LMFDB - Embedded newform 4410.2.b.c.881.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(881,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4410.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.b (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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.3
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.c.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} +4.93015i q^{11} +2.49661i q^{13} +1.00000 q^{16} +0.433546 q^{17} -7.93816i q^{19} +1.00000 q^{20} +4.93015 q^{22} +6.02734i q^{23} +1.00000 q^{25} +2.49661 q^{26} -4.71485i q^{29} +5.75858i q^{31} -1.00000i q^{32} -0.433546i q^{34} -7.12228 q^{37} -7.93816 q^{38} -1.00000i q^{40} +8.73925 q^{41} -8.32344 q^{43} -4.93015i q^{44} +6.02734 q^{46} +5.40583 q^{47} -1.00000i q^{50} -2.49661i q^{52} -7.27677i q^{53} -4.93015i q^{55} -4.71485 q^{58} -3.05505 q^{59} +3.10973i q^{61} +5.75858 q^{62} -1.00000 q^{64} -2.49661i q^{65} -14.2343 q^{67} -0.433546 q^{68} -2.47568i q^{71} -5.71644i q^{73} +7.12228i q^{74} +7.93816i q^{76} -4.32957 q^{79} -1.00000 q^{80} -8.73925i q^{82} +11.1489 q^{83} -0.433546 q^{85} +8.32344i q^{86} -4.93015 q^{88} -14.5626 q^{89} -6.02734i q^{92} -5.40583i q^{94} +7.93816i q^{95} +3.65685i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^5	$ 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	1.00000i	0.316228i
$11$	4.93015i	1.48650i	0.669016	+	0.743248i	$0.266716\pi$
$11$	4.93015i	1.48650i	−0.669016	+	0.743248i	$0.733284\pi$
$12$	0	0
$13$	2.49661i	0.692434i	0.938154	+	0.346217i	$0.112534\pi$
$13$	2.49661i	0.692434i	−0.938154	+	0.346217i	$0.887466\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.433546	0.105150	0.0525751	−	0.998617i	$-0.483257\pi$
$17$	0.433546	0.105150	0.0525751	+	0.998617i	$0.483257\pi$
$18$	0	0
$19$	− 7.93816i	− 1.82114i	−0.413357	−	0.910569i	$-0.635644\pi$
$19$	− 7.93816i	− 1.82114i	0.413357	−	0.910569i	$-0.364356\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.93015	1.05111
$23$	6.02734i	1.25679i	0.777896	+	0.628394i	$0.216287\pi$
$23$	6.02734i	1.25679i	−0.777896	+	0.628394i	$0.783713\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.49661	0.489625
$27$	0	0
$28$	0	0
$29$	− 4.71485i	− 0.875526i	−0.899090	−	0.437763i	$-0.855771\pi$
$29$	− 4.71485i	− 0.875526i	0.899090	−	0.437763i	$-0.144229\pi$
$30$	0	0
$31$	5.75858i	1.03427i	0.855903	+	0.517136i	$0.173002\pi$
$31$	5.75858i	1.03427i	−0.855903	+	0.517136i	$0.826998\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 0.433546i	− 0.0743524i
$35$	0	0
$36$	0	0
$37$	−7.12228	−1.17089	−0.585447	−	0.810710i	$-0.699081\pi$
$37$	−7.12228	−1.17089	−0.585447	+	0.810710i	$0.699081\pi$
$38$	−7.93816	−1.28774
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	8.73925	1.36484	0.682420	−	0.730960i	$-0.260927\pi$
$41$	8.73925	1.36484	0.682420	+	0.730960i	$0.260927\pi$
$42$	0	0
$43$	−8.32344	−1.26931	−0.634656	−	0.772794i	$-0.718858\pi$
$43$	−8.32344	−1.26931	−0.634656	+	0.772794i	$0.718858\pi$
$44$	− 4.93015i	− 0.743248i
$45$	0	0
$46$	6.02734	0.888683
$47$	5.40583	0.788522	0.394261	−	0.918999i	$-0.371001\pi$
$47$	5.40583	0.788522	0.394261	+	0.918999i	$0.371001\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	− 2.49661i	− 0.346217i
$53$	− 7.27677i	− 0.999541i	−0.866158	−	0.499771i	$-0.833418\pi$
$53$	− 7.27677i	− 0.999541i	0.866158	−	0.499771i	$-0.166582\pi$
$54$	0	0
$55$	− 4.93015i	− 0.664781i
$56$	0	0
$57$	0	0
$58$	−4.71485	−0.619090
$59$	−3.05505	−0.397734	−0.198867	−	0.980026i	$-0.563726\pi$
$59$	−3.05505	−0.397734	−0.198867	+	0.980026i	$0.563726\pi$
$60$	0	0
$61$	3.10973i	0.398160i	0.979983	+	0.199080i	$0.0637954\pi$
$61$	3.10973i	0.398160i	−0.979983	+	0.199080i	$0.936205\pi$
$62$	5.75858	0.731340
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 2.49661i	− 0.309666i
$66$	0	0
$67$	−14.2343	−1.73899	−0.869496	−	0.493940i	$-0.835556\pi$
$67$	−14.2343	−1.73899	−0.869496	+	0.493940i	$0.835556\pi$
$68$	−0.433546	−0.0525751
$69$	0	0
$70$	0	0
$71$	− 2.47568i	− 0.293809i	−0.989151	−	0.146905i	$-0.953069\pi$
$71$	− 2.47568i	− 0.293809i	0.989151	−	0.146905i	$-0.0469310\pi$
$72$	0	0
$73$	− 5.71644i	− 0.669059i	−0.942385	−	0.334530i	$-0.891423\pi$
$73$	− 5.71644i	− 0.669059i	0.942385	−	0.334530i	$-0.108577\pi$
$74$	7.12228i	0.827948i
$75$	0	0
$76$	7.93816i	0.910569i
$77$	0	0
$78$	0	0
$79$	−4.32957	−0.487115	−0.243557	−	0.969887i	$-0.578314\pi$
$79$	−4.32957	−0.487115	−0.243557	+	0.969887i	$0.578314\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	− 8.73925i	− 0.965088i
$83$	11.1489	1.22375	0.611877	−	0.790953i	$-0.290415\pi$
$83$	11.1489	1.22375	0.611877	+	0.790953i	$0.290415\pi$
$84$	0	0
$85$	−0.433546	−0.0470246
$86$	8.32344i	0.897540i
$87$	0	0
$88$	−4.93015	−0.525556
$89$	−14.5626	−1.54363	−0.771817	−	0.635845i	$-0.780652\pi$
$89$	−14.5626	−1.54363	−0.771817	+	0.635845i	$0.780652\pi$
$90$	0	0
$91$	0	0
$92$	− 6.02734i	− 0.628394i
$93$	0	0
$94$	− 5.40583i	− 0.557569i
$95$	7.93816i	0.814438i
$96$	0	0
$97$	3.65685i	0.371297i	0.982616	+	0.185649i	$0.0594386\pi$
$97$	3.65685i	0.371297i	−0.982616	+	0.185649i	$0.940561\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−13.7734	−1.37050	−0.685251	−	0.728307i	$-0.740307\pi$
$101$	−13.7734	−1.37050	−0.685251	+	0.728307i	$0.740307\pi$
$102$	0	0
$103$	0.253965i	0.0250239i	0.999922	+	0.0125120i	$0.00398278\pi$
$103$	0.253965i	0.0250239i	−0.999922	+	0.0125120i	$0.996017\pi$
$104$	−2.49661	−0.244812
$105$	0	0
$106$	−7.27677	−0.706782
$107$	7.31543i	0.707210i	0.935395	+	0.353605i	$0.115044\pi$
$107$	7.31543i	0.707210i	−0.935395	+	0.353605i	$0.884956\pi$
$108$	0	0
$109$	−12.0588	−1.15503	−0.577514	−	0.816381i	$-0.695977\pi$
$109$	−12.0588	−1.15503	−0.577514	+	0.816381i	$0.695977\pi$
$110$	−4.93015	−0.470071
$111$	0	0
$112$	0	0
$113$	− 15.4753i	− 1.45579i	−0.685686	−	0.727897i	$-0.740498\pi$
$113$	− 15.4753i	− 1.45579i	0.685686	−	0.727897i	$-0.259502\pi$
$114$	0	0
$115$	− 6.02734i	− 0.562052i
$116$	4.71485i	0.437763i
$117$	0	0
$118$	3.05505i	0.281240i
$119$	0	0
$120$	0	0
$121$	−13.3064	−1.20967
$122$	3.10973	0.281542
$123$	0	0
$124$	− 5.75858i	− 0.517136i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.98324	−0.708398	−0.354199	−	0.935170i	$-0.615246\pi$
$127$	−7.98324	−0.708398	−0.354199	+	0.935170i	$0.615246\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−2.49661	−0.218967
$131$	−7.68592	−0.671522	−0.335761	−	0.941947i	$-0.608993\pi$
$131$	−7.68592	−0.671522	−0.335761	+	0.941947i	$0.608993\pi$
$132$	0	0
$133$	0	0
$134$	14.2343i	1.22965i
$135$	0	0
$136$	0.433546i	0.0371762i
$137$	18.7547i	1.60232i	0.598447	+	0.801162i	$0.295785\pi$
$137$	18.7547i	1.60232i	−0.598447	+	0.801162i	$0.704215\pi$
$138$	0	0
$139$	− 12.3746i	− 1.04960i	−0.851224	−	0.524802i	$-0.824139\pi$
$139$	− 12.3746i	− 1.04960i	0.851224	−	0.524802i	$-0.175861\pi$
$140$	0	0
$141$	0	0
$142$	−2.47568	−0.207755
$143$	−12.3086	−1.02930
$144$	0	0
$145$	4.71485i	0.391547i
$146$	−5.71644	−0.473096
$147$	0	0
$148$	7.12228	0.585447
$149$	− 6.40621i	− 0.524817i	−0.964957	−	0.262408i	$-0.915483\pi$
$149$	− 6.40621i	− 0.524817i	0.964957	−	0.262408i	$-0.0845168\pi$
$150$	0	0
$151$	−7.38144	−0.600693	−0.300346	−	0.953830i	$-0.597102\pi$
$151$	−7.38144	−0.600693	−0.300346	+	0.953830i	$0.597102\pi$
$152$	7.93816	0.643870
$153$	0	0
$154$	0	0
$155$	− 5.75858i	− 0.462540i
$156$	0	0
$157$	12.9764i	1.03563i	0.855492	+	0.517817i	$0.173255\pi$
$157$	12.9764i	1.03563i	−0.855492	+	0.517817i	$0.826745\pi$
$158$	4.32957i	0.344442i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	−10.4377	−0.817544	−0.408772	−	0.912636i	$-0.634043\pi$
$163$	−10.4377	−0.817544	−0.408772	+	0.912636i	$0.634043\pi$
$164$	−8.73925	−0.682420
$165$	0	0
$166$	− 11.1489i	− 0.865324i
$167$	6.79589	0.525882	0.262941	−	0.964812i	$-0.415308\pi$
$167$	6.79589	0.525882	0.262941	+	0.964812i	$0.415308\pi$
$168$	0	0
$169$	6.76696	0.520535
$170$	0.433546i	0.0332514i
$171$	0	0
$172$	8.32344	0.634656
$173$	16.2059	1.23211	0.616055	−	0.787703i	$-0.288730\pi$
$173$	16.2059	1.23211	0.616055	+	0.787703i	$0.288730\pi$
$174$	0	0
$175$	0	0
$176$	4.93015i	0.371624i
$177$	0	0
$178$	14.5626i	1.09151i
$179$	14.1219i	1.05552i	0.849393	+	0.527760i	$0.176968\pi$
$179$	14.1219i	1.05552i	−0.849393	+	0.527760i	$0.823032\pi$
$180$	0	0
$181$	16.2645i	1.20893i	0.796630	+	0.604467i	$0.206614\pi$
$181$	16.2645i	1.20893i	−0.796630	+	0.604467i	$0.793386\pi$
$182$	0	0
$183$	0	0
$184$	−6.02734	−0.444341
$185$	7.12228	0.523640
$186$	0	0
$187$	2.13744i	0.156305i
$188$	−5.40583	−0.394261
$189$	0	0
$190$	7.93816	0.575895
$191$	− 9.21077i	− 0.666468i	−0.942844	−	0.333234i	$-0.891860\pi$
$191$	− 9.21077i	− 0.666468i	0.942844	−	0.333234i	$-0.108140\pi$
$192$	0	0
$193$	−0.974011	−0.0701108	−0.0350554	−	0.999385i	$-0.511161\pi$
$193$	−0.974011	−0.0701108	−0.0350554	+	0.999385i	$0.511161\pi$
$194$	3.65685	0.262547
$195$	0	0
$196$	0	0
$197$	− 5.34502i	− 0.380817i	−0.981705	−	0.190409i	$-0.939019\pi$
$197$	− 5.34502i	− 0.380817i	0.981705	−	0.190409i	$-0.0609813\pi$
$198$	0	0
$199$	− 22.9977i	− 1.63026i	−0.579277	−	0.815130i	$-0.696665\pi$
$199$	− 22.9977i	− 1.63026i	0.579277	−	0.815130i	$-0.303335\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	13.7734i	0.969091i
$203$	0	0
$204$	0	0
$205$	−8.73925	−0.610375
$206$	0.253965	0.0176946
$207$	0	0
$208$	2.49661i	0.173108i
$209$	39.1363	2.70712
$210$	0	0
$211$	−8.32222	−0.572925	−0.286463	−	0.958091i	$-0.592479\pi$
$211$	−8.32222	−0.572925	−0.286463	+	0.958091i	$0.592479\pi$
$212$	7.27677i	0.499771i
$213$	0	0
$214$	7.31543	0.500073
$215$	8.32344	0.567654
$216$	0	0
$217$	0	0
$218$	12.0588i	0.816728i
$219$	0	0
$220$	4.93015i	0.332391i
$221$	1.08239i	0.0728096i
$222$	0	0
$223$	17.9191i	1.19995i	0.800017	+	0.599977i	$0.204824\pi$
$223$	17.9191i	1.19995i	−0.800017	+	0.599977i	$0.795176\pi$
$224$	0	0
$225$	0	0
$226$	−15.4753	−1.02940
$227$	−8.37592	−0.555930	−0.277965	−	0.960591i	$-0.589660\pi$
$227$	−8.37592	−0.555930	−0.277965	+	0.960591i	$0.589660\pi$
$228$	0	0
$229$	− 2.01867i	− 0.133398i	−0.997773	−	0.0666988i	$-0.978753\pi$
$229$	− 2.01867i	− 0.133398i	0.997773	−	0.0666988i	$-0.0212467\pi$
$230$	−6.02734	−0.397431
$231$	0	0
$232$	4.71485	0.309545
$233$	22.3037i	1.46117i	0.682824	+	0.730583i	$0.260752\pi$
$233$	22.3037i	1.46117i	−0.682824	+	0.730583i	$0.739248\pi$
$234$	0	0
$235$	−5.40583	−0.352638
$236$	3.05505	0.198867
$237$	0	0
$238$	0	0
$239$	18.4653i	1.19442i	0.802085	+	0.597210i	$0.203724\pi$
$239$	18.4653i	1.19442i	−0.802085	+	0.597210i	$0.796276\pi$
$240$	0	0
$241$	− 24.2229i	− 1.56034i	−0.625570	−	0.780168i	$-0.715133\pi$
$241$	− 24.2229i	− 1.56034i	0.625570	−	0.780168i	$-0.284867\pi$
$242$	13.3064i	0.855367i
$243$	0	0
$244$	− 3.10973i	− 0.199080i
$245$	0	0
$246$	0	0
$247$	19.8185	1.26102
$248$	−5.75858	−0.365670
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	−10.8503	−0.684867	−0.342433	−	0.939542i	$-0.611251\pi$
$251$	−10.8503	−0.684867	−0.342433	+	0.939542i	$0.611251\pi$
$252$	0	0
$253$	−29.7157	−1.86821
$254$	7.98324i	0.500913i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.93052	−0.369936	−0.184968	−	0.982745i	$-0.559218\pi$
$257$	−5.93052	−0.369936	−0.184968	+	0.982745i	$0.559218\pi$
$258$	0	0
$259$	0	0
$260$	2.49661i	0.154833i
$261$	0	0
$262$	7.68592i	0.474838i
$263$	25.5085i	1.57292i	0.617639	+	0.786462i	$0.288089\pi$
$263$	25.5085i	1.57292i	−0.617639	+	0.786462i	$0.711911\pi$
$264$	0	0
$265$	7.27677i	0.447008i
$266$	0	0
$267$	0	0
$268$	14.2343	0.869496
$269$	−13.8571	−0.844883	−0.422442	−	0.906390i	$-0.638827\pi$
$269$	−13.8571	−0.844883	−0.422442	+	0.906390i	$0.638827\pi$
$270$	0	0
$271$	21.1841i	1.28684i	0.765512	+	0.643422i	$0.222486\pi$
$271$	21.1841i	1.28684i	−0.765512	+	0.643422i	$0.777514\pi$
$272$	0.433546	0.0262876
$273$	0	0
$274$	18.7547	1.13301
$275$	4.93015i	0.297299i
$276$	0	0
$277$	−24.1492	−1.45099	−0.725494	−	0.688229i	$-0.758388\pi$
$277$	−24.1492	−1.45099	−0.725494	+	0.688229i	$0.758388\pi$
$278$	−12.3746	−0.742182
$279$	0	0
$280$	0	0
$281$	− 6.60180i	− 0.393830i	−0.980421	−	0.196915i	$-0.936908\pi$
$281$	− 6.60180i	− 0.393830i	0.980421	−	0.196915i	$-0.0630924\pi$
$282$	0	0
$283$	− 7.27035i	− 0.432178i	−0.976374	−	0.216089i	$-0.930670\pi$
$283$	− 7.27035i	− 0.432178i	0.976374	−	0.216089i	$-0.0693301\pi$
$284$	2.47568i	0.146905i
$285$	0	0
$286$	12.3086i	0.727825i
$287$	0	0
$288$	0	0
$289$	−16.8120	−0.988943
$290$	4.71485	0.276866
$291$	0	0
$292$	5.71644i	0.334530i
$293$	−17.9150	−1.04660	−0.523302	−	0.852147i	$-0.675300\pi$
$293$	−17.9150	−1.04660	−0.523302	+	0.852147i	$0.675300\pi$
$294$	0	0
$295$	3.05505	0.177872
$296$	− 7.12228i	− 0.413974i
$297$	0	0
$298$	−6.40621	−0.371102
$299$	−15.0479	−0.870242
$300$	0	0
$301$	0	0
$302$	7.38144i	0.424754i
$303$	0	0
$304$	− 7.93816i	− 0.455285i
$305$	− 3.10973i	− 0.178063i
$306$	0	0
$307$	− 31.9470i	− 1.82331i	−0.410954	−	0.911656i	$-0.634804\pi$
$307$	− 31.9470i	− 1.82331i	0.410954	−	0.911656i	$-0.365196\pi$
$308$	0	0
$309$	0	0
$310$	−5.75858	−0.327065
$311$	−4.83750	−0.274309	−0.137155	−	0.990550i	$-0.543796\pi$
$311$	−4.83750	−0.274309	−0.137155	+	0.990550i	$0.543796\pi$
$312$	0	0
$313$	− 10.3929i	− 0.587442i	−0.955891	−	0.293721i	$-0.905106\pi$
$313$	− 10.3929i	− 0.587442i	0.955891	−	0.293721i	$-0.0948937\pi$
$314$	12.9764	0.732303
$315$	0	0
$316$	4.32957	0.243557
$317$	22.2581i	1.25014i	0.780568	+	0.625070i	$0.214930\pi$
$317$	22.2581i	1.25014i	−0.780568	+	0.625070i	$0.785070\pi$
$318$	0	0
$319$	23.2449	1.30147
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	− 3.44155i	− 0.191493i
$324$	0	0
$325$	2.49661i	0.138487i
$326$	10.4377i	0.578091i
$327$	0	0
$328$	8.73925i	0.482544i
$329$	0	0
$330$	0	0
$331$	21.2760	1.16944	0.584718	−	0.811237i	$-0.301205\pi$
$331$	21.2760	1.16944	0.584718	+	0.811237i	$0.301205\pi$
$332$	−11.1489	−0.611877
$333$	0	0
$334$	− 6.79589i	− 0.371855i
$335$	14.2343	0.777701
$336$	0	0
$337$	9.65367	0.525869	0.262934	−	0.964814i	$-0.415310\pi$
$337$	9.65367	0.525869	0.262934	+	0.964814i	$0.415310\pi$
$338$	− 6.76696i	− 0.368074i
$339$	0	0
$340$	0.433546	0.0235123
$341$	−28.3907	−1.53744
$342$	0	0
$343$	0	0
$344$	− 8.32344i	− 0.448770i
$345$	0	0
$346$	− 16.2059i	− 0.871234i
$347$	− 1.73418i	− 0.0930958i	−0.998916	−	0.0465479i	$-0.985178\pi$
$347$	− 1.73418i	− 0.0930958i	0.998916	−	0.0465479i	$-0.0148220\pi$
$348$	0	0
$349$	− 28.7035i	− 1.53646i	−0.640173	−	0.768231i	$-0.721137\pi$
$349$	− 28.7035i	− 1.53646i	0.640173	−	0.768231i	$-0.278863\pi$
$350$	0	0
$351$	0	0
$352$	4.93015	0.262778
$353$	−8.84044	−0.470529	−0.235265	−	0.971931i	$-0.575596\pi$
$353$	−8.84044	−0.470529	−0.235265	+	0.971931i	$0.575596\pi$
$354$	0	0
$355$	2.47568i	0.131396i
$356$	14.5626	0.771817
$357$	0	0
$358$	14.1219	0.746366
$359$	− 1.22753i	− 0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$359$	− 1.22753i	− 0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$360$	0	0
$361$	−44.0144	−2.31655
$362$	16.2645	0.854845
$363$	0	0
$364$	0	0
$365$	5.71644i	0.299212i
$366$	0	0
$367$	8.24587i	0.430431i	0.976567	+	0.215215i	$0.0690454\pi$
$367$	8.24587i	0.430431i	−0.976567	+	0.215215i	$0.930955\pi$
$368$	6.02734i	0.314197i
$369$	0	0
$370$	− 7.12228i	− 0.370269i
$371$	0	0
$372$	0	0
$373$	35.5657	1.84152	0.920761	−	0.390127i	$-0.127569\pi$
$373$	35.5657	1.84152	0.920761	+	0.390127i	$0.127569\pi$
$374$	2.13744	0.110525
$375$	0	0
$376$	5.40583i	0.278785i
$377$	11.7711	0.606244
$378$	0	0
$379$	−22.5323	−1.15740	−0.578702	−	0.815539i	$-0.696441\pi$
$379$	−22.5323	−1.15740	−0.578702	+	0.815539i	$0.696441\pi$
$380$	− 7.93816i	− 0.407219i
$381$	0	0
$382$	−9.21077	−0.471264
$383$	−37.0578	−1.89356	−0.946782	−	0.321876i	$-0.895687\pi$
$383$	−37.0578	−1.89356	−0.946782	+	0.321876i	$0.895687\pi$
$384$	0	0
$385$	0	0
$386$	0.974011i	0.0495758i
$387$	0	0
$388$	− 3.65685i	− 0.185649i
$389$	− 36.5542i	− 1.85337i	−0.375835	−	0.926686i	$-0.622644\pi$
$389$	− 36.5542i	− 1.85337i	0.375835	−	0.926686i	$-0.377356\pi$
$390$	0	0
$391$	2.61313i	0.132151i
$392$	0	0
$393$	0	0
$394$	−5.34502	−0.269278
$395$	4.32957	0.217844
$396$	0	0
$397$	10.9545i	0.549793i	0.961474	+	0.274896i	$0.0886436\pi$
$397$	10.9545i	0.549793i	−0.961474	+	0.274896i	$0.911356\pi$
$398$	−22.9977	−1.15277
$399$	0	0
$400$	1.00000	0.0500000
$401$	11.1393i	0.556271i	0.960542	+	0.278136i	$0.0897165\pi$
$401$	11.1393i	0.556271i	−0.960542	+	0.278136i	$0.910284\pi$
$402$	0	0
$403$	−14.3769	−0.716164
$404$	13.7734	0.685251
$405$	0	0
$406$	0	0
$407$	− 35.1139i	− 1.74053i
$408$	0	0
$409$	− 35.1346i	− 1.73729i	−0.495432	−	0.868646i	$-0.664990\pi$
$409$	− 35.1346i	− 1.73729i	0.495432	−	0.868646i	$-0.335010\pi$
$410$	8.73925i	0.431601i
$411$	0	0
$412$	− 0.253965i	− 0.0125120i
$413$	0	0
$414$	0	0
$415$	−11.1489	−0.547279
$416$	2.49661	0.122406
$417$	0	0
$418$	− 39.1363i	− 1.91422i
$419$	13.3073	0.650106	0.325053	−	0.945696i	$-0.394618\pi$
$419$	13.3073	0.650106	0.325053	+	0.945696i	$0.394618\pi$
$420$	0	0
$421$	−26.1394	−1.27395	−0.636977	−	0.770883i	$-0.719815\pi$
$421$	−26.1394	−1.27395	−0.636977	+	0.770883i	$0.719815\pi$
$422$	8.32222i	0.405119i
$423$	0	0
$424$	7.27677	0.353391
$425$	0.433546	0.0210300
$426$	0	0
$427$	0	0
$428$	− 7.31543i	− 0.353605i
$429$	0	0
$430$	− 8.32344i	− 0.401392i
$431$	− 2.40743i	− 0.115962i	−0.998318	−	0.0579808i	$-0.981534\pi$
$431$	− 2.40743i	− 0.115962i	0.998318	−	0.0579808i	$-0.0184662\pi$
$432$	0	0
$433$	15.8558i	0.761981i	0.924579	+	0.380990i	$0.124417\pi$
$433$	15.8558i	0.761981i	−0.924579	+	0.380990i	$0.875583\pi$
$434$	0	0
$435$	0	0
$436$	12.0588	0.577514
$437$	47.8460	2.28878
$438$	0	0
$439$	35.9230i	1.71451i	0.514893	+	0.857255i	$0.327832\pi$
$439$	35.9230i	1.71451i	−0.514893	+	0.857255i	$0.672168\pi$
$440$	4.93015	0.235036
$441$	0	0
$442$	1.08239	0.0514841
$443$	17.9342i	0.852079i	0.904705	+	0.426039i	$0.140091\pi$
$443$	17.9342i	0.852079i	−0.904705	+	0.426039i	$0.859909\pi$
$444$	0	0
$445$	14.5626	0.690334
$446$	17.9191	0.848496
$447$	0	0
$448$	0	0
$449$	37.3484i	1.76258i	0.472576	+	0.881290i	$0.343324\pi$
$449$	37.3484i	1.76258i	−0.472576	+	0.881290i	$0.656676\pi$
$450$	0	0
$451$	43.0858i	2.02883i
$452$	15.4753i	0.727897i
$453$	0	0
$454$	8.37592i	0.393102i
$455$	0	0
$456$	0	0
$457$	−23.1322	−1.08208	−0.541038	−	0.840998i	$-0.681969\pi$
$457$	−23.1322	−1.08208	−0.541038	+	0.840998i	$0.681969\pi$
$458$	−2.01867	−0.0943264
$459$	0	0
$460$	6.02734i	0.281026i
$461$	−22.0630	−1.02758	−0.513788	−	0.857917i	$-0.671758\pi$
$461$	−22.0630	−1.02758	−0.513788	+	0.857917i	$0.671758\pi$
$462$	0	0
$463$	−0.146069	−0.00678839	−0.00339419	−	0.999994i	$-0.501080\pi$
$463$	−0.146069	−0.00678839	−0.00339419	+	0.999994i	$0.501080\pi$
$464$	− 4.71485i	− 0.218881i
$465$	0	0
$466$	22.3037	1.03320
$467$	2.19119	0.101396	0.0506981	−	0.998714i	$-0.483855\pi$
$467$	2.19119	0.101396	0.0506981	+	0.998714i	$0.483855\pi$
$468$	0	0
$469$	0	0
$470$	5.40583i	0.249352i
$471$	0	0
$472$	− 3.05505i	− 0.140620i
$473$	− 41.0358i	− 1.88683i
$474$	0	0
$475$	− 7.93816i	− 0.364228i
$476$	0	0
$477$	0	0
$478$	18.4653	0.844583
$479$	9.48194	0.433241	0.216620	−	0.976256i	$-0.430497\pi$
$479$	9.48194	0.433241	0.216620	+	0.976256i	$0.430497\pi$
$480$	0	0
$481$	− 17.7815i	− 0.810767i
$482$	−24.2229	−1.10332
$483$	0	0
$484$	13.3064	0.604836
$485$	− 3.65685i	− 0.166049i
$486$	0	0
$487$	−12.2240	−0.553921	−0.276960	−	0.960881i	$-0.589327\pi$
$487$	−12.2240	−0.553921	−0.276960	+	0.960881i	$0.589327\pi$
$488$	−3.10973	−0.140771
$489$	0	0
$490$	0	0
$491$	− 28.6434i	− 1.29266i	−0.763058	−	0.646329i	$-0.776303\pi$
$491$	− 28.6434i	− 1.29266i	0.763058	−	0.646329i	$-0.223697\pi$
$492$	0	0
$493$	− 2.04410i	− 0.0920617i
$494$	− 19.8185i	− 0.891674i
$495$	0	0
$496$	5.75858i	0.258568i
$497$	0	0
$498$	0	0
$499$	−19.4817	−0.872120	−0.436060	−	0.899918i	$-0.643626\pi$
$499$	−19.4817	−0.872120	−0.436060	+	0.899918i	$0.643626\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	10.8503i	0.484274i
$503$	12.0239	0.536118	0.268059	−	0.963403i	$-0.413618\pi$
$503$	12.0239	0.536118	0.268059	+	0.963403i	$0.413618\pi$
$504$	0	0
$505$	13.7734	0.612907
$506$	29.7157i	1.32102i
$507$	0	0
$508$	7.98324	0.354199
$509$	34.0475	1.50913	0.754565	−	0.656226i	$-0.227848\pi$
$509$	34.0475	1.50913	0.754565	+	0.656226i	$0.227848\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	5.93052i	0.261584i
$515$	− 0.253965i	− 0.0111910i
$516$	0	0
$517$	26.6516i	1.17213i
$518$	0	0
$519$	0	0
$520$	2.49661	0.109483
$521$	6.13063	0.268588	0.134294	−	0.990942i	$-0.457123\pi$
$521$	6.13063	0.268588	0.134294	+	0.990942i	$0.457123\pi$
$522$	0	0
$523$	12.2636i	0.536248i	0.963384	+	0.268124i	$0.0864037\pi$
$523$	12.2636i	0.536248i	−0.963384	+	0.268124i	$0.913596\pi$
$524$	7.68592	0.335761
$525$	0	0
$526$	25.5085	1.11222
$527$	2.49661i	0.108754i
$528$	0	0
$529$	−13.3288	−0.579514
$530$	7.27677	0.316083
$531$	0	0
$532$	0	0
$533$	21.8185i	0.945062i
$534$	0	0
$535$	− 7.31543i	− 0.316274i
$536$	− 14.2343i	− 0.614826i
$537$	0	0
$538$	13.8571i	0.597423i
$539$	0	0
$540$	0	0
$541$	28.9323	1.24390	0.621948	−	0.783058i	$-0.286341\pi$
$541$	28.9323	1.24390	0.621948	+	0.783058i	$0.286341\pi$
$542$	21.1841	0.909936
$543$	0	0
$544$	− 0.433546i	− 0.0185881i
$545$	12.0588	0.516544
$546$	0	0
$547$	41.8985	1.79145	0.895724	−	0.444610i	$-0.146658\pi$
$547$	41.8985	1.79145	0.895724	+	0.444610i	$0.146658\pi$
$548$	− 18.7547i	− 0.801162i
$549$	0	0
$550$	4.93015	0.210222
$551$	−37.4272	−1.59445
$552$	0	0
$553$	0	0
$554$	24.1492i	1.02600i
$555$	0	0
$556$	12.3746i	0.524802i
$557$	− 2.91726i	− 0.123608i	−0.998088	−	0.0618042i	$-0.980315\pi$
$557$	− 2.91726i	− 0.123608i	0.998088	−	0.0618042i	$-0.0196854\pi$
$558$	0	0
$559$	− 20.7803i	− 0.878915i
$560$	0	0
$561$	0	0
$562$	−6.60180	−0.278480
$563$	−35.8132	−1.50935	−0.754673	−	0.656101i	$-0.772205\pi$
$563$	−35.8132	−1.50935	−0.754673	+	0.656101i	$0.772205\pi$
$564$	0	0
$565$	15.4753i	0.651051i
$566$	−7.27035	−0.305596
$567$	0	0
$568$	2.47568	0.103877
$569$	3.29053i	0.137946i	0.997619	+	0.0689731i	$0.0219723\pi$
$569$	3.29053i	0.137946i	−0.997619	+	0.0689731i	$0.978028\pi$
$570$	0	0
$571$	38.9712	1.63089	0.815447	−	0.578832i	$-0.196491\pi$
$571$	38.9712	1.63089	0.815447	+	0.578832i	$0.196491\pi$
$572$	12.3086	0.514650
$573$	0	0
$574$	0	0
$575$	6.02734i	0.251357i
$576$	0	0
$577$	− 33.2701i	− 1.38505i	−0.721392	−	0.692527i	$-0.756497\pi$
$577$	− 33.2701i	− 1.38505i	0.721392	−	0.692527i	$-0.243503\pi$
$578$	16.8120i	0.699289i
$579$	0	0
$580$	− 4.71485i	− 0.195773i
$581$	0	0
$582$	0	0
$583$	35.8756	1.48581
$584$	5.71644	0.236548
$585$	0	0
$586$	17.9150i	0.740061i
$587$	1.75413	0.0724007	0.0362004	−	0.999345i	$-0.488475\pi$
$587$	1.75413	0.0724007	0.0362004	+	0.999345i	$0.488475\pi$
$588$	0	0
$589$	45.7125	1.88355
$590$	− 3.05505i	− 0.125775i
$591$	0	0
$592$	−7.12228	−0.292724
$593$	15.6922	0.644403	0.322201	−	0.946671i	$-0.395577\pi$
$593$	15.6922	0.644403	0.322201	+	0.946671i	$0.395577\pi$
$594$	0	0
$595$	0	0
$596$	6.40621i	0.262408i
$597$	0	0
$598$	15.0479i	0.615354i
$599$	4.46979i	0.182631i	0.995822	+	0.0913154i	$0.0291072\pi$
$599$	4.46979i	0.182631i	−0.995822	+	0.0913154i	$0.970893\pi$
$600$	0	0
$601$	− 21.7714i	− 0.888074i	−0.896008	−	0.444037i	$-0.853546\pi$
$601$	− 21.7714i	− 0.888074i	0.896008	−	0.444037i	$-0.146454\pi$
$602$	0	0
$603$	0	0
$604$	7.38144	0.300346
$605$	13.3064	0.540982
$606$	0	0
$607$	− 23.3475i	− 0.947644i	−0.880621	−	0.473822i	$-0.842874\pi$
$607$	− 23.3475i	− 0.947644i	0.880621	−	0.473822i	$-0.157126\pi$
$608$	−7.93816	−0.321935
$609$	0	0
$610$	−3.10973	−0.125909
$611$	13.4962i	0.545999i
$612$	0	0
$613$	−6.20812	−0.250743	−0.125372	−	0.992110i	$-0.540012\pi$
$613$	−6.20812	−0.250743	−0.125372	+	0.992110i	$0.540012\pi$
$614$	−31.9470	−1.28928
$615$	0	0
$616$	0	0
$617$	− 21.9911i	− 0.885329i	−0.896687	−	0.442664i	$-0.854033\pi$
$617$	− 21.9911i	− 0.885329i	0.896687	−	0.442664i	$-0.145967\pi$
$618$	0	0
$619$	− 31.7753i	− 1.27716i	−0.769556	−	0.638579i	$-0.779523\pi$
$619$	− 31.7753i	− 1.27716i	0.769556	−	0.638579i	$-0.220477\pi$
$620$	5.75858i	0.231270i
$621$	0	0
$622$	4.83750i	0.193966i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.3929	−0.415384
$627$	0	0
$628$	− 12.9764i	− 0.517817i
$629$	−3.08783	−0.123120
$630$	0	0
$631$	−9.57900	−0.381334	−0.190667	−	0.981655i	$-0.561065\pi$
$631$	−9.57900	−0.381334	−0.190667	+	0.981655i	$0.561065\pi$
$632$	− 4.32957i	− 0.172221i
$633$	0	0
$634$	22.2581	0.883983
$635$	7.98324	0.316805
$636$	0	0
$637$	0	0
$638$	− 23.2449i	− 0.920275i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	38.3429i	1.51445i	0.653151	+	0.757227i	$0.273446\pi$
$641$	38.3429i	1.51445i	−0.653151	+	0.757227i	$0.726554\pi$
$642$	0	0
$643$	43.8496i	1.72926i	0.502409	+	0.864630i	$0.332447\pi$
$643$	43.8496i	1.72926i	−0.502409	+	0.864630i	$0.667553\pi$
$644$	0	0
$645$	0	0
$646$	−3.44155	−0.135406
$647$	32.8161	1.29013	0.645066	−	0.764127i	$-0.276830\pi$
$647$	32.8161	1.29013	0.645066	+	0.764127i	$0.276830\pi$
$648$	0	0
$649$	− 15.0619i	− 0.591230i
$650$	2.49661	0.0979249
$651$	0	0
$652$	10.4377	0.408772
$653$	− 28.4633i	− 1.11385i	−0.830562	−	0.556927i	$-0.811980\pi$
$653$	− 28.4633i	− 1.11385i	0.830562	−	0.556927i	$-0.188020\pi$
$654$	0	0
$655$	7.68592	0.300314
$656$	8.73925	0.341210
$657$	0	0
$658$	0	0
$659$	18.6211i	0.725375i	0.931911	+	0.362688i	$0.118141\pi$
$659$	18.6211i	0.725375i	−0.931911	+	0.362688i	$0.881859\pi$
$660$	0	0
$661$	− 2.08449i	− 0.0810773i	−0.999178	−	0.0405386i	$-0.987093\pi$
$661$	− 2.08449i	− 0.0810773i	0.999178	−	0.0405386i	$-0.0129074\pi$
$662$	− 21.2760i	− 0.826916i
$663$	0	0
$664$	11.1489i	0.432662i
$665$	0	0
$666$	0	0
$667$	28.4180	1.10035
$668$	−6.79589	−0.262941
$669$	0	0
$670$	− 14.2343i	− 0.549917i
$671$	−15.3314	−0.591864
$672$	0	0
$673$	−46.8855	−1.80730	−0.903652	−	0.428267i	$-0.859124\pi$
$673$	−46.8855	−1.80730	−0.903652	+	0.428267i	$0.859124\pi$
$674$	− 9.65367i	− 0.371845i
$675$	0	0
$676$	−6.76696	−0.260268
$677$	−0.369290	−0.0141930	−0.00709648	−	0.999975i	$-0.502259\pi$
$677$	−0.369290	−0.0141930	−0.00709648	+	0.999975i	$0.502259\pi$
$678$	0	0
$679$	0	0
$680$	− 0.433546i	− 0.0166257i
$681$	0	0
$682$	28.3907i	1.08713i
$683$	− 45.4137i	− 1.73771i	−0.495071	−	0.868853i	$-0.664858\pi$
$683$	− 45.4137i	− 1.73771i	0.495071	−	0.868853i	$-0.335142\pi$
$684$	0	0
$685$	− 18.7547i	− 0.716582i
$686$	0	0
$687$	0	0
$688$	−8.32344	−0.317328
$689$	18.1672	0.692116
$690$	0	0
$691$	− 22.6205i	− 0.860525i	−0.902704	−	0.430263i	$-0.858421\pi$
$691$	− 22.6205i	− 0.860525i	0.902704	−	0.430263i	$-0.141579\pi$
$692$	−16.2059	−0.616055
$693$	0	0
$694$	−1.73418	−0.0658286
$695$	12.3746i	0.469397i
$696$	0	0
$697$	3.78886	0.143513
$698$	−28.7035	−1.08644
$699$	0	0
$700$	0	0
$701$	− 10.6392i	− 0.401839i	−0.979608	−	0.200919i	$-0.935607\pi$
$701$	− 10.6392i	− 0.401839i	0.979608	−	0.200919i	$-0.0643929\pi$
$702$	0	0
$703$	56.5378i	2.13236i
$704$	− 4.93015i	− 0.185812i
$705$	0	0
$706$	8.84044i	0.332714i
$707$	0	0
$708$	0	0
$709$	−5.92718	−0.222600	−0.111300	−	0.993787i	$-0.535501\pi$
$709$	−5.92718	−0.222600	−0.111300	+	0.993787i	$0.535501\pi$
$710$	2.47568	0.0929107
$711$	0	0
$712$	− 14.5626i	− 0.545757i
$713$	−34.7089	−1.29986
$714$	0	0
$715$	12.3086	0.460317
$716$	− 14.1219i	− 0.527760i
$717$	0	0
$718$	−1.22753	−0.0458109
$719$	−36.1546	−1.34834	−0.674170	−	0.738576i	$-0.735498\pi$
$719$	−36.1546	−1.34834	−0.674170	+	0.738576i	$0.735498\pi$
$720$	0	0
$721$	0	0
$722$	44.0144i	1.63805i
$723$	0	0
$724$	− 16.2645i	− 0.604467i
$725$	− 4.71485i	− 0.175105i
$726$	0	0
$727$	− 18.3665i	− 0.681176i	−0.940213	−	0.340588i	$-0.889374\pi$
$727$	− 18.3665i	− 0.681176i	0.940213	−	0.340588i	$-0.110626\pi$
$728$	0	0
$729$	0	0
$730$	5.71644	0.211575
$731$	−3.60859	−0.133469
$732$	0	0
$733$	38.1497i	1.40909i	0.709658	+	0.704546i	$0.248849\pi$
$733$	38.1497i	1.40909i	−0.709658	+	0.704546i	$0.751151\pi$
$734$	8.24587	0.304361
$735$	0	0
$736$	6.02734	0.222171
$737$	− 70.1771i	− 2.58500i
$738$	0	0
$739$	37.5913	1.38282	0.691409	−	0.722463i	$-0.256990\pi$
$739$	37.5913	1.38282	0.691409	+	0.722463i	$0.256990\pi$
$740$	−7.12228	−0.261820
$741$	0	0
$742$	0	0
$743$	− 17.3548i	− 0.636686i	−0.947976	−	0.318343i	$-0.896874\pi$
$743$	− 17.3548i	− 0.636686i	0.947976	−	0.318343i	$-0.103126\pi$
$744$	0	0
$745$	6.40621i	0.234705i
$746$	− 35.5657i	− 1.30215i
$747$	0	0
$748$	− 2.13744i	− 0.0781527i
$749$	0	0
$750$	0	0
$751$	20.6533	0.753649	0.376825	−	0.926285i	$-0.377016\pi$
$751$	20.6533	0.753649	0.376825	+	0.926285i	$0.377016\pi$
$752$	5.40583	0.197130
$753$	0	0
$754$	− 11.7711i	− 0.428679i
$755$	7.38144	0.268638
$756$	0	0
$757$	−48.9613	−1.77953	−0.889765	−	0.456420i	$-0.849132\pi$
$757$	−48.9613	−1.77953	−0.889765	+	0.456420i	$0.849132\pi$
$758$	22.5323i	0.818409i
$759$	0	0
$760$	−7.93816	−0.287947
$761$	29.5763	1.07214	0.536071	−	0.844173i	$-0.319908\pi$
$761$	29.5763	1.07214	0.536071	+	0.844173i	$0.319908\pi$
$762$	0	0
$763$	0	0
$764$	9.21077i	0.333234i
$765$	0	0
$766$	37.0578i	1.33895i
$767$	− 7.62726i	− 0.275404i
$768$	0	0
$769$	− 44.9308i	− 1.62025i	−0.586260	−	0.810123i	$-0.699400\pi$
$769$	− 44.9308i	− 1.62025i	0.586260	−	0.810123i	$-0.300600\pi$
$770$	0	0
$771$	0	0
$772$	0.974011	0.0350554
$773$	26.1557	0.940756	0.470378	−	0.882465i	$-0.344118\pi$
$773$	26.1557	0.940756	0.470378	+	0.882465i	$0.344118\pi$
$774$	0	0
$775$	5.75858i	0.206854i
$776$	−3.65685	−0.131273
$777$	0	0
$778$	−36.5542	−1.31053
$779$	− 69.3735i	− 2.48556i
$780$	0	0
$781$	12.2055	0.436746
$782$	2.61313	0.0934452
$783$	0	0
$784$	0	0
$785$	− 12.9764i	− 0.463149i
$786$	0	0
$787$	52.1219i	1.85795i	0.370148	+	0.928973i	$0.379307\pi$
$787$	52.1219i	1.85795i	−0.370148	+	0.928973i	$0.620693\pi$
$788$	5.34502i	0.190409i
$789$	0	0
$790$	− 4.32957i	− 0.154039i
$791$	0	0
$792$	0	0
$793$	−7.76377	−0.275700
$794$	10.9545	0.388762
$795$	0	0
$796$	22.9977i	0.815130i
$797$	−41.7083	−1.47739	−0.738693	−	0.674042i	$-0.764557\pi$
$797$	−41.7083	−1.47739	−0.738693	+	0.674042i	$0.764557\pi$
$798$	0	0
$799$	2.34367	0.0829132
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	11.1393	0.393343
$803$	28.1829	0.994554
$804$	0	0
$805$	0	0
$806$	14.3769i	0.506405i
$807$	0	0
$808$	− 13.7734i	− 0.484546i
$809$	− 24.4771i	− 0.860568i	−0.902694	−	0.430284i	$-0.858413\pi$
$809$	− 24.4771i	− 0.860568i	0.902694	−	0.430284i	$-0.141587\pi$
$810$	0	0
$811$	− 23.5571i	− 0.827202i	−0.910458	−	0.413601i	$-0.864271\pi$
$811$	− 23.5571i	− 0.827202i	0.910458	−	0.413601i	$-0.135729\pi$
$812$	0	0
$813$	0	0
$814$	−35.1139	−1.23074
$815$	10.4377	0.365617
$816$	0	0
$817$	66.0728i	2.31159i
$818$	−35.1346	−1.22845
$819$	0	0
$820$	8.73925	0.305188
$821$	− 17.2652i	− 0.602560i	−0.953536	−	0.301280i	$-0.902586\pi$
$821$	− 17.2652i	− 0.602560i	0.953536	−	0.301280i	$-0.0974138\pi$
$822$	0	0
$823$	25.5641	0.891110	0.445555	−	0.895255i	$-0.353006\pi$
$823$	25.5641	0.891110	0.445555	+	0.895255i	$0.353006\pi$
$824$	−0.253965	−0.00884729
$825$	0	0
$826$	0	0
$827$	− 18.4885i	− 0.642907i	−0.946925	−	0.321453i	$-0.895829\pi$
$827$	− 18.4885i	− 0.642907i	0.946925	−	0.321453i	$-0.104171\pi$
$828$	0	0
$829$	− 48.6776i	− 1.69064i	−0.534257	−	0.845322i	$-0.679409\pi$
$829$	− 48.6776i	− 1.69064i	0.534257	−	0.845322i	$-0.320591\pi$
$830$	11.1489i	0.386985i
$831$	0	0
$832$	− 2.49661i	− 0.0865542i
$833$	0	0
$834$	0	0
$835$	−6.79589	−0.235182
$836$	−39.1363	−1.35356
$837$	0	0
$838$	− 13.3073i	− 0.459694i
$839$	47.7240	1.64762	0.823808	−	0.566869i	$-0.191845\pi$
$839$	47.7240	1.64762	0.823808	+	0.566869i	$0.191845\pi$
$840$	0	0
$841$	6.77019	0.233455
$842$	26.1394i	0.900822i
$843$	0	0
$844$	8.32222	0.286463
$845$	−6.76696	−0.232790
$846$	0	0
$847$	0	0
$848$	− 7.27677i	− 0.249885i
$849$	0	0
$850$	− 0.433546i	− 0.0148705i
$851$	− 42.9284i	− 1.47157i
$852$	0	0
$853$	9.75158i	0.333888i	0.985966	+	0.166944i	$0.0533899\pi$
$853$	9.75158i	0.333888i	−0.985966	+	0.166944i	$0.946610\pi$
$854$	0	0
$855$	0	0
$856$	−7.31543	−0.250036
$857$	−28.8161	−0.984338	−0.492169	−	0.870500i	$-0.663796\pi$
$857$	−28.8161	−0.984338	−0.492169	+	0.870500i	$0.663796\pi$
$858$	0	0
$859$	0.245341i	0.00837094i	0.999991	+	0.00418547i	$0.00133228\pi$
$859$	0.245341i	0.00837094i	−0.999991	+	0.00418547i	$0.998668\pi$
$860$	−8.32344	−0.283827
$861$	0	0
$862$	−2.40743	−0.0819972
$863$	44.4808i	1.51414i	0.653332	+	0.757072i	$0.273371\pi$
$863$	44.4808i	1.51414i	−0.653332	+	0.757072i	$0.726629\pi$
$864$	0	0
$865$	−16.2059	−0.551017
$866$	15.8558	0.538802
$867$	0	0
$868$	0	0
$869$	− 21.3454i	− 0.724094i
$870$	0	0
$871$	− 35.5373i	− 1.20414i
$872$	− 12.0588i	− 0.408364i
$873$	0	0
$874$	− 47.8460i	− 1.61841i
$875$	0	0
$876$	0	0
$877$	0.699363	0.0236158	0.0118079	−	0.999930i	$-0.496241\pi$
$877$	0.699363	0.0236158	0.0118079	+	0.999930i	$0.496241\pi$
$878$	35.9230	1.21234
$879$	0	0
$880$	− 4.93015i	− 0.166195i
$881$	−28.5190	−0.960831	−0.480415	−	0.877041i	$-0.659514\pi$
$881$	−28.5190	−0.960831	−0.480415	+	0.877041i	$0.659514\pi$
$882$	0	0
$883$	−35.7837	−1.20422	−0.602109	−	0.798414i	$-0.705673\pi$
$883$	−35.7837	−1.20422	−0.602109	+	0.798414i	$0.705673\pi$
$884$	− 1.08239i	− 0.0364048i
$885$	0	0
$886$	17.9342	0.602511
$887$	12.9356	0.434335	0.217167	−	0.976134i	$-0.430318\pi$
$887$	12.9356	0.434335	0.217167	+	0.976134i	$0.430318\pi$
$888$	0	0
$889$	0	0
$890$	− 14.5626i	− 0.488140i
$891$	0	0
$892$	− 17.9191i	− 0.599977i
$893$	− 42.9124i	− 1.43601i
$894$	0	0
$895$	− 14.1219i	− 0.472043i
$896$	0	0
$897$	0	0
$898$	37.3484	1.24633
$899$	27.1508	0.905531
$900$	0	0
$901$	− 3.15481i	− 0.105102i
$902$	43.0858	1.43460
$903$	0	0
$904$	15.4753	0.514701
$905$	− 16.2645i	− 0.540652i
$906$	0	0
$907$	27.0619	0.898577	0.449288	−	0.893387i	$-0.351678\pi$
$907$	27.0619	0.898577	0.449288	+	0.893387i	$0.351678\pi$
$908$	8.37592	0.277965
$909$	0	0
$910$	0	0
$911$	13.4239i	0.444752i	0.974961	+	0.222376i	$0.0713813\pi$
$911$	13.4239i	0.444752i	−0.974961	+	0.222376i	$0.928619\pi$
$912$	0	0
$913$	54.9659i	1.81911i
$914$	23.1322i	0.765144i
$915$	0	0
$916$	2.01867i	0.0666988i
$917$	0	0
$918$	0	0
$919$	6.52585	0.215268	0.107634	−	0.994191i	$-0.465673\pi$
$919$	6.52585	0.215268	0.107634	+	0.994191i	$0.465673\pi$
$920$	6.02734	0.198715
$921$	0	0
$922$	22.0630i	0.726606i
$923$	6.18080	0.203443
$924$	0	0
$925$	−7.12228	−0.234179
$926$	0.146069i	0.00480011i
$927$	0	0
$928$	−4.71485	−0.154773
$929$	−30.1718	−0.989904	−0.494952	−	0.868920i	$-0.664814\pi$
$929$	−30.1718	−0.989904	−0.494952	+	0.868920i	$0.664814\pi$
$930$	0	0
$931$	0	0
$932$	− 22.3037i	− 0.730583i
$933$	0	0
$934$	− 2.19119i	− 0.0716980i
$935$	− 2.13744i	− 0.0699019i
$936$	0	0
$937$	41.3157i	1.34972i	0.737944	+	0.674862i	$0.235797\pi$
$937$	41.3157i	1.34972i	−0.737944	+	0.674862i	$0.764203\pi$
$938$	0	0
$939$	0	0
$940$	5.40583	0.176319
$941$	20.1531	0.656971	0.328485	−	0.944509i	$-0.393462\pi$
$941$	20.1531	0.656971	0.328485	+	0.944509i	$0.393462\pi$
$942$	0	0
$943$	52.6744i	1.71531i
$944$	−3.05505	−0.0994335
$945$	0	0
$946$	−41.0358	−1.33419
$947$	43.5824i	1.41624i	0.706094	+	0.708118i	$0.250456\pi$
$947$	43.5824i	1.41624i	−0.706094	+	0.708118i	$0.749544\pi$
$948$	0	0
$949$	14.2717	0.463279
$950$	−7.93816	−0.257548
$951$	0	0
$952$	0	0
$953$	− 2.37154i	− 0.0768218i	−0.999262	−	0.0384109i	$-0.987770\pi$
$953$	− 2.37154i	− 0.0768218i	0.999262	−	0.0384109i	$-0.0122296\pi$
$954$	0	0
$955$	9.21077i	0.298053i
$956$	− 18.4653i	− 0.597210i
$957$	0	0
$958$	− 9.48194i	− 0.306348i
$959$	0	0
$960$	0	0
$961$	−2.16123	−0.0697169
$962$	−17.7815	−0.573299
$963$	0	0
$964$	24.2229i	0.780168i
$965$	0.974011	0.0313545
$966$	0	0
$967$	9.66518	0.310811	0.155406	−	0.987851i	$-0.450332\pi$
$967$	9.66518	0.310811	0.155406	+	0.987851i	$0.450332\pi$
$968$	− 13.3064i	− 0.427684i
$969$	0	0
$970$	−3.65685	−0.117415
$971$	31.8662	1.02263	0.511317	−	0.859392i	$-0.329158\pi$
$971$	31.8662	1.02263	0.511317	+	0.859392i	$0.329158\pi$
$972$	0	0
$973$	0	0
$974$	12.2240i	0.391681i
$975$	0	0
$976$	3.10973i	0.0995401i
$977$	27.1553i	0.868775i	0.900726	+	0.434387i	$0.143035\pi$
$977$	27.1553i	0.868775i	−0.900726	+	0.434387i	$0.856965\pi$
$978$	0	0
$979$	− 71.7959i	− 2.29461i
$980$	0	0
$981$	0	0
$982$	−28.6434	−0.914048
$983$	−10.9098	−0.347967	−0.173984	−	0.984749i	$-0.555664\pi$
$983$	−10.9098	−0.347967	−0.173984	+	0.984749i	$0.555664\pi$
$984$	0	0
$985$	5.34502i	0.170307i
$986$	−2.04410	−0.0650975
$987$	0	0
$988$	−19.8185	−0.630509
$989$	− 50.1682i	− 1.59526i
$990$	0	0
$991$	−15.0265	−0.477333	−0.238667	−	0.971102i	$-0.576710\pi$
$991$	−15.0265	−0.477333	−0.238667	+	0.971102i	$0.576710\pi$
$992$	5.75858	0.182835
$993$	0	0
$994$	0	0
$995$	22.9977i	0.729075i
$996$	0	0
$997$	35.7199i	1.13126i	0.824659	+	0.565630i	$0.191367\pi$
$997$	35.7199i	1.13126i	−0.824659	+	0.565630i	$0.808633\pi$
$998$	19.4817i	0.616682i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.c.881.3	✓	8
3.2	odd	2		4410.2.b.f.881.6	yes	8
7.6	odd	2		4410.2.b.f.881.3	yes	8
21.20	even	2	inner	4410.2.b.c.881.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.c.881.3	✓	8	1.1	even	1	trivial
4410.2.b.c.881.6	yes	8	21.20	even	2	inner
4410.2.b.f.881.3	yes	8	7.6	odd	2
4410.2.b.f.881.6	yes	8	3.2	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} +4.93015i q^{11} +2.49661i q^{13} +1.00000 q^{16} +0.433546 q^{17} -7.93816i q^{19} +1.00000 q^{20} +4.93015 q^{22} +6.02734i q^{23} +1.00000 q^{25} +2.49661 q^{26} -4.71485i q^{29} +5.75858i q^{31} -1.00000i q^{32} -0.433546i q^{34} -7.12228 q^{37} -7.93816 q^{38} -1.00000i q^{40} +8.73925 q^{41} -8.32344 q^{43} -4.93015i q^{44} +6.02734 q^{46} +5.40583 q^{47} -1.00000i q^{50} -2.49661i q^{52} -7.27677i q^{53} -4.93015i q^{55} -4.71485 q^{58} -3.05505 q^{59} +3.10973i q^{61} +5.75858 q^{62} -1.00000 q^{64} -2.49661i q^{65} -14.2343 q^{67} -0.433546 q^{68} -2.47568i q^{71} -5.71644i q^{73} +7.12228i q^{74} +7.93816i q^{76} -4.32957 q^{79} -1.00000 q^{80} -8.73925i q^{82} +11.1489 q^{83} -0.433546 q^{85} +8.32344i q^{86} -4.93015 q^{88} -14.5626 q^{89} -6.02734i q^{92} -5.40583i q^{94} +7.93816i q^{95} +3.65685i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^5	\( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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