LMFDB - Embedded newform 4212.2.b.i.649.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(649,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 75x^{12} + 877x^{10} - 1707x^{8} + 34683x^{6} + 343849x^{4} + 12936x^{2} + 76176 $ x^16 - 6x^14 + 75x^12 + 877x^10 - 1707x^8 + 34683x^6 + 343849x^4 + 12936*x^2 + 76176
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.11
Root			$-0.481191 + 0.489116i$ of defining polynomial
Character	$\chi$	$=$	4212.649
Dual form			4212.2.b.i.649.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46208i q^{5} -2.26664i q^{7} +5.62029i q^{11} +(-3.17749 + 1.70399i) q^{13} -4.83263 q^{17} +1.39107i q^{19} +4.74426 q^{23} +2.86233 q^{25} -6.47631 q^{29} -0.875576i q^{31} +3.31401 q^{35} -10.0004i q^{37} +4.44070i q^{41} +0.0153803 q^{43} +1.69435i q^{47} +1.86233 q^{49} -6.38795 q^{53} -8.21730 q^{55} +0.514758i q^{59} -11.2327 q^{61} +(-2.49136 - 4.64574i) q^{65} -5.94036i q^{67} -5.44248i q^{71} -15.4092i q^{73} +12.7392 q^{77} -14.7253 q^{79} -16.5741i q^{83} -7.06568i q^{85} -2.69189i q^{89} +(3.86233 + 7.20223i) q^{91} -2.03385 q^{95} +2.78213i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{13} - 24 q^{25} - 24 q^{43} - 40 q^{49} + 36 q^{55} + 12 q^{61} - 16 q^{79} - 8 q^{91}+O(q^{100}) $ 16 * q - 2 * q^13 - 24 * q^25 - 24 * q^43 - 40 * q^49 + 36 * q^55 + 12 * q^61 - 16 * q^79 - 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.46208i			0.653861i	0.945048	+	0.326931i	$0.106014\pi$
$5$			1.46208i			0.653861i	−0.945048	+	0.326931i	$0.893986\pi$
$6$		0			0
$7$		−	2.26664i		−	0.856711i	−0.903610	−	0.428355i	$-0.859093\pi$
$7$		−	2.26664i		−	0.856711i	0.903610	−	0.428355i	$-0.140907\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.62029i			1.69458i	0.531130	+	0.847291i	$0.321768\pi$
$11$			5.62029i			1.69458i	−0.531130	+	0.847291i	$0.678232\pi$
$12$		0			0
$13$	−3.17749	+	1.70399i	−0.881277	+	0.472600i
$13$	−3.17749	+	1.70399i	−0.881277	+	0.472600i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.83263			−1.17208			−0.586042	−	0.810280i	$-0.699315\pi$
$17$	−4.83263			−1.17208			−0.586042	+	0.810280i	$0.699315\pi$
$18$		0			0
$19$			1.39107i			0.319133i	0.987187	+	0.159566i	$0.0510096\pi$
$19$			1.39107i			0.319133i	−0.987187	+	0.159566i	$0.948990\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.74426			0.989247			0.494624	−	0.869107i	$-0.335306\pi$
$23$	4.74426			0.989247			0.494624	+	0.869107i	$0.335306\pi$
$24$		0			0
$25$	2.86233			0.572465
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.47631			−1.20262			−0.601311	−	0.799015i	$-0.705355\pi$
$29$	−6.47631			−1.20262			−0.601311	+	0.799015i	$0.705355\pi$
$30$		0			0
$31$		−	0.875576i		−	0.157258i	−0.996904	−	0.0786291i	$-0.974946\pi$
$31$		−	0.875576i		−	0.157258i	0.996904	−	0.0786291i	$-0.0250543\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.31401			0.560170
$36$		0			0
$37$		−	10.0004i		−	1.64405i	−0.569450	−	0.822026i	$-0.692844\pi$
$37$		−	10.0004i		−	1.64405i	0.569450	−	0.822026i	$-0.307156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.44070i			0.693521i	0.937954	+	0.346760i	$0.112718\pi$
$41$			4.44070i			0.693521i	−0.937954	+	0.346760i	$0.887282\pi$
$42$		0			0
$43$	0.0153803			0.00234548			0.00117274	−	0.999999i	$-0.499627\pi$
$43$	0.0153803			0.00234548			0.00117274	+	0.999999i	$0.499627\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.69435i			0.247146i	0.992335	+	0.123573i	$0.0394353\pi$
$47$			1.69435i			0.247146i	−0.992335	+	0.123573i	$0.960565\pi$
$48$		0			0
$49$	1.86233			0.266047
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.38795			−0.877452			−0.438726	−	0.898621i	$-0.644570\pi$
$53$	−6.38795			−0.877452			−0.438726	+	0.898621i	$0.644570\pi$
$54$		0			0
$55$	−8.21730			−1.10802
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.514758i			0.0670157i	0.999438	+	0.0335079i	$0.0106679\pi$
$59$			0.514758i			0.0670157i	−0.999438	+	0.0335079i	$0.989332\pi$
$60$		0			0
$61$	−11.2327			−1.43820			−0.719099	−	0.694908i	$-0.755445\pi$
$61$	−11.2327			−1.43820			−0.719099	+	0.694908i	$0.755445\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.49136	−	4.64574i	−0.309015	−	0.576233i
$66$		0			0
$67$		−	5.94036i		−	0.725731i	−0.931842	−	0.362866i	$-0.881798\pi$
$67$		−	5.94036i		−	0.725731i	0.931842	−	0.362866i	$-0.118202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.44248i		−	0.645904i	−0.946415	−	0.322952i	$-0.895325\pi$
$71$		−	5.44248i		−	0.645904i	0.946415	−	0.322952i	$-0.104675\pi$
$72$		0			0
$73$		−	15.4092i		−	1.80352i	−0.432242	−	0.901758i	$-0.642277\pi$
$73$		−	15.4092i		−	1.80352i	0.432242	−	0.901758i	$-0.357723\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.7392			1.45177
$78$		0			0
$79$	−14.7253			−1.65673			−0.828365	−	0.560189i	$-0.810729\pi$
$79$	−14.7253			−1.65673			−0.828365	+	0.560189i	$0.810729\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.5741i		−	1.81925i	−0.415433	−	0.909624i	$-0.636370\pi$
$83$		−	16.5741i		−	1.81925i	0.415433	−	0.909624i	$-0.363630\pi$
$84$		0			0
$85$		−	7.06568i		−	0.766381i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.69189i		−	0.285339i	−0.989770	−	0.142670i	$-0.954431\pi$
$89$		−	2.69189i		−	0.285339i	0.989770	−	0.142670i	$-0.0455687\pi$
$90$		0			0
$91$	3.86233	+	7.20223i	0.404882	+	0.754999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.03385			−0.208669
$96$		0			0
$97$			2.78213i			0.282483i	0.989975	+	0.141241i	$0.0451094\pi$
$97$			2.78213i			0.282483i	−0.989975	+	0.141241i	$0.954891\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.67753			0.365928			0.182964	−	0.983120i	$-0.441431\pi$
$101$	3.67753			0.365928			0.182964	+	0.983120i	$0.441431\pi$
$102$		0			0
$103$	−6.21730			−0.612609			−0.306305	−	0.951934i	$-0.599093\pi$
$103$	−6.21730			−0.612609			−0.306305	+	0.951934i	$0.599093\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.63141			−0.737756			−0.368878	−	0.929478i	$-0.620258\pi$
$107$	−7.63141			−0.737756			−0.368878	+	0.929478i	$0.620258\pi$
$108$		0			0
$109$			16.0613i			1.53839i	0.639013	+	0.769196i	$0.279343\pi$
$109$			16.0613i			1.53839i	−0.639013	+	0.769196i	$0.720657\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.1077			1.32714			0.663571	−	0.748113i	$-0.269040\pi$
$113$	14.1077			1.32714			0.663571	+	0.748113i	$0.269040\pi$
$114$		0			0
$115$			6.93648i			0.646830i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.9538i			1.00414i
$120$		0			0
$121$	−20.5877			−1.87161
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.4953i			1.02817i
$126$		0			0
$127$	−16.7100			−1.48277			−0.741384	−	0.671081i	$-0.765830\pi$
$127$	−16.7100			−1.48277			−0.741384	+	0.671081i	$0.765830\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.0955			0.969419			0.484710	−	0.874675i	$-0.338925\pi$
$131$	11.0955			0.969419			0.484710	+	0.874675i	$0.338925\pi$
$132$		0			0
$133$	3.15305			0.273404
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	15.7357i		−	1.34440i	−0.740372	−	0.672198i	$-0.765350\pi$
$137$		−	15.7357i		−	1.34440i	0.740372	−	0.672198i	$-0.234650\pi$
$138$		0			0
$139$	−8.20192			−0.695678			−0.347839	−	0.937554i	$-0.613084\pi$
$139$	−8.20192			−0.695678			−0.347839	+	0.937554i	$0.613084\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.57689	−	17.8584i	−0.800860	−	1.49340i
$144$		0			0
$145$		−	9.46888i		−	0.786348i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			19.6030i			1.60594i	0.596021	+	0.802969i	$0.296748\pi$
$149$			19.6030i			1.60594i	−0.596021	+	0.802969i	$0.703252\pi$
$150$		0			0
$151$		−	1.61460i		−	0.131394i	−0.997840	−	0.0656971i	$-0.979073\pi$
$151$		−	1.61460i		−	0.131394i	0.997840	−	0.0656971i	$-0.0209271\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.28016			0.102825
$156$		0			0
$157$	11.2327			0.896466			0.448233	−	0.893917i	$-0.352054\pi$
$157$	11.2327			0.896466			0.448233	+	0.893917i	$0.352054\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	10.7536i		−	0.847499i
$162$		0			0
$163$			10.4027i			0.814801i	0.913250	+	0.407400i	$0.133565\pi$
$163$			10.4027i			0.814801i	−0.913250	+	0.407400i	$0.866435\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			16.6831i			1.29097i	0.763771	+	0.645487i	$0.223346\pi$
$167$			16.6831i			1.29097i	−0.763771	+	0.645487i	$0.776654\pi$
$168$		0			0
$169$	7.19287	−	10.8288i	0.553298	−	0.832984i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.4935			−0.797808			−0.398904	−	0.916993i	$-0.630609\pi$
$173$	−10.4935			−0.797808			−0.398904	+	0.916993i	$0.630609\pi$
$174$		0			0
$175$		−	6.48788i		−	0.490437i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.6969			−1.32273			−0.661364	−	0.750065i	$-0.730022\pi$
$179$	−17.6969			−1.32273			−0.661364	+	0.750065i	$0.730022\pi$
$180$		0			0
$181$	−5.35498			−0.398032			−0.199016	−	0.979996i	$-0.563775\pi$
$181$	−5.35498			−0.398032			−0.199016	+	0.979996i	$0.563775\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.6213			1.07498
$186$		0			0
$187$		−	27.1608i		−	1.98619i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.38510			0.606724			0.303362	−	0.952875i	$-0.401891\pi$
$191$	8.38510			0.606724			0.303362	+	0.952875i	$0.401891\pi$
$192$		0			0
$193$			23.2562i			1.67402i	0.547191	+	0.837008i	$0.315697\pi$
$193$			23.2562i			1.67402i	−0.547191	+	0.837008i	$0.684303\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.63988i			0.116837i	0.998292	+	0.0584184i	$0.0186058\pi$
$197$			1.63988i			0.116837i	−0.998292	+	0.0584184i	$0.981394\pi$
$198$		0			0
$199$	−20.7253			−1.46918			−0.734590	−	0.678511i	$-0.762625\pi$
$199$	−20.7253			−1.46918			−0.734590	+	0.678511i	$0.762625\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			14.6795i			1.03030i
$204$		0			0
$205$	−6.49265			−0.453466
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.81820			−0.540796
$210$		0			0
$211$	−0.508031			−0.0349743			−0.0174871	−	0.999847i	$-0.505567\pi$
$211$	−0.508031			−0.0349743			−0.0174871	+	0.999847i	$0.505567\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.0224872i			0.00153362i
$216$		0			0
$217$	−1.98462			−0.134725
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15.3556	−	8.23473i	1.03293	−	0.553928i
$222$		0			0
$223$			8.85905i			0.593246i	0.954995	+	0.296623i	$0.0958605\pi$
$223$			8.85905i			0.593246i	−0.954995	+	0.296623i	$0.904140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			9.59645i			0.636939i	0.947933	+	0.318469i	$0.103169\pi$
$227$			9.59645i			0.636939i	−0.947933	+	0.318469i	$0.896831\pi$
$228$		0			0
$229$		−	5.88214i		−	0.388703i	−0.980932	−	0.194351i	$-0.937740\pi$
$229$		−	5.88214i		−	0.388703i	0.980932	−	0.194351i	$-0.0622602\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.76590			0.246712			0.123356	−	0.992362i	$-0.460634\pi$
$233$	3.76590			0.246712			0.123356	+	0.992362i	$0.460634\pi$
$234$		0			0
$235$	−2.47727			−0.161599
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	21.4794i		−	1.38939i	−0.719307	−	0.694693i	$-0.755540\pi$
$239$		−	21.4794i		−	1.38939i	0.719307	−	0.694693i	$-0.244460\pi$
$240$		0			0
$241$		−	26.5349i		−	1.70927i	−0.519233	−	0.854633i	$-0.673782\pi$
$241$		−	26.5349i		−	1.70927i	0.519233	−	0.854633i	$-0.326218\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			2.72287i			0.173958i
$246$		0			0
$247$	−2.37036	−	4.42010i	−0.150822	−	0.281244i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.5330			−0.917313			−0.458657	−	0.888614i	$-0.651669\pi$
$251$	−14.5330			−0.917313			−0.458657	+	0.888614i	$0.651669\pi$
$252$		0			0
$253$			26.6641i			1.67636i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.878755			−0.0548152			−0.0274076	−	0.999624i	$-0.508725\pi$
$257$	−0.878755			−0.0548152			−0.0274076	+	0.999624i	$0.508725\pi$
$258$		0			0
$259$	−22.6673			−1.40848
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.7774			−1.15786			−0.578932	−	0.815376i	$-0.696530\pi$
$263$	−18.7774			−1.15786			−0.578932	+	0.815376i	$0.696530\pi$
$264$		0			0
$265$		−	9.33968i		−	0.573732i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.8909			−0.785972			−0.392986	−	0.919544i	$-0.628558\pi$
$269$	−12.8909			−0.785972			−0.392986	+	0.919544i	$0.628558\pi$
$270$		0			0
$271$		−	32.7411i		−	1.98888i	−0.105318	−	0.994439i	$-0.533586\pi$
$271$		−	32.7411i		−	1.98888i	0.105318	−	0.994439i	$-0.466414\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			16.0871i			0.970089i
$276$		0			0
$277$	−0.660403			−0.0396798			−0.0198399	−	0.999803i	$-0.506316\pi$
$277$	−0.660403			−0.0396798			−0.0198399	+	0.999803i	$0.506316\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.4679i		−	0.803429i	−0.915765	−	0.401715i	$-0.868414\pi$
$281$		−	13.4679i		−	0.803429i	0.915765	−	0.401715i	$-0.131586\pi$
$282$		0			0
$283$	−24.4346			−1.45249			−0.726243	−	0.687438i	$-0.758735\pi$
$283$	−24.4346			−1.45249			−0.726243	+	0.687438i	$0.758735\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0655			0.594147
$288$		0			0
$289$	6.35430			0.373782
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.6542i		−	0.797688i	−0.917019	−	0.398844i	$-0.869411\pi$
$293$		−	13.6542i		−	0.797688i	0.917019	−	0.398844i	$-0.130589\pi$
$294$		0			0
$295$	−0.752616			−0.0438190
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.0748	+	8.08415i	−0.871801	+	0.467519i
$300$		0			0
$301$		−	0.0348617i		−	0.00200940i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	16.4231i		−	0.940382i
$306$		0			0
$307$		−	28.5358i		−	1.62863i	−0.580426	−	0.814313i	$-0.697114\pi$
$307$		−	28.5358i		−	1.62863i	0.580426	−	0.814313i	$-0.302886\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.9865			0.849805			0.424903	−	0.905239i	$-0.360308\pi$
$311$	14.9865			0.849805			0.424903	+	0.905239i	$0.360308\pi$
$312$		0			0
$313$	1.00000			0.0565233			0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000			0.0565233			0.0282617	+	0.999601i	$0.491003\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.03062i		−	0.226382i	−0.993573	−	0.113191i	$-0.963893\pi$
$317$		−	4.03062i		−	0.226382i	0.993573	−	0.113191i	$-0.0361073\pi$
$318$		0			0
$319$		−	36.3988i		−	2.03794i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6.72251i		−	0.374050i
$324$		0			0
$325$	−9.09501	+	4.87736i	−0.504500	+	0.270547i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.84048			0.211733
$330$		0			0
$331$			8.41453i			0.462504i	0.972894	+	0.231252i	$0.0742823\pi$
$331$			8.41453i			0.462504i	−0.972894	+	0.231252i	$0.925718\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.68528			0.474527
$336$		0			0
$337$	9.74003			0.530573			0.265287	−	0.964170i	$-0.414533\pi$
$337$	9.74003			0.530573			0.265287	+	0.964170i	$0.414533\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.92099			0.266487
$342$		0			0
$343$		−	20.0877i		−	1.08464i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.46410			−0.185963			−0.0929814	−	0.995668i	$-0.529640\pi$
$347$	−3.46410			−0.185963			−0.0929814	+	0.995668i	$0.529640\pi$
$348$		0			0
$349$			25.9091i			1.38688i	0.720513	+	0.693441i	$0.243906\pi$
$349$			25.9091i			1.38688i	−0.720513	+	0.693441i	$0.756094\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	12.4704i		−	0.663732i	−0.943327	−	0.331866i	$-0.892322\pi$
$353$		−	12.4704i		−	0.663732i	0.943327	−	0.331866i	$-0.107678\pi$
$354$		0			0
$355$	7.95734			0.422332
$356$		0			0
$357$		0			0
$358$		0			0
$359$			33.0794i			1.74586i	0.487842	+	0.872932i	$0.337784\pi$
$359$			33.0794i			1.74586i	−0.487842	+	0.872932i	$0.662216\pi$
$360$		0			0
$361$	17.0649			0.898154
$362$		0			0
$363$		0			0
$364$		0			0
$365$	22.5295			1.17925
$366$		0			0
$367$	−3.00068			−0.156634			−0.0783171	−	0.996928i	$-0.524955\pi$
$367$	−3.00068			−0.156634			−0.0783171	+	0.996928i	$0.524955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			14.4792i			0.751723i
$372$		0			0
$373$	9.55690			0.494838			0.247419	−	0.968909i	$-0.420418\pi$
$373$	9.55690			0.494838			0.247419	+	0.968909i	$0.420418\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.5784	−	11.0355i	1.05984	−	0.568359i
$378$		0			0
$379$			13.7713i			0.707383i	0.935362	+	0.353691i	$0.115074\pi$
$379$			13.7713i			0.707383i	−0.935362	+	0.353691i	$0.884926\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	10.8348i		−	0.553630i	−0.960923	−	0.276815i	$-0.910721\pi$
$383$		−	10.8348i		−	0.553630i	0.960923	−	0.276815i	$-0.0892789\pi$
$384$		0			0
$385$			18.6257i			0.949254i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−34.4138			−1.74485			−0.872425	−	0.488749i	$-0.837453\pi$
$389$	−34.4138			−1.74485			−0.872425	+	0.488749i	$0.837453\pi$
$390$		0			0
$391$	−22.9273			−1.15948
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	21.5296i		−	1.08327i
$396$		0			0
$397$		−	10.3445i		−	0.519173i	−0.965720	−	0.259587i	$-0.916414\pi$
$397$		−	10.3445i		−	0.519173i	0.965720	−	0.259587i	$-0.0835863\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.0652i			0.502634i	0.967905	+	0.251317i	$0.0808637\pi$
$401$			10.0652i			0.502634i	−0.967905	+	0.251317i	$0.919136\pi$
$402$		0			0
$403$	1.49197	+	2.78213i	0.0743203	+	0.138588i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	56.2050			2.78598
$408$		0			0
$409$		−	22.0365i		−	1.08964i	−0.838555	−	0.544818i	$-0.816599\pi$
$409$		−	22.0365i		−	1.08964i	0.838555	−	0.544818i	$-0.183401\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.16677			0.0574131
$414$		0			0
$415$	24.2327			1.18954
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.35507			0.359319			0.179659	−	0.983729i	$-0.442500\pi$
$419$	7.35507			0.359319			0.179659	+	0.983729i	$0.442500\pi$
$420$		0			0
$421$		−	9.50374i		−	0.463184i	−0.972813	−	0.231592i	$-0.925607\pi$
$421$		−	9.50374i		−	0.463184i	0.972813	−	0.231592i	$-0.0743934\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.8326			−0.670978
$426$		0			0
$427$			25.4605i			1.23212i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.0610i			0.484621i	0.970199	+	0.242310i	$0.0779052\pi$
$431$			10.0610i			0.484621i	−0.970199	+	0.242310i	$0.922095\pi$
$432$		0			0
$433$	5.74072			0.275881			0.137941	−	0.990441i	$-0.455952\pi$
$433$	5.74072			0.275881			0.137941	+	0.990441i	$0.455952\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.59959i			0.315701i
$438$		0			0
$439$	−19.7093			−0.940672			−0.470336	−	0.882487i	$-0.655867\pi$
$439$	−19.7093			−0.940672			−0.470336	+	0.882487i	$0.655867\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.6875			1.03040			0.515201	−	0.857069i	$-0.327717\pi$
$443$	21.6875			1.03040			0.515201	+	0.857069i	$0.327717\pi$
$444$		0			0
$445$	3.93575			0.186572
$446$		0			0
$447$		0			0
$448$		0			0
$449$			10.1112i			0.477177i	0.971121	+	0.238589i	$0.0766848\pi$
$449$			10.1112i			0.477177i	−0.971121	+	0.238589i	$0.923315\pi$
$450$		0			0
$451$	−24.9580			−1.17523
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.5302	+	5.64702i	−0.493665	+	0.264737i
$456$		0			0
$457$			3.93947i			0.184281i	0.995746	+	0.0921403i	$0.0293708\pi$
$457$			3.93947i			0.184281i	−0.995746	+	0.0921403i	$0.970629\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			34.3781i			1.60115i	0.599235	+	0.800573i	$0.295472\pi$
$461$			34.3781i			1.60115i	−0.599235	+	0.800573i	$0.704528\pi$
$462$		0			0
$463$			9.39055i			0.436416i	0.975902	+	0.218208i	$0.0700211\pi$
$463$			9.39055i			0.436416i	−0.975902	+	0.218208i	$0.929979\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.6179			−1.04663			−0.523315	−	0.852139i	$-0.675305\pi$
$467$	−22.6179			−1.04663			−0.523315	+	0.852139i	$0.675305\pi$
$468$		0			0
$469$	−13.4647			−0.621742
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.0864419i			0.00397460i
$474$		0			0
$475$			3.98169i			0.182692i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.10375i		−	0.187505i	−0.995596	−	0.0937525i	$-0.970114\pi$
$479$		−	4.10375i		−	0.187505i	0.995596	−	0.0937525i	$-0.0298862\pi$
$480$		0			0
$481$	17.0405	+	31.7761i	0.776980	+	1.44886i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.06770			−0.184705
$486$		0			0
$487$		−	2.19567i		−	0.0994955i	−0.998762	−	0.0497478i	$-0.984158\pi$
$487$		−	2.19567i		−	0.0994955i	0.998762	−	0.0497478i	$-0.0158417\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.9519			−1.12606			−0.563032	−	0.826435i	$-0.690365\pi$
$491$	−24.9519			−1.12606			−0.563032	+	0.826435i	$0.690365\pi$
$492$		0			0
$493$	31.2976			1.40957
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.3362			−0.553353
$498$		0			0
$499$			8.13604i			0.364219i	0.983278	+	0.182110i	$0.0582926\pi$
$499$			8.13604i			0.364219i	−0.983278	+	0.182110i	$0.941707\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.2738			−0.814791			−0.407395	−	0.913252i	$-0.633563\pi$
$503$	−18.2738			−0.814791			−0.407395	+	0.913252i	$0.633563\pi$
$504$		0			0
$505$			5.37684i			0.239266i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.4135i			0.594541i	0.954793	+	0.297271i	$0.0960764\pi$
$509$			13.4135i			0.594541i	−0.954793	+	0.297271i	$0.903924\pi$
$510$		0			0
$511$	−34.9273			−1.54509
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	9.09019i		−	0.400561i
$516$		0			0
$517$	−9.52273			−0.418809
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.540255			−0.0236690			−0.0118345	−	0.999930i	$-0.503767\pi$
$521$	−0.540255			−0.0236690			−0.0118345	+	0.999930i	$0.503767\pi$
$522$		0			0
$523$	39.4507			1.72506			0.862529	−	0.506008i	$-0.168879\pi$
$523$	39.4507			1.72506			0.862529	+	0.506008i	$0.168879\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.23134i			0.184320i
$528$		0			0
$529$	−0.491969			−0.0213900
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.56689	−	14.1103i	−0.327758	−	0.611184i
$534$		0			0
$535$		−	11.1577i		−	0.482390i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			10.4668i			0.450838i
$540$		0			0
$541$			15.5618i			0.669054i	0.942386	+	0.334527i	$0.108577\pi$
$541$			15.5618i			0.669054i	−0.942386	+	0.334527i	$0.891423\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−23.4829			−1.00590
$546$		0			0
$547$	33.4507			1.43025			0.715124	−	0.698998i	$-0.246370\pi$
$547$	33.4507			1.43025			0.715124	+	0.698998i	$0.246370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.00899i		−	0.383796i
$552$		0			0
$553$			33.3771i			1.41934i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	10.8994i		−	0.461822i	−0.972975	−	0.230911i	$-0.925829\pi$
$557$		−	10.8994i		−	0.461822i	0.972975	−	0.230911i	$-0.0741706\pi$
$558$		0			0
$559$	−0.0488708	+	0.0262078i	−0.00206701	+	0.00110847i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.0043			−0.843080			−0.421540	−	0.906810i	$-0.638510\pi$
$563$	−20.0043			−0.843080			−0.421540	+	0.906810i	$0.638510\pi$
$564$		0			0
$565$			20.6266i			0.867767i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.63205			−0.194186			−0.0970929	−	0.995275i	$-0.530954\pi$
$569$	−4.63205			−0.194186			−0.0970929	+	0.995275i	$0.530954\pi$
$570$		0			0
$571$	22.4192			0.938216			0.469108	−	0.883141i	$-0.344576\pi$
$571$	22.4192			0.938216			0.469108	+	0.883141i	$0.344576\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.5796			0.566310
$576$		0			0
$577$			16.6609i			0.693603i	0.937939	+	0.346801i	$0.112732\pi$
$577$			16.6609i			0.693603i	−0.937939	+	0.346801i	$0.887268\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−37.5677			−1.55857
$582$		0			0
$583$		−	35.9021i		−	1.48691i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			34.5650i			1.42665i	0.700834	+	0.713324i	$0.252811\pi$
$587$			34.5650i			1.42665i	−0.700834	+	0.713324i	$0.747189\pi$
$588$		0			0
$589$	1.21799			0.0501862
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.62097i		−	0.148695i	−0.997232	−	0.0743476i	$-0.976313\pi$
$593$		−	3.62097i		−	0.148695i	0.997232	−	0.0743476i	$-0.0236874\pi$
$594$		0			0
$595$	−16.0154			−0.656567
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−44.6588			−1.82471			−0.912355	−	0.409400i	$-0.865738\pi$
$599$	−44.6588			−1.82471			−0.912355	+	0.409400i	$0.865738\pi$
$600$		0			0
$601$	46.1292			1.88165			0.940824	−	0.338895i	$-0.110053\pi$
$601$	46.1292			1.88165			0.940824	+	0.338895i	$0.110053\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	30.1008i		−	1.22377i
$606$		0			0
$607$	−19.5087			−0.791834			−0.395917	−	0.918286i	$-0.629573\pi$
$607$	−19.5087			−0.791834			−0.395917	+	0.918286i	$0.629573\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.88714	−	5.38377i	−0.116801	−	0.217804i
$612$		0			0
$613$			0.907597i			0.0366575i	0.999832	+	0.0183287i	$0.00583455\pi$
$613$			0.907597i			0.0366575i	−0.999832	+	0.0183287i	$0.994165\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			41.3414i			1.66434i	0.554520	+	0.832170i	$0.312902\pi$
$617$			41.3414i			1.66434i	−0.554520	+	0.832170i	$0.687098\pi$
$618$		0			0
$619$		−	45.2738i		−	1.81971i	−0.414928	−	0.909854i	$-0.636193\pi$
$619$		−	45.2738i		−	1.81971i	0.414928	−	0.909854i	$-0.363807\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.10155			−0.244453
$624$		0			0
$625$	−2.49545			−0.0998180
$626$		0			0
$627$		0			0
$628$		0			0
$629$			48.3281i			1.92697i
$630$		0			0
$631$			16.3980i			0.652794i	0.945233	+	0.326397i	$0.105835\pi$
$631$			16.3980i			0.652794i	−0.945233	+	0.326397i	$0.894165\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	24.4313i		−	0.969525i
$636$		0			0
$637$	−5.91752	+	3.17338i	−0.234461	+	0.125734i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.0146			1.10651			0.553255	−	0.833012i	$-0.313385\pi$
$641$	28.0146			1.10651			0.553255	+	0.833012i	$0.313385\pi$
$642$		0			0
$643$			29.3926i			1.15913i	0.814926	+	0.579565i	$0.196777\pi$
$643$			29.3926i			1.15913i	−0.814926	+	0.579565i	$0.803223\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	26.7852			1.05303			0.526517	−	0.850164i	$-0.323498\pi$
$647$	26.7852			1.05303			0.526517	+	0.850164i	$0.323498\pi$
$648$		0			0
$649$	−2.89309			−0.113564
$650$		0			0
$651$		0			0
$652$		0			0
$653$	48.1734			1.88517			0.942585	−	0.333965i	$-0.108387\pi$
$653$	48.1734			1.88517			0.942585	+	0.333965i	$0.108387\pi$
$654$		0			0
$655$			16.2225i			0.633866i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	41.0152			1.59772			0.798862	−	0.601514i	$-0.205436\pi$
$659$	41.0152			1.59772			0.798862	+	0.601514i	$0.205436\pi$
$660$		0			0
$661$		−	2.46142i		−	0.0957383i	−0.998854	−	0.0478692i	$-0.984757\pi$
$661$		−	2.46142i		−	0.0957383i	0.998854	−	0.0478692i	$-0.0152431\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			4.61001i			0.178769i
$666$		0			0
$667$	−30.7253			−1.18969
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	63.1309i		−	2.43714i
$672$		0			0
$673$	−16.7554			−0.645874			−0.322937	−	0.946421i	$-0.604670\pi$
$673$	−16.7554			−0.645874			−0.322937	+	0.946421i	$0.604670\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.4894			−1.17180			−0.585901	−	0.810383i	$-0.699259\pi$
$677$	−30.4894			−1.17180			−0.585901	+	0.810383i	$0.699259\pi$
$678$		0			0
$679$	6.30611			0.242006
$680$		0			0
$681$		0			0
$682$		0			0
$683$			46.5702i			1.78196i	0.454042	+	0.890980i	$0.349981\pi$
$683$			46.5702i			1.78196i	−0.454042	+	0.890980i	$0.650019\pi$
$684$		0			0
$685$	23.0069			0.879048
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.2976	−	10.8850i	0.773278	−	0.414684i
$690$		0			0
$691$			15.7139i			0.597787i	0.954287	+	0.298893i	$0.0966175\pi$
$691$			15.7139i			0.597787i	−0.954287	+	0.298893i	$0.903383\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11.9919i		−	0.454877i
$696$		0			0
$697$		−	21.4603i		−	0.812865i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.6840			1.00784			0.503920	−	0.863750i	$-0.331891\pi$
$701$	26.6840			1.00784			0.503920	+	0.863750i	$0.331891\pi$
$702$		0			0
$703$	13.9112			0.524671
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.33566i		−	0.313495i
$708$		0			0
$709$			37.8842i			1.42277i	0.702803	+	0.711385i	$0.251932\pi$
$709$			37.8842i			1.42277i	−0.702803	+	0.711385i	$0.748068\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.15396i		−	0.155567i
$714$		0			0
$715$	26.1104	−	14.0022i	0.976473	−	0.523651i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.9259			1.15334			0.576670	−	0.816977i	$-0.304352\pi$
$719$	30.9259			1.15334			0.576670	+	0.816977i	$0.304352\pi$
$720$		0			0
$721$			14.0924i			0.524829i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.5373			−0.688459
$726$		0			0
$727$	21.2474			0.788022			0.394011	−	0.919106i	$-0.371087\pi$
$727$	21.2474			0.788022			0.394011	+	0.919106i	$0.371087\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.0743274			−0.00274910
$732$		0			0
$733$			16.2935i			0.601813i	0.953654	+	0.300907i	$0.0972892\pi$
$733$			16.2935i			0.601813i	−0.953654	+	0.300907i	$0.902711\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	33.3866			1.22981
$738$		0			0
$739$			35.8644i			1.31929i	0.751576	+	0.659647i	$0.229294\pi$
$739$			35.8644i			1.31929i	−0.751576	+	0.659647i	$0.770706\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	34.3369i		−	1.25970i	−0.776717	−	0.629850i	$-0.783116\pi$
$743$		−	34.3369i		−	1.25970i	0.776717	−	0.629850i	$-0.216884\pi$
$744$		0			0
$745$	−28.6611			−1.05006
$746$		0			0
$747$		0			0
$748$		0			0
$749$			17.2977i			0.632043i
$750$		0			0
$751$	14.9580			0.545826			0.272913	−	0.962039i	$-0.412013\pi$
$751$	14.9580			0.545826			0.272913	+	0.962039i	$0.412013\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.36067			0.0859136
$756$		0			0
$757$	31.5723			1.14751			0.573757	−	0.819026i	$-0.305485\pi$
$757$	31.5723			1.14751			0.573757	+	0.819026i	$0.305485\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.9918i			1.08720i	0.839343	+	0.543602i	$0.182940\pi$
$761$			29.9918i			1.08720i	−0.839343	+	0.543602i	$0.817060\pi$
$762$		0			0
$763$	36.4052			1.31796
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.877139	−	1.63564i	−0.0316717	−	0.0590594i
$768$		0			0
$769$		−	26.1064i		−	0.941422i	−0.882288	−	0.470711i	$-0.843997\pi$
$769$		−	26.1064i		−	0.941422i	0.882288	−	0.470711i	$-0.156003\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			10.9858i			0.395133i	0.980290	+	0.197566i	$0.0633038\pi$
$773$			10.9858i			0.395133i	−0.980290	+	0.197566i	$0.936696\pi$
$774$		0			0
$775$		−	2.50619i		−	0.0900249i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.17731			−0.221325
$780$		0			0
$781$	30.5883			1.09454
$782$		0			0
$783$		0			0
$784$		0			0
$785$			16.4231i			0.586164i
$786$		0			0
$787$		−	17.0500i		−	0.607769i	−0.952709	−	0.303884i	$-0.901716\pi$
$787$		−	17.0500i		−	0.607769i	0.952709	−	0.303884i	$-0.0982836\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	31.9772i		−	1.13698i
$792$		0			0
$793$	35.6917	−	19.1403i	1.26745	−	0.679693i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−49.7287			−1.76148			−0.880741	−	0.473599i	$-0.842955\pi$
$797$	−49.7287			−1.76148			−0.880741	+	0.473599i	$0.842955\pi$
$798$		0			0
$799$		−	8.18816i		−	0.289676i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	86.6044			3.05620
$804$		0			0
$805$	15.7225			0.554147
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.8362			1.40057			0.700284	−	0.713864i	$-0.253057\pi$
$809$	39.8362			1.40057			0.700284	+	0.713864i	$0.253057\pi$
$810$		0			0
$811$		−	41.7046i		−	1.46445i	−0.681064	−	0.732224i	$-0.738482\pi$
$811$		−	41.7046i		−	1.46445i	0.681064	−	0.732224i	$-0.261518\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.2095			−0.532767
$816$		0			0
$817$			0.0213951i			0.000748518i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	11.4953i		−	0.401190i	−0.979674	−	0.200595i	$-0.935712\pi$
$821$		−	11.4953i		−	0.401190i	0.979674	−	0.200595i	$-0.0642875\pi$
$822$		0			0
$823$	−19.7093			−0.687022			−0.343511	−	0.939149i	$-0.611616\pi$
$823$	−19.7093			−0.687022			−0.343511	+	0.939149i	$0.611616\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	52.0404i		−	1.80962i	−0.425812	−	0.904812i	$-0.640011\pi$
$827$		−	52.0404i		−	1.80962i	0.425812	−	0.904812i	$-0.359989\pi$
$828$		0			0
$829$	−13.0915			−0.454688			−0.227344	−	0.973815i	$-0.573004\pi$
$829$	−13.0915			−0.454688			−0.227344	+	0.973815i	$0.573004\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.99993			−0.311829
$834$		0			0
$835$	−24.3919			−0.844118
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1.58542i		−	0.0547348i	−0.999625	−	0.0273674i	$-0.991288\pi$
$839$		−	1.58542i		−	0.0547348i	0.999625	−	0.0273674i	$-0.00871239\pi$
$840$		0			0
$841$	12.9426			0.446298
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.8325	+	10.5165i	0.544656	+	0.361780i
$846$		0			0
$847$			46.6649i			1.60342i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	47.4444i		−	1.62637i
$852$		0			0
$853$		−	8.61667i		−	0.295029i	−0.989060	−	0.147515i	$-0.952873\pi$
$853$		−	8.61667i		−	0.295029i	0.989060	−	0.147515i	$-0.0471273\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−34.2158			−1.16879			−0.584394	−	0.811470i	$-0.698668\pi$
$857$	−34.2158			−1.16879			−0.584394	+	0.811470i	$0.698668\pi$
$858$		0			0
$859$	−0.0888022			−0.00302989			−0.00151495	−	0.999999i	$-0.500482\pi$
$859$	−0.0888022			−0.00302989			−0.00151495	+	0.999999i	$0.500482\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	16.2874i		−	0.554430i	−0.960808	−	0.277215i	$-0.910589\pi$
$863$		−	16.2874i		−	0.554430i	0.960808	−	0.277215i	$-0.0894114\pi$
$864$		0			0
$865$		−	15.3424i		−	0.521656i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	82.7607i		−	2.80746i
$870$		0			0
$871$	10.1223	+	18.8754i	0.342981	+	0.639570i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	26.0558			0.880848
$876$		0			0
$877$		−	42.7226i		−	1.44264i	−0.692602	−	0.721319i	$-0.743536\pi$
$877$		−	42.7226i		−	1.44264i	0.692602	−	0.721319i	$-0.256464\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.2625			−0.918498			−0.459249	−	0.888308i	$-0.651881\pi$
$881$	−27.2625			−0.918498			−0.459249	+	0.888308i	$0.651881\pi$
$882$		0			0
$883$	−14.1593			−0.476497			−0.238249	−	0.971204i	$-0.576573\pi$
$883$	−14.1593			−0.476497			−0.238249	+	0.971204i	$0.576573\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.1860			−1.11428			−0.557138	−	0.830420i	$-0.688101\pi$
$887$	−33.1860			−1.11428			−0.557138	+	0.830420i	$0.688101\pi$
$888$		0			0
$889$			37.8755i			1.27030i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.35695			−0.0788724
$894$		0			0
$895$		−	25.8742i		−	0.864880i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			5.67051i			0.189122i
$900$		0			0
$901$	30.8706			1.02845
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7.82940i		−	0.260258i
$906$		0			0
$907$	22.6519			0.752144			0.376072	−	0.926590i	$-0.377274\pi$
$907$	22.6519			0.752144			0.376072	+	0.926590i	$0.377274\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−20.3077			−0.672824			−0.336412	−	0.941715i	$-0.609213\pi$
$911$	−20.3077			−0.672824			−0.336412	+	0.941715i	$0.609213\pi$
$912$		0			0
$913$	93.1515			3.08286
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	25.1496i		−	0.830512i
$918$		0			0
$919$	−3.98462			−0.131440			−0.0657202	−	0.997838i	$-0.520934\pi$
$919$	−3.98462			−0.131440			−0.0657202	+	0.997838i	$0.520934\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.27391	+	17.2934i	0.305255	+	0.569220i
$924$		0			0
$925$		−	28.6244i		−	0.941163i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	0.819733i		−	0.0268946i	−0.999910	−	0.0134473i	$-0.995719\pi$
$929$		−	0.819733i		−	0.0268946i	0.999910	−	0.0134473i	$-0.00428053\pi$
$930$		0			0
$931$			2.59062i			0.0849042i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	39.7112			1.29869
$936$		0			0
$937$	29.8043			0.973664			0.486832	−	0.873496i	$-0.338152\pi$
$937$	29.8043			0.973664			0.486832	+	0.873496i	$0.338152\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			29.8604i			0.973422i	0.873563	+	0.486711i	$0.161804\pi$
$941$			29.8604i			0.973422i	−0.873563	+	0.486711i	$0.838196\pi$
$942$		0			0
$943$			21.0678i			0.686063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			13.0129i			0.422862i	0.977393	+	0.211431i	$0.0678123\pi$
$947$			13.0129i			0.422862i	−0.977393	+	0.211431i	$0.932188\pi$
$948$		0			0
$949$	26.2571	+	48.9627i	0.852342	+	1.58940i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.4857			−0.598809			−0.299405	−	0.954126i	$-0.596788\pi$
$953$	−18.4857			−0.598809			−0.299405	+	0.954126i	$0.596788\pi$
$954$		0			0
$955$			12.2597i			0.396713i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−35.6673			−1.15176
$960$		0			0
$961$	30.2334			0.975270
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34.0023			−1.09457
$966$		0			0
$967$			49.5487i			1.59338i	0.604389	+	0.796690i	$0.293417\pi$
$967$			49.5487i			1.59338i	−0.604389	+	0.796690i	$0.706583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.2752			−1.35668			−0.678338	−	0.734750i	$-0.737300\pi$
$971$	−42.2752			−1.35668			−0.678338	+	0.734750i	$0.737300\pi$
$972$		0			0
$973$			18.5908i			0.595995i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.3687i		−	0.651651i	−0.945430	−	0.325826i	$-0.894358\pi$
$977$		−	20.3687i		−	0.651651i	0.945430	−	0.325826i	$-0.105642\pi$
$978$		0			0
$979$	15.1292			0.483531
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.28003i			0.0408265i	0.999792	+	0.0204132i	$0.00649819\pi$
$983$			1.28003i			0.0408265i	−0.999792	+	0.0204132i	$0.993502\pi$
$984$		0			0
$985$	−2.39764			−0.0763951
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.0729683			0.00232026
$990$		0			0
$991$	−15.4199			−0.489830			−0.244915	−	0.969545i	$-0.578760\pi$
$991$	−15.4199			−0.489830			−0.244915	+	0.969545i	$0.578760\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	30.3021i		−	0.960640i
$996$		0			0
$997$	−8.49197			−0.268943			−0.134472	−	0.990917i	$-0.542934\pi$
$997$	−8.49197			−0.268943			−0.134472	+	0.990917i	$0.542934\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.b.i.649.11	yes	16
3.2	odd	2	inner	4212.2.b.i.649.5	✓	16
13.12	even	2	inner	4212.2.b.i.649.6	yes	16
39.38	odd	2	inner	4212.2.b.i.649.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4212.2.b.i.649.5	✓	16	3.2	odd	2	inner
4212.2.b.i.649.6	yes	16	13.12	even	2	inner
4212.2.b.i.649.11	yes	16	1.1	even	1	trivial
4212.2.b.i.649.12	yes	16	39.38	odd	2	inner

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:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.46208i q^{5} -2.26664i q^{7} +5.62029i q^{11} +(-3.17749 + 1.70399i) q^{13} -4.83263 q^{17} +1.39107i q^{19} +4.74426 q^{23} +2.86233 q^{25} -6.47631 q^{29} -0.875576i q^{31} +3.31401 q^{35} -10.0004i q^{37} +4.44070i q^{41} +0.0153803 q^{43} +1.69435i q^{47} +1.86233 q^{49} -6.38795 q^{53} -8.21730 q^{55} +0.514758i q^{59} -11.2327 q^{61} +(-2.49136 - 4.64574i) q^{65} -5.94036i q^{67} -5.44248i q^{71} -15.4092i q^{73} +12.7392 q^{77} -14.7253 q^{79} -16.5741i q^{83} -7.06568i q^{85} -2.69189i q^{89} +(3.86233 + 7.20223i) q^{91} -2.03385 q^{95} +2.78213i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{13} - 24 q^{25} - 24 q^{43} - 40 q^{49} + 36 q^{55} + 12 q^{61} - 16 q^{79} - 8 q^{91}+O(q^{100}) \) 16 * q - 2 * q^13 - 24 * q^25 - 24 * q^43 - 40 * q^49 + 36 * q^55 + 12 * q^61 - 16 * q^79 - 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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