LMFDB - Embedded newform 2496.2.c.f.961.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(961,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.961"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,6,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$19.9306603445$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2496.961
Dual form			2496.2.c.f.961.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} -2.00000 q^{17} +6.00000i q^{19} +2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +10.0000i q^{31} -4.00000 q^{35} -8.00000i q^{37} +(-3.00000 - 2.00000i) q^{39} +10.0000i q^{41} -4.00000 q^{43} -2.00000i q^{45} -12.0000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -6.00000i q^{57} -4.00000i q^{59} -2.00000 q^{61} -2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +2.00000i q^{67} -4.00000 q^{69} -4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} +4.00000i q^{83} +4.00000i q^{85} -10.0000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} -10.0000i q^{93} +12.0000 q^{95} -12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 20 q^{29} - 8 q^{35} - 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 8 q^{65} - 8 q^{69} - 2 q^{75}+ \cdots + 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 20 * q^29 - 8 * q^35 - 6 * q^39 - 8 * q^43 + 6 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^61 + 8 * q^65 - 8 * q^69 - 2 * q^75 + 2 * q^81 - 20 * q^87 + 8 * q^91 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$			2.00000i			0.516398i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$		0			0
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$	4.00000	−	6.00000i	0.496139	−	0.744208i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$	−10.0000			−1.07211
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	4.00000	−	6.00000i	0.419314	−	0.628971i
$92$		0			0
$93$		−	10.0000i		−	1.03695i
$94$		0			0
$95$	12.0000			1.23117
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$	4.00000			0.390360
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$		0			0
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$		0			0
$119$			4.00000i			0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		−	10.0000i		−	0.901670i
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$		0			0
$135$			2.00000i			0.172133i
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$			12.0000i			1.01058i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	20.0000i		−	1.66091i
$146$		0			0
$147$	−3.00000			−0.247436
$148$		0			0
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	20.0000			1.60644
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$		−	8.00000i		−	0.630488i
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$			6.00000i			0.458831i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	−16.0000			−1.17634
$186$		0			0
$187$		0			0
$188$		0			0
$189$			2.00000i			0.145479i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−4.00000	+	6.00000i	−0.286446	+	0.429669i
$196$		0			0
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$		0			0
$203$		−	20.0000i		−	1.40372i
$204$		0			0
$205$	20.0000			1.39686
$206$		0			0
$207$	4.00000			0.278019
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$	20.0000			1.35769
$218$		0			0
$219$			4.00000i			0.270295i
$220$		0			0
$221$	−6.00000	−	4.00000i	−0.403604	−	0.269069i
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	−12.0000	+	18.0000i	−0.763542	+	1.14531i
$248$		0			0
$249$		−	4.00000i		−	0.253490i
$250$		0			0
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	4.00000i		−	0.250490i
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$	10.0000			0.618984
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		0			0
$267$			6.00000i			0.367194i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$		−	10.0000i		−	0.607457i	−0.952759	−	0.303728i	$-0.901768\pi$
$271$		−	10.0000i		−	0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$		0			0
$273$	−4.00000	+	6.00000i	−0.242091	+	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$			10.0000i			0.598684i
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$	−12.0000			−0.710819
$286$		0			0
$287$	20.0000			1.18056
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			12.0000i			0.703452i
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	+	8.00000i	0.693978	+	0.462652i
$300$		0			0
$301$			8.00000i			0.461112i
$302$		0			0
$303$	2.00000			0.114897
$304$		0			0
$305$			4.00000i			0.229039i
$306$		0			0
$307$			2.00000i			0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$	−28.0000			−1.58773			−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000			−1.58773			−0.793867	+	0.608091i	$0.791935\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$	−4.00000			−0.225374
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	3.00000	+	2.00000i	0.166410	+	0.110940i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$	−24.0000			−1.32316
$330$		0			0
$331$			10.0000i			0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$			10.0000i			0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$		0			0
$333$		−	8.00000i		−	0.438397i
$334$		0			0
$335$	4.00000			0.218543
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	−14.0000			−0.760376
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$			8.00000i			0.430706i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$		0			0
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	4.00000i		−	0.211702i
$358$		0			0
$359$			4.00000i			0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	−8.00000			−0.418739
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$			10.0000i			0.520579i
$370$		0			0
$371$		−	12.0000i		−	0.623009i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$		0			0
$377$	30.0000	+	20.0000i	1.54508	+	1.03005i
$378$		0			0
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$	−8.00000			−0.403547
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		0			0
$403$	−20.0000	+	30.0000i	−0.996271	+	1.49441i
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		0			0
$411$			2.00000i			0.0986527i
$412$		0			0
$413$	−8.00000			−0.393654
$414$		0			0
$415$	8.00000			0.392705
$416$		0			0
$417$	−20.0000			−0.979404
$418$		0			0
$419$	−40.0000			−1.95413			−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000			−1.95413			−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$		0			0
$423$		−	12.0000i		−	0.583460i
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$			4.00000i			0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$			20.0000i			0.958927i
$436$		0			0
$437$			24.0000i			1.14808i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	16.0000			0.760183			0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000			0.760183			0.380091	+	0.924949i	$0.375893\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$			14.0000i			0.662177i
$448$		0			0
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	10.0000i		−	0.469841i
$454$		0			0
$455$	−12.0000	−	8.00000i	−0.562569	−	0.375046i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$	−20.0000			−0.927478
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$		0			0
$474$		0			0
$475$			6.00000i			0.275299i
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$		0			0
$481$	16.0000	−	24.0000i	0.729537	−	1.09431i
$482$		0			0
$483$			8.00000i			0.364013i
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		0			0
$489$		−	14.0000i		−	0.633102i
$490$		0			0
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$	−20.0000			−0.900755
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$			4.00000i			0.177998i
$506$		0			0
$507$	−5.00000	−	12.0000i	−0.222058	−	0.532939i
$508$		0			0
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$			32.0000i			1.41009i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$			2.00000i			0.0872872i
$526$		0			0
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	−20.0000	+	30.0000i	−0.866296	+	1.29944i
$534$		0			0
$535$			16.0000i			0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$	22.0000			0.944110
$544$		0			0
$545$	−8.00000			−0.342682
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$			60.0000i			2.55609i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	16.0000			0.679162
$556$		0			0
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	−12.0000	−	8.00000i	−0.507546	−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.0000			0.674320			0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000			0.674320			0.337160	+	0.941447i	$0.390534\pi$
$564$		0			0
$565$		−	28.0000i		−	1.17797i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$	−12.0000			−0.501307
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		0			0
$579$		−	16.0000i		−	0.664937i
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$		0			0
$584$		0			0
$585$	4.00000	−	6.00000i	0.165380	−	0.248069i
$586$		0			0
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	−60.0000			−2.47226
$590$		0			0
$591$		−	22.0000i		−	0.904959i
$592$		0			0
$593$			26.0000i			1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$			26.0000i			1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$		0			0
$595$	8.00000			0.327968
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$		−	22.0000i		−	0.894427i
$606$		0			0
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$		0			0
$609$			20.0000i			0.810441i
$610$		0			0
$611$	24.0000	−	36.0000i	0.970936	−	1.45640i
$612$		0			0
$613$		−	16.0000i		−	0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$		0			0
$615$	−20.0000			−0.806478
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		0			0
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$	−4.00000			−0.160514
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			16.0000i			0.637962i
$630$		0			0
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$	12.0000			0.476957
$634$		0			0
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$	9.00000	+	6.00000i	0.356593	+	0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$		−	6.00000i		−	0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$		−	6.00000i		−	0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$		0			0
$645$		−	8.00000i		−	0.315000i
$646$		0			0
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−20.0000			−0.783862
$652$		0			0
$653$	26.0000			1.01746			0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000			1.01746			0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$		−	16.0000i		−	0.625172i
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$			40.0000i			1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$			40.0000i			1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$		0			0
$663$	6.00000	+	4.00000i	0.233021	+	0.155347i
$664$		0			0
$665$		−	24.0000i		−	0.930680i
$666$		0			0
$667$	40.0000			1.54881
$668$		0			0
$669$			14.0000i			0.541271i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−4.00000			−0.152832
$686$		0			0
$687$			4.00000i			0.152610i
$688$		0			0
$689$	18.0000	+	12.0000i	0.685745	+	0.457164i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	40.0000i		−	1.51729i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$	48.0000			1.81035
$704$		0			0
$705$	24.0000			0.903892
$706$		0			0
$707$			4.00000i			0.150435i
$708$		0			0
$709$			36.0000i			1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$			36.0000i			1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			40.0000i			1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$		0			0
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$			32.0000i			1.19174i
$722$		0			0
$723$		−	20.0000i		−	0.743808i
$724$		0			0
$725$	10.0000			0.371391
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	8.00000			0.295891
$732$		0			0
$733$			44.0000i			1.62518i	0.582838	+	0.812589i	$0.301942\pi$
$733$			44.0000i			1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$		0			0
$735$			6.00000i			0.221313i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			26.0000i			0.956425i	0.878244	+	0.478213i	$0.158715\pi$
$739$			26.0000i			0.956425i	−0.878244	+	0.478213i	$0.841285\pi$
$740$		0			0
$741$	12.0000	−	18.0000i	0.440831	−	0.661247i
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−28.0000			−1.02584
$746$		0			0
$747$			4.00000i			0.146352i
$748$		0			0
$749$			16.0000i			0.584627i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$	−28.0000			−1.02038
$754$		0			0
$755$	20.0000			0.727875
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−8.00000			−0.289619
$764$		0			0
$765$			4.00000i			0.144620i
$766$		0			0
$767$	8.00000	−	12.0000i	0.288863	−	0.433295i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$		0			0
$777$	16.0000			0.573997
$778$		0			0
$779$	−60.0000			−2.14972
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−10.0000			−0.357371
$784$		0			0
$785$		−	4.00000i		−	0.142766i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$		−	28.0000i		−	0.995565i
$792$		0			0
$793$	−6.00000	−	4.00000i	−0.213066	−	0.142044i
$794$		0			0
$795$			12.0000i			0.425596i
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		0			0
$801$		−	6.00000i		−	0.212000i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−16.0000			−0.563926
$806$		0			0
$807$	−10.0000			−0.352017
$808$		0			0
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$		0			0
$811$		−	10.0000i		−	0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$		−	10.0000i		−	0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$		0			0
$813$			10.0000i			0.350715i
$814$		0			0
$815$	28.0000			0.980797
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$	4.00000	−	6.00000i	0.139771	−	0.209657i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	24.0000			0.836587			0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000			0.836587			0.418294	+	0.908312i	$0.362628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$	−2.00000			−0.0693792
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$	−24.0000			−0.830554
$836$		0			0
$837$		−	10.0000i		−	0.345651i
$838$		0			0
$839$		−	16.0000i		−	0.552381i	−0.961103	−	0.276191i	$-0.910928\pi$
$839$		−	16.0000i		−	0.552381i	0.961103	−	0.276191i	$-0.0890721\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$			10.0000i			0.344418i
$844$		0			0
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$		0			0
$847$		−	22.0000i		−	0.755929i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		−	32.0000i		−	1.09695i
$852$		0			0
$853$		−	56.0000i		−	1.91740i	−0.284413	−	0.958702i	$-0.591799\pi$
$853$		−	56.0000i		−	1.91740i	0.284413	−	0.958702i	$-0.408201\pi$
$854$		0			0
$855$	12.0000			0.410391
$856$		0			0
$857$	−22.0000			−0.751506			−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000			−0.751506			−0.375753	+	0.926720i	$0.622616\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$	−20.0000			−0.681598
$862$		0			0
$863$		−	44.0000i		−	1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$		−	44.0000i		−	1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$		0			0
$865$		−	12.0000i		−	0.408012i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−4.00000	+	6.00000i	−0.135535	+	0.203302i
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$	−24.0000			−0.811348
$876$		0			0
$877$		−	8.00000i		−	0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$		−	8.00000i		−	0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$		0			0
$879$		−	14.0000i		−	0.472208i
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	36.0000			1.21150			0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000			1.21150			0.605748	+	0.795656i	$0.292874\pi$
$884$		0			0
$885$	8.00000			0.268917
$886$		0			0
$887$	−12.0000			−0.402921			−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000			−0.402921			−0.201460	+	0.979497i	$0.564569\pi$
$888$		0			0
$889$		−	16.0000i		−	0.536623i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	72.0000			2.40939
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−12.0000	−	8.00000i	−0.400668	−	0.267112i
$898$		0			0
$899$			100.000i			3.33519i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$			44.0000i			1.46261i
$906$		0			0
$907$	−28.0000			−0.929725			−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000			−0.929725			−0.464862	+	0.885383i	$0.653896\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	4.00000i		−	0.132236i
$916$		0			0
$917$		−	16.0000i		−	0.528367i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		−	2.00000i		−	0.0659022i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	8.00000i		−	0.263038i
$926$		0			0
$927$	−16.0000			−0.525509
$928$		0			0
$929$		−	6.00000i		−	0.196854i	−0.995144	−	0.0984268i	$-0.968619\pi$
$929$		−	6.00000i		−	0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$		0			0
$931$			18.0000i			0.589926i
$932$		0			0
$933$	28.0000			0.916679
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	26.0000			0.848478
$940$		0			0
$941$			10.0000i			0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$			10.0000i			0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		0			0
$945$	4.00000			0.130120
$946$		0			0
$947$			52.0000i			1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$			52.0000i			1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$		0			0
$949$	8.00000	−	12.0000i	0.259691	−	0.389536i
$950$		0			0
$951$			18.0000i			0.583690i
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		−	24.0000i		−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.00000			−0.129167
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$	−8.00000			−0.257796
$964$		0			0
$965$	32.0000			1.03012
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		0			0
$969$			12.0000i			0.385496i
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		−	40.0000i		−	1.28234i
$974$		0			0
$975$	−3.00000	−	2.00000i	−0.0960769	−	0.0640513i
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	44.0000			1.40196
$986$		0			0
$987$	24.0000			0.763928
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	42.0000			1.33015			0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000			1.33015			0.665077	+	0.746775i	$0.268399\pi$
$998$		0			0
$999$			8.00000i			0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.c.f.961.1		2
4.3	odd	2		2496.2.c.m.961.1		2
8.3	odd	2		624.2.c.a.337.2		2
8.5	even	2		78.2.b.a.25.2	yes	2
13.12	even	2	inner	2496.2.c.f.961.2		2
24.5	odd	2		234.2.b.a.181.1		2
24.11	even	2		1872.2.c.b.1585.1		2
40.13	odd	4		1950.2.f.g.649.2		2
40.29	even	2		1950.2.b.c.1351.1		2
40.37	odd	4		1950.2.f.d.649.1		2
52.51	odd	2		2496.2.c.m.961.2		2
56.13	odd	2		3822.2.c.d.883.2		2
104.5	odd	4		1014.2.a.g.1.1		1
104.21	odd	4		1014.2.a.b.1.1		1
104.29	even	6		1014.2.i.c.823.1		4
104.37	odd	12		1014.2.e.e.529.1		2
104.45	odd	12		1014.2.e.b.991.1		2
104.51	odd	2		624.2.c.a.337.1		2
104.61	even	6		1014.2.i.c.361.2		4
104.69	even	6		1014.2.i.c.361.1		4
104.77	even	2		78.2.b.a.25.1	✓	2
104.83	even	4		8112.2.a.j.1.1		1
104.85	odd	12		1014.2.e.e.991.1		2
104.93	odd	12		1014.2.e.b.529.1		2
104.99	even	4		8112.2.a.g.1.1		1
104.101	even	6		1014.2.i.c.823.2		4
312.5	even	4		3042.2.a.c.1.1		1
312.77	odd	2		234.2.b.a.181.2		2
312.125	even	4		3042.2.a.n.1.1		1
312.155	even	2		1872.2.c.b.1585.2		2
520.77	odd	4		1950.2.f.g.649.1		2
520.389	even	2		1950.2.b.c.1351.2		2
520.493	odd	4		1950.2.f.d.649.2		2
728.181	odd	2		3822.2.c.d.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	104.77	even	2
78.2.b.a.25.2	yes	2	8.5	even	2
234.2.b.a.181.1		2	24.5	odd	2
234.2.b.a.181.2		2	312.77	odd	2
624.2.c.a.337.1		2	104.51	odd	2
624.2.c.a.337.2		2	8.3	odd	2
1014.2.a.b.1.1		1	104.21	odd	4
1014.2.a.g.1.1		1	104.5	odd	4
1014.2.e.b.529.1		2	104.93	odd	12
1014.2.e.b.991.1		2	104.45	odd	12
1014.2.e.e.529.1		2	104.37	odd	12
1014.2.e.e.991.1		2	104.85	odd	12
1014.2.i.c.361.1		4	104.69	even	6
1014.2.i.c.361.2		4	104.61	even	6
1014.2.i.c.823.1		4	104.29	even	6
1014.2.i.c.823.2		4	104.101	even	6
1872.2.c.b.1585.1		2	24.11	even	2
1872.2.c.b.1585.2		2	312.155	even	2
1950.2.b.c.1351.1		2	40.29	even	2
1950.2.b.c.1351.2		2	520.389	even	2
1950.2.f.d.649.1		2	40.37	odd	4
1950.2.f.d.649.2		2	520.493	odd	4
1950.2.f.g.649.1		2	520.77	odd	4
1950.2.f.g.649.2		2	40.13	odd	4
2496.2.c.f.961.1		2	1.1	even	1	trivial
2496.2.c.f.961.2		2	13.12	even	2	inner
2496.2.c.m.961.1		2	4.3	odd	2
2496.2.c.m.961.2		2	52.51	odd	2
3042.2.a.c.1.1		1	312.5	even	4
3042.2.a.n.1.1		1	312.125	even	4
3822.2.c.d.883.1		2	728.181	odd	2
3822.2.c.d.883.2		2	56.13	odd	2
8112.2.a.g.1.1		1	104.99	even	4
8112.2.a.j.1.1		1	104.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} -2.00000 q^{17} +6.00000i q^{19} +2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +10.0000i q^{31} -4.00000 q^{35} -8.00000i q^{37} +(-3.00000 - 2.00000i) q^{39} +10.0000i q^{41} -4.00000 q^{43} -2.00000i q^{45} -12.0000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -6.00000i q^{57} -4.00000i q^{59} -2.00000 q^{61} -2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +2.00000i q^{67} -4.00000 q^{69} -4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} +4.00000i q^{83} +4.00000i q^{85} -10.0000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} -10.0000i q^{93} +12.0000 q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 20 q^{29} - 8 q^{35} - 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 8 q^{65} - 8 q^{69} - 2 q^{75}+ \cdots + 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 20 * q^29 - 8 * q^35 - 6 * q^39 - 8 * q^43 + 6 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^61 + 8 * q^65 - 8 * q^69 - 2 * q^75 + 2 * q^81 - 20 * q^87 + 8 * q^91 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more