LMFDB - Embedded newform 5376.2.c.m.2689.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5376 = 2^{8} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5376.c (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:	$42.9275761266$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5376.2689
Dual form			5376.2.c.m.2689.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +6.00000i q^{13} +2.00000 q^{15} -2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} -8.00000 q^{31} -4.00000 q^{33} +2.00000i q^{35} -10.0000i q^{37} -6.00000 q^{39} +2.00000 q^{41} -8.00000i q^{43} +2.00000i q^{45} +1.00000 q^{49} -2.00000i q^{51} -10.0000i q^{53} +8.00000 q^{55} -4.00000 q^{57} +12.0000i q^{59} -10.0000i q^{61} +1.00000 q^{63} +12.0000 q^{65} -8.00000i q^{67} -4.00000i q^{69} +12.0000 q^{71} -2.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} +1.00000 q^{81} +12.0000i q^{83} +4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} -6.00000i q^{91} -8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{7} - 2 q^{9} + 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 16 q^{31} - 8 q^{33} - 12 q^{39} + 4 q^{41} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} + 24 q^{65} + 24 q^{71} - 4 q^{73} + 2 q^{81}+ \cdots + 4 q^{97}+O(q^{100})$ 2 * q - 2 * q^7 - 2 * q^9 + 4 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 16 * q^31 - 8 * q^33 - 12 * q^39 + 4 * q^41 + 2 * q^49 + 16 * q^55 - 8 * q^57 + 2 * q^63 + 24 * q^65 + 24 * q^71 - 4 * q^73 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 16 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$ .

$n$	$1793$	$2815$	$4609$	$5125$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	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
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	− 10.0000i	− 1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	− 10.0000i	− 1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	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
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 6.00000i	− 0.628971i
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−5.00000	−0.454545
$122$	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	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	− 20.0000i	− 1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$	− 20.0000i	− 1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	− 20.0000i	− 1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	− 20.0000i	− 1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	16.0000i	1.28515i
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	8.00000i	0.622799i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	− 20.0000i	− 1.49487i	−0.664335	−	0.747435i	$-0.731285\pi$
$179$	− 20.0000i	− 1.49487i	0.664335	−	0.747435i	$-0.268715\pi$
$180$	0	0
$181$	2.00000i	0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$	2.00000i	0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	12.0000i	0.859338i
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	− 4.00000i	− 0.279372i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	− 16.0000i	− 1.10149i	−0.834675	−	0.550743i	$-0.814345\pi$
$211$	− 16.0000i	− 1.10149i	0.834675	−	0.550743i	$-0.185655\pi$
$212$	0	0
$213$	12.0000i	0.822226i
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	18.0000i	1.18947i	0.803921	+	0.594737i	$0.202744\pi$
$229$	18.0000i	1.18947i	−0.803921	+	0.594737i	$0.797256\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	4.00000i	0.252478i	0.992000	+	0.126239i	$0.0402906\pi$
$251$	4.00000i	0.252478i	−0.992000	+	0.126239i	$0.959709\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−20.0000	−1.22859
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	2.00000i	0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$	2.00000i	0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	0	0
$285$	8.00000i	0.473879i
$286$	0	0
$287$	−2.00000	−0.118056
$288$	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	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	− 24.0000i	− 1.38796i
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	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
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	− 14.0000i	− 0.760376i
$340$	0	0
$341$	− 32.0000i	− 1.73290i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	0	0
$349$	− 18.0000i	− 0.963518i	−0.876304	−	0.481759i	$-0.839998\pi$
$349$	− 18.0000i	− 0.963518i	0.876304	−	0.481759i	$-0.160002\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	− 24.0000i	− 1.27379i
$356$	0	0
$357$	2.00000i	0.105851i
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	4.00000i	0.209370i
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	10.0000i	0.519174i
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000i	0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$	8.00000i	0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	38.0000i	1.92668i	0.268290	+	0.963338i	$0.413542\pi$
$389$	38.0000i	1.92668i	−0.268290	+	0.963338i	$0.586458\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	20.0000	1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	− 48.0000i	− 2.39105i
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	12.0000i	0.586238i	0.956076	+	0.293119i	$0.0946933\pi$
$419$	12.0000i	0.586238i	−0.956076	+	0.293119i	$0.905307\pi$
$420$	0	0
$421$	− 34.0000i	− 1.65706i	−0.559946	−	0.828529i	$-0.689178\pi$
$421$	− 34.0000i	− 1.65706i	0.559946	−	0.828529i	$-0.310822\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	− 24.0000i	− 1.15873i
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	8.00000i	0.376705i
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	26.0000i	1.21094i	0.795868	+	0.605470i	$0.207015\pi$
$461$	26.0000i	1.21094i	−0.795868	+	0.605470i	$0.792985\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	0	0
$469$	8.00000i	0.369406i
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	36.0000i	1.62466i	0.583200	+	0.812329i	$0.301800\pi$
$491$	36.0000i	1.62466i	−0.583200	+	0.812329i	$0.698200\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	4.00000i	0.172292i
$540$	0	0
$541$	26.0000i	1.11783i	0.829226	+	0.558914i	$0.188782\pi$
$541$	26.0000i	1.11783i	−0.829226	+	0.558914i	$0.811218\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	0	0
$549$	10.0000i	0.426790i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 20.0000i	− 0.848953i
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	48.0000	2.03018
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	28.0000i	1.17797i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	− 16.0000i	− 0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	− 16.0000i	− 0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	− 14.0000i	− 0.581820i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	40.0000	1.65663
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	− 32.0000i	− 1.31854i
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	− 4.00000i	− 0.163984i
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	− 16.0000i	− 0.629999i
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	8.00000i	0.313545i
$652$	0	0
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	−40.0000	−1.56293
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	44.0000i	1.71400i	0.515319	+	0.856998i	$0.327673\pi$
$659$	44.0000i	1.71400i	−0.515319	+	0.856998i	$0.672327\pi$
$660$	0	0
$661$	18.0000i	0.700119i	0.936727	+	0.350059i	$0.113839\pi$
$661$	18.0000i	0.700119i	−0.936727	+	0.350059i	$0.886161\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	− 28.0000i	− 1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$	− 28.0000i	− 1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	−18.0000	−0.686743
$688$	0	0
$689$	60.0000	2.28582
$690$	0	0
$691$	12.0000i	0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$	12.0000i	0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	−40.0000	−1.51729
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	14.0000i	0.529529i
$700$	0	0
$701$	− 30.0000i	− 1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$	− 30.0000i	− 1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0000i	0.376089i
$708$	0	0
$709$	38.0000i	1.42712i	0.700594	+	0.713560i	$0.252918\pi$
$709$	38.0000i	1.42712i	−0.700594	+	0.713560i	$0.747082\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	48.0000i	1.79510i
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 30.0000i	− 1.11571i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000i	0.591781i
$732$	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
$735$	2.00000	0.0737711
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	32.0000i	1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$739$	32.0000i	1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	−72.0000	−2.59977
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	8.00000i	0.286630i
$780$	0	0
$781$	48.0000i	1.71758i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	− 12.0000i	− 0.427211i
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	60.0000	2.13066
$794$	0	0
$795$	− 20.0000i	− 0.709327i
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	32.0000	1.12091
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	6.00000i	0.209657i
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	14.0000i	0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$	14.0000i	0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	46.0000i	1.58245i
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	40.0000i	1.37118i
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	− 12.0000i	− 0.409435i	−0.978821	−	0.204717i	$-0.934372\pi$
$859$	− 12.0000i	− 0.409435i	0.978821	−	0.204717i	$-0.0656275\pi$
$860$	0	0
$861$	− 2.00000i	− 0.0681598i
$862$	0	0
$863$	−52.0000	−1.77010	−0.885050	−	0.465495i	$-0.845876\pi$
$863$	−52.0000	−1.77010	−0.885050	+	0.465495i	$0.845876\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	24.0000i	0.806751i
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−40.0000	−1.33705
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	20.0000i	0.666297i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	− 48.0000i	− 1.59381i	−0.604102	−	0.796907i	$-0.706468\pi$
$907$	− 48.0000i	− 1.59381i	0.604102	−	0.796907i	$-0.293532\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	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
$913$	−48.0000	−1.58857
$914$	0	0
$915$	− 20.0000i	− 0.661180i
$916$	0	0
$917$	20.0000i	0.660458i
$918$	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
$921$	−4.00000	−0.131804
$922$	0	0
$923$	72.0000i	2.36991i
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	14.0000i	0.456873i
$940$	0	0
$941$	− 6.00000i	− 0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$	− 6.00000i	− 0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	0	0
$949$	− 12.0000i	− 0.389536i
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	24.0000i	0.776622i
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	28.0000i	0.901352i
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	− 28.0000i	− 0.898563i	−0.893390	−	0.449281i	$-0.851680\pi$
$971$	− 28.0000i	− 0.898563i	0.893390	−	0.449281i	$-0.148320\pi$
$972$	0	0
$973$	20.0000i	0.641171i
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000i	1.01754i
$990$	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
$993$	−8.00000	−0.253872
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	− 62.0000i	− 1.96356i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	− 62.0000i	− 1.96356i	0.190022	−	0.981780i	$-0.439144\pi$
$998$	0	0
$999$	−10.0000	−0.316386

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.m.2689.2		2
4.3	odd	2		5376.2.c.s.2689.1		2
8.3	odd	2		5376.2.c.s.2689.2		2
8.5	even	2	inner	5376.2.c.m.2689.1		2
16.3	odd	4		672.2.a.f.1.1	yes	1
16.5	even	4		1344.2.a.r.1.1		1
16.11	odd	4		1344.2.a.h.1.1		1
16.13	even	4		672.2.a.b.1.1	✓	1
48.5	odd	4		4032.2.a.n.1.1		1
48.11	even	4		4032.2.a.f.1.1		1
48.29	odd	4		2016.2.a.l.1.1		1
48.35	even	4		2016.2.a.k.1.1		1
112.13	odd	4		4704.2.a.be.1.1		1
112.27	even	4		9408.2.a.cb.1.1		1
112.69	odd	4		9408.2.a.g.1.1		1
112.83	even	4		4704.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.a.b.1.1	✓	1	16.13	even	4
672.2.a.f.1.1	yes	1	16.3	odd	4
1344.2.a.h.1.1		1	16.11	odd	4
1344.2.a.r.1.1		1	16.5	even	4
2016.2.a.k.1.1		1	48.35	even	4
2016.2.a.l.1.1		1	48.29	odd	4
4032.2.a.f.1.1		1	48.11	even	4
4032.2.a.n.1.1		1	48.5	odd	4
4704.2.a.m.1.1		1	112.83	even	4
4704.2.a.be.1.1		1	112.13	odd	4
5376.2.c.m.2689.1		2	8.5	even	2	inner
5376.2.c.m.2689.2		2	1.1	even	1	trivial
5376.2.c.s.2689.1		2	4.3	odd	2
5376.2.c.s.2689.2		2	8.3	odd	2
9408.2.a.g.1.1		1	112.69	odd	4
9408.2.a.cb.1.1		1	112.27	even	4

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5376.2.c.m.2689.2

Level	$5376$
Weight	$2$
Character	5376.2689
Analytic conductor	$42.928$
Analytic rank	$1$
Dimension	$2$
Inner twists	$2$