LMFDB - Embedded newform 1512.2.k.a.377.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(377,1512)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1512.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.0733807856$
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} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 $ x^16 - 24x^14 + 230x^12 - 1052x^10 + 2139x^8 - 1244x^6 + 1134x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.10
Root			$0.713245 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1512.377
Dual form			1512.2.k.a.377.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.465643 q^{5} +(0.656211 + 2.56308i) q^{7} +O(q^{10})$	$q+0.465643 q^{5} +(0.656211 + 2.56308i) q^{7} -5.28908i q^{11} -4.50044i q^{13} -3.15854 q^{17} -1.42649i q^{19} -2.27727i q^{23} -4.78318 q^{25} -6.09560i q^{29} +2.76839i q^{31} +(0.305560 + 1.19348i) q^{35} -0.613036 q^{37} +6.93370 q^{41} +3.08379 q^{43} +9.30643 q^{47} +(-6.13877 + 3.36385i) q^{49} -12.1912i q^{53} -2.46283i q^{55} +3.62968 q^{59} -2.27318i q^{61} -2.09560i q^{65} +13.3613 q^{67} -7.38468i q^{71} +10.8175i q^{73} +(13.5563 - 3.47075i) q^{77} +1.17014 q^{79} -15.0124 q^{83} -1.47075 q^{85} -5.39129 q^{89} +(11.5350 - 2.95324i) q^{91} -0.664236i q^{95} -10.4528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{7}+O(q^{10}) $ 16 * q - 2 * q^7	$ 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) $ 16 * q - 2 * q^7 - 12 * q^25 - 8 * q^37 + 8 * q^43 + 2 * q^49 - 28 * q^67 - 44 * q^79 + 16 * q^85 + 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.465643			0.208242			0.104121	−	0.994565i	$-0.466797\pi$
$5$	0.465643			0.208242			0.104121	+	0.994565i	$0.466797\pi$
$6$		0			0
$7$	0.656211	+	2.56308i	0.248025	+	0.968754i
$7$	0.656211	+	2.56308i	0.248025	+	0.968754i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.28908i		−	1.59472i	−0.603505	−	0.797359i	$-0.706230\pi$
$11$		−	5.28908i		−	1.59472i	0.603505	−	0.797359i	$-0.293770\pi$
$12$		0			0
$13$		−	4.50044i		−	1.24820i	−0.781346	−	0.624098i	$-0.785466\pi$
$13$		−	4.50044i		−	1.24820i	0.781346	−	0.624098i	$-0.214534\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.15854			−0.766059			−0.383029	−	0.923736i	$-0.625119\pi$
$17$	−3.15854			−0.766059			−0.383029	+	0.923736i	$0.625119\pi$
$18$		0			0
$19$		−	1.42649i		−	0.327259i	−0.986522	−	0.163630i	$-0.947680\pi$
$19$		−	1.42649i		−	0.327259i	0.986522	−	0.163630i	$-0.0523202\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.27727i		−	0.474844i	−0.971407	−	0.237422i	$-0.923698\pi$
$23$		−	2.27727i		−	0.474844i	0.971407	−	0.237422i	$-0.0763024\pi$
$24$		0			0
$25$	−4.78318			−0.956635
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.09560i		−	1.13192i	−0.824431	−	0.565962i	$-0.808505\pi$
$29$		−	6.09560i		−	1.13192i	0.824431	−	0.565962i	$-0.191495\pi$
$30$		0			0
$31$			2.76839i			0.497217i	0.968604	+	0.248608i	$0.0799732\pi$
$31$			2.76839i			0.497217i	−0.968604	+	0.248608i	$0.920027\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.305560	+	1.19348i	0.0516491	+	0.201735i
$36$		0			0
$37$	−0.613036			−0.100783			−0.0503913	−	0.998730i	$-0.516047\pi$
$37$	−0.613036			−0.100783			−0.0503913	+	0.998730i	$0.516047\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.93370			1.08286			0.541431	−	0.840745i	$-0.317883\pi$
$41$	6.93370			1.08286			0.541431	+	0.840745i	$0.317883\pi$
$42$		0			0
$43$	3.08379			0.470274			0.235137	−	0.971962i	$-0.424446\pi$
$43$	3.08379			0.470274			0.235137	+	0.971962i	$0.424446\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.30643			1.35748			0.678741	−	0.734377i	$-0.262526\pi$
$47$	9.30643			1.35748			0.678741	+	0.734377i	$0.262526\pi$
$48$		0			0
$49$	−6.13877	+	3.36385i	−0.876968	+	0.480549i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	12.1912i		−	1.67459i	−0.546752	−	0.837295i	$-0.684136\pi$
$53$		−	12.1912i		−	1.67459i	0.546752	−	0.837295i	$-0.315864\pi$
$54$		0			0
$55$		−	2.46283i		−	0.332087i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.62968			0.472545			0.236272	−	0.971687i	$-0.424074\pi$
$59$	3.62968			0.472545			0.236272	+	0.971687i	$0.424074\pi$
$60$		0			0
$61$		−	2.27318i		−	0.291051i	−0.989354	−	0.145526i	$-0.953513\pi$
$61$		−	2.27318i		−	0.291051i	0.989354	−	0.145526i	$-0.0464873\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.09560i		−	0.259927i
$66$		0			0
$67$	13.3613			1.63235			0.816174	−	0.577807i	$-0.196091\pi$
$67$	13.3613			1.63235			0.816174	+	0.577807i	$0.196091\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.38468i		−	0.876400i	−0.898877	−	0.438200i	$-0.855616\pi$
$71$		−	7.38468i		−	0.876400i	0.898877	−	0.438200i	$-0.144384\pi$
$72$		0			0
$73$			10.8175i			1.26609i	0.774113	+	0.633047i	$0.218196\pi$
$73$			10.8175i			1.26609i	−0.774113	+	0.633047i	$0.781804\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.5563	−	3.47075i	1.54489	−	0.395529i
$78$		0			0
$79$	1.17014			0.131651			0.0658255	−	0.997831i	$-0.479032\pi$
$79$	1.17014			0.131651			0.0658255	+	0.997831i	$0.479032\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.0124			−1.64782			−0.823912	−	0.566717i	$-0.808213\pi$
$83$	−15.0124			−1.64782			−0.823912	+	0.566717i	$0.808213\pi$
$84$		0			0
$85$	−1.47075			−0.159526
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.39129			−0.571476			−0.285738	−	0.958308i	$-0.592239\pi$
$89$	−5.39129			−0.571476			−0.285738	+	0.958308i	$0.592239\pi$
$90$		0			0
$91$	11.5350	−	2.95324i	1.20920	−	0.309583i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.664236i		−	0.0681492i
$96$		0			0
$97$		−	10.4528i		−	1.06132i	−0.847583	−	0.530662i	$-0.821943\pi$
$97$		−	10.4528i		−	1.06132i	0.847583	−	0.530662i	$-0.178057\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.0867			−1.00367			−0.501834	−	0.864964i	$-0.667341\pi$
$101$	−10.0867			−1.00367			−0.501834	+	0.864964i	$0.667341\pi$
$102$		0			0
$103$			2.35778i			0.232319i	0.993231	+	0.116159i	$0.0370583\pi$
$103$			2.35778i			0.232319i	−0.993231	+	0.116159i	$0.962942\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.7204i		−	1.42308i	−0.702646	−	0.711539i	$-0.747998\pi$
$107$		−	14.7204i		−	1.42308i	0.702646	−	0.711539i	$-0.252002\pi$
$108$		0			0
$109$	−16.9862			−1.62698			−0.813491	−	0.581578i	$-0.802435\pi$
$109$	−16.9862			−1.62698			−0.813491	+	0.581578i	$0.802435\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.94151i		−	0.653002i	−0.945197	−	0.326501i	$-0.894130\pi$
$113$		−	6.94151i		−	0.653002i	0.945197	−	0.326501i	$-0.105870\pi$
$114$		0			0
$115$		−	1.06040i		−	0.0988825i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.07267	−	8.09560i	−0.190001	−	0.742122i
$120$		0			0
$121$	−16.9744			−1.54312
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.55547			−0.407454
$126$		0			0
$127$	−3.85772			−0.342317			−0.171159	−	0.985243i	$-0.554751\pi$
$127$	−3.85772			−0.342317			−0.171159	+	0.985243i	$0.554751\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.7375			1.63711			0.818553	−	0.574431i	$-0.194776\pi$
$131$	18.7375			1.63711			0.818553	+	0.574431i	$0.194776\pi$
$132$		0			0
$133$	3.65621	−	0.936079i	0.317034	−	0.0811683i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.86953i			0.586903i	0.955974	+	0.293452i	$0.0948040\pi$
$137$			6.86953i			0.586903i	−0.955974	+	0.293452i	$0.905196\pi$
$138$		0			0
$139$			16.6007i			1.40806i	0.710172	+	0.704028i	$0.248617\pi$
$139$			16.6007i			1.40806i	−0.710172	+	0.704028i	$0.751383\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−23.8032			−1.99052
$144$		0			0
$145$		−	2.83838i		−	0.235714i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.22607i			0.264290i	0.991230	+	0.132145i	$0.0421865\pi$
$149$			3.22607i			0.264290i	−0.991230	+	0.132145i	$0.957814\pi$
$150$		0			0
$151$	2.68758			0.218712			0.109356	−	0.994003i	$-0.465121\pi$
$151$	2.68758			0.218712			0.109356	+	0.994003i	$0.465121\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.28908i			0.103541i
$156$		0			0
$157$		−	17.0259i		−	1.35882i	−0.733760	−	0.679409i	$-0.762236\pi$
$157$		−	17.0259i		−	1.35882i	0.733760	−	0.679409i	$-0.237764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.83683	−	1.49437i	0.460007	−	0.117773i
$162$		0			0
$163$	15.9862			1.25213			0.626067	−	0.779769i	$-0.284664\pi$
$163$	15.9862			1.25213			0.626067	+	0.779769i	$0.284664\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	18.6421			1.44257			0.721284	−	0.692640i	$-0.243552\pi$
$167$	18.6421			1.44257			0.721284	+	0.692640i	$0.243552\pi$
$168$		0			0
$169$	−7.25393			−0.557995
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.2322			0.777941			0.388971	−	0.921250i	$-0.372831\pi$
$173$	10.2322			0.777941			0.388971	+	0.921250i	$0.372831\pi$
$174$		0			0
$175$	−3.13877	−	12.2597i	−0.237269	−	0.926744i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			13.6620i			1.02114i	0.859836	+	0.510571i	$0.170566\pi$
$179$			13.6620i			1.02114i	−0.859836	+	0.510571i	$0.829434\pi$
$180$		0			0
$181$			11.3033i			0.840165i	0.907486	+	0.420083i	$0.137999\pi$
$181$			11.3033i			0.840165i	−0.907486	+	0.420083i	$0.862001\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.285456			−0.0209872
$186$		0			0
$187$			16.7058i			1.22165i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.66424i			0.626922i	0.949601	+	0.313461i	$0.101489\pi$
$191$			8.66424i			0.626922i	−0.949601	+	0.313461i	$0.898511\pi$
$192$		0			0
$193$	16.2775			1.17168			0.585842	−	0.810425i	$-0.300764\pi$
$193$	16.2775			1.17168			0.585842	+	0.810425i	$0.300764\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.9066i			1.70328i	0.524130	+	0.851639i	$0.324391\pi$
$197$			23.9066i			1.70328i	−0.524130	+	0.851639i	$0.675609\pi$
$198$		0			0
$199$		−	25.3806i		−	1.79919i	−0.436729	−	0.899593i	$-0.643863\pi$
$199$		−	25.3806i		−	1.79919i	0.436729	−	0.899593i	$-0.356137\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.6235	−	4.00000i	1.09656	−	0.280745i
$204$		0			0
$205$	3.22863			0.225497
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.54482			−0.521886
$210$		0			0
$211$	16.4451			1.13213			0.566065	−	0.824361i	$-0.308465\pi$
$211$	16.4451			1.13213			0.566065	+	0.824361i	$0.308465\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.43595			0.0979307
$216$		0			0
$217$	−7.09560	+	1.81665i	−0.481681	+	0.123322i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			14.2148i			0.956192i
$222$		0			0
$223$			22.8464i			1.52991i	0.644084	+	0.764954i	$0.277239\pi$
$223$			22.8464i			1.52991i	−0.644084	+	0.764954i	$0.722761\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.7190			−1.04331			−0.521653	−	0.853158i	$-0.674684\pi$
$227$	−15.7190			−1.04331			−0.521653	+	0.853158i	$0.674684\pi$
$228$		0			0
$229$		−	1.25145i		−	0.0826983i	−0.999145	−	0.0413492i	$-0.986834\pi$
$229$		−	1.25145i		−	0.0826983i	0.999145	−	0.0413492i	$-0.0131656\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	4.33348			0.282685
$236$		0			0
$237$		0			0
$238$		0			0
$239$			14.9116i			0.964554i	0.876019	+	0.482277i	$0.160190\pi$
$239$			14.9116i			0.964554i	−0.876019	+	0.482277i	$0.839810\pi$
$240$		0			0
$241$			0.966217i			0.0622395i	0.999516	+	0.0311198i	$0.00990733\pi$
$241$			0.966217i			0.0622395i	−0.999516	+	0.0311198i	$0.990093\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.85848	+	1.56635i	−0.182622	+	0.100071i
$246$		0			0
$247$	−6.41983			−0.408484
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.2532			−1.21525			−0.607626	−	0.794224i	$-0.707878\pi$
$251$	−19.2532			−1.21525			−0.607626	+	0.794224i	$0.707878\pi$
$252$		0			0
$253$	−12.0447			−0.757242
$254$		0			0
$255$		0			0
$256$		0			0
$257$	20.5437			1.28148			0.640739	−	0.767758i	$-0.278628\pi$
$257$	20.5437			1.28148			0.640739	+	0.767758i	$0.278628\pi$
$258$		0			0
$259$	−0.402281	−	1.57126i	−0.0249965	−	0.0976334i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7.89031i			0.486537i	0.969959	+	0.243269i	$0.0782197\pi$
$263$			7.89031i			0.486537i	−0.969959	+	0.243269i	$0.921780\pi$
$264$		0			0
$265$		−	5.67675i		−	0.348720i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−31.0278			−1.89180			−0.945900	−	0.324458i	$-0.894818\pi$
$269$	−31.0278			−1.89180			−0.945900	+	0.324458i	$0.894818\pi$
$270$		0			0
$271$		−	0.0605334i		−	0.00367714i	−0.999998	−	0.00183857i	$-0.999415\pi$
$271$		−	0.0605334i		−	0.00367714i	0.999998	−	0.00183857i	$-0.000585236\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			25.2986i			1.52556i
$276$		0			0
$277$	−10.6994			−0.642864			−0.321432	−	0.946933i	$-0.604164\pi$
$277$	−10.6994			−0.642864			−0.321432	+	0.946933i	$0.604164\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.92802i		−	0.472946i	−0.971638	−	0.236473i	$-0.924009\pi$
$281$		−	7.92802i		−	0.472946i	0.971638	−	0.236473i	$-0.0759915\pi$
$282$		0			0
$283$			7.60944i			0.452334i	0.974089	+	0.226167i	$0.0726196\pi$
$283$			7.60944i			0.452334i	−0.974089	+	0.226167i	$0.927380\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.54997	+	17.7716i	0.268576	+	1.04903i
$288$		0			0
$289$	−7.02362			−0.413154
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.655636			−0.0383026			−0.0191513	−	0.999817i	$-0.506096\pi$
$293$	−0.655636			−0.0383026			−0.0191513	+	0.999817i	$0.506096\pi$
$294$		0			0
$295$	1.69014			0.0984036
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.2487			−0.592699
$300$		0			0
$301$	2.02362	+	7.90400i	0.116639	+	0.455579i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.05849i		−	0.0606091i
$306$		0			0
$307$			4.45077i			0.254019i	0.991902	+	0.127009i	$0.0405379\pi$
$307$			4.45077i			0.254019i	−0.991902	+	0.127009i	$0.959462\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−23.1628			−1.31344			−0.656722	−	0.754133i	$-0.728058\pi$
$311$	−23.1628			−1.31344			−0.656722	+	0.754133i	$0.728058\pi$
$312$		0			0
$313$		−	23.5435i		−	1.33076i	−0.746505	−	0.665380i	$-0.768270\pi$
$313$		−	23.5435i		−	1.33076i	0.746505	−	0.665380i	$-0.231730\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.3217i			0.972882i	0.873713	+	0.486441i	$0.161705\pi$
$317$			17.3217i			0.972882i	−0.873713	+	0.486441i	$0.838295\pi$
$318$		0			0
$319$	−32.2401			−1.80510
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.50563i			0.250700i
$324$		0			0
$325$			21.5264i			1.19407i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.10699	+	23.8531i	0.336689	+	1.31507i
$330$		0			0
$331$	3.48000			0.191278			0.0956391	−	0.995416i	$-0.469511\pi$
$331$	3.48000			0.191278			0.0956391	+	0.995416i	$0.469511\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.22162			0.339923
$336$		0			0
$337$	5.94151			0.323655			0.161827	−	0.986819i	$-0.448261\pi$
$337$	5.94151			0.323655			0.161827	+	0.986819i	$0.448261\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.6422			0.792920
$342$		0			0
$343$	−12.6501	−	13.5268i	−0.683043	−	0.730378i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			3.04440i			0.163432i	0.996656	+	0.0817159i	$0.0260400\pi$
$347$			3.04440i			0.163432i	−0.996656	+	0.0817159i	$0.973960\pi$
$348$		0			0
$349$		−	25.5607i		−	1.36823i	−0.729373	−	0.684116i	$-0.760188\pi$
$349$		−	25.5607i		−	1.36823i	0.729373	−	0.684116i	$-0.239812\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−28.0385			−1.49234			−0.746169	−	0.665757i	$-0.768109\pi$
$353$	−28.0385			−1.49234			−0.746169	+	0.665757i	$0.768109\pi$
$354$		0			0
$355$		−	3.43863i		−	0.182503i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.33805i		−	0.334509i	−0.985914	−	0.167255i	$-0.946510\pi$
$359$		−	6.33805i		−	0.334509i	0.985914	−	0.167255i	$-0.0534902\pi$
$360$		0			0
$361$	16.9651			0.892901
$362$		0			0
$363$		0			0
$364$		0			0
$365$			5.03711i			0.263654i
$366$		0			0
$367$			2.69255i			0.140550i	0.997528	+	0.0702750i	$0.0223877\pi$
$367$			2.69255i			0.140550i	−0.997528	+	0.0702750i	$0.977612\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	31.2470	−	8.00000i	1.62227	−	0.415339i
$372$		0			0
$373$	15.5247			0.803837			0.401919	−	0.915675i	$-0.368343\pi$
$373$	15.5247			0.803837			0.401919	+	0.915675i	$0.368343\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−27.4329			−1.41286
$378$		0			0
$379$	28.9513			1.48713			0.743564	−	0.668664i	$-0.233134\pi$
$379$	28.9513			1.48713			0.743564	+	0.668664i	$0.233134\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.5816			0.898375			0.449188	−	0.893437i	$-0.351713\pi$
$383$	17.5816			0.898375			0.449188	+	0.893437i	$0.351713\pi$
$384$		0			0
$385$	6.31242	−	1.61613i	0.321711	−	0.0823658i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			2.16758i			0.109901i	0.998489	+	0.0549503i	$0.0175000\pi$
$389$			2.16758i			0.109901i	−0.998489	+	0.0549503i	$0.982500\pi$
$390$		0			0
$391$			7.19286i			0.363758i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.544868			0.0274153
$396$		0			0
$397$			28.6843i			1.43962i	0.694169	+	0.719812i	$0.255772\pi$
$397$			28.6843i			1.43962i	−0.694169	+	0.719812i	$0.744228\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			29.0843i			1.45240i	0.687482	+	0.726201i	$0.258716\pi$
$401$			29.0843i			1.45240i	−0.687482	+	0.726201i	$0.741284\pi$
$402$		0			0
$403$	12.4589			0.620624
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.24240i			0.160720i
$408$		0			0
$409$			24.6739i			1.22005i	0.792383	+	0.610024i	$0.208840\pi$
$409$			24.6739i			1.22005i	−0.792383	+	0.610024i	$0.791160\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.38184	+	9.30317i	0.117203	+	0.457779i
$414$		0			0
$415$	−6.99042			−0.343146
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.5895			−1.15242			−0.576210	−	0.817301i	$-0.695469\pi$
$419$	−23.5895			−1.15242			−0.576210	+	0.817301i	$0.695469\pi$
$420$		0			0
$421$	−5.38696			−0.262545			−0.131272	−	0.991346i	$-0.541906\pi$
$421$	−5.38696			−0.262545			−0.131272	+	0.991346i	$0.541906\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.1079			0.732839
$426$		0			0
$427$	5.82635	−	1.49169i	0.281957	−	0.0721878i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	7.33576i		−	0.353351i	−0.984269	−	0.176676i	$-0.943466\pi$
$431$		−	7.33576i		−	0.353351i	0.984269	−	0.176676i	$-0.0565344\pi$
$432$		0			0
$433$		−	19.9270i		−	0.957633i	−0.877915	−	0.478816i	$-0.841066\pi$
$433$		−	19.9270i		−	0.957633i	0.877915	−	0.478816i	$-0.158934\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.24851			−0.155397
$438$		0			0
$439$		−	10.6943i		−	0.510409i	−0.966887	−	0.255205i	$-0.917857\pi$
$439$		−	10.6943i		−	0.510409i	0.966887	−	0.255205i	$-0.0821428\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			23.6760i			1.12488i	0.826837	+	0.562441i	$0.190138\pi$
$443$			23.6760i			1.12488i	−0.826837	+	0.562441i	$0.809862\pi$
$444$		0			0
$445$	−2.51042			−0.119005
$446$		0			0
$447$		0			0
$448$		0			0
$449$			33.4409i			1.57817i	0.614282	+	0.789087i	$0.289446\pi$
$449$			33.4409i			1.57817i	−0.614282	+	0.789087i	$0.710554\pi$
$450$		0			0
$451$		−	36.6729i		−	1.72686i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.37119	−	1.37516i	0.251805	−	0.0644683i
$456$		0			0
$457$	11.5385			0.539748			0.269874	−	0.962896i	$-0.413018\pi$
$457$	11.5385			0.539748			0.269874	+	0.962896i	$0.413018\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.87564			−0.227081			−0.113541	−	0.993533i	$-0.536219\pi$
$461$	−4.87564			−0.227081			−0.113541	+	0.993533i	$0.536219\pi$
$462$		0			0
$463$	−40.8275			−1.89742			−0.948708	−	0.316154i	$-0.897608\pi$
$463$	−40.8275			−1.89742			−0.948708	+	0.316154i	$0.897608\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.27002			−0.197593			−0.0987964	−	0.995108i	$-0.531499\pi$
$467$	−4.27002			−0.197593			−0.0987964	+	0.995108i	$0.531499\pi$
$468$		0			0
$469$	8.76786	+	34.2462i	0.404862	+	1.58134i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16.3104i		−	0.749954i
$474$		0			0
$475$			6.82315i			0.313068i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.6067			−1.39846			−0.699228	−	0.714899i	$-0.746473\pi$
$479$	−30.6067			−1.39846			−0.699228	+	0.714899i	$0.746473\pi$
$480$		0			0
$481$			2.75893i			0.125796i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	4.86730i		−	0.221012i
$486$		0			0
$487$	−13.9626			−0.632704			−0.316352	−	0.948642i	$-0.602458\pi$
$487$	−13.9626			−0.632704			−0.316352	+	0.948642i	$0.602458\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	16.3729i		−	0.738897i	−0.929251	−	0.369449i	$-0.879547\pi$
$491$		−	16.3729i		−	0.738897i	0.929251	−	0.369449i	$-0.120453\pi$
$492$		0			0
$493$			19.2532i			0.867120i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.9275	−	4.84591i	0.849016	−	0.217369i
$498$		0			0
$499$	29.2986			1.31159			0.655793	−	0.754941i	$-0.272334\pi$
$499$	29.2986			1.31159			0.655793	+	0.754941i	$0.272334\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.4284			0.821681			0.410840	−	0.911707i	$-0.365235\pi$
$503$	18.4284			0.821681			0.410840	+	0.911707i	$0.365235\pi$
$504$		0			0
$505$	−4.69683			−0.209006
$506$		0			0
$507$		0			0
$508$		0			0
$509$	30.6804			1.35989			0.679943	−	0.733265i	$-0.262005\pi$
$509$	30.6804			1.35989			0.679943	+	0.733265i	$0.262005\pi$
$510$		0			0
$511$	−27.7262	+	7.09858i	−1.22653	+	0.314023i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.09788i			0.0483785i
$516$		0			0
$517$		−	49.2225i		−	2.16480i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.0125			−0.482468			−0.241234	−	0.970467i	$-0.577552\pi$
$521$	−11.0125			−0.482468			−0.241234	+	0.970467i	$0.577552\pi$
$522$		0			0
$523$		−	13.4459i		−	0.587949i	−0.955813	−	0.293975i	$-0.905022\pi$
$523$		−	13.4459i		−	0.587949i	0.955813	−	0.293975i	$-0.0949781\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.74406i		−	0.380897i
$528$		0			0
$529$	17.8140			0.774523
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	31.2047i		−	1.35162i
$534$		0			0
$535$		−	6.85448i		−	0.296345i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.7917	+	32.4685i	0.766341	+	1.39852i
$540$		0			0
$541$	−6.63665			−0.285332			−0.142666	−	0.989771i	$-0.545567\pi$
$541$	−6.63665			−0.285332			−0.142666	+	0.989771i	$0.545567\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.90950			−0.338806
$546$		0			0
$547$	−5.31242			−0.227143			−0.113571	−	0.993530i	$-0.536229\pi$
$547$	−5.31242			−0.227143			−0.113571	+	0.993530i	$0.536229\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.69531			−0.370433
$552$		0			0
$553$	0.767859	+	2.99916i	0.0326527	+	0.127537i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	15.9066i		−	0.673986i	−0.941507	−	0.336993i	$-0.890590\pi$
$557$		−	15.9066i		−	0.673986i	0.941507	−	0.336993i	$-0.109410\pi$
$558$		0			0
$559$		−	13.8784i		−	0.586994i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.61560			−0.405249			−0.202625	−	0.979256i	$-0.564947\pi$
$563$	−9.61560			−0.405249			−0.202625	+	0.979256i	$0.564947\pi$
$564$		0			0
$565$		−	3.23227i		−	0.135983i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			19.7154i			0.826514i	0.910614	+	0.413257i	$0.135609\pi$
$569$			19.7154i			0.826514i	−0.910614	+	0.413257i	$0.864391\pi$
$570$		0			0
$571$	−5.76969			−0.241454			−0.120727	−	0.992686i	$-0.538523\pi$
$571$	−5.76969			−0.241454			−0.120727	+	0.992686i	$0.538523\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			10.8926i			0.454253i
$576$		0			0
$577$			32.0587i			1.33462i	0.744780	+	0.667310i	$0.232554\pi$
$577$			32.0587i			1.33462i	−0.744780	+	0.667310i	$0.767446\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.85130	−	38.4780i	−0.408701	−	1.59634i
$582$		0			0
$583$	−64.4802			−2.67050
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.2298			1.00007			0.500035	−	0.866005i	$-0.333320\pi$
$587$	24.2298			1.00007			0.500035	+	0.866005i	$0.333320\pi$
$588$		0			0
$589$	3.94908			0.162719
$590$		0			0
$591$		0			0
$592$		0			0
$593$	44.7443			1.83743			0.918713	−	0.394925i	$-0.129230\pi$
$593$	44.7443			1.83743			0.918713	+	0.394925i	$0.129230\pi$
$594$		0			0
$595$	−0.965125	−	3.76966i	−0.0395663	−	0.154541i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			7.60574i			0.310762i	0.987855	+	0.155381i	$0.0496606\pi$
$599$			7.60574i			0.310762i	−0.987855	+	0.155381i	$0.950339\pi$
$600$		0			0
$601$		−	23.3138i		−	0.950990i	−0.879718	−	0.475495i	$-0.842269\pi$
$601$		−	23.3138i		−	0.950990i	0.879718	−	0.475495i	$-0.157731\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.90400			−0.321343
$606$		0			0
$607$			13.3948i			0.543680i	0.962342	+	0.271840i	$0.0876322\pi$
$607$			13.3948i			0.543680i	−0.962342	+	0.271840i	$0.912368\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	41.8830i		−	1.69441i
$612$		0			0
$613$	41.1231			1.66095			0.830474	−	0.557058i	$-0.188070\pi$
$613$	41.1231			1.66095			0.830474	+	0.557058i	$0.188070\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.0607i		−	0.767356i	−0.923467	−	0.383678i	$-0.874657\pi$
$617$		−	19.0607i		−	0.767356i	0.923467	−	0.383678i	$-0.125343\pi$
$618$		0			0
$619$			10.5929i			0.425766i	0.977078	+	0.212883i	$0.0682854\pi$
$619$			10.5929i			0.425766i	−0.977078	+	0.212883i	$0.931715\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.53783	−	13.8183i	−0.141740	−	0.553620i
$624$		0			0
$625$	21.7947			0.871786
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.93630			0.0772053
$630$		0			0
$631$	−26.9277			−1.07197			−0.535987	−	0.844226i	$-0.680060\pi$
$631$	−26.9277			−1.07197			−0.535987	+	0.844226i	$0.680060\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.79632			−0.0712848
$636$		0			0
$637$	15.1388	+	27.6272i	0.599820	+	1.09463i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			11.0821i			0.437717i	0.975757	+	0.218859i	$0.0702333\pi$
$641$			11.0821i			0.437717i	−0.975757	+	0.218859i	$0.929767\pi$
$642$		0			0
$643$			34.4756i			1.35958i	0.733405	+	0.679792i	$0.237930\pi$
$643$			34.4756i			1.35958i	−0.733405	+	0.679792i	$0.762070\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−35.1859			−1.38330			−0.691650	−	0.722233i	$-0.743116\pi$
$647$	−35.1859			−1.38330			−0.691650	+	0.722233i	$0.743116\pi$
$648$		0			0
$649$		−	19.1977i		−	0.753575i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	32.4082i		−	1.26823i	−0.773238	−	0.634116i	$-0.781364\pi$
$653$		−	32.4082i		−	1.26823i	0.773238	−	0.634116i	$-0.218636\pi$
$654$		0			0
$655$	8.72501			0.340914
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	5.09788i		−	0.198585i	−0.995058	−	0.0992927i	$-0.968342\pi$
$659$		−	5.09788i		−	0.198585i	0.995058	−	0.0992927i	$-0.0316580\pi$
$660$		0			0
$661$		−	20.4587i		−	0.795752i	−0.917439	−	0.397876i	$-0.869747\pi$
$661$		−	20.4587i		−	0.795752i	0.917439	−	0.397876i	$-0.130253\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.70249	−	0.435879i	0.0660198	−	0.0169027i
$666$		0			0
$667$	−13.8813			−0.537487
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0230			−0.464144
$672$		0			0
$673$	43.7926			1.68808			0.844041	−	0.536278i	$-0.180170\pi$
$673$	43.7926			1.68808			0.844041	+	0.536278i	$0.180170\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.0624			−0.578895			−0.289448	−	0.957194i	$-0.593472\pi$
$677$	−15.0624			−0.578895			−0.289448	+	0.957194i	$0.593472\pi$
$678$		0			0
$679$	26.7915	−	6.85927i	1.02816	−	0.263235i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	38.5894i		−	1.47658i	−0.674483	−	0.738291i	$-0.735633\pi$
$683$		−	38.5894i		−	1.47658i	0.674483	−	0.738291i	$-0.264367\pi$
$684$		0			0
$685$			3.19875i			0.122218i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−54.8657			−2.09022
$690$		0			0
$691$			40.1079i			1.52578i	0.646530	+	0.762889i	$0.276220\pi$
$691$			40.1079i			1.52578i	−0.646530	+	0.762889i	$0.723780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.73002i			0.293216i
$696$		0			0
$697$	−21.9004			−0.829536
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.4195i		−	0.506848i	−0.967355	−	0.253424i	$-0.918443\pi$
$701$		−	13.4195i		−	0.506848i	0.967355	−	0.253424i	$-0.0815567\pi$
$702$		0			0
$703$			0.874490i			0.0329820i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.61903	−	25.8531i	−0.248934	−	0.972308i
$708$		0			0
$709$	33.2165			1.24747			0.623736	−	0.781635i	$-0.285614\pi$
$709$	33.2165			1.24747			0.623736	+	0.781635i	$0.285614\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.30437			0.236100
$714$		0			0
$715$	−11.0838			−0.414510
$716$		0			0
$717$		0			0
$718$		0			0
$719$	43.0856			1.60682			0.803411	−	0.595425i	$-0.203016\pi$
$719$	43.0856			1.60682			0.803411	+	0.595425i	$0.203016\pi$
$720$		0			0
$721$	−6.04318	+	1.54720i	−0.225060	+	0.0576207i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			29.1563i			1.08284i
$726$		0			0
$727$			22.3861i			0.830256i	0.909763	+	0.415128i	$0.136263\pi$
$727$			22.3861i			0.830256i	−0.909763	+	0.415128i	$0.863737\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.74028			−0.360257
$732$		0			0
$733$		−	35.0569i		−	1.29486i	−0.762127	−	0.647428i	$-0.775845\pi$
$733$		−	35.0569i		−	1.29486i	0.762127	−	0.647428i	$-0.224155\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	70.6692i		−	2.60313i
$738$		0			0
$739$	2.21481			0.0814733			0.0407366	−	0.999170i	$-0.487030\pi$
$739$	2.21481			0.0814733			0.0407366	+	0.999170i	$0.487030\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	28.0516i		−	1.02911i	−0.857456	−	0.514557i	$-0.827956\pi$
$743$		−	28.0516i		−	1.02911i	0.857456	−	0.514557i	$-0.172044\pi$
$744$		0			0
$745$			1.50220i			0.0550363i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	37.7297	−	9.65972i	1.37861	−	0.352958i
$750$		0			0
$751$	9.62741			0.351309			0.175655	−	0.984452i	$-0.443796\pi$
$751$	9.62741			0.351309			0.175655	+	0.984452i	$0.443796\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.25145			0.0455450
$756$		0			0
$757$	34.0351			1.23703			0.618513	−	0.785774i	$-0.287735\pi$
$757$	34.0351			1.23703			0.618513	+	0.785774i	$0.287735\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−29.3692			−1.06463			−0.532316	−	0.846546i	$-0.678678\pi$
$761$	−29.3692			−1.06463			−0.532316	+	0.846546i	$0.678678\pi$
$762$		0			0
$763$	−11.1465	−	43.5370i	−0.403531	−	1.57614i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	16.3352i		−	0.589828i
$768$		0			0
$769$			41.2653i			1.48807i	0.668143	+	0.744033i	$0.267089\pi$
$769$			41.2653i			1.48807i	−0.668143	+	0.744033i	$0.732911\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	42.3880			1.52459			0.762296	−	0.647229i	$-0.224072\pi$
$773$	42.3880			1.52459			0.762296	+	0.647229i	$0.224072\pi$
$774$		0			0
$775$		−	13.2417i		−	0.475655i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	9.89086i		−	0.354377i
$780$		0			0
$781$	−39.0582			−1.39761
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7.92802i		−	0.282963i
$786$		0			0
$787$		−	33.2592i		−	1.18556i	−0.805364	−	0.592781i	$-0.798030\pi$
$787$		−	33.2592i		−	1.18556i	0.805364	−	0.592781i	$-0.201970\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	17.7917	−	4.55510i	0.632598	−	0.161961i
$792$		0			0
$793$	−10.2303			−0.363289
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.1338			−0.890287			−0.445143	−	0.895459i	$-0.646847\pi$
$797$	−25.1338			−0.890287			−0.445143	+	0.895459i	$0.646847\pi$
$798$		0			0
$799$	−29.3948			−1.03991
$800$		0			0
$801$		0			0
$802$		0			0
$803$	57.2147			2.01906
$804$		0			0
$805$	2.71788	−	0.695844i	0.0957928	−	0.0245253i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			20.5016i			0.720798i	0.932798	+	0.360399i	$0.117359\pi$
$809$			20.5016i			0.720798i	−0.932798	+	0.360399i	$0.882641\pi$
$810$		0			0
$811$			33.6507i			1.18164i	0.806804	+	0.590819i	$0.201195\pi$
$811$			33.6507i			1.18164i	−0.806804	+	0.590819i	$0.798805\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.44386			0.260747
$816$		0			0
$817$		−	4.39900i		−	0.153901i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			18.9629i			0.661810i	0.943664	+	0.330905i	$0.107354\pi$
$821$			18.9629i			0.661810i	−0.943664	+	0.330905i	$0.892646\pi$
$822$		0			0
$823$	−50.2027			−1.74996			−0.874978	−	0.484163i	$-0.839124\pi$
$823$	−50.2027			−1.74996			−0.874978	+	0.484163i	$0.839124\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			42.8071i			1.48855i	0.667875	+	0.744274i	$0.267204\pi$
$827$			42.8071i			1.48855i	−0.667875	+	0.744274i	$0.732796\pi$
$828$		0			0
$829$		−	0.746078i		−	0.0259124i	−0.999916	−	0.0129562i	$-0.995876\pi$
$829$		−	0.746078i		−	0.0259124i	0.999916	−	0.0129562i	$-0.00412420\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.3896	−	10.6248i	0.671809	−	0.368129i
$834$		0			0
$835$	8.68056			0.300403
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.0608			−0.450909			−0.225454	−	0.974254i	$-0.572387\pi$
$839$	−13.0608			−0.450909			−0.225454	+	0.974254i	$0.572387\pi$
$840$		0			0
$841$	−8.15632			−0.281252
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.37774			−0.116198
$846$		0			0
$847$	−11.1388	−	43.5067i	−0.382733	−	1.49491i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.39605i			0.0478560i
$852$		0			0
$853$		−	8.16725i		−	0.279641i	−0.990177	−	0.139821i	$-0.955347\pi$
$853$		−	8.16725i		−	0.279641i	0.990177	−	0.139821i	$-0.0446526\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.4676			0.938274			0.469137	−	0.883125i	$-0.344565\pi$
$857$	27.4676			0.938274			0.469137	+	0.883125i	$0.344565\pi$
$858$		0			0
$859$			7.36429i			0.251266i	0.992077	+	0.125633i	$0.0400962\pi$
$859$			7.36429i			0.251266i	−0.992077	+	0.125633i	$0.959904\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	44.6035i		−	1.51832i	−0.650904	−	0.759160i	$-0.725610\pi$
$863$		−	44.6035i		−	1.51832i	0.650904	−	0.759160i	$-0.274390\pi$
$864$		0			0
$865$	4.76457			0.162000
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	6.18897i		−	0.209946i
$870$		0			0
$871$		−	60.1319i		−	2.03749i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.98935	−	11.6760i	−0.101059	−	0.394722i
$876$		0			0
$877$	−15.4564			−0.521925			−0.260963	−	0.965349i	$-0.584040\pi$
$877$	−15.4564			−0.521925			−0.260963	+	0.965349i	$0.584040\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	53.0086			1.78591			0.892953	−	0.450150i	$-0.148629\pi$
$881$	53.0086			1.78591			0.892953	+	0.450150i	$0.148629\pi$
$882$		0			0
$883$	−12.6178			−0.424624			−0.212312	−	0.977202i	$-0.568099\pi$
$883$	−12.6178			−0.424624			−0.212312	+	0.977202i	$0.568099\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.4205			0.417038			0.208519	−	0.978018i	$-0.433136\pi$
$887$	12.4205			0.417038			0.208519	+	0.978018i	$0.433136\pi$
$888$		0			0
$889$	−2.53148	−	9.88765i	−0.0849030	−	0.331621i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	13.2755i		−	0.444249i
$894$		0			0
$895$			6.36160i			0.212645i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.8750			0.562812
$900$		0			0
$901$			38.5064i			1.28283i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			5.26329i			0.174958i
$906$		0			0
$907$	49.8675			1.65582			0.827912	−	0.560858i	$-0.189529\pi$
$907$	49.8675			1.65582			0.827912	+	0.560858i	$0.189529\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	58.5337i		−	1.93931i	−0.244481	−	0.969654i	$-0.578617\pi$
$911$		−	58.5337i		−	1.93931i	0.244481	−	0.969654i	$-0.421383\pi$
$912$		0			0
$913$			79.4018i			2.62782i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.2958	+	48.0258i	0.406043	+	1.58595i
$918$		0			0
$919$	−4.38239			−0.144562			−0.0722809	−	0.997384i	$-0.523028\pi$
$919$	−4.38239			−0.144562			−0.0722809	+	0.997384i	$0.523028\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−33.2343			−1.09392
$924$		0			0
$925$	2.93226			0.0964121
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.76966			0.123679			0.0618393	−	0.998086i	$-0.480303\pi$
$929$	3.76966			0.123679			0.0618393	+	0.998086i	$0.480303\pi$
$930$		0			0
$931$	4.79849	+	8.75690i	0.157264	+	0.286996i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			7.77894i			0.254398i
$936$		0			0
$937$		−	3.10444i		−	0.101418i	−0.998713	−	0.0507088i	$-0.983852\pi$
$937$		−	3.10444i		−	0.101418i	0.998713	−	0.0507088i	$-0.0161481\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41.8549			−1.36443			−0.682216	−	0.731151i	$-0.738984\pi$
$941$	−41.8549			−1.36443			−0.682216	+	0.731151i	$0.738984\pi$
$942$		0			0
$943$		−	15.7899i		−	0.514191i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.3267i		−	0.530548i	−0.964173	−	0.265274i	$-0.914538\pi$
$947$		−	16.3267i		−	0.530548i	0.964173	−	0.265274i	$-0.0854624\pi$
$948$		0			0
$949$	48.6836			1.58034
$950$		0			0
$951$		0			0
$952$		0			0
$953$			12.9393i			0.419144i	0.977793	+	0.209572i	$0.0672071\pi$
$953$			12.9393i			0.419144i	−0.977793	+	0.209572i	$0.932793\pi$
$954$		0			0
$955$			4.03444i			0.130552i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.6072	+	4.50786i	−0.568565	+	0.145566i
$960$		0			0
$961$	23.3360			0.752775
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.57953			0.243994
$966$		0			0
$967$	24.4451			0.786102			0.393051	−	0.919517i	$-0.371420\pi$
$967$	24.4451			0.786102			0.393051	+	0.919517i	$0.371420\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.52357			0.209351			0.104676	−	0.994506i	$-0.466620\pi$
$971$	6.52357			0.209351			0.104676	+	0.994506i	$0.466620\pi$
$972$		0			0
$973$	−42.5490	+	10.8936i	−1.36406	+	0.349232i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	59.2778i		−	1.89646i	−0.317579	−	0.948232i	$-0.602870\pi$
$977$		−	59.2778i		−	1.89646i	0.317579	−	0.948232i	$-0.397130\pi$
$978$		0			0
$979$			28.5150i			0.911343i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	30.4222			0.970318			0.485159	−	0.874426i	$-0.338762\pi$
$983$	30.4222			0.970318			0.485159	+	0.874426i	$0.338762\pi$
$984$		0			0
$985$			11.1320i			0.354694i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	7.02263i		−	0.223307i
$990$		0			0
$991$	6.41983			0.203933			0.101966	−	0.994788i	$-0.467487\pi$
$991$	6.41983			0.203933			0.101966	+	0.994788i	$0.467487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	11.8183i		−	0.374666i
$996$		0			0
$997$		−	31.8582i		−	1.00896i	−0.863424	−	0.504479i	$-0.831685\pi$
$997$		−	31.8582i		−	1.00896i	0.863424	−	0.504479i	$-0.168315\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.k.a.377.10	yes	16
3.2	odd	2	inner	1512.2.k.a.377.8	yes	16
4.3	odd	2		3024.2.k.k.1889.9		16
7.6	odd	2	inner	1512.2.k.a.377.7	✓	16
12.11	even	2		3024.2.k.k.1889.7		16
21.20	even	2	inner	1512.2.k.a.377.9	yes	16
28.27	even	2		3024.2.k.k.1889.8		16
84.83	odd	2		3024.2.k.k.1889.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.a.377.7	✓	16	7.6	odd	2	inner
1512.2.k.a.377.8	yes	16	3.2	odd	2	inner
1512.2.k.a.377.9	yes	16	21.20	even	2	inner
1512.2.k.a.377.10	yes	16	1.1	even	1	trivial
3024.2.k.k.1889.7		16	12.11	even	2
3024.2.k.k.1889.8		16	28.27	even	2
3024.2.k.k.1889.9		16	4.3	odd	2
3024.2.k.k.1889.10		16	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.465643 q^{5} +(0.656211 + 2.56308i) q^{7} +O(q^{10})\)	\(q+0.465643 q^{5} +(0.656211 + 2.56308i) q^{7} -5.28908i q^{11} -4.50044i q^{13} -3.15854 q^{17} -1.42649i q^{19} -2.27727i q^{23} -4.78318 q^{25} -6.09560i q^{29} +2.76839i q^{31} +(0.305560 + 1.19348i) q^{35} -0.613036 q^{37} +6.93370 q^{41} +3.08379 q^{43} +9.30643 q^{47} +(-6.13877 + 3.36385i) q^{49} -12.1912i q^{53} -2.46283i q^{55} +3.62968 q^{59} -2.27318i q^{61} -2.09560i q^{65} +13.3613 q^{67} -7.38468i q^{71} +10.8175i q^{73} +(13.5563 - 3.47075i) q^{77} +1.17014 q^{79} -15.0124 q^{83} -1.47075 q^{85} -5.39129 q^{89} +(11.5350 - 2.95324i) q^{91} -0.664236i q^{95} -10.4528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{7}+O(q^{10}) \) 16 * q - 2 * q^7	\( 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) \) 16 * q - 2 * q^7 - 12 * q^25 - 8 * q^37 + 8 * q^43 + 2 * q^49 - 28 * q^67 - 44 * q^79 + 16 * q^85 + 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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