LMFDB - Embedded newform 8550.2.a.cq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} -2.62222 q^{11} -5.80642 q^{13} +4.42864 q^{14} +1.00000 q^{16} -3.80642 q^{17} +1.00000 q^{19} -2.62222 q^{22} -2.62222 q^{23} -5.80642 q^{26} +4.42864 q^{28} -3.37778 q^{29} -4.42864 q^{31} +1.00000 q^{32} -3.80642 q^{34} -5.80642 q^{37} +1.00000 q^{38} -5.67307 q^{41} +10.9906 q^{43} -2.62222 q^{44} -2.62222 q^{46} +2.62222 q^{47} +12.6128 q^{49} -5.80642 q^{52} -6.00000 q^{53} +4.42864 q^{56} -3.37778 q^{58} +1.05086 q^{59} +4.75557 q^{61} -4.42864 q^{62} +1.00000 q^{64} -15.6128 q^{67} -3.80642 q^{68} -15.6128 q^{71} -11.6128 q^{73} -5.80642 q^{74} +1.00000 q^{76} -11.6128 q^{77} +4.42864 q^{79} -5.67307 q^{82} +11.9081 q^{83} +10.9906 q^{86} -2.62222 q^{88} -12.4286 q^{89} -25.7146 q^{91} -2.62222 q^{92} +2.62222 q^{94} -7.37778 q^{97} +12.6128 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} - 8 q^{11} - 4 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{22} - 8 q^{23} - 4 q^{26} - 10 q^{29} + 3 q^{32} + 2 q^{34} - 4 q^{37} + 3 q^{38} - 4 q^{41} + 6 q^{43} - 8 q^{44} - 8 q^{46} + 8 q^{47} + 11 q^{49} - 4 q^{52} - 18 q^{53} - 10 q^{58} - 10 q^{59} + 14 q^{61} + 3 q^{64} - 20 q^{67} + 2 q^{68} - 20 q^{71} - 8 q^{73} - 4 q^{74} + 3 q^{76} - 8 q^{77} - 4 q^{82} - 4 q^{83} + 6 q^{86} - 8 q^{88} - 24 q^{89} - 24 q^{91} - 8 q^{92} + 8 q^{94} - 22 q^{97} + 11 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 - 8 * q^11 - 4 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^19 - 8 * q^22 - 8 * q^23 - 4 * q^26 - 10 * q^29 + 3 * q^32 + 2 * q^34 - 4 * q^37 + 3 * q^38 - 4 * q^41 + 6 * q^43 - 8 * q^44 - 8 * q^46 + 8 * q^47 + 11 * q^49 - 4 * q^52 - 18 * q^53 - 10 * q^58 - 10 * q^59 + 14 * q^61 + 3 * q^64 - 20 * q^67 + 2 * q^68 - 20 * q^71 - 8 * q^73 - 4 * q^74 + 3 * q^76 - 8 * q^77 - 4 * q^82 - 4 * q^83 + 6 * q^86 - 8 * q^88 - 24 * q^89 - 24 * q^91 - 8 * q^92 + 8 * q^94 - 22 * 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
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−2.62222	−0.790628	−0.395314	−	0.918546i	$-0.629364\pi$
$11$	−2.62222	−0.790628	−0.395314	+	0.918546i	$0.629364\pi$
$12$	0	0
$13$	−5.80642	−1.61041	−0.805206	−	0.592995i	$-0.797945\pi$
$13$	−5.80642	−1.61041	−0.805206	+	0.592995i	$0.797945\pi$
$14$	4.42864	1.18360
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.80642	−0.923193	−0.461597	−	0.887090i	$-0.652723\pi$
$17$	−3.80642	−0.923193	−0.461597	+	0.887090i	$0.652723\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.62222	−0.559058
$23$	−2.62222	−0.546770	−0.273385	−	0.961905i	$-0.588143\pi$
$23$	−2.62222	−0.546770	−0.273385	+	0.961905i	$0.588143\pi$
$24$	0	0
$25$	0	0
$26$	−5.80642	−1.13873
$27$	0	0
$28$	4.42864	0.836934
$29$	−3.37778	−0.627239	−0.313619	−	0.949549i	$-0.601542\pi$
$29$	−3.37778	−0.627239	−0.313619	+	0.949549i	$0.601542\pi$
$30$	0	0
$31$	−4.42864	−0.795407	−0.397704	−	0.917514i	$-0.630193\pi$
$31$	−4.42864	−0.795407	−0.397704	+	0.917514i	$0.630193\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.80642	−0.652796
$35$	0	0
$36$	0	0
$37$	−5.80642	−0.954570	−0.477285	−	0.878749i	$-0.658379\pi$
$37$	−5.80642	−0.954570	−0.477285	+	0.878749i	$0.658379\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−5.67307	−0.885985	−0.442992	−	0.896525i	$-0.646083\pi$
$41$	−5.67307	−0.885985	−0.442992	+	0.896525i	$0.646083\pi$
$42$	0	0
$43$	10.9906	1.67606	0.838028	−	0.545627i	$-0.183709\pi$
$43$	10.9906	1.67606	0.838028	+	0.545627i	$0.183709\pi$
$44$	−2.62222	−0.395314
$45$	0	0
$46$	−2.62222	−0.386625
$47$	2.62222	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$47$	2.62222	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	−5.80642	−0.805206
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	4.42864	0.591802
$57$	0	0
$58$	−3.37778	−0.443525
$59$	1.05086	0.136810	0.0684048	−	0.997658i	$-0.478209\pi$
$59$	1.05086	0.136810	0.0684048	+	0.997658i	$0.478209\pi$
$60$	0	0
$61$	4.75557	0.608888	0.304444	−	0.952530i	$-0.401529\pi$
$61$	4.75557	0.608888	0.304444	+	0.952530i	$0.401529\pi$
$62$	−4.42864	−0.562438
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−15.6128	−1.90741	−0.953706	−	0.300739i	$-0.902767\pi$
$67$	−15.6128	−1.90741	−0.953706	+	0.300739i	$0.902767\pi$
$68$	−3.80642	−0.461597
$69$	0	0
$70$	0	0
$71$	−15.6128	−1.85290	−0.926452	−	0.376413i	$-0.877157\pi$
$71$	−15.6128	−1.85290	−0.926452	+	0.376413i	$0.877157\pi$
$72$	0	0
$73$	−11.6128	−1.35918	−0.679591	−	0.733592i	$-0.737843\pi$
$73$	−11.6128	−1.35918	−0.679591	+	0.733592i	$0.737843\pi$
$74$	−5.80642	−0.674983
$75$	0	0
$76$	1.00000	0.114708
$77$	−11.6128	−1.32341
$78$	0	0
$79$	4.42864	0.498261	0.249130	−	0.968470i	$-0.419855\pi$
$79$	4.42864	0.498261	0.249130	+	0.968470i	$0.419855\pi$
$80$	0	0
$81$	0	0
$82$	−5.67307	−0.626486
$83$	11.9081	1.30709	0.653544	−	0.756889i	$-0.273282\pi$
$83$	11.9081	1.30709	0.653544	+	0.756889i	$0.273282\pi$
$84$	0	0
$85$	0	0
$86$	10.9906	1.18515
$87$	0	0
$88$	−2.62222	−0.279529
$89$	−12.4286	−1.31743	−0.658717	−	0.752391i	$-0.728900\pi$
$89$	−12.4286	−1.31743	−0.658717	+	0.752391i	$0.728900\pi$
$90$	0	0
$91$	−25.7146	−2.69562
$92$	−2.62222	−0.273385
$93$	0	0
$94$	2.62222	0.270461
$95$	0	0
$96$	0	0
$97$	−7.37778	−0.749101	−0.374550	−	0.927207i	$-0.622203\pi$
$97$	−7.37778	−0.749101	−0.374550	+	0.927207i	$0.622203\pi$
$98$	12.6128	1.27409
$99$	0	0
$100$	0	0
$101$	−17.6731	−1.75854	−0.879268	−	0.476327i	$-0.841968\pi$
$101$	−17.6731	−1.75854	−0.879268	+	0.476327i	$0.841968\pi$
$102$	0	0
$103$	1.18421	0.116684	0.0583418	−	0.998297i	$-0.481419\pi$
$103$	1.18421	0.116684	0.0583418	+	0.998297i	$0.481419\pi$
$104$	−5.80642	−0.569367
$105$	0	0
$106$	−6.00000	−0.582772
$107$	4.85728	0.469571	0.234785	−	0.972047i	$-0.424561\pi$
$107$	4.85728	0.469571	0.234785	+	0.972047i	$0.424561\pi$
$108$	0	0
$109$	−6.04149	−0.578670	−0.289335	−	0.957228i	$-0.593434\pi$
$109$	−6.04149	−0.578670	−0.289335	+	0.957228i	$0.593434\pi$
$110$	0	0
$111$	0	0
$112$	4.42864	0.418467
$113$	9.34614	0.879211	0.439606	−	0.898191i	$-0.355118\pi$
$113$	9.34614	0.879211	0.439606	+	0.898191i	$0.355118\pi$
$114$	0	0
$115$	0	0
$116$	−3.37778	−0.313619
$117$	0	0
$118$	1.05086	0.0967391
$119$	−16.8573	−1.54530
$120$	0	0
$121$	−4.12399	−0.374908
$122$	4.75557	0.430549
$123$	0	0
$124$	−4.42864	−0.397704
$125$	0	0
$126$	0	0
$127$	−6.42864	−0.570450	−0.285225	−	0.958461i	$-0.592068\pi$
$127$	−6.42864	−0.570450	−0.285225	+	0.958461i	$0.592068\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	15.0923	1.31862	0.659312	−	0.751869i	$-0.270848\pi$
$131$	15.0923	1.31862	0.659312	+	0.751869i	$0.270848\pi$
$132$	0	0
$133$	4.42864	0.384012
$134$	−15.6128	−1.34874
$135$	0	0
$136$	−3.80642	−0.326398
$137$	−7.53972	−0.644162	−0.322081	−	0.946712i	$-0.604382\pi$
$137$	−7.53972	−0.644162	−0.322081	+	0.946712i	$0.604382\pi$
$138$	0	0
$139$	0.387152	0.0328378	0.0164189	−	0.999865i	$-0.494773\pi$
$139$	0.387152	0.0328378	0.0164189	+	0.999865i	$0.494773\pi$
$140$	0	0
$141$	0	0
$142$	−15.6128	−1.31020
$143$	15.2257	1.27324
$144$	0	0
$145$	0	0
$146$	−11.6128	−0.961086
$147$	0	0
$148$	−5.80642	−0.477285
$149$	14.5303	1.19037	0.595186	−	0.803588i	$-0.297078\pi$
$149$	14.5303	1.19037	0.595186	+	0.803588i	$0.297078\pi$
$150$	0	0
$151$	4.69535	0.382102	0.191051	−	0.981580i	$-0.438810\pi$
$151$	4.69535	0.382102	0.191051	+	0.981580i	$0.438810\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−11.6128	−0.935790
$155$	0	0
$156$	0	0
$157$	21.5210	1.71756	0.858781	−	0.512343i	$-0.171222\pi$
$157$	21.5210	1.71756	0.858781	+	0.512343i	$0.171222\pi$
$158$	4.42864	0.352324
$159$	0	0
$160$	0	0
$161$	−11.6128	−0.915221
$162$	0	0
$163$	8.23506	0.645020	0.322510	−	0.946566i	$-0.395473\pi$
$163$	8.23506	0.645020	0.322510	+	0.946566i	$0.395473\pi$
$164$	−5.67307	−0.442992
$165$	0	0
$166$	11.9081	0.924250
$167$	8.47013	0.655438	0.327719	−	0.944775i	$-0.393720\pi$
$167$	8.47013	0.655438	0.327719	+	0.944775i	$0.393720\pi$
$168$	0	0
$169$	20.7146	1.59343
$170$	0	0
$171$	0	0
$172$	10.9906	0.838028
$173$	−22.4701	−1.70837	−0.854186	−	0.519967i	$-0.825944\pi$
$173$	−22.4701	−1.70837	−0.854186	+	0.519967i	$0.825944\pi$
$174$	0	0
$175$	0	0
$176$	−2.62222	−0.197657
$177$	0	0
$178$	−12.4286	−0.931566
$179$	2.94914	0.220429	0.110215	−	0.993908i	$-0.464846\pi$
$179$	2.94914	0.220429	0.110215	+	0.993908i	$0.464846\pi$
$180$	0	0
$181$	22.8988	1.70205	0.851026	−	0.525124i	$-0.175981\pi$
$181$	22.8988	1.70205	0.851026	+	0.525124i	$0.175981\pi$
$182$	−25.7146	−1.90609
$183$	0	0
$184$	−2.62222	−0.193312
$185$	0	0
$186$	0	0
$187$	9.98126	0.729902
$188$	2.62222	0.191245
$189$	0	0
$190$	0	0
$191$	9.05086	0.654897	0.327448	−	0.944869i	$-0.393811\pi$
$191$	9.05086	0.654897	0.327448	+	0.944869i	$0.393811\pi$
$192$	0	0
$193$	−18.7239	−1.34778	−0.673889	−	0.738833i	$-0.735377\pi$
$193$	−18.7239	−1.34778	−0.673889	+	0.738833i	$0.735377\pi$
$194$	−7.37778	−0.529694
$195$	0	0
$196$	12.6128	0.900918
$197$	0.888922	0.0633331	0.0316665	−	0.999498i	$-0.489919\pi$
$197$	0.888922	0.0633331	0.0316665	+	0.999498i	$0.489919\pi$
$198$	0	0
$199$	21.7146	1.53930	0.769652	−	0.638464i	$-0.220430\pi$
$199$	21.7146	1.53930	0.769652	+	0.638464i	$0.220430\pi$
$200$	0	0
$201$	0	0
$202$	−17.6731	−1.24347
$203$	−14.9590	−1.04992
$204$	0	0
$205$	0	0
$206$	1.18421	0.0825077
$207$	0	0
$208$	−5.80642	−0.402603
$209$	−2.62222	−0.181382
$210$	0	0
$211$	−3.61285	−0.248719	−0.124359	−	0.992237i	$-0.539688\pi$
$211$	−3.61285	−0.248719	−0.124359	+	0.992237i	$0.539688\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	4.85728	0.332037
$215$	0	0
$216$	0	0
$217$	−19.6128	−1.33141
$218$	−6.04149	−0.409181
$219$	0	0
$220$	0	0
$221$	22.1017	1.48672
$222$	0	0
$223$	21.3876	1.43222	0.716111	−	0.697987i	$-0.245921\pi$
$223$	21.3876	1.43222	0.716111	+	0.697987i	$0.245921\pi$
$224$	4.42864	0.295901
$225$	0	0
$226$	9.34614	0.621696
$227$	8.47013	0.562182	0.281091	−	0.959681i	$-0.409304\pi$
$227$	8.47013	0.562182	0.281091	+	0.959681i	$0.409304\pi$
$228$	0	0
$229$	−16.9590	−1.12068	−0.560341	−	0.828262i	$-0.689330\pi$
$229$	−16.9590	−1.12068	−0.560341	+	0.828262i	$0.689330\pi$
$230$	0	0
$231$	0	0
$232$	−3.37778	−0.221762
$233$	−17.9081	−1.17320	−0.586600	−	0.809876i	$-0.699534\pi$
$233$	−17.9081	−1.17320	−0.586600	+	0.809876i	$0.699534\pi$
$234$	0	0
$235$	0	0
$236$	1.05086	0.0684048
$237$	0	0
$238$	−16.8573	−1.09270
$239$	28.2766	1.82906	0.914529	−	0.404520i	$-0.132562\pi$
$239$	28.2766	1.82906	0.914529	+	0.404520i	$0.132562\pi$
$240$	0	0
$241$	−11.5111	−0.741498	−0.370749	−	0.928733i	$-0.620899\pi$
$241$	−11.5111	−0.741498	−0.370749	+	0.928733i	$0.620899\pi$
$242$	−4.12399	−0.265100
$243$	0	0
$244$	4.75557	0.304444
$245$	0	0
$246$	0	0
$247$	−5.80642	−0.369454
$248$	−4.42864	−0.281219
$249$	0	0
$250$	0	0
$251$	7.74620	0.488936	0.244468	−	0.969657i	$-0.421387\pi$
$251$	7.74620	0.488936	0.244468	+	0.969657i	$0.421387\pi$
$252$	0	0
$253$	6.87601	0.432291
$254$	−6.42864	−0.403369
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.8796	−0.865783	−0.432891	−	0.901446i	$-0.642507\pi$
$257$	−13.8796	−0.865783	−0.432891	+	0.901446i	$0.642507\pi$
$258$	0	0
$259$	−25.7146	−1.59782
$260$	0	0
$261$	0	0
$262$	15.0923	0.932408
$263$	−18.6222	−1.14830	−0.574148	−	0.818752i	$-0.694666\pi$
$263$	−18.6222	−1.14830	−0.574148	+	0.818752i	$0.694666\pi$
$264$	0	0
$265$	0	0
$266$	4.42864	0.271537
$267$	0	0
$268$	−15.6128	−0.953706
$269$	−15.8479	−0.966264	−0.483132	−	0.875547i	$-0.660501\pi$
$269$	−15.8479	−0.966264	−0.483132	+	0.875547i	$0.660501\pi$
$270$	0	0
$271$	1.51114	0.0917951	0.0458975	−	0.998946i	$-0.485385\pi$
$271$	1.51114	0.0917951	0.0458975	+	0.998946i	$0.485385\pi$
$272$	−3.80642	−0.230798
$273$	0	0
$274$	−7.53972	−0.455491
$275$	0	0
$276$	0	0
$277$	−1.05086	−0.0631398	−0.0315699	−	0.999502i	$-0.510051\pi$
$277$	−1.05086	−0.0631398	−0.0315699	+	0.999502i	$0.510051\pi$
$278$	0.387152	0.0232199
$279$	0	0
$280$	0	0
$281$	−3.45091	−0.205864	−0.102932	−	0.994688i	$-0.532822\pi$
$281$	−3.45091	−0.205864	−0.102932	+	0.994688i	$0.532822\pi$
$282$	0	0
$283$	−15.5812	−0.926206	−0.463103	−	0.886304i	$-0.653264\pi$
$283$	−15.5812	−0.926206	−0.463103	+	0.886304i	$0.653264\pi$
$284$	−15.6128	−0.926452
$285$	0	0
$286$	15.2257	0.900314
$287$	−25.1240	−1.48302
$288$	0	0
$289$	−2.51114	−0.147714
$290$	0	0
$291$	0	0
$292$	−11.6128	−0.679591
$293$	7.12399	0.416188	0.208094	−	0.978109i	$-0.433274\pi$
$293$	7.12399	0.416188	0.208094	+	0.978109i	$0.433274\pi$
$294$	0	0
$295$	0	0
$296$	−5.80642	−0.337492
$297$	0	0
$298$	14.5303	0.841721
$299$	15.2257	0.880525
$300$	0	0
$301$	48.6735	2.80550
$302$	4.69535	0.270187
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−1.12399	−0.0641492	−0.0320746	−	0.999485i	$-0.510211\pi$
$307$	−1.12399	−0.0641492	−0.0320746	+	0.999485i	$0.510211\pi$
$308$	−11.6128	−0.661703
$309$	0	0
$310$	0	0
$311$	−14.9491	−0.847688	−0.423844	−	0.905735i	$-0.639320\pi$
$311$	−14.9491	−0.847688	−0.423844	+	0.905735i	$0.639320\pi$
$312$	0	0
$313$	−3.14272	−0.177637	−0.0888185	−	0.996048i	$-0.528309\pi$
$313$	−3.14272	−0.177637	−0.0888185	+	0.996048i	$0.528309\pi$
$314$	21.5210	1.21450
$315$	0	0
$316$	4.42864	0.249130
$317$	10.5906	0.594826	0.297413	−	0.954749i	$-0.403876\pi$
$317$	10.5906	0.594826	0.297413	+	0.954749i	$0.403876\pi$
$318$	0	0
$319$	8.85728	0.495912
$320$	0	0
$321$	0	0
$322$	−11.6128	−0.647159
$323$	−3.80642	−0.211795
$324$	0	0
$325$	0	0
$326$	8.23506	0.456098
$327$	0	0
$328$	−5.67307	−0.313243
$329$	11.6128	0.640237
$330$	0	0
$331$	−15.1427	−0.832319	−0.416160	−	0.909292i	$-0.636624\pi$
$331$	−15.1427	−0.832319	−0.416160	+	0.909292i	$0.636624\pi$
$332$	11.9081	0.653544
$333$	0	0
$334$	8.47013	0.463465
$335$	0	0
$336$	0	0
$337$	−11.1111	−0.605259	−0.302629	−	0.953108i	$-0.597864\pi$
$337$	−11.1111	−0.605259	−0.302629	+	0.953108i	$0.597864\pi$
$338$	20.7146	1.12672
$339$	0	0
$340$	0	0
$341$	11.6128	0.628871
$342$	0	0
$343$	24.8573	1.34217
$344$	10.9906	0.592575
$345$	0	0
$346$	−22.4701	−1.20800
$347$	−14.1936	−0.761951	−0.380976	−	0.924585i	$-0.624412\pi$
$347$	−14.1936	−0.761951	−0.380976	+	0.924585i	$0.624412\pi$
$348$	0	0
$349$	32.3684	1.73264	0.866321	−	0.499488i	$-0.166479\pi$
$349$	32.3684	1.73264	0.866321	+	0.499488i	$0.166479\pi$
$350$	0	0
$351$	0	0
$352$	−2.62222	−0.139765
$353$	−11.8064	−0.628393	−0.314196	−	0.949358i	$-0.601735\pi$
$353$	−11.8064	−0.628393	−0.314196	+	0.949358i	$0.601735\pi$
$354$	0	0
$355$	0	0
$356$	−12.4286	−0.658717
$357$	0	0
$358$	2.94914	0.155867
$359$	10.8287	0.571517	0.285758	−	0.958302i	$-0.407755\pi$
$359$	10.8287	0.571517	0.285758	+	0.958302i	$0.407755\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.8988	1.20353
$363$	0	0
$364$	−25.7146	−1.34781
$365$	0	0
$366$	0	0
$367$	1.46965	0.0767151	0.0383576	−	0.999264i	$-0.487787\pi$
$367$	1.46965	0.0767151	0.0383576	+	0.999264i	$0.487787\pi$
$368$	−2.62222	−0.136692
$369$	0	0
$370$	0	0
$371$	−26.5718	−1.37954
$372$	0	0
$373$	−24.3783	−1.26226	−0.631129	−	0.775678i	$-0.717408\pi$
$373$	−24.3783	−1.26226	−0.631129	+	0.775678i	$0.717408\pi$
$374$	9.98126	0.516119
$375$	0	0
$376$	2.62222	0.135230
$377$	19.6128	1.01011
$378$	0	0
$379$	18.9590	0.973858	0.486929	−	0.873442i	$-0.338117\pi$
$379$	18.9590	0.973858	0.486929	+	0.873442i	$0.338117\pi$
$380$	0	0
$381$	0	0
$382$	9.05086	0.463082
$383$	−26.1017	−1.33374	−0.666868	−	0.745176i	$-0.732365\pi$
$383$	−26.1017	−1.33374	−0.666868	+	0.745176i	$0.732365\pi$
$384$	0	0
$385$	0	0
$386$	−18.7239	−0.953023
$387$	0	0
$388$	−7.37778	−0.374550
$389$	3.57136	0.181075	0.0905376	−	0.995893i	$-0.471141\pi$
$389$	3.57136	0.181075	0.0905376	+	0.995893i	$0.471141\pi$
$390$	0	0
$391$	9.98126	0.504774
$392$	12.6128	0.637045
$393$	0	0
$394$	0.888922	0.0447832
$395$	0	0
$396$	0	0
$397$	−31.4193	−1.57689	−0.788444	−	0.615107i	$-0.789113\pi$
$397$	−31.4193	−1.57689	−0.788444	+	0.615107i	$0.789113\pi$
$398$	21.7146	1.08845
$399$	0	0
$400$	0	0
$401$	−12.0415	−0.601323	−0.300662	−	0.953731i	$-0.597207\pi$
$401$	−12.0415	−0.601323	−0.300662	+	0.953731i	$0.597207\pi$
$402$	0	0
$403$	25.7146	1.28093
$404$	−17.6731	−0.879268
$405$	0	0
$406$	−14.9590	−0.742402
$407$	15.2257	0.754710
$408$	0	0
$409$	26.4701	1.30886	0.654432	−	0.756121i	$-0.272908\pi$
$409$	26.4701	1.30886	0.654432	+	0.756121i	$0.272908\pi$
$410$	0	0
$411$	0	0
$412$	1.18421	0.0583418
$413$	4.65386	0.229001
$414$	0	0
$415$	0	0
$416$	−5.80642	−0.284683
$417$	0	0
$418$	−2.62222	−0.128257
$419$	25.9684	1.26864	0.634319	−	0.773072i	$-0.281281\pi$
$419$	25.9684	1.26864	0.634319	+	0.773072i	$0.281281\pi$
$420$	0	0
$421$	−29.2672	−1.42640	−0.713198	−	0.700963i	$-0.752754\pi$
$421$	−29.2672	−1.42640	−0.713198	+	0.700963i	$0.752754\pi$
$422$	−3.61285	−0.175871
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	21.0607	1.01920
$428$	4.85728	0.234785
$429$	0	0
$430$	0	0
$431$	−14.8385	−0.714747	−0.357374	−	0.933961i	$-0.616328\pi$
$431$	−14.8385	−0.714747	−0.357374	+	0.933961i	$0.616328\pi$
$432$	0	0
$433$	5.27607	0.253552	0.126776	−	0.991931i	$-0.459537\pi$
$433$	5.27607	0.253552	0.126776	+	0.991931i	$0.459537\pi$
$434$	−19.6128	−0.941447
$435$	0	0
$436$	−6.04149	−0.289335
$437$	−2.62222	−0.125438
$438$	0	0
$439$	31.8578	1.52049	0.760244	−	0.649638i	$-0.225079\pi$
$439$	31.8578	1.52049	0.760244	+	0.649638i	$0.225079\pi$
$440$	0	0
$441$	0	0
$442$	22.1017	1.05127
$443$	−4.94914	−0.235141	−0.117570	−	0.993065i	$-0.537511\pi$
$443$	−4.94914	−0.235141	−0.117570	+	0.993065i	$0.537511\pi$
$444$	0	0
$445$	0	0
$446$	21.3876	1.01273
$447$	0	0
$448$	4.42864	0.209234
$449$	−9.75605	−0.460416	−0.230208	−	0.973141i	$-0.573941\pi$
$449$	−9.75605	−0.460416	−0.230208	+	0.973141i	$0.573941\pi$
$450$	0	0
$451$	14.8760	0.700484
$452$	9.34614	0.439606
$453$	0	0
$454$	8.47013	0.397523
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−16.9590	−0.792442
$459$	0	0
$460$	0	0
$461$	36.1245	1.68248	0.841242	−	0.540659i	$-0.181825\pi$
$461$	36.1245	1.68248	0.841242	+	0.540659i	$0.181825\pi$
$462$	0	0
$463$	−35.1209	−1.63221	−0.816104	−	0.577905i	$-0.803870\pi$
$463$	−35.1209	−1.63221	−0.816104	+	0.577905i	$0.803870\pi$
$464$	−3.37778	−0.156810
$465$	0	0
$466$	−17.9081	−0.829578
$467$	−4.56199	−0.211104	−0.105552	−	0.994414i	$-0.533661\pi$
$467$	−4.56199	−0.211104	−0.105552	+	0.994414i	$0.533661\pi$
$468$	0	0
$469$	−69.1437	−3.19276
$470$	0	0
$471$	0	0
$472$	1.05086	0.0483695
$473$	−28.8198	−1.32514
$474$	0	0
$475$	0	0
$476$	−16.8573	−0.772652
$477$	0	0
$478$	28.2766	1.29334
$479$	−37.4005	−1.70887	−0.854437	−	0.519555i	$-0.826098\pi$
$479$	−37.4005	−1.70887	−0.854437	+	0.519555i	$0.826098\pi$
$480$	0	0
$481$	33.7146	1.53725
$482$	−11.5111	−0.524318
$483$	0	0
$484$	−4.12399	−0.187454
$485$	0	0
$486$	0	0
$487$	20.9175	0.947862	0.473931	−	0.880562i	$-0.342835\pi$
$487$	20.9175	0.947862	0.473931	+	0.880562i	$0.342835\pi$
$488$	4.75557	0.215274
$489$	0	0
$490$	0	0
$491$	15.8666	0.716052	0.358026	−	0.933712i	$-0.383450\pi$
$491$	15.8666	0.716052	0.358026	+	0.933712i	$0.383450\pi$
$492$	0	0
$493$	12.8573	0.579063
$494$	−5.80642	−0.261243
$495$	0	0
$496$	−4.42864	−0.198852
$497$	−69.1437	−3.10152
$498$	0	0
$499$	16.7368	0.749244	0.374622	−	0.927178i	$-0.377773\pi$
$499$	16.7368	0.749244	0.374622	+	0.927178i	$0.377773\pi$
$500$	0	0
$501$	0	0
$502$	7.74620	0.345730
$503$	21.9684	0.979521	0.489760	−	0.871857i	$-0.337084\pi$
$503$	21.9684	0.979521	0.489760	+	0.871857i	$0.337084\pi$
$504$	0	0
$505$	0	0
$506$	6.87601	0.305676
$507$	0	0
$508$	−6.42864	−0.285225
$509$	15.2573	0.676270	0.338135	−	0.941098i	$-0.390204\pi$
$509$	15.2573	0.676270	0.338135	+	0.941098i	$0.390204\pi$
$510$	0	0
$511$	−51.4291	−2.27509
$512$	1.00000	0.0441942
$513$	0	0
$514$	−13.8796	−0.612201
$515$	0	0
$516$	0	0
$517$	−6.87601	−0.302407
$518$	−25.7146	−1.12983
$519$	0	0
$520$	0	0
$521$	18.3269	0.802917	0.401459	−	0.915877i	$-0.368503\pi$
$521$	18.3269	0.802917	0.401459	+	0.915877i	$0.368503\pi$
$522$	0	0
$523$	−24.8573	−1.08693	−0.543466	−	0.839431i	$-0.682888\pi$
$523$	−24.8573	−1.08693	−0.543466	+	0.839431i	$0.682888\pi$
$524$	15.0923	0.659312
$525$	0	0
$526$	−18.6222	−0.811967
$527$	16.8573	0.734315
$528$	0	0
$529$	−16.1240	−0.701043
$530$	0	0
$531$	0	0
$532$	4.42864	0.192006
$533$	32.9403	1.42680
$534$	0	0
$535$	0	0
$536$	−15.6128	−0.674372
$537$	0	0
$538$	−15.8479	−0.683252
$539$	−33.0736	−1.42458
$540$	0	0
$541$	−1.34614	−0.0578751	−0.0289376	−	0.999581i	$-0.509212\pi$
$541$	−1.34614	−0.0578751	−0.0289376	+	0.999581i	$0.509212\pi$
$542$	1.51114	0.0649089
$543$	0	0
$544$	−3.80642	−0.163199
$545$	0	0
$546$	0	0
$547$	8.59057	0.367306	0.183653	−	0.982991i	$-0.441208\pi$
$547$	8.59057	0.367306	0.183653	+	0.982991i	$0.441208\pi$
$548$	−7.53972	−0.322081
$549$	0	0
$550$	0	0
$551$	−3.37778	−0.143898
$552$	0	0
$553$	19.6128	0.834023
$554$	−1.05086	−0.0446466
$555$	0	0
$556$	0.387152	0.0164189
$557$	−5.86665	−0.248578	−0.124289	−	0.992246i	$-0.539665\pi$
$557$	−5.86665	−0.248578	−0.124289	+	0.992246i	$0.539665\pi$
$558$	0	0
$559$	−63.8163	−2.69914
$560$	0	0
$561$	0	0
$562$	−3.45091	−0.145568
$563$	31.4291	1.32458	0.662290	−	0.749248i	$-0.269585\pi$
$563$	31.4291	1.32458	0.662290	+	0.749248i	$0.269585\pi$
$564$	0	0
$565$	0	0
$566$	−15.5812	−0.654927
$567$	0	0
$568$	−15.6128	−0.655101
$569$	7.95851	0.333638	0.166819	−	0.985988i	$-0.446650\pi$
$569$	7.95851	0.333638	0.166819	+	0.985988i	$0.446650\pi$
$570$	0	0
$571$	−30.8385	−1.29055	−0.645276	−	0.763949i	$-0.723258\pi$
$571$	−30.8385	−1.29055	−0.645276	+	0.763949i	$0.723258\pi$
$572$	15.2257	0.636618
$573$	0	0
$574$	−25.1240	−1.04865
$575$	0	0
$576$	0	0
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	−2.51114	−0.104450
$579$	0	0
$580$	0	0
$581$	52.7368	2.18789
$582$	0	0
$583$	15.7333	0.651606
$584$	−11.6128	−0.480543
$585$	0	0
$586$	7.12399	0.294289
$587$	17.8064	0.734950	0.367475	−	0.930033i	$-0.380222\pi$
$587$	17.8064	0.734950	0.367475	+	0.930033i	$0.380222\pi$
$588$	0	0
$589$	−4.42864	−0.182479
$590$	0	0
$591$	0	0
$592$	−5.80642	−0.238643
$593$	−13.3176	−0.546887	−0.273443	−	0.961888i	$-0.588163\pi$
$593$	−13.3176	−0.546887	−0.273443	+	0.961888i	$0.588163\pi$
$594$	0	0
$595$	0	0
$596$	14.5303	0.595186
$597$	0	0
$598$	15.2257	0.622625
$599$	43.8163	1.79028	0.895142	−	0.445781i	$-0.147074\pi$
$599$	43.8163	1.79028	0.895142	+	0.445781i	$0.147074\pi$
$600$	0	0
$601$	−9.73329	−0.397029	−0.198515	−	0.980098i	$-0.563612\pi$
$601$	−9.73329	−0.397029	−0.198515	+	0.980098i	$0.563612\pi$
$602$	48.6735	1.98379
$603$	0	0
$604$	4.69535	0.191051
$605$	0	0
$606$	0	0
$607$	31.9398	1.29640	0.648198	−	0.761472i	$-0.275523\pi$
$607$	31.9398	1.29640	0.648198	+	0.761472i	$0.275523\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−15.2257	−0.615966
$612$	0	0
$613$	−11.8064	−0.476857	−0.238428	−	0.971160i	$-0.576632\pi$
$613$	−11.8064	−0.476857	−0.238428	+	0.971160i	$0.576632\pi$
$614$	−1.12399	−0.0453603
$615$	0	0
$616$	−11.6128	−0.467895
$617$	26.5620	1.06935	0.534673	−	0.845059i	$-0.320435\pi$
$617$	26.5620	1.06935	0.534673	+	0.845059i	$0.320435\pi$
$618$	0	0
$619$	7.34614	0.295266	0.147633	−	0.989042i	$-0.452834\pi$
$619$	7.34614	0.295266	0.147633	+	0.989042i	$0.452834\pi$
$620$	0	0
$621$	0	0
$622$	−14.9491	−0.599406
$623$	−55.0420	−2.20521
$624$	0	0
$625$	0	0
$626$	−3.14272	−0.125608
$627$	0	0
$628$	21.5210	0.858781
$629$	22.1017	0.881253
$630$	0	0
$631$	−30.7556	−1.22436	−0.612180	−	0.790718i	$-0.709707\pi$
$631$	−30.7556	−1.22436	−0.612180	+	0.790718i	$0.709707\pi$
$632$	4.42864	0.176162
$633$	0	0
$634$	10.5906	0.420605
$635$	0	0
$636$	0	0
$637$	−73.2355	−2.90170
$638$	8.85728	0.350663
$639$	0	0
$640$	0	0
$641$	39.3876	1.55572	0.777859	−	0.628439i	$-0.216306\pi$
$641$	39.3876	1.55572	0.777859	+	0.628439i	$0.216306\pi$
$642$	0	0
$643$	−24.1146	−0.950988	−0.475494	−	0.879719i	$-0.657731\pi$
$643$	−24.1146	−0.950988	−0.475494	+	0.879719i	$0.657731\pi$
$644$	−11.6128	−0.457610
$645$	0	0
$646$	−3.80642	−0.149762
$647$	−47.6829	−1.87461	−0.937304	−	0.348512i	$-0.886687\pi$
$647$	−47.6829	−1.87461	−0.937304	+	0.348512i	$0.886687\pi$
$648$	0	0
$649$	−2.75557	−0.108166
$650$	0	0
$651$	0	0
$652$	8.23506	0.322510
$653$	21.7462	0.850995	0.425497	−	0.904960i	$-0.360099\pi$
$653$	21.7462	0.850995	0.425497	+	0.904960i	$0.360099\pi$
$654$	0	0
$655$	0	0
$656$	−5.67307	−0.221496
$657$	0	0
$658$	11.6128	0.452716
$659$	16.1936	0.630812	0.315406	−	0.948957i	$-0.397859\pi$
$659$	16.1936	0.630812	0.315406	+	0.948957i	$0.397859\pi$
$660$	0	0
$661$	−22.5116	−0.875600	−0.437800	−	0.899072i	$-0.644242\pi$
$661$	−22.5116	−0.875600	−0.437800	+	0.899072i	$0.644242\pi$
$662$	−15.1427	−0.588539
$663$	0	0
$664$	11.9081	0.462125
$665$	0	0
$666$	0	0
$667$	8.85728	0.342955
$668$	8.47013	0.327719
$669$	0	0
$670$	0	0
$671$	−12.4701	−0.481404
$672$	0	0
$673$	−9.66323	−0.372490	−0.186245	−	0.982503i	$-0.559632\pi$
$673$	−9.66323	−0.372490	−0.186245	+	0.982503i	$0.559632\pi$
$674$	−11.1111	−0.427983
$675$	0	0
$676$	20.7146	0.796714
$677$	−6.85728	−0.263547	−0.131773	−	0.991280i	$-0.542067\pi$
$677$	−6.85728	−0.263547	−0.131773	+	0.991280i	$0.542067\pi$
$678$	0	0
$679$	−32.6735	−1.25390
$680$	0	0
$681$	0	0
$682$	11.6128	0.444679
$683$	30.3051	1.15959	0.579797	−	0.814761i	$-0.303132\pi$
$683$	30.3051	1.15959	0.579797	+	0.814761i	$0.303132\pi$
$684$	0	0
$685$	0	0
$686$	24.8573	0.949055
$687$	0	0
$688$	10.9906	0.419014
$689$	34.8385	1.32724
$690$	0	0
$691$	7.61285	0.289606	0.144803	−	0.989460i	$-0.453745\pi$
$691$	7.61285	0.289606	0.144803	+	0.989460i	$0.453745\pi$
$692$	−22.4701	−0.854186
$693$	0	0
$694$	−14.1936	−0.538781
$695$	0	0
$696$	0	0
$697$	21.5941	0.817935
$698$	32.3684	1.22516
$699$	0	0
$700$	0	0
$701$	32.3654	1.22242	0.611211	−	0.791467i	$-0.290683\pi$
$701$	32.3654	1.22242	0.611211	+	0.791467i	$0.290683\pi$
$702$	0	0
$703$	−5.80642	−0.218993
$704$	−2.62222	−0.0988285
$705$	0	0
$706$	−11.8064	−0.444341
$707$	−78.2677	−2.94356
$708$	0	0
$709$	−49.8992	−1.87401	−0.937003	−	0.349322i	$-0.886412\pi$
$709$	−49.8992	−1.87401	−0.937003	+	0.349322i	$0.886412\pi$
$710$	0	0
$711$	0	0
$712$	−12.4286	−0.465783
$713$	11.6128	0.434905
$714$	0	0
$715$	0	0
$716$	2.94914	0.110215
$717$	0	0
$718$	10.8287	0.404123
$719$	26.4415	0.986103	0.493052	−	0.870000i	$-0.335881\pi$
$719$	26.4415	0.986103	0.493052	+	0.870000i	$0.335881\pi$
$720$	0	0
$721$	5.24443	0.195313
$722$	1.00000	0.0372161
$723$	0	0
$724$	22.8988	0.851026
$725$	0	0
$726$	0	0
$727$	−48.1245	−1.78484	−0.892419	−	0.451208i	$-0.850993\pi$
$727$	−48.1245	−1.78484	−0.892419	+	0.451208i	$0.850993\pi$
$728$	−25.7146	−0.953045
$729$	0	0
$730$	0	0
$731$	−41.8350	−1.54732
$732$	0	0
$733$	14.0286	0.518157	0.259079	−	0.965856i	$-0.416581\pi$
$733$	14.0286	0.518157	0.259079	+	0.965856i	$0.416581\pi$
$734$	1.46965	0.0542458
$735$	0	0
$736$	−2.62222	−0.0966562
$737$	40.9403	1.50805
$738$	0	0
$739$	10.1847	0.374650	0.187325	−	0.982298i	$-0.440018\pi$
$739$	10.1847	0.374650	0.187325	+	0.982298i	$0.440018\pi$
$740$	0	0
$741$	0	0
$742$	−26.5718	−0.975483
$743$	−42.9590	−1.57601	−0.788006	−	0.615667i	$-0.788887\pi$
$743$	−42.9590	−1.57601	−0.788006	+	0.615667i	$0.788887\pi$
$744$	0	0
$745$	0	0
$746$	−24.3783	−0.892552
$747$	0	0
$748$	9.98126	0.364951
$749$	21.5111	0.786000
$750$	0	0
$751$	−7.18421	−0.262155	−0.131078	−	0.991372i	$-0.541844\pi$
$751$	−7.18421	−0.262155	−0.131078	+	0.991372i	$0.541844\pi$
$752$	2.62222	0.0956224
$753$	0	0
$754$	19.6128	0.714258
$755$	0	0
$756$	0	0
$757$	−41.2543	−1.49941	−0.749706	−	0.661771i	$-0.769805\pi$
$757$	−41.2543	−1.49941	−0.749706	+	0.661771i	$0.769805\pi$
$758$	18.9590	0.688621
$759$	0	0
$760$	0	0
$761$	−44.3051	−1.60606	−0.803030	−	0.595939i	$-0.796780\pi$
$761$	−44.3051	−1.60606	−0.803030	+	0.595939i	$0.796780\pi$
$762$	0	0
$763$	−26.7556	−0.968617
$764$	9.05086	0.327448
$765$	0	0
$766$	−26.1017	−0.943093
$767$	−6.10171	−0.220320
$768$	0	0
$769$	−6.59057	−0.237662	−0.118831	−	0.992914i	$-0.537915\pi$
$769$	−6.59057	−0.237662	−0.118831	+	0.992914i	$0.537915\pi$
$770$	0	0
$771$	0	0
$772$	−18.7239	−0.673889
$773$	−8.83854	−0.317900	−0.158950	−	0.987287i	$-0.550811\pi$
$773$	−8.83854	−0.317900	−0.158950	+	0.987287i	$0.550811\pi$
$774$	0	0
$775$	0	0
$776$	−7.37778	−0.264847
$777$	0	0
$778$	3.57136	0.128039
$779$	−5.67307	−0.203259
$780$	0	0
$781$	40.9403	1.46496
$782$	9.98126	0.356929
$783$	0	0
$784$	12.6128	0.450459
$785$	0	0
$786$	0	0
$787$	16.9403	0.603855	0.301927	−	0.953331i	$-0.402370\pi$
$787$	16.9403	0.603855	0.301927	+	0.953331i	$0.402370\pi$
$788$	0.888922	0.0316665
$789$	0	0
$790$	0	0
$791$	41.3907	1.47168
$792$	0	0
$793$	−27.6128	−0.980561
$794$	−31.4193	−1.11503
$795$	0	0
$796$	21.7146	0.769652
$797$	−40.1847	−1.42341	−0.711707	−	0.702476i	$-0.752078\pi$
$797$	−40.1847	−1.42341	−0.711707	+	0.702476i	$0.752078\pi$
$798$	0	0
$799$	−9.98126	−0.353112
$800$	0	0
$801$	0	0
$802$	−12.0415	−0.425200
$803$	30.4514	1.07461
$804$	0	0
$805$	0	0
$806$	25.7146	0.905757
$807$	0	0
$808$	−17.6731	−0.621736
$809$	−29.1052	−1.02329	−0.511643	−	0.859198i	$-0.670963\pi$
$809$	−29.1052	−1.02329	−0.511643	+	0.859198i	$0.670963\pi$
$810$	0	0
$811$	38.7753	1.36158	0.680792	−	0.732477i	$-0.261636\pi$
$811$	38.7753	1.36158	0.680792	+	0.732477i	$0.261636\pi$
$812$	−14.9590	−0.524958
$813$	0	0
$814$	15.2257	0.533660
$815$	0	0
$816$	0	0
$817$	10.9906	0.384514
$818$	26.4701	0.925506
$819$	0	0
$820$	0	0
$821$	−2.53035	−0.0883098	−0.0441549	−	0.999025i	$-0.514060\pi$
$821$	−2.53035	−0.0883098	−0.0441549	+	0.999025i	$0.514060\pi$
$822$	0	0
$823$	24.0415	0.838034	0.419017	−	0.907978i	$-0.362375\pi$
$823$	24.0415	0.838034	0.419017	+	0.907978i	$0.362375\pi$
$824$	1.18421	0.0412538
$825$	0	0
$826$	4.65386	0.161928
$827$	−6.57184	−0.228525	−0.114263	−	0.993451i	$-0.536451\pi$
$827$	−6.57184	−0.228525	−0.114263	+	0.993451i	$0.536451\pi$
$828$	0	0
$829$	−28.7338	−0.997965	−0.498983	−	0.866612i	$-0.666293\pi$
$829$	−28.7338	−0.997965	−0.498983	+	0.866612i	$0.666293\pi$
$830$	0	0
$831$	0	0
$832$	−5.80642	−0.201302
$833$	−48.0098	−1.66344
$834$	0	0
$835$	0	0
$836$	−2.62222	−0.0906912
$837$	0	0
$838$	25.9684	0.897062
$839$	50.5718	1.74593	0.872967	−	0.487780i	$-0.162193\pi$
$839$	50.5718	1.74593	0.872967	+	0.487780i	$0.162193\pi$
$840$	0	0
$841$	−17.5906	−0.606571
$842$	−29.2672	−1.00861
$843$	0	0
$844$	−3.61285	−0.124359
$845$	0	0
$846$	0	0
$847$	−18.2636	−0.627546
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	15.2257	0.521930
$852$	0	0
$853$	31.2355	1.06948	0.534742	−	0.845015i	$-0.320409\pi$
$853$	31.2355	1.06948	0.534742	+	0.845015i	$0.320409\pi$
$854$	21.0607	0.720682
$855$	0	0
$856$	4.85728	0.166018
$857$	27.3274	0.933486	0.466743	−	0.884393i	$-0.345427\pi$
$857$	27.3274	0.933486	0.466743	+	0.884393i	$0.345427\pi$
$858$	0	0
$859$	−34.7753	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$859$	−34.7753	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$860$	0	0
$861$	0	0
$862$	−14.8385	−0.505403
$863$	21.2444	0.723169	0.361584	−	0.932339i	$-0.382236\pi$
$863$	21.2444	0.723169	0.361584	+	0.932339i	$0.382236\pi$
$864$	0	0
$865$	0	0
$866$	5.27607	0.179288
$867$	0	0
$868$	−19.6128	−0.665703
$869$	−11.6128	−0.393939
$870$	0	0
$871$	90.6548	3.07172
$872$	−6.04149	−0.204591
$873$	0	0
$874$	−2.62222	−0.0886978
$875$	0	0
$876$	0	0
$877$	−22.3970	−0.756293	−0.378146	−	0.925746i	$-0.623438\pi$
$877$	−22.3970	−0.756293	−0.378146	+	0.925746i	$0.623438\pi$
$878$	31.8578	1.07515
$879$	0	0
$880$	0	0
$881$	20.3684	0.686229	0.343115	−	0.939294i	$-0.388518\pi$
$881$	20.3684	0.686229	0.343115	+	0.939294i	$0.388518\pi$
$882$	0	0
$883$	−21.6829	−0.729688	−0.364844	−	0.931069i	$-0.618878\pi$
$883$	−21.6829	−0.729688	−0.364844	+	0.931069i	$0.618878\pi$
$884$	22.1017	0.743361
$885$	0	0
$886$	−4.94914	−0.166270
$887$	−7.14272	−0.239829	−0.119915	−	0.992784i	$-0.538262\pi$
$887$	−7.14272	−0.239829	−0.119915	+	0.992784i	$0.538262\pi$
$888$	0	0
$889$	−28.4701	−0.954857
$890$	0	0
$891$	0	0
$892$	21.3876	0.716111
$893$	2.62222	0.0877491
$894$	0	0
$895$	0	0
$896$	4.42864	0.147950
$897$	0	0
$898$	−9.75605	−0.325563
$899$	14.9590	0.498910
$900$	0	0
$901$	22.8385	0.760862
$902$	14.8760	0.495317
$903$	0	0
$904$	9.34614	0.310848
$905$	0	0
$906$	0	0
$907$	−15.3461	−0.509560	−0.254780	−	0.966999i	$-0.582003\pi$
$907$	−15.3461	−0.509560	−0.254780	+	0.966999i	$0.582003\pi$
$908$	8.47013	0.281091
$909$	0	0
$910$	0	0
$911$	−12.1204	−0.401568	−0.200784	−	0.979636i	$-0.564349\pi$
$911$	−12.1204	−0.401568	−0.200784	+	0.979636i	$0.564349\pi$
$912$	0	0
$913$	−31.2257	−1.03342
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−16.9590	−0.560341
$917$	66.8385	2.20720
$918$	0	0
$919$	−12.2667	−0.404641	−0.202321	−	0.979319i	$-0.564848\pi$
$919$	−12.2667	−0.404641	−0.202321	+	0.979319i	$0.564848\pi$
$920$	0	0
$921$	0	0
$922$	36.1245	1.18970
$923$	90.6548	2.98394
$924$	0	0
$925$	0	0
$926$	−35.1209	−1.15415
$927$	0	0
$928$	−3.37778	−0.110881
$929$	40.1847	1.31842	0.659208	−	0.751960i	$-0.270892\pi$
$929$	40.1847	1.31842	0.659208	+	0.751960i	$0.270892\pi$
$930$	0	0
$931$	12.6128	0.413369
$932$	−17.9081	−0.586600
$933$	0	0
$934$	−4.56199	−0.149273
$935$	0	0
$936$	0	0
$937$	24.7368	0.808117	0.404059	−	0.914733i	$-0.367599\pi$
$937$	24.7368	0.808117	0.404059	+	0.914733i	$0.367599\pi$
$938$	−69.1437	−2.25762
$939$	0	0
$940$	0	0
$941$	1.66323	0.0542196	0.0271098	−	0.999632i	$-0.491370\pi$
$941$	1.66323	0.0542196	0.0271098	+	0.999632i	$0.491370\pi$
$942$	0	0
$943$	14.8760	0.484430
$944$	1.05086	0.0342024
$945$	0	0
$946$	−28.8198	−0.937013
$947$	14.8671	0.483117	0.241558	−	0.970386i	$-0.422341\pi$
$947$	14.8671	0.483117	0.241558	+	0.970386i	$0.422341\pi$
$948$	0	0
$949$	67.4291	2.18884
$950$	0	0
$951$	0	0
$952$	−16.8573	−0.546348
$953$	−8.83854	−0.286308	−0.143154	−	0.989700i	$-0.545725\pi$
$953$	−8.83854	−0.286308	−0.143154	+	0.989700i	$0.545725\pi$
$954$	0	0
$955$	0	0
$956$	28.2766	0.914529
$957$	0	0
$958$	−37.4005	−1.20836
$959$	−33.3907	−1.07824
$960$	0	0
$961$	−11.3872	−0.367327
$962$	33.7146	1.08700
$963$	0	0
$964$	−11.5111	−0.370749
$965$	0	0
$966$	0	0
$967$	−1.55262	−0.0499290	−0.0249645	−	0.999688i	$-0.507947\pi$
$967$	−1.55262	−0.0499290	−0.0249645	+	0.999688i	$0.507947\pi$
$968$	−4.12399	−0.132550
$969$	0	0
$970$	0	0
$971$	−44.5433	−1.42946	−0.714731	−	0.699400i	$-0.753451\pi$
$971$	−44.5433	−1.42946	−0.714731	+	0.699400i	$0.753451\pi$
$972$	0	0
$973$	1.71456	0.0549662
$974$	20.9175	0.670240
$975$	0	0
$976$	4.75557	0.152222
$977$	15.8350	0.506607	0.253303	−	0.967387i	$-0.418483\pi$
$977$	15.8350	0.506607	0.253303	+	0.967387i	$0.418483\pi$
$978$	0	0
$979$	32.5906	1.04160
$980$	0	0
$981$	0	0
$982$	15.8666	0.506325
$983$	6.87601	0.219311	0.109655	−	0.993970i	$-0.465025\pi$
$983$	6.87601	0.219311	0.109655	+	0.993970i	$0.465025\pi$
$984$	0	0
$985$	0	0
$986$	12.8573	0.409459
$987$	0	0
$988$	−5.80642	−0.184727
$989$	−28.8198	−0.916417
$990$	0	0
$991$	−29.8765	−0.949058	−0.474529	−	0.880240i	$-0.657382\pi$
$991$	−29.8765	−0.949058	−0.474529	+	0.880240i	$0.657382\pi$
$992$	−4.42864	−0.140609
$993$	0	0
$994$	−69.1437	−2.19310
$995$	0	0
$996$	0	0
$997$	−5.90813	−0.187112	−0.0935562	−	0.995614i	$-0.529823\pi$
$997$	−5.90813	−0.187112	−0.0935562	+	0.995614i	$0.529823\pi$
$998$	16.7368	0.529795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cq.1.3		3
3.2	odd	2		2850.2.a.bl.1.3		3
5.2	odd	4		1710.2.d.f.1369.4		6
5.3	odd	4		1710.2.d.f.1369.1		6
5.4	even	2		8550.2.a.ce.1.1		3
15.2	even	4		570.2.d.c.229.3	✓	6
15.8	even	4		570.2.d.c.229.6	yes	6
15.14	odd	2		2850.2.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.c.229.3	✓	6	15.2	even	4
570.2.d.c.229.6	yes	6	15.8	even	4
1710.2.d.f.1369.1		6	5.3	odd	4
1710.2.d.f.1369.4		6	5.2	odd	4
2850.2.a.bl.1.3		3	3.2	odd	2
2850.2.a.bm.1.1		3	15.14	odd	2
8550.2.a.ce.1.1		3	5.4	even	2
8550.2.a.cq.1.3		3	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} -2.62222 q^{11} -5.80642 q^{13} +4.42864 q^{14} +1.00000 q^{16} -3.80642 q^{17} +1.00000 q^{19} -2.62222 q^{22} -2.62222 q^{23} -5.80642 q^{26} +4.42864 q^{28} -3.37778 q^{29} -4.42864 q^{31} +1.00000 q^{32} -3.80642 q^{34} -5.80642 q^{37} +1.00000 q^{38} -5.67307 q^{41} +10.9906 q^{43} -2.62222 q^{44} -2.62222 q^{46} +2.62222 q^{47} +12.6128 q^{49} -5.80642 q^{52} -6.00000 q^{53} +4.42864 q^{56} -3.37778 q^{58} +1.05086 q^{59} +4.75557 q^{61} -4.42864 q^{62} +1.00000 q^{64} -15.6128 q^{67} -3.80642 q^{68} -15.6128 q^{71} -11.6128 q^{73} -5.80642 q^{74} +1.00000 q^{76} -11.6128 q^{77} +4.42864 q^{79} -5.67307 q^{82} +11.9081 q^{83} +10.9906 q^{86} -2.62222 q^{88} -12.4286 q^{89} -25.7146 q^{91} -2.62222 q^{92} +2.62222 q^{94} -7.37778 q^{97} +12.6128 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} - 8 q^{11} - 4 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{22} - 8 q^{23} - 4 q^{26} - 10 q^{29} + 3 q^{32} + 2 q^{34} - 4 q^{37} + 3 q^{38} - 4 q^{41} + 6 q^{43} - 8 q^{44} - 8 q^{46} + 8 q^{47} + 11 q^{49} - 4 q^{52} - 18 q^{53} - 10 q^{58} - 10 q^{59} + 14 q^{61} + 3 q^{64} - 20 q^{67} + 2 q^{68} - 20 q^{71} - 8 q^{73} - 4 q^{74} + 3 q^{76} - 8 q^{77} - 4 q^{82} - 4 q^{83} + 6 q^{86} - 8 q^{88} - 24 q^{89} - 24 q^{91} - 8 q^{92} + 8 q^{94} - 22 q^{97} + 11 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 - 8 * q^11 - 4 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^19 - 8 * q^22 - 8 * q^23 - 4 * q^26 - 10 * q^29 + 3 * q^32 + 2 * q^34 - 4 * q^37 + 3 * q^38 - 4 * q^41 + 6 * q^43 - 8 * q^44 - 8 * q^46 + 8 * q^47 + 11 * q^49 - 4 * q^52 - 18 * q^53 - 10 * q^58 - 10 * q^59 + 14 * q^61 + 3 * q^64 - 20 * q^67 + 2 * q^68 - 20 * q^71 - 8 * q^73 - 4 * q^74 + 3 * q^76 - 8 * q^77 - 4 * q^82 - 4 * q^83 + 6 * q^86 - 8 * q^88 - 24 * q^89 - 24 * q^91 - 8 * q^92 + 8 * q^94 - 22 * q^97 + 11 * q^98

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