LMFDB - Embedded newform 1568.2.t.c.753.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,2,Mod(177,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.177"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-2,0,0,0,0,0,-8,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			753.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1568.753
Dual form			1568.2.t.c.177.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{3} +(1.22474 - 0.707107i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.44949 - 1.41421i) q^{11} +4.24264i q^{13} -2.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(-3.67423 + 2.12132i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-1.50000 + 2.59808i) q^{25} +5.65685i q^{27} -2.82843i q^{29} +(-2.00000 + 3.46410i) q^{31} +(2.00000 + 3.46410i) q^{33} +(-7.34847 + 4.24264i) q^{37} +(3.00000 - 5.19615i) q^{39} +6.00000 q^{41} -8.48528i q^{43} +(-1.22474 - 0.707107i) q^{45} +(-7.34847 + 4.24264i) q^{51} +(-4.89898 - 2.82843i) q^{53} -4.00000 q^{55} +6.00000 q^{57} +(1.22474 + 0.707107i) q^{59} +(-11.0227 + 6.36396i) q^{61} +(3.00000 + 5.19615i) q^{65} +8.48528i q^{69} +(-1.00000 + 1.73205i) q^{73} +(3.67423 - 2.12132i) q^{75} +(4.00000 + 6.92820i) q^{79} +(2.50000 - 4.33013i) q^{81} +15.5563i q^{83} -8.48528i q^{85} +(-2.00000 + 3.46410i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(4.89898 - 2.82843i) q^{93} +(-3.00000 + 5.19615i) q^{95} -10.0000 q^{97} +2.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9} - 8 q^{15} + 12 q^{17} - 12 q^{23} - 6 q^{25} - 8 q^{31} + 8 q^{33} + 12 q^{39} + 24 q^{41} - 16 q^{55} + 24 q^{57} + 12 q^{65} - 4 q^{73} + 16 q^{79} + 10 q^{81} - 8 q^{87} - 12 q^{89} - 12 q^{95}+ \cdots - 40 q^{97}+O(q^{100}) $ 4 * q - 2 * q^9 - 8 * q^15 + 12 * q^17 - 12 * q^23 - 6 * q^25 - 8 * q^31 + 8 * q^33 + 12 * q^39 + 24 * q^41 - 16 * q^55 + 24 * q^57 + 12 * q^65 - 4 * q^73 + 16 * q^79 + 10 * q^81 - 8 * q^87 - 12 * q^89 - 12 * q^95 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−1.22474	−	0.707107i	−0.707107	−	0.408248i	0.102882	−	0.994694i	$-0.467194\pi$
$3$	−1.22474	−	0.707107i	−0.707107	−	0.408248i	−0.809989	+	0.586445i	$0.800527\pi$
$4$		0			0
$5$	1.22474	−	0.707107i	0.547723	−	0.316228i	−0.200480	−	0.979698i	$-0.564250\pi$
$5$	1.22474	−	0.707107i	0.547723	−	0.316228i	0.748203	+	0.663470i	$0.230917\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−2.44949	−	1.41421i	−0.738549	−	0.426401i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	−2.44949	−	1.41421i	−0.738549	−	0.426401i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$	−2.00000			−0.516398
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−3.67423	+	2.12132i	−0.842927	+	0.486664i	−0.858258	−	0.513218i	$-0.828453\pi$
$19$	−3.67423	+	2.12132i	−0.842927	+	0.486664i	0.0153309	+	0.999882i	$0.495120\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$	2.00000	+	3.46410i	0.348155	+	0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.34847	+	4.24264i	−1.20808	+	0.697486i	−0.962340	−	0.271850i	$-0.912365\pi$
$37$	−7.34847	+	4.24264i	−1.20808	+	0.697486i	−0.245741	+	0.969335i	$0.579031\pi$
$38$		0			0
$39$	3.00000	−	5.19615i	0.480384	−	0.832050i
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$	−1.22474	−	0.707107i	−0.182574	−	0.105409i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−7.34847	+	4.24264i	−1.02899	+	0.594089i
$52$		0			0
$53$	−4.89898	−	2.82843i	−0.672927	−	0.388514i	0.124258	−	0.992250i	$-0.460345\pi$
$53$	−4.89898	−	2.82843i	−0.672927	−	0.388514i	−0.797185	+	0.603736i	$0.793678\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	1.22474	+	0.707107i	0.159448	+	0.0920575i	0.577601	−	0.816319i	$-0.303989\pi$
$59$	1.22474	+	0.707107i	0.159448	+	0.0920575i	−0.418153	+	0.908377i	$0.637322\pi$
$60$		0			0
$61$	−11.0227	+	6.36396i	−1.41131	+	0.814822i	−0.995512	−	0.0946341i	$-0.969832\pi$
$61$	−11.0227	+	6.36396i	−1.41131	+	0.814822i	−0.415800	+	0.909456i	$0.636499\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$			8.48528i			1.02151i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$	3.67423	−	2.12132i	0.424264	−	0.244949i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	$-0.0180781\pi$
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	−0.548352	+	0.836247i	$0.684745\pi$
$80$		0			0
$81$	2.50000	−	4.33013i	0.277778	−	0.481125i
$82$		0			0
$83$			15.5563i			1.70753i	0.520658	+	0.853766i	$0.325687\pi$
$83$			15.5563i			1.70753i	−0.520658	+	0.853766i	$0.674313\pi$
$84$		0			0
$85$		−	8.48528i		−	0.920358i
$86$		0			0
$87$	−2.00000	+	3.46410i	−0.214423	+	0.371391i
$88$		0			0
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.89898	−	2.82843i	0.508001	−	0.293294i
$94$		0			0
$95$	−3.00000	+	5.19615i	−0.307794	+	0.533114i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$			2.82843i			0.284268i
$100$		0			0
$101$	−8.57321	−	4.94975i	−0.853067	−	0.492518i	0.00861771	−	0.999963i	$-0.497257\pi$
$101$	−8.57321	−	4.94975i	−0.853067	−	0.492518i	−0.861684	+	0.507445i	$0.830590\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.89898	+	2.82843i	−0.473602	+	0.273434i	−0.717746	−	0.696305i	$-0.754826\pi$
$107$	−4.89898	+	2.82843i	−0.473602	+	0.273434i	0.244144	+	0.969739i	$0.421493\pi$
$108$		0			0
$109$	−7.34847	−	4.24264i	−0.703856	−	0.406371i	0.104926	−	0.994480i	$-0.466539\pi$
$109$	−7.34847	−	4.24264i	−0.703856	−	0.406371i	−0.808782	+	0.588109i	$0.799873\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	−7.34847	−	4.24264i	−0.685248	−	0.395628i
$116$		0			0
$117$	3.67423	−	2.12132i	0.339683	−	0.196116i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.50000	−	2.59808i	−0.136364	−	0.236189i
$122$		0			0
$123$	−7.34847	−	4.24264i	−0.662589	−	0.382546i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$	−6.00000	+	10.3923i	−0.528271	+	0.914991i
$130$		0			0
$131$	−1.22474	+	0.707107i	−0.107006	+	0.0617802i	−0.552548	−	0.833481i	$-0.686344\pi$
$131$	−1.22474	+	0.707107i	−0.107006	+	0.0617802i	0.445542	+	0.895261i	$0.353011\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	4.00000	+	6.92820i	0.344265	+	0.596285i
$136$		0			0
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	0.708942	+	0.705266i	$0.249173\pi$
$138$		0			0
$139$		−	4.24264i		−	0.359856i	−0.983680	−	0.179928i	$-0.942414\pi$
$139$		−	4.24264i		−	0.359856i	0.983680	−	0.179928i	$-0.0575865\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	10.3923i	0.501745	−	0.869048i
$144$		0			0
$145$	−2.00000	−	3.46410i	−0.166091	−	0.287678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.79796	+	5.65685i	−0.802680	+	0.463428i	−0.844407	−	0.535701i	$-0.820047\pi$
$149$	−9.79796	+	5.65685i	−0.802680	+	0.463428i	0.0417274	+	0.999129i	$0.486714\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$			5.65685i			0.454369i
$156$		0			0
$157$	11.0227	+	6.36396i	0.879708	+	0.507899i	0.870562	−	0.492059i	$-0.163756\pi$
$157$	11.0227	+	6.36396i	0.879708	+	0.507899i	0.00914557	+	0.999958i	$0.497089\pi$
$158$		0			0
$159$	4.00000	+	6.92820i	0.317221	+	0.549442i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.34847	−	4.24264i	0.575577	−	0.332309i	−0.183797	−	0.982964i	$-0.558839\pi$
$163$	7.34847	−	4.24264i	0.575577	−	0.332309i	0.759374	+	0.650655i	$0.225506\pi$
$164$		0			0
$165$	4.89898	+	2.82843i	0.381385	+	0.220193i
$166$		0			0
$167$	−24.0000			−1.85718			−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000			−1.85718			−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	3.67423	+	2.12132i	0.280976	+	0.162221i
$172$		0			0
$173$	8.57321	−	4.94975i	0.651809	−	0.376322i	−0.137340	−	0.990524i	$-0.543855\pi$
$173$	8.57321	−	4.94975i	0.651809	−	0.376322i	0.789149	+	0.614202i	$0.210522\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.00000	−	1.73205i	−0.0751646	−	0.130189i
$178$		0			0
$179$	4.89898	+	2.82843i	0.366167	+	0.211407i	0.671783	−	0.740748i	$-0.265529\pi$
$179$	4.89898	+	2.82843i	0.366167	+	0.211407i	−0.305616	+	0.952155i	$0.598862\pi$
$180$		0			0
$181$		−	12.7279i		−	0.946059i	−0.881047	−	0.473029i	$-0.843160\pi$
$181$		−	12.7279i		−	0.946059i	0.881047	−	0.473029i	$-0.156840\pi$
$182$		0			0
$183$	18.0000			1.33060
$184$		0			0
$185$	−6.00000	+	10.3923i	−0.441129	+	0.764057i
$186$		0			0
$187$	−14.6969	+	8.48528i	−1.07475	+	0.620505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	−0.785022	−	0.619467i	$-0.787349\pi$
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	0.928986	+	0.370116i	$0.120682\pi$
$194$		0			0
$195$		−	8.48528i		−	0.607644i
$196$		0			0
$197$			5.65685i			0.403034i	0.979485	+	0.201517i	$0.0645872\pi$
$197$			5.65685i			0.403034i	−0.979485	+	0.201517i	$0.935413\pi$
$198$		0			0
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	−0.256391	−	0.966573i	$-0.582534\pi$
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	0.965272	−	0.261245i	$-0.0841331\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	7.34847	−	4.24264i	0.513239	−	0.296319i
$206$		0			0
$207$	−3.00000	+	5.19615i	−0.208514	+	0.361158i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$		−	16.9706i		−	1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$		−	16.9706i		−	1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.00000	−	10.3923i	−0.409197	−	0.708749i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.44949	−	1.41421i	0.165521	−	0.0955637i
$220$		0			0
$221$	22.0454	+	12.7279i	1.48293	+	0.856173i
$222$		0			0
$223$	28.0000			1.87502			0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000			1.87502			0.937509	+	0.347960i	$0.113126\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$		0			0
$227$	8.57321	+	4.94975i	0.569024	+	0.328526i	0.756760	−	0.653693i	$-0.226781\pi$
$227$	8.57321	+	4.94975i	0.569024	+	0.328526i	−0.187735	+	0.982220i	$0.560115\pi$
$228$		0			0
$229$	−3.67423	+	2.12132i	−0.242800	+	0.140181i	−0.616463	−	0.787384i	$-0.711435\pi$
$229$	−3.67423	+	2.12132i	−0.242800	+	0.140181i	0.373663	+	0.927565i	$0.378102\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	11.3137i		−	0.734904i
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$		0			0
$243$	8.57321	−	4.94975i	0.549972	−	0.317526i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.00000	−	15.5885i	−0.572656	−	0.991870i
$248$		0			0
$249$	11.0000	−	19.0526i	0.697097	−	1.20741i
$250$		0			0
$251$		−	18.3848i		−	1.16044i	−0.814461	−	0.580218i	$-0.802967\pi$
$251$		−	18.3848i		−	1.16044i	0.814461	−	0.580218i	$-0.197033\pi$
$252$		0			0
$253$			16.9706i			1.06693i
$254$		0			0
$255$	−6.00000	+	10.3923i	−0.375735	+	0.650791i
$256$		0			0
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.44949	+	1.41421i	−0.151620	+	0.0875376i
$262$		0			0
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	−0.212565	−	0.977147i	$-0.568182\pi$
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	0.952517	−	0.304487i	$-0.0984850\pi$
$264$		0			0
$265$	−8.00000			−0.491436
$266$		0			0
$267$			8.48528i			0.519291i
$268$		0			0
$269$	6.12372	+	3.53553i	0.373370	+	0.215565i	0.674930	−	0.737882i	$-0.264174\pi$
$269$	6.12372	+	3.53553i	0.373370	+	0.215565i	−0.301560	+	0.953447i	$0.597507\pi$
$270$		0			0
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	0.991658	+	0.128897i	$0.0411435\pi$
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	−0.384201	+	0.923249i	$0.625523\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.34847	−	4.24264i	0.443129	−	0.255841i
$276$		0			0
$277$	−14.6969	−	8.48528i	−0.883053	−	0.509831i	−0.0113895	−	0.999935i	$-0.503625\pi$
$277$	−14.6969	−	8.48528i	−0.883053	−	0.509831i	−0.871664	+	0.490104i	$0.836959\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		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
$283$	−11.0227	−	6.36396i	−0.655232	−	0.378298i	0.135226	−	0.990815i	$-0.456824\pi$
$283$	−11.0227	−	6.36396i	−0.655232	−	0.378298i	−0.790458	+	0.612517i	$0.790157\pi$
$284$		0			0
$285$	7.34847	−	4.24264i	0.435286	−	0.251312i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$	12.2474	+	7.07107i	0.717958	+	0.414513i
$292$		0			0
$293$		−	24.0416i		−	1.40453i	−0.711917	−	0.702264i	$-0.752173\pi$
$293$		−	24.0416i		−	1.40453i	0.711917	−	0.702264i	$-0.247827\pi$
$294$		0			0
$295$	2.00000			0.116445
$296$		0			0
$297$	8.00000	−	13.8564i	0.464207	−	0.804030i
$298$		0			0
$299$	22.0454	−	12.7279i	1.27492	−	0.736075i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	7.00000	+	12.1244i	0.402139	+	0.696526i
$304$		0			0
$305$	−9.00000	+	15.5885i	−0.515339	+	0.892592i
$306$		0			0
$307$		−	12.7279i		−	0.726421i	−0.931707	−	0.363210i	$-0.881681\pi$
$307$		−	12.7279i		−	0.726421i	0.931707	−	0.363210i	$-0.118319\pi$
$308$		0			0
$309$			5.65685i			0.321807i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.5959	−	11.3137i	1.10062	−	0.635441i	0.164234	−	0.986421i	$-0.447485\pi$
$317$	19.5959	−	11.3137i	1.10062	−	0.635441i	0.936383	+	0.350980i	$0.114152\pi$
$318$		0			0
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$			25.4558i			1.41640i
$324$		0			0
$325$	−11.0227	−	6.36396i	−0.611430	−	0.353009i
$326$		0			0
$327$	6.00000	+	10.3923i	0.331801	+	0.574696i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.0454	−	12.7279i	1.21173	−	0.699590i	0.248590	−	0.968609i	$-0.420033\pi$
$331$	22.0454	−	12.7279i	1.21173	−	0.699590i	0.963135	+	0.269019i	$0.0866994\pi$
$332$		0			0
$333$	7.34847	+	4.24264i	0.402694	+	0.232495i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	32.0000			1.74315			0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000			1.74315			0.871576	+	0.490261i	$0.163099\pi$
$338$		0			0
$339$	14.6969	+	8.48528i	0.798228	+	0.460857i
$340$		0			0
$341$	9.79796	−	5.65685i	0.530589	−	0.306336i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	6.00000	+	10.3923i	0.323029	+	0.559503i
$346$		0			0
$347$	12.2474	+	7.07107i	0.657477	+	0.379595i	0.791315	−	0.611408i	$-0.209397\pi$
$347$	12.2474	+	7.07107i	0.657477	+	0.379595i	−0.133838	+	0.991003i	$0.542730\pi$
$348$		0			0
$349$		−	4.24264i		−	0.227103i	−0.993532	−	0.113552i	$-0.963777\pi$
$349$		−	4.24264i		−	0.227103i	0.993532	−	0.113552i	$-0.0362227\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.0000	−	25.9808i	−0.791670	−	1.37121i	−0.924932	−	0.380131i	$-0.875879\pi$
$359$	−15.0000	−	25.9808i	−0.791670	−	1.37121i	0.133263	−	0.991081i	$-0.457455\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.0263158	+	0.0455803i
$362$		0			0
$363$			4.24264i			0.222681i
$364$		0			0
$365$			2.82843i			0.148047i
$366$		0			0
$367$	−14.0000	+	24.2487i	−0.730794	+	1.26577i	0.225750	+	0.974185i	$0.427517\pi$
$367$	−14.0000	+	24.2487i	−0.730794	+	1.26577i	−0.956544	+	0.291587i	$0.905817\pi$
$368$		0			0
$369$	−3.00000	−	5.19615i	−0.156174	−	0.270501i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	29.3939	−	16.9706i	1.52196	−	0.878702i	0.522294	−	0.852766i	$-0.325076\pi$
$373$	29.3939	−	16.9706i	1.52196	−	0.878702i	0.999664	−	0.0259367i	$-0.00825685\pi$
$374$		0			0
$375$	8.00000	−	13.8564i	0.413118	−	0.715542i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$			25.4558i			1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$			25.4558i			1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$		0			0
$381$	2.44949	+	1.41421i	0.125491	+	0.0724524i
$382$		0			0
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	−0.990702	−	0.136047i	$-0.956560\pi$
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	0.377531	−	0.925997i	$-0.376773\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.34847	+	4.24264i	−0.373544	+	0.215666i
$388$		0			0
$389$	2.44949	+	1.41421i	0.124194	+	0.0717035i	0.560810	−	0.827945i	$-0.310490\pi$
$389$	2.44949	+	1.41421i	0.124194	+	0.0717035i	−0.436616	+	0.899648i	$0.643823\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$		0			0
$393$	2.00000			0.100887
$394$		0			0
$395$	9.79796	+	5.65685i	0.492989	+	0.284627i
$396$		0			0
$397$	−18.3712	+	10.6066i	−0.922023	+	0.532330i	−0.884280	−	0.466957i	$-0.845350\pi$
$397$	−18.3712	+	10.6066i	−0.922023	+	0.532330i	−0.0377429	+	0.999287i	$0.512017\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	−14.6969	−	8.48528i	−0.732107	−	0.422682i
$404$		0			0
$405$		−	7.07107i		−	0.351364i
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	−0.454759	−	0.890614i	$-0.650275\pi$
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	0.998674	−	0.0514740i	$-0.0163919\pi$
$410$		0			0
$411$	7.34847	−	4.24264i	0.362473	−	0.209274i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	11.0000	+	19.0526i	0.539969	+	0.935253i
$416$		0			0
$417$	−3.00000	+	5.19615i	−0.146911	+	0.254457i
$418$		0			0
$419$		−	26.8701i		−	1.31269i	−0.754462	−	0.656344i	$-0.772102\pi$
$419$		−	26.8701i		−	1.31269i	0.754462	−	0.656344i	$-0.227898\pi$
$420$		0			0
$421$		−	16.9706i		−	0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$		−	16.9706i		−	0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.00000	+	15.5885i	0.436564	+	0.756151i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−14.6969	+	8.48528i	−0.709575	+	0.409673i
$430$		0			0
$431$	−3.00000	+	5.19615i	−0.144505	+	0.250290i	−0.929188	−	0.369607i	$-0.879492\pi$
$431$	−3.00000	+	5.19615i	−0.144505	+	0.250290i	0.784683	+	0.619897i	$0.212826\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$			5.65685i			0.271225i
$436$		0			0
$437$	22.0454	+	12.7279i	1.05457	+	0.608859i
$438$		0			0
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.5959	+	11.3137i	−0.931030	+	0.537531i	−0.887137	−	0.461506i	$-0.847309\pi$
$443$	−19.5959	+	11.3137i	−0.931030	+	0.537531i	−0.0438929	+	0.999036i	$0.513976\pi$
$444$		0			0
$445$	−7.34847	−	4.24264i	−0.348351	−	0.201120i
$446$		0			0
$447$	16.0000			0.756774
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	−14.6969	−	8.48528i	−0.692052	−	0.399556i
$452$		0			0
$453$	12.2474	−	7.07107i	0.575435	−	0.332228i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	0.981921	+	0.189292i	$0.0606194\pi$
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	−0.327028	+	0.945015i	$0.606047\pi$
$458$		0			0
$459$	29.3939	+	16.9706i	1.37199	+	0.792118i
$460$		0			0
$461$		−	15.5563i		−	0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$		−	15.5563i		−	0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$		0			0
$465$	4.00000	−	6.92820i	0.185496	−	0.321288i
$466$		0			0
$467$	6.12372	−	3.53553i	0.283372	−	0.163605i	−0.351577	−	0.936159i	$-0.614354\pi$
$467$	6.12372	−	3.53553i	0.283372	−	0.163605i	0.634949	+	0.772554i	$0.281021\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−9.00000	−	15.5885i	−0.414698	−	0.718278i
$472$		0			0
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$		−	12.7279i		−	0.583997i
$476$		0			0
$477$			5.65685i			0.259010i
$478$		0			0
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	−0.969920	−	0.243426i	$-0.921729\pi$
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	0.695773	+	0.718262i	$0.255062\pi$
$480$		0			0
$481$	−18.0000	−	31.1769i	−0.820729	−	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.2474	+	7.07107i	−0.556128	+	0.321081i
$486$		0			0
$487$	1.00000	−	1.73205i	0.0453143	−	0.0784867i	−0.842479	−	0.538730i	$-0.818904\pi$
$487$	1.00000	−	1.73205i	0.0453143	−	0.0784867i	0.887793	+	0.460243i	$0.152238\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$		−	39.5980i		−	1.78703i	−0.449032	−	0.893516i	$-0.648231\pi$
$491$		−	39.5980i		−	1.78703i	0.449032	−	0.893516i	$-0.351769\pi$
$492$		0			0
$493$	−14.6969	−	8.48528i	−0.661917	−	0.382158i
$494$		0			0
$495$	2.00000	+	3.46410i	0.0898933	+	0.155700i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.6969	−	8.48528i	0.657925	−	0.379853i	−0.133561	−	0.991041i	$-0.542641\pi$
$499$	14.6969	−	8.48528i	0.657925	−	0.379853i	0.791486	+	0.611187i	$0.209308\pi$
$500$		0			0
$501$	29.3939	+	16.9706i	1.31322	+	0.758189i
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$	−14.0000			−0.622992
$506$		0			0
$507$	6.12372	+	3.53553i	0.271964	+	0.157019i
$508$		0			0
$509$	1.22474	−	0.707107i	0.0542859	−	0.0313420i	−0.472611	−	0.881271i	$-0.656689\pi$
$509$	1.22474	−	0.707107i	0.0542859	−	0.0313420i	0.526897	+	0.849929i	$0.323355\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.0000	−	20.7846i	−0.529813	−	0.917663i
$514$		0			0
$515$	−4.89898	−	2.82843i	−0.215875	−	0.124635i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−14.0000			−0.614532
$520$		0			0
$521$	3.00000	−	5.19615i	0.131432	−	0.227648i	−0.792797	−	0.609486i	$-0.791376\pi$
$521$	3.00000	−	5.19615i	0.131432	−	0.227648i	0.924229	+	0.381839i	$0.124709\pi$
$522$		0			0
$523$	−25.7196	+	14.8492i	−1.12464	+	0.649312i	−0.942582	−	0.333975i	$-0.891610\pi$
$523$	−25.7196	+	14.8492i	−1.12464	+	0.649312i	−0.182060	+	0.983287i	$0.558276\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.0000	+	20.7846i	0.522728	+	0.905392i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		−	1.41421i		−	0.0613716i
$532$		0			0
$533$			25.4558i			1.10262i
$534$		0			0
$535$	−4.00000	+	6.92820i	−0.172935	+	0.299532i
$536$		0			0
$537$	−4.00000	−	6.92820i	−0.172613	−	0.298974i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	14.6969	−	8.48528i	0.631871	−	0.364811i	−0.149605	−	0.988746i	$-0.547800\pi$
$541$	14.6969	−	8.48528i	0.631871	−	0.364811i	0.781476	+	0.623935i	$0.214467\pi$
$542$		0			0
$543$	−9.00000	+	15.5885i	−0.386227	+	0.668965i
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		0			0
$549$	11.0227	+	6.36396i	0.470438	+	0.271607i
$550$		0			0
$551$	6.00000	+	10.3923i	0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	14.6969	−	8.48528i	0.623850	−	0.360180i
$556$		0			0
$557$	−4.89898	−	2.82843i	−0.207576	−	0.119844i	0.392608	−	0.919706i	$-0.371573\pi$
$557$	−4.89898	−	2.82843i	−0.207576	−	0.119844i	−0.600185	+	0.799862i	$0.704906\pi$
$558$		0			0
$559$	36.0000			1.52264
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	1.22474	+	0.707107i	0.0516168	+	0.0298010i	0.525586	−	0.850740i	$-0.323846\pi$
$563$	1.22474	+	0.707107i	0.0516168	+	0.0298010i	−0.473970	+	0.880541i	$0.657179\pi$
$564$		0			0
$565$	−14.6969	+	8.48528i	−0.618305	+	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−22.0454	−	12.7279i	−0.922572	−	0.532647i	−0.0381170	−	0.999273i	$-0.512136\pi$
$571$	−22.0454	−	12.7279i	−0.922572	−	0.532647i	−0.884455	+	0.466626i	$0.845469\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000			0.750652
$576$		0			0
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	0.134380	+	0.990930i	$0.457096\pi$
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	−0.925361	+	0.379088i	$0.876238\pi$
$578$		0			0
$579$	−4.89898	+	2.82843i	−0.203595	+	0.117545i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.00000	+	13.8564i	0.331326	+	0.573874i
$584$		0			0
$585$	3.00000	−	5.19615i	0.124035	−	0.214834i
$586$		0			0
$587$			41.0122i			1.69275i	0.532584	+	0.846377i	$0.321221\pi$
$587$			41.0122i			1.69275i	−0.532584	+	0.846377i	$0.678779\pi$
$588$		0			0
$589$		−	16.9706i		−	0.699260i
$590$		0			0
$591$	4.00000	−	6.92820i	0.164538	−	0.284988i
$592$		0			0
$593$	15.0000	+	25.9808i	0.615976	+	1.06690i	0.990212	+	0.139569i	$0.0445716\pi$
$593$	15.0000	+	25.9808i	0.615976	+	1.06690i	−0.374236	+	0.927333i	$0.622095\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−24.4949	+	14.1421i	−1.00251	+	0.578799i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.67423	−	2.12132i	−0.149379	−	0.0862439i
$606$		0			0
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	0.983262	+	0.182199i	$0.0583216\pi$
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	−0.333842	+	0.942629i	$0.608345\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7.34847	−	4.24264i	−0.296802	−	0.171359i	0.344203	−	0.938895i	$-0.388149\pi$
$613$	−7.34847	−	4.24264i	−0.296802	−	0.171359i	−0.641005	+	0.767536i	$0.721482\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		0			0
$617$	24.0000			0.966204			0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000			0.966204			0.483102	+	0.875564i	$0.339510\pi$
$618$		0			0
$619$	3.67423	+	2.12132i	0.147680	+	0.0852631i	0.572019	−	0.820240i	$-0.306160\pi$
$619$	3.67423	+	2.12132i	0.147680	+	0.0852631i	−0.424339	+	0.905503i	$0.639494\pi$
$620$		0			0
$621$	29.3939	−	16.9706i	1.17954	−	0.681005i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$	−14.6969	−	8.48528i	−0.586939	−	0.338869i
$628$		0			0
$629$			50.9117i			2.02998i
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$	−12.0000	+	20.7846i	−0.476957	+	0.826114i
$634$		0			0
$635$	−2.44949	+	1.41421i	−0.0972050	+	0.0561214i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.198194	−	0.980163i	$-0.436492\pi$
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.749749	−	0.661723i	$-0.230174\pi$
$642$		0			0
$643$			21.2132i			0.836567i	0.908317	+	0.418284i	$0.137368\pi$
$643$			21.2132i			0.836567i	−0.908317	+	0.418284i	$0.862632\pi$
$644$		0			0
$645$			16.9706i			0.668215i
$646$		0			0
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	−0.723645	−	0.690172i	$-0.757535\pi$
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	0.959529	+	0.281609i	$0.0908680\pi$
$648$		0			0
$649$	−2.00000	−	3.46410i	−0.0785069	−	0.135978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.8434	+	18.3848i	−1.24613	+	0.719452i	−0.970335	−	0.241765i	$-0.922274\pi$
$653$	−31.8434	+	18.3848i	−1.24613	+	0.719452i	−0.275792	+	0.961217i	$0.588940\pi$
$654$		0			0
$655$	−1.00000	+	1.73205i	−0.0390732	+	0.0676768i
$656$		0			0
$657$	2.00000			0.0780274
$658$		0			0
$659$			2.82843i			0.110180i	0.998481	+	0.0550899i	$0.0175446\pi$
$659$			2.82843i			0.110180i	−0.998481	+	0.0550899i	$0.982455\pi$
$660$		0			0
$661$	−33.0681	−	19.0919i	−1.28620	−	0.742588i	−0.308226	−	0.951313i	$-0.599735\pi$
$661$	−33.0681	−	19.0919i	−1.28620	−	0.742588i	−0.977974	+	0.208725i	$0.933069\pi$
$662$		0			0
$663$	−18.0000	−	31.1769i	−0.699062	−	1.21081i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−14.6969	+	8.48528i	−0.569068	+	0.328551i
$668$		0			0
$669$	−34.2929	−	19.7990i	−1.32584	−	0.765473i
$670$		0			0
$671$	36.0000			1.38976
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	−14.6969	−	8.48528i	−0.565685	−	0.326599i
$676$		0			0
$677$	8.57321	−	4.94975i	0.329495	−	0.190234i	−0.326122	−	0.945328i	$-0.605742\pi$
$677$	8.57321	−	4.94975i	0.329495	−	0.190234i	0.655617	+	0.755094i	$0.272409\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7.00000	−	12.1244i	−0.268241	−	0.464606i
$682$		0			0
$683$	4.89898	+	2.82843i	0.187454	+	0.108227i	0.590790	−	0.806825i	$-0.298816\pi$
$683$	4.89898	+	2.82843i	0.187454	+	0.108227i	−0.403336	+	0.915052i	$0.632149\pi$
$684$		0			0
$685$			8.48528i			0.324206i
$686$		0			0
$687$	6.00000			0.228914
$688$		0			0
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	−11.0227	+	6.36396i	−0.419323	+	0.242096i	−0.694788	−	0.719215i	$-0.744502\pi$
$691$	−11.0227	+	6.36396i	−0.419323	+	0.242096i	0.275464	+	0.961311i	$0.411168\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.00000	−	5.19615i	−0.113796	−	0.197101i
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$		0			0
$699$			8.48528i			0.320943i
$700$		0			0
$701$		−	19.7990i		−	0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$		−	19.7990i		−	0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$		0			0
$703$	18.0000	−	31.1769i	0.678883	−	1.17586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	22.0454	−	12.7279i	0.827933	−	0.478007i	−0.0252116	−	0.999682i	$-0.508026\pi$
$709$	22.0454	−	12.7279i	0.827933	−	0.478007i	0.853144	+	0.521675i	$0.174693\pi$
$710$		0			0
$711$	4.00000	−	6.92820i	0.150012	−	0.259828i
$712$		0			0
$713$	24.0000			0.898807
$714$		0			0
$715$		−	16.9706i		−	0.634663i
$716$		0			0
$717$	7.34847	+	4.24264i	0.274434	+	0.158444i
$718$		0			0
$719$	12.0000	+	20.7846i	0.447524	+	0.775135i	0.998224	−	0.0595683i	$-0.0189724\pi$
$719$	12.0000	+	20.7846i	0.447524	+	0.775135i	−0.550700	+	0.834703i	$0.685639\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−12.2474	+	7.07107i	−0.455488	+	0.262976i
$724$		0			0
$725$	7.34847	+	4.24264i	0.272915	+	0.157568i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	−44.0908	−	25.4558i	−1.63076	−	0.941518i
$732$		0			0
$733$	−25.7196	+	14.8492i	−0.949977	+	0.548469i	−0.893074	−	0.449910i	$-0.851456\pi$
$733$	−25.7196	+	14.8492i	−0.949977	+	0.548469i	−0.0569030	+	0.998380i	$0.518123\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−36.7423	−	21.2132i	−1.35159	−	0.780340i	−0.363117	−	0.931744i	$-0.618287\pi$
$739$	−36.7423	−	21.2132i	−1.35159	−	0.780340i	−0.988472	+	0.151403i	$0.951621\pi$
$740$		0			0
$741$			25.4558i			0.935144i
$742$		0			0
$743$	30.0000			1.10059			0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000			1.10059			0.550297	+	0.834969i	$0.314515\pi$
$744$		0			0
$745$	−8.00000	+	13.8564i	−0.293097	+	0.507659i
$746$		0			0
$747$	13.4722	−	7.77817i	0.492922	−	0.284589i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11.0000	−	19.0526i	−0.401396	−	0.695238i	0.592499	−	0.805571i	$-0.298141\pi$
$751$	−11.0000	−	19.0526i	−0.401396	−	0.695238i	−0.993895	+	0.110333i	$0.964808\pi$
$752$		0			0
$753$	−13.0000	+	22.5167i	−0.473746	+	0.820553i
$754$		0			0
$755$			14.1421i			0.514685i
$756$		0			0
$757$		−	25.4558i		−	0.925208i	−0.886565	−	0.462604i	$-0.846915\pi$
$757$		−	25.4558i		−	0.925208i	0.886565	−	0.462604i	$-0.153085\pi$
$758$		0			0
$759$	12.0000	−	20.7846i	0.435572	−	0.754434i
$760$		0			0
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	−0.998684	−	0.0512772i	$-0.983671\pi$
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	0.454935	−	0.890525i	$-0.349663\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−7.34847	+	4.24264i	−0.265684	+	0.153393i
$766$		0			0
$767$	−3.00000	+	5.19615i	−0.108324	+	0.187622i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		−	8.48528i		−	0.305590i
$772$		0			0
$773$	28.1691	+	16.2635i	1.01317	+	0.584956i	0.912119	−	0.409925i	$-0.134445\pi$
$773$	28.1691	+	16.2635i	1.01317	+	0.584956i	0.101054	+	0.994881i	$0.467778\pi$
$774$		0			0
$775$	−6.00000	−	10.3923i	−0.215526	−	0.373303i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.0454	+	12.7279i	−0.789859	+	0.456025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	16.0000			0.571793
$784$		0			0
$785$	18.0000			0.642448
$786$		0			0
$787$	33.0681	+	19.0919i	1.17875	+	0.680552i	0.955725	−	0.294260i	$-0.0950733\pi$
$787$	33.0681	+	19.0919i	1.17875	+	0.680552i	0.223026	+	0.974813i	$0.428407\pi$
$788$		0			0
$789$	−29.3939	+	16.9706i	−1.04645	+	0.604168i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−27.0000	−	46.7654i	−0.958798	−	1.66069i
$794$		0			0
$795$	9.79796	+	5.65685i	0.347498	+	0.200628i
$796$		0			0
$797$			26.8701i			0.951786i	0.879503	+	0.475893i	$0.157875\pi$
$797$			26.8701i			0.951786i	−0.879503	+	0.475893i	$0.842125\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−3.00000	+	5.19615i	−0.106000	+	0.183597i
$802$		0			0
$803$	4.89898	−	2.82843i	0.172881	−	0.0998130i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.00000	−	8.66025i	−0.176008	−	0.304855i
$808$		0			0
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	−0.952012	−	0.306062i	$-0.900989\pi$
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	0.741063	+	0.671436i	$0.234322\pi$
$810$		0			0
$811$			12.7279i			0.446938i	0.974711	+	0.223469i	$0.0717381\pi$
$811$			12.7279i			0.446938i	−0.974711	+	0.223469i	$0.928262\pi$
$812$		0			0
$813$		−	28.2843i		−	0.991973i
$814$		0			0
$815$	6.00000	−	10.3923i	0.210171	−	0.364027i
$816$		0			0
$817$	18.0000	+	31.1769i	0.629740	+	1.09074i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.5959	−	11.3137i	0.683902	−	0.394851i	−0.117421	−	0.993082i	$-0.537463\pi$
$821$	19.5959	−	11.3137i	0.683902	−	0.394851i	0.801324	+	0.598231i	$0.204129\pi$
$822$		0			0
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	0.997686	−	0.0679910i	$-0.0216589\pi$
$824$		0			0
$825$	−12.0000			−0.417786
$826$		0			0
$827$			45.2548i			1.57366i	0.617167	+	0.786832i	$0.288280\pi$
$827$			45.2548i			1.57366i	−0.617167	+	0.786832i	$0.711720\pi$
$828$		0			0
$829$	−18.3712	−	10.6066i	−0.638057	−	0.368383i	0.145809	−	0.989313i	$-0.453422\pi$
$829$	−18.3712	−	10.6066i	−0.638057	−	0.368383i	−0.783866	+	0.620930i	$0.786755\pi$
$830$		0			0
$831$	12.0000	+	20.7846i	0.416275	+	0.721010i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−29.3939	+	16.9706i	−1.01722	+	0.587291i
$836$		0			0
$837$	−19.5959	−	11.3137i	−0.677334	−	0.391059i
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	7.34847	+	4.24264i	0.253095	+	0.146124i
$844$		0			0
$845$	−6.12372	+	3.53553i	−0.210663	+	0.121626i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	9.00000	+	15.5885i	0.308879	+	0.534994i
$850$		0			0
$851$	44.0908	+	25.4558i	1.51141	+	0.872615i
$852$		0			0
$853$		−	4.24264i		−	0.145265i	−0.997359	−	0.0726326i	$-0.976860\pi$
$853$		−	4.24264i		−	0.145265i	0.997359	−	0.0726326i	$-0.0231401\pi$
$854$		0			0
$855$	6.00000			0.205196
$856$		0			0
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	0.244701	+	0.969599i	$0.421310\pi$
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	−0.962048	+	0.272882i	$0.912023\pi$
$858$		0			0
$859$	−40.4166	+	23.3345i	−1.37900	+	0.796164i	−0.992039	−	0.125934i	$-0.959807\pi$
$859$	−40.4166	+	23.3345i	−1.37900	+	0.796164i	−0.386957	+	0.922098i	$0.626474\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	0.994720	−	0.102624i	$-0.0327240\pi$
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	−0.586235	+	0.810141i	$0.699391\pi$
$864$		0			0
$865$	7.00000	−	12.1244i	0.238007	−	0.412240i
$866$		0			0
$867$			26.8701i			0.912555i
$868$		0			0
$869$		−	22.6274i		−	0.767583i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.00000	+	8.66025i	0.169224	+	0.293105i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36.7423	+	21.2132i	−1.24070	+	0.716319i	−0.969237	−	0.246130i	$-0.920841\pi$
$877$	−36.7423	+	21.2132i	−1.24070	+	0.716319i	−0.271464	+	0.962449i	$0.587508\pi$
$878$		0			0
$879$	−17.0000	+	29.4449i	−0.573396	+	0.993151i
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	50.9117i		−	1.71331i	−0.515886	−	0.856657i	$-0.672537\pi$
$883$		−	50.9117i		−	1.71331i	0.515886	−	0.856657i	$-0.327463\pi$
$884$		0			0
$885$	−2.44949	−	1.41421i	−0.0823387	−	0.0475383i
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−12.2474	+	7.07107i	−0.410305	+	0.236890i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	8.00000			0.267411
$896$		0			0
$897$	−36.0000			−1.20201
$898$		0			0
$899$	9.79796	+	5.65685i	0.326780	+	0.188667i
$900$		0			0
$901$	−29.3939	+	16.9706i	−0.979252	+	0.565371i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.00000	−	15.5885i	−0.299170	−	0.518178i
$906$		0			0
$907$	−29.3939	−	16.9706i	−0.976008	−	0.563498i	−0.0749452	−	0.997188i	$-0.523878\pi$
$907$	−29.3939	−	16.9706i	−0.976008	−	0.563498i	−0.901062	+	0.433689i	$0.857212\pi$
$908$		0			0
$909$			9.89949i			0.328346i
$910$		0			0
$911$	−30.0000			−0.993944			−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000			−0.993944			−0.496972	+	0.867766i	$0.665555\pi$
$912$		0			0
$913$	22.0000	−	38.1051i	0.728094	−	1.26110i
$914$		0			0
$915$	22.0454	−	12.7279i	0.728799	−	0.420772i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	0.703378	−	0.710816i	$-0.251674\pi$
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	−0.967274	+	0.253735i	$0.918341\pi$
$920$		0			0
$921$	−9.00000	+	15.5885i	−0.296560	+	0.513657i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	25.4558i		−	0.836983i
$926$		0			0
$927$	−2.00000	+	3.46410i	−0.0656886	+	0.113776i
$928$		0			0
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	0.679770	−	0.733426i	$-0.262080\pi$
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	−0.975050	+	0.221985i	$0.928746\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	+	20.7846i	−0.392442	+	0.679729i
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$		−	14.1421i		−	0.461511i
$940$		0			0
$941$	−45.3156	−	26.1630i	−1.47725	−	0.852888i	−0.477575	−	0.878591i	$-0.658484\pi$
$941$	−45.3156	−	26.1630i	−1.47725	−	0.852888i	−0.999670	+	0.0257029i	$0.991818\pi$
$942$		0			0
$943$	−18.0000	−	31.1769i	−0.586161	−	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.44949	−	1.41421i	0.0795977	−	0.0459558i	−0.459673	−	0.888088i	$-0.652033\pi$
$947$	2.44949	−	1.41421i	0.0795977	−	0.0459558i	0.539271	+	0.842133i	$0.318700\pi$
$948$		0			0
$949$	−7.34847	−	4.24264i	−0.238541	−	0.137722i
$950$		0			0
$951$	−32.0000			−1.03767
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	9.79796	−	5.65685i	0.316723	−	0.182860i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$	4.89898	+	2.82843i	0.157867	+	0.0911448i
$964$		0			0
$965$		−	5.65685i		−	0.182101i
$966$		0			0
$967$	22.0000			0.707472			0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000			0.707472			0.353736	+	0.935345i	$0.384911\pi$
$968$		0			0
$969$	18.0000	−	31.1769i	0.578243	−	1.00155i
$970$		0			0
$971$	28.1691	−	16.2635i	0.903990	−	0.521919i	0.0254978	−	0.999675i	$-0.491883\pi$
$971$	28.1691	−	16.2635i	0.903990	−	0.521919i	0.878493	+	0.477756i	$0.158550\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	9.00000	+	15.5885i	0.288231	+	0.499230i
$976$		0			0
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	−0.685381	−	0.728184i	$-0.740364\pi$
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	0.973317	+	0.229465i	$0.0736978\pi$
$978$		0			0
$979$			16.9706i			0.542382i
$980$		0			0
$981$			8.48528i			0.270914i
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	4.00000	+	6.92820i	0.127451	+	0.220751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−44.0908	+	25.4558i	−1.40201	+	0.809449i
$990$		0			0
$991$	−8.00000	+	13.8564i	−0.254128	+	0.440163i	−0.964658	−	0.263504i	$-0.915122\pi$
$991$	−8.00000	+	13.8564i	−0.254128	+	0.440163i	0.710530	+	0.703667i	$0.248455\pi$
$992$		0			0
$993$	−36.0000			−1.14243
$994$		0			0
$995$		−	28.2843i		−	0.896672i
$996$		0			0
$997$	18.3712	+	10.6066i	0.581821	+	0.335914i	0.761857	−	0.647746i	$-0.224288\pi$
$997$	18.3712	+	10.6066i	0.581821	+	0.335914i	−0.180036	+	0.983660i	$0.557621\pi$
$998$		0			0
$999$	−24.0000	−	41.5692i	−0.759326	−	1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.t.c.753.1		4
4.3	odd	2		392.2.p.a.165.1		4
7.2	even	3	inner	1568.2.t.c.177.2		4
7.3	odd	6		1568.2.b.a.785.1		2
7.4	even	3		224.2.b.a.113.2		2
7.5	odd	6		1568.2.t.b.177.1		4
7.6	odd	2		1568.2.t.b.753.2		4
8.3	odd	2		392.2.p.a.165.2		4
8.5	even	2	inner	1568.2.t.c.753.2		4
21.11	odd	6		2016.2.c.a.1009.1		2
28.3	even	6		392.2.b.b.197.1		2
28.11	odd	6		56.2.b.a.29.1	✓	2
28.19	even	6		392.2.p.b.373.2		4
28.23	odd	6		392.2.p.a.373.2		4
28.27	even	2		392.2.p.b.165.1		4
56.3	even	6		392.2.b.b.197.2		2
56.5	odd	6		1568.2.t.b.177.2		4
56.11	odd	6		56.2.b.a.29.2	yes	2
56.13	odd	2		1568.2.t.b.753.1		4
56.19	even	6		392.2.p.b.373.1		4
56.27	even	2		392.2.p.b.165.2		4
56.37	even	6	inner	1568.2.t.c.177.1		4
56.45	odd	6		1568.2.b.a.785.2		2
56.51	odd	6		392.2.p.a.373.1		4
56.53	even	6		224.2.b.a.113.1		2
84.11	even	6		504.2.c.a.253.2		2
112.11	odd	12		1792.2.a.n.1.1		2
112.53	even	12		1792.2.a.p.1.2		2
112.67	odd	12		1792.2.a.n.1.2		2
112.109	even	12		1792.2.a.p.1.1		2
168.11	even	6		504.2.c.a.253.1		2
168.53	odd	6		2016.2.c.a.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.a.29.1	✓	2	28.11	odd	6
56.2.b.a.29.2	yes	2	56.11	odd	6
224.2.b.a.113.1		2	56.53	even	6
224.2.b.a.113.2		2	7.4	even	3
392.2.b.b.197.1		2	28.3	even	6
392.2.b.b.197.2		2	56.3	even	6
392.2.p.a.165.1		4	4.3	odd	2
392.2.p.a.165.2		4	8.3	odd	2
392.2.p.a.373.1		4	56.51	odd	6
392.2.p.a.373.2		4	28.23	odd	6
392.2.p.b.165.1		4	28.27	even	2
392.2.p.b.165.2		4	56.27	even	2
392.2.p.b.373.1		4	56.19	even	6
392.2.p.b.373.2		4	28.19	even	6
504.2.c.a.253.1		2	168.11	even	6
504.2.c.a.253.2		2	84.11	even	6
1568.2.b.a.785.1		2	7.3	odd	6
1568.2.b.a.785.2		2	56.45	odd	6
1568.2.t.b.177.1		4	7.5	odd	6
1568.2.t.b.177.2		4	56.5	odd	6
1568.2.t.b.753.1		4	56.13	odd	2
1568.2.t.b.753.2		4	7.6	odd	2
1568.2.t.c.177.1		4	56.37	even	6	inner
1568.2.t.c.177.2		4	7.2	even	3	inner
1568.2.t.c.753.1		4	1.1	even	1	trivial
1568.2.t.c.753.2		4	8.5	even	2	inner
1792.2.a.n.1.1		2	112.11	odd	12
1792.2.a.n.1.2		2	112.67	odd	12
1792.2.a.p.1.1		2	112.109	even	12
1792.2.a.p.1.2		2	112.53	even	12
2016.2.c.a.1009.1		2	21.11	odd	6
2016.2.c.a.1009.2		2	168.53	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{3} +(1.22474 - 0.707107i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.44949 - 1.41421i) q^{11} +4.24264i q^{13} -2.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(-3.67423 + 2.12132i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-1.50000 + 2.59808i) q^{25} +5.65685i q^{27} -2.82843i q^{29} +(-2.00000 + 3.46410i) q^{31} +(2.00000 + 3.46410i) q^{33} +(-7.34847 + 4.24264i) q^{37} +(3.00000 - 5.19615i) q^{39} +6.00000 q^{41} -8.48528i q^{43} +(-1.22474 - 0.707107i) q^{45} +(-7.34847 + 4.24264i) q^{51} +(-4.89898 - 2.82843i) q^{53} -4.00000 q^{55} +6.00000 q^{57} +(1.22474 + 0.707107i) q^{59} +(-11.0227 + 6.36396i) q^{61} +(3.00000 + 5.19615i) q^{65} +8.48528i q^{69} +(-1.00000 + 1.73205i) q^{73} +(3.67423 - 2.12132i) q^{75} +(4.00000 + 6.92820i) q^{79} +(2.50000 - 4.33013i) q^{81} +15.5563i q^{83} -8.48528i q^{85} +(-2.00000 + 3.46410i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(4.89898 - 2.82843i) q^{93} +(-3.00000 + 5.19615i) q^{95} -10.0000 q^{97} +2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9} - 8 q^{15} + 12 q^{17} - 12 q^{23} - 6 q^{25} - 8 q^{31} + 8 q^{33} + 12 q^{39} + 24 q^{41} - 16 q^{55} + 24 q^{57} + 12 q^{65} - 4 q^{73} + 16 q^{79} + 10 q^{81} - 8 q^{87} - 12 q^{89} - 12 q^{95}+ \cdots - 40 q^{97}+O(q^{100}) \) 4 * q - 2 * q^9 - 8 * q^15 + 12 * q^17 - 12 * q^23 - 6 * q^25 - 8 * q^31 + 8 * q^33 + 12 * q^39 + 24 * q^41 - 16 * q^55 + 24 * q^57 + 12 * q^65 - 4 * q^73 + 16 * q^79 + 10 * q^81 - 8 * q^87 - 12 * q^89 - 12 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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