LMFDB - Embedded newform 5376.2.c.q.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:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
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.q.2689.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +6.00000i q^{13} -4.00000 q^{15} -4.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} -2.00000 q^{23} -11.0000 q^{25} -1.00000i q^{27} +2.00000i q^{29} -2.00000 q^{33} +4.00000i q^{35} +2.00000i q^{37} -6.00000 q^{39} -4.00000i q^{43} -4.00000i q^{45} +12.0000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} -8.00000i q^{59} -6.00000i q^{61} -1.00000 q^{63} -24.0000 q^{65} +8.00000i q^{67} -2.00000i q^{69} -14.0000 q^{71} +2.00000 q^{73} -11.0000i q^{75} +2.00000i q^{77} +12.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -16.0000i q^{85} -2.00000 q^{87} +6.00000i q^{91} -16.0000 q^{95} -2.00000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{7} - 2 q^{9} - 8 q^{15} - 8 q^{17} - 4 q^{23} - 22 q^{25} - 4 q^{33} - 12 q^{39} + 24 q^{47} + 2 q^{49} - 16 q^{55} - 8 q^{57} - 2 q^{63} - 48 q^{65} - 28 q^{71} + 4 q^{73} + 24 q^{79} + 2 q^{81} - 4 q^{87} - 32 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 - 8 * q^15 - 8 * q^17 - 4 * q^23 - 22 * q^25 - 4 * q^33 - 12 * q^39 + 24 * q^47 + 2 * q^49 - 16 * q^55 - 8 * q^57 - 2 * q^63 - 48 * q^65 - 28 * q^71 + 4 * q^73 + 24 * q^79 + 2 * q^81 - 4 * q^87 - 32 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

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
$5$	4.00000i	1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$	4.00000i	1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	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
$13$	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
$15$	−4.00000	−1.03280
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	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
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−11.0000	−2.20000
$26$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	4.00000i	0.676123i
$36$	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
$39$	−6.00000	−0.960769
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	0	0
$61$	− 6.00000i	− 0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$	− 6.00000i	− 0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−24.0000	−2.97683
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	− 2.00000i	− 0.240772i
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	− 11.0000i	− 1.27017i
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	− 16.0000i	− 1.73544i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	6.00000i	0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.0000	−1.64157
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	16.0000i	1.59206i	0.605257	+	0.796030i	$0.293070\pi$
$101$	16.0000i	1.59206i	−0.605257	+	0.796030i	$0.706930\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	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
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 24.0000i	− 2.14663i
$126$	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
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	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
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	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
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	0	0
$181$	− 6.00000i	− 0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$	− 6.00000i	− 0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	− 24.0000i	− 1.71868i
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	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
$205$	0	0
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	− 14.0000i	− 0.959264i
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	− 24.0000i	− 1.61441i
$222$	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
$225$	11.0000	0.733333
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	48.0000i	3.13117i
$236$	0	0
$237$	12.0000i	0.779484i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	4.00000i	0.255551i
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	− 24.0000i	− 1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$	− 24.0000i	− 1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	0	0
$255$	16.0000	1.00196
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.0000i	1.21942i	0.792624	+	0.609711i	$0.208714\pi$
$269$	20.0000i	1.21942i	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	− 22.0000i	− 1.32665i
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 16.0000i	− 0.947758i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	− 4.00000i	− 0.230556i
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	16.0000i	0.910208i
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	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
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	− 66.0000i	− 3.66102i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	−32.0000	−1.74835
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	10.0000i	0.543125i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	0	0
$349$	− 22.0000i	− 1.17763i	−0.808267	−	0.588817i	$-0.799594\pi$
$349$	− 22.0000i	− 1.17763i	0.808267	−	0.588817i	$-0.200406\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	− 56.0000i	− 2.97217i
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	8.00000i	0.418739i
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	12.0000i	0.614779i
$382$	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
$387$	4.00000i	0.203331i
$388$	0	0
$389$	22.0000i	1.11544i	0.830028	+	0.557722i	$0.188325\pi$
$389$	22.0000i	1.11544i	−0.830028	+	0.557722i	$0.811675\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	48.0000i	2.41514i
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.00000i	0.198762i
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	2.00000i	0.0986527i
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	44.0000	2.13431
$426$	0	0
$427$	− 6.00000i	− 0.290360i
$428$	0	0
$429$	− 12.0000i	− 0.579365i
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	− 8.00000i	− 0.383571i
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	38.0000i	1.80543i	0.430234	+	0.902717i	$0.358431\pi$
$443$	38.0000i	1.80543i	−0.430234	+	0.902717i	$0.641569\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	−24.0000	−1.12514
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	4.00000i	0.186704i
$460$	0	0
$461$	− 40.0000i	− 1.86299i	−0.363760	−	0.931493i	$-0.618507\pi$
$461$	− 40.0000i	− 1.86299i	0.363760	−	0.931493i	$-0.381493\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	8.00000i	0.369406i
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	− 44.0000i	− 2.01886i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	− 2.00000i	− 0.0910032i
$484$	0	0
$485$	− 8.00000i	− 0.363261i
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	− 42.0000i	− 1.89543i	−0.319113	−	0.947717i	$-0.603385\pi$
$491$	− 42.0000i	− 1.89543i	0.319113	−	0.947717i	$-0.396615\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−14.0000	−0.627986
$498$	0	0
$499$	20.0000i	0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$	20.0000i	0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$	0	0
$501$	− 4.00000i	− 0.178707i
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−64.0000	−2.84796
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	36.0000i	1.59567i	0.602875	+	0.797836i	$0.294022\pi$
$509$	36.0000i	1.59567i	−0.602875	+	0.797836i	$0.705978\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	64.0000i	2.82018i
$516$	0	0
$517$	24.0000i	1.05552i
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	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
$525$	− 11.0000i	− 0.480079i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−72.0000	−3.11283
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	2.00000i	0.0861461i
$540$	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
$543$	6.00000	0.257485
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	− 24.0000i	− 1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$	− 24.0000i	− 1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$	0	0
$549$	6.00000i	0.256074i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	− 8.00000i	− 0.339581i
$556$	0	0
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 8.00000i	− 0.337160i	−0.985688	−	0.168580i	$-0.946082\pi$
$563$	− 8.00000i	− 0.337160i	0.985688	−	0.168580i	$-0.0539181\pi$
$564$	0	0
$565$	40.0000i	1.68281i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	− 24.0000i	− 1.00437i	−0.864761	−	0.502184i	$-0.832530\pi$
$571$	− 24.0000i	− 1.00437i	0.864761	−	0.502184i	$-0.167470\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	22.0000	0.917463
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	− 10.0000i	− 0.415586i
$580$	0	0
$581$	4.00000i	0.165948i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	32.0000i	1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$	32.0000i	1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	22.0000	0.904959
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	28.0000i	1.13836i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	72.0000i	2.91281i
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	2.00000i	0.0802572i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	− 8.00000i	− 0.318981i
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	48.0000i	1.90482i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	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
$645$	16.0000i	0.629999i
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	− 2.00000i	− 0.0779089i	−0.999241	−	0.0389545i	$-0.987597\pi$
$659$	− 2.00000i	− 0.0779089i	0.999241	−	0.0389545i	$-0.0124027\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	11.0000i	0.423390i
$676$	0	0
$677$	− 24.0000i	− 0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$	− 24.0000i	− 0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	0	0
$685$	8.00000i	0.305664i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	28.0000i	1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$	28.0000i	1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 14.0000i	− 0.529529i
$700$	0	0
$701$	50.0000i	1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$	50.0000i	1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−48.0000	−1.80778
$706$	0	0
$707$	16.0000i	0.601742i
$708$	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
$711$	−12.0000	−0.450035
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 48.0000i	− 1.79510i
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	− 22.0000i	− 0.817059i
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000i	0.591781i
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	18.0000i	0.657706i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−32.0000	−1.16000	−0.580000	−	0.814617i	$-0.696947\pi$
$761$	−32.0000	−1.16000	−0.580000	+	0.814617i	$0.696947\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	16.0000i	0.578481i
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	− 24.0000i	− 0.864339i
$772$	0	0
$773$	− 8.00000i	− 0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$	− 8.00000i	− 0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 28.0000i	− 1.00192i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	56.0000	1.99873
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	− 18.0000i	− 0.640817i
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	24.0000i	0.851192i
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	−20.0000	−0.704033
$808$	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
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	0	0
$815$	−64.0000	−2.24182
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	− 6.00000i	− 0.209657i
$820$	0	0
$821$	2.00000i	0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$	2.00000i	0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	22.0000	0.765942
$826$	0	0
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	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
$831$	22.0000	0.763172
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	10.0000i	0.344418i
$844$	0	0
$845$	− 92.0000i	− 3.16490i
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 4.00000i	− 0.137118i
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	0	0
$859$	44.0000i	1.50126i	0.660722	+	0.750630i	$0.270250\pi$
$859$	44.0000i	1.50126i	−0.660722	+	0.750630i	$0.729750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.0000	−0.476566	−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000	−0.476566	−0.238283	+	0.971196i	$0.576585\pi$
$864$	0	0
$865$	64.0000	2.17607
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	− 24.0000i	− 0.811348i
$876$	0	0
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	0	0
$885$	32.0000i	1.07567i
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	48.0000i	1.60626i
$894$	0	0
$895$	−72.0000	−2.40669
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	0	0
$900$	0	0
$901$	24.0000i	0.799556i
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	− 16.0000i	− 0.530687i
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	24.0000i	0.793416i
$916$	0	0
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	− 84.0000i	− 2.76489i
$924$	0	0
$925$	− 22.0000i	− 0.723356i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	− 4.00000i	− 0.130954i
$934$	0	0
$935$	32.0000	1.04651
$936$	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
$939$	− 10.0000i	− 0.326338i
$940$	0	0
$941$	− 28.0000i	− 0.912774i	−0.889781	−	0.456387i	$-0.849143\pi$
$941$	− 28.0000i	− 0.912774i	0.889781	−	0.456387i	$-0.150857\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	− 10.0000i	− 0.324956i	−0.986712	−	0.162478i	$-0.948051\pi$
$947$	− 10.0000i	− 0.324956i	0.986712	−	0.162478i	$-0.0519487\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	− 72.0000i	− 2.32987i
$956$	0	0
$957$	− 4.00000i	− 0.129302i
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 18.0000i	− 0.580042i
$964$	0	0
$965$	− 40.0000i	− 1.28765i
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	36.0000i	1.15529i	0.816286	+	0.577647i	$0.196029\pi$
$971$	36.0000i	1.15529i	−0.816286	+	0.577647i	$0.803971\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	0	0
$975$	66.0000	2.11369
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	88.0000	2.80391
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	8.00000i	0.254385i
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	64.0000i	2.02894i
$996$	0	0
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.q.2689.2		2
4.3	odd	2		5376.2.c.p.2689.1		2
8.3	odd	2		5376.2.c.p.2689.2		2
8.5	even	2	inner	5376.2.c.q.2689.1		2
16.3	odd	4		336.2.a.f.1.1		1
16.5	even	4		1344.2.a.k.1.1		1
16.11	odd	4		1344.2.a.a.1.1		1
16.13	even	4		84.2.a.a.1.1	✓	1
48.5	odd	4		4032.2.a.bm.1.1		1
48.11	even	4		4032.2.a.bn.1.1		1
48.29	odd	4		252.2.a.a.1.1		1
48.35	even	4		1008.2.a.a.1.1		1
80.13	odd	4		2100.2.k.i.1849.1		2
80.19	odd	4		8400.2.a.e.1.1		1
80.29	even	4		2100.2.a.r.1.1		1
80.77	odd	4		2100.2.k.i.1849.2		2
112.3	even	12		2352.2.q.z.961.1		2
112.13	odd	4		588.2.a.d.1.1		1
112.19	even	12		2352.2.q.z.1537.1		2
112.27	even	4		9408.2.a.df.1.1		1
112.45	odd	12		588.2.i.d.373.1		2
112.51	odd	12		2352.2.q.b.1537.1		2
112.61	odd	12		588.2.i.d.361.1		2
112.67	odd	12		2352.2.q.b.961.1		2
112.69	odd	4		9408.2.a.bn.1.1		1
112.83	even	4		2352.2.a.a.1.1		1
112.93	even	12		588.2.i.e.361.1		2
112.109	even	12		588.2.i.e.373.1		2
144.13	even	12		2268.2.j.a.1513.1		2
144.29	odd	12		2268.2.j.n.757.1		2
144.61	even	12		2268.2.j.a.757.1		2
144.77	odd	12		2268.2.j.n.1513.1		2
240.29	odd	4		6300.2.a.w.1.1		1
240.77	even	4		6300.2.k.g.6049.1		2
240.173	even	4		6300.2.k.g.6049.2		2
336.83	odd	4		7056.2.a.cd.1.1		1
336.125	even	4		1764.2.a.k.1.1		1
336.173	even	12		1764.2.k.a.361.1		2
336.221	odd	12		1764.2.k.k.1549.1		2
336.269	even	12		1764.2.k.a.1549.1		2
336.317	odd	12		1764.2.k.k.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.a.1.1	✓	1	16.13	even	4
252.2.a.a.1.1		1	48.29	odd	4
336.2.a.f.1.1		1	16.3	odd	4
588.2.a.d.1.1		1	112.13	odd	4
588.2.i.d.361.1		2	112.61	odd	12
588.2.i.d.373.1		2	112.45	odd	12
588.2.i.e.361.1		2	112.93	even	12
588.2.i.e.373.1		2	112.109	even	12
1008.2.a.a.1.1		1	48.35	even	4
1344.2.a.a.1.1		1	16.11	odd	4
1344.2.a.k.1.1		1	16.5	even	4
1764.2.a.k.1.1		1	336.125	even	4
1764.2.k.a.361.1		2	336.173	even	12
1764.2.k.a.1549.1		2	336.269	even	12
1764.2.k.k.361.1		2	336.317	odd	12
1764.2.k.k.1549.1		2	336.221	odd	12
2100.2.a.r.1.1		1	80.29	even	4
2100.2.k.i.1849.1		2	80.13	odd	4
2100.2.k.i.1849.2		2	80.77	odd	4
2268.2.j.a.757.1		2	144.61	even	12
2268.2.j.a.1513.1		2	144.13	even	12
2268.2.j.n.757.1		2	144.29	odd	12
2268.2.j.n.1513.1		2	144.77	odd	12
2352.2.a.a.1.1		1	112.83	even	4
2352.2.q.b.961.1		2	112.67	odd	12
2352.2.q.b.1537.1		2	112.51	odd	12
2352.2.q.z.961.1		2	112.3	even	12
2352.2.q.z.1537.1		2	112.19	even	12
4032.2.a.bm.1.1		1	48.5	odd	4
4032.2.a.bn.1.1		1	48.11	even	4
5376.2.c.p.2689.1		2	4.3	odd	2
5376.2.c.p.2689.2		2	8.3	odd	2
5376.2.c.q.2689.1		2	8.5	even	2	inner
5376.2.c.q.2689.2		2	1.1	even	1	trivial
6300.2.a.w.1.1		1	240.29	odd	4
6300.2.k.g.6049.1		2	240.77	even	4
6300.2.k.g.6049.2		2	240.173	even	4
7056.2.a.cd.1.1		1	336.83	odd	4
8400.2.a.e.1.1		1	80.19	odd	4
9408.2.a.bn.1.1		1	112.69	odd	4
9408.2.a.df.1.1		1	112.27	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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +6.00000i q^{13} -4.00000 q^{15} -4.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} -2.00000 q^{23} -11.0000 q^{25} -1.00000i q^{27} +2.00000i q^{29} -2.00000 q^{33} +4.00000i q^{35} +2.00000i q^{37} -6.00000 q^{39} -4.00000i q^{43} -4.00000i q^{45} +12.0000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} -8.00000i q^{59} -6.00000i q^{61} -1.00000 q^{63} -24.0000 q^{65} +8.00000i q^{67} -2.00000i q^{69} -14.0000 q^{71} +2.00000 q^{73} -11.0000i q^{75} +2.00000i q^{77} +12.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -16.0000i q^{85} -2.00000 q^{87} +6.00000i q^{91} -16.0000 q^{95} -2.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} - 8 q^{15} - 8 q^{17} - 4 q^{23} - 22 q^{25} - 4 q^{33} - 12 q^{39} + 24 q^{47} + 2 q^{49} - 16 q^{55} - 8 q^{57} - 2 q^{63} - 48 q^{65} - 28 q^{71} + 4 q^{73} + 24 q^{79} + 2 q^{81} - 4 q^{87} - 32 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 - 8 * q^15 - 8 * q^17 - 4 * q^23 - 22 * q^25 - 4 * q^33 - 12 * q^39 + 24 * q^47 + 2 * q^49 - 16 * q^55 - 8 * q^57 - 2 * q^63 - 48 * q^65 - 28 * q^71 + 4 * q^73 + 24 * q^79 + 2 * q^81 - 4 * q^87 - 32 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more