LMFDB - Embedded newform 5850.2.e.bi.5149.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1950)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.4
Root			$2.70156i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bi.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.70156i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.70156i q^{7} -1.00000i q^{8} +1.70156 q^{11} -1.00000i q^{13} -1.70156 q^{14} +1.00000 q^{16} +5.70156i q^{17} -4.70156 q^{19} +1.70156i q^{22} +1.00000 q^{26} -1.70156i q^{28} +6.40312 q^{29} +9.10469 q^{31} +1.00000i q^{32} -5.70156 q^{34} -4.70156i q^{37} -4.70156i q^{38} -2.70156 q^{41} +1.40312i q^{43} -1.70156 q^{44} +7.00000i q^{47} +4.10469 q^{49} +1.00000i q^{52} +10.4031i q^{53} +1.70156 q^{56} +6.40312i q^{58} -3.70156 q^{59} -5.10469 q^{61} +9.10469i q^{62} -1.00000 q^{64} -6.40312i q^{67} -5.70156i q^{68} -4.70156 q^{71} -12.0000i q^{73} +4.70156 q^{74} +4.70156 q^{76} +2.89531i q^{77} -0.701562 q^{79} -2.70156i q^{82} +4.29844i q^{83} -1.40312 q^{86} -1.70156i q^{88} -1.40312 q^{89} +1.70156 q^{91} -7.00000 q^{94} +15.4031i q^{97} +4.10469i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 6 q^{11} + 6 q^{14} + 4 q^{16} - 6 q^{19} + 4 q^{26} - 2 q^{31} - 10 q^{34} + 2 q^{41} + 6 q^{44} - 22 q^{49} - 6 q^{56} - 2 q^{59} + 18 q^{61} - 4 q^{64} - 6 q^{71} + 6 q^{74} + 6 q^{76} + 10 q^{79} + 20 q^{86} + 20 q^{89} - 6 q^{91} - 28 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 6 * q^11 + 6 * q^14 + 4 * q^16 - 6 * q^19 + 4 * q^26 - 2 * q^31 - 10 * q^34 + 2 * q^41 + 6 * q^44 - 22 * q^49 - 6 * q^56 - 2 * q^59 + 18 * q^61 - 4 * q^64 - 6 * q^71 + 6 * q^74 + 6 * q^76 + 10 * q^79 + 20 * q^86 + 20 * q^89 - 6 * q^91 - 28 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.70156i	0.643130i	0.946888	+	0.321565i	$0.104209\pi$
$7$	1.70156i	0.643130i	−0.946888	+	0.321565i	$0.895791\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	1.70156	0.513040	0.256520	−	0.966539i	$-0.417424\pi$
$11$	1.70156	0.513040	0.256520	+	0.966539i	$0.417424\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−1.70156	−0.454762
$15$	0	0
$16$	1.00000	0.250000
$17$	5.70156i	1.38283i	0.722457	+	0.691416i	$0.243013\pi$
$17$	5.70156i	1.38283i	−0.722457	+	0.691416i	$0.756987\pi$
$18$	0	0
$19$	−4.70156	−1.07861	−0.539306	−	0.842110i	$-0.681313\pi$
$19$	−4.70156	−1.07861	−0.539306	+	0.842110i	$0.681313\pi$
$20$	0	0
$21$	0	0
$22$	1.70156i	0.362774i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 1.70156i	− 0.321565i
$29$	6.40312	1.18903	0.594515	−	0.804084i	$-0.297344\pi$
$29$	6.40312	1.18903	0.594515	+	0.804084i	$0.297344\pi$
$30$	0	0
$31$	9.10469	1.63525	0.817625	−	0.575751i	$-0.195290\pi$
$31$	9.10469	1.63525	0.817625	+	0.575751i	$0.195290\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−5.70156	−0.977810
$35$	0	0
$36$	0	0
$37$	− 4.70156i	− 0.772932i	−0.922304	−	0.386466i	$-0.873696\pi$
$37$	− 4.70156i	− 0.772932i	0.922304	−	0.386466i	$-0.126304\pi$
$38$	− 4.70156i	− 0.762694i
$39$	0	0
$40$	0	0
$41$	−2.70156	−0.421913	−0.210957	−	0.977495i	$-0.567658\pi$
$41$	−2.70156	−0.421913	−0.210957	+	0.977495i	$0.567658\pi$
$42$	0	0
$43$	1.40312i	0.213974i	0.994260	+	0.106987i	$0.0341204\pi$
$43$	1.40312i	0.213974i	−0.994260	+	0.106987i	$0.965880\pi$
$44$	−1.70156	−0.256520
$45$	0	0
$46$	0	0
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	0	0
$49$	4.10469	0.586384
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	10.4031i	1.42898i	0.699646	+	0.714490i	$0.253341\pi$
$53$	10.4031i	1.42898i	−0.699646	+	0.714490i	$0.746659\pi$
$54$	0	0
$55$	0	0
$56$	1.70156	0.227381
$57$	0	0
$58$	6.40312i	0.840771i
$59$	−3.70156	−0.481902	−0.240951	−	0.970537i	$-0.577459\pi$
$59$	−3.70156	−0.481902	−0.240951	+	0.970537i	$0.577459\pi$
$60$	0	0
$61$	−5.10469	−0.653588	−0.326794	−	0.945096i	$-0.605968\pi$
$61$	−5.10469	−0.653588	−0.326794	+	0.945096i	$0.605968\pi$
$62$	9.10469i	1.15630i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 6.40312i	− 0.782266i	−0.920334	−	0.391133i	$-0.872083\pi$
$67$	− 6.40312i	− 0.782266i	0.920334	−	0.391133i	$-0.127917\pi$
$68$	− 5.70156i	− 0.691416i
$69$	0	0
$70$	0	0
$71$	−4.70156	−0.557973	−0.278986	−	0.960295i	$-0.589998\pi$
$71$	−4.70156	−0.557973	−0.278986	+	0.960295i	$0.589998\pi$
$72$	0	0
$73$	− 12.0000i	− 1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$	− 12.0000i	− 1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$	4.70156	0.546545
$75$	0	0
$76$	4.70156	0.539306
$77$	2.89531i	0.329952i
$78$	0	0
$79$	−0.701562	−0.0789319	−0.0394660	−	0.999221i	$-0.512566\pi$
$79$	−0.701562	−0.0789319	−0.0394660	+	0.999221i	$0.512566\pi$
$80$	0	0
$81$	0	0
$82$	− 2.70156i	− 0.298338i
$83$	4.29844i	0.471815i	0.971776	+	0.235907i	$0.0758062\pi$
$83$	4.29844i	0.471815i	−0.971776	+	0.235907i	$0.924194\pi$
$84$	0	0
$85$	0	0
$86$	−1.40312	−0.151303
$87$	0	0
$88$	− 1.70156i	− 0.181387i
$89$	−1.40312	−0.148731	−0.0743654	−	0.997231i	$-0.523693\pi$
$89$	−1.40312	−0.148731	−0.0743654	+	0.997231i	$0.523693\pi$
$90$	0	0
$91$	1.70156	0.178372
$92$	0	0
$93$	0	0
$94$	−7.00000	−0.721995
$95$	0	0
$96$	0	0
$97$	15.4031i	1.56395i	0.623310	+	0.781975i	$0.285788\pi$
$97$	15.4031i	1.56395i	−0.623310	+	0.781975i	$0.714212\pi$
$98$	4.10469i	0.414636i
$99$	0	0
$100$	0	0
$101$	−0.298438	−0.0296957	−0.0148478	−	0.999890i	$-0.504726\pi$
$101$	−0.298438	−0.0296957	−0.0148478	+	0.999890i	$0.504726\pi$
$102$	0	0
$103$	− 11.4031i	− 1.12358i	−0.827279	−	0.561792i	$-0.810112\pi$
$103$	− 11.4031i	− 1.12358i	0.827279	−	0.561792i	$-0.189888\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.4031	−1.01044
$107$	− 0.104686i	− 0.0101204i	−0.999987	−	0.00506021i	$-0.998389\pi$
$107$	− 0.104686i	− 0.0101204i	0.999987	−	0.00506021i	$-0.00161072\pi$
$108$	0	0
$109$	−10.7016	−1.02502	−0.512512	−	0.858680i	$-0.671285\pi$
$109$	−10.7016	−1.02502	−0.512512	+	0.858680i	$0.671285\pi$
$110$	0	0
$111$	0	0
$112$	1.70156i	0.160783i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−6.40312	−0.594515
$117$	0	0
$118$	− 3.70156i	− 0.340756i
$119$	−9.70156	−0.889341
$120$	0	0
$121$	−8.10469	−0.736790
$122$	− 5.10469i	− 0.462157i
$123$	0	0
$124$	−9.10469	−0.817625
$125$	0	0
$126$	0	0
$127$	12.7016i	1.12708i	0.826088	+	0.563541i	$0.190561\pi$
$127$	12.7016i	1.12708i	−0.826088	+	0.563541i	$0.809439\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	9.50781	0.830701	0.415351	−	0.909661i	$-0.363659\pi$
$131$	9.50781	0.830701	0.415351	+	0.909661i	$0.363659\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	6.40312	0.553146
$135$	0	0
$136$	5.70156	0.488905
$137$	7.29844i	0.623548i	0.950156	+	0.311774i	$0.100923\pi$
$137$	7.29844i	0.623548i	−0.950156	+	0.311774i	$0.899077\pi$
$138$	0	0
$139$	3.40312	0.288649	0.144325	−	0.989530i	$-0.453899\pi$
$139$	3.40312	0.288649	0.144325	+	0.989530i	$0.453899\pi$
$140$	0	0
$141$	0	0
$142$	− 4.70156i	− 0.394546i
$143$	− 1.70156i	− 0.142292i
$144$	0	0
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	4.70156i	0.386466i
$149$	19.4031	1.58957	0.794783	−	0.606894i	$-0.207585\pi$
$149$	19.4031	1.58957	0.794783	+	0.606894i	$0.207585\pi$
$150$	0	0
$151$	5.10469	0.415413	0.207707	−	0.978191i	$-0.433400\pi$
$151$	5.10469	0.415413	0.207707	+	0.978191i	$0.433400\pi$
$152$	4.70156i	0.381347i
$153$	0	0
$154$	−2.89531	−0.233311
$155$	0	0
$156$	0	0
$157$	16.2984i	1.30076i	0.759610	+	0.650378i	$0.225390\pi$
$157$	16.2984i	1.30076i	−0.759610	+	0.650378i	$0.774610\pi$
$158$	− 0.701562i	− 0.0558133i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.80625i	0.533107i	0.963820	+	0.266553i	$0.0858848\pi$
$163$	6.80625i	0.533107i	−0.963820	+	0.266553i	$0.914115\pi$
$164$	2.70156	0.210957
$165$	0	0
$166$	−4.29844	−0.333623
$167$	18.1047i	1.40098i	0.713661	+	0.700491i	$0.247036\pi$
$167$	18.1047i	1.40098i	−0.713661	+	0.700491i	$0.752964\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 1.40312i	− 0.106987i
$173$	25.8062i	1.96201i	0.193976	+	0.981006i	$0.437862\pi$
$173$	25.8062i	1.96201i	−0.193976	+	0.981006i	$0.562138\pi$
$174$	0	0
$175$	0	0
$176$	1.70156	0.128260
$177$	0	0
$178$	− 1.40312i	− 0.105169i
$179$	14.2094	1.06206	0.531029	−	0.847354i	$-0.321805\pi$
$179$	14.2094	1.06206	0.531029	+	0.847354i	$0.321805\pi$
$180$	0	0
$181$	17.7016	1.31575	0.657873	−	0.753129i	$-0.271456\pi$
$181$	17.7016	1.31575	0.657873	+	0.753129i	$0.271456\pi$
$182$	1.70156i	0.126128i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.70156i	0.709448i
$188$	− 7.00000i	− 0.510527i
$189$	0	0
$190$	0	0
$191$	−12.8062	−0.926628	−0.463314	−	0.886194i	$-0.653340\pi$
$191$	−12.8062	−0.926628	−0.463314	+	0.886194i	$0.653340\pi$
$192$	0	0
$193$	− 22.2094i	− 1.59867i	−0.600889	−	0.799333i	$-0.705186\pi$
$193$	− 22.2094i	− 1.59867i	0.600889	−	0.799333i	$-0.294814\pi$
$194$	−15.4031	−1.10588
$195$	0	0
$196$	−4.10469	−0.293192
$197$	7.40312i	0.527451i	0.964598	+	0.263725i	$0.0849513\pi$
$197$	7.40312i	0.527451i	−0.964598	+	0.263725i	$0.915049\pi$
$198$	0	0
$199$	−14.7016	−1.04217	−0.521083	−	0.853506i	$-0.674472\pi$
$199$	−14.7016	−1.04217	−0.521083	+	0.853506i	$0.674472\pi$
$200$	0	0
$201$	0	0
$202$	− 0.298438i	− 0.0209980i
$203$	10.8953i	0.764701i
$204$	0	0
$205$	0	0
$206$	11.4031	0.794493
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−18.8062	−1.29468	−0.647338	−	0.762203i	$-0.724118\pi$
$211$	−18.8062	−1.29468	−0.647338	+	0.762203i	$0.724118\pi$
$212$	− 10.4031i	− 0.714490i
$213$	0	0
$214$	0.104686	0.00715621
$215$	0	0
$216$	0	0
$217$	15.4922i	1.05168i
$218$	− 10.7016i	− 0.724801i
$219$	0	0
$220$	0	0
$221$	5.70156	0.383529
$222$	0	0
$223$	− 1.40312i	− 0.0939601i	−0.998896	−	0.0469801i	$-0.985040\pi$
$223$	− 1.40312i	− 0.0939601i	0.998896	−	0.0469801i	$-0.0149597\pi$
$224$	−1.70156	−0.113690
$225$	0	0
$226$	−14.0000	−0.931266
$227$	3.10469i	0.206065i	0.994678	+	0.103033i	$0.0328546\pi$
$227$	3.10469i	0.206065i	−0.994678	+	0.103033i	$0.967145\pi$
$228$	0	0
$229$	−9.29844	−0.614458	−0.307229	−	0.951636i	$-0.599402\pi$
$229$	−9.29844	−0.614458	−0.307229	+	0.951636i	$0.599402\pi$
$230$	0	0
$231$	0	0
$232$	− 6.40312i	− 0.420386i
$233$	− 18.2094i	− 1.19294i	−0.802637	−	0.596468i	$-0.796570\pi$
$233$	− 18.2094i	− 1.19294i	0.802637	−	0.596468i	$-0.203430\pi$
$234$	0	0
$235$	0	0
$236$	3.70156	0.240951
$237$	0	0
$238$	− 9.70156i	− 0.628859i
$239$	−13.1047	−0.847672	−0.423836	−	0.905739i	$-0.639317\pi$
$239$	−13.1047	−0.847672	−0.423836	+	0.905739i	$0.639317\pi$
$240$	0	0
$241$	−15.4031	−0.992202	−0.496101	−	0.868265i	$-0.665236\pi$
$241$	−15.4031	−0.992202	−0.496101	+	0.868265i	$0.665236\pi$
$242$	− 8.10469i	− 0.520989i
$243$	0	0
$244$	5.10469	0.326794
$245$	0	0
$246$	0	0
$247$	4.70156i	0.299153i
$248$	− 9.10469i	− 0.578148i
$249$	0	0
$250$	0	0
$251$	−6.70156	−0.422999	−0.211499	−	0.977378i	$-0.567835\pi$
$251$	−6.70156	−0.422999	−0.211499	+	0.977378i	$0.567835\pi$
$252$	0	0
$253$	0	0
$254$	−12.7016	−0.796967
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 21.7016i	− 1.35371i	−0.736118	−	0.676853i	$-0.763343\pi$
$257$	− 21.7016i	− 1.35371i	0.736118	−	0.676853i	$-0.236657\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	9.50781i	0.587395i
$263$	− 2.00000i	− 0.123325i	−0.998097	−	0.0616626i	$-0.980360\pi$
$263$	− 2.00000i	− 0.123325i	0.998097	−	0.0616626i	$-0.0196403\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	6.40312i	0.391133i
$269$	−7.20937	−0.439563	−0.219782	−	0.975549i	$-0.570534\pi$
$269$	−7.20937	−0.439563	−0.219782	+	0.975549i	$0.570534\pi$
$270$	0	0
$271$	−12.5078	−0.759795	−0.379898	−	0.925029i	$-0.624041\pi$
$271$	−12.5078	−0.759795	−0.379898	+	0.925029i	$0.624041\pi$
$272$	5.70156i	0.345708i
$273$	0	0
$274$	−7.29844	−0.440915
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	3.40312i	0.204106i
$279$	0	0
$280$	0	0
$281$	−22.9109	−1.36675	−0.683376	−	0.730067i	$-0.739489\pi$
$281$	−22.9109	−1.36675	−0.683376	+	0.730067i	$0.739489\pi$
$282$	0	0
$283$	8.59688i	0.511031i	0.966805	+	0.255516i	$0.0822452\pi$
$283$	8.59688i	0.511031i	−0.966805	+	0.255516i	$0.917755\pi$
$284$	4.70156	0.278986
$285$	0	0
$286$	1.70156	0.100615
$287$	− 4.59688i	− 0.271345i
$288$	0	0
$289$	−15.5078	−0.912224
$290$	0	0
$291$	0	0
$292$	12.0000i	0.702247i
$293$	− 8.59688i	− 0.502235i	−0.967957	−	0.251117i	$-0.919202\pi$
$293$	− 8.59688i	− 0.502235i	0.967957	−	0.251117i	$-0.0807980\pi$
$294$	0	0
$295$	0	0
$296$	−4.70156	−0.273273
$297$	0	0
$298$	19.4031i	1.12399i
$299$	0	0
$300$	0	0
$301$	−2.38750	−0.137613
$302$	5.10469i	0.293742i
$303$	0	0
$304$	−4.70156	−0.269653
$305$	0	0
$306$	0	0
$307$	− 0.701562i	− 0.0400403i	−0.999800	−	0.0200201i	$-0.993627\pi$
$307$	− 0.701562i	− 0.0400403i	0.999800	−	0.0200201i	$-0.00637303\pi$
$308$	− 2.89531i	− 0.164976i
$309$	0	0
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	17.2094i	0.972731i	0.873755	+	0.486366i	$0.161678\pi$
$313$	17.2094i	0.972731i	−0.873755	+	0.486366i	$0.838322\pi$
$314$	−16.2984	−0.919774
$315$	0	0
$316$	0.701562	0.0394660
$317$	26.8062i	1.50559i	0.658256	+	0.752794i	$0.271295\pi$
$317$	26.8062i	1.50559i	−0.658256	+	0.752794i	$0.728705\pi$
$318$	0	0
$319$	10.8953	0.610020
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 26.8062i	− 1.49154i
$324$	0	0
$325$	0	0
$326$	−6.80625	−0.376963
$327$	0	0
$328$	2.70156i	0.149169i
$329$	−11.9109	−0.656671
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 4.29844i	− 0.235907i
$333$	0	0
$334$	−18.1047	−0.990644
$335$	0	0
$336$	0	0
$337$	11.1047i	0.604911i	0.953164	+	0.302455i	$0.0978063\pi$
$337$	11.1047i	0.604911i	−0.953164	+	0.302455i	$0.902194\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	15.4922	0.838949
$342$	0	0
$343$	18.8953i	1.02025i
$344$	1.40312	0.0756514
$345$	0	0
$346$	−25.8062	−1.38735
$347$	− 21.5078i	− 1.15460i	−0.816532	−	0.577300i	$-0.804106\pi$
$347$	− 21.5078i	− 1.15460i	0.816532	−	0.577300i	$-0.195894\pi$
$348$	0	0
$349$	−1.40312	−0.0751075	−0.0375538	−	0.999295i	$-0.511957\pi$
$349$	−1.40312	−0.0751075	−0.0375538	+	0.999295i	$0.511957\pi$
$350$	0	0
$351$	0	0
$352$	1.70156i	0.0906936i
$353$	17.5078i	0.931847i	0.884825	+	0.465923i	$0.154278\pi$
$353$	17.5078i	0.931847i	−0.884825	+	0.465923i	$0.845722\pi$
$354$	0	0
$355$	0	0
$356$	1.40312	0.0743654
$357$	0	0
$358$	14.2094i	0.750989i
$359$	16.6125	0.876774	0.438387	−	0.898786i	$-0.355550\pi$
$359$	16.6125	0.876774	0.438387	+	0.898786i	$0.355550\pi$
$360$	0	0
$361$	3.10469	0.163405
$362$	17.7016i	0.930373i
$363$	0	0
$364$	−1.70156	−0.0891861
$365$	0	0
$366$	0	0
$367$	8.70156i	0.454218i	0.973869	+	0.227109i	$0.0729273\pi$
$367$	8.70156i	0.454218i	−0.973869	+	0.227109i	$0.927073\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.7016	−0.919019
$372$	0	0
$373$	11.7016i	0.605884i	0.953009	+	0.302942i	$0.0979688\pi$
$373$	11.7016i	0.605884i	−0.953009	+	0.302942i	$0.902031\pi$
$374$	−9.70156	−0.501656
$375$	0	0
$376$	7.00000	0.360997
$377$	− 6.40312i	− 0.329778i
$378$	0	0
$379$	33.1047	1.70047	0.850237	−	0.526400i	$-0.176459\pi$
$379$	33.1047	1.70047	0.850237	+	0.526400i	$0.176459\pi$
$380$	0	0
$381$	0	0
$382$	− 12.8062i	− 0.655225i
$383$	− 0.492189i	− 0.0251497i	−0.999921	−	0.0125749i	$-0.995997\pi$
$383$	− 0.492189i	− 0.0251497i	0.999921	−	0.0125749i	$-0.00400281\pi$
$384$	0	0
$385$	0	0
$386$	22.2094	1.13043
$387$	0	0
$388$	− 15.4031i	− 0.781975i
$389$	−7.89531	−0.400308	−0.200154	−	0.979764i	$-0.564144\pi$
$389$	−7.89531	−0.400308	−0.200154	+	0.979764i	$0.564144\pi$
$390$	0	0
$391$	0	0
$392$	− 4.10469i	− 0.207318i
$393$	0	0
$394$	−7.40312	−0.372964
$395$	0	0
$396$	0	0
$397$	− 22.9109i	− 1.14987i	−0.818200	−	0.574933i	$-0.805028\pi$
$397$	− 22.9109i	− 1.14987i	0.818200	−	0.574933i	$-0.194972\pi$
$398$	− 14.7016i	− 0.736923i
$399$	0	0
$400$	0	0
$401$	20.2094	1.00921	0.504604	−	0.863351i	$-0.331639\pi$
$401$	20.2094	1.00921	0.504604	+	0.863351i	$0.331639\pi$
$402$	0	0
$403$	− 9.10469i	− 0.453537i
$404$	0.298438	0.0148478
$405$	0	0
$406$	−10.8953	−0.540725
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	−16.5969	−0.820663	−0.410331	−	0.911937i	$-0.634587\pi$
$409$	−16.5969	−0.820663	−0.410331	+	0.911937i	$0.634587\pi$
$410$	0	0
$411$	0	0
$412$	11.4031i	0.561792i
$413$	− 6.29844i	− 0.309926i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	−34.9109	−1.70551	−0.852755	−	0.522310i	$-0.825070\pi$
$419$	−34.9109	−1.70551	−0.852755	+	0.522310i	$0.825070\pi$
$420$	0	0
$421$	−13.4031	−0.653228	−0.326614	−	0.945158i	$-0.605908\pi$
$421$	−13.4031	−0.653228	−0.326614	+	0.945158i	$0.605908\pi$
$422$	− 18.8062i	− 0.915474i
$423$	0	0
$424$	10.4031	0.505220
$425$	0	0
$426$	0	0
$427$	− 8.68594i	− 0.420342i
$428$	0.104686i	0.00506021i
$429$	0	0
$430$	0	0
$431$	22.3141	1.07483	0.537415	−	0.843318i	$-0.319401\pi$
$431$	22.3141	1.07483	0.537415	+	0.843318i	$0.319401\pi$
$432$	0	0
$433$	8.10469i	0.389486i	0.980854	+	0.194743i	$0.0623873\pi$
$433$	8.10469i	0.389486i	−0.980854	+	0.194743i	$0.937613\pi$
$434$	−15.4922	−0.743649
$435$	0	0
$436$	10.7016	0.512512
$437$	0	0
$438$	0	0
$439$	16.7016	0.797122	0.398561	−	0.917142i	$-0.369510\pi$
$439$	16.7016	0.797122	0.398561	+	0.917142i	$0.369510\pi$
$440$	0	0
$441$	0	0
$442$	5.70156i	0.271196i
$443$	− 0.492189i	− 0.0233846i	−0.999932	−	0.0116923i	$-0.996278\pi$
$443$	− 0.492189i	− 0.0233846i	0.999932	−	0.0116923i	$-0.00372186\pi$
$444$	0	0
$445$	0	0
$446$	1.40312	0.0664399
$447$	0	0
$448$	− 1.70156i	− 0.0803913i
$449$	8.70156	0.410652	0.205326	−	0.978694i	$-0.434175\pi$
$449$	8.70156	0.410652	0.205326	+	0.978694i	$0.434175\pi$
$450$	0	0
$451$	−4.59688	−0.216458
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−3.10469	−0.145710
$455$	0	0
$456$	0	0
$457$	15.4031i	0.720528i	0.932850	+	0.360264i	$0.117313\pi$
$457$	15.4031i	0.720528i	−0.932850	+	0.360264i	$0.882687\pi$
$458$	− 9.29844i	− 0.434487i
$459$	0	0
$460$	0	0
$461$	2.20937	0.102901	0.0514504	−	0.998676i	$-0.483616\pi$
$461$	2.20937	0.102901	0.0514504	+	0.998676i	$0.483616\pi$
$462$	0	0
$463$	− 27.7016i	− 1.28740i	−0.765278	−	0.643700i	$-0.777398\pi$
$463$	− 27.7016i	− 1.28740i	0.765278	−	0.643700i	$-0.222602\pi$
$464$	6.40312	0.297258
$465$	0	0
$466$	18.2094	0.843533
$467$	36.7016i	1.69835i	0.528115	+	0.849173i	$0.322899\pi$
$467$	36.7016i	1.69835i	−0.528115	+	0.849173i	$0.677101\pi$
$468$	0	0
$469$	10.8953	0.503099
$470$	0	0
$471$	0	0
$472$	3.70156i	0.170378i
$473$	2.38750i	0.109778i
$474$	0	0
$475$	0	0
$476$	9.70156	0.444670
$477$	0	0
$478$	− 13.1047i	− 0.599394i
$479$	10.6125	0.484897	0.242449	−	0.970164i	$-0.422049\pi$
$479$	10.6125	0.484897	0.242449	+	0.970164i	$0.422049\pi$
$480$	0	0
$481$	−4.70156	−0.214373
$482$	− 15.4031i	− 0.701593i
$483$	0	0
$484$	8.10469	0.368395
$485$	0	0
$486$	0	0
$487$	27.1047i	1.22823i	0.789216	+	0.614115i	$0.210487\pi$
$487$	27.1047i	1.22823i	−0.789216	+	0.614115i	$0.789513\pi$
$488$	5.10469i	0.231078i
$489$	0	0
$490$	0	0
$491$	−20.5969	−0.929524	−0.464762	−	0.885436i	$-0.653860\pi$
$491$	−20.5969	−0.929524	−0.464762	+	0.885436i	$0.653860\pi$
$492$	0	0
$493$	36.5078i	1.64423i
$494$	−4.70156	−0.211533
$495$	0	0
$496$	9.10469	0.408812
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	21.2094	0.949462	0.474731	−	0.880131i	$-0.342545\pi$
$499$	21.2094	0.949462	0.474731	+	0.880131i	$0.342545\pi$
$500$	0	0
$501$	0	0
$502$	− 6.70156i	− 0.299105i
$503$	− 26.2094i	− 1.16862i	−0.811531	−	0.584309i	$-0.801366\pi$
$503$	− 26.2094i	− 1.16862i	0.811531	−	0.584309i	$-0.198634\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 12.7016i	− 0.563541i
$509$	7.40312	0.328138	0.164069	−	0.986449i	$-0.447538\pi$
$509$	7.40312	0.328138	0.164069	+	0.986449i	$0.447538\pi$
$510$	0	0
$511$	20.4187	0.903272
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	21.7016	0.957215
$515$	0	0
$516$	0	0
$517$	11.9109i	0.523842i
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	−2.20937	−0.0967944	−0.0483972	−	0.998828i	$-0.515411\pi$
$521$	−2.20937	−0.0967944	−0.0483972	+	0.998828i	$0.515411\pi$
$522$	0	0
$523$	0.806248i	0.0352548i	0.999845	+	0.0176274i	$0.00561126\pi$
$523$	0.806248i	0.0352548i	−0.999845	+	0.0176274i	$0.994389\pi$
$524$	−9.50781	−0.415351
$525$	0	0
$526$	2.00000	0.0872041
$527$	51.9109i	2.26128i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	8.00000i	0.346844i
$533$	2.70156i	0.117018i
$534$	0	0
$535$	0	0
$536$	−6.40312	−0.276573
$537$	0	0
$538$	− 7.20937i	− 0.310818i
$539$	6.98438	0.300838
$540$	0	0
$541$	−40.2094	−1.72874	−0.864368	−	0.502860i	$-0.832281\pi$
$541$	−40.2094	−1.72874	−0.864368	+	0.502860i	$0.832281\pi$
$542$	− 12.5078i	− 0.537256i
$543$	0	0
$544$	−5.70156	−0.244452
$545$	0	0
$546$	0	0
$547$	− 19.6125i	− 0.838570i	−0.907855	−	0.419285i	$-0.862281\pi$
$547$	− 19.6125i	− 0.838570i	0.907855	−	0.419285i	$-0.137719\pi$
$548$	− 7.29844i	− 0.311774i
$549$	0	0
$550$	0	0
$551$	−30.1047	−1.28250
$552$	0	0
$553$	− 1.19375i	− 0.0507635i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−3.40312	−0.144325
$557$	− 8.80625i	− 0.373133i	−0.982442	−	0.186566i	$-0.940264\pi$
$557$	− 8.80625i	− 0.373133i	0.982442	−	0.186566i	$-0.0597359\pi$
$558$	0	0
$559$	1.40312	0.0593458
$560$	0	0
$561$	0	0
$562$	− 22.9109i	− 0.966439i
$563$	0.492189i	0.0207433i	0.999946	+	0.0103717i	$0.00330146\pi$
$563$	0.492189i	0.0207433i	−0.999946	+	0.0103717i	$0.996699\pi$
$564$	0	0
$565$	0	0
$566$	−8.59688	−0.361354
$567$	0	0
$568$	4.70156i	0.197273i
$569$	−17.7016	−0.742088	−0.371044	−	0.928615i	$-0.621000\pi$
$569$	−17.7016	−0.742088	−0.371044	+	0.928615i	$0.621000\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	1.70156i	0.0711459i
$573$	0	0
$574$	4.59688	0.191870
$575$	0	0
$576$	0	0
$577$	14.2094i	0.591544i	0.955259	+	0.295772i	$0.0955769\pi$
$577$	14.2094i	0.591544i	−0.955259	+	0.295772i	$0.904423\pi$
$578$	− 15.5078i	− 0.645040i
$579$	0	0
$580$	0	0
$581$	−7.31406	−0.303438
$582$	0	0
$583$	17.7016i	0.733124i
$584$	−12.0000	−0.496564
$585$	0	0
$586$	8.59688	0.355134
$587$	24.5078i	1.01155i	0.862667	+	0.505773i	$0.168793\pi$
$587$	24.5078i	1.01155i	−0.862667	+	0.505773i	$0.831207\pi$
$588$	0	0
$589$	−42.8062	−1.76380
$590$	0	0
$591$	0	0
$592$	− 4.70156i	− 0.193233i
$593$	26.9109i	1.10510i	0.833480	+	0.552550i	$0.186345\pi$
$593$	26.9109i	1.10510i	−0.833480	+	0.552550i	$0.813655\pi$
$594$	0	0
$595$	0	0
$596$	−19.4031	−0.794783
$597$	0	0
$598$	0	0
$599$	−30.2094	−1.23432	−0.617161	−	0.786837i	$-0.711717\pi$
$599$	−30.2094	−1.23432	−0.617161	+	0.786837i	$0.711717\pi$
$600$	0	0
$601$	−40.6125	−1.65662	−0.828309	−	0.560271i	$-0.810697\pi$
$601$	−40.6125	−1.65662	−0.828309	+	0.560271i	$0.810697\pi$
$602$	− 2.38750i	− 0.0973074i
$603$	0	0
$604$	−5.10469	−0.207707
$605$	0	0
$606$	0	0
$607$	15.8953i	0.645171i	0.946540	+	0.322585i	$0.104552\pi$
$607$	15.8953i	0.645171i	−0.946540	+	0.322585i	$0.895448\pi$
$608$	− 4.70156i	− 0.190674i
$609$	0	0
$610$	0	0
$611$	7.00000	0.283190
$612$	0	0
$613$	12.8062i	0.517240i	0.965979	+	0.258620i	$0.0832677\pi$
$613$	12.8062i	0.517240i	−0.965979	+	0.258620i	$0.916732\pi$
$614$	0.701562	0.0283127
$615$	0	0
$616$	2.89531	0.116656
$617$	− 7.89531i	− 0.317853i	−0.987290	−	0.158927i	$-0.949197\pi$
$617$	− 7.89531i	− 0.317853i	0.987290	−	0.158927i	$-0.0508033\pi$
$618$	0	0
$619$	38.8062	1.55975	0.779877	−	0.625932i	$-0.215281\pi$
$619$	38.8062	1.55975	0.779877	+	0.625932i	$0.215281\pi$
$620$	0	0
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	− 2.38750i	− 0.0956533i
$624$	0	0
$625$	0	0
$626$	−17.2094	−0.687825
$627$	0	0
$628$	− 16.2984i	− 0.650378i
$629$	26.8062	1.06884
$630$	0	0
$631$	−17.6125	−0.701142	−0.350571	−	0.936536i	$-0.614013\pi$
$631$	−17.6125	−0.701142	−0.350571	+	0.936536i	$0.614013\pi$
$632$	0.701562i	0.0279066i
$633$	0	0
$634$	−26.8062	−1.06461
$635$	0	0
$636$	0	0
$637$	− 4.10469i	− 0.162634i
$638$	10.8953i	0.431350i
$639$	0	0
$640$	0	0
$641$	26.5078	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$641$	26.5078	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$642$	0	0
$643$	− 16.7016i	− 0.658645i	−0.944217	−	0.329323i	$-0.893180\pi$
$643$	− 16.7016i	− 0.658645i	0.944217	−	0.329323i	$-0.106820\pi$
$644$	0	0
$645$	0	0
$646$	26.8062	1.05468
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−6.29844	−0.247235
$650$	0	0
$651$	0	0
$652$	− 6.80625i	− 0.266553i
$653$	27.7016i	1.08405i	0.840364	+	0.542023i	$0.182341\pi$
$653$	27.7016i	1.08405i	−0.840364	+	0.542023i	$0.817659\pi$
$654$	0	0
$655$	0	0
$656$	−2.70156	−0.105478
$657$	0	0
$658$	− 11.9109i	− 0.464337i
$659$	38.3141	1.49250	0.746252	−	0.665664i	$-0.231851\pi$
$659$	38.3141	1.49250	0.746252	+	0.665664i	$0.231851\pi$
$660$	0	0
$661$	43.1203	1.67719	0.838593	−	0.544759i	$-0.183379\pi$
$661$	43.1203	1.67719	0.838593	+	0.544759i	$0.183379\pi$
$662$	− 12.0000i	− 0.466393i
$663$	0	0
$664$	4.29844	0.166812
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 18.1047i	− 0.700491i
$669$	0	0
$670$	0	0
$671$	−8.68594	−0.335317
$672$	0	0
$673$	− 28.0156i	− 1.07992i	−0.841690	−	0.539961i	$-0.818439\pi$
$673$	− 28.0156i	− 1.07992i	0.841690	−	0.539961i	$-0.181561\pi$
$674$	−11.1047	−0.427737
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 19.6125i	− 0.753769i	−0.926260	−	0.376885i	$-0.876995\pi$
$677$	− 19.6125i	− 0.753769i	0.926260	−	0.376885i	$-0.123005\pi$
$678$	0	0
$679$	−26.2094	−1.00582
$680$	0	0
$681$	0	0
$682$	15.4922i	0.593227i
$683$	− 38.2984i	− 1.46545i	−0.680525	−	0.732724i	$-0.738248\pi$
$683$	− 38.2984i	− 1.46545i	0.680525	−	0.732724i	$-0.261752\pi$
$684$	0	0
$685$	0	0
$686$	−18.8953	−0.721426
$687$	0	0
$688$	1.40312i	0.0534936i
$689$	10.4031	0.396327
$690$	0	0
$691$	−47.8062	−1.81864	−0.909318	−	0.416103i	$-0.863396\pi$
$691$	−47.8062	−1.81864	−0.909318	+	0.416103i	$0.863396\pi$
$692$	− 25.8062i	− 0.981006i
$693$	0	0
$694$	21.5078	0.816425
$695$	0	0
$696$	0	0
$697$	− 15.4031i	− 0.583435i
$698$	− 1.40312i	− 0.0531090i
$699$	0	0
$700$	0	0
$701$	13.9109	0.525409	0.262704	−	0.964876i	$-0.415386\pi$
$701$	13.9109	0.525409	0.262704	+	0.964876i	$0.415386\pi$
$702$	0	0
$703$	22.1047i	0.833694i
$704$	−1.70156	−0.0641300
$705$	0	0
$706$	−17.5078	−0.658915
$707$	− 0.507811i	− 0.0190982i
$708$	0	0
$709$	35.6125	1.33746	0.668728	−	0.743507i	$-0.266839\pi$
$709$	35.6125	1.33746	0.668728	+	0.743507i	$0.266839\pi$
$710$	0	0
$711$	0	0
$712$	1.40312i	0.0525843i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−14.2094	−0.531029
$717$	0	0
$718$	16.6125i	0.619973i
$719$	53.0156	1.97715	0.988575	−	0.150733i	$-0.0481633\pi$
$719$	53.0156	1.97715	0.988575	+	0.150733i	$0.0481633\pi$
$720$	0	0
$721$	19.4031	0.722610
$722$	3.10469i	0.115544i
$723$	0	0
$724$	−17.7016	−0.657873
$725$	0	0
$726$	0	0
$727$	− 46.2094i	− 1.71381i	−0.515474	−	0.856905i	$-0.672384\pi$
$727$	− 46.2094i	− 1.71381i	0.515474	−	0.856905i	$-0.327616\pi$
$728$	− 1.70156i	− 0.0630641i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	3.08907i	0.114097i	0.998371	+	0.0570486i	$0.0181690\pi$
$733$	3.08907i	0.114097i	−0.998371	+	0.0570486i	$0.981831\pi$
$734$	−8.70156	−0.321181
$735$	0	0
$736$	0	0
$737$	− 10.8953i	− 0.401334i
$738$	0	0
$739$	46.6125	1.71467	0.857334	−	0.514760i	$-0.172119\pi$
$739$	46.6125	1.71467	0.857334	+	0.514760i	$0.172119\pi$
$740$	0	0
$741$	0	0
$742$	− 17.7016i	− 0.649845i
$743$	− 20.6125i	− 0.756199i	−0.925765	−	0.378100i	$-0.876578\pi$
$743$	− 20.6125i	− 0.756199i	0.925765	−	0.378100i	$-0.123422\pi$
$744$	0	0
$745$	0	0
$746$	−11.7016	−0.428425
$747$	0	0
$748$	− 9.70156i	− 0.354724i
$749$	0.178130	0.00650874
$750$	0	0
$751$	−23.5078	−0.857812	−0.428906	−	0.903349i	$-0.641101\pi$
$751$	−23.5078	−0.857812	−0.428906	+	0.903349i	$0.641101\pi$
$752$	7.00000i	0.255264i
$753$	0	0
$754$	6.40312	0.233188
$755$	0	0
$756$	0	0
$757$	37.1047i	1.34859i	0.738461	+	0.674296i	$0.235553\pi$
$757$	37.1047i	1.34859i	−0.738461	+	0.674296i	$0.764447\pi$
$758$	33.1047i	1.20242i
$759$	0	0
$760$	0	0
$761$	−43.7172	−1.58475	−0.792373	−	0.610036i	$-0.791155\pi$
$761$	−43.7172	−1.58475	−0.792373	+	0.610036i	$0.791155\pi$
$762$	0	0
$763$	− 18.2094i	− 0.659224i
$764$	12.8062	0.463314
$765$	0	0
$766$	0.492189	0.0177835
$767$	3.70156i	0.133656i
$768$	0	0
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	0	0
$772$	22.2094i	0.799333i
$773$	− 39.4031i	− 1.41723i	−0.705594	−	0.708616i	$-0.749320\pi$
$773$	− 39.4031i	− 1.41723i	0.705594	−	0.708616i	$-0.250680\pi$
$774$	0	0
$775$	0	0
$776$	15.4031	0.552940
$777$	0	0
$778$	− 7.89531i	− 0.283061i
$779$	12.7016	0.455081
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	4.10469	0.146596
$785$	0	0
$786$	0	0
$787$	− 53.1047i	− 1.89298i	−0.322737	−	0.946489i	$-0.604603\pi$
$787$	− 53.1047i	− 1.89298i	0.322737	−	0.946489i	$-0.395397\pi$
$788$	− 7.40312i	− 0.263725i
$789$	0	0
$790$	0	0
$791$	−23.8219	−0.847008
$792$	0	0
$793$	5.10469i	0.181273i
$794$	22.9109	0.813079
$795$	0	0
$796$	14.7016	0.521083
$797$	− 41.9109i	− 1.48456i	−0.670089	−	0.742281i	$-0.733744\pi$
$797$	− 41.9109i	− 1.48456i	0.670089	−	0.742281i	$-0.266256\pi$
$798$	0	0
$799$	−39.9109	−1.41195
$800$	0	0
$801$	0	0
$802$	20.2094i	0.713618i
$803$	− 20.4187i	− 0.720562i
$804$	0	0
$805$	0	0
$806$	9.10469	0.320699
$807$	0	0
$808$	0.298438i	0.0104990i
$809$	32.8062	1.15341	0.576703	−	0.816954i	$-0.304339\pi$
$809$	32.8062	1.15341	0.576703	+	0.816954i	$0.304339\pi$
$810$	0	0
$811$	42.2984	1.48530	0.742650	−	0.669680i	$-0.233569\pi$
$811$	42.2984	1.48530	0.742650	+	0.669680i	$0.233569\pi$
$812$	− 10.8953i	− 0.382351i
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	− 6.59688i	− 0.230795i
$818$	− 16.5969i	− 0.580296i
$819$	0	0
$820$	0	0
$821$	33.4031	1.16578	0.582889	−	0.812552i	$-0.301922\pi$
$821$	33.4031	1.16578	0.582889	+	0.812552i	$0.301922\pi$
$822$	0	0
$823$	− 2.10469i	− 0.0733648i	−0.999327	−	0.0366824i	$-0.988321\pi$
$823$	− 2.10469i	− 0.0733648i	0.999327	−	0.0366824i	$-0.0116790\pi$
$824$	−11.4031	−0.397247
$825$	0	0
$826$	6.29844	0.219151
$827$	14.2984i	0.497205i	0.968606	+	0.248603i	$0.0799713\pi$
$827$	14.2984i	0.497205i	−0.968606	+	0.248603i	$0.920029\pi$
$828$	0	0
$829$	−26.5078	−0.920654	−0.460327	−	0.887749i	$-0.652268\pi$
$829$	−26.5078	−0.920654	−0.460327	+	0.887749i	$0.652268\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	23.4031i	0.810870i
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	0	0
$838$	− 34.9109i	− 1.20598i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	− 13.4031i	− 0.461902i
$843$	0	0
$844$	18.8062	0.647338
$845$	0	0
$846$	0	0
$847$	− 13.7906i	− 0.473852i
$848$	10.4031i	0.357245i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 31.5078i	− 1.07881i	−0.842047	−	0.539403i	$-0.818650\pi$
$853$	− 31.5078i	− 1.07881i	0.842047	−	0.539403i	$-0.181350\pi$
$854$	8.68594	0.297227
$855$	0	0
$856$	−0.104686	−0.00357811
$857$	43.6125i	1.48977i	0.667190	+	0.744887i	$0.267497\pi$
$857$	43.6125i	1.48977i	−0.667190	+	0.744887i	$0.732503\pi$
$858$	0	0
$859$	−32.2094	−1.09897	−0.549485	−	0.835504i	$-0.685176\pi$
$859$	−32.2094	−1.09897	−0.549485	+	0.835504i	$0.685176\pi$
$860$	0	0
$861$	0	0
$862$	22.3141i	0.760020i
$863$	48.2250i	1.64160i	0.571217	+	0.820799i	$0.306471\pi$
$863$	48.2250i	1.64160i	−0.571217	+	0.820799i	$0.693529\pi$
$864$	0	0
$865$	0	0
$866$	−8.10469	−0.275408
$867$	0	0
$868$	− 15.4922i	− 0.525839i
$869$	−1.19375	−0.0404952
$870$	0	0
$871$	−6.40312	−0.216962
$872$	10.7016i	0.362401i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 27.7172i	− 0.935943i	−0.883744	−	0.467971i	$-0.844985\pi$
$877$	− 27.7172i	− 0.935943i	0.883744	−	0.467971i	$-0.155015\pi$
$878$	16.7016i	0.563650i
$879$	0	0
$880$	0	0
$881$	−30.7172	−1.03489	−0.517444	−	0.855717i	$-0.673116\pi$
$881$	−30.7172	−1.03489	−0.517444	+	0.855717i	$0.673116\pi$
$882$	0	0
$883$	− 5.61250i	− 0.188876i	−0.995531	−	0.0944378i	$-0.969895\pi$
$883$	− 5.61250i	− 0.188876i	0.995531	−	0.0944378i	$-0.0301053\pi$
$884$	−5.70156	−0.191764
$885$	0	0
$886$	0.492189	0.0165354
$887$	− 20.0000i	− 0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$	− 20.0000i	− 0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$	0	0
$889$	−21.6125	−0.724860
$890$	0	0
$891$	0	0
$892$	1.40312i	0.0469801i
$893$	− 32.9109i	− 1.10132i
$894$	0	0
$895$	0	0
$896$	1.70156	0.0568452
$897$	0	0
$898$	8.70156i	0.290375i
$899$	58.2984	1.94436
$900$	0	0
$901$	−59.3141	−1.97604
$902$	− 4.59688i	− 0.153059i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 39.0156i	− 1.29549i	−0.761856	−	0.647746i	$-0.775712\pi$
$907$	− 39.0156i	− 1.29549i	0.761856	−	0.647746i	$-0.224288\pi$
$908$	− 3.10469i	− 0.103033i
$909$	0	0
$910$	0	0
$911$	−3.79063	−0.125589	−0.0627945	−	0.998026i	$-0.520001\pi$
$911$	−3.79063	−0.125589	−0.0627945	+	0.998026i	$0.520001\pi$
$912$	0	0
$913$	7.31406i	0.242060i
$914$	−15.4031	−0.509490
$915$	0	0
$916$	9.29844	0.307229
$917$	16.1781i	0.534249i
$918$	0	0
$919$	24.1047	0.795140	0.397570	−	0.917572i	$-0.369854\pi$
$919$	24.1047	0.795140	0.397570	+	0.917572i	$0.369854\pi$
$920$	0	0
$921$	0	0
$922$	2.20937i	0.0727618i
$923$	4.70156i	0.154754i
$924$	0	0
$925$	0	0
$926$	27.7016	0.910330
$927$	0	0
$928$	6.40312i	0.210193i
$929$	58.7016	1.92594	0.962968	−	0.269616i	$-0.0868968\pi$
$929$	58.7016	1.92594	0.962968	+	0.269616i	$0.0868968\pi$
$930$	0	0
$931$	−19.2984	−0.632481
$932$	18.2094i	0.596468i
$933$	0	0
$934$	−36.7016	−1.20091
$935$	0	0
$936$	0	0
$937$	29.9109i	0.977148i	0.872523	+	0.488574i	$0.162483\pi$
$937$	29.9109i	0.977148i	−0.872523	+	0.488574i	$0.837517\pi$
$938$	10.8953i	0.355745i
$939$	0	0
$940$	0	0
$941$	48.4187	1.57841	0.789203	−	0.614132i	$-0.210494\pi$
$941$	48.4187	1.57841	0.789203	+	0.614132i	$0.210494\pi$
$942$	0	0
$943$	0	0
$944$	−3.70156	−0.120476
$945$	0	0
$946$	−2.38750	−0.0776244
$947$	15.4922i	0.503429i	0.967802	+	0.251714i	$0.0809943\pi$
$947$	15.4922i	0.503429i	−0.967802	+	0.251714i	$0.919006\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	9.70156i	0.314429i
$953$	37.7016i	1.22127i	0.791911	+	0.610637i	$0.209086\pi$
$953$	37.7016i	1.22127i	−0.791911	+	0.610637i	$0.790914\pi$
$954$	0	0
$955$	0	0
$956$	13.1047	0.423836
$957$	0	0
$958$	10.6125i	0.342874i
$959$	−12.4187	−0.401022
$960$	0	0
$961$	51.8953	1.67404
$962$	− 4.70156i	− 0.151584i
$963$	0	0
$964$	15.4031	0.496101
$965$	0	0
$966$	0	0
$967$	− 49.1047i	− 1.57910i	−0.613686	−	0.789550i	$-0.710314\pi$
$967$	− 49.1047i	− 1.57910i	0.613686	−	0.789550i	$-0.289686\pi$
$968$	8.10469i	0.260494i
$969$	0	0
$970$	0	0
$971$	56.1047	1.80049	0.900243	−	0.435389i	$-0.143389\pi$
$971$	56.1047	1.80049	0.900243	+	0.435389i	$0.143389\pi$
$972$	0	0
$973$	5.79063i	0.185639i
$974$	−27.1047	−0.868490
$975$	0	0
$976$	−5.10469	−0.163397
$977$	− 10.5969i	− 0.339024i	−0.985528	−	0.169512i	$-0.945781\pi$
$977$	− 10.5969i	− 0.339024i	0.985528	−	0.169512i	$-0.0542192\pi$
$978$	0	0
$979$	−2.38750	−0.0763049
$980$	0	0
$981$	0	0
$982$	− 20.5969i	− 0.657273i
$983$	− 24.5078i	− 0.781678i	−0.920459	−	0.390839i	$-0.872185\pi$
$983$	− 24.5078i	− 0.781678i	0.920459	−	0.390839i	$-0.127815\pi$
$984$	0	0
$985$	0	0
$986$	−36.5078	−1.16265
$987$	0	0
$988$	− 4.70156i	− 0.149577i
$989$	0	0
$990$	0	0
$991$	10.1047	0.320986	0.160493	−	0.987037i	$-0.448692\pi$
$991$	10.1047	0.320986	0.160493	+	0.987037i	$0.448692\pi$
$992$	9.10469i	0.289074i
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	42.5078i	1.34624i	0.739535	+	0.673118i	$0.235045\pi$
$997$	42.5078i	1.34624i	−0.739535	+	0.673118i	$0.764955\pi$
$998$	21.2094i	0.671371i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bi.5149.4		4
3.2	odd	2		1950.2.e.p.1249.2		4
5.2	odd	4		5850.2.a.cg.1.1		2
5.3	odd	4		5850.2.a.cj.1.2		2
5.4	even	2	inner	5850.2.e.bi.5149.1		4
15.2	even	4		1950.2.a.bg.1.1	yes	2
15.8	even	4		1950.2.a.bc.1.2	✓	2
15.14	odd	2		1950.2.e.p.1249.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.bc.1.2	✓	2	15.8	even	4
1950.2.a.bg.1.1	yes	2	15.2	even	4
1950.2.e.p.1249.2		4	3.2	odd	2
1950.2.e.p.1249.3		4	15.14	odd	2
5850.2.a.cg.1.1		2	5.2	odd	4
5850.2.a.cj.1.2		2	5.3	odd	4
5850.2.e.bi.5149.1		4	5.4	even	2	inner
5850.2.e.bi.5149.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.70156i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.70156i q^{7} -1.00000i q^{8} +1.70156 q^{11} -1.00000i q^{13} -1.70156 q^{14} +1.00000 q^{16} +5.70156i q^{17} -4.70156 q^{19} +1.70156i q^{22} +1.00000 q^{26} -1.70156i q^{28} +6.40312 q^{29} +9.10469 q^{31} +1.00000i q^{32} -5.70156 q^{34} -4.70156i q^{37} -4.70156i q^{38} -2.70156 q^{41} +1.40312i q^{43} -1.70156 q^{44} +7.00000i q^{47} +4.10469 q^{49} +1.00000i q^{52} +10.4031i q^{53} +1.70156 q^{56} +6.40312i q^{58} -3.70156 q^{59} -5.10469 q^{61} +9.10469i q^{62} -1.00000 q^{64} -6.40312i q^{67} -5.70156i q^{68} -4.70156 q^{71} -12.0000i q^{73} +4.70156 q^{74} +4.70156 q^{76} +2.89531i q^{77} -0.701562 q^{79} -2.70156i q^{82} +4.29844i q^{83} -1.40312 q^{86} -1.70156i q^{88} -1.40312 q^{89} +1.70156 q^{91} -7.00000 q^{94} +15.4031i q^{97} +4.10469i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 6 q^{11} + 6 q^{14} + 4 q^{16} - 6 q^{19} + 4 q^{26} - 2 q^{31} - 10 q^{34} + 2 q^{41} + 6 q^{44} - 22 q^{49} - 6 q^{56} - 2 q^{59} + 18 q^{61} - 4 q^{64} - 6 q^{71} + 6 q^{74} + 6 q^{76} + 10 q^{79} + 20 q^{86} + 20 q^{89} - 6 q^{91} - 28 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 6 * q^11 + 6 * q^14 + 4 * q^16 - 6 * q^19 + 4 * q^26 - 2 * q^31 - 10 * q^34 + 2 * q^41 + 6 * q^44 - 22 * q^49 - 6 * q^56 - 2 * q^59 + 18 * q^61 - 4 * q^64 - 6 * q^71 + 6 * q^74 + 6 * q^76 + 10 * q^79 + 20 * q^86 + 20 * q^89 - 6 * q^91 - 28 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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