LMFDB - Embedded newform 882.3.c.e.685.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.4
Root			$1.84776i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.e.685.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} +2.82843 q^{8} +0.448342i q^{10} +2.82843 q^{11} -3.11586i q^{13} +4.00000 q^{16} -17.9749i q^{17} -18.7402i q^{19} +0.634051i q^{20} +4.00000 q^{22} +19.5147 q^{23} +24.8995 q^{25} -4.40649i q^{26} +43.5563 q^{29} +36.5838i q^{31} +5.65685 q^{32} -25.4203i q^{34} -13.6985 q^{37} -26.5027i q^{38} +0.896683i q^{40} +53.9564i q^{41} +7.59798 q^{43} +5.65685 q^{44} +27.5980 q^{46} -56.7459i q^{47} +35.2132 q^{50} -6.23172i q^{52} +20.2254 q^{53} +0.896683i q^{55} +61.5980 q^{58} -89.5891i q^{59} -74.4900i q^{61} +51.7373i q^{62} +8.00000 q^{64} +0.987807 q^{65} +4.00000 q^{67} -35.9497i q^{68} +131.799 q^{71} +33.8712i q^{73} -19.3726 q^{74} -37.4804i q^{76} -75.5980 q^{79} +1.26810i q^{80} +76.3059i q^{82} +131.245i q^{83} +5.69848 q^{85} +10.7452 q^{86} +8.00000 q^{88} -111.305i q^{89} +39.0294 q^{92} -80.2509i q^{94} +5.94113 q^{95} +148.106i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} + 16 q^{22} + 112 q^{23} + 60 q^{25} + 112 q^{29} + 64 q^{37} - 128 q^{43} - 48 q^{46} + 56 q^{50} - 168 q^{53} + 88 q^{58} + 32 q^{64} + 168 q^{65} + 16 q^{67} + 448 q^{71} - 168 q^{74} - 144 q^{79} - 96 q^{85} + 224 q^{86} + 32 q^{88} + 224 q^{92} - 112 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 + 16 * q^22 + 112 * q^23 + 60 * q^25 + 112 * q^29 + 64 * q^37 - 128 * q^43 - 48 * q^46 + 56 * q^50 - 168 * q^53 + 88 * q^58 + 32 * q^64 + 168 * q^65 + 16 * q^67 + 448 * q^71 - 168 * q^74 - 144 * q^79 - 96 * q^85 + 224 * q^86 + 32 * q^88 + 224 * q^92 - 112 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0.317025i	0.0634051i	0.999497	+	0.0317025i	$0.0100929\pi$
$5$	0.317025i	0.0634051i	−0.999497	+	0.0317025i	$0.989907\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	0	0
$10$	0.448342i	0.0448342i
$11$	2.82843	0.257130	0.128565	−	0.991701i	$-0.458963\pi$
$11$	2.82843	0.257130	0.128565	+	0.991701i	$0.458963\pi$
$12$	0	0
$13$	− 3.11586i	− 0.239682i	−0.992793	−	0.119841i	$-0.961762\pi$
$13$	− 3.11586i	− 0.239682i	0.992793	−	0.119841i	$-0.0382384\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 17.9749i	− 1.05734i	−0.848826	−	0.528672i	$-0.822690\pi$
$17$	− 17.9749i	− 1.05734i	0.848826	−	0.528672i	$-0.177310\pi$
$18$	0	0
$19$	− 18.7402i	− 0.986328i	−0.869936	−	0.493164i	$-0.835840\pi$
$19$	− 18.7402i	− 0.986328i	0.869936	−	0.493164i	$-0.164160\pi$
$20$	0.634051i	0.0317025i
$21$	0	0
$22$	4.00000	0.181818
$23$	19.5147	0.848466	0.424233	−	0.905553i	$-0.360544\pi$
$23$	19.5147	0.848466	0.424233	+	0.905553i	$0.360544\pi$
$24$	0	0
$25$	24.8995	0.995980
$26$	− 4.40649i	− 0.169480i
$27$	0	0
$28$	0	0
$29$	43.5563	1.50194	0.750972	−	0.660335i	$-0.229586\pi$
$29$	43.5563	1.50194	0.750972	+	0.660335i	$0.229586\pi$
$30$	0	0
$31$	36.5838i	1.18012i	0.807359	+	0.590061i	$0.200896\pi$
$31$	36.5838i	1.18012i	−0.807359	+	0.590061i	$0.799104\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	− 25.4203i	− 0.747655i
$35$	0	0
$36$	0	0
$37$	−13.6985	−0.370229	−0.185115	−	0.982717i	$-0.559266\pi$
$37$	−13.6985	−0.370229	−0.185115	+	0.982717i	$0.559266\pi$
$38$	− 26.5027i	− 0.697439i
$39$	0	0
$40$	0.896683i	0.0224171i
$41$	53.9564i	1.31601i	0.753013	+	0.658005i	$0.228600\pi$
$41$	53.9564i	1.31601i	−0.753013	+	0.658005i	$0.771400\pi$
$42$	0	0
$43$	7.59798	0.176697	0.0883486	−	0.996090i	$-0.471841\pi$
$43$	7.59798	0.176697	0.0883486	+	0.996090i	$0.471841\pi$
$44$	5.65685	0.128565
$45$	0	0
$46$	27.5980	0.599956
$47$	− 56.7459i	− 1.20736i	−0.797227	−	0.603680i	$-0.793700\pi$
$47$	− 56.7459i	− 1.20736i	0.797227	−	0.603680i	$-0.206300\pi$
$48$	0	0
$49$	0	0
$50$	35.2132	0.704264
$51$	0	0
$52$	− 6.23172i	− 0.119841i
$53$	20.2254	0.381611	0.190806	−	0.981628i	$-0.438890\pi$
$53$	20.2254	0.381611	0.190806	+	0.981628i	$0.438890\pi$
$54$	0	0
$55$	0.896683i	0.0163033i
$56$	0	0
$57$	0	0
$58$	61.5980	1.06203
$59$	− 89.5891i	− 1.51846i	−0.650823	−	0.759230i	$-0.725576\pi$
$59$	− 89.5891i	− 1.51846i	0.650823	−	0.759230i	$-0.274424\pi$
$60$	0	0
$61$	− 74.4900i	− 1.22115i	−0.791959	−	0.610574i	$-0.790939\pi$
$61$	− 74.4900i	− 1.22115i	0.791959	−	0.610574i	$-0.209061\pi$
$62$	51.7373i	0.834472i
$63$	0	0
$64$	8.00000	0.125000
$65$	0.987807	0.0151970
$66$	0	0
$67$	4.00000	0.0597015	0.0298507	−	0.999554i	$-0.490497\pi$
$67$	4.00000	0.0597015	0.0298507	+	0.999554i	$0.490497\pi$
$68$	− 35.9497i	− 0.528672i
$69$	0	0
$70$	0	0
$71$	131.799	1.85632	0.928162	−	0.372177i	$-0.121388\pi$
$71$	131.799	1.85632	0.928162	+	0.372177i	$0.121388\pi$
$72$	0	0
$73$	33.8712i	0.463989i	0.972717	+	0.231994i	$0.0745251\pi$
$73$	33.8712i	0.463989i	−0.972717	+	0.231994i	$0.925475\pi$
$74$	−19.3726	−0.261792
$75$	0	0
$76$	− 37.4804i	− 0.493164i
$77$	0	0
$78$	0	0
$79$	−75.5980	−0.956936	−0.478468	−	0.878105i	$-0.658808\pi$
$79$	−75.5980	−0.956936	−0.478468	+	0.878105i	$0.658808\pi$
$80$	1.26810i	0.0158513i
$81$	0	0
$82$	76.3059i	0.930560i
$83$	131.245i	1.58127i	0.612289	+	0.790634i	$0.290249\pi$
$83$	131.245i	1.58127i	−0.612289	+	0.790634i	$0.709751\pi$
$84$	0	0
$85$	5.69848	0.0670410
$86$	10.7452	0.124944
$87$	0	0
$88$	8.00000	0.0909091
$89$	− 111.305i	− 1.25061i	−0.780379	−	0.625306i	$-0.784974\pi$
$89$	− 111.305i	− 1.25061i	0.780379	−	0.625306i	$-0.215026\pi$
$90$	0	0
$91$	0	0
$92$	39.0294	0.424233
$93$	0	0
$94$	− 80.2509i	− 0.853733i
$95$	5.94113	0.0625382
$96$	0	0
$97$	148.106i	1.52686i	0.645888	+	0.763432i	$0.276487\pi$
$97$	148.106i	1.52686i	−0.645888	+	0.763432i	$0.723513\pi$
$98$	0	0
$99$	0	0
$100$	49.7990	0.497990
$101$	− 100.019i	− 0.990288i	−0.868811	−	0.495144i	$-0.835115\pi$
$101$	− 100.019i	− 0.990288i	0.868811	−	0.495144i	$-0.164885\pi$
$102$	0	0
$103$	6.18666i	0.0600647i	0.999549	+	0.0300323i	$0.00956103\pi$
$103$	6.18666i	0.0600647i	−0.999549	+	0.0300323i	$0.990439\pi$
$104$	− 8.81298i	− 0.0847402i
$105$	0	0
$106$	28.6030	0.269840
$107$	42.1421	0.393852	0.196926	−	0.980418i	$-0.436904\pi$
$107$	42.1421	0.393852	0.196926	+	0.980418i	$0.436904\pi$
$108$	0	0
$109$	−120.693	−1.10728	−0.553640	−	0.832756i	$-0.686761\pi$
$109$	−120.693	−1.10728	−0.553640	+	0.832756i	$0.686761\pi$
$110$	1.26810i	0.0115282i
$111$	0	0
$112$	0	0
$113$	104.794	0.927380	0.463690	−	0.885998i	$-0.346525\pi$
$113$	104.794	0.927380	0.463690	+	0.885998i	$0.346525\pi$
$114$	0	0
$115$	6.18666i	0.0537970i
$116$	87.1127	0.750972
$117$	0	0
$118$	− 126.698i	− 1.07371i
$119$	0	0
$120$	0	0
$121$	−113.000	−0.933884
$122$	− 105.345i	− 0.863482i
$123$	0	0
$124$	73.1675i	0.590061i
$125$	15.8194i	0.126555i
$126$	0	0
$127$	−166.392	−1.31017	−0.655086	−	0.755554i	$-0.727368\pi$
$127$	−166.392	−1.31017	−0.655086	+	0.755554i	$0.727368\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	1.39697	0.0107459
$131$	65.6864i	0.501423i	0.968062	+	0.250711i	$0.0806645\pi$
$131$	65.6864i	0.501423i	−0.968062	+	0.250711i	$0.919336\pi$
$132$	0	0
$133$	0	0
$134$	5.65685	0.0422153
$135$	0	0
$136$	− 50.8406i	− 0.373828i
$137$	−3.95837	−0.0288932	−0.0144466	−	0.999896i	$-0.504599\pi$
$137$	−3.95837	−0.0288932	−0.0144466	+	0.999896i	$0.504599\pi$
$138$	0	0
$139$	96.3010i	0.692813i	0.938084	+	0.346407i	$0.112598\pi$
$139$	96.3010i	0.692813i	−0.938084	+	0.346407i	$0.887402\pi$
$140$	0	0
$141$	0	0
$142$	186.392	1.31262
$143$	− 8.81298i	− 0.0616293i
$144$	0	0
$145$	13.8085i	0.0952308i
$146$	47.9011i	0.328090i
$147$	0	0
$148$	−27.3970	−0.185115
$149$	120.912	0.811488	0.405744	−	0.913987i	$-0.367012\pi$
$149$	120.912	0.811488	0.405744	+	0.913987i	$0.367012\pi$
$150$	0	0
$151$	−162.392	−1.07544	−0.537722	−	0.843122i	$-0.680715\pi$
$151$	−162.392	−1.07544	−0.537722	+	0.843122i	$0.680715\pi$
$152$	− 53.0054i	− 0.348719i
$153$	0	0
$154$	0	0
$155$	−11.5980	−0.0748257
$156$	0	0
$157$	− 66.0617i	− 0.420775i	−0.977618	−	0.210387i	$-0.932527\pi$
$157$	− 66.0617i	− 0.420775i	0.977618	−	0.210387i	$-0.0674725\pi$
$158$	−106.912	−0.676656
$159$	0	0
$160$	1.79337i	0.0112085i
$161$	0	0
$162$	0	0
$163$	−209.990	−1.28828	−0.644141	−	0.764907i	$-0.722785\pi$
$163$	−209.990	−1.28828	−0.644141	+	0.764907i	$0.722785\pi$
$164$	107.913i	0.658005i
$165$	0	0
$166$	185.609i	1.11813i
$167$	− 152.739i	− 0.914606i	−0.889311	−	0.457303i	$-0.848815\pi$
$167$	− 152.739i	− 0.914606i	0.889311	−	0.457303i	$-0.151185\pi$
$168$	0	0
$169$	159.291	0.942553
$170$	8.05887	0.0474051
$171$	0	0
$172$	15.1960	0.0883486
$173$	35.5690i	0.205601i	0.994702	+	0.102800i	$0.0327803\pi$
$173$	35.5690i	0.205601i	−0.994702	+	0.102800i	$0.967220\pi$
$174$	0	0
$175$	0	0
$176$	11.3137	0.0642824
$177$	0	0
$178$	− 157.408i	− 0.884317i
$179$	63.9167	0.357077	0.178538	−	0.983933i	$-0.442863\pi$
$179$	63.9167	0.357077	0.178538	+	0.983933i	$0.442863\pi$
$180$	0	0
$181$	183.748i	1.01518i	0.861598	+	0.507591i	$0.169464\pi$
$181$	183.748i	1.01518i	−0.861598	+	0.507591i	$0.830536\pi$
$182$	0	0
$183$	0	0
$184$	55.1960	0.299978
$185$	− 4.34277i	− 0.0234744i
$186$	0	0
$187$	− 50.8406i	− 0.271875i
$188$	− 113.492i	− 0.603680i
$189$	0	0
$190$	8.40202	0.0442212
$191$	−154.426	−0.808515	−0.404258	−	0.914645i	$-0.632470\pi$
$191$	−154.426	−0.808515	−0.404258	+	0.914645i	$0.632470\pi$
$192$	0	0
$193$	−23.1960	−0.120186	−0.0600932	−	0.998193i	$-0.519140\pi$
$193$	−23.1960	−0.120186	−0.0600932	+	0.998193i	$0.519140\pi$
$194$	209.453i	1.07966i
$195$	0	0
$196$	0	0
$197$	−202.510	−1.02797	−0.513984	−	0.857800i	$-0.671831\pi$
$197$	−202.510	−1.02797	−0.513984	+	0.857800i	$0.671831\pi$
$198$	0	0
$199$	230.083i	1.15619i	0.815968	+	0.578097i	$0.196204\pi$
$199$	230.083i	1.15619i	−0.815968	+	0.578097i	$0.803796\pi$
$200$	70.4264	0.352132
$201$	0	0
$202$	− 141.448i	− 0.700239i
$203$	0	0
$204$	0	0
$205$	−17.1056	−0.0834417
$206$	8.74926i	0.0424721i
$207$	0	0
$208$	− 12.4634i	− 0.0599204i
$209$	− 53.0054i	− 0.253614i
$210$	0	0
$211$	−172.402	−0.817071	−0.408536	−	0.912742i	$-0.633960\pi$
$211$	−172.402	−0.817071	−0.408536	+	0.912742i	$0.633960\pi$
$212$	40.4508	0.190806
$213$	0	0
$214$	59.5980	0.278495
$215$	2.40875i	0.0112035i
$216$	0	0
$217$	0	0
$218$	−170.686	−0.782965
$219$	0	0
$220$	1.79337i	0.00815166i
$221$	−56.0071	−0.253426
$222$	0	0
$223$	55.3240i	0.248090i	0.992277	+	0.124045i	$0.0395867\pi$
$223$	55.3240i	0.248090i	−0.992277	+	0.124045i	$0.960413\pi$
$224$	0	0
$225$	0	0
$226$	148.201	0.655757
$227$	369.578i	1.62810i	0.580796	+	0.814049i	$0.302741\pi$
$227$	369.578i	1.62810i	−0.580796	+	0.814049i	$0.697259\pi$
$228$	0	0
$229$	− 294.396i	− 1.28557i	−0.766046	−	0.642786i	$-0.777779\pi$
$229$	− 294.396i	− 1.28557i	0.766046	−	0.642786i	$-0.222221\pi$
$230$	8.74926i	0.0380403i
$231$	0	0
$232$	123.196	0.531017
$233$	−174.787	−0.750158	−0.375079	−	0.926993i	$-0.622384\pi$
$233$	−174.787	−0.750158	−0.375079	+	0.926993i	$0.622384\pi$
$234$	0	0
$235$	17.9899	0.0765528
$236$	− 179.178i	− 0.759230i
$237$	0	0
$238$	0	0
$239$	−323.279	−1.35263	−0.676316	−	0.736611i	$-0.736425\pi$
$239$	−323.279	−1.35263	−0.676316	+	0.736611i	$0.736425\pi$
$240$	0	0
$241$	− 165.501i	− 0.686726i	−0.939203	−	0.343363i	$-0.888434\pi$
$241$	− 165.501i	− 0.686726i	0.939203	−	0.343363i	$-0.111566\pi$
$242$	−159.806	−0.660356
$243$	0	0
$244$	− 148.980i	− 0.610574i
$245$	0	0
$246$	0	0
$247$	−58.3919	−0.236405
$248$	103.475i	0.417236i
$249$	0	0
$250$	22.3720i	0.0894881i
$251$	− 99.7339i	− 0.397346i	−0.980066	−	0.198673i	$-0.936337\pi$
$251$	− 99.7339i	− 0.397346i	0.980066	−	0.198673i	$-0.0636632\pi$
$252$	0	0
$253$	55.1960	0.218166
$254$	−235.314	−0.926432
$255$	0	0
$256$	16.0000	0.0625000
$257$	379.976i	1.47851i	0.673427	+	0.739254i	$0.264821\pi$
$257$	379.976i	1.47851i	−0.673427	+	0.739254i	$0.735179\pi$
$258$	0	0
$259$	0	0
$260$	1.97561	0.00759851
$261$	0	0
$262$	92.8946i	0.354559i
$263$	−233.622	−0.888298	−0.444149	−	0.895953i	$-0.646494\pi$
$263$	−233.622	−0.888298	−0.444149	+	0.895953i	$0.646494\pi$
$264$	0	0
$265$	6.41196i	0.0241961i
$266$	0	0
$267$	0	0
$268$	8.00000	0.0298507
$269$	465.286i	1.72969i	0.502040	+	0.864845i	$0.332583\pi$
$269$	465.286i	1.72969i	−0.502040	+	0.864845i	$0.667417\pi$
$270$	0	0
$271$	141.671i	0.522773i	0.965234	+	0.261386i	$0.0841797\pi$
$271$	141.671i	0.522773i	−0.965234	+	0.261386i	$0.915820\pi$
$272$	− 71.8994i	− 0.264336i
$273$	0	0
$274$	−5.59798	−0.0204306
$275$	70.4264	0.256096
$276$	0	0
$277$	314.000	1.13357	0.566787	−	0.823864i	$-0.308186\pi$
$277$	314.000	1.13357	0.566787	+	0.823864i	$0.308186\pi$
$278$	136.190i	0.489893i
$279$	0	0
$280$	0	0
$281$	−323.002	−1.14947	−0.574737	−	0.818338i	$-0.694896\pi$
$281$	−323.002	−1.14947	−0.574737	+	0.818338i	$0.694896\pi$
$282$	0	0
$283$	534.500i	1.88869i	0.328955	+	0.944346i	$0.393304\pi$
$283$	534.500i	1.88869i	−0.328955	+	0.944346i	$0.606696\pi$
$284$	263.598	0.928162
$285$	0	0
$286$	− 12.4634i	− 0.0435785i
$287$	0	0
$288$	0	0
$289$	−34.0955	−0.117977
$290$	19.5281i	0.0673383i
$291$	0	0
$292$	67.7424i	0.231994i
$293$	− 392.246i	− 1.33872i	−0.742936	−	0.669362i	$-0.766567\pi$
$293$	− 392.246i	− 1.33872i	0.742936	−	0.669362i	$-0.233433\pi$
$294$	0	0
$295$	28.4020	0.0962780
$296$	−38.7452	−0.130896
$297$	0	0
$298$	170.995	0.573809
$299$	− 60.8051i	− 0.203362i
$300$	0	0
$301$	0	0
$302$	−229.657	−0.760453
$303$	0	0
$304$	− 74.9609i	− 0.246582i
$305$	23.6152	0.0774270
$306$	0	0
$307$	− 541.583i	− 1.76411i	−0.471143	−	0.882057i	$-0.656158\pi$
$307$	− 541.583i	− 1.76411i	0.471143	−	0.882057i	$-0.343842\pi$
$308$	0	0
$309$	0	0
$310$	−16.4020	−0.0529097
$311$	− 57.8866i	− 0.186130i	−0.995660	−	0.0930652i	$-0.970333\pi$
$311$	− 57.8866i	− 0.186130i	0.995660	−	0.0930652i	$-0.0296665\pi$
$312$	0	0
$313$	− 461.981i	− 1.47598i	−0.674813	−	0.737989i	$-0.735776\pi$
$313$	− 461.981i	− 1.47598i	0.674813	−	0.737989i	$-0.264224\pi$
$314$	− 93.4253i	− 0.297533i
$315$	0	0
$316$	−151.196	−0.478468
$317$	−415.196	−1.30977	−0.654883	−	0.755730i	$-0.727282\pi$
$317$	−415.196	−1.30977	−0.654883	+	0.755730i	$0.727282\pi$
$318$	0	0
$319$	123.196	0.386194
$320$	2.53620i	0.00792563i
$321$	0	0
$322$	0	0
$323$	−336.853	−1.04289
$324$	0	0
$325$	− 77.5834i	− 0.238718i
$326$	−296.971	−0.910953
$327$	0	0
$328$	152.612i	0.465280i
$329$	0	0
$330$	0	0
$331$	76.4020	0.230822	0.115411	−	0.993318i	$-0.463182\pi$
$331$	76.4020	0.230822	0.115411	+	0.993318i	$0.463182\pi$
$332$	262.491i	0.790634i
$333$	0	0
$334$	− 216.006i	− 0.646724i
$335$	1.26810i	0.00378538i
$336$	0	0
$337$	376.291	1.11659	0.558296	−	0.829642i	$-0.311455\pi$
$337$	376.291	1.11659	0.558296	+	0.829642i	$0.311455\pi$
$338$	225.272	0.666485
$339$	0	0
$340$	11.3970	0.0335205
$341$	103.475i	0.303444i
$342$	0	0
$343$	0	0
$344$	21.4903	0.0624719
$345$	0	0
$346$	50.3021i	0.145382i
$347$	78.3289	0.225732	0.112866	−	0.993610i	$-0.463997\pi$
$347$	78.3289	0.225732	0.112866	+	0.993610i	$0.463997\pi$
$348$	0	0
$349$	376.710i	1.07940i	0.841858	+	0.539700i	$0.181462\pi$
$349$	376.710i	1.07940i	−0.841858	+	0.539700i	$0.818538\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.0454545
$353$	213.352i	0.604396i	0.953245	+	0.302198i	$0.0977203\pi$
$353$	213.352i	0.604396i	−0.953245	+	0.302198i	$0.902280\pi$
$354$	0	0
$355$	41.7836i	0.117700i
$356$	− 222.609i	− 0.625306i
$357$	0	0
$358$	90.3919	0.252491
$359$	28.0143	0.0780342	0.0390171	−	0.999239i	$-0.487577\pi$
$359$	28.0143	0.0780342	0.0390171	+	0.999239i	$0.487577\pi$
$360$	0	0
$361$	9.80404	0.0271580
$362$	259.859i	0.717842i
$363$	0	0
$364$	0	0
$365$	−10.7380	−0.0294192
$366$	0	0
$367$	− 149.742i	− 0.408015i	−0.978969	−	0.204008i	$-0.934603\pi$
$367$	− 149.742i	− 0.408015i	0.978969	−	0.204008i	$-0.0653967\pi$
$368$	78.0589	0.212117
$369$	0	0
$370$	− 6.14160i	− 0.0165989i
$371$	0	0
$372$	0	0
$373$	231.196	0.619828	0.309914	−	0.950765i	$-0.399700\pi$
$373$	231.196	0.619828	0.309914	+	0.950765i	$0.399700\pi$
$374$	− 71.8994i	− 0.192244i
$375$	0	0
$376$	− 160.502i	− 0.426866i
$377$	− 135.716i	− 0.359988i
$378$	0	0
$379$	−585.186	−1.54403	−0.772013	−	0.635607i	$-0.780750\pi$
$379$	−585.186	−1.54403	−0.772013	+	0.635607i	$0.780750\pi$
$380$	11.8823	0.0312691
$381$	0	0
$382$	−218.392	−0.571707
$383$	533.858i	1.39388i	0.717127	+	0.696942i	$0.245457\pi$
$383$	533.858i	1.39388i	−0.717127	+	0.696942i	$0.754543\pi$
$384$	0	0
$385$	0	0
$386$	−32.8040	−0.0849846
$387$	0	0
$388$	296.212i	0.763432i
$389$	19.7918	0.0508788	0.0254394	−	0.999676i	$-0.491902\pi$
$389$	19.7918	0.0508788	0.0254394	+	0.999676i	$0.491902\pi$
$390$	0	0
$391$	− 350.774i	− 0.897121i
$392$	0	0
$393$	0	0
$394$	−286.392	−0.726883
$395$	− 23.9665i	− 0.0606746i
$396$	0	0
$397$	367.340i	0.925291i	0.886543	+	0.462645i	$0.153100\pi$
$397$	367.340i	0.925291i	−0.886543	+	0.462645i	$0.846900\pi$
$398$	325.386i	0.817552i
$399$	0	0
$400$	99.5980	0.248995
$401$	692.943	1.72804	0.864019	−	0.503459i	$-0.167940\pi$
$401$	692.943	1.72804	0.864019	+	0.503459i	$0.167940\pi$
$402$	0	0
$403$	113.990	0.282853
$404$	− 200.038i	− 0.495144i
$405$	0	0
$406$	0	0
$407$	−38.7452	−0.0951970
$408$	0	0
$409$	− 290.899i	− 0.711245i	−0.934630	−	0.355623i	$-0.884269\pi$
$409$	− 290.899i	− 0.711245i	0.934630	−	0.355623i	$-0.115731\pi$
$410$	−24.1909	−0.0590022
$411$	0	0
$412$	12.3733i	0.0300323i
$413$	0	0
$414$	0	0
$415$	−41.6081	−0.100260
$416$	− 17.6260i	− 0.0423701i
$417$	0	0
$418$	− 74.9609i	− 0.179332i
$419$	− 624.333i	− 1.49005i	−0.667034	−	0.745027i	$-0.732437\pi$
$419$	− 624.333i	− 1.49005i	0.667034	−	0.745027i	$-0.267563\pi$
$420$	0	0
$421$	753.980	1.79093	0.895463	−	0.445136i	$-0.146845\pi$
$421$	753.980	1.79093	0.895463	+	0.445136i	$0.146845\pi$
$422$	−243.813	−0.577757
$423$	0	0
$424$	57.2061	0.134920
$425$	− 447.565i	− 1.05309i
$426$	0	0
$427$	0	0
$428$	84.2843	0.196926
$429$	0	0
$430$	3.40649i	0.00792207i
$431$	1.15137	0.00267139	0.00133570	−	0.999999i	$-0.499575\pi$
$431$	1.15137	0.00267139	0.00133570	+	0.999999i	$0.499575\pi$
$432$	0	0
$433$	193.834i	0.447655i	0.974629	+	0.223827i	$0.0718552\pi$
$433$	193.834i	0.447655i	−0.974629	+	0.223827i	$0.928145\pi$
$434$	0	0
$435$	0	0
$436$	−241.387	−0.553640
$437$	− 365.710i	− 0.836865i
$438$	0	0
$439$	444.295i	1.01206i	0.862515	+	0.506031i	$0.168888\pi$
$439$	444.295i	1.01206i	−0.862515	+	0.506031i	$0.831112\pi$
$440$	2.53620i	0.00576410i
$441$	0	0
$442$	−79.2061	−0.179199
$443$	−403.897	−0.911730	−0.455865	−	0.890049i	$-0.650670\pi$
$443$	−403.897	−0.911730	−0.455865	+	0.890049i	$0.650670\pi$
$444$	0	0
$445$	35.2864	0.0792952
$446$	78.2399i	0.175426i
$447$	0	0
$448$	0	0
$449$	242.520	0.540133	0.270067	−	0.962842i	$-0.412954\pi$
$449$	242.520	0.540133	0.270067	+	0.962842i	$0.412954\pi$
$450$	0	0
$451$	152.612i	0.338385i
$452$	209.588	0.463690
$453$	0	0
$454$	522.663i	1.15124i
$455$	0	0
$456$	0	0
$457$	744.764	1.62968	0.814840	−	0.579686i	$-0.196825\pi$
$457$	744.764	1.62968	0.814840	+	0.579686i	$0.196825\pi$
$458$	− 416.339i	− 0.909036i
$459$	0	0
$460$	12.3733i	0.0268985i
$461$	764.329i	1.65798i	0.559263	+	0.828990i	$0.311084\pi$
$461$	764.329i	1.65798i	−0.559263	+	0.828990i	$0.688916\pi$
$462$	0	0
$463$	−658.774	−1.42284	−0.711419	−	0.702768i	$-0.751947\pi$
$463$	−658.774	−1.42284	−0.711419	+	0.702768i	$0.751947\pi$
$464$	174.225	0.375486
$465$	0	0
$466$	−247.186	−0.530442
$467$	− 65.6226i	− 0.140520i	−0.997529	−	0.0702598i	$-0.977617\pi$
$467$	− 65.6226i	− 0.140520i	0.997529	−	0.0702598i	$-0.0223828\pi$
$468$	0	0
$469$	0	0
$470$	25.4416	0.0541310
$471$	0	0
$472$	− 253.396i	− 0.536857i
$473$	21.4903	0.0454341
$474$	0	0
$475$	− 466.622i	− 0.982362i
$476$	0	0
$477$	0	0
$478$	−457.186	−0.956456
$479$	248.478i	0.518743i	0.965778	+	0.259371i	$0.0835153\pi$
$479$	248.478i	0.518743i	−0.965778	+	0.259371i	$0.916485\pi$
$480$	0	0
$481$	42.6826i	0.0887371i
$482$	− 234.054i	− 0.485589i
$483$	0	0
$484$	−226.000	−0.466942
$485$	−46.9533	−0.0968110
$486$	0	0
$487$	−764.402	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$487$	−764.402	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$488$	− 210.690i	− 0.431741i
$489$	0	0
$490$	0	0
$491$	782.621	1.59393	0.796967	−	0.604023i	$-0.206436\pi$
$491$	782.621	1.59393	0.796967	+	0.604023i	$0.206436\pi$
$492$	0	0
$493$	− 782.919i	− 1.58807i
$494$	−82.5786	−0.167163
$495$	0	0
$496$	146.335i	0.295030i
$497$	0	0
$498$	0	0
$499$	−131.980	−0.264489	−0.132244	−	0.991217i	$-0.542218\pi$
$499$	−131.980	−0.264489	−0.132244	+	0.991217i	$0.542218\pi$
$500$	31.6388i	0.0632776i
$501$	0	0
$502$	− 141.045i	− 0.280966i
$503$	756.910i	1.50479i	0.658712	+	0.752395i	$0.271102\pi$
$503$	756.910i	1.50479i	−0.658712	+	0.752395i	$0.728898\pi$
$504$	0	0
$505$	31.7086	0.0627893
$506$	78.0589	0.154267
$507$	0	0
$508$	−332.784	−0.655086
$509$	− 233.418i	− 0.458582i	−0.973358	−	0.229291i	$-0.926359\pi$
$509$	− 233.418i	− 0.458582i	0.973358	−	0.229291i	$-0.0736407\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	0.0441942
$513$	0	0
$514$	537.368i	1.04546i
$515$	−1.96133	−0.00380840
$516$	0	0
$517$	− 160.502i	− 0.310448i
$518$	0	0
$519$	0	0
$520$	2.79394	0.00537296
$521$	− 596.691i	− 1.14528i	−0.819807	−	0.572640i	$-0.805919\pi$
$521$	− 596.691i	− 1.14528i	0.819807	−	0.572640i	$-0.194081\pi$
$522$	0	0
$523$	815.152i	1.55861i	0.626646	+	0.779304i	$0.284427\pi$
$523$	815.152i	1.55861i	−0.626646	+	0.779304i	$0.715573\pi$
$524$	131.373i	0.250711i
$525$	0	0
$526$	−330.392	−0.628122
$527$	657.588	1.24779
$528$	0	0
$529$	−148.176	−0.280105
$530$	9.06789i	0.0171092i
$531$	0	0
$532$	0	0
$533$	168.121	0.315423
$534$	0	0
$535$	13.3601i	0.0249722i
$536$	11.3137	0.0211077
$537$	0	0
$538$	658.014i	1.22307i
$539$	0	0
$540$	0	0
$541$	233.568	0.431733	0.215867	−	0.976423i	$-0.430742\pi$
$541$	233.568	0.431733	0.215867	+	0.976423i	$0.430742\pi$
$542$	200.354i	0.369656i
$543$	0	0
$544$	− 101.681i	− 0.186914i
$545$	− 38.2629i	− 0.0702071i
$546$	0	0
$547$	−27.1758	−0.0496815	−0.0248407	−	0.999691i	$-0.507908\pi$
$547$	−27.1758	−0.0496815	−0.0248407	+	0.999691i	$0.507908\pi$
$548$	−7.91674	−0.0144466
$549$	0	0
$550$	99.5980	0.181087
$551$	− 816.256i	− 1.48141i
$552$	0	0
$553$	0	0
$554$	444.063	0.801558
$555$	0	0
$556$	192.602i	0.346407i
$557$	−290.745	−0.521984	−0.260992	−	0.965341i	$-0.584050\pi$
$557$	−290.745	−0.521984	−0.260992	+	0.965341i	$0.584050\pi$
$558$	0	0
$559$	− 23.6742i	− 0.0423511i
$560$	0	0
$561$	0	0
$562$	−456.794	−0.812801
$563$	− 976.476i	− 1.73442i	−0.497946	−	0.867208i	$-0.665912\pi$
$563$	− 976.476i	− 1.73442i	0.497946	−	0.867208i	$-0.334088\pi$
$564$	0	0
$565$	33.2223i	0.0588006i
$566$	755.897i	1.33551i
$567$	0	0
$568$	372.784	0.656310
$569$	−635.252	−1.11644	−0.558218	−	0.829694i	$-0.688515\pi$
$569$	−635.252	−1.11644	−0.558218	+	0.829694i	$0.688515\pi$
$570$	0	0
$571$	453.186	0.793671	0.396835	−	0.917890i	$-0.370108\pi$
$571$	453.186	0.793671	0.396835	+	0.917890i	$0.370108\pi$
$572$	− 17.6260i	− 0.0308146i
$573$	0	0
$574$	0	0
$575$	485.907	0.845055
$576$	0	0
$577$	− 1093.00i	− 1.89429i	−0.320807	−	0.947144i	$-0.603954\pi$
$577$	− 1093.00i	− 1.89429i	0.320807	−	0.947144i	$-0.396046\pi$
$578$	−48.2183	−0.0834226
$579$	0	0
$580$	27.6169i	0.0476154i
$581$	0	0
$582$	0	0
$583$	57.2061	0.0981236
$584$	95.8022i	0.164045i
$585$	0	0
$586$	− 554.720i	− 0.946621i
$587$	883.401i	1.50494i	0.658625	+	0.752471i	$0.271138\pi$
$587$	883.401i	1.50494i	−0.658625	+	0.752471i	$0.728862\pi$
$588$	0	0
$589$	685.588	1.16399
$590$	40.1665	0.0680789
$591$	0	0
$592$	−54.7939	−0.0925573
$593$	− 100.462i	− 0.169413i	−0.996406	−	0.0847065i	$-0.973005\pi$
$593$	− 100.462i	− 0.169413i	0.996406	−	0.0847065i	$-0.0269953\pi$
$594$	0	0
$595$	0	0
$596$	241.823	0.405744
$597$	0	0
$598$	− 85.9915i	− 0.143798i
$599$	1034.05	1.72630	0.863149	−	0.504949i	$-0.168489\pi$
$599$	1034.05	1.72630	0.863149	+	0.504949i	$0.168489\pi$
$600$	0	0
$601$	401.682i	0.668357i	0.942510	+	0.334178i	$0.108459\pi$
$601$	401.682i	0.668357i	−0.942510	+	0.334178i	$0.891541\pi$
$602$	0	0
$603$	0	0
$604$	−324.784	−0.537722
$605$	− 35.8239i	− 0.0592130i
$606$	0	0
$607$	876.217i	1.44352i	0.692143	+	0.721760i	$0.256667\pi$
$607$	876.217i	1.44352i	−0.692143	+	0.721760i	$0.743333\pi$
$608$	− 106.011i	− 0.174360i
$609$	0	0
$610$	33.3970	0.0547491
$611$	−176.812	−0.289382
$612$	0	0
$613$	−85.4773	−0.139441	−0.0697205	−	0.997567i	$-0.522211\pi$
$613$	−85.4773	−0.139441	−0.0697205	+	0.997567i	$0.522211\pi$
$614$	− 765.914i	− 1.24742i
$615$	0	0
$616$	0	0
$617$	522.115	0.846215	0.423108	−	0.906079i	$-0.360939\pi$
$617$	522.115	0.846215	0.423108	+	0.906079i	$0.360939\pi$
$618$	0	0
$619$	653.218i	1.05528i	0.849468	+	0.527640i	$0.176923\pi$
$619$	653.218i	1.05528i	−0.849468	+	0.527640i	$0.823077\pi$
$620$	−23.1960	−0.0374128
$621$	0	0
$622$	− 81.8640i	− 0.131614i
$623$	0	0
$624$	0	0
$625$	617.472	0.987956
$626$	− 653.340i	− 1.04367i
$627$	0	0
$628$	− 132.123i	− 0.210387i
$629$	246.228i	0.391460i
$630$	0	0
$631$	−728.764	−1.15493	−0.577467	−	0.816414i	$-0.695959\pi$
$631$	−728.764	−1.15493	−0.577467	+	0.816414i	$0.695959\pi$
$632$	−213.823	−0.338328
$633$	0	0
$634$	−587.176	−0.926145
$635$	− 52.7505i	− 0.0830716i
$636$	0	0
$637$	0	0
$638$	174.225	0.273081
$639$	0	0
$640$	3.58673i	0.00560427i
$641$	−190.606	−0.297357	−0.148679	−	0.988886i	$-0.547502\pi$
$641$	−190.606	−0.297357	−0.148679	+	0.988886i	$0.547502\pi$
$642$	0	0
$643$	− 910.647i	− 1.41625i	−0.706089	−	0.708124i	$-0.749542\pi$
$643$	− 910.647i	− 1.41625i	0.706089	−	0.708124i	$-0.250458\pi$
$644$	0	0
$645$	0	0
$646$	−476.382	−0.737433
$647$	− 355.757i	− 0.549856i	−0.961465	−	0.274928i	$-0.911346\pi$
$647$	− 355.757i	− 0.549856i	0.961465	−	0.274928i	$-0.0886539\pi$
$648$	0	0
$649$	− 253.396i	− 0.390441i
$650$	− 109.719i	− 0.168799i
$651$	0	0
$652$	−419.980	−0.644141
$653$	1130.75	1.73163	0.865813	−	0.500367i	$-0.166802\pi$
$653$	1130.75	1.73163	0.865813	+	0.500367i	$0.166802\pi$
$654$	0	0
$655$	−20.8242	−0.0317927
$656$	215.826i	0.329003i
$657$	0	0
$658$	0	0
$659$	520.388	0.789663	0.394831	−	0.918754i	$-0.370803\pi$
$659$	520.388	0.789663	0.394831	+	0.918754i	$0.370803\pi$
$660$	0	0
$661$	− 351.243i	− 0.531381i	−0.964058	−	0.265691i	$-0.914400\pi$
$661$	− 351.243i	− 0.531381i	0.964058	−	0.265691i	$-0.0855999\pi$
$662$	108.049	0.163216
$663$	0	0
$664$	371.218i	0.559063i
$665$	0	0
$666$	0	0
$667$	849.990	1.27435
$668$	− 305.479i	− 0.457303i
$669$	0	0
$670$	1.79337i	0.00267667i
$671$	− 210.690i	− 0.313993i
$672$	0	0
$673$	397.075	0.590008	0.295004	−	0.955496i	$-0.404679\pi$
$673$	397.075	0.590008	0.295004	+	0.955496i	$0.404679\pi$
$674$	532.156	0.789550
$675$	0	0
$676$	318.583	0.471276
$677$	− 361.210i	− 0.533545i	−0.963760	−	0.266772i	$-0.914043\pi$
$677$	− 361.210i	− 0.533545i	0.963760	−	0.266772i	$-0.0859572\pi$
$678$	0	0
$679$	0	0
$680$	16.1177	0.0237026
$681$	0	0
$682$	146.335i	0.214568i
$683$	−938.455	−1.37402	−0.687010	−	0.726648i	$-0.741077\pi$
$683$	−938.455	−1.37402	−0.687010	+	0.726648i	$0.741077\pi$
$684$	0	0
$685$	− 1.25490i	− 0.00183198i
$686$	0	0
$687$	0	0
$688$	30.3919	0.0441743
$689$	− 63.0195i	− 0.0914652i
$690$	0	0
$691$	− 381.888i	− 0.552660i	−0.961063	−	0.276330i	$-0.910882\pi$
$691$	− 381.888i	− 0.552660i	0.961063	−	0.276330i	$-0.0891182\pi$
$692$	71.1379i	0.102800i
$693$	0	0
$694$	110.774	0.159616
$695$	−30.5299	−0.0439279
$696$	0	0
$697$	969.859	1.39148
$698$	532.749i	0.763251i
$699$	0	0
$700$	0	0
$701$	713.325	1.01758	0.508791	−	0.860890i	$-0.330093\pi$
$701$	713.325	1.01758	0.508791	+	0.860890i	$0.330093\pi$
$702$	0	0
$703$	256.713i	0.365167i
$704$	22.6274	0.0321412
$705$	0	0
$706$	301.725i	0.427372i
$707$	0	0
$708$	0	0
$709$	−388.894	−0.548511	−0.274256	−	0.961657i	$-0.588431\pi$
$709$	−388.894	−0.548511	−0.274256	+	0.961657i	$0.588431\pi$
$710$	59.0910i	0.0832267i
$711$	0	0
$712$	− 314.817i	− 0.442158i
$713$	713.922i	1.00129i
$714$	0	0
$715$	2.79394	0.00390761
$716$	127.833	0.178538
$717$	0	0
$718$	39.6182	0.0551785
$719$	− 460.435i	− 0.640383i	−0.947353	−	0.320192i	$-0.896253\pi$
$719$	− 460.435i	− 0.640383i	0.947353	−	0.320192i	$-0.103747\pi$
$720$	0	0
$721$	0	0
$722$	13.8650	0.0192036
$723$	0	0
$724$	367.496i	0.507591i
$725$	1084.53	1.49590
$726$	0	0
$727$	− 1005.79i	− 1.38347i	−0.722150	−	0.691737i	$-0.756846\pi$
$727$	− 1005.79i	− 1.38347i	0.722150	−	0.691737i	$-0.243154\pi$
$728$	0	0
$729$	0	0
$730$	−15.1859	−0.0208025
$731$	− 136.573i	− 0.186830i
$732$	0	0
$733$	− 779.623i	− 1.06361i	−0.846868	−	0.531803i	$-0.821515\pi$
$733$	− 779.623i	− 1.06361i	0.846868	−	0.531803i	$-0.178485\pi$
$734$	− 211.767i	− 0.288510i
$735$	0	0
$736$	110.392	0.149989
$737$	11.3137	0.0153510
$738$	0	0
$739$	72.3818	0.0979456	0.0489728	−	0.998800i	$-0.484405\pi$
$739$	72.3818	0.0979456	0.0489728	+	0.998800i	$0.484405\pi$
$740$	− 8.68553i	− 0.0117372i
$741$	0	0
$742$	0	0
$743$	−145.358	−0.195637	−0.0978185	−	0.995204i	$-0.531186\pi$
$743$	−145.358	−0.195637	−0.0978185	+	0.995204i	$0.531186\pi$
$744$	0	0
$745$	38.3321i	0.0514524i
$746$	326.960	0.438285
$747$	0	0
$748$	− 101.681i	− 0.135937i
$749$	0	0
$750$	0	0
$751$	4.02020	0.00535313	0.00267657	−	0.999996i	$-0.499148\pi$
$751$	4.02020	0.00535313	0.00267657	+	0.999996i	$0.499148\pi$
$752$	− 226.984i	− 0.301840i
$753$	0	0
$754$	− 191.931i	− 0.254550i
$755$	− 51.4824i	− 0.0681885i
$756$	0	0
$757$	705.678	0.932204	0.466102	−	0.884731i	$-0.345658\pi$
$757$	705.678	0.932204	0.466102	+	0.884731i	$0.345658\pi$
$758$	−827.578	−1.09179
$759$	0	0
$760$	16.8040	0.0221106
$761$	883.814i	1.16138i	0.814123	+	0.580692i	$0.197218\pi$
$761$	883.814i	1.16138i	−0.814123	+	0.580692i	$0.802782\pi$
$762$	0	0
$763$	0	0
$764$	−308.853	−0.404258
$765$	0	0
$766$	754.989i	0.985625i
$767$	−279.147	−0.363947
$768$	0	0
$769$	52.5686i	0.0683597i	0.999416	+	0.0341799i	$0.0108819\pi$
$769$	52.5686i	0.0683597i	−0.999416	+	0.0341799i	$0.989118\pi$
$770$	0	0
$771$	0	0
$772$	−46.3919	−0.0600932
$773$	− 613.176i	− 0.793242i	−0.917982	−	0.396621i	$-0.870183\pi$
$773$	− 613.176i	− 0.793242i	0.917982	−	0.396621i	$-0.129817\pi$
$774$	0	0
$775$	910.917i	1.17538i
$776$	418.907i	0.539828i
$777$	0	0
$778$	27.9899	0.0359767
$779$	1011.16	1.29802
$780$	0	0
$781$	372.784	0.477316
$782$	− 496.070i	− 0.634360i
$783$	0	0
$784$	0	0
$785$	20.9432	0.0266793
$786$	0	0
$787$	1058.60i	1.34511i	0.740048	+	0.672554i	$0.234803\pi$
$787$	1058.60i	1.34511i	−0.740048	+	0.672554i	$0.765197\pi$
$788$	−405.019	−0.513984
$789$	0	0
$790$	− 33.8937i	− 0.0429034i
$791$	0	0
$792$	0	0
$793$	−232.101	−0.292687
$794$	519.498i	0.654279i
$795$	0	0
$796$	460.165i	0.578097i
$797$	− 1128.99i	− 1.41655i	−0.705934	−	0.708277i	$-0.749473\pi$
$797$	− 1128.99i	− 1.41655i	0.705934	−	0.708277i	$-0.250527\pi$
$798$	0	0
$799$	−1020.00	−1.27660
$800$	140.853	0.176066
$801$	0	0
$802$	979.970	1.22191
$803$	95.8022i	0.119305i
$804$	0	0
$805$	0	0
$806$	161.206	0.200008
$807$	0	0
$808$	− 282.897i	− 0.350120i
$809$	604.118	0.746746	0.373373	−	0.927681i	$-0.378201\pi$
$809$	604.118	0.746746	0.373373	+	0.927681i	$0.378201\pi$
$810$	0	0
$811$	− 194.215i	− 0.239476i	−0.992806	−	0.119738i	$-0.961795\pi$
$811$	− 194.215i	− 0.239476i	0.992806	−	0.119738i	$-0.0382055\pi$
$812$	0	0
$813$	0	0
$814$	−54.7939	−0.0673144
$815$	− 66.5721i	− 0.0816836i
$816$	0	0
$817$	− 142.388i	− 0.174281i
$818$	− 411.394i	− 0.502926i
$819$	0	0
$820$	−34.2111	−0.0417209
$821$	−776.098	−0.945308	−0.472654	−	0.881248i	$-0.656704\pi$
$821$	−776.098	−0.945308	−0.472654	+	0.881248i	$0.656704\pi$
$822$	0	0
$823$	−148.804	−0.180807	−0.0904034	−	0.995905i	$-0.528816\pi$
$823$	−148.804	−0.180807	−0.0904034	+	0.995905i	$0.528816\pi$
$824$	17.4985i	0.0212361i
$825$	0	0
$826$	0	0
$827$	−1018.50	−1.23156	−0.615782	−	0.787916i	$-0.711160\pi$
$827$	−1018.50	−1.23156	−0.615782	+	0.787916i	$0.711160\pi$
$828$	0	0
$829$	− 310.313i	− 0.374322i	−0.982329	−	0.187161i	$-0.940071\pi$
$829$	− 310.313i	− 0.374322i	0.982329	−	0.187161i	$-0.0599286\pi$
$830$	−58.8427	−0.0708948
$831$	0	0
$832$	− 24.9269i	− 0.0299602i
$833$	0	0
$834$	0	0
$835$	48.4222	0.0579907
$836$	− 106.011i	− 0.126807i
$837$	0	0
$838$	− 882.940i	− 1.05363i
$839$	− 331.790i	− 0.395459i	−0.980257	−	0.197729i	$-0.936643\pi$
$839$	− 331.790i	− 0.395459i	0.980257	−	0.197729i	$-0.0633568\pi$
$840$	0	0
$841$	1056.16	1.25583
$842$	1066.29	1.26638
$843$	0	0
$844$	−344.804	−0.408536
$845$	50.4994i	0.0597626i
$846$	0	0
$847$	0	0
$848$	80.9016	0.0954028
$849$	0	0
$850$	− 632.952i	− 0.744650i
$851$	−267.322	−0.314127
$852$	0	0
$853$	358.597i	0.420395i	0.977659	+	0.210197i	$0.0674106\pi$
$853$	358.597i	0.420395i	−0.977659	+	0.210197i	$0.932589\pi$
$854$	0	0
$855$	0	0
$856$	119.196	0.139248
$857$	832.835i	0.971803i	0.874014	+	0.485901i	$0.161509\pi$
$857$	832.835i	0.971803i	−0.874014	+	0.485901i	$0.838491\pi$
$858$	0	0
$859$	− 1550.24i	− 1.80470i	−0.431002	−	0.902351i	$-0.641840\pi$
$859$	− 1550.24i	− 1.80470i	0.431002	−	0.902351i	$-0.358160\pi$
$860$	4.81750i	0.00560175i
$861$	0	0
$862$	1.62828	0.00188896
$863$	−917.249	−1.06286	−0.531430	−	0.847102i	$-0.678345\pi$
$863$	−917.249	−1.06286	−0.531430	+	0.847102i	$0.678345\pi$
$864$	0	0
$865$	−11.2763	−0.0130361
$866$	274.123i	0.316540i
$867$	0	0
$868$	0	0
$869$	−213.823	−0.246057
$870$	0	0
$871$	− 12.4634i	− 0.0143093i
$872$	−341.373	−0.391482
$873$	0	0
$874$	− 517.192i	− 0.591753i
$875$	0	0
$876$	0	0
$877$	1490.26	1.69927	0.849636	−	0.527370i	$-0.176822\pi$
$877$	1490.26	1.69927	0.849636	+	0.527370i	$0.176822\pi$
$878$	628.328i	0.715636i
$879$	0	0
$880$	3.58673i	0.00407583i
$881$	156.004i	0.177076i	0.996073	+	0.0885378i	$0.0282194\pi$
$881$	156.004i	0.177076i	−0.996073	+	0.0885378i	$0.971781\pi$
$882$	0	0
$883$	−61.1859	−0.0692932	−0.0346466	−	0.999400i	$-0.511031\pi$
$883$	−61.1859	−0.0692932	−0.0346466	+	0.999400i	$0.511031\pi$
$884$	−112.014	−0.126713
$885$	0	0
$886$	−571.196	−0.644691
$887$	880.738i	0.992940i	0.868054	+	0.496470i	$0.165371\pi$
$887$	880.738i	0.992940i	−0.868054	+	0.496470i	$0.834629\pi$
$888$	0	0
$889$	0	0
$890$	49.9025	0.0560702
$891$	0	0
$892$	110.648i	0.124045i
$893$	−1063.43	−1.19085
$894$	0	0
$895$	20.2632i	0.0226405i
$896$	0	0
$897$	0	0
$898$	342.975	0.381932
$899$	1593.46i	1.77248i
$900$	0	0
$901$	− 363.549i	− 0.403495i
$902$	215.826i	0.239275i
$903$	0	0
$904$	296.402	0.327878
$905$	−58.2527	−0.0643677
$906$	0	0
$907$	1288.76	1.42091	0.710454	−	0.703744i	$-0.248490\pi$
$907$	1288.76	1.42091	0.710454	+	0.703744i	$0.248490\pi$
$908$	739.156i	0.814049i
$909$	0	0
$910$	0	0
$911$	−788.563	−0.865601	−0.432801	−	0.901490i	$-0.642475\pi$
$911$	−788.563	−0.865601	−0.432801	+	0.901490i	$0.642475\pi$
$912$	0	0
$913$	371.218i	0.406591i
$914$	1053.25	1.15236
$915$	0	0
$916$	− 588.792i	− 0.642786i
$917$	0	0
$918$	0	0
$919$	−1497.59	−1.62958	−0.814792	−	0.579753i	$-0.803149\pi$
$919$	−1497.59	−1.62958	−0.814792	+	0.579753i	$0.803149\pi$
$920$	17.4985i	0.0190201i
$921$	0	0
$922$	1080.92i	1.17237i
$923$	− 410.667i	− 0.444927i
$924$	0	0
$925$	−341.085	−0.368741
$926$	−931.647	−1.00610
$927$	0	0
$928$	246.392	0.265509
$929$	− 1000.22i	− 1.07667i	−0.842732	−	0.538334i	$-0.819054\pi$
$929$	− 1000.22i	− 1.07667i	0.842732	−	0.538334i	$-0.180946\pi$
$930$	0	0
$931$	0	0
$932$	−349.574	−0.375079
$933$	0	0
$934$	− 92.8044i	− 0.0993623i
$935$	16.1177	0.0172382
$936$	0	0
$937$	776.352i	0.828550i	0.910152	+	0.414275i	$0.135965\pi$
$937$	776.352i	0.828550i	−0.910152	+	0.414275i	$0.864035\pi$
$938$	0	0
$939$	0	0
$940$	35.9798	0.0382764
$941$	− 3.45542i	− 0.00367207i	−0.999998	−	0.00183603i	$-0.999416\pi$
$941$	− 3.45542i	− 0.00367207i	0.999998	−	0.00183603i	$-0.000584428\pi$
$942$	0	0
$943$	1052.94i	1.11659i
$944$	− 358.356i	− 0.379615i
$945$	0	0
$946$	30.3919	0.0321268
$947$	−890.045	−0.939857	−0.469929	−	0.882704i	$-0.655720\pi$
$947$	−890.045	−0.939857	−0.469929	+	0.882704i	$0.655720\pi$
$948$	0	0
$949$	105.538	0.111210
$950$	− 659.903i	− 0.694635i
$951$	0	0
$952$	0	0
$953$	−1399.43	−1.46845	−0.734224	−	0.678907i	$-0.762454\pi$
$953$	−1399.43	−1.46845	−0.734224	+	0.678907i	$0.762454\pi$
$954$	0	0
$955$	− 48.9571i	− 0.0512640i
$956$	−646.558	−0.676316
$957$	0	0
$958$	351.401i	0.366806i
$959$	0	0
$960$	0	0
$961$	−377.372	−0.392686
$962$	60.3623i	0.0627466i
$963$	0	0
$964$	− 331.002i	− 0.343363i
$965$	− 7.35371i	− 0.00762042i
$966$	0	0
$967$	−404.382	−0.418182	−0.209091	−	0.977896i	$-0.567050\pi$
$967$	−404.382	−0.418182	−0.209091	+	0.977896i	$0.567050\pi$
$968$	−319.612	−0.330178
$969$	0	0
$970$	−66.4020	−0.0684557
$971$	629.341i	0.648137i	0.946034	+	0.324069i	$0.105051\pi$
$971$	629.341i	0.648137i	−0.946034	+	0.324069i	$0.894949\pi$
$972$	0	0
$973$	0	0
$974$	−1081.03	−1.10988
$975$	0	0
$976$	− 297.960i	− 0.305287i
$977$	−835.530	−0.855200	−0.427600	−	0.903968i	$-0.640641\pi$
$977$	−835.530	−0.855200	−0.427600	+	0.903968i	$0.640641\pi$
$978$	0	0
$979$	− 314.817i	− 0.321570i
$980$	0	0
$981$	0	0
$982$	1106.79	1.12708
$983$	585.085i	0.595204i	0.954690	+	0.297602i	$0.0961868\pi$
$983$	585.085i	0.595204i	−0.954690	+	0.297602i	$0.903813\pi$
$984$	0	0
$985$	− 64.2007i	− 0.0651784i
$986$	− 1107.21i	− 1.12294i
$987$	0	0
$988$	−116.784	−0.118202
$989$	148.272	0.149922
$990$	0	0
$991$	−1859.92	−1.87681	−0.938405	−	0.345537i	$-0.887697\pi$
$991$	−1859.92	−1.87681	−0.938405	+	0.345537i	$0.887697\pi$
$992$	206.949i	0.208618i
$993$	0	0
$994$	0	0
$995$	−72.9420	−0.0733085
$996$	0	0
$997$	− 137.481i	− 0.137895i	−0.997620	−	0.0689473i	$-0.978036\pi$
$997$	− 137.481i	− 0.137895i	0.997620	−	0.0689473i	$-0.0219640\pi$
$998$	−186.648	−0.187022
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.e.685.4	yes	4
3.2	odd	2		882.3.c.d.685.1	✓	4
7.2	even	3		882.3.n.g.325.2		8
7.3	odd	6		882.3.n.g.19.2		8
7.4	even	3		882.3.n.g.19.1		8
7.5	odd	6		882.3.n.g.325.1		8
7.6	odd	2	inner	882.3.c.e.685.3	yes	4
21.2	odd	6		882.3.n.h.325.3		8
21.5	even	6		882.3.n.h.325.4		8
21.11	odd	6		882.3.n.h.19.4		8
21.17	even	6		882.3.n.h.19.3		8
21.20	even	2		882.3.c.d.685.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.3.c.d.685.1	✓	4	3.2	odd	2
882.3.c.d.685.2	yes	4	21.20	even	2
882.3.c.e.685.3	yes	4	7.6	odd	2	inner
882.3.c.e.685.4	yes	4	1.1	even	1	trivial
882.3.n.g.19.1		8	7.4	even	3
882.3.n.g.19.2		8	7.3	odd	6
882.3.n.g.325.1		8	7.5	odd	6
882.3.n.g.325.2		8	7.2	even	3
882.3.n.h.19.3		8	21.17	even	6
882.3.n.h.19.4		8	21.11	odd	6
882.3.n.h.325.3		8	21.2	odd	6
882.3.n.h.325.4		8	21.5	even	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} +2.82843 q^{8} +0.448342i q^{10} +2.82843 q^{11} -3.11586i q^{13} +4.00000 q^{16} -17.9749i q^{17} -18.7402i q^{19} +0.634051i q^{20} +4.00000 q^{22} +19.5147 q^{23} +24.8995 q^{25} -4.40649i q^{26} +43.5563 q^{29} +36.5838i q^{31} +5.65685 q^{32} -25.4203i q^{34} -13.6985 q^{37} -26.5027i q^{38} +0.896683i q^{40} +53.9564i q^{41} +7.59798 q^{43} +5.65685 q^{44} +27.5980 q^{46} -56.7459i q^{47} +35.2132 q^{50} -6.23172i q^{52} +20.2254 q^{53} +0.896683i q^{55} +61.5980 q^{58} -89.5891i q^{59} -74.4900i q^{61} +51.7373i q^{62} +8.00000 q^{64} +0.987807 q^{65} +4.00000 q^{67} -35.9497i q^{68} +131.799 q^{71} +33.8712i q^{73} -19.3726 q^{74} -37.4804i q^{76} -75.5980 q^{79} +1.26810i q^{80} +76.3059i q^{82} +131.245i q^{83} +5.69848 q^{85} +10.7452 q^{86} +8.00000 q^{88} -111.305i q^{89} +39.0294 q^{92} -80.2509i q^{94} +5.94113 q^{95} +148.106i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} + 16 q^{22} + 112 q^{23} + 60 q^{25} + 112 q^{29} + 64 q^{37} - 128 q^{43} - 48 q^{46} + 56 q^{50} - 168 q^{53} + 88 q^{58} + 32 q^{64} + 168 q^{65} + 16 q^{67} + 448 q^{71} - 168 q^{74} - 144 q^{79} - 96 q^{85} + 224 q^{86} + 32 q^{88} + 224 q^{92} - 112 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 + 16 * q^22 + 112 * q^23 + 60 * q^25 + 112 * q^29 + 64 * q^37 - 128 * q^43 - 48 * q^46 + 56 * q^50 - 168 * q^53 + 88 * q^58 + 32 * q^64 + 168 * q^65 + 16 * q^67 + 448 * q^71 - 168 * q^74 - 144 * q^79 - 96 * q^85 + 224 * q^86 + 32 * q^88 + 224 * q^92 - 112 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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