LMFDB - Embedded newform 2816.2.c.l.1409.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.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:	$22.4858732092$
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 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.l.1409.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} -1.00000i q^{11} -2.00000i q^{13} -3.00000 q^{15} -8.00000 q^{17} +6.00000i q^{19} +4.00000i q^{21} -5.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} +4.00000i q^{29} +1.00000 q^{31} +1.00000 q^{33} +12.0000i q^{35} -3.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +6.00000i q^{43} +6.00000i q^{45} -12.0000 q^{47} +9.00000 q^{49} -8.00000i q^{51} +6.00000i q^{53} +3.00000 q^{55} -6.00000 q^{57} -3.00000i q^{59} +8.00000 q^{63} +6.00000 q^{65} +11.0000i q^{67} -5.00000i q^{69} +5.00000 q^{71} +10.0000 q^{73} -4.00000i q^{75} -4.00000i q^{77} -2.00000 q^{79} +1.00000 q^{81} -2.00000i q^{83} -24.0000i q^{85} -4.00000 q^{87} +5.00000 q^{89} -8.00000i q^{91} +1.00000i q^{93} -18.0000 q^{95} +13.0000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 8 * q^7 + 4 * q^9	$ 2 q + 8 q^{7} + 4 q^{9} - 6 q^{15} - 16 q^{17} - 10 q^{23} - 8 q^{25} + 2 q^{31} + 2 q^{33} + 4 q^{39} + 12 q^{41} - 24 q^{47} + 18 q^{49} + 6 q^{55} - 12 q^{57} + 16 q^{63} + 12 q^{65} + 10 q^{71} + 20 q^{73} - 4 q^{79} + 2 q^{81} - 8 q^{87} + 10 q^{89} - 36 q^{95} + 26 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 + 4 * q^9 - 6 * q^15 - 16 * q^17 - 10 * q^23 - 8 * q^25 + 2 * q^31 + 2 * q^33 + 4 * q^39 + 12 * q^41 - 24 * q^47 + 18 * q^49 + 6 * q^55 - 12 * q^57 + 16 * q^63 + 12 * q^65 + 10 * q^71 + 20 * q^73 - 4 * q^79 + 2 * q^81 - 8 * q^87 + 10 * q^89 - 36 * q^95 + 26 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	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
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	4.00000i	0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$	4.00000i	0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	12.0000i	2.02837i
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	6.00000i	0.894427i
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	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
$55$	3.00000	0.404520
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	− 3.00000i	− 0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$	− 3.00000i	− 0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	8.00000	1.00791
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	− 5.00000i	− 0.601929i
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	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
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	− 24.0000i	− 2.60317i
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	−18.0000	−1.84676
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	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
$105$	−12.0000	−1.17108
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	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
$111$	3.00000	0.284747
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	− 15.0000i	− 1.39876i
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−32.0000	−2.93344
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	3.00000i	0.268328i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	− 8.00000i	− 0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$	− 8.00000i	− 0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	0	0
$133$	24.0000i	2.08106i
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	8.00000i	0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$	8.00000i	0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	− 4.00000i	− 0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$	− 4.00000i	− 0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	−16.0000	−1.29352
$154$	0	0
$155$	3.00000i	0.240966i
$156$	0	0
$157$	− 23.0000i	− 1.83560i	−0.397043	−	0.917800i	$-0.629964\pi$
$157$	− 23.0000i	− 1.83560i	0.397043	−	0.917800i	$-0.370036\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	3.00000i	0.233550i
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	12.0000i	0.917663i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	− 13.0000i	− 0.971666i	−0.874052	−	0.485833i	$-0.838516\pi$
$179$	− 13.0000i	− 0.971666i	0.874052	−	0.485833i	$-0.161484\pi$
$180$	0	0
$181$	− 7.00000i	− 0.520306i	−0.965567	−	0.260153i	$-0.916227\pi$
$181$	− 7.00000i	− 0.520306i	0.965567	−	0.260153i	$-0.0837730\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	20.0000i	1.45479i
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	0	0
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	0	0
$195$	6.00000i	0.429669i
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	−11.0000	−0.775880
$202$	0	0
$203$	16.0000i	1.12298i
$204$	0	0
$205$	18.0000i	1.25717i
$206$	0	0
$207$	−10.0000	−0.695048
$208$	0	0
$209$	6.00000	0.415029
$210$	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
$213$	5.00000i	0.342594i
$214$	0	0
$215$	−18.0000	−1.22759
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	16.0000i	1.07628i
$222$	0	0
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 27.0000i	− 1.78421i	−0.451828	−	0.892105i	$-0.649228\pi$
$229$	− 27.0000i	− 1.78421i	0.451828	−	0.892105i	$-0.350772\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	− 36.0000i	− 2.34838i
$236$	0	0
$237$	− 2.00000i	− 0.129914i
$238$	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
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	27.0000i	1.72497i
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	− 15.0000i	− 0.946792i	−0.880850	−	0.473396i	$-0.843028\pi$
$251$	− 15.0000i	− 0.946792i	0.880850	−	0.473396i	$-0.156972\pi$
$252$	0	0
$253$	5.00000i	0.314347i
$254$	0	0
$255$	24.0000	1.50294
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	− 12.0000i	− 0.745644i
$260$	0	0
$261$	8.00000i	0.495188i
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	5.00000i	0.305995i
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	0	0
$285$	− 18.0000i	− 1.06623i
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	13.0000i	0.762073i
$292$	0	0
$293$	− 22.0000i	− 1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	− 22.0000i	− 1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	10.0000i	0.578315i
$300$	0	0
$301$	24.0000i	1.38334i
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	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
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	0	0
$315$	24.0000i	1.35225i
$316$	0	0
$317$	− 19.0000i	− 1.06715i	−0.845754	−	0.533573i	$-0.820849\pi$
$317$	− 19.0000i	− 1.06715i	0.845754	−	0.533573i	$-0.179151\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	− 48.0000i	− 2.67079i
$324$	0	0
$325$	8.00000i	0.443760i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	− 13.0000i	− 0.714545i	−0.934000	−	0.357272i	$-0.883707\pi$
$331$	− 13.0000i	− 0.714545i	0.934000	−	0.357272i	$-0.116293\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	−33.0000	−1.80298
$336$	0	0
$337$	36.0000	1.96104	0.980522	−	0.196407i	$-0.0629273\pi$
$337$	36.0000	1.96104	0.980522	+	0.196407i	$0.0629273\pi$
$338$	0	0
$339$	1.00000i	0.0543125i
$340$	0	0
$341$	− 1.00000i	− 0.0541530i
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	15.0000	0.807573
$346$	0	0
$347$	− 30.0000i	− 1.61048i	−0.592946	−	0.805242i	$-0.702035\pi$
$347$	− 30.0000i	− 1.61048i	0.592946	−	0.805242i	$-0.297965\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	15.0000i	0.796117i
$356$	0	0
$357$	− 32.0000i	− 1.69362i
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	30.0000i	1.57027i
$366$	0	0
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	− 17.0000i	− 0.873231i	−0.899648	−	0.436616i	$-0.856177\pi$
$379$	− 17.0000i	− 0.873231i	0.899648	−	0.436616i	$-0.143823\pi$
$380$	0	0
$381$	2.00000i	0.102463i
$382$	0	0
$383$	13.0000	0.664269	0.332134	−	0.943232i	$-0.392231\pi$
$383$	13.0000	0.664269	0.332134	+	0.943232i	$0.392231\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	− 17.0000i	− 0.861934i	−0.902368	−	0.430967i	$-0.858172\pi$
$389$	− 17.0000i	− 0.861934i	0.902368	−	0.430967i	$-0.141828\pi$
$390$	0	0
$391$	40.0000	2.02289
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	− 6.00000i	− 0.301893i
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	0	0
$405$	3.00000i	0.149071i
$406$	0	0
$407$	−3.00000	−0.148704
$408$	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
$411$	3.00000i	0.147979i
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	8.00000i	0.390826i	0.980721	+	0.195413i	$0.0626047\pi$
$419$	8.00000i	0.390826i	−0.980721	+	0.195413i	$0.937395\pi$
$420$	0	0
$421$	34.0000i	1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$	34.0000i	1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$	0	0
$423$	−24.0000	−1.16692
$424$	0	0
$425$	32.0000	1.55223
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.00000i	− 0.0965609i
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	− 12.0000i	− 0.575356i
$436$	0	0
$437$	− 30.0000i	− 1.43509i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	21.0000i	0.997740i	0.866677	+	0.498870i	$0.166252\pi$
$443$	21.0000i	0.997740i	−0.866677	+	0.498870i	$0.833748\pi$
$444$	0	0
$445$	15.0000i	0.711068i
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	− 6.00000i	− 0.282529i
$452$	0	0
$453$	10.0000i	0.469841i
$454$	0	0
$455$	24.0000	1.12514
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	− 40.0000i	− 1.86704i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	7.00000	0.325318	0.162659	−	0.986682i	$-0.447993\pi$
$463$	7.00000	0.325318	0.162659	+	0.986682i	$0.447993\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	− 5.00000i	− 0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$	− 5.00000i	− 0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$	0	0
$469$	44.0000i	2.03173i
$470$	0	0
$471$	23.0000	1.05978
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	− 20.0000i	− 0.910032i
$484$	0	0
$485$	39.0000i	1.77090i
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	− 32.0000i	− 1.44121i
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	20.0000	0.897123
$498$	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
$501$	− 18.0000i	− 0.804181i
$502$	0	0
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	19.0000i	0.842160i	0.907023	+	0.421080i	$0.138349\pi$
$509$	19.0000i	0.842160i	−0.907023	+	0.421080i	$0.861651\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	−30.0000	−1.32453
$514$	0	0
$515$	− 48.0000i	− 2.11513i
$516$	0	0
$517$	12.0000i	0.527759i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	0	0
$523$	− 18.0000i	− 0.787085i	−0.919306	−	0.393543i	$-0.871249\pi$
$523$	− 18.0000i	− 0.787085i	0.919306	−	0.393543i	$-0.128751\pi$
$524$	0	0
$525$	− 16.0000i	− 0.698297i
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−48.0000	−2.07522
$536$	0	0
$537$	13.0000	0.560991
$538$	0	0
$539$	− 9.00000i	− 0.387657i
$540$	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
$543$	7.00000	0.300399
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	− 44.0000i	− 1.88130i	−0.339372	−	0.940652i	$-0.610215\pi$
$547$	− 44.0000i	− 1.88130i	0.339372	−	0.940652i	$-0.389785\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	9.00000i	0.382029i
$556$	0	0
$557$	36.0000i	1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$	36.0000i	1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	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
$565$	3.00000i	0.126211i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	11.0000i	0.459532i
$574$	0	0
$575$	20.0000	0.834058
$576$	0	0
$577$	−27.0000	−1.12402	−0.562012	−	0.827129i	$-0.689973\pi$
$577$	−27.0000	−1.12402	−0.562012	+	0.827129i	$0.689973\pi$
$578$	0	0
$579$	− 12.0000i	− 0.498703i
$580$	0	0
$581$	− 8.00000i	− 0.331896i
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	6.00000i	0.247226i
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	− 96.0000i	− 3.93562i
$596$	0	0
$597$	24.0000i	0.982255i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	0	0
$603$	22.0000i	0.895909i
$604$	0	0
$605$	− 3.00000i	− 0.121967i
$606$	0	0
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	0	0
$609$	−16.0000	−0.648353
$610$	0	0
$611$	24.0000i	0.970936i
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	−18.0000	−0.725830
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	− 29.0000i	− 1.16561i	−0.812613	−	0.582804i	$-0.801955\pi$
$619$	− 29.0000i	− 1.16561i	0.812613	−	0.582804i	$-0.198045\pi$
$620$	0	0
$621$	− 25.0000i	− 1.00322i
$622$	0	0
$623$	20.0000	0.801283
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	6.00000i	0.239617i
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	6.00000i	0.238103i
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	35.0000	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$641$	35.0000	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	− 18.0000i	− 0.708749i
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	4.00000i	0.156772i
$652$	0	0
$653$	− 13.0000i	− 0.508729i	−0.967108	−	0.254365i	$-0.918134\pi$
$653$	− 13.0000i	− 0.508729i	0.967108	−	0.254365i	$-0.0818663\pi$
$654$	0	0
$655$	24.0000	0.937758
$656$	0	0
$657$	20.0000	0.780274
$658$	0	0
$659$	50.0000i	1.94772i	0.227142	+	0.973862i	$0.427062\pi$
$659$	50.0000i	1.94772i	−0.227142	+	0.973862i	$0.572938\pi$
$660$	0	0
$661$	19.0000i	0.739014i	0.929228	+	0.369507i	$0.120473\pi$
$661$	19.0000i	0.739014i	−0.929228	+	0.369507i	$0.879527\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	0	0
$665$	−72.0000	−2.79204
$666$	0	0
$667$	− 20.0000i	− 0.774403i
$668$	0	0
$669$	13.0000i	0.502609i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	52.0000	1.99558
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	9.00000i	0.343872i
$686$	0	0
$687$	27.0000	1.03011
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	11.0000i	0.418460i	0.977866	+	0.209230i	$0.0670957\pi$
$691$	11.0000i	0.418460i	−0.977866	+	0.209230i	$0.932904\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	−48.0000	−1.81813
$698$	0	0
$699$	12.0000i	0.453882i
$700$	0	0
$701$	36.0000i	1.35970i	0.733351	+	0.679851i	$0.237955\pi$
$701$	36.0000i	1.35970i	−0.733351	+	0.679851i	$0.762045\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	36.0000	1.35584
$706$	0	0
$707$	16.0000i	0.601742i
$708$	0	0
$709$	5.00000i	0.187779i	0.995583	+	0.0938895i	$0.0299300\pi$
$709$	5.00000i	0.187779i	−0.995583	+	0.0938895i	$0.970070\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	− 6.00000i	− 0.224387i
$716$	0	0
$717$	6.00000i	0.224074i
$718$	0	0
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	0	0
$723$	− 20.0000i	− 0.743808i
$724$	0	0
$725$	− 16.0000i	− 0.594225i
$726$	0	0
$727$	27.0000	1.00137	0.500687	−	0.865628i	$-0.333081\pi$
$727$	27.0000	1.00137	0.500687	+	0.865628i	$0.333081\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 48.0000i	− 1.77534i
$732$	0	0
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	0	0
$735$	−27.0000	−0.995910
$736$	0	0
$737$	11.0000	0.405190
$738$	0	0
$739$	− 34.0000i	− 1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$	− 34.0000i	− 1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$	0	0
$741$	12.0000i	0.440831i
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	64.0000i	2.33851i
$750$	0	0
$751$	43.0000	1.56909	0.784546	−	0.620070i	$-0.212896\pi$
$751$	43.0000	1.56909	0.784546	+	0.620070i	$0.212896\pi$
$752$	0	0
$753$	15.0000	0.546630
$754$	0	0
$755$	30.0000i	1.09181i
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	0	0
$759$	−5.00000	−0.181489
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	− 48.0000i	− 1.73544i
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	− 10.0000i	− 0.360141i
$772$	0	0
$773$	− 34.0000i	− 1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	− 34.0000i	− 1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	36.0000i	1.28983i
$780$	0	0
$781$	− 5.00000i	− 0.178914i
$782$	0	0
$783$	−20.0000	−0.714742
$784$	0	0
$785$	69.0000	2.46272
$786$	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
$791$	4.00000	0.142224
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 18.0000i	− 0.638394i
$796$	0	0
$797$	− 51.0000i	− 1.80651i	−0.429101	−	0.903256i	$-0.641170\pi$
$797$	− 51.0000i	− 1.80651i	0.429101	−	0.903256i	$-0.358830\pi$
$798$	0	0
$799$	96.0000	3.39624
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	− 60.0000i	− 2.11472i
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	14.0000i	0.491606i	0.969320	+	0.245803i	$0.0790517\pi$
$811$	14.0000i	0.491606i	−0.969320	+	0.245803i	$0.920948\pi$
$812$	0	0
$813$	− 22.0000i	− 0.771574i
$814$	0	0
$815$	−60.0000	−2.10171
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	− 16.0000i	− 0.559085i
$820$	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
$823$	15.0000	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$823$	15.0000	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	42.0000i	1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$	42.0000i	1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$	0	0
$829$	− 7.00000i	− 0.243120i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$	− 7.00000i	− 0.243120i	0.992584	−	0.121560i	$-0.0387897\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	−72.0000	−2.49465
$834$	0	0
$835$	− 54.0000i	− 1.86875i
$836$	0	0
$837$	5.00000i	0.172825i
$838$	0	0
$839$	55.0000	1.89881	0.949405	−	0.314053i	$-0.101687\pi$
$839$	55.0000	1.89881	0.949405	+	0.314053i	$0.101687\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	27.0000i	0.928828i
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	15.0000i	0.514193i
$852$	0	0
$853$	− 38.0000i	− 1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$	− 38.0000i	− 1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$	0	0
$855$	−36.0000	−1.23117
$856$	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
$859$	− 15.0000i	− 0.511793i	−0.966704	−	0.255897i	$-0.917629\pi$
$859$	− 15.0000i	− 0.511793i	0.966704	−	0.255897i	$-0.0823707\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	47.0000i	1.59620i
$868$	0	0
$869$	2.00000i	0.0678454i
$870$	0	0
$871$	22.0000	0.745442
$872$	0	0
$873$	26.0000	0.879967
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	0	0
$879$	22.0000	0.742042
$880$	0	0
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	9.00000i	0.302532i
$886$	0	0
$887$	14.0000	0.470074	0.235037	−	0.971986i	$-0.424479\pi$
$887$	14.0000	0.470074	0.235037	+	0.971986i	$0.424479\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	− 72.0000i	− 2.40939i
$894$	0	0
$895$	39.0000	1.30363
$896$	0	0
$897$	−10.0000	−0.333890
$898$	0	0
$899$	4.00000i	0.133407i
$900$	0	0
$901$	− 48.0000i	− 1.59911i
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	21.0000	0.698064
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	8.00000i	0.265343i
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−2.00000	−0.0661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 32.0000i	− 1.05673i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	12.0000i	0.394558i
$926$	0	0
$927$	−32.0000	−1.05102
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	54.0000i	1.76978i
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	−24.0000	−0.784884
$936$	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
$939$	25.0000i	0.815844i
$940$	0	0
$941$	− 42.0000i	− 1.36916i	−0.728937	−	0.684580i	$-0.759985\pi$
$941$	− 42.0000i	− 1.36916i	0.728937	−	0.684580i	$-0.240015\pi$
$942$	0	0
$943$	−30.0000	−0.976934
$944$	0	0
$945$	−60.0000	−1.95180
$946$	0	0
$947$	27.0000i	0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$	27.0000i	0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$	0	0
$949$	− 20.0000i	− 0.649227i
$950$	0	0
$951$	19.0000	0.616117
$952$	0	0
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	33.0000i	1.06785i
$956$	0	0
$957$	4.00000i	0.129302i
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	32.0000i	1.03119i
$964$	0	0
$965$	− 36.0000i	− 1.15888i
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	51.0000i	1.63667i	0.574743	+	0.818334i	$0.305102\pi$
$971$	51.0000i	1.63667i	−0.574743	+	0.818334i	$0.694898\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	0	0
$979$	− 5.00000i	− 0.159801i
$980$	0	0
$981$	4.00000i	0.127710i
$982$	0	0
$983$	59.0000	1.88181	0.940904	−	0.338674i	$-0.109978\pi$
$983$	59.0000	1.88181	0.940904	+	0.338674i	$0.109978\pi$
$984$	0	0
$985$	−54.0000	−1.72058
$986$	0	0
$987$	− 48.0000i	− 1.52786i
$988$	0	0
$989$	− 30.0000i	− 0.953945i
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	13.0000	0.412543
$994$	0	0
$995$	72.0000i	2.28255i
$996$	0	0
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	0	0
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.l.1409.2		2
4.3	odd	2		2816.2.c.b.1409.1		2
8.3	odd	2		2816.2.c.b.1409.2		2
8.5	even	2	inner	2816.2.c.l.1409.1		2
16.3	odd	4		704.2.a.j.1.1		1
16.5	even	4		352.2.a.d.1.1	yes	1
16.11	odd	4		352.2.a.b.1.1	✓	1
16.13	even	4		704.2.a.e.1.1		1
48.5	odd	4		3168.2.a.y.1.1		1
48.11	even	4		3168.2.a.z.1.1		1
48.29	odd	4		6336.2.a.g.1.1		1
48.35	even	4		6336.2.a.l.1.1		1
80.59	odd	4		8800.2.a.p.1.1		1
80.69	even	4		8800.2.a.m.1.1		1
176.21	odd	4		3872.2.a.i.1.1		1
176.43	even	4		3872.2.a.d.1.1		1
176.109	odd	4		7744.2.a.o.1.1		1
176.131	even	4		7744.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.b.1.1	✓	1	16.11	odd	4
352.2.a.d.1.1	yes	1	16.5	even	4
704.2.a.e.1.1		1	16.13	even	4
704.2.a.j.1.1		1	16.3	odd	4
2816.2.c.b.1409.1		2	4.3	odd	2
2816.2.c.b.1409.2		2	8.3	odd	2
2816.2.c.l.1409.1		2	8.5	even	2	inner
2816.2.c.l.1409.2		2	1.1	even	1	trivial
3168.2.a.y.1.1		1	48.5	odd	4
3168.2.a.z.1.1		1	48.11	even	4
3872.2.a.d.1.1		1	176.43	even	4
3872.2.a.i.1.1		1	176.21	odd	4
6336.2.a.g.1.1		1	48.29	odd	4
6336.2.a.l.1.1		1	48.35	even	4
7744.2.a.o.1.1		1	176.109	odd	4
7744.2.a.ba.1.1		1	176.131	even	4
8800.2.a.m.1.1		1	80.69	even	4
8800.2.a.p.1.1		1	80.59	odd	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} -1.00000i q^{11} -2.00000i q^{13} -3.00000 q^{15} -8.00000 q^{17} +6.00000i q^{19} +4.00000i q^{21} -5.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} +4.00000i q^{29} +1.00000 q^{31} +1.00000 q^{33} +12.0000i q^{35} -3.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +6.00000i q^{43} +6.00000i q^{45} -12.0000 q^{47} +9.00000 q^{49} -8.00000i q^{51} +6.00000i q^{53} +3.00000 q^{55} -6.00000 q^{57} -3.00000i q^{59} +8.00000 q^{63} +6.00000 q^{65} +11.0000i q^{67} -5.00000i q^{69} +5.00000 q^{71} +10.0000 q^{73} -4.00000i q^{75} -4.00000i q^{77} -2.00000 q^{79} +1.00000 q^{81} -2.00000i q^{83} -24.0000i q^{85} -4.00000 q^{87} +5.00000 q^{89} -8.00000i q^{91} +1.00000i q^{93} -18.0000 q^{95} +13.0000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 8 * q^7 + 4 * q^9	\( 2 q + 8 q^{7} + 4 q^{9} - 6 q^{15} - 16 q^{17} - 10 q^{23} - 8 q^{25} + 2 q^{31} + 2 q^{33} + 4 q^{39} + 12 q^{41} - 24 q^{47} + 18 q^{49} + 6 q^{55} - 12 q^{57} + 16 q^{63} + 12 q^{65} + 10 q^{71} + 20 q^{73} - 4 q^{79} + 2 q^{81} - 8 q^{87} + 10 q^{89} - 36 q^{95} + 26 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 + 4 * q^9 - 6 * q^15 - 16 * q^17 - 10 * q^23 - 8 * q^25 + 2 * q^31 + 2 * q^33 + 4 * q^39 + 12 * q^41 - 24 * q^47 + 18 * q^49 + 6 * q^55 - 12 * q^57 + 16 * q^63 + 12 * q^65 + 10 * q^71 + 20 * q^73 - 4 * q^79 + 2 * q^81 - 8 * q^87 + 10 * q^89 - 36 * q^95 + 26 * q^97

Introduction

L-functions

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