LMFDB - Embedded newform 192.2.c.b.191.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,2,Mod(191,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.53312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.2.c.b.191.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 + 1.00000i) q^{3} +2.82843i q^{5} -2.00000i q^{7} +(1.00000 + 2.82843i) q^{9} -2.82843 q^{11} +2.00000 q^{13} +(-2.82843 + 4.00000i) q^{15} -6.00000i q^{19} +(2.00000 - 2.82843i) q^{21} +5.65685 q^{23} -3.00000 q^{25} +(-1.41421 + 5.00000i) q^{27} -2.82843i q^{29} -2.00000i q^{31} +(-4.00000 - 2.82843i) q^{33} +5.65685 q^{35} -6.00000 q^{37} +(2.82843 + 2.00000i) q^{39} -5.65685i q^{41} +2.00000i q^{43} +(-8.00000 + 2.82843i) q^{45} -11.3137 q^{47} +3.00000 q^{49} -8.48528i q^{53} -8.00000i q^{55} +(6.00000 - 8.48528i) q^{57} -2.82843 q^{59} +2.00000 q^{61} +(5.65685 - 2.00000i) q^{63} +5.65685i q^{65} +2.00000i q^{67} +(8.00000 + 5.65685i) q^{69} -5.65685 q^{71} -6.00000 q^{73} +(-4.24264 - 3.00000i) q^{75} +5.65685i q^{77} +14.0000i q^{79} +(-7.00000 + 5.65685i) q^{81} +2.82843 q^{83} +(2.82843 - 4.00000i) q^{87} +16.9706i q^{89} -4.00000i q^{91} +(2.00000 - 2.82843i) q^{93} +16.9706 q^{95} +10.0000 q^{97} +(-2.82843 - 8.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} + 8 q^{13} + 8 q^{21} - 12 q^{25} - 16 q^{33} - 24 q^{37} - 32 q^{45} + 12 q^{49} + 24 q^{57} + 8 q^{61} + 32 q^{69} - 24 q^{73} - 28 q^{81} + 8 q^{93} + 40 q^{97}+O(q^{100}) $ 4 * q + 4 * q^9 + 8 * q^13 + 8 * q^21 - 12 * q^25 - 16 * q^33 - 24 * q^37 - 32 * q^45 + 12 * q^49 + 24 * q^57 + 8 * q^61 + 32 * q^69 - 24 * q^73 - 28 * q^81 + 8 * q^93 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.41421	+	1.00000i	0.816497	+	0.577350i
$3$	1.41421	+	1.00000i	0.816497	+	0.577350i
$4$		0			0
$5$			2.82843i			1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$			2.82843i			1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000	+	2.82843i	0.333333	+	0.942809i
$10$		0			0
$11$	−2.82843			−0.852803			−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843			−0.852803			−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	−2.82843	+	4.00000i	−0.730297	+	1.03280i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$	2.00000	−	2.82843i	0.436436	−	0.617213i
$22$		0			0
$23$	5.65685			1.17954			0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685			1.17954			0.589768	+	0.807573i	$0.299219\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$	−1.41421	+	5.00000i	−0.272166	+	0.962250i
$28$		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
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$	−4.00000	−	2.82843i	−0.696311	−	0.492366i
$34$		0			0
$35$	5.65685			0.956183
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$	2.82843	+	2.00000i	0.452911	+	0.320256i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	−8.00000	+	2.82843i	−1.19257	+	0.421637i
$46$		0			0
$47$	−11.3137			−1.65027			−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137			−1.65027			−0.825137	+	0.564933i	$0.808902\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.48528i		−	1.16554i	−0.812636	−	0.582772i	$-0.801968\pi$
$53$		−	8.48528i		−	1.16554i	0.812636	−	0.582772i	$-0.198032\pi$
$54$		0			0
$55$		−	8.00000i		−	1.07872i
$56$		0			0
$57$	6.00000	−	8.48528i	0.794719	−	1.12390i
$58$		0			0
$59$	−2.82843			−0.368230			−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843			−0.368230			−0.184115	+	0.982905i	$0.558942\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$	5.65685	−	2.00000i	0.712697	−	0.251976i
$64$		0			0
$65$			5.65685i			0.701646i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	8.00000	+	5.65685i	0.963087	+	0.681005i
$70$		0			0
$71$	−5.65685			−0.671345			−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685			−0.671345			−0.335673	+	0.941979i	$0.608964\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	−4.24264	−	3.00000i	−0.489898	−	0.346410i
$76$		0			0
$77$			5.65685i			0.644658i
$78$		0			0
$79$			14.0000i			1.57512i	0.616236	+	0.787562i	$0.288657\pi$
$79$			14.0000i			1.57512i	−0.616236	+	0.787562i	$0.711343\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	2.82843			0.310460			0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843			0.310460			0.155230	+	0.987878i	$0.450388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.82843	−	4.00000i	0.303239	−	0.428845i
$88$		0			0
$89$			16.9706i			1.79888i	0.437048	+	0.899438i	$0.356024\pi$
$89$			16.9706i			1.79888i	−0.437048	+	0.899438i	$0.643976\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$		0			0
$93$	2.00000	−	2.82843i	0.207390	−	0.293294i
$94$		0			0
$95$	16.9706			1.74114
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	−2.82843	−	8.00000i	−0.284268	−	0.804030i
$100$		0			0
$101$			2.82843i			0.281439i	0.990050	+	0.140720i	$0.0449416\pi$
$101$			2.82843i			0.281439i	−0.990050	+	0.140720i	$0.955058\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$	8.00000	+	5.65685i	0.780720	+	0.552052i
$106$		0			0
$107$	8.48528			0.820303			0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528			0.820303			0.410152	+	0.912017i	$0.365476\pi$
$108$		0			0
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$		0			0
$111$	−8.48528	−	6.00000i	−0.805387	−	0.569495i
$112$		0			0
$113$		−	11.3137i		−	1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$		−	11.3137i		−	1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$		0			0
$115$			16.0000i			1.49201i
$116$		0			0
$117$	2.00000	+	5.65685i	0.184900	+	0.522976i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	5.65685	−	8.00000i	0.510061	−	0.721336i
$124$		0			0
$125$			5.65685i			0.505964i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$		0			0
$129$	−2.00000	+	2.82843i	−0.176090	+	0.249029i
$130$		0			0
$131$	−19.7990			−1.72985			−0.864923	−	0.501905i	$-0.832633\pi$
$131$	−19.7990			−1.72985			−0.864923	+	0.501905i	$0.832633\pi$
$132$		0			0
$133$	−12.0000			−1.04053
$134$		0			0
$135$	−14.1421	−	4.00000i	−1.21716	−	0.344265i
$136$		0			0
$137$		−	5.65685i		−	0.483298i	−0.970364	−	0.241649i	$-0.922312\pi$
$137$		−	5.65685i		−	0.483298i	0.970364	−	0.241649i	$-0.0776882\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$		0			0
$141$	−16.0000	−	11.3137i	−1.34744	−	0.952786i
$142$		0			0
$143$	−5.65685			−0.473050
$144$		0			0
$145$	8.00000			0.664364
$146$		0			0
$147$	4.24264	+	3.00000i	0.349927	+	0.247436i
$148$		0			0
$149$			14.1421i			1.15857i	0.815125	+	0.579284i	$0.196668\pi$
$149$			14.1421i			1.15857i	−0.815125	+	0.579284i	$0.803332\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.65685			0.454369
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	8.48528	−	12.0000i	0.672927	−	0.951662i
$160$		0			0
$161$		−	11.3137i		−	0.891645i
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$	8.00000	−	11.3137i	0.622799	−	0.880771i
$166$		0			0
$167$	−5.65685			−0.437741			−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685			−0.437741			−0.218870	+	0.975754i	$0.570237\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	16.9706	−	6.00000i	1.29777	−	0.458831i
$172$		0			0
$173$		−	14.1421i		−	1.07521i	−0.843198	−	0.537603i	$-0.819330\pi$
$173$		−	14.1421i		−	1.07521i	0.843198	−	0.537603i	$-0.180670\pi$
$174$		0			0
$175$			6.00000i			0.453557i
$176$		0			0
$177$	−4.00000	−	2.82843i	−0.300658	−	0.212598i
$178$		0			0
$179$	−8.48528			−0.634220			−0.317110	−	0.948389i	$-0.602712\pi$
$179$	−8.48528			−0.634220			−0.317110	+	0.948389i	$0.602712\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	2.82843	+	2.00000i	0.209083	+	0.147844i
$184$		0			0
$185$		−	16.9706i		−	1.24770i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.0000	+	2.82843i	0.727393	+	0.205738i
$190$		0			0
$191$	22.6274			1.63726			0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274			1.63726			0.818631	+	0.574320i	$0.194733\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	−5.65685	+	8.00000i	−0.405096	+	0.572892i
$196$		0			0
$197$		−	19.7990i		−	1.41062i	−0.708899	−	0.705310i	$-0.750808\pi$
$197$		−	19.7990i		−	1.41062i	0.708899	−	0.705310i	$-0.249192\pi$
$198$		0			0
$199$		−	2.00000i		−	0.141776i	−0.997484	−	0.0708881i	$-0.977417\pi$
$199$		−	2.00000i		−	0.141776i	0.997484	−	0.0708881i	$-0.0225833\pi$
$200$		0			0
$201$	−2.00000	+	2.82843i	−0.141069	+	0.199502i
$202$		0			0
$203$	−5.65685			−0.397033
$204$		0			0
$205$	16.0000			1.11749
$206$		0			0
$207$	5.65685	+	16.0000i	0.393179	+	1.11208i
$208$		0			0
$209$			16.9706i			1.17388i
$210$		0			0
$211$			2.00000i			0.137686i	0.997628	+	0.0688428i	$0.0219307\pi$
$211$			2.00000i			0.137686i	−0.997628	+	0.0688428i	$0.978069\pi$
$212$		0			0
$213$	−8.00000	−	5.65685i	−0.548151	−	0.387601i
$214$		0			0
$215$	−5.65685			−0.385794
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	−8.48528	−	6.00000i	−0.573382	−	0.405442i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	2.00000i		−	0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$		−	2.00000i		−	0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$		0			0
$225$	−3.00000	−	8.48528i	−0.200000	−	0.565685i
$226$		0			0
$227$	14.1421			0.938647			0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421			0.938647			0.469323	+	0.883026i	$0.344498\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$	−5.65685	+	8.00000i	−0.372194	+	0.526361i
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		−	32.0000i		−	2.08745i
$236$		0			0
$237$	−14.0000	+	19.7990i	−0.909398	+	1.28608i
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	−15.5563	+	1.00000i	−0.997940	+	0.0641500i
$244$		0			0
$245$			8.48528i			0.542105i
$246$		0			0
$247$		−	12.0000i		−	0.763542i
$248$		0			0
$249$	4.00000	+	2.82843i	0.253490	+	0.179244i
$250$		0			0
$251$	8.48528			0.535586			0.267793	−	0.963476i	$-0.413706\pi$
$251$	8.48528			0.535586			0.267793	+	0.963476i	$0.413706\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$			22.6274i			1.41146i	0.708481	+	0.705730i	$0.249381\pi$
$257$			22.6274i			1.41146i	−0.708481	+	0.705730i	$0.750619\pi$
$258$		0			0
$259$			12.0000i			0.745644i
$260$		0			0
$261$	8.00000	−	2.82843i	0.495188	−	0.175075i
$262$		0			0
$263$	−5.65685			−0.348817			−0.174408	−	0.984673i	$-0.555801\pi$
$263$	−5.65685			−0.348817			−0.174408	+	0.984673i	$0.555801\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$	−16.9706	+	24.0000i	−1.03858	+	1.46878i
$268$		0			0
$269$			8.48528i			0.517357i	0.965964	+	0.258678i	$0.0832870\pi$
$269$			8.48528i			0.517357i	−0.965964	+	0.258678i	$0.916713\pi$
$270$		0			0
$271$		−	18.0000i		−	1.09342i	−0.837321	−	0.546711i	$-0.815880\pi$
$271$		−	18.0000i		−	1.09342i	0.837321	−	0.546711i	$-0.184120\pi$
$272$		0			0
$273$	4.00000	−	5.65685i	0.242091	−	0.342368i
$274$		0			0
$275$	8.48528			0.511682
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$	5.65685	−	2.00000i	0.338667	−	0.119737i
$280$		0			0
$281$		−	5.65685i		−	0.337460i	−0.985662	−	0.168730i	$-0.946033\pi$
$281$		−	5.65685i		−	0.337460i	0.985662	−	0.168730i	$-0.0539665\pi$
$282$		0			0
$283$			10.0000i			0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$			10.0000i			0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$		0			0
$285$	24.0000	+	16.9706i	1.42164	+	1.00525i
$286$		0			0
$287$	−11.3137			−0.667827
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	14.1421	+	10.0000i	0.829027	+	0.586210i
$292$		0			0
$293$			2.82843i			0.165238i	0.996581	+	0.0826192i	$0.0263285\pi$
$293$			2.82843i			0.165238i	−0.996581	+	0.0826192i	$0.973671\pi$
$294$		0			0
$295$		−	8.00000i		−	0.465778i
$296$		0			0
$297$	4.00000	−	14.1421i	0.232104	−	0.820610i
$298$		0			0
$299$	11.3137			0.654289
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$	−2.82843	+	4.00000i	−0.162489	+	0.229794i
$304$		0			0
$305$			5.65685i			0.323911i
$306$		0			0
$307$		−	30.0000i		−	1.71219i	−0.516818	−	0.856095i	$-0.672884\pi$
$307$		−	30.0000i		−	1.71219i	0.516818	−	0.856095i	$-0.327116\pi$
$308$		0			0
$309$	−14.0000	+	19.7990i	−0.796432	+	1.12633i
$310$		0			0
$311$	5.65685			0.320771			0.160385	−	0.987054i	$-0.448726\pi$
$311$	5.65685			0.320771			0.160385	+	0.987054i	$0.448726\pi$
$312$		0			0
$313$	2.00000			0.113047			0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000			0.113047			0.0565233	+	0.998401i	$0.481998\pi$
$314$		0			0
$315$	5.65685	+	16.0000i	0.318728	+	0.901498i
$316$		0			0
$317$		−	2.82843i		−	0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$		−	2.82843i		−	0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$		0			0
$319$			8.00000i			0.447914i
$320$		0			0
$321$	12.0000	+	8.48528i	0.669775	+	0.473602i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$	25.4558	+	18.0000i	1.40771	+	0.995402i
$328$		0			0
$329$			22.6274i			1.24749i
$330$		0			0
$331$			10.0000i			0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$			10.0000i			0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$		0			0
$333$	−6.00000	−	16.9706i	−0.328798	−	0.929981i
$334$		0			0
$335$	−5.65685			−0.309067
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	11.3137	−	16.0000i	0.614476	−	0.869001i
$340$		0			0
$341$			5.65685i			0.306336i
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	−16.0000	+	22.6274i	−0.861411	+	1.21822i
$346$		0			0
$347$	−14.1421			−0.759190			−0.379595	−	0.925153i	$-0.623937\pi$
$347$	−14.1421			−0.759190			−0.379595	+	0.925153i	$0.623937\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−2.82843	+	10.0000i	−0.150970	+	0.533761i
$352$		0			0
$353$		−	22.6274i		−	1.20434i	−0.798369	−	0.602168i	$-0.794304\pi$
$353$		−	22.6274i		−	1.20434i	0.798369	−	0.602168i	$-0.205696\pi$
$354$		0			0
$355$		−	16.0000i		−	0.849192i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.9706			0.895672			0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706			0.895672			0.447836	+	0.894116i	$0.352195\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−4.24264	−	3.00000i	−0.222681	−	0.157459i
$364$		0			0
$365$		−	16.9706i		−	0.888280i
$366$		0			0
$367$			14.0000i			0.730794i	0.930852	+	0.365397i	$0.119067\pi$
$367$			14.0000i			0.730794i	−0.930852	+	0.365397i	$0.880933\pi$
$368$		0			0
$369$	16.0000	−	5.65685i	0.832927	−	0.294484i
$370$		0			0
$371$	−16.9706			−0.881068
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$	−5.65685	+	8.00000i	−0.292119	+	0.413118i
$376$		0			0
$377$		−	5.65685i		−	0.291343i
$378$		0			0
$379$			2.00000i			0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$			2.00000i			0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$		0			0
$381$	18.0000	−	25.4558i	0.922168	−	1.30414i
$382$		0			0
$383$	−22.6274			−1.15621			−0.578103	−	0.815963i	$-0.696207\pi$
$383$	−22.6274			−1.15621			−0.578103	+	0.815963i	$0.696207\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$	−5.65685	+	2.00000i	−0.287554	+	0.101666i
$388$		0			0
$389$			2.82843i			0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$			2.82843i			0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−28.0000	−	19.7990i	−1.41241	−	0.998727i
$394$		0			0
$395$	−39.5980			−1.99239
$396$		0			0
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$		0			0
$399$	−16.9706	−	12.0000i	−0.849591	−	0.600751i
$400$		0			0
$401$		−	22.6274i		−	1.12996i	−0.825105	−	0.564980i	$-0.808884\pi$
$401$		−	22.6274i		−	1.12996i	0.825105	−	0.564980i	$-0.191116\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	−16.0000	−	19.7990i	−0.795046	−	0.983820i
$406$		0			0
$407$	16.9706			0.841200
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$	5.65685	−	8.00000i	0.279032	−	0.394611i
$412$		0			0
$413$			5.65685i			0.278356i
$414$		0			0
$415$			8.00000i			0.392705i
$416$		0			0
$417$	−10.0000	+	14.1421i	−0.489702	+	0.692543i
$418$		0			0
$419$	36.7696			1.79631			0.898155	−	0.439679i	$-0.144908\pi$
$419$	36.7696			1.79631			0.898155	+	0.439679i	$0.144908\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$	−11.3137	−	32.0000i	−0.550091	−	1.55589i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$	−8.00000	−	5.65685i	−0.386244	−	0.273115i
$430$		0			0
$431$	33.9411			1.63489			0.817443	−	0.576009i	$-0.195391\pi$
$431$	33.9411			1.63489			0.817443	+	0.576009i	$0.195391\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		0			0
$435$	11.3137	+	8.00000i	0.542451	+	0.383571i
$436$		0			0
$437$		−	33.9411i		−	1.62362i
$438$		0			0
$439$			14.0000i			0.668184i	0.942541	+	0.334092i	$0.108430\pi$
$439$			14.0000i			0.668184i	−0.942541	+	0.334092i	$0.891570\pi$
$440$		0			0
$441$	3.00000	+	8.48528i	0.142857	+	0.404061i
$442$		0			0
$443$	−14.1421			−0.671913			−0.335957	−	0.941877i	$-0.609060\pi$
$443$	−14.1421			−0.671913			−0.335957	+	0.941877i	$0.609060\pi$
$444$		0			0
$445$	−48.0000			−2.27542
$446$		0			0
$447$	−14.1421	+	20.0000i	−0.668900	+	0.945968i
$448$		0			0
$449$		−	33.9411i		−	1.60178i	−0.598811	−	0.800890i	$-0.704360\pi$
$449$		−	33.9411i		−	1.60178i	0.598811	−	0.800890i	$-0.295640\pi$
$450$		0			0
$451$			16.0000i			0.753411i
$452$		0			0
$453$	−14.0000	+	19.7990i	−0.657777	+	0.930238i
$454$		0			0
$455$	11.3137			0.530395
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.1127i			1.44906i	0.689242	+	0.724531i	$0.257944\pi$
$461$			31.1127i			1.44906i	−0.689242	+	0.724531i	$0.742056\pi$
$462$		0			0
$463$		−	2.00000i		−	0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$		−	2.00000i		−	0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$		0			0
$465$	8.00000	+	5.65685i	0.370991	+	0.262330i
$466$		0			0
$467$	25.4558			1.17796			0.588978	−	0.808149i	$-0.299530\pi$
$467$	25.4558			1.17796			0.588978	+	0.808149i	$0.299530\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−19.7990	−	14.0000i	−0.912289	−	0.645086i
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$			18.0000i			0.825897i
$476$		0			0
$477$	24.0000	−	8.48528i	1.09888	−	0.388514i
$478$		0			0
$479$	−22.6274			−1.03387			−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274			−1.03387			−0.516937	+	0.856024i	$0.672928\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$	11.3137	−	16.0000i	0.514792	−	0.728025i
$484$		0			0
$485$			28.2843i			1.28432i
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$	6.00000	−	8.48528i	0.271329	−	0.383718i
$490$		0			0
$491$	31.1127			1.40410			0.702048	−	0.712129i	$-0.252269\pi$
$491$	31.1127			1.40410			0.702048	+	0.712129i	$0.252269\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	22.6274	−	8.00000i	1.01703	−	0.359573i
$496$		0			0
$497$			11.3137i			0.507489i
$498$		0			0
$499$			34.0000i			1.52205i	0.648723	+	0.761025i	$0.275303\pi$
$499$			34.0000i			1.52205i	−0.648723	+	0.761025i	$0.724697\pi$
$500$		0			0
$501$	−8.00000	−	5.65685i	−0.357414	−	0.252730i
$502$		0			0
$503$	5.65685			0.252227			0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685			0.252227			0.126113	+	0.992016i	$0.459750\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	−12.7279	−	9.00000i	−0.565267	−	0.399704i
$508$		0			0
$509$		−	2.82843i		−	0.125368i	−0.998033	−	0.0626839i	$-0.980034\pi$
$509$		−	2.82843i		−	0.125368i	0.998033	−	0.0626839i	$-0.0199660\pi$
$510$		0			0
$511$			12.0000i			0.530849i
$512$		0			0
$513$	30.0000	+	8.48528i	1.32453	+	0.374634i
$514$		0			0
$515$	−39.5980			−1.74490
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	14.1421	−	20.0000i	0.620771	−	0.877903i
$520$		0			0
$521$		−	5.65685i		−	0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$		−	5.65685i		−	0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		0			0
$525$	−6.00000	+	8.48528i	−0.261861	+	0.370328i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.00000			0.391304
$530$		0			0
$531$	−2.82843	−	8.00000i	−0.122743	−	0.347170i
$532$		0			0
$533$		−	11.3137i		−	0.490051i
$534$		0			0
$535$			24.0000i			1.03761i
$536$		0			0
$537$	−12.0000	−	8.48528i	−0.517838	−	0.366167i
$538$		0			0
$539$	−8.48528			−0.365487
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$	−8.48528	−	6.00000i	−0.364138	−	0.257485i
$544$		0			0
$545$			50.9117i			2.18082i
$546$		0			0
$547$		−	38.0000i		−	1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$		−	38.0000i		−	1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$		0			0
$549$	2.00000	+	5.65685i	0.0853579	+	0.241429i
$550$		0			0
$551$	−16.9706			−0.722970
$552$		0			0
$553$	28.0000			1.19068
$554$		0			0
$555$	16.9706	−	24.0000i	0.720360	−	1.01874i
$556$		0			0
$557$			8.48528i			0.359533i	0.983709	+	0.179766i	$0.0575342\pi$
$557$			8.48528i			0.359533i	−0.983709	+	0.179766i	$0.942466\pi$
$558$		0			0
$559$			4.00000i			0.169182i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.82843			0.119204			0.0596020	−	0.998222i	$-0.481017\pi$
$563$	2.82843			0.119204			0.0596020	+	0.998222i	$0.481017\pi$
$564$		0			0
$565$	32.0000			1.34625
$566$		0			0
$567$	11.3137	+	14.0000i	0.475131	+	0.587945i
$568$		0			0
$569$			5.65685i			0.237148i	0.992945	+	0.118574i	$0.0378322\pi$
$569$			5.65685i			0.237148i	−0.992945	+	0.118574i	$0.962168\pi$
$570$		0			0
$571$		−	22.0000i		−	0.920671i	−0.887745	−	0.460336i	$-0.847729\pi$
$571$		−	22.0000i		−	0.920671i	0.887745	−	0.460336i	$-0.152271\pi$
$572$		0			0
$573$	32.0000	+	22.6274i	1.33682	+	0.945274i
$574$		0			0
$575$	−16.9706			−0.707721
$576$		0			0
$577$	−6.00000			−0.249783			−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000			−0.249783			−0.124892	+	0.992170i	$0.539858\pi$
$578$		0			0
$579$	−19.7990	−	14.0000i	−0.822818	−	0.581820i
$580$		0			0
$581$		−	5.65685i		−	0.234686i
$582$		0			0
$583$			24.0000i			0.993978i
$584$		0			0
$585$	−16.0000	+	5.65685i	−0.661519	+	0.233882i
$586$		0			0
$587$	−14.1421			−0.583708			−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421			−0.583708			−0.291854	+	0.956463i	$0.594272\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	19.7990	−	28.0000i	0.814422	−	1.15177i
$592$		0			0
$593$			33.9411i			1.39379i	0.717171	+	0.696897i	$0.245437\pi$
$593$			33.9411i			1.39379i	−0.717171	+	0.696897i	$0.754563\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.00000	−	2.82843i	0.0818546	−	0.115760i
$598$		0			0
$599$	−39.5980			−1.61793			−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980			−1.61793			−0.808965	+	0.587857i	$0.799972\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	−5.65685	+	2.00000i	−0.230365	+	0.0814463i
$604$		0			0
$605$		−	8.48528i		−	0.344976i
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$		0			0
$609$	−8.00000	−	5.65685i	−0.324176	−	0.229227i
$610$		0			0
$611$	−22.6274			−0.915407
$612$		0			0
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$		0			0
$615$	22.6274	+	16.0000i	0.912426	+	0.645182i
$616$		0			0
$617$			28.2843i			1.13868i	0.822102	+	0.569341i	$0.192802\pi$
$617$			28.2843i			1.13868i	−0.822102	+	0.569341i	$0.807198\pi$
$618$		0			0
$619$			10.0000i			0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$			10.0000i			0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$		0			0
$621$	−8.00000	+	28.2843i	−0.321029	+	1.13501i
$622$		0			0
$623$	33.9411			1.35982
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	−16.9706	+	24.0000i	−0.677739	+	0.958468i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$		0			0
$633$	−2.00000	+	2.82843i	−0.0794929	+	0.112420i
$634$		0			0
$635$	50.9117			2.02037
$636$		0			0
$637$	6.00000			0.237729
$638$		0			0
$639$	−5.65685	−	16.0000i	−0.223782	−	0.632950i
$640$		0			0
$641$		−	11.3137i		−	0.446865i	−0.974719	−	0.223432i	$-0.928274\pi$
$641$		−	11.3137i		−	0.446865i	0.974719	−	0.223432i	$-0.0717262\pi$
$642$		0			0
$643$			26.0000i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$			26.0000i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$		0			0
$645$	−8.00000	−	5.65685i	−0.315000	−	0.222738i
$646$		0			0
$647$	16.9706			0.667182			0.333591	−	0.942718i	$-0.391740\pi$
$647$	16.9706			0.667182			0.333591	+	0.942718i	$0.391740\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$	−5.65685	−	4.00000i	−0.221710	−	0.156772i
$652$		0			0
$653$		−	14.1421i		−	0.553425i	−0.960953	−	0.276712i	$-0.910755\pi$
$653$		−	14.1421i		−	0.553425i	0.960953	−	0.276712i	$-0.0892449\pi$
$654$		0			0
$655$		−	56.0000i		−	2.18810i
$656$		0			0
$657$	−6.00000	−	16.9706i	−0.234082	−	0.662085i
$658$		0			0
$659$	14.1421			0.550899			0.275450	−	0.961315i	$-0.411173\pi$
$659$	14.1421			0.550899			0.275450	+	0.961315i	$0.411173\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	33.9411i		−	1.31618i
$666$		0			0
$667$		−	16.0000i		−	0.619522i
$668$		0			0
$669$	2.00000	−	2.82843i	0.0773245	−	0.109353i
$670$		0			0
$671$	−5.65685			−0.218380
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$	4.24264	−	15.0000i	0.163299	−	0.577350i
$676$		0			0
$677$			2.82843i			0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$			2.82843i			0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$		0			0
$679$		−	20.0000i		−	0.767530i
$680$		0			0
$681$	20.0000	+	14.1421i	0.766402	+	0.541928i
$682$		0			0
$683$	42.4264			1.62340			0.811701	−	0.584074i	$-0.198542\pi$
$683$	42.4264			1.62340			0.811701	+	0.584074i	$0.198542\pi$
$684$		0			0
$685$	16.0000			0.611329
$686$		0			0
$687$	14.1421	+	10.0000i	0.539556	+	0.381524i
$688$		0			0
$689$		−	16.9706i		−	0.646527i
$690$		0			0
$691$			26.0000i			0.989087i	0.869153	+	0.494543i	$0.164665\pi$
$691$			26.0000i			0.989087i	−0.869153	+	0.494543i	$0.835335\pi$
$692$		0			0
$693$	−16.0000	+	5.65685i	−0.607790	+	0.214886i
$694$		0			0
$695$	−28.2843			−1.07288
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−5.65685	+	8.00000i	−0.213962	+	0.302588i
$700$		0			0
$701$		−	25.4558i		−	0.961454i	−0.876870	−	0.480727i	$-0.840373\pi$
$701$		−	25.4558i		−	0.961454i	0.876870	−	0.480727i	$-0.159627\pi$
$702$		0			0
$703$			36.0000i			1.35777i
$704$		0			0
$705$	32.0000	−	45.2548i	1.20519	−	1.70440i
$706$		0			0
$707$	5.65685			0.212748
$708$		0			0
$709$	−38.0000			−1.42712			−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000			−1.42712			−0.713560	+	0.700594i	$0.752918\pi$
$710$		0			0
$711$	−39.5980	+	14.0000i	−1.48504	+	0.525041i
$712$		0			0
$713$		−	11.3137i		−	0.423702i
$714$		0			0
$715$		−	16.0000i		−	0.598366i
$716$		0			0
$717$	−16.0000	−	11.3137i	−0.597531	−	0.422518i
$718$		0			0
$719$	33.9411			1.26579			0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411			1.26579			0.632895	+	0.774237i	$0.281866\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$		0			0
$723$	2.82843	+	2.00000i	0.105190	+	0.0743808i
$724$		0			0
$725$			8.48528i			0.315135i
$726$		0			0
$727$		−	34.0000i		−	1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$		−	34.0000i		−	1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$		0			0
$729$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−46.0000			−1.69905			−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000			−1.69905			−0.849524	+	0.527549i	$0.823111\pi$
$734$		0			0
$735$	−8.48528	+	12.0000i	−0.312984	+	0.442627i
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$	12.0000	−	16.9706i	0.440831	−	0.623429i
$742$		0			0
$743$	−5.65685			−0.207530			−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685			−0.207530			−0.103765	+	0.994602i	$0.533089\pi$
$744$		0			0
$745$	−40.0000			−1.46549
$746$		0			0
$747$	2.82843	+	8.00000i	0.103487	+	0.292705i
$748$		0			0
$749$		−	16.9706i		−	0.620091i
$750$		0			0
$751$		−	34.0000i		−	1.24068i	−0.784334	−	0.620339i	$-0.786995\pi$
$751$		−	34.0000i		−	1.24068i	0.784334	−	0.620339i	$-0.213005\pi$
$752$		0			0
$753$	12.0000	+	8.48528i	0.437304	+	0.309221i
$754$		0			0
$755$	−39.5980			−1.44112
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$	−22.6274	−	16.0000i	−0.821323	−	0.580763i
$760$		0			0
$761$			28.2843i			1.02530i	0.858596	+	0.512652i	$0.171337\pi$
$761$			28.2843i			1.02530i	−0.858596	+	0.512652i	$0.828663\pi$
$762$		0			0
$763$		−	36.0000i		−	1.30329i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.65685			−0.204257
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$	−22.6274	+	32.0000i	−0.814907	+	1.15245i
$772$		0			0
$773$			25.4558i			0.915583i	0.889060	+	0.457792i	$0.151359\pi$
$773$			25.4558i			0.915583i	−0.889060	+	0.457792i	$0.848641\pi$
$774$		0			0
$775$			6.00000i			0.215526i
$776$		0			0
$777$	−12.0000	+	16.9706i	−0.430498	+	0.608816i
$778$		0			0
$779$	−33.9411			−1.21607
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$	14.1421	+	4.00000i	0.505399	+	0.142948i
$784$		0			0
$785$		−	39.5980i		−	1.41331i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		0			0
$789$	−8.00000	−	5.65685i	−0.284808	−	0.201389i
$790$		0			0
$791$	−22.6274			−0.804538
$792$		0			0
$793$	4.00000			0.142044
$794$		0			0
$795$	33.9411	+	24.0000i	1.20377	+	0.851192i
$796$		0			0
$797$		−	2.82843i		−	0.100188i	−0.998745	−	0.0500940i	$-0.984048\pi$
$797$		−	2.82843i		−	0.100188i	0.998745	−	0.0500940i	$-0.0159521\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−48.0000	+	16.9706i	−1.69600	+	0.599625i
$802$		0			0
$803$	16.9706			0.598878
$804$		0			0
$805$	32.0000			1.12785
$806$		0			0
$807$	−8.48528	+	12.0000i	−0.298696	+	0.422420i
$808$		0			0
$809$		−	50.9117i		−	1.78996i	−0.446107	−	0.894980i	$-0.647190\pi$
$809$		−	50.9117i		−	1.78996i	0.446107	−	0.894980i	$-0.352810\pi$
$810$		0			0
$811$		−	30.0000i		−	1.05344i	−0.850038	−	0.526721i	$-0.823421\pi$
$811$		−	30.0000i		−	1.05344i	0.850038	−	0.526721i	$-0.176579\pi$
$812$		0			0
$813$	18.0000	−	25.4558i	0.631288	−	0.892775i
$814$		0			0
$815$	16.9706			0.594453
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$	11.3137	−	4.00000i	0.395333	−	0.139771i
$820$		0			0
$821$			14.1421i			0.493564i	0.969071	+	0.246782i	$0.0793731\pi$
$821$			14.1421i			0.493564i	−0.969071	+	0.246782i	$0.920627\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$		0			0
$825$	12.0000	+	8.48528i	0.417786	+	0.295420i
$826$		0			0
$827$	−25.4558			−0.885186			−0.442593	−	0.896723i	$-0.645941\pi$
$827$	−25.4558			−0.885186			−0.442593	+	0.896723i	$0.645941\pi$
$828$		0			0
$829$	18.0000			0.625166			0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000			0.625166			0.312583	+	0.949890i	$0.398806\pi$
$830$		0			0
$831$	14.1421	+	10.0000i	0.490585	+	0.346896i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	16.0000i		−	0.553703i
$836$		0			0
$837$	10.0000	+	2.82843i	0.345651	+	0.0977647i
$838$		0			0
$839$	39.5980			1.36707			0.683537	−	0.729916i	$-0.260441\pi$
$839$	39.5980			1.36707			0.683537	+	0.729916i	$0.260441\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	5.65685	−	8.00000i	0.194832	−	0.275535i
$844$		0			0
$845$		−	25.4558i		−	0.875708i
$846$		0			0
$847$			6.00000i			0.206162i
$848$		0			0
$849$	−10.0000	+	14.1421i	−0.343199	+	0.485357i
$850$		0			0
$851$	−33.9411			−1.16349
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$	16.9706	+	48.0000i	0.580381	+	1.64157i
$856$		0			0
$857$			28.2843i			0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$			28.2843i			0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$		0			0
$859$			2.00000i			0.0682391i	0.999418	+	0.0341196i	$0.0108627\pi$
$859$			2.00000i			0.0682391i	−0.999418	+	0.0341196i	$0.989137\pi$
$860$		0			0
$861$	−16.0000	−	11.3137i	−0.545279	−	0.385570i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	40.0000			1.36004
$866$		0			0
$867$	24.0416	+	17.0000i	0.816497	+	0.577350i
$868$		0			0
$869$		−	39.5980i		−	1.34327i
$870$		0			0
$871$			4.00000i			0.135535i
$872$		0			0
$873$	10.0000	+	28.2843i	0.338449	+	0.957278i
$874$		0			0
$875$	11.3137			0.382473
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$	−2.82843	+	4.00000i	−0.0954005	+	0.134917i
$880$		0			0
$881$			33.9411i			1.14351i	0.820426	+	0.571753i	$0.193736\pi$
$881$			33.9411i			1.14351i	−0.820426	+	0.571753i	$0.806264\pi$
$882$		0			0
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$	8.00000	−	11.3137i	0.268917	−	0.380306i
$886$		0			0
$887$	5.65685			0.189939			0.0949693	−	0.995480i	$-0.469725\pi$
$887$	5.65685			0.189939			0.0949693	+	0.995480i	$0.469725\pi$
$888$		0			0
$889$	−36.0000			−1.20740
$890$		0			0
$891$	19.7990	−	16.0000i	0.663291	−	0.536020i
$892$		0			0
$893$			67.8823i			2.27159i
$894$		0			0
$895$		−	24.0000i		−	0.802232i
$896$		0			0
$897$	16.0000	+	11.3137i	0.534224	+	0.377754i
$898$		0			0
$899$	−5.65685			−0.188667
$900$		0			0
$901$		0			0
$902$		0			0
$903$	5.65685	+	4.00000i	0.188248	+	0.133112i
$904$		0			0
$905$		−	16.9706i		−	0.564121i
$906$		0			0
$907$			2.00000i			0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$			2.00000i			0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$		0			0
$909$	−8.00000	+	2.82843i	−0.265343	+	0.0938130i
$910$		0			0
$911$	11.3137			0.374840			0.187420	−	0.982280i	$-0.439987\pi$
$911$	11.3137			0.374840			0.187420	+	0.982280i	$0.439987\pi$
$912$		0			0
$913$	−8.00000			−0.264761
$914$		0			0
$915$	−5.65685	+	8.00000i	−0.187010	+	0.264472i
$916$		0			0
$917$			39.5980i			1.30764i
$918$		0			0
$919$			30.0000i			0.989609i	0.869004	+	0.494804i	$0.164760\pi$
$919$			30.0000i			0.989609i	−0.869004	+	0.494804i	$0.835240\pi$
$920$		0			0
$921$	30.0000	−	42.4264i	0.988534	−	1.39800i
$922$		0			0
$923$	−11.3137			−0.372395
$924$		0			0
$925$	18.0000			0.591836
$926$		0			0
$927$	−39.5980	+	14.0000i	−1.30057	+	0.459820i
$928$		0			0
$929$			11.3137i			0.371191i	0.982626	+	0.185595i	$0.0594214\pi$
$929$			11.3137i			0.371191i	−0.982626	+	0.185595i	$0.940579\pi$
$930$		0			0
$931$		−	18.0000i		−	0.589926i
$932$		0			0
$933$	8.00000	+	5.65685i	0.261908	+	0.185197i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−30.0000			−0.980057			−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000			−0.980057			−0.490029	+	0.871706i	$0.663014\pi$
$938$		0			0
$939$	2.82843	+	2.00000i	0.0923022	+	0.0652675i
$940$		0			0
$941$			53.7401i			1.75188i	0.482422	+	0.875939i	$0.339757\pi$
$941$			53.7401i			1.75188i	−0.482422	+	0.875939i	$0.660243\pi$
$942$		0			0
$943$		−	32.0000i		−	1.04206i
$944$		0			0
$945$	−8.00000	+	28.2843i	−0.260240	+	0.920087i
$946$		0			0
$947$	14.1421			0.459558			0.229779	−	0.973243i	$-0.426200\pi$
$947$	14.1421			0.459558			0.229779	+	0.973243i	$0.426200\pi$
$948$		0			0
$949$	−12.0000			−0.389536
$950$		0			0
$951$	2.82843	−	4.00000i	0.0917180	−	0.129709i
$952$		0			0
$953$			50.9117i			1.64919i	0.565723	+	0.824596i	$0.308597\pi$
$953$			50.9117i			1.64919i	−0.565723	+	0.824596i	$0.691403\pi$
$954$		0			0
$955$			64.0000i			2.07099i
$956$		0			0
$957$	−8.00000	+	11.3137i	−0.258603	+	0.365720i
$958$		0			0
$959$	−11.3137			−0.365339
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	8.48528	+	24.0000i	0.273434	+	0.773389i
$964$		0			0
$965$		−	39.5980i		−	1.27470i
$966$		0			0
$967$		−	50.0000i		−	1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$		−	50.0000i		−	1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.7990			0.635380			0.317690	−	0.948195i	$-0.397093\pi$
$971$	19.7990			0.635380			0.317690	+	0.948195i	$0.397093\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$	−8.48528	−	6.00000i	−0.271746	−	0.192154i
$976$		0			0
$977$		−	45.2548i		−	1.44783i	−0.689889	−	0.723915i	$-0.742341\pi$
$977$		−	45.2548i		−	1.44783i	0.689889	−	0.723915i	$-0.257659\pi$
$978$		0			0
$979$		−	48.0000i		−	1.53409i
$980$		0			0
$981$	18.0000	+	50.9117i	0.574696	+	1.62549i
$982$		0			0
$983$	28.2843			0.902128			0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843			0.902128			0.451064	+	0.892492i	$0.351045\pi$
$984$		0			0
$985$	56.0000			1.78431
$986$		0			0
$987$	−22.6274	+	32.0000i	−0.720239	+	1.01857i
$988$		0			0
$989$			11.3137i			0.359755i
$990$		0			0
$991$		−	18.0000i		−	0.571789i	−0.958261	−	0.285894i	$-0.907709\pi$
$991$		−	18.0000i		−	0.571789i	0.958261	−	0.285894i	$-0.0922907\pi$
$992$		0			0
$993$	−10.0000	+	14.1421i	−0.317340	+	0.448787i
$994$		0			0
$995$	5.65685			0.179334
$996$		0			0
$997$	−38.0000			−1.20347			−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000			−1.20347			−0.601736	+	0.798695i	$0.705524\pi$
$998$		0			0
$999$	8.48528	−	30.0000i	0.268462	−	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.2.c.b.191.4		4
3.2	odd	2	inner	192.2.c.b.191.2		4
4.3	odd	2	inner	192.2.c.b.191.1		4
8.3	odd	2		96.2.c.a.95.4	yes	4
8.5	even	2		96.2.c.a.95.1	✓	4
12.11	even	2	inner	192.2.c.b.191.3		4
16.3	odd	4		768.2.f.g.383.2		4
16.5	even	4		768.2.f.g.383.1		4
16.11	odd	4		768.2.f.a.383.3		4
16.13	even	4		768.2.f.a.383.4		4
24.5	odd	2		96.2.c.a.95.3	yes	4
24.11	even	2		96.2.c.a.95.2	yes	4
40.3	even	4		2400.2.o.a.2399.3		4
40.13	odd	4		2400.2.o.h.2399.2		4
40.19	odd	2		2400.2.h.c.1151.1		4
40.27	even	4		2400.2.o.h.2399.1		4
40.29	even	2		2400.2.h.c.1151.4		4
40.37	odd	4		2400.2.o.a.2399.4		4
48.5	odd	4		768.2.f.g.383.4		4
48.11	even	4		768.2.f.a.383.2		4
48.29	odd	4		768.2.f.a.383.1		4
48.35	even	4		768.2.f.g.383.3		4
72.5	odd	6		2592.2.s.e.1727.4		8
72.11	even	6		2592.2.s.e.863.2		8
72.13	even	6		2592.2.s.e.1727.2		8
72.29	odd	6		2592.2.s.e.863.1		8
72.43	odd	6		2592.2.s.e.863.4		8
72.59	even	6		2592.2.s.e.1727.3		8
72.61	even	6		2592.2.s.e.863.3		8
72.67	odd	6		2592.2.s.e.1727.1		8
120.29	odd	2		2400.2.h.c.1151.2		4
120.53	even	4		2400.2.o.h.2399.3		4
120.59	even	2		2400.2.h.c.1151.3		4
120.77	even	4		2400.2.o.a.2399.1		4
120.83	odd	4		2400.2.o.a.2399.2		4
120.107	odd	4		2400.2.o.h.2399.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.c.a.95.1	✓	4	8.5	even	2
96.2.c.a.95.2	yes	4	24.11	even	2
96.2.c.a.95.3	yes	4	24.5	odd	2
96.2.c.a.95.4	yes	4	8.3	odd	2
192.2.c.b.191.1		4	4.3	odd	2	inner
192.2.c.b.191.2		4	3.2	odd	2	inner
192.2.c.b.191.3		4	12.11	even	2	inner
192.2.c.b.191.4		4	1.1	even	1	trivial
768.2.f.a.383.1		4	48.29	odd	4
768.2.f.a.383.2		4	48.11	even	4
768.2.f.a.383.3		4	16.11	odd	4
768.2.f.a.383.4		4	16.13	even	4
768.2.f.g.383.1		4	16.5	even	4
768.2.f.g.383.2		4	16.3	odd	4
768.2.f.g.383.3		4	48.35	even	4
768.2.f.g.383.4		4	48.5	odd	4
2400.2.h.c.1151.1		4	40.19	odd	2
2400.2.h.c.1151.2		4	120.29	odd	2
2400.2.h.c.1151.3		4	120.59	even	2
2400.2.h.c.1151.4		4	40.29	even	2
2400.2.o.a.2399.1		4	120.77	even	4
2400.2.o.a.2399.2		4	120.83	odd	4
2400.2.o.a.2399.3		4	40.3	even	4
2400.2.o.a.2399.4		4	40.37	odd	4
2400.2.o.h.2399.1		4	40.27	even	4
2400.2.o.h.2399.2		4	40.13	odd	4
2400.2.o.h.2399.3		4	120.53	even	4
2400.2.o.h.2399.4		4	120.107	odd	4
2592.2.s.e.863.1		8	72.29	odd	6
2592.2.s.e.863.2		8	72.11	even	6
2592.2.s.e.863.3		8	72.61	even	6
2592.2.s.e.863.4		8	72.43	odd	6
2592.2.s.e.1727.1		8	72.67	odd	6
2592.2.s.e.1727.2		8	72.13	even	6
2592.2.s.e.1727.3		8	72.59	even	6
2592.2.s.e.1727.4		8	72.5	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.00000i) q^{3} +2.82843i q^{5} -2.00000i q^{7} +(1.00000 + 2.82843i) q^{9} -2.82843 q^{11} +2.00000 q^{13} +(-2.82843 + 4.00000i) q^{15} -6.00000i q^{19} +(2.00000 - 2.82843i) q^{21} +5.65685 q^{23} -3.00000 q^{25} +(-1.41421 + 5.00000i) q^{27} -2.82843i q^{29} -2.00000i q^{31} +(-4.00000 - 2.82843i) q^{33} +5.65685 q^{35} -6.00000 q^{37} +(2.82843 + 2.00000i) q^{39} -5.65685i q^{41} +2.00000i q^{43} +(-8.00000 + 2.82843i) q^{45} -11.3137 q^{47} +3.00000 q^{49} -8.48528i q^{53} -8.00000i q^{55} +(6.00000 - 8.48528i) q^{57} -2.82843 q^{59} +2.00000 q^{61} +(5.65685 - 2.00000i) q^{63} +5.65685i q^{65} +2.00000i q^{67} +(8.00000 + 5.65685i) q^{69} -5.65685 q^{71} -6.00000 q^{73} +(-4.24264 - 3.00000i) q^{75} +5.65685i q^{77} +14.0000i q^{79} +(-7.00000 + 5.65685i) q^{81} +2.82843 q^{83} +(2.82843 - 4.00000i) q^{87} +16.9706i q^{89} -4.00000i q^{91} +(2.00000 - 2.82843i) q^{93} +16.9706 q^{95} +10.0000 q^{97} +(-2.82843 - 8.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} + 8 q^{13} + 8 q^{21} - 12 q^{25} - 16 q^{33} - 24 q^{37} - 32 q^{45} + 12 q^{49} + 24 q^{57} + 8 q^{61} + 32 q^{69} - 24 q^{73} - 28 q^{81} + 8 q^{93} + 40 q^{97}+O(q^{100}) \) 4 * q + 4 * q^9 + 8 * q^13 + 8 * q^21 - 12 * q^25 - 16 * q^33 - 24 * q^37 - 32 * q^45 + 12 * q^49 + 24 * q^57 + 8 * q^61 + 32 * q^69 - 24 * q^73 - 28 * q^81 + 8 * q^93 + 40 * q^97

Introduction

L-functions

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