LMFDB - Embedded newform 4368.2.h.f.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not 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:	$34.8786556029$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.f.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +(2.00000 - 3.00000i) q^{13} +2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{35} +2.00000i q^{37} +(-2.00000 + 3.00000i) q^{39} -4.00000 q^{43} -2.00000i q^{45} +8.00000i q^{47} -1.00000 q^{49} +2.00000 q^{51} +4.00000 q^{53} -4.00000i q^{57} -6.00000i q^{59} +12.0000 q^{61} +1.00000i q^{63} +(-6.00000 - 4.00000i) q^{65} -2.00000i q^{67} -6.00000 q^{69} +14.0000i q^{73} -1.00000 q^{75} +1.00000 q^{81} -14.0000i q^{83} +4.00000i q^{85} -4.00000i q^{89} +(3.00000 + 2.00000i) q^{91} +8.00000 q^{95} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 12 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{35} - 4 q^{39} - 8 q^{43} - 2 q^{49} + 4 q^{51} + 8 q^{53} + 24 q^{61} - 12 q^{65} - 12 q^{69} - 2 q^{75} + 2 q^{81} + 6 q^{91} + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 12 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^35 - 4 * q^39 - 8 * q^43 - 2 * q^49 + 4 * q^51 + 8 * q^53 + 24 * q^61 - 12 * q^65 - 12 * q^69 - 2 * q^75 + 2 * q^81 + 6 * q^91 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		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
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		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
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	−2.00000	+	3.00000i	−0.320256	+	0.480384i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		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
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	12.0000			1.53644			0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000			1.53644			0.768221	+	0.640184i	$0.221142\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−6.00000	−	4.00000i	−0.744208	−	0.496139i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		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
$75$	−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
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$		0			0
$91$	3.00000	+	2.00000i	0.314485	+	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$	−2.00000			−0.195180
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		−	2.00000i		−	0.189832i
$112$		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
$115$		−	12.0000i		−	1.11901i
$116$		0			0
$117$	2.00000	−	3.00000i	0.184900	−	0.277350i
$118$		0			0
$119$		−	2.00000i		−	0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$			2.00000i			0.172133i
$136$		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
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$	−4.00000			−0.317221
$160$		0			0
$161$			6.00000i			0.472866i
$162$		0			0
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$		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
$171$			4.00000i			0.305888i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	12.0000			0.891953			0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000			0.891953			0.445976	+	0.895045i	$0.352856\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$	4.00000			0.294086
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	−2.00000			−0.144715			−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000			−0.144715			−0.0723575	+	0.997379i	$0.523052\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$	6.00000	+	4.00000i	0.429669	+	0.286446i
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			2.00000i			0.141069i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	6.00000			0.417029
$208$		0			0
$209$		0			0
$210$		0			0
$211$	28.0000			1.92760			0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000			1.92760			0.963800	+	0.266627i	$0.0859092\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	14.0000i		−	0.946032i
$220$		0			0
$221$	−4.00000	+	6.00000i	−0.269069	+	0.403604i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	22.0000i		−	1.46019i	−0.683345	−	0.730096i	$-0.739475\pi$
$227$		−	22.0000i		−	1.46019i	0.683345	−	0.730096i	$-0.260525\pi$
$228$		0			0
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		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
$235$	16.0000			1.04372
$236$		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
$241$		−	10.0000i		−	0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$		−	10.0000i		−	0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			2.00000i			0.127775i
$246$		0			0
$247$	12.0000	+	8.00000i	0.763542	+	0.509028i
$248$		0			0
$249$			14.0000i			0.887214i
$250$		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
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$		−	8.00000i		−	0.491436i
$266$		0			0
$267$			4.00000i			0.244796i
$268$		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
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−3.00000	−	2.00000i	−0.181568	−	0.121046i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$		0			0
$283$	16.0000			0.951101			0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000			0.951101			0.475551	+	0.879688i	$0.342249\pi$
$284$		0			0
$285$	−8.00000			−0.473879
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	2.00000i		−	0.117242i
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	18.0000i	0.693978	−	1.04097i
$300$		0			0
$301$		−	4.00000i		−	0.230556i
$302$		0			0
$303$	−2.00000			−0.114897
$304$		0			0
$305$		−	24.0000i		−	1.37424i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$		0			0
$315$	2.00000			0.112687
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	2.00000	−	3.00000i	0.110940	−	0.166410i
$326$		0			0
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$		0			0
$333$			2.00000i			0.109599i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	−14.0000			−0.760376
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			12.0000i			0.646058i
$346$		0			0
$347$	32.0000			1.71785			0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000			1.71785			0.858925	+	0.512101i	$0.171133\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	−2.00000	+	3.00000i	−0.106752	+	0.160128i
$352$		0			0
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			2.00000i			0.105851i
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	28.0000			1.46559
$366$		0			0
$367$	22.0000			1.14839			0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000			1.14839			0.574195	+	0.818718i	$0.305315\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			4.00000i			0.207670i
$372$		0			0
$373$	34.0000			1.76045			0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000			1.76045			0.880227	+	0.474554i	$0.157390\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$		0			0
$399$	4.00000			0.200250
$400$		0			0
$401$			30.0000i			1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$			30.0000i			1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	14.0000i		−	0.692255i	−0.938187	−	0.346128i	$-0.887496\pi$
$409$		−	14.0000i		−	0.692255i	0.938187	−	0.346128i	$-0.112504\pi$
$410$		0			0
$411$		−	2.00000i		−	0.0986527i
$412$		0			0
$413$	6.00000			0.295241
$414$		0			0
$415$	−28.0000			−1.37447
$416$		0			0
$417$	20.0000			0.979404
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$			12.0000i			0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		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
$435$		0			0
$436$		0			0
$437$			24.0000i			1.14808i
$438$		0			0
$439$	−30.0000			−1.43182			−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000			−1.43182			−0.715911	+	0.698192i	$0.753988\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	−8.00000			−0.379236
$446$		0			0
$447$		−	6.00000i		−	0.283790i
$448$		0			0
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.00000	−	6.00000i	0.187523	−	0.281284i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$		−	10.0000i		−	0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$		−	10.0000i		−	0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$		0			0
$466$		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
$469$	2.00000			0.0923514
$470$		0			0
$471$	−8.00000			−0.368621
$472$		0			0
$473$		0			0
$474$		0			0
$475$			4.00000i			0.183533i
$476$		0			0
$477$	4.00000			0.183147
$478$		0			0
$479$		−	16.0000i		−	0.731059i	−0.930800	−	0.365529i	$-0.880888\pi$
$479$		−	16.0000i		−	0.731059i	0.930800	−	0.365529i	$-0.119112\pi$
$480$		0			0
$481$	6.00000	+	4.00000i	0.273576	+	0.182384i
$482$		0			0
$483$		−	6.00000i		−	0.273009i
$484$		0			0
$485$	4.00000			0.181631
$486$		0			0
$487$			28.0000i			1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$			28.0000i			1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$		0			0
$489$			14.0000i			0.633102i
$490$		0			0
$491$	8.00000			0.361035			0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000			0.361035			0.180517	+	0.983572i	$0.442223\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$		−	4.00000i		−	0.177998i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$		0			0
$509$			26.0000i			1.15243i	0.817298	+	0.576215i	$0.195471\pi$
$509$			26.0000i			1.15243i	−0.817298	+	0.576215i	$0.804529\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$			28.0000i			1.23383i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		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
$525$		−	1.00000i		−	0.0436436i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	24.0000i		−	1.03761i
$536$		0			0
$537$	−20.0000			−0.863064
$538$		0			0
$539$		0			0
$540$		0			0
$541$			30.0000i			1.28980i	0.764267	+	0.644900i	$0.223101\pi$
$541$			30.0000i			1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$		0			0
$543$	−12.0000			−0.514969
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$	12.0000			0.513083			0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000			0.513083			0.256541	+	0.966533i	$0.417417\pi$
$548$		0			0
$549$	12.0000			0.512148
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.00000			−0.169791
$556$		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
$559$	−8.00000	+	12.0000i	−0.338364	+	0.507546i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$		−	28.0000i		−	1.17797i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$	2.00000			0.0835512
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		0			0
$579$		−	4.00000i		−	0.166234i
$580$		0			0
$581$	14.0000			0.580818
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−6.00000	−	4.00000i	−0.248069	−	0.165380i
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			18.0000i			0.740421i
$592$		0			0
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	−4.00000			−0.163984
$596$		0			0
$597$	−10.0000			−0.409273
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	−18.0000			−0.734235			−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000			−0.734235			−0.367118	+	0.930175i	$0.619655\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$		−	22.0000i		−	0.894427i
$606$		0			0
$607$	−18.0000			−0.730597			−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000			−0.730597			−0.365299	+	0.930890i	$0.619033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000	+	16.0000i	0.970936	+	0.647291i
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$		−	16.0000i		−	0.643094i	−0.946894	−	0.321547i	$-0.895797\pi$
$619$		−	16.0000i		−	0.643094i	0.946894	−	0.321547i	$-0.104203\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$		0			0
$623$	4.00000			0.160257
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$	−28.0000			−1.11290
$634$		0			0
$635$		−	24.0000i		−	0.952411i
$636$		0			0
$637$	−2.00000	+	3.00000i	−0.0792429	+	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$		−	8.00000i		−	0.315000i
$646$		0			0
$647$	−28.0000			−1.10079			−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000			−1.10079			−0.550397	+	0.834903i	$0.685524\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.00000			0.156532			0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000			0.156532			0.0782660	+	0.996933i	$0.475062\pi$
$654$		0			0
$655$			24.0000i			0.937758i
$656$		0			0
$657$			14.0000i			0.546192i
$658$		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
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$		0			0
$663$	4.00000	−	6.00000i	0.155347	−	0.233021i
$664$		0			0
$665$			8.00000i			0.310227i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	16.0000i		−	0.618596i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	−22.0000			−0.845529			−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000			−0.845529			−0.422764	+	0.906240i	$0.638940\pi$
$678$		0			0
$679$	−2.00000			−0.0767530
$680$		0			0
$681$			22.0000i			0.843042i
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$	4.00000			0.152832
$686$		0			0
$687$			14.0000i			0.534133i
$688$		0			0
$689$	8.00000	−	12.0000i	0.304776	−	0.457164i
$690$		0			0
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			40.0000i			1.51729i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$	−16.0000			−0.602595
$706$		0			0
$707$			2.00000i			0.0752177i
$708$		0			0
$709$			46.0000i			1.72757i	0.503864	+	0.863783i	$0.331911\pi$
$709$			46.0000i			1.72757i	−0.503864	+	0.863783i	$0.668089\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		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	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$		−	14.0000i		−	0.521387i
$722$		0			0
$723$			10.0000i			0.371904i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−38.0000			−1.40934			−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000			−1.40934			−0.704671	+	0.709534i	$0.748905\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	8.00000			0.295891
$732$		0			0
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		0			0
$735$		−	2.00000i		−	0.0737711i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$	−12.0000	−	8.00000i	−0.440831	−	0.293887i
$742$		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
$745$	12.0000			0.439646
$746$		0			0
$747$		−	14.0000i		−	0.512233i
$748$		0			0
$749$			12.0000i			0.438470i
$750$		0			0
$751$	−12.0000			−0.437886			−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000			−0.437886			−0.218943	+	0.975738i	$0.570261\pi$
$752$		0			0
$753$	−28.0000			−1.02038
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	40.0000i		−	1.45000i	−0.688749	−	0.724999i	$-0.741840\pi$
$761$		−	40.0000i		−	1.45000i	0.688749	−	0.724999i	$-0.258160\pi$
$762$		0			0
$763$	−6.00000			−0.217215
$764$		0			0
$765$			4.00000i			0.144620i
$766$		0			0
$767$	−18.0000	−	12.0000i	−0.649942	−	0.433295i
$768$		0			0
$769$			26.0000i			0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$			26.0000i			0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$		−	26.0000i		−	0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.00000			0.0717496
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	16.0000i		−	0.571064i
$786$		0			0
$787$			48.0000i			1.71102i	0.517790	+	0.855508i	$0.326755\pi$
$787$			48.0000i			1.71102i	−0.517790	+	0.855508i	$0.673245\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		0			0
$791$			14.0000i			0.497783i
$792$		0			0
$793$	24.0000	−	36.0000i	0.852265	−	1.27840i
$794$		0			0
$795$			8.00000i			0.283731i
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$		−	16.0000i		−	0.566039i
$800$		0			0
$801$		−	4.00000i		−	0.141333i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	12.0000			0.422944
$806$		0			0
$807$	−10.0000			−0.352017
$808$		0			0
$809$	50.0000			1.75791			0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$		0			0
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−28.0000			−0.980797
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$		0			0
$819$	3.00000	+	2.00000i	0.104828	+	0.0698857i
$820$		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
$823$	−4.00000			−0.139431			−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000			−0.139431			−0.0697156	+	0.997567i	$0.522209\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$		0			0
$829$	−40.0000			−1.38926			−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000			−1.38926			−0.694629	+	0.719368i	$0.744431\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	−24.0000			−0.830554
$836$		0			0
$837$		0			0
$838$		0			0
$839$			4.00000i			0.138095i	0.997613	+	0.0690477i	$0.0219961\pi$
$839$			4.00000i			0.138095i	−0.997613	+	0.0690477i	$0.978004\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$			30.0000i			1.03325i
$844$		0			0
$845$	−24.0000	+	10.0000i	−0.825625	+	0.344010i
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$			12.0000i			0.411355i
$852$		0			0
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$		0			0
$855$	8.00000			0.273594
$856$		0			0
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.0000i			0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$			16.0000i			0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$		0			0
$865$			12.0000i			0.408012i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−6.00000	−	4.00000i	−0.203302	−	0.135535i
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$	12.0000			0.405674
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		0			0
$879$			6.00000i			0.202375i
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$	12.0000			0.403376
$886$		0			0
$887$	−8.00000			−0.268614			−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000			−0.268614			−0.134307	+	0.990940i	$0.542881\pi$
$888$		0			0
$889$			12.0000i			0.402467i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.0000			−1.07084
$894$		0			0
$895$		−	40.0000i		−	1.33705i
$896$		0			0
$897$	−12.0000	+	18.0000i	−0.400668	+	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$			4.00000i			0.133112i
$904$		0			0
$905$		−	24.0000i		−	0.797787i
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$		0			0
$909$	2.00000			0.0663358
$910$		0			0
$911$	−22.0000			−0.728893			−0.364446	−	0.931224i	$-0.618742\pi$
$911$	−22.0000			−0.728893			−0.364446	+	0.931224i	$0.618742\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			24.0000i			0.793416i
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		0			0
$921$		−	28.0000i		−	0.922631i
$922$		0			0
$923$		0			0
$924$		0			0
$925$			2.00000i			0.0657596i
$926$		0			0
$927$	−14.0000			−0.459820
$928$		0			0
$929$		−	24.0000i		−	0.787414i	−0.919236	−	0.393707i	$-0.871192\pi$
$929$		−	24.0000i		−	0.787414i	0.919236	−	0.393707i	$-0.128808\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$		0			0
$933$	12.0000			0.392862
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$	6.00000			0.195803
$940$		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
$943$		0			0
$944$		0			0
$945$	−2.00000			−0.0650600
$946$		0			0
$947$		−	32.0000i		−	1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$		−	32.0000i		−	1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$		0			0
$949$	42.0000	+	28.0000i	1.36338	+	0.908918i
$950$		0			0
$951$		−	2.00000i		−	0.0648544i
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$			4.00000i			0.129437i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.00000			−0.0645834
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	12.0000			0.386695
$964$		0			0
$965$	8.00000			0.257529
$966$		0			0
$967$			28.0000i			0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$			28.0000i			0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$		0			0
$969$			8.00000i			0.256997i
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$		−	20.0000i		−	0.641171i
$974$		0			0
$975$	−2.00000	+	3.00000i	−0.0640513	+	0.0960769i
$976$		0			0
$977$			42.0000i			1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			6.00000i			0.191565i
$982$		0			0
$983$		−	4.00000i		−	0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$		−	4.00000i		−	0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$	8.00000			0.254643
$988$		0			0
$989$	−24.0000			−0.763156
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$			10.0000i			0.317340i
$994$		0			0
$995$		−	20.0000i		−	0.634043i
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$		−	2.00000i		−	0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.f.337.1		2
4.3	odd	2		546.2.c.b.337.1	✓	2
12.11	even	2		1638.2.c.b.883.2		2
13.12	even	2	inner	4368.2.h.f.337.2		2
28.27	even	2		3822.2.c.c.883.1		2
52.31	even	4		7098.2.a.k.1.1		1
52.47	even	4		7098.2.a.bc.1.1		1
52.51	odd	2		546.2.c.b.337.2	yes	2
156.155	even	2		1638.2.c.b.883.1		2
364.363	even	2		3822.2.c.c.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.b.337.1	✓	2	4.3	odd	2
546.2.c.b.337.2	yes	2	52.51	odd	2
1638.2.c.b.883.1		2	156.155	even	2
1638.2.c.b.883.2		2	12.11	even	2
3822.2.c.c.883.1		2	28.27	even	2
3822.2.c.c.883.2		2	364.363	even	2
4368.2.h.f.337.1		2	1.1	even	1	trivial
4368.2.h.f.337.2		2	13.12	even	2	inner
7098.2.a.k.1.1		1	52.31	even	4
7098.2.a.bc.1.1		1	52.47	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +(2.00000 - 3.00000i) q^{13} +2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{35} +2.00000i q^{37} +(-2.00000 + 3.00000i) q^{39} -4.00000 q^{43} -2.00000i q^{45} +8.00000i q^{47} -1.00000 q^{49} +2.00000 q^{51} +4.00000 q^{53} -4.00000i q^{57} -6.00000i q^{59} +12.0000 q^{61} +1.00000i q^{63} +(-6.00000 - 4.00000i) q^{65} -2.00000i q^{67} -6.00000 q^{69} +14.0000i q^{73} -1.00000 q^{75} +1.00000 q^{81} -14.0000i q^{83} +4.00000i q^{85} -4.00000i q^{89} +(3.00000 + 2.00000i) q^{91} +8.00000 q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 12 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{35} - 4 q^{39} - 8 q^{43} - 2 q^{49} + 4 q^{51} + 8 q^{53} + 24 q^{61} - 12 q^{65} - 12 q^{69} - 2 q^{75} + 2 q^{81} + 6 q^{91} + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 12 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^35 - 4 * q^39 - 8 * q^43 - 2 * q^49 + 4 * q^51 + 8 * q^53 + 24 * q^61 - 12 * q^65 - 12 * q^69 - 2 * q^75 + 2 * q^81 + 6 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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