LMFDB - Embedded newform 756.2.j.a.253.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(253,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("756.253");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.j (of order $3$, degree $2$, not 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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			253.1
Root			$0.500000 - 1.41036i$ of defining polynomial
Character	$\chi$	$=$	756.253
Dual form			756.2.j.a.505.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.97141 + 3.41458i) q^{5} +(-0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q+(-1.97141 + 3.41458i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(0.471410 + 0.816506i) q^{11} +(0.500000 - 0.866025i) q^{13} -5.60301 q^{17} +1.28263 q^{19} +(-2.33009 + 4.03584i) q^{23} +(-5.27292 - 9.13296i) q^{25} +(-3.83009 - 6.63392i) q^{29} +(-3.91423 + 6.77965i) q^{31} +3.94282 q^{35} -9.82846 q^{37} +(0.471410 - 0.816506i) q^{41} +(-4.63160 - 8.02217i) q^{43} +(-2.64132 - 4.57489i) q^{47} +(-0.500000 + 0.866025i) q^{49} +9.22545 q^{53} -3.71737 q^{55} +(-4.77292 + 8.26693i) q^{59} +(5.27292 + 9.13296i) q^{61} +(1.97141 + 3.41458i) q^{65} +(0.858685 - 1.48729i) q^{67} -3.54583 q^{71} +2.71737 q^{73} +(0.471410 - 0.816506i) q^{77} +(8.18715 + 14.1806i) q^{79} +(-0.198495 - 0.343803i) q^{83} +(11.0458 - 19.1319i) q^{85} -5.50808 q^{89} -1.00000 q^{91} +(-2.52859 + 4.37965i) q^{95} +(6.77292 + 11.7310i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - 3 q^{7}+O(q^{10}) $ 6 * q - 3 * q^5 - 3 * q^7	$ 6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67} + 30 q^{71} + 18 q^{73} - 6 q^{77} - 3 q^{79} - 18 q^{83} + 15 q^{85} - 12 q^{89} - 6 q^{91} - 24 q^{95} + 15 q^{97}+O(q^{100}) $ 6 * q - 3 * q^5 - 3 * q^7 - 6 * q^11 + 3 * q^13 + 6 * q^19 - 6 * q^23 - 6 * q^25 - 15 * q^29 + 3 * q^31 + 6 * q^35 - 6 * q^37 - 6 * q^41 - 3 * q^43 - 15 * q^47 - 3 * q^49 + 36 * q^53 - 24 * q^55 - 3 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 + 30 * q^71 + 18 * q^73 - 6 * q^77 - 3 * q^79 - 18 * q^83 + 15 * q^85 - 12 * q^89 - 6 * q^91 - 24 * q^95 + 15 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$5$	−1.97141	+	3.41458i	−0.881641	+	1.52705i	−0.0321260	+	0.999484i	$0.510228\pi$
$5$	−1.97141	+	3.41458i	−0.881641	+	1.52705i	−0.849515	+	0.527564i	$0.823106\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.471410	+	0.816506i	0.142135	+	0.246186i	0.928301	−	0.371831i	$-0.121270\pi$
$11$	0.471410	+	0.816506i	0.142135	+	0.246186i	−0.786165	+	0.618017i	$0.787936\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.60301			−1.35893			−0.679465	−	0.733708i	$-0.737788\pi$
$17$	−5.60301			−1.35893			−0.679465	+	0.733708i	$0.737788\pi$
$18$		0			0
$19$	1.28263			0.294256			0.147128	−	0.989117i	$-0.452997\pi$
$19$	1.28263			0.294256			0.147128	+	0.989117i	$0.452997\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.33009	+	4.03584i	−0.485858	+	0.841531i	−0.999868	−	0.0162531i	$-0.994826\pi$
$23$	−2.33009	+	4.03584i	−0.485858	+	0.841531i	0.514010	+	0.857784i	$0.328160\pi$
$24$		0			0
$25$	−5.27292	−	9.13296i	−1.05458	−	1.82659i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.83009	−	6.63392i	−0.711231	−	1.23189i	−0.964395	−	0.264465i	$-0.914805\pi$
$29$	−3.83009	−	6.63392i	−0.711231	−	1.23189i	0.253165	−	0.967423i	$-0.418529\pi$
$30$		0			0
$31$	−3.91423	+	6.77965i	−0.703016	+	1.21766i	0.264386	+	0.964417i	$0.414831\pi$
$31$	−3.91423	+	6.77965i	−0.703016	+	1.21766i	−0.967403	+	0.253243i	$0.918503\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.94282			0.666458
$36$		0			0
$37$	−9.82846			−1.61579			−0.807894	−	0.589327i	$-0.799393\pi$
$37$	−9.82846			−1.61579			−0.807894	+	0.589327i	$0.799393\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.471410	−	0.816506i	0.0736219	−	0.127517i	−0.826864	−	0.562402i	$-0.809878\pi$
$41$	0.471410	−	0.816506i	0.0736219	−	0.127517i	0.900486	+	0.434885i	$0.143211\pi$
$42$		0			0
$43$	−4.63160	−	8.02217i	−0.706312	−	1.22337i	−0.966216	−	0.257734i	$-0.917024\pi$
$43$	−4.63160	−	8.02217i	−0.706312	−	1.22337i	0.259903	−	0.965635i	$-0.416309\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.64132	−	4.57489i	−0.385275	−	0.667317i	0.606532	−	0.795059i	$-0.292560\pi$
$47$	−2.64132	−	4.57489i	−0.385275	−	0.667317i	−0.991807	+	0.127743i	$0.959227\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.22545			1.26721			0.633607	−	0.773655i	$-0.281574\pi$
$53$	9.22545			1.26721			0.633607	+	0.773655i	$0.281574\pi$
$54$		0			0
$55$	−3.71737			−0.501250
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.77292	+	8.26693i	−0.621381	+	1.07626i	0.367848	+	0.929886i	$0.380095\pi$
$59$	−4.77292	+	8.26693i	−0.621381	+	1.07626i	−0.989229	+	0.146377i	$0.953239\pi$
$60$		0			0
$61$	5.27292	+	9.13296i	0.675128	+	1.16936i	0.976432	+	0.215827i	$0.0692448\pi$
$61$	5.27292	+	9.13296i	0.675128	+	1.16936i	−0.301304	+	0.953528i	$0.597422\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.97141	+	3.41458i	0.244523	+	0.423527i
$66$		0			0
$67$	0.858685	−	1.48729i	0.104905	−	0.181701i	−0.808794	−	0.588092i	$-0.799879\pi$
$67$	0.858685	−	1.48729i	0.104905	−	0.181701i	0.913699	+	0.406391i	$0.133213\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.54583			−0.420813			−0.210406	−	0.977614i	$-0.567479\pi$
$71$	−3.54583			−0.420813			−0.210406	+	0.977614i	$0.567479\pi$
$72$		0			0
$73$	2.71737			0.318044			0.159022	−	0.987275i	$-0.449166\pi$
$73$	2.71737			0.318044			0.159022	+	0.987275i	$0.449166\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.471410	−	0.816506i	0.0537222	−	0.0930495i
$78$		0			0
$79$	8.18715	+	14.1806i	0.921126	+	1.59544i	0.797676	+	0.603086i	$0.206062\pi$
$79$	8.18715	+	14.1806i	0.921126	+	1.59544i	0.123449	+	0.992351i	$0.460604\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.198495	−	0.343803i	−0.0217877	−	0.0377373i	0.854926	−	0.518750i	$-0.173602\pi$
$83$	−0.198495	−	0.343803i	−0.0217877	−	0.0377373i	−0.876714	+	0.481013i	$0.840269\pi$
$84$		0			0
$85$	11.0458	−	19.1319i	1.19809	−	2.07515i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.50808			−0.583855			−0.291928	−	0.956440i	$-0.594297\pi$
$89$	−5.50808			−0.583855			−0.291928	+	0.956440i	$0.594297\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.52859	+	4.37965i	−0.259428	+	0.449342i
$96$		0			0
$97$	6.77292	+	11.7310i	0.687685	+	1.19111i	0.972585	+	0.232549i	$0.0747064\pi$
$97$	6.77292	+	11.7310i	0.687685	+	1.19111i	−0.284899	+	0.958557i	$0.591960\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.30150	−	2.25427i	−0.129505	−	0.224309i	0.793980	−	0.607944i	$-0.208005\pi$
$101$	−1.30150	−	2.25427i	−0.129505	−	0.224309i	−0.923485	+	0.383635i	$0.874672\pi$
$102$		0			0
$103$	−7.99028	+	13.8396i	−0.787306	+	1.36365i	0.140305	+	0.990108i	$0.455192\pi$
$103$	−7.99028	+	13.8396i	−0.787306	+	1.36365i	−0.927612	+	0.373546i	$0.878142\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.8856			1.05235			0.526177	−	0.850375i	$-0.323625\pi$
$107$	10.8856			1.05235			0.526177	+	0.850375i	$0.323625\pi$
$108$		0			0
$109$	−3.28263			−0.314419			−0.157209	−	0.987565i	$-0.550250\pi$
$109$	−3.28263			−0.314419			−0.157209	+	0.987565i	$0.550250\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.22545	+	5.58664i	−0.303425	+	0.525547i	−0.976909	−	0.213654i	$-0.931463\pi$
$113$	−3.22545	+	5.58664i	−0.303425	+	0.525547i	0.673485	+	0.739201i	$0.264797\pi$
$114$		0			0
$115$	−9.18715	−	15.9126i	−0.856706	−	1.48386i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.80150	+	4.85235i	0.256814	+	0.444814i
$120$		0			0
$121$	5.05555	−	8.75646i	0.459595	−	0.796042i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	21.8662			1.95577
$126$		0			0
$127$	−0.828460			−0.0735140			−0.0367570	−	0.999324i	$-0.511703\pi$
$127$	−0.828460			−0.0735140			−0.0367570	+	0.999324i	$0.511703\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.71574	−	4.70379i	0.237275	−	0.410972i	−0.722656	−	0.691207i	$-0.757079\pi$
$131$	2.71574	−	4.70379i	0.237275	−	0.410972i	0.959931	+	0.280235i	$0.0904125\pi$
$132$		0			0
$133$	−0.641315	−	1.11079i	−0.0556091	−	0.0963177i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.5744	−	18.3154i	−0.903434	−	1.56479i	−0.823006	−	0.568033i	$-0.807705\pi$
$137$	−10.5744	−	18.3154i	−0.903434	−	1.56479i	−0.0804276	−	0.996760i	$-0.525629\pi$
$138$		0			0
$139$	0.923945	−	1.60032i	0.0783680	−	0.135737i	−0.824178	−	0.566331i	$-0.808362\pi$
$139$	0.923945	−	1.60032i	0.0783680	−	0.135737i	0.902546	+	0.430594i	$0.141696\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.942820			0.0788426
$144$		0			0
$145$	30.2028			2.50820
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.07442	−	10.5212i	0.497636	−	0.861931i	−0.502360	−	0.864658i	$-0.667535\pi$
$149$	6.07442	−	10.5212i	0.497636	−	0.861931i	0.999996	+	0.00272771i	$0.000868258\pi$
$150$		0			0
$151$	4.49028	+	7.77740i	0.365414	+	0.632916i	0.988843	−	0.148965i	$-0.0475940\pi$
$151$	4.49028	+	7.77740i	0.365414	+	0.632916i	−0.623428	+	0.781880i	$0.714261\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.4331	−	26.7309i	−1.23962	−	2.14708i
$156$		0			0
$157$	5.90451	−	10.2269i	0.471232	−	0.816197i	−0.528227	−	0.849103i	$-0.677143\pi$
$157$	5.90451	−	10.2269i	0.471232	−	0.816197i	0.999458	+	0.0329062i	$0.0104763\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.66019			0.367274
$162$		0			0
$163$	−12.2826			−0.962050			−0.481025	−	0.876707i	$-0.659735\pi$
$163$	−12.2826			−0.962050			−0.481025	+	0.876707i	$0.659735\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4428	+	19.8195i	−0.885472	+	1.53368i	−0.0403003	+	0.999188i	$0.512831\pi$
$167$	−11.4428	+	19.8195i	−0.885472	+	1.53368i	−0.845172	+	0.534495i	$0.820502\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.63160	+	16.6824i	0.732277	+	1.26834i	0.955908	+	0.293667i	$0.0948757\pi$
$173$	9.63160	+	16.6824i	0.732277	+	1.26834i	−0.223631	+	0.974674i	$0.571791\pi$
$174$		0			0
$175$	−5.27292	+	9.13296i	−0.398595	+	0.690387i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.83173			0.361140			0.180570	−	0.983562i	$-0.442206\pi$
$179$	4.83173			0.361140			0.180570	+	0.983562i	$0.442206\pi$
$180$		0			0
$181$	3.26320			0.242552			0.121276	−	0.992619i	$-0.461301\pi$
$181$	3.26320			0.242552			0.121276	+	0.992619i	$0.461301\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.3759	−	33.5601i	1.42455	−	2.46739i
$186$		0			0
$187$	−2.64132	−	4.57489i	−0.193152	−	0.334549i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.34897	+	12.7288i	0.531753	+	0.921023i	0.999313	+	0.0370616i	$0.0117998\pi$
$191$	7.34897	+	12.7288i	0.531753	+	0.921023i	−0.467560	+	0.883961i	$0.654867\pi$
$192$		0			0
$193$	−5.05555	+	8.75646i	−0.363906	+	0.630304i	−0.988600	−	0.150566i	$-0.951890\pi$
$193$	−5.05555	+	8.75646i	−0.363906	+	0.630304i	0.624694	+	0.780870i	$0.285224\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.6375			−1.18537			−0.592686	−	0.805434i	$-0.701933\pi$
$197$	−16.6375			−1.18537			−0.592686	+	0.805434i	$0.701933\pi$
$198$		0			0
$199$	13.1111			0.929421			0.464710	−	0.885463i	$-0.346158\pi$
$199$	13.1111			0.929421			0.464710	+	0.885463i	$0.346158\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.83009	+	6.63392i	−0.268820	+	0.465610i
$204$		0			0
$205$	1.85868	+	3.21934i	0.129816	+	0.224848i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.604645	+	1.04728i	0.0418242	+	0.0724416i
$210$		0			0
$211$	−1.06526	+	1.84509i	−0.0733355	+	0.127021i	−0.900361	−	0.435143i	$-0.856698\pi$
$211$	−1.06526	+	1.84509i	−0.0733355	+	0.127021i	0.827026	+	0.562164i	$0.190031\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	36.5231			2.49086
$216$		0			0
$217$	7.82846			0.531431
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.80150	+	4.85235i	−0.188450	+	0.326404i
$222$		0			0
$223$	7.04583	+	12.2037i	0.471824	+	0.817223i	0.999480	−	0.0322352i	$-0.0102626\pi$
$223$	7.04583	+	12.2037i	0.471824	+	0.817223i	−0.527657	+	0.849458i	$0.676929\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.39536	+	4.14888i	0.158985	+	0.275371i	0.934503	−	0.355955i	$-0.115844\pi$
$227$	2.39536	+	4.14888i	0.158985	+	0.275371i	−0.775518	+	0.631326i	$0.782511\pi$
$228$		0			0
$229$	−9.04583	+	15.6678i	−0.597765	+	1.03536i	0.395385	+	0.918516i	$0.370611\pi$
$229$	−9.04583	+	15.6678i	−0.597765	+	1.03536i	−0.993150	+	0.116844i	$0.962722\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.14884			0.206287			0.103144	−	0.994666i	$-0.467110\pi$
$233$	3.14884			0.206287			0.103144	+	0.994666i	$0.467110\pi$
$234$		0			0
$235$	20.8285			1.35870
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.16019	+	10.6698i	−0.398470	+	0.690170i	−0.993537	−	0.113506i	$-0.963792\pi$
$239$	−6.16019	+	10.6698i	−0.398470	+	0.690170i	0.595068	+	0.803676i	$0.297125\pi$
$240$		0			0
$241$	−7.27292	−	12.5971i	−0.468490	−	0.811448i	0.530862	−	0.847458i	$-0.321868\pi$
$241$	−7.27292	−	12.5971i	−0.468490	−	0.811448i	−0.999351	+	0.0360106i	$0.988535\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.97141	−	3.41458i	−0.125949	−	0.218150i
$246$		0			0
$247$	0.641315	−	1.11079i	0.0408059	−	0.0706779i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.28263			0.144078			0.0720392	−	0.997402i	$-0.477049\pi$
$251$	2.28263			0.144078			0.0720392	+	0.997402i	$0.477049\pi$
$252$		0			0
$253$	−4.39372			−0.276231
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.396990	−	0.687607i	0.0247636	−	0.0428917i	−0.853378	−	0.521293i	$-0.825450\pi$
$257$	0.396990	−	0.687607i	0.0247636	−	0.0428917i	0.878142	+	0.478401i	$0.158783\pi$
$258$		0			0
$259$	4.91423	+	8.51170i	0.305355	+	0.528891i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.24433	+	9.08344i	0.323379	+	0.560109i	0.981183	−	0.193080i	$-0.0618478\pi$
$263$	5.24433	+	9.08344i	0.323379	+	0.560109i	−0.657804	+	0.753189i	$0.728514\pi$
$264$		0			0
$265$	−18.1871	+	31.5011i	−1.11723	+	1.93509i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.3398			−0.630429			−0.315215	−	0.949020i	$-0.602077\pi$
$269$	−10.3398			−0.630429			−0.315215	+	0.949020i	$0.602077\pi$
$270$		0			0
$271$	−30.8285			−1.87270			−0.936348	−	0.351074i	$-0.885817\pi$
$271$	−30.8285			−1.87270			−0.936348	+	0.351074i	$0.885817\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.97141	−	8.61073i	0.299787	−	0.519247i
$276$		0			0
$277$	−7.12188	−	12.3355i	−0.427913	−	0.741166i	0.568775	−	0.822493i	$-0.307418\pi$
$277$	−7.12188	−	12.3355i	−0.427913	−	0.741166i	−0.996687	+	0.0813269i	$0.974084\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.46169	−	7.72788i	−0.266162	−	0.461007i	0.701705	−	0.712468i	$-0.252422\pi$
$281$	−4.46169	−	7.72788i	−0.266162	−	0.461007i	−0.967868	+	0.251461i	$0.919089\pi$
$282$		0			0
$283$	11.6316	−	20.1465i	0.691427	−	1.19759i	−0.279944	−	0.960016i	$-0.590316\pi$
$283$	11.6316	−	20.1465i	0.691427	−	1.19759i	0.971370	−	0.237570i	$-0.0763509\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.942820			−0.0556529
$288$		0			0
$289$	14.3937			0.846689
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.9412	−	20.6827i	0.697611	−	1.20830i	−0.271681	−	0.962387i	$-0.587580\pi$
$293$	11.9412	−	20.6827i	0.697611	−	1.20830i	0.969292	−	0.245911i	$-0.0790871\pi$
$294$		0			0
$295$	−18.8187	−	32.5950i	−1.09567	−	1.89776i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.33009	+	4.03584i	0.134753	+	0.233399i
$300$		0			0
$301$	−4.63160	+	8.02217i	−0.266961	+	0.462390i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−41.5803			−2.38088
$306$		0			0
$307$	−7.37429			−0.420873			−0.210436	−	0.977608i	$-0.567489\pi$
$307$	−7.37429			−0.420873			−0.210436	+	0.977608i	$0.567489\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.58577	−	7.94279i	0.260035	−	0.450394i	−0.706216	−	0.707997i	$-0.749599\pi$
$311$	4.58577	−	7.94279i	0.260035	−	0.450394i	0.966251	+	0.257603i	$0.0829325\pi$
$312$		0			0
$313$	−8.05555	−	13.9526i	−0.455326	−	0.788648i	0.543381	−	0.839486i	$-0.317144\pi$
$313$	−8.05555	−	13.9526i	−0.455326	−	0.788648i	−0.998707	+	0.0508381i	$0.983811\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.13160	−	3.69204i	−0.119723	−	0.207366i	0.799935	−	0.600086i	$-0.204867\pi$
$317$	−2.13160	−	3.69204i	−0.119723	−	0.207366i	−0.919658	+	0.392721i	$0.871534\pi$
$318$		0			0
$319$	3.61109	−	6.25459i	0.202182	−	0.350190i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.18659			−0.399873
$324$		0			0
$325$	−10.5458			−0.584977
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.64132	+	4.57489i	−0.145620	+	0.252222i
$330$		0			0
$331$	6.14132	+	10.6371i	0.337557	+	0.584666i	0.983973	−	0.178319i	$-0.0570660\pi$
$331$	6.14132	+	10.6371i	0.337557	+	0.584666i	−0.646415	+	0.762986i	$0.723733\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.38564	+	5.86410i	0.184977	+	0.320390i
$336$		0			0
$337$	13.4903	−	23.3659i	0.734863	−	1.27282i	−0.219921	−	0.975518i	$-0.570580\pi$
$337$	13.4903	−	23.3659i	0.734863	−	1.27282i	0.954784	−	0.297302i	$-0.0960867\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.38083			−0.399694
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.7524	+	18.6237i	−0.577219	+	0.999773i	0.418577	+	0.908181i	$0.362529\pi$
$347$	−10.7524	+	18.6237i	−0.577219	+	0.999773i	−0.995797	+	0.0915921i	$0.970804\pi$
$348$		0			0
$349$	2.54583	+	4.40951i	0.136275	+	0.236035i	0.926084	−	0.377318i	$-0.123154\pi$
$349$	2.54583	+	4.40951i	0.136275	+	0.236035i	−0.789809	+	0.613353i	$0.789820\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.78426	+	8.28659i	0.254641	+	0.441051i	0.964798	−	0.262993i	$-0.0847095\pi$
$353$	4.78426	+	8.28659i	0.254641	+	0.441051i	−0.710157	+	0.704043i	$0.751376\pi$
$354$		0			0
$355$	6.99028	−	12.1075i	0.371006	−	0.642601i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.9623			−0.578565			−0.289283	−	0.957244i	$-0.593417\pi$
$359$	−10.9623			−0.578565			−0.289283	+	0.957244i	$0.593417\pi$
$360$		0			0
$361$	−17.3549			−0.913414
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.35705	+	9.27868i	−0.280401	+	0.485668i
$366$		0			0
$367$	−2.34897	−	4.06853i	−0.122615	−	0.212376i	0.798183	−	0.602415i	$-0.205795\pi$
$367$	−2.34897	−	4.06853i	−0.122615	−	0.212376i	−0.920798	+	0.390039i	$0.872461\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.61273	−	7.98947i	−0.239481	−	0.414793i
$372$		0			0
$373$	−4.63160	+	8.02217i	−0.239815	+	0.415372i	−0.960661	−	0.277723i	$-0.910420\pi$
$373$	−4.63160	+	8.02217i	−0.239815	+	0.415372i	0.720846	+	0.693095i	$0.243754\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.66019			−0.394520
$378$		0			0
$379$	−13.5458			−0.695803			−0.347901	−	0.937531i	$-0.613106\pi$
$379$	−13.5458			−0.695803			−0.347901	+	0.937531i	$0.613106\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.13968	−	10.6342i	0.313723	−	0.543384i	−0.665442	−	0.746449i	$-0.731757\pi$
$383$	6.13968	−	10.6342i	0.313723	−	0.543384i	0.979165	+	0.203065i	$0.0650903\pi$
$384$		0			0
$385$	1.85868	+	3.21934i	0.0947274	+	0.164073i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.75567	−	6.50502i	−0.190420	−	0.329818i	0.754969	−	0.655760i	$-0.227652\pi$
$389$	−3.75567	−	6.50502i	−0.190420	−	0.329818i	−0.945390	+	0.325943i	$0.894318\pi$
$390$		0			0
$391$	13.0555	−	22.6129i	0.660247	−	1.14358i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−64.5609			−3.24841
$396$		0			0
$397$	−22.9201			−1.15033			−0.575164	−	0.818038i	$-0.695062\pi$
$397$	−22.9201			−1.15033			−0.575164	+	0.818038i	$0.695062\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.5631	+	18.2958i	−0.527495	+	0.913647i	0.471992	+	0.881603i	$0.343535\pi$
$401$	−10.5631	+	18.2958i	−0.527495	+	0.913647i	−0.999486	+	0.0320445i	$0.989798\pi$
$402$		0			0
$403$	3.91423	+	6.77965i	0.194982	+	0.337718i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.63323	−	8.02500i	−0.229661	−	0.397784i
$408$		0			0
$409$	−1.71737	+	2.97457i	−0.0849185	+	0.147083i	−0.905356	−	0.424652i	$-0.860396\pi$
$409$	−1.71737	+	2.97457i	−0.0849185	+	0.147083i	0.820438	+	0.571736i	$0.193730\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.54583			0.469720
$414$		0			0
$415$	1.56526			0.0768356
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.2157	+	17.6942i	−0.499071	+	0.864417i	−0.999999	−	0.00107202i	$-0.999659\pi$
$419$	−10.2157	+	17.6942i	−0.499071	+	0.864417i	0.500928	+	0.865489i	$0.332992\pi$
$420$		0			0
$421$	−7.90451	−	13.6910i	−0.385242	−	0.667260i	0.606560	−	0.795037i	$-0.292549\pi$
$421$	−7.90451	−	13.6910i	−0.385242	−	0.667260i	−0.991803	+	0.127778i	$0.959216\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	29.5442	+	51.1720i	1.43310	+	2.48221i
$426$		0			0
$427$	5.27292	−	9.13296i	0.255174	−	0.441975i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.7518			1.28859			0.644296	−	0.764777i	$-0.277151\pi$
$431$	26.7518			1.28859			0.644296	+	0.764777i	$0.277151\pi$
$432$		0			0
$433$	−14.4347			−0.693689			−0.346845	−	0.937923i	$-0.612747\pi$
$433$	−14.4347			−0.693689			−0.346845	+	0.937923i	$0.612747\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.98865	+	5.17649i	−0.142967	+	0.247625i
$438$		0			0
$439$	19.3187	+	33.4610i	0.922033	+	1.59701i	0.796264	+	0.604949i	$0.206807\pi$
$439$	19.3187	+	33.4610i	0.922033	+	1.59701i	0.125769	+	0.992060i	$0.459860\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.28263	+	9.14978i	0.250985	+	0.434719i	0.963797	−	0.266636i	$-0.0859121\pi$
$443$	5.28263	+	9.14978i	0.250985	+	0.434719i	−0.712812	+	0.701355i	$0.752579\pi$
$444$		0			0
$445$	10.8587	−	18.8078i	0.514751	−	0.891575i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.37429			0.442400			0.221200	−	0.975228i	$-0.429003\pi$
$449$	9.37429			0.442400			0.221200	+	0.975228i	$0.429003\pi$
$450$		0			0
$451$	0.888910			0.0418571
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.97141	−	3.41458i	0.0924211	−	0.160078i
$456$		0			0
$457$	−10.3587	−	17.9418i	−0.484559	−	0.839281i	0.515284	−	0.857020i	$-0.327687\pi$
$457$	−10.3587	−	17.9418i	−0.484559	−	0.839281i	−0.999843	+	0.0177391i	$0.994353\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.6300	+	30.5360i	0.821109	+	1.42220i	0.904857	+	0.425716i	$0.139978\pi$
$461$	17.6300	+	30.5360i	0.821109	+	1.42220i	−0.0837475	+	0.996487i	$0.526689\pi$
$462$		0			0
$463$	−3.55555	+	6.15838i	−0.165240	+	0.286204i	−0.936741	−	0.350024i	$-0.886173\pi$
$463$	−3.55555	+	6.15838i	−0.165240	+	0.286204i	0.771500	+	0.636229i	$0.219507\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.03448			−0.232968			−0.116484	−	0.993193i	$-0.537162\pi$
$467$	−5.03448			−0.232968			−0.116484	+	0.993193i	$0.537162\pi$
$468$		0			0
$469$	−1.71737			−0.0793008
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.36677	−	7.56346i	0.200784	−	0.347768i
$474$		0			0
$475$	−6.76320	−	11.7142i	−0.310317	−	0.537485i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.13323	+	5.42692i	0.143161	+	0.247962i	0.928685	−	0.370869i	$-0.120940\pi$
$479$	3.13323	+	5.42692i	0.143161	+	0.247962i	−0.785524	+	0.618831i	$0.787607\pi$
$480$		0			0
$481$	−4.91423	+	8.51170i	−0.224070	+	0.388100i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−53.4088			−2.42517
$486$		0			0
$487$	14.0722			0.637674			0.318837	−	0.947810i	$-0.396708\pi$
$487$	14.0722			0.637674			0.318837	+	0.947810i	$0.396708\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.97141	−	3.41458i	0.0889685	−	0.154098i	−0.818107	−	0.575066i	$-0.804976\pi$
$491$	1.97141	−	3.41458i	0.0889685	−	0.154098i	0.907075	+	0.420968i	$0.138310\pi$
$492$		0			0
$493$	21.4601	+	37.1699i	0.966512	+	1.67405i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.77292	+	3.07078i	0.0795261	+	0.137743i
$498$		0			0
$499$	−9.82846	+	17.0234i	−0.439982	+	0.762072i	−0.997688	−	0.0679674i	$-0.978349\pi$
$499$	−9.82846	+	17.0234i	−0.439982	+	0.762072i	0.557705	+	0.830039i	$0.311682\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.2632			0.457613			0.228807	−	0.973472i	$-0.426518\pi$
$503$	10.2632			0.457613			0.228807	+	0.973472i	$0.426518\pi$
$504$		0			0
$505$	10.2632			0.456706
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.669905	−	1.16031i	0.0296930	−	0.0514298i	−0.850797	−	0.525494i	$-0.823880\pi$
$509$	0.669905	−	1.16031i	0.0296930	−	0.0514298i	0.880490	+	0.474065i	$0.157214\pi$
$510$		0			0
$511$	−1.35868	−	2.35331i	−0.0601047	−	0.104104i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−31.5043	−	54.5670i	−1.38824	−	2.40451i
$516$		0			0
$517$	2.49028	−	4.31330i	0.109523	−	0.189699i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.7940			−0.691947			−0.345973	−	0.938244i	$-0.612451\pi$
$521$	−15.7940			−0.691947			−0.345973	+	0.938244i	$0.612451\pi$
$522$		0			0
$523$	−10.6764			−0.466844			−0.233422	−	0.972376i	$-0.574992\pi$
$523$	−10.6764			−0.466844			−0.233422	+	0.972376i	$0.574992\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.9315	−	37.9864i	0.955350	−	1.65471i
$528$		0			0
$529$	0.641315	+	1.11079i	0.0278833	+	0.0482952i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.471410	−	0.816506i	−0.0204190	−	0.0353668i
$534$		0			0
$535$	−21.4601	+	37.1699i	−0.927799	+	1.60700i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.942820			−0.0406101
$540$		0			0
$541$	16.6569			0.716137			0.358068	−	0.933695i	$-0.383435\pi$
$541$	16.6569			0.716137			0.358068	+	0.933695i	$0.383435\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.47141	−	11.2088i	0.277205	−	0.480133i
$546$		0			0
$547$	−7.81875	−	13.5425i	−0.334305	−	0.579034i	0.649046	−	0.760749i	$-0.275168\pi$
$547$	−7.81875	−	13.5425i	−0.334305	−	0.579034i	−0.983351	+	0.181715i	$0.941835\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.91260	−	8.50886i	−0.209284	−	0.362490i
$552$		0			0
$553$	8.18715	−	14.1806i	0.348153	−	0.603018i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.7907			0.711445			0.355723	−	0.934592i	$-0.384235\pi$
$557$	16.7907			0.711445			0.355723	+	0.934592i	$0.384235\pi$
$558$		0			0
$559$	−9.26320			−0.391792
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.58577	−	13.1389i	0.319702	−	0.553740i	−0.660724	−	0.750629i	$-0.729751\pi$
$563$	7.58577	−	13.1389i	0.319702	−	0.553740i	0.980426	+	0.196889i	$0.0630838\pi$
$564$		0			0
$565$	−12.7174	−	22.0271i	−0.535024	−	0.926688i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.7632	+	30.7668i	0.744672	+	1.28981i	0.950348	+	0.311190i	$0.100727\pi$
$569$	17.7632	+	30.7668i	0.744672	+	1.28981i	−0.205676	+	0.978620i	$0.565939\pi$
$570$		0			0
$571$	0.772915	−	1.33873i	0.0323455	−	0.0560240i	−0.849400	−	0.527750i	$-0.823036\pi$
$571$	0.772915	−	1.33873i	0.0323455	−	0.0560240i	0.881745	+	0.471726i	$0.156369\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	49.1456			2.04951
$576$		0			0
$577$	−1.67635			−0.0697874			−0.0348937	−	0.999391i	$-0.511109\pi$
$577$	−1.67635			−0.0697874			−0.0348937	+	0.999391i	$0.511109\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.198495	+	0.343803i	−0.00823496	+	0.0142634i
$582$		0			0
$583$	4.34897	+	7.53264i	0.180116	+	0.311970i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.7346	−	34.1813i	−0.814535	−	1.41082i	−0.909662	−	0.415350i	$-0.863659\pi$
$587$	−19.7346	−	34.1813i	−0.814535	−	1.41082i	0.0951271	−	0.995465i	$-0.469674\pi$
$588$		0			0
$589$	−5.02051	+	8.69578i	−0.206866	+	0.358303i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.8317			−0.937587			−0.468793	−	0.883308i	$-0.655311\pi$
$593$	−22.8317			−0.937587			−0.468793	+	0.883308i	$0.655311\pi$
$594$		0			0
$595$	−22.0917			−0.905670
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.6375	−	39.2093i	0.924943	−	1.60205i	0.133290	−	0.991077i	$-0.457446\pi$
$599$	22.6375	−	39.2093i	0.924943	−	1.60205i	0.791653	−	0.610971i	$-0.209221\pi$
$600$		0			0
$601$	−13.5253	−	23.4265i	−0.551709	−	0.955589i	−0.998151	−	0.0607761i	$-0.980642\pi$
$601$	−13.5253	−	23.4265i	−0.551709	−	0.955589i	0.446442	−	0.894813i	$-0.352691\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.9331	+	34.5252i	0.810396	+	1.40365i
$606$		0			0
$607$	9.32846	−	16.1574i	0.378631	−	0.655807i	−0.612233	−	0.790678i	$-0.709728\pi$
$607$	9.32846	−	16.1574i	0.378631	−	0.655807i	0.990863	+	0.134870i	$0.0430618\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.28263			−0.213712
$612$		0			0
$613$	−28.9611			−1.16973			−0.584865	−	0.811131i	$-0.698852\pi$
$613$	−28.9611			−1.16973			−0.584865	+	0.811131i	$0.698852\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.8457	−	32.6417i	0.758699	−	1.31411i	−0.184815	−	0.982773i	$-0.559169\pi$
$617$	18.8457	−	32.6417i	0.758699	−	1.31411i	0.943514	−	0.331332i	$-0.107498\pi$
$618$		0			0
$619$	−0.00971516	−	0.0168272i	−0.000390485	−	0.000676340i	0.865830	−	0.500338i	$-0.166791\pi$
$619$	−0.00971516	−	0.0168272i	−0.000390485	−	0.000676340i	−0.866221	+	0.499662i	$0.833458\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.75404	+	4.77014i	0.110338	+	0.191112i
$624$		0			0
$625$	−16.7427	+	28.9992i	−0.669708	+	1.15997i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	55.0690			2.19574
$630$		0			0
$631$	28.4854			1.13399			0.566993	−	0.823723i	$-0.308107\pi$
$631$	28.4854			1.13399			0.566993	+	0.823723i	$0.308107\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.63323	−	2.82885i	0.0648129	−	0.112259i
$636$		0			0
$637$	0.500000	+	0.866025i	0.0198107	+	0.0343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.2696	+	38.5722i	0.879598	+	1.52351i	0.851783	+	0.523896i	$0.175522\pi$
$641$	22.2696	+	38.5722i	0.879598	+	1.52351i	0.0278156	+	0.999613i	$0.491145\pi$
$642$		0			0
$643$	−4.50972	+	7.81106i	−0.177846	+	0.308038i	−0.941142	−	0.338010i	$-0.890246\pi$
$643$	−4.50972	+	7.81106i	−0.177846	+	0.308038i	0.763297	+	0.646048i	$0.223579\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−42.8252			−1.68363			−0.841816	−	0.539765i	$-0.818513\pi$
$647$	−42.8252			−1.68363			−0.841816	+	0.539765i	$0.818513\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.80150	−	4.85235i	0.109631	−	0.189887i	−0.805990	−	0.591930i	$-0.798366\pi$
$653$	2.80150	−	4.85235i	0.109631	−	0.189887i	0.915621	+	0.402043i	$0.131700\pi$
$654$		0			0
$655$	10.7077	+	18.5462i	0.418383	+	0.724660i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.26376	−	5.65299i	−0.127138	−	0.220209i	0.795429	−	0.606047i	$-0.207246\pi$
$659$	−3.26376	−	5.65299i	−0.127138	−	0.220209i	−0.922567	+	0.385838i	$0.873912\pi$
$660$		0			0
$661$	5.50972	−	9.54311i	0.214303	−	0.371184i	−0.738754	−	0.673976i	$-0.764585\pi$
$661$	5.50972	−	9.54311i	0.214303	−	0.371184i	0.953057	+	0.302792i	$0.0979186\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.05718			0.196109
$666$		0			0
$667$	35.6979			1.38223
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.97141	+	8.61073i	−0.191919	+	0.332414i
$672$		0			0
$673$	1.67743	+	2.90539i	0.0646602	+	0.111995i	0.896543	−	0.442956i	$-0.146070\pi$
$673$	1.67743	+	2.90539i	0.0646602	+	0.111995i	−0.831883	+	0.554951i	$0.812737\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.3187	−	35.1931i	−0.780913	−	1.35258i	−0.931411	−	0.363970i	$-0.881421\pi$
$677$	−20.3187	−	35.1931i	−0.780913	−	1.35258i	0.150498	−	0.988610i	$-0.451912\pi$
$678$		0			0
$679$	6.77292	−	11.7310i	0.259921	−	0.450196i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14.0755			0.538584			0.269292	−	0.963059i	$-0.413210\pi$
$683$	14.0755			0.538584			0.269292	+	0.963059i	$0.413210\pi$
$684$		0			0
$685$	83.3861			3.18602
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.61273	−	7.98947i	0.175731	−	0.304375i
$690$		0			0
$691$	21.9601	+	38.0359i	0.835400	+	1.44696i	0.893704	+	0.448656i	$0.148097\pi$
$691$	21.9601	+	38.0359i	0.835400	+	1.44696i	−0.0583042	+	0.998299i	$0.518569\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.64295	+	6.30977i	0.138185	+	0.239343i
$696$		0			0
$697$	−2.64132	+	4.57489i	−0.100047	+	0.173286i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.69578			−0.290666			−0.145333	−	0.989383i	$-0.546425\pi$
$701$	−7.69578			−0.290666			−0.145333	+	0.989383i	$0.546425\pi$
$702$		0			0
$703$	−12.6063			−0.475455
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.30150	+	2.25427i	−0.0489481	+	0.0847807i
$708$		0			0
$709$	−0.111090	−	0.192414i	−0.00417209	−	0.00722626i	0.863932	−	0.503609i	$-0.167995\pi$
$709$	−0.111090	−	0.192414i	−0.00417209	−	0.00722626i	−0.868104	+	0.496383i	$0.834661\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.2411	−	31.5944i	−0.683133	−	1.18322i
$714$		0			0
$715$	−1.85868	+	3.21934i	−0.0695109	+	0.120396i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.64403			0.0986056			0.0493028	−	0.998784i	$-0.484300\pi$
$719$	2.64403			0.0986056			0.0493028	+	0.998784i	$0.484300\pi$
$720$		0			0
$721$	15.9806			0.595148
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.3915	+	69.9602i	−1.50010	+	2.59826i
$726$		0			0
$727$	−12.7427	−	22.0710i	−0.472600	−	0.818568i	0.526908	−	0.849922i	$-0.323351\pi$
$727$	−12.7427	−	22.0710i	−0.472600	−	0.818568i	−0.999508	+	0.0313547i	$0.990018\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	25.9509	+	44.9483i	0.959829	+	1.66247i
$732$		0			0
$733$	−0.934740	+	1.61902i	−0.0345254	+	0.0597997i	−0.882772	−	0.469802i	$-0.844325\pi$
$733$	−0.934740	+	1.61902i	−0.0345254	+	0.0597997i	0.848246	+	0.529602i	$0.177659\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.61917			0.0596429
$738$		0			0
$739$	51.1639			1.88209			0.941047	−	0.338276i	$-0.109844\pi$
$739$	51.1639			1.88209			0.941047	+	0.338276i	$0.109844\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.3668	+	33.5442i	−0.710498	+	1.23062i	0.254173	+	0.967159i	$0.418197\pi$
$743$	−19.3668	+	33.5442i	−0.710498	+	1.23062i	−0.964671	+	0.263459i	$0.915137\pi$
$744$		0			0
$745$	23.9503	+	41.4832i	0.877473	+	1.51983i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.44282	−	9.42724i	−0.198876	−	0.344464i
$750$		0			0
$751$	−3.09549	+	5.36154i	−0.112956	+	0.195645i	−0.916961	−	0.398977i	$-0.869365\pi$
$751$	−3.09549	+	5.36154i	−0.112956	+	0.195645i	0.804005	+	0.594623i	$0.202699\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−35.4088			−1.28866
$756$		0			0
$757$	−13.7174			−0.498566			−0.249283	−	0.968431i	$-0.580195\pi$
$757$	−13.7174			−0.498566			−0.249283	+	0.968431i	$0.580195\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.6602	−	34.0524i	0.712681	−	1.23440i	−0.251166	−	0.967944i	$-0.580814\pi$
$761$	19.6602	−	34.0524i	0.712681	−	1.23440i	0.963847	−	0.266456i	$-0.0858528\pi$
$762$		0			0
$763$	1.64132	+	2.84284i	0.0594196	+	0.102918i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.77292	+	8.26693i	0.172340	+	0.298502i
$768$		0			0
$769$	−11.3646	+	19.6840i	−0.409817	+	0.709824i	−0.994869	−	0.101172i	$-0.967741\pi$
$769$	−11.3646	+	19.6840i	−0.409817	+	0.709824i	0.585052	+	0.810996i	$0.301074\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.7324			1.03343			0.516717	−	0.856156i	$-0.327154\pi$
$773$	28.7324			1.03343			0.516717	+	0.856156i	$0.327154\pi$
$774$		0			0
$775$	82.5576			2.96556
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.604645	−	1.04728i	0.0216636	−	0.0375225i
$780$		0			0
$781$	−1.67154	−	2.89519i	−0.0598124	−	0.103598i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.2804	+	40.3229i	0.830915	+	1.43919i
$786$		0			0
$787$	27.0059	−	46.7756i	0.962656	−	1.66737i	0.246871	−	0.969048i	$-0.420598\pi$
$787$	27.0059	−	46.7756i	0.962656	−	1.66737i	0.715785	−	0.698321i	$-0.246069\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.45090			0.229368
$792$		0			0
$793$	10.5458			0.374493
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.103565	+	0.179381i	−0.00366848	+	0.00635399i	−0.867854	−	0.496820i	$-0.834501\pi$
$797$	−0.103565	+	0.179381i	−0.00366848	+	0.00635399i	0.864185	+	0.503174i	$0.167834\pi$
$798$		0			0
$799$	14.7993	+	25.6332i	0.523562	+	0.906836i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.28100	+	2.21875i	0.0452053	+	0.0782980i
$804$		0			0
$805$	−9.18715	+	15.9126i	−0.323804	+	0.560846i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	36.0539			1.26759			0.633794	−	0.773502i	$-0.281497\pi$
$809$	36.0539			1.26759			0.633794	+	0.773502i	$0.281497\pi$
$810$		0			0
$811$	−2.01943			−0.0709118			−0.0354559	−	0.999371i	$-0.511288\pi$
$811$	−2.01943			−0.0709118			−0.0354559	+	0.999371i	$0.511288\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.2141	−	41.9401i	0.848183	−	1.46910i
$816$		0			0
$817$	−5.94063	−	10.2895i	−0.207836	−	0.359983i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.1871	−	26.3049i	−0.530035	−	0.918048i	−0.999386	−	0.0350359i	$-0.988845\pi$
$821$	−15.1871	−	26.3049i	−0.530035	−	0.918048i	0.469351	−	0.883012i	$-0.344488\pi$
$822$		0			0
$823$	6.56526	−	11.3714i	0.228851	−	0.396381i	−0.728617	−	0.684921i	$-0.759837\pi$
$823$	6.56526	−	11.3714i	0.228851	−	0.396381i	0.957468	+	0.288540i	$0.0931699\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.3236			0.393762			0.196881	−	0.980427i	$-0.436919\pi$
$827$	11.3236			0.393762			0.196881	+	0.980427i	$0.436919\pi$
$828$		0			0
$829$	−11.0604			−0.384145			−0.192073	−	0.981381i	$-0.561521\pi$
$829$	−11.0604			−0.384145			−0.192073	+	0.981381i	$0.561521\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.80150	−	4.85235i	0.0970664	−	0.168124i
$834$		0			0
$835$	−45.1170	−	78.1449i	−1.56134	−	2.70432i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.27128	+	9.13013i	0.181985	+	0.315207i	0.942556	−	0.334047i	$-0.108414\pi$
$839$	5.27128	+	9.13013i	0.181985	+	0.315207i	−0.760572	+	0.649254i	$0.775081\pi$
$840$		0			0
$841$	−14.8393	+	25.7023i	−0.511698	+	0.886288i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−47.3138			−1.62765
$846$		0			0
$847$	−10.1111			−0.347421
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22.9012	−	39.6661i	0.785045	−	1.35974i
$852$		0			0
$853$	13.6413	+	23.6275i	0.467070	+	0.808989i	0.999292	−	0.0376160i	$-0.0119764\pi$
$853$	13.6413	+	23.6275i	0.467070	+	0.808989i	−0.532223	+	0.846604i	$0.678643\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.21574	−	2.10571i	−0.0415287	−	0.0719299i	0.844514	−	0.535534i	$-0.179889\pi$
$857$	−1.21574	−	2.10571i	−0.0415287	−	0.0719299i	−0.886043	+	0.463604i	$0.846556\pi$
$858$		0			0
$859$	15.8743	−	27.4951i	0.541624	−	0.938120i	−0.457187	−	0.889370i	$-0.651143\pi$
$859$	15.8743	−	27.4951i	0.541624	−	0.938120i	0.998811	−	0.0487495i	$-0.0155236\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.8252			0.947181			0.473590	−	0.880745i	$-0.342958\pi$
$863$	27.8252			0.947181			0.473590	+	0.880745i	$0.342958\pi$
$864$		0			0
$865$	−75.9513			−2.58242
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.71900	+	13.3697i	−0.261849	+	0.453536i
$870$		0			0
$871$	−0.858685	−	1.48729i	−0.0290954	−	0.0503948i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.9331	−	18.9367i	−0.369606	−	0.640177i
$876$		0			0
$877$	2.29342	−	3.97233i	0.0774434	−	0.134136i	−0.824703	−	0.565566i	$-0.808658\pi$
$877$	2.29342	−	3.97233i	0.0774434	−	0.134136i	0.902146	+	0.431430i	$0.141991\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−57.4465			−1.93542			−0.967711	−	0.252062i	$-0.918891\pi$
$881$	−57.4465			−1.93542			−0.967711	+	0.252062i	$0.918891\pi$
$882$		0			0
$883$	36.3937			1.22475			0.612373	−	0.790569i	$-0.290215\pi$
$883$	36.3937			1.22475			0.612373	+	0.790569i	$0.290215\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.7238	+	37.6268i	−0.729414	+	1.26338i	0.227717	+	0.973727i	$0.426874\pi$
$887$	−21.7238	+	37.6268i	−0.729414	+	1.26338i	−0.957131	+	0.289655i	$0.906459\pi$
$888$		0			0
$889$	0.414230	+	0.717468i	0.0138928	+	0.0240631i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.38783	−	5.86789i	−0.113369	−	0.196362i
$894$		0			0
$895$	−9.52532	+	16.4983i	−0.318396	+	0.551479i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	59.9675			2.00003
$900$		0			0
$901$	−51.6903			−1.72205
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.43310	+	11.1425i	−0.213844	+	0.370388i
$906$		0			0
$907$	−11.1569	−	19.3244i	−0.370459	−	0.641655i	0.619177	−	0.785252i	$-0.287466\pi$
$907$	−11.1569	−	19.3244i	−0.370459	−	0.641655i	−0.989636	+	0.143597i	$0.954133\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.67799	−	13.2987i	−0.254383	−	0.440604i	0.710345	−	0.703854i	$-0.248539\pi$
$911$	−7.67799	−	13.2987i	−0.254383	−	0.440604i	−0.964728	+	0.263250i	$0.915206\pi$
$912$		0			0
$913$	0.187145	−	0.324145i	0.00619360	−	0.0107276i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.43147			−0.179363
$918$		0			0
$919$	−36.8285			−1.21486			−0.607429	−	0.794374i	$-0.707799\pi$
$919$	−36.8285			−1.21486			−0.607429	+	0.794374i	$0.707799\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.77292	+	3.07078i	−0.0583562	+	0.101076i
$924$		0			0
$925$	51.8246	+	89.7629i	1.70398	+	2.95139i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.8646	−	36.1385i	−0.684545	−	1.18567i	−0.973580	−	0.228347i	$-0.926668\pi$
$929$	−20.8646	−	36.1385i	−0.684545	−	1.18567i	0.289035	−	0.957318i	$-0.406666\pi$
$930$		0			0
$931$	−0.641315	+	1.11079i	−0.0210183	+	0.0364047i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20.8285			0.681163
$936$		0			0
$937$	−25.3743			−0.828942			−0.414471	−	0.910063i	$-0.636033\pi$
$937$	−25.3743			−0.828942			−0.414471	+	0.910063i	$0.636033\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.5712	−	40.8264i	0.768398	−	1.33090i	−0.170034	−	0.985438i	$-0.554388\pi$
$941$	23.5712	−	40.8264i	0.768398	−	1.33090i	0.938431	−	0.345465i	$-0.112279\pi$
$942$		0			0
$943$	2.19686	+	3.80507i	0.0715396	+	0.123910i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.2535	−	29.8839i	−0.560663	−	0.971097i	−0.997439	−	0.0715263i	$-0.977213\pi$
$947$	−17.2535	−	29.8839i	−0.560663	−	0.971097i	0.436776	−	0.899570i	$-0.356120\pi$
$948$		0			0
$949$	1.35868	−	2.35331i	0.0441048	−	0.0763917i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	44.1833			1.43124			0.715619	−	0.698491i	$-0.246145\pi$
$953$	44.1833			1.43124			0.715619	+	0.698491i	$0.246145\pi$
$954$		0			0
$955$	−57.9513			−1.87526
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.5744	+	18.3154i	−0.341466	+	0.591436i
$960$		0			0
$961$	−15.1424	−	26.2274i	−0.488464	−	0.846045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.9331	−	34.5252i	−0.641669	−	1.11140i
$966$		0			0
$967$	−21.2330	+	36.7766i	−0.682806	+	1.18266i	0.291314	+	0.956627i	$0.405907\pi$
$967$	−21.2330	+	36.7766i	−0.682806	+	1.18266i	−0.974121	+	0.226028i	$0.927426\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.05391			−0.194279			−0.0971396	−	0.995271i	$-0.530969\pi$
$971$	−6.05391			−0.194279			−0.0971396	+	0.995271i	$0.530969\pi$
$972$		0			0
$973$	−1.84789			−0.0592406
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.0264	+	38.1508i	−0.704687	+	1.22055i	0.262117	+	0.965036i	$0.415579\pi$
$977$	−22.0264	+	38.1508i	−0.704687	+	1.22055i	−0.966804	+	0.255518i	$0.917754\pi$
$978$		0			0
$979$	−2.59656	−	4.49738i	−0.0829866	−	0.143737i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.23517	−	3.87142i	−0.0712907	−	0.123479i	0.828177	−	0.560467i	$-0.189378\pi$
$983$	−2.23517	−	3.87142i	−0.0712907	−	0.123479i	−0.899467	+	0.436988i	$0.856045\pi$
$984$		0			0
$985$	32.7993	−	56.8101i	1.04507	−	1.81012i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43.1683			1.37267
$990$		0			0
$991$	−31.5048			−1.00078			−0.500392	−	0.865799i	$-0.666811\pi$
$991$	−31.5048			−1.00078			−0.500392	+	0.865799i	$0.666811\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−25.8473	+	44.7689i	−0.819416	+	1.41927i
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.j.a.253.1		6
3.2	odd	2		252.2.j.b.85.1	✓	6
4.3	odd	2		3024.2.r.i.1009.1		6
7.2	even	3		5292.2.l.g.361.3		6
7.3	odd	6		5292.2.i.g.1549.3		6
7.4	even	3		5292.2.i.d.1549.1		6
7.5	odd	6		5292.2.l.d.361.1		6
7.6	odd	2		5292.2.j.e.1765.3		6
9.2	odd	6		252.2.j.b.169.1	yes	6
9.4	even	3		2268.2.a.j.1.3		3
9.5	odd	6		2268.2.a.g.1.1		3
9.7	even	3	inner	756.2.j.a.505.1		6
12.11	even	2		1008.2.r.g.337.3		6
21.2	odd	6		1764.2.l.d.949.2		6
21.5	even	6		1764.2.l.g.949.2		6
21.11	odd	6		1764.2.i.f.373.3		6
21.17	even	6		1764.2.i.e.373.1		6
21.20	even	2		1764.2.j.d.589.3		6
36.7	odd	6		3024.2.r.i.2017.1		6
36.11	even	6		1008.2.r.g.673.3		6
36.23	even	6		9072.2.a.bt.1.1		3
36.31	odd	6		9072.2.a.bz.1.3		3
63.2	odd	6		1764.2.i.f.1537.3		6
63.11	odd	6		1764.2.l.d.961.2		6
63.16	even	3		5292.2.i.d.2125.1		6
63.20	even	6		1764.2.j.d.1177.3		6
63.25	even	3		5292.2.l.g.3313.3		6
63.34	odd	6		5292.2.j.e.3529.3		6
63.38	even	6		1764.2.l.g.961.2		6
63.47	even	6		1764.2.i.e.1537.1		6
63.52	odd	6		5292.2.l.d.3313.1		6
63.61	odd	6		5292.2.i.g.2125.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.1	✓	6	3.2	odd	2
252.2.j.b.169.1	yes	6	9.2	odd	6
756.2.j.a.253.1		6	1.1	even	1	trivial
756.2.j.a.505.1		6	9.7	even	3	inner
1008.2.r.g.337.3		6	12.11	even	2
1008.2.r.g.673.3		6	36.11	even	6
1764.2.i.e.373.1		6	21.17	even	6
1764.2.i.e.1537.1		6	63.47	even	6
1764.2.i.f.373.3		6	21.11	odd	6
1764.2.i.f.1537.3		6	63.2	odd	6
1764.2.j.d.589.3		6	21.20	even	2
1764.2.j.d.1177.3		6	63.20	even	6
1764.2.l.d.949.2		6	21.2	odd	6
1764.2.l.d.961.2		6	63.11	odd	6
1764.2.l.g.949.2		6	21.5	even	6
1764.2.l.g.961.2		6	63.38	even	6
2268.2.a.g.1.1		3	9.5	odd	6
2268.2.a.j.1.3		3	9.4	even	3
3024.2.r.i.1009.1		6	4.3	odd	2
3024.2.r.i.2017.1		6	36.7	odd	6
5292.2.i.d.1549.1		6	7.4	even	3
5292.2.i.d.2125.1		6	63.16	even	3
5292.2.i.g.1549.3		6	7.3	odd	6
5292.2.i.g.2125.3		6	63.61	odd	6
5292.2.j.e.1765.3		6	7.6	odd	2
5292.2.j.e.3529.3		6	63.34	odd	6
5292.2.l.d.361.1		6	7.5	odd	6
5292.2.l.d.3313.1		6	63.52	odd	6
5292.2.l.g.361.3		6	7.2	even	3
5292.2.l.g.3313.3		6	63.25	even	3
9072.2.a.bt.1.1		3	36.23	even	6
9072.2.a.bz.1.3		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.97141 + 3.41458i) q^{5} +(-0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(-1.97141 + 3.41458i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(0.471410 + 0.816506i) q^{11} +(0.500000 - 0.866025i) q^{13} -5.60301 q^{17} +1.28263 q^{19} +(-2.33009 + 4.03584i) q^{23} +(-5.27292 - 9.13296i) q^{25} +(-3.83009 - 6.63392i) q^{29} +(-3.91423 + 6.77965i) q^{31} +3.94282 q^{35} -9.82846 q^{37} +(0.471410 - 0.816506i) q^{41} +(-4.63160 - 8.02217i) q^{43} +(-2.64132 - 4.57489i) q^{47} +(-0.500000 + 0.866025i) q^{49} +9.22545 q^{53} -3.71737 q^{55} +(-4.77292 + 8.26693i) q^{59} +(5.27292 + 9.13296i) q^{61} +(1.97141 + 3.41458i) q^{65} +(0.858685 - 1.48729i) q^{67} -3.54583 q^{71} +2.71737 q^{73} +(0.471410 - 0.816506i) q^{77} +(8.18715 + 14.1806i) q^{79} +(-0.198495 - 0.343803i) q^{83} +(11.0458 - 19.1319i) q^{85} -5.50808 q^{89} -1.00000 q^{91} +(-2.52859 + 4.37965i) q^{95} +(6.77292 + 11.7310i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - 3 q^{7}+O(q^{10}) \) 6 * q - 3 * q^5 - 3 * q^7	\( 6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67} + 30 q^{71} + 18 q^{73} - 6 q^{77} - 3 q^{79} - 18 q^{83} + 15 q^{85} - 12 q^{89} - 6 q^{91} - 24 q^{95} + 15 q^{97}+O(q^{100}) \) 6 * q - 3 * q^5 - 3 * q^7 - 6 * q^11 + 3 * q^13 + 6 * q^19 - 6 * q^23 - 6 * q^25 - 15 * q^29 + 3 * q^31 + 6 * q^35 - 6 * q^37 - 6 * q^41 - 3 * q^43 - 15 * q^47 - 3 * q^49 + 36 * q^53 - 24 * q^55 - 3 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 + 30 * q^71 + 18 * q^73 - 6 * q^77 - 3 * q^79 - 18 * q^83 + 15 * q^85 - 12 * q^89 - 6 * q^91 - 24 * q^95 + 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more