LMFDB - Embedded newform 3456.2.p.e.2879.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3456,2,Mod(575,3456)] 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("3456.575"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.9349208943630483456.9
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2879.6
Root			$0.500000 + 1.74530i$ of defining polynomial
Character	$\chi$	$=$	3456.2879
Dual form			3456.2.p.e.575.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.158919 - 0.275255i) q^{5} +(2.93038 - 1.69185i) q^{7} +(-0.642559 + 0.370982i) q^{11} +(-2.59808 - 1.50000i) q^{13} +4.24264i q^{17} -2.57024 q^{19} +(-2.02166 + 3.50162i) q^{23} +(2.44949 + 4.24264i) q^{25} +(4.01229 + 6.94949i) q^{29} +(4.24755 + 2.45233i) q^{31} -1.07547i q^{35} -7.34847i q^{37} +(0.825765 + 0.476756i) q^{41} +(3.50162 + 6.06499i) q^{43} +(5.15627 + 8.93092i) q^{47} +(2.22474 - 3.85337i) q^{49} +5.51399 q^{53} +0.235824i q^{55} +(3.50162 + 2.02166i) q^{59} +(-7.79423 + 4.50000i) q^{61} +(-0.825765 + 0.476756i) q^{65} +(6.36068 - 11.0170i) q^{67} +13.5389 q^{71} -6.44949 q^{73} +(-1.25529 + 2.17423i) q^{77} +(8.29088 - 4.78674i) q^{79} +(2.21650 - 1.27970i) q^{83} +(1.16781 + 0.674235i) q^{85} -14.6349i q^{89} -10.1511 q^{91} +(-0.408459 + 0.707471i) q^{95} +(-5.62372 - 9.74058i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 72 q^{41} + 16 q^{49} - 72 q^{65} - 64 q^{73} + 8 q^{97}+O(q^{100}) $ 16 * q + 72 * q^41 + 16 * q^49 - 72 * q^65 - 64 * q^73 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.158919	−	0.275255i	0.0710706	−	0.123098i	−0.828300	−	0.560285i	$-0.810692\pi$
$5$	0.158919	−	0.275255i	0.0710706	−	0.123098i	0.899371	+	0.437187i	$0.144025\pi$
$6$		0			0
$7$	2.93038	−	1.69185i	1.10758	−	0.639461i	0.169378	−	0.985551i	$-0.445824\pi$
$7$	2.93038	−	1.69185i	1.10758	−	0.639461i	0.938201	+	0.346090i	$0.112491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.642559	+	0.370982i	−0.193739	+	0.111855i	−0.593732	−	0.804663i	$-0.702346\pi$
$11$	−0.642559	+	0.370982i	−0.193739	+	0.111855i	0.399993	+	0.916518i	$0.369013\pi$
$12$		0			0
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	0.0943882	−	0.995535i	$-0.469911\pi$
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	−0.814965	+	0.579510i	$0.803244\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$	−2.57024			−0.589653			−0.294827	−	0.955551i	$-0.595262\pi$
$19$	−2.57024			−0.589653			−0.294827	+	0.955551i	$0.595262\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.02166	+	3.50162i	−0.421546	+	0.730139i	−0.996091	−	0.0883343i	$-0.971846\pi$
$23$	−2.02166	+	3.50162i	−0.421546	+	0.730139i	0.574545	+	0.818473i	$0.305179\pi$
$24$		0			0
$25$	2.44949	+	4.24264i	0.489898	+	0.848528i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.01229	+	6.94949i	0.745064	+	1.29049i	0.950165	+	0.311747i	$0.100914\pi$
$29$	4.01229	+	6.94949i	0.745064	+	1.29049i	−0.205102	+	0.978741i	$0.565752\pi$
$30$		0			0
$31$	4.24755	+	2.45233i	0.762883	+	0.440451i	0.830330	−	0.557272i	$-0.188152\pi$
$31$	4.24755	+	2.45233i	0.762883	+	0.440451i	−0.0674468	+	0.997723i	$0.521485\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.07547i		−	0.181787i
$36$		0			0
$37$		−	7.34847i		−	1.20808i	−0.796954	−	0.604040i	$-0.793557\pi$
$37$		−	7.34847i		−	1.20808i	0.796954	−	0.604040i	$-0.206443\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.825765	+	0.476756i	0.128963	+	0.0744568i	0.563094	−	0.826393i	$-0.309611\pi$
$41$	0.825765	+	0.476756i	0.128963	+	0.0744568i	−0.434131	+	0.900850i	$0.642944\pi$
$42$		0			0
$43$	3.50162	+	6.06499i	0.533992	+	0.924902i	0.999211	+	0.0397062i	$0.0126422\pi$
$43$	3.50162	+	6.06499i	0.533992	+	0.924902i	−0.465219	+	0.885196i	$0.654024\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.15627	+	8.93092i	0.752119	+	1.30271i	0.946794	+	0.321841i	$0.104302\pi$
$47$	5.15627	+	8.93092i	0.752119	+	1.30271i	−0.194675	+	0.980868i	$0.562365\pi$
$48$		0			0
$49$	2.22474	−	3.85337i	0.317821	−	0.550482i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.51399			0.757405			0.378702	−	0.925519i	$-0.376370\pi$
$53$	5.51399			0.757405			0.378702	+	0.925519i	$0.376370\pi$
$54$		0			0
$55$			0.235824i			0.0317985i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.50162	+	2.02166i	0.455872	+	0.263198i	0.710307	−	0.703892i	$-0.248556\pi$
$59$	3.50162	+	2.02166i	0.455872	+	0.263198i	−0.254435	+	0.967090i	$0.581889\pi$
$60$		0			0
$61$	−7.79423	+	4.50000i	−0.997949	+	0.576166i	−0.907641	−	0.419748i	$-0.862118\pi$
$61$	−7.79423	+	4.50000i	−0.997949	+	0.576166i	−0.0903080	+	0.995914i	$0.528785\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.825765	+	0.476756i	−0.102424	+	0.0591343i
$66$		0			0
$67$	6.36068	−	11.0170i	0.777081	−	1.34594i	−0.156536	−	0.987672i	$-0.550033\pi$
$67$	6.36068	−	11.0170i	0.777081	−	1.34594i	0.933617	−	0.358272i	$-0.116634\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.5389			1.60678			0.803389	−	0.595455i	$-0.203028\pi$
$71$	13.5389			1.60678			0.803389	+	0.595455i	$0.203028\pi$
$72$		0			0
$73$	−6.44949			−0.754856			−0.377428	−	0.926039i	$-0.623191\pi$
$73$	−6.44949			−0.754856			−0.377428	+	0.926039i	$0.623191\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.25529	+	2.17423i	−0.143054	+	0.247777i
$78$		0			0
$79$	8.29088	−	4.78674i	0.932797	−	0.538550i	0.0451016	−	0.998982i	$-0.485639\pi$
$79$	8.29088	−	4.78674i	0.932797	−	0.538550i	0.887695	+	0.460432i	$0.152306\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.21650	−	1.27970i	0.243293	−	0.140465i	−0.373396	−	0.927672i	$-0.621807\pi$
$83$	2.21650	−	1.27970i	0.243293	−	0.140465i	0.616689	+	0.787207i	$0.288474\pi$
$84$		0			0
$85$	1.16781	+	0.674235i	0.126667	+	0.0731310i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.6349i		−	1.55130i	−0.631163	−	0.775651i	$-0.717422\pi$
$89$		−	14.6349i		−	1.55130i	0.631163	−	0.775651i	$-0.282578\pi$
$90$		0			0
$91$	−10.1511			−1.06413
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.408459	+	0.707471i	−0.0419070	+	0.0725850i
$96$		0			0
$97$	−5.62372	−	9.74058i	−0.571003	−	0.989006i	−0.996463	−	0.0840287i	$-0.973221\pi$
$97$	−5.62372	−	9.74058i	−0.571003	−	0.989006i	0.425461	−	0.904977i	$-0.360112\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.158919	−	0.275255i	−0.0158130	−	0.0273889i	0.858011	−	0.513632i	$-0.171700\pi$
$101$	−0.158919	−	0.275255i	−0.0158130	−	0.0273889i	−0.873824	+	0.486243i	$0.838367\pi$
$102$		0			0
$103$	13.2429	+	7.64580i	1.30486	+	0.753363i	0.981234	−	0.192821i	$-0.0617636\pi$
$103$	13.2429	+	7.64580i	1.30486	+	0.753363i	0.323629	+	0.946184i	$0.395097\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.75366i			0.846248i	0.906072	+	0.423124i	$0.139067\pi$
$107$			8.75366i			0.846248i	−0.906072	+	0.423124i	$0.860933\pi$
$108$		0			0
$109$		−	13.3485i		−	1.27855i	−0.768978	−	0.639276i	$-0.779234\pi$
$109$		−	13.3485i		−	1.27855i	0.768978	−	0.639276i	$-0.220766\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.50000	−	2.59808i	−0.423324	−	0.244406i	0.273174	−	0.961965i	$-0.411926\pi$
$113$	−4.50000	−	2.59808i	−0.423324	−	0.244406i	−0.696499	+	0.717558i	$0.745260\pi$
$114$		0			0
$115$	0.642559	+	1.11295i	0.0599190	+	0.103783i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.17793	+	12.4325i	0.658000	+	1.13969i
$120$		0			0
$121$	−5.22474	+	9.04952i	−0.474977	+	0.822684i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.14626			0.281410
$126$		0			0
$127$		−	1.62694i		−	0.144368i	−0.997391	−	0.0721839i	$-0.977003\pi$
$127$		−	1.62694i		−	0.144368i	0.997391	−	0.0721839i	$-0.0229969\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−16.2230	−	9.36635i	−1.41741	−	0.818341i	−0.421338	−	0.906904i	$-0.638440\pi$
$131$	−16.2230	−	9.36635i	−1.41741	−	0.818341i	−0.996071	+	0.0885620i	$0.971773\pi$
$132$		0			0
$133$	−7.53177	+	4.34847i	−0.653087	+	0.377060i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.52270	−	3.76588i	0.557272	−	0.321741i	−0.194778	−	0.980847i	$-0.562399\pi$
$137$	6.52270	−	3.76588i	0.557272	−	0.321741i	0.752050	+	0.659106i	$0.229065\pi$
$138$		0			0
$139$	−6.64951	+	11.5173i	−0.564004	+	0.976883i	0.433138	+	0.901328i	$0.357406\pi$
$139$	−6.64951	+	11.5173i	−0.564004	+	0.976883i	−0.997142	+	0.0755556i	$0.975927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.22589			0.186138
$144$		0			0
$145$	2.55051			0.211808
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.91591	+	5.05051i	−0.238881	+	0.413754i	−0.960393	−	0.278647i	$-0.910114\pi$
$149$	−2.91591	+	5.05051i	−0.238881	+	0.413754i	0.721513	+	0.692401i	$0.243447\pi$
$150$		0			0
$151$	14.1516	−	8.17045i	1.15164	−	0.664902i	0.202356	−	0.979312i	$-0.435140\pi$
$151$	14.1516	−	8.17045i	1.15164	−	0.664902i	0.949287	+	0.314410i	$0.101807\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.35003	−	0.779441i	0.108437	−	0.0626062i
$156$		0			0
$157$	19.3543	+	11.1742i	1.54464	+	0.891801i	0.998536	+	0.0540865i	$0.0172247\pi$
$157$	19.3543	+	11.1742i	1.54464	+	0.891801i	0.546108	+	0.837715i	$0.316109\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			13.6814i			1.07825i
$162$		0			0
$163$	11.4362			0.895756			0.447878	−	0.894095i	$-0.352180\pi$
$163$	11.4362			0.895756			0.447878	+	0.894095i	$0.352180\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.19959	+	15.9342i	−0.711886	+	1.23302i	0.252262	+	0.967659i	$0.418825\pi$
$167$	−9.19959	+	15.9342i	−0.711886	+	1.23302i	−0.964148	+	0.265364i	$0.914508\pi$
$168$		0			0
$169$	−2.00000	−	3.46410i	−0.153846	−	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.59771	−	16.6237i	−0.729701	−	1.26388i	−0.957010	−	0.290056i	$-0.906326\pi$
$173$	−9.59771	−	16.6237i	−0.729701	−	1.26388i	0.227309	−	0.973823i	$-0.427007\pi$
$174$		0			0
$175$	14.3559	+	8.28836i	1.08520	+	0.626541i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.7760i		−	1.70236i	−0.524874	−	0.851180i	$-0.675888\pi$
$179$		−	22.7760i		−	1.70236i	0.524874	−	0.851180i	$-0.324112\pi$
$180$		0			0
$181$			4.65153i			0.345746i	0.984944	+	0.172873i	$0.0553049\pi$
$181$			4.65153i			0.345746i	−0.984944	+	0.172873i	$0.944695\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.02270	−	1.16781i	−0.148712	−	0.0858590i
$186$		0			0
$187$	−1.57394	−	2.72615i	−0.115098	−	0.199356i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.8345	+	22.2299i	0.928669	+	1.60850i	0.785552	+	0.618796i	$0.212379\pi$
$191$	12.8345	+	22.2299i	0.928669	+	1.60850i	0.143117	+	0.989706i	$0.454287\pi$
$192$		0			0
$193$	1.27526	−	2.20881i	0.0917949	−	0.158993i	−0.816472	−	0.577386i	$-0.804073\pi$
$193$	1.27526	−	2.20881i	0.0917949	−	0.158993i	0.908266	+	0.418392i	$0.137406\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.0280			−0.785711			−0.392855	−	0.919600i	$-0.628513\pi$
$197$	−11.0280			−0.785711			−0.392855	+	0.919600i	$0.628513\pi$
$198$		0			0
$199$		−	5.24648i		−	0.371913i	−0.982558	−	0.185956i	$-0.940462\pi$
$199$		−	5.24648i		−	0.371913i	0.982558	−	0.185956i	$-0.0595383\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	23.5151	+	13.5764i	1.65043	+	0.952878i
$204$		0			0
$205$	0.262459	−	0.151531i	0.0183309	−	0.0105834i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.65153	−	0.953512i	0.114239	−	0.0659558i
$210$		0			0
$211$	5.07556	−	8.79114i	0.349416	−	0.605207i	−0.636730	−	0.771087i	$-0.719713\pi$
$211$	5.07556	−	8.79114i	0.349416	−	0.605207i	0.986146	+	0.165880i	$0.0530466\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.22589			0.151805
$216$		0			0
$217$	16.5959			1.12660
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.36396	−	11.0227i	0.428086	−	0.741467i
$222$		0			0
$223$	16.4693	−	9.50857i	1.10287	−	0.636741i	0.165895	−	0.986143i	$-0.446949\pi$
$223$	16.4693	−	9.50857i	1.10287	−	0.636741i	0.936973	+	0.349403i	$0.113615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.7863	+	7.38216i	−0.848655	+	0.489971i	−0.860197	−	0.509962i	$-0.829659\pi$
$227$	−12.7863	+	7.38216i	−0.848655	+	0.489971i	0.0115418	+	0.999933i	$0.496326\pi$
$228$		0			0
$229$	16.4938	+	9.52270i	1.08994	+	0.629278i	0.933561	−	0.358419i	$-0.116684\pi$
$229$	16.4938	+	9.52270i	1.08994	+	0.629278i	0.156381	+	0.987697i	$0.450017\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			8.48528i			0.555889i	0.960597	+	0.277945i	$0.0896532\pi$
$233$			8.48528i			0.555889i	−0.960597	+	0.277945i	$0.910347\pi$
$234$		0			0
$235$	3.27771			0.213814
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.02166	+	3.50162i	−0.130770	+	0.226501i	−0.923974	−	0.382456i	$-0.875078\pi$
$239$	−2.02166	+	3.50162i	−0.130770	+	0.226501i	0.793203	+	0.608957i	$0.208412\pi$
$240$		0			0
$241$	7.84847	+	13.5939i	0.505564	+	0.875663i	0.999979	+	0.00643712i	$0.00204901\pi$
$241$	7.84847	+	13.5939i	0.505564	+	0.875663i	−0.494415	+	0.869226i	$0.664618\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.707107	−	1.22474i	−0.0451754	−	0.0782461i
$246$		0			0
$247$	6.67767	+	3.85536i	0.424890	+	0.245310i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			24.2599i			1.53127i	0.643273	+	0.765637i	$0.277576\pi$
$251$			24.2599i			1.53127i	−0.643273	+	0.765637i	$0.722424\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.84847	+	1.64456i	0.177683	+	0.102585i	0.586203	−	0.810164i	$-0.300622\pi$
$257$	2.84847	+	1.64456i	0.177683	+	0.102585i	−0.408521	+	0.912749i	$0.633955\pi$
$258$		0			0
$259$	−12.4325	−	21.5338i	−0.772521	−	1.33804i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.61320	−	2.79415i	−0.0994744	−	0.172295i	0.811993	−	0.583667i	$-0.198383\pi$
$263$	−1.61320	−	2.79415i	−0.0994744	−	0.172295i	−0.911467	+	0.411373i	$0.865049\pi$
$264$		0			0
$265$	0.876276	−	1.51775i	0.0538292	−	0.0932349i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.4634			−1.06476			−0.532380	−	0.846505i	$-0.678702\pi$
$269$	−17.4634			−1.06476			−0.532380	+	0.846505i	$0.678702\pi$
$270$		0			0
$271$			4.19718i			0.254961i	0.991841	+	0.127480i	$0.0406890\pi$
$271$			4.19718i			0.254961i	−0.991841	+	0.127480i	$0.959311\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.14789	−	1.81743i	−0.189825	−	0.109595i
$276$		0			0
$277$	−16.4938	+	9.52270i	−0.991017	+	0.572164i	−0.905578	−	0.424180i	$-0.860563\pi$
$277$	−16.4938	+	9.52270i	−0.991017	+	0.572164i	−0.0854387	+	0.996343i	$0.527229\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.8712	+	13.2047i	−1.36438	+	0.787725i	−0.990203	−	0.139633i	$-0.955408\pi$
$281$	−22.8712	+	13.2047i	−1.36438	+	0.787725i	−0.374176	+	0.927358i	$0.622075\pi$
$282$		0			0
$283$	−11.7900	+	20.4208i	−0.700842	+	1.21389i	0.267330	+	0.963605i	$0.413859\pi$
$283$	−11.7900	+	20.4208i	−0.700842	+	1.21389i	−0.968171	+	0.250288i	$0.919475\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.22641			0.190449
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.30518	−	5.72474i	0.193091	−	0.334443i	−0.753182	−	0.657812i	$-0.771482\pi$
$293$	3.30518	−	5.72474i	0.193091	−	0.334443i	0.946273	+	0.323369i	$0.104815\pi$
$294$		0			0
$295$	1.11295	−	0.642559i	0.0647982	−	0.0374113i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.5049	−	6.06499i	0.607512	−	0.350747i
$300$		0			0
$301$	20.5222	+	11.8485i	1.18288	+	0.682934i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2.86054i			0.163794i
$306$		0			0
$307$	17.7320			1.01202			0.506010	−	0.862528i	$-0.331120\pi$
$307$	17.7320			1.01202			0.506010	+	0.862528i	$0.331120\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.65653	−	9.79739i	0.320752	−	0.555559i	−0.659891	−	0.751361i	$-0.729398\pi$
$311$	5.65653	−	9.79739i	0.320752	−	0.555559i	0.980643	+	0.195802i	$0.0627310\pi$
$312$		0			0
$313$	4.05051	+	7.01569i	0.228948	+	0.396550i	0.957497	−	0.288444i	$-0.0931380\pi$
$313$	4.05051	+	7.01569i	0.228948	+	0.396550i	−0.728548	+	0.684994i	$0.759805\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.96240	−	3.39898i	−0.110219	−	0.190906i	0.805639	−	0.592406i	$-0.201822\pi$
$317$	−1.96240	−	3.39898i	−0.110219	−	0.190906i	−0.915859	+	0.401501i	$0.868489\pi$
$318$		0			0
$319$	−5.15627	−	2.97697i	−0.288696	−	0.166679i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10.9046i		−	0.606748i
$324$		0			0
$325$		−	14.6969i		−	0.815239i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	30.2196	+	17.4473i	1.66606	+	0.961902i
$330$		0			0
$331$	9.50857	+	16.4693i	0.522638	+	0.905236i	0.999653	+	0.0263407i	$0.00838547\pi$
$331$	9.50857	+	16.4693i	0.522638	+	0.905236i	−0.477015	+	0.878895i	$0.658281\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.02166	−	3.50162i	−0.110455	−	0.191314i
$336$		0			0
$337$	0.848469	−	1.46959i	0.0462191	−	0.0800538i	−0.841990	−	0.539493i	$-0.818616\pi$
$337$	0.848469	−	1.46959i	0.0462191	−	0.0800538i	0.888209	+	0.459439i	$0.151949\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.63907			−0.197067
$342$		0			0
$343$			8.63019i			0.465986i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.6490	−	8.45763i	−0.786402	−	0.454029i	0.0522924	−	0.998632i	$-0.483347\pi$
$347$	−14.6490	−	8.45763i	−0.786402	−	0.454029i	−0.838694	+	0.544602i	$0.816681\pi$
$348$		0			0
$349$	−14.1582	+	8.17423i	−0.757871	+	0.437557i	−0.828531	−	0.559944i	$-0.810823\pi$
$349$	−14.1582	+	8.17423i	−0.757871	+	0.437557i	0.0706601	+	0.997500i	$0.477489\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.1742	−	9.91555i	0.914092	−	0.527751i	0.0323467	−	0.999477i	$-0.489702\pi$
$353$	17.1742	−	9.91555i	0.914092	−	0.527751i	0.881746	+	0.471725i	$0.156369\pi$
$354$		0			0
$355$	2.15159	−	3.72666i	0.114195	−	0.197791i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.08665			0.426797			0.213398	−	0.976965i	$-0.431547\pi$
$359$	8.08665			0.426797			0.213398	+	0.976965i	$0.431547\pi$
$360$		0			0
$361$	−12.3939			−0.652309
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.02494	+	1.77526i	−0.0536480	+	0.0929211i
$366$		0			0
$367$	−11.9257	+	6.88533i	−0.622519	+	0.359411i	−0.777849	−	0.628451i	$-0.783689\pi$
$367$	−11.9257	+	6.88533i	−0.622519	+	0.359411i	0.155330	+	0.987863i	$0.450356\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.1581	−	9.32887i	0.838886	−	0.484331i
$372$		0			0
$373$	25.7183	+	14.8485i	1.33164	+	0.768825i	0.985551	−	0.169376i	$-0.0541752\pi$
$373$	25.7183	+	14.8485i	1.33164	+	0.768825i	0.346092	+	0.938201i	$0.387509\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	24.0737i		−	1.23986i
$378$		0			0
$379$	−8.28836			−0.425745			−0.212872	−	0.977080i	$-0.568282\pi$
$379$	−8.28836			−0.425745			−0.212872	+	0.977080i	$0.568282\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.15627	−	8.93092i	0.263473	−	0.456349i	−0.703689	−	0.710508i	$-0.748465\pi$
$383$	5.15627	−	8.93092i	0.263473	−	0.456349i	0.967162	+	0.254159i	$0.0817987\pi$
$384$		0			0
$385$	0.398979	+	0.691053i	0.0203339	+	0.0352193i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.11243	+	1.92679i	0.0564025	+	0.0976919i	0.892848	−	0.450358i	$-0.148704\pi$
$389$	1.11243	+	1.92679i	0.0564025	+	0.0976919i	−0.836446	+	0.548050i	$0.815370\pi$
$390$		0			0
$391$	−14.8561	−	8.57719i	−0.751306	−	0.433767i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.04281i		−	0.153100i
$396$		0			0
$397$		−	16.0454i		−	0.805296i	−0.915355	−	0.402648i	$-0.868090\pi$
$397$		−	16.0454i		−	0.805296i	0.915355	−	0.402648i	$-0.131910\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−33.5227	−	19.3543i	−1.67404	−	0.966510i	−0.965337	−	0.261007i	$-0.915946\pi$
$401$	−33.5227	−	19.3543i	−1.67404	−	0.966510i	−0.708707	−	0.705503i	$-0.750721\pi$
$402$		0			0
$403$	−7.35698	−	12.7427i	−0.366477	−	0.634757i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.72615	+	4.72183i	0.135130	+	0.234052i
$408$		0			0
$409$	6.29796	−	10.9084i	0.311414	−	0.539385i	−0.667255	−	0.744830i	$-0.732531\pi$
$409$	6.29796	−	10.9084i	0.311414	−	0.539385i	0.978669	+	0.205445i	$0.0658641\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.6814			0.673219
$414$		0			0
$415$		−	0.813472i		−	0.0399317i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.22345	−	4.74781i	−0.401742	−	0.231946i	0.285493	−	0.958381i	$-0.407843\pi$
$419$	−8.22345	−	4.74781i	−0.401742	−	0.231946i	−0.687235	+	0.726435i	$0.741176\pi$
$420$		0			0
$421$	−10.1298	+	5.84847i	−0.493698	+	0.285037i	−0.726108	−	0.687581i	$-0.758673\pi$
$421$	−10.1298	+	5.84847i	−0.493698	+	0.285037i	0.232409	+	0.972618i	$0.425339\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.0000	+	10.3923i	−0.873128	+	0.504101i
$426$		0			0
$427$	−15.2267	+	26.3734i	−0.736871	+	1.27630i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.08665			0.389520			0.194760	−	0.980851i	$-0.437607\pi$
$431$	8.08665			0.389520			0.194760	+	0.980851i	$0.437607\pi$
$432$		0			0
$433$	−17.5505			−0.843424			−0.421712	−	0.906730i	$-0.638571\pi$
$433$	−17.5505			−0.843424			−0.421712	+	0.906730i	$0.638571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.19615	−	9.00000i	0.248566	−	0.430528i
$438$		0			0
$439$	−15.5606	+	8.98392i	−0.742667	+	0.428779i	−0.823038	−	0.567986i	$-0.807723\pi$
$439$	−15.5606	+	8.98392i	−0.742667	+	0.428779i	0.0803710	+	0.996765i	$0.474389\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.5146	+	18.1950i	−1.49730	+	0.864469i	−0.999995	−	0.00310536i	$-0.999012\pi$
$443$	−31.5146	+	18.1950i	−1.49730	+	0.864469i	−0.497308	+	0.867574i	$0.665678\pi$
$444$		0			0
$445$	−4.02834	−	2.32577i	−0.190962	−	0.110252i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	10.8209i		−	0.510670i	−0.966853	−	0.255335i	$-0.917814\pi$
$449$		−	10.8209i		−	0.510670i	0.966853	−	0.255335i	$-0.0821857\pi$
$450$		0			0
$451$	−0.707471			−0.0333135
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.61320	+	2.79415i	−0.0756281	+	0.130992i
$456$		0			0
$457$	−18.0732	−	31.3037i	−0.845429	−	1.46433i	−0.885248	−	0.465120i	$-0.846011\pi$
$457$	−18.0732	−	31.3037i	−0.845429	−	1.46433i	0.0398186	−	0.999207i	$-0.487322\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.4477	+	18.0959i	0.486597	+	0.842811i	0.999881	−	0.0154078i	$-0.00490464\pi$
$461$	10.4477	+	18.0959i	0.486597	+	0.842811i	−0.513284	+	0.858219i	$0.671571\pi$
$462$		0			0
$463$	−8.69934	−	5.02256i	−0.404292	−	0.233418i	0.284042	−	0.958812i	$-0.408324\pi$
$463$	−8.69934	−	5.02256i	−0.404292	−	0.233418i	−0.688334	+	0.725393i	$0.741658\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.5063i			0.717545i	0.933425	+	0.358773i	$0.116805\pi$
$467$			15.5063i			0.717545i	−0.933425	+	0.358773i	$0.883195\pi$
$468$		0			0
$469$		−	43.0454i		−	1.98765i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.50000	−	2.59808i	−0.206910	−	0.119460i
$474$		0			0
$475$	−6.29577	−	10.9046i	−0.288870	−	0.500337i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.9691	−	27.6592i	−0.729645	−	1.26378i	−0.957033	−	0.289978i	$-0.906352\pi$
$479$	−15.9691	−	27.6592i	−0.729645	−	1.26378i	0.227388	−	0.973804i	$-0.426981\pi$
$480$		0			0
$481$	−11.0227	+	19.0919i	−0.502592	+	0.870515i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.57486			−0.162326
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	32.0922	+	18.5285i	1.44830	+	0.836178i	0.998380	−	0.0568900i	$-0.0181184\pi$
$491$	32.0922	+	18.5285i	1.44830	+	0.836178i	0.449922	+	0.893068i	$0.351452\pi$
$492$		0			0
$493$	−29.4842	+	17.0227i	−1.32790	+	0.766664i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	39.6742	−	22.9059i	1.77963	−	1.02747i
$498$		0			0
$499$	6.07186	−	10.5168i	0.271814	−	0.470795i	−0.697513	−	0.716573i	$-0.745710\pi$
$499$	6.07186	−	10.5168i	0.271814	−	0.470795i	0.969326	+	0.245777i	$0.0790432\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−33.3471			−1.48687			−0.743437	−	0.668806i	$-0.766806\pi$
$503$	−33.3471			−1.48687			−0.743437	+	0.668806i	$0.766806\pi$
$504$		0			0
$505$	−0.101021			−0.00449535
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.8865	+	34.4444i	−0.881453	+	1.52672i	−0.0317262	+	0.999497i	$0.510100\pi$
$509$	−19.8865	+	34.4444i	−0.881453	+	1.52672i	−0.849726	+	0.527224i	$0.823233\pi$
$510$		0			0
$511$	−18.8994	+	10.9116i	−0.836062	+	0.482701i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.20909	−	2.43012i	0.185475	−	0.107084i
$516$		0			0
$517$	−6.62642	−	3.82577i	−0.291430	−	0.168257i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			1.90702i			0.0835482i	0.999127	+	0.0417741i	$0.0133010\pi$
$521$			1.90702i			0.0835482i	−0.999127	+	0.0417741i	$0.986699\pi$
$522$		0			0
$523$	13.4288			0.587202			0.293601	−	0.955928i	$-0.405146\pi$
$523$	13.4288			0.587202			0.293601	+	0.955928i	$0.405146\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.4043	+	18.0208i	−0.453220	+	0.785000i
$528$		0			0
$529$	3.32577	+	5.76039i	0.144598	+	0.250452i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.43027	−	2.47730i	−0.0619518	−	0.107304i
$534$		0			0
$535$	2.40949	+	1.39112i	0.104171	+	0.0601433i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			3.30136i			0.142200i
$540$		0			0
$541$		−	20.6969i		−	0.889831i	−0.895573	−	0.444915i	$-0.853234\pi$
$541$		−	20.6969i		−	0.889831i	0.895573	−	0.444915i	$-0.146766\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.67423	−	2.12132i	−0.157387	−	0.0908674i
$546$		0			0
$547$	−3.79045	−	6.56524i	−0.162068	−	0.280710i	0.773542	−	0.633745i	$-0.218483\pi$
$547$	−3.79045	−	6.56524i	−0.162068	−	0.280710i	−0.935610	+	0.353035i	$0.885150\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.3125	−	17.8618i	−0.439329	−	0.760940i
$552$		0			0
$553$	16.1969	−	28.0539i	0.688764	−	1.19297i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.7839			1.47384			0.736920	−	0.675980i	$-0.236279\pi$
$557$	34.7839			1.47384			0.736920	+	0.675980i	$0.236279\pi$
$558$		0			0
$559$		−	21.0097i		−	0.888617i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.9268	−	10.3500i	−0.755523	−	0.436201i	0.0721633	−	0.997393i	$-0.477010\pi$
$563$	−17.9268	−	10.3500i	−0.755523	−	0.436201i	−0.827686	+	0.561192i	$0.810343\pi$
$564$		0			0
$565$	−1.43027	+	0.825765i	−0.0601718	+	0.0347402i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.5000	+	7.79423i	−0.565949	+	0.326751i	−0.755530	−	0.655114i	$-0.772621\pi$
$569$	−13.5000	+	7.79423i	−0.565949	+	0.326751i	0.189580	+	0.981865i	$0.439287\pi$
$570$		0			0
$571$	14.6490	−	25.3729i	0.613043	−	1.06182i	−0.377681	−	0.925936i	$-0.623278\pi$
$571$	14.6490	−	25.3729i	0.613043	−	1.06182i	0.990724	−	0.135887i	$-0.0433883\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.8082			−0.826057
$576$		0			0
$577$	−5.34847			−0.222660			−0.111330	−	0.993784i	$-0.535511\pi$
$577$	−5.34847			−0.222660			−0.111330	+	0.993784i	$0.535511\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.33013	−	7.50000i	0.179644	−	0.311152i
$582$		0			0
$583$	−3.54307	+	2.04559i	−0.146739	+	0.0847197i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	39.8030	−	22.9802i	1.64284	−	0.948496i	0.663028	−	0.748595i	$-0.269271\pi$
$587$	39.8030	−	22.9802i	1.64284	−	0.948496i	0.979816	−	0.199901i	$-0.0640622\pi$
$588$		0			0
$589$	−10.9172	−	6.30306i	−0.449836	−	0.259713i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			16.5420i			0.679297i	0.940552	+	0.339649i	$0.110308\pi$
$593$			16.5420i			0.679297i	−0.940552	+	0.339649i	$0.889692\pi$
$594$		0			0
$595$	4.56283			0.187058
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.5121	−	33.7960i	0.797244	−	1.38087i	−0.124160	−	0.992262i	$-0.539624\pi$
$599$	19.5121	−	33.7960i	0.797244	−	1.38087i	0.921404	−	0.388605i	$-0.127043\pi$
$600$		0			0
$601$	11.6237	+	20.1329i	0.474142	+	0.821237i	0.999562	−	0.0296056i	$-0.00942514\pi$
$601$	11.6237	+	20.1329i	0.474142	+	0.821237i	−0.525420	+	0.850843i	$0.676092\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.66062	+	2.87628i	0.0675137	+	0.116937i
$606$		0			0
$607$	−3.33884	−	1.92768i	−0.135519	−	0.0782421i	0.430708	−	0.902492i	$-0.358264\pi$
$607$	−3.33884	−	1.92768i	−0.135519	−	0.0782421i	−0.566227	+	0.824249i	$0.691597\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	30.9376i		−	1.25160i
$612$		0			0
$613$		−	4.04541i		−	0.163392i	−0.996657	−	0.0816962i	$-0.973966\pi$
$613$		−	4.04541i		−	0.163392i	0.996657	−	0.0816962i	$-0.0260337\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.8712	+	13.2047i	0.920759	+	0.531600i	0.883877	−	0.467719i	$-0.154924\pi$
$617$	22.8712	+	13.2047i	0.920759	+	0.531600i	0.0368817	+	0.999320i	$0.488258\pi$
$618$		0			0
$619$	19.0820	+	33.0511i	0.766972	+	1.32844i	0.939197	+	0.343377i	$0.111571\pi$
$619$	19.0820	+	33.0511i	0.766972	+	1.32844i	−0.172225	+	0.985058i	$0.555096\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−24.7602	−	42.8859i	−0.991997	−	1.71819i
$624$		0			0
$625$	−11.7474	+	20.3472i	−0.469898	+	0.813887i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	31.1769			1.24310
$630$		0			0
$631$		−	9.33766i		−	0.371726i	−0.982576	−	0.185863i	$-0.940492\pi$
$631$		−	9.33766i		−	0.371726i	0.982576	−	0.185863i	$-0.0595081\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.447824	−	0.258552i	−0.0177714	−	0.0102603i
$636$		0			0
$637$	−11.5601	+	6.67423i	−0.458028	+	0.264443i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.1969	+	5.88721i	−0.402755	+	0.232531i	−0.687672	−	0.726022i	$-0.741367\pi$
$641$	−10.1969	+	5.88721i	−0.402755	+	0.232531i	0.284917	+	0.958552i	$0.408034\pi$
$642$		0			0
$643$	6.36068	−	11.0170i	0.250841	−	0.434469i	−0.712917	−	0.701249i	$-0.752626\pi$
$643$	6.36068	−	11.0170i	0.250841	−	0.434469i	0.963758	+	0.266780i	$0.0859596\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.6251			−0.810856			−0.405428	−	0.914127i	$-0.632877\pi$
$647$	−20.6251			−0.810856			−0.405428	+	0.914127i	$0.632877\pi$
$648$		0			0
$649$	−3.00000			−0.117760
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.13021	−	5.42168i	0.122495	−	0.212167i	−0.798256	−	0.602318i	$-0.794244\pi$
$653$	3.13021	−	5.42168i	0.122495	−	0.212167i	0.920751	+	0.390151i	$0.127577\pi$
$654$		0			0
$655$	−5.15627	+	2.97697i	−0.201472	+	0.116320i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.8002	+	14.3184i	−0.966078	+	0.557765i	−0.898038	−	0.439917i	$-0.855008\pi$
$659$	−24.8002	+	14.3184i	−0.966078	+	0.557765i	−0.0680394	+	0.997683i	$0.521674\pi$
$660$		0			0
$661$	−3.76588	−	2.17423i	−0.146476	−	0.0845679i	0.424971	−	0.905207i	$-0.360284\pi$
$661$	−3.76588	−	2.17423i	−0.146476	−	0.0845679i	−0.571447	+	0.820639i	$0.693618\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.76421i			0.107192i
$666$		0			0
$667$	−32.4460			−1.25631
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.33884	−	5.78304i	0.128894	−	0.223252i
$672$		0			0
$673$	2.62372	+	4.54442i	0.101137	+	0.175175i	0.912153	−	0.409849i	$-0.134419\pi$
$673$	2.62372	+	4.54442i	0.101137	+	0.175175i	−0.811016	+	0.585024i	$0.801085\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.3763	+	17.9722i	0.398792	+	0.690728i	0.993577	−	0.113157i	$-0.0360963\pi$
$677$	10.3763	+	17.9722i	0.398792	+	0.690728i	−0.594785	+	0.803885i	$0.702763\pi$
$678$		0			0
$679$	−32.9593	−	19.0290i	−1.26486	−	0.730268i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			18.9912i			0.726680i	0.931657	+	0.363340i	$0.118364\pi$
$683$			18.9912i			0.726680i	−0.931657	+	0.363340i	$0.881636\pi$
$684$		0			0
$685$		−	2.39388i		−	0.0914653i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.3258	−	8.27098i	−0.545768	−	0.315099i
$690$		0			0
$691$	23.2262	+	40.2290i	0.883567	+	1.53038i	0.847347	+	0.531040i	$0.178199\pi$
$691$	23.2262	+	40.2290i	0.883567	+	1.53038i	0.0362206	+	0.999344i	$0.488468\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.11346	+	3.66062i	0.0801681	+	0.138855i
$696$		0			0
$697$	−2.02270	+	3.50343i	−0.0766154	+	0.132702i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−32.8769			−1.24174			−0.620871	−	0.783913i	$-0.713221\pi$
$701$	−32.8769			−1.24174			−0.620871	+	0.783913i	$0.713221\pi$
$702$		0			0
$703$			18.8873i			0.712349i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.931383	−	0.537734i	−0.0350283	−	0.0202236i
$708$		0			0
$709$	3.76588	−	2.17423i	0.141431	−	0.0816551i	−0.427615	−	0.903961i	$-0.640646\pi$
$709$	3.76588	−	2.17423i	0.141431	−	0.0816551i	0.569046	+	0.822306i	$0.307313\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17.1742	+	9.91555i	−0.643180	+	0.371340i
$714$		0			0
$715$	0.353736	−	0.612688i	0.0132290	−	0.0229132i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.90357			0.332047			0.166023	−	0.986122i	$-0.446907\pi$
$719$	8.90357			0.332047			0.166023	+	0.986122i	$0.446907\pi$
$720$		0			0
$721$	51.7423			1.92699
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−19.6561	+	34.0454i	−0.730010	+	1.26441i
$726$		0			0
$727$	−23.0552	+	13.3109i	−0.855070	+	0.493675i	−0.862358	−	0.506299i	$-0.831013\pi$
$727$	−23.0552	+	13.3109i	−0.855070	+	0.493675i	0.00728826	+	0.999973i	$0.497680\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−25.7316	+	14.8561i	−0.951716	+	0.549473i
$732$		0			0
$733$	12.9904	+	7.50000i	0.479811	+	0.277019i	0.720338	−	0.693624i	$-0.243987\pi$
$733$	12.9904	+	7.50000i	0.479811	+	0.277019i	−0.240527	+	0.970642i	$0.577320\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9.43879i			0.347682i
$738$		0			0
$739$	−31.1609			−1.14627			−0.573135	−	0.819461i	$-0.694273\pi$
$739$	−31.1609			−1.14627			−0.573135	+	0.819461i	$0.694273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.60805	−	16.6416i	0.352485	−	0.610522i	−0.634199	−	0.773170i	$-0.718670\pi$
$743$	9.60805	−	16.6416i	0.352485	−	0.610522i	0.986684	+	0.162647i	$0.0520033\pi$
$744$		0			0
$745$	0.926786	+	1.60524i	0.0339548	+	0.0588115i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.8099	+	25.6515i	0.541143	+	0.937287i
$750$		0			0
$751$	−42.5467	−	24.5643i	−1.55255	−	0.896366i	−0.997933	−	0.0642608i	$-0.979531\pi$
$751$	−42.5467	−	24.5643i	−1.55255	−	0.896366i	−0.554618	−	0.832105i	$-0.687136\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	5.19375i		−	0.189020i
$756$		0			0
$757$		−	12.0000i		−	0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$		−	12.0000i		−	0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.4773	+	11.8226i	0.742301	+	0.428568i	0.822905	−	0.568178i	$-0.192352\pi$
$761$	20.4773	+	11.8226i	0.742301	+	0.428568i	−0.0806041	+	0.996746i	$0.525685\pi$
$762$		0			0
$763$	−22.5837	−	39.1161i	−0.817584	−	1.41610i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.06499	−	10.5049i	−0.218994	−	0.379309i
$768$		0			0
$769$	6.29796	−	10.9084i	0.227110	−	0.393366i	−0.729840	−	0.683618i	$-0.760406\pi$
$769$	6.29796	−	10.9084i	0.227110	−	0.393366i	0.956950	+	0.290251i	$0.0937389\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.1346			0.796126			0.398063	−	0.917358i	$-0.369682\pi$
$773$	22.1346			0.796126			0.398063	+	0.917358i	$0.369682\pi$
$774$		0			0
$775$			24.0278i			0.863104i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.12241	−	1.22538i	−0.0760434	−	0.0439037i
$780$		0			0
$781$	−8.69958	+	5.02270i	−0.311295	+	0.179726i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.15153	−	3.55159i	0.219558	−	0.126762i
$786$		0			0
$787$	2.92397	−	5.06447i	0.104228	−	0.180529i	−0.809194	−	0.587541i	$-0.800096\pi$
$787$	2.92397	−	5.06447i	0.104228	−	0.180529i	0.913423	+	0.407012i	$0.133429\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.5823			−0.625154
$792$		0			0
$793$	27.0000			0.958798
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.7470	−	39.3990i	0.805740	−	1.39558i	−0.110050	−	0.993926i	$-0.535101\pi$
$797$	22.7470	−	39.3990i	0.805740	−	1.39558i	0.915790	−	0.401657i	$-0.131566\pi$
$798$		0			0
$799$	−37.8907	+	21.8762i	−1.34048	+	0.773924i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.14418	−	2.39264i	0.146245	−	0.0844346i
$804$		0			0
$805$	3.76588	+	2.17423i	0.132730	+	0.0766317i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.14966i		−	0.216211i	−0.994139	−	0.108105i	$-0.965522\pi$
$809$		−	6.14966i		−	0.216211i	0.994139	−	0.108105i	$-0.0344784\pi$
$810$		0			0
$811$	−19.1470			−0.672341			−0.336170	−	0.941801i	$-0.609132\pi$
$811$	−19.1470			−0.672341			−0.336170	+	0.941801i	$0.609132\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.81743	−	3.14789i	0.0636619	−	0.110266i
$816$		0			0
$817$	−9.00000	−	15.5885i	−0.314870	−	0.545371i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.5441	−	19.9949i	−0.402890	−	0.697827i	0.591183	−	0.806537i	$-0.298661\pi$
$821$	−11.5441	−	19.9949i	−0.402890	−	0.697827i	−0.994073	+	0.108711i	$0.965328\pi$
$822$		0			0
$823$	−31.7339	−	18.3216i	−1.10617	−	0.638650i	−0.168339	−	0.985729i	$-0.553840\pi$
$823$	−31.7339	−	18.3216i	−1.10617	−	0.638650i	−0.937836	+	0.347079i	$0.887174\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.0779i		−	0.941591i	−0.882242	−	0.470795i	$-0.843967\pi$
$827$		−	27.0779i		−	0.941591i	0.882242	−	0.470795i	$-0.156033\pi$
$828$		0			0
$829$		−	49.3485i		−	1.71394i	−0.515364	−	0.856972i	$-0.672343\pi$
$829$		−	49.3485i		−	1.71394i	0.515364	−	0.856972i	$-0.327657\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.3485	+	9.43879i	0.566441	+	0.327035i
$834$		0			0
$835$	2.92397	+	5.06447i	0.101188	+	0.175263i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.1037	−	33.0885i	−0.659532	−	1.14234i	−0.980737	−	0.195333i	$-0.937421\pi$
$839$	−19.1037	−	33.0885i	−0.659532	−	1.14234i	0.321205	−	0.947010i	$-0.395912\pi$
$840$		0			0
$841$	−17.6969	+	30.6520i	−0.610239	+	1.05697i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.27135			−0.0437357
$846$		0			0
$847$			35.3580i			1.21492i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	25.7316	+	14.8561i	0.882066	+	0.509261i
$852$		0			0
$853$	−41.3068	+	23.8485i	−1.41432	+	0.816556i	−0.995791	−	0.0916492i	$-0.970786\pi$
$853$	−41.3068	+	23.8485i	−1.41432	+	0.816556i	−0.418525	+	0.908205i	$0.637453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.82577	+	5.67291i	−0.335642	+	0.193783i	−0.658343	−	0.752718i	$-0.728742\pi$
$857$	−9.82577	+	5.67291i	−0.335642	+	0.193783i	0.322701	+	0.946501i	$0.395409\pi$
$858$		0			0
$859$	17.0895	−	29.5998i	0.583085	−	1.00993i	−0.412026	−	0.911172i	$-0.635179\pi$
$859$	17.0895	−	29.5998i	0.583085	−	1.00993i	0.995111	−	0.0987606i	$-0.0314878\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.816917			0.0278082			0.0139041	−	0.999903i	$-0.495574\pi$
$863$	0.816917			0.0278082			0.0139041	+	0.999903i	$0.495574\pi$
$864$		0			0
$865$	−6.10102			−0.207441
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.55159	+	6.15153i	−0.120479	+	0.208676i
$870$		0			0
$871$	−33.0511	+	19.0820i	−1.11989	+	0.646571i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.21975	−	5.32302i	0.311684	−	0.179951i
$876$		0			0
$877$	38.4462	+	22.1969i	1.29824	+	0.749537i	0.980099	−	0.198507i	$-0.0636093\pi$
$877$	38.4462	+	22.1969i	1.29824	+	0.749537i	0.318137	+	0.948045i	$0.396943\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			35.4196i			1.19332i	0.802496	+	0.596658i	$0.203505\pi$
$881$			35.4196i			1.19332i	−0.802496	+	0.596658i	$0.796495\pi$
$882$		0			0
$883$	20.3023			0.683225			0.341613	−	0.939841i	$-0.389027\pi$
$883$	20.3023			0.683225			0.341613	+	0.939841i	$0.389027\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.29139	−	16.0932i	0.311974	−	0.540356i	−0.666815	−	0.745223i	$-0.732343\pi$
$887$	9.29139	−	16.0932i	0.311974	−	0.540356i	0.978790	+	0.204867i	$0.0656763\pi$
$888$		0			0
$889$	−2.75255	−	4.76756i	−0.0923176	−	0.159899i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.2528	−	22.9546i	−0.443489	−	0.768146i
$894$		0			0
$895$	−6.26922	−	3.61953i	−0.209557	−	0.120988i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			39.3578i			1.31266i
$900$		0			0
$901$			23.3939i			0.779363i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.28036	+	0.739215i	0.0425605	+	0.0245723i
$906$		0			0
$907$	21.2336	+	36.7777i	0.705051	+	1.22118i	0.966673	+	0.256014i	$0.0824094\pi$
$907$	21.2336	+	36.7777i	0.705051	+	1.22118i	−0.261622	+	0.965171i	$0.584257\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.65601	−	8.06445i	−0.154261	−	0.267187i	0.778529	−	0.627609i	$-0.215966\pi$
$911$	−4.65601	−	8.06445i	−0.154261	−	0.267187i	−0.932790	+	0.360422i	$0.882633\pi$
$912$		0			0
$913$	−0.949490	+	1.64456i	−0.0314235	+	0.0544271i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−63.3860			−2.09319
$918$		0			0
$919$		−	35.2520i		−	1.16286i	−0.813597	−	0.581429i	$-0.802494\pi$
$919$		−	35.2520i		−	1.16286i	0.813597	−	0.581429i	$-0.197506\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.1752	−	20.3084i	−1.15781	−	0.668460i
$924$		0			0
$925$	31.1769	−	18.0000i	1.02509	−	0.591836i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.87117	−	2.81237i	0.159818	−	0.0922710i	−0.417958	−	0.908466i	$-0.637254\pi$
$929$	4.87117	−	2.81237i	0.159818	−	0.0922710i	0.577776	+	0.816195i	$0.303921\pi$
$930$		0			0
$931$	−5.71812	+	9.90408i	−0.187404	+	0.324593i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.00052			−0.0327204
$936$		0			0
$937$	−8.85357			−0.289234			−0.144617	−	0.989488i	$-0.546195\pi$
$937$	−8.85357			−0.289234			−0.144617	+	0.989488i	$0.546195\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.94426	−	12.0278i	0.226376	−	0.392095i	−0.730355	−	0.683068i	$-0.760645\pi$
$941$	6.94426	−	12.0278i	0.226376	−	0.392095i	0.956731	+	0.290972i	$0.0939788\pi$
$942$		0			0
$943$	−3.33884	+	1.92768i	−0.108727	+	0.0627738i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.6629	−	15.3939i	0.866429	−	0.500233i	0.000269172	−	1.00000i	$-0.499914\pi$
$947$	26.6629	−	15.3939i	0.866429	−	0.500233i	0.866160	+	0.499767i	$0.166581\pi$
$948$		0			0
$949$	16.7563	+	9.67423i	0.543931	+	0.314039i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	43.9048i		−	1.42222i	−0.703082	−	0.711109i	$-0.748193\pi$
$953$		−	43.9048i		−	1.42222i	0.703082	−	0.711109i	$-0.251807\pi$
$954$		0			0
$955$	8.15854			0.264004
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.7427	−	22.0709i	0.411482	−	0.712708i
$960$		0			0
$961$	−3.47219	−	6.01402i	−0.112006	−	0.194001i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.405324	−	0.702041i	−0.0130478	−	0.0225995i
$966$		0			0
$967$	27.2821	+	15.7513i	0.877334	+	0.506529i	0.869778	−	0.493443i	$-0.164262\pi$
$967$	27.2821	+	15.7513i	0.877334	+	0.506529i	0.00755547	+	0.999971i	$0.497595\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	11.2381i		−	0.360648i	−0.983607	−	0.180324i	$-0.942285\pi$
$971$		−	11.2381i		−	0.360648i	0.983607	−	0.180324i	$-0.0577146\pi$
$972$		0			0
$973$			45.0000i			1.44263i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−47.8485	−	27.6253i	−1.53081	−	0.883812i	−0.999325	−	0.0367438i	$-0.988301\pi$
$977$	−47.8485	−	27.6253i	−1.53081	−	0.883812i	−0.531483	−	0.847069i	$-0.678365\pi$
$978$		0			0
$979$	5.42930	+	9.40382i	0.173521	+	0.300547i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.29088	−	14.3602i	−0.264438	−	0.458020i	0.702978	−	0.711211i	$-0.251853\pi$
$983$	−8.29088	−	14.3602i	−0.264438	−	0.458020i	−0.967416	+	0.253191i	$0.918520\pi$
$984$		0			0
$985$	−1.75255	+	3.03551i	−0.0558409	+	0.0967193i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−28.3164			−0.900408
$990$		0			0
$991$		−	30.1116i		−	0.956525i	−0.878217	−	0.478263i	$-0.841267\pi$
$991$		−	30.1116i		−	0.956525i	0.878217	−	0.478263i	$-0.158733\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.44412	−	0.833763i	−0.0457817	−	0.0264321i
$996$		0			0
$997$	−16.4938	+	9.52270i	−0.522364	+	0.301587i	−0.737901	−	0.674909i	$-0.764183\pi$
$997$	−16.4938	+	9.52270i	−0.522364	+	0.301587i	0.215537	+	0.976496i	$0.430850\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.p.e.2879.6		16
3.2	odd	2		1152.2.p.e.959.3	yes	16
4.3	odd	2	inner	3456.2.p.e.2879.5		16
8.3	odd	2	inner	3456.2.p.e.2879.3		16
8.5	even	2	inner	3456.2.p.e.2879.4		16
9.2	odd	6	inner	3456.2.p.e.575.3		16
9.7	even	3		1152.2.p.e.191.4	yes	16
12.11	even	2		1152.2.p.e.959.5	yes	16
24.5	odd	2		1152.2.p.e.959.6	yes	16
24.11	even	2		1152.2.p.e.959.4	yes	16
36.7	odd	6		1152.2.p.e.191.6	yes	16
36.11	even	6	inner	3456.2.p.e.575.4		16
72.11	even	6	inner	3456.2.p.e.575.6		16
72.29	odd	6	inner	3456.2.p.e.575.5		16
72.43	odd	6		1152.2.p.e.191.3	✓	16
72.61	even	6		1152.2.p.e.191.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.p.e.191.3	✓	16	72.43	odd	6
1152.2.p.e.191.4	yes	16	9.7	even	3
1152.2.p.e.191.5	yes	16	72.61	even	6
1152.2.p.e.191.6	yes	16	36.7	odd	6
1152.2.p.e.959.3	yes	16	3.2	odd	2
1152.2.p.e.959.4	yes	16	24.11	even	2
1152.2.p.e.959.5	yes	16	12.11	even	2
1152.2.p.e.959.6	yes	16	24.5	odd	2
3456.2.p.e.575.3		16	9.2	odd	6	inner
3456.2.p.e.575.4		16	36.11	even	6	inner
3456.2.p.e.575.5		16	72.29	odd	6	inner
3456.2.p.e.575.6		16	72.11	even	6	inner
3456.2.p.e.2879.3		16	8.3	odd	2	inner
3456.2.p.e.2879.4		16	8.5	even	2	inner
3456.2.p.e.2879.5		16	4.3	odd	2	inner
3456.2.p.e.2879.6		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.158919 - 0.275255i) q^{5} +(2.93038 - 1.69185i) q^{7} +(-0.642559 + 0.370982i) q^{11} +(-2.59808 - 1.50000i) q^{13} +4.24264i q^{17} -2.57024 q^{19} +(-2.02166 + 3.50162i) q^{23} +(2.44949 + 4.24264i) q^{25} +(4.01229 + 6.94949i) q^{29} +(4.24755 + 2.45233i) q^{31} -1.07547i q^{35} -7.34847i q^{37} +(0.825765 + 0.476756i) q^{41} +(3.50162 + 6.06499i) q^{43} +(5.15627 + 8.93092i) q^{47} +(2.22474 - 3.85337i) q^{49} +5.51399 q^{53} +0.235824i q^{55} +(3.50162 + 2.02166i) q^{59} +(-7.79423 + 4.50000i) q^{61} +(-0.825765 + 0.476756i) q^{65} +(6.36068 - 11.0170i) q^{67} +13.5389 q^{71} -6.44949 q^{73} +(-1.25529 + 2.17423i) q^{77} +(8.29088 - 4.78674i) q^{79} +(2.21650 - 1.27970i) q^{83} +(1.16781 + 0.674235i) q^{85} -14.6349i q^{89} -10.1511 q^{91} +(-0.408459 + 0.707471i) q^{95} +(-5.62372 - 9.74058i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 72 q^{41} + 16 q^{49} - 72 q^{65} - 64 q^{73} + 8 q^{97}+O(q^{100}) \) 16 * q + 72 * q^41 + 16 * q^49 - 72 * q^65 - 64 * q^73 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more