LMFDB - Embedded newform 6600.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.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:	$52.7012653340$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1320)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.52543 q^{7} +1.00000 q^{9} +1.00000 q^{11} -0.903212 q^{13} +2.90321 q^{17} -2.00000 q^{19} +1.52543 q^{21} +8.85728 q^{23} -1.00000 q^{27} -6.62222 q^{29} -10.2351 q^{31} -1.00000 q^{33} -8.56199 q^{37} +0.903212 q^{39} +2.23506 q^{41} +12.5763 q^{43} -8.56199 q^{47} -4.67307 q^{49} -2.90321 q^{51} +3.18421 q^{53} +2.00000 q^{57} +12.0415 q^{59} -13.6128 q^{61} -1.52543 q^{63} +14.3684 q^{67} -8.85728 q^{69} +6.91750 q^{71} +9.76049 q^{73} -1.52543 q^{77} -8.75557 q^{79} +1.00000 q^{81} +7.09679 q^{83} +6.62222 q^{87} -14.9906 q^{89} +1.37778 q^{91} +10.2351 q^{93} +7.61285 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{27} - 20 q^{29} - 4 q^{31} - 3 q^{33} - 12 q^{37} - 4 q^{39} - 20 q^{41} + 18 q^{43} - 12 q^{47} - q^{49}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^27 - 20 * q^29 - 4 * q^31 - 3 * q^33 - 12 * q^37 - 4 * q^39 - 20 * q^41 + 18 * q^43 - 12 * q^47 - q^49 - 2 * q^51 - 4 * q^53 + 6 * q^57 - 4 * q^59 - 14 * q^61 + 2 * q^63 + 16 * q^67 + 8 * q^71 - 4 * q^73 + 2 * q^77 - 26 * q^79 + 3 * q^81 + 28 * q^83 + 20 * q^87 - 18 * q^89 + 4 * q^91 + 4 * q^93 - 4 * q^97 + 3 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.903212	−0.250506	−0.125253	−	0.992125i	$-0.539974\pi$
$13$	−0.903212	−0.250506	−0.125253	+	0.992125i	$0.539974\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.90321	0.704132	0.352066	−	0.935975i	$-0.385479\pi$
$17$	2.90321	0.704132	0.352066	+	0.935975i	$0.385479\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.52543	0.332876
$22$	0	0
$23$	8.85728	1.84687	0.923435	−	0.383754i	$-0.125369\pi$
$23$	8.85728	1.84687	0.923435	+	0.383754i	$0.125369\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.62222	−1.22971	−0.614857	−	0.788638i	$-0.710786\pi$
$29$	−6.62222	−1.22971	−0.614857	+	0.788638i	$0.710786\pi$
$30$	0	0
$31$	−10.2351	−1.83827	−0.919136	−	0.393941i	$-0.871112\pi$
$31$	−10.2351	−1.83827	−0.919136	+	0.393941i	$0.871112\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.56199	−1.40758	−0.703791	−	0.710407i	$-0.748511\pi$
$37$	−8.56199	−1.40758	−0.703791	+	0.710407i	$0.748511\pi$
$38$	0	0
$39$	0.903212	0.144630
$40$	0	0
$41$	2.23506	0.349058	0.174529	−	0.984652i	$-0.444160\pi$
$41$	2.23506	0.349058	0.174529	+	0.984652i	$0.444160\pi$
$42$	0	0
$43$	12.5763	1.91787	0.958933	−	0.283634i	$-0.0915401\pi$
$43$	12.5763	1.91787	0.958933	+	0.283634i	$0.0915401\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.56199	−1.24889	−0.624447	−	0.781067i	$-0.714676\pi$
$47$	−8.56199	−1.24889	−0.624447	+	0.781067i	$0.714676\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	−2.90321	−0.406531
$52$	0	0
$53$	3.18421	0.437385	0.218692	−	0.975794i	$-0.429821\pi$
$53$	3.18421	0.437385	0.218692	+	0.975794i	$0.429821\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	12.0415	1.56767	0.783834	−	0.620970i	$-0.213261\pi$
$59$	12.0415	1.56767	0.783834	+	0.620970i	$0.213261\pi$
$60$	0	0
$61$	−13.6128	−1.74295	−0.871473	−	0.490443i	$-0.836835\pi$
$61$	−13.6128	−1.74295	−0.871473	+	0.490443i	$0.836835\pi$
$62$	0	0
$63$	−1.52543	−0.192186
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.3684	1.75538	0.877691	−	0.479227i	$-0.159083\pi$
$67$	14.3684	1.75538	0.877691	+	0.479227i	$0.159083\pi$
$68$	0	0
$69$	−8.85728	−1.06629
$70$	0	0
$71$	6.91750	0.820956	0.410478	−	0.911870i	$-0.365362\pi$
$71$	6.91750	0.820956	0.410478	+	0.911870i	$0.365362\pi$
$72$	0	0
$73$	9.76049	1.14238	0.571190	−	0.820818i	$-0.306482\pi$
$73$	9.76049	1.14238	0.571190	+	0.820818i	$0.306482\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.52543	−0.173839
$78$	0	0
$79$	−8.75557	−0.985078	−0.492539	−	0.870290i	$-0.663931\pi$
$79$	−8.75557	−0.985078	−0.492539	+	0.870290i	$0.663931\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.09679	0.778974	0.389487	−	0.921032i	$-0.372652\pi$
$83$	7.09679	0.778974	0.389487	+	0.921032i	$0.372652\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.62222	0.709976
$88$	0	0
$89$	−14.9906	−1.58900	−0.794502	−	0.607262i	$-0.792268\pi$
$89$	−14.9906	−1.58900	−0.794502	+	0.607262i	$0.792268\pi$
$90$	0	0
$91$	1.37778	0.144431
$92$	0	0
$93$	10.2351	1.06133
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.61285	0.772968	0.386484	−	0.922296i	$-0.373690\pi$
$97$	7.61285	0.772968	0.386484	+	0.922296i	$0.373690\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	10.2351	1.01843	0.509213	−	0.860640i	$-0.329936\pi$
$101$	10.2351	1.01843	0.509213	+	0.860640i	$0.329936\pi$
$102$	0	0
$103$	−5.80642	−0.572124	−0.286062	−	0.958211i	$-0.592346\pi$
$103$	−5.80642	−0.572124	−0.286062	+	0.958211i	$0.592346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.00492	−0.677191	−0.338596	−	0.940932i	$-0.609952\pi$
$107$	−7.00492	−0.677191	−0.338596	+	0.940932i	$0.609952\pi$
$108$	0	0
$109$	9.61285	0.920744	0.460372	−	0.887726i	$-0.347716\pi$
$109$	9.61285	0.920744	0.460372	+	0.887726i	$0.347716\pi$
$110$	0	0
$111$	8.56199	0.812668
$112$	0	0
$113$	−11.4795	−1.07990	−0.539950	−	0.841697i	$-0.681557\pi$
$113$	−11.4795	−1.07990	−0.539950	+	0.841697i	$0.681557\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.903212	−0.0835020
$118$	0	0
$119$	−4.42864	−0.405973
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.23506	−0.201529
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.4336	−1.54698	−0.773489	−	0.633809i	$-0.781490\pi$
$127$	−17.4336	−1.54698	−0.773489	+	0.633809i	$0.781490\pi$
$128$	0	0
$129$	−12.5763	−1.10728
$130$	0	0
$131$	1.24443	0.108726	0.0543632	−	0.998521i	$-0.482687\pi$
$131$	1.24443	0.108726	0.0543632	+	0.998521i	$0.482687\pi$
$132$	0	0
$133$	3.05086	0.264543
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.69535	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$137$	−4.69535	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$138$	0	0
$139$	14.4701	1.22734	0.613670	−	0.789563i	$-0.289693\pi$
$139$	14.4701	1.22734	0.613670	+	0.789563i	$0.289693\pi$
$140$	0	0
$141$	8.56199	0.721050
$142$	0	0
$143$	−0.903212	−0.0755304
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.67307	0.385428
$148$	0	0
$149$	−11.4795	−0.940437	−0.470218	−	0.882550i	$-0.655825\pi$
$149$	−11.4795	−0.940437	−0.470218	+	0.882550i	$0.655825\pi$
$150$	0	0
$151$	−3.24443	−0.264028	−0.132014	−	0.991248i	$-0.542144\pi$
$151$	−3.24443	−0.264028	−0.132014	+	0.991248i	$0.542144\pi$
$152$	0	0
$153$	2.90321	0.234711
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.90813	−0.631138	−0.315569	−	0.948903i	$-0.602195\pi$
$157$	−7.90813	−0.631138	−0.315569	+	0.948903i	$0.602195\pi$
$158$	0	0
$159$	−3.18421	−0.252524
$160$	0	0
$161$	−13.5111	−1.06483
$162$	0	0
$163$	−9.41927	−0.737774	−0.368887	−	0.929474i	$-0.620261\pi$
$163$	−9.41927	−0.737774	−0.368887	+	0.929474i	$0.620261\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.3733	1.34439	0.672195	−	0.740374i	$-0.265352\pi$
$167$	17.3733	1.34439	0.672195	+	0.740374i	$0.265352\pi$
$168$	0	0
$169$	−12.1842	−0.937247
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−13.5669	−1.03147	−0.515737	−	0.856747i	$-0.672482\pi$
$173$	−13.5669	−1.03147	−0.515737	+	0.856747i	$0.672482\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0415	−0.905094
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	4.32693	0.321618	0.160809	−	0.986986i	$-0.448590\pi$
$181$	4.32693	0.321618	0.160809	+	0.986986i	$0.448590\pi$
$182$	0	0
$183$	13.6128	1.00629
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.90321	0.212304
$188$	0	0
$189$	1.52543	0.110959
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−15.5669	−1.12053	−0.560266	−	0.828313i	$-0.689301\pi$
$193$	−15.5669	−1.12053	−0.560266	+	0.828313i	$0.689301\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.3412	−1.02177	−0.510885	−	0.859649i	$-0.670682\pi$
$197$	−14.3412	−1.02177	−0.510885	+	0.859649i	$0.670682\pi$
$198$	0	0
$199$	1.24443	0.0882154	0.0441077	−	0.999027i	$-0.485956\pi$
$199$	1.24443	0.0882154	0.0441077	+	0.999027i	$0.485956\pi$
$200$	0	0
$201$	−14.3684	−1.01347
$202$	0	0
$203$	10.1017	0.709001
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.85728	0.615623
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−24.9590	−1.71825	−0.859124	−	0.511768i	$-0.828991\pi$
$211$	−24.9590	−1.71825	−0.859124	+	0.511768i	$0.828991\pi$
$212$	0	0
$213$	−6.91750	−0.473979
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.6128	1.05987
$218$	0	0
$219$	−9.76049	−0.659553
$220$	0	0
$221$	−2.62222	−0.176389
$222$	0	0
$223$	−1.53972	−0.103107	−0.0515536	−	0.998670i	$-0.516417\pi$
$223$	−1.53972	−0.103107	−0.0515536	+	0.998670i	$0.516417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.3002	−1.01551	−0.507755	−	0.861501i	$-0.669525\pi$
$227$	−15.3002	−1.01551	−0.507755	+	0.861501i	$0.669525\pi$
$228$	0	0
$229$	15.7146	1.03845	0.519224	−	0.854638i	$-0.326221\pi$
$229$	15.7146	1.03845	0.519224	+	0.854638i	$0.326221\pi$
$230$	0	0
$231$	1.52543	0.100366
$232$	0	0
$233$	−29.0049	−1.90018	−0.950088	−	0.311983i	$-0.899007\pi$
$233$	−29.0049	−1.90018	−0.950088	+	0.311983i	$0.899007\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.75557	0.568735
$238$	0	0
$239$	−15.6128	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$239$	−15.6128	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$240$	0	0
$241$	5.61285	0.361555	0.180778	−	0.983524i	$-0.442139\pi$
$241$	5.61285	0.361555	0.180778	+	0.983524i	$0.442139\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.80642	0.114940
$248$	0	0
$249$	−7.09679	−0.449741
$250$	0	0
$251$	−2.91750	−0.184151	−0.0920755	−	0.995752i	$-0.529350\pi$
$251$	−2.91750	−0.184151	−0.0920755	+	0.995752i	$0.529350\pi$
$252$	0	0
$253$	8.85728	0.556852
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0923	0.691921	0.345961	−	0.938249i	$-0.387553\pi$
$257$	11.0923	0.691921	0.345961	+	0.938249i	$0.387553\pi$
$258$	0	0
$259$	13.0607	0.811552
$260$	0	0
$261$	−6.62222	−0.409905
$262$	0	0
$263$	5.46520	0.336999	0.168499	−	0.985702i	$-0.446108\pi$
$263$	5.46520	0.336999	0.168499	+	0.985702i	$0.446108\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.9906	0.917412
$268$	0	0
$269$	−10.8573	−0.661980	−0.330990	−	0.943634i	$-0.607383\pi$
$269$	−10.8573	−0.661980	−0.330990	+	0.943634i	$0.607383\pi$
$270$	0	0
$271$	−0.101710	−0.00617846	−0.00308923	−	0.999995i	$-0.500983\pi$
$271$	−0.101710	−0.00617846	−0.00308923	+	0.999995i	$0.500983\pi$
$272$	0	0
$273$	−1.37778	−0.0833873
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.19850	0.552684	0.276342	−	0.961059i	$-0.410878\pi$
$277$	9.19850	0.552684	0.276342	+	0.961059i	$0.410878\pi$
$278$	0	0
$279$	−10.2351	−0.612757
$280$	0	0
$281$	−1.11108	−0.0662814	−0.0331407	−	0.999451i	$-0.510551\pi$
$281$	−1.11108	−0.0662814	−0.0331407	+	0.999451i	$0.510551\pi$
$282$	0	0
$283$	15.7190	0.934398	0.467199	−	0.884152i	$-0.345263\pi$
$283$	15.7190	0.934398	0.467199	+	0.884152i	$0.345263\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.40943	−0.201252
$288$	0	0
$289$	−8.57136	−0.504198
$290$	0	0
$291$	−7.61285	−0.446273
$292$	0	0
$293$	−6.43309	−0.375825	−0.187912	−	0.982186i	$-0.560172\pi$
$293$	−6.43309	−0.375825	−0.187912	+	0.982186i	$0.560172\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−19.1842	−1.10576
$302$	0	0
$303$	−10.2351	−0.587989
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.7190	0.897131	0.448565	−	0.893750i	$-0.351935\pi$
$307$	15.7190	0.897131	0.448565	+	0.893750i	$0.351935\pi$
$308$	0	0
$309$	5.80642	0.330316
$310$	0	0
$311$	24.8988	1.41188	0.705940	−	0.708272i	$-0.250525\pi$
$311$	24.8988	1.41188	0.705940	+	0.708272i	$0.250525\pi$
$312$	0	0
$313$	17.1526	0.969520	0.484760	−	0.874647i	$-0.338907\pi$
$313$	17.1526	0.969520	0.484760	+	0.874647i	$0.338907\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.3368	0.692902	0.346451	−	0.938068i	$-0.387387\pi$
$317$	12.3368	0.692902	0.346451	+	0.938068i	$0.387387\pi$
$318$	0	0
$319$	−6.62222	−0.370773
$320$	0	0
$321$	7.00492	0.390977
$322$	0	0
$323$	−5.80642	−0.323078
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.61285	−0.531592
$328$	0	0
$329$	13.0607	0.720060
$330$	0	0
$331$	−9.37778	−0.515450	−0.257725	−	0.966218i	$-0.582973\pi$
$331$	−9.37778	−0.515450	−0.257725	+	0.966218i	$0.582973\pi$
$332$	0	0
$333$	−8.56199	−0.469194
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0558	−0.983561	−0.491780	−	0.870719i	$-0.663654\pi$
$337$	−18.0558	−0.983561	−0.491780	+	0.870719i	$0.663654\pi$
$338$	0	0
$339$	11.4795	0.623481
$340$	0	0
$341$	−10.2351	−0.554260
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.5161	−0.886629	−0.443314	−	0.896366i	$-0.646197\pi$
$347$	−16.5161	−0.886629	−0.443314	+	0.896366i	$0.646197\pi$
$348$	0	0
$349$	−8.36842	−0.447951	−0.223976	−	0.974595i	$-0.571904\pi$
$349$	−8.36842	−0.447951	−0.223976	+	0.974595i	$0.571904\pi$
$350$	0	0
$351$	0.903212	0.0482099
$352$	0	0
$353$	−11.7748	−0.626708	−0.313354	−	0.949636i	$-0.601453\pi$
$353$	−11.7748	−0.626708	−0.313354	+	0.949636i	$0.601453\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.42864	0.234388
$358$	0	0
$359$	−12.2034	−0.644072	−0.322036	−	0.946727i	$-0.604367\pi$
$359$	−12.2034	−0.644072	−0.322036	+	0.946727i	$0.604367\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.6637	0.974237	0.487119	−	0.873336i	$-0.338048\pi$
$367$	18.6637	0.974237	0.487119	+	0.873336i	$0.338048\pi$
$368$	0	0
$369$	2.23506	0.116353
$370$	0	0
$371$	−4.85728	−0.252177
$372$	0	0
$373$	−34.3225	−1.77715	−0.888575	−	0.458731i	$-0.848304\pi$
$373$	−34.3225	−1.77715	−0.888575	+	0.458731i	$0.848304\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.98126	0.308051
$378$	0	0
$379$	−11.7333	−0.602699	−0.301349	−	0.953514i	$-0.597437\pi$
$379$	−11.7333	−0.602699	−0.301349	+	0.953514i	$0.597437\pi$
$380$	0	0
$381$	17.4336	0.893148
$382$	0	0
$383$	−18.6637	−0.953671	−0.476835	−	0.878993i	$-0.658216\pi$
$383$	−18.6637	−0.953671	−0.476835	+	0.878993i	$0.658216\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.5763	0.639288
$388$	0	0
$389$	−15.1240	−0.766816	−0.383408	−	0.923579i	$-0.625250\pi$
$389$	−15.1240	−0.766816	−0.383408	+	0.923579i	$0.625250\pi$
$390$	0	0
$391$	25.7146	1.30044
$392$	0	0
$393$	−1.24443	−0.0627733
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.1847	1.71568	0.857840	−	0.513917i	$-0.171806\pi$
$397$	34.1847	1.71568	0.857840	+	0.513917i	$0.171806\pi$
$398$	0	0
$399$	−3.05086	−0.152734
$400$	0	0
$401$	17.0923	0.853551	0.426775	−	0.904358i	$-0.359649\pi$
$401$	17.0923	0.853551	0.426775	+	0.904358i	$0.359649\pi$
$402$	0	0
$403$	9.24443	0.460498
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.56199	−0.424402
$408$	0	0
$409$	−27.1240	−1.34119	−0.670597	−	0.741822i	$-0.733962\pi$
$409$	−27.1240	−1.34119	−0.670597	+	0.741822i	$0.733962\pi$
$410$	0	0
$411$	4.69535	0.231604
$412$	0	0
$413$	−18.3684	−0.903851
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.4701	−0.708605
$418$	0	0
$419$	14.4889	0.707827	0.353914	−	0.935278i	$-0.384851\pi$
$419$	14.4889	0.707827	0.353914	+	0.935278i	$0.384851\pi$
$420$	0	0
$421$	−7.67307	−0.373963	−0.186981	−	0.982363i	$-0.559870\pi$
$421$	−7.67307	−0.373963	−0.186981	+	0.982363i	$0.559870\pi$
$422$	0	0
$423$	−8.56199	−0.416298
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.7654	1.00491
$428$	0	0
$429$	0.903212	0.0436075
$430$	0	0
$431$	−8.38715	−0.403995	−0.201997	−	0.979386i	$-0.564743\pi$
$431$	−8.38715	−0.403995	−0.201997	+	0.979386i	$0.564743\pi$
$432$	0	0
$433$	−31.3461	−1.50640	−0.753200	−	0.657792i	$-0.771491\pi$
$433$	−31.3461	−1.50640	−0.753200	+	0.657792i	$0.771491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.7146	−0.847402
$438$	0	0
$439$	−37.5496	−1.79214	−0.896071	−	0.443910i	$-0.853591\pi$
$439$	−37.5496	−1.79214	−0.896071	+	0.443910i	$0.853591\pi$
$440$	0	0
$441$	−4.67307	−0.222527
$442$	0	0
$443$	10.9304	0.519319	0.259660	−	0.965700i	$-0.416390\pi$
$443$	10.9304	0.519319	0.259660	+	0.965700i	$0.416390\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.4795	0.542961
$448$	0	0
$449$	−11.2573	−0.531267	−0.265633	−	0.964074i	$-0.585581\pi$
$449$	−11.2573	−0.531267	−0.265633	+	0.964074i	$0.585581\pi$
$450$	0	0
$451$	2.23506	0.105245
$452$	0	0
$453$	3.24443	0.152437
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.3733	−1.37403	−0.687013	−	0.726645i	$-0.741079\pi$
$457$	−29.3733	−1.37403	−0.687013	+	0.726645i	$0.741079\pi$
$458$	0	0
$459$	−2.90321	−0.135510
$460$	0	0
$461$	−31.9496	−1.48804	−0.744021	−	0.668156i	$-0.767084\pi$
$461$	−31.9496	−1.48804	−0.744021	+	0.668156i	$0.767084\pi$
$462$	0	0
$463$	−34.4800	−1.60242	−0.801210	−	0.598383i	$-0.795810\pi$
$463$	−34.4800	−1.60242	−0.801210	+	0.598383i	$0.795810\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.6128	−0.537379	−0.268689	−	0.963227i	$-0.586590\pi$
$467$	−11.6128	−0.537379	−0.268689	+	0.963227i	$0.586590\pi$
$468$	0	0
$469$	−21.9180	−1.01208
$470$	0	0
$471$	7.90813	0.364388
$472$	0	0
$473$	12.5763	0.578258
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.18421	0.145795
$478$	0	0
$479$	−41.0607	−1.87611	−0.938056	−	0.346485i	$-0.887375\pi$
$479$	−41.0607	−1.87611	−0.938056	+	0.346485i	$0.887375\pi$
$480$	0	0
$481$	7.73329	0.352608
$482$	0	0
$483$	13.5111	0.614778
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.47013	−0.202561	−0.101280	−	0.994858i	$-0.532294\pi$
$487$	−4.47013	−0.202561	−0.101280	+	0.994858i	$0.532294\pi$
$488$	0	0
$489$	9.41927	0.425954
$490$	0	0
$491$	−21.5941	−0.974529	−0.487264	−	0.873255i	$-0.662005\pi$
$491$	−21.5941	−0.974529	−0.487264	+	0.873255i	$0.662005\pi$
$492$	0	0
$493$	−19.2257	−0.865882
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.5521	−0.473329
$498$	0	0
$499$	−26.8385	−1.20146	−0.600729	−	0.799453i	$-0.705123\pi$
$499$	−26.8385	−1.20146	−0.600729	+	0.799453i	$0.705123\pi$
$500$	0	0
$501$	−17.3733	−0.776184
$502$	0	0
$503$	17.7605	0.791901	0.395951	−	0.918272i	$-0.370415\pi$
$503$	17.7605	0.791901	0.395951	+	0.918272i	$0.370415\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.1842	0.541120
$508$	0	0
$509$	18.0830	0.801514	0.400757	−	0.916184i	$-0.368747\pi$
$509$	18.0830	0.801514	0.400757	+	0.916184i	$0.368747\pi$
$510$	0	0
$511$	−14.8889	−0.658647
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8.56199	−0.376556
$518$	0	0
$519$	13.5669	0.595521
$520$	0	0
$521$	2.40006	0.105149	0.0525743	−	0.998617i	$-0.483257\pi$
$521$	2.40006	0.105149	0.0525743	+	0.998617i	$0.483257\pi$
$522$	0	0
$523$	19.7101	0.861863	0.430932	−	0.902385i	$-0.358185\pi$
$523$	19.7101	0.861863	0.430932	+	0.902385i	$0.358185\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.7146	−1.29439
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	12.0415	0.522556
$532$	0	0
$533$	−2.01874	−0.0874412
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−4.67307	−0.201283
$540$	0	0
$541$	−2.59057	−0.111377	−0.0556887	−	0.998448i	$-0.517735\pi$
$541$	−2.59057	−0.111377	−0.0556887	+	0.998448i	$0.517735\pi$
$542$	0	0
$543$	−4.32693	−0.185686
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.2810	1.03818	0.519090	−	0.854719i	$-0.326271\pi$
$547$	24.2810	1.03818	0.519090	+	0.854719i	$0.326271\pi$
$548$	0	0
$549$	−13.6128	−0.580982
$550$	0	0
$551$	13.2444	0.564232
$552$	0	0
$553$	13.3560	0.567954
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−45.1798	−1.91433	−0.957164	−	0.289546i	$-0.906496\pi$
$557$	−45.1798	−1.91433	−0.957164	+	0.289546i	$0.906496\pi$
$558$	0	0
$559$	−11.3590	−0.480437
$560$	0	0
$561$	−2.90321	−0.122574
$562$	0	0
$563$	−2.97634	−0.125438	−0.0627189	−	0.998031i	$-0.519977\pi$
$563$	−2.97634	−0.125438	−0.0627189	+	0.998031i	$0.519977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.52543	−0.0640619
$568$	0	0
$569$	39.1753	1.64231	0.821157	−	0.570702i	$-0.193329\pi$
$569$	39.1753	1.64231	0.821157	+	0.570702i	$0.193329\pi$
$570$	0	0
$571$	−11.1240	−0.465524	−0.232762	−	0.972534i	$-0.574776\pi$
$571$	−11.1240	−0.465524	−0.232762	+	0.972534i	$0.574776\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.19358	0.257842	0.128921	−	0.991655i	$-0.458849\pi$
$577$	6.19358	0.257842	0.128921	+	0.991655i	$0.458849\pi$
$578$	0	0
$579$	15.5669	0.646939
$580$	0	0
$581$	−10.8256	−0.449123
$582$	0	0
$583$	3.18421	0.131876
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.22570	0.298236	0.149118	−	0.988819i	$-0.452357\pi$
$587$	7.22570	0.298236	0.149118	+	0.988819i	$0.452357\pi$
$588$	0	0
$589$	20.4701	0.843457
$590$	0	0
$591$	14.3412	0.589919
$592$	0	0
$593$	21.5955	0.886821	0.443410	−	0.896319i	$-0.353768\pi$
$593$	21.5955	0.886821	0.443410	+	0.896319i	$0.353768\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.24443	−0.0509312
$598$	0	0
$599$	38.4514	1.57108	0.785541	−	0.618810i	$-0.212385\pi$
$599$	38.4514	1.57108	0.785541	+	0.618810i	$0.212385\pi$
$600$	0	0
$601$	22.5906	0.921489	0.460744	−	0.887533i	$-0.347583\pi$
$601$	22.5906	0.921489	0.460744	+	0.887533i	$0.347583\pi$
$602$	0	0
$603$	14.3684	0.585127
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.9733	0.932457	0.466228	−	0.884664i	$-0.345613\pi$
$607$	22.9733	0.932457	0.466228	+	0.884664i	$0.345613\pi$
$608$	0	0
$609$	−10.1017	−0.409342
$610$	0	0
$611$	7.73329	0.312856
$612$	0	0
$613$	23.5383	0.950704	0.475352	−	0.879796i	$-0.342321\pi$
$613$	23.5383	0.950704	0.475352	+	0.879796i	$0.342321\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.8256	−0.918926	−0.459463	−	0.888197i	$-0.651958\pi$
$617$	−22.8256	−0.918926	−0.459463	+	0.888197i	$0.651958\pi$
$618$	0	0
$619$	−25.3778	−1.02002	−0.510010	−	0.860169i	$-0.670358\pi$
$619$	−25.3778	−1.02002	−0.510010	+	0.860169i	$0.670358\pi$
$620$	0	0
$621$	−8.85728	−0.355430
$622$	0	0
$623$	22.8671	0.916152
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−24.8573	−0.991125
$630$	0	0
$631$	28.6735	1.14148	0.570738	−	0.821132i	$-0.306657\pi$
$631$	28.6735	1.14148	0.570738	+	0.821132i	$0.306657\pi$
$632$	0	0
$633$	24.9590	0.992031
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.22077	0.167233
$638$	0	0
$639$	6.91750	0.273652
$640$	0	0
$641$	−18.6035	−0.734793	−0.367397	−	0.930064i	$-0.619751\pi$
$641$	−18.6035	−0.734793	−0.367397	+	0.930064i	$0.619751\pi$
$642$	0	0
$643$	27.9081	1.10059	0.550295	−	0.834971i	$-0.314515\pi$
$643$	27.9081	1.10059	0.550295	+	0.834971i	$0.314515\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.5620	−0.808375	−0.404188	−	0.914676i	$-0.632446\pi$
$647$	−20.5620	−0.808375	−0.404188	+	0.914676i	$0.632446\pi$
$648$	0	0
$649$	12.0415	0.472670
$650$	0	0
$651$	−15.6128	−0.611916
$652$	0	0
$653$	−10.7685	−0.421403	−0.210702	−	0.977550i	$-0.567575\pi$
$653$	−10.7685	−0.421403	−0.210702	+	0.977550i	$0.567575\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.76049	0.380793
$658$	0	0
$659$	−8.20342	−0.319560	−0.159780	−	0.987153i	$-0.551078\pi$
$659$	−8.20342	−0.319560	−0.159780	+	0.987153i	$0.551078\pi$
$660$	0	0
$661$	16.1017	0.626284	0.313142	−	0.949706i	$-0.398618\pi$
$661$	16.1017	0.626284	0.313142	+	0.949706i	$0.398618\pi$
$662$	0	0
$663$	2.62222	0.101838
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−58.6548	−2.27112
$668$	0	0
$669$	1.53972	0.0595289
$670$	0	0
$671$	−13.6128	−0.525518
$672$	0	0
$673$	10.8845	0.419566	0.209783	−	0.977748i	$-0.432724\pi$
$673$	10.8845	0.419566	0.209783	+	0.977748i	$0.432724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.60793	0.100231	0.0501154	−	0.998743i	$-0.484041\pi$
$677$	2.60793	0.100231	0.0501154	+	0.998743i	$0.484041\pi$
$678$	0	0
$679$	−11.6128	−0.445660
$680$	0	0
$681$	15.3002	0.586305
$682$	0	0
$683$	9.60300	0.367449	0.183724	−	0.982978i	$-0.441185\pi$
$683$	9.60300	0.367449	0.183724	+	0.982978i	$0.441185\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.7146	−0.599548
$688$	0	0
$689$	−2.87601	−0.109567
$690$	0	0
$691$	26.6351	1.01325	0.506624	−	0.862167i	$-0.330893\pi$
$691$	26.6351	1.01325	0.506624	+	0.862167i	$0.330893\pi$
$692$	0	0
$693$	−1.52543	−0.0579462
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.48886	0.245783
$698$	0	0
$699$	29.0049	1.09707
$700$	0	0
$701$	−2.62222	−0.0990397	−0.0495199	−	0.998773i	$-0.515769\pi$
$701$	−2.62222	−0.0990397	−0.0495199	+	0.998773i	$0.515769\pi$
$702$	0	0
$703$	17.1240	0.645843
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.6128	−0.587182
$708$	0	0
$709$	29.1842	1.09604	0.548018	−	0.836467i	$-0.315383\pi$
$709$	29.1842	1.09604	0.548018	+	0.836467i	$0.315383\pi$
$710$	0	0
$711$	−8.75557	−0.328359
$712$	0	0
$713$	−90.6548	−3.39505
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.6128	0.583072
$718$	0	0
$719$	40.9403	1.52681	0.763407	−	0.645918i	$-0.223525\pi$
$719$	40.9403	1.52681	0.763407	+	0.645918i	$0.223525\pi$
$720$	0	0
$721$	8.85728	0.329862
$722$	0	0
$723$	−5.61285	−0.208744
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.2257	−0.416338	−0.208169	−	0.978093i	$-0.566750\pi$
$727$	−11.2257	−0.416338	−0.208169	+	0.978093i	$0.566750\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.5116	1.35043
$732$	0	0
$733$	7.74176	0.285948	0.142974	−	0.989726i	$-0.454333\pi$
$733$	7.74176	0.285948	0.142974	+	0.989726i	$0.454333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.3684	0.529267
$738$	0	0
$739$	−24.1017	−0.886596	−0.443298	−	0.896374i	$-0.646192\pi$
$739$	−24.1017	−0.886596	−0.443298	+	0.896374i	$0.646192\pi$
$740$	0	0
$741$	−1.80642	−0.0663606
$742$	0	0
$743$	8.22077	0.301591	0.150795	−	0.988565i	$-0.451817\pi$
$743$	8.22077	0.301591	0.150795	+	0.988565i	$0.451817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.09679	0.259658
$748$	0	0
$749$	10.6855	0.390440
$750$	0	0
$751$	11.8163	0.431182	0.215591	−	0.976484i	$-0.430832\pi$
$751$	11.8163	0.431182	0.215591	+	0.976484i	$0.430832\pi$
$752$	0	0
$753$	2.91750	0.106320
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.6637	−0.387579	−0.193789	−	0.981043i	$-0.562078\pi$
$757$	−10.6637	−0.387579	−0.193789	+	0.981043i	$0.562078\pi$
$758$	0	0
$759$	−8.85728	−0.321499
$760$	0	0
$761$	−42.7052	−1.54806	−0.774031	−	0.633148i	$-0.781763\pi$
$761$	−42.7052	−1.54806	−0.774031	+	0.633148i	$0.781763\pi$
$762$	0	0
$763$	−14.6637	−0.530862
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.8760	−0.392710
$768$	0	0
$769$	7.63158	0.275202	0.137601	−	0.990488i	$-0.456061\pi$
$769$	7.63158	0.275202	0.137601	+	0.990488i	$0.456061\pi$
$770$	0	0
$771$	−11.0923	−0.399481
$772$	0	0
$773$	−8.52051	−0.306461	−0.153231	−	0.988190i	$-0.548968\pi$
$773$	−8.52051	−0.306461	−0.153231	+	0.988190i	$0.548968\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.0607	−0.468550
$778$	0	0
$779$	−4.47013	−0.160159
$780$	0	0
$781$	6.91750	0.247528
$782$	0	0
$783$	6.62222	0.236659
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.3131	1.47265	0.736327	−	0.676626i	$-0.236559\pi$
$787$	41.3131	1.47265	0.736327	+	0.676626i	$0.236559\pi$
$788$	0	0
$789$	−5.46520	−0.194566
$790$	0	0
$791$	17.5111	0.622624
$792$	0	0
$793$	12.2953	0.436618
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−48.7239	−1.72589	−0.862945	−	0.505298i	$-0.831383\pi$
$797$	−48.7239	−1.72589	−0.862945	+	0.505298i	$0.831383\pi$
$798$	0	0
$799$	−24.8573	−0.879387
$800$	0	0
$801$	−14.9906	−0.529668
$802$	0	0
$803$	9.76049	0.344440
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.8573	0.382194
$808$	0	0
$809$	−52.4197	−1.84298	−0.921490	−	0.388402i	$-0.873027\pi$
$809$	−52.4197	−1.84298	−0.921490	+	0.388402i	$0.873027\pi$
$810$	0	0
$811$	−22.8573	−0.802628	−0.401314	−	0.915941i	$-0.631446\pi$
$811$	−22.8573	−0.802628	−0.401314	+	0.915941i	$0.631446\pi$
$812$	0	0
$813$	0.101710	0.00356713
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25.1526	−0.879977
$818$	0	0
$819$	1.37778	0.0481437
$820$	0	0
$821$	52.5402	1.83367	0.916833	−	0.399272i	$-0.130737\pi$
$821$	52.5402	1.83367	0.916833	+	0.399272i	$0.130737\pi$
$822$	0	0
$823$	−24.8859	−0.867467	−0.433733	−	0.901041i	$-0.642804\pi$
$823$	−24.8859	−0.867467	−0.433733	+	0.901041i	$0.642804\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.60793	0.299327	0.149663	−	0.988737i	$-0.452181\pi$
$827$	8.60793	0.299327	0.149663	+	0.988737i	$0.452181\pi$
$828$	0	0
$829$	6.16193	0.214013	0.107006	−	0.994258i	$-0.465873\pi$
$829$	6.16193	0.214013	0.107006	+	0.994258i	$0.465873\pi$
$830$	0	0
$831$	−9.19850	−0.319092
$832$	0	0
$833$	−13.5669	−0.470066
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.2351	0.353776
$838$	0	0
$839$	−46.4929	−1.60511	−0.802556	−	0.596577i	$-0.796527\pi$
$839$	−46.4929	−1.60511	−0.802556	+	0.596577i	$0.796527\pi$
$840$	0	0
$841$	14.8537	0.512198
$842$	0	0
$843$	1.11108	0.0382676
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.52543	−0.0524143
$848$	0	0
$849$	−15.7190	−0.539475
$850$	0	0
$851$	−75.8360	−2.59962
$852$	0	0
$853$	0.699791	0.0239604	0.0119802	−	0.999928i	$-0.496186\pi$
$853$	0.699791	0.0239604	0.0119802	+	0.999928i	$0.496186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.0558	−0.548455	−0.274227	−	0.961665i	$-0.588422\pi$
$857$	−16.0558	−0.548455	−0.274227	+	0.961665i	$0.588422\pi$
$858$	0	0
$859$	1.11108	0.0379095	0.0189547	−	0.999820i	$-0.493966\pi$
$859$	1.11108	0.0379095	0.0189547	+	0.999820i	$0.493966\pi$
$860$	0	0
$861$	3.40943	0.116193
$862$	0	0
$863$	43.0134	1.46419	0.732096	−	0.681201i	$-0.238542\pi$
$863$	43.0134	1.46419	0.732096	+	0.681201i	$0.238542\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.57136	0.291099
$868$	0	0
$869$	−8.75557	−0.297012
$870$	0	0
$871$	−12.9777	−0.439733
$872$	0	0
$873$	7.61285	0.257656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.03704	−0.271392	−0.135696	−	0.990751i	$-0.543327\pi$
$877$	−8.03704	−0.271392	−0.135696	+	0.990751i	$0.543327\pi$
$878$	0	0
$879$	6.43309	0.216983
$880$	0	0
$881$	−17.9367	−0.604303	−0.302152	−	0.953260i	$-0.597705\pi$
$881$	−17.9367	−0.604303	−0.302152	+	0.953260i	$0.597705\pi$
$882$	0	0
$883$	−3.22570	−0.108553	−0.0542766	−	0.998526i	$-0.517285\pi$
$883$	−3.22570	−0.108553	−0.0542766	+	0.998526i	$0.517285\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.43662	0.0482371	0.0241186	−	0.999709i	$-0.492322\pi$
$887$	1.43662	0.0482371	0.0241186	+	0.999709i	$0.492322\pi$
$888$	0	0
$889$	26.5936	0.891922
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	17.1240	0.573032
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	67.7788	2.26055
$900$	0	0
$901$	9.24443	0.307977
$902$	0	0
$903$	19.1842	0.638410
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.4380	0.645428	0.322714	−	0.946496i	$-0.395405\pi$
$907$	19.4380	0.645428	0.322714	+	0.946496i	$0.395405\pi$
$908$	0	0
$909$	10.2351	0.339476
$910$	0	0
$911$	16.6735	0.552419	0.276210	−	0.961097i	$-0.410922\pi$
$911$	16.6735	0.552419	0.276210	+	0.961097i	$0.410922\pi$
$912$	0	0
$913$	7.09679	0.234869
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.89829	−0.0626871
$918$	0	0
$919$	−31.4479	−1.03737	−0.518684	−	0.854966i	$-0.673578\pi$
$919$	−31.4479	−1.03737	−0.518684	+	0.854966i	$0.673578\pi$
$920$	0	0
$921$	−15.7190	−0.517959
$922$	0	0
$923$	−6.24797	−0.205654
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.80642	−0.190708
$928$	0	0
$929$	−44.3881	−1.45633	−0.728163	−	0.685404i	$-0.759626\pi$
$929$	−44.3881	−1.45633	−0.728163	+	0.685404i	$0.759626\pi$
$930$	0	0
$931$	9.34614	0.306307
$932$	0	0
$933$	−24.8988	−0.815149
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.6686	0.707883	0.353942	−	0.935268i	$-0.384841\pi$
$937$	21.6686	0.707883	0.353942	+	0.935268i	$0.384841\pi$
$938$	0	0
$939$	−17.1526	−0.559753
$940$	0	0
$941$	15.4795	0.504617	0.252309	−	0.967647i	$-0.418810\pi$
$941$	15.4795	0.504617	0.252309	+	0.967647i	$0.418810\pi$
$942$	0	0
$943$	19.7966	0.644665
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.0098	1.62510	0.812551	−	0.582890i	$-0.198078\pi$
$947$	50.0098	1.62510	0.812551	+	0.582890i	$0.198078\pi$
$948$	0	0
$949$	−8.81579	−0.286173
$950$	0	0
$951$	−12.3368	−0.400047
$952$	0	0
$953$	−10.9032	−0.353190	−0.176595	−	0.984284i	$-0.556508\pi$
$953$	−10.9032	−0.353190	−0.176595	+	0.984284i	$0.556508\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.62222	0.214066
$958$	0	0
$959$	7.16241	0.231286
$960$	0	0
$961$	73.7565	2.37924
$962$	0	0
$963$	−7.00492	−0.225730
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.6815	−1.40470	−0.702352	−	0.711830i	$-0.747867\pi$
$967$	−43.6815	−1.40470	−0.702352	+	0.711830i	$0.747867\pi$
$968$	0	0
$969$	5.80642	0.186529
$970$	0	0
$971$	−2.55215	−0.0819023	−0.0409512	−	0.999161i	$-0.513039\pi$
$971$	−2.55215	−0.0819023	−0.0409512	+	0.999161i	$0.513039\pi$
$972$	0	0
$973$	−22.0731	−0.707632
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.40990	0.205071	0.102535	−	0.994729i	$-0.467304\pi$
$977$	6.40990	0.205071	0.102535	+	0.994729i	$0.467304\pi$
$978$	0	0
$979$	−14.9906	−0.479103
$980$	0	0
$981$	9.61285	0.306915
$982$	0	0
$983$	−40.1748	−1.28138	−0.640689	−	0.767800i	$-0.721351\pi$
$983$	−40.1748	−1.28138	−0.640689	+	0.767800i	$0.721351\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.0607	−0.415727
$988$	0	0
$989$	111.392	3.54205
$990$	0	0
$991$	18.2351	0.579256	0.289628	−	0.957139i	$-0.406468\pi$
$991$	18.2351	0.579256	0.289628	+	0.957139i	$0.406468\pi$
$992$	0	0
$993$	9.37778	0.297595
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.16638	0.131951	0.0659753	−	0.997821i	$-0.478984\pi$
$997$	4.16638	0.131951	0.0659753	+	0.997821i	$0.478984\pi$
$998$	0	0
$999$	8.56199	0.270889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.a.bq.1.1		3
5.2	odd	4		1320.2.d.b.529.5	yes	6
5.3	odd	4		1320.2.d.b.529.2	✓	6
5.4	even	2		6600.2.a.bu.1.3		3
15.2	even	4		3960.2.d.d.3169.4		6
15.8	even	4		3960.2.d.d.3169.3		6
20.3	even	4		2640.2.d.f.529.5		6
20.7	even	4		2640.2.d.f.529.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.d.b.529.2	✓	6	5.3	odd	4
1320.2.d.b.529.5	yes	6	5.2	odd	4
2640.2.d.f.529.2		6	20.7	even	4
2640.2.d.f.529.5		6	20.3	even	4
3960.2.d.d.3169.3		6	15.8	even	4
3960.2.d.d.3169.4		6	15.2	even	4
6600.2.a.bq.1.1		3	1.1	even	1	trivial
6600.2.a.bu.1.3		3	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.52543 q^{7} +1.00000 q^{9} +1.00000 q^{11} -0.903212 q^{13} +2.90321 q^{17} -2.00000 q^{19} +1.52543 q^{21} +8.85728 q^{23} -1.00000 q^{27} -6.62222 q^{29} -10.2351 q^{31} -1.00000 q^{33} -8.56199 q^{37} +0.903212 q^{39} +2.23506 q^{41} +12.5763 q^{43} -8.56199 q^{47} -4.67307 q^{49} -2.90321 q^{51} +3.18421 q^{53} +2.00000 q^{57} +12.0415 q^{59} -13.6128 q^{61} -1.52543 q^{63} +14.3684 q^{67} -8.85728 q^{69} +6.91750 q^{71} +9.76049 q^{73} -1.52543 q^{77} -8.75557 q^{79} +1.00000 q^{81} +7.09679 q^{83} +6.62222 q^{87} -14.9906 q^{89} +1.37778 q^{91} +10.2351 q^{93} +7.61285 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{27} - 20 q^{29} - 4 q^{31} - 3 q^{33} - 12 q^{37} - 4 q^{39} - 20 q^{41} + 18 q^{43} - 12 q^{47} - q^{49}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^27 - 20 * q^29 - 4 * q^31 - 3 * q^33 - 12 * q^37 - 4 * q^39 - 20 * q^41 + 18 * q^43 - 12 * q^47 - q^49 - 2 * q^51 - 4 * q^53 + 6 * q^57 - 4 * q^59 - 14 * q^61 + 2 * q^63 + 16 * q^67 + 8 * q^71 - 4 * q^73 + 2 * q^77 - 26 * q^79 + 3 * q^81 + 28 * q^83 + 20 * q^87 - 18 * q^89 + 4 * q^91 + 4 * q^93 - 4 * q^97 + 3 * q^99

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