LMFDB - Embedded newform 672.2.c.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(337,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, 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("672.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	672.337
Dual form			672.2.c.a.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.73205i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.73205i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.73205i q^{11} +1.46410i q^{13} +2.73205 q^{15} +6.19615 q^{17} +7.46410i q^{19} -1.00000i q^{21} +6.73205 q^{23} -2.46410 q^{25} +1.00000i q^{27} +3.46410i q^{29} -2.00000 q^{31} -4.73205 q^{33} +2.73205i q^{35} +4.92820i q^{37} +1.46410 q^{39} +7.66025 q^{41} -0.535898i q^{43} -2.73205i q^{45} +1.00000 q^{49} -6.19615i q^{51} -10.3923i q^{53} +12.9282 q^{55} +7.46410 q^{57} -1.07180i q^{59} -6.92820i q^{61} -1.00000 q^{63} -4.00000 q^{65} +6.00000i q^{67} -6.73205i q^{69} -10.7321 q^{71} +8.53590 q^{73} +2.46410i q^{75} -4.73205i q^{77} -8.39230 q^{79} +1.00000 q^{81} -5.46410i q^{83} +16.9282i q^{85} +3.46410 q^{87} +3.26795 q^{89} +1.46410i q^{91} +2.00000i q^{93} -20.3923 q^{95} -11.4641 q^{97} +4.73205i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 4 * q^9	$ 4 q + 4 q^{7} - 4 q^{9} + 4 q^{15} + 4 q^{17} + 20 q^{23} + 4 q^{25} - 8 q^{31} - 12 q^{33} - 8 q^{39} - 4 q^{41} + 4 q^{49} + 24 q^{55} + 16 q^{57} - 4 q^{63} - 16 q^{65} - 36 q^{71} + 48 q^{73} + 8 q^{79} + 4 q^{81} + 20 q^{89} - 40 q^{95} - 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^9 + 4 * q^15 + 4 * q^17 + 20 * q^23 + 4 * q^25 - 8 * q^31 - 12 * q^33 - 8 * q^39 - 4 * q^41 + 4 * q^49 + 24 * q^55 + 16 * q^57 - 4 * q^63 - 16 * q^65 - 36 * q^71 + 48 * q^73 + 8 * q^79 + 4 * q^81 + 20 * q^89 - 40 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	2.73205i	1.22181i	0.791704	+	0.610905i	$0.209194\pi$
$5$	2.73205i	1.22181i	−0.791704	+	0.610905i	$0.790806\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.73205i	− 1.42677i	−0.700774	−	0.713384i	$-0.747162\pi$
$11$	− 4.73205i	− 1.42677i	0.700774	−	0.713384i	$-0.252838\pi$
$12$	0	0
$13$	1.46410i	0.406069i	0.979172	+	0.203034i	$0.0650803\pi$
$13$	1.46410i	0.406069i	−0.979172	+	0.203034i	$0.934920\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	6.19615	1.50279	0.751394	−	0.659854i	$-0.229382\pi$
$17$	6.19615	1.50279	0.751394	+	0.659854i	$0.229382\pi$
$18$	0	0
$19$	7.46410i	1.71238i	0.516659	+	0.856191i	$0.327175\pi$
$19$	7.46410i	1.71238i	−0.516659	+	0.856191i	$0.672825\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	6.73205	1.40373	0.701865	−	0.712310i	$-0.252351\pi$
$23$	6.73205	1.40373	0.701865	+	0.712310i	$0.252351\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−4.73205	−0.823744
$34$	0	0
$35$	2.73205i	0.461801i
$36$	0	0
$37$	4.92820i	0.810192i	0.914274	+	0.405096i	$0.132762\pi$
$37$	4.92820i	0.810192i	−0.914274	+	0.405096i	$0.867238\pi$
$38$	0	0
$39$	1.46410	0.234444
$40$	0	0
$41$	7.66025	1.19633	0.598165	−	0.801373i	$-0.295897\pi$
$41$	7.66025	1.19633	0.598165	+	0.801373i	$0.295897\pi$
$42$	0	0
$43$	− 0.535898i	− 0.0817237i	−0.999165	−	0.0408619i	$-0.986990\pi$
$43$	− 0.535898i	− 0.0817237i	0.999165	−	0.0408619i	$-0.0130104\pi$
$44$	0	0
$45$	− 2.73205i	− 0.407270i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 6.19615i	− 0.867635i
$52$	0	0
$53$	− 10.3923i	− 1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$	− 10.3923i	− 1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$	0	0
$55$	12.9282	1.74324
$56$	0	0
$57$	7.46410	0.988644
$58$	0	0
$59$	− 1.07180i	− 0.139536i	−0.997563	−	0.0697680i	$-0.977774\pi$
$59$	− 1.07180i	− 0.139536i	0.997563	−	0.0697680i	$-0.0222259\pi$
$60$	0	0
$61$	− 6.92820i	− 0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	− 6.92820i	− 0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	0	0
$69$	− 6.73205i	− 0.810444i
$70$	0	0
$71$	−10.7321	−1.27366	−0.636830	−	0.771004i	$-0.719755\pi$
$71$	−10.7321	−1.27366	−0.636830	+	0.771004i	$0.719755\pi$
$72$	0	0
$73$	8.53590	0.999051	0.499526	−	0.866299i	$-0.333508\pi$
$73$	8.53590	0.999051	0.499526	+	0.866299i	$0.333508\pi$
$74$	0	0
$75$	2.46410i	0.284530i
$76$	0	0
$77$	− 4.73205i	− 0.539267i
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.46410i	− 0.599763i	−0.953976	−	0.299882i	$-0.903053\pi$
$83$	− 5.46410i	− 0.599763i	0.953976	−	0.299882i	$-0.0969472\pi$
$84$	0	0
$85$	16.9282i	1.83612i
$86$	0	0
$87$	3.46410	0.371391
$88$	0	0
$89$	3.26795	0.346402	0.173201	−	0.984887i	$-0.444589\pi$
$89$	3.26795	0.346402	0.173201	+	0.984887i	$0.444589\pi$
$90$	0	0
$91$	1.46410i	0.153480i
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	−20.3923	−2.09221
$96$	0	0
$97$	−11.4641	−1.16400	−0.582002	−	0.813188i	$-0.697730\pi$
$97$	−11.4641	−1.16400	−0.582002	+	0.813188i	$0.697730\pi$
$98$	0	0
$99$	4.73205i	0.475589i
$100$	0	0
$101$	− 4.19615i	− 0.417533i	−0.977966	−	0.208766i	$-0.933055\pi$
$101$	− 4.19615i	− 0.417533i	0.977966	−	0.208766i	$-0.0669448\pi$
$102$	0	0
$103$	−12.9282	−1.27385	−0.636927	−	0.770924i	$-0.719795\pi$
$103$	−12.9282	−1.27385	−0.636927	+	0.770924i	$0.719795\pi$
$104$	0	0
$105$	2.73205	0.266621
$106$	0	0
$107$	− 9.12436i	− 0.882085i	−0.897486	−	0.441042i	$-0.854609\pi$
$107$	− 9.12436i	− 0.882085i	0.897486	−	0.441042i	$-0.145391\pi$
$108$	0	0
$109$	1.07180i	0.102660i	0.998682	+	0.0513298i	$0.0163460\pi$
$109$	1.07180i	0.102660i	−0.998682	+	0.0513298i	$0.983654\pi$
$110$	0	0
$111$	4.92820	0.467764
$112$	0	0
$113$	−10.3923	−0.977626	−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923	−0.977626	−0.488813	+	0.872389i	$0.662570\pi$
$114$	0	0
$115$	18.3923i	1.71509i
$116$	0	0
$117$	− 1.46410i	− 0.135356i
$118$	0	0
$119$	6.19615	0.568000
$120$	0	0
$121$	−11.3923	−1.03566
$122$	0	0
$123$	− 7.66025i	− 0.690702i
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	17.4641	1.54969	0.774844	−	0.632152i	$-0.217828\pi$
$127$	17.4641	1.54969	0.774844	+	0.632152i	$0.217828\pi$
$128$	0	0
$129$	−0.535898	−0.0471832
$130$	0	0
$131$	6.53590i	0.571044i	0.958372	+	0.285522i	$0.0921670\pi$
$131$	6.53590i	0.571044i	−0.958372	+	0.285522i	$0.907833\pi$
$132$	0	0
$133$	7.46410i	0.647220i
$134$	0	0
$135$	−2.73205	−0.235137
$136$	0	0
$137$	−15.4641	−1.32119	−0.660594	−	0.750744i	$-0.729695\pi$
$137$	−15.4641	−1.32119	−0.660594	+	0.750744i	$0.729695\pi$
$138$	0	0
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.92820	0.579365
$144$	0	0
$145$	−9.46410	−0.785951
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	3.46410i	0.283790i	0.989882	+	0.141895i	$0.0453196\pi$
$149$	3.46410i	0.283790i	−0.989882	+	0.141895i	$0.954680\pi$
$150$	0	0
$151$	−20.7846	−1.69143	−0.845714	−	0.533637i	$-0.820825\pi$
$151$	−20.7846	−1.69143	−0.845714	+	0.533637i	$0.820825\pi$
$152$	0	0
$153$	−6.19615	−0.500929
$154$	0	0
$155$	− 5.46410i	− 0.438887i
$156$	0	0
$157$	− 1.07180i	− 0.0855387i	−0.999085	−	0.0427693i	$-0.986382\pi$
$157$	− 1.07180i	− 0.0855387i	0.999085	−	0.0427693i	$-0.0136181\pi$
$158$	0	0
$159$	−10.3923	−0.824163
$160$	0	0
$161$	6.73205	0.530560
$162$	0	0
$163$	7.07180i	0.553906i	0.960883	+	0.276953i	$0.0893246\pi$
$163$	7.07180i	0.553906i	−0.960883	+	0.276953i	$0.910675\pi$
$164$	0	0
$165$	− 12.9282i	− 1.00646i
$166$	0	0
$167$	6.53590	0.505763	0.252882	−	0.967497i	$-0.418622\pi$
$167$	6.53590	0.505763	0.252882	+	0.967497i	$0.418622\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	− 7.46410i	− 0.570794i
$172$	0	0
$173$	− 5.66025i	− 0.430341i	−0.976576	−	0.215171i	$-0.930969\pi$
$173$	− 5.66025i	− 0.430341i	0.976576	−	0.215171i	$-0.0690307\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	−1.07180	−0.0805612
$178$	0	0
$179$	4.33975i	0.324368i	0.986761	+	0.162184i	$0.0518538\pi$
$179$	4.33975i	0.324368i	−0.986761	+	0.162184i	$0.948146\pi$
$180$	0	0
$181$	5.46410i	0.406143i	0.979164	+	0.203072i	$0.0650925\pi$
$181$	5.46410i	0.406143i	−0.979164	+	0.203072i	$0.934908\pi$
$182$	0	0
$183$	−6.92820	−0.512148
$184$	0	0
$185$	−13.4641	−0.989900
$186$	0	0
$187$	− 29.3205i	− 2.14413i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	11.1244	0.804930	0.402465	−	0.915435i	$-0.368153\pi$
$191$	11.1244	0.804930	0.402465	+	0.915435i	$0.368153\pi$
$192$	0	0
$193$	−23.3205	−1.67865	−0.839323	−	0.543632i	$-0.817049\pi$
$193$	−23.3205	−1.67865	−0.839323	+	0.543632i	$0.817049\pi$
$194$	0	0
$195$	4.00000i	0.286446i
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	2.92820	0.207575	0.103787	−	0.994600i	$-0.466904\pi$
$199$	2.92820	0.207575	0.103787	+	0.994600i	$0.466904\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	3.46410i	0.243132i
$204$	0	0
$205$	20.9282i	1.46169i
$206$	0	0
$207$	−6.73205	−0.467910
$208$	0	0
$209$	35.3205	2.44317
$210$	0	0
$211$	− 14.3923i	− 0.990807i	−0.868663	−	0.495404i	$-0.835020\pi$
$211$	− 14.3923i	− 0.990807i	0.868663	−	0.495404i	$-0.164980\pi$
$212$	0	0
$213$	10.7321i	0.735348i
$214$	0	0
$215$	1.46410	0.0998509
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	− 8.53590i	− 0.576803i
$220$	0	0
$221$	9.07180i	0.610235i
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	0	0
$227$	− 1.46410i	− 0.0971758i	−0.998819	−	0.0485879i	$-0.984528\pi$
$227$	− 1.46410i	− 0.0971758i	0.998819	−	0.0485879i	$-0.0154721\pi$
$228$	0	0
$229$	− 27.3205i	− 1.80539i	−0.430281	−	0.902695i	$-0.641586\pi$
$229$	− 27.3205i	− 1.80539i	0.430281	−	0.902695i	$-0.358414\pi$
$230$	0	0
$231$	−4.73205	−0.311346
$232$	0	0
$233$	−23.8564	−1.56289	−0.781443	−	0.623977i	$-0.785516\pi$
$233$	−23.8564	−1.56289	−0.781443	+	0.623977i	$0.785516\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.39230i	0.545139i
$238$	0	0
$239$	27.1244	1.75453	0.877264	−	0.480008i	$-0.159366\pi$
$239$	27.1244	1.75453	0.877264	+	0.480008i	$0.159366\pi$
$240$	0	0
$241$	18.3923	1.18475	0.592376	−	0.805661i	$-0.298190\pi$
$241$	18.3923	1.18475	0.592376	+	0.805661i	$0.298190\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	2.73205i	0.174544i
$246$	0	0
$247$	−10.9282	−0.695345
$248$	0	0
$249$	−5.46410	−0.346273
$250$	0	0
$251$	− 7.60770i	− 0.480193i	−0.970749	−	0.240097i	$-0.922821\pi$
$251$	− 7.60770i	− 0.480193i	0.970749	−	0.240097i	$-0.0771792\pi$
$252$	0	0
$253$	− 31.8564i	− 2.00280i
$254$	0	0
$255$	16.9282	1.06009
$256$	0	0
$257$	−22.9808	−1.43350	−0.716750	−	0.697330i	$-0.754371\pi$
$257$	−22.9808	−1.43350	−0.716750	+	0.697330i	$0.754371\pi$
$258$	0	0
$259$	4.92820i	0.306224i
$260$	0	0
$261$	− 3.46410i	− 0.214423i
$262$	0	0
$263$	−6.73205	−0.415116	−0.207558	−	0.978223i	$-0.566552\pi$
$263$	−6.73205	−0.415116	−0.207558	+	0.978223i	$0.566552\pi$
$264$	0	0
$265$	28.3923	1.74413
$266$	0	0
$267$	− 3.26795i	− 0.199995i
$268$	0	0
$269$	− 3.80385i	− 0.231925i	−0.993254	−	0.115962i	$-0.963005\pi$
$269$	− 3.80385i	− 0.231925i	0.993254	−	0.115962i	$-0.0369952\pi$
$270$	0	0
$271$	−4.14359	−0.251705	−0.125853	−	0.992049i	$-0.540167\pi$
$271$	−4.14359	−0.251705	−0.125853	+	0.992049i	$0.540167\pi$
$272$	0	0
$273$	1.46410	0.0886115
$274$	0	0
$275$	11.6603i	0.703140i
$276$	0	0
$277$	20.9282i	1.25745i	0.777626	+	0.628727i	$0.216424\pi$
$277$	20.9282i	1.25745i	−0.777626	+	0.628727i	$0.783576\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	25.3205i	1.50515i	0.658508	+	0.752574i	$0.271188\pi$
$283$	25.3205i	1.50515i	−0.658508	+	0.752574i	$0.728812\pi$
$284$	0	0
$285$	20.3923i	1.20794i
$286$	0	0
$287$	7.66025	0.452170
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	11.4641i	0.672038i
$292$	0	0
$293$	27.1244i	1.58462i	0.610118	+	0.792311i	$0.291122\pi$
$293$	27.1244i	1.58462i	−0.610118	+	0.792311i	$0.708878\pi$
$294$	0	0
$295$	2.92820	0.170487
$296$	0	0
$297$	4.73205	0.274581
$298$	0	0
$299$	9.85641i	0.570011i
$300$	0	0
$301$	− 0.535898i	− 0.0308887i
$302$	0	0
$303$	−4.19615	−0.241063
$304$	0	0
$305$	18.9282	1.08383
$306$	0	0
$307$	22.3923i	1.27800i	0.769208	+	0.638998i	$0.220651\pi$
$307$	22.3923i	1.27800i	−0.769208	+	0.638998i	$0.779349\pi$
$308$	0	0
$309$	12.9282i	0.735460i
$310$	0	0
$311$	−5.85641	−0.332086	−0.166043	−	0.986118i	$-0.553099\pi$
$311$	−5.85641	−0.332086	−0.166043	+	0.986118i	$0.553099\pi$
$312$	0	0
$313$	29.7128	1.67947	0.839734	−	0.542998i	$-0.182711\pi$
$313$	29.7128	1.67947	0.839734	+	0.542998i	$0.182711\pi$
$314$	0	0
$315$	− 2.73205i	− 0.153934i
$316$	0	0
$317$	− 18.3923i	− 1.03301i	−0.856283	−	0.516507i	$-0.827232\pi$
$317$	− 18.3923i	− 1.03301i	0.856283	−	0.516507i	$-0.172768\pi$
$318$	0	0
$319$	16.3923	0.917793
$320$	0	0
$321$	−9.12436	−0.509272
$322$	0	0
$323$	46.2487i	2.57335i
$324$	0	0
$325$	− 3.60770i	− 0.200119i
$326$	0	0
$327$	1.07180	0.0592705
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 5.60770i	− 0.308227i	−0.988053	−	0.154113i	$-0.950748\pi$
$331$	− 5.60770i	− 0.308227i	0.988053	−	0.154113i	$-0.0492521\pi$
$332$	0	0
$333$	− 4.92820i	− 0.270064i
$334$	0	0
$335$	−16.3923	−0.895607
$336$	0	0
$337$	−0.143594	−0.00782204	−0.00391102	−	0.999992i	$-0.501245\pi$
$337$	−0.143594	−0.00782204	−0.00391102	+	0.999992i	$0.501245\pi$
$338$	0	0
$339$	10.3923i	0.564433i
$340$	0	0
$341$	9.46410i	0.512510i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	18.3923	0.990208
$346$	0	0
$347$	− 15.6603i	− 0.840686i	−0.907365	−	0.420343i	$-0.861910\pi$
$347$	− 15.6603i	− 0.840686i	0.907365	−	0.420343i	$-0.138090\pi$
$348$	0	0
$349$	− 26.9282i	− 1.44143i	−0.693230	−	0.720717i	$-0.743813\pi$
$349$	− 26.9282i	− 1.44143i	0.693230	−	0.720717i	$-0.256187\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	0	0
$353$	21.1244	1.12434	0.562168	−	0.827023i	$-0.309967\pi$
$353$	21.1244	1.12434	0.562168	+	0.827023i	$0.309967\pi$
$354$	0	0
$355$	− 29.3205i	− 1.55617i
$356$	0	0
$357$	− 6.19615i	− 0.327935i
$358$	0	0
$359$	−10.3397	−0.545711	−0.272855	−	0.962055i	$-0.587968\pi$
$359$	−10.3397	−0.545711	−0.272855	+	0.962055i	$0.587968\pi$
$360$	0	0
$361$	−36.7128	−1.93225
$362$	0	0
$363$	11.3923i	0.597941i
$364$	0	0
$365$	23.3205i	1.22065i
$366$	0	0
$367$	5.07180	0.264746	0.132373	−	0.991200i	$-0.457740\pi$
$367$	5.07180	0.264746	0.132373	+	0.991200i	$0.457740\pi$
$368$	0	0
$369$	−7.66025	−0.398777
$370$	0	0
$371$	− 10.3923i	− 0.539542i
$372$	0	0
$373$	− 24.7846i	− 1.28330i	−0.766998	−	0.641649i	$-0.778251\pi$
$373$	− 24.7846i	− 1.28330i	0.766998	−	0.641649i	$-0.221749\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−5.07180	−0.261211
$378$	0	0
$379$	− 4.53590i	− 0.232993i	−0.993191	−	0.116497i	$-0.962834\pi$
$379$	− 4.53590i	− 0.232993i	0.993191	−	0.116497i	$-0.0371665\pi$
$380$	0	0
$381$	− 17.4641i	− 0.894713i
$382$	0	0
$383$	0.679492	0.0347204	0.0173602	−	0.999849i	$-0.494474\pi$
$383$	0.679492	0.0347204	0.0173602	+	0.999849i	$0.494474\pi$
$384$	0	0
$385$	12.9282	0.658882
$386$	0	0
$387$	0.535898i	0.0272412i
$388$	0	0
$389$	− 24.5359i	− 1.24402i	−0.783010	−	0.622010i	$-0.786316\pi$
$389$	− 24.5359i	− 1.24402i	0.783010	−	0.622010i	$-0.213684\pi$
$390$	0	0
$391$	41.7128	2.10951
$392$	0	0
$393$	6.53590	0.329692
$394$	0	0
$395$	− 22.9282i	− 1.15364i
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	7.46410	0.373672
$400$	0	0
$401$	20.5359	1.02551	0.512757	−	0.858534i	$-0.328624\pi$
$401$	20.5359	1.02551	0.512757	+	0.858534i	$0.328624\pi$
$402$	0	0
$403$	− 2.92820i	− 0.145864i
$404$	0	0
$405$	2.73205i	0.135757i
$406$	0	0
$407$	23.3205	1.15595
$408$	0	0
$409$	−11.4641	−0.566863	−0.283432	−	0.958992i	$-0.591473\pi$
$409$	−11.4641	−0.566863	−0.283432	+	0.958992i	$0.591473\pi$
$410$	0	0
$411$	15.4641i	0.762788i
$412$	0	0
$413$	− 1.07180i	− 0.0527397i
$414$	0	0
$415$	14.9282	0.732797
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	27.7128i	1.35386i	0.736048	+	0.676930i	$0.236690\pi$
$419$	27.7128i	1.35386i	−0.736048	+	0.676930i	$0.763310\pi$
$420$	0	0
$421$	− 4.92820i	− 0.240186i	−0.992763	−	0.120093i	$-0.961681\pi$
$421$	− 4.92820i	− 0.240186i	0.992763	−	0.120093i	$-0.0383193\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.2679	−0.740604
$426$	0	0
$427$	− 6.92820i	− 0.335279i
$428$	0	0
$429$	− 6.92820i	− 0.334497i
$430$	0	0
$431$	−12.5885	−0.606365	−0.303182	−	0.952933i	$-0.598049\pi$
$431$	−12.5885	−0.606365	−0.303182	+	0.952933i	$0.598049\pi$
$432$	0	0
$433$	−16.9282	−0.813518	−0.406759	−	0.913536i	$-0.633341\pi$
$433$	−16.9282	−0.813518	−0.406759	+	0.913536i	$0.633341\pi$
$434$	0	0
$435$	9.46410i	0.453769i
$436$	0	0
$437$	50.2487i	2.40372i
$438$	0	0
$439$	27.7128	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$439$	27.7128	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 11.6603i	− 0.553995i	−0.960871	−	0.276998i	$-0.910661\pi$
$443$	− 11.6603i	− 0.553995i	0.960871	−	0.276998i	$-0.0893394\pi$
$444$	0	0
$445$	8.92820i	0.423237i
$446$	0	0
$447$	3.46410	0.163846
$448$	0	0
$449$	20.5359	0.969149	0.484574	−	0.874750i	$-0.338974\pi$
$449$	20.5359	0.969149	0.484574	+	0.874750i	$0.338974\pi$
$450$	0	0
$451$	− 36.2487i	− 1.70689i
$452$	0	0
$453$	20.7846i	0.976546i
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	8.92820	0.417644	0.208822	−	0.977954i	$-0.433037\pi$
$457$	8.92820	0.417644	0.208822	+	0.977954i	$0.433037\pi$
$458$	0	0
$459$	6.19615i	0.289212i
$460$	0	0
$461$	− 15.1244i	− 0.704411i	−0.935923	−	0.352206i	$-0.885432\pi$
$461$	− 15.1244i	− 0.704411i	0.935923	−	0.352206i	$-0.114568\pi$
$462$	0	0
$463$	−37.4641	−1.74110	−0.870552	−	0.492076i	$-0.836238\pi$
$463$	−37.4641	−1.74110	−0.870552	+	0.492076i	$0.836238\pi$
$464$	0	0
$465$	−5.46410	−0.253392
$466$	0	0
$467$	10.5359i	0.487543i	0.969833	+	0.243772i	$0.0783847\pi$
$467$	10.5359i	0.487543i	−0.969833	+	0.243772i	$0.921615\pi$
$468$	0	0
$469$	6.00000i	0.277054i
$470$	0	0
$471$	−1.07180	−0.0493858
$472$	0	0
$473$	−2.53590	−0.116601
$474$	0	0
$475$	− 18.3923i	− 0.843897i
$476$	0	0
$477$	10.3923i	0.475831i
$478$	0	0
$479$	−1.07180	−0.0489716	−0.0244858	−	0.999700i	$-0.507795\pi$
$479$	−1.07180	−0.0489716	−0.0244858	+	0.999700i	$0.507795\pi$
$480$	0	0
$481$	−7.21539	−0.328993
$482$	0	0
$483$	− 6.73205i	− 0.306319i
$484$	0	0
$485$	− 31.3205i	− 1.42219i
$486$	0	0
$487$	−2.92820	−0.132690	−0.0663448	−	0.997797i	$-0.521134\pi$
$487$	−2.92820	−0.132690	−0.0663448	+	0.997797i	$0.521134\pi$
$488$	0	0
$489$	7.07180	0.319798
$490$	0	0
$491$	− 3.26795i	− 0.147480i	−0.997277	−	0.0737402i	$-0.976506\pi$
$491$	− 3.26795i	− 0.147480i	0.997277	−	0.0737402i	$-0.0234936\pi$
$492$	0	0
$493$	21.4641i	0.966695i
$494$	0	0
$495$	−12.9282	−0.581080
$496$	0	0
$497$	−10.7321	−0.481398
$498$	0	0
$499$	24.2487i	1.08552i	0.839887	+	0.542761i	$0.182621\pi$
$499$	24.2487i	1.08552i	−0.839887	+	0.542761i	$0.817379\pi$
$500$	0	0
$501$	− 6.53590i	− 0.292002i
$502$	0	0
$503$	−25.1769	−1.12258	−0.561292	−	0.827618i	$-0.689695\pi$
$503$	−25.1769	−1.12258	−0.561292	+	0.827618i	$0.689695\pi$
$504$	0	0
$505$	11.4641	0.510146
$506$	0	0
$507$	− 10.8564i	− 0.482150i
$508$	0	0
$509$	19.5167i	0.865061i	0.901619	+	0.432530i	$0.142379\pi$
$509$	19.5167i	0.865061i	−0.901619	+	0.432530i	$0.857621\pi$
$510$	0	0
$511$	8.53590	0.377606
$512$	0	0
$513$	−7.46410	−0.329548
$514$	0	0
$515$	− 35.3205i	− 1.55641i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5.66025	−0.248458
$520$	0	0
$521$	13.5167	0.592176	0.296088	−	0.955161i	$-0.404318\pi$
$521$	13.5167	0.592176	0.296088	+	0.955161i	$0.404318\pi$
$522$	0	0
$523$	− 34.6410i	− 1.51475i	−0.652983	−	0.757373i	$-0.726483\pi$
$523$	− 34.6410i	− 1.51475i	0.652983	−	0.757373i	$-0.273517\pi$
$524$	0	0
$525$	2.46410i	0.107542i
$526$	0	0
$527$	−12.3923	−0.539817
$528$	0	0
$529$	22.3205	0.970457
$530$	0	0
$531$	1.07180i	0.0465120i
$532$	0	0
$533$	11.2154i	0.485792i
$534$	0	0
$535$	24.9282	1.07774
$536$	0	0
$537$	4.33975	0.187274
$538$	0	0
$539$	− 4.73205i	− 0.203824i
$540$	0	0
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	0	0
$543$	5.46410	0.234487
$544$	0	0
$545$	−2.92820	−0.125430
$546$	0	0
$547$	− 34.7846i	− 1.48728i	−0.668579	−	0.743641i	$-0.733097\pi$
$547$	− 34.7846i	− 1.48728i	0.668579	−	0.743641i	$-0.266903\pi$
$548$	0	0
$549$	6.92820i	0.295689i
$550$	0	0
$551$	−25.8564	−1.10152
$552$	0	0
$553$	−8.39230	−0.356877
$554$	0	0
$555$	13.4641i	0.571519i
$556$	0	0
$557$	39.4641i	1.67215i	0.548617	+	0.836074i	$0.315155\pi$
$557$	39.4641i	1.67215i	−0.548617	+	0.836074i	$0.684845\pi$
$558$	0	0
$559$	0.784610	0.0331855
$560$	0	0
$561$	−29.3205	−1.23791
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 28.3923i	− 1.19447i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−11.0718	−0.464154	−0.232077	−	0.972697i	$-0.574552\pi$
$569$	−11.0718	−0.464154	−0.232077	+	0.972697i	$0.574552\pi$
$570$	0	0
$571$	28.9282i	1.21061i	0.795995	+	0.605304i	$0.206948\pi$
$571$	28.9282i	1.21061i	−0.795995	+	0.605304i	$0.793052\pi$
$572$	0	0
$573$	− 11.1244i	− 0.464727i
$574$	0	0
$575$	−16.5885	−0.691786
$576$	0	0
$577$	−45.7128	−1.90305	−0.951525	−	0.307572i	$-0.900483\pi$
$577$	−45.7128	−1.90305	−0.951525	+	0.307572i	$0.900483\pi$
$578$	0	0
$579$	23.3205i	0.969167i
$580$	0	0
$581$	− 5.46410i	− 0.226689i
$582$	0	0
$583$	−49.1769	−2.03670
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	11.6077i	0.479101i	0.970884	+	0.239550i	$0.0770000\pi$
$587$	11.6077i	0.479101i	−0.970884	+	0.239550i	$0.923000\pi$
$588$	0	0
$589$	− 14.9282i	− 0.615106i
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−9.12436	−0.374692	−0.187346	−	0.982294i	$-0.559989\pi$
$593$	−9.12436	−0.374692	−0.187346	+	0.982294i	$0.559989\pi$
$594$	0	0
$595$	16.9282i	0.693989i
$596$	0	0
$597$	− 2.92820i	− 0.119843i
$598$	0	0
$599$	21.2679	0.868985	0.434492	−	0.900675i	$-0.356928\pi$
$599$	21.2679	0.868985	0.434492	+	0.900675i	$0.356928\pi$
$600$	0	0
$601$	−23.8564	−0.973123	−0.486562	−	0.873646i	$-0.661749\pi$
$601$	−23.8564	−0.973123	−0.486562	+	0.873646i	$0.661749\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	0	0
$605$	− 31.1244i	− 1.26538i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	3.46410	0.140372
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 40.0000i	− 1.61558i	−0.589467	−	0.807792i	$-0.700662\pi$
$613$	− 40.0000i	− 1.61558i	0.589467	−	0.807792i	$-0.299338\pi$
$614$	0	0
$615$	20.9282	0.843907
$616$	0	0
$617$	−6.39230	−0.257345	−0.128672	−	0.991687i	$-0.541072\pi$
$617$	−6.39230	−0.257345	−0.128672	+	0.991687i	$0.541072\pi$
$618$	0	0
$619$	− 25.8564i	− 1.03926i	−0.854392	−	0.519628i	$-0.826070\pi$
$619$	− 25.8564i	− 1.03926i	0.854392	−	0.519628i	$-0.173930\pi$
$620$	0	0
$621$	6.73205i	0.270148i
$622$	0	0
$623$	3.26795	0.130928
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	− 35.3205i	− 1.41057i
$628$	0	0
$629$	30.5359i	1.21755i
$630$	0	0
$631$	7.32051	0.291425	0.145712	−	0.989327i	$-0.453453\pi$
$631$	7.32051	0.291425	0.145712	+	0.989327i	$0.453453\pi$
$632$	0	0
$633$	−14.3923	−0.572043
$634$	0	0
$635$	47.7128i	1.89343i
$636$	0	0
$637$	1.46410i	0.0580098i
$638$	0	0
$639$	10.7321	0.424553
$640$	0	0
$641$	28.2487	1.11576	0.557879	−	0.829923i	$-0.311616\pi$
$641$	28.2487	1.11576	0.557879	+	0.829923i	$0.311616\pi$
$642$	0	0
$643$	− 2.39230i	− 0.0943433i	−0.998887	−	0.0471716i	$-0.984979\pi$
$643$	− 2.39230i	− 0.0943433i	0.998887	−	0.0471716i	$-0.0150208\pi$
$644$	0	0
$645$	− 1.46410i	− 0.0576489i
$646$	0	0
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	0	0
$649$	−5.07180	−0.199085
$650$	0	0
$651$	2.00000i	0.0783862i
$652$	0	0
$653$	− 0.928203i	− 0.0363234i	−0.999835	−	0.0181617i	$-0.994219\pi$
$653$	− 0.928203i	− 0.0363234i	0.999835	−	0.0181617i	$-0.00578137\pi$
$654$	0	0
$655$	−17.8564	−0.697708
$656$	0	0
$657$	−8.53590	−0.333017
$658$	0	0
$659$	− 23.6603i	− 0.921673i	−0.887485	−	0.460836i	$-0.847550\pi$
$659$	− 23.6603i	− 0.921673i	0.887485	−	0.460836i	$-0.152450\pi$
$660$	0	0
$661$	7.21539i	0.280646i	0.990106	+	0.140323i	$0.0448141\pi$
$661$	7.21539i	0.280646i	−0.990106	+	0.140323i	$0.955186\pi$
$662$	0	0
$663$	9.07180	0.352319
$664$	0	0
$665$	−20.3923	−0.790780
$666$	0	0
$667$	23.3205i	0.902974i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	−32.7846	−1.26564
$672$	0	0
$673$	−1.46410	−0.0564370	−0.0282185	−	0.999602i	$-0.508983\pi$
$673$	−1.46410	−0.0564370	−0.0282185	+	0.999602i	$0.508983\pi$
$674$	0	0
$675$	− 2.46410i	− 0.0948433i
$676$	0	0
$677$	0.588457i	0.0226163i	0.999936	+	0.0113081i	$0.00359956\pi$
$677$	0.588457i	0.0226163i	−0.999936	+	0.0113081i	$0.996400\pi$
$678$	0	0
$679$	−11.4641	−0.439952
$680$	0	0
$681$	−1.46410	−0.0561045
$682$	0	0
$683$	− 5.80385i	− 0.222078i	−0.993816	−	0.111039i	$-0.964582\pi$
$683$	− 5.80385i	− 0.222078i	0.993816	−	0.111039i	$-0.0354179\pi$
$684$	0	0
$685$	− 42.2487i	− 1.61424i
$686$	0	0
$687$	−27.3205	−1.04234
$688$	0	0
$689$	15.2154	0.579660
$690$	0	0
$691$	16.0000i	0.608669i	0.952565	+	0.304334i	$0.0984340\pi$
$691$	16.0000i	0.608669i	−0.952565	+	0.304334i	$0.901566\pi$
$692$	0	0
$693$	4.73205i	0.179756i
$694$	0	0
$695$	21.8564	0.829061
$696$	0	0
$697$	47.4641	1.79783
$698$	0	0
$699$	23.8564i	0.902332i
$700$	0	0
$701$	− 8.92820i	− 0.337214i	−0.985683	−	0.168607i	$-0.946073\pi$
$701$	− 8.92820i	− 0.337214i	0.985683	−	0.168607i	$-0.0539268\pi$
$702$	0	0
$703$	−36.7846	−1.38736
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.19615i	− 0.157813i
$708$	0	0
$709$	− 9.07180i	− 0.340698i	−0.985384	−	0.170349i	$-0.945510\pi$
$709$	− 9.07180i	− 0.340698i	0.985384	−	0.170349i	$-0.0544896\pi$
$710$	0	0
$711$	8.39230	0.314736
$712$	0	0
$713$	−13.4641	−0.504235
$714$	0	0
$715$	18.9282i	0.707875i
$716$	0	0
$717$	− 27.1244i	− 1.01298i
$718$	0	0
$719$	36.3923	1.35720	0.678602	−	0.734506i	$-0.262586\pi$
$719$	36.3923	1.35720	0.678602	+	0.734506i	$0.262586\pi$
$720$	0	0
$721$	−12.9282	−0.481471
$722$	0	0
$723$	− 18.3923i	− 0.684017i
$724$	0	0
$725$	− 8.53590i	− 0.317015i
$726$	0	0
$727$	−45.7128	−1.69539	−0.847697	−	0.530480i	$-0.822012\pi$
$727$	−45.7128	−1.69539	−0.847697	+	0.530480i	$0.822012\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.32051i	− 0.122813i
$732$	0	0
$733$	− 7.32051i	− 0.270389i	−0.990819	−	0.135195i	$-0.956834\pi$
$733$	− 7.32051i	− 0.270389i	0.990819	−	0.135195i	$-0.0431659\pi$
$734$	0	0
$735$	2.73205	0.100773
$736$	0	0
$737$	28.3923	1.04584
$738$	0	0
$739$	29.7128i	1.09300i	0.837458	+	0.546502i	$0.184041\pi$
$739$	29.7128i	1.09300i	−0.837458	+	0.546502i	$0.815959\pi$
$740$	0	0
$741$	10.9282i	0.401458i
$742$	0	0
$743$	−30.0526	−1.10252	−0.551261	−	0.834333i	$-0.685853\pi$
$743$	−30.0526	−1.10252	−0.551261	+	0.834333i	$0.685853\pi$
$744$	0	0
$745$	−9.46410	−0.346738
$746$	0	0
$747$	5.46410i	0.199921i
$748$	0	0
$749$	− 9.12436i	− 0.333397i
$750$	0	0
$751$	34.9282	1.27455	0.637274	−	0.770637i	$-0.280062\pi$
$751$	34.9282	1.27455	0.637274	+	0.770637i	$0.280062\pi$
$752$	0	0
$753$	−7.60770	−0.277240
$754$	0	0
$755$	− 56.7846i	− 2.06660i
$756$	0	0
$757$	− 1.85641i	− 0.0674722i	−0.999431	−	0.0337361i	$-0.989259\pi$
$757$	− 1.85641i	− 0.0674722i	0.999431	−	0.0337361i	$-0.0107406\pi$
$758$	0	0
$759$	−31.8564	−1.15631
$760$	0	0
$761$	−14.5885	−0.528831	−0.264416	−	0.964409i	$-0.585179\pi$
$761$	−14.5885	−0.528831	−0.264416	+	0.964409i	$0.585179\pi$
$762$	0	0
$763$	1.07180i	0.0388016i
$764$	0	0
$765$	− 16.9282i	− 0.612040i
$766$	0	0
$767$	1.56922	0.0566612
$768$	0	0
$769$	37.7128	1.35996	0.679979	−	0.733231i	$-0.261989\pi$
$769$	37.7128	1.35996	0.679979	+	0.733231i	$0.261989\pi$
$770$	0	0
$771$	22.9808i	0.827632i
$772$	0	0
$773$	− 36.9808i	− 1.33011i	−0.746796	−	0.665053i	$-0.768409\pi$
$773$	− 36.9808i	− 1.33011i	0.746796	−	0.665053i	$-0.231591\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	4.92820	0.176798
$778$	0	0
$779$	57.1769i	2.04858i
$780$	0	0
$781$	50.7846i	1.81722i
$782$	0	0
$783$	−3.46410	−0.123797
$784$	0	0
$785$	2.92820	0.104512
$786$	0	0
$787$	13.8564i	0.493928i	0.969025	+	0.246964i	$0.0794329\pi$
$787$	13.8564i	0.493928i	−0.969025	+	0.246964i	$0.920567\pi$
$788$	0	0
$789$	6.73205i	0.239667i
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	10.1436	0.360210
$794$	0	0
$795$	− 28.3923i	− 1.00697i
$796$	0	0
$797$	− 45.6603i	− 1.61737i	−0.588242	−	0.808685i	$-0.700180\pi$
$797$	− 45.6603i	− 1.61737i	0.588242	−	0.808685i	$-0.299820\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−3.26795	−0.115467
$802$	0	0
$803$	− 40.3923i	− 1.42541i
$804$	0	0
$805$	18.3923i	0.648244i
$806$	0	0
$807$	−3.80385	−0.133902
$808$	0	0
$809$	−32.5359	−1.14390	−0.571951	−	0.820288i	$-0.693813\pi$
$809$	−32.5359	−1.14390	−0.571951	+	0.820288i	$0.693813\pi$
$810$	0	0
$811$	1.07180i	0.0376359i	0.999823	+	0.0188179i	$0.00599029\pi$
$811$	1.07180i	0.0376359i	−0.999823	+	0.0188179i	$0.994010\pi$
$812$	0	0
$813$	4.14359i	0.145322i
$814$	0	0
$815$	−19.3205	−0.676768
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	− 1.46410i	− 0.0511599i
$820$	0	0
$821$	14.3923i	0.502295i	0.967949	+	0.251147i	$0.0808079\pi$
$821$	14.3923i	0.502295i	−0.967949	+	0.251147i	$0.919192\pi$
$822$	0	0
$823$	15.3205	0.534039	0.267020	−	0.963691i	$-0.413961\pi$
$823$	15.3205	0.534039	0.267020	+	0.963691i	$0.413961\pi$
$824$	0	0
$825$	11.6603	0.405958
$826$	0	0
$827$	12.3397i	0.429095i	0.976714	+	0.214548i	$0.0688277\pi$
$827$	12.3397i	0.429095i	−0.976714	+	0.214548i	$0.931172\pi$
$828$	0	0
$829$	52.1051i	1.80969i	0.425746	+	0.904843i	$0.360012\pi$
$829$	52.1051i	1.80969i	−0.425746	+	0.904843i	$0.639988\pi$
$830$	0	0
$831$	20.9282	0.725991
$832$	0	0
$833$	6.19615	0.214684
$834$	0	0
$835$	17.8564i	0.617946i
$836$	0	0
$837$	− 2.00000i	− 0.0691301i
$838$	0	0
$839$	−1.75129	−0.0604612	−0.0302306	−	0.999543i	$-0.509624\pi$
$839$	−1.75129	−0.0604612	−0.0302306	+	0.999543i	$0.509624\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	29.6603i	1.02034i
$846$	0	0
$847$	−11.3923	−0.391444
$848$	0	0
$849$	25.3205	0.868998
$850$	0	0
$851$	33.1769i	1.13729i
$852$	0	0
$853$	15.2154i	0.520965i	0.965479	+	0.260483i	$0.0838816\pi$
$853$	15.2154i	0.520965i	−0.965479	+	0.260483i	$0.916118\pi$
$854$	0	0
$855$	20.3923	0.697402
$856$	0	0
$857$	−16.7321	−0.571556	−0.285778	−	0.958296i	$-0.592252\pi$
$857$	−16.7321	−0.571556	−0.285778	+	0.958296i	$0.592252\pi$
$858$	0	0
$859$	− 36.5359i	− 1.24659i	−0.781987	−	0.623294i	$-0.785794\pi$
$859$	− 36.5359i	− 1.24659i	0.781987	−	0.623294i	$-0.214206\pi$
$860$	0	0
$861$	− 7.66025i	− 0.261061i
$862$	0	0
$863$	0.588457	0.0200313	0.0100157	−	0.999950i	$-0.496812\pi$
$863$	0.588457	0.0200313	0.0100157	+	0.999950i	$0.496812\pi$
$864$	0	0
$865$	15.4641	0.525795
$866$	0	0
$867$	− 21.3923i	− 0.726521i
$868$	0	0
$869$	39.7128i	1.34716i
$870$	0	0
$871$	−8.78461	−0.297655
$872$	0	0
$873$	11.4641	0.388001
$874$	0	0
$875$	6.92820i	0.234216i
$876$	0	0
$877$	54.6410i	1.84510i	0.385882	+	0.922548i	$0.373897\pi$
$877$	54.6410i	1.84510i	−0.385882	+	0.922548i	$0.626103\pi$
$878$	0	0
$879$	27.1244	0.914882
$880$	0	0
$881$	−44.0526	−1.48417	−0.742084	−	0.670307i	$-0.766163\pi$
$881$	−44.0526	−1.48417	−0.742084	+	0.670307i	$0.766163\pi$
$882$	0	0
$883$	− 14.3923i	− 0.484340i	−0.970234	−	0.242170i	$-0.922141\pi$
$883$	− 14.3923i	− 0.484340i	0.970234	−	0.242170i	$-0.0778591\pi$
$884$	0	0
$885$	− 2.92820i	− 0.0984305i
$886$	0	0
$887$	−33.4641	−1.12361	−0.561807	−	0.827268i	$-0.689894\pi$
$887$	−33.4641	−1.12361	−0.561807	+	0.827268i	$0.689894\pi$
$888$	0	0
$889$	17.4641	0.585727
$890$	0	0
$891$	− 4.73205i	− 0.158530i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−11.8564	−0.396316
$896$	0	0
$897$	9.85641	0.329096
$898$	0	0
$899$	− 6.92820i	− 0.231069i
$900$	0	0
$901$	− 64.3923i	− 2.14522i
$902$	0	0
$903$	−0.535898	−0.0178336
$904$	0	0
$905$	−14.9282	−0.496230
$906$	0	0
$907$	23.1769i	0.769577i	0.923005	+	0.384788i	$0.125726\pi$
$907$	23.1769i	0.769577i	−0.923005	+	0.384788i	$0.874274\pi$
$908$	0	0
$909$	4.19615i	0.139178i
$910$	0	0
$911$	25.2679	0.837165	0.418582	−	0.908179i	$-0.362527\pi$
$911$	25.2679	0.837165	0.418582	+	0.908179i	$0.362527\pi$
$912$	0	0
$913$	−25.8564	−0.855722
$914$	0	0
$915$	− 18.9282i	− 0.625747i
$916$	0	0
$917$	6.53590i	0.215834i
$918$	0	0
$919$	37.8564	1.24877	0.624384	−	0.781118i	$-0.285350\pi$
$919$	37.8564	1.24877	0.624384	+	0.781118i	$0.285350\pi$
$920$	0	0
$921$	22.3923	0.737852
$922$	0	0
$923$	− 15.7128i	− 0.517194i
$924$	0	0
$925$	− 12.1436i	− 0.399279i
$926$	0	0
$927$	12.9282	0.424618
$928$	0	0
$929$	49.1244	1.61172	0.805859	−	0.592108i	$-0.201704\pi$
$929$	49.1244	1.61172	0.805859	+	0.592108i	$0.201704\pi$
$930$	0	0
$931$	7.46410i	0.244626i
$932$	0	0
$933$	5.85641i	0.191730i
$934$	0	0
$935$	80.1051	2.61972
$936$	0	0
$937$	8.92820	0.291672	0.145836	−	0.989309i	$-0.453413\pi$
$937$	8.92820	0.291672	0.145836	+	0.989309i	$0.453413\pi$
$938$	0	0
$939$	− 29.7128i	− 0.969641i
$940$	0	0
$941$	− 20.1962i	− 0.658376i	−0.944264	−	0.329188i	$-0.893225\pi$
$941$	− 20.1962i	− 0.658376i	0.944264	−	0.329188i	$-0.106775\pi$
$942$	0	0
$943$	51.5692	1.67932
$944$	0	0
$945$	−2.73205	−0.0888736
$946$	0	0
$947$	− 50.1962i	− 1.63116i	−0.578647	−	0.815578i	$-0.696419\pi$
$947$	− 50.1962i	− 1.63116i	0.578647	−	0.815578i	$-0.303581\pi$
$948$	0	0
$949$	12.4974i	0.405684i
$950$	0	0
$951$	−18.3923	−0.596411
$952$	0	0
$953$	−12.9282	−0.418786	−0.209393	−	0.977832i	$-0.567149\pi$
$953$	−12.9282	−0.418786	−0.209393	+	0.977832i	$0.567149\pi$
$954$	0	0
$955$	30.3923i	0.983472i
$956$	0	0
$957$	− 16.3923i	− 0.529888i
$958$	0	0
$959$	−15.4641	−0.499362
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	9.12436i	0.294028i
$964$	0	0
$965$	− 63.7128i	− 2.05099i
$966$	0	0
$967$	−21.1769	−0.681004	−0.340502	−	0.940244i	$-0.610597\pi$
$967$	−21.1769	−0.681004	−0.340502	+	0.940244i	$0.610597\pi$
$968$	0	0
$969$	46.2487	1.48572
$970$	0	0
$971$	− 32.7846i	− 1.05211i	−0.850451	−	0.526054i	$-0.823671\pi$
$971$	− 32.7846i	− 1.05211i	0.850451	−	0.526054i	$-0.176329\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	−3.60770	−0.115539
$976$	0	0
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	− 15.4641i	− 0.494235i
$980$	0	0
$981$	− 1.07180i	− 0.0342198i
$982$	0	0
$983$	12.7846	0.407766	0.203883	−	0.978995i	$-0.434644\pi$
$983$	12.7846	0.407766	0.203883	+	0.978995i	$0.434644\pi$
$984$	0	0
$985$	60.1051	1.91511
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 3.60770i	− 0.114718i
$990$	0	0
$991$	−4.78461	−0.151988	−0.0759941	−	0.997108i	$-0.524213\pi$
$991$	−4.78461	−0.151988	−0.0759941	+	0.997108i	$0.524213\pi$
$992$	0	0
$993$	−5.60770	−0.177955
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	24.0000i	0.760088i	0.924968	+	0.380044i	$0.124091\pi$
$997$	24.0000i	0.760088i	−0.924968	+	0.380044i	$0.875909\pi$
$998$	0	0
$999$	−4.92820	−0.155921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.c.a.337.2		4
3.2	odd	2		2016.2.c.d.1009.1		4
4.3	odd	2		168.2.c.a.85.1	✓	4
7.6	odd	2		4704.2.c.b.2353.3		4
8.3	odd	2		168.2.c.a.85.2	yes	4
8.5	even	2	inner	672.2.c.a.337.3		4
12.11	even	2		504.2.c.b.253.4		4
16.3	odd	4		5376.2.a.u.1.2		2
16.5	even	4		5376.2.a.o.1.1		2
16.11	odd	4		5376.2.a.y.1.1		2
16.13	even	4		5376.2.a.bc.1.2		2
24.5	odd	2		2016.2.c.d.1009.4		4
24.11	even	2		504.2.c.b.253.3		4
28.27	even	2		1176.2.c.b.589.1		4
56.13	odd	2		4704.2.c.b.2353.2		4
56.27	even	2		1176.2.c.b.589.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.a.85.1	✓	4	4.3	odd	2
168.2.c.a.85.2	yes	4	8.3	odd	2
504.2.c.b.253.3		4	24.11	even	2
504.2.c.b.253.4		4	12.11	even	2
672.2.c.a.337.2		4	1.1	even	1	trivial
672.2.c.a.337.3		4	8.5	even	2	inner
1176.2.c.b.589.1		4	28.27	even	2
1176.2.c.b.589.2		4	56.27	even	2
2016.2.c.d.1009.1		4	3.2	odd	2
2016.2.c.d.1009.4		4	24.5	odd	2
4704.2.c.b.2353.2		4	56.13	odd	2
4704.2.c.b.2353.3		4	7.6	odd	2
5376.2.a.o.1.1		2	16.5	even	4
5376.2.a.u.1.2		2	16.3	odd	4
5376.2.a.y.1.1		2	16.11	odd	4
5376.2.a.bc.1.2		2	16.13	even	4

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.73205i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.73205i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.73205i q^{11} +1.46410i q^{13} +2.73205 q^{15} +6.19615 q^{17} +7.46410i q^{19} -1.00000i q^{21} +6.73205 q^{23} -2.46410 q^{25} +1.00000i q^{27} +3.46410i q^{29} -2.00000 q^{31} -4.73205 q^{33} +2.73205i q^{35} +4.92820i q^{37} +1.46410 q^{39} +7.66025 q^{41} -0.535898i q^{43} -2.73205i q^{45} +1.00000 q^{49} -6.19615i q^{51} -10.3923i q^{53} +12.9282 q^{55} +7.46410 q^{57} -1.07180i q^{59} -6.92820i q^{61} -1.00000 q^{63} -4.00000 q^{65} +6.00000i q^{67} -6.73205i q^{69} -10.7321 q^{71} +8.53590 q^{73} +2.46410i q^{75} -4.73205i q^{77} -8.39230 q^{79} +1.00000 q^{81} -5.46410i q^{83} +16.9282i q^{85} +3.46410 q^{87} +3.26795 q^{89} +1.46410i q^{91} +2.00000i q^{93} -20.3923 q^{95} -11.4641 q^{97} +4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 4 * q^9	\( 4 q + 4 q^{7} - 4 q^{9} + 4 q^{15} + 4 q^{17} + 20 q^{23} + 4 q^{25} - 8 q^{31} - 12 q^{33} - 8 q^{39} - 4 q^{41} + 4 q^{49} + 24 q^{55} + 16 q^{57} - 4 q^{63} - 16 q^{65} - 36 q^{71} + 48 q^{73} + 8 q^{79} + 4 q^{81} + 20 q^{89} - 40 q^{95} - 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^9 + 4 * q^15 + 4 * q^17 + 20 * q^23 + 4 * q^25 - 8 * q^31 - 12 * q^33 - 8 * q^39 - 4 * q^41 + 4 * q^49 + 24 * q^55 + 16 * q^57 - 4 * q^63 - 16 * q^65 - 36 * q^71 + 48 * q^73 + 8 * q^79 + 4 * q^81 + 20 * q^89 - 40 * q^95 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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