LMFDB - Embedded newform 5376.2.c.ba.2689.1

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 2688)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +4.00000i q^{13} +2.00000 q^{15} -8.00000 q^{17} -4.00000i q^{19} -1.00000i q^{21} -2.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} -4.00000 q^{33} +2.00000i q^{35} -2.00000i q^{37} +4.00000 q^{39} +12.0000 q^{41} -6.00000i q^{43} -2.00000i q^{45} +8.00000 q^{47} +1.00000 q^{49} +8.00000i q^{51} +6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} +4.00000i q^{59} +4.00000i q^{61} -1.00000 q^{63} -8.00000 q^{65} +10.0000i q^{67} +2.00000i q^{69} +6.00000 q^{71} +10.0000 q^{73} -1.00000i q^{75} -4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -16.0000i q^{85} +6.00000 q^{87} -16.0000 q^{89} +4.00000i q^{91} +8.00000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{15} - 16 q^{17} - 4 q^{23} + 2 q^{25} - 8 q^{33} + 8 q^{39} + 24 q^{41} + 16 q^{47} + 2 q^{49} + 16 q^{55} - 8 q^{57} - 2 q^{63} - 16 q^{65} + 12 q^{71} + 20 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} - 32 q^{89} + 16 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 - 16 * q^17 - 4 * q^23 + 2 * q^25 - 8 * q^33 + 8 * q^39 + 24 * q^41 + 16 * q^47 + 2 * q^49 + 16 * q^55 - 8 * q^57 - 2 * q^63 - 16 * q^65 + 12 * q^71 + 20 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 - 32 * q^89 + 16 * q^95 + 20 * 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
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	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	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	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
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.00000i	1.12022i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	2.00000i	0.240772i
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	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
$85$	− 16.0000i	− 1.73544i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	4.00000i	0.419314i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	−2.00000	−0.172133
$136$	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
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	16.0000	1.33799
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.0000i	1.91541i	0.287754	+	0.957704i	$0.407091\pi$
$157$	24.0000i	1.91541i	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	32.0000i	2.34007i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−26.0000	−1.88129	−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000	−1.88129	−0.940647	+	0.339387i	$0.889781\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	8.00000i	0.572892i
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	24.0000i	1.67623i
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	− 10.0000i	− 0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$	− 10.0000i	− 0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$	0	0
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	− 32.0000i	− 2.15255i
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	16.0000i	1.04372i
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	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
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	16.0000i	0.979184i
$268$	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
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	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
$295$	−8.00000	−0.465778
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 6.00000i	− 0.345834i
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	16.0000i	0.910208i
$310$	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
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	32.0000i	1.78053i
$324$	0	0
$325$	4.00000i	0.221880i
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	34.0000i	1.86881i	0.356214	+	0.934405i	$0.384068\pi$
$331$	34.0000i	1.86881i	−0.356214	+	0.934405i	$0.615932\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−20.0000	−1.09272
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	− 2.00000i	− 0.108625i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	− 16.0000i	− 0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$	− 16.0000i	− 0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	12.0000i	0.636894i
$356$	0	0
$357$	8.00000i	0.423405i
$358$	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
$363$	5.00000i	0.262432i
$364$	0	0
$365$	20.0000i	1.04685i
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	− 38.0000i	− 1.96757i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	− 38.0000i	− 1.96757i	0.179364	−	0.983783i	$-0.442596\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	2.00000i	0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$	2.00000i	0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$	0	0
$381$	− 20.0000i	− 1.02463i
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	10.0000i	0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$	10.0000i	0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	16.0000i	0.805047i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	− 16.0000i	− 0.772487i
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	0	0
$445$	− 32.0000i	− 1.51695i
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	− 48.0000i	− 2.26023i
$452$	0	0
$453$	20.0000i	0.939682i
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	− 8.00000i	− 0.373408i
$460$	0	0
$461$	14.0000i	0.652045i	0.945362	+	0.326023i	$0.105709\pi$
$461$	14.0000i	0.652045i	−0.945362	+	0.326023i	$0.894291\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	0	0
$469$	10.0000i	0.461757i
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	20.0000i	0.908153i
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	− 48.0000i	− 2.16181i
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	14.0000i	0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$	14.0000i	0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$	0	0
$501$	− 4.00000i	− 0.178707i
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	− 22.0000i	− 0.975133i	−0.873086	−	0.487566i	$-0.837885\pi$
$509$	− 22.0000i	− 0.975133i	0.873086	−	0.487566i	$-0.162115\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	48.0000i	2.07911i
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	0	0
$549$	− 4.00000i	− 0.170716i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	− 4.00000i	− 0.169791i
$556$	0	0
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	32.0000	1.35104
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	4.00000i	0.168281i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	− 26.0000i	− 1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$571$	− 26.0000i	− 1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$572$	0	0
$573$	26.0000i	1.08617i
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	2.00000i	0.0831172i
$580$	0	0
$581$	12.0000i	0.497844i
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	− 10.0000i	− 0.407231i
$604$	0	0
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	32.0000i	1.29458i
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	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
$621$	− 2.00000i	− 0.0802572i
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	16.0000i	0.638978i
$628$	0	0
$629$	16.0000i	0.637962i
$630$	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
$633$	−10.0000	−0.397464
$634$	0	0
$635$	40.0000i	1.58735i
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 24.0000i	− 0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$	− 24.0000i	− 0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$	0	0
$645$	− 12.0000i	− 0.472500i
$646$	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
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	28.0000i	1.08907i	0.838737	+	0.544537i	$0.183295\pi$
$661$	28.0000i	1.08907i	−0.838737	+	0.544537i	$0.816705\pi$
$662$	0	0
$663$	−32.0000	−1.24278
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	− 24.0000i	− 0.927894i
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	50.0000i	1.92166i	0.277145	+	0.960828i	$0.410612\pi$
$677$	50.0000i	1.92166i	−0.277145	+	0.960828i	$0.589388\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	4.00000i	0.152832i
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	− 24.0000i	− 0.913003i	−0.889723	−	0.456502i	$-0.849102\pi$
$691$	− 24.0000i	− 0.913003i	0.889723	−	0.456502i	$-0.150898\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	−96.0000	−3.63626
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	34.0000i	1.28416i	0.766637	+	0.642081i	$0.221929\pi$
$701$	34.0000i	1.28416i	−0.766637	+	0.642081i	$0.778071\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	− 34.0000i	− 1.27690i	−0.769665	−	0.638448i	$-0.779577\pi$
$709$	− 34.0000i	− 1.27690i	0.769665	−	0.638448i	$-0.220423\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	32.0000i	1.19673i
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000i	1.77534i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	40.0000	1.47342
$738$	0	0
$739$	− 18.0000i	− 0.662141i	−0.943606	−	0.331070i	$-0.892590\pi$
$739$	− 18.0000i	− 0.662141i	0.943606	−	0.331070i	$-0.107410\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	8.00000i	0.292314i
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	− 40.0000i	− 1.45575i
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−28.0000	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$761$	−28.0000	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$762$	0	0
$763$	− 6.00000i	− 0.217215i
$764$	0	0
$765$	16.0000i	0.578481i
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	24.0000i	0.864339i
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	− 48.0000i	− 1.71978i
$780$	0	0
$781$	− 24.0000i	− 0.858788i
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−48.0000	−1.71319
$786$	0	0
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	0	0
$789$	− 18.0000i	− 0.640817i
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	54.0000i	1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$	54.0000i	1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	0	0
$799$	−64.0000	−2.26416
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	− 4.00000i	− 0.140981i
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	− 12.0000i	− 0.421377i	−0.977553	−	0.210688i	$-0.932429\pi$
$811$	− 12.0000i	− 0.421377i	0.977553	−	0.210688i	$-0.0675706\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	−28.0000	−0.980797
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	− 4.00000i	− 0.139771i
$820$	0	0
$821$	22.0000i	0.767805i	0.923374	+	0.383903i	$0.125420\pi$
$821$	22.0000i	0.767805i	−0.923374	+	0.383903i	$0.874580\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	32.0000i	1.11141i	0.831381	+	0.555703i	$0.187551\pi$
$829$	32.0000i	1.11141i	−0.831381	+	0.555703i	$0.812449\pi$
$830$	0	0
$831$	−30.0000	−1.04069
$832$	0	0
$833$	−8.00000	−0.277184
$834$	0	0
$835$	8.00000i	0.276851i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	− 6.00000i	− 0.206406i
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.00000i	0.137118i
$852$	0	0
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	− 28.0000i	− 0.955348i	−0.878537	−	0.477674i	$-0.841480\pi$
$859$	− 28.0000i	− 0.955348i	0.878537	−	0.477674i	$-0.158520\pi$
$860$	0	0
$861$	− 12.0000i	− 0.408959i
$862$	0	0
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	− 47.0000i	− 1.59620i
$868$	0	0
$869$	− 32.0000i	− 1.08553i
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	58.0000i	1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$	58.0000i	1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$	0	0
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	− 26.0000i	− 0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$	− 26.0000i	− 0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$	0	0
$885$	8.00000i	0.268917i
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	− 32.0000i	− 1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 48.0000i	− 1.59911i
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−40.0000	−1.32964
$906$	0	0
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	0	0
$909$	− 6.00000i	− 0.199007i
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	8.00000i	0.264472i
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	− 2.00000i	− 0.0657596i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	−64.0000	−2.09302
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	− 2.00000i	− 0.0652675i
$940$	0	0
$941$	38.0000i	1.23876i	0.785090	+	0.619382i	$0.212617\pi$
$941$	38.0000i	1.23876i	−0.785090	+	0.619382i	$0.787383\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	40.0000i	1.29983i	0.760009	+	0.649913i	$0.225195\pi$
$947$	40.0000i	1.29983i	−0.760009	+	0.649913i	$0.774805\pi$
$948$	0	0
$949$	40.0000i	1.29845i
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	0	0
$955$	− 52.0000i	− 1.68268i
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	− 44.0000i	− 1.41203i	−0.708198	−	0.706014i	$-0.750492\pi$
$971$	− 44.0000i	− 1.41203i	0.708198	−	0.706014i	$-0.249508\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	64.0000i	2.04545i
$980$	0	0
$981$	6.00000i	0.191565i
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	12.0000i	0.381578i
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	34.0000	1.07896
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	60.0000i	1.90022i	0.311916	+	0.950110i	$0.399029\pi$
$997$	60.0000i	1.90022i	−0.311916	+	0.950110i	$0.600971\pi$
$998$	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.ba.2689.1		2
4.3	odd	2		5376.2.c.c.2689.2		2
8.3	odd	2		5376.2.c.c.2689.1		2
8.5	even	2	inner	5376.2.c.ba.2689.2		2
16.3	odd	4		2688.2.a.k.1.1	yes	1
16.5	even	4		2688.2.a.b.1.1	✓	1
16.11	odd	4		2688.2.a.o.1.1	yes	1
16.13	even	4		2688.2.a.v.1.1	yes	1
48.5	odd	4		8064.2.a.w.1.1		1
48.11	even	4		8064.2.a.z.1.1		1
48.29	odd	4		8064.2.a.d.1.1		1
48.35	even	4		8064.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2688.2.a.b.1.1	✓	1	16.5	even	4
2688.2.a.k.1.1	yes	1	16.3	odd	4
2688.2.a.o.1.1	yes	1	16.11	odd	4
2688.2.a.v.1.1	yes	1	16.13	even	4
5376.2.c.c.2689.1		2	8.3	odd	2
5376.2.c.c.2689.2		2	4.3	odd	2
5376.2.c.ba.2689.1		2	1.1	even	1	trivial
5376.2.c.ba.2689.2		2	8.5	even	2	inner
8064.2.a.d.1.1		1	48.29	odd	4
8064.2.a.e.1.1		1	48.35	even	4
8064.2.a.w.1.1		1	48.5	odd	4
8064.2.a.z.1.1		1	48.11	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +4.00000i q^{13} +2.00000 q^{15} -8.00000 q^{17} -4.00000i q^{19} -1.00000i q^{21} -2.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} -4.00000 q^{33} +2.00000i q^{35} -2.00000i q^{37} +4.00000 q^{39} +12.0000 q^{41} -6.00000i q^{43} -2.00000i q^{45} +8.00000 q^{47} +1.00000 q^{49} +8.00000i q^{51} +6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} +4.00000i q^{59} +4.00000i q^{61} -1.00000 q^{63} -8.00000 q^{65} +10.0000i q^{67} +2.00000i q^{69} +6.00000 q^{71} +10.0000 q^{73} -1.00000i q^{75} -4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -16.0000i q^{85} +6.00000 q^{87} -16.0000 q^{89} +4.00000i q^{91} +8.00000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} - 16 q^{17} - 4 q^{23} + 2 q^{25} - 8 q^{33} + 8 q^{39} + 24 q^{41} + 16 q^{47} + 2 q^{49} + 16 q^{55} - 8 q^{57} - 2 q^{63} - 16 q^{65} + 12 q^{71} + 20 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} - 32 q^{89} + 16 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 - 16 * q^17 - 4 * q^23 + 2 * q^25 - 8 * q^33 + 8 * q^39 + 24 * q^41 + 16 * q^47 + 2 * q^49 + 16 * q^55 - 8 * q^57 - 2 * q^63 - 16 * q^65 + 12 * q^71 + 20 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 - 32 * q^89 + 16 * q^95 + 20 * q^97

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