LMFDB - Embedded newform 8820.2.d.a.881.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(881,8820)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("8820.881");

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.d (of order $2$, degree $1$, 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:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 $ x^12 - 2x^11 - 9x^10 + 58x^9 - 78x^8 - 298x^7 + 1341x^6 - 2086x^5 - 3822x^4 + 19894x^3 - 21609x^2 - 33614*x + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.9
Root			$2.61827 - 0.380350i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.a.881.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} +2.49275i q^{11} +6.80937i q^{13} +7.80234 q^{17} -2.46203i q^{19} +3.62609i q^{23} +1.00000 q^{25} -2.55005i q^{29} -4.25932i q^{31} +1.85102 q^{37} -4.23654 q^{41} +8.70309 q^{43} +1.03235 q^{47} -0.0572980i q^{53} -2.49275i q^{55} +11.1879 q^{59} +3.10930i q^{61} -6.80937i q^{65} +10.3681 q^{67} +8.13806i q^{71} -4.15463i q^{73} -4.16865 q^{79} -11.7066 q^{83} -7.80234 q^{85} -9.96401 q^{89} +2.46203i q^{95} +4.91533i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5}+O(q^{10}) $ 12 * q - 12 * q^5	$ 12 q - 12 q^{5} + 12 q^{25} + 4 q^{37} + 8 q^{41} + 36 q^{43} - 32 q^{47} - 4 q^{67} - 28 q^{79} - 40 q^{83} - 40 q^{89}+O(q^{100}) $ 12 * q - 12 * q^5 + 12 * q^25 + 4 * q^37 + 8 * q^41 + 36 * q^43 - 32 * q^47 - 4 * q^67 - 28 * q^79 - 40 * q^83 - 40 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$.

$n$	$1081$	$4411$	$7057$	$7841$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	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
$11$	2.49275i	0.751593i	0.926702	+	0.375796i	$0.122631\pi$
$11$	2.49275i	0.751593i	−0.926702	+	0.375796i	$0.877369\pi$
$12$	0	0
$13$	6.80937i	1.88858i	0.329116	+	0.944289i	$0.393249\pi$
$13$	6.80937i	1.88858i	−0.329116	+	0.944289i	$0.606751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.80234	1.89234	0.946172	−	0.323664i	$-0.104915\pi$
$17$	7.80234	1.89234	0.946172	+	0.323664i	$0.104915\pi$
$18$	0	0
$19$	− 2.46203i	− 0.564829i	−0.959292	−	0.282415i	$-0.908865\pi$
$19$	− 2.46203i	− 0.564829i	0.959292	−	0.282415i	$-0.0911354\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.62609i	0.756093i	0.925787	+	0.378046i	$0.123404\pi$
$23$	3.62609i	0.756093i	−0.925787	+	0.378046i	$0.876596\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.55005i	− 0.473532i	−0.971567	−	0.236766i	$-0.923913\pi$
$29$	− 2.55005i	− 0.473532i	0.971567	−	0.236766i	$-0.0760875\pi$
$30$	0	0
$31$	− 4.25932i	− 0.764996i	−0.923956	−	0.382498i	$-0.875064\pi$
$31$	− 4.25932i	− 0.764996i	0.923956	−	0.382498i	$-0.124936\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.85102	0.304305	0.152153	−	0.988357i	$-0.451379\pi$
$37$	1.85102	0.304305	0.152153	+	0.988357i	$0.451379\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.23654	−0.661636	−0.330818	−	0.943695i	$-0.607325\pi$
$41$	−4.23654	−0.661636	−0.330818	+	0.943695i	$0.607325\pi$
$42$	0	0
$43$	8.70309	1.32721	0.663605	−	0.748084i	$-0.269026\pi$
$43$	8.70309	1.32721	0.663605	+	0.748084i	$0.269026\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.03235	0.150584	0.0752919	−	0.997162i	$-0.476011\pi$
$47$	1.03235	0.150584	0.0752919	+	0.997162i	$0.476011\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.0572980i	− 0.00787049i	−0.999992	−	0.00393524i	$-0.998747\pi$
$53$	− 0.0572980i	− 0.00787049i	0.999992	−	0.00393524i	$-0.00125263\pi$
$54$	0	0
$55$	− 2.49275i	− 0.336122i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.1879	1.45654	0.728268	−	0.685293i	$-0.240326\pi$
$59$	11.1879	1.45654	0.728268	+	0.685293i	$0.240326\pi$
$60$	0	0
$61$	3.10930i	0.398105i	0.979989	+	0.199052i	$0.0637864\pi$
$61$	3.10930i	0.398105i	−0.979989	+	0.199052i	$0.936214\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.80937i	− 0.844598i
$66$	0	0
$67$	10.3681	1.26667	0.633335	−	0.773878i	$-0.281686\pi$
$67$	10.3681	1.26667	0.633335	+	0.773878i	$0.281686\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.13806i	0.965810i	0.875673	+	0.482905i	$0.160418\pi$
$71$	8.13806i	0.965810i	−0.875673	+	0.482905i	$0.839582\pi$
$72$	0	0
$73$	− 4.15463i	− 0.486262i	−0.969993	−	0.243131i	$-0.921825\pi$
$73$	− 4.15463i	− 0.486262i	0.969993	−	0.243131i	$-0.0781745\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.16865	−0.469009	−0.234505	−	0.972115i	$-0.575347\pi$
$79$	−4.16865	−0.469009	−0.234505	+	0.972115i	$0.575347\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.7066	−1.28496	−0.642481	−	0.766302i	$-0.722095\pi$
$83$	−11.7066	−1.28496	−0.642481	+	0.766302i	$0.722095\pi$
$84$	0	0
$85$	−7.80234	−0.846282
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.96401	−1.05618	−0.528091	−	0.849188i	$-0.677092\pi$
$89$	−9.96401	−1.05618	−0.528091	+	0.849188i	$0.677092\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.46203i	0.252599i
$96$	0	0
$97$	4.91533i	0.499076i	0.968365	+	0.249538i	$0.0802787\pi$
$97$	4.91533i	0.499076i	−0.968365	+	0.249538i	$0.919721\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.5251	−1.94282	−0.971410	−	0.237410i	$-0.923702\pi$
$101$	−19.5251	−1.94282	−0.971410	+	0.237410i	$0.923702\pi$
$102$	0	0
$103$	18.6886i	1.84144i	0.390224	+	0.920720i	$0.372398\pi$
$103$	18.6886i	1.84144i	−0.390224	+	0.920720i	$0.627602\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.62954i	0.834249i	0.908849	+	0.417124i	$0.136962\pi$
$107$	8.62954i	0.834249i	−0.908849	+	0.417124i	$0.863038\pi$
$108$	0	0
$109$	15.2593	1.46157	0.730787	−	0.682606i	$-0.239153\pi$
$109$	15.2593	1.46157	0.730787	+	0.682606i	$0.239153\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 15.1622i	− 1.42634i	−0.700991	−	0.713171i	$-0.747259\pi$
$113$	− 15.1622i	− 1.42634i	0.700991	−	0.713171i	$-0.252741\pi$
$114$	0	0
$115$	− 3.62609i	− 0.338135i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.78620	0.435109
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.7942	1.22403	0.612017	−	0.790844i	$-0.290358\pi$
$127$	13.7942	1.22403	0.612017	+	0.790844i	$0.290358\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.0506	−1.14023	−0.570116	−	0.821564i	$-0.693102\pi$
$131$	−13.0506	−1.14023	−0.570116	+	0.821564i	$0.693102\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.85597i	− 0.842052i	−0.907049	−	0.421026i	$-0.861670\pi$
$137$	− 9.85597i	− 0.842052i	0.907049	−	0.421026i	$-0.138330\pi$
$138$	0	0
$139$	− 14.9603i	− 1.26892i	−0.772957	−	0.634458i	$-0.781223\pi$
$139$	− 14.9603i	− 1.26892i	0.772957	−	0.634458i	$-0.218777\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.9741	−1.41944
$144$	0	0
$145$	2.55005i	0.211770i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.7571i	− 1.04511i	−0.852607	−	0.522553i	$-0.824980\pi$
$149$	− 12.7571i	− 1.04511i	0.852607	−	0.522553i	$-0.175020\pi$
$150$	0	0
$151$	−13.7228	−1.11674	−0.558371	−	0.829591i	$-0.688573\pi$
$151$	−13.7228	−1.11674	−0.558371	+	0.829591i	$0.688573\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.25932i	0.342117i
$156$	0	0
$157$	12.0973i	0.965473i	0.875766	+	0.482736i	$0.160357\pi$
$157$	12.0973i	0.965473i	−0.875766	+	0.482736i	$0.839643\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.6499	0.912491	0.456245	−	0.889854i	$-0.349194\pi$
$163$	11.6499	0.912491	0.456245	+	0.889854i	$0.349194\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.88337	0.145739	0.0728697	−	0.997341i	$-0.476784\pi$
$167$	1.88337	0.145739	0.0728697	+	0.997341i	$0.476784\pi$
$168$	0	0
$169$	−33.3675	−2.56673
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.95988	−0.301064	−0.150532	−	0.988605i	$-0.548099\pi$
$173$	−3.95988	−0.301064	−0.150532	+	0.988605i	$0.548099\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.12414i	0.233509i	0.993161	+	0.116755i	$0.0372491\pi$
$179$	3.12414i	0.233509i	−0.993161	+	0.116755i	$0.962751\pi$
$180$	0	0
$181$	− 0.118397i	− 0.00880041i	−0.999990	−	0.00440021i	$-0.998599\pi$
$181$	− 0.118397i	− 0.00880041i	0.999990	−	0.00440021i	$-0.00140063\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.85102	−0.136089
$186$	0	0
$187$	19.4493i	1.42227i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.59215i	0.694063i	0.937853	+	0.347032i	$0.112810\pi$
$191$	9.59215i	0.694063i	−0.937853	+	0.347032i	$0.887190\pi$
$192$	0	0
$193$	4.45035	0.320343	0.160172	−	0.987089i	$-0.448795\pi$
$193$	4.45035	0.320343	0.160172	+	0.987089i	$0.448795\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.4602i	− 1.31523i	−0.753352	−	0.657617i	$-0.771565\pi$
$197$	− 18.4602i	− 1.31523i	0.753352	−	0.657617i	$-0.228435\pi$
$198$	0	0
$199$	25.0373i	1.77485i	0.460956	+	0.887423i	$0.347507\pi$
$199$	25.0373i	1.77485i	−0.460956	+	0.887423i	$0.652493\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.23654	0.295893
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.13723	0.424521
$210$	0	0
$211$	−3.37077	−0.232054	−0.116027	−	0.993246i	$-0.537016\pi$
$211$	−3.37077	−0.232054	−0.116027	+	0.993246i	$0.537016\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.70309	−0.593546
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	53.1290i	3.57384i
$222$	0	0
$223$	14.2272i	0.952727i	0.879249	+	0.476363i	$0.158045\pi$
$223$	14.2272i	0.952727i	−0.879249	+	0.476363i	$0.841955\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.16720	−0.342959	−0.171479	−	0.985188i	$-0.554855\pi$
$227$	−5.16720	−0.342959	−0.171479	+	0.985188i	$0.554855\pi$
$228$	0	0
$229$	0.0799743i	0.00528485i	0.999997	+	0.00264243i	$0.000841111\pi$
$229$	0.0799743i	0.00528485i	−0.999997	+	0.00264243i	$0.999159\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.6802i	1.87890i	0.342686	+	0.939450i	$0.388663\pi$
$233$	28.6802i	1.87890i	−0.342686	+	0.939450i	$0.611337\pi$
$234$	0	0
$235$	−1.03235	−0.0673432
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.5801i	0.684368i	0.939633	+	0.342184i	$0.111167\pi$
$239$	10.5801i	0.684368i	−0.939633	+	0.342184i	$0.888833\pi$
$240$	0	0
$241$	10.6682i	0.687199i	0.939116	+	0.343599i	$0.111646\pi$
$241$	10.6682i	0.687199i	−0.939116	+	0.343599i	$0.888354\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.7649	1.06672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.0904	1.33122	0.665608	−	0.746301i	$-0.268172\pi$
$251$	21.0904	1.33122	0.665608	+	0.746301i	$0.268172\pi$
$252$	0	0
$253$	−9.03895	−0.568274
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.32409	0.207351	0.103676	−	0.994611i	$-0.466940\pi$
$257$	3.32409	0.207351	0.103676	+	0.994611i	$0.466940\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 14.8298i	− 0.914442i	−0.889353	−	0.457221i	$-0.848845\pi$
$263$	− 14.8298i	− 0.914442i	0.889353	−	0.457221i	$-0.151155\pi$
$264$	0	0
$265$	0.0572980i	0.00351979i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.0666	1.71125	0.855625	−	0.517596i	$-0.173173\pi$
$269$	28.0666	1.71125	0.855625	+	0.517596i	$0.173173\pi$
$270$	0	0
$271$	2.76980i	0.168253i	0.996455	+	0.0841266i	$0.0268100\pi$
$271$	2.76980i	0.168253i	−0.996455	+	0.0841266i	$0.973190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.49275i	0.150319i
$276$	0	0
$277$	8.26844	0.496802	0.248401	−	0.968657i	$-0.420095\pi$
$277$	8.26844	0.496802	0.248401	+	0.968657i	$0.420095\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.909217i	− 0.0542393i	−0.999632	−	0.0271197i	$-0.991366\pi$
$281$	− 0.909217i	− 0.0542393i	0.999632	−	0.0271197i	$-0.00863351\pi$
$282$	0	0
$283$	− 20.9924i	− 1.24787i	−0.781477	−	0.623934i	$-0.785533\pi$
$283$	− 20.9924i	− 1.24787i	0.781477	−	0.623934i	$-0.214467\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	43.8764	2.58097
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.21335	−0.246147	−0.123073	−	0.992398i	$-0.539275\pi$
$293$	−4.21335	−0.246147	−0.123073	+	0.992398i	$0.539275\pi$
$294$	0	0
$295$	−11.1879	−0.651382
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.6914	−1.42794
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.10930i	− 0.178038i
$306$	0	0
$307$	1.81626i	0.103660i	0.998656	+	0.0518298i	$0.0165053\pi$
$307$	1.81626i	0.103660i	−0.998656	+	0.0518298i	$0.983495\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.16249	0.236033	0.118017	−	0.993012i	$-0.462346\pi$
$311$	4.16249	0.236033	0.118017	+	0.993012i	$0.462346\pi$
$312$	0	0
$313$	18.1613i	1.02654i	0.858228	+	0.513269i	$0.171566\pi$
$313$	18.1613i	1.02654i	−0.858228	+	0.513269i	$0.828434\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.8702i	1.50918i	0.656195	+	0.754591i	$0.272165\pi$
$317$	26.8702i	1.50918i	−0.656195	+	0.754591i	$0.727835\pi$
$318$	0	0
$319$	6.35663	0.355903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 19.2096i	− 1.06885i
$324$	0	0
$325$	6.80937i	0.377716i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	33.8557	1.86088	0.930439	−	0.366446i	$-0.119426\pi$
$331$	33.8557	1.86088	0.930439	+	0.366446i	$0.119426\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.3681	−0.566472
$336$	0	0
$337$	−27.4405	−1.49478	−0.747391	−	0.664385i	$-0.768694\pi$
$337$	−27.4405	−1.49478	−0.747391	+	0.664385i	$0.768694\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.6174	0.574966
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.3962i	1.36334i	0.731659	+	0.681670i	$0.238746\pi$
$347$	25.3962i	1.36334i	−0.731659	+	0.681670i	$0.761254\pi$
$348$	0	0
$349$	9.13149i	0.488797i	0.969675	+	0.244399i	$0.0785906\pi$
$349$	9.13149i	0.488797i	−0.969675	+	0.244399i	$0.921409\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.2703	−1.87725	−0.938625	−	0.344940i	$-0.887899\pi$
$353$	−35.2703	−1.87725	−0.938625	+	0.344940i	$0.887899\pi$
$354$	0	0
$355$	− 8.13806i	− 0.431923i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.0277i	1.37369i	0.726804	+	0.686845i	$0.241005\pi$
$359$	26.0277i	1.37369i	−0.726804	+	0.686845i	$0.758995\pi$
$360$	0	0
$361$	12.9384	0.680968
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.15463i	0.217463i
$366$	0	0
$367$	12.8504i	0.670787i	0.942078	+	0.335394i	$0.108869\pi$
$367$	12.8504i	0.670787i	−0.942078	+	0.335394i	$0.891131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.250678	−0.0129796	−0.00648982	−	0.999979i	$-0.502066\pi$
$373$	−0.250678	−0.0129796	−0.00648982	+	0.999979i	$0.502066\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.3642	0.894303
$378$	0	0
$379$	18.7504	0.963144	0.481572	−	0.876407i	$-0.340066\pi$
$379$	18.7504	0.963144	0.481572	+	0.876407i	$0.340066\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.7565	1.36719	0.683596	−	0.729861i	$-0.260415\pi$
$383$	26.7565	1.36719	0.683596	+	0.729861i	$0.260415\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 10.1139i	− 0.512796i	−0.966571	−	0.256398i	$-0.917464\pi$
$389$	− 10.1139i	− 0.512796i	0.966571	−	0.256398i	$-0.0825357\pi$
$390$	0	0
$391$	28.2920i	1.43079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.16865	0.209747
$396$	0	0
$397$	− 10.3156i	− 0.517724i	−0.965914	−	0.258862i	$-0.916653\pi$
$397$	− 10.3156i	− 0.517724i	0.965914	−	0.258862i	$-0.0833474\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.4351i	1.46992i	0.678111	+	0.734960i	$0.262799\pi$
$401$	29.4351i	1.46992i	−0.678111	+	0.734960i	$0.737201\pi$
$402$	0	0
$403$	29.0033	1.44476
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.61412i	0.228713i
$408$	0	0
$409$	10.3019i	0.509395i	0.967021	+	0.254698i	$0.0819760\pi$
$409$	10.3019i	0.509395i	−0.967021	+	0.254698i	$0.918024\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.7066	0.574652
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.7293	−1.40352	−0.701759	−	0.712415i	$-0.747601\pi$
$419$	−28.7293	−1.40352	−0.701759	+	0.712415i	$0.747601\pi$
$420$	0	0
$421$	3.83262	0.186790	0.0933951	−	0.995629i	$-0.470228\pi$
$421$	3.83262	0.186790	0.0933951	+	0.995629i	$0.470228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.80234	0.378469
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 31.9282i	− 1.53793i	−0.639292	−	0.768964i	$-0.720773\pi$
$431$	− 31.9282i	− 1.53793i	0.639292	−	0.768964i	$-0.279227\pi$
$432$	0	0
$433$	− 37.0451i	− 1.78027i	−0.455693	−	0.890137i	$-0.650608\pi$
$433$	− 37.0451i	− 1.78027i	0.455693	−	0.890137i	$-0.349392\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.92756	0.427063
$438$	0	0
$439$	17.1417i	0.818130i	0.912505	+	0.409065i	$0.134145\pi$
$439$	17.1417i	0.818130i	−0.912505	+	0.409065i	$0.865855\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.67512i	− 0.127099i	−0.997979	−	0.0635493i	$-0.979758\pi$
$443$	− 2.67512i	− 0.127099i	0.997979	−	0.0635493i	$-0.0202420\pi$
$444$	0	0
$445$	9.96401	0.472339
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.3398i	1.29024i	0.764080	+	0.645122i	$0.223193\pi$
$449$	27.3398i	1.29024i	−0.764080	+	0.645122i	$0.776807\pi$
$450$	0	0
$451$	− 10.5606i	− 0.497281i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.2076	−0.617827	−0.308914	−	0.951090i	$-0.599965\pi$
$457$	−13.2076	−0.617827	−0.308914	+	0.951090i	$0.599965\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.8322	0.504505	0.252252	−	0.967661i	$-0.418829\pi$
$461$	10.8322	0.504505	0.252252	+	0.967661i	$0.418829\pi$
$462$	0	0
$463$	−7.92734	−0.368415	−0.184207	−	0.982887i	$-0.558972\pi$
$463$	−7.92734	−0.368415	−0.184207	+	0.982887i	$0.558972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.1946	−1.62861	−0.814306	−	0.580436i	$-0.802882\pi$
$467$	−35.1946	−1.62861	−0.814306	+	0.580436i	$0.802882\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21.6946i	0.997520i
$474$	0	0
$475$	− 2.46203i	− 0.112966i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.5175	−0.937469	−0.468734	−	0.883339i	$-0.655290\pi$
$479$	−20.5175	−0.937469	−0.468734	+	0.883339i	$0.655290\pi$
$480$	0	0
$481$	12.6042i	0.574704i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.91533i	− 0.223193i
$486$	0	0
$487$	−11.5668	−0.524141	−0.262071	−	0.965049i	$-0.584405\pi$
$487$	−11.5668	−0.524141	−0.262071	+	0.965049i	$0.584405\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 27.3496i	− 1.23427i	−0.786857	−	0.617136i	$-0.788293\pi$
$491$	− 27.3496i	− 1.23427i	0.786857	−	0.617136i	$-0.211707\pi$
$492$	0	0
$493$	− 19.8963i	− 0.896086i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.2674	0.728228	0.364114	−	0.931354i	$-0.381372\pi$
$499$	16.2674	0.728228	0.364114	+	0.931354i	$0.381372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.42605	−0.375699	−0.187849	−	0.982198i	$-0.560152\pi$
$503$	−8.42605	−0.375699	−0.187849	+	0.982198i	$0.560152\pi$
$504$	0	0
$505$	19.5251	0.868855
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.494132	0.0219020	0.0109510	−	0.999940i	$-0.496514\pi$
$509$	0.494132	0.0219020	0.0109510	+	0.999940i	$0.496514\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 18.6886i	− 0.823517i
$516$	0	0
$517$	2.57339i	0.113178i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.9349	1.35528	0.677641	−	0.735393i	$-0.263003\pi$
$521$	30.9349	1.35528	0.677641	+	0.735393i	$0.263003\pi$
$522$	0	0
$523$	39.2996i	1.71845i	0.511597	+	0.859225i	$0.329054\pi$
$523$	39.2996i	1.71845i	−0.511597	+	0.859225i	$0.670946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 33.2326i	− 1.44764i
$528$	0	0
$529$	9.85145	0.428324
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 28.8481i	− 1.24955i
$534$	0	0
$535$	− 8.62954i	− 0.373087i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−35.0467	−1.50678	−0.753388	−	0.657576i	$-0.771582\pi$
$541$	−35.0467	−1.50678	−0.753388	+	0.657576i	$0.771582\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.2593	−0.653635
$546$	0	0
$547$	5.91544	0.252926	0.126463	−	0.991971i	$-0.459637\pi$
$547$	5.91544	0.252926	0.126463	+	0.991971i	$0.459637\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.27830	−0.267465
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.2759i	0.859118i	0.903039	+	0.429559i	$0.141331\pi$
$557$	20.2759i	0.859118i	−0.903039	+	0.429559i	$0.858669\pi$
$558$	0	0
$559$	59.2626i	2.50654i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.6012	−0.783945	−0.391973	−	0.919977i	$-0.628207\pi$
$563$	−18.6012	−0.783945	−0.391973	+	0.919977i	$0.628207\pi$
$564$	0	0
$565$	15.1622i	0.637879i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 26.3492i	− 1.10461i	−0.833641	−	0.552307i	$-0.813747\pi$
$569$	− 26.3492i	− 1.10461i	0.833641	−	0.552307i	$-0.186253\pi$
$570$	0	0
$571$	19.1942	0.803252	0.401626	−	0.915804i	$-0.368445\pi$
$571$	19.1942	0.803252	0.401626	+	0.915804i	$0.368445\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.62609i	0.151219i
$576$	0	0
$577$	− 9.60332i	− 0.399792i	−0.979817	−	0.199896i	$-0.935940\pi$
$577$	− 9.60332i	− 0.399792i	0.979817	−	0.199896i	$-0.0640604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.142830	0.00591540
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.3348	−0.756757	−0.378378	−	0.925651i	$-0.623518\pi$
$587$	−18.3348	−0.756757	−0.378378	+	0.925651i	$0.623518\pi$
$588$	0	0
$589$	−10.4866	−0.432092
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.01920	−0.370374	−0.185187	−	0.982703i	$-0.559289\pi$
$593$	−9.01920	−0.370374	−0.185187	+	0.982703i	$0.559289\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.6561i	1.25257i	0.779592	+	0.626287i	$0.215426\pi$
$599$	30.6561i	1.25257i	−0.779592	+	0.626287i	$0.784574\pi$
$600$	0	0
$601$	− 19.7777i	− 0.806750i	−0.915035	−	0.403375i	$-0.867837\pi$
$601$	− 19.7777i	− 0.806750i	0.915035	−	0.403375i	$-0.132163\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.78620	−0.194587
$606$	0	0
$607$	31.6679i	1.28536i	0.766135	+	0.642680i	$0.222177\pi$
$607$	31.6679i	1.28536i	−0.766135	+	0.642680i	$0.777823\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.02966i	0.284390i
$612$	0	0
$613$	42.5337	1.71792	0.858960	−	0.512043i	$-0.171111\pi$
$613$	42.5337	1.71792	0.858960	+	0.512043i	$0.171111\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.6510i	0.952152i	0.879404	+	0.476076i	$0.157941\pi$
$617$	23.6510i	0.952152i	−0.879404	+	0.476076i	$0.842059\pi$
$618$	0	0
$619$	− 2.06167i	− 0.0828655i	−0.999141	−	0.0414328i	$-0.986808\pi$
$619$	− 2.06167i	− 0.0828655i	0.999141	−	0.0414328i	$-0.0131922\pi$
$620$	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
$629$	14.4422	0.575850
$630$	0	0
$631$	−26.6728	−1.06183	−0.530914	−	0.847426i	$-0.678151\pi$
$631$	−26.6728	−1.06183	−0.530914	+	0.847426i	$0.678151\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.7942	−0.547405
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.0170i	0.988113i	0.869430	+	0.494057i	$0.164486\pi$
$641$	25.0170i	0.988113i	−0.869430	+	0.494057i	$0.835514\pi$
$642$	0	0
$643$	− 12.0500i	− 0.475206i	−0.971362	−	0.237603i	$-0.923638\pi$
$643$	− 12.0500i	− 0.475206i	0.971362	−	0.237603i	$-0.0763618\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.7120	−0.657015	−0.328507	−	0.944501i	$-0.606546\pi$
$647$	−16.7120	−0.657015	−0.328507	+	0.944501i	$0.606546\pi$
$648$	0	0
$649$	27.8885i	1.09472i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.8808i	− 1.44326i	−0.692280	−	0.721629i	$-0.743394\pi$
$653$	− 36.8808i	− 1.44326i	0.692280	−	0.721629i	$-0.256606\pi$
$654$	0	0
$655$	13.0506	0.509928
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.3948i	0.794471i	0.917717	+	0.397235i	$0.130030\pi$
$659$	20.3948i	0.794471i	−0.917717	+	0.397235i	$0.869970\pi$
$660$	0	0
$661$	25.6607i	0.998086i	0.866577	+	0.499043i	$0.166315\pi$
$661$	25.6607i	0.998086i	−0.866577	+	0.499043i	$0.833685\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.24671	0.358034
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.75070	−0.299213
$672$	0	0
$673$	21.9694	0.846858	0.423429	−	0.905929i	$-0.360826\pi$
$673$	21.9694	0.846858	0.423429	+	0.905929i	$0.360826\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.5266	−1.36540	−0.682699	−	0.730700i	$-0.739194\pi$
$677$	−35.5266	−1.36540	−0.682699	+	0.730700i	$0.739194\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 44.8631i	− 1.71664i	−0.513116	−	0.858319i	$-0.671509\pi$
$683$	− 44.8631i	− 1.71664i	0.513116	−	0.858319i	$-0.328491\pi$
$684$	0	0
$685$	9.85597i	0.376577i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.390163	0.0148640
$690$	0	0
$691$	18.6842i	0.710781i	0.934718	+	0.355390i	$0.115652\pi$
$691$	18.6842i	0.710781i	−0.934718	+	0.355390i	$0.884348\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.9603i	0.567477i
$696$	0	0
$697$	−33.0549	−1.25204
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 32.7324i	− 1.23628i	−0.786066	−	0.618142i	$-0.787886\pi$
$701$	− 32.7324i	− 1.23628i	0.786066	−	0.618142i	$-0.212114\pi$
$702$	0	0
$703$	− 4.55726i	− 0.171880i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.8987	−0.597089	−0.298545	−	0.954396i	$-0.596501\pi$
$709$	−15.8987	−0.597089	−0.298545	+	0.954396i	$0.596501\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.4447	0.578408
$714$	0	0
$715$	16.9741	0.634794
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.1334	−0.900024	−0.450012	−	0.893022i	$-0.648580\pi$
$719$	−24.1334	−0.900024	−0.450012	+	0.893022i	$0.648580\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.55005i	− 0.0947064i
$726$	0	0
$727$	28.8371i	1.06951i	0.845007	+	0.534755i	$0.179596\pi$
$727$	28.8371i	1.06951i	−0.845007	+	0.534755i	$0.820404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	67.9045	2.51154
$732$	0	0
$733$	− 29.9940i	− 1.10785i	−0.832565	−	0.553927i	$-0.813128\pi$
$733$	− 29.9940i	− 1.10785i	0.832565	−	0.553927i	$-0.186872\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.8452i	0.952019i
$738$	0	0
$739$	−53.6812	−1.97470	−0.987348	−	0.158567i	$-0.949313\pi$
$739$	−53.6812	−1.97470	−0.987348	+	0.158567i	$0.949313\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.2416i	1.14614i	0.819505	+	0.573071i	$0.194248\pi$
$743$	31.2416i	1.14614i	−0.819505	+	0.573071i	$0.805752\pi$
$744$	0	0
$745$	12.7571i	0.467386i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.2981	−1.17858	−0.589288	−	0.807923i	$-0.700592\pi$
$751$	−32.2981	−1.17858	−0.589288	+	0.807923i	$0.700592\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.7228	0.499422
$756$	0	0
$757$	−39.3069	−1.42864	−0.714318	−	0.699822i	$-0.753263\pi$
$757$	−39.3069	−1.42864	−0.714318	+	0.699822i	$0.753263\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.8177	−1.47964	−0.739820	−	0.672805i	$-0.765089\pi$
$761$	−40.8177	−1.47964	−0.739820	+	0.672805i	$0.765089\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	76.1822i	2.75078i
$768$	0	0
$769$	− 20.3239i	− 0.732897i	−0.930438	−	0.366449i	$-0.880574\pi$
$769$	− 20.3239i	− 0.732897i	0.930438	−	0.366449i	$-0.119426\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.04787	−0.217527	−0.108763	−	0.994068i	$-0.534689\pi$
$773$	−6.04787	−0.217527	−0.108763	+	0.994068i	$0.534689\pi$
$774$	0	0
$775$	− 4.25932i	− 0.152999i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.4305i	0.373711i
$780$	0	0
$781$	−20.2861	−0.725895
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 12.0973i	− 0.431772i
$786$	0	0
$787$	− 32.6539i	− 1.16399i	−0.813193	−	0.581994i	$-0.802273\pi$
$787$	− 32.6539i	− 1.16399i	0.813193	−	0.581994i	$-0.197727\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−21.1724	−0.751852
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.923420	−0.0327092	−0.0163546	−	0.999866i	$-0.505206\pi$
$797$	−0.923420	−0.0327092	−0.0163546	+	0.999866i	$0.505206\pi$
$798$	0	0
$799$	8.05475	0.284957
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.3564	0.365471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 7.65715i	− 0.269211i	−0.990899	−	0.134605i	$-0.957023\pi$
$809$	− 7.65715i	− 0.269211i	0.990899	−	0.134605i	$-0.0429767\pi$
$810$	0	0
$811$	50.4421i	1.77126i	0.464389	+	0.885632i	$0.346274\pi$
$811$	50.4421i	1.77126i	−0.464389	+	0.885632i	$0.653726\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.6499	−0.408078
$816$	0	0
$817$	− 21.4273i	− 0.749646i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.6200i	− 0.649842i	−0.945741	−	0.324921i	$-0.894662\pi$
$821$	− 18.6200i	− 0.649842i	0.945741	−	0.324921i	$-0.105338\pi$
$822$	0	0
$823$	8.87664	0.309420	0.154710	−	0.987960i	$-0.450556\pi$
$823$	8.87664	0.309420	0.154710	+	0.987960i	$0.450556\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.06144i	− 0.141230i	−0.997504	−	0.0706150i	$-0.977504\pi$
$827$	− 4.06144i	− 0.141230i	0.997504	−	0.0706150i	$-0.0224962\pi$
$828$	0	0
$829$	− 27.5658i	− 0.957400i	−0.877978	−	0.478700i	$-0.841108\pi$
$829$	− 27.5658i	− 0.957400i	0.877978	−	0.478700i	$-0.158892\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.88337	−0.0651766
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.0792	−1.34916	−0.674582	−	0.738200i	$-0.735676\pi$
$839$	−39.0792	−1.34916	−0.674582	+	0.738200i	$0.735676\pi$
$840$	0	0
$841$	22.4973	0.775767
$842$	0	0
$843$	0	0
$844$	0	0
$845$	33.3675	1.14788
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.71196i	0.230083i
$852$	0	0
$853$	− 26.4630i	− 0.906074i	−0.891492	−	0.453037i	$-0.850340\pi$
$853$	− 26.4630i	− 0.906074i	0.891492	−	0.453037i	$-0.149660\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.44101	0.185861	0.0929307	−	0.995673i	$-0.470377\pi$
$857$	5.44101	0.185861	0.0929307	+	0.995673i	$0.470377\pi$
$858$	0	0
$859$	− 34.8369i	− 1.18862i	−0.804237	−	0.594309i	$-0.797426\pi$
$859$	− 34.8369i	− 1.18862i	0.804237	−	0.594309i	$-0.202574\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 20.2478i	− 0.689244i	−0.938741	−	0.344622i	$-0.888007\pi$
$863$	− 20.2478i	− 0.689244i	0.938741	−	0.344622i	$-0.111993\pi$
$864$	0	0
$865$	3.95988	0.134640
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 10.3914i	− 0.352504i
$870$	0	0
$871$	70.6004i	2.39220i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.09346	0.205761	0.102881	−	0.994694i	$-0.467194\pi$
$877$	6.09346	0.205761	0.102881	+	0.994694i	$0.467194\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.3020	0.582918	0.291459	−	0.956583i	$-0.405859\pi$
$881$	17.3020	0.582918	0.291459	+	0.956583i	$0.405859\pi$
$882$	0	0
$883$	−24.0929	−0.810792	−0.405396	−	0.914141i	$-0.632866\pi$
$883$	−24.0929	−0.810792	−0.405396	+	0.914141i	$0.632866\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.4849	1.25862	0.629310	−	0.777155i	$-0.283338\pi$
$887$	37.4849	1.25862	0.629310	+	0.777155i	$0.283338\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.54168i	− 0.0850542i
$894$	0	0
$895$	− 3.12414i	− 0.104429i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.8615	−0.362250
$900$	0	0
$901$	− 0.447058i	− 0.0148937i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.118397i	0.00393566i
$906$	0	0
$907$	16.8036	0.557954	0.278977	−	0.960298i	$-0.410005\pi$
$907$	16.8036	0.557954	0.278977	+	0.960298i	$0.410005\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.0890i	1.42760i	0.700348	+	0.713802i	$0.253028\pi$
$911$	43.0890i	1.42760i	−0.700348	+	0.713802i	$0.746972\pi$
$912$	0	0
$913$	− 29.1815i	− 0.965767i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	28.8847	0.952817	0.476409	−	0.879224i	$-0.341938\pi$
$919$	28.8847	0.952817	0.476409	+	0.879224i	$0.341938\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−55.4150	−1.82401
$924$	0	0
$925$	1.85102	0.0608610
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.82804	−0.0599761	−0.0299881	−	0.999550i	$-0.509547\pi$
$929$	−1.82804	−0.0599761	−0.0299881	+	0.999550i	$0.509547\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 19.4493i	− 0.636059i
$936$	0	0
$937$	− 19.7243i	− 0.644366i	−0.946677	−	0.322183i	$-0.895583\pi$
$937$	− 19.7243i	− 0.644366i	0.946677	−	0.322183i	$-0.104417\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34.9537	1.13946	0.569729	−	0.821833i	$-0.307048\pi$
$941$	34.9537	1.13946	0.569729	+	0.821833i	$0.307048\pi$
$942$	0	0
$943$	− 15.3621i	− 0.500258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.5241i	0.764432i	0.924073	+	0.382216i	$0.124839\pi$
$947$	23.5241i	0.764432i	−0.924073	+	0.382216i	$0.875161\pi$
$948$	0	0
$949$	28.2904	0.918344
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.27335i	− 0.0736410i	−0.999322	−	0.0368205i	$-0.988277\pi$
$953$	− 2.27335i	− 0.0736410i	0.999322	−	0.0368205i	$-0.0117230\pi$
$954$	0	0
$955$	− 9.59215i	− 0.310395i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.8582	0.414781
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.45035	−0.143262
$966$	0	0
$967$	32.0779	1.03156	0.515778	−	0.856722i	$-0.327503\pi$
$967$	32.0779	1.03156	0.515778	+	0.856722i	$0.327503\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.62313	−0.212546	−0.106273	−	0.994337i	$-0.533892\pi$
$971$	−6.62313	−0.212546	−0.106273	+	0.994337i	$0.533892\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.3070i	− 1.25754i	−0.777591	−	0.628771i	$-0.783558\pi$
$977$	− 39.3070i	− 1.25754i	0.777591	−	0.628771i	$-0.216442\pi$
$978$	0	0
$979$	− 24.8378i	− 0.793819i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.0207	−0.574772	−0.287386	−	0.957815i	$-0.592786\pi$
$983$	−18.0207	−0.574772	−0.287386	+	0.957815i	$0.592786\pi$
$984$	0	0
$985$	18.4602i	0.588190i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.5582i	1.00349i
$990$	0	0
$991$	−0.466136	−0.0148073	−0.00740365	−	0.999973i	$-0.502357\pi$
$991$	−0.466136	−0.0148073	−0.00740365	+	0.999973i	$0.502357\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 25.0373i	− 0.793735i
$996$	0	0
$997$	− 23.3689i	− 0.740099i	−0.929012	−	0.370050i	$-0.879341\pi$
$997$	− 23.3689i	− 0.740099i	0.929012	−	0.370050i	$-0.120659\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.a.881.9		12
3.2	odd	2		8820.2.d.b.881.4		12
7.2	even	3		1260.2.cg.b.521.6	yes	12
7.3	odd	6		1260.2.cg.a.341.6	✓	12
7.6	odd	2		8820.2.d.b.881.9		12
21.2	odd	6		1260.2.cg.a.521.6	yes	12
21.17	even	6		1260.2.cg.b.341.6	yes	12
21.20	even	2	inner	8820.2.d.a.881.4		12
35.2	odd	12		6300.2.dd.b.4049.8		24
35.3	even	12		6300.2.dd.c.1349.8		24
35.9	even	6		6300.2.ch.b.4301.1		12
35.17	even	12		6300.2.dd.c.1349.5		24
35.23	odd	12		6300.2.dd.b.4049.5		24
35.24	odd	6		6300.2.ch.c.1601.1		12
105.2	even	12		6300.2.dd.c.4049.8		24
105.17	odd	12		6300.2.dd.b.1349.5		24
105.23	even	12		6300.2.dd.c.4049.5		24
105.38	odd	12		6300.2.dd.b.1349.8		24
105.44	odd	6		6300.2.ch.c.4301.1		12
105.59	even	6		6300.2.ch.b.1601.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.6	✓	12	7.3	odd	6
1260.2.cg.a.521.6	yes	12	21.2	odd	6
1260.2.cg.b.341.6	yes	12	21.17	even	6
1260.2.cg.b.521.6	yes	12	7.2	even	3
6300.2.ch.b.1601.1		12	105.59	even	6
6300.2.ch.b.4301.1		12	35.9	even	6
6300.2.ch.c.1601.1		12	35.24	odd	6
6300.2.ch.c.4301.1		12	105.44	odd	6
6300.2.dd.b.1349.5		24	105.17	odd	12
6300.2.dd.b.1349.8		24	105.38	odd	12
6300.2.dd.b.4049.5		24	35.23	odd	12
6300.2.dd.b.4049.8		24	35.2	odd	12
6300.2.dd.c.1349.5		24	35.17	even	12
6300.2.dd.c.1349.8		24	35.3	even	12
6300.2.dd.c.4049.5		24	105.23	even	12
6300.2.dd.c.4049.8		24	105.2	even	12
8820.2.d.a.881.4		12	21.20	even	2	inner
8820.2.d.a.881.9		12	1.1	even	1	trivial
8820.2.d.b.881.4		12	3.2	odd	2
8820.2.d.b.881.9		12	7.6	odd	2

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} +2.49275i q^{11} +6.80937i q^{13} +7.80234 q^{17} -2.46203i q^{19} +3.62609i q^{23} +1.00000 q^{25} -2.55005i q^{29} -4.25932i q^{31} +1.85102 q^{37} -4.23654 q^{41} +8.70309 q^{43} +1.03235 q^{47} -0.0572980i q^{53} -2.49275i q^{55} +11.1879 q^{59} +3.10930i q^{61} -6.80937i q^{65} +10.3681 q^{67} +8.13806i q^{71} -4.15463i q^{73} -4.16865 q^{79} -11.7066 q^{83} -7.80234 q^{85} -9.96401 q^{89} +2.46203i q^{95} +4.91533i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5}+O(q^{10}) \) 12 * q - 12 * q^5	\( 12 q - 12 q^{5} + 12 q^{25} + 4 q^{37} + 8 q^{41} + 36 q^{43} - 32 q^{47} - 4 q^{67} - 28 q^{79} - 40 q^{83} - 40 q^{89}+O(q^{100}) \) 12 * q - 12 * q^5 + 12 * q^25 + 4 * q^37 + 8 * q^41 + 36 * q^43 - 32 * q^47 - 4 * q^67 - 28 * q^79 - 40 * q^83 - 40 * q^89

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