LMFDB - Embedded newform 6048.2.c.c.3025.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.3
Root			$0.178197 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.c.3025.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.356394i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.356394i q^{5} +1.00000 q^{7} -5.96816i q^{11} -2.87298i q^{13} +3.16228 q^{17} +0.127017i q^{19} +2.80588 q^{23} +4.87298 q^{25} +2.44949i q^{29} +7.00000 q^{31} -0.356394i q^{35} +1.87298i q^{37} -6.99208 q^{41} +1.12702i q^{43} +7.34847 q^{47} +1.00000 q^{49} -2.12702 q^{55} -6.63568i q^{59} +13.7460i q^{61} -1.02391 q^{65} -8.61895i q^{67} -5.96816 q^{71} +0.872983 q^{73} -5.96816i q^{77} -12.6190 q^{79} -6.63568i q^{83} -1.12702i q^{85} -6.99208 q^{89} -2.87298i q^{91} +0.0452680 q^{95} -9.74597 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 8 q^{25} + 56 q^{31} + 8 q^{49} - 48 q^{55} - 24 q^{73} - 8 q^{79} - 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 8 * q^25 + 56 * q^31 + 8 * q^49 - 48 * q^55 - 24 * q^73 - 8 * q^79 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 0.356394i	− 0.159384i	−0.996820	−	0.0796921i	$-0.974606\pi$
$5$	− 0.356394i	− 0.159384i	0.996820	−	0.0796921i	$-0.0253937\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.96816i	− 1.79947i	−0.436438	−	0.899734i	$-0.643760\pi$
$11$	− 5.96816i	− 1.79947i	0.436438	−	0.899734i	$-0.356240\pi$
$12$	0	0
$13$	− 2.87298i	− 0.796822i	−0.917207	−	0.398411i	$-0.869562\pi$
$13$	− 2.87298i	− 0.796822i	0.917207	−	0.398411i	$-0.130438\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16228	0.766965	0.383482	−	0.923548i	$-0.374725\pi$
$17$	3.16228	0.766965	0.383482	+	0.923548i	$0.374725\pi$
$18$	0	0
$19$	0.127017i	0.0291396i	0.999894	+	0.0145698i	$0.00463788\pi$
$19$	0.127017i	0.0291396i	−0.999894	+	0.0145698i	$0.995362\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.80588	0.585067	0.292534	−	0.956255i	$-0.405502\pi$
$23$	2.80588	0.585067	0.292534	+	0.956255i	$0.405502\pi$
$24$	0	0
$25$	4.87298	0.974597
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.44949i	0.454859i	0.973795	+	0.227429i	$0.0730321\pi$
$29$	2.44949i	0.454859i	−0.973795	+	0.227429i	$0.926968\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.356394i	− 0.0602416i
$36$	0	0
$37$	1.87298i	0.307917i	0.988077	+	0.153958i	$0.0492021\pi$
$37$	1.87298i	0.307917i	−0.988077	+	0.153958i	$0.950798\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.99208	−1.09198	−0.545989	−	0.837792i	$-0.683846\pi$
$41$	−6.99208	−1.09198	−0.545989	+	0.837792i	$0.683846\pi$
$42$	0	0
$43$	1.12702i	0.171868i	0.996301	+	0.0859342i	$0.0273875\pi$
$43$	1.12702i	0.171868i	−0.996301	+	0.0859342i	$0.972613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.34847	1.07188	0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847	1.07188	0.535942	+	0.844255i	$0.319956\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−2.12702	−0.286807
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.63568i	− 0.863892i	−0.901899	−	0.431946i	$-0.857827\pi$
$59$	− 6.63568i	− 0.863892i	0.901899	−	0.431946i	$-0.142173\pi$
$60$	0	0
$61$	13.7460i	1.75999i	0.474982	+	0.879995i	$0.342454\pi$
$61$	13.7460i	1.75999i	−0.474982	+	0.879995i	$0.657546\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.02391	−0.127001
$66$	0	0
$67$	− 8.61895i	− 1.05297i	−0.850184	−	0.526486i	$-0.823509\pi$
$67$	− 8.61895i	− 1.05297i	0.850184	−	0.526486i	$-0.176491\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.96816	−0.708290	−0.354145	−	0.935190i	$-0.615228\pi$
$71$	−5.96816	−0.708290	−0.354145	+	0.935190i	$0.615228\pi$
$72$	0	0
$73$	0.872983	0.102175	0.0510875	−	0.998694i	$-0.483731\pi$
$73$	0.872983	0.102175	0.0510875	+	0.998694i	$0.483731\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.96816i	− 0.680135i
$78$	0	0
$79$	−12.6190	−1.41974	−0.709871	−	0.704331i	$-0.751247\pi$
$79$	−12.6190	−1.41974	−0.709871	+	0.704331i	$0.751247\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.63568i	− 0.728361i	−0.931328	−	0.364180i	$-0.881349\pi$
$83$	− 6.63568i	− 0.728361i	0.931328	−	0.364180i	$-0.118651\pi$
$84$	0	0
$85$	− 1.12702i	− 0.122242i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.99208	−0.741158	−0.370579	−	0.928801i	$-0.620841\pi$
$89$	−6.99208	−0.741158	−0.370579	+	0.928801i	$0.620841\pi$
$90$	0	0
$91$	− 2.87298i	− 0.301170i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.0452680	0.00464440
$96$	0	0
$97$	−9.74597	−0.989553	−0.494776	−	0.869020i	$-0.664750\pi$
$97$	−9.74597	−0.989553	−0.494776	+	0.869020i	$0.664750\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 11.5347i	− 1.14774i	−0.818946	−	0.573871i	$-0.805441\pi$
$101$	− 11.5347i	− 1.14774i	0.818946	−	0.573871i	$-0.194559\pi$
$102$	0	0
$103$	10.7460	1.05883	0.529416	−	0.848363i	$-0.322411\pi$
$103$	10.7460	1.05883	0.529416	+	0.848363i	$0.322411\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.3858i	− 1.39073i	−0.718657	−	0.695364i	$-0.755243\pi$
$107$	− 14.3858i	− 1.39073i	0.718657	−	0.695364i	$-0.244757\pi$
$108$	0	0
$109$	12.7460i	1.22084i	0.792077	+	0.610421i	$0.209000\pi$
$109$	12.7460i	1.22084i	−0.792077	+	0.610421i	$0.791000\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.8353	1.58373	0.791866	−	0.610695i	$-0.209110\pi$
$113$	16.8353	1.58373	0.791866	+	0.610695i	$0.209110\pi$
$114$	0	0
$115$	− 1.00000i	− 0.0932505i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.16228	0.289886
$120$	0	0
$121$	−24.6190	−2.23809
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.51867i	− 0.314720i
$126$	0	0
$127$	−10.8730	−0.964821	−0.482411	−	0.875945i	$-0.660239\pi$
$127$	−10.8730	−0.964821	−0.482411	+	0.875945i	$0.660239\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 8.77405i	− 0.766592i	−0.923626	−	0.383296i	$-0.874789\pi$
$131$	− 8.77405i	− 0.766592i	0.923626	−	0.383296i	$-0.125211\pi$
$132$	0	0
$133$	0.127017i	0.0110137i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.58785	−0.391967	−0.195983	−	0.980607i	$-0.562790\pi$
$137$	−4.58785	−0.391967	−0.195983	+	0.980607i	$0.562790\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.1464	−1.43386
$144$	0	0
$145$	0.872983	0.0724973
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.79796i	0.802680i	0.915929	+	0.401340i	$0.131455\pi$
$149$	9.79796i	0.802680i	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.49476i	− 0.200384i
$156$	0	0
$157$	− 7.12702i	− 0.568798i	−0.958706	−	0.284399i	$-0.908206\pi$
$157$	− 7.12702i	− 0.568798i	0.958706	−	0.284399i	$-0.0917940\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.80588	0.221135
$162$	0	0
$163$	− 22.8730i	− 1.79155i	−0.444507	−	0.895775i	$-0.646621\pi$
$163$	− 22.8730i	− 1.79155i	0.444507	−	0.895775i	$-0.353379\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.42558	−0.110314	−0.0551572	−	0.998478i	$-0.517566\pi$
$167$	−1.42558	−0.110314	−0.0551572	+	0.998478i	$0.517566\pi$
$168$	0	0
$169$	4.74597	0.365074
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 18.2156i	− 1.38491i	−0.721462	−	0.692454i	$-0.756530\pi$
$173$	− 18.2156i	− 1.38491i	0.721462	−	0.692454i	$-0.243470\pi$
$174$	0	0
$175$	4.87298	0.368363
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.77405i	− 0.655803i	−0.944712	−	0.327901i	$-0.893659\pi$
$179$	− 8.77405i	− 0.655803i	0.944712	−	0.327901i	$-0.106341\pi$
$180$	0	0
$181$	21.7460i	1.61636i	0.588932	+	0.808182i	$0.299549\pi$
$181$	21.7460i	1.61636i	−0.588932	+	0.808182i	$0.700451\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.667520	0.0490770
$186$	0	0
$187$	− 18.8730i	− 1.38013i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.7795	1.57591	0.787956	−	0.615731i	$-0.211139\pi$
$191$	21.7795	1.57591	0.787956	+	0.615731i	$0.211139\pi$
$192$	0	0
$193$	−9.49193	−0.683244	−0.341622	−	0.939837i	$-0.610976\pi$
$193$	−9.49193	−0.683244	−0.341622	+	0.939837i	$0.610976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.04783i	0.145902i	0.997336	+	0.0729508i	$0.0232416\pi$
$197$	2.04783i	0.145902i	−0.997336	+	0.0729508i	$0.976758\pi$
$198$	0	0
$199$	−0.127017	−0.00900397	−0.00450199	−	0.999990i	$-0.501433\pi$
$199$	−0.127017	−0.00900397	−0.00450199	+	0.999990i	$0.501433\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.44949i	0.171920i
$204$	0	0
$205$	2.49193i	0.174044i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.758056	0.0524358
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.401662	0.0273931
$216$	0	0
$217$	7.00000	0.475191
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 9.08517i	− 0.611135i
$222$	0	0
$223$	−7.61895	−0.510203	−0.255101	−	0.966914i	$-0.582109\pi$
$223$	−7.61895	−0.510203	−0.255101	+	0.966914i	$0.582109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.9205i	1.72040i	0.509955	+	0.860201i	$0.329662\pi$
$227$	25.9205i	1.72040i	−0.509955	+	0.860201i	$0.670338\pi$
$228$	0	0
$229$	15.7460i	1.04052i	0.854007	+	0.520261i	$0.174165\pi$
$229$	15.7460i	1.04052i	−0.854007	+	0.520261i	$0.825835\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.61177	−0.367639	−0.183820	−	0.982960i	$-0.558846\pi$
$233$	−5.61177	−0.367639	−0.183820	+	0.982960i	$0.558846\pi$
$234$	0	0
$235$	− 2.61895i	− 0.170841i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.2474	0.792222	0.396111	−	0.918203i	$-0.370360\pi$
$239$	12.2474	0.792222	0.396111	+	0.918203i	$0.370360\pi$
$240$	0	0
$241$	28.6190	1.84351	0.921754	−	0.387774i	$-0.126756\pi$
$241$	28.6190	1.84351	0.921754	+	0.387774i	$0.126756\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.356394i	− 0.0227692i
$246$	0	0
$247$	0.364917	0.0232191
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16.8353i	− 1.06263i	−0.847173	−	0.531317i	$-0.821697\pi$
$251$	− 16.8353i	− 1.06263i	0.847173	−	0.531317i	$-0.178303\pi$
$252$	0	0
$253$	− 16.7460i	− 1.05281i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0428	1.25024	0.625119	−	0.780529i	$-0.285050\pi$
$257$	20.0428	1.25024	0.625119	+	0.780529i	$0.285050\pi$
$258$	0	0
$259$	1.87298i	0.116382i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.80588	−0.173018	−0.0865091	−	0.996251i	$-0.527571\pi$
$263$	−2.80588	−0.173018	−0.0865091	+	0.996251i	$0.527571\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.5560i	− 0.643612i	−0.946806	−	0.321806i	$-0.895710\pi$
$269$	− 10.5560i	− 0.643612i	0.946806	−	0.321806i	$-0.104290\pi$
$270$	0	0
$271$	19.7460	1.19948	0.599741	−	0.800194i	$-0.295270\pi$
$271$	19.7460	1.19948	0.599741	+	0.800194i	$0.295270\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 29.0828i	− 1.75376i
$276$	0	0
$277$	5.87298i	0.352873i	0.984312	+	0.176437i	$0.0564571\pi$
$277$	5.87298i	0.352873i	−0.984312	+	0.176437i	$0.943543\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.0454	−1.31512	−0.657559	−	0.753403i	$-0.728411\pi$
$281$	−22.0454	−1.31512	−0.657559	+	0.753403i	$0.728411\pi$
$282$	0	0
$283$	11.4919i	0.683125i	0.939859	+	0.341562i	$0.110956\pi$
$283$	11.4919i	0.683125i	−0.939859	+	0.341562i	$0.889044\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.99208	−0.412729
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 31.8434i	− 1.86031i	−0.367168	−	0.930155i	$-0.619673\pi$
$293$	− 31.8434i	− 1.86031i	0.367168	−	0.930155i	$-0.380327\pi$
$294$	0	0
$295$	−2.36492	−0.137691
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.06126i	− 0.466195i
$300$	0	0
$301$	1.12702i	0.0649602i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.89898	0.280515
$306$	0	0
$307$	29.6190i	1.69044i	0.534416	+	0.845221i	$0.320531\pi$
$307$	29.6190i	1.69044i	−0.534416	+	0.845221i	$0.679469\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.7104	1.17438	0.587189	−	0.809450i	$-0.300235\pi$
$311$	20.7104	1.17438	0.587189	+	0.809450i	$0.300235\pi$
$312$	0	0
$313$	2.87298	0.162391	0.0811953	−	0.996698i	$-0.474126\pi$
$313$	2.87298	0.162391	0.0811953	+	0.996698i	$0.474126\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.6730i	− 0.767954i	−0.923343	−	0.383977i	$-0.874554\pi$
$317$	− 13.6730i	− 0.767954i	0.923343	−	0.383977i	$-0.125446\pi$
$318$	0	0
$319$	14.6190	0.818504
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.401662i	0.0223491i
$324$	0	0
$325$	− 14.0000i	− 0.776580i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.34847	0.405134
$330$	0	0
$331$	− 10.0000i	− 0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$	− 10.0000i	− 0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.07174	−0.167827
$336$	0	0
$337$	−3.25403	−0.177258	−0.0886292	−	0.996065i	$-0.528249\pi$
$337$	−3.25403	−0.177258	−0.0886292	+	0.996065i	$0.528249\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 41.7771i	− 2.26236i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.4765i	1.95816i	0.203475	+	0.979080i	$0.434777\pi$
$347$	36.4765i	1.95816i	−0.203475	+	0.979080i	$0.565223\pi$
$348$	0	0
$349$	− 23.4919i	− 1.25749i	−0.777610	−	0.628747i	$-0.783568\pi$
$349$	− 23.4919i	− 1.25749i	0.777610	−	0.628747i	$-0.216432\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.3166	0.708773	0.354386	−	0.935099i	$-0.384690\pi$
$353$	13.3166	0.708773	0.354386	+	0.935099i	$0.384690\pi$
$354$	0	0
$355$	2.12702i	0.112890i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.1733	−1.53971	−0.769854	−	0.638221i	$-0.779671\pi$
$359$	−29.1733	−1.53971	−0.769854	+	0.638221i	$0.779671\pi$
$360$	0	0
$361$	18.9839	0.999151
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.311126i	− 0.0162851i
$366$	0	0
$367$	−21.8730	−1.14176	−0.570880	−	0.821033i	$-0.693398\pi$
$367$	−21.8730	−1.14176	−0.570880	+	0.821033i	$0.693398\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	28.2379i	1.46210i	0.682322	+	0.731052i	$0.260970\pi$
$373$	28.2379i	1.46210i	−0.682322	+	0.731052i	$0.739030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.03734	0.362442
$378$	0	0
$379$	− 8.87298i	− 0.455775i	−0.973688	−	0.227887i	$-0.926818\pi$
$379$	− 8.87298i	− 0.455775i	0.973688	−	0.227887i	$-0.0731818\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.4576	−0.892039	−0.446020	−	0.895023i	$-0.647159\pi$
$383$	−17.4576	−0.892039	−0.446020	+	0.895023i	$0.647159\pi$
$384$	0	0
$385$	−2.12702	−0.108403
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.4097i	− 0.781304i	−0.920538	−	0.390652i	$-0.872250\pi$
$389$	− 15.4097i	− 0.781304i	0.920538	−	0.390652i	$-0.127750\pi$
$390$	0	0
$391$	8.87298	0.448726
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.49732i	0.226285i
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.5242	0.825178	0.412589	−	0.910917i	$-0.364625\pi$
$401$	16.5242	0.825178	0.412589	+	0.910917i	$0.364625\pi$
$402$	0	0
$403$	− 20.1109i	− 1.00179i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.1783	0.554086
$408$	0	0
$409$	−12.3649	−0.611406	−0.305703	−	0.952127i	$-0.598891\pi$
$409$	−12.3649	−0.611406	−0.305703	+	0.952127i	$0.598891\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.63568i	− 0.326521i
$414$	0	0
$415$	−2.36492	−0.116089
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 28.7716i	− 1.40559i	−0.711394	−	0.702793i	$-0.751936\pi$
$419$	− 28.7716i	− 1.40559i	0.711394	−	0.702793i	$-0.248064\pi$
$420$	0	0
$421$	− 19.1109i	− 0.931407i	−0.884941	−	0.465704i	$-0.845801\pi$
$421$	− 19.1109i	− 0.931407i	0.884941	−	0.465704i	$-0.154199\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.4097	0.747482
$426$	0	0
$427$	13.7460i	0.665214i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−23.1146	−1.11339	−0.556695	−	0.830717i	$-0.687931\pi$
$431$	−23.1146	−1.11339	−0.556695	+	0.830717i	$0.687931\pi$
$432$	0	0
$433$	−4.87298	−0.234181	−0.117090	−	0.993121i	$-0.537357\pi$
$433$	−4.87298	−0.234181	−0.117090	+	0.993121i	$0.537357\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.356394i	0.0170486i
$438$	0	0
$439$	−25.7460	−1.22879	−0.614394	−	0.788999i	$-0.710599\pi$
$439$	−25.7460	−1.22879	−0.614394	+	0.788999i	$0.710599\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 27.7024i	− 1.31618i	−0.752938	−	0.658091i	$-0.771364\pi$
$443$	− 27.7024i	− 1.31618i	0.752938	−	0.658091i	$-0.228636\pi$
$444$	0	0
$445$	2.49193i	0.118129i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.9100	−1.45873	−0.729366	−	0.684123i	$-0.760185\pi$
$449$	−30.9100	−1.45873	−0.729366	+	0.684123i	$0.760185\pi$
$450$	0	0
$451$	41.7298i	1.96498i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.02391	−0.0480018
$456$	0	0
$457$	9.25403	0.432885	0.216443	−	0.976295i	$-0.430555\pi$
$457$	9.25403	0.432885	0.216443	+	0.976295i	$0.430555\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.9389i	0.649198i	0.945852	+	0.324599i	$0.105229\pi$
$461$	13.9389i	0.649198i	−0.945852	+	0.324599i	$0.894771\pi$
$462$	0	0
$463$	−0.508067	−0.0236119	−0.0118059	−	0.999930i	$-0.503758\pi$
$463$	−0.508067	−0.0236119	−0.0118059	+	0.999930i	$0.503758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.401662i	− 0.0185867i	−0.999957	−	0.00929335i	$-0.997042\pi$
$467$	− 0.401662i	− 0.0185867i	0.999957	−	0.00929335i	$-0.00295821\pi$
$468$	0	0
$469$	− 8.61895i	− 0.397986i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.72622	0.309272
$474$	0	0
$475$	0.618950i	0.0283994i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.712788	0.0325681	0.0162841	−	0.999867i	$-0.494816\pi$
$479$	0.712788	0.0325681	0.0162841	+	0.999867i	$0.494816\pi$
$480$	0	0
$481$	5.38105	0.245355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.47340i	0.157719i
$486$	0	0
$487$	−26.1109	−1.18320	−0.591599	−	0.806233i	$-0.701503\pi$
$487$	−26.1109	−1.18320	−0.591599	+	0.806233i	$0.701503\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.4072i	0.605057i	0.953140	+	0.302528i	$0.0978307\pi$
$491$	13.4072i	0.605057i	−0.953140	+	0.302528i	$0.902169\pi$
$492$	0	0
$493$	7.74597i	0.348861i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.96816	−0.267709
$498$	0	0
$499$	34.3649i	1.53838i	0.639017	+	0.769192i	$0.279341\pi$
$499$	34.3649i	1.53838i	−0.639017	+	0.769192i	$0.720659\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.34847	−0.327652	−0.163826	−	0.986489i	$-0.552384\pi$
$503$	−7.34847	−0.327652	−0.163826	+	0.986489i	$0.552384\pi$
$504$	0	0
$505$	−4.11088	−0.182932
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 0.311126i	− 0.0137904i	−0.999976	−	0.00689521i	$-0.997805\pi$
$509$	− 0.311126i	− 0.0137904i	0.999976	−	0.00689521i	$-0.00219483\pi$
$510$	0	0
$511$	0.872983	0.0386185
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.82980i	− 0.168761i
$516$	0	0
$517$	− 43.8569i	− 1.92882i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.8408	−1.30735	−0.653675	−	0.756776i	$-0.726774\pi$
$521$	−29.8408	−1.30735	−0.653675	+	0.756776i	$0.726774\pi$
$522$	0	0
$523$	− 34.7460i	− 1.51934i	−0.650312	−	0.759668i	$-0.725362\pi$
$523$	− 34.7460i	− 1.51934i	0.650312	−	0.759668i	$-0.274638\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1359	0.964257
$528$	0	0
$529$	−15.1270	−0.657696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	20.0881i	0.870113i
$534$	0	0
$535$	−5.12702	−0.221660
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.96816i	− 0.257067i
$540$	0	0
$541$	− 30.8569i	− 1.32664i	−0.748336	−	0.663320i	$-0.769147\pi$
$541$	− 30.8569i	− 1.32664i	0.748336	−	0.663320i	$-0.230853\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.54259	0.194583
$546$	0	0
$547$	39.2379i	1.67769i	0.544369	+	0.838846i	$0.316769\pi$
$547$	39.2379i	1.67769i	−0.544369	+	0.838846i	$0.683231\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.311126	−0.0132544
$552$	0	0
$553$	−12.6190	−0.536612
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.17571i	− 0.388787i	−0.980924	−	0.194394i	$-0.937726\pi$
$557$	− 9.17571i	− 0.388787i	0.980924	−	0.194394i	$-0.0622739\pi$
$558$	0	0
$559$	3.23790	0.136949
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.34847i	0.309701i	0.987938	+	0.154851i	$0.0494896\pi$
$563$	7.34847i	0.309701i	−0.987938	+	0.154851i	$0.950510\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.52123	−0.231462	−0.115731	−	0.993281i	$-0.536921\pi$
$569$	−5.52123	−0.231462	−0.115731	+	0.993281i	$0.536921\pi$
$570$	0	0
$571$	− 32.9839i	− 1.38033i	−0.723651	−	0.690166i	$-0.757538\pi$
$571$	− 32.9839i	− 1.38033i	0.723651	−	0.690166i	$-0.242462\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.6730	0.570205
$576$	0	0
$577$	−22.2540	−0.926448	−0.463224	−	0.886241i	$-0.653307\pi$
$577$	−22.2540	−0.926448	−0.463224	+	0.886241i	$0.653307\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.63568i	− 0.275294i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.8568i	− 1.56252i	−0.624208	−	0.781259i	$-0.714578\pi$
$587$	− 37.8568i	− 1.56252i	0.624208	−	0.781259i	$-0.285422\pi$
$588$	0	0
$589$	0.889117i	0.0366354i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.8035	−0.936426	−0.468213	−	0.883616i	$-0.655102\pi$
$593$	−22.8035	−0.936426	−0.468213	+	0.883616i	$0.655102\pi$
$594$	0	0
$595$	− 1.12702i	− 0.0462032i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	41.3755	1.69056	0.845278	−	0.534327i	$-0.179435\pi$
$599$	41.3755	1.69056	0.845278	+	0.534327i	$0.179435\pi$
$600$	0	0
$601$	32.1109	1.30983	0.654915	−	0.755702i	$-0.272704\pi$
$601$	32.1109	1.30983	0.654915	+	0.755702i	$0.272704\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.77405i	0.356716i
$606$	0	0
$607$	−15.2379	−0.618487	−0.309244	−	0.950983i	$-0.600076\pi$
$607$	−15.2379	−0.618487	−0.309244	+	0.950983i	$0.600076\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21.1120i	− 0.854101i
$612$	0	0
$613$	40.2379i	1.62519i	0.582826	+	0.812597i	$0.301947\pi$
$613$	40.2379i	1.62519i	−0.582826	+	0.812597i	$0.698053\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.7209	0.632898	0.316449	−	0.948610i	$-0.397509\pi$
$617$	15.7209	0.632898	0.316449	+	0.948610i	$0.397509\pi$
$618$	0	0
$619$	0.491933i	0.0197725i	0.999951	+	0.00988624i	$0.00314694\pi$
$619$	0.491933i	0.0197725i	−0.999951	+	0.00988624i	$0.996853\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.99208	−0.280132
$624$	0	0
$625$	23.1109	0.924435
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.92289i	0.236161i
$630$	0	0
$631$	8.87298	0.353228	0.176614	−	0.984280i	$-0.443486\pi$
$631$	8.87298	0.353228	0.176614	+	0.984280i	$0.443486\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.87507i	0.153777i
$636$	0	0
$637$	− 2.87298i	− 0.113832i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.1703	0.717685	0.358843	−	0.933398i	$-0.383171\pi$
$641$	18.1703	0.717685	0.358843	+	0.933398i	$0.383171\pi$
$642$	0	0
$643$	8.38105i	0.330516i	0.986250	+	0.165258i	$0.0528457\pi$
$643$	8.38105i	0.330516i	−0.986250	+	0.165258i	$0.947154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.4232	−0.842231	−0.421116	−	0.907007i	$-0.638361\pi$
$647$	−21.4232	−0.842231	−0.421116	+	0.907007i	$0.638361\pi$
$648$	0	0
$649$	−39.6028	−1.55455
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.311126i	0.0121753i	0.999981	+	0.00608765i	$0.00193777\pi$
$653$	0.311126i	0.0121753i	−0.999981	+	0.00608765i	$0.998062\pi$
$654$	0	0
$655$	−3.12702	−0.122183
$656$	0	0
$657$	0	0
$658$	0	0
$659$	35.3620i	1.37751i	0.724994	+	0.688755i	$0.241842\pi$
$659$	35.3620i	1.37751i	−0.724994	+	0.688755i	$0.758158\pi$
$660$	0	0
$661$	− 45.8569i	− 1.78362i	−0.452405	−	0.891812i	$-0.649434\pi$
$661$	− 45.8569i	− 1.78362i	0.452405	−	0.891812i	$-0.350566\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.0452680	0.00175542
$666$	0	0
$667$	6.87298i	0.266123i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	82.0381	3.16705
$672$	0	0
$673$	29.4919	1.13683	0.568415	−	0.822742i	$-0.307557\pi$
$673$	29.4919	1.13683	0.568415	+	0.822742i	$0.307557\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.9045i	− 0.688125i	−0.938947	−	0.344063i	$-0.888197\pi$
$677$	− 17.9045i	− 0.688125i	0.938947	−	0.344063i	$-0.111803\pi$
$678$	0	0
$679$	−9.74597	−0.374016
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.44157i	0.361271i	0.983550	+	0.180636i	$0.0578155\pi$
$683$	9.44157i	0.361271i	−0.983550	+	0.180636i	$0.942184\pi$
$684$	0	0
$685$	1.63508i	0.0624733i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	22.0000i	0.836919i	0.908235	+	0.418460i	$0.137430\pi$
$691$	22.0000i	0.836919i	−0.908235	+	0.418460i	$0.862570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.98952	−0.189263
$696$	0	0
$697$	−22.1109	−0.837509
$698$	0	0
$699$	0	0
$700$	0	0
$701$	48.7692i	1.84199i	0.389577	+	0.920994i	$0.372621\pi$
$701$	48.7692i	1.84199i	−0.389577	+	0.920994i	$0.627379\pi$
$702$	0	0
$703$	−0.237900	−0.00897257
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 11.5347i	− 0.433806i
$708$	0	0
$709$	21.2540i	0.798212i	0.916905	+	0.399106i	$0.130679\pi$
$709$	21.2540i	0.798212i	−0.916905	+	0.399106i	$0.869321\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.6412	0.735568
$714$	0	0
$715$	6.11088i	0.228534i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.8329	−1.37363	−0.686817	−	0.726830i	$-0.740993\pi$
$719$	−36.8329	−1.37363	−0.686817	+	0.726830i	$0.740993\pi$
$720$	0	0
$721$	10.7460	0.400201
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11.9363i	0.443304i
$726$	0	0
$727$	−3.49193	−0.129509	−0.0647543	−	0.997901i	$-0.520626\pi$
$727$	−3.49193	−0.129509	−0.0647543	+	0.997901i	$0.520626\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.56394i	0.131817i
$732$	0	0
$733$	8.36492i	0.308965i	0.987996	+	0.154483i	$0.0493711\pi$
$733$	8.36492i	0.308965i	−0.987996	+	0.154483i	$0.950629\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−51.4393	−1.89479
$738$	0	0
$739$	0.254033i	0.00934477i	0.999989	+	0.00467238i	$0.00148727\pi$
$739$	0.254033i	0.00934477i	−0.999989	+	0.00467238i	$0.998513\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.3037	−1.40523	−0.702614	−	0.711571i	$-0.747984\pi$
$743$	−38.3037	−1.40523	−0.702614	+	0.711571i	$0.747984\pi$
$744$	0	0
$745$	3.49193	0.127935
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 14.3858i	− 0.525646i
$750$	0	0
$751$	51.3488	1.87374	0.936872	−	0.349673i	$-0.113707\pi$
$751$	51.3488	1.87374	0.936872	+	0.349673i	$0.113707\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 4.27673i	− 0.155646i
$756$	0	0
$757$	− 25.7460i	− 0.935753i	−0.883794	−	0.467877i	$-0.845019\pi$
$757$	− 25.7460i	− 0.935753i	0.883794	−	0.467877i	$-0.154981\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.6945	1.25768	0.628838	−	0.777537i	$-0.283531\pi$
$761$	34.6945	1.25768	0.628838	+	0.777537i	$0.283531\pi$
$762$	0	0
$763$	12.7460i	0.461435i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.0642	−0.688368
$768$	0	0
$769$	11.6351	0.419572	0.209786	−	0.977747i	$-0.432723\pi$
$769$	11.6351	0.419572	0.209786	+	0.977747i	$0.432723\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0001i	0.791290i	0.918404	+	0.395645i	$0.129479\pi$
$773$	22.0001i	0.791290i	−0.918404	+	0.395645i	$0.870521\pi$
$774$	0	0
$775$	34.1109	1.22530
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.888110i	− 0.0318198i
$780$	0	0
$781$	35.6190i	1.27455i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.54003	−0.0906574
$786$	0	0
$787$	− 4.25403i	− 0.151640i	−0.997122	−	0.0758200i	$-0.975843\pi$
$787$	− 4.25403i	− 0.151640i	0.997122	−	0.0758200i	$-0.0241574\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.8353	0.598594
$792$	0	0
$793$	39.4919	1.40240
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.2664i	1.10751i	0.832679	+	0.553756i	$0.186806\pi$
$797$	31.2664i	1.10751i	−0.832679	+	0.553756i	$0.813194\pi$
$798$	0	0
$799$	23.2379	0.822098
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 5.21011i	− 0.183861i
$804$	0	0
$805$	− 1.00000i	− 0.0352454i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.58785	0.161300	0.0806502	−	0.996742i	$-0.474300\pi$
$809$	4.58785	0.161300	0.0806502	+	0.996742i	$0.474300\pi$
$810$	0	0
$811$	41.9839i	1.47425i	0.675755	+	0.737126i	$0.263818\pi$
$811$	41.9839i	1.47425i	−0.675755	+	0.737126i	$0.736182\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.15179	−0.285545
$816$	0	0
$817$	−0.143150	−0.00500818
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.1359i	0.772550i	0.922384	+	0.386275i	$0.126238\pi$
$821$	22.1359i	0.772550i	−0.922384	+	0.386275i	$0.873762\pi$
$822$	0	0
$823$	37.4919	1.30689	0.653443	−	0.756975i	$-0.273324\pi$
$823$	37.4919	1.30689	0.653443	+	0.756975i	$0.273324\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.1519i	1.04849i	0.851569	+	0.524243i	$0.175652\pi$
$827$	30.1519i	1.04849i	−0.851569	+	0.524243i	$0.824348\pi$
$828$	0	0
$829$	− 2.36492i	− 0.0821370i	−0.999156	−	0.0410685i	$-0.986924\pi$
$829$	− 2.36492i	− 0.0821370i	0.999156	−	0.0410685i	$-0.0130762\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.16228	0.109566
$834$	0	0
$835$	0.508067i	0.0175824i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.5457	−1.29622	−0.648110	−	0.761547i	$-0.724440\pi$
$839$	−37.5457	−1.29622	−0.648110	+	0.761547i	$0.724440\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.69143i	− 0.0581871i
$846$	0	0
$847$	−24.6190	−0.845917
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.25537i	0.180152i
$852$	0	0
$853$	− 5.23790i	− 0.179342i	−0.995971	−	0.0896711i	$-0.971418\pi$
$853$	− 5.23790i	− 0.179342i	0.995971	−	0.0896711i	$-0.0285816\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.6095	1.62631	0.813155	−	0.582048i	$-0.197748\pi$
$857$	47.6095	1.62631	0.813155	+	0.582048i	$0.197748\pi$
$858$	0	0
$859$	− 31.8730i	− 1.08749i	−0.839250	−	0.543746i	$-0.817005\pi$
$859$	− 31.8730i	− 1.08749i	0.839250	−	0.543746i	$-0.182995\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.3221	0.896016	0.448008	−	0.894030i	$-0.352134\pi$
$863$	26.3221	0.896016	0.448008	+	0.894030i	$0.352134\pi$
$864$	0	0
$865$	−6.49193	−0.220732
$866$	0	0
$867$	0	0
$868$	0	0
$869$	75.3119i	2.55478i
$870$	0	0
$871$	−24.7621	−0.839032
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.51867i	− 0.118953i
$876$	0	0
$877$	− 13.4919i	− 0.455590i	−0.973709	−	0.227795i	$-0.926848\pi$
$877$	− 13.4919i	− 0.455590i	0.973709	−	0.227795i	$-0.0731516\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.4233	−1.46297	−0.731484	−	0.681859i	$-0.761172\pi$
$881$	−43.4233	−1.46297	−0.731484	+	0.681859i	$0.761172\pi$
$882$	0	0
$883$	− 15.2379i	− 0.512796i	−0.966571	−	0.256398i	$-0.917464\pi$
$883$	− 15.2379i	− 0.512796i	0.966571	−	0.256398i	$-0.0825358\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0564	1.61358	0.806788	−	0.590841i	$-0.201204\pi$
$887$	48.0564	1.61358	0.806788	+	0.590841i	$0.201204\pi$
$888$	0	0
$889$	−10.8730	−0.364668
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.933378i	0.0312343i
$894$	0	0
$895$	−3.12702	−0.104525
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.1464i	0.571865i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.75013	0.257623
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33.3595	−1.10525	−0.552624	−	0.833430i	$-0.686374\pi$
$911$	−33.3595	−1.10525	−0.552624	+	0.833430i	$0.686374\pi$
$912$	0	0
$913$	−39.6028	−1.31066
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.77405i	− 0.289744i
$918$	0	0
$919$	14.5081	0.478577	0.239288	−	0.970949i	$-0.423086\pi$
$919$	14.5081	0.478577	0.239288	+	0.970949i	$0.423086\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17.1464i	0.564382i
$924$	0	0
$925$	9.12702i	0.300094i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.4710	−0.770058	−0.385029	−	0.922904i	$-0.625809\pi$
$929$	−23.4710	−0.770058	−0.385029	+	0.922904i	$0.625809\pi$
$930$	0	0
$931$	0.127017i	0.00416280i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.72622	−0.219971
$936$	0	0
$937$	41.4919	1.35548	0.677741	−	0.735301i	$-0.262959\pi$
$937$	41.4919	1.35548	0.677741	+	0.735301i	$0.262959\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 11.4894i	− 0.374544i	−0.982308	−	0.187272i	$-0.940036\pi$
$941$	− 11.4894i	− 0.374544i	0.982308	−	0.187272i	$-0.0599645\pi$
$942$	0	0
$943$	−19.6190	−0.638881
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.00256i	− 0.0650745i	−0.999471	−	0.0325372i	$-0.989641\pi$
$947$	− 2.00256i	− 0.0650745i	0.999471	−	0.0325372i	$-0.0103587\pi$
$948$	0	0
$949$	− 2.50807i	− 0.0814153i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0.311126	0.0100784	0.00503918	−	0.999987i	$-0.498396\pi$
$953$	0.311126	0.0100784	0.00503918	+	0.999987i	$0.498396\pi$
$954$	0	0
$955$	− 7.76210i	− 0.251176i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.58785	−0.148150
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.38287i	0.108898i
$966$	0	0
$967$	−25.2379	−0.811596	−0.405798	−	0.913963i	$-0.633006\pi$
$967$	−25.2379	−0.811596	−0.405798	+	0.913963i	$0.633006\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 13.7636i	− 0.441694i	−0.975309	−	0.220847i	$-0.929118\pi$
$971$	− 13.7636i	− 0.441694i	0.975309	−	0.220847i	$-0.0708821\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.1599	−0.740949	−0.370475	−	0.928843i	$-0.620805\pi$
$977$	−23.1599	−0.740949	−0.370475	+	0.928843i	$0.620805\pi$
$978$	0	0
$979$	41.7298i	1.33369i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.6682	1.71175	0.855875	−	0.517183i	$-0.173019\pi$
$983$	53.6682	1.71175	0.855875	+	0.517183i	$0.173019\pi$
$984$	0	0
$985$	0.729833	0.0232544
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.16228i	0.100555i
$990$	0	0
$991$	−17.3810	−0.552127	−0.276064	−	0.961139i	$-0.589030\pi$
$991$	−17.3810	−0.552127	−0.276064	+	0.961139i	$0.589030\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.0452680i	0.00143509i
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.c.3025.3		8
3.2	odd	2	inner	6048.2.c.c.3025.5		8
4.3	odd	2		1512.2.c.c.757.2	yes	8
8.3	odd	2		1512.2.c.c.757.1	✓	8
8.5	even	2	inner	6048.2.c.c.3025.6		8
12.11	even	2		1512.2.c.c.757.7	yes	8
24.5	odd	2	inner	6048.2.c.c.3025.4		8
24.11	even	2		1512.2.c.c.757.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.c.757.1	✓	8	8.3	odd	2
1512.2.c.c.757.2	yes	8	4.3	odd	2
1512.2.c.c.757.7	yes	8	12.11	even	2
1512.2.c.c.757.8	yes	8	24.11	even	2
6048.2.c.c.3025.3		8	1.1	even	1	trivial
6048.2.c.c.3025.4		8	24.5	odd	2	inner
6048.2.c.c.3025.5		8	3.2	odd	2	inner
6048.2.c.c.3025.6		8	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.356394i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.356394i q^{5} +1.00000 q^{7} -5.96816i q^{11} -2.87298i q^{13} +3.16228 q^{17} +0.127017i q^{19} +2.80588 q^{23} +4.87298 q^{25} +2.44949i q^{29} +7.00000 q^{31} -0.356394i q^{35} +1.87298i q^{37} -6.99208 q^{41} +1.12702i q^{43} +7.34847 q^{47} +1.00000 q^{49} -2.12702 q^{55} -6.63568i q^{59} +13.7460i q^{61} -1.02391 q^{65} -8.61895i q^{67} -5.96816 q^{71} +0.872983 q^{73} -5.96816i q^{77} -12.6190 q^{79} -6.63568i q^{83} -1.12702i q^{85} -6.99208 q^{89} -2.87298i q^{91} +0.0452680 q^{95} -9.74597 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 8 q^{25} + 56 q^{31} + 8 q^{49} - 48 q^{55} - 24 q^{73} - 8 q^{79} - 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 8 * q^25 + 56 * q^31 + 8 * q^49 - 48 * q^55 - 24 * q^73 - 8 * q^79 - 16 * q^97

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