LMFDB - Embedded newform 1350.4.c.i.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.i.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -22.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} -22.0000i q^{7} -8.00000i q^{8} -12.0000 q^{11} -38.0000i q^{13} +44.0000 q^{14} +16.0000 q^{16} -105.000i q^{17} +157.000 q^{19} -24.0000i q^{22} +117.000i q^{23} +76.0000 q^{26} +88.0000i q^{28} -66.0000 q^{29} -25.0000 q^{31} +32.0000i q^{32} +210.000 q^{34} +314.000i q^{37} +314.000i q^{38} -504.000 q^{41} -380.000i q^{43} +48.0000 q^{44} -234.000 q^{46} -252.000i q^{47} -141.000 q^{49} +152.000i q^{52} -3.00000i q^{53} -176.000 q^{56} -132.000i q^{58} +318.000 q^{59} +293.000 q^{61} -50.0000i q^{62} -64.0000 q^{64} -322.000i q^{67} +420.000i q^{68} -120.000 q^{71} -44.0000i q^{73} -628.000 q^{74} -628.000 q^{76} +264.000i q^{77} -917.000 q^{79} -1008.00i q^{82} -309.000i q^{83} +760.000 q^{86} +96.0000i q^{88} -1272.00 q^{89} -836.000 q^{91} -468.000i q^{92} +504.000 q^{94} +1328.00i q^{97} -282.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 24 q^{11} + 88 q^{14} + 32 q^{16} + 314 q^{19} + 152 q^{26} - 132 q^{29} - 50 q^{31} + 420 q^{34} - 1008 q^{41} + 96 q^{44} - 468 q^{46} - 282 q^{49} - 352 q^{56} + 636 q^{59} + 586 q^{61} - 128 q^{64} - 240 q^{71} - 1256 q^{74} - 1256 q^{76} - 1834 q^{79} + 1520 q^{86} - 2544 q^{89} - 1672 q^{91} + 1008 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 24 * q^11 + 88 * q^14 + 32 * q^16 + 314 * q^19 + 152 * q^26 - 132 * q^29 - 50 * q^31 + 420 * q^34 - 1008 * q^41 + 96 * q^44 - 468 * q^46 - 282 * q^49 - 352 * q^56 + 636 * q^59 + 586 * q^61 - 128 * q^64 - 240 * q^71 - 1256 * q^74 - 1256 * q^76 - 1834 * q^79 + 1520 * q^86 - 2544 * q^89 - 1672 * q^91 + 1008 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 22.0000i	− 1.18789i	−0.804506	−	0.593944i	$-0.797570\pi$
$7$	− 22.0000i	− 1.18789i	0.804506	−	0.593944i	$-0.202430\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	− 38.0000i	− 0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$	− 38.0000i	− 0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$	44.0000	0.839964
$15$	0	0
$16$	16.0000	0.250000
$17$	− 105.000i	− 1.49801i	−0.662562	−	0.749007i	$-0.730531\pi$
$17$	− 105.000i	− 1.49801i	0.662562	−	0.749007i	$-0.269469\pi$
$18$	0	0
$19$	157.000	1.89570	0.947849	−	0.318719i	$-0.103253\pi$
$19$	157.000	1.89570	0.947849	+	0.318719i	$0.103253\pi$
$20$	0	0
$21$	0	0
$22$	− 24.0000i	− 0.232583i
$23$	117.000i	1.06070i	0.847778	+	0.530352i	$0.177940\pi$
$23$	117.000i	1.06070i	−0.847778	+	0.530352i	$0.822060\pi$
$24$	0	0
$25$	0	0
$26$	76.0000	0.573263
$27$	0	0
$28$	88.0000i	0.593944i
$29$	−66.0000	−0.422617	−0.211308	−	0.977419i	$-0.567772\pi$
$29$	−66.0000	−0.422617	−0.211308	+	0.977419i	$0.567772\pi$
$30$	0	0
$31$	−25.0000	−0.144843	−0.0724215	−	0.997374i	$-0.523073\pi$
$31$	−25.0000	−0.144843	−0.0724215	+	0.997374i	$0.523073\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	210.000	1.05926
$35$	0	0
$36$	0	0
$37$	314.000i	1.39517i	0.716502	+	0.697585i	$0.245742\pi$
$37$	314.000i	1.39517i	−0.716502	+	0.697585i	$0.754258\pi$
$38$	314.000i	1.34046i
$39$	0	0
$40$	0	0
$41$	−504.000	−1.91979	−0.959897	−	0.280352i	$-0.909549\pi$
$41$	−504.000	−1.91979	−0.959897	+	0.280352i	$0.909549\pi$
$42$	0	0
$43$	− 380.000i	− 1.34766i	−0.738886	−	0.673831i	$-0.764648\pi$
$43$	− 380.000i	− 1.34766i	0.738886	−	0.673831i	$-0.235352\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	−234.000	−0.750031
$47$	− 252.000i	− 0.782085i	−0.920373	−	0.391042i	$-0.872115\pi$
$47$	− 252.000i	− 0.782085i	0.920373	−	0.391042i	$-0.127885\pi$
$48$	0	0
$49$	−141.000	−0.411079
$50$	0	0
$51$	0	0
$52$	152.000i	0.405358i
$53$	− 3.00000i	− 0.00777513i	−0.999992	−	0.00388756i	$-0.998763\pi$
$53$	− 3.00000i	− 0.00777513i	0.999992	−	0.00388756i	$-0.00123745\pi$
$54$	0	0
$55$	0	0
$56$	−176.000	−0.419982
$57$	0	0
$58$	− 132.000i	− 0.298835i
$59$	318.000	0.701696	0.350848	−	0.936432i	$-0.385893\pi$
$59$	318.000	0.701696	0.350848	+	0.936432i	$0.385893\pi$
$60$	0	0
$61$	293.000	0.614997	0.307498	−	0.951549i	$-0.400508\pi$
$61$	293.000	0.614997	0.307498	+	0.951549i	$0.400508\pi$
$62$	− 50.0000i	− 0.102419i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 322.000i	− 0.587143i	−0.955937	−	0.293571i	$-0.905156\pi$
$67$	− 322.000i	− 0.587143i	0.955937	−	0.293571i	$-0.0948438\pi$
$68$	420.000i	0.749007i
$69$	0	0
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	− 44.0000i	− 0.0705453i	−0.999378	−	0.0352727i	$-0.988770\pi$
$73$	− 44.0000i	− 0.0705453i	0.999378	−	0.0352727i	$-0.0112300\pi$
$74$	−628.000	−0.986534
$75$	0	0
$76$	−628.000	−0.947849
$77$	264.000i	0.390722i
$78$	0	0
$79$	−917.000	−1.30596	−0.652978	−	0.757377i	$-0.726481\pi$
$79$	−917.000	−1.30596	−0.652978	+	0.757377i	$0.726481\pi$
$80$	0	0
$81$	0	0
$82$	− 1008.00i	− 1.35750i
$83$	− 309.000i	− 0.408640i	−0.978904	−	0.204320i	$-0.934502\pi$
$83$	− 309.000i	− 0.408640i	0.978904	−	0.204320i	$-0.0654984\pi$
$84$	0	0
$85$	0	0
$86$	760.000	0.952941
$87$	0	0
$88$	96.0000i	0.116291i
$89$	−1272.00	−1.51496	−0.757482	−	0.652856i	$-0.773570\pi$
$89$	−1272.00	−1.51496	−0.757482	+	0.652856i	$0.773570\pi$
$90$	0	0
$91$	−836.000	−0.963040
$92$	− 468.000i	− 0.530352i
$93$	0	0
$94$	504.000	0.553017
$95$	0	0
$96$	0	0
$97$	1328.00i	1.39008i	0.718970	+	0.695041i	$0.244614\pi$
$97$	1328.00i	1.39008i	−0.718970	+	0.695041i	$0.755386\pi$
$98$	− 282.000i	− 0.290677i
$99$	0	0
$100$	0	0
$101$	−492.000	−0.484711	−0.242356	−	0.970187i	$-0.577920\pi$
$101$	−492.000	−0.484711	−0.242356	+	0.970187i	$0.577920\pi$
$102$	0	0
$103$	− 548.000i	− 0.524233i	−0.965036	−	0.262117i	$-0.915579\pi$
$103$	− 548.000i	− 0.524233i	0.965036	−	0.262117i	$-0.0844205\pi$
$104$	−304.000	−0.286631
$105$	0	0
$106$	6.00000	0.00549784
$107$	732.000i	0.661356i	0.943744	+	0.330678i	$0.107277\pi$
$107$	732.000i	0.661356i	−0.943744	+	0.330678i	$0.892723\pi$
$108$	0	0
$109$	907.000	0.797017	0.398508	−	0.917165i	$-0.369528\pi$
$109$	907.000	0.797017	0.398508	+	0.917165i	$0.369528\pi$
$110$	0	0
$111$	0	0
$112$	− 352.000i	− 0.296972i
$113$	1542.00i	1.28371i	0.766826	+	0.641855i	$0.221835\pi$
$113$	1542.00i	1.28371i	−0.766826	+	0.641855i	$0.778165\pi$
$114$	0	0
$115$	0	0
$116$	264.000	0.211308
$117$	0	0
$118$	636.000i	0.496174i
$119$	−2310.00	−1.77947
$120$	0	0
$121$	−1187.00	−0.891811
$122$	586.000i	0.434868i
$123$	0	0
$124$	100.000	0.0724215
$125$	0	0
$126$	0	0
$127$	− 2554.00i	− 1.78449i	−0.451547	−	0.892247i	$-0.649128\pi$
$127$	− 2554.00i	− 1.78449i	0.451547	−	0.892247i	$-0.350872\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−150.000	−0.100042	−0.0500212	−	0.998748i	$-0.515929\pi$
$131$	−150.000	−0.100042	−0.0500212	+	0.998748i	$0.515929\pi$
$132$	0	0
$133$	− 3454.00i	− 2.25188i
$134$	644.000	0.415173
$135$	0	0
$136$	−840.000	−0.529628
$137$	1653.00i	1.03084i	0.856937	+	0.515421i	$0.172364\pi$
$137$	1653.00i	1.03084i	−0.856937	+	0.515421i	$0.827636\pi$
$138$	0	0
$139$	−1124.00	−0.685874	−0.342937	−	0.939358i	$-0.611422\pi$
$139$	−1124.00	−0.685874	−0.342937	+	0.939358i	$0.611422\pi$
$140$	0	0
$141$	0	0
$142$	− 240.000i	− 0.141833i
$143$	456.000i	0.266662i
$144$	0	0
$145$	0	0
$146$	88.0000	0.0498831
$147$	0	0
$148$	− 1256.00i	− 0.697585i
$149$	−1608.00	−0.884111	−0.442055	−	0.896988i	$-0.645751\pi$
$149$	−1608.00	−0.884111	−0.442055	+	0.896988i	$0.645751\pi$
$150$	0	0
$151$	−2488.00	−1.34086	−0.670432	−	0.741971i	$-0.733891\pi$
$151$	−2488.00	−1.34086	−0.670432	+	0.741971i	$0.733891\pi$
$152$	− 1256.00i	− 0.670231i
$153$	0	0
$154$	−528.000	−0.276282
$155$	0	0
$156$	0	0
$157$	− 2968.00i	− 1.50874i	−0.656449	−	0.754370i	$-0.727942\pi$
$157$	− 2968.00i	− 1.50874i	0.656449	−	0.754370i	$-0.272058\pi$
$158$	− 1834.00i	− 0.923451i
$159$	0	0
$160$	0	0
$161$	2574.00	1.26000
$162$	0	0
$163$	− 3170.00i	− 1.52327i	−0.648004	−	0.761637i	$-0.724396\pi$
$163$	− 3170.00i	− 1.52327i	0.648004	−	0.761637i	$-0.275604\pi$
$164$	2016.00	0.959897
$165$	0	0
$166$	618.000	0.288952
$167$	327.000i	0.151521i	0.997126	+	0.0757605i	$0.0241385\pi$
$167$	327.000i	0.151521i	−0.997126	+	0.0757605i	$0.975862\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	0	0
$172$	1520.00i	0.673831i
$173$	− 1305.00i	− 0.573510i	−0.958004	−	0.286755i	$-0.907423\pi$
$173$	− 1305.00i	− 0.573510i	0.958004	−	0.286755i	$-0.0925766\pi$
$174$	0	0
$175$	0	0
$176$	−192.000	−0.0822304
$177$	0	0
$178$	− 2544.00i	− 1.07124i
$179$	4044.00	1.68862	0.844309	−	0.535856i	$-0.180011\pi$
$179$	4044.00	1.68862	0.844309	+	0.535856i	$0.180011\pi$
$180$	0	0
$181$	−1051.00	−0.431603	−0.215802	−	0.976437i	$-0.569236\pi$
$181$	−1051.00	−0.431603	−0.215802	+	0.976437i	$0.569236\pi$
$182$	− 1672.00i	− 0.680972i
$183$	0	0
$184$	936.000	0.375015
$185$	0	0
$186$	0	0
$187$	1260.00i	0.492729i
$188$	1008.00i	0.391042i
$189$	0	0
$190$	0	0
$191$	−2598.00	−0.984213	−0.492106	−	0.870535i	$-0.663773\pi$
$191$	−2598.00	−0.984213	−0.492106	+	0.870535i	$0.663773\pi$
$192$	0	0
$193$	− 4370.00i	− 1.62984i	−0.579572	−	0.814921i	$-0.696780\pi$
$193$	− 4370.00i	− 1.62984i	0.579572	−	0.814921i	$-0.303220\pi$
$194$	−2656.00	−0.982937
$195$	0	0
$196$	564.000	0.205539
$197$	− 2943.00i	− 1.06437i	−0.846629	−	0.532183i	$-0.821372\pi$
$197$	− 2943.00i	− 1.06437i	0.846629	−	0.532183i	$-0.178628\pi$
$198$	0	0
$199$	4768.00	1.69847	0.849233	−	0.528019i	$-0.177065\pi$
$199$	4768.00	1.69847	0.849233	+	0.528019i	$0.177065\pi$
$200$	0	0
$201$	0	0
$202$	− 984.000i	− 0.342743i
$203$	1452.00i	0.502022i
$204$	0	0
$205$	0	0
$206$	1096.00	0.370689
$207$	0	0
$208$	− 608.000i	− 0.202679i
$209$	−1884.00	−0.623536
$210$	0	0
$211$	−1267.00	−0.413383	−0.206692	−	0.978406i	$-0.566270\pi$
$211$	−1267.00	−0.413383	−0.206692	+	0.978406i	$0.566270\pi$
$212$	12.0000i	0.00388756i
$213$	0	0
$214$	−1464.00	−0.467649
$215$	0	0
$216$	0	0
$217$	550.000i	0.172057i
$218$	1814.00i	0.563576i
$219$	0	0
$220$	0	0
$221$	−3990.00	−1.21446
$222$	0	0
$223$	2986.00i	0.896670i	0.893866	+	0.448335i	$0.147983\pi$
$223$	2986.00i	0.896670i	−0.893866	+	0.448335i	$0.852017\pi$
$224$	704.000	0.209991
$225$	0	0
$226$	−3084.00	−0.907720
$227$	5409.00i	1.58153i	0.612118	+	0.790766i	$0.290318\pi$
$227$	5409.00i	1.58153i	−0.612118	+	0.790766i	$0.709682\pi$
$228$	0	0
$229$	−4331.00	−1.24978	−0.624892	−	0.780711i	$-0.714857\pi$
$229$	−4331.00	−1.24978	−0.624892	+	0.780711i	$0.714857\pi$
$230$	0	0
$231$	0	0
$232$	528.000i	0.149418i
$233$	− 2586.00i	− 0.727101i	−0.931575	−	0.363550i	$-0.881564\pi$
$233$	− 2586.00i	− 0.727101i	0.931575	−	0.363550i	$-0.118436\pi$
$234$	0	0
$235$	0	0
$236$	−1272.00	−0.350848
$237$	0	0
$238$	− 4620.00i	− 1.25828i
$239$	−510.000	−0.138030	−0.0690150	−	0.997616i	$-0.521986\pi$
$239$	−510.000	−0.138030	−0.0690150	+	0.997616i	$0.521986\pi$
$240$	0	0
$241$	−205.000	−0.0547934	−0.0273967	−	0.999625i	$-0.508722\pi$
$241$	−205.000	−0.0547934	−0.0273967	+	0.999625i	$0.508722\pi$
$242$	− 2374.00i	− 0.630605i
$243$	0	0
$244$	−1172.00	−0.307498
$245$	0	0
$246$	0	0
$247$	− 5966.00i	− 1.53687i
$248$	200.000i	0.0512097i
$249$	0	0
$250$	0	0
$251$	−4680.00	−1.17689	−0.588444	−	0.808538i	$-0.700259\pi$
$251$	−4680.00	−1.17689	−0.588444	+	0.808538i	$0.700259\pi$
$252$	0	0
$253$	− 1404.00i	− 0.348888i
$254$	5108.00	1.26183
$255$	0	0
$256$	256.000	0.0625000
$257$	− 6159.00i	− 1.49489i	−0.664321	−	0.747447i	$-0.731279\pi$
$257$	− 6159.00i	− 1.49489i	0.664321	−	0.747447i	$-0.268721\pi$
$258$	0	0
$259$	6908.00	1.65731
$260$	0	0
$261$	0	0
$262$	− 300.000i	− 0.0707407i
$263$	6240.00i	1.46302i	0.681829	+	0.731511i	$0.261185\pi$
$263$	6240.00i	1.46302i	−0.681829	+	0.731511i	$0.738815\pi$
$264$	0	0
$265$	0	0
$266$	6908.00	1.59232
$267$	0	0
$268$	1288.00i	0.293571i
$269$	7758.00	1.75841	0.879207	−	0.476439i	$-0.158073\pi$
$269$	7758.00	1.75841	0.879207	+	0.476439i	$0.158073\pi$
$270$	0	0
$271$	−7345.00	−1.64641	−0.823205	−	0.567745i	$-0.807816\pi$
$271$	−7345.00	−1.64641	−0.823205	+	0.567745i	$0.807816\pi$
$272$	− 1680.00i	− 0.374504i
$273$	0	0
$274$	−3306.00	−0.728915
$275$	0	0
$276$	0	0
$277$	− 3004.00i	− 0.651599i	−0.945439	−	0.325799i	$-0.894367\pi$
$277$	− 3004.00i	− 0.651599i	0.945439	−	0.325799i	$-0.105633\pi$
$278$	− 2248.00i	− 0.484986i
$279$	0	0
$280$	0	0
$281$	2046.00	0.434356	0.217178	−	0.976132i	$-0.430315\pi$
$281$	2046.00	0.434356	0.217178	+	0.976132i	$0.430315\pi$
$282$	0	0
$283$	5488.00i	1.15275i	0.817186	+	0.576374i	$0.195533\pi$
$283$	5488.00i	1.15275i	−0.817186	+	0.576374i	$0.804467\pi$
$284$	480.000	0.100291
$285$	0	0
$286$	−912.000	−0.188558
$287$	11088.0i	2.28050i
$288$	0	0
$289$	−6112.00	−1.24405
$290$	0	0
$291$	0	0
$292$	176.000i	0.0352727i
$293$	− 333.000i	− 0.0663961i	−0.999449	−	0.0331981i	$-0.989431\pi$
$293$	− 333.000i	− 0.0663961i	0.999449	−	0.0331981i	$-0.0105692\pi$
$294$	0	0
$295$	0	0
$296$	2512.00	0.493267
$297$	0	0
$298$	− 3216.00i	− 0.625161i
$299$	4446.00	0.859929
$300$	0	0
$301$	−8360.00	−1.60087
$302$	− 4976.00i	− 0.948135i
$303$	0	0
$304$	2512.00	0.473925
$305$	0	0
$306$	0	0
$307$	2918.00i	0.542472i	0.962513	+	0.271236i	$0.0874325\pi$
$307$	2918.00i	0.542472i	−0.962513	+	0.271236i	$0.912568\pi$
$308$	− 1056.00i	− 0.195361i
$309$	0	0
$310$	0	0
$311$	−5754.00	−1.04913	−0.524565	−	0.851370i	$-0.675772\pi$
$311$	−5754.00	−1.04913	−0.524565	+	0.851370i	$0.675772\pi$
$312$	0	0
$313$	− 3368.00i	− 0.608213i	−0.952638	−	0.304106i	$-0.901642\pi$
$313$	− 3368.00i	− 0.608213i	0.952638	−	0.304106i	$-0.0983578\pi$
$314$	5936.00	1.06684
$315$	0	0
$316$	3668.00	0.652978
$317$	2871.00i	0.508680i	0.967115	+	0.254340i	$0.0818582\pi$
$317$	2871.00i	0.508680i	−0.967115	+	0.254340i	$0.918142\pi$
$318$	0	0
$319$	792.000	0.139008
$320$	0	0
$321$	0	0
$322$	5148.00i	0.890953i
$323$	− 16485.0i	− 2.83978i
$324$	0	0
$325$	0	0
$326$	6340.00	1.07712
$327$	0	0
$328$	4032.00i	0.678750i
$329$	−5544.00	−0.929029
$330$	0	0
$331$	−10540.0	−1.75024	−0.875122	−	0.483902i	$-0.839219\pi$
$331$	−10540.0	−1.75024	−0.875122	+	0.483902i	$0.839219\pi$
$332$	1236.00i	0.204320i
$333$	0	0
$334$	−654.000	−0.107142
$335$	0	0
$336$	0	0
$337$	5006.00i	0.809182i	0.914498	+	0.404591i	$0.132586\pi$
$337$	5006.00i	0.809182i	−0.914498	+	0.404591i	$0.867414\pi$
$338$	1506.00i	0.242354i
$339$	0	0
$340$	0	0
$341$	300.000	0.0476420
$342$	0	0
$343$	− 4444.00i	− 0.699573i
$344$	−3040.00	−0.476470
$345$	0	0
$346$	2610.00	0.405533
$347$	− 36.0000i	− 0.00556940i	−0.999996	−	0.00278470i	$-0.999114\pi$
$347$	− 36.0000i	− 0.00556940i	0.999996	−	0.00278470i	$-0.000886398\pi$
$348$	0	0
$349$	6715.00	1.02993	0.514965	−	0.857211i	$-0.327805\pi$
$349$	6715.00	1.02993	0.514965	+	0.857211i	$0.327805\pi$
$350$	0	0
$351$	0	0
$352$	− 384.000i	− 0.0581456i
$353$	− 12822.0i	− 1.93328i	−0.256148	−	0.966638i	$-0.582453\pi$
$353$	− 12822.0i	− 1.93328i	0.256148	−	0.966638i	$-0.417547\pi$
$354$	0	0
$355$	0	0
$356$	5088.00	0.757482
$357$	0	0
$358$	8088.00i	1.19403i
$359$	−5478.00	−0.805342	−0.402671	−	0.915345i	$-0.631918\pi$
$359$	−5478.00	−0.805342	−0.402671	+	0.915345i	$0.631918\pi$
$360$	0	0
$361$	17790.0	2.59367
$362$	− 2102.00i	− 0.305190i
$363$	0	0
$364$	3344.00	0.481520
$365$	0	0
$366$	0	0
$367$	− 2446.00i	− 0.347902i	−0.984754	−	0.173951i	$-0.944347\pi$
$367$	− 2446.00i	− 0.347902i	0.984754	−	0.173951i	$-0.0556535\pi$
$368$	1872.00i	0.265176i
$369$	0	0
$370$	0	0
$371$	−66.0000	−0.00923598
$372$	0	0
$373$	− 11696.0i	− 1.62358i	−0.583948	−	0.811791i	$-0.698493\pi$
$373$	− 11696.0i	− 1.62358i	0.583948	−	0.811791i	$-0.301507\pi$
$374$	−2520.00	−0.348412
$375$	0	0
$376$	−2016.00	−0.276509
$377$	2508.00i	0.342622i
$378$	0	0
$379$	2095.00	0.283939	0.141970	−	0.989871i	$-0.454656\pi$
$379$	2095.00	0.283939	0.141970	+	0.989871i	$0.454656\pi$
$380$	0	0
$381$	0	0
$382$	− 5196.00i	− 0.695944i
$383$	11313.0i	1.50931i	0.656119	+	0.754657i	$0.272197\pi$
$383$	11313.0i	1.50931i	−0.656119	+	0.754657i	$0.727803\pi$
$384$	0	0
$385$	0	0
$386$	8740.00	1.15247
$387$	0	0
$388$	− 5312.00i	− 0.695041i
$389$	−2124.00	−0.276841	−0.138420	−	0.990374i	$-0.544203\pi$
$389$	−2124.00	−0.276841	−0.138420	+	0.990374i	$0.544203\pi$
$390$	0	0
$391$	12285.0	1.58895
$392$	1128.00i	0.145338i
$393$	0	0
$394$	5886.00	0.752620
$395$	0	0
$396$	0	0
$397$	6410.00i	0.810349i	0.914239	+	0.405175i	$0.132789\pi$
$397$	6410.00i	0.810349i	−0.914239	+	0.405175i	$0.867211\pi$
$398$	9536.00i	1.20100i
$399$	0	0
$400$	0	0
$401$	−9882.00	−1.23063	−0.615316	−	0.788280i	$-0.710972\pi$
$401$	−9882.00	−1.23063	−0.615316	+	0.788280i	$0.710972\pi$
$402$	0	0
$403$	950.000i	0.117426i
$404$	1968.00	0.242356
$405$	0	0
$406$	−2904.00	−0.354983
$407$	− 3768.00i	− 0.458901i
$408$	0	0
$409$	−5897.00	−0.712929	−0.356464	−	0.934309i	$-0.616018\pi$
$409$	−5897.00	−0.712929	−0.356464	+	0.934309i	$0.616018\pi$
$410$	0	0
$411$	0	0
$412$	2192.00i	0.262117i
$413$	− 6996.00i	− 0.833537i
$414$	0	0
$415$	0	0
$416$	1216.00	0.143316
$417$	0	0
$418$	− 3768.00i	− 0.440906i
$419$	−6852.00	−0.798907	−0.399454	−	0.916753i	$-0.630800\pi$
$419$	−6852.00	−0.798907	−0.399454	+	0.916753i	$0.630800\pi$
$420$	0	0
$421$	323.000	0.0373921	0.0186960	−	0.999825i	$-0.494049\pi$
$421$	323.000	0.0373921	0.0186960	+	0.999825i	$0.494049\pi$
$422$	− 2534.00i	− 0.292306i
$423$	0	0
$424$	−24.0000	−0.00274892
$425$	0	0
$426$	0	0
$427$	− 6446.00i	− 0.730548i
$428$	− 2928.00i	− 0.330678i
$429$	0	0
$430$	0	0
$431$	10242.0	1.14464	0.572320	−	0.820030i	$-0.306044\pi$
$431$	10242.0	1.14464	0.572320	+	0.820030i	$0.306044\pi$
$432$	0	0
$433$	14398.0i	1.59798i	0.601347	+	0.798988i	$0.294631\pi$
$433$	14398.0i	1.59798i	−0.601347	+	0.798988i	$0.705369\pi$
$434$	−1100.00	−0.121663
$435$	0	0
$436$	−3628.00	−0.398508
$437$	18369.0i	2.01077i
$438$	0	0
$439$	−4079.00	−0.443463	−0.221731	−	0.975108i	$-0.571171\pi$
$439$	−4079.00	−0.443463	−0.221731	+	0.975108i	$0.571171\pi$
$440$	0	0
$441$	0	0
$442$	− 7980.00i	− 0.858755i
$443$	− 5781.00i	− 0.620008i	−0.950735	−	0.310004i	$-0.899670\pi$
$443$	− 5781.00i	− 0.620008i	0.950735	−	0.310004i	$-0.100330\pi$
$444$	0	0
$445$	0	0
$446$	−5972.00	−0.634041
$447$	0	0
$448$	1408.00i	0.148486i
$449$	15078.0	1.58480	0.792400	−	0.610002i	$-0.208832\pi$
$449$	15078.0	1.58480	0.792400	+	0.610002i	$0.208832\pi$
$450$	0	0
$451$	6048.00	0.631462
$452$	− 6168.00i	− 0.641855i
$453$	0	0
$454$	−10818.0	−1.11831
$455$	0	0
$456$	0	0
$457$	4268.00i	0.436868i	0.975852	+	0.218434i	$0.0700948\pi$
$457$	4268.00i	0.436868i	−0.975852	+	0.218434i	$0.929905\pi$
$458$	− 8662.00i	− 0.883731i
$459$	0	0
$460$	0	0
$461$	−5634.00	−0.569201	−0.284600	−	0.958646i	$-0.591861\pi$
$461$	−5634.00	−0.569201	−0.284600	+	0.958646i	$0.591861\pi$
$462$	0	0
$463$	− 4526.00i	− 0.454300i	−0.973860	−	0.227150i	$-0.927059\pi$
$463$	− 4526.00i	− 0.454300i	0.973860	−	0.227150i	$-0.0729408\pi$
$464$	−1056.00	−0.105654
$465$	0	0
$466$	5172.00	0.514138
$467$	969.000i	0.0960171i	0.998847	+	0.0480085i	$0.0152875\pi$
$467$	969.000i	0.0960171i	−0.998847	+	0.0480085i	$0.984713\pi$
$468$	0	0
$469$	−7084.00	−0.697460
$470$	0	0
$471$	0	0
$472$	− 2544.00i	− 0.248087i
$473$	4560.00i	0.443275i
$474$	0	0
$475$	0	0
$476$	9240.00	0.889737
$477$	0	0
$478$	− 1020.00i	− 0.0976019i
$479$	−9756.00	−0.930612	−0.465306	−	0.885150i	$-0.654056\pi$
$479$	−9756.00	−0.930612	−0.465306	+	0.885150i	$0.654056\pi$
$480$	0	0
$481$	11932.0	1.13109
$482$	− 410.000i	− 0.0387448i
$483$	0	0
$484$	4748.00	0.445905
$485$	0	0
$486$	0	0
$487$	8768.00i	0.815844i	0.913017	+	0.407922i	$0.133746\pi$
$487$	8768.00i	0.815844i	−0.913017	+	0.407922i	$0.866254\pi$
$488$	− 2344.00i	− 0.217434i
$489$	0	0
$490$	0	0
$491$	−2274.00	−0.209011	−0.104505	−	0.994524i	$-0.533326\pi$
$491$	−2274.00	−0.209011	−0.104505	+	0.994524i	$0.533326\pi$
$492$	0	0
$493$	6930.00i	0.633086i
$494$	11932.0	1.08673
$495$	0	0
$496$	−400.000	−0.0362107
$497$	2640.00i	0.238270i
$498$	0	0
$499$	1969.00	0.176642	0.0883212	−	0.996092i	$-0.471850\pi$
$499$	1969.00	0.176642	0.0883212	+	0.996092i	$0.471850\pi$
$500$	0	0
$501$	0	0
$502$	− 9360.00i	− 0.832186i
$503$	10701.0i	0.948577i	0.880370	+	0.474288i	$0.157295\pi$
$503$	10701.0i	0.948577i	−0.880370	+	0.474288i	$0.842705\pi$
$504$	0	0
$505$	0	0
$506$	2808.00	0.246701
$507$	0	0
$508$	10216.0i	0.892247i
$509$	−12420.0	−1.08155	−0.540773	−	0.841169i	$-0.681868\pi$
$509$	−12420.0	−1.08155	−0.540773	+	0.841169i	$0.681868\pi$
$510$	0	0
$511$	−968.000	−0.0838000
$512$	512.000i	0.0441942i
$513$	0	0
$514$	12318.0	1.05705
$515$	0	0
$516$	0	0
$517$	3024.00i	0.257244i
$518$	13816.0i	1.17189i
$519$	0	0
$520$	0	0
$521$	−18816.0	−1.58223	−0.791117	−	0.611665i	$-0.790500\pi$
$521$	−18816.0	−1.58223	−0.791117	+	0.611665i	$0.790500\pi$
$522$	0	0
$523$	16798.0i	1.40445i	0.711957	+	0.702223i	$0.247809\pi$
$523$	16798.0i	1.40445i	−0.711957	+	0.702223i	$0.752191\pi$
$524$	600.000	0.0500212
$525$	0	0
$526$	−12480.0	−1.03451
$527$	2625.00i	0.216977i
$528$	0	0
$529$	−1522.00	−0.125092
$530$	0	0
$531$	0	0
$532$	13816.0i	1.12594i
$533$	19152.0i	1.55641i
$534$	0	0
$535$	0	0
$536$	−2576.00	−0.207586
$537$	0	0
$538$	15516.0i	1.24339i
$539$	1692.00	0.135213
$540$	0	0
$541$	−5890.00	−0.468079	−0.234040	−	0.972227i	$-0.575195\pi$
$541$	−5890.00	−0.468079	−0.234040	+	0.972227i	$0.575195\pi$
$542$	− 14690.0i	− 1.16419i
$543$	0	0
$544$	3360.00	0.264814
$545$	0	0
$546$	0	0
$547$	5516.00i	0.431165i	0.976486	+	0.215582i	$0.0691650\pi$
$547$	5516.00i	0.431165i	−0.976486	+	0.215582i	$0.930835\pi$
$548$	− 6612.00i	− 0.515421i
$549$	0	0
$550$	0	0
$551$	−10362.0	−0.801154
$552$	0	0
$553$	20174.0i	1.55133i
$554$	6008.00	0.460750
$555$	0	0
$556$	4496.00	0.342937
$557$	4146.00i	0.315389i	0.987488	+	0.157694i	$0.0504061\pi$
$557$	4146.00i	0.315389i	−0.987488	+	0.157694i	$0.949594\pi$
$558$	0	0
$559$	−14440.0	−1.09257
$560$	0	0
$561$	0	0
$562$	4092.00i	0.307136i
$563$	21444.0i	1.60525i	0.596483	+	0.802626i	$0.296564\pi$
$563$	21444.0i	1.60525i	−0.596483	+	0.802626i	$0.703436\pi$
$564$	0	0
$565$	0	0
$566$	−10976.0	−0.815116
$567$	0	0
$568$	960.000i	0.0709167i
$569$	−14778.0	−1.08880	−0.544399	−	0.838826i	$-0.683242\pi$
$569$	−14778.0	−1.08880	−0.544399	+	0.838826i	$0.683242\pi$
$570$	0	0
$571$	9131.00	0.669213	0.334606	−	0.942358i	$-0.391397\pi$
$571$	9131.00	0.669213	0.334606	+	0.942358i	$0.391397\pi$
$572$	− 1824.00i	− 0.133331i
$573$	0	0
$574$	−22176.0	−1.61256
$575$	0	0
$576$	0	0
$577$	− 2344.00i	− 0.169120i	−0.996418	−	0.0845598i	$-0.973052\pi$
$577$	− 2344.00i	− 0.169120i	0.996418	−	0.0845598i	$-0.0269484\pi$
$578$	− 12224.0i	− 0.879674i
$579$	0	0
$580$	0	0
$581$	−6798.00	−0.485419
$582$	0	0
$583$	36.0000i	0.00255741i
$584$	−352.000	−0.0249415
$585$	0	0
$586$	666.000	0.0469492
$587$	− 26829.0i	− 1.88646i	−0.332142	−	0.943229i	$-0.607771\pi$
$587$	− 26829.0i	− 1.88646i	0.332142	−	0.943229i	$-0.392229\pi$
$588$	0	0
$589$	−3925.00	−0.274579
$590$	0	0
$591$	0	0
$592$	5024.00i	0.348792i
$593$	− 8181.00i	− 0.566532i	−0.959041	−	0.283266i	$-0.908582\pi$
$593$	− 8181.00i	− 0.566532i	0.959041	−	0.283266i	$-0.0914179\pi$
$594$	0	0
$595$	0	0
$596$	6432.00	0.442055
$597$	0	0
$598$	8892.00i	0.608062i
$599$	−24078.0	−1.64240	−0.821202	−	0.570637i	$-0.806696\pi$
$599$	−24078.0	−1.64240	−0.821202	+	0.570637i	$0.806696\pi$
$600$	0	0
$601$	22565.0	1.53152	0.765762	−	0.643124i	$-0.222362\pi$
$601$	22565.0	1.53152	0.765762	+	0.643124i	$0.222362\pi$
$602$	− 16720.0i	− 1.13199i
$603$	0	0
$604$	9952.00	0.670432
$605$	0	0
$606$	0	0
$607$	− 14716.0i	− 0.984026i	−0.870588	−	0.492013i	$-0.836261\pi$
$607$	− 14716.0i	− 0.984026i	0.870588	−	0.492013i	$-0.163739\pi$
$608$	5024.00i	0.335115i
$609$	0	0
$610$	0	0
$611$	−9576.00	−0.634048
$612$	0	0
$613$	7552.00i	0.497590i	0.968556	+	0.248795i	$0.0800345\pi$
$613$	7552.00i	0.497590i	−0.968556	+	0.248795i	$0.919966\pi$
$614$	−5836.00	−0.383586
$615$	0	0
$616$	2112.00	0.138141
$617$	− 3981.00i	− 0.259755i	−0.991530	−	0.129878i	$-0.958542\pi$
$617$	− 3981.00i	− 0.259755i	0.991530	−	0.129878i	$-0.0414585\pi$
$618$	0	0
$619$	−13928.0	−0.904384	−0.452192	−	0.891921i	$-0.649358\pi$
$619$	−13928.0	−0.904384	−0.452192	+	0.891921i	$0.649358\pi$
$620$	0	0
$621$	0	0
$622$	− 11508.0i	− 0.741847i
$623$	27984.0i	1.79961i
$624$	0	0
$625$	0	0
$626$	6736.00	0.430071
$627$	0	0
$628$	11872.0i	0.754370i
$629$	32970.0	2.08998
$630$	0	0
$631$	18605.0	1.17378	0.586889	−	0.809668i	$-0.300353\pi$
$631$	18605.0	1.17378	0.586889	+	0.809668i	$0.300353\pi$
$632$	7336.00i	0.461725i
$633$	0	0
$634$	−5742.00	−0.359691
$635$	0	0
$636$	0	0
$637$	5358.00i	0.333268i
$638$	1584.00i	0.0982934i
$639$	0	0
$640$	0	0
$641$	14928.0	0.919845	0.459922	−	0.887959i	$-0.347877\pi$
$641$	14928.0	0.919845	0.459922	+	0.887959i	$0.347877\pi$
$642$	0	0
$643$	6082.00i	0.373018i	0.982453	+	0.186509i	$0.0597174\pi$
$643$	6082.00i	0.373018i	−0.982453	+	0.186509i	$0.940283\pi$
$644$	−10296.0	−0.629999
$645$	0	0
$646$	32970.0	2.00803
$647$	4875.00i	0.296223i	0.988971	+	0.148111i	$0.0473194\pi$
$647$	4875.00i	0.296223i	−0.988971	+	0.148111i	$0.952681\pi$
$648$	0	0
$649$	−3816.00	−0.230803
$650$	0	0
$651$	0	0
$652$	12680.0i	0.761637i
$653$	5157.00i	0.309049i	0.987989	+	0.154525i	$0.0493846\pi$
$653$	5157.00i	0.309049i	−0.987989	+	0.154525i	$0.950615\pi$
$654$	0	0
$655$	0	0
$656$	−8064.00	−0.479949
$657$	0	0
$658$	− 11088.0i	− 0.656923i
$659$	1506.00	0.0890219	0.0445109	−	0.999009i	$-0.485827\pi$
$659$	1506.00	0.0890219	0.0445109	+	0.999009i	$0.485827\pi$
$660$	0	0
$661$	−2386.00	−0.140400	−0.0702002	−	0.997533i	$-0.522364\pi$
$661$	−2386.00	−0.140400	−0.0702002	+	0.997533i	$0.522364\pi$
$662$	− 21080.0i	− 1.23761i
$663$	0	0
$664$	−2472.00	−0.144476
$665$	0	0
$666$	0	0
$667$	− 7722.00i	− 0.448271i
$668$	− 1308.00i	− 0.0757605i
$669$	0	0
$670$	0	0
$671$	−3516.00	−0.202286
$672$	0	0
$673$	− 21158.0i	− 1.21186i	−0.795518	−	0.605929i	$-0.792801\pi$
$673$	− 21158.0i	− 1.21186i	0.795518	−	0.605929i	$-0.207199\pi$
$674$	−10012.0	−0.572178
$675$	0	0
$676$	−3012.00	−0.171370
$677$	26826.0i	1.52291i	0.648220	+	0.761453i	$0.275514\pi$
$677$	26826.0i	1.52291i	−0.648220	+	0.761453i	$0.724486\pi$
$678$	0	0
$679$	29216.0	1.65126
$680$	0	0
$681$	0	0
$682$	600.000i	0.0336880i
$683$	− 32493.0i	− 1.82037i	−0.414206	−	0.910183i	$-0.635941\pi$
$683$	− 32493.0i	− 1.82037i	0.414206	−	0.910183i	$-0.364059\pi$
$684$	0	0
$685$	0	0
$686$	8888.00	0.494673
$687$	0	0
$688$	− 6080.00i	− 0.336915i
$689$	−114.000	−0.00630342
$690$	0	0
$691$	−7531.00	−0.414606	−0.207303	−	0.978277i	$-0.566469\pi$
$691$	−7531.00	−0.414606	−0.207303	+	0.978277i	$0.566469\pi$
$692$	5220.00i	0.286755i
$693$	0	0
$694$	72.0000	0.00393816
$695$	0	0
$696$	0	0
$697$	52920.0i	2.87588i
$698$	13430.0i	0.728271i
$699$	0	0
$700$	0	0
$701$	24306.0	1.30959	0.654797	−	0.755805i	$-0.272754\pi$
$701$	24306.0	1.30959	0.654797	+	0.755805i	$0.272754\pi$
$702$	0	0
$703$	49298.0i	2.64482i
$704$	768.000	0.0411152
$705$	0	0
$706$	25644.0	1.36703
$707$	10824.0i	0.575783i
$708$	0	0
$709$	27454.0	1.45424	0.727120	−	0.686510i	$-0.240858\pi$
$709$	27454.0	1.45424	0.727120	+	0.686510i	$0.240858\pi$
$710$	0	0
$711$	0	0
$712$	10176.0i	0.535620i
$713$	− 2925.00i	− 0.153635i
$714$	0	0
$715$	0	0
$716$	−16176.0	−0.844309
$717$	0	0
$718$	− 10956.0i	− 0.569463i
$719$	5334.00	0.276668	0.138334	−	0.990386i	$-0.455825\pi$
$719$	5334.00	0.276668	0.138334	+	0.990386i	$0.455825\pi$
$720$	0	0
$721$	−12056.0	−0.622731
$722$	35580.0i	1.83400i
$723$	0	0
$724$	4204.00	0.215802
$725$	0	0
$726$	0	0
$727$	26048.0i	1.32884i	0.747359	+	0.664420i	$0.231321\pi$
$727$	26048.0i	1.32884i	−0.747359	+	0.664420i	$0.768679\pi$
$728$	6688.00i	0.340486i
$729$	0	0
$730$	0	0
$731$	−39900.0	−2.01882
$732$	0	0
$733$	33136.0i	1.66972i	0.550461	+	0.834861i	$0.314452\pi$
$733$	33136.0i	1.66972i	−0.550461	+	0.834861i	$0.685548\pi$
$734$	4892.00	0.246004
$735$	0	0
$736$	−3744.00	−0.187508
$737$	3864.00i	0.193124i
$738$	0	0
$739$	11599.0	0.577370	0.288685	−	0.957424i	$-0.406782\pi$
$739$	11599.0	0.577370	0.288685	+	0.957424i	$0.406782\pi$
$740$	0	0
$741$	0	0
$742$	− 132.000i	− 0.00653083i
$743$	− 26424.0i	− 1.30471i	−0.757912	−	0.652357i	$-0.773780\pi$
$743$	− 26424.0i	− 1.30471i	0.757912	−	0.652357i	$-0.226220\pi$
$744$	0	0
$745$	0	0
$746$	23392.0	1.14805
$747$	0	0
$748$	− 5040.00i	− 0.246365i
$749$	16104.0	0.785617
$750$	0	0
$751$	13661.0	0.663778	0.331889	−	0.943319i	$-0.392314\pi$
$751$	13661.0	0.663778	0.331889	+	0.943319i	$0.392314\pi$
$752$	− 4032.00i	− 0.195521i
$753$	0	0
$754$	−5016.00	−0.242270
$755$	0	0
$756$	0	0
$757$	− 22846.0i	− 1.09690i	−0.836184	−	0.548449i	$-0.815218\pi$
$757$	− 22846.0i	− 1.09690i	0.836184	−	0.548449i	$-0.184782\pi$
$758$	4190.00i	0.200775i
$759$	0	0
$760$	0	0
$761$	20862.0	0.993754	0.496877	−	0.867821i	$-0.334480\pi$
$761$	20862.0	0.993754	0.496877	+	0.867821i	$0.334480\pi$
$762$	0	0
$763$	− 19954.0i	− 0.946767i
$764$	10392.0	0.492106
$765$	0	0
$766$	−22626.0	−1.06725
$767$	− 12084.0i	− 0.568876i
$768$	0	0
$769$	22219.0	1.04192	0.520961	−	0.853581i	$-0.325574\pi$
$769$	22219.0	1.04192	0.520961	+	0.853581i	$0.325574\pi$
$770$	0	0
$771$	0	0
$772$	17480.0i	0.814921i
$773$	− 17619.0i	− 0.819808i	−0.912129	−	0.409904i	$-0.865562\pi$
$773$	− 17619.0i	− 0.819808i	0.912129	−	0.409904i	$-0.134438\pi$
$774$	0	0
$775$	0	0
$776$	10624.0	0.491468
$777$	0	0
$778$	− 4248.00i	− 0.195756i
$779$	−79128.0	−3.63935
$780$	0	0
$781$	1440.00	0.0659760
$782$	24570.0i	1.12356i
$783$	0	0
$784$	−2256.00	−0.102770
$785$	0	0
$786$	0	0
$787$	15584.0i	0.705857i	0.935650	+	0.352929i	$0.114814\pi$
$787$	15584.0i	0.705857i	−0.935650	+	0.352929i	$0.885186\pi$
$788$	11772.0i	0.532183i
$789$	0	0
$790$	0	0
$791$	33924.0	1.52490
$792$	0	0
$793$	− 11134.0i	− 0.498588i
$794$	−12820.0	−0.573003
$795$	0	0
$796$	−19072.0	−0.849233
$797$	− 13023.0i	− 0.578793i	−0.957209	−	0.289397i	$-0.906545\pi$
$797$	− 13023.0i	− 0.578793i	0.957209	−	0.289397i	$-0.0934547\pi$
$798$	0	0
$799$	−26460.0	−1.17157
$800$	0	0
$801$	0	0
$802$	− 19764.0i	− 0.870188i
$803$	528.000i	0.0232039i
$804$	0	0
$805$	0	0
$806$	−1900.00	−0.0830331
$807$	0	0
$808$	3936.00i	0.171371i
$809$	25872.0	1.12436	0.562182	−	0.827013i	$-0.309962\pi$
$809$	25872.0	1.12436	0.562182	+	0.827013i	$0.309962\pi$
$810$	0	0
$811$	22052.0	0.954809	0.477405	−	0.878684i	$-0.341578\pi$
$811$	22052.0	0.954809	0.477405	+	0.878684i	$0.341578\pi$
$812$	− 5808.00i	− 0.251011i
$813$	0	0
$814$	7536.00	0.324492
$815$	0	0
$816$	0	0
$817$	− 59660.0i	− 2.55476i
$818$	− 11794.0i	− 0.504117i
$819$	0	0
$820$	0	0
$821$	1914.00	0.0813630	0.0406815	−	0.999172i	$-0.487047\pi$
$821$	1914.00	0.0813630	0.0406815	+	0.999172i	$0.487047\pi$
$822$	0	0
$823$	11068.0i	0.468780i	0.972143	+	0.234390i	$0.0753093\pi$
$823$	11068.0i	0.468780i	−0.972143	+	0.234390i	$0.924691\pi$
$824$	−4384.00	−0.185345
$825$	0	0
$826$	13992.0	0.589399
$827$	− 21405.0i	− 0.900030i	−0.893021	−	0.450015i	$-0.851419\pi$
$827$	− 21405.0i	− 0.900030i	0.893021	−	0.450015i	$-0.148581\pi$
$828$	0	0
$829$	40042.0	1.67758	0.838791	−	0.544453i	$-0.183263\pi$
$829$	40042.0	1.67758	0.838791	+	0.544453i	$0.183263\pi$
$830$	0	0
$831$	0	0
$832$	2432.00i	0.101339i
$833$	14805.0i	0.615802i
$834$	0	0
$835$	0	0
$836$	7536.00	0.311768
$837$	0	0
$838$	− 13704.0i	− 0.564913i
$839$	−17904.0	−0.736728	−0.368364	−	0.929682i	$-0.620082\pi$
$839$	−17904.0	−0.736728	−0.368364	+	0.929682i	$0.620082\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	646.000i	0.0264402i
$843$	0	0
$844$	5068.00	0.206692
$845$	0	0
$846$	0	0
$847$	26114.0i	1.05937i
$848$	− 48.0000i	− 0.00194378i
$849$	0	0
$850$	0	0
$851$	−36738.0	−1.47986
$852$	0	0
$853$	21580.0i	0.866219i	0.901341	+	0.433110i	$0.142584\pi$
$853$	21580.0i	0.866219i	−0.901341	+	0.433110i	$0.857416\pi$
$854$	12892.0	0.516575
$855$	0	0
$856$	5856.00	0.233825
$857$	17151.0i	0.683625i	0.939768	+	0.341813i	$0.111041\pi$
$857$	17151.0i	0.683625i	−0.939768	+	0.341813i	$0.888959\pi$
$858$	0	0
$859$	−33425.0	−1.32764	−0.663822	−	0.747891i	$-0.731067\pi$
$859$	−33425.0	−1.32764	−0.663822	+	0.747891i	$0.731067\pi$
$860$	0	0
$861$	0	0
$862$	20484.0i	0.809383i
$863$	− 6603.00i	− 0.260450i	−0.991484	−	0.130225i	$-0.958430\pi$
$863$	− 6603.00i	− 0.260450i	0.991484	−	0.130225i	$-0.0415700\pi$
$864$	0	0
$865$	0	0
$866$	−28796.0	−1.12994
$867$	0	0
$868$	− 2200.00i	− 0.0860286i
$869$	11004.0	0.429557
$870$	0	0
$871$	−12236.0	−0.476006
$872$	− 7256.00i	− 0.281788i
$873$	0	0
$874$	−36738.0	−1.42183
$875$	0	0
$876$	0	0
$877$	− 43384.0i	− 1.67044i	−0.549918	−	0.835219i	$-0.685341\pi$
$877$	− 43384.0i	− 1.67044i	0.549918	−	0.835219i	$-0.314659\pi$
$878$	− 8158.00i	− 0.313575i
$879$	0	0
$880$	0	0
$881$	12726.0	0.486663	0.243331	−	0.969943i	$-0.421760\pi$
$881$	12726.0	0.486663	0.243331	+	0.969943i	$0.421760\pi$
$882$	0	0
$883$	− 2786.00i	− 0.106179i	−0.998590	−	0.0530897i	$-0.983093\pi$
$883$	− 2786.00i	− 0.106179i	0.998590	−	0.0530897i	$-0.0169069\pi$
$884$	15960.0	0.607232
$885$	0	0
$886$	11562.0	0.438412
$887$	− 4389.00i	− 0.166142i	−0.996544	−	0.0830711i	$-0.973527\pi$
$887$	− 4389.00i	− 0.166142i	0.996544	−	0.0830711i	$-0.0264729\pi$
$888$	0	0
$889$	−56188.0	−2.11978
$890$	0	0
$891$	0	0
$892$	− 11944.0i	− 0.448335i
$893$	− 39564.0i	− 1.48260i
$894$	0	0
$895$	0	0
$896$	−2816.00	−0.104995
$897$	0	0
$898$	30156.0i	1.12062i
$899$	1650.00	0.0612131
$900$	0	0
$901$	−315.000	−0.0116472
$902$	12096.0i	0.446511i
$903$	0	0
$904$	12336.0	0.453860
$905$	0	0
$906$	0	0
$907$	− 24868.0i	− 0.910395i	−0.890390	−	0.455198i	$-0.849569\pi$
$907$	− 24868.0i	− 0.910395i	0.890390	−	0.455198i	$-0.150431\pi$
$908$	− 21636.0i	− 0.790766i
$909$	0	0
$910$	0	0
$911$	−126.000	−0.00458240	−0.00229120	−	0.999997i	$-0.500729\pi$
$911$	−126.000	−0.00458240	−0.00229120	+	0.999997i	$0.500729\pi$
$912$	0	0
$913$	3708.00i	0.134411i
$914$	−8536.00	−0.308912
$915$	0	0
$916$	17324.0	0.624892
$917$	3300.00i	0.118839i
$918$	0	0
$919$	16144.0	0.579479	0.289740	−	0.957106i	$-0.406431\pi$
$919$	16144.0	0.579479	0.289740	+	0.957106i	$0.406431\pi$
$920$	0	0
$921$	0	0
$922$	− 11268.0i	− 0.402486i
$923$	4560.00i	0.162616i
$924$	0	0
$925$	0	0
$926$	9052.00	0.321239
$927$	0	0
$928$	− 2112.00i	− 0.0747088i
$929$	42192.0	1.49007	0.745035	−	0.667026i	$-0.232433\pi$
$929$	42192.0	1.49007	0.745035	+	0.667026i	$0.232433\pi$
$930$	0	0
$931$	−22137.0	−0.779281
$932$	10344.0i	0.363550i
$933$	0	0
$934$	−1938.00	−0.0678943
$935$	0	0
$936$	0	0
$937$	3272.00i	0.114079i	0.998372	+	0.0570393i	$0.0181660\pi$
$937$	3272.00i	0.114079i	−0.998372	+	0.0570393i	$0.981834\pi$
$938$	− 14168.0i	− 0.493179i
$939$	0	0
$940$	0	0
$941$	20838.0	0.721891	0.360945	−	0.932587i	$-0.382454\pi$
$941$	20838.0	0.721891	0.360945	+	0.932587i	$0.382454\pi$
$942$	0	0
$943$	− 58968.0i	− 2.03633i
$944$	5088.00	0.175424
$945$	0	0
$946$	−9120.00	−0.313443
$947$	13353.0i	0.458199i	0.973403	+	0.229099i	$0.0735781\pi$
$947$	13353.0i	0.458199i	−0.973403	+	0.229099i	$0.926422\pi$
$948$	0	0
$949$	−1672.00	−0.0571922
$950$	0	0
$951$	0	0
$952$	18480.0i	0.629139i
$953$	− 13098.0i	− 0.445211i	−0.974909	−	0.222605i	$-0.928544\pi$
$953$	− 13098.0i	− 0.445211i	0.974909	−	0.222605i	$-0.0714561\pi$
$954$	0	0
$955$	0	0
$956$	2040.00	0.0690150
$957$	0	0
$958$	− 19512.0i	− 0.658042i
$959$	36366.0	1.22452
$960$	0	0
$961$	−29166.0	−0.979021
$962$	23864.0i	0.799799i
$963$	0	0
$964$	820.000	0.0273967
$965$	0	0
$966$	0	0
$967$	19826.0i	0.659319i	0.944100	+	0.329659i	$0.106934\pi$
$967$	19826.0i	0.659319i	−0.944100	+	0.329659i	$0.893066\pi$
$968$	9496.00i	0.315303i
$969$	0	0
$970$	0	0
$971$	23322.0	0.770792	0.385396	−	0.922751i	$-0.374065\pi$
$971$	23322.0	0.770792	0.385396	+	0.922751i	$0.374065\pi$
$972$	0	0
$973$	24728.0i	0.814741i
$974$	−17536.0	−0.576889
$975$	0	0
$976$	4688.00	0.153749
$977$	− 47346.0i	− 1.55039i	−0.631721	−	0.775196i	$-0.717651\pi$
$977$	− 47346.0i	− 1.55039i	0.631721	−	0.775196i	$-0.282349\pi$
$978$	0	0
$979$	15264.0	0.498304
$980$	0	0
$981$	0	0
$982$	− 4548.00i	− 0.147793i
$983$	33033.0i	1.07181i	0.844278	+	0.535905i	$0.180029\pi$
$983$	33033.0i	1.07181i	−0.844278	+	0.535905i	$0.819971\pi$
$984$	0	0
$985$	0	0
$986$	−13860.0	−0.447660
$987$	0	0
$988$	23864.0i	0.768436i
$989$	44460.0	1.42947
$990$	0	0
$991$	6017.00	0.192872	0.0964361	−	0.995339i	$-0.469256\pi$
$991$	6017.00	0.192872	0.0964361	+	0.995339i	$0.469256\pi$
$992$	− 800.000i	− 0.0256049i
$993$	0	0
$994$	−5280.00	−0.168482
$995$	0	0
$996$	0	0
$997$	− 34216.0i	− 1.08689i	−0.839444	−	0.543446i	$-0.817119\pi$
$997$	− 34216.0i	− 1.08689i	0.839444	−	0.543446i	$-0.182881\pi$
$998$	3938.00i	0.124905i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.i.649.2		2
3.2	odd	2		1350.4.c.l.649.1		2
5.2	odd	4		1350.4.a.l.1.1		1
5.3	odd	4		270.4.a.g.1.1	yes	1
5.4	even	2	inner	1350.4.c.i.649.1		2
15.2	even	4		1350.4.a.z.1.1		1
15.8	even	4		270.4.a.c.1.1	✓	1
15.14	odd	2		1350.4.c.l.649.2		2
20.3	even	4		2160.4.a.i.1.1		1
45.13	odd	12		810.4.e.k.541.1		2
45.23	even	12		810.4.e.s.541.1		2
45.38	even	12		810.4.e.s.271.1		2
45.43	odd	12		810.4.e.k.271.1		2
60.23	odd	4		2160.4.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.c.1.1	✓	1	15.8	even	4
270.4.a.g.1.1	yes	1	5.3	odd	4
810.4.e.k.271.1		2	45.43	odd	12
810.4.e.k.541.1		2	45.13	odd	12
810.4.e.s.271.1		2	45.38	even	12
810.4.e.s.541.1		2	45.23	even	12
1350.4.a.l.1.1		1	5.2	odd	4
1350.4.a.z.1.1		1	15.2	even	4
1350.4.c.i.649.1		2	5.4	even	2	inner
1350.4.c.i.649.2		2	1.1	even	1	trivial
1350.4.c.l.649.1		2	3.2	odd	2
1350.4.c.l.649.2		2	15.14	odd	2
2160.4.a.i.1.1		1	20.3	even	4
2160.4.a.r.1.1		1	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -22.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} -22.0000i q^{7} -8.00000i q^{8} -12.0000 q^{11} -38.0000i q^{13} +44.0000 q^{14} +16.0000 q^{16} -105.000i q^{17} +157.000 q^{19} -24.0000i q^{22} +117.000i q^{23} +76.0000 q^{26} +88.0000i q^{28} -66.0000 q^{29} -25.0000 q^{31} +32.0000i q^{32} +210.000 q^{34} +314.000i q^{37} +314.000i q^{38} -504.000 q^{41} -380.000i q^{43} +48.0000 q^{44} -234.000 q^{46} -252.000i q^{47} -141.000 q^{49} +152.000i q^{52} -3.00000i q^{53} -176.000 q^{56} -132.000i q^{58} +318.000 q^{59} +293.000 q^{61} -50.0000i q^{62} -64.0000 q^{64} -322.000i q^{67} +420.000i q^{68} -120.000 q^{71} -44.0000i q^{73} -628.000 q^{74} -628.000 q^{76} +264.000i q^{77} -917.000 q^{79} -1008.00i q^{82} -309.000i q^{83} +760.000 q^{86} +96.0000i q^{88} -1272.00 q^{89} -836.000 q^{91} -468.000i q^{92} +504.000 q^{94} +1328.00i q^{97} -282.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 24 q^{11} + 88 q^{14} + 32 q^{16} + 314 q^{19} + 152 q^{26} - 132 q^{29} - 50 q^{31} + 420 q^{34} - 1008 q^{41} + 96 q^{44} - 468 q^{46} - 282 q^{49} - 352 q^{56} + 636 q^{59} + 586 q^{61} - 128 q^{64} - 240 q^{71} - 1256 q^{74} - 1256 q^{76} - 1834 q^{79} + 1520 q^{86} - 2544 q^{89} - 1672 q^{91} + 1008 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 24 * q^11 + 88 * q^14 + 32 * q^16 + 314 * q^19 + 152 * q^26 - 132 * q^29 - 50 * q^31 + 420 * q^34 - 1008 * q^41 + 96 * q^44 - 468 * q^46 - 282 * q^49 - 352 * q^56 + 636 * q^59 + 586 * q^61 - 128 * q^64 - 240 * q^71 - 1256 * q^74 - 1256 * q^76 - 1834 * q^79 + 1520 * q^86 - 2544 * q^89 - 1672 * q^91 + 1008 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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