LMFDB - Embedded newform 4608.2.k.bb.1153.2

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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1153.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.bb.3457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +(3.00000 + 3.00000i) q^{13} +4.00000 q^{17} +(-5.65685 - 5.65685i) q^{19} -5.65685i q^{23} +3.00000i q^{25} +(1.00000 + 1.00000i) q^{29} +2.82843 q^{31} +(2.82843 + 2.82843i) q^{35} +(3.00000 - 3.00000i) q^{37} +4.00000i q^{41} +11.3137 q^{47} -1.00000 q^{49} +(-5.00000 + 5.00000i) q^{53} +(8.48528 - 8.48528i) q^{59} +(-1.00000 - 1.00000i) q^{61} +6.00000 q^{65} +(-2.82843 - 2.82843i) q^{67} -11.3137i q^{71} +14.0000i q^{73} -8.48528 q^{79} +(11.3137 + 11.3137i) q^{83} +(4.00000 - 4.00000i) q^{85} +6.00000i q^{89} +(-8.48528 + 8.48528i) q^{91} -11.3137 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} + 12 q^{13} + 16 q^{17} + 4 q^{29} + 12 q^{37} - 4 q^{49} - 20 q^{53} - 4 q^{61} + 24 q^{65} + 16 q^{85} + 64 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 12 * q^13 + 16 * q^17 + 4 * q^29 + 12 * q^37 - 4 * q^49 - 20 * q^53 - 4 * q^61 + 24 * q^65 + 16 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	0.894427	+	0.447214i	$0.147584\pi$
$6$		0			0
$7$			2.82843i			1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$			2.82843i			1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	0.987797	−	0.155747i	$-0.0497784\pi$
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	−0.155747	+	0.987797i	$0.549778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−5.65685	−	5.65685i	−1.29777	−	1.29777i	−0.929861	−	0.367910i	$-0.880073\pi$
$19$	−5.65685	−	5.65685i	−1.29777	−	1.29777i	−0.367910	−	0.929861i	$-0.619927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.65685i		−	1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$		−	5.65685i		−	1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	+	1.00000i	0.185695	+	0.185695i	0.793832	−	0.608137i	$-0.208083\pi$
$29$	1.00000	+	1.00000i	0.185695	+	0.185695i	−0.608137	+	0.793832i	$0.708083\pi$
$30$		0			0
$31$	2.82843			0.508001			0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843			0.508001			0.254000	+	0.967204i	$0.418254\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.82843	+	2.82843i	0.478091	+	0.478091i
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	11.3137			1.65027			0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137			1.65027			0.825137	+	0.564933i	$0.191098\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	+	5.00000i	−0.686803	+	0.686803i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−5.00000	+	5.00000i	−0.686803	+	0.686803i	0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.48528	−	8.48528i	1.10469	−	1.10469i	0.110853	−	0.993837i	$-0.464642\pi$
$59$	8.48528	−	8.48528i	1.10469	−	1.10469i	0.993837	−	0.110853i	$-0.0353582\pi$
$60$		0			0
$61$	−1.00000	−	1.00000i	−0.128037	−	0.128037i	0.640184	−	0.768221i	$-0.278858\pi$
$61$	−1.00000	−	1.00000i	−0.128037	−	0.128037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	−2.82843	−	2.82843i	−0.345547	−	0.345547i	0.512901	−	0.858448i	$-0.328571\pi$
$67$	−2.82843	−	2.82843i	−0.345547	−	0.345547i	−0.858448	+	0.512901i	$0.828571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.3137i		−	1.34269i	−0.741145	−	0.671345i	$-0.765717\pi$
$71$		−	11.3137i		−	1.34269i	0.741145	−	0.671345i	$-0.234283\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.48528			−0.954669			−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528			−0.954669			−0.477334	+	0.878722i	$0.658397\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.3137	+	11.3137i	1.24184	+	1.24184i	0.959237	+	0.282604i	$0.0911983\pi$
$83$	11.3137	+	11.3137i	1.24184	+	1.24184i	0.282604	+	0.959237i	$0.408802\pi$
$84$		0			0
$85$	4.00000	−	4.00000i	0.433861	−	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−8.48528	+	8.48528i	−0.889499	+	0.889499i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.3137			−1.16076
$96$		0			0
$97$	16.0000			1.62455			0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000			1.62455			0.812277	+	0.583272i	$0.198228\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.0000	−	11.0000i	1.09454	−	1.09454i	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	11.0000	−	11.0000i	1.09454	−	1.09454i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$		0			0
$103$			19.7990i			1.95085i	0.220326	+	0.975426i	$0.429288\pi$
$103$			19.7990i			1.95085i	−0.220326	+	0.975426i	$0.570712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	−0.986151	−	0.165848i	$-0.946964\pi$
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	0.165848	+	0.986151i	$0.446964\pi$
$108$		0			0
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	0.548683	−	0.836031i	$-0.315129\pi$
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	−0.836031	+	0.548683i	$0.815129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	−5.65685	−	5.65685i	−0.527504	−	0.527504i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			11.3137i			1.03713i
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.00000	+	8.00000i	0.715542	+	0.715542i
$126$		0			0
$127$	−8.48528			−0.752947			−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528			−0.752947			−0.376473	+	0.926427i	$0.622863\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.82843	−	2.82843i	−0.247121	−	0.247121i	0.572667	−	0.819788i	$-0.305909\pi$
$131$	−2.82843	−	2.82843i	−0.247121	−	0.247121i	−0.819788	+	0.572667i	$0.805909\pi$
$132$		0			0
$133$	16.0000	−	16.0000i	1.38738	−	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	2.82843	−	2.82843i	0.239904	−	0.239904i	−0.576906	−	0.816810i	$-0.695740\pi$
$139$	2.82843	−	2.82843i	0.239904	−	0.239904i	0.816810	+	0.576906i	$0.195740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000			0.166091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	−	9.00000i	0.737309	−	0.737309i	−0.234748	−	0.972056i	$-0.575426\pi$
$149$	9.00000	−	9.00000i	0.737309	−	0.737309i	0.972056	+	0.234748i	$0.0754264\pi$
$150$		0			0
$151$			2.82843i			0.230174i	0.993355	+	0.115087i	$0.0367147\pi$
$151$			2.82843i			0.230174i	−0.993355	+	0.115087i	$0.963285\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.82843	−	2.82843i	0.227185	−	0.227185i
$156$		0			0
$157$	15.0000	+	15.0000i	1.19713	+	1.19713i	0.975022	+	0.222108i	$0.0712939\pi$
$157$	15.0000	+	15.0000i	1.19713	+	1.19713i	0.222108	+	0.975022i	$0.428706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$	11.3137	+	11.3137i	0.886158	+	0.886158i	0.994152	−	0.107994i	$-0.0344426\pi$
$163$	11.3137	+	11.3137i	0.886158	+	0.886158i	−0.107994	+	0.994152i	$0.534443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11.3137i		−	0.875481i	−0.899101	−	0.437741i	$-0.855779\pi$
$167$		−	11.3137i		−	0.875481i	0.899101	−	0.437741i	$-0.144221\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	0.491011	−	0.871154i	$-0.336628\pi$
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	−0.871154	+	0.491011i	$0.836628\pi$
$174$		0			0
$175$	−8.48528			−0.641427
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.82843	−	2.82843i	−0.211407	−	0.211407i	0.593458	−	0.804865i	$-0.297762\pi$
$179$	−2.82843	−	2.82843i	−0.211407	−	0.211407i	−0.804865	+	0.593458i	$0.797762\pi$
$180$		0			0
$181$	9.00000	−	9.00000i	0.668965	−	0.668965i	−0.288512	−	0.957476i	$-0.593160\pi$
$181$	9.00000	−	9.00000i	0.668965	−	0.668965i	0.957476	+	0.288512i	$0.0931604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.00000i		−	0.441129i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.65685			0.409316			0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685			0.409316			0.204658	+	0.978834i	$0.434392\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$		−	8.48528i		−	0.601506i	−0.953702	−	0.300753i	$-0.902762\pi$
$199$		−	8.48528i		−	0.601506i	0.953702	−	0.300753i	$-0.0972379\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.82843	+	2.82843i	−0.198517	+	0.198517i
$204$		0			0
$205$	4.00000	+	4.00000i	0.279372	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	8.48528	+	8.48528i	0.584151	+	0.584151i	0.936041	−	0.351890i	$-0.114461\pi$
$211$	8.48528	+	8.48528i	0.584151	+	0.584151i	−0.351890	+	0.936041i	$0.614461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			8.00000i			0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000	+	12.0000i	0.807207	+	0.807207i
$222$		0			0
$223$	25.4558			1.70465			0.852325	−	0.523013i	$-0.175192\pi$
$223$	25.4558			1.70465			0.852325	+	0.523013i	$0.175192\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.65685	−	5.65685i	−0.375459	−	0.375459i	0.494002	−	0.869461i	$-0.335534\pi$
$227$	−5.65685	−	5.65685i	−0.375459	−	0.375459i	−0.869461	+	0.494002i	$0.835534\pi$
$228$		0			0
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	11.3137	−	11.3137i	0.738025	−	0.738025i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.65685			−0.365911			−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685			−0.365911			−0.182956	+	0.983121i	$0.558567\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.00000	+	1.00000i	−0.0638877	+	0.0638877i
$246$		0			0
$247$		−	33.9411i		−	2.15962i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.65685	+	5.65685i	−0.357057	+	0.357057i	−0.862727	−	0.505670i	$-0.831245\pi$
$251$	−5.65685	+	5.65685i	−0.357057	+	0.357057i	0.505670	+	0.862727i	$0.331245\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$	8.48528	+	8.48528i	0.527250	+	0.527250i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			28.2843i			1.74408i	0.489432	+	0.872041i	$0.337204\pi$
$263$			28.2843i			1.74408i	−0.489432	+	0.872041i	$0.662796\pi$
$264$		0			0
$265$			10.0000i			0.614295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.0000	−	15.0000i	−0.914566	−	0.914566i	0.0820612	−	0.996627i	$-0.473850\pi$
$269$	−15.0000	−	15.0000i	−0.914566	−	0.914566i	−0.996627	+	0.0820612i	$0.973850\pi$
$270$		0			0
$271$	8.48528			0.515444			0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528			0.515444			0.257722	+	0.966219i	$0.417028\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.00000	+	1.00000i	−0.0600842	+	0.0600842i	−0.736510	−	0.676426i	$-0.763528\pi$
$277$	−1.00000	+	1.00000i	−0.0600842	+	0.0600842i	0.676426	+	0.736510i	$0.263528\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.0000i		−	1.31241i	−0.754583	−	0.656205i	$-0.772161\pi$
$281$		−	22.0000i		−	1.31241i	0.754583	−	0.656205i	$-0.227839\pi$
$282$		0			0
$283$	−2.82843	+	2.82843i	−0.168133	+	0.168133i	−0.786158	−	0.618026i	$-0.787933\pi$
$283$	−2.82843	+	2.82843i	−0.168133	+	0.168133i	0.618026	+	0.786158i	$0.287933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.3137			−0.667827
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.0000	−	13.0000i	0.759468	−	0.759468i	−0.216757	−	0.976226i	$-0.569548\pi$
$293$	13.0000	−	13.0000i	0.759468	−	0.759468i	0.976226	+	0.216757i	$0.0695481\pi$
$294$		0			0
$295$		−	16.9706i		−	0.988064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.9706	−	16.9706i	0.981433	−	0.981433i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$	−2.82843	−	2.82843i	−0.161427	−	0.161427i	0.621772	−	0.783199i	$-0.286413\pi$
$307$	−2.82843	−	2.82843i	−0.161427	−	0.161427i	−0.783199	+	0.621772i	$0.786413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.2843i			1.60385i	0.597422	+	0.801927i	$0.296192\pi$
$311$			28.2843i			1.60385i	−0.597422	+	0.801927i	$0.703808\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	−	15.0000i	−0.842484	−	0.842484i	0.146697	−	0.989181i	$-0.453136\pi$
$317$	−15.0000	−	15.0000i	−0.842484	−	0.842484i	−0.989181	+	0.146697i	$0.953136\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−22.6274	−	22.6274i	−1.25902	−	1.25902i
$324$		0			0
$325$	−9.00000	+	9.00000i	−0.499230	+	0.499230i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			32.0000i			1.76422i
$330$		0			0
$331$	−2.82843	+	2.82843i	−0.155464	+	0.155464i	−0.780553	−	0.625089i	$-0.785063\pi$
$331$	−2.82843	+	2.82843i	−0.155464	+	0.155464i	0.625089	+	0.780553i	$0.285063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.65685			−0.309067
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−22.6274	+	22.6274i	−1.21470	+	1.21470i	−0.245241	+	0.969462i	$0.578867\pi$
$347$	−22.6274	+	22.6274i	−1.21470	+	1.21470i	−0.969462	+	0.245241i	$0.921133\pi$
$348$		0			0
$349$	9.00000	+	9.00000i	0.481759	+	0.481759i	0.905693	−	0.423934i	$-0.139351\pi$
$349$	9.00000	+	9.00000i	0.481759	+	0.481759i	−0.423934	+	0.905693i	$0.639351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$	−11.3137	−	11.3137i	−0.600469	−	0.600469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	11.3137i		−	0.597115i	−0.954392	−	0.298557i	$-0.903495\pi$
$359$		−	11.3137i		−	0.597115i	0.954392	−	0.298557i	$-0.0965054\pi$
$360$		0			0
$361$			45.0000i			2.36842i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0000	+	14.0000i	0.732793	+	0.732793i
$366$		0			0
$367$	−2.82843			−0.147643			−0.0738213	−	0.997271i	$-0.523519\pi$
$367$	−2.82843			−0.147643			−0.0738213	+	0.997271i	$0.523519\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.1421	−	14.1421i	−0.734223	−	0.734223i
$372$		0			0
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	−0.625161	−	0.780496i	$-0.714967\pi$
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	0.780496	+	0.625161i	$0.214967\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.00000i			0.309016i
$378$		0			0
$379$	11.3137	−	11.3137i	0.581146	−	0.581146i	−0.354072	−	0.935218i	$-0.615203\pi$
$379$	11.3137	−	11.3137i	0.581146	−	0.581146i	0.935218	+	0.354072i	$0.115203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	28.2843			1.44526			0.722629	−	0.691236i	$-0.242933\pi$
$383$	28.2843			1.44526			0.722629	+	0.691236i	$0.242933\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$		−	22.6274i		−	1.14432i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.48528	+	8.48528i	−0.426941	+	0.426941i
$396$		0			0
$397$	−15.0000	−	15.0000i	−0.752828	−	0.752828i	0.222178	−	0.975006i	$-0.428683\pi$
$397$	−15.0000	−	15.0000i	−0.752828	−	0.752828i	−0.975006	+	0.222178i	$0.928683\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−20.0000			−0.998752			−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000			−0.998752			−0.499376	+	0.866385i	$0.666437\pi$
$402$		0			0
$403$	8.48528	+	8.48528i	0.422682	+	0.422682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	32.0000i		−	1.58230i	−0.611623	−	0.791149i	$-0.709483\pi$
$409$		−	32.0000i		−	1.58230i	0.611623	−	0.791149i	$-0.290517\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.0000	+	24.0000i	1.18096	+	1.18096i
$414$		0			0
$415$	22.6274			1.11074
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.65685	−	5.65685i	−0.276355	−	0.276355i	0.555297	−	0.831652i	$-0.312605\pi$
$419$	−5.65685	−	5.65685i	−0.276355	−	0.276355i	−0.831652	+	0.555297i	$0.812605\pi$
$420$		0			0
$421$	9.00000	−	9.00000i	0.438633	−	0.438633i	−0.452919	−	0.891552i	$-0.649617\pi$
$421$	9.00000	−	9.00000i	0.438633	−	0.438633i	0.891552	+	0.452919i	$0.149617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			12.0000i			0.582086i
$426$		0			0
$427$	2.82843	−	2.82843i	0.136877	−	0.136877i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.3137			−0.544962			−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137			−0.544962			−0.272481	+	0.962161i	$0.587844\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−32.0000	+	32.0000i	−1.53077	+	1.53077i
$438$		0			0
$439$		−	2.82843i		−	0.134993i	−0.997719	−	0.0674967i	$-0.978499\pi$
$439$		−	2.82843i		−	0.134993i	0.997719	−	0.0674967i	$-0.0215012\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.9706	−	16.9706i	0.806296	−	0.806296i	−0.177775	−	0.984071i	$-0.556890\pi$
$443$	16.9706	−	16.9706i	0.806296	−	0.806296i	0.984071	+	0.177775i	$0.0568900\pi$
$444$		0			0
$445$	6.00000	+	6.00000i	0.284427	+	0.284427i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$			16.9706i			0.795592i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.0000	−	11.0000i	−0.512321	−	0.512321i	0.402916	−	0.915237i	$-0.367997\pi$
$461$	−11.0000	−	11.0000i	−0.512321	−	0.512321i	−0.915237	+	0.402916i	$0.867997\pi$
$462$		0			0
$463$	31.1127			1.44593			0.722965	−	0.690885i	$-0.242779\pi$
$463$	31.1127			1.44593			0.722965	+	0.690885i	$0.242779\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.65685	−	5.65685i	−0.261768	−	0.261768i	0.564004	−	0.825772i	$-0.309260\pi$
$467$	−5.65685	−	5.65685i	−0.261768	−	0.261768i	−0.825772	+	0.564004i	$0.809260\pi$
$468$		0			0
$469$	8.00000	−	8.00000i	0.369406	−	0.369406i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	16.9706	−	16.9706i	0.778663	−	0.778663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−28.2843			−1.29234			−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843			−1.29234			−0.646171	+	0.763193i	$0.723631\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	−	16.0000i	0.726523	−	0.726523i
$486$		0			0
$487$		−	2.82843i		−	0.128168i	−0.997944	−	0.0640841i	$-0.979587\pi$
$487$		−	2.82843i		−	0.128168i	0.997944	−	0.0640841i	$-0.0204126\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.48528	+	8.48528i	−0.382935	+	0.382935i	−0.872159	−	0.489223i	$-0.837280\pi$
$491$	−8.48528	+	8.48528i	−0.382935	+	0.382935i	0.489223	+	0.872159i	$0.337280\pi$
$492$		0			0
$493$	4.00000	+	4.00000i	0.180151	+	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	32.0000			1.43540
$498$		0			0
$499$	19.7990	+	19.7990i	0.886325	+	0.886325i	0.994168	−	0.107843i	$-0.0343945\pi$
$499$	19.7990	+	19.7990i	0.886325	+	0.886325i	−0.107843	+	0.994168i	$0.534395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	11.3137i		−	0.504453i	−0.967668	−	0.252227i	$-0.918837\pi$
$503$		−	11.3137i		−	0.504453i	0.967668	−	0.252227i	$-0.0811629\pi$
$504$		0			0
$505$		−	22.0000i		−	0.978987i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.0000	+	23.0000i	1.01946	+	1.01946i	0.999807	+	0.0196502i	$0.00625524\pi$
$509$	23.0000	+	23.0000i	1.01946	+	1.01946i	0.0196502	+	0.999807i	$0.493745\pi$
$510$		0			0
$511$	−39.5980			−1.75171
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.7990	+	19.7990i	0.872448	+	0.872448i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.0000i		−	0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$		−	12.0000i		−	0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$		0			0
$523$	−5.65685	+	5.65685i	−0.247357	+	0.247357i	−0.819885	−	0.572528i	$-0.805963\pi$
$523$	−5.65685	+	5.65685i	−0.247357	+	0.247357i	0.572528	+	0.819885i	$0.305963\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11.3137			0.492833
$528$		0			0
$529$	−9.00000			−0.391304
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000	+	12.0000i	−0.519778	+	0.519778i
$534$		0			0
$535$			16.9706i			0.733701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.00000	+	3.00000i	0.128980	+	0.128980i	0.768650	−	0.639670i	$-0.220929\pi$
$541$	3.00000	+	3.00000i	0.128980	+	0.128980i	−0.639670	+	0.768650i	$0.720929\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000			−0.257012
$546$		0			0
$547$	5.65685	+	5.65685i	0.241870	+	0.241870i	0.817623	−	0.575754i	$-0.195291\pi$
$547$	5.65685	+	5.65685i	0.241870	+	0.241870i	−0.575754	+	0.817623i	$0.695291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.3137i		−	0.481980i
$552$		0			0
$553$		−	24.0000i		−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.0000	−	21.0000i	−0.889799	−	0.889799i	0.104705	−	0.994503i	$-0.466610\pi$
$557$	−21.0000	−	21.0000i	−0.889799	−	0.889799i	−0.994503	+	0.104705i	$0.966610\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.65685	+	5.65685i	0.238408	+	0.238408i	0.816191	−	0.577783i	$-0.196082\pi$
$563$	5.65685	+	5.65685i	0.238408	+	0.238408i	−0.577783	+	0.816191i	$0.696082\pi$
$564$		0			0
$565$	−18.0000	+	18.0000i	−0.757266	+	0.757266i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	20.0000i		−	0.838444i	−0.907884	−	0.419222i	$-0.862303\pi$
$569$		−	20.0000i		−	0.838444i	0.907884	−	0.419222i	$-0.137697\pi$
$570$		0			0
$571$	−14.1421	+	14.1421i	−0.591830	+	0.591830i	−0.938125	−	0.346296i	$-0.887439\pi$
$571$	−14.1421	+	14.1421i	−0.591830	+	0.591830i	0.346296	+	0.938125i	$0.387439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.9706			0.707721
$576$		0			0
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.0000	+	32.0000i	−1.32758	+	1.32758i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.82843	−	2.82843i	0.116742	−	0.116742i	−0.646323	−	0.763064i	$-0.723694\pi$
$587$	2.82843	−	2.82843i	0.116742	−	0.116742i	0.763064	+	0.646323i	$0.223694\pi$
$588$		0			0
$589$	−16.0000	−	16.0000i	−0.659269	−	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$	11.3137	+	11.3137i	0.463817	+	0.463817i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	39.5980i		−	1.61793i	−0.587857	−	0.808965i	$-0.700028\pi$
$599$		−	39.5980i		−	1.61793i	0.587857	−	0.808965i	$-0.299972\pi$
$600$		0			0
$601$		−	14.0000i		−	0.571072i	−0.958368	−	0.285536i	$-0.907828\pi$
$601$		−	14.0000i		−	0.571072i	0.958368	−	0.285536i	$-0.0921716\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	+	11.0000i	0.447214	+	0.447214i
$606$		0			0
$607$	−42.4264			−1.72203			−0.861017	−	0.508576i	$-0.830172\pi$
$607$	−42.4264			−1.72203			−0.861017	+	0.508576i	$0.830172\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	33.9411	+	33.9411i	1.37311	+	1.37311i
$612$		0			0
$613$	−11.0000	+	11.0000i	−0.444286	+	0.444286i	−0.893449	−	0.449164i	$-0.851722\pi$
$613$	−11.0000	+	11.0000i	−0.444286	+	0.444286i	0.449164	+	0.893449i	$0.351722\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$		0			0
$619$	−31.1127	+	31.1127i	−1.25052	+	1.25052i	−0.295040	+	0.955485i	$0.595333\pi$
$619$	−31.1127	+	31.1127i	−1.25052	+	1.25052i	−0.955485	+	0.295040i	$0.904667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.9706			−0.679911
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000	−	12.0000i	0.478471	−	0.478471i
$630$		0			0
$631$		−	8.48528i		−	0.337794i	−0.985634	−	0.168897i	$-0.945980\pi$
$631$		−	8.48528i		−	0.337794i	0.985634	−	0.168897i	$-0.0540205\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.48528	+	8.48528i	−0.336728	+	0.336728i
$636$		0			0
$637$	−3.00000	−	3.00000i	−0.118864	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$	−11.3137	−	11.3137i	−0.446169	−	0.446169i	0.447910	−	0.894079i	$-0.352169\pi$
$643$	−11.3137	−	11.3137i	−0.446169	−	0.446169i	−0.894079	+	0.447910i	$0.852169\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			39.5980i			1.55676i	0.627795	+	0.778379i	$0.283958\pi$
$647$			39.5980i			1.55676i	−0.627795	+	0.778379i	$0.716042\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	−	3.00000i	−0.117399	−	0.117399i	0.645967	−	0.763366i	$-0.276455\pi$
$653$	−3.00000	−	3.00000i	−0.117399	−	0.117399i	−0.763366	+	0.645967i	$0.776455\pi$
$654$		0			0
$655$	−5.65685			−0.221032
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.1421	−	14.1421i	−0.550899	−	0.550899i	0.375801	−	0.926700i	$-0.377368\pi$
$659$	−14.1421	−	14.1421i	−0.550899	−	0.550899i	−0.926700	+	0.375801i	$0.877368\pi$
$660$		0			0
$661$	−19.0000	+	19.0000i	−0.739014	+	0.739014i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−19.0000	+	19.0000i	−0.739014	+	0.739014i	0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	32.0000i		−	1.24091i
$666$		0			0
$667$	5.65685	−	5.65685i	0.219034	−	0.219034i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.0000	−	23.0000i	0.883962	−	0.883962i	−0.109973	−	0.993935i	$-0.535076\pi$
$677$	23.0000	−	23.0000i	0.883962	−	0.883962i	0.993935	+	0.109973i	$0.0350764\pi$
$678$		0			0
$679$			45.2548i			1.73672i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.6274	−	22.6274i	0.865814	−	0.865814i	−0.126192	−	0.992006i	$-0.540275\pi$
$683$	22.6274	−	22.6274i	0.865814	−	0.865814i	0.992006	+	0.126192i	$0.0402755\pi$
$684$		0			0
$685$	12.0000	+	12.0000i	0.458496	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.0000			−1.14291
$690$		0			0
$691$	−28.2843	−	28.2843i	−1.07598	−	1.07598i	−0.996865	−	0.0791192i	$-0.974789\pi$
$691$	−28.2843	−	28.2843i	−1.07598	−	1.07598i	−0.0791192	−	0.996865i	$-0.525211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	5.65685i		−	0.214577i
$696$		0			0
$697$			16.0000i			0.606043i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−33.0000	−	33.0000i	−1.24639	−	1.24639i	−0.957302	−	0.289091i	$-0.906647\pi$
$701$	−33.0000	−	33.0000i	−1.24639	−	1.24639i	−0.289091	−	0.957302i	$-0.593353\pi$
$702$		0			0
$703$	−33.9411			−1.28011
$704$		0			0
$705$		0			0
$706$		0			0
$707$	31.1127	+	31.1127i	1.17011	+	1.17011i
$708$		0			0
$709$	−25.0000	+	25.0000i	−0.938895	+	0.938895i	−0.998238	−	0.0593429i	$-0.981099\pi$
$709$	−25.0000	+	25.0000i	−0.938895	+	0.938895i	0.0593429	+	0.998238i	$0.481099\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.9706			−0.632895			−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706			−0.632895			−0.316448	+	0.948610i	$0.602490\pi$
$720$		0			0
$721$	−56.0000			−2.08555
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	+	3.00000i	−0.111417	+	0.111417i
$726$		0			0
$727$			42.4264i			1.57351i	0.617266	+	0.786754i	$0.288240\pi$
$727$			42.4264i			1.57351i	−0.617266	+	0.786754i	$0.711760\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	0.649529	−	0.760337i	$-0.274966\pi$
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	−0.760337	+	0.649529i	$0.774966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−25.4558	−	25.4558i	−0.936408	−	0.936408i	0.0616872	−	0.998096i	$-0.480352\pi$
$739$	−25.4558	−	25.4558i	−0.936408	−	0.936408i	−0.998096	+	0.0616872i	$0.980352\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.3137i			0.415060i	0.978229	+	0.207530i	$0.0665424\pi$
$743$			11.3137i			0.415060i	−0.978229	+	0.207530i	$0.933458\pi$
$744$		0			0
$745$		−	18.0000i		−	0.659469i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−24.0000	−	24.0000i	−0.876941	−	0.876941i
$750$		0			0
$751$	8.48528			0.309632			0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528			0.309632			0.154816	+	0.987943i	$0.450521\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.82843	+	2.82843i	0.102937	+	0.102937i
$756$		0			0
$757$	−15.0000	+	15.0000i	−0.545184	+	0.545184i	−0.925044	−	0.379860i	$-0.875972\pi$
$757$	−15.0000	+	15.0000i	−0.545184	+	0.545184i	0.379860	+	0.925044i	$0.375972\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			36.0000i			1.30500i	0.757789	+	0.652499i	$0.226280\pi$
$761$			36.0000i			1.30500i	−0.757789	+	0.652499i	$0.773720\pi$
$762$		0			0
$763$	8.48528	−	8.48528i	0.307188	−	0.307188i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	50.9117			1.83831
$768$		0			0
$769$	−32.0000			−1.15395			−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000			−1.15395			−0.576975	+	0.816762i	$0.695767\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.0000	+	13.0000i	−0.467578	+	0.467578i	−0.901129	−	0.433551i	$-0.857260\pi$
$773$	−13.0000	+	13.0000i	−0.467578	+	0.467578i	0.433551	+	0.901129i	$0.357260\pi$
$774$		0			0
$775$			8.48528i			0.304800i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	22.6274	−	22.6274i	0.810711	−	0.810711i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.0000			1.07075
$786$		0			0
$787$	−22.6274	−	22.6274i	−0.806580	−	0.806580i	0.177534	−	0.984115i	$-0.443188\pi$
$787$	−22.6274	−	22.6274i	−0.806580	−	0.806580i	−0.984115	+	0.177534i	$0.943188\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	50.9117i		−	1.81021i
$792$		0			0
$793$		−	6.00000i		−	0.213066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.0000	+	35.0000i	1.23976	+	1.23976i	0.960097	+	0.279666i	$0.0902238\pi$
$797$	35.0000	+	35.0000i	1.23976	+	1.23976i	0.279666	+	0.960097i	$0.409776\pi$
$798$		0			0
$799$	45.2548			1.60100
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	16.0000	−	16.0000i	0.563926	−	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4.00000i		−	0.140633i	−0.997525	−	0.0703163i	$-0.977599\pi$
$809$		−	4.00000i		−	0.140633i	0.997525	−	0.0703163i	$-0.0224008\pi$
$810$		0			0
$811$	−28.2843	+	28.2843i	−0.993195	+	0.993195i	−0.999977	−	0.00678191i	$-0.997841\pi$
$811$	−28.2843	+	28.2843i	−0.993195	+	0.993195i	0.00678191	+	0.999977i	$0.497841\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	22.6274			0.792604
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.00000	+	5.00000i	−0.174501	+	0.174501i	−0.788954	−	0.614453i	$-0.789377\pi$
$821$	−5.00000	+	5.00000i	−0.174501	+	0.174501i	0.614453	+	0.788954i	$0.289377\pi$
$822$		0			0
$823$			25.4558i			0.887335i	0.896191	+	0.443667i	$0.146323\pi$
$823$			25.4558i			0.887335i	−0.896191	+	0.443667i	$0.853677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.1421	+	14.1421i	−0.491770	+	0.491770i	−0.908864	−	0.417093i	$-0.863049\pi$
$827$	−14.1421	+	14.1421i	−0.491770	+	0.491770i	0.417093	+	0.908864i	$0.363049\pi$
$828$		0			0
$829$	−37.0000	−	37.0000i	−1.28506	−	1.28506i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−37.0000	−	37.0000i	−1.28506	−	1.28506i	−0.347314	−	0.937749i	$-0.612906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.00000			−0.138592
$834$		0			0
$835$	−11.3137	−	11.3137i	−0.391527	−	0.391527i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	45.2548i		−	1.56237i	−0.624299	−	0.781185i	$-0.714615\pi$
$839$		−	45.2548i		−	1.56237i	0.624299	−	0.781185i	$-0.285385\pi$
$840$		0			0
$841$		−	27.0000i		−	0.931034i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.00000	+	5.00000i	0.172005	+	0.172005i
$846$		0			0
$847$	−31.1127			−1.06904
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.9706	−	16.9706i	−0.581743	−	0.581743i
$852$		0			0
$853$	11.0000	−	11.0000i	0.376633	−	0.376633i	−0.493253	−	0.869886i	$-0.664192\pi$
$853$	11.0000	−	11.0000i	0.376633	−	0.376633i	0.869886	+	0.493253i	$0.164192\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20.0000i		−	0.683187i	−0.939848	−	0.341593i	$-0.889033\pi$
$857$		−	20.0000i		−	0.683187i	0.939848	−	0.341593i	$-0.110967\pi$
$858$		0			0
$859$	−5.65685	+	5.65685i	−0.193009	+	0.193009i	−0.796995	−	0.603986i	$-0.793578\pi$
$859$	−5.65685	+	5.65685i	−0.193009	+	0.193009i	0.603986	+	0.796995i	$0.293578\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.65685			−0.192562			−0.0962808	−	0.995354i	$-0.530695\pi$
$863$	−5.65685			−0.192562			−0.0962808	+	0.995354i	$0.530695\pi$
$864$		0			0
$865$	−10.0000			−0.340010
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	16.9706i		−	0.575026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6274	+	22.6274i	−0.764946	+	0.764946i
$876$		0			0
$877$	−23.0000	−	23.0000i	−0.776655	−	0.776655i	0.202606	−	0.979260i	$-0.435059\pi$
$877$	−23.0000	−	23.0000i	−0.776655	−	0.776655i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	16.9706	+	16.9706i	0.571105	+	0.571105i	0.932437	−	0.361332i	$-0.117678\pi$
$883$	16.9706	+	16.9706i	0.571105	+	0.571105i	−0.361332	+	0.932437i	$0.617678\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.9706i		−	0.569816i	−0.958555	−	0.284908i	$-0.908037\pi$
$887$		−	16.9706i		−	0.569816i	0.958555	−	0.284908i	$-0.0919630\pi$
$888$		0			0
$889$		−	24.0000i		−	0.804934i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−64.0000	−	64.0000i	−2.14168	−	2.14168i
$894$		0			0
$895$	−5.65685			−0.189088
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.82843	+	2.82843i	0.0943333	+	0.0943333i
$900$		0			0
$901$	−20.0000	+	20.0000i	−0.666297	+	0.666297i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	18.0000i		−	0.598340i
$906$		0			0
$907$	22.6274	−	22.6274i	0.751331	−	0.751331i	−0.223397	−	0.974728i	$-0.571714\pi$
$907$	22.6274	−	22.6274i	0.751331	−	0.751331i	0.974728	+	0.223397i	$0.0717145\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.65685			−0.187420			−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685			−0.187420			−0.0937100	+	0.995600i	$0.529873\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.00000	−	8.00000i	0.264183	−	0.264183i
$918$		0			0
$919$			2.82843i			0.0933012i	0.998911	+	0.0466506i	$0.0148547\pi$
$919$			2.82843i			0.0933012i	−0.998911	+	0.0466506i	$0.985145\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33.9411	−	33.9411i	1.11719	−	1.11719i
$924$		0			0
$925$	9.00000	+	9.00000i	0.295918	+	0.295918i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.0000			0.656179			0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000			0.656179			0.328089	+	0.944647i	$0.393595\pi$
$930$		0			0
$931$	5.65685	+	5.65685i	0.185396	+	0.185396i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	42.0000i		−	1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$		−	42.0000i		−	1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.0000	+	37.0000i	1.20617	+	1.20617i	0.972259	+	0.233906i	$0.0751509\pi$
$941$	37.0000	+	37.0000i	1.20617	+	1.20617i	0.233906	+	0.972259i	$0.424849\pi$
$942$		0			0
$943$	22.6274			0.736850
$944$		0			0
$945$		0			0
$946$		0			0
$947$	36.7696	+	36.7696i	1.19485	+	1.19485i	0.975689	+	0.219161i	$0.0703321\pi$
$947$	36.7696	+	36.7696i	1.19485	+	1.19485i	0.219161	+	0.975689i	$0.429668\pi$
$948$		0			0
$949$	−42.0000	+	42.0000i	−1.36338	+	1.36338i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	36.0000i		−	1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$		−	36.0000i		−	1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$		0			0
$955$	5.65685	−	5.65685i	0.183052	−	0.183052i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33.9411			−1.09602
$960$		0			0
$961$	−23.0000			−0.741935
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	42.4264i		−	1.36434i	−0.731193	−	0.682171i	$-0.761036\pi$
$967$		−	42.4264i		−	1.36434i	0.731193	−	0.682171i	$-0.238964\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.9706	−	16.9706i	0.544611	−	0.544611i	−0.380266	−	0.924877i	$-0.624168\pi$
$971$	16.9706	−	16.9706i	0.544611	−	0.544611i	0.924877	+	0.380266i	$0.124168\pi$
$972$		0			0
$973$	8.00000	+	8.00000i	0.256468	+	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−36.0000			−1.15174			−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000			−1.15174			−0.575871	+	0.817541i	$0.695337\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.9706i		−	0.541277i	−0.962681	−	0.270638i	$-0.912765\pi$
$983$		−	16.9706i		−	0.541277i	0.962681	−	0.270638i	$-0.0872348\pi$
$984$		0			0
$985$		−	10.0000i		−	0.318626i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	25.4558			0.808632			0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558			0.808632			0.404316	+	0.914619i	$0.367510\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.48528	−	8.48528i	−0.269002	−	0.269002i
$996$		0			0
$997$	−5.00000	+	5.00000i	−0.158352	+	0.158352i	−0.781836	−	0.623484i	$-0.785717\pi$
$997$	−5.00000	+	5.00000i	−0.158352	+	0.158352i	0.623484	+	0.781836i	$0.285717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bb.1153.2		4
3.2	odd	2		1536.2.j.b.1153.2	yes	4
4.3	odd	2	inner	4608.2.k.bb.1153.1		4
8.3	odd	2		4608.2.k.y.1153.1		4
8.5	even	2		4608.2.k.y.1153.2		4
12.11	even	2		1536.2.j.b.1153.1	yes	4
16.3	odd	4		4608.2.k.y.3457.2		4
16.5	even	4	inner	4608.2.k.bb.3457.1		4
16.11	odd	4	inner	4608.2.k.bb.3457.2		4
16.13	even	4		4608.2.k.y.3457.1		4
24.5	odd	2		1536.2.j.c.1153.1	yes	4
24.11	even	2		1536.2.j.c.1153.2	yes	4
32.5	even	8		9216.2.a.h.1.2		2
32.11	odd	8		9216.2.a.h.1.1		2
32.21	even	8		9216.2.a.i.1.1		2
32.27	odd	8		9216.2.a.i.1.2		2
48.5	odd	4		1536.2.j.b.385.2	yes	4
48.11	even	4		1536.2.j.b.385.1	✓	4
48.29	odd	4		1536.2.j.c.385.1	yes	4
48.35	even	4		1536.2.j.c.385.2	yes	4
96.5	odd	8		3072.2.a.b.1.1		2
96.11	even	8		3072.2.a.b.1.2		2
96.29	odd	8		3072.2.d.c.1537.3		4
96.35	even	8		3072.2.d.c.1537.1		4
96.53	odd	8		3072.2.a.h.1.2		2
96.59	even	8		3072.2.a.h.1.1		2
96.77	odd	8		3072.2.d.c.1537.2		4
96.83	even	8		3072.2.d.c.1537.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.b.385.1	✓	4	48.11	even	4
1536.2.j.b.385.2	yes	4	48.5	odd	4
1536.2.j.b.1153.1	yes	4	12.11	even	2
1536.2.j.b.1153.2	yes	4	3.2	odd	2
1536.2.j.c.385.1	yes	4	48.29	odd	4
1536.2.j.c.385.2	yes	4	48.35	even	4
1536.2.j.c.1153.1	yes	4	24.5	odd	2
1536.2.j.c.1153.2	yes	4	24.11	even	2
3072.2.a.b.1.1		2	96.5	odd	8
3072.2.a.b.1.2		2	96.11	even	8
3072.2.a.h.1.1		2	96.59	even	8
3072.2.a.h.1.2		2	96.53	odd	8
3072.2.d.c.1537.1		4	96.35	even	8
3072.2.d.c.1537.2		4	96.77	odd	8
3072.2.d.c.1537.3		4	96.29	odd	8
3072.2.d.c.1537.4		4	96.83	even	8
4608.2.k.y.1153.1		4	8.3	odd	2
4608.2.k.y.1153.2		4	8.5	even	2
4608.2.k.y.3457.1		4	16.13	even	4
4608.2.k.y.3457.2		4	16.3	odd	4
4608.2.k.bb.1153.1		4	4.3	odd	2	inner
4608.2.k.bb.1153.2		4	1.1	even	1	trivial
4608.2.k.bb.3457.1		4	16.5	even	4	inner
4608.2.k.bb.3457.2		4	16.11	odd	4	inner
9216.2.a.h.1.1		2	32.11	odd	8
9216.2.a.h.1.2		2	32.5	even	8
9216.2.a.i.1.1		2	32.21	even	8
9216.2.a.i.1.2		2	32.27	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)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +(3.00000 + 3.00000i) q^{13} +4.00000 q^{17} +(-5.65685 - 5.65685i) q^{19} -5.65685i q^{23} +3.00000i q^{25} +(1.00000 + 1.00000i) q^{29} +2.82843 q^{31} +(2.82843 + 2.82843i) q^{35} +(3.00000 - 3.00000i) q^{37} +4.00000i q^{41} +11.3137 q^{47} -1.00000 q^{49} +(-5.00000 + 5.00000i) q^{53} +(8.48528 - 8.48528i) q^{59} +(-1.00000 - 1.00000i) q^{61} +6.00000 q^{65} +(-2.82843 - 2.82843i) q^{67} -11.3137i q^{71} +14.0000i q^{73} -8.48528 q^{79} +(11.3137 + 11.3137i) q^{83} +(4.00000 - 4.00000i) q^{85} +6.00000i q^{89} +(-8.48528 + 8.48528i) q^{91} -11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} + 12 q^{13} + 16 q^{17} + 4 q^{29} + 12 q^{37} - 4 q^{49} - 20 q^{53} - 4 q^{61} + 24 q^{65} + 16 q^{85} + 64 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 12 * q^13 + 16 * q^17 + 4 * q^29 + 12 * q^37 - 4 * q^49 - 20 * q^53 - 4 * q^61 + 24 * q^65 + 16 * q^85 + 64 * q^97

Introduction

L-functions

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Database

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