0
$811$	24.2369	0.851072	0.425536	−	0.904942i	$-0.360086\pi$
$811$	24.2369	0.851072	0.425536	+	0.904942i	$0.360086\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.0893977	−0.00313146
$816$	0	0
$817$	17.4462	0.610366
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.3278	−0.883945	−0.441972	−	0.897029i	$-0.645721\pi$
$821$	−25.3278	−0.883945	−0.441972	+	0.897029i	$0.645721\pi$
$822$	0	0
$823$	−45.6530	−1.59136	−0.795682	−	0.605715i	$-0.792887\pi$
$823$	−45.6530	−1.59136	−0.795682	+	0.605715i	$0.792887\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.2801	1.88750	0.943752	−	0.330655i	$-0.107270\pi$
$827$	54.2801	1.88750	0.943752	+	0.330655i	$0.107270\pi$
$828$	0	0
$829$	−22.4311	−0.779063	−0.389531	−	0.921013i	$-0.627363\pi$
$829$	−22.4311	−0.779063	−0.389531	+	0.921013i	$0.627363\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.8736	−0.653933
$834$	0	0
$835$	0.147625	0.00510878
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.1945	−0.938857	−0.469428	−	0.882970i	$-0.655540\pi$
$839$	−27.1945	−0.938857	−0.469428	+	0.882970i	$0.655540\pi$
$840$	0	0
$841$	−18.0114	−0.621082
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.0931066	−0.00320297
$846$	0	0
$847$	−5.56976	−0.191379
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.05119	0.138873
$852$	0	0
$853$	44.9837	1.54021	0.770106	−	0.637916i	$-0.220203\pi$
$853$	44.9837	1.54021	0.770106	+	0.637916i	$0.220203\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.1998	0.416737	0.208369	−	0.978050i	$-0.433185\pi$
$857$	12.1998	0.416737	0.208369	+	0.978050i	$0.433185\pi$
$858$	0	0
$859$	45.3846	1.54850	0.774251	−	0.632879i	$-0.218127\pi$
$859$	45.3846	1.54850	0.774251	+	0.632879i	$0.218127\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.90855	0.235170	0.117585	−	0.993063i	$-0.462485\pi$
$863$	6.90855	0.235170	0.117585	+	0.993063i	$0.462485\pi$
$864$	0	0
$865$	−0.0424924	−0.00144478
$866$	0	0
$867$	0	0
$868$	0	0
$869$	53.2293	1.80568
$870$	0	0
$871$	2.13615	0.0723807
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.185297	0.00626419
$876$	0	0
$877$	−41.2205	−1.39192	−0.695959	−	0.718082i	$-0.745020\pi$
$877$	−41.2205	−1.39192	−0.695959	+	0.718082i	$0.745020\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37.5860	1.26631	0.633153	−	0.774027i	$-0.281760\pi$
$881$	37.5860	1.26631	0.633153	+	0.774027i	$0.281760\pi$
$882$	0	0
$883$	−31.2987	−1.05328	−0.526642	−	0.850087i	$-0.676549\pi$
$883$	−31.2987	−1.05328	−0.526642	+	0.850087i	$0.676549\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.2915	1.15139	0.575697	−	0.817663i	$-0.304731\pi$
$887$	34.2915	1.15139	0.575697	+	0.817663i	$0.304731\pi$
$888$	0	0
$889$	4.57378	0.153400
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.1737	0.808942
$894$	0	0
$895$	−0.133654	−0.00446757
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.3814	−0.879869
$900$	0	0
$901$	−45.9256	−1.53000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.150451	−0.00500117
$906$	0	0
$907$	21.6830	0.719972	0.359986	−	0.932958i	$-0.382782\pi$
$907$	21.6830	0.719972	0.359986	+	0.932958i	$0.382782\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.7392	−0.919042	−0.459521	−	0.888167i	$-0.651979\pi$
$911$	−27.7392	−0.919042	−0.459521	+	0.888167i	$0.651979\pi$
$912$	0	0
$913$	−55.9469	−1.85157
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−32.2317	−1.06439
$918$	0	0
$919$	−34.2970	−1.13135	−0.565676	−	0.824628i	$-0.691385\pi$
$919$	−34.2970	−1.13135	−0.565676	+	0.824628i	$0.691385\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.65696	−0.120370
$924$	0	0
$925$	−29.2990	−0.963345
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.8883	−1.17746	−0.588728	−	0.808331i	$-0.700371\pi$
$929$	−35.8883	−1.17746	−0.588728	+	0.808331i	$0.700371\pi$
$930$	0	0
$931$	−6.98466	−0.228913
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.226365	−0.00740292
$936$	0	0
$937$	19.9545	0.651885	0.325943	−	0.945390i	$-0.394318\pi$
$937$	19.9545	0.651885	0.325943	+	0.945390i	$0.394318\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.22250	−0.105050	−0.0525252	−	0.998620i	$-0.516727\pi$
$941$	−3.22250	−0.105050	−0.0525252	+	0.998620i	$0.516727\pi$
$942$	0	0
$943$	−2.91492	−0.0949227
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.5164	−0.991650	−0.495825	−	0.868423i	$-0.665134\pi$
$947$	−30.5164	−0.991650	−0.495825	+	0.868423i	$0.665134\pi$
$948$	0	0
$949$	23.1179	0.750438
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.8299	−0.642354	−0.321177	−	0.947019i	$-0.604078\pi$
$953$	−19.8299	−0.642354	−0.321177	+	0.947019i	$0.604078\pi$
$954$	0	0
$955$	−0.0993747	−0.00321569
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.62818	−0.0525768
$960$	0	0
$961$	32.3363	1.04311
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.0557019	0.00179311
$966$	0	0
$967$	−16.9836	−0.546155	−0.273078	−	0.961992i	$-0.588042\pi$
$967$	−16.9836	−0.546155	−0.273078	+	0.961992i	$0.588042\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.8069	−1.18119	−0.590595	−	0.806968i	$-0.701107\pi$
$971$	−36.8069	−1.18119	−0.590595	+	0.806968i	$0.701107\pi$
$972$	0	0
$973$	−0.213594	−0.00684750
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.8780	−1.46777	−0.733883	−	0.679276i	$-0.762294\pi$
$977$	−45.8780	−1.46777	−0.733883	+	0.679276i	$0.762294\pi$
$978$	0	0
$979$	15.9148	0.508640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7.77903	−0.248112	−0.124056	−	0.992275i	$-0.539590\pi$
$983$	−7.77903	−0.248112	−0.124056	+	0.992275i	$0.539590\pi$
$984$	0	0
$985$	−0.100866	−0.00321387
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.81375	0.153068
$990$	0	0
$991$	7.74180	0.245927	0.122963	−	0.992411i	$-0.460760\pi$
$991$	7.74180	0.245927	0.122963	+	0.992411i	$0.460760\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.0787517	−0.00249660
$996$	0	0
$997$	10.4168	0.329905	0.164952	−	0.986302i	$-0.447253\pi$
$997$	10.4168	0.329905	0.164952	+	0.986302i	$0.447253\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5832.2.a.h.1.7		12
3.2	odd	2		5832.2.a.i.1.6		12
27.4	even	9		216.2.q.a.97.4	yes	24
27.7	even	9		216.2.q.a.49.4	✓	24
27.20	odd	18		648.2.q.a.361.3		24
27.23	odd	18		648.2.q.a.289.3		24
108.7	odd	18		432.2.u.e.49.1		24
108.31	odd	18		432.2.u.e.97.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.q.a.49.4	✓	24	27.7	even	9
216.2.q.a.97.4	yes	24	27.4	even	9
432.2.u.e.49.1		24	108.7	odd	18
432.2.u.e.97.1		24	108.31	odd	18
648.2.q.a.289.3		24	27.23	odd	18
648.2.q.a.361.3		24	27.20	odd	18
5832.2.a.h.1.7		12	1.1	even	1	trivial
5832.2.a.i.1.6		12	3.2	odd	2

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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:	\( N \)	\(=\)	\( 5832 = 2^{3} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5832.a (trivial)

\(f(q)\)	\(=\)	\(q+0.00902835 q^{5} -2.05241 q^{7} +O(q^{10})\)	\(q+0.00902835 q^{5} -2.05241 q^{7} -3.70321 q^{11} -1.63930 q^{13} +6.77052 q^{17} +2.50560 q^{19} +0.691342 q^{23} -4.99992 q^{25} -3.31491 q^{29} +7.95841 q^{31} -0.0185299 q^{35} +5.85989 q^{37} -4.21631 q^{41} +6.96290 q^{43} +9.64787 q^{47} -2.78762 q^{49} -6.78317 q^{53} -0.0334339 q^{55} +2.85946 q^{59} -11.9259 q^{61} -0.0148002 q^{65} -1.30309 q^{67} +2.23080 q^{71} -14.1023 q^{73} +7.60050 q^{77} -14.3738 q^{79} +15.1077 q^{83} +0.0611266 q^{85} -4.29757 q^{89} +3.36451 q^{91} +0.0226214 q^{95} -6.91857 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{5} - 6 q^{7}+O(q^{10}) \) 12 * q - 3 * q^5 - 6 * q^7	\( 12 q - 3 q^{5} - 6 q^{7} + 9 q^{11} - 9 q^{13} - 6 q^{17} - 9 q^{19} + 3 q^{25} - 9 q^{29} - 12 q^{31} + 18 q^{35} - 15 q^{37} - 12 q^{41} - 18 q^{43} - 6 q^{49} - 9 q^{53} - 27 q^{55} + 27 q^{59} - 27 q^{61} - 12 q^{65} - 30 q^{67} - 36 q^{73} - 12 q^{77} - 24 q^{79} + 42 q^{83} - 36 q^{85} - 9 q^{89} - 39 q^{91} + 6 q^{95} - 42 q^{97}+O(q^{100}) \) 12 * q - 3 * q^5 - 6 * q^7 + 9 * q^11 - 9 * q^13 - 6 * q^17 - 9 * q^19 + 3 * q^25 - 9 * q^29 - 12 * q^31 + 18 * q^35 - 15 * q^37 - 12 * q^41 - 18 * q^43 - 6 * q^49 - 9 * q^53 - 27 * q^55 + 27 * q^59 - 27 * q^61 - 12 * q^65 - 30 * q^67 - 36 * q^73 - 12 * q^77 - 24 * q^79 + 42 * q^83 - 36 * q^85 - 9 * q^89 - 39 * q^91 + 6 * q^95 - 42 * q^97

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