LMFDB - Embedded newform 5850.2.a.cj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(1,5850)]

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, 0, 0]))

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

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

chi := DirichletCharacter("5850.1");

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.a (trivial)

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:	yes
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	5850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.70156 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.70156 q^{7} +1.00000 q^{8} +1.70156 q^{11} +1.00000 q^{13} +1.70156 q^{14} +1.00000 q^{16} +5.70156 q^{17} +4.70156 q^{19} +1.70156 q^{22} +1.00000 q^{26} +1.70156 q^{28} -6.40312 q^{29} +9.10469 q^{31} +1.00000 q^{32} +5.70156 q^{34} -4.70156 q^{37} +4.70156 q^{38} -2.70156 q^{41} -1.40312 q^{43} +1.70156 q^{44} +7.00000 q^{47} -4.10469 q^{49} +1.00000 q^{52} -10.4031 q^{53} +1.70156 q^{56} -6.40312 q^{58} +3.70156 q^{59} -5.10469 q^{61} +9.10469 q^{62} +1.00000 q^{64} -6.40312 q^{67} +5.70156 q^{68} -4.70156 q^{71} +12.0000 q^{73} -4.70156 q^{74} +4.70156 q^{76} +2.89531 q^{77} +0.701562 q^{79} -2.70156 q^{82} -4.29844 q^{83} -1.40312 q^{86} +1.70156 q^{88} +1.40312 q^{89} +1.70156 q^{91} +7.00000 q^{94} +15.4031 q^{97} -4.10469 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} + 3 q^{19} - 3 q^{22} + 2 q^{26} - 3 q^{28} - q^{31} + 2 q^{32} + 5 q^{34} - 3 q^{37} + 3 q^{38} + q^{41} + 10 q^{43} - 3 q^{44} + 14 q^{47} + 11 q^{49} + 2 q^{52} - 8 q^{53} - 3 q^{56} + q^{59} + 9 q^{61} - q^{62} + 2 q^{64} + 5 q^{68} - 3 q^{71} + 24 q^{73} - 3 q^{74} + 3 q^{76} + 25 q^{77} - 5 q^{79} + q^{82} - 15 q^{83} + 10 q^{86} - 3 q^{88} - 10 q^{89} - 3 q^{91} + 14 q^{94} + 18 q^{97} + 11 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^7 + 2 * q^8 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 + 3 * q^19 - 3 * q^22 + 2 * q^26 - 3 * q^28 - q^31 + 2 * q^32 + 5 * q^34 - 3 * q^37 + 3 * q^38 + q^41 + 10 * q^43 - 3 * q^44 + 14 * q^47 + 11 * q^49 + 2 * q^52 - 8 * q^53 - 3 * q^56 + q^59 + 9 * q^61 - q^62 + 2 * q^64 + 5 * q^68 - 3 * q^71 + 24 * q^73 - 3 * q^74 + 3 * q^76 + 25 * q^77 - 5 * q^79 + q^82 - 15 * q^83 + 10 * q^86 - 3 * q^88 - 10 * q^89 - 3 * q^91 + 14 * q^94 + 18 * q^97 + 11 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.70156	0.643130	0.321565	−	0.946888i	$-0.395791\pi$
$7$	1.70156	0.643130	0.321565	+	0.946888i	$0.395791\pi$
$8$	1.00000	0.353553
$9$	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
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.70156	0.454762
$15$	0	0
$16$	1.00000	0.250000
$17$	5.70156	1.38283	0.691416	−	0.722457i	$-0.256987\pi$
$17$	5.70156	1.38283	0.691416	+	0.722457i	$0.256987\pi$
$18$	0	0
$19$	4.70156	1.07861	0.539306	−	0.842110i	$-0.318687\pi$
$19$	4.70156	1.07861	0.539306	+	0.842110i	$0.318687\pi$
$20$	0	0
$21$	0	0
$22$	1.70156	0.362774
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	1.70156	0.321565
$29$	−6.40312	−1.18903	−0.594515	−	0.804084i	$-0.702656\pi$
$29$	−6.40312	−1.18903	−0.594515	+	0.804084i	$0.702656\pi$
$30$	0	0
$31$	9.10469	1.63525	0.817625	−	0.575751i	$-0.195290\pi$
$31$	9.10469	1.63525	0.817625	+	0.575751i	$0.195290\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.70156	0.977810
$35$	0	0
$36$	0	0
$37$	−4.70156	−0.772932	−0.386466	−	0.922304i	$-0.626304\pi$
$37$	−4.70156	−0.772932	−0.386466	+	0.922304i	$0.626304\pi$
$38$	4.70156	0.762694
$39$	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
$43$	−1.40312	−0.213974	−0.106987	−	0.994260i	$-0.534120\pi$
$43$	−1.40312	−0.213974	−0.106987	+	0.994260i	$0.534120\pi$
$44$	1.70156	0.256520
$45$	0	0
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	−10.4031	−1.42898	−0.714490	−	0.699646i	$-0.753341\pi$
$53$	−10.4031	−1.42898	−0.714490	+	0.699646i	$0.753341\pi$
$54$	0	0
$55$	0	0
$56$	1.70156	0.227381
$57$	0	0
$58$	−6.40312	−0.840771
$59$	3.70156	0.481902	0.240951	−	0.970537i	$-0.422541\pi$
$59$	3.70156	0.481902	0.240951	+	0.970537i	$0.422541\pi$
$60$	0	0
$61$	−5.10469	−0.653588	−0.326794	−	0.945096i	$-0.605968\pi$
$61$	−5.10469	−0.653588	−0.326794	+	0.945096i	$0.605968\pi$
$62$	9.10469	1.15630
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−6.40312	−0.782266	−0.391133	−	0.920334i	$-0.627917\pi$
$67$	−6.40312	−0.782266	−0.391133	+	0.920334i	$0.627917\pi$
$68$	5.70156	0.691416
$69$	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
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−4.70156	−0.546545
$75$	0	0
$76$	4.70156	0.539306
$77$	2.89531	0.329952
$78$	0	0
$79$	0.701562	0.0789319	0.0394660	−	0.999221i	$-0.487434\pi$
$79$	0.701562	0.0789319	0.0394660	+	0.999221i	$0.487434\pi$
$80$	0	0
$81$	0	0
$82$	−2.70156	−0.298338
$83$	−4.29844	−0.471815	−0.235907	−	0.971776i	$-0.575806\pi$
$83$	−4.29844	−0.471815	−0.235907	+	0.971776i	$0.575806\pi$
$84$	0	0
$85$	0	0
$86$	−1.40312	−0.151303
$87$	0	0
$88$	1.70156	0.181387
$89$	1.40312	0.148731	0.0743654	−	0.997231i	$-0.476307\pi$
$89$	1.40312	0.148731	0.0743654	+	0.997231i	$0.476307\pi$
$90$	0	0
$91$	1.70156	0.178372
$92$	0	0
$93$	0	0
$94$	7.00000	0.721995
$95$	0	0
$96$	0	0
$97$	15.4031	1.56395	0.781975	−	0.623310i	$-0.214212\pi$
$97$	15.4031	1.56395	0.781975	+	0.623310i	$0.214212\pi$
$98$	−4.10469	−0.414636
$99$	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
$103$	11.4031	1.12358	0.561792	−	0.827279i	$-0.310112\pi$
$103$	11.4031	1.12358	0.561792	+	0.827279i	$0.310112\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.4031	−1.01044
$107$	−0.104686	−0.0101204	−0.00506021	−	0.999987i	$-0.501611\pi$
$107$	−0.104686	−0.0101204	−0.00506021	+	0.999987i	$0.501611\pi$
$108$	0	0
$109$	10.7016	1.02502	0.512512	−	0.858680i	$-0.328715\pi$
$109$	10.7016	1.02502	0.512512	+	0.858680i	$0.328715\pi$
$110$	0	0
$111$	0	0
$112$	1.70156	0.160783
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−6.40312	−0.594515
$117$	0	0
$118$	3.70156	0.340756
$119$	9.70156	0.889341
$120$	0	0
$121$	−8.10469	−0.736790
$122$	−5.10469	−0.462157
$123$	0	0
$124$	9.10469	0.817625
$125$	0	0
$126$	0	0
$127$	12.7016	1.12708	0.563541	−	0.826088i	$-0.309439\pi$
$127$	12.7016	1.12708	0.563541	+	0.826088i	$0.309439\pi$
$128$	1.00000	0.0883883
$129$	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
$133$	8.00000	0.693688
$134$	−6.40312	−0.553146
$135$	0	0
$136$	5.70156	0.488905
$137$	7.29844	0.623548	0.311774	−	0.950156i	$-0.399077\pi$
$137$	7.29844	0.623548	0.311774	+	0.950156i	$0.399077\pi$
$138$	0	0
$139$	−3.40312	−0.288649	−0.144325	−	0.989530i	$-0.546101\pi$
$139$	−3.40312	−0.288649	−0.144325	+	0.989530i	$0.546101\pi$
$140$	0	0
$141$	0	0
$142$	−4.70156	−0.394546
$143$	1.70156	0.142292
$144$	0	0
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	−4.70156	−0.386466
$149$	−19.4031	−1.58957	−0.794783	−	0.606894i	$-0.792415\pi$
$149$	−19.4031	−1.58957	−0.794783	+	0.606894i	$0.792415\pi$
$150$	0	0
$151$	5.10469	0.415413	0.207707	−	0.978191i	$-0.433400\pi$
$151$	5.10469	0.415413	0.207707	+	0.978191i	$0.433400\pi$
$152$	4.70156	0.381347
$153$	0	0
$154$	2.89531	0.233311
$155$	0	0
$156$	0	0
$157$	16.2984	1.30076	0.650378	−	0.759610i	$-0.274610\pi$
$157$	16.2984	1.30076	0.650378	+	0.759610i	$0.274610\pi$
$158$	0.701562	0.0558133
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.80625	−0.533107	−0.266553	−	0.963820i	$-0.585885\pi$
$163$	−6.80625	−0.533107	−0.266553	+	0.963820i	$0.585885\pi$
$164$	−2.70156	−0.210957
$165$	0	0
$166$	−4.29844	−0.333623
$167$	18.1047	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$167$	18.1047	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−1.40312	−0.106987
$173$	−25.8062	−1.96201	−0.981006	−	0.193976i	$-0.937862\pi$
$173$	−25.8062	−1.96201	−0.981006	+	0.193976i	$0.937862\pi$
$174$	0	0
$175$	0	0
$176$	1.70156	0.128260
$177$	0	0
$178$	1.40312	0.105169
$179$	−14.2094	−1.06206	−0.531029	−	0.847354i	$-0.678195\pi$
$179$	−14.2094	−1.06206	−0.531029	+	0.847354i	$0.678195\pi$
$180$	0	0
$181$	17.7016	1.31575	0.657873	−	0.753129i	$-0.271456\pi$
$181$	17.7016	1.31575	0.657873	+	0.753129i	$0.271456\pi$
$182$	1.70156	0.126128
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.70156	0.709448
$188$	7.00000	0.510527
$189$	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
$193$	22.2094	1.59867	0.799333	−	0.600889i	$-0.205186\pi$
$193$	22.2094	1.59867	0.799333	+	0.600889i	$0.205186\pi$
$194$	15.4031	1.10588
$195$	0	0
$196$	−4.10469	−0.293192
$197$	7.40312	0.527451	0.263725	−	0.964598i	$-0.415049\pi$
$197$	7.40312	0.527451	0.263725	+	0.964598i	$0.415049\pi$
$198$	0	0
$199$	14.7016	1.04217	0.521083	−	0.853506i	$-0.325528\pi$
$199$	14.7016	1.04217	0.521083	+	0.853506i	$0.325528\pi$
$200$	0	0
$201$	0	0
$202$	−0.298438	−0.0209980
$203$	−10.8953	−0.764701
$204$	0	0
$205$	0	0
$206$	11.4031	0.794493
$207$	0	0
$208$	1.00000	0.0693375
$209$	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	$0.724118\pi$
$212$	−10.4031	−0.714490
$213$	0	0
$214$	−0.104686	−0.00715621
$215$	0	0
$216$	0	0
$217$	15.4922	1.05168
$218$	10.7016	0.724801
$219$	0	0
$220$	0	0
$221$	5.70156	0.383529
$222$	0	0
$223$	1.40312	0.0939601	0.0469801	−	0.998896i	$-0.485040\pi$
$223$	1.40312	0.0939601	0.0469801	+	0.998896i	$0.485040\pi$
$224$	1.70156	0.113690
$225$	0	0
$226$	−14.0000	−0.931266
$227$	3.10469	0.206065	0.103033	−	0.994678i	$-0.467145\pi$
$227$	3.10469	0.206065	0.103033	+	0.994678i	$0.467145\pi$
$228$	0	0
$229$	9.29844	0.614458	0.307229	−	0.951636i	$-0.400598\pi$
$229$	9.29844	0.614458	0.307229	+	0.951636i	$0.400598\pi$
$230$	0	0
$231$	0	0
$232$	−6.40312	−0.420386
$233$	18.2094	1.19294	0.596468	−	0.802637i	$-0.296570\pi$
$233$	18.2094	1.19294	0.596468	+	0.802637i	$0.296570\pi$
$234$	0	0
$235$	0	0
$236$	3.70156	0.240951
$237$	0	0
$238$	9.70156	0.628859
$239$	13.1047	0.847672	0.423836	−	0.905739i	$-0.360683\pi$
$239$	13.1047	0.847672	0.423836	+	0.905739i	$0.360683\pi$
$240$	0	0
$241$	−15.4031	−0.992202	−0.496101	−	0.868265i	$-0.665236\pi$
$241$	−15.4031	−0.992202	−0.496101	+	0.868265i	$0.665236\pi$
$242$	−8.10469	−0.520989
$243$	0	0
$244$	−5.10469	−0.326794
$245$	0	0
$246$	0	0
$247$	4.70156	0.299153
$248$	9.10469	0.578148
$249$	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
$257$	−21.7016	−1.35371	−0.676853	−	0.736118i	$-0.736657\pi$
$257$	−21.7016	−1.35371	−0.676853	+	0.736118i	$0.736657\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	9.50781	0.587395
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	−6.40312	−0.391133
$269$	7.20937	0.439563	0.219782	−	0.975549i	$-0.429466\pi$
$269$	7.20937	0.439563	0.219782	+	0.975549i	$0.429466\pi$
$270$	0	0
$271$	−12.5078	−0.759795	−0.379898	−	0.925029i	$-0.624041\pi$
$271$	−12.5078	−0.759795	−0.379898	+	0.925029i	$0.624041\pi$
$272$	5.70156	0.345708
$273$	0	0
$274$	7.29844	0.440915
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−3.40312	−0.204106
$279$	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
$283$	−8.59688	−0.511031	−0.255516	−	0.966805i	$-0.582245\pi$
$283$	−8.59688	−0.511031	−0.255516	+	0.966805i	$0.582245\pi$
$284$	−4.70156	−0.278986
$285$	0	0
$286$	1.70156	0.100615
$287$	−4.59688	−0.271345
$288$	0	0
$289$	15.5078	0.912224
$290$	0	0
$291$	0	0
$292$	12.0000	0.702247
$293$	8.59688	0.502235	0.251117	−	0.967957i	$-0.419202\pi$
$293$	8.59688	0.502235	0.251117	+	0.967957i	$0.419202\pi$
$294$	0	0
$295$	0	0
$296$	−4.70156	−0.273273
$297$	0	0
$298$	−19.4031	−1.12399
$299$	0	0
$300$	0	0
$301$	−2.38750	−0.137613
$302$	5.10469	0.293742
$303$	0	0
$304$	4.70156	0.269653
$305$	0	0
$306$	0	0
$307$	−0.701562	−0.0400403	−0.0200201	−	0.999800i	$-0.506373\pi$
$307$	−0.701562	−0.0400403	−0.0200201	+	0.999800i	$0.506373\pi$
$308$	2.89531	0.164976
$309$	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
$313$	−17.2094	−0.972731	−0.486366	−	0.873755i	$-0.661678\pi$
$313$	−17.2094	−0.972731	−0.486366	+	0.873755i	$0.661678\pi$
$314$	16.2984	0.919774
$315$	0	0
$316$	0.701562	0.0394660
$317$	26.8062	1.50559	0.752794	−	0.658256i	$-0.228705\pi$
$317$	26.8062	1.50559	0.752794	+	0.658256i	$0.228705\pi$
$318$	0	0
$319$	−10.8953	−0.610020
$320$	0	0
$321$	0	0
$322$	0	0
$323$	26.8062	1.49154
$324$	0	0
$325$	0	0
$326$	−6.80625	−0.376963
$327$	0	0
$328$	−2.70156	−0.149169
$329$	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	$0.606978\pi$
$332$	−4.29844	−0.235907
$333$	0	0
$334$	18.1047	0.990644
$335$	0	0
$336$	0	0
$337$	11.1047	0.604911	0.302455	−	0.953164i	$-0.402194\pi$
$337$	11.1047	0.604911	0.302455	+	0.953164i	$0.402194\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	15.4922	0.838949
$342$	0	0
$343$	−18.8953	−1.02025
$344$	−1.40312	−0.0756514
$345$	0	0
$346$	−25.8062	−1.38735
$347$	−21.5078	−1.15460	−0.577300	−	0.816532i	$-0.695894\pi$
$347$	−21.5078	−1.15460	−0.577300	+	0.816532i	$0.695894\pi$
$348$	0	0
$349$	1.40312	0.0751075	0.0375538	−	0.999295i	$-0.488043\pi$
$349$	1.40312	0.0751075	0.0375538	+	0.999295i	$0.488043\pi$
$350$	0	0
$351$	0	0
$352$	1.70156	0.0906936
$353$	−17.5078	−0.931847	−0.465923	−	0.884825i	$-0.654278\pi$
$353$	−17.5078	−0.931847	−0.465923	+	0.884825i	$0.654278\pi$
$354$	0	0
$355$	0	0
$356$	1.40312	0.0743654
$357$	0	0
$358$	−14.2094	−0.750989
$359$	−16.6125	−0.876774	−0.438387	−	0.898786i	$-0.644450\pi$
$359$	−16.6125	−0.876774	−0.438387	+	0.898786i	$0.644450\pi$
$360$	0	0
$361$	3.10469	0.163405
$362$	17.7016	0.930373
$363$	0	0
$364$	1.70156	0.0891861
$365$	0	0
$366$	0	0
$367$	8.70156	0.454218	0.227109	−	0.973869i	$-0.427073\pi$
$367$	8.70156	0.454218	0.227109	+	0.973869i	$0.427073\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.7016	−0.919019
$372$	0	0
$373$	−11.7016	−0.605884	−0.302942	−	0.953009i	$-0.597969\pi$
$373$	−11.7016	−0.605884	−0.302942	+	0.953009i	$0.597969\pi$
$374$	9.70156	0.501656
$375$	0	0
$376$	7.00000	0.360997
$377$	−6.40312	−0.329778
$378$	0	0
$379$	−33.1047	−1.70047	−0.850237	−	0.526400i	$-0.823541\pi$
$379$	−33.1047	−1.70047	−0.850237	+	0.526400i	$0.823541\pi$
$380$	0	0
$381$	0	0
$382$	−12.8062	−0.655225
$383$	0.492189	0.0251497	0.0125749	−	0.999921i	$-0.495997\pi$
$383$	0.492189	0.0251497	0.0125749	+	0.999921i	$0.495997\pi$
$384$	0	0
$385$	0	0
$386$	22.2094	1.13043
$387$	0	0
$388$	15.4031	0.781975
$389$	7.89531	0.400308	0.200154	−	0.979764i	$-0.435856\pi$
$389$	7.89531	0.400308	0.200154	+	0.979764i	$0.435856\pi$
$390$	0	0
$391$	0	0
$392$	−4.10469	−0.207318
$393$	0	0
$394$	7.40312	0.372964
$395$	0	0
$396$	0	0
$397$	−22.9109	−1.14987	−0.574933	−	0.818200i	$-0.694972\pi$
$397$	−22.9109	−1.14987	−0.574933	+	0.818200i	$0.694972\pi$
$398$	14.7016	0.736923
$399$	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
$403$	9.10469	0.453537
$404$	−0.298438	−0.0148478
$405$	0	0
$406$	−10.8953	−0.540725
$407$	−8.00000	−0.396545
$408$	0	0
$409$	16.5969	0.820663	0.410331	−	0.911937i	$-0.365413\pi$
$409$	16.5969	0.820663	0.410331	+	0.911937i	$0.365413\pi$
$410$	0	0
$411$	0	0
$412$	11.4031	0.561792
$413$	6.29844	0.309926
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	8.00000	0.391293
$419$	34.9109	1.70551	0.852755	−	0.522310i	$-0.174930\pi$
$419$	34.9109	1.70551	0.852755	+	0.522310i	$0.174930\pi$
$420$	0	0
$421$	−13.4031	−0.653228	−0.326614	−	0.945158i	$-0.605908\pi$
$421$	−13.4031	−0.653228	−0.326614	+	0.945158i	$0.605908\pi$
$422$	−18.8062	−0.915474
$423$	0	0
$424$	−10.4031	−0.505220
$425$	0	0
$426$	0	0
$427$	−8.68594	−0.420342
$428$	−0.104686	−0.00506021
$429$	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
$433$	−8.10469	−0.389486	−0.194743	−	0.980854i	$-0.562387\pi$
$433$	−8.10469	−0.389486	−0.194743	+	0.980854i	$0.562387\pi$
$434$	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	$-0.630490\pi$
$439$	−16.7016	−0.797122	−0.398561	+	0.917142i	$0.630490\pi$
$440$	0	0
$441$	0	0
$442$	5.70156	0.271196
$443$	0.492189	0.0233846	0.0116923	−	0.999932i	$-0.496278\pi$
$443$	0.492189	0.0233846	0.0116923	+	0.999932i	$0.496278\pi$
$444$	0	0
$445$	0	0
$446$	1.40312	0.0664399
$447$	0	0
$448$	1.70156	0.0803913
$449$	−8.70156	−0.410652	−0.205326	−	0.978694i	$-0.565825\pi$
$449$	−8.70156	−0.410652	−0.205326	+	0.978694i	$0.565825\pi$
$450$	0	0
$451$	−4.59688	−0.216458
$452$	−14.0000	−0.658505
$453$	0	0
$454$	3.10469	0.145710
$455$	0	0
$456$	0	0
$457$	15.4031	0.720528	0.360264	−	0.932850i	$-0.382687\pi$
$457$	15.4031	0.720528	0.360264	+	0.932850i	$0.382687\pi$
$458$	9.29844	0.434487
$459$	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
$463$	27.7016	1.28740	0.643700	−	0.765278i	$-0.277398\pi$
$463$	27.7016	1.28740	0.643700	+	0.765278i	$0.277398\pi$
$464$	−6.40312	−0.297258
$465$	0	0
$466$	18.2094	0.843533
$467$	36.7016	1.69835	0.849173	−	0.528115i	$-0.177101\pi$
$467$	36.7016	1.69835	0.849173	+	0.528115i	$0.177101\pi$
$468$	0	0
$469$	−10.8953	−0.503099
$470$	0	0
$471$	0	0
$472$	3.70156	0.170378
$473$	−2.38750	−0.109778
$474$	0	0
$475$	0	0
$476$	9.70156	0.444670
$477$	0	0
$478$	13.1047	0.599394
$479$	−10.6125	−0.484897	−0.242449	−	0.970164i	$-0.577951\pi$
$479$	−10.6125	−0.484897	−0.242449	+	0.970164i	$0.577951\pi$
$480$	0	0
$481$	−4.70156	−0.214373
$482$	−15.4031	−0.701593
$483$	0	0
$484$	−8.10469	−0.368395
$485$	0	0
$486$	0	0
$487$	27.1047	1.22823	0.614115	−	0.789216i	$-0.289513\pi$
$487$	27.1047	1.22823	0.614115	+	0.789216i	$0.289513\pi$
$488$	−5.10469	−0.231078
$489$	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
$493$	−36.5078	−1.64423
$494$	4.70156	0.211533
$495$	0	0
$496$	9.10469	0.408812
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−21.2094	−0.949462	−0.474731	−	0.880131i	$-0.657455\pi$
$499$	−21.2094	−0.949462	−0.474731	+	0.880131i	$0.657455\pi$
$500$	0	0
$501$	0	0
$502$	−6.70156	−0.299105
$503$	26.2094	1.16862	0.584309	−	0.811531i	$-0.301366\pi$
$503$	26.2094	1.16862	0.584309	+	0.811531i	$0.301366\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	12.7016	0.563541
$509$	−7.40312	−0.328138	−0.164069	−	0.986449i	$-0.552462\pi$
$509$	−7.40312	−0.328138	−0.164069	+	0.986449i	$0.552462\pi$
$510$	0	0
$511$	20.4187	0.903272
$512$	1.00000	0.0441942
$513$	0	0
$514$	−21.7016	−0.957215
$515$	0	0
$516$	0	0
$517$	11.9109	0.523842
$518$	−8.00000	−0.351500
$519$	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
$523$	−0.806248	−0.0352548	−0.0176274	−	0.999845i	$-0.505611\pi$
$523$	−0.806248	−0.0352548	−0.0176274	+	0.999845i	$0.505611\pi$
$524$	9.50781	0.415351
$525$	0	0
$526$	2.00000	0.0872041
$527$	51.9109	2.26128
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	−2.70156	−0.117018
$534$	0	0
$535$	0	0
$536$	−6.40312	−0.276573
$537$	0	0
$538$	7.20937	0.310818
$539$	−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	$0.832281\pi$
$542$	−12.5078	−0.537256
$543$	0	0
$544$	5.70156	0.244452
$545$	0	0
$546$	0	0
$547$	−19.6125	−0.838570	−0.419285	−	0.907855i	$-0.637719\pi$
$547$	−19.6125	−0.838570	−0.419285	+	0.907855i	$0.637719\pi$
$548$	7.29844	0.311774
$549$	0	0
$550$	0	0
$551$	−30.1047	−1.28250
$552$	0	0
$553$	1.19375	0.0507635
$554$	22.0000	0.934690
$555$	0	0
$556$	−3.40312	−0.144325
$557$	−8.80625	−0.373133	−0.186566	−	0.982442i	$-0.559736\pi$
$557$	−8.80625	−0.373133	−0.186566	+	0.982442i	$0.559736\pi$
$558$	0	0
$559$	−1.40312	−0.0593458
$560$	0	0
$561$	0	0
$562$	−22.9109	−0.966439
$563$	−0.492189	−0.0207433	−0.0103717	−	0.999946i	$-0.503301\pi$
$563$	−0.492189	−0.0207433	−0.0103717	+	0.999946i	$0.503301\pi$
$564$	0	0
$565$	0	0
$566$	−8.59688	−0.361354
$567$	0	0
$568$	−4.70156	−0.197273
$569$	17.7016	0.742088	0.371044	−	0.928615i	$-0.379000\pi$
$569$	17.7016	0.742088	0.371044	+	0.928615i	$0.379000\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	1.70156	0.0711459
$573$	0	0
$574$	−4.59688	−0.191870
$575$	0	0
$576$	0	0
$577$	14.2094	0.591544	0.295772	−	0.955259i	$-0.404423\pi$
$577$	14.2094	0.591544	0.295772	+	0.955259i	$0.404423\pi$
$578$	15.5078	0.645040
$579$	0	0
$580$	0	0
$581$	−7.31406	−0.303438
$582$	0	0
$583$	−17.7016	−0.733124
$584$	12.0000	0.496564
$585$	0	0
$586$	8.59688	0.355134
$587$	24.5078	1.01155	0.505773	−	0.862667i	$-0.331207\pi$
$587$	24.5078	1.01155	0.505773	+	0.862667i	$0.331207\pi$
$588$	0	0
$589$	42.8062	1.76380
$590$	0	0
$591$	0	0
$592$	−4.70156	−0.193233
$593$	−26.9109	−1.10510	−0.552550	−	0.833480i	$-0.686345\pi$
$593$	−26.9109	−1.10510	−0.552550	+	0.833480i	$0.686345\pi$
$594$	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	$-0.288283\pi$
$599$	30.2094	1.23432	0.617161	+	0.786837i	$0.288283\pi$
$600$	0	0
$601$	−40.6125	−1.65662	−0.828309	−	0.560271i	$-0.810697\pi$
$601$	−40.6125	−1.65662	−0.828309	+	0.560271i	$0.810697\pi$
$602$	−2.38750	−0.0973074
$603$	0	0
$604$	5.10469	0.207707
$605$	0	0
$606$	0	0
$607$	15.8953	0.645171	0.322585	−	0.946540i	$-0.395448\pi$
$607$	15.8953	0.645171	0.322585	+	0.946540i	$0.395448\pi$
$608$	4.70156	0.190674
$609$	0	0
$610$	0	0
$611$	7.00000	0.283190
$612$	0	0
$613$	−12.8062	−0.517240	−0.258620	−	0.965979i	$-0.583268\pi$
$613$	−12.8062	−0.517240	−0.258620	+	0.965979i	$0.583268\pi$
$614$	−0.701562	−0.0283127
$615$	0	0
$616$	2.89531	0.116656
$617$	−7.89531	−0.317853	−0.158927	−	0.987290i	$-0.550803\pi$
$617$	−7.89531	−0.317853	−0.158927	+	0.987290i	$0.550803\pi$
$618$	0	0
$619$	−38.8062	−1.55975	−0.779877	−	0.625932i	$-0.784719\pi$
$619$	−38.8062	−1.55975	−0.779877	+	0.625932i	$0.784719\pi$
$620$	0	0
$621$	0	0
$622$	−30.0000	−1.20289
$623$	2.38750	0.0956533
$624$	0	0
$625$	0	0
$626$	−17.2094	−0.687825
$627$	0	0
$628$	16.2984	0.650378
$629$	−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	$0.614013\pi$
$632$	0.701562	0.0279066
$633$	0	0
$634$	26.8062	1.06461
$635$	0	0
$636$	0	0
$637$	−4.10469	−0.162634
$638$	−10.8953	−0.431350
$639$	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
$643$	16.7016	0.658645	0.329323	−	0.944217i	$-0.393180\pi$
$643$	16.7016	0.658645	0.329323	+	0.944217i	$0.393180\pi$
$644$	0	0
$645$	0	0
$646$	26.8062	1.05468
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	6.29844	0.247235
$650$	0	0
$651$	0	0
$652$	−6.80625	−0.266553
$653$	−27.7016	−1.08405	−0.542023	−	0.840364i	$-0.682341\pi$
$653$	−27.7016	−1.08405	−0.542023	+	0.840364i	$0.682341\pi$
$654$	0	0
$655$	0	0
$656$	−2.70156	−0.105478
$657$	0	0
$658$	11.9109	0.464337
$659$	−38.3141	−1.49250	−0.746252	−	0.665664i	$-0.768149\pi$
$659$	−38.3141	−1.49250	−0.746252	+	0.665664i	$0.768149\pi$
$660$	0	0
$661$	43.1203	1.67719	0.838593	−	0.544759i	$-0.183379\pi$
$661$	43.1203	1.67719	0.838593	+	0.544759i	$0.183379\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−4.29844	−0.166812
$665$	0	0
$666$	0	0
$667$	0	0
$668$	18.1047	0.700491
$669$	0	0
$670$	0	0
$671$	−8.68594	−0.335317
$672$	0	0
$673$	28.0156	1.07992	0.539961	−	0.841690i	$-0.318439\pi$
$673$	28.0156	1.07992	0.539961	+	0.841690i	$0.318439\pi$
$674$	11.1047	0.427737
$675$	0	0
$676$	1.00000	0.0384615
$677$	−19.6125	−0.753769	−0.376885	−	0.926260i	$-0.623005\pi$
$677$	−19.6125	−0.753769	−0.376885	+	0.926260i	$0.623005\pi$
$678$	0	0
$679$	26.2094	1.00582
$680$	0	0
$681$	0	0
$682$	15.4922	0.593227
$683$	38.2984	1.46545	0.732724	−	0.680525i	$-0.238248\pi$
$683$	38.2984	1.46545	0.732724	+	0.680525i	$0.238248\pi$
$684$	0	0
$685$	0	0
$686$	−18.8953	−0.721426
$687$	0	0
$688$	−1.40312	−0.0534936
$689$	−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	$0.863396\pi$
$692$	−25.8062	−0.981006
$693$	0	0
$694$	−21.5078	−0.816425
$695$	0	0
$696$	0	0
$697$	−15.4031	−0.583435
$698$	1.40312	0.0531090
$699$	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
$703$	−22.1047	−0.833694
$704$	1.70156	0.0641300
$705$	0	0
$706$	−17.5078	−0.658915
$707$	−0.507811	−0.0190982
$708$	0	0
$709$	−35.6125	−1.33746	−0.668728	−	0.743507i	$-0.733161\pi$
$709$	−35.6125	−1.33746	−0.668728	+	0.743507i	$0.733161\pi$
$710$	0	0
$711$	0	0
$712$	1.40312	0.0525843
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−14.2094	−0.531029
$717$	0	0
$718$	−16.6125	−0.619973
$719$	−53.0156	−1.97715	−0.988575	−	0.150733i	$-0.951837\pi$
$719$	−53.0156	−1.97715	−0.988575	+	0.150733i	$0.951837\pi$
$720$	0	0
$721$	19.4031	0.722610
$722$	3.10469	0.115544
$723$	0	0
$724$	17.7016	0.657873
$725$	0	0
$726$	0	0
$727$	−46.2094	−1.71381	−0.856905	−	0.515474i	$-0.827616\pi$
$727$	−46.2094	−1.71381	−0.856905	+	0.515474i	$0.827616\pi$
$728$	1.70156	0.0630641
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−3.08907	−0.114097	−0.0570486	−	0.998371i	$-0.518169\pi$
$733$	−3.08907	−0.114097	−0.0570486	+	0.998371i	$0.518169\pi$
$734$	8.70156	0.321181
$735$	0	0
$736$	0	0
$737$	−10.8953	−0.401334
$738$	0	0
$739$	−46.6125	−1.71467	−0.857334	−	0.514760i	$-0.827881\pi$
$739$	−46.6125	−1.71467	−0.857334	+	0.514760i	$0.827881\pi$
$740$	0	0
$741$	0	0
$742$	−17.7016	−0.649845
$743$	20.6125	0.756199	0.378100	−	0.925765i	$-0.376578\pi$
$743$	20.6125	0.756199	0.378100	+	0.925765i	$0.376578\pi$
$744$	0	0
$745$	0	0
$746$	−11.7016	−0.428425
$747$	0	0
$748$	9.70156	0.354724
$749$	−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	$0.641101\pi$
$752$	7.00000	0.255264
$753$	0	0
$754$	−6.40312	−0.233188
$755$	0	0
$756$	0	0
$757$	37.1047	1.34859	0.674296	−	0.738461i	$-0.264447\pi$
$757$	37.1047	1.34859	0.674296	+	0.738461i	$0.264447\pi$
$758$	−33.1047	−1.20242
$759$	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
$763$	18.2094	0.659224
$764$	−12.8062	−0.463314
$765$	0	0
$766$	0.492189	0.0177835
$767$	3.70156	0.133656
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	0	0
$772$	22.2094	0.799333
$773$	39.4031	1.41723	0.708616	−	0.705594i	$-0.249320\pi$
$773$	39.4031	1.41723	0.708616	+	0.705594i	$0.249320\pi$
$774$	0	0
$775$	0	0
$776$	15.4031	0.552940
$777$	0	0
$778$	7.89531	0.283061
$779$	−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
$787$	−53.1047	−1.89298	−0.946489	−	0.322737i	$-0.895397\pi$
$787$	−53.1047	−1.89298	−0.946489	+	0.322737i	$0.895397\pi$
$788$	7.40312	0.263725
$789$	0	0
$790$	0	0
$791$	−23.8219	−0.847008
$792$	0	0
$793$	−5.10469	−0.181273
$794$	−22.9109	−0.813079
$795$	0	0
$796$	14.7016	0.521083
$797$	−41.9109	−1.48456	−0.742281	−	0.670089i	$-0.766256\pi$
$797$	−41.9109	−1.48456	−0.742281	+	0.670089i	$0.766256\pi$
$798$	0	0
$799$	39.9109	1.41195
$800$	0	0
$801$	0	0
$802$	20.2094	0.713618
$803$	20.4187	0.720562
$804$	0	0
$805$	0	0
$806$	9.10469	0.320699
$807$	0	0
$808$	−0.298438	−0.0104990
$809$	−32.8062	−1.15341	−0.576703	−	0.816954i	$-0.695661\pi$
$809$	−32.8062	−1.15341	−0.576703	+	0.816954i	$0.695661\pi$
$810$	0	0
$811$	42.2984	1.48530	0.742650	−	0.669680i	$-0.233569\pi$
$811$	42.2984	1.48530	0.742650	+	0.669680i	$0.233569\pi$
$812$	−10.8953	−0.382351
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	−6.59688	−0.230795
$818$	16.5969	0.580296
$819$	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
$823$	2.10469	0.0733648	0.0366824	−	0.999327i	$-0.488321\pi$
$823$	2.10469	0.0733648	0.0366824	+	0.999327i	$0.488321\pi$
$824$	11.4031	0.397247
$825$	0	0
$826$	6.29844	0.219151
$827$	14.2984	0.497205	0.248603	−	0.968606i	$-0.420029\pi$
$827$	14.2984	0.497205	0.248603	+	0.968606i	$0.420029\pi$
$828$	0	0
$829$	26.5078	0.920654	0.460327	−	0.887749i	$-0.347732\pi$
$829$	26.5078	0.920654	0.460327	+	0.887749i	$0.347732\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	−23.4031	−0.810870
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	0	0
$838$	34.9109	1.20598
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	−13.4031	−0.461902
$843$	0	0
$844$	−18.8062	−0.647338
$845$	0	0
$846$	0	0
$847$	−13.7906	−0.473852
$848$	−10.4031	−0.357245
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	31.5078	1.07881	0.539403	−	0.842047i	$-0.318650\pi$
$853$	31.5078	1.07881	0.539403	+	0.842047i	$0.318650\pi$
$854$	−8.68594	−0.297227
$855$	0	0
$856$	−0.104686	−0.00357811
$857$	43.6125	1.48977	0.744887	−	0.667190i	$-0.232503\pi$
$857$	43.6125	1.48977	0.744887	+	0.667190i	$0.232503\pi$
$858$	0	0
$859$	32.2094	1.09897	0.549485	−	0.835504i	$-0.314824\pi$
$859$	32.2094	1.09897	0.549485	+	0.835504i	$0.314824\pi$
$860$	0	0
$861$	0	0
$862$	22.3141	0.760020
$863$	−48.2250	−1.64160	−0.820799	−	0.571217i	$-0.806471\pi$
$863$	−48.2250	−1.64160	−0.820799	+	0.571217i	$0.806471\pi$
$864$	0	0
$865$	0	0
$866$	−8.10469	−0.275408
$867$	0	0
$868$	15.4922	0.525839
$869$	1.19375	0.0404952
$870$	0	0
$871$	−6.40312	−0.216962
$872$	10.7016	0.362401
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.7172	−0.935943	−0.467971	−	0.883744i	$-0.655015\pi$
$877$	−27.7172	−0.935943	−0.467971	+	0.883744i	$0.655015\pi$
$878$	−16.7016	−0.563650
$879$	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
$883$	5.61250	0.188876	0.0944378	−	0.995531i	$-0.469895\pi$
$883$	5.61250	0.188876	0.0944378	+	0.995531i	$0.469895\pi$
$884$	5.70156	0.191764
$885$	0	0
$886$	0.492189	0.0165354
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	21.6125	0.724860
$890$	0	0
$891$	0	0
$892$	1.40312	0.0469801
$893$	32.9109	1.10132
$894$	0	0
$895$	0	0
$896$	1.70156	0.0568452
$897$	0	0
$898$	−8.70156	−0.290375
$899$	−58.2984	−1.94436
$900$	0	0
$901$	−59.3141	−1.97604
$902$	−4.59688	−0.153059
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	0	0
$907$	−39.0156	−1.29549	−0.647746	−	0.761856i	$-0.724288\pi$
$907$	−39.0156	−1.29549	−0.647746	+	0.761856i	$0.724288\pi$
$908$	3.10469	0.103033
$909$	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
$913$	−7.31406	−0.242060
$914$	15.4031	0.509490
$915$	0	0
$916$	9.29844	0.307229
$917$	16.1781	0.534249
$918$	0	0
$919$	−24.1047	−0.795140	−0.397570	−	0.917572i	$-0.630146\pi$
$919$	−24.1047	−0.795140	−0.397570	+	0.917572i	$0.630146\pi$
$920$	0	0
$921$	0	0
$922$	2.20937	0.0727618
$923$	−4.70156	−0.154754
$924$	0	0
$925$	0	0
$926$	27.7016	0.910330
$927$	0	0
$928$	−6.40312	−0.210193
$929$	−58.7016	−1.92594	−0.962968	−	0.269616i	$-0.913103\pi$
$929$	−58.7016	−1.92594	−0.962968	+	0.269616i	$0.913103\pi$
$930$	0	0
$931$	−19.2984	−0.632481
$932$	18.2094	0.596468
$933$	0	0
$934$	36.7016	1.20091
$935$	0	0
$936$	0	0
$937$	29.9109	0.977148	0.488574	−	0.872523i	$-0.337517\pi$
$937$	29.9109	0.977148	0.488574	+	0.872523i	$0.337517\pi$
$938$	−10.8953	−0.355745
$939$	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
$947$	15.4922	0.503429	0.251714	−	0.967802i	$-0.419006\pi$
$947$	15.4922	0.503429	0.251714	+	0.967802i	$0.419006\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	9.70156	0.314429
$953$	−37.7016	−1.22127	−0.610637	−	0.791911i	$-0.709086\pi$
$953$	−37.7016	−1.22127	−0.610637	+	0.791911i	$0.709086\pi$
$954$	0	0
$955$	0	0
$956$	13.1047	0.423836
$957$	0	0
$958$	−10.6125	−0.342874
$959$	12.4187	0.401022
$960$	0	0
$961$	51.8953	1.67404
$962$	−4.70156	−0.151584
$963$	0	0
$964$	−15.4031	−0.496101
$965$	0	0
$966$	0	0
$967$	−49.1047	−1.57910	−0.789550	−	0.613686i	$-0.789686\pi$
$967$	−49.1047	−1.57910	−0.789550	+	0.613686i	$0.789686\pi$
$968$	−8.10469	−0.260494
$969$	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
$973$	−5.79063	−0.185639
$974$	27.1047	0.868490
$975$	0	0
$976$	−5.10469	−0.163397
$977$	−10.5969	−0.339024	−0.169512	−	0.985528i	$-0.554219\pi$
$977$	−10.5969	−0.339024	−0.169512	+	0.985528i	$0.554219\pi$
$978$	0	0
$979$	2.38750	0.0763049
$980$	0	0
$981$	0	0
$982$	−20.5969	−0.657273
$983$	24.5078	0.781678	0.390839	−	0.920459i	$-0.372185\pi$
$983$	24.5078	0.781678	0.390839	+	0.920459i	$0.372185\pi$
$984$	0	0
$985$	0	0
$986$	−36.5078	−1.16265
$987$	0	0
$988$	4.70156	0.149577
$989$	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	$0.448692\pi$
$992$	9.10469	0.289074
$993$	0	0
$994$	−8.00000	−0.253745
$995$	0	0
$996$	0	0
$997$	42.5078	1.34624	0.673118	−	0.739535i	$-0.264955\pi$
$997$	42.5078	1.34624	0.673118	+	0.739535i	$0.264955\pi$
$998$	−21.2094	−0.671371
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.bc.1.2	✓	2	3.2	odd	2
1950.2.a.bg.1.1	yes	2	15.14	odd	2
1950.2.e.p.1249.2		4	15.2	even	4
1950.2.e.p.1249.3		4	15.8	even	4
5850.2.a.cg.1.1		2	5.4	even	2
5850.2.a.cj.1.2		2	1.1	even	1	trivial
5850.2.e.bi.5149.1		4	5.3	odd	4
5850.2.e.bi.5149.4		4	5.2	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.70156 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.70156 q^{7} +1.00000 q^{8} +1.70156 q^{11} +1.00000 q^{13} +1.70156 q^{14} +1.00000 q^{16} +5.70156 q^{17} +4.70156 q^{19} +1.70156 q^{22} +1.00000 q^{26} +1.70156 q^{28} -6.40312 q^{29} +9.10469 q^{31} +1.00000 q^{32} +5.70156 q^{34} -4.70156 q^{37} +4.70156 q^{38} -2.70156 q^{41} -1.40312 q^{43} +1.70156 q^{44} +7.00000 q^{47} -4.10469 q^{49} +1.00000 q^{52} -10.4031 q^{53} +1.70156 q^{56} -6.40312 q^{58} +3.70156 q^{59} -5.10469 q^{61} +9.10469 q^{62} +1.00000 q^{64} -6.40312 q^{67} +5.70156 q^{68} -4.70156 q^{71} +12.0000 q^{73} -4.70156 q^{74} +4.70156 q^{76} +2.89531 q^{77} +0.701562 q^{79} -2.70156 q^{82} -4.29844 q^{83} -1.40312 q^{86} +1.70156 q^{88} +1.40312 q^{89} +1.70156 q^{91} +7.00000 q^{94} +15.4031 q^{97} -4.10469 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 3 q^{7} + 2 q^{8} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} + 3 q^{19} - 3 q^{22} + 2 q^{26} - 3 q^{28} - q^{31} + 2 q^{32} + 5 q^{34} - 3 q^{37} + 3 q^{38} + q^{41} + 10 q^{43} - 3 q^{44} + 14 q^{47} + 11 q^{49} + 2 q^{52} - 8 q^{53} - 3 q^{56} + q^{59} + 9 q^{61} - q^{62} + 2 q^{64} + 5 q^{68} - 3 q^{71} + 24 q^{73} - 3 q^{74} + 3 q^{76} + 25 q^{77} - 5 q^{79} + q^{82} - 15 q^{83} + 10 q^{86} - 3 q^{88} - 10 q^{89} - 3 q^{91} + 14 q^{94} + 18 q^{97} + 11 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^7 + 2 * q^8 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 + 3 * q^19 - 3 * q^22 + 2 * q^26 - 3 * q^28 - q^31 + 2 * q^32 + 5 * q^34 - 3 * q^37 + 3 * q^38 + q^41 + 10 * q^43 - 3 * q^44 + 14 * q^47 + 11 * q^49 + 2 * q^52 - 8 * q^53 - 3 * q^56 + q^59 + 9 * q^61 - q^62 + 2 * q^64 + 5 * q^68 - 3 * q^71 + 24 * q^73 - 3 * q^74 + 3 * q^76 + 25 * q^77 - 5 * q^79 + q^82 - 15 * q^83 + 10 * q^86 - 3 * q^88 - 10 * q^89 - 3 * q^91 + 14 * q^94 + 18 * q^97 + 11 * q^98

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