LMFDB - Embedded newform 1224.2.w.j.217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(217,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1224.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1224.w (of order $4$, degree $2$, 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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.269485056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{3} + 81x^{2} - 72x + 32 $ x^6 - 8x^3 + 81x^2 - 72*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			217.2
Root			$0.467090 + 0.467090i$ of defining polynomial
Character	$\chi$	$=$	1224.217
Dual form			1224.2.w.j.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.467090 - 0.467090i) q^{5} +(-1.00000 + 1.00000i) q^{7} +(1.53291 - 1.53291i) q^{11} -5.56365 q^{13} +(4.09656 + 0.467090i) q^{17} -5.69529i q^{19} +(-6.09656 + 6.09656i) q^{23} -4.56365i q^{25} +(-4.56365 - 4.56365i) q^{31} +0.934181 q^{35} +(-3.56365 - 3.56365i) q^{37} +(5.03074 - 5.03074i) q^{41} -1.69529i q^{43} -11.2589 q^{47} +5.00000i q^{49} -11.2589i q^{53} -1.43201 q^{55} +3.06582i q^{59} +(5.69529 - 5.69529i) q^{61} +(2.59873 + 2.59873i) q^{65} -15.1273 q^{67} +(9.49783 + 9.49783i) q^{71} +(-6.69529 - 6.69529i) q^{73} +3.06582i q^{77} +(-1.13164 + 1.13164i) q^{79} -2.13164i q^{83} +(-1.69529 - 2.13164i) q^{85} -12.9342 q^{89} +(5.56365 - 5.56365i) q^{91} +(-2.66022 + 2.66022i) q^{95} +(-8.43201 - 8.43201i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} + 12 q^{11} - 6 q^{17} - 6 q^{23} + 6 q^{31} + 12 q^{37} - 6 q^{41} - 12 q^{47} + 36 q^{55} + 12 q^{61} + 24 q^{65} - 24 q^{67} + 18 q^{71} - 18 q^{73} - 18 q^{79} + 12 q^{85} - 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 + 12 * q^11 - 6 * q^17 - 6 * q^23 + 6 * q^31 + 12 * q^37 - 6 * q^41 - 12 * q^47 + 36 * q^55 + 12 * q^61 + 24 * q^65 - 24 * q^67 + 18 * q^71 - 18 * q^73 - 18 * q^79 + 12 * q^85 - 72 * q^89 + 48 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	−0.467090	−	0.467090i	−0.208889	−	0.208889i	0.594906	−	0.803795i	$-0.297189\pi$
$5$	−0.467090	−	0.467090i	−0.208889	−	0.208889i	−0.803795	+	0.594906i	$0.797189\pi$
$6$		0			0
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	0.492403	+	0.870367i	$0.336119\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.53291	−	1.53291i	0.462190	−	0.462190i	−0.437183	−	0.899373i	$-0.644024\pi$
$11$	1.53291	−	1.53291i	0.462190	−	0.462190i	0.899373	+	0.437183i	$0.144024\pi$
$12$		0			0
$13$	−5.56365			−1.54308			−0.771540	−	0.636181i	$-0.780513\pi$
$13$	−5.56365			−1.54308			−0.771540	+	0.636181i	$0.780513\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.09656	+	0.467090i	0.993562	+	0.113286i
$17$	4.09656	+	0.467090i	0.993562	+	0.113286i
$18$		0			0
$19$		−	5.69529i		−	1.30659i	−0.757104	−	0.653295i	$-0.773386\pi$
$19$		−	5.69529i		−	1.30659i	0.757104	−	0.653295i	$-0.226614\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.09656	+	6.09656i	−1.27122	+	1.27122i	−0.325773	+	0.945448i	$0.605625\pi$
$23$	−6.09656	+	6.09656i	−1.27122	+	1.27122i	−0.945448	+	0.325773i	$0.894375\pi$
$24$		0			0
$25$		−	4.56365i		−	0.912731i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	−4.56365	−	4.56365i	−0.819656	−	0.819656i	0.166402	−	0.986058i	$-0.446785\pi$
$31$	−4.56365	−	4.56365i	−0.819656	−	0.819656i	−0.986058	+	0.166402i	$0.946785\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.934181			0.157905
$36$		0			0
$37$	−3.56365	−	3.56365i	−0.585861	−	0.585861i	0.350647	−	0.936508i	$-0.385962\pi$
$37$	−3.56365	−	3.56365i	−0.585861	−	0.585861i	−0.936508	+	0.350647i	$0.885962\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.03074	−	5.03074i	0.785670	−	0.785670i	−0.195111	−	0.980781i	$-0.562507\pi$
$41$	5.03074	−	5.03074i	0.785670	−	0.785670i	0.980781	+	0.195111i	$0.0625067\pi$
$42$		0			0
$43$		−	1.69529i		−	0.258530i	−0.991610	−	0.129265i	$-0.958738\pi$
$43$		−	1.69529i		−	0.258530i	0.991610	−	0.129265i	$-0.0412617\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.2589			−1.64229			−0.821143	−	0.570723i	$-0.806663\pi$
$47$	−11.2589			−1.64229			−0.821143	+	0.570723i	$0.806663\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	11.2589i		−	1.54654i	−0.634080	−	0.773268i	$-0.718621\pi$
$53$		−	11.2589i		−	1.54654i	0.634080	−	0.773268i	$-0.281379\pi$
$54$		0			0
$55$	−1.43201			−0.193093
$56$		0			0
$57$		0			0
$58$		0			0
$59$			3.06582i			0.399136i	0.979884	+	0.199568i	$0.0639538\pi$
$59$			3.06582i			0.399136i	−0.979884	+	0.199568i	$0.936046\pi$
$60$		0			0
$61$	5.69529	−	5.69529i	0.729207	−	0.729207i	−0.241255	−	0.970462i	$-0.577559\pi$
$61$	5.69529	−	5.69529i	0.729207	−	0.729207i	0.970462	+	0.241255i	$0.0775589\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.59873	+	2.59873i	0.322333	+	0.322333i
$66$		0			0
$67$	−15.1273			−1.84809			−0.924047	−	0.382278i	$-0.875140\pi$
$67$	−15.1273			−1.84809			−0.924047	+	0.382278i	$0.875140\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.49783	+	9.49783i	1.12719	+	1.12719i	0.990633	+	0.136553i	$0.0436023\pi$
$71$	9.49783	+	9.49783i	1.12719	+	1.12719i	0.136553	+	0.990633i	$0.456398\pi$
$72$		0			0
$73$	−6.69529	−	6.69529i	−0.783625	−	0.783625i	0.196816	−	0.980440i	$-0.436940\pi$
$73$	−6.69529	−	6.69529i	−0.783625	−	0.783625i	−0.980440	+	0.196816i	$0.936940\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.06582i			0.349383i
$78$		0			0
$79$	−1.13164	+	1.13164i	−0.127319	+	0.127319i	−0.767895	−	0.640576i	$-0.778696\pi$
$79$	−1.13164	+	1.13164i	−0.127319	+	0.127319i	0.640576	+	0.767895i	$0.278696\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.13164i		−	0.233978i	−0.993133	−	0.116989i	$-0.962676\pi$
$83$		−	2.13164i		−	0.233978i	0.993133	−	0.116989i	$-0.0373242\pi$
$84$		0			0
$85$	−1.69529	−	2.13164i	−0.183880	−	0.231209i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.9342			−1.37102			−0.685510	−	0.728063i	$-0.740421\pi$
$89$	−12.9342			−1.37102			−0.685510	+	0.728063i	$0.740421\pi$
$90$		0			0
$91$	5.56365	−	5.56365i	0.583229	−	0.583229i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.66022	+	2.66022i	−0.272932	+	0.272932i
$96$		0			0
$97$	−8.43201	−	8.43201i	−0.856141	−	0.856141i	0.134740	−	0.990881i	$-0.456980\pi$
$97$	−8.43201	−	8.43201i	−0.856141	−	0.856141i	−0.990881	+	0.134740i	$0.956980\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.12731			0.908201			0.454100	−	0.890950i	$-0.349961\pi$
$101$	9.12731			0.908201			0.454100	+	0.890950i	$0.349961\pi$
$102$		0			0
$103$	4.30471			0.424156			0.212078	−	0.977253i	$-0.431977\pi$
$103$	4.30471			0.424156			0.212078	+	0.977253i	$0.431977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.46709	−	6.46709i	−0.625197	−	0.625197i	0.321659	−	0.946856i	$-0.395760\pi$
$107$	−6.46709	−	6.46709i	−0.625197	−	0.625197i	−0.946856	+	0.321659i	$0.895760\pi$
$108$		0			0
$109$	−6.00000	+	6.00000i	−0.574696	+	0.574696i	−0.933437	−	0.358741i	$-0.883206\pi$
$109$	−6.00000	+	6.00000i	−0.574696	+	0.574696i	0.358741	+	0.933437i	$0.383206\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.0966	−	12.0966i	1.13795	−	1.13795i	0.149132	−	0.988817i	$-0.452352\pi$
$113$	12.0966	−	12.0966i	1.13795	−	1.13795i	0.988817	−	0.149132i	$-0.0476478\pi$
$114$		0			0
$115$	5.69529			0.531089
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.56365	+	3.62947i	−0.418349	+	0.332713i
$120$		0			0
$121$			6.30038i			0.572761i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.46709	+	4.46709i	−0.399549	+	0.399549i
$126$		0			0
$127$			5.69529i			0.505375i	0.967548	+	0.252688i	$0.0813145\pi$
$127$			5.69529i			0.505375i	−0.967548	+	0.252688i	$0.918685\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.53291	+	1.53291i	0.133931	+	0.133931i	0.770894	−	0.636963i	$-0.219810\pi$
$131$	1.53291	+	1.53291i	0.133931	+	0.133931i	−0.636963	+	0.770894i	$0.719810\pi$
$132$		0			0
$133$	5.69529	+	5.69529i	0.493844	+	0.493844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3248			0.882104			0.441052	−	0.897482i	$-0.354605\pi$
$137$	10.3248			0.882104			0.441052	+	0.897482i	$0.354605\pi$
$138$		0			0
$139$	5.56365	+	5.56365i	0.471903	+	0.471903i	0.902530	−	0.430627i	$-0.141708\pi$
$139$	5.56365	+	5.56365i	0.471903	+	0.471903i	−0.430627	+	0.902530i	$0.641708\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.52858	+	8.52858i	−0.713195	+	0.713195i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.3248			0.845838			0.422919	−	0.906168i	$-0.361006\pi$
$149$	10.3248			0.845838			0.422919	+	0.906168i	$0.361006\pi$
$150$		0			0
$151$		−	15.1273i		−	1.23104i	−0.788120	−	0.615521i	$-0.788945\pi$
$151$		−	15.1273i		−	1.23104i	0.788120	−	0.615521i	$-0.211055\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			4.26328i			0.342435i
$156$		0			0
$157$	5.56365			0.444028			0.222014	−	0.975043i	$-0.428737\pi$
$157$	5.56365			0.444028			0.222014	+	0.975043i	$0.428737\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12.1931i		−	0.960953i
$162$		0			0
$163$	−15.1273	+	15.1273i	−1.18486	+	1.18486i	−0.206393	+	0.978469i	$0.566172\pi$
$163$	−15.1273	+	15.1273i	−1.18486	+	1.18486i	−0.978469	+	0.206393i	$0.933828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.29402	−	3.29402i	−0.254899	−	0.254899i	0.568077	−	0.822976i	$-0.307688\pi$
$167$	−3.29402	−	3.29402i	−0.254899	−	0.254899i	−0.822976	+	0.568077i	$0.807688\pi$
$168$		0			0
$169$	17.9542			1.38110
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.59873	+	2.59873i	0.197578	+	0.197578i	0.798961	−	0.601383i	$-0.205383\pi$
$173$	2.59873	+	2.59873i	0.197578	+	0.197578i	−0.601383	+	0.798961i	$0.705383\pi$
$174$		0			0
$175$	4.56365	+	4.56365i	0.344980	+	0.344980i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9.86836i			0.737596i	0.929510	+	0.368798i	$0.120231\pi$
$179$			9.86836i			0.737596i	−0.929510	+	0.368798i	$0.879769\pi$
$180$		0			0
$181$	9.43201	−	9.43201i	0.701076	−	0.701076i	−0.263565	−	0.964642i	$-0.584898\pi$
$181$	9.43201	−	9.43201i	0.701076	−	0.701076i	0.964642	+	0.263565i	$0.0848985\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			3.32910i			0.244760i
$186$		0			0
$187$	6.99567	−	5.56365i	0.511574	−	0.406855i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.3248			−1.03650			−0.518252	−	0.855228i	$-0.673417\pi$
$191$	−14.3248			−1.03650			−0.518252	+	0.855228i	$0.673417\pi$
$192$		0			0
$193$	−11.9542	+	11.9542i	−0.860485	+	0.860485i	−0.991394	−	0.130910i	$-0.958210\pi$
$193$	−11.9542	+	11.9542i	−0.860485	+	0.860485i	0.130910	+	0.991394i	$0.458210\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.6602	−	16.6602i	1.18699	−	1.18699i	0.209096	−	0.977895i	$-0.432948\pi$
$197$	16.6602	−	16.6602i	1.18699	−	1.18699i	0.977895	−	0.209096i	$-0.0670522\pi$
$198$		0			0
$199$	0.695292	+	0.695292i	0.0492879	+	0.0492879i	0.731321	−	0.682033i	$-0.238904\pi$
$199$	0.695292	+	0.695292i	0.0492879	+	0.0492879i	−0.682033	+	0.731321i	$0.738904\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.69962			−0.328236
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.73037	−	8.73037i	−0.603892	−	0.603892i
$210$		0			0
$211$	6.00000	−	6.00000i	0.413057	−	0.413057i	−0.469745	−	0.882802i	$-0.655654\pi$
$211$	6.00000	−	6.00000i	0.413057	−	0.413057i	0.882802	+	0.469745i	$0.155654\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.791854	+	0.791854i	−0.0540040	+	0.0540040i
$216$		0			0
$217$	9.12731			0.619602
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.7919	−	2.59873i	−1.53315	−	0.174809i
$222$		0			0
$223$			3.43201i			0.229825i	0.993376	+	0.114912i	$0.0366587\pi$
$223$			3.43201i			0.229825i	−0.993376	+	0.114912i	$0.963341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.66022	−	2.66022i	0.176565	−	0.176565i	−0.613292	−	0.789856i	$-0.710155\pi$
$227$	2.66022	−	2.66022i	0.176565	−	0.176565i	0.789856	+	0.613292i	$0.210155\pi$
$228$		0			0
$229$			1.39058i			0.0918923i	0.998944	+	0.0459462i	$0.0146303\pi$
$229$			1.39058i			0.0918923i	−0.998944	+	0.0459462i	$0.985370\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.837618	+	0.837618i	0.0548742	+	0.0548742i	0.734011	−	0.679137i	$-0.237646\pi$
$233$	0.837618	+	0.837618i	0.0548742	+	0.0548742i	−0.679137	+	0.734011i	$0.737646\pi$
$234$		0			0
$235$	5.25894	+	5.25894i	0.343056	+	0.343056i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.39058			−0.0899494			−0.0449747	−	0.998988i	$-0.514321\pi$
$239$	−1.39058			−0.0899494			−0.0449747	+	0.998988i	$0.514321\pi$
$240$		0			0
$241$	10.2589	+	10.2589i	0.660837	+	0.660837i	0.955577	−	0.294741i	$-0.0952332\pi$
$241$	10.2589	+	10.2589i	0.660837	+	0.660837i	−0.294741	+	0.955577i	$0.595233\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.33545	−	2.33545i	0.149207	−	0.149207i
$246$		0			0
$247$			31.6866i			2.01617i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.4521			0.722848			0.361424	−	0.932402i	$-0.382291\pi$
$251$	11.4521			0.722848			0.361424	+	0.932402i	$0.382291\pi$
$252$		0			0
$253$			18.6910i			1.17509i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			31.1888i			1.94550i	0.231849	+	0.972752i	$0.425522\pi$
$257$			31.1888i			1.94550i	−0.231849	+	0.972752i	$0.574478\pi$
$258$		0			0
$259$	7.12731			0.442869
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.9298i			0.982277i	0.871082	+	0.491138i	$0.163419\pi$
$263$			15.9298i			0.982277i	−0.871082	+	0.491138i	$0.836581\pi$
$264$		0			0
$265$	−5.25894	+	5.25894i	−0.323054	+	0.323054i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.53291	+	7.53291i	0.459290	+	0.459290i	0.898422	−	0.439133i	$-0.144714\pi$
$269$	7.53291	+	7.53291i	0.459290	+	0.459290i	−0.439133	+	0.898422i	$0.644714\pi$
$270$		0			0
$271$	19.9499			1.21187			0.605935	−	0.795514i	$-0.292799\pi$
$271$	19.9499			1.21187			0.605935	+	0.795514i	$0.292799\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.99567	−	6.99567i	−0.421855	−	0.421855i
$276$		0			0
$277$	11.2589	+	11.2589i	0.676484	+	0.676484i	0.959203	−	0.282718i	$-0.0912362\pi$
$277$	11.2589	+	11.2589i	0.676484	+	0.676484i	−0.282718	+	0.959203i	$0.591236\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.9342i			0.771589i	0.922585	+	0.385794i	$0.126073\pi$
$281$			12.9342i			0.771589i	−0.922585	+	0.385794i	$0.873927\pi$
$282$		0			0
$283$	−10.8226	+	10.8226i	−0.643337	+	0.643337i	−0.951374	−	0.308038i	$-0.900328\pi$
$283$	−10.8226	+	10.8226i	−0.643337	+	0.643337i	0.308038	+	0.951374i	$0.400328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.0615i			0.593911i
$288$		0			0
$289$	16.5637	+	3.82693i	0.974333	+	0.225114i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.67524			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$293$	−9.67524			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$294$		0			0
$295$	1.43201	−	1.43201i	0.0833751	−	0.0833751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	33.9192	−	33.9192i	1.96160	−	1.96160i
$300$		0			0
$301$	1.69529	+	1.69529i	0.0977150	+	0.0977150i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.32043			−0.304647
$306$		0			0
$307$	22.8640			1.30492			0.652460	−	0.757824i	$-0.273737\pi$
$307$	22.8640			1.30492			0.652460	+	0.757824i	$0.273737\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−21.4978	−	21.4978i	−1.21903	−	1.21903i	−0.967972	−	0.251058i	$-0.919221\pi$
$311$	−21.4978	−	21.4978i	−1.21903	−	1.21903i	−0.251058	−	0.967972i	$-0.580779\pi$
$312$		0			0
$313$	6.69529	−	6.69529i	0.378440	−	0.378440i	−0.492099	−	0.870539i	$-0.663770\pi$
$313$	6.69529	−	6.69529i	0.378440	−	0.378440i	0.870539	+	0.492099i	$0.163770\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.39058	+	1.39058i	−0.0781029	+	0.0781029i	−0.745079	−	0.666976i	$-0.767588\pi$
$317$	−1.39058	+	1.39058i	−0.0781029	+	0.0781029i	0.666976	+	0.745079i	$0.267588\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.66022	−	23.3311i	0.148018	−	1.29818i
$324$		0			0
$325$			25.3906i			1.40842i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.2589	−	11.2589i	0.620726	−	0.620726i
$330$		0			0
$331$		−	1.43201i		−	0.0787106i	−0.999225	−	0.0393553i	$-0.987470\pi$
$331$		−	1.43201i		−	0.0787106i	0.999225	−	0.0393553i	$-0.0125304\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.06582	+	7.06582i	0.386047	+	0.386047i
$336$		0			0
$337$	−1.13164	−	1.13164i	−0.0616443	−	0.0616443i	0.675613	−	0.737257i	$-0.263879\pi$
$337$	−1.13164	−	1.13164i	−0.0616443	−	0.0616443i	−0.737257	+	0.675613i	$0.763879\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.9913			−0.757673
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.5837	−	17.5837i	0.943943	−	0.943943i	−0.0545672	−	0.998510i	$-0.517378\pi$
$347$	17.5837	−	17.5837i	0.943943	−	0.943943i	0.998510	+	0.0545672i	$0.0173779\pi$
$348$		0			0
$349$		−	21.8269i		−	1.16837i	−0.811621	−	0.584184i	$-0.801414\pi$
$349$		−	21.8269i		−	1.16837i	0.811621	−	0.584184i	$-0.198586\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.1931			−1.28767			−0.643835	−	0.765164i	$-0.722658\pi$
$353$	−24.1931			−1.28767			−0.643835	+	0.765164i	$0.722658\pi$
$354$		0			0
$355$		−	8.87269i		−	0.470914i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			25.5837i			1.35026i	0.737700	+	0.675128i	$0.235912\pi$
$359$			25.5837i			1.35026i	−0.737700	+	0.675128i	$0.764088\pi$
$360$		0			0
$361$	−13.4363			−0.707176
$362$		0			0
$363$		0			0
$364$		0			0
$365$			6.25461i			0.327381i
$366$		0			0
$367$	15.3863	−	15.3863i	0.803156	−	0.803156i	−0.180432	−	0.983588i	$-0.557749\pi$
$367$	15.3863	−	15.3863i	0.803156	−	0.803156i	0.983588	+	0.180432i	$0.0577495\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.2589	+	11.2589i	0.584535	+	0.584535i
$372$		0			0
$373$	−19.6452			−1.01719			−0.508595	−	0.861006i	$-0.669835\pi$
$373$	−19.6452			−1.01719			−0.508595	+	0.861006i	$0.669835\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.56365	−	3.56365i	−0.183053	−	0.183053i	0.609632	−	0.792685i	$-0.291317\pi$
$379$	−3.56365	−	3.56365i	−0.183053	−	0.183053i	−0.792685	+	0.609632i	$0.791317\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			35.2589i			1.80165i	0.434185	+	0.900824i	$0.357037\pi$
$383$			35.2589i			1.80165i	−0.434185	+	0.900824i	$0.642963\pi$
$384$		0			0
$385$	1.43201	−	1.43201i	0.0729822	−	0.0729822i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	22.3248i		−	1.13191i	−0.824436	−	0.565955i	$-0.808507\pi$
$389$		−	22.3248i		−	1.13191i	0.824436	−	0.565955i	$-0.191493\pi$
$390$		0			0
$391$	−27.8226	+	22.1273i	−1.40705	+	1.11903i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.05715			0.0531912
$396$		0			0
$397$	24.6495	−	24.6495i	1.23712	−	1.23712i	0.275953	−	0.961171i	$-0.411006\pi$
$397$	24.6495	−	24.6495i	1.23712	−	1.23712i	0.961171	−	0.275953i	$-0.0889935\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.35551	+	3.35551i	−0.167566	+	0.167566i	−0.785909	−	0.618343i	$-0.787804\pi$
$401$	−3.35551	+	3.35551i	−0.167566	+	0.167566i	0.618343	+	0.785909i	$0.287804\pi$
$402$		0			0
$403$	25.3906	+	25.3906i	1.26480	+	1.26480i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.9255			−0.541558
$408$		0			0
$409$	−23.5637			−1.16515			−0.582574	−	0.812778i	$-0.697954\pi$
$409$	−23.5637			−1.16515			−0.582574	+	0.812778i	$0.697954\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.06582	−	3.06582i	−0.150859	−	0.150859i
$414$		0			0
$415$	−0.995668	+	0.995668i	−0.0488754	+	0.0488754i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.7153	−	19.7153i	0.963158	−	0.963158i	−0.0361873	−	0.999345i	$-0.511521\pi$
$419$	19.7153	−	19.7153i	0.963158	−	0.963158i	0.999345	+	0.0361873i	$0.0115213\pi$
$420$		0			0
$421$	−2.17307			−0.105909			−0.0529545	−	0.998597i	$-0.516864\pi$
$421$	−2.17307			−0.105909			−0.0529545	+	0.998597i	$0.516864\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.13164	−	18.6953i	0.103400	−	0.906855i
$426$		0			0
$427$			11.3906i			0.551229i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.6295	+	11.6295i	−0.560172	+	0.560172i	−0.929356	−	0.369184i	$-0.879637\pi$
$431$	−11.6295	+	11.6295i	−0.560172	+	0.560172i	0.369184	+	0.929356i	$0.379637\pi$
$432$		0			0
$433$		−	4.69962i		−	0.225850i	−0.993604	−	0.112925i	$-0.963978\pi$
$433$		−	4.69962i		−	0.225850i	0.993604	−	0.112925i	$-0.0360219\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	34.7217	+	34.7217i	1.66096	+	1.66096i
$438$		0			0
$439$	−4.56365	−	4.56365i	−0.217811	−	0.217811i	0.589764	−	0.807575i	$-0.299221\pi$
$439$	−4.56365	−	4.56365i	−0.217811	−	0.217811i	−0.807575	+	0.589764i	$0.799221\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.3863			0.968580			0.484290	−	0.874908i	$-0.339078\pi$
$443$	20.3863			0.968580			0.484290	+	0.874908i	$0.339078\pi$
$444$		0			0
$445$	6.04143	+	6.04143i	0.286391	+	0.286391i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−26.1474	+	26.1474i	−1.23397	+	1.23397i	−0.271544	+	0.962426i	$0.587534\pi$
$449$	−26.1474	+	26.1474i	−1.23397	+	1.23397i	−0.962426	+	0.271544i	$0.912466\pi$
$450$		0			0
$451$		−	15.4234i		−	0.726257i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.19746			−0.243661
$456$		0			0
$457$			8.43635i			0.394636i	0.980340	+	0.197318i	$0.0632231\pi$
$457$			8.43635i			0.394636i	−0.980340	+	0.197318i	$0.936777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.6666i			1.10226i	0.834419	+	0.551131i	$0.185804\pi$
$461$			23.6666i			1.10226i	−0.834419	+	0.551131i	$0.814196\pi$
$462$		0			0
$463$	−8.78117			−0.408095			−0.204048	−	0.978961i	$-0.565410\pi$
$463$	−8.78117			−0.408095			−0.204048	+	0.978961i	$0.565410\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	38.9255i		−	1.80126i	−0.434589	−	0.900629i	$-0.643106\pi$
$467$		−	38.9255i		−	1.80126i	0.434589	−	0.900629i	$-0.356894\pi$
$468$		0			0
$469$	15.1273	−	15.1273i	0.698514	−	0.698514i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.59873	−	2.59873i	−0.119490	−	0.119490i
$474$		0			0
$475$	−25.9913			−1.19256
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.6846	−	16.6846i	−0.762339	−	0.762339i	0.214406	−	0.976745i	$-0.431219\pi$
$479$	−16.6846	−	16.6846i	−0.762339	−	0.762339i	−0.976745	+	0.214406i	$0.931219\pi$
$480$		0			0
$481$	19.8269	+	19.8269i	0.904030	+	0.904030i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			7.87703i			0.357677i
$486$		0			0
$487$	29.6495	−	29.6495i	1.34355	−	1.34355i	0.451049	−	0.892499i	$-0.351050\pi$
$487$	29.6495	−	29.6495i	1.34355	−	1.34355i	0.892499	−	0.451049i	$-0.148950\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	6.25461i		−	0.282267i	−0.989991	−	0.141133i	$-0.954925\pi$
$491$		−	6.25461i		−	0.282267i	0.989991	−	0.141133i	$-0.0450746\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.9957			−0.852072
$498$		0			0
$499$	−14.3863	+	14.3863i	−0.644017	+	0.644017i	−0.951541	−	0.307523i	$-0.900500\pi$
$499$	−14.3863	+	14.3863i	−0.644017	+	0.644017i	0.307523	+	0.951541i	$0.400500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.8333	−	17.8333i	0.795147	−	0.795147i	−0.187179	−	0.982326i	$-0.559934\pi$
$503$	17.8333	−	17.8333i	0.795147	−	0.795147i	0.982326	+	0.187179i	$0.0599344\pi$
$504$		0			0
$505$	−4.26328	−	4.26328i	−0.189713	−	0.189713i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.34614			0.281288			0.140644	−	0.990060i	$-0.455083\pi$
$509$	6.34614			0.281288			0.140644	+	0.990060i	$0.455083\pi$
$510$		0			0
$511$	13.3906			0.592365
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.01069	−	2.01069i	−0.0886015	−	0.0886015i
$516$		0			0
$517$	−17.2589	+	17.2589i	−0.759048	+	0.759048i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.2282	−	18.2282i	0.798592	−	0.798592i	−0.184282	−	0.982874i	$-0.558996\pi$
$521$	18.2282	−	18.2282i	0.798592	−	0.798592i	0.982874	+	0.184282i	$0.0589958\pi$
$522$		0			0
$523$	−29.9085			−1.30781			−0.653903	−	0.756578i	$-0.726870\pi$
$523$	−29.9085			−1.30781			−0.653903	+	0.756578i	$0.726870\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.5637	−	20.8269i	−0.721524	−	0.907235i
$528$		0			0
$529$		−	51.3362i		−	2.23201i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−27.9893	+	27.9893i	−1.21235	+	1.21235i
$534$		0			0
$535$			6.04143i			0.261194i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.66455	+	7.66455i	0.330135	+	0.330135i
$540$		0			0
$541$	−18.6910	−	18.6910i	−0.803587	−	0.803587i	0.180067	−	0.983654i	$-0.442368\pi$
$541$	−18.6910	−	18.6910i	−0.803587	−	0.803587i	−0.983654	+	0.180067i	$0.942368\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.60508			0.240095
$546$		0			0
$547$	−24.6910	−	24.6910i	−1.05571	−	1.05571i	−0.998354	−	0.0573554i	$-0.981733\pi$
$547$	−24.6910	−	24.6910i	−1.05571	−	1.05571i	−0.0573554	−	0.998354i	$-0.518267\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	2.26328i		−	0.0962443i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−44.8427			−1.90004			−0.950022	−	0.312183i	$-0.898940\pi$
$557$	−44.8427			−1.90004			−0.950022	+	0.312183i	$0.898940\pi$
$558$		0			0
$559$			9.43201i			0.398932i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	30.9041i		−	1.30245i	−0.758883	−	0.651227i	$-0.774255\pi$
$563$		−	30.9041i		−	1.30245i	0.758883	−	0.651227i	$-0.225745\pi$
$564$		0			0
$565$	−11.3004			−0.475410
$566$		0			0
$567$		0			0
$568$		0			0
$569$			15.5222i			0.650725i	0.945589	+	0.325363i	$0.105486\pi$
$569$			15.5222i			0.650725i	−0.945589	+	0.325363i	$0.894514\pi$
$570$		0			0
$571$	−9.86836	+	9.86836i	−0.412978	+	0.412978i	−0.882775	−	0.469797i	$-0.844327\pi$
$571$	−9.86836	+	9.86836i	−0.412978	+	0.412978i	0.469797	+	0.882775i	$0.344327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	27.8226	+	27.8226i	1.16028	+	1.16028i
$576$		0			0
$577$	0.436347			0.0181654			0.00908268	−	0.999959i	$-0.497109\pi$
$577$	0.436347			0.0181654			0.00908268	+	0.999959i	$0.497109\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.13164	+	2.13164i	0.0884353	+	0.0884353i
$582$		0			0
$583$	−17.2589	−	17.2589i	−0.714792	−	0.714792i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			15.6137i			0.644448i	0.946663	+	0.322224i	$0.104430\pi$
$587$			15.6137i			0.644448i	−0.946663	+	0.322224i	$0.895570\pi$
$588$		0			0
$589$	−25.9913	+	25.9913i	−1.07095	+	1.07095i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			21.1273i			0.867594i	0.901011	+	0.433797i	$0.142827\pi$
$593$			21.1273i			0.867594i	−0.901011	+	0.433797i	$0.857173\pi$
$594$		0			0
$595$	3.82693	+	0.436347i	0.156889	+	0.0178885i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.8025			−0.768251			−0.384126	−	0.923281i	$-0.625497\pi$
$599$	−18.8025			−0.768251			−0.384126	+	0.923281i	$0.625497\pi$
$600$		0			0
$601$	1.69096	−	1.69096i	0.0689757	−	0.0689757i	−0.671777	−	0.740753i	$-0.734469\pi$
$601$	1.69096	−	1.69096i	0.0689757	−	0.0689757i	0.740753	+	0.671777i	$0.234469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.94285	−	2.94285i	0.119644	−	0.119644i
$606$		0			0
$607$	−26.6452	−	26.6452i	−1.08150	−	1.08150i	−0.996370	−	0.0851248i	$-0.972871\pi$
$607$	−26.6452	−	26.6452i	−1.08150	−	1.08150i	−0.0851248	−	0.996370i	$-0.527129\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	62.6409			2.53418
$612$		0			0
$613$	−35.8183			−1.44669			−0.723343	−	0.690489i	$-0.757395\pi$
$613$	−35.8183			−1.44669			−0.723343	+	0.690489i	$0.757395\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.4363	−	19.4363i	−0.782478	−	0.782478i	0.197770	−	0.980248i	$-0.436630\pi$
$617$	−19.4363	−	19.4363i	−0.782478	−	0.782478i	−0.980248	+	0.197770i	$0.936630\pi$
$618$		0			0
$619$	31.9499	−	31.9499i	1.28418	−	1.28418i	0.345906	−	0.938269i	$-0.387572\pi$
$619$	31.9499	−	31.9499i	1.28418	−	1.28418i	0.938269	−	0.345906i	$-0.112428\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.9342	−	12.9342i	0.518197	−	0.518197i
$624$		0			0
$625$	−18.6452			−0.745808
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.9342	−	16.2633i	−0.515720	−	0.648459i
$630$		0			0
$631$		−	21.6866i		−	0.863331i	−0.902034	−	0.431666i	$-0.857926\pi$
$631$		−	21.6866i		−	0.863331i	0.902034	−	0.431666i	$-0.142074\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.66022	−	2.66022i	0.105567	−	0.105567i
$636$		0			0
$637$		−	27.8183i		−	1.10220i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.35551	−	3.35551i	−0.132535	−	0.132535i	0.637727	−	0.770262i	$-0.279875\pi$
$641$	−3.35551	−	3.35551i	−0.132535	−	0.132535i	−0.770262	+	0.637727i	$0.779875\pi$
$642$		0			0
$643$	5.60508	+	5.60508i	0.221043	+	0.221043i	0.808938	−	0.587895i	$-0.200043\pi$
$643$	5.60508	+	5.60508i	0.221043	+	0.221043i	−0.587895	+	0.808938i	$0.700043\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.67524			0.0658603			0.0329302	−	0.999458i	$-0.489516\pi$
$647$	1.67524			0.0658603			0.0329302	+	0.999458i	$0.489516\pi$
$648$		0			0
$649$	4.69962	+	4.69962i	0.184476	+	0.184476i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.5987	−	18.5987i	0.727825	−	0.727825i	−0.242362	−	0.970186i	$-0.577922\pi$
$653$	18.5987	−	18.5987i	0.727825	−	0.727825i	0.970186	+	0.242362i	$0.0779220\pi$
$654$		0			0
$655$		−	1.43201i		−	0.0559534i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−34.8025			−1.35571			−0.677857	−	0.735194i	$-0.737091\pi$
$659$	−34.8025			−1.35571			−0.677857	+	0.735194i	$0.737091\pi$
$660$		0			0
$661$		−	35.2088i		−	1.36947i	−0.728794	−	0.684733i	$-0.759919\pi$
$661$		−	35.2088i		−	1.36947i	0.728794	−	0.684733i	$-0.240081\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	5.32043i		−	0.206317i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	17.4607i		−	0.674064i
$672$		0			0
$673$	2.17740	−	2.17740i	0.0839327	−	0.0839327i	−0.663894	−	0.747827i	$-0.731097\pi$
$673$	2.17740	−	2.17740i	0.0839327	−	0.0839327i	0.747827	+	0.663894i	$0.231097\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.79185	+	6.79185i	0.261032	+	0.261032i	0.825473	−	0.564441i	$-0.190908\pi$
$677$	6.79185	+	6.79185i	0.261032	+	0.261032i	−0.564441	+	0.825473i	$0.690908\pi$
$678$		0			0
$679$	16.8640			0.647182
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.1781	+	21.1781i	0.810358	+	0.810358i	0.984687	−	0.174330i	$-0.0557758\pi$
$683$	21.1781	+	21.1781i	0.810358	+	0.810358i	−0.174330	+	0.984687i	$0.555776\pi$
$684$		0			0
$685$	−4.82260	−	4.82260i	−0.184262	−	0.184262i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			62.6409i			2.38643i
$690$		0			0
$691$	4.51789	−	4.51789i	0.171869	−	0.171869i	−0.615931	−	0.787800i	$-0.711220\pi$
$691$	4.51789	−	4.51789i	0.171869	−	0.171869i	0.787800	+	0.615931i	$0.211220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	5.19746i		−	0.197151i
$696$		0			0
$697$	22.9586	−	18.2589i	0.869618	−	0.691607i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−37.2503			−1.40692			−0.703462	−	0.710733i	$-0.748363\pi$
$701$	−37.2503			−1.40692			−0.703462	+	0.710733i	$0.748363\pi$
$702$		0			0
$703$	−20.2960	+	20.2960i	−0.765480	+	0.765480i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.12731	+	9.12731i	−0.343268	+	0.343268i
$708$		0			0
$709$	3.86836	+	3.86836i	0.145279	+	0.145279i	0.776006	−	0.630726i	$-0.217243\pi$
$709$	3.86836	+	3.86836i	0.145279	+	0.145279i	−0.630726	+	0.776006i	$0.717243\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	55.6452			2.08393
$714$		0			0
$715$	7.96723			0.297958
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.42133	+	4.42133i	0.164888	+	0.164888i	0.784728	−	0.619840i	$-0.212803\pi$
$719$	4.42133	+	4.42133i	0.164888	+	0.164888i	−0.619840	+	0.784728i	$0.712803\pi$
$720$		0			0
$721$	−4.30471	+	4.30471i	−0.160316	+	0.160316i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	18.5179			0.686791			0.343395	−	0.939191i	$-0.388423\pi$
$727$	18.5179			0.686791			0.343395	+	0.939191i	$0.388423\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.791854	−	6.94487i	0.0292878	−	0.256865i
$732$		0			0
$733$			13.3906i			0.494592i	0.968940	+	0.247296i	$0.0795421\pi$
$733$			13.3906i			0.494592i	−0.968940	+	0.247296i	$0.920458\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−23.1888	+	23.1888i	−0.854170	+	0.854170i
$738$		0			0
$739$			14.9141i			0.548625i	0.961641	+	0.274312i	$0.0884503\pi$
$739$			14.9141i			0.548625i	−0.961641	+	0.274312i	$0.911550\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.56799	+	5.56799i	0.204270	+	0.204270i	0.801826	−	0.597557i	$-0.203862\pi$
$743$	5.56799	+	5.56799i	0.204270	+	0.204270i	−0.597557	+	0.801826i	$0.703862\pi$
$744$		0			0
$745$	−4.82260	−	4.82260i	−0.176686	−	0.176686i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.9342			0.472605
$750$		0			0
$751$	−16.5637	−	16.5637i	−0.604416	−	0.604416i	0.337065	−	0.941481i	$-0.390566\pi$
$751$	−16.5637	−	16.5637i	−0.604416	−	0.604416i	−0.941481	+	0.337065i	$0.890566\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.06582	+	7.06582i	−0.257151	+	0.257151i
$756$		0			0
$757$		−	9.90979i		−	0.360178i	−0.983650	−	0.180089i	$-0.942361\pi$
$757$		−	9.90979i		−	0.360178i	0.983650	−	0.180089i	$-0.0576385\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.00433			−0.181407			−0.0907034	−	0.995878i	$-0.528912\pi$
$761$	−5.00433			−0.181407			−0.0907034	+	0.995878i	$0.528912\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	17.0572i		−	0.615898i
$768$		0			0
$769$	−22.0815			−0.796281			−0.398140	−	0.917324i	$-0.630344\pi$
$769$	−22.0815			−0.796281			−0.398140	+	0.917324i	$0.630344\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	43.6452i		−	1.56981i	−0.619617	−	0.784904i	$-0.712712\pi$
$773$		−	43.6452i		−	1.56981i	0.619617	−	0.784904i	$-0.287288\pi$
$774$		0			0
$775$	−20.8269	+	20.8269i	−0.748125	+	0.748125i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.6516	−	28.6516i	−1.02655	−	1.02655i
$780$		0			0
$781$	29.1186			1.04195
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.59873	−	2.59873i	−0.0927526	−	0.0927526i
$786$		0			0
$787$	−35.2589	−	35.2589i	−1.25685	−	1.25685i	−0.952591	−	0.304255i	$-0.901592\pi$
$787$	−35.2589	−	35.2589i	−1.25685	−	1.25685i	−0.304255	−	0.952591i	$-0.598408\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			24.1931i			0.860209i
$792$		0			0
$793$	−31.6866	+	31.6866i	−1.12523	+	1.12523i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			17.5136i			0.620362i	0.950678	+	0.310181i	$0.100390\pi$
$797$			17.5136i			0.620362i	−0.950678	+	0.310181i	$0.899610\pi$
$798$		0			0
$799$	−46.1230	−	5.25894i	−1.63171	−	0.186048i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.5266			−0.724366
$804$		0			0
$805$	−5.69529	+	5.69529i	−0.200733	+	0.200733i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	38.6145	−	38.6145i	1.35761	−	1.35761i	0.480759	−	0.876853i	$-0.340361\pi$
$809$	38.6145	−	38.6145i	1.35761	−	1.35761i	0.876853	−	0.480759i	$-0.159639\pi$
$810$		0			0
$811$	−1.78550	−	1.78550i	−0.0626973	−	0.0626973i	0.675063	−	0.737760i	$-0.264116\pi$
$811$	−1.78550	−	1.78550i	−0.0626973	−	0.0626973i	−0.737760	+	0.675063i	$0.764116\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.1316			0.495010
$816$		0			0
$817$	−9.65518			−0.337792
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.7260	−	23.7260i	−0.828044	−	0.828044i	0.159202	−	0.987246i	$-0.449108\pi$
$821$	−23.7260	−	23.7260i	−0.828044	−	0.828044i	−0.987246	+	0.159202i	$0.949108\pi$
$822$		0			0
$823$	−11.6495	+	11.6495i	−0.406077	+	0.406077i	−0.880368	−	0.474291i	$-0.842704\pi$
$823$	−11.6495	+	11.6495i	−0.406077	+	0.406077i	0.474291	+	0.880368i	$0.342704\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.59873	−	8.59873i	0.299007	−	0.299007i	−0.541618	−	0.840625i	$-0.682188\pi$
$827$	8.59873	−	8.59873i	0.299007	−	0.299007i	0.840625	+	0.541618i	$0.182188\pi$
$828$		0			0
$829$	27.3819			0.951013			0.475507	−	0.879712i	$-0.342265\pi$
$829$	27.3819			0.951013			0.475507	+	0.879712i	$0.342265\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.33545	+	20.4828i	−0.0809186	+	0.709687i
$834$		0			0
$835$			3.07721i			0.106491i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.7016	+	15.7016i	−0.542081	+	0.542081i	−0.924139	−	0.382057i	$-0.875216\pi$
$839$	−15.7016	+	15.7016i	−0.542081	+	0.542081i	0.382057	+	0.924139i	$0.375216\pi$
$840$		0			0
$841$		−	29.0000i		−	1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.38625	−	8.38625i	−0.288496	−	0.288496i
$846$		0			0
$847$	−6.30038	−	6.30038i	−0.216483	−	0.216483i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	43.4521			1.48952
$852$		0			0
$853$	33.5637	+	33.5637i	1.14920	+	1.14920i	0.986711	+	0.162487i	$0.0519515\pi$
$853$	33.5637	+	33.5637i	1.14920	+	1.14920i	0.162487	+	0.986711i	$0.448048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.8183	−	22.8183i	0.779457	−	0.779457i	−0.200281	−	0.979738i	$-0.564186\pi$
$857$	22.8183	−	22.8183i	0.779457	−	0.779457i	0.979738	+	0.200281i	$0.0641856\pi$
$858$		0			0
$859$			4.60942i			0.157271i	0.996903	+	0.0786356i	$0.0250564\pi$
$859$			4.60942i			0.157271i	−0.996903	+	0.0786356i	$0.974944\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.934181			0.0317999			0.0158999	−	0.999874i	$-0.494939\pi$
$863$	0.934181			0.0317999			0.0158999	+	0.999874i	$0.494939\pi$
$864$		0			0
$865$		−	2.42768i		−	0.0825437i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			3.46940i			0.117691i
$870$		0			0
$871$	84.1631			2.85176
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	8.93418i		−	0.302030i
$876$		0			0
$877$	−2.82260	+	2.82260i	−0.0953124	+	0.0953124i	−0.753155	−	0.657843i	$-0.771469\pi$
$877$	−2.82260	+	2.82260i	−0.0953124	+	0.0953124i	0.657843	+	0.753155i	$0.271469\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.1474	−	18.1474i	−0.611400	−	0.611400i	0.331911	−	0.943311i	$-0.392307\pi$
$881$	−18.1474	−	18.1474i	−0.611400	−	0.611400i	−0.943311	+	0.331911i	$0.892307\pi$
$882$		0			0
$883$	−10.2218			−0.343992			−0.171996	−	0.985098i	$-0.555022\pi$
$883$	−10.2218			−0.343992			−0.171996	+	0.985098i	$0.555022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.0307	+	27.0307i	0.907604	+	0.907604i	0.996078	−	0.0884749i	$-0.0281993\pi$
$887$	27.0307	+	27.0307i	0.907604	+	0.907604i	−0.0884749	+	0.996078i	$0.528199\pi$
$888$		0			0
$889$	−5.69529	−	5.69529i	−0.191014	−	0.191014i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			64.1230i			2.14579i
$894$		0			0
$895$	4.60942	−	4.60942i	0.154076	−	0.154076i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	5.25894	−	46.1230i	0.175201	−	1.53658i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.81121			−0.292894
$906$		0			0
$907$	28.9542	−	28.9542i	0.961410	−	0.961410i	−0.0378729	−	0.999283i	$-0.512058\pi$
$907$	28.9542	−	28.9542i	0.961410	−	0.961410i	0.999283	+	0.0378729i	$0.0120582\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.51285	−	4.51285i	0.149518	−	0.149518i	−0.628385	−	0.777903i	$-0.716284\pi$
$911$	4.51285	−	4.51285i	0.149518	−	0.149518i	0.777903	+	0.628385i	$0.216284\pi$
$912$		0			0
$913$	−3.26761	−	3.26761i	−0.108142	−	0.108142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.06582			−0.101242
$918$		0			0
$919$	4.05010			0.133600			0.0668002	−	0.997766i	$-0.478721\pi$
$919$	4.05010			0.133600			0.0668002	+	0.997766i	$0.478721\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−52.8427	−	52.8427i	−1.73934	−	1.73934i
$924$		0			0
$925$	−16.2633	+	16.2633i	−0.534733	+	0.534733i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.552965	+	0.552965i	−0.0181422	+	0.0181422i	−0.716120	−	0.697977i	$-0.754084\pi$
$929$	−0.552965	+	0.552965i	−0.0181422	+	0.0181422i	0.697977	+	0.716120i	$0.254084\pi$
$930$		0			0
$931$	28.4765			0.933278
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.86634	−	0.668880i	−0.191850	−	0.0218747i
$936$		0			0
$937$			43.0358i			1.40592i	0.711231	+	0.702959i	$0.248138\pi$
$937$			43.0358i			1.40592i	−0.711231	+	0.702959i	$0.751862\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.2589	+	23.2589i	−0.758220	+	0.758220i	−0.975998	−	0.217778i	$-0.930119\pi$
$941$	−23.2589	+	23.2589i	−0.758220	+	0.758220i	0.217778	+	0.975998i	$0.430119\pi$
$942$		0			0
$943$			61.3405i			1.99752i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.7982	+	33.7982i	1.09829	+	1.09829i	0.994610	+	0.103684i	$0.0330631\pi$
$947$	33.7982	+	33.7982i	1.09829	+	1.09829i	0.103684	+	0.994610i	$0.466937\pi$
$948$		0			0
$949$	37.2503	+	37.2503i	1.20920	+	1.20920i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.60508			−0.181566			−0.0907832	−	0.995871i	$-0.528937\pi$
$953$	−5.60508			−0.181566			−0.0907832	+	0.995871i	$0.528937\pi$
$954$		0			0
$955$	6.69096	+	6.69096i	0.216514	+	0.216514i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.3248	+	10.3248i	−0.333404	+	0.333404i
$960$		0			0
$961$			10.6539i			0.343673i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.1674			0.359492
$966$		0			0
$967$			27.0859i			0.871023i	0.900183	+	0.435512i	$0.143432\pi$
$967$			27.0859i			0.871023i	−0.900183	+	0.435512i	$0.856568\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			46.3776i			1.48833i	0.667997	+	0.744164i	$0.267152\pi$
$971$			46.3776i			1.48833i	−0.667997	+	0.744164i	$0.732848\pi$
$972$		0			0
$973$	−11.1273			−0.356725
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.45640i			0.142573i	0.997456	+	0.0712865i	$0.0227105\pi$
$977$			4.45640i			0.142573i	−0.997456	+	0.0712865i	$0.977290\pi$
$978$		0			0
$979$	−19.8269	+	19.8269i	−0.633671	+	0.633671i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.8119	−	17.8119i	−0.568112	−	0.568112i	0.363487	−	0.931599i	$-0.381586\pi$
$983$	−17.8119	−	17.8119i	−0.568112	−	0.568112i	−0.931599	+	0.363487i	$0.881586\pi$
$984$		0			0
$985$	−15.5637			−0.495899
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.3355	+	10.3355i	0.328648	+	0.328648i
$990$		0			0
$991$	11.2132	+	11.2132i	0.356198	+	0.356198i	0.862410	−	0.506211i	$-0.168954\pi$
$991$	11.2132	+	11.2132i	0.356198	+	0.356198i	−0.506211	+	0.862410i	$0.668954\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	0.649528i		−	0.0205914i
$996$		0			0
$997$	14.0815	−	14.0815i	0.445967	−	0.445967i	−0.448044	−	0.894011i	$-0.647879\pi$
$997$	14.0815	−	14.0815i	0.445967	−	0.445967i	0.894011	+	0.448044i	$0.147879\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.w.j.217.2	yes	6
3.2	odd	2		1224.2.w.i.217.2	✓	6
4.3	odd	2		2448.2.be.v.1441.2		6
12.11	even	2		2448.2.be.w.1441.2		6
17.4	even	4	inner	1224.2.w.j.361.2	yes	6
51.38	odd	4		1224.2.w.i.361.2	yes	6
68.55	odd	4		2448.2.be.v.1585.2		6
204.191	even	4		2448.2.be.w.1585.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.2.w.i.217.2	✓	6	3.2	odd	2
1224.2.w.i.361.2	yes	6	51.38	odd	4
1224.2.w.j.217.2	yes	6	1.1	even	1	trivial
1224.2.w.j.361.2	yes	6	17.4	even	4	inner
2448.2.be.v.1441.2		6	4.3	odd	2
2448.2.be.v.1585.2		6	68.55	odd	4
2448.2.be.w.1441.2		6	12.11	even	2
2448.2.be.w.1585.2		6	204.191	even	4

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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1224.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.467090 - 0.467090i) q^{5} +(-1.00000 + 1.00000i) q^{7} +(1.53291 - 1.53291i) q^{11} -5.56365 q^{13} +(4.09656 + 0.467090i) q^{17} -5.69529i q^{19} +(-6.09656 + 6.09656i) q^{23} -4.56365i q^{25} +(-4.56365 - 4.56365i) q^{31} +0.934181 q^{35} +(-3.56365 - 3.56365i) q^{37} +(5.03074 - 5.03074i) q^{41} -1.69529i q^{43} -11.2589 q^{47} +5.00000i q^{49} -11.2589i q^{53} -1.43201 q^{55} +3.06582i q^{59} +(5.69529 - 5.69529i) q^{61} +(2.59873 + 2.59873i) q^{65} -15.1273 q^{67} +(9.49783 + 9.49783i) q^{71} +(-6.69529 - 6.69529i) q^{73} +3.06582i q^{77} +(-1.13164 + 1.13164i) q^{79} -2.13164i q^{83} +(-1.69529 - 2.13164i) q^{85} -12.9342 q^{89} +(5.56365 - 5.56365i) q^{91} +(-2.66022 + 2.66022i) q^{95} +(-8.43201 - 8.43201i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} + 12 q^{11} - 6 q^{17} - 6 q^{23} + 6 q^{31} + 12 q^{37} - 6 q^{41} - 12 q^{47} + 36 q^{55} + 12 q^{61} + 24 q^{65} - 24 q^{67} + 18 q^{71} - 18 q^{73} - 18 q^{79} + 12 q^{85} - 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 + 12 * q^11 - 6 * q^17 - 6 * q^23 + 6 * q^31 + 12 * q^37 - 6 * q^41 - 12 * q^47 + 36 * q^55 + 12 * q^61 + 24 * q^65 - 24 * q^67 + 18 * q^71 - 18 * q^73 - 18 * q^79 + 12 * q^85 - 72 * q^89 + 48 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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