LMFDB - Embedded newform 4368.2.h.j.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 1092)
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.j.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +4.00000i q^{31} +2.00000i q^{33} +2.00000 q^{35} +(3.00000 + 2.00000i) q^{39} +2.00000i q^{41} +8.00000 q^{43} -2.00000i q^{45} +10.0000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} +4.00000i q^{57} +6.00000i q^{59} +2.00000 q^{61} +1.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +6.00000i q^{71} -4.00000i q^{73} +1.00000 q^{75} -2.00000 q^{77} +1.00000 q^{81} +6.00000i q^{83} +4.00000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +(-2.00000 + 3.00000i) q^{91} +4.00000i q^{93} +8.00000 q^{95} +4.00000i q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{13} - 4 q^{17} + 2 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{35} + 6 q^{39} + 16 q^{43} - 2 q^{49} - 4 q^{51} - 20 q^{53} + 8 q^{55} + 4 q^{61} + 8 q^{65} + 2 q^{75} - 4 q^{77} + 2 q^{81} - 12 q^{87} - 4 q^{91} + 16 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 2 * q^25 + 2 * q^27 - 12 * q^29 + 4 * q^35 + 6 * q^39 + 16 * q^43 - 2 * q^49 - 4 * q^51 - 20 * q^53 + 8 * q^55 + 4 * q^61 + 8 * q^65 + 2 * q^75 - 4 * q^77 + 2 * q^81 - 12 * q^87 - 4 * 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
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		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
$19$			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
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	3.00000	+	2.00000i	0.480384	+	0.320256i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$		0			0
$47$			10.0000i			1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$			10.0000i			1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$			4.00000i			0.529813i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	4.00000	−	6.00000i	0.496139	−	0.744208i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		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
$77$	−2.00000			−0.227921
$78$		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
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$	−6.00000			−0.643268
$88$		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
$91$	−2.00000	+	3.00000i	−0.209657	+	0.314485i
$92$		0			0
$93$			4.00000i			0.414781i
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$			4.00000i			0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$			4.00000i			0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$		0			0
$99$			2.00000i			0.201008i
$100$		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
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	2.00000			0.195180
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$		0			0
$119$		−	2.00000i		−	0.183340i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$			2.00000i			0.180334i
$124$		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
$129$	8.00000			0.704361
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$		−	2.00000i		−	0.172133i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			10.0000i			0.842152i
$142$		0			0
$143$	−4.00000	+	6.00000i	−0.334497	+	0.501745i
$144$		0			0
$145$			12.0000i			0.996546i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$		−	4.00000i		−	0.325515i	−0.986666	−	0.162758i	$-0.947961\pi$
$151$		−	4.00000i		−	0.325515i	0.986666	−	0.162758i	$-0.0520389\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	18.0000			1.43656			0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000			1.43656			0.718278	+	0.695756i	$0.244931\pi$
$158$		0			0
$159$	−10.0000			−0.793052
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	4.00000			0.311400
$166$		0			0
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	4.00000i		−	0.292509i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$		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
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$	4.00000			0.279372
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$			6.00000i			0.411113i
$214$		0			0
$215$		−	16.0000i		−	1.09119i
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	−6.00000	−	4.00000i	−0.403604	−	0.269069i
$222$		0			0
$223$		−	28.0000i		−	1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$		−	28.0000i		−	1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$			18.0000i			1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$			18.0000i			1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$		0			0
$229$		−	28.0000i		−	1.85029i	−0.379611	−	0.925146i	$-0.623942\pi$
$229$		−	28.0000i		−	1.85029i	0.379611	−	0.925146i	$-0.376058\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	20.0000			1.30466
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.0000i		−	0.646846i	−0.946254	−	0.323423i	$-0.895166\pi$
$239$		−	10.0000i		−	0.646846i	0.946254	−	0.323423i	$-0.104834\pi$
$240$		0			0
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			2.00000i			0.127775i
$246$		0			0
$247$	−8.00000	+	12.0000i	−0.509028	+	0.763542i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			4.00000i			0.250490i
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$			20.0000i			1.22859i
$266$		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
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	−2.00000	+	3.00000i	−0.121046	+	0.181568i
$274$		0			0
$275$			2.00000i			0.120605i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$			4.00000i			0.239474i
$280$		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
$283$	−28.0000			−1.66443			−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000			−1.66443			−0.832214	+	0.554455i	$0.812927\pi$
$284$		0			0
$285$	8.00000			0.473879
$286$		0			0
$287$	−2.00000			−0.118056
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			4.00000i			0.234484i
$292$		0			0
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$		0			0
$300$		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
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	34.0000			1.92179			0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000			1.92179			0.960897	+	0.276907i	$0.0893093\pi$
$314$		0			0
$315$	2.00000			0.112687
$316$		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
$319$		−	12.0000i		−	0.671871i
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	3.00000	+	2.00000i	0.166410	+	0.110940i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$	−10.0000			−0.551318
$330$		0			0
$331$			32.0000i			1.75888i	0.476011	+	0.879440i	$0.342082\pi$
$331$			32.0000i			1.75888i	−0.476011	+	0.879440i	$0.657918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	−2.00000			−0.108625
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		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
$349$			8.00000i			0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$			8.00000i			0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$		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
$355$	12.0000			0.636894
$356$		0			0
$357$		−	2.00000i		−	0.105851i
$358$		0			0
$359$			18.0000i			0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$			18.0000i			0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$	−8.00000			−0.418739
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$			2.00000i			0.104116i
$370$		0			0
$371$		−	10.0000i		−	0.519174i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$		−	12.0000i		−	0.619677i
$376$		0			0
$377$	−18.0000	−	12.0000i	−0.927047	−	0.618031i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$		−	38.0000i		−	1.94171i	−0.239669	−	0.970855i	$-0.577039\pi$
$383$		−	38.0000i		−	1.94171i	0.239669	−	0.970855i	$-0.422961\pi$
$384$		0			0
$385$			4.00000i			0.203859i
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	−22.0000			−1.11544			−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000			−1.11544			−0.557722	+	0.830028i	$0.688325\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	32.0000i		−	1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$		−	32.0000i		−	1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$		0			0
$399$	−4.00000			−0.200250
$400$		0			0
$401$		−	22.0000i		−	1.09863i	−0.835616	−	0.549314i	$-0.814889\pi$
$401$		−	22.0000i		−	1.09863i	0.835616	−	0.549314i	$-0.185111\pi$
$402$		0			0
$403$	−8.00000	+	12.0000i	−0.398508	+	0.597763i
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	36.0000i		−	1.78009i	−0.455877	−	0.890043i	$-0.650674\pi$
$409$		−	36.0000i		−	1.78009i	0.455877	−	0.890043i	$-0.349326\pi$
$410$		0			0
$411$			6.00000i			0.295958i
$412$		0			0
$413$	−6.00000			−0.295241
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$			28.0000i			1.36464i	0.731055	+	0.682318i	$0.239028\pi$
$421$			28.0000i			1.36464i	−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$			10.0000i			0.486217i
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$			2.00000i			0.0967868i
$428$		0			0
$429$	−4.00000	+	6.00000i	−0.193122	+	0.289683i
$430$		0			0
$431$		−	30.0000i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$		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
$435$			12.0000i			0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−12.0000			−0.572729			−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000			−0.572729			−0.286364	+	0.958121i	$0.592447\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$		−	6.00000i		−	0.283790i
$448$		0			0
$449$		−	30.0000i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$		−	30.0000i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$		0			0
$451$	−4.00000			−0.188353
$452$		0			0
$453$		−	4.00000i		−	0.187936i
$454$		0			0
$455$	6.00000	+	4.00000i	0.281284	+	0.187523i
$456$		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
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		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
$465$	8.00000			0.370991
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	18.0000			0.829396
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$			4.00000i			0.183533i
$476$		0			0
$477$	−10.0000			−0.457869
$478$		0			0
$479$			6.00000i			0.274147i	0.990561	+	0.137073i	$0.0437697\pi$
$479$			6.00000i			0.274147i	−0.990561	+	0.137073i	$0.956230\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000			0.363261
$486$		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
$489$		−	4.00000i		−	0.180886i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$	4.00000			0.179787
$496$		0			0
$497$	−6.00000			−0.269137
$498$		0			0
$499$		−	8.00000i		−	0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$		−	8.00000i		−	0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$		0			0
$501$			2.00000i			0.0893534i
$502$		0			0
$503$	−40.0000			−1.78351			−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000			−1.78351			−0.891756	+	0.452517i	$0.850526\pi$
$504$		0			0
$505$			4.00000i			0.177998i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$		0			0
$509$			14.0000i			0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$			14.0000i			0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$			8.00000i			0.352522i
$516$		0			0
$517$	−20.0000			−0.879599
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$			1.00000i			0.0436436i
$526$		0			0
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$			6.00000i			0.260378i
$532$		0			0
$533$	−4.00000	+	6.00000i	−0.173259	+	0.259889i
$534$		0			0
$535$			8.00000i			0.345870i
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$		−	2.00000i		−	0.0861461i
$540$		0			0
$541$		−	4.00000i		−	0.171973i	−0.996296	−	0.0859867i	$-0.972596\pi$
$541$		−	4.00000i		−	0.171973i	0.996296	−	0.0859867i	$-0.0274043\pi$
$542$		0			0
$543$	10.0000			0.429141
$544$		0			0
$545$	8.00000			0.342682
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	2.00000			0.0853579
$550$		0			0
$551$		−	24.0000i		−	1.02243i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		0			0
$559$	24.0000	+	16.0000i	1.01509	+	0.676728i
$560$		0			0
$561$		−	4.00000i		−	0.168880i
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$			4.00000i			0.168281i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		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
$573$	−16.0000			−0.668410
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$			24.0000i			0.997406i
$580$		0			0
$581$	−6.00000			−0.248922
$582$		0			0
$583$		−	20.0000i		−	0.828315i
$584$		0			0
$585$	4.00000	−	6.00000i	0.165380	−	0.248069i
$586$		0			0
$587$		−	26.0000i		−	1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$		−	26.0000i		−	1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$			22.0000i			0.904959i
$592$		0			0
$593$		−	46.0000i		−	1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$		−	46.0000i		−	1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$		0			0
$595$	−4.00000			−0.163984
$596$		0			0
$597$	−8.00000			−0.327418
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	14.0000i		−	0.569181i
$606$		0			0
$607$	20.0000			0.811775			0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000			0.811775			0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		−	6.00000i		−	0.243132i
$610$		0			0
$611$	−20.0000	+	30.0000i	−0.809113	+	1.21367i
$612$		0			0
$613$		−	8.00000i		−	0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$		−	8.00000i		−	0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$		0			0
$615$	4.00000			0.161296
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000			0.240385
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−8.00000			−0.319489
$628$		0			0
$629$		0			0
$630$		0			0
$631$			4.00000i			0.159237i	0.996825	+	0.0796187i	$0.0253703\pi$
$631$			4.00000i			0.159237i	−0.996825	+	0.0796187i	$0.974630\pi$
$632$		0			0
$633$	20.0000			0.794929
$634$		0			0
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$	−3.00000	−	2.00000i	−0.118864	−	0.0792429i
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$			16.0000i			0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$			16.0000i			0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$		0			0
$645$		−	16.0000i		−	0.629999i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$		0			0
$651$	−4.00000			−0.156772
$652$		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
$655$		−	24.0000i		−	0.937758i
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$		−	16.0000i		−	0.622328i	−0.950356	−	0.311164i	$-0.899281\pi$
$661$		−	16.0000i		−	0.622328i	0.950356	−	0.311164i	$-0.100719\pi$
$662$		0			0
$663$	−6.00000	−	4.00000i	−0.233021	−	0.155347i
$664$		0			0
$665$			8.00000i			0.310227i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	28.0000i		−	1.08254i
$670$		0			0
$671$			4.00000i			0.154418i
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	−10.0000			−0.384331			−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000			−0.384331			−0.192166	+	0.981363i	$0.561551\pi$
$678$		0			0
$679$	−4.00000			−0.153506
$680$		0			0
$681$			18.0000i			0.689761i
$682$		0			0
$683$		−	46.0000i		−	1.76014i	−0.474843	−	0.880071i	$-0.657495\pi$
$683$		−	46.0000i		−	1.76014i	0.474843	−	0.880071i	$-0.342505\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		−	28.0000i		−	1.06827i
$688$		0			0
$689$	−30.0000	−	20.0000i	−1.14291	−	0.761939i
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		0			0
$693$	−2.00000			−0.0759737
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	4.00000i		−	0.151511i
$698$		0			0
$699$	18.0000			0.680823
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	20.0000			0.753244
$706$		0			0
$707$		−	2.00000i		−	0.0752177i
$708$		0			0
$709$			4.00000i			0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$			4.00000i			0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	12.0000	+	8.00000i	0.448775	+	0.299183i
$716$		0			0
$717$		−	10.0000i		−	0.373457i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$		−	4.00000i		−	0.148968i
$722$		0			0
$723$			4.00000i			0.148762i
$724$		0			0
$725$	−6.00000			−0.222834
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−16.0000			−0.591781
$732$		0			0
$733$		−	28.0000i		−	1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$733$		−	28.0000i		−	1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$734$		0			0
$735$			2.00000i			0.0737711i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			40.0000i			1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$			40.0000i			1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$		0			0
$741$	−8.00000	+	12.0000i	−0.293887	+	0.440831i
$742$		0			0
$743$		−	42.0000i		−	1.54083i	−0.637542	−	0.770415i	$-0.720049\pi$
$743$		−	42.0000i		−	1.54083i	0.637542	−	0.770415i	$-0.279951\pi$
$744$		0			0
$745$	−12.0000			−0.439646
$746$		0			0
$747$			6.00000i			0.219529i
$748$		0			0
$749$		−	4.00000i		−	0.146157i
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$	4.00000			0.145768
$754$		0			0
$755$	−8.00000			−0.291150
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		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
$763$	−4.00000			−0.144810
$764$		0			0
$765$			4.00000i			0.144620i
$766$		0			0
$767$	−12.0000	+	18.0000i	−0.433295	+	0.649942i
$768$		0			0
$769$		−	8.00000i		−	0.288487i	−0.989542	−	0.144244i	$-0.953925\pi$
$769$		−	8.00000i		−	0.288487i	0.989542	−	0.144244i	$-0.0460749\pi$
$770$		0			0
$771$	2.00000			0.0720282
$772$		0			0
$773$			34.0000i			1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$			34.0000i			1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$			4.00000i			0.143684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		−	36.0000i		−	1.28490i
$786$		0			0
$787$		−	8.00000i		−	0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$		−	8.00000i		−	0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$		0			0
$789$	−16.0000			−0.569615
$790$		0			0
$791$		−	2.00000i		−	0.0711118i
$792$		0			0
$793$	6.00000	+	4.00000i	0.213066	+	0.142044i
$794$		0			0
$795$			20.0000i			0.709327i
$796$		0			0
$797$	26.0000			0.920967			0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000			0.920967			0.460484	+	0.887668i	$0.347676\pi$
$798$		0			0
$799$		−	20.0000i		−	0.707549i
$800$		0			0
$801$		−	6.00000i		−	0.212000i
$802$		0			0
$803$	8.00000			0.282314
$804$		0			0
$805$		0			0
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$	14.0000			0.492214			0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000			0.492214			0.246107	+	0.969243i	$0.420849\pi$
$810$		0			0
$811$			12.0000i			0.421377i	0.977553	+	0.210688i	$0.0675706\pi$
$811$			12.0000i			0.421377i	−0.977553	+	0.210688i	$0.932429\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	−8.00000			−0.280228
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$	−2.00000	+	3.00000i	−0.0698857	+	0.104828i
$820$		0			0
$821$			18.0000i			0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$			18.0000i			0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$		0			0
$823$	−28.0000			−0.976019			−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000			−0.976019			−0.488009	+	0.872838i	$0.662277\pi$
$824$		0			0
$825$			2.00000i			0.0696311i
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	4.00000			0.138426
$836$		0			0
$837$			4.00000i			0.138260i
$838$		0			0
$839$		−	34.0000i		−	1.17381i	−0.809656	−	0.586905i	$-0.800346\pi$
$839$		−	34.0000i		−	1.17381i	0.809656	−	0.586905i	$-0.199654\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	10.0000i		−	0.344418i
$844$		0			0
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$	−28.0000			−0.960958
$850$		0			0
$851$		0			0
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$	8.00000			0.273594
$856$		0			0
$857$	−46.0000			−1.57133			−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000			−1.57133			−0.785665	+	0.618652i	$0.787679\pi$
$858$		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
$861$	−2.00000			−0.0681598
$862$		0			0
$863$		−	54.0000i		−	1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$		−	54.0000i		−	1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$		0			0
$865$		−	28.0000i		−	0.952029i
$866$		0			0
$867$	−13.0000			−0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$			4.00000i			0.135379i
$874$		0			0
$875$	12.0000			0.405674
$876$		0			0
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$		−	2.00000i		−	0.0674583i
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		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
$887$	16.0000			0.537227			0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000			0.537227			0.268614	+	0.963248i	$0.413434\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$			2.00000i			0.0670025i
$892$		0			0
$893$	−40.0000			−1.33855
$894$		0			0
$895$		−	24.0000i		−	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	24.0000i		−	0.800445i
$900$		0			0
$901$	20.0000			0.666297
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$		−	20.0000i		−	0.664822i
$906$		0			0
$907$	−36.0000			−1.19536			−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000			−1.19536			−0.597680	+	0.801735i	$0.703911\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		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
$913$	−12.0000			−0.397142
$914$		0			0
$915$		−	4.00000i		−	0.132236i
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$			8.00000i			0.263609i
$922$		0			0
$923$	−12.0000	+	18.0000i	−0.394985	+	0.592477i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$			34.0000i			1.11550i	0.830008	+	0.557752i	$0.188336\pi$
$929$			34.0000i			1.11550i	−0.830008	+	0.557752i	$0.811664\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$		0			0
$933$	−8.00000			−0.261908
$934$		0			0
$935$	−8.00000			−0.261628
$936$		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
$939$	34.0000			1.10955
$940$		0			0
$941$			42.0000i			1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$			42.0000i			1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	2.00000			0.0650600
$946$		0			0
$947$			38.0000i			1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$			38.0000i			1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$		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
$955$			32.0000i			1.03550i
$956$		0			0
$957$		−	12.0000i		−	0.387905i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−4.00000			−0.128898
$964$		0			0
$965$	48.0000			1.54517
$966$		0			0
$967$		−	60.0000i		−	1.92947i	−0.263223	−	0.964735i	$-0.584786\pi$
$967$		−	60.0000i		−	1.92947i	0.263223	−	0.964735i	$-0.415214\pi$
$968$		0			0
$969$		−	8.00000i		−	0.256997i
$970$		0			0
$971$	52.0000			1.66876			0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000			1.66876			0.834380	+	0.551190i	$0.185826\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	3.00000	+	2.00000i	0.0960769	+	0.0640513i
$976$		0			0
$977$			30.0000i			0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$			30.0000i			0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$		0			0
$981$			4.00000i			0.127710i
$982$		0			0
$983$		−	42.0000i		−	1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$		−	42.0000i		−	1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$		0			0
$985$	44.0000			1.40196
$986$		0			0
$987$	−10.0000			−0.318304
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−36.0000			−1.14358			−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000			−1.14358			−0.571789	+	0.820401i	$0.693750\pi$
$992$		0			0
$993$			32.0000i			1.01549i
$994$		0			0
$995$			16.0000i			0.507234i
$996$		0			0
$997$	22.0000			0.696747			0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000			0.696747			0.348373	+	0.937356i	$0.386734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.j.337.1		2
4.3	odd	2		1092.2.e.a.337.1	✓	2
12.11	even	2		3276.2.e.d.2521.2		2
13.12	even	2	inner	4368.2.h.j.337.2		2
28.27	even	2		7644.2.e.f.4705.2		2
52.51	odd	2		1092.2.e.a.337.2	yes	2
156.155	even	2		3276.2.e.d.2521.1		2
364.363	even	2		7644.2.e.f.4705.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.e.a.337.1	✓	2	4.3	odd	2
1092.2.e.a.337.2	yes	2	52.51	odd	2
3276.2.e.d.2521.1		2	156.155	even	2
3276.2.e.d.2521.2		2	12.11	even	2
4368.2.h.j.337.1		2	1.1	even	1	trivial
4368.2.h.j.337.2		2	13.12	even	2	inner
7644.2.e.f.4705.1		2	364.363	even	2
7644.2.e.f.4705.2		2	28.27	even	2

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +4.00000i q^{31} +2.00000i q^{33} +2.00000 q^{35} +(3.00000 + 2.00000i) q^{39} +2.00000i q^{41} +8.00000 q^{43} -2.00000i q^{45} +10.0000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} +4.00000i q^{57} +6.00000i q^{59} +2.00000 q^{61} +1.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +6.00000i q^{71} -4.00000i q^{73} +1.00000 q^{75} -2.00000 q^{77} +1.00000 q^{81} +6.00000i q^{83} +4.00000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +(-2.00000 + 3.00000i) q^{91} +4.00000i q^{93} +8.00000 q^{95} +4.00000i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 2 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{35} + 6 q^{39} + 16 q^{43} - 2 q^{49} - 4 q^{51} - 20 q^{53} + 8 q^{55} + 4 q^{61} + 8 q^{65} + 2 q^{75} - 4 q^{77} + 2 q^{81} - 12 q^{87} - 4 q^{91} + 16 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 2 * q^25 + 2 * q^27 - 12 * q^29 + 4 * q^35 + 6 * q^39 + 16 * q^43 - 2 * q^49 - 4 * q^51 - 20 * q^53 + 8 * q^55 + 4 * q^61 + 8 * q^65 + 2 * q^75 - 4 * q^77 + 2 * q^81 - 12 * q^87 - 4 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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