LMFDB - Embedded newform 4608.2.k.u.1153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1153.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.u.3457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{5} +4.00000i q^{7} +(4.00000 - 4.00000i) q^{11} +(3.00000 + 3.00000i) q^{13} +6.00000 q^{17} +(4.00000 + 4.00000i) q^{19} -8.00000i q^{23} +3.00000i q^{25} +(3.00000 + 3.00000i) q^{29} -4.00000 q^{31} +(4.00000 + 4.00000i) q^{35} +(1.00000 - 1.00000i) q^{37} -2.00000i q^{41} +(4.00000 - 4.00000i) q^{43} -8.00000 q^{47} -9.00000 q^{49} +(-7.00000 + 7.00000i) q^{53} -8.00000i q^{55} +(-3.00000 - 3.00000i) q^{61} +6.00000 q^{65} +(-8.00000 - 8.00000i) q^{67} +10.0000i q^{73} +(16.0000 + 16.0000i) q^{77} +12.0000 q^{79} +(-4.00000 - 4.00000i) q^{83} +(6.00000 - 6.00000i) q^{85} -16.0000i q^{89} +(-12.0000 + 12.0000i) q^{91} +8.00000 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 8 q^{11} + 6 q^{13} + 12 q^{17} + 8 q^{19} + 6 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{37} + 8 q^{43} - 16 q^{47} - 18 q^{49} - 14 q^{53} - 6 q^{61} + 12 q^{65} - 16 q^{67} + 32 q^{77} + 24 q^{79}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 8 * q^11 + 6 * q^13 + 12 * q^17 + 8 * q^19 + 6 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^37 + 8 * q^43 - 16 * q^47 - 18 * q^49 - 14 * q^53 - 6 * q^61 + 12 * q^65 - 16 * q^67 + 32 * q^77 + 24 * q^79 - 8 * q^83 + 12 * q^85 - 24 * q^91 + 16 * q^95 + 16 * q^97

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
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.00000	−	4.00000i	1.20605	−	1.20605i	0.233748	−	0.972297i	$-0.424901\pi$
$11$	4.00000	−	4.00000i	1.20605	−	1.20605i	0.972297	−	0.233748i	$-0.0750991\pi$
$12$		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
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	4.00000	+	4.00000i	0.917663	+	0.917663i	0.996859	−	0.0791961i	$-0.0252353\pi$
$19$	4.00000	+	4.00000i	0.917663	+	0.917663i	−0.0791961	+	0.996859i	$0.525235\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	0.928477	−	0.371391i	$-0.121119\pi$
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000	+	4.00000i	0.676123	+	0.676123i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$	4.00000	−	4.00000i	0.609994	−	0.609994i	−0.332950	−	0.942944i	$-0.608044\pi$
$43$	4.00000	−	4.00000i	0.609994	−	0.609994i	0.942944	+	0.332950i	$0.108044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	−0.999287	−	0.0377628i	$-0.987977\pi$
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	0.0377628	+	0.999287i	$0.487977\pi$
$54$		0			0
$55$		−	8.00000i		−	1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−3.00000	−	3.00000i	−0.384111	−	0.384111i	0.488470	−	0.872581i	$-0.337555\pi$
$61$	−3.00000	−	3.00000i	−0.384111	−	0.384111i	−0.872581	+	0.488470i	$0.837555\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	−8.00000	−	8.00000i	−0.977356	−	0.977356i	0.0223937	−	0.999749i	$-0.492871\pi$
$67$	−8.00000	−	8.00000i	−0.977356	−	0.977356i	−0.999749	+	0.0223937i	$0.992871\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.0000	+	16.0000i	1.82337	+	1.82337i
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000	−	4.00000i	−0.439057	−	0.439057i	0.452638	−	0.891695i	$-0.350483\pi$
$83$	−4.00000	−	4.00000i	−0.439057	−	0.439057i	−0.891695	+	0.452638i	$0.850483\pi$
$84$		0			0
$85$	6.00000	−	6.00000i	0.650791	−	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.0000i		−	1.69600i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$		−	16.0000i		−	1.69600i	0.529999	−	0.847998i	$-0.322192\pi$
$90$		0			0
$91$	−12.0000	+	12.0000i	−1.25794	+	1.25794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.00000	+	7.00000i	−0.696526	+	0.696526i	−0.963660	−	0.267133i	$-0.913924\pi$
$101$	−7.00000	+	7.00000i	−0.696526	+	0.696526i	0.267133	+	0.963660i	$0.413924\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	+	8.00000i	−0.773389	+	0.773389i	−0.978697	−	0.205308i	$-0.934180\pi$
$107$	−8.00000	+	8.00000i	−0.773389	+	0.773389i	0.205308	+	0.978697i	$0.434180\pi$
$108$		0			0
$109$	5.00000	+	5.00000i	0.478913	+	0.478913i	0.904784	−	0.425871i	$-0.140032\pi$
$109$	5.00000	+	5.00000i	0.478913	+	0.478913i	−0.425871	+	0.904784i	$0.640032\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$	−8.00000	−	8.00000i	−0.746004	−	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			24.0000i			2.20008i
$120$		0			0
$121$		−	21.0000i		−	1.90909i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.00000	+	8.00000i	0.715542	+	0.715542i
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	−	8.00000i	−0.698963	−	0.698963i	0.265224	−	0.964187i	$-0.414554\pi$
$131$	−8.00000	−	8.00000i	−0.698963	−	0.698963i	−0.964187	+	0.265224i	$0.914554\pi$
$132$		0			0
$133$	−16.0000	+	16.0000i	−1.38738	+	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	24.0000			2.00698
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.00000	−	1.00000i	0.0819232	−	0.0819232i	−0.664958	−	0.746881i	$-0.731550\pi$
$149$	1.00000	−	1.00000i	0.0819232	−	0.0819232i	0.746881	+	0.664958i	$0.231550\pi$
$150$		0			0
$151$			4.00000i			0.325515i	0.986666	+	0.162758i	$0.0520389\pi$
$151$			4.00000i			0.325515i	−0.986666	+	0.162758i	$0.947961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000	+	4.00000i	−0.321288	+	0.321288i
$156$		0			0
$157$	−3.00000	−	3.00000i	−0.239426	−	0.239426i	0.577186	−	0.816612i	$-0.304151\pi$
$157$	−3.00000	−	3.00000i	−0.239426	−	0.239426i	−0.816612	+	0.577186i	$0.804151\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	32.0000			2.52195
$162$		0			0
$163$	12.0000	+	12.0000i	0.939913	+	0.939913i	0.998294	−	0.0583818i	$-0.0185941\pi$
$163$	12.0000	+	12.0000i	0.939913	+	0.939913i	−0.0583818	+	0.998294i	$0.518594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$	−12.0000			−0.907115
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.00000	+	8.00000i	0.597948	+	0.597948i	0.939766	−	0.341818i	$-0.111043\pi$
$179$	8.00000	+	8.00000i	0.597948	+	0.597948i	−0.341818	+	0.939766i	$0.611043\pi$
$180$		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
$185$		−	2.00000i		−	0.147043i
$186$		0			0
$187$	24.0000	−	24.0000i	1.75505	−	1.75505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$	8.00000			0.575853			0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000			0.575853			0.287926	+	0.957653i	$0.407034\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.00000	+	1.00000i	−0.0712470	+	0.0712470i	−0.741832	−	0.670585i	$-0.766043\pi$
$197$	−1.00000	+	1.00000i	−0.0712470	+	0.0712470i	0.670585	+	0.741832i	$0.266043\pi$
$198$		0			0
$199$			4.00000i			0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$			4.00000i			0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0000	+	12.0000i	−0.842235	+	0.842235i
$204$		0			0
$205$	−2.00000	−	2.00000i	−0.139686	−	0.139686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	32.0000			2.21349
$210$		0			0
$211$	16.0000	+	16.0000i	1.10149	+	1.10149i	0.994232	+	0.107254i	$0.0342057\pi$
$211$	16.0000	+	16.0000i	1.10149	+	1.10149i	0.107254	+	0.994232i	$0.465794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.0000	+	18.0000i	1.21081	+	1.21081i
$222$		0			0
$223$	12.0000			0.803579			0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000			0.803579			0.401790	+	0.915732i	$0.368388\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	−	12.0000i	−0.796468	−	0.796468i	0.186069	−	0.982537i	$-0.440425\pi$
$227$	−12.0000	−	12.0000i	−0.796468	−	0.796468i	−0.982537	+	0.186069i	$0.940425\pi$
$228$		0			0
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	−0.899498	−	0.436925i	$-0.856068\pi$
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	0.436925	+	0.899498i	$0.356068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	−8.00000	+	8.00000i	−0.521862	+	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−24.0000			−1.54598			−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000			−1.54598			−0.772988	+	0.634421i	$0.781239\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.00000	+	9.00000i	−0.574989	+	0.574989i
$246$		0			0
$247$			24.0000i			1.52708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.00000	−	4.00000i	0.252478	−	0.252478i	−0.569508	−	0.821986i	$-0.692866\pi$
$251$	4.00000	−	4.00000i	0.252478	−	0.252478i	0.821986	+	0.569508i	$0.192866\pi$
$252$		0			0
$253$	−32.0000	−	32.0000i	−2.01182	−	2.01182i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	4.00000	+	4.00000i	0.248548	+	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$			14.0000i			0.860013i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000	+	3.00000i	0.182913	+	0.182913i	0.792624	−	0.609711i	$-0.208714\pi$
$269$	3.00000	+	3.00000i	0.182913	+	0.182913i	−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	+	12.0000i	0.723627	+	0.723627i
$276$		0			0
$277$	15.0000	−	15.0000i	0.901263	−	0.901263i	−0.0942828	−	0.995545i	$-0.530056\pi$
$277$	15.0000	−	15.0000i	0.901263	−	0.901263i	0.995545	+	0.0942828i	$0.0300558\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	−0.903705	−	0.428155i	$-0.859164\pi$
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	0.428155	+	0.903705i	$0.359164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.00000i	−0.0584206	+	0.0584206i	−0.735714	−	0.677293i	$-0.763153\pi$
$293$	−1.00000	+	1.00000i	−0.0584206	+	0.0584206i	0.677293	+	0.735714i	$0.263153\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.0000	−	24.0000i	1.38796	−	1.38796i
$300$		0			0
$301$	16.0000	+	16.0000i	0.922225	+	0.922225i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.00000			−0.343559
$306$		0			0
$307$	−24.0000	−	24.0000i	−1.36975	−	1.36975i	−0.860787	−	0.508965i	$-0.830028\pi$
$307$	−24.0000	−	24.0000i	−1.36975	−	1.36975i	−0.508965	−	0.860787i	$-0.669972\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.0000i		−	0.907277i	−0.891186	−	0.453638i	$-0.850126\pi$
$311$		−	16.0000i		−	0.907277i	0.891186	−	0.453638i	$-0.149874\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.0000	+	19.0000i	1.06715	+	1.06715i	0.997577	+	0.0695692i	$0.0221625\pi$
$317$	19.0000	+	19.0000i	1.06715	+	1.06715i	0.0695692	+	0.997577i	$0.477838\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000	+	24.0000i	1.33540	+	1.33540i
$324$		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
$331$	−16.0000	+	16.0000i	−0.879440	+	0.879440i	−0.993477	−	0.114037i	$-0.963622\pi$
$331$	−16.0000	+	16.0000i	−0.879440	+	0.879440i	0.114037	+	0.993477i	$0.463622\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.0000			−0.874173
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000	+	16.0000i	−0.866449	+	0.866449i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.0000	−	20.0000i	1.07366	−	1.07366i	0.0765939	−	0.997062i	$-0.475596\pi$
$347$	20.0000	−	20.0000i	1.07366	−	1.07366i	0.997062	−	0.0765939i	$-0.0244045\pi$
$348$		0			0
$349$	11.0000	+	11.0000i	0.588817	+	0.588817i	0.937311	−	0.348494i	$-0.113307\pi$
$349$	11.0000	+	11.0000i	0.588817	+	0.588817i	−0.348494	+	0.937311i	$0.613307\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.0000			0.851594			0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000			0.851594			0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	16.0000i		−	0.844448i	−0.906492	−	0.422224i	$-0.861250\pi$
$359$		−	16.0000i		−	0.844448i	0.906492	−	0.422224i	$-0.138750\pi$
$360$		0			0
$361$			13.0000i			0.684211i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.0000	+	10.0000i	0.523424	+	0.523424i
$366$		0			0
$367$	20.0000			1.04399			0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000			1.04399			0.521996	+	0.852948i	$0.325188\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−28.0000	−	28.0000i	−1.45369	−	1.45369i
$372$		0			0
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	−0.979263	−	0.202593i	$-0.935063\pi$
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	0.202593	+	0.979263i	$0.435063\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			18.0000i			0.927047i
$378$		0			0
$379$	12.0000	−	12.0000i	0.616399	−	0.616399i	−0.328207	−	0.944606i	$-0.606444\pi$
$379$	12.0000	−	12.0000i	0.616399	−	0.616399i	0.944606	+	0.328207i	$0.106444\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$	32.0000			1.63087
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.0000	+	25.0000i	−1.26755	+	1.26755i	−0.320201	+	0.947350i	$0.603750\pi$
$389$	−25.0000	+	25.0000i	−1.26755	+	1.26755i	−0.947350	+	0.320201i	$0.896250\pi$
$390$		0			0
$391$		−	48.0000i		−	2.42746i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000	−	12.0000i	0.603786	−	0.603786i
$396$		0			0
$397$	27.0000	+	27.0000i	1.35509	+	1.35509i	0.879862	+	0.475229i	$0.157635\pi$
$397$	27.0000	+	27.0000i	1.35509	+	1.35509i	0.475229	+	0.879862i	$0.342365\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$	−12.0000	−	12.0000i	−0.597763	−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.00000i		−	0.396545i
$408$		0			0
$409$		−	8.00000i		−	0.395575i	−0.980245	−	0.197787i	$-0.936624\pi$
$409$		−	8.00000i		−	0.395575i	0.980245	−	0.197787i	$-0.0633755\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−8.00000			−0.392705
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000	+	12.0000i	0.586238	+	0.586238i	0.936611	−	0.350372i	$-0.113945\pi$
$419$	12.0000	+	12.0000i	0.586238	+	0.586238i	−0.350372	+	0.936611i	$0.613945\pi$
$420$		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
$425$			18.0000i			0.873128i
$426$		0			0
$427$	12.0000	−	12.0000i	0.580721	−	0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$	8.00000			0.384455			0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000			0.384455			0.192228	+	0.981350i	$0.438429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	32.0000	−	32.0000i	1.53077	−	1.53077i
$438$		0			0
$439$		−	20.0000i		−	0.954548i	−0.878755	−	0.477274i	$-0.841625\pi$
$439$		−	20.0000i		−	0.954548i	0.878755	−	0.477274i	$-0.158375\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	12.0000i	−0.570137	+	0.570137i	−0.932167	−	0.362029i	$-0.882084\pi$
$443$	−12.0000	+	12.0000i	−0.570137	+	0.570137i	0.362029	+	0.932167i	$0.382084\pi$
$444$		0			0
$445$	−16.0000	−	16.0000i	−0.758473	−	0.758473i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	−8.00000	−	8.00000i	−0.376705	−	0.376705i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			24.0000i			1.12514i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.0000	−	19.0000i	−0.884918	−	0.884918i	0.109111	−	0.994030i	$-0.465200\pi$
$461$	−19.0000	−	19.0000i	−0.884918	−	0.884918i	−0.994030	+	0.109111i	$0.965200\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.00000	−	4.00000i	−0.185098	−	0.185098i	0.608475	−	0.793573i	$-0.291782\pi$
$467$	−4.00000	−	4.00000i	−0.185098	−	0.185098i	−0.793573	+	0.608475i	$0.791782\pi$
$468$		0			0
$469$	32.0000	−	32.0000i	1.47762	−	1.47762i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	32.0000i		−	1.47136i
$474$		0			0
$475$	−12.0000	+	12.0000i	−0.550598	+	0.550598i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.0000			−0.731059			−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000			−0.731059			−0.365529	+	0.930800i	$0.619112\pi$
$480$		0			0
$481$	6.00000			0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	−	8.00000i	0.363261	−	0.363261i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000	+	8.00000i	−0.361035	+	0.361035i	−0.864194	−	0.503159i	$-0.832171\pi$
$491$	−8.00000	+	8.00000i	−0.361035	+	0.361035i	0.503159	+	0.864194i	$0.332171\pi$
$492$		0			0
$493$	18.0000	+	18.0000i	0.810679	+	0.810679i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	8.00000	+	8.00000i	0.358129	+	0.358129i	0.863123	−	0.504994i	$-0.168505\pi$
$499$	8.00000	+	8.00000i	0.358129	+	0.358129i	−0.504994	+	0.863123i	$0.668505\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$		0			0
$505$			14.0000i			0.622992i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.0000	−	11.0000i	−0.487566	−	0.487566i	0.419971	−	0.907537i	$-0.362040\pi$
$509$	−11.0000	−	11.0000i	−0.487566	−	0.487566i	−0.907537	+	0.419971i	$0.862040\pi$
$510$		0			0
$511$	−40.0000			−1.76950
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	−	4.00000i	−0.176261	−	0.176261i
$516$		0			0
$517$	−32.0000	+	32.0000i	−1.40736	+	1.40736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			30.0000i			1.31432i	0.753749	+	0.657162i	$0.228243\pi$
$521$			30.0000i			1.31432i	−0.753749	+	0.657162i	$0.771757\pi$
$522$		0			0
$523$	12.0000	−	12.0000i	0.524723	−	0.524723i	−0.394271	−	0.918994i	$-0.629003\pi$
$523$	12.0000	−	12.0000i	0.524723	−	0.524723i	0.918994	+	0.394271i	$0.129003\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−24.0000			−1.04546
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.00000	−	6.00000i	0.259889	−	0.259889i
$534$		0			0
$535$			16.0000i			0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−36.0000	+	36.0000i	−1.55063	+	1.55063i
$540$		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
$545$	10.0000			0.428353
$546$		0			0
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	0.915470	−	0.402387i	$-0.131819\pi$
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	−0.402387	+	0.915470i	$0.631819\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			24.0000i			1.02243i
$552$		0			0
$553$			48.0000i			2.04117i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.0000	+	11.0000i	0.466085	+	0.466085i	0.900644	−	0.434559i	$-0.143096\pi$
$557$	11.0000	+	11.0000i	0.466085	+	0.466085i	−0.434559	+	0.900644i	$0.643096\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.0000	+	20.0000i	0.842900	+	0.842900i	0.989235	−	0.146336i	$-0.0467479\pi$
$563$	20.0000	+	20.0000i	0.842900	+	0.842900i	−0.146336	+	0.989235i	$0.546748\pi$
$564$		0			0
$565$	16.0000	−	16.0000i	0.673125	−	0.673125i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			18.0000i			0.754599i	0.926091	+	0.377300i	$0.123147\pi$
$569$			18.0000i			0.754599i	−0.926091	+	0.377300i	$0.876853\pi$
$570$		0			0
$571$	−24.0000	+	24.0000i	−1.00437	+	1.00437i	−0.00437833	+	0.999990i	$0.501394\pi$
$571$	−24.0000	+	24.0000i	−1.00437	+	1.00437i	−0.999990	+	0.00437833i	$0.998606\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0000	−	16.0000i	0.663792	−	0.663792i
$582$		0			0
$583$			56.0000i			2.31928i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.0000	+	16.0000i	−0.660391	+	0.660391i	−0.955472	−	0.295081i	$-0.904653\pi$
$587$	−16.0000	+	16.0000i	−0.660391	+	0.660391i	0.295081	+	0.955472i	$0.404653\pi$
$588$		0			0
$589$	−16.0000	−	16.0000i	−0.659269	−	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.0000			−0.657041			−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000			−0.657041			−0.328521	+	0.944497i	$0.606550\pi$
$594$		0			0
$595$	24.0000	+	24.0000i	0.983904	+	0.983904i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			40.0000i			1.63436i	0.576386	+	0.817178i	$0.304463\pi$
$599$			40.0000i			1.63436i	−0.576386	+	0.817178i	$0.695537\pi$
$600$		0			0
$601$		−	38.0000i		−	1.55005i	−0.631929	−	0.775026i	$-0.717737\pi$
$601$		−	38.0000i		−	1.55005i	0.631929	−	0.775026i	$-0.282263\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−21.0000	−	21.0000i	−0.853771	−	0.853771i
$606$		0			0
$607$	12.0000			0.487065			0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000			0.487065			0.243532	+	0.969893i	$0.421694\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000	−	24.0000i	−0.970936	−	0.970936i
$612$		0			0
$613$	7.00000	−	7.00000i	0.282727	−	0.282727i	−0.551468	−	0.834196i	$-0.685932\pi$
$613$	7.00000	−	7.00000i	0.282727	−	0.282727i	0.834196	+	0.551468i	$0.185932\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.0000i		−	1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$		−	32.0000i		−	1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	64.0000			2.56411
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.00000	−	6.00000i	0.239236	−	0.239236i
$630$		0			0
$631$			20.0000i			0.796187i	0.917345	+	0.398094i	$0.130328\pi$
$631$			20.0000i			0.796187i	−0.917345	+	0.398094i	$0.869672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.00000	+	4.00000i	−0.158735	+	0.158735i
$636$		0			0
$637$	−27.0000	−	27.0000i	−1.06978	−	1.06978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.0000			−0.868948			−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000			−0.868948			−0.434474	+	0.900684i	$0.643066\pi$
$642$		0			0
$643$	−12.0000	−	12.0000i	−0.473234	−	0.473234i	0.429726	−	0.902959i	$-0.358610\pi$
$643$	−12.0000	−	12.0000i	−0.473234	−	0.473234i	−0.902959	+	0.429726i	$0.858610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.0000	−	19.0000i	−0.743527	−	0.743527i	0.229728	−	0.973255i	$-0.426216\pi$
$653$	−19.0000	−	19.0000i	−0.743527	−	0.743527i	−0.973255	+	0.229728i	$0.926216\pi$
$654$		0			0
$655$	−16.0000			−0.625172
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.00000	−	8.00000i	−0.311636	−	0.311636i	0.533907	−	0.845543i	$-0.320723\pi$
$659$	−8.00000	−	8.00000i	−0.311636	−	0.311636i	−0.845543	+	0.533907i	$0.820723\pi$
$660$		0			0
$661$	7.00000	−	7.00000i	0.272268	−	0.272268i	−0.557744	−	0.830013i	$-0.688333\pi$
$661$	7.00000	−	7.00000i	0.272268	−	0.272268i	0.830013	+	0.557744i	$0.188333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			32.0000i			1.24091i
$666$		0			0
$667$	24.0000	−	24.0000i	0.929284	−	0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.0000			−0.926510
$672$		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
$677$	−9.00000	+	9.00000i	−0.345898	+	0.345898i	−0.858579	−	0.512681i	$-0.828652\pi$
$677$	−9.00000	+	9.00000i	−0.345898	+	0.345898i	0.512681	+	0.858579i	$0.328652\pi$
$678$		0			0
$679$			32.0000i			1.22805i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000	+	12.0000i	−0.459167	+	0.459167i	−0.898382	−	0.439215i	$-0.855257\pi$
$683$	−12.0000	+	12.0000i	−0.459167	+	0.459167i	0.439215	+	0.898382i	$0.355257\pi$
$684$		0			0
$685$	2.00000	+	2.00000i	0.0764161	+	0.0764161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−42.0000			−1.60007
$690$		0			0
$691$	36.0000	+	36.0000i	1.36950	+	1.36950i	0.861145	+	0.508360i	$0.169748\pi$
$691$	36.0000	+	36.0000i	1.36950	+	1.36950i	0.508360	+	0.861145i	$0.330252\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.0000	−	11.0000i	−0.415464	−	0.415464i	0.468173	−	0.883637i	$-0.344913\pi$
$701$	−11.0000	−	11.0000i	−0.415464	−	0.415464i	−0.883637	+	0.468173i	$0.844913\pi$
$702$		0			0
$703$	8.00000			0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.0000	−	28.0000i	−1.05305	−	1.05305i
$708$		0			0
$709$	−33.0000	+	33.0000i	−1.23934	+	1.23934i	−0.279070	+	0.960271i	$0.590026\pi$
$709$	−33.0000	+	33.0000i	−1.23934	+	1.23934i	−0.960271	+	0.279070i	$0.909974\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$	24.0000	−	24.0000i	0.897549	−	0.897549i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.00000	+	9.00000i	−0.334252	+	0.334252i
$726$		0			0
$727$		−	52.0000i		−	1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$		−	52.0000i		−	1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24.0000	−	24.0000i	0.887672	−	0.887672i
$732$		0			0
$733$	37.0000	+	37.0000i	1.36663	+	1.36663i	0.865201	+	0.501425i	$0.167191\pi$
$733$	37.0000	+	37.0000i	1.36663	+	1.36663i	0.501425	+	0.865201i	$0.332809\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−64.0000			−2.35747
$738$		0			0
$739$	−24.0000	−	24.0000i	−0.882854	−	0.882854i	0.110970	−	0.993824i	$-0.464604\pi$
$739$	−24.0000	−	24.0000i	−0.882854	−	0.882854i	−0.993824	+	0.110970i	$0.964604\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$		−	2.00000i		−	0.0732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.0000	−	32.0000i	−1.16925	−	1.16925i
$750$		0			0
$751$	−28.0000			−1.02173			−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000			−1.02173			−0.510867	+	0.859660i	$0.670676\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.00000	+	4.00000i	0.145575	+	0.145575i
$756$		0			0
$757$	−7.00000	+	7.00000i	−0.254419	+	0.254419i	−0.822780	−	0.568360i	$-0.807578\pi$
$757$	−7.00000	+	7.00000i	−0.254419	+	0.254419i	0.568360	+	0.822780i	$0.307578\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	18.0000i		−	0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$		−	18.0000i		−	0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$		0			0
$763$	−20.0000	+	20.0000i	−0.724049	+	0.724049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.00000	−	9.00000i	0.323708	−	0.323708i	−0.526480	−	0.850188i	$-0.676489\pi$
$773$	9.00000	−	9.00000i	0.323708	−	0.323708i	0.850188	+	0.526480i	$0.176489\pi$
$774$		0			0
$775$		−	12.0000i		−	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.00000	−	8.00000i	0.286630	−	0.286630i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.00000			−0.214149
$786$		0			0
$787$	−12.0000	−	12.0000i	−0.427754	−	0.427754i	0.460109	−	0.887863i	$-0.347810\pi$
$787$	−12.0000	−	12.0000i	−0.427754	−	0.427754i	−0.887863	+	0.460109i	$0.847810\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			64.0000i			2.27558i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5.00000	−	5.00000i	−0.177109	−	0.177109i	0.612985	−	0.790094i	$-0.289968\pi$
$797$	−5.00000	−	5.00000i	−0.177109	−	0.177109i	−0.790094	+	0.612985i	$0.789968\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$		0			0
$802$		0			0
$803$	40.0000	+	40.0000i	1.41157	+	1.41157i
$804$		0			0
$805$	32.0000	−	32.0000i	1.12785	−	1.12785i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			2.00000i			0.0703163i	0.999382	+	0.0351581i	$0.0111935\pi$
$809$			2.00000i			0.0703163i	−0.999382	+	0.0351581i	$0.988807\pi$
$810$		0			0
$811$	−4.00000	+	4.00000i	−0.140459	+	0.140459i	−0.773840	−	0.633381i	$-0.781667\pi$
$811$	−4.00000	+	4.00000i	−0.140459	+	0.140459i	0.633381	+	0.773840i	$0.281667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.0000			0.840683
$816$		0			0
$817$	32.0000			1.11954
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−39.0000	+	39.0000i	−1.36111	+	1.36111i	−0.488603	+	0.872506i	$0.662493\pi$
$821$	−39.0000	+	39.0000i	−1.36111	+	1.36111i	−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$		−	44.0000i		−	1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$		−	44.0000i		−	1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	32.0000	−	32.0000i	1.11275	−	1.11275i	0.119972	−	0.992777i	$-0.461720\pi$
$827$	32.0000	−	32.0000i	1.11275	−	1.11275i	0.992777	−	0.119972i	$-0.0382804\pi$
$828$		0			0
$829$	−29.0000	−	29.0000i	−1.00721	−	1.00721i	−0.999974	−	0.00723783i	$-0.997696\pi$
$829$	−29.0000	−	29.0000i	−1.00721	−	1.00721i	−0.00723783	−	0.999974i	$-0.502304\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−54.0000			−1.87099
$834$		0			0
$835$	8.00000	+	8.00000i	0.276851	+	0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	48.0000i		−	1.65714i	−0.559883	−	0.828572i	$-0.689154\pi$
$839$		−	48.0000i		−	1.65714i	0.559883	−	0.828572i	$-0.310846\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.00000	+	5.00000i	0.172005	+	0.172005i
$846$		0			0
$847$	84.0000			2.88627
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000	−	8.00000i	−0.274236	−	0.274236i
$852$		0			0
$853$	1.00000	−	1.00000i	0.0342393	−	0.0342393i	−0.689780	−	0.724019i	$-0.742293\pi$
$853$	1.00000	−	1.00000i	0.0342393	−	0.0342393i	0.724019	+	0.689780i	$0.242293\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$	12.0000	−	12.0000i	0.409435	−	0.409435i	−0.472107	−	0.881541i	$-0.656506\pi$
$859$	12.0000	−	12.0000i	0.409435	−	0.409435i	0.881541	+	0.472107i	$0.156506\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	22.0000			0.748022
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.0000	−	48.0000i	1.62829	−	1.62829i
$870$		0			0
$871$		−	48.0000i		−	1.62642i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−32.0000	+	32.0000i	−1.08180	+	1.08180i
$876$		0			0
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	0.617630	−	0.786468i	$-0.288093\pi$
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	−0.786468	+	0.617630i	$0.788093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−4.00000	−	4.00000i	−0.134611	−	0.134611i	0.636591	−	0.771202i	$-0.280344\pi$
$883$	−4.00000	−	4.00000i	−0.134611	−	0.134611i	−0.771202	+	0.636591i	$0.780344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$		0			0
$889$		−	16.0000i		−	0.536623i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.0000	−	32.0000i	−1.07084	−	1.07084i
$894$		0			0
$895$	16.0000			0.534821
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	−	12.0000i	−0.400222	−	0.400222i
$900$		0			0
$901$	−42.0000	+	42.0000i	−1.39922	+	1.39922i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	18.0000i		−	0.598340i
$906$		0			0
$907$	−28.0000	+	28.0000i	−0.929725	+	0.929725i	−0.997688	−	0.0679631i	$-0.978350\pi$
$907$	−28.0000	+	28.0000i	−0.929725	+	0.929725i	0.0679631	+	0.997688i	$0.478350\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.0000			−1.32526			−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000			−1.32526			−0.662630	+	0.748947i	$0.730560\pi$
$912$		0			0
$913$	−32.0000			−1.05905
$914$		0			0
$915$		0			0
$916$		0			0
$917$	32.0000	−	32.0000i	1.05673	−	1.05673i
$918$		0			0
$919$			20.0000i			0.659739i	0.944027	+	0.329870i	$0.107005\pi$
$919$			20.0000i			0.659739i	−0.944027	+	0.329870i	$0.892995\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.00000	+	3.00000i	0.0986394	+	0.0986394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−42.0000			−1.37798			−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000			−1.37798			−0.688988	+	0.724773i	$0.741945\pi$
$930$		0			0
$931$	−36.0000	−	36.0000i	−1.17985	−	1.17985i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	48.0000i		−	1.56977i
$936$		0			0
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.0000	−	35.0000i	−1.14097	−	1.14097i	−0.988274	−	0.152694i	$-0.951205\pi$
$941$	−35.0000	−	35.0000i	−1.14097	−	1.14097i	−0.152694	−	0.988274i	$-0.548795\pi$
$942$		0			0
$943$	−16.0000			−0.521032
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.0000	+	40.0000i	1.29983	+	1.29983i	0.928505	+	0.371321i	$0.121095\pi$
$947$	40.0000	+	40.0000i	1.29983	+	1.29983i	0.371321	+	0.928505i	$0.378905\pi$
$948$		0			0
$949$	−30.0000	+	30.0000i	−0.973841	+	0.973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$	−24.0000	+	24.0000i	−0.776622	+	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000			−0.258333
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.00000	−	8.00000i	0.257529	−	0.257529i
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.0000	−	28.0000i	0.898563	−	0.898563i	−0.0967463	−	0.995309i	$-0.530844\pi$
$971$	28.0000	−	28.0000i	0.898563	−	0.898563i	0.995309	+	0.0967463i	$0.0308436\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.0000			−1.21573			−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000			−1.21573			−0.607864	+	0.794041i	$0.707973\pi$
$978$		0			0
$979$	−64.0000	−	64.0000i	−2.04545	−	2.04545i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	32.0000i		−	1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$		−	32.0000i		−	1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$		0			0
$985$			2.00000i			0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000	−	32.0000i	−1.01754	−	1.01754i
$990$		0			0
$991$	44.0000			1.39771			0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000			1.39771			0.698853	+	0.715265i	$0.253694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.00000	+	4.00000i	0.126809	+	0.126809i
$996$		0			0
$997$	33.0000	−	33.0000i	1.04512	−	1.04512i	0.0461877	−	0.998933i	$-0.485293\pi$
$997$	33.0000	−	33.0000i	1.04512	−	1.04512i	0.998933	−	0.0461877i	$-0.0147072\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.u.1153.1	yes	2
3.2	odd	2		4608.2.k.e.1153.1	yes	2
4.3	odd	2		4608.2.k.n.1153.1	yes	2
8.3	odd	2		4608.2.k.k.1153.1	yes	2
8.5	even	2		4608.2.k.d.1153.1	✓	2
12.11	even	2		4608.2.k.l.1153.1	yes	2
16.3	odd	4		4608.2.k.k.3457.1	yes	2
16.5	even	4	inner	4608.2.k.u.3457.1	yes	2
16.11	odd	4		4608.2.k.n.3457.1	yes	2
16.13	even	4		4608.2.k.d.3457.1	yes	2
24.5	odd	2		4608.2.k.t.1153.1	yes	2
24.11	even	2		4608.2.k.m.1153.1	yes	2
32.5	even	8		9216.2.a.a.1.2		2
32.11	odd	8		9216.2.a.t.1.1		2
32.21	even	8		9216.2.a.a.1.1		2
32.27	odd	8		9216.2.a.t.1.2		2
48.5	odd	4		4608.2.k.e.3457.1	yes	2
48.11	even	4		4608.2.k.l.3457.1	yes	2
48.29	odd	4		4608.2.k.t.3457.1	yes	2
48.35	even	4		4608.2.k.m.3457.1	yes	2
96.5	odd	8		9216.2.a.c.1.1		2
96.11	even	8		9216.2.a.v.1.2		2
96.53	odd	8		9216.2.a.c.1.2		2
96.59	even	8		9216.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.d.1153.1	✓	2	8.5	even	2
4608.2.k.d.3457.1	yes	2	16.13	even	4
4608.2.k.e.1153.1	yes	2	3.2	odd	2
4608.2.k.e.3457.1	yes	2	48.5	odd	4
4608.2.k.k.1153.1	yes	2	8.3	odd	2
4608.2.k.k.3457.1	yes	2	16.3	odd	4
4608.2.k.l.1153.1	yes	2	12.11	even	2
4608.2.k.l.3457.1	yes	2	48.11	even	4
4608.2.k.m.1153.1	yes	2	24.11	even	2
4608.2.k.m.3457.1	yes	2	48.35	even	4
4608.2.k.n.1153.1	yes	2	4.3	odd	2
4608.2.k.n.3457.1	yes	2	16.11	odd	4
4608.2.k.t.1153.1	yes	2	24.5	odd	2
4608.2.k.t.3457.1	yes	2	48.29	odd	4
4608.2.k.u.1153.1	yes	2	1.1	even	1	trivial
4608.2.k.u.3457.1	yes	2	16.5	even	4	inner
9216.2.a.a.1.1		2	32.21	even	8
9216.2.a.a.1.2		2	32.5	even	8
9216.2.a.c.1.1		2	96.5	odd	8
9216.2.a.c.1.2		2	96.53	odd	8
9216.2.a.t.1.1		2	32.11	odd	8
9216.2.a.t.1.2		2	32.27	odd	8
9216.2.a.v.1.1		2	96.59	even	8
9216.2.a.v.1.2		2	96.11	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)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{5} +4.00000i q^{7} +(4.00000 - 4.00000i) q^{11} +(3.00000 + 3.00000i) q^{13} +6.00000 q^{17} +(4.00000 + 4.00000i) q^{19} -8.00000i q^{23} +3.00000i q^{25} +(3.00000 + 3.00000i) q^{29} -4.00000 q^{31} +(4.00000 + 4.00000i) q^{35} +(1.00000 - 1.00000i) q^{37} -2.00000i q^{41} +(4.00000 - 4.00000i) q^{43} -8.00000 q^{47} -9.00000 q^{49} +(-7.00000 + 7.00000i) q^{53} -8.00000i q^{55} +(-3.00000 - 3.00000i) q^{61} +6.00000 q^{65} +(-8.00000 - 8.00000i) q^{67} +10.0000i q^{73} +(16.0000 + 16.0000i) q^{77} +12.0000 q^{79} +(-4.00000 - 4.00000i) q^{83} +(6.00000 - 6.00000i) q^{85} -16.0000i q^{89} +(-12.0000 + 12.0000i) q^{91} +8.00000 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 8 q^{11} + 6 q^{13} + 12 q^{17} + 8 q^{19} + 6 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{37} + 8 q^{43} - 16 q^{47} - 18 q^{49} - 14 q^{53} - 6 q^{61} + 12 q^{65} - 16 q^{67} + 32 q^{77} + 24 q^{79}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 8 * q^11 + 6 * q^13 + 12 * q^17 + 8 * q^19 + 6 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^37 + 8 * q^43 - 16 * q^47 - 18 * q^49 - 14 * q^53 - 6 * q^61 + 12 * q^65 - 16 * q^67 + 32 * q^77 + 24 * q^79 - 8 * q^83 + 12 * q^85 - 24 * q^91 + 16 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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