LMFDB - Embedded newform 4608.2.k.bg.3457.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3457.1
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	4608.3457
Dual form			4608.2.k.bg.1153.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23607 - 2.23607i) q^{5} -4.57649i q^{7} +O(q^{10})$	$q+(-2.23607 - 2.23607i) q^{5} -4.57649i q^{7} +(1.74806 + 1.74806i) q^{11} +(1.00000 - 1.00000i) q^{13} -6.47214 q^{17} +(-1.74806 + 1.74806i) q^{19} +5.65685i q^{23} +5.00000i q^{25} +(0.236068 - 0.236068i) q^{29} -10.2333 q^{31} +(-10.2333 + 10.2333i) q^{35} +(-1.47214 - 1.47214i) q^{37} +6.47214i q^{41} +(7.40492 + 7.40492i) q^{43} -13.9443 q^{49} +(3.76393 + 3.76393i) q^{53} -7.81758i q^{55} +(-6.32456 - 6.32456i) q^{59} +(7.47214 - 7.47214i) q^{61} -4.47214 q^{65} +(8.48528 - 8.48528i) q^{67} -3.49613i q^{71} +14.9443i q^{73} +(8.00000 - 8.00000i) q^{77} +1.08036 q^{79} +(1.74806 - 1.74806i) q^{83} +(14.4721 + 14.4721i) q^{85} +10.0000i q^{89} +(-4.57649 - 4.57649i) q^{91} +7.81758 q^{95} +4.94427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} - 16 q^{17} - 16 q^{29} + 24 q^{37} - 40 q^{49} + 48 q^{53} + 24 q^{61} + 64 q^{77} + 80 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 16 * q^17 - 16 * q^29 + 24 * q^37 - 40 * q^49 + 48 * q^53 + 24 * q^61 + 64 * q^77 + 80 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	−2.23607	−	2.23607i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$5$	−2.23607	−	2.23607i	−1.00000	−	1.00000i	−1.00000			$\pi$
$6$		0			0
$7$		−	4.57649i		−	1.72975i	−0.501986	−	0.864876i	$-0.667397\pi$
$7$		−	4.57649i		−	1.72975i	0.501986	−	0.864876i	$-0.332603\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.74806	+	1.74806i	0.527061	+	0.527061i	0.919695	−	0.392634i	$-0.128436\pi$
$11$	1.74806	+	1.74806i	0.527061	+	0.527061i	−0.392634	+	0.919695i	$0.628436\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.47214			−1.56972			−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214			−1.56972			−0.784862	+	0.619671i	$0.787266\pi$
$18$		0			0
$19$	−1.74806	+	1.74806i	−0.401033	+	0.401033i	−0.878597	−	0.477564i	$-0.841520\pi$
$19$	−1.74806	+	1.74806i	−0.401033	+	0.401033i	0.477564	+	0.878597i	$0.341520\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.65685i			1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$23$			5.65685i			1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.236068	−	0.236068i	0.0438367	−	0.0438367i	−0.684849	−	0.728685i	$-0.740132\pi$
$29$	0.236068	−	0.236068i	0.0438367	−	0.0438367i	0.728685	+	0.684849i	$0.240132\pi$
$30$		0			0
$31$	−10.2333			−1.83796			−0.918982	−	0.394301i	$-0.870987\pi$
$31$	−10.2333			−1.83796			−0.918982	+	0.394301i	$0.870987\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.2333	+	10.2333i	−1.72975	+	1.72975i
$36$		0			0
$37$	−1.47214	−	1.47214i	−0.242018	−	0.242018i	0.575667	−	0.817684i	$-0.304743\pi$
$37$	−1.47214	−	1.47214i	−0.242018	−	0.242018i	−0.817684	+	0.575667i	$0.804743\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.47214i			1.01078i	0.862892	+	0.505389i	$0.168651\pi$
$41$			6.47214i			1.01078i	−0.862892	+	0.505389i	$0.831349\pi$
$42$		0			0
$43$	7.40492	+	7.40492i	1.12924	+	1.12924i	0.990301	+	0.138938i	$0.0443690\pi$
$43$	7.40492	+	7.40492i	1.12924	+	1.12924i	0.138938	+	0.990301i	$0.455631\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−13.9443			−1.99204
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.76393	+	3.76393i	0.517016	+	0.517016i	0.916667	−	0.399651i	$-0.130869\pi$
$53$	3.76393	+	3.76393i	0.517016	+	0.517016i	−0.399651	+	0.916667i	$0.630869\pi$
$54$		0			0
$55$		−	7.81758i		−	1.05412i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.32456	−	6.32456i	−0.823387	−	0.823387i	0.163205	−	0.986592i	$-0.447817\pi$
$59$	−6.32456	−	6.32456i	−0.823387	−	0.823387i	−0.986592	+	0.163205i	$0.947817\pi$
$60$		0			0
$61$	7.47214	−	7.47214i	0.956709	−	0.956709i	−0.0423921	−	0.999101i	$-0.513498\pi$
$61$	7.47214	−	7.47214i	0.956709	−	0.956709i	0.999101	+	0.0423921i	$0.0134979\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.47214			−0.554700
$66$		0			0
$67$	8.48528	−	8.48528i	1.03664	−	1.03664i	0.0373395	−	0.999303i	$-0.488112\pi$
$67$	8.48528	−	8.48528i	1.03664	−	1.03664i	0.999303	−	0.0373395i	$-0.0118883\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.49613i		−	0.414914i	−0.978244	−	0.207457i	$-0.933481\pi$
$71$		−	3.49613i		−	0.414914i	0.978244	−	0.207457i	$-0.0665187\pi$
$72$		0			0
$73$			14.9443i			1.74909i	0.484940	+	0.874547i	$0.338841\pi$
$73$			14.9443i			1.74909i	−0.484940	+	0.874547i	$0.661159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.00000	−	8.00000i	0.911685	−	0.911685i
$78$		0			0
$79$	1.08036			0.121550			0.0607752	−	0.998151i	$-0.480643\pi$
$79$	1.08036			0.121550			0.0607752	+	0.998151i	$0.480643\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.74806	−	1.74806i	0.191875	−	0.191875i	−0.604631	−	0.796506i	$-0.706679\pi$
$83$	1.74806	−	1.74806i	0.191875	−	0.191875i	0.796506	+	0.604631i	$0.206679\pi$
$84$		0			0
$85$	14.4721	+	14.4721i	1.56972	+	1.56972i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$	−4.57649	−	4.57649i	−0.479747	−	0.479747i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.81758			0.802067
$96$		0			0
$97$	4.94427			0.502015			0.251007	−	0.967985i	$-0.419238\pi$
$97$	4.94427			0.502015			0.251007	+	0.967985i	$0.419238\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.23607	−	4.23607i	−0.421505	−	0.421505i	0.464217	−	0.885722i	$-0.346336\pi$
$101$	−4.23607	−	4.23607i	−0.421505	−	0.421505i	−0.885722	+	0.464217i	$0.846336\pi$
$102$		0			0
$103$			4.57649i			0.450935i	0.974251	+	0.225468i	$0.0723910\pi$
$103$			4.57649i			0.450935i	−0.974251	+	0.225468i	$0.927609\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.9814	−	11.9814i	−1.15829	−	1.15829i	−0.984844	−	0.173443i	$-0.944511\pi$
$107$	−11.9814	−	11.9814i	−1.15829	−	1.15829i	−0.173443	−	0.984844i	$-0.555489\pi$
$108$		0			0
$109$	−5.94427	+	5.94427i	−0.569358	+	0.569358i	−0.931949	−	0.362591i	$-0.881892\pi$
$109$	−5.94427	+	5.94427i	−0.569358	+	0.569358i	0.362591	+	0.931949i	$0.381892\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	12.6491	−	12.6491i	1.17954	−	1.17954i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			29.6197i			2.71523i
$120$		0			0
$121$		−	4.88854i		−	0.444413i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.07262			0.716329			0.358165	−	0.933658i	$-0.383403\pi$
$127$	8.07262			0.716329			0.358165	+	0.933658i	$0.383403\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.667701	−	0.667701i	0.0583373	−	0.0583373i	−0.677336	−	0.735674i	$-0.736866\pi$
$131$	0.667701	−	0.667701i	0.0583373	−	0.0583373i	0.735674	+	0.677336i	$0.236866\pi$
$132$		0			0
$133$	8.00000	+	8.00000i	0.693688	+	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.47214i		−	0.552952i	−0.961021	−	0.276476i	$-0.910833\pi$
$137$		−	6.47214i		−	0.552952i	0.961021	−	0.276476i	$-0.0891666\pi$
$138$		0			0
$139$	−0.667701	−	0.667701i	−0.0566337	−	0.0566337i	0.678223	−	0.734856i	$-0.262750\pi$
$139$	−0.667701	−	0.667701i	−0.0566337	−	0.0566337i	−0.734856	+	0.678223i	$0.762750\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.49613			0.292361
$144$		0			0
$145$	−1.05573			−0.0876734
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.7082	+	10.7082i	0.877250	+	0.877250i	0.993249	−	0.115999i	$-0.0370070\pi$
$149$	10.7082	+	10.7082i	0.877250	+	0.877250i	−0.115999	+	0.993249i	$0.537007\pi$
$150$		0			0
$151$		−	11.5687i		−	0.941451i	−0.882280	−	0.470726i	$-0.843992\pi$
$151$		−	11.5687i		−	0.941451i	0.882280	−	0.470726i	$-0.156008\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	22.8825	+	22.8825i	1.83796	+	1.83796i
$156$		0			0
$157$	−0.527864	+	0.527864i	−0.0421281	+	0.0421281i	−0.727857	−	0.685729i	$-0.759484\pi$
$157$	−0.527864	+	0.527864i	−0.0421281	+	0.0421281i	0.685729	+	0.727857i	$0.259484\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.8885			2.04030
$162$		0			0
$163$	3.90879	−	3.90879i	0.306160	−	0.306160i	−0.537258	−	0.843418i	$-0.680540\pi$
$163$	3.90879	−	3.90879i	0.306160	−	0.306160i	0.843418	+	0.537258i	$0.180540\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.81758i		−	0.604943i	−0.953159	−	0.302471i	$-0.902188\pi$
$167$		−	7.81758i		−	0.604943i	0.953159	−	0.302471i	$-0.0978116\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.70820	+	6.70820i	−0.510015	+	0.510015i	−0.914531	−	0.404516i	$-0.867440\pi$
$173$	−6.70820	+	6.70820i	−0.510015	+	0.510015i	0.404516	+	0.914531i	$0.367440\pi$
$174$		0			0
$175$	22.8825			1.72975
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.48528	−	8.48528i	0.634220	−	0.634220i	−0.314904	−	0.949124i	$-0.601972\pi$
$179$	8.48528	−	8.48528i	0.634220	−	0.634220i	0.949124	+	0.314904i	$0.101972\pi$
$180$		0			0
$181$	11.0000	+	11.0000i	0.817624	+	0.817624i	0.985763	−	0.168140i	$-0.0537759\pi$
$181$	11.0000	+	11.0000i	0.817624	+	0.817624i	−0.168140	+	0.985763i	$0.553776\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6.58359i			0.484035i
$186$		0			0
$187$	−11.3137	−	11.3137i	−0.827340	−	0.827340i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.16073			−0.156345			−0.0781723	−	0.996940i	$-0.524908\pi$
$191$	−2.16073			−0.156345			−0.0781723	+	0.996940i	$0.524908\pi$
$192$		0			0
$193$	−12.9443			−0.931749			−0.465875	−	0.884851i	$-0.654260\pi$
$193$	−12.9443			−0.931749			−0.465875	+	0.884851i	$0.654260\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.1803	+	17.1803i	1.22405	+	1.22405i	0.966180	+	0.257869i	$0.0830201\pi$
$197$	17.1803	+	17.1803i	1.22405	+	1.22405i	0.257869	+	0.966180i	$0.416980\pi$
$198$		0			0
$199$			13.7295i			0.973257i	0.873609	+	0.486628i	$0.161773\pi$
$199$			13.7295i			0.973257i	−0.873609	+	0.486628i	$0.838227\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.08036	−	1.08036i	−0.0758266	−	0.0758266i
$204$		0			0
$205$	14.4721	−	14.4721i	1.01078	−	1.01078i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.11146			−0.422738
$210$		0			0
$211$	−17.6383	+	17.6383i	−1.21427	+	1.21427i	−0.244659	+	0.969609i	$0.578676\pi$
$211$	−17.6383	+	17.6383i	−1.21427	+	1.21427i	−0.969609	+	0.244659i	$0.921324\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	33.1158i		−	2.25848i
$216$		0			0
$217$			46.8328i			3.17922i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.47214	+	6.47214i	−0.435363	+	0.435363i
$222$		0			0
$223$	−10.2333			−0.685275			−0.342638	−	0.939468i	$-0.611320\pi$
$223$	−10.2333			−0.685275			−0.342638	+	0.939468i	$0.611320\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.90879	−	3.90879i	0.259436	−	0.259436i	−0.565389	−	0.824824i	$-0.691274\pi$
$227$	3.90879	−	3.90879i	0.259436	−	0.259436i	0.824824	+	0.565389i	$0.191274\pi$
$228$		0			0
$229$	7.94427	+	7.94427i	0.524972	+	0.524972i	0.919069	−	0.394097i	$-0.128943\pi$
$229$	7.94427	+	7.94427i	0.524972	+	0.524972i	−0.394097	+	0.919069i	$0.628943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.0000i		−	0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$		−	10.0000i		−	0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.4744			−0.871589			−0.435794	−	0.900046i	$-0.643533\pi$
$239$	−13.4744			−0.871589			−0.435794	+	0.900046i	$0.643533\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	31.1803	+	31.1803i	1.99204	+	1.99204i
$246$		0			0
$247$			3.49613i			0.222453i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.9010	−	10.9010i	−0.688068	−	0.688068i	0.273737	−	0.961805i	$-0.411740\pi$
$251$	−10.9010	−	10.9010i	−0.688068	−	0.688068i	−0.961805	+	0.273737i	$0.911740\pi$
$252$		0			0
$253$	−9.88854	+	9.88854i	−0.621687	+	0.621687i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$		0			0
$259$	−6.73722	+	6.73722i	−0.418630	+	0.418630i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			12.6491i			0.779978i	0.920820	+	0.389989i	$0.127521\pi$
$263$			12.6491i			0.779978i	−0.920820	+	0.389989i	$0.872479\pi$
$264$		0			0
$265$		−	16.8328i		−	1.03403i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.23607	−	8.23607i	0.502162	−	0.502162i	−0.409947	−	0.912109i	$-0.634453\pi$
$269$	8.23607	−	8.23607i	0.502162	−	0.502162i	0.912109	+	0.409947i	$0.134453\pi$
$270$		0			0
$271$	−8.07262			−0.490377			−0.245188	−	0.969475i	$-0.578850\pi$
$271$	−8.07262			−0.490377			−0.245188	+	0.969475i	$0.578850\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.74032	+	8.74032i	−0.527061	+	0.527061i
$276$		0			0
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	0.980015	−	0.198921i	$-0.0637438\pi$
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	−0.198921	+	0.980015i	$0.563744\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			31.8885i			1.90231i	0.308712	+	0.951156i	$0.400102\pi$
$281$			31.8885i			1.90231i	−0.308712	+	0.951156i	$0.599898\pi$
$282$		0			0
$283$	18.9737	+	18.9737i	1.12787	+	1.12787i	0.990523	+	0.137344i	$0.0438566\pi$
$283$	18.9737	+	18.9737i	1.12787	+	1.12787i	0.137344	+	0.990523i	$0.456143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	29.6197			1.74839
$288$		0			0
$289$	24.8885			1.46403
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.7639	−	11.7639i	−0.687256	−	0.687256i	0.274368	−	0.961625i	$-0.411531\pi$
$293$	−11.7639	−	11.7639i	−0.687256	−	0.687256i	−0.961625	+	0.274368i	$0.911531\pi$
$294$		0			0
$295$			28.2843i			1.64677i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.65685	+	5.65685i	0.327144	+	0.327144i
$300$		0			0
$301$	33.8885	−	33.8885i	1.95330	−	1.95330i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.4164			−1.91342
$306$		0			0
$307$	−21.1344	+	21.1344i	−1.20620	+	1.20620i	−0.233956	+	0.972247i	$0.575167\pi$
$307$	−21.1344	+	21.1344i	−1.20620	+	1.20620i	−0.972247	+	0.233956i	$0.924833\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	30.9551i		−	1.75530i	−0.479301	−	0.877651i	$-0.659110\pi$
$311$		−	30.9551i		−	1.75530i	0.479301	−	0.877651i	$-0.340890\pi$
$312$		0			0
$313$			3.05573i			0.172720i	0.996264	+	0.0863600i	$0.0275235\pi$
$313$			3.05573i			0.172720i	−0.996264	+	0.0863600i	$0.972476\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.76393	+	7.76393i	−0.436066	+	0.436066i	−0.890686	−	0.454620i	$-0.849775\pi$
$317$	−7.76393	+	7.76393i	−0.436066	+	0.436066i	0.454620	+	0.890686i	$0.349775\pi$
$318$		0			0
$319$	0.825324			0.0462093
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.3137	−	11.3137i	0.629512	−	0.629512i
$324$		0			0
$325$	5.00000	+	5.00000i	0.277350	+	0.277350i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.667701	+	0.667701i	0.0367002	+	0.0367002i	0.725219	−	0.688519i	$-0.241739\pi$
$331$	0.667701	+	0.667701i	0.0367002	+	0.0367002i	−0.688519	+	0.725219i	$0.741739\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−37.9473			−2.07328
$336$		0			0
$337$	2.94427			0.160385			0.0801924	−	0.996779i	$-0.474447\pi$
$337$	2.94427			0.160385			0.0801924	+	0.996779i	$0.474447\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.8885	−	17.8885i	−0.968719	−	0.968719i
$342$		0			0
$343$			31.7804i			1.71598i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.56564	+	9.56564i	0.513511	+	0.513511i	0.915600	−	0.402090i	$-0.131716\pi$
$347$	9.56564	+	9.56564i	0.513511	+	0.513511i	−0.402090	+	0.915600i	$0.631716\pi$
$348$		0			0
$349$	−15.4721	+	15.4721i	−0.828204	+	0.828204i	−0.987268	−	0.159064i	$-0.949152\pi$
$349$	−15.4721	+	15.4721i	−0.828204	+	0.828204i	0.159064	+	0.987268i	$0.449152\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.8885			−1.27146			−0.635729	−	0.771912i	$-0.719301\pi$
$353$	−23.8885			−1.27146			−0.635729	+	0.771912i	$0.719301\pi$
$354$		0			0
$355$	−7.81758	+	7.81758i	−0.414914	+	0.414914i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	26.1235i		−	1.37875i	−0.724406	−	0.689374i	$-0.757886\pi$
$359$		−	26.1235i		−	1.37875i	0.724406	−	0.689374i	$-0.242114\pi$
$360$		0			0
$361$			12.8885i			0.678344i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	33.4164	−	33.4164i	1.74909	−	1.74909i
$366$		0			0
$367$	−26.3786			−1.37695			−0.688475	−	0.725260i	$-0.741720\pi$
$367$	−26.3786			−1.37695			−0.688475	+	0.725260i	$0.741720\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.2256	−	17.2256i	0.894309	−	0.894309i
$372$		0			0
$373$	6.52786	+	6.52786i	0.338000	+	0.338000i	0.855614	−	0.517614i	$-0.173180\pi$
$373$	6.52786	+	6.52786i	0.338000	+	0.338000i	−0.517614	+	0.855614i	$0.673180\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	0.472136i		−	0.0243162i
$378$		0			0
$379$	0.412662	+	0.412662i	0.0211970	+	0.0211970i	0.717626	−	0.696429i	$-0.245229\pi$
$379$	0.412662	+	0.412662i	0.0211970	+	0.0211970i	−0.696429	+	0.717626i	$0.745229\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.7804			−1.62390			−0.811951	−	0.583725i	$-0.801595\pi$
$383$	−31.7804			−1.62390			−0.811951	+	0.583725i	$0.801595\pi$
$384$		0			0
$385$	−35.7771			−1.82337
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.65248	−	7.65248i	−0.387996	−	0.387996i	0.485976	−	0.873972i	$-0.338464\pi$
$389$	−7.65248	−	7.65248i	−0.387996	−	0.387996i	−0.873972	+	0.485976i	$0.838464\pi$
$390$		0			0
$391$		−	36.6119i		−	1.85154i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.41577	−	2.41577i	−0.121550	−	0.121550i
$396$		0			0
$397$	−20.4164	+	20.4164i	−1.02467	+	1.02467i	−0.0249822	+	0.999688i	$0.507953\pi$
$397$	−20.4164	+	20.4164i	−1.02467	+	1.02467i	−0.999688	+	0.0249822i	$0.992047\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.4721			1.12220			0.561102	−	0.827746i	$-0.310377\pi$
$401$	22.4721			1.12220			0.561102	+	0.827746i	$0.310377\pi$
$402$		0			0
$403$	−10.2333	+	10.2333i	−0.509759	+	0.509759i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.14678i		−	0.255116i
$408$		0			0
$409$			4.94427i			0.244479i	0.992501	+	0.122239i	$0.0390075\pi$
$409$			4.94427i			0.244479i	−0.992501	+	0.122239i	$0.960992\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−28.9443	+	28.9443i	−1.42425	+	1.42425i
$414$		0			0
$415$	−7.81758			−0.383750
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.40492	−	7.40492i	0.361754	−	0.361754i	−0.502704	−	0.864458i	$-0.667662\pi$
$419$	7.40492	−	7.40492i	0.361754	−	0.361754i	0.864458	+	0.502704i	$0.167662\pi$
$420$		0			0
$421$	−5.00000	−	5.00000i	−0.243685	−	0.243685i	0.574688	−	0.818373i	$-0.305124\pi$
$421$	−5.00000	−	5.00000i	−0.243685	−	0.243685i	−0.818373	+	0.574688i	$0.805124\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	32.3607i		−	1.56972i
$426$		0			0
$427$	−34.1962	−	34.1962i	−1.65487	−	1.65487i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.99226			−0.336805			−0.168403	−	0.985718i	$-0.553861\pi$
$431$	−6.99226			−0.336805			−0.168403	+	0.985718i	$0.553861\pi$
$432$		0			0
$433$	−24.0000			−1.15337			−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.88854	−	9.88854i	−0.473033	−	0.473033i
$438$		0			0
$439$		−	25.0432i		−	1.19525i	−0.801777	−	0.597623i	$-0.796112\pi$
$439$		−	25.0432i		−	1.19525i	0.801777	−	0.597623i	$-0.203888\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.9010	+	10.9010i	0.517924	+	0.517924i	0.916943	−	0.399019i	$-0.130649\pi$
$443$	10.9010	+	10.9010i	0.517924	+	0.517924i	−0.399019	+	0.916943i	$0.630649\pi$
$444$		0			0
$445$	22.3607	−	22.3607i	1.06000	−	1.06000i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.41641			−0.161230			−0.0806151	−	0.996745i	$-0.525688\pi$
$449$	−3.41641			−0.161230			−0.0806151	+	0.996745i	$0.525688\pi$
$450$		0			0
$451$	−11.3137	+	11.3137i	−0.532742	+	0.532742i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			20.4667i			0.959493i
$456$		0			0
$457$		−	20.9443i		−	0.979732i	−0.871798	−	0.489866i	$-0.837046\pi$
$457$		−	20.9443i		−	0.979732i	0.871798	−	0.489866i	$-0.162954\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.76393	−	1.76393i	0.0821545	−	0.0821545i	−0.664835	−	0.746990i	$-0.731498\pi$
$461$	1.76393	−	1.76393i	0.0821545	−	0.0821545i	0.746990	+	0.664835i	$0.231498\pi$
$462$		0			0
$463$	−1.08036			−0.0502087			−0.0251044	−	0.999685i	$-0.507992\pi$
$463$	−1.08036			−0.0502087			−0.0251044	+	0.999685i	$0.507992\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.2148	+	22.2148i	−1.02798	+	1.02798i	−0.0283790	+	0.999597i	$0.509035\pi$
$467$	−22.2148	+	22.2148i	−1.02798	+	1.02798i	−0.999597	+	0.0283790i	$0.990965\pi$
$468$		0			0
$469$	−38.8328	−	38.8328i	−1.79313	−	1.79313i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			25.8885i			1.19036i
$474$		0			0
$475$	−8.74032	−	8.74032i	−0.401033	−	0.401033i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.15298			−0.418210			−0.209105	−	0.977893i	$-0.567055\pi$
$479$	−9.15298			−0.418210			−0.209105	+	0.977893i	$0.567055\pi$
$480$		0			0
$481$	−2.94427			−0.134247
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.0557	−	11.0557i	−0.502015	−	0.502015i
$486$		0			0
$487$		−	2.41577i		−	0.109469i	−0.998501	−	0.0547344i	$-0.982569\pi$
$487$		−	2.41577i		−	0.109469i	0.998501	−	0.0547344i	$-0.0174312\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.82068	+	9.82068i	0.443201	+	0.443201i	0.893086	−	0.449885i	$-0.148535\pi$
$491$	9.82068	+	9.82068i	0.443201	+	0.443201i	−0.449885	+	0.893086i	$0.648535\pi$
$492$		0			0
$493$	−1.52786	+	1.52786i	−0.0688115	+	0.0688115i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000			−0.717698
$498$		0			0
$499$	4.98915	−	4.98915i	0.223345	−	0.223345i	−0.586560	−	0.809906i	$-0.699518\pi$
$499$	4.98915	−	4.98915i	0.223345	−	0.223345i	0.809906	+	0.586560i	$0.199518\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			33.1158i			1.47656i	0.674494	+	0.738280i	$0.264362\pi$
$503$			33.1158i			1.47656i	−0.674494	+	0.738280i	$0.735638\pi$
$504$		0			0
$505$			18.9443i			0.843009i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.7639	−	23.7639i	1.05332	−	1.05332i	0.0548216	−	0.998496i	$-0.482541\pi$
$509$	23.7639	−	23.7639i	1.05332	−	1.05332i	0.998496	−	0.0548216i	$-0.0174590\pi$
$510$		0			0
$511$	68.3923			3.02550
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.2333	−	10.2333i	0.450935	−	0.450935i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			16.3607i			0.716774i	0.933573	+	0.358387i	$0.116673\pi$
$521$			16.3607i			0.716774i	−0.933573	+	0.358387i	$0.883327\pi$
$522$		0			0
$523$	20.0540	+	20.0540i	0.876901	+	0.876901i	0.993213	−	0.116311i	$-0.0371070\pi$
$523$	20.0540	+	20.0540i	0.876901	+	0.876901i	−0.116311	+	0.993213i	$0.537107\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	66.2316			2.88509
$528$		0			0
$529$	−9.00000			−0.391304
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.47214	+	6.47214i	0.280339	+	0.280339i
$534$		0			0
$535$			53.5825i			2.31657i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.3755	−	24.3755i	−1.04993	−	1.04993i
$540$		0			0
$541$	−16.8885	+	16.8885i	−0.726095	+	0.726095i	−0.969840	−	0.243744i	$-0.921624\pi$
$541$	−16.8885	+	16.8885i	−0.726095	+	0.726095i	0.243744	+	0.969840i	$0.421624\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	26.5836			1.13872
$546$		0			0
$547$	−8.74032	+	8.74032i	−0.373709	+	0.373709i	−0.868826	−	0.495117i	$-0.835125\pi$
$547$	−8.74032	+	8.74032i	−0.373709	+	0.373709i	0.495117	+	0.868826i	$0.335125\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			0.825324i			0.0351600i
$552$		0			0
$553$		−	4.94427i		−	0.210252i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.1803	−	11.1803i	0.473726	−	0.473726i	−0.429392	−	0.903118i	$-0.641272\pi$
$557$	11.1803	−	11.1803i	0.473726	−	0.473726i	0.903118	+	0.429392i	$0.141272\pi$
$558$		0			0
$559$	14.8098			0.626389
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.40492	−	7.40492i	0.312080	−	0.312080i	−0.533635	−	0.845715i	$-0.679174\pi$
$563$	7.40492	−	7.40492i	0.312080	−	0.312080i	0.845715	+	0.533635i	$0.179174\pi$
$564$		0			0
$565$	4.47214	+	4.47214i	0.188144	+	0.188144i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			19.4164i			0.813978i	0.913433	+	0.406989i	$0.133421\pi$
$569$			19.4164i			0.813978i	−0.913433	+	0.406989i	$0.866579\pi$
$570$		0			0
$571$	−6.32456	−	6.32456i	−0.264674	−	0.264674i	0.562276	−	0.826950i	$-0.309926\pi$
$571$	−6.32456	−	6.32456i	−0.264674	−	0.264674i	−0.826950	+	0.562276i	$0.809926\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.2843			−1.17954
$576$		0			0
$577$	9.05573			0.376995			0.188497	−	0.982074i	$-0.439638\pi$
$577$	9.05573			0.376995			0.188497	+	0.982074i	$0.439638\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.00000	−	8.00000i	−0.331896	−	0.331896i
$582$		0			0
$583$			13.1592i			0.544998i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.6460	+	10.6460i	0.439408	+	0.439408i	0.891813	−	0.452405i	$-0.149434\pi$
$587$	10.6460	+	10.6460i	0.439408	+	0.439408i	−0.452405	+	0.891813i	$0.649434\pi$
$588$		0			0
$589$	17.8885	−	17.8885i	0.737085	−	0.737085i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.8885			−1.14525			−0.572623	−	0.819819i	$-0.694074\pi$
$593$	−27.8885			−1.14525			−0.572623	+	0.819819i	$0.694074\pi$
$594$		0			0
$595$	66.2316	−	66.2316i	2.71523	−	2.71523i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	5.65685i		−	0.231133i	−0.993300	−	0.115566i	$-0.963132\pi$
$599$		−	5.65685i		−	0.231133i	0.993300	−	0.115566i	$-0.0368683\pi$
$600$		0			0
$601$			10.9443i			0.446426i	0.974770	+	0.223213i	$0.0716546\pi$
$601$			10.9443i			0.446426i	−0.974770	+	0.223213i	$0.928345\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.9311	+	10.9311i	−0.444413	+	0.444413i
$606$		0			0
$607$	−25.8685			−1.04997			−0.524985	−	0.851111i	$-0.675929\pi$
$607$	−25.8685			−1.04997			−0.524985	+	0.851111i	$0.675929\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.47214	+	1.47214i	0.0594590	+	0.0594590i	0.736211	−	0.676752i	$-0.236613\pi$
$613$	1.47214	+	1.47214i	0.0594590	+	0.0594590i	−0.676752	+	0.736211i	$0.736613\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			19.8885i			0.800683i	0.916366	+	0.400341i	$0.131108\pi$
$617$			19.8885i			0.800683i	−0.916366	+	0.400341i	$0.868892\pi$
$618$		0			0
$619$	21.1344	+	21.1344i	0.849463	+	0.849463i	0.990066	−	0.140603i	$-0.0449041\pi$
$619$	21.1344	+	21.1344i	0.849463	+	0.849463i	−0.140603	+	0.990066i	$0.544904\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	45.7649			1.83353
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.52786	+	9.52786i	0.379901	+	0.379901i
$630$		0			0
$631$		−	29.8747i		−	1.18929i	−0.803987	−	0.594647i	$-0.797292\pi$
$631$		−	29.8747i		−	1.18929i	0.803987	−	0.594647i	$-0.202708\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.0509	−	18.0509i	−0.716329	−	0.716329i
$636$		0			0
$637$	−13.9443	+	13.9443i	−0.552492	+	0.552492i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−35.4164			−1.39886			−0.699432	−	0.714699i	$-0.746564\pi$
$641$	−35.4164			−1.39886			−0.699432	+	0.714699i	$0.746564\pi$
$642$		0			0
$643$	25.7109	−	25.7109i	1.01394	−	1.01394i	0.0140368	−	0.999901i	$-0.495532\pi$
$643$	25.7109	−	25.7109i	1.01394	−	1.01394i	0.999901	−	0.0140368i	$-0.00446819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	8.32766i		−	0.327394i	−0.986511	−	0.163697i	$-0.947658\pi$
$647$		−	8.32766i		−	0.327394i	0.986511	−	0.163697i	$-0.0523419\pi$
$648$		0			0
$649$		−	22.1115i		−	0.867951i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.81966	−	4.81966i	0.188608	−	0.188608i	−0.606486	−	0.795094i	$-0.707421\pi$
$653$	4.81966	−	4.81966i	0.188608	−	0.188608i	0.795094	+	0.606486i	$0.207421\pi$
$654$		0			0
$655$	−2.98605			−0.116675
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.82843	+	2.82843i	−0.110180	+	0.110180i	−0.760047	−	0.649868i	$-0.774824\pi$
$659$	−2.82843	+	2.82843i	−0.110180	+	0.110180i	0.649868	+	0.760047i	$0.274824\pi$
$660$		0			0
$661$	−1.58359	−	1.58359i	−0.0615946	−	0.0615946i	0.675638	−	0.737233i	$-0.263868\pi$
$661$	−1.58359	−	1.58359i	−0.0615946	−	0.0615946i	−0.737233	+	0.675638i	$0.763868\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	35.7771i		−	1.38738i
$666$		0			0
$667$	1.33540	+	1.33540i	0.0517070	+	0.0517070i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	26.1235			1.00849
$672$		0			0
$673$	17.8885			0.689553			0.344776	−	0.938685i	$-0.387955\pi$
$673$	17.8885			0.689553			0.344776	+	0.938685i	$0.387955\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.6525	−	31.6525i	−1.21650	−	1.21650i	−0.968848	−	0.247655i	$-0.920340\pi$
$677$	−31.6525	−	31.6525i	−1.21650	−	1.21650i	−0.247655	−	0.968848i	$-0.579660\pi$
$678$		0			0
$679$		−	22.6274i		−	0.868361i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.5579	+	16.5579i	0.633570	+	0.633570i	0.948962	−	0.315391i	$-0.102136\pi$
$683$	16.5579	+	16.5579i	0.633570	+	0.633570i	−0.315391	+	0.948962i	$0.602136\pi$
$684$		0			0
$685$	−14.4721	+	14.4721i	−0.552952	+	0.552952i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.52786			0.286789
$690$		0			0
$691$	24.3755	−	24.3755i	0.927287	−	0.927287i	−0.0702429	−	0.997530i	$-0.522377\pi$
$691$	24.3755	−	24.3755i	0.927287	−	0.927287i	0.997530	+	0.0702429i	$0.0223774\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2.98605i			0.113267i
$696$		0			0
$697$		−	41.8885i		−	1.58664i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.81966	−	2.81966i	0.106497	−	0.106497i	−0.651850	−	0.758348i	$-0.726007\pi$
$701$	2.81966	−	2.81966i	0.106497	−	0.106497i	0.758348	+	0.651850i	$0.226007\pi$
$702$		0			0
$703$	5.14678			0.194114
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.3863	+	19.3863i	−0.729098	+	0.729098i
$708$		0			0
$709$	−6.05573	−	6.05573i	−0.227428	−	0.227428i	0.584190	−	0.811617i	$-0.301412\pi$
$709$	−6.05573	−	6.05573i	−0.227428	−	0.227428i	−0.811617	+	0.584190i	$0.801412\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	57.8885i		−	2.16794i
$714$		0			0
$715$	−7.81758	−	7.81758i	−0.292361	−	0.292361i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.1452			0.602116			0.301058	−	0.953606i	$-0.402660\pi$
$719$	16.1452			0.602116			0.301058	+	0.953606i	$0.402660\pi$
$720$		0			0
$721$	20.9443			0.780005
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.18034	+	1.18034i	0.0438367	+	0.0438367i
$726$		0			0
$727$			11.5687i			0.429061i	0.976717	+	0.214531i	$0.0688222\pi$
$727$			11.5687i			0.429061i	−0.976717	+	0.214531i	$0.931178\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−47.9256	−	47.9256i	−1.77259	−	1.77259i
$732$		0			0
$733$	−4.05573	+	4.05573i	−0.149802	+	0.149802i	−0.778029	−	0.628228i	$-0.783781\pi$
$733$	−4.05573	+	4.05573i	−0.149802	+	0.149802i	0.628228	+	0.778029i	$0.283781\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.6656			1.09275
$738$		0			0
$739$	−10.6460	+	10.6460i	−0.391620	+	0.391620i	−0.875264	−	0.483645i	$-0.839313\pi$
$739$	−10.6460	+	10.6460i	−0.391620	+	0.391620i	0.483645	+	0.875264i	$0.339313\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			14.8098i			0.543320i	0.962393	+	0.271660i	$0.0875726\pi$
$743$			14.8098i			0.543320i	−0.962393	+	0.271660i	$0.912427\pi$
$744$		0			0
$745$		−	47.8885i		−	1.75450i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−54.8328	+	54.8328i	−2.00355	+	2.00355i
$750$		0			0
$751$	−1.08036			−0.0394230			−0.0197115	−	0.999806i	$-0.506275\pi$
$751$	−1.08036			−0.0394230			−0.0197115	+	0.999806i	$0.506275\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.8685	+	25.8685i	−0.941451	+	0.941451i
$756$		0			0
$757$	7.94427	+	7.94427i	0.288739	+	0.288739i	0.836582	−	0.547842i	$-0.184551\pi$
$757$	7.94427	+	7.94427i	0.288739	+	0.288739i	−0.547842	+	0.836582i	$0.684551\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.4164i		−	0.703844i	−0.936029	−	0.351922i	$-0.885528\pi$
$761$		−	19.4164i		−	0.703844i	0.936029	−	0.351922i	$-0.114472\pi$
$762$		0			0
$763$	27.2039	+	27.2039i	0.984848	+	0.984848i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.6491			−0.456733
$768$		0			0
$769$	22.8328			0.823372			0.411686	−	0.911326i	$-0.364940\pi$
$769$	22.8328			0.823372			0.411686	+	0.911326i	$0.364940\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.70820	+	8.70820i	0.313212	+	0.313212i	0.846153	−	0.532940i	$-0.178913\pi$
$773$	8.70820	+	8.70820i	0.313212	+	0.313212i	−0.532940	+	0.846153i	$0.678913\pi$
$774$		0			0
$775$		−	51.1667i		−	1.83796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.3137	−	11.3137i	−0.405356	−	0.405356i
$780$		0			0
$781$	6.11146	−	6.11146i	0.218685	−	0.218685i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.36068			0.0842563
$786$		0			0
$787$	3.90879	−	3.90879i	0.139333	−	0.139333i	−0.634000	−	0.773333i	$-0.718588\pi$
$787$	3.90879	−	3.90879i	0.139333	−	0.139333i	0.773333	+	0.634000i	$0.218588\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			9.15298i			0.325443i
$792$		0			0
$793$		−	14.9443i		−	0.530687i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.6525	+	27.6525i	−0.979501	+	0.979501i	−0.999794	−	0.0202931i	$-0.993540\pi$
$797$	−27.6525	+	27.6525i	−0.979501	+	0.979501i	0.0202931	+	0.999794i	$0.493540\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−26.1235	+	26.1235i	−0.921880	+	0.921880i
$804$		0			0
$805$	−57.8885	−	57.8885i	−2.04030	−	2.04030i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.41641i			0.120115i	0.998195	+	0.0600573i	$0.0191283\pi$
$809$			3.41641i			0.120115i	−0.998195	+	0.0600573i	$0.980872\pi$
$810$		0			0
$811$	−23.5502	−	23.5502i	−0.826958	−	0.826958i	0.160137	−	0.987095i	$-0.448806\pi$
$811$	−23.5502	−	23.5502i	−0.826958	−	0.826958i	−0.987095	+	0.160137i	$0.948806\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.4806			−0.612320
$816$		0			0
$817$	−25.8885			−0.905725
$818$		0			0
$819$		0			0
$820$		0			0
$821$	31.5410	+	31.5410i	1.10079	+	1.10079i	0.994316	+	0.106474i	$0.0339560\pi$
$821$	31.5410	+	31.5410i	1.10079	+	1.10079i	0.106474	+	0.994316i	$0.466044\pi$
$822$		0			0
$823$		−	4.57649i		−	0.159526i	−0.996814	−	0.0797632i	$-0.974584\pi$
$823$		−	4.57649i		−	0.159526i	0.996814	−	0.0797632i	$-0.0254164\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	38.1050	+	38.1050i	1.32504	+	1.32504i	0.909639	+	0.415400i	$0.136359\pi$
$827$	38.1050	+	38.1050i	1.32504	+	1.32504i	0.415400	+	0.909639i	$0.363641\pi$
$828$		0			0
$829$	13.9443	−	13.9443i	0.484305	−	0.484305i	−0.422199	−	0.906503i	$-0.638742\pi$
$829$	13.9443	−	13.9443i	0.484305	−	0.484305i	0.906503	+	0.422199i	$0.138742\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	90.2492			3.12695
$834$		0			0
$835$	−17.4806	+	17.4806i	−0.604943	+	0.604943i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			44.4295i			1.53388i	0.641721	+	0.766939i	$0.278221\pi$
$839$			44.4295i			1.53388i	−0.641721	+	0.766939i	$0.721779\pi$
$840$		0			0
$841$			28.8885i			0.996157i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.5967	−	24.5967i	0.846154	−	0.846154i
$846$		0			0
$847$	−22.3724			−0.768724
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.32766	−	8.32766i	0.285468	−	0.285468i
$852$		0			0
$853$	−12.5279	−	12.5279i	−0.428946	−	0.428946i	0.459323	−	0.888269i	$-0.348092\pi$
$853$	−12.5279	−	12.5279i	−0.428946	−	0.428946i	−0.888269	+	0.459323i	$0.848092\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	28.5836i		−	0.976397i	−0.872733	−	0.488198i	$-0.837654\pi$
$857$		−	28.5836i		−	0.976397i	0.872733	−	0.488198i	$-0.162346\pi$
$858$		0			0
$859$	32.1931	+	32.1931i	1.09841	+	1.09841i	0.994596	+	0.103817i	$0.0331055\pi$
$859$	32.1931	+	32.1931i	1.09841	+	1.09841i	0.103817	+	0.994596i	$0.466894\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−34.4512			−1.17273			−0.586366	−	0.810046i	$-0.699442\pi$
$863$	−34.4512			−1.17273			−0.586366	+	0.810046i	$0.699442\pi$
$864$		0			0
$865$	30.0000			1.02003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.88854	+	1.88854i	0.0640645	+	0.0640645i
$870$		0			0
$871$		−	16.9706i		−	0.575026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25.3607	+	25.3607i	−0.856369	+	0.856369i	−0.990908	−	0.134539i	$-0.957045\pi$
$877$	−25.3607	+	25.3607i	−0.856369	+	0.856369i	0.134539	+	0.990908i	$0.457045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.8885			0.939589			0.469794	−	0.882776i	$-0.344328\pi$
$881$	27.8885			0.939589			0.469794	+	0.882776i	$0.344328\pi$
$882$		0			0
$883$	−13.0618	+	13.0618i	−0.439564	+	0.439564i	−0.891865	−	0.452301i	$-0.850603\pi$
$883$	−13.0618	+	13.0618i	−0.439564	+	0.439564i	0.452301	+	0.891865i	$0.350603\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			42.2688i			1.41925i	0.704581	+	0.709623i	$0.251135\pi$
$887$			42.2688i			1.41925i	−0.704581	+	0.709623i	$0.748865\pi$
$888$		0			0
$889$		−	36.9443i		−	1.23907i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−37.9473			−1.26844
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.41577	+	2.41577i	−0.0805703	+	0.0805703i
$900$		0			0
$901$	−24.3607	−	24.3607i	−0.811572	−	0.811572i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	49.1935i		−	1.63525i
$906$		0			0
$907$	−36.1993	−	36.1993i	−1.20198	−	1.20198i	−0.973564	−	0.228413i	$-0.926647\pi$
$907$	−36.1993	−	36.1993i	−1.20198	−	1.20198i	−0.228413	−	0.973564i	$-0.573353\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.16073			0.0715880			0.0357940	−	0.999359i	$-0.488604\pi$
$911$	2.16073			0.0715880			0.0357940	+	0.999359i	$0.488604\pi$
$912$		0			0
$913$	6.11146			0.202260
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.05573	−	3.05573i	−0.100909	−	0.100909i
$918$		0			0
$919$		−	34.1962i		−	1.12803i	−0.825765	−	0.564014i	$-0.809257\pi$
$919$		−	34.1962i		−	1.12803i	0.825765	−	0.564014i	$-0.190743\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.49613	−	3.49613i	−0.115076	−	0.115076i
$924$		0			0
$925$	7.36068	−	7.36068i	0.242018	−	0.242018i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.47214			−0.212344			−0.106172	−	0.994348i	$-0.533859\pi$
$929$	−6.47214			−0.212344			−0.106172	+	0.994348i	$0.533859\pi$
$930$		0			0
$931$	24.3755	−	24.3755i	0.798874	−	0.798874i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			50.5964i			1.65468i
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.7639	−	17.7639i	0.579088	−	0.579088i	−0.355564	−	0.934652i	$-0.615711\pi$
$941$	17.7639	−	17.7639i	0.579088	−	0.579088i	0.934652	+	0.355564i	$0.115711\pi$
$942$		0			0
$943$	−36.6119			−1.19225
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.7835	+	33.7835i	−1.09782	+	1.09782i	−0.103151	+	0.994666i	$0.532892\pi$
$947$	−33.7835	+	33.7835i	−1.09782	+	1.09782i	−0.994666	+	0.103151i	$0.967108\pi$
$948$		0			0
$949$	14.9443	+	14.9443i	0.485112	+	0.485112i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			9.52786i			0.308638i	0.988021	+	0.154319i	$0.0493183\pi$
$953$			9.52786i			0.308638i	−0.988021	+	0.154319i	$0.950682\pi$
$954$		0			0
$955$	4.83153	+	4.83153i	0.156345	+	0.156345i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−29.6197			−0.956469
$960$		0			0
$961$	73.7214			2.37811
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.9443	+	28.9443i	0.931749	+	0.931749i
$966$		0			0
$967$		−	18.5610i		−	0.596882i	−0.954428	−	0.298441i	$-0.903533\pi$
$967$		−	18.5610i		−	0.596882i	0.954428	−	0.298441i	$-0.0964666\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.3972	+	14.3972i	0.462027	+	0.462027i	0.899319	−	0.437292i	$-0.144062\pi$
$971$	14.3972	+	14.3972i	0.462027	+	0.462027i	−0.437292	+	0.899319i	$0.644062\pi$
$972$		0			0
$973$	−3.05573	+	3.05573i	−0.0979621	+	0.0979621i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.5836			0.402585			0.201292	−	0.979531i	$-0.435486\pi$
$977$	12.5836			0.402585			0.201292	+	0.979531i	$0.435486\pi$
$978$		0			0
$979$	−17.4806	+	17.4806i	−0.558684	+	0.558684i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	53.5825i		−	1.70902i	−0.519438	−	0.854508i	$-0.673859\pi$
$983$		−	53.5825i		−	1.70902i	0.519438	−	0.854508i	$-0.326141\pi$
$984$		0			0
$985$		−	76.8328i		−	2.44810i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41.8885	+	41.8885i	−1.33198	+	1.33198i
$990$		0			0
$991$	−17.2256			−0.547189			−0.273595	−	0.961845i	$-0.588213\pi$
$991$	−17.2256			−0.547189			−0.273595	+	0.961845i	$0.588213\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.7000	−	30.7000i	0.973257	−	0.973257i
$996$		0			0
$997$	−20.5279	−	20.5279i	−0.650124	−	0.650124i	0.302899	−	0.953023i	$-0.402046\pi$
$997$	−20.5279	−	20.5279i	−0.650124	−	0.650124i	−0.953023	+	0.302899i	$0.902046\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bg.3457.1		8
3.2	odd	2		1536.2.j.h.385.4	yes	8
4.3	odd	2	inner	4608.2.k.bg.3457.2		8
8.3	odd	2		4608.2.k.bf.3457.4		8
8.5	even	2		4608.2.k.bf.3457.3		8
12.11	even	2		1536.2.j.h.385.2	yes	8
16.3	odd	4	inner	4608.2.k.bg.1153.1		8
16.5	even	4		4608.2.k.bf.1153.4		8
16.11	odd	4		4608.2.k.bf.1153.3		8
16.13	even	4	inner	4608.2.k.bg.1153.2		8
24.5	odd	2		1536.2.j.g.385.1	✓	8
24.11	even	2		1536.2.j.g.385.3	yes	8
32.3	odd	8		9216.2.a.bj.1.4		4
32.13	even	8		9216.2.a.bj.1.1		4
32.19	odd	8		9216.2.a.bd.1.2		4
32.29	even	8		9216.2.a.bd.1.3		4
48.5	odd	4		1536.2.j.g.1153.1	yes	8
48.11	even	4		1536.2.j.g.1153.3	yes	8
48.29	odd	4		1536.2.j.h.1153.4	yes	8
48.35	even	4		1536.2.j.h.1153.2	yes	8
96.5	odd	8		3072.2.d.g.1537.8		8
96.11	even	8		3072.2.d.g.1537.5		8
96.29	odd	8		3072.2.a.q.1.1		4
96.35	even	8		3072.2.a.k.1.2		4
96.53	odd	8		3072.2.d.g.1537.2		8
96.59	even	8		3072.2.d.g.1537.3		8
96.77	odd	8		3072.2.a.k.1.3		4
96.83	even	8		3072.2.a.q.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.1	✓	8	24.5	odd	2
1536.2.j.g.385.3	yes	8	24.11	even	2
1536.2.j.g.1153.1	yes	8	48.5	odd	4
1536.2.j.g.1153.3	yes	8	48.11	even	4
1536.2.j.h.385.2	yes	8	12.11	even	2
1536.2.j.h.385.4	yes	8	3.2	odd	2
1536.2.j.h.1153.2	yes	8	48.35	even	4
1536.2.j.h.1153.4	yes	8	48.29	odd	4
3072.2.a.k.1.2		4	96.35	even	8
3072.2.a.k.1.3		4	96.77	odd	8
3072.2.a.q.1.1		4	96.29	odd	8
3072.2.a.q.1.4		4	96.83	even	8
3072.2.d.g.1537.2		8	96.53	odd	8
3072.2.d.g.1537.3		8	96.59	even	8
3072.2.d.g.1537.5		8	96.11	even	8
3072.2.d.g.1537.8		8	96.5	odd	8
4608.2.k.bf.1153.3		8	16.11	odd	4
4608.2.k.bf.1153.4		8	16.5	even	4
4608.2.k.bf.3457.3		8	8.5	even	2
4608.2.k.bf.3457.4		8	8.3	odd	2
4608.2.k.bg.1153.1		8	16.3	odd	4	inner
4608.2.k.bg.1153.2		8	16.13	even	4	inner
4608.2.k.bg.3457.1		8	1.1	even	1	trivial
4608.2.k.bg.3457.2		8	4.3	odd	2	inner
9216.2.a.bd.1.2		4	32.19	odd	8
9216.2.a.bd.1.3		4	32.29	even	8
9216.2.a.bj.1.1		4	32.13	even	8
9216.2.a.bj.1.4		4	32.3	odd	8

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.23607 - 2.23607i) q^{5} -4.57649i q^{7} +O(q^{10})\)	\(q+(-2.23607 - 2.23607i) q^{5} -4.57649i q^{7} +(1.74806 + 1.74806i) q^{11} +(1.00000 - 1.00000i) q^{13} -6.47214 q^{17} +(-1.74806 + 1.74806i) q^{19} +5.65685i q^{23} +5.00000i q^{25} +(0.236068 - 0.236068i) q^{29} -10.2333 q^{31} +(-10.2333 + 10.2333i) q^{35} +(-1.47214 - 1.47214i) q^{37} +6.47214i q^{41} +(7.40492 + 7.40492i) q^{43} -13.9443 q^{49} +(3.76393 + 3.76393i) q^{53} -7.81758i q^{55} +(-6.32456 - 6.32456i) q^{59} +(7.47214 - 7.47214i) q^{61} -4.47214 q^{65} +(8.48528 - 8.48528i) q^{67} -3.49613i q^{71} +14.9443i q^{73} +(8.00000 - 8.00000i) q^{77} +1.08036 q^{79} +(1.74806 - 1.74806i) q^{83} +(14.4721 + 14.4721i) q^{85} +10.0000i q^{89} +(-4.57649 - 4.57649i) q^{91} +7.81758 q^{95} +4.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} - 16 q^{17} - 16 q^{29} + 24 q^{37} - 40 q^{49} + 48 q^{53} + 24 q^{61} + 64 q^{77} + 80 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 16 * q^17 - 16 * q^29 + 24 * q^37 - 40 * q^49 + 48 * q^53 + 24 * q^61 + 64 * q^77 + 80 * q^85 - 32 * q^97

Introduction

L-functions

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