LMFDB - Embedded newform 672.2.c.a.337.1

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			337.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	672.337
Dual form			672.2.c.a.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.26795i q^{11} -5.46410i q^{13} -0.732051 q^{15} -4.19615 q^{17} +0.535898i q^{19} -1.00000i q^{21} +3.26795 q^{23} +4.46410 q^{25} +1.00000i q^{27} -3.46410i q^{29} -2.00000 q^{31} -1.26795 q^{33} -0.732051i q^{35} -8.92820i q^{37} -5.46410 q^{39} -9.66025 q^{41} -7.46410i q^{43} +0.732051i q^{45} +1.00000 q^{49} +4.19615i q^{51} +10.3923i q^{53} -0.928203 q^{55} +0.535898 q^{57} -14.9282i q^{59} +6.92820i q^{61} -1.00000 q^{63} -4.00000 q^{65} +6.00000i q^{67} -3.26795i q^{69} -7.26795 q^{71} +15.4641 q^{73} -4.46410i q^{75} -1.26795i q^{77} +12.3923 q^{79} +1.00000 q^{81} +1.46410i q^{83} +3.07180i q^{85} -3.46410 q^{87} +6.73205 q^{89} -5.46410i q^{91} +2.00000i q^{93} +0.392305 q^{95} -4.53590 q^{97} +1.26795i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 0.732051i	− 0.327383i	−0.986512	−	0.163692i	$-0.947660\pi$
$5$	− 0.732051i	− 0.327383i	0.986512	−	0.163692i	$-0.0523402\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.26795i	− 0.382301i	−0.981561	−	0.191151i	$-0.938778\pi$
$11$	− 1.26795i	− 0.382301i	0.981561	−	0.191151i	$-0.0612219\pi$
$12$	0	0
$13$	− 5.46410i	− 1.51547i	−0.652563	−	0.757735i	$-0.726306\pi$
$13$	− 5.46410i	− 1.51547i	0.652563	−	0.757735i	$-0.273694\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	−4.19615	−1.01772	−0.508858	−	0.860850i	$-0.669932\pi$
$17$	−4.19615	−1.01772	−0.508858	+	0.860850i	$0.669932\pi$
$18$	0	0
$19$	0.535898i	0.122944i	0.998109	+	0.0614718i	$0.0195794\pi$
$19$	0.535898i	0.122944i	−0.998109	+	0.0614718i	$0.980421\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	3.26795	0.681415	0.340707	−	0.940169i	$-0.389334\pi$
$23$	3.26795	0.681415	0.340707	+	0.940169i	$0.389334\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	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
$33$	−1.26795	−0.220722
$34$	0	0
$35$	− 0.732051i	− 0.123739i
$36$	0	0
$37$	− 8.92820i	− 1.46779i	−0.679264	−	0.733894i	$-0.737701\pi$
$37$	− 8.92820i	− 1.46779i	0.679264	−	0.733894i	$-0.262299\pi$
$38$	0	0
$39$	−5.46410	−0.874957
$40$	0	0
$41$	−9.66025	−1.50868	−0.754339	−	0.656485i	$-0.772043\pi$
$41$	−9.66025	−1.50868	−0.754339	+	0.656485i	$0.772043\pi$
$42$	0	0
$43$	− 7.46410i	− 1.13826i	−0.822246	−	0.569132i	$-0.807279\pi$
$43$	− 7.46410i	− 1.13826i	0.822246	−	0.569132i	$-0.192721\pi$
$44$	0	0
$45$	0.732051i	0.109128i
$46$	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
$51$	4.19615i	0.587579i
$52$	0	0
$53$	10.3923i	1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$	10.3923i	1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$	0	0
$55$	−0.928203	−0.125159
$56$	0	0
$57$	0.535898	0.0709815
$58$	0	0
$59$	− 14.9282i	− 1.94349i	−0.236040	−	0.971743i	$-0.575850\pi$
$59$	− 14.9282i	− 1.94349i	0.236040	−	0.971743i	$-0.424150\pi$
$60$	0	0
$61$	6.92820i	0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.92820i	0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	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
$69$	− 3.26795i	− 0.393415i
$70$	0	0
$71$	−7.26795	−0.862547	−0.431273	−	0.902221i	$-0.641936\pi$
$71$	−7.26795	−0.862547	−0.431273	+	0.902221i	$0.641936\pi$
$72$	0	0
$73$	15.4641	1.80994	0.904968	−	0.425479i	$-0.139895\pi$
$73$	15.4641	1.80994	0.904968	+	0.425479i	$0.139895\pi$
$74$	0	0
$75$	− 4.46410i	− 0.515470i
$76$	0	0
$77$	− 1.26795i	− 0.144496i
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.46410i	0.160706i	0.996766	+	0.0803530i	$0.0256048\pi$
$83$	1.46410i	0.160706i	−0.996766	+	0.0803530i	$0.974395\pi$
$84$	0	0
$85$	3.07180i	0.333183i
$86$	0	0
$87$	−3.46410	−0.371391
$88$	0	0
$89$	6.73205	0.713596	0.356798	−	0.934182i	$-0.383868\pi$
$89$	6.73205	0.713596	0.356798	+	0.934182i	$0.383868\pi$
$90$	0	0
$91$	− 5.46410i	− 0.572793i
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0.392305	0.0402496
$96$	0	0
$97$	−4.53590	−0.460551	−0.230275	−	0.973126i	$-0.573963\pi$
$97$	−4.53590	−0.460551	−0.230275	+	0.973126i	$0.573963\pi$
$98$	0	0
$99$	1.26795i	0.127434i
$100$	0	0
$101$	6.19615i	0.616540i	0.951299	+	0.308270i	$0.0997501\pi$
$101$	6.19615i	0.616540i	−0.951299	+	0.308270i	$0.900250\pi$
$102$	0	0
$103$	0.928203	0.0914586	0.0457293	−	0.998954i	$-0.485439\pi$
$103$	0.928203	0.0914586	0.0457293	+	0.998954i	$0.485439\pi$
$104$	0	0
$105$	−0.732051	−0.0714408
$106$	0	0
$107$	15.1244i	1.46213i	0.682310	+	0.731063i	$0.260976\pi$
$107$	15.1244i	1.46213i	−0.682310	+	0.731063i	$0.739024\pi$
$108$	0	0
$109$	14.9282i	1.42986i	0.699195	+	0.714931i	$0.253542\pi$
$109$	14.9282i	1.42986i	−0.699195	+	0.714931i	$0.746458\pi$
$110$	0	0
$111$	−8.92820	−0.847428
$112$	0	0
$113$	10.3923	0.977626	0.488813	−	0.872389i	$-0.337430\pi$
$113$	10.3923	0.977626	0.488813	+	0.872389i	$0.337430\pi$
$114$	0	0
$115$	− 2.39230i	− 0.223084i
$116$	0	0
$117$	5.46410i	0.505156i
$118$	0	0
$119$	−4.19615	−0.384661
$120$	0	0
$121$	9.39230	0.853846
$122$	0	0
$123$	9.66025i	0.871036i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	10.5359	0.934910	0.467455	−	0.884017i	$-0.345171\pi$
$127$	10.5359	0.934910	0.467455	+	0.884017i	$0.345171\pi$
$128$	0	0
$129$	−7.46410	−0.657178
$130$	0	0
$131$	13.4641i	1.17636i	0.808729	+	0.588182i	$0.200156\pi$
$131$	13.4641i	1.17636i	−0.808729	+	0.588182i	$0.799844\pi$
$132$	0	0
$133$	0.535898i	0.0464683i
$134$	0	0
$135$	0.732051	0.0630049
$136$	0	0
$137$	−8.53590	−0.729271	−0.364636	−	0.931150i	$-0.618806\pi$
$137$	−8.53590	−0.729271	−0.364636	+	0.931150i	$0.618806\pi$
$138$	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
$145$	−2.53590	−0.210595
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	− 3.46410i	− 0.283790i	−0.989882	−	0.141895i	$-0.954680\pi$
$149$	− 3.46410i	− 0.283790i	0.989882	−	0.141895i	$-0.0453196\pi$
$150$	0	0
$151$	20.7846	1.69143	0.845714	−	0.533637i	$-0.179175\pi$
$151$	20.7846	1.69143	0.845714	+	0.533637i	$0.179175\pi$
$152$	0	0
$153$	4.19615	0.339239
$154$	0	0
$155$	1.46410i	0.117599i
$156$	0	0
$157$	− 14.9282i	− 1.19140i	−0.803207	−	0.595700i	$-0.796875\pi$
$157$	− 14.9282i	− 1.19140i	0.803207	−	0.595700i	$-0.203125\pi$
$158$	0	0
$159$	10.3923	0.824163
$160$	0	0
$161$	3.26795	0.257550
$162$	0	0
$163$	20.9282i	1.63922i	0.572919	+	0.819612i	$0.305811\pi$
$163$	20.9282i	1.63922i	−0.572919	+	0.819612i	$0.694189\pi$
$164$	0	0
$165$	0.928203i	0.0722605i
$166$	0	0
$167$	13.4641	1.04188	0.520942	−	0.853592i	$-0.325581\pi$
$167$	13.4641	1.04188	0.520942	+	0.853592i	$0.325581\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	− 0.535898i	− 0.0409812i
$172$	0	0
$173$	11.6603i	0.886513i	0.896395	+	0.443256i	$0.146177\pi$
$173$	11.6603i	0.886513i	−0.896395	+	0.443256i	$0.853823\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	−14.9282	−1.12207
$178$	0	0
$179$	21.6603i	1.61896i	0.587145	+	0.809482i	$0.300252\pi$
$179$	21.6603i	1.61896i	−0.587145	+	0.809482i	$0.699748\pi$
$180$	0	0
$181$	− 1.46410i	− 0.108826i	−0.998519	−	0.0544129i	$-0.982671\pi$
$181$	− 1.46410i	− 0.108826i	0.998519	−	0.0544129i	$-0.0173287\pi$
$182$	0	0
$183$	6.92820	0.512148
$184$	0	0
$185$	−6.53590	−0.480529
$186$	0	0
$187$	5.32051i	0.389074i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−13.1244	−0.949645	−0.474823	−	0.880082i	$-0.657488\pi$
$191$	−13.1244	−0.949645	−0.474823	+	0.880082i	$0.657488\pi$
$192$	0	0
$193$	11.3205	0.814868	0.407434	−	0.913235i	$-0.366424\pi$
$193$	11.3205	0.814868	0.407434	+	0.913235i	$0.366424\pi$
$194$	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
$199$	−10.9282	−0.774680	−0.387340	−	0.921937i	$-0.626606\pi$
$199$	−10.9282	−0.774680	−0.387340	+	0.921937i	$0.626606\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	− 3.46410i	− 0.243132i
$204$	0	0
$205$	7.07180i	0.493916i
$206$	0	0
$207$	−3.26795	−0.227138
$208$	0	0
$209$	0.679492	0.0470014
$210$	0	0
$211$	6.39230i	0.440064i	0.975493	+	0.220032i	$0.0706162\pi$
$211$	6.39230i	0.440064i	−0.975493	+	0.220032i	$0.929384\pi$
$212$	0	0
$213$	7.26795i	0.497992i
$214$	0	0
$215$	−5.46410	−0.372649
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	− 15.4641i	− 1.04497i
$220$	0	0
$221$	22.9282i	1.54232i
$222$	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
$225$	−4.46410	−0.297607
$226$	0	0
$227$	5.46410i	0.362665i	0.983422	+	0.181333i	$0.0580410\pi$
$227$	5.46410i	0.362665i	−0.983422	+	0.181333i	$0.941959\pi$
$228$	0	0
$229$	7.32051i	0.483753i	0.970307	+	0.241876i	$0.0777629\pi$
$229$	7.32051i	0.483753i	−0.970307	+	0.241876i	$0.922237\pi$
$230$	0	0
$231$	−1.26795	−0.0834249
$232$	0	0
$233$	3.85641	0.252642	0.126321	−	0.991989i	$-0.459683\pi$
$233$	3.85641	0.252642	0.126321	+	0.991989i	$0.459683\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 12.3923i	− 0.804967i
$238$	0	0
$239$	2.87564	0.186010	0.0930050	−	0.995666i	$-0.470353\pi$
$239$	2.87564	0.186010	0.0930050	+	0.995666i	$0.470353\pi$
$240$	0	0
$241$	−2.39230	−0.154102	−0.0770510	−	0.997027i	$-0.524550\pi$
$241$	−2.39230	−0.154102	−0.0770510	+	0.997027i	$0.524550\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 0.732051i	− 0.0467690i
$246$	0	0
$247$	2.92820	0.186317
$248$	0	0
$249$	1.46410	0.0927837
$250$	0	0
$251$	− 28.3923i	− 1.79211i	−0.443947	−	0.896053i	$-0.646422\pi$
$251$	− 28.3923i	− 1.79211i	0.443947	−	0.896053i	$-0.353578\pi$
$252$	0	0
$253$	− 4.14359i	− 0.260505i
$254$	0	0
$255$	3.07180	0.192363
$256$	0	0
$257$	28.9808	1.80777	0.903885	−	0.427775i	$-0.140703\pi$
$257$	28.9808	1.80777	0.903885	+	0.427775i	$0.140703\pi$
$258$	0	0
$259$	− 8.92820i	− 0.554772i
$260$	0	0
$261$	3.46410i	0.214423i
$262$	0	0
$263$	−3.26795	−0.201510	−0.100755	−	0.994911i	$-0.532126\pi$
$263$	−3.26795	−0.201510	−0.100755	+	0.994911i	$0.532126\pi$
$264$	0	0
$265$	7.60770	0.467337
$266$	0	0
$267$	− 6.73205i	− 0.411995i
$268$	0	0
$269$	− 14.1962i	− 0.865555i	−0.901501	−	0.432777i	$-0.857534\pi$
$269$	− 14.1962i	− 0.865555i	0.901501	−	0.432777i	$-0.142466\pi$
$270$	0	0
$271$	−31.8564	−1.93514	−0.967569	−	0.252605i	$-0.918713\pi$
$271$	−31.8564	−1.93514	−0.967569	+	0.252605i	$0.918713\pi$
$272$	0	0
$273$	−5.46410	−0.330702
$274$	0	0
$275$	− 5.66025i	− 0.341326i
$276$	0	0
$277$	7.07180i	0.424903i	0.977172	+	0.212452i	$0.0681448\pi$
$277$	7.07180i	0.424903i	−0.977172	+	0.212452i	$0.931855\pi$
$278$	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
$283$	− 9.32051i	− 0.554047i	−0.960863	−	0.277023i	$-0.910652\pi$
$283$	− 9.32051i	− 0.554047i	0.960863	−	0.277023i	$-0.0893479\pi$
$284$	0	0
$285$	− 0.392305i	− 0.0232381i
$286$	0	0
$287$	−9.66025	−0.570227
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	4.53590i	0.265899i
$292$	0	0
$293$	2.87564i	0.167997i	0.996466	+	0.0839985i	$0.0267691\pi$
$293$	2.87564i	0.167997i	−0.996466	+	0.0839985i	$0.973231\pi$
$294$	0	0
$295$	−10.9282	−0.636265
$296$	0	0
$297$	1.26795	0.0735739
$298$	0	0
$299$	− 17.8564i	− 1.03266i
$300$	0	0
$301$	− 7.46410i	− 0.430224i
$302$	0	0
$303$	6.19615	0.355960
$304$	0	0
$305$	5.07180	0.290410
$306$	0	0
$307$	1.60770i	0.0917560i	0.998947	+	0.0458780i	$0.0146086\pi$
$307$	1.60770i	0.0917560i	−0.998947	+	0.0458780i	$0.985391\pi$
$308$	0	0
$309$	− 0.928203i	− 0.0528036i
$310$	0	0
$311$	21.8564	1.23936	0.619682	−	0.784853i	$-0.287262\pi$
$311$	21.8564	1.23936	0.619682	+	0.784853i	$0.287262\pi$
$312$	0	0
$313$	−25.7128	−1.45337	−0.726687	−	0.686969i	$-0.758941\pi$
$313$	−25.7128	−1.45337	−0.726687	+	0.686969i	$0.758941\pi$
$314$	0	0
$315$	0.732051i	0.0412464i
$316$	0	0
$317$	2.39230i	0.134365i	0.997741	+	0.0671826i	$0.0214010\pi$
$317$	2.39230i	0.134365i	−0.997741	+	0.0671826i	$0.978599\pi$
$318$	0	0
$319$	−4.39230	−0.245922
$320$	0	0
$321$	15.1244	0.844159
$322$	0	0
$323$	− 2.24871i	− 0.125122i
$324$	0	0
$325$	− 24.3923i	− 1.35304i
$326$	0	0
$327$	14.9282	0.825532
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 26.3923i	− 1.45065i	−0.688405	−	0.725326i	$-0.741689\pi$
$331$	− 26.3923i	− 1.45065i	0.688405	−	0.725326i	$-0.258311\pi$
$332$	0	0
$333$	8.92820i	0.489263i
$334$	0	0
$335$	4.39230	0.239977
$336$	0	0
$337$	−27.8564	−1.51744	−0.758718	−	0.651420i	$-0.774174\pi$
$337$	−27.8564	−1.51744	−0.758718	+	0.651420i	$0.774174\pi$
$338$	0	0
$339$	− 10.3923i	− 0.564433i
$340$	0	0
$341$	2.53590i	0.137327i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.39230	−0.128797
$346$	0	0
$347$	1.66025i	0.0891271i	0.999007	+	0.0445636i	$0.0141897\pi$
$347$	1.66025i	0.0891271i	−0.999007	+	0.0445636i	$0.985810\pi$
$348$	0	0
$349$	− 13.0718i	− 0.699717i	−0.936803	−	0.349859i	$-0.886230\pi$
$349$	− 13.0718i	− 0.699717i	0.936803	−	0.349859i	$-0.113770\pi$
$350$	0	0
$351$	5.46410	0.291652
$352$	0	0
$353$	−3.12436	−0.166293	−0.0831463	−	0.996537i	$-0.526497\pi$
$353$	−3.12436	−0.166293	−0.0831463	+	0.996537i	$0.526497\pi$
$354$	0	0
$355$	5.32051i	0.282383i
$356$	0	0
$357$	4.19615i	0.222084i
$358$	0	0
$359$	−27.6603	−1.45985	−0.729926	−	0.683526i	$-0.760446\pi$
$359$	−27.6603	−1.45985	−0.729926	+	0.683526i	$0.760446\pi$
$360$	0	0
$361$	18.7128	0.984885
$362$	0	0
$363$	− 9.39230i	− 0.492968i
$364$	0	0
$365$	− 11.3205i	− 0.592542i
$366$	0	0
$367$	18.9282	0.988044	0.494022	−	0.869449i	$-0.335526\pi$
$367$	18.9282	0.988044	0.494022	+	0.869449i	$0.335526\pi$
$368$	0	0
$369$	9.66025	0.502893
$370$	0	0
$371$	10.3923i	0.539542i
$372$	0	0
$373$	16.7846i	0.869074i	0.900654	+	0.434537i	$0.143088\pi$
$373$	16.7846i	0.869074i	−0.900654	+	0.434537i	$0.856912\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	−18.9282	−0.974852
$378$	0	0
$379$	− 11.4641i	− 0.588871i	−0.955671	−	0.294436i	$-0.904868\pi$
$379$	− 11.4641i	− 0.588871i	0.955671	−	0.294436i	$-0.0951317\pi$
$380$	0	0
$381$	− 10.5359i	− 0.539770i
$382$	0	0
$383$	35.3205	1.80479	0.902397	−	0.430906i	$-0.141806\pi$
$383$	35.3205	1.80479	0.902397	+	0.430906i	$0.141806\pi$
$384$	0	0
$385$	−0.928203	−0.0473056
$386$	0	0
$387$	7.46410i	0.379422i
$388$	0	0
$389$	− 31.4641i	− 1.59529i	−0.603125	−	0.797647i	$-0.706078\pi$
$389$	− 31.4641i	− 1.59529i	0.603125	−	0.797647i	$-0.293922\pi$
$390$	0	0
$391$	−13.7128	−0.693487
$392$	0	0
$393$	13.4641	0.679174
$394$	0	0
$395$	− 9.07180i	− 0.456452i
$396$	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
$399$	0.535898	0.0268285
$400$	0	0
$401$	27.4641	1.37149	0.685746	−	0.727841i	$-0.259476\pi$
$401$	27.4641	1.37149	0.685746	+	0.727841i	$0.259476\pi$
$402$	0	0
$403$	10.9282i	0.544373i
$404$	0	0
$405$	− 0.732051i	− 0.0363759i
$406$	0	0
$407$	−11.3205	−0.561137
$408$	0	0
$409$	−4.53590	−0.224286	−0.112143	−	0.993692i	$-0.535771\pi$
$409$	−4.53590	−0.224286	−0.112143	+	0.993692i	$0.535771\pi$
$410$	0	0
$411$	8.53590i	0.421045i
$412$	0	0
$413$	− 14.9282i	− 0.734569i
$414$	0	0
$415$	1.07180	0.0526124
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 27.7128i	− 1.35386i	−0.736048	−	0.676930i	$-0.763310\pi$
$419$	− 27.7128i	− 1.35386i	0.736048	−	0.676930i	$-0.236690\pi$
$420$	0	0
$421$	8.92820i	0.435134i	0.976045	+	0.217567i	$0.0698121\pi$
$421$	8.92820i	0.435134i	−0.976045	+	0.217567i	$0.930188\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.7321	−0.908638
$426$	0	0
$427$	6.92820i	0.335279i
$428$	0	0
$429$	6.92820i	0.334497i
$430$	0	0
$431$	18.5885	0.895374	0.447687	−	0.894190i	$-0.352248\pi$
$431$	18.5885	0.895374	0.447687	+	0.894190i	$0.352248\pi$
$432$	0	0
$433$	−3.07180	−0.147621	−0.0738106	−	0.997272i	$-0.523516\pi$
$433$	−3.07180	−0.147621	−0.0738106	+	0.997272i	$0.523516\pi$
$434$	0	0
$435$	2.53590i	0.121587i
$436$	0	0
$437$	1.75129i	0.0837755i
$438$	0	0
$439$	−27.7128	−1.32266	−0.661330	−	0.750095i	$-0.730008\pi$
$439$	−27.7128	−1.32266	−0.661330	+	0.750095i	$0.730008\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	5.66025i	0.268927i	0.990919	+	0.134463i	$0.0429311\pi$
$443$	5.66025i	0.268927i	−0.990919	+	0.134463i	$0.957069\pi$
$444$	0	0
$445$	− 4.92820i	− 0.233619i
$446$	0	0
$447$	−3.46410	−0.163846
$448$	0	0
$449$	27.4641	1.29611	0.648056	−	0.761593i	$-0.275582\pi$
$449$	27.4641	1.29611	0.648056	+	0.761593i	$0.275582\pi$
$450$	0	0
$451$	12.2487i	0.576769i
$452$	0	0
$453$	− 20.7846i	− 0.976546i
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−4.92820	−0.230532	−0.115266	−	0.993335i	$-0.536772\pi$
$457$	−4.92820	−0.230532	−0.115266	+	0.993335i	$0.536772\pi$
$458$	0	0
$459$	− 4.19615i	− 0.195860i
$460$	0	0
$461$	9.12436i	0.424964i	0.977165	+	0.212482i	$0.0681546\pi$
$461$	9.12436i	0.424964i	−0.977165	+	0.212482i	$0.931845\pi$
$462$	0	0
$463$	−30.5359	−1.41912	−0.709562	−	0.704643i	$-0.751107\pi$
$463$	−30.5359	−1.41912	−0.709562	+	0.704643i	$0.751107\pi$
$464$	0	0
$465$	1.46410	0.0678961
$466$	0	0
$467$	17.4641i	0.808142i	0.914728	+	0.404071i	$0.132405\pi$
$467$	17.4641i	0.808142i	−0.914728	+	0.404071i	$0.867595\pi$
$468$	0	0
$469$	6.00000i	0.277054i
$470$	0	0
$471$	−14.9282	−0.687855
$472$	0	0
$473$	−9.46410	−0.435160
$474$	0	0
$475$	2.39230i	0.109766i
$476$	0	0
$477$	− 10.3923i	− 0.475831i
$478$	0	0
$479$	−14.9282	−0.682087	−0.341044	−	0.940048i	$-0.610780\pi$
$479$	−14.9282	−0.682087	−0.341044	+	0.940048i	$0.610780\pi$
$480$	0	0
$481$	−48.7846	−2.22439
$482$	0	0
$483$	− 3.26795i	− 0.148697i
$484$	0	0
$485$	3.32051i	0.150777i
$486$	0	0
$487$	10.9282	0.495204	0.247602	−	0.968862i	$-0.420357\pi$
$487$	10.9282	0.495204	0.247602	+	0.968862i	$0.420357\pi$
$488$	0	0
$489$	20.9282	0.946406
$490$	0	0
$491$	− 6.73205i	− 0.303813i	−0.988395	−	0.151907i	$-0.951459\pi$
$491$	− 6.73205i	− 0.303813i	0.988395	−	0.151907i	$-0.0485413\pi$
$492$	0	0
$493$	14.5359i	0.654664i
$494$	0	0
$495$	0.928203	0.0417196
$496$	0	0
$497$	−7.26795	−0.326012
$498$	0	0
$499$	− 24.2487i	− 1.08552i	−0.839887	−	0.542761i	$-0.817379\pi$
$499$	− 24.2487i	− 1.08552i	0.839887	−	0.542761i	$-0.182621\pi$
$500$	0	0
$501$	− 13.4641i	− 0.601532i
$502$	0	0
$503$	37.1769	1.65764	0.828818	−	0.559518i	$-0.189014\pi$
$503$	37.1769	1.65764	0.828818	+	0.559518i	$0.189014\pi$
$504$	0	0
$505$	4.53590	0.201845
$506$	0	0
$507$	16.8564i	0.748619i
$508$	0	0
$509$	− 25.5167i	− 1.13101i	−0.824746	−	0.565503i	$-0.808682\pi$
$509$	− 25.5167i	− 1.13101i	0.824746	−	0.565503i	$-0.191318\pi$
$510$	0	0
$511$	15.4641	0.684092
$512$	0	0
$513$	−0.535898	−0.0236605
$514$	0	0
$515$	− 0.679492i	− 0.0299420i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	11.6603	0.511828
$520$	0	0
$521$	−31.5167	−1.38077	−0.690385	−	0.723442i	$-0.742559\pi$
$521$	−31.5167	−1.38077	−0.690385	+	0.723442i	$0.742559\pi$
$522$	0	0
$523$	34.6410i	1.51475i	0.652983	+	0.757373i	$0.273517\pi$
$523$	34.6410i	1.51475i	−0.652983	+	0.757373i	$0.726483\pi$
$524$	0	0
$525$	− 4.46410i	− 0.194829i
$526$	0	0
$527$	8.39230	0.365575
$528$	0	0
$529$	−12.3205	−0.535674
$530$	0	0
$531$	14.9282i	0.647829i
$532$	0	0
$533$	52.7846i	2.28636i
$534$	0	0
$535$	11.0718	0.478676
$536$	0	0
$537$	21.6603	0.934709
$538$	0	0
$539$	− 1.26795i	− 0.0546144i
$540$	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
$543$	−1.46410	−0.0628306
$544$	0	0
$545$	10.9282	0.468113
$546$	0	0
$547$	6.78461i	0.290089i	0.989425	+	0.145044i	$0.0463325\pi$
$547$	6.78461i	0.290089i	−0.989425	+	0.145044i	$0.953667\pi$
$548$	0	0
$549$	− 6.92820i	− 0.295689i
$550$	0	0
$551$	1.85641	0.0790856
$552$	0	0
$553$	12.3923	0.526974
$554$	0	0
$555$	6.53590i	0.277433i
$556$	0	0
$557$	32.5359i	1.37859i	0.724481	+	0.689295i	$0.242080\pi$
$557$	32.5359i	1.37859i	−0.724481	+	0.689295i	$0.757920\pi$
$558$	0	0
$559$	−40.7846	−1.72501
$560$	0	0
$561$	5.32051	0.224632
$562$	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
$565$	− 7.60770i	− 0.320058i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−24.9282	−1.04504	−0.522522	−	0.852626i	$-0.675009\pi$
$569$	−24.9282	−1.04504	−0.522522	+	0.852626i	$0.675009\pi$
$570$	0	0
$571$	15.0718i	0.630735i	0.948970	+	0.315368i	$0.102128\pi$
$571$	15.0718i	0.630735i	−0.948970	+	0.315368i	$0.897872\pi$
$572$	0	0
$573$	13.1244i	0.548278i
$574$	0	0
$575$	14.5885	0.608381
$576$	0	0
$577$	9.71281	0.404350	0.202175	−	0.979349i	$-0.435199\pi$
$577$	9.71281	0.404350	0.202175	+	0.979349i	$0.435199\pi$
$578$	0	0
$579$	− 11.3205i	− 0.470464i
$580$	0	0
$581$	1.46410i	0.0607412i
$582$	0	0
$583$	13.1769	0.545732
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	32.3923i	1.33697i	0.743724	+	0.668487i	$0.233058\pi$
$587$	32.3923i	1.33697i	−0.743724	+	0.668487i	$0.766942\pi$
$588$	0	0
$589$	− 1.07180i	− 0.0441626i
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	15.1244	0.621083	0.310541	−	0.950560i	$-0.399490\pi$
$593$	15.1244	0.621083	0.310541	+	0.950560i	$0.399490\pi$
$594$	0	0
$595$	3.07180i	0.125931i
$596$	0	0
$597$	10.9282i	0.447262i
$598$	0	0
$599$	24.7321	1.01052	0.505262	−	0.862966i	$-0.331396\pi$
$599$	24.7321	1.01052	0.505262	+	0.862966i	$0.331396\pi$
$600$	0	0
$601$	3.85641	0.157306	0.0786531	−	0.996902i	$-0.474938\pi$
$601$	3.85641	0.157306	0.0786531	+	0.996902i	$0.474938\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	0	0
$605$	− 6.87564i	− 0.279535i
$606$	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
$615$	7.07180	0.285162
$616$	0	0
$617$	14.3923	0.579412	0.289706	−	0.957116i	$-0.406442\pi$
$617$	14.3923	0.579412	0.289706	+	0.957116i	$0.406442\pi$
$618$	0	0
$619$	1.85641i	0.0746153i	0.999304	+	0.0373076i	$0.0118781\pi$
$619$	1.85641i	0.0746153i	−0.999304	+	0.0373076i	$0.988122\pi$
$620$	0	0
$621$	3.26795i	0.131138i
$622$	0	0
$623$	6.73205	0.269714
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	− 0.679492i	− 0.0271363i
$628$	0	0
$629$	37.4641i	1.49379i
$630$	0	0
$631$	−27.3205	−1.08761	−0.543806	−	0.839211i	$-0.683017\pi$
$631$	−27.3205	−1.08761	−0.543806	+	0.839211i	$0.683017\pi$
$632$	0	0
$633$	6.39230	0.254071
$634$	0	0
$635$	− 7.71281i	− 0.306074i
$636$	0	0
$637$	− 5.46410i	− 0.216496i
$638$	0	0
$639$	7.26795	0.287516
$640$	0	0
$641$	−20.2487	−0.799776	−0.399888	−	0.916564i	$-0.630951\pi$
$641$	−20.2487	−0.799776	−0.399888	+	0.916564i	$0.630951\pi$
$642$	0	0
$643$	18.3923i	0.725322i	0.931921	+	0.362661i	$0.118132\pi$
$643$	18.3923i	0.725322i	−0.931921	+	0.362661i	$0.881868\pi$
$644$	0	0
$645$	5.46410i	0.215149i
$646$	0	0
$647$	−11.3205	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$647$	−11.3205	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$648$	0	0
$649$	−18.9282	−0.742997
$650$	0	0
$651$	2.00000i	0.0783862i
$652$	0	0
$653$	12.9282i	0.505920i	0.967477	+	0.252960i	$0.0814041\pi$
$653$	12.9282i	0.505920i	−0.967477	+	0.252960i	$0.918596\pi$
$654$	0	0
$655$	9.85641	0.385122
$656$	0	0
$657$	−15.4641	−0.603312
$658$	0	0
$659$	− 6.33975i	− 0.246961i	−0.992347	−	0.123481i	$-0.960594\pi$
$659$	− 6.33975i	− 0.246961i	0.992347	−	0.123481i	$-0.0394057\pi$
$660$	0	0
$661$	48.7846i	1.89750i	0.316025	+	0.948751i	$0.397651\pi$
$661$	48.7846i	1.89750i	−0.316025	+	0.948751i	$0.602349\pi$
$662$	0	0
$663$	22.9282	0.890458
$664$	0	0
$665$	0.392305	0.0152129
$666$	0	0
$667$	− 11.3205i	− 0.438332i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	8.78461	0.339126
$672$	0	0
$673$	5.46410	0.210626	0.105313	−	0.994439i	$-0.466416\pi$
$673$	5.46410	0.210626	0.105313	+	0.994439i	$0.466416\pi$
$674$	0	0
$675$	4.46410i	0.171823i
$676$	0	0
$677$	− 30.5885i	− 1.17561i	−0.809003	−	0.587805i	$-0.799992\pi$
$677$	− 30.5885i	− 1.17561i	0.809003	−	0.587805i	$-0.200008\pi$
$678$	0	0
$679$	−4.53590	−0.174072
$680$	0	0
$681$	5.46410	0.209385
$682$	0	0
$683$	− 16.1962i	− 0.619729i	−0.950781	−	0.309864i	$-0.899716\pi$
$683$	− 16.1962i	− 0.619729i	0.950781	−	0.309864i	$-0.100284\pi$
$684$	0	0
$685$	6.24871i	0.238751i
$686$	0	0
$687$	7.32051	0.279295
$688$	0	0
$689$	56.7846	2.16332
$690$	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
$693$	1.26795i	0.0481654i
$694$	0	0
$695$	−5.85641	−0.222146
$696$	0	0
$697$	40.5359	1.53541
$698$	0	0
$699$	− 3.85641i	− 0.145863i
$700$	0	0
$701$	4.92820i	0.186136i	0.995660	+	0.0930678i	$0.0296673\pi$
$701$	4.92820i	0.186136i	−0.995660	+	0.0930678i	$0.970333\pi$
$702$	0	0
$703$	4.78461	0.180455
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.19615i	0.233030i
$708$	0	0
$709$	− 22.9282i	− 0.861087i	−0.902570	−	0.430543i	$-0.858322\pi$
$709$	− 22.9282i	− 0.861087i	0.902570	−	0.430543i	$-0.141678\pi$
$710$	0	0
$711$	−12.3923	−0.464748
$712$	0	0
$713$	−6.53590	−0.244771
$714$	0	0
$715$	5.07180i	0.189674i
$716$	0	0
$717$	− 2.87564i	− 0.107393i
$718$	0	0
$719$	15.6077	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$719$	15.6077	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$720$	0	0
$721$	0.928203	0.0345681
$722$	0	0
$723$	2.39230i	0.0889708i
$724$	0	0
$725$	− 15.4641i	− 0.574322i
$726$	0	0
$727$	9.71281	0.360228	0.180114	−	0.983646i	$-0.442353\pi$
$727$	9.71281	0.360228	0.180114	+	0.983646i	$0.442353\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	31.3205i	1.15843i
$732$	0	0
$733$	27.3205i	1.00911i	0.863381	+	0.504553i	$0.168343\pi$
$733$	27.3205i	1.00911i	−0.863381	+	0.504553i	$0.831657\pi$
$734$	0	0
$735$	−0.732051	−0.0270021
$736$	0	0
$737$	7.60770	0.280233
$738$	0	0
$739$	− 25.7128i	− 0.945861i	−0.881100	−	0.472931i	$-0.843196\pi$
$739$	− 25.7128i	− 0.945861i	0.881100	−	0.472931i	$-0.156804\pi$
$740$	0	0
$741$	− 2.92820i	− 0.107570i
$742$	0	0
$743$	8.05256	0.295420	0.147710	−	0.989031i	$-0.452810\pi$
$743$	8.05256	0.295420	0.147710	+	0.989031i	$0.452810\pi$
$744$	0	0
$745$	−2.53590	−0.0929081
$746$	0	0
$747$	− 1.46410i	− 0.0535687i
$748$	0	0
$749$	15.1244i	0.552632i
$750$	0	0
$751$	21.0718	0.768921	0.384460	−	0.923141i	$-0.374388\pi$
$751$	21.0718	0.768921	0.384460	+	0.923141i	$0.374388\pi$
$752$	0	0
$753$	−28.3923	−1.03467
$754$	0	0
$755$	− 15.2154i	− 0.553745i
$756$	0	0
$757$	25.8564i	0.939767i	0.882728	+	0.469884i	$0.155704\pi$
$757$	25.8564i	0.939767i	−0.882728	+	0.469884i	$0.844296\pi$
$758$	0	0
$759$	−4.14359	−0.150403
$760$	0	0
$761$	16.5885	0.601331	0.300666	−	0.953730i	$-0.402791\pi$
$761$	16.5885	0.601331	0.300666	+	0.953730i	$0.402791\pi$
$762$	0	0
$763$	14.9282i	0.540437i
$764$	0	0
$765$	− 3.07180i	− 0.111061i
$766$	0	0
$767$	−81.5692	−2.94529
$768$	0	0
$769$	−17.7128	−0.638740	−0.319370	−	0.947630i	$-0.603471\pi$
$769$	−17.7128	−0.638740	−0.319370	+	0.947630i	$0.603471\pi$
$770$	0	0
$771$	− 28.9808i	− 1.04372i
$772$	0	0
$773$	14.9808i	0.538821i	0.963025	+	0.269410i	$0.0868288\pi$
$773$	14.9808i	0.538821i	−0.963025	+	0.269410i	$0.913171\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	−8.92820	−0.320298
$778$	0	0
$779$	− 5.17691i	− 0.185482i
$780$	0	0
$781$	9.21539i	0.329753i
$782$	0	0
$783$	3.46410	0.123797
$784$	0	0
$785$	−10.9282	−0.390044
$786$	0	0
$787$	− 13.8564i	− 0.493928i	−0.969025	−	0.246964i	$-0.920567\pi$
$787$	− 13.8564i	− 0.493928i	0.969025	−	0.246964i	$-0.0794329\pi$
$788$	0	0
$789$	3.26795i	0.116342i
$790$	0	0
$791$	10.3923	0.369508
$792$	0	0
$793$	37.8564	1.34432
$794$	0	0
$795$	− 7.60770i	− 0.269817i
$796$	0	0
$797$	− 28.3397i	− 1.00385i	−0.864913	−	0.501923i	$-0.832626\pi$
$797$	− 28.3397i	− 1.00385i	0.864913	−	0.501923i	$-0.167374\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.73205	−0.237865
$802$	0	0
$803$	− 19.6077i	− 0.691941i
$804$	0	0
$805$	− 2.39230i	− 0.0843177i
$806$	0	0
$807$	−14.1962	−0.499728
$808$	0	0
$809$	−39.4641	−1.38748	−0.693742	−	0.720224i	$-0.744039\pi$
$809$	−39.4641	−1.38748	−0.693742	+	0.720224i	$0.744039\pi$
$810$	0	0
$811$	14.9282i	0.524200i	0.965041	+	0.262100i	$0.0844150\pi$
$811$	14.9282i	0.524200i	−0.965041	+	0.262100i	$0.915585\pi$
$812$	0	0
$813$	31.8564i	1.11725i
$814$	0	0
$815$	15.3205	0.536654
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	5.46410i	0.190931i
$820$	0	0
$821$	− 6.39230i	− 0.223093i	−0.993759	−	0.111546i	$-0.964420\pi$
$821$	− 6.39230i	− 0.223093i	0.993759	−	0.111546i	$-0.0355804\pi$
$822$	0	0
$823$	−19.3205	−0.673471	−0.336735	−	0.941599i	$-0.609323\pi$
$823$	−19.3205	−0.673471	−0.336735	+	0.941599i	$0.609323\pi$
$824$	0	0
$825$	−5.66025	−0.197065
$826$	0	0
$827$	29.6603i	1.03139i	0.856773	+	0.515694i	$0.172466\pi$
$827$	29.6603i	1.03139i	−0.856773	+	0.515694i	$0.827534\pi$
$828$	0	0
$829$	− 24.1051i	− 0.837205i	−0.908169	−	0.418603i	$-0.862520\pi$
$829$	− 24.1051i	− 0.837205i	0.908169	−	0.418603i	$-0.137480\pi$
$830$	0	0
$831$	7.07180	0.245318
$832$	0	0
$833$	−4.19615	−0.145388
$834$	0	0
$835$	− 9.85641i	− 0.341095i
$836$	0	0
$837$	− 2.00000i	− 0.0691301i
$838$	0	0
$839$	−50.2487	−1.73478	−0.867389	−	0.497631i	$-0.834204\pi$
$839$	−50.2487	−1.73478	−0.867389	+	0.497631i	$0.834204\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	12.3397i	0.424500i
$846$	0	0
$847$	9.39230	0.322723
$848$	0	0
$849$	−9.32051	−0.319879
$850$	0	0
$851$	− 29.1769i	− 1.00017i
$852$	0	0
$853$	56.7846i	1.94427i	0.234426	+	0.972134i	$0.424679\pi$
$853$	56.7846i	1.94427i	−0.234426	+	0.972134i	$0.575321\pi$
$854$	0	0
$855$	−0.392305	−0.0134165
$856$	0	0
$857$	−13.2679	−0.453225	−0.226612	−	0.973985i	$-0.572765\pi$
$857$	−13.2679	−0.453225	−0.226612	+	0.973985i	$0.572765\pi$
$858$	0	0
$859$	− 43.4641i	− 1.48298i	−0.670966	−	0.741488i	$-0.734120\pi$
$859$	− 43.4641i	− 1.48298i	0.670966	−	0.741488i	$-0.265880\pi$
$860$	0	0
$861$	9.66025i	0.329221i
$862$	0	0
$863$	−30.5885	−1.04124	−0.520622	−	0.853788i	$-0.674300\pi$
$863$	−30.5885	−1.04124	−0.520622	+	0.853788i	$0.674300\pi$
$864$	0	0
$865$	8.53590	0.290229
$866$	0	0
$867$	− 0.607695i	− 0.0206384i
$868$	0	0
$869$	− 15.7128i	− 0.533021i
$870$	0	0
$871$	32.7846	1.11086
$872$	0	0
$873$	4.53590	0.153517
$874$	0	0
$875$	− 6.92820i	− 0.234216i
$876$	0	0
$877$	− 14.6410i	− 0.494392i	−0.968966	−	0.247196i	$-0.920491\pi$
$877$	− 14.6410i	− 0.494392i	0.968966	−	0.247196i	$-0.0795092\pi$
$878$	0	0
$879$	2.87564	0.0969931
$880$	0	0
$881$	−5.94744	−0.200374	−0.100187	−	0.994969i	$-0.531944\pi$
$881$	−5.94744	−0.200374	−0.100187	+	0.994969i	$0.531944\pi$
$882$	0	0
$883$	6.39230i	0.215118i	0.994199	+	0.107559i	$0.0343035\pi$
$883$	6.39230i	0.215118i	−0.994199	+	0.107559i	$0.965697\pi$
$884$	0	0
$885$	10.9282i	0.367348i
$886$	0	0
$887$	−26.5359	−0.890988	−0.445494	−	0.895285i	$-0.646972\pi$
$887$	−26.5359	−0.890988	−0.445494	+	0.895285i	$0.646972\pi$
$888$	0	0
$889$	10.5359	0.353363
$890$	0	0
$891$	− 1.26795i	− 0.0424779i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	15.8564	0.530021
$896$	0	0
$897$	−17.8564	−0.596208
$898$	0	0
$899$	6.92820i	0.231069i
$900$	0	0
$901$	− 43.6077i	− 1.45278i
$902$	0	0
$903$	−7.46410	−0.248390
$904$	0	0
$905$	−1.07180	−0.0356277
$906$	0	0
$907$	− 39.1769i	− 1.30085i	−0.759571	−	0.650424i	$-0.774591\pi$
$907$	− 39.1769i	− 1.30085i	0.759571	−	0.650424i	$-0.225409\pi$
$908$	0	0
$909$	− 6.19615i	− 0.205513i
$910$	0	0
$911$	28.7321	0.951935	0.475968	−	0.879463i	$-0.342098\pi$
$911$	28.7321	0.951935	0.475968	+	0.879463i	$0.342098\pi$
$912$	0	0
$913$	1.85641	0.0614381
$914$	0	0
$915$	− 5.07180i	− 0.167668i
$916$	0	0
$917$	13.4641i	0.444624i
$918$	0	0
$919$	10.1436	0.334606	0.167303	−	0.985906i	$-0.446494\pi$
$919$	10.1436	0.334606	0.167303	+	0.985906i	$0.446494\pi$
$920$	0	0
$921$	1.60770	0.0529754
$922$	0	0
$923$	39.7128i	1.30716i
$924$	0	0
$925$	− 39.8564i	− 1.31047i
$926$	0	0
$927$	−0.928203	−0.0304862
$928$	0	0
$929$	24.8756	0.816143	0.408072	−	0.912950i	$-0.366201\pi$
$929$	24.8756	0.816143	0.408072	+	0.912950i	$0.366201\pi$
$930$	0	0
$931$	0.535898i	0.0175634i
$932$	0	0
$933$	− 21.8564i	− 0.715547i
$934$	0	0
$935$	3.89488	0.127376
$936$	0	0
$937$	−4.92820	−0.160997	−0.0804987	−	0.996755i	$-0.525651\pi$
$937$	−4.92820	−0.160997	−0.0804987	+	0.996755i	$0.525651\pi$
$938$	0	0
$939$	25.7128i	0.839106i
$940$	0	0
$941$	− 9.80385i	− 0.319596i	−0.987150	−	0.159798i	$-0.948916\pi$
$941$	− 9.80385i	− 0.319596i	0.987150	−	0.159798i	$-0.0510843\pi$
$942$	0	0
$943$	−31.5692	−1.02804
$944$	0	0
$945$	0.732051	0.0238136
$946$	0	0
$947$	− 39.8038i	− 1.29345i	−0.762723	−	0.646726i	$-0.776138\pi$
$947$	− 39.8038i	− 1.29345i	0.762723	−	0.646726i	$-0.223862\pi$
$948$	0	0
$949$	− 84.4974i	− 2.74290i
$950$	0	0
$951$	2.39230	0.0775758
$952$	0	0
$953$	0.928203	0.0300675	0.0150337	−	0.999887i	$-0.495214\pi$
$953$	0.928203	0.0300675	0.0150337	+	0.999887i	$0.495214\pi$
$954$	0	0
$955$	9.60770i	0.310898i
$956$	0	0
$957$	4.39230i	0.141983i
$958$	0	0
$959$	−8.53590	−0.275639
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	− 15.1244i	− 0.487376i
$964$	0	0
$965$	− 8.28719i	− 0.266774i
$966$	0	0
$967$	41.1769	1.32416	0.662080	−	0.749433i	$-0.269674\pi$
$967$	41.1769	1.32416	0.662080	+	0.749433i	$0.269674\pi$
$968$	0	0
$969$	−2.24871	−0.0722390
$970$	0	0
$971$	8.78461i	0.281912i	0.990016	+	0.140956i	$0.0450175\pi$
$971$	8.78461i	0.281912i	−0.990016	+	0.140956i	$0.954982\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	−24.3923	−0.781179
$976$	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
$979$	− 8.53590i	− 0.272808i
$980$	0	0
$981$	− 14.9282i	− 0.476621i
$982$	0	0
$983$	−28.7846	−0.918086	−0.459043	−	0.888414i	$-0.651808\pi$
$983$	−28.7846	−0.918086	−0.459043	+	0.888414i	$0.651808\pi$
$984$	0	0
$985$	−16.1051	−0.513152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 24.3923i	− 0.775630i
$990$	0	0
$991$	36.7846	1.16850	0.584251	−	0.811573i	$-0.301388\pi$
$991$	36.7846	1.16850	0.584251	+	0.811573i	$0.301388\pi$
$992$	0	0
$993$	−26.3923	−0.837534
$994$	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
$999$	8.92820	0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.c.a.337.1		4
3.2	odd	2		2016.2.c.d.1009.3		4
4.3	odd	2		168.2.c.a.85.4	yes	4
7.6	odd	2		4704.2.c.b.2353.4		4
8.3	odd	2		168.2.c.a.85.3	✓	4
8.5	even	2	inner	672.2.c.a.337.4		4
12.11	even	2		504.2.c.b.253.1		4
16.3	odd	4		5376.2.a.u.1.1		2
16.5	even	4		5376.2.a.o.1.2		2
16.11	odd	4		5376.2.a.y.1.2		2
16.13	even	4		5376.2.a.bc.1.1		2
24.5	odd	2		2016.2.c.d.1009.2		4
24.11	even	2		504.2.c.b.253.2		4
28.27	even	2		1176.2.c.b.589.4		4
56.13	odd	2		4704.2.c.b.2353.1		4
56.27	even	2		1176.2.c.b.589.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.a.85.3	✓	4	8.3	odd	2
168.2.c.a.85.4	yes	4	4.3	odd	2
504.2.c.b.253.1		4	12.11	even	2
504.2.c.b.253.2		4	24.11	even	2
672.2.c.a.337.1		4	1.1	even	1	trivial
672.2.c.a.337.4		4	8.5	even	2	inner
1176.2.c.b.589.3		4	56.27	even	2
1176.2.c.b.589.4		4	28.27	even	2
2016.2.c.d.1009.2		4	24.5	odd	2
2016.2.c.d.1009.3		4	3.2	odd	2
4704.2.c.b.2353.1		4	56.13	odd	2
4704.2.c.b.2353.4		4	7.6	odd	2
5376.2.a.o.1.2		2	16.5	even	4
5376.2.a.u.1.1		2	16.3	odd	4
5376.2.a.y.1.2		2	16.11	odd	4
5376.2.a.bc.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.26795i q^{11} -5.46410i q^{13} -0.732051 q^{15} -4.19615 q^{17} +0.535898i q^{19} -1.00000i q^{21} +3.26795 q^{23} +4.46410 q^{25} +1.00000i q^{27} -3.46410i q^{29} -2.00000 q^{31} -1.26795 q^{33} -0.732051i q^{35} -8.92820i q^{37} -5.46410 q^{39} -9.66025 q^{41} -7.46410i q^{43} +0.732051i q^{45} +1.00000 q^{49} +4.19615i q^{51} +10.3923i q^{53} -0.928203 q^{55} +0.535898 q^{57} -14.9282i q^{59} +6.92820i q^{61} -1.00000 q^{63} -4.00000 q^{65} +6.00000i q^{67} -3.26795i q^{69} -7.26795 q^{71} +15.4641 q^{73} -4.46410i q^{75} -1.26795i q^{77} +12.3923 q^{79} +1.00000 q^{81} +1.46410i q^{83} +3.07180i q^{85} -3.46410 q^{87} +6.73205 q^{89} -5.46410i q^{91} +2.00000i q^{93} +0.392305 q^{95} -4.53590 q^{97} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 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

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Database

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