LMFDB - Embedded newform 2475.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108 q^{2} -1.90321 q^{4} -0.903212 q^{7} +1.21432 q^{8} -1.00000 q^{11} +2.90321 q^{13} +0.280996 q^{14} +3.42864 q^{16} -2.28100 q^{17} +2.42864 q^{19} +0.311108 q^{22} -4.00000 q^{23} -0.903212 q^{26} +1.71900 q^{28} -7.05086 q^{29} -2.62222 q^{31} -3.49532 q^{32} +0.709636 q^{34} +5.80642 q^{37} -0.755569 q^{38} +10.6637 q^{41} +10.7096 q^{43} +1.90321 q^{44} +1.24443 q^{46} -0.949145 q^{47} -6.18421 q^{49} -5.52543 q^{52} -0.815792 q^{53} -1.09679 q^{56} +2.19358 q^{58} +1.67307 q^{59} -7.24443 q^{61} +0.815792 q^{62} -5.76986 q^{64} +12.8573 q^{67} +4.34122 q^{68} -9.28592 q^{71} -5.65878 q^{73} -1.80642 q^{74} -4.62222 q^{76} +0.903212 q^{77} -16.5303 q^{79} -3.31756 q^{82} -7.76049 q^{83} -3.33185 q^{86} -1.21432 q^{88} -6.13335 q^{89} -2.62222 q^{91} +7.61285 q^{92} +0.295286 q^{94} -12.4701 q^{97} +1.92396 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8} - 3 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} + q^{22} - 12 q^{23} + 4 q^{26} + 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + 4 q^{37} - 2 q^{38}+ \cdots - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8 - 3 * q^11 + 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 + q^22 - 12 * q^23 + 4 * q^26 + 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 + 12 * q^43 - q^44 + 4 * q^46 - 16 * q^47 - 5 * q^49 - 10 * q^52 - 16 * q^53 - 10 * q^56 + 20 * q^58 - 8 * q^59 - 22 * q^61 + 16 * q^62 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^71 - 10 * q^73 + 8 * q^74 - 14 * q^76 - 4 * q^77 - 10 * q^79 + 4 * q^82 + 10 * q^83 + 10 * q^86 + 3 * q^88 - 18 * q^89 - 8 * q^91 - 4 * q^92 - 12 * q^94 + 16 * q^97 - 21 * q^98

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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.903212	−0.341382	−0.170691	−	0.985325i	$-0.554600\pi$
$7$	−0.903212	−0.341382	−0.170691	+	0.985325i	$0.554600\pi$
$8$	1.21432	0.429327
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.90321	0.805206	0.402603	−	0.915375i	$-0.368106\pi$
$13$	2.90321	0.805206	0.402603	+	0.915375i	$0.368106\pi$
$14$	0.280996	0.0750994
$15$	0	0
$16$	3.42864	0.857160
$17$	−2.28100	−0.553223	−0.276611	−	0.960982i	$-0.589211\pi$
$17$	−2.28100	−0.553223	−0.276611	+	0.960982i	$0.589211\pi$
$18$	0	0
$19$	2.42864	0.557168	0.278584	−	0.960412i	$-0.410135\pi$
$19$	2.42864	0.557168	0.278584	+	0.960412i	$0.410135\pi$
$20$	0	0
$21$	0	0
$22$	0.311108	0.0663284
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	−0.903212	−0.177134
$27$	0	0
$28$	1.71900	0.324861
$29$	−7.05086	−1.30931	−0.654655	−	0.755927i	$-0.727186\pi$
$29$	−7.05086	−1.30931	−0.654655	+	0.755927i	$0.727186\pi$
$30$	0	0
$31$	−2.62222	−0.470964	−0.235482	−	0.971879i	$-0.575667\pi$
$31$	−2.62222	−0.470964	−0.235482	+	0.971879i	$0.575667\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	0.709636	0.121702
$35$	0	0
$36$	0	0
$37$	5.80642	0.954570	0.477285	−	0.878749i	$-0.341621\pi$
$37$	5.80642	0.954570	0.477285	+	0.878749i	$0.341621\pi$
$38$	−0.755569	−0.122569
$39$	0	0
$40$	0	0
$41$	10.6637	1.66539	0.832695	−	0.553731i	$-0.186797\pi$
$41$	10.6637	1.66539	0.832695	+	0.553731i	$0.186797\pi$
$42$	0	0
$43$	10.7096	1.63320	0.816602	−	0.577201i	$-0.195855\pi$
$43$	10.7096	1.63320	0.816602	+	0.577201i	$0.195855\pi$
$44$	1.90321	0.286920
$45$	0	0
$46$	1.24443	0.183481
$47$	−0.949145	−0.138447	−0.0692235	−	0.997601i	$-0.522052\pi$
$47$	−0.949145	−0.138447	−0.0692235	+	0.997601i	$0.522052\pi$
$48$	0	0
$49$	−6.18421	−0.883458
$50$	0	0
$51$	0	0
$52$	−5.52543	−0.766239
$53$	−0.815792	−0.112058	−0.0560288	−	0.998429i	$-0.517844\pi$
$53$	−0.815792	−0.112058	−0.0560288	+	0.998429i	$0.517844\pi$
$54$	0	0
$55$	0	0
$56$	−1.09679	−0.146564
$57$	0	0
$58$	2.19358	0.288031
$59$	1.67307	0.217815	0.108908	−	0.994052i	$-0.465265\pi$
$59$	1.67307	0.217815	0.108908	+	0.994052i	$0.465265\pi$
$60$	0	0
$61$	−7.24443	−0.927554	−0.463777	−	0.885952i	$-0.653506\pi$
$61$	−7.24443	−0.927554	−0.463777	+	0.885952i	$0.653506\pi$
$62$	0.815792	0.103606
$63$	0	0
$64$	−5.76986	−0.721232
$65$	0	0
$66$	0	0
$67$	12.8573	1.57077	0.785383	−	0.619010i	$-0.212466\pi$
$67$	12.8573	1.57077	0.785383	+	0.619010i	$0.212466\pi$
$68$	4.34122	0.526450
$69$	0	0
$70$	0	0
$71$	−9.28592	−1.10204	−0.551018	−	0.834493i	$-0.685760\pi$
$71$	−9.28592	−1.10204	−0.551018	+	0.834493i	$0.685760\pi$
$72$	0	0
$73$	−5.65878	−0.662310	−0.331155	−	0.943576i	$-0.607438\pi$
$73$	−5.65878	−0.662310	−0.331155	+	0.943576i	$0.607438\pi$
$74$	−1.80642	−0.209993
$75$	0	0
$76$	−4.62222	−0.530204
$77$	0.903212	0.102931
$78$	0	0
$79$	−16.5303	−1.85981	−0.929905	−	0.367800i	$-0.880111\pi$
$79$	−16.5303	−1.85981	−0.929905	+	0.367800i	$0.880111\pi$
$80$	0	0
$81$	0	0
$82$	−3.31756	−0.366363
$83$	−7.76049	−0.851825	−0.425912	−	0.904764i	$-0.640047\pi$
$83$	−7.76049	−0.851825	−0.425912	+	0.904764i	$0.640047\pi$
$84$	0	0
$85$	0	0
$86$	−3.33185	−0.359283
$87$	0	0
$88$	−1.21432	−0.129447
$89$	−6.13335	−0.650134	−0.325067	−	0.945691i	$-0.605387\pi$
$89$	−6.13335	−0.650134	−0.325067	+	0.945691i	$0.605387\pi$
$90$	0	0
$91$	−2.62222	−0.274883
$92$	7.61285	0.793694
$93$	0	0
$94$	0.295286	0.0304565
$95$	0	0
$96$	0	0
$97$	−12.4701	−1.26615	−0.633075	−	0.774091i	$-0.718207\pi$
$97$	−12.4701	−1.26615	−0.633075	+	0.774091i	$0.718207\pi$
$98$	1.92396	0.194349
$99$	0	0
$100$	0	0
$101$	−16.1748	−1.60946	−0.804728	−	0.593643i	$-0.797689\pi$
$101$	−16.1748	−1.60946	−0.804728	+	0.593643i	$0.797689\pi$
$102$	0	0
$103$	−17.1526	−1.69009	−0.845046	−	0.534693i	$-0.820427\pi$
$103$	−17.1526	−1.69009	−0.845046	+	0.534693i	$0.820427\pi$
$104$	3.52543	0.345697
$105$	0	0
$106$	0.253799	0.0246512
$107$	13.5669	1.31156	0.655782	−	0.754951i	$-0.272339\pi$
$107$	13.5669	1.31156	0.655782	+	0.754951i	$0.272339\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	−3.09679	−0.292619
$113$	−14.2351	−1.33912	−0.669561	−	0.742757i	$-0.733518\pi$
$113$	−14.2351	−1.33912	−0.669561	+	0.742757i	$0.733518\pi$
$114$	0	0
$115$	0	0
$116$	13.4193	1.24595
$117$	0	0
$118$	−0.520505	−0.0479164
$119$	2.06022	0.188860
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.25380	0.204049
$123$	0	0
$124$	4.99063	0.448172
$125$	0	0
$126$	0	0
$127$	11.0049	0.976529	0.488264	−	0.872696i	$-0.337630\pi$
$127$	11.0049	0.976529	0.488264	+	0.872696i	$0.337630\pi$
$128$	8.78568	0.776552
$129$	0	0
$130$	0	0
$131$	−1.24443	−0.108726	−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443	−0.108726	−0.0543632	+	0.998521i	$0.517313\pi$
$132$	0	0
$133$	−2.19358	−0.190207
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−2.76986	−0.237513
$137$	4.42864	0.378364	0.189182	−	0.981942i	$-0.439416\pi$
$137$	4.42864	0.378364	0.189182	+	0.981942i	$0.439416\pi$
$138$	0	0
$139$	0.917502	0.0778215	0.0389108	−	0.999243i	$-0.487611\pi$
$139$	0.917502	0.0778215	0.0389108	+	0.999243i	$0.487611\pi$
$140$	0	0
$141$	0	0
$142$	2.88892	0.242433
$143$	−2.90321	−0.242779
$144$	0	0
$145$	0	0
$146$	1.76049	0.145699
$147$	0	0
$148$	−11.0509	−0.908375
$149$	−2.19358	−0.179705	−0.0898524	−	0.995955i	$-0.528640\pi$
$149$	−2.19358	−0.179705	−0.0898524	+	0.995955i	$0.528640\pi$
$150$	0	0
$151$	−10.4286	−0.848671	−0.424335	−	0.905505i	$-0.639492\pi$
$151$	−10.4286	−0.848671	−0.424335	+	0.905505i	$0.639492\pi$
$152$	2.94914	0.239207
$153$	0	0
$154$	−0.280996	−0.0226433
$155$	0	0
$156$	0	0
$157$	5.80642	0.463403	0.231702	−	0.972787i	$-0.425571\pi$
$157$	5.80642	0.463403	0.231702	+	0.972787i	$0.425571\pi$
$158$	5.14272	0.409133
$159$	0	0
$160$	0	0
$161$	3.61285	0.284732
$162$	0	0
$163$	11.0509	0.865570	0.432785	−	0.901497i	$-0.357531\pi$
$163$	11.0509	0.865570	0.432785	+	0.901497i	$0.357531\pi$
$164$	−20.2953	−1.58480
$165$	0	0
$166$	2.41435	0.187390
$167$	−13.9541	−1.07980	−0.539899	−	0.841730i	$-0.681538\pi$
$167$	−13.9541	−1.07980	−0.539899	+	0.841730i	$0.681538\pi$
$168$	0	0
$169$	−4.57136	−0.351643
$170$	0	0
$171$	0	0
$172$	−20.3827	−1.55417
$173$	18.8430	1.43261	0.716303	−	0.697789i	$-0.245833\pi$
$173$	18.8430	1.43261	0.716303	+	0.697789i	$0.245833\pi$
$174$	0	0
$175$	0	0
$176$	−3.42864	−0.258443
$177$	0	0
$178$	1.90813	0.143021
$179$	−4.85728	−0.363050	−0.181525	−	0.983386i	$-0.558103\pi$
$179$	−4.85728	−0.363050	−0.181525	+	0.983386i	$0.558103\pi$
$180$	0	0
$181$	−16.7971	−1.24852	−0.624258	−	0.781219i	$-0.714598\pi$
$181$	−16.7971	−1.24852	−0.624258	+	0.781219i	$0.714598\pi$
$182$	0.815792	0.0604705
$183$	0	0
$184$	−4.85728	−0.358083
$185$	0	0
$186$	0	0
$187$	2.28100	0.166803
$188$	1.80642	0.131747
$189$	0	0
$190$	0	0
$191$	−16.8573	−1.21975	−0.609875	−	0.792498i	$-0.708780\pi$
$191$	−16.8573	−1.21975	−0.609875	+	0.792498i	$0.708780\pi$
$192$	0	0
$193$	−24.6178	−1.77203	−0.886013	−	0.463661i	$-0.846536\pi$
$193$	−24.6178	−1.77203	−0.886013	+	0.463661i	$0.846536\pi$
$194$	3.87955	0.278536
$195$	0	0
$196$	11.7699	0.840704
$197$	−14.8716	−1.05956	−0.529778	−	0.848137i	$-0.677725\pi$
$197$	−14.8716	−1.05956	−0.529778	+	0.848137i	$0.677725\pi$
$198$	0	0
$199$	−11.2257	−0.795768	−0.397884	−	0.917436i	$-0.630255\pi$
$199$	−11.2257	−0.795768	−0.397884	+	0.917436i	$0.630255\pi$
$200$	0	0
$201$	0	0
$202$	5.03212	0.354059
$203$	6.36842	0.446975
$204$	0	0
$205$	0	0
$206$	5.33630	0.371797
$207$	0	0
$208$	9.95407	0.690190
$209$	−2.42864	−0.167993
$210$	0	0
$211$	−11.9398	−0.821968	−0.410984	−	0.911643i	$-0.634815\pi$
$211$	−11.9398	−0.821968	−0.410984	+	0.911643i	$0.634815\pi$
$212$	1.55262	0.106635
$213$	0	0
$214$	−4.22077	−0.288526
$215$	0	0
$216$	0	0
$217$	2.36842	0.160779
$218$	3.11108	0.210709
$219$	0	0
$220$	0	0
$221$	−6.62222	−0.445458
$222$	0	0
$223$	21.8064	1.46027	0.730133	−	0.683305i	$-0.239458\pi$
$223$	21.8064	1.46027	0.730133	+	0.683305i	$0.239458\pi$
$224$	3.15701	0.210937
$225$	0	0
$226$	4.42864	0.294589
$227$	−3.19850	−0.212292	−0.106146	−	0.994351i	$-0.533851\pi$
$227$	−3.19850	−0.212292	−0.106146	+	0.994351i	$0.533851\pi$
$228$	0	0
$229$	7.12399	0.470766	0.235383	−	0.971903i	$-0.424366\pi$
$229$	7.12399	0.470766	0.235383	+	0.971903i	$0.424366\pi$
$230$	0	0
$231$	0	0
$232$	−8.56199	−0.562122
$233$	19.5254	1.27915	0.639577	−	0.768727i	$-0.279110\pi$
$233$	19.5254	1.27915	0.639577	+	0.768727i	$0.279110\pi$
$234$	0	0
$235$	0	0
$236$	−3.18421	−0.207274
$237$	0	0
$238$	−0.640951	−0.0415467
$239$	21.9813	1.42185	0.710925	−	0.703268i	$-0.248277\pi$
$239$	21.9813	1.42185	0.710925	+	0.703268i	$0.248277\pi$
$240$	0	0
$241$	5.34614	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$241$	5.34614	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$242$	−0.311108	−0.0199988
$243$	0	0
$244$	13.7877	0.882666
$245$	0	0
$246$	0	0
$247$	7.05086	0.448635
$248$	−3.18421	−0.202197
$249$	0	0
$250$	0	0
$251$	23.7748	1.50065	0.750325	−	0.661069i	$-0.229897\pi$
$251$	23.7748	1.50065	0.750325	+	0.661069i	$0.229897\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−3.42372	−0.214823
$255$	0	0
$256$	8.80642	0.550401
$257$	8.13335	0.507345	0.253672	−	0.967290i	$-0.418362\pi$
$257$	8.13335	0.507345	0.253672	+	0.967290i	$0.418362\pi$
$258$	0	0
$259$	−5.24443	−0.325873
$260$	0	0
$261$	0	0
$262$	0.387152	0.0239183
$263$	−22.9032	−1.41227	−0.706136	−	0.708076i	$-0.749563\pi$
$263$	−22.9032	−1.41227	−0.706136	+	0.708076i	$0.749563\pi$
$264$	0	0
$265$	0	0
$266$	0.682439	0.0418430
$267$	0	0
$268$	−24.4701	−1.49475
$269$	−11.8350	−0.721593	−0.360796	−	0.932645i	$-0.617495\pi$
$269$	−11.8350	−0.721593	−0.360796	+	0.932645i	$0.617495\pi$
$270$	0	0
$271$	14.8988	0.905036	0.452518	−	0.891755i	$-0.350526\pi$
$271$	14.8988	0.905036	0.452518	+	0.891755i	$0.350526\pi$
$272$	−7.82071	−0.474200
$273$	0	0
$274$	−1.37778	−0.0832350
$275$	0	0
$276$	0	0
$277$	−27.6686	−1.66245	−0.831223	−	0.555939i	$-0.812359\pi$
$277$	−27.6686	−1.66245	−0.831223	+	0.555939i	$0.812359\pi$
$278$	−0.285442	−0.0171197
$279$	0	0
$280$	0	0
$281$	−9.80642	−0.585002	−0.292501	−	0.956265i	$-0.594488\pi$
$281$	−9.80642	−0.585002	−0.292501	+	0.956265i	$0.594488\pi$
$282$	0	0
$283$	−19.0049	−1.12973	−0.564863	−	0.825185i	$-0.691071\pi$
$283$	−19.0049	−1.12973	−0.564863	+	0.825185i	$0.691071\pi$
$284$	17.6731	1.04870
$285$	0	0
$286$	0.903212	0.0534080
$287$	−9.63158	−0.568534
$288$	0	0
$289$	−11.7971	−0.693944
$290$	0	0
$291$	0	0
$292$	10.7699	0.630258
$293$	30.7511	1.79650	0.898250	−	0.439485i	$-0.144839\pi$
$293$	30.7511	1.79650	0.898250	+	0.439485i	$0.144839\pi$
$294$	0	0
$295$	0	0
$296$	7.05086	0.409823
$297$	0	0
$298$	0.682439	0.0395326
$299$	−11.6128	−0.671588
$300$	0	0
$301$	−9.67307	−0.557547
$302$	3.24443	0.186696
$303$	0	0
$304$	8.32693	0.477582
$305$	0	0
$306$	0	0
$307$	−13.4938	−0.770131	−0.385065	−	0.922889i	$-0.625821\pi$
$307$	−13.4938	−0.770131	−0.385065	+	0.922889i	$0.625821\pi$
$308$	−1.71900	−0.0979493
$309$	0	0
$310$	0	0
$311$	17.5526	0.995318	0.497659	−	0.867373i	$-0.334193\pi$
$311$	17.5526	0.995318	0.497659	+	0.867373i	$0.334193\pi$
$312$	0	0
$313$	14.3970	0.813766	0.406883	−	0.913480i	$-0.366616\pi$
$313$	14.3970	0.813766	0.406883	+	0.913480i	$0.366616\pi$
$314$	−1.80642	−0.101942
$315$	0	0
$316$	31.4608	1.76981
$317$	−29.4608	−1.65468	−0.827341	−	0.561701i	$-0.810147\pi$
$317$	−29.4608	−1.65468	−0.827341	+	0.561701i	$0.810147\pi$
$318$	0	0
$319$	7.05086	0.394772
$320$	0	0
$321$	0	0
$322$	−1.12399	−0.0626372
$323$	−5.53972	−0.308238
$324$	0	0
$325$	0	0
$326$	−3.43801	−0.190414
$327$	0	0
$328$	12.9491	0.714997
$329$	0.857279	0.0472633
$330$	0	0
$331$	−2.62222	−0.144130	−0.0720650	−	0.997400i	$-0.522959\pi$
$331$	−2.62222	−0.144130	−0.0720650	+	0.997400i	$0.522959\pi$
$332$	14.7699	0.810601
$333$	0	0
$334$	4.34122	0.237541
$335$	0	0
$336$	0	0
$337$	−5.00492	−0.272635	−0.136318	−	0.990665i	$-0.543527\pi$
$337$	−5.00492	−0.272635	−0.136318	+	0.990665i	$0.543527\pi$
$338$	1.42219	0.0773567
$339$	0	0
$340$	0	0
$341$	2.62222	0.142001
$342$	0	0
$343$	11.9081	0.642979
$344$	13.0049	0.701178
$345$	0	0
$346$	−5.86220	−0.315154
$347$	22.8113	1.22458	0.612289	−	0.790634i	$-0.290249\pi$
$347$	22.8113	1.22458	0.612289	+	0.790634i	$0.290249\pi$
$348$	0	0
$349$	21.2257	1.13619	0.568093	−	0.822965i	$-0.307681\pi$
$349$	21.2257	1.13619	0.568093	+	0.822965i	$0.307681\pi$
$350$	0	0
$351$	0	0
$352$	3.49532	0.186301
$353$	−7.18421	−0.382377	−0.191188	−	0.981553i	$-0.561234\pi$
$353$	−7.18421	−0.382377	−0.191188	+	0.981553i	$0.561234\pi$
$354$	0	0
$355$	0	0
$356$	11.6731	0.618672
$357$	0	0
$358$	1.51114	0.0798661
$359$	14.1017	0.744260	0.372130	−	0.928181i	$-0.378628\pi$
$359$	14.1017	0.744260	0.372130	+	0.928181i	$0.378628\pi$
$360$	0	0
$361$	−13.1017	−0.689564
$362$	5.22570	0.274656
$363$	0	0
$364$	4.99063	0.261580
$365$	0	0
$366$	0	0
$367$	−3.90813	−0.204003	−0.102001	−	0.994784i	$-0.532525\pi$
$367$	−3.90813	−0.204003	−0.102001	+	0.994784i	$0.532525\pi$
$368$	−13.7146	−0.714921
$369$	0	0
$370$	0	0
$371$	0.736833	0.0382545
$372$	0	0
$373$	12.9763	0.671890	0.335945	−	0.941882i	$-0.390944\pi$
$373$	12.9763	0.671890	0.335945	+	0.941882i	$0.390944\pi$
$374$	−0.709636	−0.0366944
$375$	0	0
$376$	−1.15257	−0.0594390
$377$	−20.4701	−1.05427
$378$	0	0
$379$	36.0830	1.85346	0.926729	−	0.375731i	$-0.122608\pi$
$379$	36.0830	1.85346	0.926729	+	0.375731i	$0.122608\pi$
$380$	0	0
$381$	0	0
$382$	5.24443	0.268328
$383$	−20.2953	−1.03704	−0.518520	−	0.855065i	$-0.673517\pi$
$383$	−20.2953	−1.03704	−0.518520	+	0.855065i	$0.673517\pi$
$384$	0	0
$385$	0	0
$386$	7.65878	0.389822
$387$	0	0
$388$	23.7333	1.20488
$389$	−30.4701	−1.54490	−0.772448	−	0.635078i	$-0.780968\pi$
$389$	−30.4701	−1.54490	−0.772448	+	0.635078i	$0.780968\pi$
$390$	0	0
$391$	9.12399	0.461420
$392$	−7.50961	−0.379292
$393$	0	0
$394$	4.62666	0.233088
$395$	0	0
$396$	0	0
$397$	4.97773	0.249825	0.124912	−	0.992168i	$-0.460135\pi$
$397$	4.97773	0.249825	0.124912	+	0.992168i	$0.460135\pi$
$398$	3.49240	0.175058
$399$	0	0
$400$	0	0
$401$	1.86665	0.0932159	0.0466079	−	0.998913i	$-0.485159\pi$
$401$	1.86665	0.0932159	0.0466079	+	0.998913i	$0.485159\pi$
$402$	0	0
$403$	−7.61285	−0.379223
$404$	30.7841	1.53157
$405$	0	0
$406$	−1.98126	−0.0983285
$407$	−5.80642	−0.287814
$408$	0	0
$409$	−3.63158	−0.179570	−0.0897851	−	0.995961i	$-0.528618\pi$
$409$	−3.63158	−0.179570	−0.0897851	+	0.995961i	$0.528618\pi$
$410$	0	0
$411$	0	0
$412$	32.6450	1.60830
$413$	−1.51114	−0.0743582
$414$	0	0
$415$	0	0
$416$	−10.1476	−0.497529
$417$	0	0
$418$	0.755569	0.0369561
$419$	−4.85728	−0.237294	−0.118647	−	0.992937i	$-0.537856\pi$
$419$	−4.85728	−0.237294	−0.118647	+	0.992937i	$0.537856\pi$
$420$	0	0
$421$	22.6321	1.10302	0.551510	−	0.834169i	$-0.314052\pi$
$421$	22.6321	1.10302	0.551510	+	0.834169i	$0.314052\pi$
$422$	3.71456	0.180822
$423$	0	0
$424$	−0.990632	−0.0481093
$425$	0	0
$426$	0	0
$427$	6.54326	0.316650
$428$	−25.8207	−1.24809
$429$	0	0
$430$	0	0
$431$	−1.24443	−0.0599421	−0.0299711	−	0.999551i	$-0.509542\pi$
$431$	−1.24443	−0.0599421	−0.0299711	+	0.999551i	$0.509542\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−0.736833	−0.0353691
$435$	0	0
$436$	19.0321	0.911473
$437$	−9.71456	−0.464710
$438$	0	0
$439$	−2.42864	−0.115913	−0.0579563	−	0.998319i	$-0.518458\pi$
$439$	−2.42864	−0.115913	−0.0579563	+	0.998319i	$0.518458\pi$
$440$	0	0
$441$	0	0
$442$	2.06022	0.0979948
$443$	−31.0509	−1.47527	−0.737635	−	0.675199i	$-0.764058\pi$
$443$	−31.0509	−1.47527	−0.737635	+	0.675199i	$0.764058\pi$
$444$	0	0
$445$	0	0
$446$	−6.78415	−0.321239
$447$	0	0
$448$	5.21141	0.246216
$449$	37.3590	1.76308	0.881541	−	0.472107i	$-0.156506\pi$
$449$	37.3590	1.76308	0.881541	+	0.472107i	$0.156506\pi$
$450$	0	0
$451$	−10.6637	−0.502134
$452$	27.0923	1.27432
$453$	0	0
$454$	0.995078	0.0467013
$455$	0	0
$456$	0	0
$457$	8.73822	0.408757	0.204378	−	0.978892i	$-0.434483\pi$
$457$	8.73822	0.408757	0.204378	+	0.978892i	$0.434483\pi$
$458$	−2.21633	−0.103562
$459$	0	0
$460$	0	0
$461$	−31.7877	−1.48050	−0.740250	−	0.672332i	$-0.765293\pi$
$461$	−31.7877	−1.48050	−0.740250	+	0.672332i	$0.765293\pi$
$462$	0	0
$463$	12.0919	0.561957	0.280978	−	0.959714i	$-0.409341\pi$
$463$	12.0919	0.561957	0.280978	+	0.959714i	$0.409341\pi$
$464$	−24.1748	−1.12229
$465$	0	0
$466$	−6.07451	−0.281396
$467$	15.3461	0.710135	0.355067	−	0.934841i	$-0.384458\pi$
$467$	15.3461	0.710135	0.355067	+	0.934841i	$0.384458\pi$
$468$	0	0
$469$	−11.6128	−0.536231
$470$	0	0
$471$	0	0
$472$	2.03164	0.0935139
$473$	−10.7096	−0.492430
$474$	0	0
$475$	0	0
$476$	−3.92104	−0.179721
$477$	0	0
$478$	−6.83854	−0.312788
$479$	−5.89829	−0.269500	−0.134750	−	0.990880i	$-0.543023\pi$
$479$	−5.89829	−0.269500	−0.134750	+	0.990880i	$0.543023\pi$
$480$	0	0
$481$	16.8573	0.768626
$482$	−1.66323	−0.0757579
$483$	0	0
$484$	−1.90321	−0.0865096
$485$	0	0
$486$	0	0
$487$	31.3461	1.42043	0.710215	−	0.703985i	$-0.248598\pi$
$487$	31.3461	1.42043	0.710215	+	0.703985i	$0.248598\pi$
$488$	−8.79706	−0.398224
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	16.0830	0.724341
$494$	−2.19358	−0.0986937
$495$	0	0
$496$	−8.99063	−0.403691
$497$	8.38715	0.376215
$498$	0	0
$499$	15.1427	0.677881	0.338941	−	0.940808i	$-0.389931\pi$
$499$	15.1427	0.677881	0.338941	+	0.940808i	$0.389931\pi$
$500$	0	0
$501$	0	0
$502$	−7.39652	−0.330123
$503$	−26.0370	−1.16093	−0.580467	−	0.814284i	$-0.697130\pi$
$503$	−26.0370	−1.16093	−0.580467	+	0.814284i	$0.697130\pi$
$504$	0	0
$505$	0	0
$506$	−1.24443	−0.0553217
$507$	0	0
$508$	−20.9447	−0.929271
$509$	24.5718	1.08913	0.544564	−	0.838719i	$-0.316695\pi$
$509$	24.5718	1.08913	0.544564	+	0.838719i	$0.316695\pi$
$510$	0	0
$511$	5.11108	0.226101
$512$	−20.3111	−0.897633
$513$	0	0
$514$	−2.53035	−0.111609
$515$	0	0
$516$	0	0
$517$	0.949145	0.0417433
$518$	1.63158	0.0716877
$519$	0	0
$520$	0	0
$521$	4.88892	0.214188	0.107094	−	0.994249i	$-0.465845\pi$
$521$	4.88892	0.214188	0.107094	+	0.994249i	$0.465845\pi$
$522$	0	0
$523$	−4.22077	−0.184562	−0.0922808	−	0.995733i	$-0.529416\pi$
$523$	−4.22077	−0.184562	−0.0922808	+	0.995733i	$0.529416\pi$
$524$	2.36842	0.103465
$525$	0	0
$526$	7.12537	0.310681
$527$	5.98126	0.260548
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	4.17484	0.181002
$533$	30.9590	1.34098
$534$	0	0
$535$	0	0
$536$	15.6128	0.674372
$537$	0	0
$538$	3.68196	0.158741
$539$	6.18421	0.266373
$540$	0	0
$541$	−13.6128	−0.585262	−0.292631	−	0.956225i	$-0.594531\pi$
$541$	−13.6128	−0.585262	−0.292631	+	0.956225i	$0.594531\pi$
$542$	−4.63512	−0.199096
$543$	0	0
$544$	7.97280	0.341831
$545$	0	0
$546$	0	0
$547$	−7.48394	−0.319990	−0.159995	−	0.987118i	$-0.551148\pi$
$547$	−7.48394	−0.319990	−0.159995	+	0.987118i	$0.551148\pi$
$548$	−8.42864	−0.360054
$549$	0	0
$550$	0	0
$551$	−17.1240	−0.729506
$552$	0	0
$553$	14.9304	0.634906
$554$	8.60793	0.365716
$555$	0	0
$556$	−1.74620	−0.0740554
$557$	32.2908	1.36821	0.684103	−	0.729385i	$-0.260194\pi$
$557$	32.2908	1.36821	0.684103	+	0.729385i	$0.260194\pi$
$558$	0	0
$559$	31.0923	1.31507
$560$	0	0
$561$	0	0
$562$	3.05086	0.128693
$563$	7.49378	0.315825	0.157913	−	0.987453i	$-0.449524\pi$
$563$	7.49378	0.315825	0.157913	+	0.987453i	$0.449524\pi$
$564$	0	0
$565$	0	0
$566$	5.91258	0.248524
$567$	0	0
$568$	−11.2761	−0.473134
$569$	12.9491	0.542856	0.271428	−	0.962459i	$-0.412504\pi$
$569$	12.9491	0.542856	0.271428	+	0.962459i	$0.412504\pi$
$570$	0	0
$571$	−15.2859	−0.639696	−0.319848	−	0.947469i	$-0.603632\pi$
$571$	−15.2859	−0.639696	−0.319848	+	0.947469i	$0.603632\pi$
$572$	5.52543	0.231030
$573$	0	0
$574$	2.99646	0.125070
$575$	0	0
$576$	0	0
$577$	−28.4415	−1.18404	−0.592019	−	0.805924i	$-0.701669\pi$
$577$	−28.4415	−1.18404	−0.592019	+	0.805924i	$0.701669\pi$
$578$	3.67016	0.152658
$579$	0	0
$580$	0	0
$581$	7.00937	0.290798
$582$	0	0
$583$	0.815792	0.0337866
$584$	−6.87157	−0.284348
$585$	0	0
$586$	−9.56691	−0.395206
$587$	−8.47013	−0.349600	−0.174800	−	0.984604i	$-0.555928\pi$
$587$	−8.47013	−0.349600	−0.174800	+	0.984604i	$0.555928\pi$
$588$	0	0
$589$	−6.36842	−0.262406
$590$	0	0
$591$	0	0
$592$	19.9081	0.818219
$593$	26.5763	1.09136	0.545679	−	0.837995i	$-0.316272\pi$
$593$	26.5763	1.09136	0.545679	+	0.837995i	$0.316272\pi$
$594$	0	0
$595$	0	0
$596$	4.17484	0.171008
$597$	0	0
$598$	3.61285	0.147740
$599$	8.77430	0.358508	0.179254	−	0.983803i	$-0.442632\pi$
$599$	8.77430	0.358508	0.179254	+	0.983803i	$0.442632\pi$
$600$	0	0
$601$	−41.8163	−1.70572	−0.852861	−	0.522139i	$-0.825134\pi$
$601$	−41.8163	−1.70572	−0.852861	+	0.522139i	$0.825134\pi$
$602$	3.00937	0.122653
$603$	0	0
$604$	19.8479	0.807600
$605$	0	0
$606$	0	0
$607$	−29.9353	−1.21504	−0.607519	−	0.794305i	$-0.707835\pi$
$607$	−29.9353	−1.21504	−0.607519	+	0.794305i	$0.707835\pi$
$608$	−8.48886	−0.344269
$609$	0	0
$610$	0	0
$611$	−2.75557	−0.111478
$612$	0	0
$613$	−26.1289	−1.05534	−0.527668	−	0.849450i	$-0.676934\pi$
$613$	−26.1289	−1.05534	−0.527668	+	0.849450i	$0.676934\pi$
$614$	4.19802	0.169418
$615$	0	0
$616$	1.09679	0.0441909
$617$	−3.66323	−0.147476	−0.0737380	−	0.997278i	$-0.523493\pi$
$617$	−3.66323	−0.147476	−0.0737380	+	0.997278i	$0.523493\pi$
$618$	0	0
$619$	43.2958	1.74020	0.870102	−	0.492872i	$-0.164053\pi$
$619$	43.2958	1.74020	0.870102	+	0.492872i	$0.164053\pi$
$620$	0	0
$621$	0	0
$622$	−5.46076	−0.218956
$623$	5.53972	0.221944
$624$	0	0
$625$	0	0
$626$	−4.47902	−0.179018
$627$	0	0
$628$	−11.0509	−0.440977
$629$	−13.2444	−0.528090
$630$	0	0
$631$	8.97773	0.357398	0.178699	−	0.983904i	$-0.442811\pi$
$631$	8.97773	0.357398	0.178699	+	0.983904i	$0.442811\pi$
$632$	−20.0731	−0.798466
$633$	0	0
$634$	9.16547	0.364007
$635$	0	0
$636$	0	0
$637$	−17.9541	−0.711366
$638$	−2.19358	−0.0868445
$639$	0	0
$640$	0	0
$641$	−9.21279	−0.363883	−0.181942	−	0.983309i	$-0.558238\pi$
$641$	−9.21279	−0.363883	−0.181942	+	0.983309i	$0.558238\pi$
$642$	0	0
$643$	16.3783	0.645896	0.322948	−	0.946417i	$-0.395326\pi$
$643$	16.3783	0.645896	0.322948	+	0.946417i	$0.395326\pi$
$644$	−6.87601	−0.270953
$645$	0	0
$646$	1.72345	0.0678082
$647$	9.80642	0.385530	0.192765	−	0.981245i	$-0.438254\pi$
$647$	9.80642	0.385530	0.192765	+	0.981245i	$0.438254\pi$
$648$	0	0
$649$	−1.67307	−0.0656738
$650$	0	0
$651$	0	0
$652$	−21.0321	−0.823681
$653$	33.0736	1.29427	0.647135	−	0.762375i	$-0.275967\pi$
$653$	33.0736	1.29427	0.647135	+	0.762375i	$0.275967\pi$
$654$	0	0
$655$	0	0
$656$	36.5620	1.42751
$657$	0	0
$658$	−0.266706	−0.0103973
$659$	−34.1017	−1.32841	−0.664207	−	0.747549i	$-0.731231\pi$
$659$	−34.1017	−1.32841	−0.664207	+	0.747549i	$0.731231\pi$
$660$	0	0
$661$	−5.40943	−0.210402	−0.105201	−	0.994451i	$-0.533549\pi$
$661$	−5.40943	−0.210402	−0.105201	+	0.994451i	$0.533549\pi$
$662$	0.815792	0.0317066
$663$	0	0
$664$	−9.42372	−0.365711
$665$	0	0
$666$	0	0
$667$	28.2034	1.09204
$668$	26.5575	1.02754
$669$	0	0
$670$	0	0
$671$	7.24443	0.279668
$672$	0	0
$673$	−24.1476	−0.930823	−0.465412	−	0.885094i	$-0.654094\pi$
$673$	−24.1476	−0.930823	−0.465412	+	0.885094i	$0.654094\pi$
$674$	1.55707	0.0599761
$675$	0	0
$676$	8.70027	0.334626
$677$	26.2810	1.01006	0.505030	−	0.863102i	$-0.331481\pi$
$677$	26.2810	1.01006	0.505030	+	0.863102i	$0.331481\pi$
$678$	0	0
$679$	11.2632	0.432241
$680$	0	0
$681$	0	0
$682$	−0.815792	−0.0312383
$683$	15.3176	0.586110	0.293055	−	0.956096i	$-0.405328\pi$
$683$	15.3176	0.586110	0.293055	+	0.956096i	$0.405328\pi$
$684$	0	0
$685$	0	0
$686$	−3.70471	−0.141447
$687$	0	0
$688$	36.7195	1.39992
$689$	−2.36842	−0.0902295
$690$	0	0
$691$	15.0223	0.571474	0.285737	−	0.958308i	$-0.407762\pi$
$691$	15.0223	0.571474	0.285737	+	0.958308i	$0.407762\pi$
$692$	−35.8622	−1.36328
$693$	0	0
$694$	−7.09679	−0.269390
$695$	0	0
$696$	0	0
$697$	−24.3239	−0.921332
$698$	−6.60348	−0.249945
$699$	0	0
$700$	0	0
$701$	19.9081	0.751920	0.375960	−	0.926636i	$-0.377313\pi$
$701$	19.9081	0.751920	0.375960	+	0.926636i	$0.377313\pi$
$702$	0	0
$703$	14.1017	0.531856
$704$	5.76986	0.217460
$705$	0	0
$706$	2.23506	0.0841177
$707$	14.6093	0.549440
$708$	0	0
$709$	−13.5081	−0.507306	−0.253653	−	0.967295i	$-0.581632\pi$
$709$	−13.5081	−0.507306	−0.253653	+	0.967295i	$0.581632\pi$
$710$	0	0
$711$	0	0
$712$	−7.44785	−0.279120
$713$	10.4889	0.392811
$714$	0	0
$715$	0	0
$716$	9.24443	0.345481
$717$	0	0
$718$	−4.38715	−0.163727
$719$	16.0830	0.599794	0.299897	−	0.953972i	$-0.403048\pi$
$719$	16.0830	0.599794	0.299897	+	0.953972i	$0.403048\pi$
$720$	0	0
$721$	15.4924	0.576967
$722$	4.07604	0.151695
$723$	0	0
$724$	31.9684	1.18809
$725$	0	0
$726$	0	0
$727$	23.6128	0.875752	0.437876	−	0.899035i	$-0.355731\pi$
$727$	23.6128	0.875752	0.437876	+	0.899035i	$0.355731\pi$
$728$	−3.18421	−0.118015
$729$	0	0
$730$	0	0
$731$	−24.4286	−0.903526
$732$	0	0
$733$	−30.0459	−1.10977	−0.554886	−	0.831926i	$-0.687238\pi$
$733$	−30.0459	−1.10977	−0.554886	+	0.831926i	$0.687238\pi$
$734$	1.21585	0.0448779
$735$	0	0
$736$	13.9813	0.515356
$737$	−12.8573	−0.473604
$738$	0	0
$739$	−24.4099	−0.897933	−0.448966	−	0.893549i	$-0.648208\pi$
$739$	−24.4099	−0.897933	−0.448966	+	0.893549i	$0.648208\pi$
$740$	0	0
$741$	0	0
$742$	−0.229234	−0.00841546
$743$	33.1798	1.21725	0.608624	−	0.793459i	$-0.291722\pi$
$743$	33.1798	1.21725	0.608624	+	0.793459i	$0.291722\pi$
$744$	0	0
$745$	0	0
$746$	−4.03704	−0.147807
$747$	0	0
$748$	−4.34122	−0.158731
$749$	−12.2538	−0.447744
$750$	0	0
$751$	−22.5718	−0.823658	−0.411829	−	0.911261i	$-0.635110\pi$
$751$	−22.5718	−0.823658	−0.411829	+	0.911261i	$0.635110\pi$
$752$	−3.25428	−0.118671
$753$	0	0
$754$	6.36842	0.231924
$755$	0	0
$756$	0	0
$757$	−4.94914	−0.179880	−0.0899399	−	0.995947i	$-0.528667\pi$
$757$	−4.94914	−0.179880	−0.0899399	+	0.995947i	$0.528667\pi$
$758$	−11.2257	−0.407736
$759$	0	0
$760$	0	0
$761$	−14.6637	−0.531559	−0.265779	−	0.964034i	$-0.585629\pi$
$761$	−14.6637	−0.531559	−0.265779	+	0.964034i	$0.585629\pi$
$762$	0	0
$763$	9.03212	0.326985
$764$	32.0830	1.16072
$765$	0	0
$766$	6.31402	0.228135
$767$	4.85728	0.175386
$768$	0	0
$769$	−44.5718	−1.60730	−0.803651	−	0.595101i	$-0.797112\pi$
$769$	−44.5718	−1.60730	−0.803651	+	0.595101i	$0.797112\pi$
$770$	0	0
$771$	0	0
$772$	46.8528	1.68627
$773$	17.3145	0.622759	0.311380	−	0.950286i	$-0.399209\pi$
$773$	17.3145	0.622759	0.311380	+	0.950286i	$0.399209\pi$
$774$	0	0
$775$	0	0
$776$	−15.1427	−0.543592
$777$	0	0
$778$	9.47949	0.339856
$779$	25.8983	0.927903
$780$	0	0
$781$	9.28592	0.332276
$782$	−2.83854	−0.101506
$783$	0	0
$784$	−21.2034	−0.757265
$785$	0	0
$786$	0	0
$787$	−36.5161	−1.30166	−0.650828	−	0.759225i	$-0.725578\pi$
$787$	−36.5161	−1.30166	−0.650828	+	0.759225i	$0.725578\pi$
$788$	28.3037	1.00828
$789$	0	0
$790$	0	0
$791$	12.8573	0.457152
$792$	0	0
$793$	−21.0321	−0.746872
$794$	−1.54861	−0.0549581
$795$	0	0
$796$	21.3649	0.757258
$797$	−14.3180	−0.507171	−0.253585	−	0.967313i	$-0.581610\pi$
$797$	−14.3180	−0.507171	−0.253585	+	0.967313i	$0.581610\pi$
$798$	0	0
$799$	2.16500	0.0765921
$800$	0	0
$801$	0	0
$802$	−0.580728	−0.0205062
$803$	5.65878	0.199694
$804$	0	0
$805$	0	0
$806$	2.36842	0.0834239
$807$	0	0
$808$	−19.6414	−0.690983
$809$	−32.0544	−1.12697	−0.563486	−	0.826125i	$-0.690540\pi$
$809$	−32.0544	−1.12697	−0.563486	+	0.826125i	$0.690540\pi$
$810$	0	0
$811$	−8.44738	−0.296627	−0.148314	−	0.988940i	$-0.547385\pi$
$811$	−8.44738	−0.296627	−0.148314	+	0.988940i	$0.547385\pi$
$812$	−12.1204	−0.425344
$813$	0	0
$814$	1.80642	0.0633151
$815$	0	0
$816$	0	0
$817$	26.0098	0.909969
$818$	1.12981	0.0395030
$819$	0	0
$820$	0	0
$821$	−17.2159	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$821$	−17.2159	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$822$	0	0
$823$	−12.7654	−0.444974	−0.222487	−	0.974936i	$-0.571418\pi$
$823$	−12.7654	−0.444974	−0.222487	+	0.974936i	$0.571418\pi$
$824$	−20.8287	−0.725602
$825$	0	0
$826$	0.470127	0.0163578
$827$	8.70964	0.302864	0.151432	−	0.988468i	$-0.451612\pi$
$827$	8.70964	0.302864	0.151432	+	0.988468i	$0.451612\pi$
$828$	0	0
$829$	−8.32693	−0.289206	−0.144603	−	0.989490i	$-0.546191\pi$
$829$	−8.32693	−0.289206	−0.144603	+	0.989490i	$0.546191\pi$
$830$	0	0
$831$	0	0
$832$	−16.7511	−0.580741
$833$	14.1062	0.488749
$834$	0	0
$835$	0	0
$836$	4.62222	0.159863
$837$	0	0
$838$	1.51114	0.0522014
$839$	−12.8988	−0.445315	−0.222657	−	0.974897i	$-0.571473\pi$
$839$	−12.8988	−0.445315	−0.222657	+	0.974897i	$0.571473\pi$
$840$	0	0
$841$	20.7146	0.714295
$842$	−7.04101	−0.242649
$843$	0	0
$844$	22.7239	0.782190
$845$	0	0
$846$	0	0
$847$	−0.903212	−0.0310347
$848$	−2.79706	−0.0960513
$849$	0	0
$850$	0	0
$851$	−23.2257	−0.796167
$852$	0	0
$853$	19.6686	0.673441	0.336720	−	0.941605i	$-0.390682\pi$
$853$	19.6686	0.673441	0.336720	+	0.941605i	$0.390682\pi$
$854$	−2.03566	−0.0696588
$855$	0	0
$856$	16.4746	0.563089
$857$	31.8207	1.08697	0.543487	−	0.839417i	$-0.317104\pi$
$857$	31.8207	1.08697	0.543487	+	0.839417i	$0.317104\pi$
$858$	0	0
$859$	27.8292	0.949519	0.474760	−	0.880116i	$-0.342535\pi$
$859$	27.8292	0.949519	0.474760	+	0.880116i	$0.342535\pi$
$860$	0	0
$861$	0	0
$862$	0.387152	0.0131865
$863$	−4.82870	−0.164371	−0.0821854	−	0.996617i	$-0.526190\pi$
$863$	−4.82870	−0.164371	−0.0821854	+	0.996617i	$0.526190\pi$
$864$	0	0
$865$	0	0
$866$	4.97773	0.169150
$867$	0	0
$868$	−4.50760	−0.152998
$869$	16.5303	0.560754
$870$	0	0
$871$	37.3274	1.26479
$872$	−12.1432	−0.411221
$873$	0	0
$874$	3.02227	0.102230
$875$	0	0
$876$	0	0
$877$	−21.9826	−0.742301	−0.371151	−	0.928573i	$-0.621037\pi$
$877$	−21.9826	−0.742301	−0.371151	+	0.928573i	$0.621037\pi$
$878$	0.755569	0.0254992
$879$	0	0
$880$	0	0
$881$	12.1017	0.407717	0.203858	−	0.979000i	$-0.434652\pi$
$881$	12.1017	0.407717	0.203858	+	0.979000i	$0.434652\pi$
$882$	0	0
$883$	−8.73683	−0.294018	−0.147009	−	0.989135i	$-0.546965\pi$
$883$	−8.73683	−0.294018	−0.147009	+	0.989135i	$0.546965\pi$
$884$	12.6035	0.423901
$885$	0	0
$886$	9.66016	0.324540
$887$	−19.8524	−0.666577	−0.333288	−	0.942825i	$-0.608158\pi$
$887$	−19.8524	−0.666577	−0.333288	+	0.942825i	$0.608158\pi$
$888$	0	0
$889$	−9.93978	−0.333369
$890$	0	0
$891$	0	0
$892$	−41.5022	−1.38960
$893$	−2.30513	−0.0771383
$894$	0	0
$895$	0	0
$896$	−7.93533	−0.265101
$897$	0	0
$898$	−11.6227	−0.387854
$899$	18.4889	0.616638
$900$	0	0
$901$	1.86082	0.0619928
$902$	3.31756	0.110463
$903$	0	0
$904$	−17.2859	−0.574921
$905$	0	0
$906$	0	0
$907$	−32.8287	−1.09006	−0.545030	−	0.838417i	$-0.683482\pi$
$907$	−32.8287	−1.09006	−0.545030	+	0.838417i	$0.683482\pi$
$908$	6.08742	0.202018
$909$	0	0
$910$	0	0
$911$	16.3497	0.541689	0.270845	−	0.962623i	$-0.412697\pi$
$911$	16.3497	0.541689	0.270845	+	0.962623i	$0.412697\pi$
$912$	0	0
$913$	7.76049	0.256835
$914$	−2.71853	−0.0899209
$915$	0	0
$916$	−13.5585	−0.447984
$917$	1.12399	0.0371173
$918$	0	0
$919$	20.0228	0.660490	0.330245	−	0.943895i	$-0.392869\pi$
$919$	20.0228	0.660490	0.330245	+	0.943895i	$0.392869\pi$
$920$	0	0
$921$	0	0
$922$	9.88940	0.325690
$923$	−26.9590	−0.887366
$924$	0	0
$925$	0	0
$926$	−3.76187	−0.123623
$927$	0	0
$928$	24.6450	0.809011
$929$	43.5308	1.42820	0.714100	−	0.700044i	$-0.246836\pi$
$929$	43.5308	1.42820	0.714100	+	0.700044i	$0.246836\pi$
$930$	0	0
$931$	−15.0192	−0.492235
$932$	−37.1610	−1.21725
$933$	0	0
$934$	−4.77430	−0.156220
$935$	0	0
$936$	0	0
$937$	43.4563	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$937$	43.4563	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$938$	3.61285	0.117964
$939$	0	0
$940$	0	0
$941$	23.7244	0.773393	0.386697	−	0.922207i	$-0.373616\pi$
$941$	23.7244	0.773393	0.386697	+	0.922207i	$0.373616\pi$
$942$	0	0
$943$	−42.6548	−1.38903
$944$	5.73636	0.186703
$945$	0	0
$946$	3.33185	0.108328
$947$	−11.7047	−0.380352	−0.190176	−	0.981750i	$-0.560906\pi$
$947$	−11.7047	−0.380352	−0.190176	+	0.981750i	$0.560906\pi$
$948$	0	0
$949$	−16.4286	−0.533296
$950$	0	0
$951$	0	0
$952$	2.50177	0.0810828
$953$	−46.1258	−1.49416	−0.747081	−	0.664733i	$-0.768545\pi$
$953$	−46.1258	−1.49416	−0.747081	+	0.664733i	$0.768545\pi$
$954$	0	0
$955$	0	0
$956$	−41.8350	−1.35304
$957$	0	0
$958$	1.83500	0.0592863
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−24.1240	−0.778193
$962$	−5.24443	−0.169087
$963$	0	0
$964$	−10.1748	−0.327710
$965$	0	0
$966$	0	0
$967$	17.0495	0.548274	0.274137	−	0.961691i	$-0.411608\pi$
$967$	17.0495	0.548274	0.274137	+	0.961691i	$0.411608\pi$
$968$	1.21432	0.0390297
$969$	0	0
$970$	0	0
$971$	58.1847	1.86724	0.933618	−	0.358271i	$-0.116634\pi$
$971$	58.1847	1.86724	0.933618	+	0.358271i	$0.116634\pi$
$972$	0	0
$973$	−0.828699	−0.0265669
$974$	−9.75203	−0.312475
$975$	0	0
$976$	−24.8385	−0.795062
$977$	51.7373	1.65522	0.827612	−	0.561301i	$-0.189699\pi$
$977$	51.7373	1.65522	0.827612	+	0.561301i	$0.189699\pi$
$978$	0	0
$979$	6.13335	0.196023
$980$	0	0
$981$	0	0
$982$	−2.48886	−0.0794228
$983$	26.3970	0.841933	0.420967	−	0.907076i	$-0.361691\pi$
$983$	26.3970	0.841933	0.420967	+	0.907076i	$0.361691\pi$
$984$	0	0
$985$	0	0
$986$	−5.00354	−0.159345
$987$	0	0
$988$	−13.4193	−0.426924
$989$	−42.8385	−1.36219
$990$	0	0
$991$	−23.0923	−0.733552	−0.366776	−	0.930309i	$-0.619539\pi$
$991$	−23.0923	−0.733552	−0.366776	+	0.930309i	$0.619539\pi$
$992$	9.16547	0.291004
$993$	0	0
$994$	−2.60931	−0.0827622
$995$	0	0
$996$	0	0
$997$	12.9131	0.408961	0.204480	−	0.978871i	$-0.434450\pi$
$997$	12.9131	0.408961	0.204480	+	0.978871i	$0.434450\pi$
$998$	−4.71102	−0.149125
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.ba.1.2		3
3.2	odd	2		825.2.a.l.1.2		3
5.2	odd	4		495.2.c.e.199.3		6
5.3	odd	4		495.2.c.e.199.4		6
5.4	even	2		2475.2.a.bc.1.2		3
15.2	even	4		165.2.c.b.34.4	yes	6
15.8	even	4		165.2.c.b.34.3	✓	6
15.14	odd	2		825.2.a.j.1.2		3
33.32	even	2		9075.2.a.cg.1.2		3
60.23	odd	4		2640.2.d.h.529.2		6
60.47	odd	4		2640.2.d.h.529.5		6
165.32	odd	4		1815.2.c.e.364.3		6
165.98	odd	4		1815.2.c.e.364.4		6
165.164	even	2		9075.2.a.ch.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.3	✓	6	15.8	even	4
165.2.c.b.34.4	yes	6	15.2	even	4
495.2.c.e.199.3		6	5.2	odd	4
495.2.c.e.199.4		6	5.3	odd	4
825.2.a.j.1.2		3	15.14	odd	2
825.2.a.l.1.2		3	3.2	odd	2
1815.2.c.e.364.3		6	165.32	odd	4
1815.2.c.e.364.4		6	165.98	odd	4
2475.2.a.ba.1.2		3	1.1	even	1	trivial
2475.2.a.bc.1.2		3	5.4	even	2
2640.2.d.h.529.2		6	60.23	odd	4
2640.2.d.h.529.5		6	60.47	odd	4
9075.2.a.cg.1.2		3	33.32	even	2
9075.2.a.ch.1.2		3	165.164	even	2

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.90321 q^{4} -0.903212 q^{7} +1.21432 q^{8} -1.00000 q^{11} +2.90321 q^{13} +0.280996 q^{14} +3.42864 q^{16} -2.28100 q^{17} +2.42864 q^{19} +0.311108 q^{22} -4.00000 q^{23} -0.903212 q^{26} +1.71900 q^{28} -7.05086 q^{29} -2.62222 q^{31} -3.49532 q^{32} +0.709636 q^{34} +5.80642 q^{37} -0.755569 q^{38} +10.6637 q^{41} +10.7096 q^{43} +1.90321 q^{44} +1.24443 q^{46} -0.949145 q^{47} -6.18421 q^{49} -5.52543 q^{52} -0.815792 q^{53} -1.09679 q^{56} +2.19358 q^{58} +1.67307 q^{59} -7.24443 q^{61} +0.815792 q^{62} -5.76986 q^{64} +12.8573 q^{67} +4.34122 q^{68} -9.28592 q^{71} -5.65878 q^{73} -1.80642 q^{74} -4.62222 q^{76} +0.903212 q^{77} -16.5303 q^{79} -3.31756 q^{82} -7.76049 q^{83} -3.33185 q^{86} -1.21432 q^{88} -6.13335 q^{89} -2.62222 q^{91} +7.61285 q^{92} +0.295286 q^{94} -12.4701 q^{97} +1.92396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8} - 3 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} + q^{22} - 12 q^{23} + 4 q^{26} + 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + 4 q^{37} - 2 q^{38}+ \cdots - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8 - 3 * q^11 + 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 + q^22 - 12 * q^23 + 4 * q^26 + 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 + 12 * q^43 - q^44 + 4 * q^46 - 16 * q^47 - 5 * q^49 - 10 * q^52 - 16 * q^53 - 10 * q^56 + 20 * q^58 - 8 * q^59 - 22 * q^61 + 16 * q^62 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^71 - 10 * q^73 + 8 * q^74 - 14 * q^76 - 4 * q^77 - 10 * q^79 + 4 * q^82 + 10 * q^83 + 10 * q^86 + 3 * q^88 - 18 * q^89 - 8 * q^91 - 4 * q^92 - 12 * q^94 + 16 * q^97 - 21 * q^98

Introduction

L-functions

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