LMFDB - Embedded newform 4212.2.i.n.1405.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1405,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1405"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-1,0,0,0,0,0,-7,0,-2,0,0,0,2] 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 4x^{2} + 3x + 9 $ x^4 - x^3 + 4x^2 + 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1404)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1405.1
Root			$1.15139 - 1.99426i$ of defining polynomial
Character	$\chi$	$=$	4212.1405
Dual form			4212.2.i.n.2809.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.15139 + 1.99426i) q^{5} +(-1.80278 - 3.12250i) q^{7} +(-2.65139 - 4.59234i) q^{11} +(-0.500000 + 0.866025i) q^{13} +2.30278 q^{17} -3.30278 q^{19} +(1.95416 - 3.38471i) q^{23} +(-0.151388 - 0.262211i) q^{25} +(-1.50000 - 2.59808i) q^{29} +(-1.00000 + 1.73205i) q^{31} +8.30278 q^{35} -10.2111 q^{37} +(0.802776 - 1.39045i) q^{41} +(4.65139 + 8.05644i) q^{43} +(2.75694 + 4.77516i) q^{47} +(-3.00000 + 5.19615i) q^{49} +2.30278 q^{53} +12.2111 q^{55} +(0.105551 - 0.182820i) q^{59} +(-1.80278 - 3.12250i) q^{61} +(-1.15139 - 1.99426i) q^{65} +(-0.545837 + 0.945417i) q^{67} +16.6056 q^{71} -10.9083 q^{73} +(-9.55971 + 16.5579i) q^{77} +(3.95416 + 6.84881i) q^{79} +(-5.40833 - 9.36750i) q^{83} +(-2.65139 + 4.59234i) q^{85} -4.60555 q^{89} +3.60555 q^{91} +(3.80278 - 6.58660i) q^{95} +(6.60555 + 11.4412i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} - 7 q^{11} - 2 q^{13} + 2 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{29} - 4 q^{31} + 26 q^{35} - 12 q^{37} - 4 q^{41} + 15 q^{43} - 7 q^{47} - 12 q^{49} + 2 q^{53} + 20 q^{55} - 14 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 4 * q - q^5 - 7 * q^11 - 2 * q^13 + 2 * q^17 - 6 * q^19 - 3 * q^23 + 3 * q^25 - 6 * q^29 - 4 * q^31 + 26 * q^35 - 12 * q^37 - 4 * q^41 + 15 * q^43 - 7 * q^47 - 12 * q^49 + 2 * q^53 + 20 * q^55 - 14 * q^59 - q^65 - 13 * q^67 + 52 * q^71 - 22 * q^73 - 13 * q^77 + 5 * q^79 - 7 * q^85 - 4 * q^89 + 8 * q^95 + 12 * q^97

Character values

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

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−1.15139	+	1.99426i	−0.514916	+	0.891861i	0.484934	+	0.874551i	$0.338844\pi$
$5$	−1.15139	+	1.99426i	−0.514916	+	0.891861i	−0.999850	+	0.0173104i	$0.994490\pi$
$6$		0			0
$7$	−1.80278	−	3.12250i	−0.681385	−	1.18019i	−0.974558	−	0.224134i	$-0.928045\pi$
$7$	−1.80278	−	3.12250i	−0.681385	−	1.18019i	0.293173	−	0.956059i	$-0.405289\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.65139	−	4.59234i	−0.799424	−	1.38464i	−0.919992	−	0.391937i	$-0.871805\pi$
$11$	−2.65139	−	4.59234i	−0.799424	−	1.38464i	0.120569	−	0.992705i	$-0.461528\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.30278			0.558505			0.279253	−	0.960218i	$-0.409913\pi$
$17$	2.30278			0.558505			0.279253	+	0.960218i	$0.409913\pi$
$18$		0			0
$19$	−3.30278			−0.757709			−0.378854	−	0.925456i	$-0.623682\pi$
$19$	−3.30278			−0.757709			−0.378854	+	0.925456i	$0.623682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.95416	−	3.38471i	0.407471	−	0.705761i	−0.587134	−	0.809489i	$-0.699744\pi$
$23$	1.95416	−	3.38471i	0.407471	−	0.705761i	0.994606	+	0.103729i	$0.0330773\pi$
$24$		0			0
$25$	−0.151388	−	0.262211i	−0.0302776	−	0.0524423i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	$-0.890815\pi$
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	0.762140	+	0.647412i	$0.224149\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.30278			1.40343
$36$		0			0
$37$	−10.2111			−1.67869			−0.839347	−	0.543595i	$-0.817063\pi$
$37$	−10.2111			−1.67869			−0.839347	+	0.543595i	$0.817063\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.802776	−	1.39045i	0.125372	−	0.217152i	−0.796506	−	0.604631i	$-0.793321\pi$
$41$	0.802776	−	1.39045i	0.125372	−	0.217152i	0.921878	+	0.387479i	$0.126654\pi$
$42$		0			0
$43$	4.65139	+	8.05644i	0.709330	+	1.22860i	0.965106	+	0.261860i	$0.0843357\pi$
$43$	4.65139	+	8.05644i	0.709330	+	1.22860i	−0.255776	+	0.966736i	$0.582331\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.75694	+	4.77516i	0.402141	+	0.696528i	0.993984	−	0.109525i	$-0.0349329\pi$
$47$	2.75694	+	4.77516i	0.402141	+	0.696528i	−0.591843	+	0.806053i	$0.701600\pi$
$48$		0			0
$49$	−3.00000	+	5.19615i	−0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.30278			0.316311			0.158155	−	0.987414i	$-0.449445\pi$
$53$	2.30278			0.316311			0.158155	+	0.987414i	$0.449445\pi$
$54$		0			0
$55$	12.2111			1.64654
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.105551	−	0.182820i	0.0137416	−	0.0238012i	−0.859073	−	0.511853i	$-0.828959\pi$
$59$	0.105551	−	0.182820i	0.0137416	−	0.0238012i	0.872814	+	0.488052i	$0.162292\pi$
$60$		0			0
$61$	−1.80278	−	3.12250i	−0.230822	−	0.399795i	0.727228	−	0.686396i	$-0.240808\pi$
$61$	−1.80278	−	3.12250i	−0.230822	−	0.399795i	−0.958050	+	0.286601i	$0.907475\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.15139	−	1.99426i	−0.142812	−	0.247358i
$66$		0			0
$67$	−0.545837	+	0.945417i	−0.0666845	+	0.115501i	−0.897440	−	0.441137i	$-0.854575\pi$
$67$	−0.545837	+	0.945417i	−0.0666845	+	0.115501i	0.830755	+	0.556638i	$0.187909\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.6056			1.97072			0.985358	−	0.170497i	$-0.0545373\pi$
$71$	16.6056			1.97072			0.985358	+	0.170497i	$0.0545373\pi$
$72$		0			0
$73$	−10.9083			−1.27672			−0.638362	−	0.769737i	$-0.720388\pi$
$73$	−10.9083			−1.27672			−0.638362	+	0.769737i	$0.720388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.55971	+	16.5579i	−1.08943	+	1.88695i
$78$		0			0
$79$	3.95416	+	6.84881i	0.444878	+	0.770552i	0.998044	−	0.0625198i	$-0.0199137\pi$
$79$	3.95416	+	6.84881i	0.444878	+	0.770552i	−0.553166	+	0.833071i	$0.686580\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.40833	−	9.36750i	−0.593641	−	1.02822i	−0.993737	−	0.111743i	$-0.964357\pi$
$83$	−5.40833	−	9.36750i	−0.593641	−	1.02822i	0.400096	−	0.916473i	$-0.368977\pi$
$84$		0			0
$85$	−2.65139	+	4.59234i	−0.287583	+	0.498109i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.60555			−0.488187			−0.244094	−	0.969752i	$-0.578490\pi$
$89$	−4.60555			−0.488187			−0.244094	+	0.969752i	$0.578490\pi$
$90$		0			0
$91$	3.60555			0.377964
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.80278	−	6.58660i	0.390157	−	0.675771i
$96$		0			0
$97$	6.60555	+	11.4412i	0.670692	+	1.16167i	0.977708	+	0.209968i	$0.0673361\pi$
$97$	6.60555	+	11.4412i	0.670692	+	1.16167i	−0.307016	+	0.951704i	$0.599331\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.10555	+	10.5751i	0.607525	+	1.05226i	0.991647	+	0.128982i	$0.0411709\pi$
$101$	6.10555	+	10.5751i	0.607525	+	1.05226i	−0.384122	+	0.923282i	$0.625496\pi$
$102$		0			0
$103$	−2.84861	+	4.93394i	−0.280682	+	0.486156i	−0.971553	−	0.236823i	$-0.923894\pi$
$103$	−2.84861	+	4.93394i	−0.280682	+	0.486156i	0.690871	+	0.722978i	$0.257227\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.90833			0.957874			0.478937	−	0.877849i	$-0.341022\pi$
$107$	9.90833			0.957874			0.478937	+	0.877849i	$0.341022\pi$
$108$		0			0
$109$	−9.30278			−0.891044			−0.445522	−	0.895271i	$-0.646982\pi$
$109$	−9.30278			−0.891044			−0.445522	+	0.895271i	$0.646982\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.40833	+	9.36750i	−0.508773	+	0.881220i	0.491176	+	0.871060i	$0.336567\pi$
$113$	−5.40833	+	9.36750i	−0.508773	+	0.881220i	−0.999948	+	0.0101595i	$0.996766\pi$
$114$		0			0
$115$	4.50000	+	7.79423i	0.419627	+	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.15139	−	7.19041i	−0.380557	−	0.659144i
$120$		0			0
$121$	−8.55971	+	14.8259i	−0.778156	+	1.34781i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.8167			−0.967471
$126$		0			0
$127$	22.0278			1.95465			0.977324	−	0.211751i	$-0.0679166\pi$
$127$	22.0278			1.95465			0.977324	+	0.211751i	$0.0679166\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.71110	+	13.3560i	−0.673722	+	1.16692i	0.303118	+	0.952953i	$0.401972\pi$
$131$	−7.71110	+	13.3560i	−0.673722	+	1.16692i	−0.976841	+	0.213968i	$0.931361\pi$
$132$		0			0
$133$	5.95416	+	10.3129i	0.516291	+	0.894243i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.697224	−	1.20763i	−0.0595679	−	0.103175i	0.834704	−	0.550699i	$-0.185639\pi$
$137$	−0.697224	−	1.20763i	−0.0595679	−	0.103175i	−0.894272	+	0.447525i	$0.852306\pi$
$138$		0			0
$139$	−7.10555	+	12.3072i	−0.602685	+	1.04388i	0.389728	+	0.920930i	$0.372569\pi$
$139$	−7.10555	+	12.3072i	−0.602685	+	1.04388i	−0.992413	+	0.122951i	$0.960764\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.30278			0.443440
$144$		0			0
$145$	6.90833			0.573705
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.86249	−	15.3503i	0.726044	−	1.25754i	−0.232500	−	0.972597i	$-0.574690\pi$
$149$	8.86249	−	15.3503i	0.726044	−	1.25754i	0.958543	−	0.284948i	$-0.0919762\pi$
$150$		0			0
$151$	5.80278	+	10.0507i	0.472223	+	0.817915i	0.999495	−	0.0317822i	$-0.0101183\pi$
$151$	5.80278	+	10.0507i	0.472223	+	0.817915i	−0.527272	+	0.849697i	$0.676785\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.30278	−	3.98852i	−0.184963	−	0.320366i
$156$		0			0
$157$	4.86249	−	8.42208i	0.388069	−	0.672155i	−0.604121	−	0.796893i	$-0.706476\pi$
$157$	4.86249	−	8.42208i	0.388069	−	0.672155i	0.992190	+	0.124738i	$0.0398089\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.0917			−1.11058
$162$		0			0
$163$	2.90833			0.227798			0.113899	−	0.993492i	$-0.463666\pi$
$163$	2.90833			0.227798			0.113899	+	0.993492i	$0.463666\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.50000	+	12.9904i	−0.580367	+	1.00523i	0.415068	+	0.909790i	$0.363758\pi$
$167$	−7.50000	+	12.9904i	−0.580367	+	1.00523i	−0.995436	+	0.0954356i	$0.969576\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.34861	+	10.9961i	0.482676	+	0.836019i	0.999802	−	0.0198898i	$-0.00633153\pi$
$173$	6.34861	+	10.9961i	0.482676	+	0.836019i	−0.517126	+	0.855909i	$0.672998\pi$
$174$		0			0
$175$	−0.545837	+	0.945417i	−0.0412614	+	0.0714668i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.7250			−1.54906			−0.774529	−	0.632539i	$-0.782013\pi$
$179$	−20.7250			−1.54906			−0.774529	+	0.632539i	$0.782013\pi$
$180$		0			0
$181$	8.90833			0.662151			0.331075	−	0.943604i	$-0.392589\pi$
$181$	8.90833			0.662151			0.331075	+	0.943604i	$0.392589\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.7569	−	20.3636i	0.864387	−	1.49716i
$186$		0			0
$187$	−6.10555	−	10.5751i	−0.446482	−	0.773330i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.84861	−	8.39804i	−0.350833	−	0.607661i	0.635562	−	0.772049i	$-0.280768\pi$
$191$	−4.84861	−	8.39804i	−0.350833	−	0.607661i	−0.986396	+	0.164388i	$0.947435\pi$
$192$		0			0
$193$	−12.1653	+	21.0709i	−0.875675	+	1.51671i	−0.0196331	+	0.999807i	$0.506250\pi$
$193$	−12.1653	+	21.0709i	−0.875675	+	1.51671i	−0.856042	+	0.516906i	$0.827084\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.8167			−0.984396			−0.492198	−	0.870483i	$-0.663806\pi$
$197$	−13.8167			−0.984396			−0.492198	+	0.870483i	$0.663806\pi$
$198$		0			0
$199$	−21.9361			−1.55501			−0.777504	−	0.628878i	$-0.783514\pi$
$199$	−21.9361			−1.55501			−0.777504	+	0.628878i	$0.783514\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.40833	+	9.36750i	−0.379590	+	0.657469i
$204$		0			0
$205$	1.84861	+	3.20189i	0.129113	+	0.223630i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.75694	+	15.1675i	0.605730	+	1.04916i
$210$		0			0
$211$	−2.74306	+	4.75112i	−0.188840	+	0.327081i	−0.944864	−	0.327464i	$-0.893806\pi$
$211$	−2.74306	+	4.75112i	−0.188840	+	0.327081i	0.756024	+	0.654544i	$0.227139\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−21.4222			−1.46098
$216$		0			0
$217$	7.21110			0.489522
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.15139	+	1.99426i	−0.0774507	+	0.134149i
$222$		0			0
$223$	12.9542	+	22.4373i	0.867475	+	1.50251i	0.864569	+	0.502515i	$0.167592\pi$
$223$	12.9542	+	22.4373i	0.867475	+	1.50251i	0.00290640	+	0.999996i	$0.499075\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.0597	−	19.1560i	−0.734059	−	1.27143i	−0.955135	−	0.296171i	$-0.904290\pi$
$227$	−11.0597	−	19.1560i	−0.734059	−	1.27143i	0.221076	−	0.975257i	$-0.429043\pi$
$228$		0			0
$229$	−11.2569	+	19.4976i	−0.743879	+	1.28844i	0.206837	+	0.978375i	$0.433683\pi$
$229$	−11.2569	+	19.4976i	−0.743879	+	1.28844i	−0.950716	+	0.310062i	$0.899650\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$	−12.6972			−0.828276
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.4083	−	24.9560i	0.931997	−	1.61427i	0.152094	−	0.988366i	$-0.451398\pi$
$239$	14.4083	−	24.9560i	0.931997	−	1.61427i	0.779903	−	0.625900i	$-0.215268\pi$
$240$		0			0
$241$	−1.34861	−	2.33586i	−0.0868717	−	0.150466i	0.819316	−	0.573343i	$-0.194354\pi$
$241$	−1.34861	−	2.33586i	−0.0868717	−	0.150466i	−0.906187	+	0.422877i	$0.861020\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.90833	−	11.9656i	−0.441357	−	0.764452i
$246$		0			0
$247$	1.65139	−	2.86029i	0.105075	−	0.181996i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.51388			0.348033			0.174016	−	0.984743i	$-0.444325\pi$
$251$	5.51388			0.348033			0.174016	+	0.984743i	$0.444325\pi$
$252$		0			0
$253$	−20.7250			−1.30297
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0458	−	17.3999i	0.626642	−	1.08538i	−0.361579	−	0.932342i	$-0.617762\pi$
$257$	10.0458	−	17.3999i	0.626642	−	1.08538i	0.988221	−	0.153034i	$-0.0489045\pi$
$258$		0			0
$259$	18.4083	+	31.8842i	1.14384	+	1.98119i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.84861	+	8.39804i	0.298978	+	0.517845i	0.975902	−	0.218207i	$-0.0700209\pi$
$263$	4.84861	+	8.39804i	0.298978	+	0.517845i	−0.676924	+	0.736053i	$0.736688\pi$
$264$		0			0
$265$	−2.65139	+	4.59234i	−0.162873	+	0.282105i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	20.5139			1.25075			0.625377	−	0.780323i	$-0.284945\pi$
$269$	20.5139			1.25075			0.625377	+	0.780323i	$0.284945\pi$
$270$		0			0
$271$	3.18335			0.193375			0.0966873	−	0.995315i	$-0.469175\pi$
$271$	3.18335			0.193375			0.0966873	+	0.995315i	$0.469175\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.802776	+	1.39045i	−0.0484092	+	0.0838472i
$276$		0			0
$277$	−14.1194	−	24.4556i	−0.848354	−	1.46939i	−0.882676	−	0.469982i	$-0.844260\pi$
$277$	−14.1194	−	24.4556i	−0.848354	−	1.46939i	0.0343218	−	0.999411i	$-0.489073\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.6056	+	18.3694i	0.632674	+	1.09582i	0.987003	+	0.160703i	$0.0513761\pi$
$281$	10.6056	+	18.3694i	0.632674	+	1.09582i	−0.354329	+	0.935121i	$0.615291\pi$
$282$		0			0
$283$	9.36249	−	16.2163i	0.556542	−	0.963960i	−0.441239	−	0.897389i	$-0.645461\pi$
$283$	9.36249	−	16.2163i	0.556542	−	0.963960i	0.997782	−	0.0665701i	$-0.0212056\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.78890			−0.341708
$288$		0			0
$289$	−11.6972			−0.688072
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.0139	+	27.7369i	−0.935541	+	1.62040i	−0.161874	+	0.986811i	$0.551754\pi$
$293$	−16.0139	+	27.7369i	−0.935541	+	1.62040i	−0.773667	+	0.633593i	$0.781580\pi$
$294$		0			0
$295$	0.243061	+	0.420994i	0.0141516	+	0.0245112i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.95416	+	3.38471i	0.113012	+	0.195743i
$300$		0			0
$301$	16.7708	−	29.0479i	0.966654	−	1.67429i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.30278			0.475416
$306$		0			0
$307$	−21.9361			−1.25196			−0.625979	−	0.779840i	$-0.715301\pi$
$307$	−21.9361			−1.25196			−0.625979	+	0.779840i	$0.715301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.45416	+	16.3751i	−0.536096	+	0.928546i	0.463013	+	0.886352i	$0.346768\pi$
$311$	−9.45416	+	16.3751i	−0.536096	+	0.928546i	−0.999109	+	0.0421947i	$0.986565\pi$
$312$		0			0
$313$	−1.69722	−	2.93968i	−0.0959328	−	0.166160i	0.814065	−	0.580774i	$-0.197250\pi$
$313$	−1.69722	−	2.93968i	−0.0959328	−	0.166160i	−0.909998	+	0.414614i	$0.863917\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.5458	+	19.9980i	0.648479	+	1.12320i	0.983486	+	0.180983i	$0.0579278\pi$
$317$	11.5458	+	19.9980i	0.648479	+	1.12320i	−0.335007	+	0.942215i	$0.608739\pi$
$318$		0			0
$319$	−7.95416	+	13.7770i	−0.445348	+	0.771365i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.60555			−0.423184
$324$		0			0
$325$	0.302776			0.0167950
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.94029	−	17.2171i	0.548026	−	0.949208i
$330$		0			0
$331$	−6.40833	−	11.0995i	−0.352234	−	0.610086i	0.634407	−	0.772999i	$-0.281244\pi$
$331$	−6.40833	−	11.0995i	−0.352234	−	0.610086i	−0.986640	+	0.162913i	$0.947911\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.25694	−	2.17708i	−0.0686739	−	0.118947i
$336$		0			0
$337$	−0.197224	+	0.341603i	−0.0107435	+	0.0186083i	−0.871347	−	0.490667i	$-0.836753\pi$
$337$	−0.197224	+	0.341603i	−0.0107435	+	0.0186083i	0.860604	+	0.509275i	$0.170086\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.6056			0.574323
$342$		0			0
$343$	−3.60555			−0.194681
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.31665	−	10.9408i	0.339096	−	0.587331i	−0.645167	−	0.764041i	$-0.723212\pi$
$347$	6.31665	−	10.9408i	0.339096	−	0.587331i	0.984263	+	0.176710i	$0.0565456\pi$
$348$		0			0
$349$	−5.15139	−	8.92247i	−0.275747	−	0.477609i	0.694576	−	0.719419i	$-0.255592\pi$
$349$	−5.15139	−	8.92247i	−0.275747	−	0.477609i	−0.970323	+	0.241811i	$0.922259\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.8028	+	22.1751i	0.681423	+	1.18026i	0.974547	+	0.224185i	$0.0719719\pi$
$353$	12.8028	+	22.1751i	0.681423	+	1.18026i	−0.293124	+	0.956075i	$0.594695\pi$
$354$		0			0
$355$	−19.1194	+	33.1158i	−1.01475	+	1.75761i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.8167			0.887549			0.443775	−	0.896138i	$-0.353639\pi$
$359$	16.8167			0.887549			0.443775	+	0.896138i	$0.353639\pi$
$360$		0			0
$361$	−8.09167			−0.425878
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.5597	−	21.7541i	0.657406	−	1.13866i
$366$		0			0
$367$	0.288897	+	0.500385i	0.0150803	+	0.0261199i	0.873467	−	0.486883i	$-0.161866\pi$
$367$	0.288897	+	0.500385i	0.0150803	+	0.0261199i	−0.858387	+	0.513003i	$0.828533\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.15139	−	7.19041i	−0.215529	−	0.373308i
$372$		0			0
$373$	1.75694	−	3.04311i	0.0909709	−	0.157566i	−0.816949	−	0.576710i	$-0.804336\pi$
$373$	1.75694	−	3.04311i	0.0909709	−	0.157566i	0.907920	+	0.419144i	$0.137670\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.00000			0.154508
$378$		0			0
$379$	34.0278			1.74789			0.873944	−	0.486026i	$-0.161554\pi$
$379$	34.0278			1.74789			0.873944	+	0.486026i	$0.161554\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.00000	+	5.19615i	−0.153293	+	0.265511i	−0.932436	−	0.361335i	$-0.882321\pi$
$383$	−3.00000	+	5.19615i	−0.153293	+	0.265511i	0.779143	+	0.626846i	$0.215654\pi$
$384$		0			0
$385$	−22.0139	−	38.1292i	−1.12193	−	1.94324i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.51388	+	9.55032i	0.279565	+	0.484220i	0.971277	−	0.237953i	$-0.0764765\pi$
$389$	5.51388	+	9.55032i	0.279565	+	0.484220i	−0.691712	+	0.722174i	$0.743143\pi$
$390$		0			0
$391$	4.50000	−	7.79423i	0.227575	−	0.394171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−18.2111			−0.916300
$396$		0			0
$397$	−9.78890			−0.491291			−0.245645	−	0.969360i	$-0.579000\pi$
$397$	−9.78890			−0.491291			−0.245645	+	0.969360i	$0.579000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.90833	−	11.9656i	0.344985	−	0.597532i	−0.640366	−	0.768070i	$-0.721217\pi$
$401$	6.90833	−	11.9656i	0.344985	−	0.597532i	0.985351	+	0.170538i	$0.0545505\pi$
$402$		0			0
$403$	−1.00000	−	1.73205i	−0.0498135	−	0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	27.0736	+	46.8928i	1.34199	+	2.32439i
$408$		0			0
$409$	−8.81665	+	15.2709i	−0.435955	+	0.755097i	−0.997373	−	0.0724356i	$-0.976923\pi$
$409$	−8.81665	+	15.2709i	−0.435955	+	0.755097i	0.561418	+	0.827533i	$0.310256\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.761141			−0.0374533
$414$		0			0
$415$	24.9083			1.22270
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.4222	−	26.7120i	0.753424	−	1.30497i	−0.192730	−	0.981252i	$-0.561734\pi$
$419$	15.4222	−	26.7120i	0.753424	−	1.30497i	0.946154	−	0.323717i	$-0.104933\pi$
$420$		0			0
$421$	7.51388	+	13.0144i	0.366204	+	0.634284i	0.988969	−	0.148125i	$-0.0473239\pi$
$421$	7.51388	+	13.0144i	0.366204	+	0.634284i	−0.622765	+	0.782409i	$0.713991\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.348612	−	0.603814i	−0.0169102	−	0.0292893i
$426$		0			0
$427$	−6.50000	+	11.2583i	−0.314557	+	0.544829i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	41.2389			1.98641			0.993203	−	0.116395i	$-0.0371339\pi$
$431$	41.2389			1.98641			0.993203	+	0.116395i	$0.0371339\pi$
$432$		0			0
$433$	14.4861			0.696159			0.348079	−	0.937465i	$-0.386834\pi$
$433$	14.4861			0.696159			0.348079	+	0.937465i	$0.386834\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.45416	+	11.1789i	−0.308745	+	0.534761i
$438$		0			0
$439$	6.60555	+	11.4412i	0.315266	+	0.546056i	0.979494	−	0.201474i	$-0.0645730\pi$
$439$	6.60555	+	11.4412i	0.315266	+	0.546056i	−0.664228	+	0.747530i	$0.731240\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.80278	+	16.9789i	0.465744	+	0.806692i	0.999235	−	0.0391137i	$-0.0124535\pi$
$443$	9.80278	+	16.9789i	0.465744	+	0.806692i	−0.533491	+	0.845806i	$0.679120\pi$
$444$		0			0
$445$	5.30278	−	9.18468i	0.251376	−	0.435395i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.6333			1.44568			0.722838	−	0.691018i	$-0.242837\pi$
$449$	30.6333			1.44568			0.722838	+	0.691018i	$0.242837\pi$
$450$		0			0
$451$	−8.51388			−0.400903
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.15139	+	7.19041i	−0.194620	+	0.337092i
$456$		0			0
$457$	15.7111	+	27.2124i	0.734934	+	1.27294i	0.954752	+	0.297403i	$0.0961204\pi$
$457$	15.7111	+	27.2124i	0.734934	+	1.27294i	−0.219818	+	0.975541i	$0.570546\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.0458	−	22.5961i	−0.607605	−	1.05240i	−0.991634	−	0.129082i	$-0.958797\pi$
$461$	−13.0458	−	22.5961i	−0.607605	−	1.05240i	0.384029	−	0.923321i	$-0.374536\pi$
$462$		0			0
$463$	−0.165266	+	0.286249i	−0.00768056	+	0.0133031i	−0.869840	−	0.493334i	$-0.835778\pi$
$463$	−0.165266	+	0.286249i	−0.00768056	+	0.0133031i	0.862160	+	0.506637i	$0.169111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−29.2389			−1.35301			−0.676507	−	0.736437i	$-0.736507\pi$
$467$	−29.2389			−1.35301			−0.676507	+	0.736437i	$0.736507\pi$
$468$		0			0
$469$	3.93608			0.181751
$470$		0			0
$471$		0			0
$472$		0			0
$473$	24.6653	−	42.7215i	1.13411	−	1.96434i
$474$		0			0
$475$	0.500000	+	0.866025i	0.0229416	+	0.0397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.74306	−	13.4114i	−0.353789	−	0.612781i	0.633121	−	0.774053i	$-0.281774\pi$
$479$	−7.74306	−	13.4114i	−0.353789	−	0.612781i	−0.986910	+	0.161272i	$0.948440\pi$
$480$		0			0
$481$	5.10555	−	8.84307i	0.232793	−	0.403209i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.4222			−1.38140
$486$		0			0
$487$	−1.21110			−0.0548803			−0.0274401	−	0.999623i	$-0.508736\pi$
$487$	−1.21110			−0.0548803			−0.0274401	+	0.999623i	$0.508736\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.95416	+	3.38471i	−0.0881902	+	0.152750i	−0.906746	−	0.421677i	$-0.861442\pi$
$491$	−1.95416	+	3.38471i	−0.0881902	+	0.152750i	0.818556	+	0.574427i	$0.194775\pi$
$492$		0			0
$493$	−3.45416	−	5.98279i	−0.155568	−	0.269451i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−29.9361	−	51.8508i	−1.34282	−	2.32583i
$498$		0			0
$499$	−0.651388	+	1.12824i	−0.0291601	+	0.0505068i	−0.880237	−	0.474534i	$-0.842617\pi$
$499$	−0.651388	+	1.12824i	−0.0291601	+	0.0505068i	0.851077	+	0.525041i	$0.175950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.02776			−0.0904132			−0.0452066	−	0.998978i	$-0.514395\pi$
$503$	−2.02776			−0.0904132			−0.0452066	+	0.998978i	$0.514395\pi$
$504$		0			0
$505$	−28.1194			−1.25130
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.9083	+	22.3579i	−0.572152	+	0.990996i	0.424193	+	0.905572i	$0.360558\pi$
$509$	−12.9083	+	22.3579i	−0.572152	+	0.990996i	−0.996345	+	0.0854238i	$0.972776\pi$
$510$		0			0
$511$	19.6653	+	34.0612i	0.869940	+	1.50678i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.55971	−	11.3618i	−0.289056	−	0.500659i
$516$		0			0
$517$	14.6194	−	25.3216i	0.642962	−	1.11364i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.8444			−1.21989			−0.609943	−	0.792445i	$-0.708808\pi$
$521$	−27.8444			−1.21989			−0.609943	+	0.792445i	$0.708808\pi$
$522$		0			0
$523$	42.3305			1.85098			0.925492	−	0.378766i	$-0.123651\pi$
$523$	42.3305			1.85098			0.925492	+	0.378766i	$0.123651\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.30278	+	3.98852i	−0.100310	+	0.173743i
$528$		0			0
$529$	3.86249	+	6.69003i	0.167934	+	0.290871i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.802776	+	1.39045i	0.0347721	+	0.0602270i
$534$		0			0
$535$	−11.4083	+	19.7598i	−0.493225	+	0.854291i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	31.8167			1.37044
$540$		0			0
$541$	−3.09167			−0.132921			−0.0664607	−	0.997789i	$-0.521171\pi$
$541$	−3.09167			−0.132921			−0.0664607	+	0.997789i	$0.521171\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.7111	−	18.5522i	0.458813	−	0.794688i
$546$		0			0
$547$	−6.54584	−	11.3377i	−0.279880	−	0.484766i	0.691475	−	0.722401i	$-0.256961\pi$
$547$	−6.54584	−	11.3377i	−0.279880	−	0.484766i	−0.971355	+	0.237635i	$0.923628\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.95416	+	8.58086i	0.211054	+	0.365557i
$552$		0			0
$553$	14.2569	−	24.6937i	0.606267	−	1.05008i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.8167			1.47523			0.737614	−	0.675222i	$-0.235952\pi$
$557$	34.8167			1.47523			0.737614	+	0.675222i	$0.235952\pi$
$558$		0			0
$559$	−9.30278			−0.393465
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.28890	−	7.42859i	0.180755	−	0.313078i	−0.761383	−	0.648303i	$-0.775479\pi$
$563$	4.28890	−	7.42859i	0.180755	−	0.313078i	0.942138	+	0.335225i	$0.108812\pi$
$564$		0			0
$565$	−12.4542	−	21.5712i	−0.523951	−	0.907509i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.316654	−	0.548461i	−0.0132748	−	0.0229927i	0.859312	−	0.511452i	$-0.170892\pi$
$569$	−0.316654	−	0.548461i	−0.0132748	−	0.0229927i	−0.872587	+	0.488460i	$0.837559\pi$
$570$		0			0
$571$	−1.21110	+	2.09769i	−0.0506831	+	0.0877856i	−0.890254	−	0.455465i	$-0.849473\pi$
$571$	−1.21110	+	2.09769i	−0.0506831	+	0.0877856i	0.839571	+	0.543250i	$0.182807\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.18335			−0.0493489
$576$		0			0
$577$	−5.88057			−0.244811			−0.122406	−	0.992480i	$-0.539061\pi$
$577$	−5.88057			−0.244811			−0.122406	+	0.992480i	$0.539061\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.5000	+	33.7750i	−0.808996	+	1.40122i
$582$		0			0
$583$	−6.10555	−	10.5751i	−0.252866	−	0.437977i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.8486	−	23.9865i	−0.571593	−	0.990029i	−0.996403	−	0.0847459i	$-0.972992\pi$
$587$	−13.8486	−	23.9865i	−0.571593	−	0.990029i	0.424809	−	0.905283i	$-0.360341\pi$
$588$		0			0
$589$	3.30278	−	5.72058i	0.136088	−	0.235712i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.72498			0.111902			0.0559508	−	0.998434i	$-0.482181\pi$
$593$	2.72498			0.111902			0.0559508	+	0.998434i	$0.482181\pi$
$594$		0			0
$595$	19.1194			0.783820
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.7250	+	30.7006i	−0.724223	+	1.25439i	0.235070	+	0.971978i	$0.424468\pi$
$599$	−17.7250	+	30.7006i	−0.724223	+	1.25439i	−0.959293	+	0.282413i	$0.908865\pi$
$600$		0			0
$601$	8.77082	+	15.1915i	0.357769	+	0.619674i	0.987588	−	0.157068i	$-0.0502042\pi$
$601$	8.77082	+	15.1915i	0.357769	+	0.619674i	−0.629819	+	0.776742i	$0.716871\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.7111	−	34.1406i	−0.801370	−	1.38801i
$606$		0			0
$607$	9.95416	−	17.2411i	0.404027	−	0.699795i	−0.590181	−	0.807271i	$-0.700943\pi$
$607$	9.95416	−	17.2411i	0.404027	−	0.699795i	0.994208	+	0.107476i	$0.0342768\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.51388			−0.223068
$612$		0			0
$613$	29.4222			1.18835			0.594176	−	0.804335i	$-0.297478\pi$
$613$	29.4222			1.18835			0.594176	+	0.804335i	$0.297478\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.0458	+	17.3999i	−0.404430	+	0.700494i	−0.994255	−	0.107038i	$-0.965863\pi$
$617$	−10.0458	+	17.3999i	−0.404430	+	0.700494i	0.589825	+	0.807531i	$0.299197\pi$
$618$		0			0
$619$	6.81665	+	11.8068i	0.273984	+	0.474555i	0.969878	−	0.243590i	$-0.0783250\pi$
$619$	6.81665	+	11.8068i	0.273984	+	0.474555i	−0.695894	+	0.718145i	$0.744992\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.30278	+	14.3808i	0.332644	+	0.576156i
$624$		0			0
$625$	13.2111	−	22.8823i	0.528444	−	0.915292i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.5139			−0.937560
$630$		0			0
$631$	−7.42221			−0.295473			−0.147737	−	0.989027i	$-0.547199\pi$
$631$	−7.42221			−0.295473			−0.147737	+	0.989027i	$0.547199\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.3625	+	43.9291i	−1.00648	+	1.74327i
$636$		0			0
$637$	−3.00000	−	5.19615i	−0.118864	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.60555	−	7.97705i	−0.181908	−	0.315074i	0.760622	−	0.649195i	$-0.224894\pi$
$641$	−4.60555	−	7.97705i	−0.181908	−	0.315074i	−0.942530	+	0.334121i	$0.891561\pi$
$642$		0			0
$643$	−2.50000	+	4.33013i	−0.0985904	+	0.170764i	−0.911101	−	0.412182i	$-0.864767\pi$
$643$	−2.50000	+	4.33013i	−0.0985904	+	0.170764i	0.812511	+	0.582946i	$0.198100\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	34.5416			1.35797			0.678986	−	0.734151i	$-0.262420\pi$
$647$	34.5416			1.35797			0.678986	+	0.734151i	$0.262420\pi$
$648$		0			0
$649$	−1.11943			−0.0439415
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.1194	−	17.5274i	0.396004	−	0.685899i	−0.597225	−	0.802074i	$-0.703730\pi$
$653$	10.1194	−	17.5274i	0.396004	−	0.685899i	0.993229	+	0.116175i	$0.0370633\pi$
$654$		0			0
$655$	−17.7569	−	30.7559i	−0.693821	−	1.20173i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.5000	−	28.5788i	−0.642749	−	1.11327i	−0.984817	−	0.173598i	$-0.944461\pi$
$659$	−16.5000	−	28.5788i	−0.642749	−	1.11327i	0.342068	−	0.939675i	$-0.388873\pi$
$660$		0			0
$661$	0.288897	−	0.500385i	0.0112368	−	0.0194627i	−0.860352	−	0.509700i	$-0.829756\pi$
$661$	0.288897	−	0.500385i	0.0112368	−	0.0194627i	0.871589	+	0.490237i	$0.163090\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.4222			−1.06339
$666$		0			0
$667$	−11.7250			−0.453993
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.55971	+	16.5579i	−0.369049	+	0.639211i
$672$		0			0
$673$	−4.48612	−	7.77019i	−0.172927	−	0.299519i	0.766515	−	0.642227i	$-0.221989\pi$
$673$	−4.48612	−	7.77019i	−0.172927	−	0.299519i	−0.939442	+	0.342708i	$0.888656\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.5278	+	21.6987i	0.481481	+	0.833949i	0.999774	−	0.0212537i	$-0.00676578\pi$
$677$	12.5278	+	21.6987i	0.481481	+	0.833949i	−0.518293	+	0.855203i	$0.673432\pi$
$678$		0			0
$679$	23.8167	−	41.2517i	0.913999	−	1.58309i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.3944			0.512524			0.256262	−	0.966607i	$-0.417509\pi$
$683$	13.3944			0.512524			0.256262	+	0.966607i	$0.417509\pi$
$684$		0			0
$685$	3.21110			0.122690
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.15139	+	1.99426i	−0.0438644	+	0.0759753i
$690$		0			0
$691$	0.880571	+	1.52519i	0.0334985	+	0.0580211i	0.882289	−	0.470709i	$-0.156002\pi$
$691$	0.880571	+	1.52519i	0.0334985	+	0.0580211i	−0.848790	+	0.528730i	$0.822668\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.3625	−	28.3407i	−0.620665	−	1.07502i
$696$		0			0
$697$	1.84861	−	3.20189i	0.0700212	−	0.121280i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	49.6056			1.87358			0.936788	−	0.349898i	$-0.113784\pi$
$701$	49.6056			1.87358			0.936788	+	0.349898i	$0.113784\pi$
$702$		0			0
$703$	33.7250			1.27196
$704$		0			0
$705$		0			0
$706$		0			0
$707$	22.0139	−	38.1292i	0.827917	−	1.43399i
$708$		0			0
$709$	−13.2431	−	22.9377i	−0.497354	−	0.861442i	0.502642	−	0.864495i	$-0.332361\pi$
$709$	−13.2431	−	22.9377i	−0.497354	−	0.861442i	−0.999995	+	0.00305312i	$0.999028\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.90833	+	6.76942i	0.146368	+	0.253517i
$714$		0			0
$715$	−6.10555	+	10.5751i	−0.228335	+	0.395487i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−37.8167			−1.41032			−0.705162	−	0.709047i	$-0.749126\pi$
$719$	−37.8167			−1.41032			−0.705162	+	0.709047i	$0.749126\pi$
$720$		0			0
$721$	20.5416			0.765010
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.454163	+	0.786634i	−0.0168672	+	0.0292149i
$726$		0			0
$727$	−4.31665	−	7.47666i	−0.160096	−	0.277294i	0.774807	−	0.632198i	$-0.217847\pi$
$727$	−4.31665	−	7.47666i	−0.160096	−	0.277294i	−0.934903	+	0.354904i	$0.884514\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.7111	+	18.5522i	0.396164	+	0.686177i
$732$		0			0
$733$	4.78890	−	8.29461i	0.176882	−	0.306369i	−0.763929	−	0.645300i	$-0.776732\pi$
$733$	4.78890	−	8.29461i	0.176882	−	0.306369i	0.940811	+	0.338932i	$0.110066\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.78890			0.213237
$738$		0			0
$739$	−48.9361			−1.80014			−0.900071	−	0.435742i	$-0.856486\pi$
$739$	−48.9361			−1.80014			−0.900071	+	0.435742i	$0.856486\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.54584	−	4.40952i	0.0933977	−	0.161770i	−0.815541	−	0.578699i	$-0.803561\pi$
$743$	2.54584	−	4.40952i	0.0933977	−	0.161770i	0.908939	+	0.416930i	$0.136894\pi$
$744$		0			0
$745$	20.4083	+	35.3483i	0.747703	+	1.29506i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.8625	−	30.9387i	−0.652681	−	1.13048i
$750$		0			0
$751$	−15.2708	+	26.4498i	−0.557240	+	0.965168i	0.440485	+	0.897760i	$0.354806\pi$
$751$	−15.2708	+	26.4498i	−0.557240	+	0.965168i	−0.997725	+	0.0674083i	$0.978527\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−26.7250			−0.972622
$756$		0			0
$757$	−11.3305			−0.411815			−0.205908	−	0.978571i	$-0.566015\pi$
$757$	−11.3305			−0.411815			−0.205908	+	0.978571i	$0.566015\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.3944	+	18.0037i	−0.376798	+	0.652634i	−0.990594	−	0.136831i	$-0.956308\pi$
$761$	−10.3944	+	18.0037i	−0.376798	+	0.652634i	0.613796	+	0.789465i	$0.289642\pi$
$762$		0			0
$763$	16.7708	+	29.0479i	0.607144	+	1.05160i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.105551	+	0.182820i	0.00381124	+	0.00660125i
$768$		0			0
$769$	−9.40833	+	16.2957i	−0.339273	+	0.587638i	−0.984296	−	0.176525i	$-0.943514\pi$
$769$	−9.40833	+	16.2957i	−0.339273	+	0.587638i	0.645023	+	0.764163i	$0.276848\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.2750			0.441502			0.220751	−	0.975330i	$-0.429149\pi$
$773$	12.2750			0.441502			0.220751	+	0.975330i	$0.429149\pi$
$774$		0			0
$775$	0.605551			0.0217520
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.65139	+	4.59234i	−0.0949958	+	0.164538i
$780$		0			0
$781$	−44.0278	−	76.2583i	−1.57544	−	2.72874i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.1972	+	19.3942i	0.399646	+	0.692207i
$786$		0			0
$787$	24.9222	−	43.1665i	0.888381	−	1.53872i	0.0465915	−	0.998914i	$-0.485164\pi$
$787$	24.9222	−	43.1665i	0.888381	−	1.53872i	0.841789	−	0.539806i	$-0.181503\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	39.0000			1.38668
$792$		0			0
$793$	3.60555			0.128037
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.7111	+	44.5329i	−0.910734	+	1.57744i	−0.0977041	+	0.995216i	$0.531150\pi$
$797$	−25.7111	+	44.5329i	−0.910734	+	1.57744i	−0.813030	+	0.582222i	$0.802183\pi$
$798$		0			0
$799$	6.34861	+	10.9961i	0.224598	+	0.389015i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	28.9222	+	50.0947i	1.02064	+	1.76780i
$804$		0			0
$805$	16.2250	−	28.1025i	0.571855	−	0.990483i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.1472			−0.848970			−0.424485	−	0.905435i	$-0.639545\pi$
$809$	−24.1472			−0.848970			−0.424485	+	0.905435i	$0.639545\pi$
$810$		0			0
$811$	−37.4222			−1.31407			−0.657036	−	0.753859i	$-0.728190\pi$
$811$	−37.4222			−1.31407			−0.657036	+	0.753859i	$0.728190\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.34861	+	5.79997i	−0.117297	+	0.203164i
$816$		0			0
$817$	−15.3625	−	26.6086i	−0.537466	−	0.930918i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.09167	−	8.81904i	−0.177701	−	0.307786i	0.763392	−	0.645936i	$-0.223533\pi$
$821$	−5.09167	−	8.81904i	−0.177701	−	0.307786i	−0.941093	+	0.338149i	$0.890199\pi$
$822$		0			0
$823$	22.4083	−	38.8124i	0.781105	−	1.35291i	−0.150193	−	0.988657i	$-0.547990\pi$
$823$	22.4083	−	38.8124i	0.781105	−	1.35291i	0.931298	−	0.364257i	$-0.118677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.6056			0.473111			0.236556	−	0.971618i	$-0.423981\pi$
$827$	13.6056			0.473111			0.236556	+	0.971618i	$0.423981\pi$
$828$		0			0
$829$	−19.2111			−0.667229			−0.333615	−	0.942710i	$-0.608268\pi$
$829$	−19.2111			−0.667229			−0.333615	+	0.942710i	$0.608268\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.90833	+	11.9656i	−0.239359	+	0.414583i
$834$		0			0
$835$	−17.2708	−	29.9139i	−0.597681	−	1.03521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.5597	−	21.7541i	−0.433610	−	0.751034i	0.563572	−	0.826067i	$-0.309427\pi$
$839$	−12.5597	−	21.7541i	−0.433610	−	0.751034i	−0.997181	+	0.0750336i	$0.976094\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.30278			0.0792179
$846$		0			0
$847$	61.7250			2.12090
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19.9542	+	34.5616i	−0.684020	+	1.18476i
$852$		0			0
$853$	−20.5736	−	35.6345i	−0.704426	−	1.22010i	−0.966898	−	0.255163i	$-0.917871\pi$
$853$	−20.5736	−	35.6345i	−0.704426	−	1.22010i	0.262472	−	0.964940i	$-0.415462\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.36249	+	12.7522i	0.251498	+	0.435607i	0.963938	−	0.266125i	$-0.0857435\pi$
$857$	7.36249	+	12.7522i	0.251498	+	0.435607i	−0.712441	+	0.701732i	$0.752410\pi$
$858$		0			0
$859$	−25.5278	+	44.2154i	−0.870996	+	1.50861i	−0.0100279	+	0.999950i	$0.503192\pi$
$859$	−25.5278	+	44.2154i	−0.870996	+	1.50861i	−0.860968	+	0.508659i	$0.830141\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.3944			−0.864437			−0.432219	−	0.901769i	$-0.642269\pi$
$863$	−25.3944			−0.864437			−0.432219	+	0.901769i	$0.642269\pi$
$864$		0			0
$865$	−29.2389			−0.994151
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.9680	−	36.3177i	0.711292	−	1.23199i
$870$		0			0
$871$	−0.545837	−	0.945417i	−0.0184950	−	0.0320342i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.5000	+	33.7750i	0.659220	+	1.14180i
$876$		0			0
$877$	3.28890	−	5.69654i	0.111058	−	0.192358i	−0.805139	−	0.593086i	$-0.797909\pi$
$877$	3.28890	−	5.69654i	0.111058	−	0.192358i	0.916197	+	0.400728i	$0.131243\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33.4222			−1.12602			−0.563011	−	0.826449i	$-0.690357\pi$
$881$	−33.4222			−1.12602			−0.563011	+	0.826449i	$0.690357\pi$
$882$		0			0
$883$	−3.93608			−0.132460			−0.0662299	−	0.997804i	$-0.521097\pi$
$883$	−3.93608			−0.132460			−0.0662299	+	0.997804i	$0.521097\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.2250	+	28.1025i	−0.544782	+	0.943589i	0.453839	+	0.891084i	$0.350054\pi$
$887$	−16.2250	+	28.1025i	−0.544782	+	0.943589i	−0.998621	+	0.0525056i	$0.983279\pi$
$888$		0			0
$889$	−39.7111	−	68.7816i	−1.33187	−	2.30686i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.10555	−	15.7713i	−0.304706	−	0.527766i
$894$		0			0
$895$	23.8625	−	41.3310i	0.797635	−	1.38154i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.00000			0.200111
$900$		0			0
$901$	5.30278			0.176661
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.2569	+	17.7655i	−0.340952	+	0.590547i
$906$		0			0
$907$	−21.0278	−	36.4211i	−0.698215	−	1.20934i	−0.969085	−	0.246728i	$-0.920645\pi$
$907$	−21.0278	−	36.4211i	−0.698215	−	1.20934i	0.270870	−	0.962616i	$-0.412689\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.83473	+	6.64195i	0.127050	+	0.220058i	0.922533	−	0.385919i	$-0.126116\pi$
$911$	3.83473	+	6.64195i	0.127050	+	0.220058i	−0.795482	+	0.605977i	$0.792782\pi$
$912$		0			0
$913$	−28.6791	+	49.6737i	−0.949141	+	1.64396i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	55.6056			1.83626
$918$		0			0
$919$	38.0000			1.25350			0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000			1.25350			0.626752	+	0.779219i	$0.284384\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.30278	+	14.3808i	−0.273289	+	0.473351i
$924$		0			0
$925$	1.54584	+	2.67747i	0.0508268	+	0.0880346i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.10555	+	5.37897i	0.101890	+	0.176478i	0.912463	−	0.409159i	$-0.134178\pi$
$929$	3.10555	+	5.37897i	0.101890	+	0.176478i	−0.810573	+	0.585637i	$0.800844\pi$
$930$		0			0
$931$	9.90833	−	17.1617i	0.324732	−	0.562453i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28.1194			0.919604
$936$		0			0
$937$	−45.7250			−1.49377			−0.746885	−	0.664953i	$-0.768451\pi$
$937$	−45.7250			−1.49377			−0.746885	+	0.664953i	$0.768451\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.30278	−	14.3808i	0.270663	−	0.468802i	−0.698369	−	0.715738i	$-0.746091\pi$
$941$	8.30278	−	14.3808i	0.270663	−	0.468802i	0.969032	+	0.246936i	$0.0794238\pi$
$942$		0			0
$943$	−3.13751	−	5.43433i	−0.102171	−	0.176966i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.9222	−	34.5063i	−0.647385	−	1.12130i	−0.983745	−	0.179570i	$-0.942529\pi$
$947$	−19.9222	−	34.5063i	−0.647385	−	1.12130i	0.336361	−	0.941733i	$-0.390804\pi$
$948$		0			0
$949$	5.45416	−	9.44689i	0.177050	−	0.306659i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$	22.3305			0.722599
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.51388	+	4.35416i	−0.0811774	+	0.140603i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−28.0139	−	48.5215i	−0.901799	−	1.56196i
$966$		0			0
$967$	24.3305	−	42.1417i	0.782417	−	1.35519i	−0.148113	−	0.988970i	$-0.547320\pi$
$967$	24.3305	−	42.1417i	0.782417	−	1.35519i	0.930530	−	0.366216i	$-0.119347\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.2111			0.873246			0.436623	−	0.899645i	$-0.356174\pi$
$971$	27.2111			0.873246			0.436623	+	0.899645i	$0.356174\pi$
$972$		0			0
$973$	51.2389			1.64264
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.90833	+	11.9656i	−0.221017	+	0.382813i	−0.955117	−	0.296229i	$-0.904271\pi$
$977$	−6.90833	+	11.9656i	−0.221017	+	0.382813i	0.734100	+	0.679041i	$0.237604\pi$
$978$		0			0
$979$	12.2111	+	21.1503i	0.390269	+	0.675965i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.7250	+	35.8967i	0.661024	+	1.14493i	0.980347	+	0.197281i	$0.0632113\pi$
$983$	20.7250	+	35.8967i	0.661024	+	1.14493i	−0.319323	+	0.947646i	$0.603455\pi$
$984$		0			0
$985$	15.9083	−	27.5540i	0.506881	−	0.877944i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.3583			1.15613
$990$		0			0
$991$	18.3944			0.584319			0.292159	−	0.956370i	$-0.405626\pi$
$991$	18.3944			0.584319			0.292159	+	0.956370i	$0.405626\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25.2569	−	43.7463i	0.800699	−	1.38685i
$996$		0			0
$997$	−3.75694	−	6.50721i	−0.118983	−	0.206085i	0.800382	−	0.599491i	$-0.204630\pi$
$997$	−3.75694	−	6.50721i	−0.118983	−	0.206085i	−0.919365	+	0.393405i	$0.871297\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.n.1405.1		4
3.2	odd	2		4212.2.i.s.1405.2		4
9.2	odd	6		4212.2.i.s.2809.2		4
9.4	even	3		1404.2.a.i.1.2	yes	2
9.5	odd	6		1404.2.a.f.1.1	✓	2
9.7	even	3	inner	4212.2.i.n.2809.1		4
36.23	even	6		5616.2.a.bp.1.1		2
36.31	odd	6		5616.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1404.2.a.f.1.1	✓	2	9.5	odd	6
1404.2.a.i.1.2	yes	2	9.4	even	3
4212.2.i.n.1405.1		4	1.1	even	1	trivial
4212.2.i.n.2809.1		4	9.7	even	3	inner
4212.2.i.s.1405.2		4	3.2	odd	2
4212.2.i.s.2809.2		4	9.2	odd	6
5616.2.a.bp.1.1		2	36.23	even	6
5616.2.a.bt.1.2		2	36.31	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.15139 + 1.99426i) q^{5} +(-1.80278 - 3.12250i) q^{7} +(-2.65139 - 4.59234i) q^{11} +(-0.500000 + 0.866025i) q^{13} +2.30278 q^{17} -3.30278 q^{19} +(1.95416 - 3.38471i) q^{23} +(-0.151388 - 0.262211i) q^{25} +(-1.50000 - 2.59808i) q^{29} +(-1.00000 + 1.73205i) q^{31} +8.30278 q^{35} -10.2111 q^{37} +(0.802776 - 1.39045i) q^{41} +(4.65139 + 8.05644i) q^{43} +(2.75694 + 4.77516i) q^{47} +(-3.00000 + 5.19615i) q^{49} +2.30278 q^{53} +12.2111 q^{55} +(0.105551 - 0.182820i) q^{59} +(-1.80278 - 3.12250i) q^{61} +(-1.15139 - 1.99426i) q^{65} +(-0.545837 + 0.945417i) q^{67} +16.6056 q^{71} -10.9083 q^{73} +(-9.55971 + 16.5579i) q^{77} +(3.95416 + 6.84881i) q^{79} +(-5.40833 - 9.36750i) q^{83} +(-2.65139 + 4.59234i) q^{85} -4.60555 q^{89} +3.60555 q^{91} +(3.80278 - 6.58660i) q^{95} +(6.60555 + 11.4412i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} - 7 q^{11} - 2 q^{13} + 2 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{29} - 4 q^{31} + 26 q^{35} - 12 q^{37} - 4 q^{41} + 15 q^{43} - 7 q^{47} - 12 q^{49} + 2 q^{53} + 20 q^{55} - 14 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 4 * q - q^5 - 7 * q^11 - 2 * q^13 + 2 * q^17 - 6 * q^19 - 3 * q^23 + 3 * q^25 - 6 * q^29 - 4 * q^31 + 26 * q^35 - 12 * q^37 - 4 * q^41 + 15 * q^43 - 7 * q^47 - 12 * q^49 + 2 * q^53 + 20 * q^55 - 14 * q^59 - q^65 - 13 * q^67 + 52 * q^71 - 22 * q^73 - 13 * q^77 + 5 * q^79 - 7 * q^85 - 4 * q^89 + 8 * q^95 + 12 * q^97

Introduction

L-functions

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