LMFDB - Embedded newform 8820.2.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(1,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8820.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

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:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +O(q^{10})$	$q+1.00000 q^{5} +4.57649 q^{11} -1.74806 q^{13} +2.47214 q^{17} -7.73877 q^{19} +5.99070 q^{23} +1.00000 q^{25} +3.49613 q^{29} -7.07107 q^{31} -10.9443 q^{37} -12.4721 q^{41} -10.4721 q^{43} -8.47214 q^{47} -11.6476 q^{53} +4.57649 q^{55} -4.94427 q^{59} +4.91034 q^{61} -1.74806 q^{65} +4.94427 q^{67} -7.40492 q^{71} +3.90879 q^{73} +10.9443 q^{79} +9.41641 q^{83} +2.47214 q^{85} -6.00000 q^{89} -7.73877 q^{95} +13.7295 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} - 8 q^{17} + 4 q^{25} - 8 q^{37} - 32 q^{41} - 24 q^{43} - 16 q^{47} + 16 q^{59} - 16 q^{67} + 8 q^{79} - 16 q^{83} - 8 q^{85} - 24 q^{89}+O(q^{100}) $ 4 * q + 4 * q^5 - 8 * q^17 + 4 * q^25 - 8 * q^37 - 32 * q^41 - 24 * q^43 - 16 * q^47 + 16 * q^59 - 16 * q^67 + 8 * q^79 - 16 * q^83 - 8 * q^85 - 24 * q^89

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.57649	1.37986	0.689932	−	0.723874i	$-0.257640\pi$
$11$	4.57649	1.37986	0.689932	+	0.723874i	$0.257640\pi$
$12$	0	0
$13$	−1.74806	−0.484826	−0.242413	−	0.970173i	$-0.577939\pi$
$13$	−1.74806	−0.484826	−0.242413	+	0.970173i	$0.577939\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	−7.73877	−1.77540	−0.887698	−	0.460427i	$-0.847696\pi$
$19$	−7.73877	−1.77540	−0.887698	+	0.460427i	$0.847696\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.99070	1.24915	0.624574	−	0.780966i	$-0.285273\pi$
$23$	5.99070	1.24915	0.624574	+	0.780966i	$0.285273\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.49613	0.649215	0.324607	−	0.945849i	$-0.394768\pi$
$29$	3.49613	0.649215	0.324607	+	0.945849i	$0.394768\pi$
$30$	0	0
$31$	−7.07107	−1.27000	−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107	−1.27000	−0.635001	+	0.772512i	$0.719000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.47214	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$47$	−8.47214	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.6476	−1.59992	−0.799958	−	0.600056i	$-0.795145\pi$
$53$	−11.6476	−1.59992	−0.799958	+	0.600056i	$0.795145\pi$
$54$	0	0
$55$	4.57649	0.617094
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	0	0
$61$	4.91034	0.628705	0.314352	−	0.949306i	$-0.398213\pi$
$61$	4.91034	0.628705	0.314352	+	0.949306i	$0.398213\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.74806	−0.216821
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.40492	−0.878802	−0.439401	−	0.898291i	$-0.644809\pi$
$71$	−7.40492	−0.878802	−0.439401	+	0.898291i	$0.644809\pi$
$72$	0	0
$73$	3.90879	0.457489	0.228745	−	0.973486i	$-0.426538\pi$
$73$	3.90879	0.457489	0.228745	+	0.973486i	$0.426538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.9443	1.23133	0.615663	−	0.788009i	$-0.288888\pi$
$79$	10.9443	1.23133	0.615663	+	0.788009i	$0.288888\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.41641	1.03359	0.516793	−	0.856111i	$-0.327126\pi$
$83$	9.41641	1.03359	0.516793	+	0.856111i	$0.327126\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.73877	−0.793981
$96$	0	0
$97$	13.7295	1.39402	0.697008	−	0.717063i	$-0.254514\pi$
$97$	13.7295	1.39402	0.697008	+	0.717063i	$0.254514\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.472136	0.0469793	0.0234896	−	0.999724i	$-0.492522\pi$
$101$	0.472136	0.0469793	0.0234896	+	0.999724i	$0.492522\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9799	1.06146	0.530731	−	0.847540i	$-0.321917\pi$
$107$	10.9799	1.06146	0.530731	+	0.847540i	$0.321917\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.6398	−1.75349	−0.876743	−	0.480959i	$-0.840289\pi$
$113$	−18.6398	−1.75349	−0.876743	+	0.480959i	$0.840289\pi$
$114$	0	0
$115$	5.99070	0.558636
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.94427	0.904025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4721	0.929252	0.464626	−	0.885507i	$-0.346189\pi$
$127$	10.4721	0.929252	0.464626	+	0.885507i	$0.346189\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.4164	0.997456	0.498728	−	0.866758i	$-0.333801\pi$
$131$	11.4164	0.997456	0.498728	+	0.866758i	$0.333801\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.16228	0.270172	0.135086	−	0.990834i	$-0.456869\pi$
$137$	3.16228	0.270172	0.135086	+	0.990834i	$0.456869\pi$
$138$	0	0
$139$	−11.2349	−0.952932	−0.476466	−	0.879193i	$-0.658082\pi$
$139$	−11.2349	−0.952932	−0.476466	+	0.879193i	$0.658082\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	3.49613	0.290338
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.9706	1.39028	0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706	1.39028	0.695141	+	0.718873i	$0.255342\pi$
$150$	0	0
$151$	−11.8885	−0.967476	−0.483738	−	0.875213i	$-0.660721\pi$
$151$	−11.8885	−0.967476	−0.483738	+	0.875213i	$0.660721\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.07107	−0.567962
$156$	0	0
$157$	−7.40492	−0.590977	−0.295488	−	0.955346i	$-0.595482\pi$
$157$	−7.40492	−0.590977	−0.295488	+	0.955346i	$0.595482\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.52786	−0.119672	−0.0598358	−	0.998208i	$-0.519058\pi$
$163$	−1.52786	−0.119672	−0.0598358	+	0.998208i	$0.519058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.9443	−1.93025	−0.965123	−	0.261797i	$-0.915685\pi$
$167$	−24.9443	−1.93025	−0.965123	+	0.261797i	$0.915685\pi$
$168$	0	0
$169$	−9.94427	−0.764944
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.4721	−1.70852	−0.854262	−	0.519842i	$-0.825991\pi$
$173$	−22.4721	−1.70852	−0.854262	+	0.519842i	$0.825991\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.40492	−0.553470	−0.276735	−	0.960946i	$-0.589252\pi$
$179$	−7.40492	−0.553470	−0.276735	+	0.960946i	$0.589252\pi$
$180$	0	0
$181$	4.24264	0.315353	0.157676	−	0.987491i	$-0.449600\pi$
$181$	4.24264	0.315353	0.157676	+	0.987491i	$0.449600\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.9443	−0.804639
$186$	0	0
$187$	11.3137	0.827340
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.5471	1.55909	0.779545	−	0.626346i	$-0.215450\pi$
$191$	21.5471	1.55909	0.779545	+	0.626346i	$0.215450\pi$
$192$	0	0
$193$	−16.4721	−1.18569	−0.592845	−	0.805316i	$-0.701995\pi$
$193$	−16.4721	−1.18569	−0.592845	+	0.805316i	$0.701995\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.65841	0.474392	0.237196	−	0.971462i	$-0.423772\pi$
$197$	6.65841	0.474392	0.237196	+	0.971462i	$0.423772\pi$
$198$	0	0
$199$	−10.5672	−0.749089	−0.374544	−	0.927209i	$-0.622201\pi$
$199$	−10.5672	−0.749089	−0.374544	+	0.927209i	$0.622201\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.4721	−0.871092
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−35.4164	−2.44980
$210$	0	0
$211$	4.94427	0.340378	0.170189	−	0.985411i	$-0.445562\pi$
$211$	4.94427	0.340378	0.170189	+	0.985411i	$0.445562\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.4721	−0.714194
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.32145	−0.290692
$222$	0	0
$223$	−4.98915	−0.334098	−0.167049	−	0.985949i	$-0.553424\pi$
$223$	−4.98915	−0.334098	−0.167049	+	0.985949i	$0.553424\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	13.3956	0.885208	0.442604	−	0.896717i	$-0.354055\pi$
$229$	13.3956	0.885208	0.442604	+	0.896717i	$0.354055\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.82688	0.119683	0.0598413	−	0.998208i	$-0.480941\pi$
$233$	1.82688	0.119683	0.0598413	+	0.998208i	$0.480941\pi$
$234$	0	0
$235$	−8.47214	−0.552661
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.3941	−0.801706	−0.400853	−	0.916142i	$-0.631286\pi$
$239$	−12.3941	−0.801706	−0.400853	+	0.916142i	$0.631286\pi$
$240$	0	0
$241$	17.5595	1.13110	0.565552	−	0.824713i	$-0.308663\pi$
$241$	17.5595	1.13110	0.565552	+	0.824713i	$0.308663\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.5279	0.860757
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9443	−1.57447	−0.787234	−	0.616654i	$-0.788488\pi$
$251$	−24.9443	−1.57447	−0.787234	+	0.616654i	$0.788488\pi$
$252$	0	0
$253$	27.4164	1.72365
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.8885	0.741587	0.370793	−	0.928715i	$-0.379086\pi$
$257$	11.8885	0.741587	0.370793	+	0.928715i	$0.379086\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.16228	−0.194994	−0.0974972	−	0.995236i	$-0.531084\pi$
$263$	−3.16228	−0.194994	−0.0974972	+	0.995236i	$0.531084\pi$
$264$	0	0
$265$	−11.6476	−0.715504
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.41641	0.330244	0.165122	−	0.986273i	$-0.447198\pi$
$269$	5.41641	0.330244	0.165122	+	0.986273i	$0.447198\pi$
$270$	0	0
$271$	4.24264	0.257722	0.128861	−	0.991663i	$-0.458868\pi$
$271$	4.24264	0.257722	0.128861	+	0.991663i	$0.458868\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.57649	0.275973
$276$	0	0
$277$	16.4721	0.989715	0.494857	−	0.868974i	$-0.335220\pi$
$277$	16.4721	0.989715	0.494857	+	0.868974i	$0.335220\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.6491	0.754583	0.377291	−	0.926095i	$-0.376856\pi$
$281$	12.6491	0.754583	0.377291	+	0.926095i	$0.376856\pi$
$282$	0	0
$283$	21.1344	1.25631	0.628155	−	0.778089i	$-0.283811\pi$
$283$	21.1344	1.25631	0.628155	+	0.778089i	$0.283811\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.05573	−0.295359	−0.147679	−	0.989035i	$-0.547180\pi$
$293$	−5.05573	−0.295359	−0.147679	+	0.989035i	$0.547180\pi$
$294$	0	0
$295$	−4.94427	−0.287867
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.4721	−0.605619
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.91034	0.281165
$306$	0	0
$307$	−6.99226	−0.399069	−0.199535	−	0.979891i	$-0.563943\pi$
$307$	−6.99226	−0.399069	−0.199535	+	0.979891i	$0.563943\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.4164	1.32782	0.663911	−	0.747811i	$-0.268895\pi$
$311$	23.4164	1.32782	0.663911	+	0.747811i	$0.268895\pi$
$312$	0	0
$313$	−31.3677	−1.77301	−0.886505	−	0.462720i	$-0.846874\pi$
$313$	−31.3677	−1.77301	−0.886505	+	0.462720i	$0.846874\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.6212	1.71986	0.859930	−	0.510413i	$-0.170507\pi$
$317$	30.6212	1.71986	0.859930	+	0.510413i	$0.170507\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.1313	−1.06449
$324$	0	0
$325$	−1.74806	−0.0969651
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.0557	0.607678	0.303839	−	0.952723i	$-0.401732\pi$
$331$	11.0557	0.607678	0.303839	+	0.952723i	$0.401732\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.94427	0.270134
$336$	0	0
$337$	−25.4164	−1.38452	−0.692260	−	0.721648i	$-0.743385\pi$
$337$	−25.4164	−1.38452	−0.692260	+	0.721648i	$0.743385\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−32.3607	−1.75243
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.1406	−0.705424	−0.352712	−	0.935732i	$-0.614740\pi$
$347$	−13.1406	−0.705424	−0.352712	+	0.935732i	$0.614740\pi$
$348$	0	0
$349$	14.7310	0.788534	0.394267	−	0.918996i	$-0.370999\pi$
$349$	14.7310	0.788534	0.394267	+	0.918996i	$0.370999\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.9443	−1.22120	−0.610600	−	0.791939i	$-0.709072\pi$
$353$	−22.9443	−1.22120	−0.610600	+	0.791939i	$0.709072\pi$
$354$	0	0
$355$	−7.40492	−0.393012
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.5548	0.768173	0.384086	−	0.923297i	$-0.374516\pi$
$359$	14.5548	0.768173	0.384086	+	0.923297i	$0.374516\pi$
$360$	0	0
$361$	40.8885	2.15203
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.90879	0.204595
$366$	0	0
$367$	−6.32456	−0.330139	−0.165070	−	0.986282i	$-0.552785\pi$
$367$	−6.32456	−0.330139	−0.165070	+	0.986282i	$0.552785\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.11146	−0.314756
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.4164	−0.685546	−0.342773	−	0.939418i	$-0.611366\pi$
$383$	−13.4164	−0.685546	−0.342773	+	0.939418i	$0.611366\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.6383	−0.894295	−0.447148	−	0.894460i	$-0.647560\pi$
$389$	−17.6383	−0.894295	−0.447148	+	0.894460i	$0.647560\pi$
$390$	0	0
$391$	14.8098	0.748966
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.9443	0.550666
$396$	0	0
$397$	−17.2256	−0.864528	−0.432264	−	0.901747i	$-0.642285\pi$
$397$	−17.2256	−0.864528	−0.432264	+	0.901747i	$0.642285\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.4558	1.27120	0.635602	−	0.772017i	$-0.280752\pi$
$401$	25.4558	1.27120	0.635602	+	0.772017i	$0.280752\pi$
$402$	0	0
$403$	12.3607	0.615729
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−50.0864	−2.48269
$408$	0	0
$409$	9.89949	0.489499	0.244749	−	0.969586i	$-0.421294\pi$
$409$	9.89949	0.489499	0.244749	+	0.969586i	$0.421294\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.41641	0.462233
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.8885	0.873913	0.436956	−	0.899483i	$-0.356056\pi$
$419$	17.8885	0.873913	0.436956	+	0.899483i	$0.356056\pi$
$420$	0	0
$421$	−8.94427	−0.435917	−0.217959	−	0.975958i	$-0.569940\pi$
$421$	−8.94427	−0.435917	−0.217959	+	0.975958i	$0.569940\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.47214	0.119916
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.57339	0.123956	0.0619779	−	0.998078i	$-0.480259\pi$
$431$	2.57339	0.123956	0.0619779	+	0.998078i	$0.480259\pi$
$432$	0	0
$433$	−27.8716	−1.33942	−0.669712	−	0.742621i	$-0.733582\pi$
$433$	−27.8716	−1.33942	−0.669712	+	0.742621i	$0.733582\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−46.3607	−2.21773
$438$	0	0
$439$	10.5672	0.504345	0.252172	−	0.967682i	$-0.418855\pi$
$439$	10.5672	0.504345	0.252172	+	0.967682i	$0.418855\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.1034	1.76284	0.881418	−	0.472337i	$-0.156590\pi$
$443$	37.1034	1.76284	0.881418	+	0.472337i	$0.156590\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.1452	−0.761941	−0.380970	−	0.924587i	$-0.624410\pi$
$449$	−16.1452	−0.761941	−0.380970	+	0.924587i	$0.624410\pi$
$450$	0	0
$451$	−57.0786	−2.68773
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.7771	1.58003	0.790013	−	0.613090i	$-0.210074\pi$
$457$	33.7771	1.58003	0.790013	+	0.613090i	$0.210074\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.0557	−0.608066	−0.304033	−	0.952662i	$-0.598333\pi$
$461$	−13.0557	−0.608066	−0.304033	+	0.952662i	$0.598333\pi$
$462$	0	0
$463$	−19.4164	−0.902357	−0.451178	−	0.892434i	$-0.648996\pi$
$463$	−19.4164	−0.902357	−0.451178	+	0.892434i	$0.648996\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.9443	−0.969185	−0.484593	−	0.874740i	$-0.661032\pi$
$467$	−20.9443	−0.969185	−0.484593	+	0.874740i	$0.661032\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−47.9256	−2.20362
$474$	0	0
$475$	−7.73877	−0.355079
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.52786	−0.435339	−0.217670	−	0.976023i	$-0.569846\pi$
$479$	−9.52786	−0.435339	−0.217670	+	0.976023i	$0.569846\pi$
$480$	0	0
$481$	19.1313	0.872312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.7295	0.623423
$486$	0	0
$487$	−16.3607	−0.741373	−0.370687	−	0.928758i	$-0.620878\pi$
$487$	−16.3607	−0.741373	−0.370687	+	0.928758i	$0.620878\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.57339	−0.116135	−0.0580677	−	0.998313i	$-0.518494\pi$
$491$	−2.57339	−0.116135	−0.0580677	+	0.998313i	$0.518494\pi$
$492$	0	0
$493$	8.64290	0.389257
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−26.9443	−1.20619	−0.603096	−	0.797669i	$-0.706066\pi$
$499$	−26.9443	−1.20619	−0.603096	+	0.797669i	$0.706066\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−43.3050	−1.93087	−0.965436	−	0.260640i	$-0.916067\pi$
$503$	−43.3050	−1.93087	−0.965436	+	0.260640i	$0.916067\pi$
$504$	0	0
$505$	0.472136	0.0210098
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.52786	−0.156370	−0.0781849	−	0.996939i	$-0.524912\pi$
$509$	−3.52786	−0.156370	−0.0781849	+	0.996939i	$0.524912\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.82843	−0.124635
$516$	0	0
$517$	−38.7727	−1.70522
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.3607	−1.50537	−0.752684	−	0.658382i	$-0.771241\pi$
$521$	−34.3607	−1.50537	−0.752684	+	0.658382i	$0.771241\pi$
$522$	0	0
$523$	20.3091	0.888054	0.444027	−	0.896014i	$-0.353550\pi$
$523$	20.3091	0.888054	0.444027	+	0.896014i	$0.353550\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.4806	−0.761469
$528$	0	0
$529$	12.8885	0.560371
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.8021	0.944353
$534$	0	0
$535$	10.9799	0.474701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.0557	−1.15261
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.3153	−0.521814	−0.260907	−	0.965364i	$-0.584022\pi$
$557$	−12.3153	−0.521814	−0.260907	+	0.965364i	$0.584022\pi$
$558$	0	0
$559$	18.3060	0.774260
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.5279	1.49732	0.748660	−	0.662954i	$-0.230697\pi$
$563$	35.5279	1.49732	0.748660	+	0.662954i	$0.230697\pi$
$564$	0	0
$565$	−18.6398	−0.784183
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.7990	−0.830017	−0.415008	−	0.909818i	$-0.636221\pi$
$569$	−19.7990	−0.830017	−0.415008	+	0.909818i	$0.636221\pi$
$570$	0	0
$571$	25.8885	1.08340	0.541701	−	0.840571i	$-0.317781\pi$
$571$	25.8885	1.08340	0.541701	+	0.840571i	$0.317781\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.99070	0.249830
$576$	0	0
$577$	−2.57339	−0.107132	−0.0535658	−	0.998564i	$-0.517059\pi$
$577$	−2.57339	−0.107132	−0.0535658	+	0.998564i	$0.517059\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−53.3050	−2.20767
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.3607	−0.757826	−0.378913	−	0.925432i	$-0.623702\pi$
$587$	−18.3607	−0.757826	−0.378913	+	0.925432i	$0.623702\pi$
$588$	0	0
$589$	54.7214	2.25475
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.8328	1.01976	0.509881	−	0.860245i	$-0.329690\pi$
$593$	24.8328	1.01976	0.509881	+	0.860245i	$0.329690\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.7140	1.13236	0.566181	−	0.824281i	$-0.308420\pi$
$599$	27.7140	1.13236	0.566181	+	0.824281i	$0.308420\pi$
$600$	0	0
$601$	9.23179	0.376573	0.188286	−	0.982114i	$-0.439707\pi$
$601$	9.23179	0.376573	0.188286	+	0.982114i	$0.439707\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.94427	0.404292
$606$	0	0
$607$	−14.1421	−0.574012	−0.287006	−	0.957929i	$-0.592660\pi$
$607$	−14.1421	−0.574012	−0.287006	+	0.957929i	$0.592660\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.8098	0.599142
$612$	0	0
$613$	−43.3050	−1.74907	−0.874535	−	0.484962i	$-0.838833\pi$
$613$	−43.3050	−1.74907	−0.874535	+	0.484962i	$0.838833\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.81913	0.355045	0.177522	−	0.984117i	$-0.443192\pi$
$617$	8.81913	0.355045	0.177522	+	0.984117i	$0.443192\pi$
$618$	0	0
$619$	6.91344	0.277875	0.138937	−	0.990301i	$-0.455631\pi$
$619$	6.91344	0.277875	0.138937	+	0.990301i	$0.455631\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.0557	−1.07878
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.4721	0.415574
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.81758	−0.308776	−0.154388	−	0.988010i	$-0.549341\pi$
$641$	−7.81758	−0.308776	−0.154388	+	0.988010i	$0.549341\pi$
$642$	0	0
$643$	23.1375	0.912454	0.456227	−	0.889863i	$-0.349201\pi$
$643$	23.1375	0.912454	0.456227	+	0.889863i	$0.349201\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.8885	1.48955	0.744776	−	0.667314i	$-0.232556\pi$
$647$	37.8885	1.48955	0.744776	+	0.667314i	$0.232556\pi$
$648$	0	0
$649$	−22.6274	−0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.15143	0.318990	0.159495	−	0.987199i	$-0.449013\pi$
$653$	8.15143	0.318990	0.159495	+	0.987199i	$0.449013\pi$
$654$	0	0
$655$	11.4164	0.446076
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.3600	−1.49429	−0.747147	−	0.664659i	$-0.768577\pi$
$659$	−38.3600	−1.49429	−0.747147	+	0.664659i	$0.768577\pi$
$660$	0	0
$661$	−47.3367	−1.84119	−0.920593	−	0.390523i	$-0.872294\pi$
$661$	−47.3367	−1.84119	−0.920593	+	0.390523i	$0.872294\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.9443	0.810965
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.4721	0.867527
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.4419	1.54747	0.773733	−	0.633511i	$-0.218387\pi$
$683$	40.4419	1.54747	0.773733	+	0.633511i	$0.218387\pi$
$684$	0	0
$685$	3.16228	0.120824
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.3607	0.775680
$690$	0	0
$691$	29.6985	1.12978	0.564892	−	0.825165i	$-0.308918\pi$
$691$	29.6985	1.12978	0.564892	+	0.825165i	$0.308918\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.2349	−0.426164
$696$	0	0
$697$	−30.8328	−1.16788
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−37.9473	−1.43325	−0.716625	−	0.697458i	$-0.754314\pi$
$701$	−37.9473	−1.43325	−0.716625	+	0.697458i	$0.754314\pi$
$702$	0	0
$703$	84.6952	3.19434
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.9443	1.08702	0.543512	−	0.839401i	$-0.317094\pi$
$709$	28.9443	1.08702	0.543512	+	0.839401i	$0.317094\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−42.3607	−1.58642
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.8328	0.702346	0.351173	−	0.936311i	$-0.385783\pi$
$719$	18.8328	0.702346	0.351173	+	0.936311i	$0.385783\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.49613	0.129843
$726$	0	0
$727$	16.9706	0.629403	0.314702	−	0.949191i	$-0.398096\pi$
$727$	16.9706	0.629403	0.314702	+	0.949191i	$0.398096\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.8885	−0.957522
$732$	0	0
$733$	3.08347	0.113890	0.0569452	−	0.998377i	$-0.481864\pi$
$733$	3.08347	0.113890	0.0569452	+	0.998377i	$0.481864\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.6274	0.833492
$738$	0	0
$739$	−17.8885	−0.658041	−0.329020	−	0.944323i	$-0.606718\pi$
$739$	−17.8885	−0.658041	−0.329020	+	0.944323i	$0.606718\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.4791	0.604559	0.302280	−	0.953219i	$-0.402252\pi$
$743$	16.4791	0.604559	0.302280	+	0.953219i	$0.402252\pi$
$744$	0	0
$745$	16.9706	0.621753
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.11146	0.0770481	0.0385241	−	0.999258i	$-0.487734\pi$
$751$	2.11146	0.0770481	0.0385241	+	0.999258i	$0.487734\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.8885	−0.432668
$756$	0	0
$757$	−12.4721	−0.453307	−0.226654	−	0.973975i	$-0.572779\pi$
$757$	−12.4721	−0.453307	−0.226654	+	0.973975i	$0.572779\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.88854	−0.285959	−0.142980	−	0.989726i	$-0.545668\pi$
$761$	−7.88854	−0.285959	−0.142980	+	0.989726i	$0.545668\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.64290	0.312077
$768$	0	0
$769$	21.8809	0.789046	0.394523	−	0.918886i	$-0.370910\pi$
$769$	21.8809	0.789046	0.394523	+	0.918886i	$0.370910\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.583592	−0.0209904	−0.0104952	−	0.999945i	$-0.503341\pi$
$773$	−0.583592	−0.0209904	−0.0104952	+	0.999945i	$0.503341\pi$
$774$	0	0
$775$	−7.07107	−0.254000
$776$	0	0
$777$	0	0
$778$	0	0
$779$	96.5190	3.45815
$780$	0	0
$781$	−33.8885	−1.21263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.40492	−0.264293
$786$	0	0
$787$	−27.6166	−0.984424	−0.492212	−	0.870475i	$-0.663812\pi$
$787$	−27.6166	−0.984424	−0.492212	+	0.870475i	$0.663812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.58359	−0.304812
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.4721	−0.512629	−0.256315	−	0.966593i	$-0.582508\pi$
$797$	−14.4721	−0.512629	−0.256315	+	0.966593i	$0.582508\pi$
$798$	0	0
$799$	−20.9443	−0.740955
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.8885	0.631273
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.2951	−0.819013	−0.409506	−	0.912307i	$-0.634299\pi$
$809$	−23.2951	−0.819013	−0.409506	+	0.912307i	$0.634299\pi$
$810$	0	0
$811$	−5.57804	−0.195872	−0.0979358	−	0.995193i	$-0.531224\pi$
$811$	−5.57804	−0.195872	−0.0979358	+	0.995193i	$0.531224\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.52786	−0.0535187
$816$	0	0
$817$	81.0414	2.83528
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−52.0895	−1.81793	−0.908967	−	0.416867i	$-0.863128\pi$
$821$	−52.0895	−1.81793	−0.908967	+	0.416867i	$0.863128\pi$
$822$	0	0
$823$	−21.5279	−0.750414	−0.375207	−	0.926941i	$-0.622428\pi$
$823$	−21.5279	−0.750414	−0.375207	+	0.926941i	$0.622428\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.3075	0.671388	0.335694	−	0.941971i	$-0.391029\pi$
$827$	19.3075	0.671388	0.335694	+	0.941971i	$0.391029\pi$
$828$	0	0
$829$	8.40647	0.291969	0.145984	−	0.989287i	$-0.453365\pi$
$829$	8.40647	0.291969	0.145984	+	0.989287i	$0.453365\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.9443	−0.863232
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.47214	0.223443	0.111721	−	0.993740i	$-0.464364\pi$
$839$	6.47214	0.223443	0.111721	+	0.993740i	$0.464364\pi$
$840$	0	0
$841$	−16.7771	−0.578520
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.94427	−0.342093
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−65.5639	−2.24750
$852$	0	0
$853$	−27.7140	−0.948909	−0.474454	−	0.880280i	$-0.657355\pi$
$853$	−27.7140	−0.948909	−0.474454	+	0.880280i	$0.657355\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.5279	−0.872015	−0.436008	−	0.899943i	$-0.643608\pi$
$857$	−25.5279	−0.872015	−0.436008	+	0.899943i	$0.643608\pi$
$858$	0	0
$859$	31.8592	1.08702	0.543511	−	0.839402i	$-0.317095\pi$
$859$	31.8592	1.08702	0.543511	+	0.839402i	$0.317095\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.00155	0.0340932	0.0170466	−	0.999855i	$-0.494574\pi$
$863$	1.00155	0.0340932	0.0170466	+	0.999855i	$0.494574\pi$
$864$	0	0
$865$	−22.4721	−0.764076
$866$	0	0
$867$	0	0
$868$	0	0
$869$	50.0864	1.69906
$870$	0	0
$871$	−8.64290	−0.292854
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.5279	0.389268	0.194634	−	0.980876i	$-0.437648\pi$
$877$	11.5279	0.389268	0.194634	+	0.980876i	$0.437648\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	27.8885	0.939589	0.469794	−	0.882776i	$-0.344328\pi$
$881$	27.8885	0.939589	0.469794	+	0.882776i	$0.344328\pi$
$882$	0	0
$883$	−6.11146	−0.205667	−0.102833	−	0.994699i	$-0.532791\pi$
$883$	−6.11146	−0.205667	−0.102833	+	0.994699i	$0.532791\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.63932	0.0550430	0.0275215	−	0.999621i	$-0.491239\pi$
$887$	1.63932	0.0550430	0.0275215	+	0.999621i	$0.491239\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	65.5639	2.19401
$894$	0	0
$895$	−7.40492	−0.247519
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24.7214	−0.824504
$900$	0	0
$901$	−28.7943	−0.959279
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.24264	0.141030
$906$	0	0
$907$	23.0557	0.765553	0.382776	−	0.923841i	$-0.374968\pi$
$907$	23.0557	0.765553	0.382776	+	0.923841i	$0.374968\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.6783	−1.34773	−0.673867	−	0.738853i	$-0.735368\pi$
$911$	−40.6783	−1.34773	−0.673867	+	0.738853i	$0.735368\pi$
$912$	0	0
$913$	43.0941	1.42621
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.9443	0.426066
$924$	0	0
$925$	−10.9443	−0.359845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−52.8328	−1.73339	−0.866694	−	0.498840i	$-0.833760\pi$
$929$	−52.8328	−1.73339	−0.866694	+	0.498840i	$0.833760\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.3137	0.369998
$936$	0	0
$937$	45.5099	1.48674	0.743371	−	0.668879i	$-0.233226\pi$
$937$	45.5099	1.48674	0.743371	+	0.668879i	$0.233226\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−74.7169	−2.43312
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.4760	−0.470406	−0.235203	−	0.971946i	$-0.575576\pi$
$947$	−14.4760	−0.470406	−0.235203	+	0.971946i	$0.575576\pi$
$948$	0	0
$949$	−6.83282	−0.221803
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29.2858	0.948661	0.474330	−	0.880347i	$-0.342690\pi$
$953$	29.2858	0.948661	0.474330	+	0.880347i	$0.342690\pi$
$954$	0	0
$955$	21.5471	0.697246
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.4721	−0.530257
$966$	0	0
$967$	26.8328	0.862885	0.431443	−	0.902140i	$-0.358005\pi$
$967$	26.8328	0.862885	0.431443	+	0.902140i	$0.358005\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.4721	1.36300	0.681498	−	0.731820i	$-0.261329\pi$
$971$	42.4721	1.36300	0.681498	+	0.731820i	$0.261329\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.0848	−1.57036	−0.785181	−	0.619266i	$-0.787430\pi$
$977$	−49.0848	−1.57036	−0.785181	+	0.619266i	$0.787430\pi$
$978$	0	0
$979$	−27.4589	−0.877592
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.9443	−0.540438	−0.270219	−	0.962799i	$-0.587096\pi$
$983$	−16.9443	−0.540438	−0.270219	+	0.962799i	$0.587096\pi$
$984$	0	0
$985$	6.65841	0.212154
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62.7355	−1.99487
$990$	0	0
$991$	52.7214	1.67475	0.837375	−	0.546629i	$-0.184089\pi$
$991$	52.7214	1.67475	0.837375	+	0.546629i	$0.184089\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.5672	−0.335003
$996$	0	0
$997$	−27.2039	−0.861556	−0.430778	−	0.902458i	$-0.641761\pi$
$997$	−27.2039	−0.861556	−0.430778	+	0.902458i	$0.641761\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bs.1.4	yes	4
3.2	odd	2		8820.2.a.br.1.1	✓	4
7.6	odd	2		8820.2.a.br.1.4	yes	4
21.20	even	2	inner	8820.2.a.bs.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8820.2.a.br.1.1	✓	4	3.2	odd	2
8820.2.a.br.1.4	yes	4	7.6	odd	2
8820.2.a.bs.1.1	yes	4	21.20	even	2	inner
8820.2.a.bs.1.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +O(q^{10})\)	\(q+1.00000 q^{5} +4.57649 q^{11} -1.74806 q^{13} +2.47214 q^{17} -7.73877 q^{19} +5.99070 q^{23} +1.00000 q^{25} +3.49613 q^{29} -7.07107 q^{31} -10.9443 q^{37} -12.4721 q^{41} -10.4721 q^{43} -8.47214 q^{47} -11.6476 q^{53} +4.57649 q^{55} -4.94427 q^{59} +4.91034 q^{61} -1.74806 q^{65} +4.94427 q^{67} -7.40492 q^{71} +3.90879 q^{73} +10.9443 q^{79} +9.41641 q^{83} +2.47214 q^{85} -6.00000 q^{89} -7.73877 q^{95} +13.7295 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} - 8 q^{17} + 4 q^{25} - 8 q^{37} - 32 q^{41} - 24 q^{43} - 16 q^{47} + 16 q^{59} - 16 q^{67} + 8 q^{79} - 16 q^{83} - 8 q^{85} - 24 q^{89}+O(q^{100}) \) 4 * q + 4 * q^5 - 8 * q^17 + 4 * q^25 - 8 * q^37 - 32 * q^41 - 24 * q^43 - 16 * q^47 + 16 * q^59 - 16 * q^67 + 8 * q^79 - 16 * q^83 - 8 * q^85 - 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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