LMFDB - Embedded newform 4608.2.k.be.3457.3

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:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3457.3
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4608.3457
Dual form			4608.2.k.be.1153.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.234633 - 0.234633i) q^{5} +3.08239i q^{7} +O(q^{10})$	$q+(-0.234633 - 0.234633i) q^{5} +3.08239i q^{7} +(-2.61313 - 2.61313i) q^{11} +(-3.28130 + 3.28130i) q^{13} +6.52395 q^{17} +(-0.613126 + 0.613126i) q^{19} +4.00000i q^{23} -4.88989i q^{25} +(3.46088 - 3.46088i) q^{29} +6.14386 q^{31} +(0.723231 - 0.723231i) q^{35} +(2.57900 + 2.57900i) q^{37} +3.92856i q^{41} +(-2.77791 - 2.77791i) q^{43} -1.65685 q^{47} -2.50114 q^{49} +(-0.399418 - 0.399418i) q^{53} +1.22625i q^{55} +(4.66364 + 4.66364i) q^{59} +(-10.4689 + 10.4689i) q^{61} +1.53981 q^{65} +(-9.88989 + 9.88989i) q^{67} -7.49207i q^{71} -5.62408i q^{73} +(8.05468 - 8.05468i) q^{77} -14.9040 q^{79} +(-6.61313 + 6.61313i) q^{83} +(-1.53073 - 1.53073i) q^{85} +18.1094i q^{89} +(-10.1143 - 10.1143i) q^{91} +0.287719 q^{95} -17.0479 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 8 q^{13} + 16 q^{19} - 8 q^{29} + 16 q^{31} + 32 q^{35} - 8 q^{37} + 16 q^{43} + 32 q^{47} - 8 q^{49} + 8 q^{53} + 32 q^{59} - 8 q^{61} + 16 q^{65} - 32 q^{67} - 48 q^{79} - 32 q^{83} - 48 q^{91} - 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^13 + 16 * q^19 - 8 * q^29 + 16 * q^31 + 32 * q^35 - 8 * q^37 + 16 * q^43 + 32 * q^47 - 8 * q^49 + 8 * q^53 + 32 * q^59 - 8 * q^61 + 16 * q^65 - 32 * q^67 - 48 * q^79 - 32 * q^83 - 48 * q^91 - 64 * q^95 - 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
$5$	−0.234633	−	0.234633i	−0.104931	−	0.104931i	0.652692	−	0.757623i	$-0.273640\pi$
$5$	−0.234633	−	0.234633i	−0.104931	−	0.104931i	−0.757623	+	0.652692i	$0.773640\pi$
$6$		0			0
$7$			3.08239i			1.16503i	0.812818	+	0.582517i	$0.197932\pi$
$7$			3.08239i			1.16503i	−0.812818	+	0.582517i	$0.802068\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.61313	−	2.61313i	−0.787887	−	0.787887i	0.193260	−	0.981148i	$-0.438094\pi$
$11$	−2.61313	−	2.61313i	−0.787887	−	0.787887i	−0.981148	+	0.193260i	$0.938094\pi$
$12$		0			0
$13$	−3.28130	+	3.28130i	−0.910070	+	0.910070i	−0.996277	−	0.0862071i	$-0.972525\pi$
$13$	−3.28130	+	3.28130i	−0.910070	+	0.910070i	0.0862071	+	0.996277i	$0.472525\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.52395			1.58229			0.791145	−	0.611629i	$-0.209486\pi$
$17$	6.52395			1.58229			0.791145	+	0.611629i	$0.209486\pi$
$18$		0			0
$19$	−0.613126	+	0.613126i	−0.140661	+	0.140661i	−0.773931	−	0.633270i	$-0.781712\pi$
$19$	−0.613126	+	0.613126i	−0.140661	+	0.140661i	0.633270	+	0.773931i	$0.281712\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$		−	4.88989i		−	0.977979i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.46088	−	3.46088i	0.642670	−	0.642670i	−0.308541	−	0.951211i	$-0.599841\pi$
$29$	3.46088	−	3.46088i	0.642670	−	0.642670i	0.951211	+	0.308541i	$0.0998407\pi$
$30$		0			0
$31$	6.14386			1.10347			0.551735	−	0.834020i	$-0.313966\pi$
$31$	6.14386			1.10347			0.551735	+	0.834020i	$0.313966\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.723231	−	0.723231i	0.122248	−	0.122248i
$36$		0			0
$37$	2.57900	+	2.57900i	0.423985	+	0.423985i	0.886573	−	0.462588i	$-0.153079\pi$
$37$	2.57900	+	2.57900i	0.423985	+	0.423985i	−0.462588	+	0.886573i	$0.653079\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.92856i			0.613538i	0.951784	+	0.306769i	$0.0992479\pi$
$41$			3.92856i			0.613538i	−0.951784	+	0.306769i	$0.900752\pi$
$42$		0			0
$43$	−2.77791	−	2.77791i	−0.423627	−	0.423627i	0.462823	−	0.886451i	$-0.346836\pi$
$43$	−2.77791	−	2.77791i	−0.423627	−	0.423627i	−0.886451	+	0.462823i	$0.846836\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.65685			−0.241677			−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685			−0.241677			−0.120839	+	0.992672i	$0.538558\pi$
$48$		0			0
$49$	−2.50114			−0.357306
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.399418	−	0.399418i	−0.0548642	−	0.0548642i	0.679142	−	0.734007i	$-0.262352\pi$
$53$	−0.399418	−	0.399418i	−0.0548642	−	0.0548642i	−0.734007	+	0.679142i	$0.762352\pi$
$54$		0			0
$55$			1.22625i			0.165348i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.66364	+	4.66364i	0.607155	+	0.607155i	0.942201	−	0.335047i	$-0.108752\pi$
$59$	4.66364	+	4.66364i	0.607155	+	0.607155i	−0.335047	+	0.942201i	$0.608752\pi$
$60$		0			0
$61$	−10.4689	+	10.4689i	−1.34040	+	1.34040i	−0.444749	+	0.895655i	$0.646707\pi$
$61$	−10.4689	+	10.4689i	−1.34040	+	1.34040i	−0.895655	+	0.444749i	$0.853293\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.53981			0.190989
$66$		0			0
$67$	−9.88989	+	9.88989i	−1.20824	+	1.20824i	−0.236647	+	0.971596i	$0.576049\pi$
$67$	−9.88989	+	9.88989i	−1.20824	+	1.20824i	−0.971596	+	0.236647i	$0.923951\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.49207i		−	0.889145i	−0.895743	−	0.444573i	$-0.853356\pi$
$71$		−	7.49207i		−	0.889145i	0.895743	−	0.444573i	$-0.146644\pi$
$72$		0			0
$73$		−	5.62408i		−	0.658248i	−0.944287	−	0.329124i	$-0.893247\pi$
$73$		−	5.62408i		−	0.658248i	0.944287	−	0.329124i	$-0.106753\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.05468	−	8.05468i	0.917916	−	0.917916i
$78$		0			0
$79$	−14.9040			−1.67683			−0.838417	−	0.545029i	$-0.816519\pi$
$79$	−14.9040			−1.67683			−0.838417	+	0.545029i	$0.816519\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.61313	+	6.61313i	−0.725885	+	0.725885i	−0.969797	−	0.243912i	$-0.921569\pi$
$83$	−6.61313	+	6.61313i	−0.725885	+	0.725885i	0.243912	+	0.969797i	$0.421569\pi$
$84$		0			0
$85$	−1.53073	−	1.53073i	−0.166031	−	0.166031i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			18.1094i			1.91959i	0.280705	+	0.959794i	$0.409432\pi$
$89$			18.1094i			1.91959i	−0.280705	+	0.959794i	$0.590568\pi$
$90$		0			0
$91$	−10.1143	−	10.1143i	−1.06026	−	1.06026i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.287719			0.0295194
$96$		0			0
$97$	−17.0479			−1.73095			−0.865476	−	0.500951i	$-0.832984\pi$
$97$	−17.0479			−1.73095			−0.865476	+	0.500951i	$0.832984\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.59379	+	8.59379i	0.855114	+	0.855114i	0.990758	−	0.135643i	$-0.0433101\pi$
$101$	8.59379	+	8.59379i	0.855114	+	0.855114i	−0.135643	+	0.990758i	$0.543310\pi$
$102$		0			0
$103$			17.1043i			1.68534i	0.538433	+	0.842668i	$0.319016\pi$
$103$			17.1043i			1.68534i	−0.538433	+	0.842668i	$0.680984\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.9932	−	12.9932i	−1.25610	−	1.25610i	−0.952939	−	0.303162i	$-0.901958\pi$
$107$	−12.9932	−	12.9932i	−1.25610	−	1.25610i	−0.303162	−	0.952939i	$-0.598042\pi$
$108$		0			0
$109$	5.54712	−	5.54712i	0.531318	−	0.531318i	−0.389647	−	0.920964i	$-0.627403\pi$
$109$	5.54712	−	5.54712i	0.531318	−	0.531318i	0.920964	+	0.389647i	$0.127403\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.65685			0.344008			0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685			0.344008			0.172004	+	0.985096i	$0.444976\pi$
$114$		0			0
$115$	0.938533	−	0.938533i	0.0875186	−	0.0875186i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			20.1094i			1.84342i
$120$		0			0
$121$			2.65685i			0.241532i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.32050	+	2.32050i	−0.207552	+	0.207552i
$126$		0			0
$127$	−0.638213			−0.0566322			−0.0283161	−	0.999599i	$-0.509015\pi$
$127$	−0.638213			−0.0566322			−0.0283161	+	0.999599i	$0.509015\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.72739	+	6.72739i	−0.587775	+	0.587775i	−0.937028	−	0.349253i	$-0.886435\pi$
$131$	−6.72739	+	6.72739i	−0.587775	+	0.587775i	0.349253	+	0.937028i	$0.386435\pi$
$132$		0			0
$133$	−1.88989	−	1.88989i	−0.163875	−	0.163875i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	13.9150i		−	1.18884i	−0.804156	−	0.594419i	$-0.797382\pi$
$137$		−	13.9150i		−	1.18884i	0.804156	−	0.594419i	$-0.202618\pi$
$138$		0			0
$139$	−0.0546790	−	0.0546790i	−0.00463781	−	0.00463781i	0.704784	−	0.709422i	$-0.251044\pi$
$139$	−0.0546790	−	0.0546790i	−0.00463781	−	0.00463781i	−0.709422	+	0.704784i	$0.751044\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.1489			1.43407
$144$		0			0
$145$	−1.62408			−0.134872
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.89149	−	9.89149i	−0.810342	−	0.810342i	0.174343	−	0.984685i	$-0.444220\pi$
$149$	−9.89149	−	9.89149i	−0.810342	−	0.810342i	−0.984685	+	0.174343i	$0.944220\pi$
$150$		0			0
$151$			18.5828i			1.51225i	0.654430	+	0.756123i	$0.272909\pi$
$151$			18.5828i			1.51225i	−0.654430	+	0.756123i	$0.727091\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.44155	−	1.44155i	−0.115788	−	0.115788i
$156$		0			0
$157$	3.75057	−	3.75057i	0.299328	−	0.299328i	−0.541423	−	0.840751i	$-0.682114\pi$
$157$	3.75057	−	3.75057i	0.299328	−	0.299328i	0.840751	+	0.541423i	$0.182114\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.3296			−0.971706
$162$		0			0
$163$	−9.40878	+	9.40878i	−0.736952	+	0.736952i	−0.971987	−	0.235035i	$-0.924480\pi$
$163$	−9.40878	+	9.40878i	−0.736952	+	0.736952i	0.235035	+	0.971987i	$0.424480\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.9310i			1.23278i	0.787442	+	0.616389i	$0.211405\pi$
$167$			15.9310i			1.23278i	−0.787442	+	0.616389i	$0.788595\pi$
$168$		0			0
$169$		−	8.53392i		−	0.656455i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.9462	−	13.9462i	1.06031	−	1.06031i	0.0622466	−	0.998061i	$-0.480173\pi$
$173$	13.9462	−	13.9462i	1.06031	−	1.06031i	0.998061	−	0.0622466i	$-0.0198265\pi$
$174$		0			0
$175$	15.0726			1.13938
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.23304	+	4.23304i	−0.316392	+	0.316392i	−0.847380	−	0.530987i	$-0.821821\pi$
$179$	−4.23304	+	4.23304i	−0.316392	+	0.316392i	0.530987	+	0.847380i	$0.321821\pi$
$180$		0			0
$181$	−6.93816	−	6.93816i	−0.515709	−	0.515709i	0.400561	−	0.916270i	$-0.368815\pi$
$181$	−6.93816	−	6.93816i	−0.515709	−	0.515709i	−0.916270	+	0.400561i	$0.868815\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.21024i		−	0.0889784i
$186$		0			0
$187$	−17.0479	−	17.0479i	−1.24667	−	1.24667i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.2900			−1.39578			−0.697888	−	0.716207i	$-0.745877\pi$
$191$	−19.2900			−1.39578			−0.697888	+	0.716207i	$0.745877\pi$
$192$		0			0
$193$	−0.155713			−0.0112084			−0.00560422	−	0.999984i	$-0.501784\pi$
$193$	−0.155713			−0.0112084			−0.00560422	+	0.999984i	$0.501784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.33476	−	9.33476i	−0.665074	−	0.665074i	0.291497	−	0.956572i	$-0.405847\pi$
$197$	−9.33476	−	9.33476i	−0.665074	−	0.665074i	−0.956572	+	0.291497i	$0.905847\pi$
$198$		0			0
$199$			23.1144i			1.63854i	0.573409	+	0.819269i	$0.305620\pi$
$199$			23.1144i			1.63854i	−0.573409	+	0.819269i	$0.694380\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.6678	+	10.6678i	0.748733	+	0.748733i
$204$		0			0
$205$	0.921770	−	0.921770i	0.0643792	−	0.0643792i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.20435			0.221650
$210$		0			0
$211$	−2.39782	+	2.39782i	−0.165073	+	0.165073i	−0.784810	−	0.619737i	$-0.787239\pi$
$211$	−2.39782	+	2.39782i	−0.165073	+	0.165073i	0.619737	+	0.784810i	$0.287239\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			1.30358i			0.0889034i
$216$		0			0
$217$			18.9378i			1.28558i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−21.4071	+	21.4071i	−1.43999	+	1.43999i
$222$		0			0
$223$	10.5326			0.705316			0.352658	−	0.935752i	$-0.385278\pi$
$223$	10.5326			0.705316			0.352658	+	0.935752i	$0.385278\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.5995	−	16.5995i	1.10175	−	1.10175i	0.107551	−	0.994200i	$-0.465699\pi$
$227$	16.5995	−	16.5995i	1.10175	−	1.10175i	0.994200	−	0.107551i	$-0.0343010\pi$
$228$		0			0
$229$	−2.21984	−	2.21984i	−0.146691	−	0.146691i	0.629947	−	0.776638i	$-0.283077\pi$
$229$	−2.21984	−	2.21984i	−0.146691	−	0.146691i	−0.776638	+	0.629947i	$0.783077\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.98414i		−	0.588571i	−0.955718	−	0.294285i	$-0.904918\pi$
$233$		−	8.98414i		−	0.588571i	0.955718	−	0.294285i	$-0.0950816\pi$
$234$		0			0
$235$	0.388753	+	0.388753i	0.0253594	+	0.0253594i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.69870			−0.109880			−0.0549400	−	0.998490i	$-0.517497\pi$
$239$	−1.69870			−0.109880			−0.0549400	+	0.998490i	$0.517497\pi$
$240$		0			0
$241$	14.3288			0.923001			0.461500	−	0.887140i	$-0.347311\pi$
$241$	14.3288			0.923001			0.461500	+	0.887140i	$0.347311\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.586851	+	0.586851i	0.0374925	+	0.0374925i
$246$		0			0
$247$		−	4.02371i		−	0.256022i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.26998	−	8.26998i	−0.521997	−	0.521997i	0.396177	−	0.918174i	$-0.370337\pi$
$251$	−8.26998	−	8.26998i	−0.521997	−	0.521997i	−0.918174	+	0.396177i	$0.870337\pi$
$252$		0			0
$253$	10.4525	−	10.4525i	0.657143	−	0.657143i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.98642			−0.248666			−0.124333	−	0.992241i	$-0.539679\pi$
$257$	−3.98642			−0.248666			−0.124333	+	0.992241i	$0.539679\pi$
$258$		0			0
$259$	−7.94948	+	7.94948i	−0.493957	+	0.493957i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	11.8635i		−	0.731534i	−0.930706	−	0.365767i	$-0.880807\pi$
$263$		−	11.8635i		−	0.731534i	0.930706	−	0.365767i	$-0.119193\pi$
$264$		0			0
$265$			0.187433i			0.0115139i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.1236	+	15.1236i	−0.922104	+	0.922104i	−0.997178	−	0.0750742i	$-0.976081\pi$
$269$	−15.1236	+	15.1236i	−0.922104	+	0.922104i	0.0750742	+	0.997178i	$0.476081\pi$
$270$		0			0
$271$	−12.0530			−0.732165			−0.366082	−	0.930582i	$-0.619301\pi$
$271$	−12.0530			−0.732165			−0.366082	+	0.930582i	$0.619301\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.7779	+	12.7779i	−0.770537	+	0.770537i
$276$		0			0
$277$	3.40424	+	3.40424i	0.204541	+	0.204541i	0.801942	−	0.597401i	$-0.203800\pi$
$277$	3.40424	+	3.40424i	0.204541	+	0.204541i	−0.597401	+	0.801942i	$0.703800\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.5754i		−	1.10812i	−0.832477	−	0.554059i	$-0.813078\pi$
$281$		−	18.5754i		−	1.10812i	0.832477	−	0.554059i	$-0.186922\pi$
$282$		0			0
$283$	−0.993212	−	0.993212i	−0.0590403	−	0.0590403i	0.676970	−	0.736010i	$-0.263293\pi$
$283$	−0.993212	−	0.993212i	−0.0590403	−	0.0590403i	−0.736010	+	0.676970i	$0.763293\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.1094			−0.714793
$288$		0			0
$289$	25.5619			1.50364
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.7131	+	19.7131i	1.15165	+	1.15165i	0.986221	+	0.165432i	$0.0529018\pi$
$293$	19.7131	+	19.7131i	1.15165	+	1.15165i	0.165432	+	0.986221i	$0.447098\pi$
$294$		0			0
$295$		−	2.18849i		−	0.127419i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.1252	−	13.1252i	−0.759051	−	0.759051i
$300$		0			0
$301$	8.56261	−	8.56261i	0.493541	−	0.493541i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.91270			0.281300
$306$		0			0
$307$	−12.3424	+	12.3424i	−0.704418	+	0.704418i	−0.965356	−	0.260938i	$-0.915968\pi$
$307$	−12.3424	+	12.3424i	−0.704418	+	0.704418i	0.260938	+	0.965356i	$0.415968\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	0.466081i		−	0.0264290i	−0.999913	−	0.0132145i	$-0.995794\pi$
$311$		−	0.466081i		−	0.0264290i	0.999913	−	0.0132145i	$-0.00420643\pi$
$312$		0			0
$313$			2.49886i			0.141244i	0.997503	+	0.0706219i	$0.0224984\pi$
$313$			2.49886i			0.141244i	−0.997503	+	0.0706219i	$0.977502\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.59608	+	7.59608i	−0.426638	+	0.426638i	−0.887482	−	0.460843i	$-0.847547\pi$
$317$	−7.59608	+	7.59608i	−0.426638	+	0.426638i	0.460843	+	0.887482i	$0.347547\pi$
$318$		0			0
$319$	−18.0875			−1.01270
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.00000	+	4.00000i	−0.222566	+	0.222566i
$324$		0			0
$325$	16.0452	+	16.0452i	0.890029	+	0.890029i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	5.10707i		−	0.281562i
$330$		0			0
$331$	3.72511	+	3.72511i	0.204751	+	0.204751i	0.802032	−	0.597281i	$-0.203752\pi$
$331$	3.72511	+	3.72511i	0.204751	+	0.204751i	−0.597281	+	0.802032i	$0.703752\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.64099			0.253565
$336$		0			0
$337$	27.4740			1.49660			0.748302	−	0.663359i	$-0.230870\pi$
$337$	27.4740			1.49660			0.748302	+	0.663359i	$0.230870\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0547	−	16.0547i	−0.869410	−	0.869410i
$342$		0			0
$343$			13.8672i			0.748761i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.71644	−	5.71644i	−0.306875	−	0.306875i	0.536821	−	0.843696i	$-0.319625\pi$
$347$	−5.71644	−	5.71644i	−0.306875	−	0.306875i	−0.843696	+	0.536821i	$0.819625\pi$
$348$		0			0
$349$	−3.51753	+	3.51753i	−0.188289	+	0.188289i	−0.794956	−	0.606667i	$-0.792506\pi$
$349$	−3.51753	+	3.51753i	−0.188289	+	0.188289i	0.606667	+	0.794956i	$0.292506\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.0799			−1.22842			−0.614210	−	0.789143i	$-0.710525\pi$
$353$	−23.0799			−1.22842			−0.614210	+	0.789143i	$0.710525\pi$
$354$		0			0
$355$	−1.75789	+	1.75789i	−0.0932990	+	0.0932990i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			29.2583i			1.54419i	0.635505	+	0.772097i	$0.280792\pi$
$359$			29.2583i			1.54419i	−0.635505	+	0.772097i	$0.719208\pi$
$360$		0			0
$361$			18.2482i			0.960429i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.31959	+	1.31959i	−0.0690707	+	0.0690707i
$366$		0			0
$367$	13.1698			0.687461			0.343730	−	0.939068i	$-0.388309\pi$
$367$	13.1698			0.687461			0.343730	+	0.939068i	$0.388309\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.23116	−	1.23116i	0.0639187	−	0.0639187i
$372$		0			0
$373$	10.8120	+	10.8120i	0.559826	+	0.559826i	0.929258	−	0.369432i	$-0.120448\pi$
$373$	10.8120	+	10.8120i	0.559826	+	0.559826i	−0.369432	+	0.929258i	$0.620448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			22.7124i			1.16975i
$378$		0			0
$379$	4.53580	+	4.53580i	0.232988	+	0.232988i	0.813939	−	0.580951i	$-0.197319\pi$
$379$	4.53580	+	4.53580i	0.232988	+	0.232988i	−0.580951	+	0.813939i	$0.697319\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	34.0721			1.74100			0.870501	−	0.492167i	$-0.163795\pi$
$383$	34.0721			1.74100			0.870501	+	0.492167i	$0.163795\pi$
$384$		0			0
$385$	−3.77979			−0.192636
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.9941	+	22.9941i	1.16584	+	1.16584i	0.983174	+	0.182671i	$0.0584743\pi$
$389$	22.9941	+	22.9941i	1.16584	+	1.16584i	0.182671	+	0.983174i	$0.441526\pi$
$390$		0			0
$391$			26.0958i			1.31972i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.49698	+	3.49698i	0.175952	+	0.175952i
$396$		0			0
$397$	3.17438	−	3.17438i	0.159318	−	0.159318i	−0.622947	−	0.782264i	$-0.714065\pi$
$397$	3.17438	−	3.17438i	0.159318	−	0.159318i	0.782264	+	0.622947i	$0.214065\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.21024			−0.160312			−0.0801558	−	0.996782i	$-0.525542\pi$
$401$	−3.21024			−0.160312			−0.0801558	+	0.996782i	$0.525542\pi$
$402$		0			0
$403$	−20.1599	+	20.1599i	−1.00423	+	1.00423i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.4785i		−	0.668104i
$408$		0			0
$409$			14.0456i			0.694511i	0.937771	+	0.347255i	$0.112886\pi$
$409$			14.0456i			0.694511i	−0.937771	+	0.347255i	$0.887114\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.3752	+	14.3752i	−0.707356	+	0.707356i
$414$		0			0
$415$	3.10332			0.152336
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.0521	+	19.0521i	−0.930754	+	0.930754i	−0.997753	−	0.0669993i	$-0.978657\pi$
$419$	−19.0521	+	19.0521i	−0.930754	+	0.930754i	0.0669993	+	0.997753i	$0.478657\pi$
$420$		0			0
$421$	4.04598	+	4.04598i	0.197189	+	0.197189i	0.798794	−	0.601605i	$-0.205472\pi$
$421$	4.04598	+	4.04598i	0.197189	+	0.197189i	−0.601605	+	0.798794i	$0.705472\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	31.9014i		−	1.54745i
$426$		0			0
$427$	−32.2692	−	32.2692i	−1.56162	−	1.56162i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.9547			−1.34653			−0.673265	−	0.739401i	$-0.735109\pi$
$431$	−27.9547			−1.34653			−0.673265	+	0.739401i	$0.735109\pi$
$432$		0			0
$433$	1.81151			0.0870556			0.0435278	−	0.999052i	$-0.486140\pi$
$433$	1.81151			0.0870556			0.0435278	+	0.999052i	$0.486140\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.45250	−	2.45250i	−0.117319	−	0.117319i
$438$		0			0
$439$		−	4.45985i		−	0.212857i	−0.994320	−	0.106429i	$-0.966058\pi$
$439$		−	4.45985i		−	0.212857i	0.994320	−	0.106429i	$-0.0339415\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.85295	−	1.85295i	−0.0880365	−	0.0880365i	0.661717	−	0.749754i	$-0.269828\pi$
$443$	−1.85295	−	1.85295i	−0.0880365	−	0.0880365i	−0.749754	+	0.661717i	$0.769828\pi$
$444$		0			0
$445$	4.24906	−	4.24906i	0.201425	−	0.201425i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−27.9787			−1.32040			−0.660199	−	0.751091i	$-0.729528\pi$
$449$	−27.9787			−1.32040			−0.660199	+	0.751091i	$0.729528\pi$
$450$		0			0
$451$	10.2658	−	10.2658i	0.483398	−	0.483398i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.74628i			0.222509i
$456$		0			0
$457$			14.3933i			0.673291i	0.941631	+	0.336646i	$0.109292\pi$
$457$			14.3933i			0.673291i	−0.941631	+	0.336646i	$0.890708\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.50724	+	5.50724i	−0.256498	+	0.256498i	−0.823628	−	0.567130i	$-0.808054\pi$
$461$	−5.50724	+	5.50724i	−0.256498	+	0.256498i	0.567130	+	0.823628i	$0.308054\pi$
$462$		0			0
$463$	−15.6642			−0.727977			−0.363989	−	0.931403i	$-0.618585\pi$
$463$	−15.6642			−0.727977			−0.363989	+	0.931403i	$0.618585\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.73002	−	3.73002i	0.172605	−	0.172605i	−0.615518	−	0.788123i	$-0.711053\pi$
$467$	3.73002	−	3.73002i	0.172605	−	0.172605i	0.788123	+	0.615518i	$0.211053\pi$
$468$		0			0
$469$	−30.4845	−	30.4845i	−1.40764	−	1.40764i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			14.5181i			0.667541i
$474$		0			0
$475$	2.99812	+	2.99812i	0.137563	+	0.137563i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.9310			−0.545141			−0.272571	−	0.962136i	$-0.587874\pi$
$479$	−11.9310			−0.545141			−0.272571	+	0.962136i	$0.587874\pi$
$480$		0			0
$481$	−16.9250			−0.771712
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	4.00000i	0.181631	+	0.181631i
$486$		0			0
$487$		−	37.1782i		−	1.68470i	−0.538928	−	0.842352i	$-0.681170\pi$
$487$		−	37.1782i		−	1.68470i	0.538928	−	0.842352i	$-0.318830\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.7470	+	17.7470i	0.800911	+	0.800911i	0.983238	−	0.182327i	$-0.0583628\pi$
$491$	17.7470	+	17.7470i	0.800911	+	0.800911i	−0.182327	+	0.983238i	$0.558363\pi$
$492$		0			0
$493$	22.5786	−	22.5786i	1.01689	−	1.01689i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	23.0935			1.03588
$498$		0			0
$499$	23.2255	−	23.2255i	1.03972	−	1.03972i	0.0405384	−	0.999178i	$-0.487093\pi$
$499$	23.2255	−	23.2255i	1.03972	−	1.03972i	0.999178	−	0.0405384i	$-0.0129073\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			8.35327i			0.372454i	0.982507	+	0.186227i	$0.0596260\pi$
$503$			8.35327i			0.372454i	−0.982507	+	0.186227i	$0.940374\pi$
$504$		0			0
$505$		−	4.03278i		−	0.179456i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.60287	−	6.60287i	0.292667	−	0.292667i	−0.545466	−	0.838133i	$-0.683647\pi$
$509$	6.60287	−	6.60287i	0.292667	−	0.292667i	0.838133	+	0.545466i	$0.183647\pi$
$510$		0			0
$511$	17.3356			0.766882
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.01323	−	4.01323i	0.176844	−	0.176844i
$516$		0			0
$517$	4.32957	+	4.32957i	0.190414	+	0.190414i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	4.38287i		−	0.192017i	−0.995381	−	0.0960084i	$-0.969392\pi$
$521$		−	4.38287i		−	0.192017i	0.995381	−	0.0960084i	$-0.0306076\pi$
$522$		0			0
$523$	−6.52567	−	6.52567i	−0.285348	−	0.285348i	0.549890	−	0.835237i	$-0.314670\pi$
$523$	−6.52567	−	6.52567i	−0.285348	−	0.285348i	−0.835237	+	0.549890i	$0.814670\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	40.0822			1.74601
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.8908	−	12.8908i	−0.558362	−	0.558362i
$534$		0			0
$535$			6.09728i			0.263608i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.53580	+	6.53580i	0.281517	+	0.281517i
$540$		0			0
$541$	−26.1418	+	26.1418i	−1.12392	+	1.12392i	−0.132776	+	0.991146i	$0.542389\pi$
$541$	−26.1418	+	26.1418i	−1.12392	+	1.12392i	−0.991146	+	0.132776i	$0.957611\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.60308			−0.111504
$546$		0			0
$547$	28.7643	−	28.7643i	1.22987	−	1.22987i	0.265863	−	0.964011i	$-0.414343\pi$
$547$	28.7643	−	28.7643i	1.22987	−	1.22987i	0.964011	−	0.265863i	$-0.0856570\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			4.24392i			0.180797i
$552$		0			0
$553$		−	45.9401i		−	1.95357i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	28.0144	−	28.0144i	1.18701	−	1.18701i	0.209119	−	0.977890i	$-0.432940\pi$
$557$	28.0144	−	28.0144i	1.18701	−	1.18701i	0.977890	−	0.209119i	$-0.0670596\pi$
$558$		0			0
$559$	18.2303			0.771061
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.7142	−	14.7142i	0.620128	−	0.620128i	−0.325436	−	0.945564i	$-0.605511\pi$
$563$	14.7142	−	14.7142i	0.620128	−	0.620128i	0.945564	+	0.325436i	$0.105511\pi$
$564$		0			0
$565$	−0.858019	−	0.858019i	−0.0360971	−	0.0360971i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			15.7585i			0.660632i	0.943870	+	0.330316i	$0.107155\pi$
$569$			15.7585i			0.660632i	−0.943870	+	0.330316i	$0.892845\pi$
$570$		0			0
$571$	17.4593	+	17.4593i	0.730649	+	0.730649i	0.970748	−	0.240100i	$-0.0771801\pi$
$571$	17.4593	+	17.4593i	0.730649	+	0.730649i	−0.240100	+	0.970748i	$0.577180\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19.5596			0.815691
$576$		0			0
$577$	−21.0924			−0.878090			−0.439045	−	0.898465i	$-0.644683\pi$
$577$	−21.0924			−0.878090			−0.439045	+	0.898465i	$0.644683\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20.3842	−	20.3842i	−0.845681	−	0.845681i
$582$		0			0
$583$			2.08746i			0.0864536i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.1776	−	10.1776i	−0.420075	−	0.420075i	0.465154	−	0.885230i	$-0.345999\pi$
$587$	−10.1776	−	10.1776i	−0.420075	−	0.420075i	−0.885230	+	0.465154i	$0.845999\pi$
$588$		0			0
$589$	−3.76696	+	3.76696i	−0.155215	+	0.155215i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.1931			0.911360			0.455680	−	0.890144i	$-0.349396\pi$
$593$	22.1931			0.911360			0.455680	+	0.890144i	$0.349396\pi$
$594$		0			0
$595$	4.71832	−	4.71832i	0.193432	−	0.193432i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	37.4366i		−	1.52962i	−0.644256	−	0.764810i	$-0.722833\pi$
$599$		−	37.4366i		−	1.52962i	0.644256	−	0.764810i	$-0.277167\pi$
$600$		0			0
$601$			1.68963i			0.0689215i	0.999406	+	0.0344608i	$0.0109714\pi$
$601$			1.68963i			0.0689215i	−0.999406	+	0.0344608i	$0.989029\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.623386	−	0.623386i	0.0253442	−	0.0253442i
$606$		0			0
$607$	−36.8841			−1.49708			−0.748539	−	0.663090i	$-0.769245\pi$
$607$	−36.8841			−1.49708			−0.748539	+	0.663090i	$0.769245\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.43664	−	5.43664i	0.219943	−	0.219943i
$612$		0			0
$613$	9.51753	+	9.51753i	0.384409	+	0.384409i	0.872688	−	0.488278i	$-0.162375\pi$
$613$	9.51753	+	9.51753i	0.384409	+	0.384409i	−0.488278	+	0.872688i	$0.662375\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.3160i		−	0.817891i	−0.912559	−	0.408946i	$-0.865897\pi$
$617$		−	20.3160i		−	0.817891i	0.912559	−	0.408946i	$-0.134103\pi$
$618$		0			0
$619$	27.5969	+	27.5969i	1.10921	+	1.10921i	0.993254	+	0.115960i	$0.0369945\pi$
$619$	27.5969	+	27.5969i	1.10921	+	1.10921i	0.115960	+	0.993254i	$0.463006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−55.8201			−2.23639
$624$		0			0
$625$	−23.3605			−0.934422
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.8252	+	16.8252i	0.670866	+	0.670866i
$630$		0			0
$631$			0.629888i			0.0250755i	0.999921	+	0.0125377i	$0.00399099\pi$
$631$			0.629888i			0.0250755i	−0.999921	+	0.0125377i	$0.996009\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.149746	+	0.149746i	0.00594248	+	0.00594248i
$636$		0			0
$637$	8.20701	−	8.20701i	0.325173	−	0.325173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.52166			−0.139097			−0.0695486	−	0.997579i	$-0.522156\pi$
$641$	−3.52166			−0.139097			−0.0695486	+	0.997579i	$0.522156\pi$
$642$		0			0
$643$	6.95627	−	6.95627i	0.274329	−	0.274329i	−0.556511	−	0.830840i	$-0.687860\pi$
$643$	6.95627	−	6.95627i	0.274329	−	0.274329i	0.830840	+	0.556511i	$0.187860\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	30.7549i		−	1.20910i	−0.796567	−	0.604550i	$-0.793353\pi$
$647$		−	30.7549i		−	1.20910i	0.796567	−	0.604550i	$-0.206647\pi$
$648$		0			0
$649$		−	24.3734i		−	0.956739i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.52479	−	6.52479i	0.255335	−	0.255335i	−0.567819	−	0.823154i	$-0.692213\pi$
$653$	6.52479	−	6.52479i	0.255335	−	0.255335i	0.823154	+	0.567819i	$0.192213\pi$
$654$		0			0
$655$	3.15694			0.123352
$656$		0			0
$657$		0			0
$658$		0			0
$659$	32.9242	−	32.9242i	1.28255	−	1.28255i	0.343330	−	0.939215i	$-0.388445\pi$
$659$	32.9242	−	32.9242i	1.28255	−	1.28255i	0.939215	−	0.343330i	$-0.111555\pi$
$660$		0			0
$661$	−19.7506	−	19.7506i	−0.768208	−	0.768208i	0.209583	−	0.977791i	$-0.432789\pi$
$661$	−19.7506	−	19.7506i	−0.768208	−	0.768208i	−0.977791	+	0.209583i	$0.932789\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.886864i			0.0343911i
$666$		0			0
$667$	13.8435	+	13.8435i	0.536024	+	0.536024i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	54.7131			2.11217
$672$		0			0
$673$	6.34315			0.244510			0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315			0.244510			0.122255	+	0.992499i	$0.460987\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.2680	−	22.2680i	−0.855830	−	0.855830i	0.135013	−	0.990844i	$-0.456892\pi$
$677$	−22.2680	−	22.2680i	−0.855830	−	0.855830i	−0.990844	+	0.135013i	$0.956892\pi$
$678$		0			0
$679$		−	52.5483i		−	2.01662i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	19.8039	+	19.8039i	0.757775	+	0.757775i	0.975917	−	0.218142i	$-0.0699996\pi$
$683$	19.8039	+	19.8039i	0.757775	+	0.757775i	−0.218142	+	0.975917i	$0.570000\pi$
$684$		0			0
$685$	−3.26492	+	3.26492i	−0.124746	+	0.124746i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.62122			0.0998606
$690$		0			0
$691$	−27.0302	+	27.0302i	−1.02828	+	1.02828i	−0.0286870	+	0.999588i	$0.509133\pi$
$691$	−27.0302	+	27.0302i	−1.02828	+	1.02828i	−0.999588	+	0.0286870i	$0.990867\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.0256590i			0.000973301i
$696$		0			0
$697$			25.6297i			0.970794i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.39035	+	6.39035i	−0.241360	+	0.241360i	−0.817413	−	0.576053i	$-0.804592\pi$
$701$	−6.39035	+	6.39035i	−0.241360	+	0.241360i	0.576053	+	0.817413i	$0.304592\pi$
$702$		0			0
$703$	−3.16250			−0.119276
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−26.4894	+	26.4894i	−0.996238	+	0.996238i
$708$		0			0
$709$	−28.2327	−	28.2327i	−1.06030	−	1.06030i	−0.998061	−	0.0622388i	$-0.980176\pi$
$709$	−28.2327	−	28.2327i	−1.06030	−	1.06030i	−0.0622388	−	0.998061i	$-0.519824\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			24.5754i			0.920357i
$714$		0			0
$715$	−4.02371	−	4.02371i	−0.150478	−	0.150478i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.3036			−1.09284			−0.546420	−	0.837512i	$-0.684010\pi$
$719$	−29.3036			−1.09284			−0.546420	+	0.837512i	$0.684010\pi$
$720$		0			0
$721$	−52.7221			−1.96348
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.9234	−	16.9234i	−0.628518	−	0.628518i
$726$		0			0
$727$			12.6201i			0.468052i	0.972230	+	0.234026i	$0.0751902\pi$
$727$			12.6201i			0.468052i	−0.972230	+	0.234026i	$0.924810\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.1229	−	18.1229i	−0.670301	−	0.670301i
$732$		0			0
$733$	−7.01549	+	7.01549i	−0.259123	+	0.259123i	−0.824697	−	0.565574i	$-0.808655\pi$
$733$	−7.01549	+	7.01549i	−0.259123	+	0.259123i	0.565574	+	0.824697i	$0.308655\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	51.6871			1.90392
$738$		0			0
$739$	−14.1840	+	14.1840i	−0.521766	+	0.521766i	−0.918105	−	0.396338i	$-0.870281\pi$
$739$	−14.1840	+	14.1840i	−0.521766	+	0.521766i	0.396338	+	0.918105i	$0.370281\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.0969i		−	0.407107i	−0.979064	−	0.203554i	$-0.934751\pi$
$743$		−	11.0969i		−	0.407107i	0.979064	−	0.203554i	$-0.0652492\pi$
$744$		0			0
$745$			4.64174i			0.170060i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	40.0502	−	40.0502i	1.46340	−	1.46340i
$750$		0			0
$751$	37.3294			1.36217			0.681084	−	0.732205i	$-0.261509\pi$
$751$	37.3294			1.36217			0.681084	+	0.732205i	$0.261509\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.36014	−	4.36014i	0.158682	−	0.158682i
$756$		0			0
$757$	18.9054	+	18.9054i	0.687128	+	0.687128i	0.961596	−	0.274468i	$-0.0885019\pi$
$757$	18.9054	+	18.9054i	0.687128	+	0.687128i	−0.274468	+	0.961596i	$0.588502\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			27.1767i			0.985155i	0.870269	+	0.492578i	$0.163945\pi$
$761$			27.1767i			0.985155i	−0.870269	+	0.492578i	$0.836055\pi$
$762$		0			0
$763$	17.0984	+	17.0984i	0.619004	+	0.619004i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.6057			−1.10511
$768$		0			0
$769$	−8.34877			−0.301064			−0.150532	−	0.988605i	$-0.548099\pi$
$769$	−8.34877			−0.301064			−0.150532	+	0.988605i	$0.548099\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.6575	+	14.6575i	0.527195	+	0.527195i	0.919735	−	0.392540i	$-0.128403\pi$
$773$	14.6575	+	14.6575i	0.527195	+	0.527195i	−0.392540	+	0.919735i	$0.628403\pi$
$774$		0			0
$775$		−	30.0428i		−	1.07917i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.40870	−	2.40870i	−0.0863007	−	0.0863007i
$780$		0			0
$781$	−19.5777	+	19.5777i	−0.700546	+	0.700546i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.76002			−0.0628177
$786$		0			0
$787$	15.3533	−	15.3533i	0.547288	−	0.547288i	−0.378368	−	0.925655i	$-0.623514\pi$
$787$	15.3533	−	15.3533i	0.547288	−	0.547288i	0.925655	+	0.378368i	$0.123514\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.2719i			0.400781i
$792$		0			0
$793$		−	68.7032i		−	2.43972i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.4828	−	19.4828i	0.690116	−	0.690116i	−0.272142	−	0.962257i	$-0.587732\pi$
$797$	19.4828	−	19.4828i	0.690116	−	0.690116i	0.962257	+	0.272142i	$0.0877319\pi$
$798$		0			0
$799$	−10.8092			−0.382403
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.6964	+	14.6964i	−0.518625	+	0.518625i
$804$		0			0
$805$	2.89293	+	2.89293i	0.101962	+	0.101962i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.8535i			0.662854i	0.943481	+	0.331427i	$0.107530\pi$
$809$			18.8535i			0.662854i	−0.943481	+	0.331427i	$0.892470\pi$
$810$		0			0
$811$	4.79981	+	4.79981i	0.168544	+	0.168544i	0.786339	−	0.617795i	$-0.211974\pi$
$811$	4.79981	+	4.79981i	0.168544	+	0.168544i	−0.617795	+	0.786339i	$0.711974\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.41522			0.154658
$816$		0			0
$817$	3.40642			0.119175
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.3257	+	11.3257i	0.395269	+	0.395269i	0.876561	−	0.481291i	$-0.159832\pi$
$821$	11.3257	+	11.3257i	0.395269	+	0.395269i	−0.481291	+	0.876561i	$0.659832\pi$
$822$		0			0
$823$		−	28.5609i		−	0.995570i	−0.867301	−	0.497785i	$-0.834147\pi$
$823$		−	28.5609i		−	0.995570i	0.867301	−	0.497785i	$-0.165853\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.5332	+	13.5332i	0.470594	+	0.470594i	0.902107	−	0.431512i	$-0.142020\pi$
$827$	13.5332	+	13.5332i	0.470594	+	0.470594i	−0.431512	+	0.902107i	$0.642020\pi$
$828$		0			0
$829$	−25.5200	+	25.5200i	−0.886345	+	0.886345i	−0.994170	−	0.107825i	$-0.965611\pi$
$829$	−25.5200	+	25.5200i	−0.886345	+	0.886345i	0.107825	+	0.994170i	$0.465611\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.3173			−0.565361
$834$		0			0
$835$	3.73794	−	3.73794i	0.129357	−	0.129357i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			18.8767i			0.651697i	0.945422	+	0.325849i	$0.105650\pi$
$839$			18.8767i			0.651697i	−0.945422	+	0.325849i	$0.894350\pi$
$840$		0			0
$841$			5.04455i			0.173950i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.00234	+	2.00234i	−0.0688826	+	0.0688826i
$846$		0			0
$847$	−8.18947			−0.281393
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10.3160	+	10.3160i	−0.353628	+	0.353628i
$852$		0			0
$853$	18.5982	+	18.5982i	0.636790	+	0.636790i	0.949762	−	0.312972i	$-0.101325\pi$
$853$	18.5982	+	18.5982i	0.636790	+	0.636790i	−0.312972	+	0.949762i	$0.601325\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			24.8671i			0.849444i	0.905324	+	0.424722i	$0.139628\pi$
$857$			24.8671i			0.849444i	−0.905324	+	0.424722i	$0.860372\pi$
$858$		0			0
$859$	11.0302	+	11.0302i	0.376344	+	0.376344i	0.869781	−	0.493437i	$-0.164260\pi$
$859$	11.0302	+	11.0302i	0.376344	+	0.376344i	−0.493437	+	0.869781i	$0.664260\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−43.8654			−1.49320			−0.746598	−	0.665275i	$-0.768314\pi$
$863$	−43.8654			−1.49320			−0.746598	+	0.665275i	$0.768314\pi$
$864$		0			0
$865$	−6.54447			−0.222519
$866$		0			0
$867$		0			0
$868$		0			0
$869$	38.9461	+	38.9461i	1.32116	+	1.32116i
$870$		0			0
$871$		−	64.9035i		−	2.19917i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.15268	−	7.15268i	−0.241805	−	0.241805i
$876$		0			0
$877$	−32.2023	+	32.2023i	−1.08740	+	1.08740i	−0.0915994	+	0.995796i	$0.529198\pi$
$877$	−32.2023	+	32.2023i	−1.08740	+	1.08740i	−0.995796	+	0.0915994i	$0.970802\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−19.8589			−0.669064			−0.334532	−	0.942384i	$-0.608578\pi$
$881$	−19.8589			−0.669064			−0.334532	+	0.942384i	$0.608578\pi$
$882$		0			0
$883$	17.8714	−	17.8714i	0.601421	−	0.601421i	−0.339269	−	0.940689i	$-0.610180\pi$
$883$	17.8714	−	17.8714i	0.601421	−	0.601421i	0.940689	+	0.339269i	$0.110180\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.07012i			0.0695079i	0.999396	+	0.0347539i	$0.0110648\pi$
$887$			2.07012i			0.0695079i	−0.999396	+	0.0347539i	$0.988935\pi$
$888$		0			0
$889$		−	1.96722i		−	0.0659785i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.01586	−	1.01586i	0.0339945	−	0.0339945i
$894$		0			0
$895$	1.98642			0.0663988
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.2632	−	21.2632i	0.709167	−	0.709167i
$900$		0			0
$901$	−2.60578	−	2.60578i	−0.0868111	−	0.0868111i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			3.25584i			0.108228i
$906$		0			0
$907$	−5.73002	−	5.73002i	−0.190262	−	0.190262i	0.605547	−	0.795809i	$-0.292954\pi$
$907$	−5.73002	−	5.73002i	−0.190262	−	0.190262i	−0.795809	+	0.605547i	$0.792954\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.23034			0.206420			0.103210	−	0.994660i	$-0.467089\pi$
$911$	6.23034			0.206420			0.103210	+	0.994660i	$0.467089\pi$
$912$		0			0
$913$	34.5619			1.14383
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.7365	−	20.7365i	−0.684778	−	0.684778i
$918$		0			0
$919$		−	15.1680i		−	0.500348i	−0.968201	−	0.250174i	$-0.919512\pi$
$919$		−	15.1680i		−	0.500348i	0.968201	−	0.250174i	$-0.0804878\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.5838	+	24.5838i	0.809184	+	0.809184i
$924$		0			0
$925$	12.6110	−	12.6110i	0.414648	−	0.414648i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.16951			0.300842			0.150421	−	0.988622i	$-0.451937\pi$
$929$	9.16951			0.300842			0.150421	+	0.988622i	$0.451937\pi$
$930$		0			0
$931$	1.53351	−	1.53351i	0.0502589	−	0.0502589i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			8.00000i			0.261628i
$936$		0			0
$937$			19.3183i			0.631101i	0.948909	+	0.315550i	$0.102189\pi$
$937$			19.3183i			0.631101i	−0.948909	+	0.315550i	$0.897811\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.38112	−	7.38112i	0.240618	−	0.240618i	−0.576488	−	0.817106i	$-0.695577\pi$
$941$	7.38112	−	7.38112i	0.240618	−	0.240618i	0.817106	+	0.576488i	$0.195577\pi$
$942$		0			0
$943$	−15.7142			−0.511726
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.6425	−	25.6425i	0.833270	−	0.833270i	−0.154692	−	0.987963i	$-0.549439\pi$
$947$	25.6425	−	25.6425i	0.833270	−	0.833270i	0.987963	+	0.154692i	$0.0494387\pi$
$948$		0			0
$949$	18.4543	+	18.4543i	0.599052	+	0.599052i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	47.9271i		−	1.55251i	−0.630419	−	0.776255i	$-0.717117\pi$
$953$		−	47.9271i		−	1.55251i	0.630419	−	0.776255i	$-0.282883\pi$
$954$		0			0
$955$	4.52607	+	4.52607i	0.146460	+	0.146460i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.8914			1.38504
$960$		0			0
$961$	6.74701			0.217646
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.0365353	+	0.0365353i	0.00117611	+	0.00117611i
$966$		0			0
$967$			20.7211i			0.666346i	0.942866	+	0.333173i	$0.108119\pi$
$967$			20.7211i			0.666346i	−0.942866	+	0.333173i	$0.891881\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.9329	+	26.9329i	0.864317	+	0.864317i	0.991836	−	0.127519i	$-0.0407014\pi$
$971$	26.9329	+	26.9329i	0.864317	+	0.864317i	−0.127519	+	0.991836i	$0.540701\pi$
$972$		0			0
$973$	0.168542	−	0.168542i	0.00540321	−	0.00540321i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.7363			1.36726			0.683628	−	0.729831i	$-0.260401\pi$
$977$	42.7363			1.36726			0.683628	+	0.729831i	$0.260401\pi$
$978$		0			0
$979$	47.3220	−	47.3220i	1.51242	−	1.51242i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	42.0031i		−	1.33969i	−0.742501	−	0.669845i	$-0.766361\pi$
$983$		−	42.0031i		−	1.33969i	0.742501	−	0.669845i	$-0.233639\pi$
$984$		0			0
$985$			4.38049i			0.139574i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.1116	−	11.1116i	0.353330	−	0.353330i
$990$		0			0
$991$	11.1918			0.355518			0.177759	−	0.984074i	$-0.443115\pi$
$991$	11.1918			0.355518			0.177759	+	0.984074i	$0.443115\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.42341	−	5.42341i	0.171934	−	0.171934i
$996$		0			0
$997$	36.9980	+	36.9980i	1.17174	+	1.17174i	0.981795	+	0.189942i	$0.0608301\pi$
$997$	36.9980	+	36.9980i	1.17174	+	1.17174i	0.189942	+	0.981795i	$0.439170\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.be.3457.3		8
3.2	odd	2		1536.2.j.j.385.1	yes	8
4.3	odd	2		4608.2.k.bc.3457.3		8
8.3	odd	2		4608.2.k.bj.3457.2		8
8.5	even	2		4608.2.k.bh.3457.2		8
12.11	even	2		1536.2.j.i.385.3	yes	8
16.3	odd	4		4608.2.k.bc.1153.3		8
16.5	even	4		4608.2.k.bh.1153.2		8
16.11	odd	4		4608.2.k.bj.1153.2		8
16.13	even	4	inner	4608.2.k.be.1153.3		8
24.5	odd	2		1536.2.j.e.385.4	✓	8
24.11	even	2		1536.2.j.f.385.2	yes	8
32.3	odd	8		9216.2.a.z.1.2		4
32.13	even	8		9216.2.a.bm.1.3		4
32.19	odd	8		9216.2.a.ba.1.3		4
32.29	even	8		9216.2.a.bl.1.2		4
48.5	odd	4		1536.2.j.e.1153.4	yes	8
48.11	even	4		1536.2.j.f.1153.2	yes	8
48.29	odd	4		1536.2.j.j.1153.1	yes	8
48.35	even	4		1536.2.j.i.1153.3	yes	8
96.5	odd	8		3072.2.d.e.1537.2		8
96.11	even	8		3072.2.d.j.1537.3		8
96.29	odd	8		3072.2.a.m.1.3		4
96.35	even	8		3072.2.a.p.1.3		4
96.53	odd	8		3072.2.d.e.1537.7		8
96.59	even	8		3072.2.d.j.1537.6		8
96.77	odd	8		3072.2.a.s.1.2		4
96.83	even	8		3072.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.4	✓	8	24.5	odd	2
1536.2.j.e.1153.4	yes	8	48.5	odd	4
1536.2.j.f.385.2	yes	8	24.11	even	2
1536.2.j.f.1153.2	yes	8	48.11	even	4
1536.2.j.i.385.3	yes	8	12.11	even	2
1536.2.j.i.1153.3	yes	8	48.35	even	4
1536.2.j.j.385.1	yes	8	3.2	odd	2
1536.2.j.j.1153.1	yes	8	48.29	odd	4
3072.2.a.j.1.2		4	96.83	even	8
3072.2.a.m.1.3		4	96.29	odd	8
3072.2.a.p.1.3		4	96.35	even	8
3072.2.a.s.1.2		4	96.77	odd	8
3072.2.d.e.1537.2		8	96.5	odd	8
3072.2.d.e.1537.7		8	96.53	odd	8
3072.2.d.j.1537.3		8	96.11	even	8
3072.2.d.j.1537.6		8	96.59	even	8
4608.2.k.bc.1153.3		8	16.3	odd	4
4608.2.k.bc.3457.3		8	4.3	odd	2
4608.2.k.be.1153.3		8	16.13	even	4	inner
4608.2.k.be.3457.3		8	1.1	even	1	trivial
4608.2.k.bh.1153.2		8	16.5	even	4
4608.2.k.bh.3457.2		8	8.5	even	2
4608.2.k.bj.1153.2		8	16.11	odd	4
4608.2.k.bj.3457.2		8	8.3	odd	2
9216.2.a.z.1.2		4	32.3	odd	8
9216.2.a.ba.1.3		4	32.19	odd	8
9216.2.a.bl.1.2		4	32.29	even	8
9216.2.a.bm.1.3		4	32.13	even	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)

\(f(q)\)	\(=\)	\(q+(-0.234633 - 0.234633i) q^{5} +3.08239i q^{7} +O(q^{10})\)	\(q+(-0.234633 - 0.234633i) q^{5} +3.08239i q^{7} +(-2.61313 - 2.61313i) q^{11} +(-3.28130 + 3.28130i) q^{13} +6.52395 q^{17} +(-0.613126 + 0.613126i) q^{19} +4.00000i q^{23} -4.88989i q^{25} +(3.46088 - 3.46088i) q^{29} +6.14386 q^{31} +(0.723231 - 0.723231i) q^{35} +(2.57900 + 2.57900i) q^{37} +3.92856i q^{41} +(-2.77791 - 2.77791i) q^{43} -1.65685 q^{47} -2.50114 q^{49} +(-0.399418 - 0.399418i) q^{53} +1.22625i q^{55} +(4.66364 + 4.66364i) q^{59} +(-10.4689 + 10.4689i) q^{61} +1.53981 q^{65} +(-9.88989 + 9.88989i) q^{67} -7.49207i q^{71} -5.62408i q^{73} +(8.05468 - 8.05468i) q^{77} -14.9040 q^{79} +(-6.61313 + 6.61313i) q^{83} +(-1.53073 - 1.53073i) q^{85} +18.1094i q^{89} +(-10.1143 - 10.1143i) q^{91} +0.287719 q^{95} -17.0479 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 8 q^{13} + 16 q^{19} - 8 q^{29} + 16 q^{31} + 32 q^{35} - 8 q^{37} + 16 q^{43} + 32 q^{47} - 8 q^{49} + 8 q^{53} + 32 q^{59} - 8 q^{61} + 16 q^{65} - 32 q^{67} - 48 q^{79} - 32 q^{83} - 48 q^{91} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^13 + 16 * q^19 - 8 * q^29 + 16 * q^31 + 32 * q^35 - 8 * q^37 + 16 * q^43 + 32 * q^47 - 8 * q^49 + 8 * q^53 + 32 * q^59 - 8 * q^61 + 16 * q^65 - 32 * q^67 - 48 * q^79 - 32 * q^83 - 48 * q^91 - 64 * q^95 - 32 * q^97

Introduction

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