LMFDB - Embedded newform 2475.2.a.y.1.3

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:	$0$
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			1.3
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)

$f(q)$	$=$	$q+1.21432 q^{2} -0.525428 q^{4} +4.90321 q^{7} -3.06668 q^{8} +1.00000 q^{11} +4.14764 q^{13} +5.95407 q^{14} -2.67307 q^{16} +5.33185 q^{17} -5.18421 q^{19} +1.21432 q^{22} -4.00000 q^{23} +5.03657 q^{26} -2.57628 q^{28} -1.80642 q^{29} +2.62222 q^{31} +2.88739 q^{32} +6.47457 q^{34} +5.80642 q^{37} -6.29529 q^{38} -1.80642 q^{41} +4.90321 q^{43} -0.525428 q^{44} -4.85728 q^{46} -7.05086 q^{47} +17.0415 q^{49} -2.17929 q^{52} +7.18421 q^{53} -15.0366 q^{56} -2.19358 q^{58} -1.67307 q^{59} +0.755569 q^{61} +3.18421 q^{62} +8.85236 q^{64} -4.85728 q^{67} -2.80150 q^{68} -0.428639 q^{71} +12.7096 q^{73} +7.05086 q^{74} +2.72393 q^{76} +4.90321 q^{77} -6.42864 q^{79} -2.19358 q^{82} +2.90321 q^{83} +5.95407 q^{86} -3.06668 q^{88} -0.622216 q^{89} +20.3368 q^{91} +2.10171 q^{92} -8.56199 q^{94} -2.75557 q^{97} +20.6938 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} + 3 q^{11} + 6 q^{13} - 2 q^{14} + 5 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 8 q^{26} + 12 q^{28} + 8 q^{29} + 8 q^{31} - 11 q^{32} + 26 q^{34}+ \cdots + 29 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 + 3 * q^11 + 6 * q^13 - 2 * q^14 + 5 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 8 * q^26 + 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 4 * q^37 - 6 * q^38 + 8 * q^41 + 8 * q^43 + 5 * q^44 + 12 * q^46 - 8 * q^47 + 11 * q^49 - 26 * q^52 + 8 * q^53 - 38 * q^56 - 20 * q^58 + 8 * q^59 + 2 * q^61 - 4 * q^62 + 33 * q^64 + 12 * q^67 - 28 * q^68 + 12 * q^71 + 18 * q^73 + 8 * q^74 - 18 * q^76 + 8 * q^77 - 6 * q^79 - 20 * q^82 + 2 * q^83 - 2 * q^86 - 9 * q^88 - 2 * q^89 + 8 * q^91 - 20 * q^92 - 12 * q^94 - 8 * q^97 + 29 * 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 $

$2$	1.21432	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$2$	1.21432	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$3$	0	0
$3$	0	0
$4$	−0.525428	−0.262714
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.90321	1.85324	0.926620	−	0.375999i	$-0.122700\pi$
$7$	4.90321	1.85324	0.926620	+	0.375999i	$0.122700\pi$
$8$	−3.06668	−1.08423
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.14764	1.15035	0.575175	−	0.818031i	$-0.304934\pi$
$13$	4.14764	1.15035	0.575175	+	0.818031i	$0.304934\pi$
$14$	5.95407	1.59129
$15$	0	0
$16$	−2.67307	−0.668268
$17$	5.33185	1.29316	0.646582	−	0.762845i	$-0.276198\pi$
$17$	5.33185	1.29316	0.646582	+	0.762845i	$0.276198\pi$
$18$	0	0
$19$	−5.18421	−1.18934	−0.594669	−	0.803970i	$-0.702717\pi$
$19$	−5.18421	−1.18934	−0.594669	+	0.803970i	$0.702717\pi$
$20$	0	0
$21$	0	0
$22$	1.21432	0.258894
$23$	−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
$26$	5.03657	0.987752
$27$	0	0
$28$	−2.57628	−0.486872
$29$	−1.80642	−0.335444	−0.167722	−	0.985834i	$-0.553641\pi$
$29$	−1.80642	−0.335444	−0.167722	+	0.985834i	$0.553641\pi$
$30$	0	0
$31$	2.62222	0.470964	0.235482	−	0.971879i	$-0.424333\pi$
$31$	2.62222	0.470964	0.235482	+	0.971879i	$0.424333\pi$
$32$	2.88739	0.510423
$33$	0	0
$34$	6.47457	1.11038
$35$	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	$0.341621\pi$
$38$	−6.29529	−1.02123
$39$	0	0
$40$	0	0
$41$	−1.80642	−0.282116	−0.141058	−	0.990001i	$-0.545050\pi$
$41$	−1.80642	−0.282116	−0.141058	+	0.990001i	$0.545050\pi$
$42$	0	0
$43$	4.90321	0.747733	0.373866	−	0.927483i	$-0.378032\pi$
$43$	4.90321	0.747733	0.373866	+	0.927483i	$0.378032\pi$
$44$	−0.525428	−0.0792112
$45$	0	0
$46$	−4.85728	−0.716167
$47$	−7.05086	−1.02847	−0.514236	−	0.857648i	$-0.671925\pi$
$47$	−7.05086	−1.02847	−0.514236	+	0.857648i	$0.671925\pi$
$48$	0	0
$49$	17.0415	2.43450
$50$	0	0
$51$	0	0
$52$	−2.17929	−0.302213
$53$	7.18421	0.986827	0.493413	−	0.869795i	$-0.335749\pi$
$53$	7.18421	0.986827	0.493413	+	0.869795i	$0.335749\pi$
$54$	0	0
$55$	0	0
$56$	−15.0366	−2.00935
$57$	0	0
$58$	−2.19358	−0.288031
$59$	−1.67307	−0.217815	−0.108908	−	0.994052i	$-0.534735\pi$
$59$	−1.67307	−0.217815	−0.108908	+	0.994052i	$0.534735\pi$
$60$	0	0
$61$	0.755569	0.0967407	0.0483703	−	0.998829i	$-0.484597\pi$
$61$	0.755569	0.0967407	0.0483703	+	0.998829i	$0.484597\pi$
$62$	3.18421	0.404395
$63$	0	0
$64$	8.85236	1.10654
$65$	0	0
$66$	0	0
$67$	−4.85728	−0.593411	−0.296706	−	0.954969i	$-0.595888\pi$
$67$	−4.85728	−0.593411	−0.296706	+	0.954969i	$0.595888\pi$
$68$	−2.80150	−0.339732
$69$	0	0
$70$	0	0
$71$	−0.428639	−0.0508701	−0.0254351	−	0.999676i	$-0.508097\pi$
$71$	−0.428639	−0.0508701	−0.0254351	+	0.999676i	$0.508097\pi$
$72$	0	0
$73$	12.7096	1.48755	0.743775	−	0.668430i	$-0.233033\pi$
$73$	12.7096	1.48755	0.743775	+	0.668430i	$0.233033\pi$
$74$	7.05086	0.819645
$75$	0	0
$76$	2.72393	0.312456
$77$	4.90321	0.558773
$78$	0	0
$79$	−6.42864	−0.723278	−0.361639	−	0.932318i	$-0.617783\pi$
$79$	−6.42864	−0.723278	−0.361639	+	0.932318i	$0.617783\pi$
$80$	0	0
$81$	0	0
$82$	−2.19358	−0.242240
$83$	2.90321	0.318669	0.159334	−	0.987225i	$-0.449065\pi$
$83$	2.90321	0.318669	0.159334	+	0.987225i	$0.449065\pi$
$84$	0	0
$85$	0	0
$86$	5.95407	0.642044
$87$	0	0
$88$	−3.06668	−0.326909
$89$	−0.622216	−0.0659547	−0.0329774	−	0.999456i	$-0.510499\pi$
$89$	−0.622216	−0.0659547	−0.0329774	+	0.999456i	$0.510499\pi$
$90$	0	0
$91$	20.3368	2.13187
$92$	2.10171	0.219118
$93$	0	0
$94$	−8.56199	−0.883102
$95$	0	0
$96$	0	0
$97$	−2.75557	−0.279786	−0.139893	−	0.990167i	$-0.544676\pi$
$97$	−2.75557	−0.279786	−0.139893	+	0.990167i	$0.544676\pi$
$98$	20.6938	2.09039
$99$	0	0
$100$	0	0
$101$	17.8064	1.77181	0.885903	−	0.463871i	$-0.153540\pi$
$101$	17.8064	1.77181	0.885903	+	0.463871i	$0.153540\pi$
$102$	0	0
$103$	4.94914	0.487654	0.243827	−	0.969819i	$-0.421597\pi$
$103$	4.94914	0.487654	0.243827	+	0.969819i	$0.421597\pi$
$104$	−12.7195	−1.24725
$105$	0	0
$106$	8.72393	0.847343
$107$	−11.1985	−1.08260	−0.541300	−	0.840830i	$-0.682068\pi$
$107$	−11.1985	−1.08260	−0.541300	+	0.840830i	$0.682068\pi$
$108$	0	0
$109$	15.7146	1.50518	0.752591	−	0.658488i	$-0.228804\pi$
$109$	15.7146	1.50518	0.752591	+	0.658488i	$0.228804\pi$
$110$	0	0
$111$	0	0
$112$	−13.1066	−1.23846
$113$	1.76494	0.166031	0.0830156	−	0.996548i	$-0.473545\pi$
$113$	1.76494	0.166031	0.0830156	+	0.996548i	$0.473545\pi$
$114$	0	0
$115$	0	0
$116$	0.949145	0.0881259
$117$	0	0
$118$	−2.03164	−0.187028
$119$	26.1432	2.39654
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.917502	0.0830667
$123$	0	0
$124$	−1.37778	−0.123729
$125$	0	0
$126$	0	0
$127$	18.7096	1.66021	0.830106	−	0.557606i	$-0.188280\pi$
$127$	18.7096	1.66021	0.830106	+	0.557606i	$0.188280\pi$
$128$	4.97481	0.439715
$129$	0	0
$130$	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
$133$	−25.4193	−2.20413
$134$	−5.89829	−0.509535
$135$	0	0
$136$	−16.3511	−1.40209
$137$	−18.7971	−1.60594	−0.802970	−	0.596019i	$-0.796748\pi$
$137$	−18.7971	−1.60594	−0.802970	+	0.596019i	$0.796748\pi$
$138$	0	0
$139$	14.0415	1.19098	0.595492	−	0.803361i	$-0.296957\pi$
$139$	14.0415	1.19098	0.595492	+	0.803361i	$0.296957\pi$
$140$	0	0
$141$	0	0
$142$	−0.520505	−0.0436798
$143$	4.14764	0.346843
$144$	0	0
$145$	0	0
$146$	15.4336	1.27729
$147$	0	0
$148$	−3.05086	−0.250779
$149$	3.05086	0.249936	0.124968	−	0.992161i	$-0.460117\pi$
$149$	3.05086	0.249936	0.124968	+	0.992161i	$0.460117\pi$
$150$	0	0
$151$	−0.326929	−0.0266051	−0.0133026	−	0.999912i	$-0.504234\pi$
$151$	−0.326929	−0.0266051	−0.0133026	+	0.999912i	$0.504234\pi$
$152$	15.8983	1.28952
$153$	0	0
$154$	5.95407	0.479792
$155$	0	0
$156$	0	0
$157$	−19.9081	−1.58884	−0.794421	−	0.607367i	$-0.792226\pi$
$157$	−19.9081	−1.58884	−0.794421	+	0.607367i	$0.792226\pi$
$158$	−7.80642	−0.621046
$159$	0	0
$160$	0	0
$161$	−19.6128	−1.54571
$162$	0	0
$163$	12.1748	0.953607	0.476804	−	0.879010i	$-0.341795\pi$
$163$	12.1748	0.953607	0.476804	+	0.879010i	$0.341795\pi$
$164$	0.949145	0.0741158
$165$	0	0
$166$	3.52543	0.273626
$167$	−13.0049	−1.00635	−0.503176	−	0.864184i	$-0.667835\pi$
$167$	−13.0049	−1.00635	−0.503176	+	0.864184i	$0.667835\pi$
$168$	0	0
$169$	4.20294	0.323303
$170$	0	0
$171$	0	0
$172$	−2.57628	−0.196440
$173$	−13.8938	−1.05633	−0.528165	−	0.849142i	$-0.677120\pi$
$173$	−13.8938	−1.05633	−0.528165	+	0.849142i	$0.677120\pi$
$174$	0	0
$175$	0	0
$176$	−2.67307	−0.201490
$177$	0	0
$178$	−0.755569	−0.0566323
$179$	−12.8573	−0.960998	−0.480499	−	0.876995i	$-0.659544\pi$
$179$	−12.8573	−0.960998	−0.480499	+	0.876995i	$0.659544\pi$
$180$	0	0
$181$	0.917502	0.0681974	0.0340987	−	0.999418i	$-0.489144\pi$
$181$	0.917502	0.0681974	0.0340987	+	0.999418i	$0.489144\pi$
$182$	24.6953	1.83054
$183$	0	0
$184$	12.2667	0.904314
$185$	0	0
$186$	0	0
$187$	5.33185	0.389904
$188$	3.70471	0.270194
$189$	0	0
$190$	0	0
$191$	−14.3684	−1.03966	−0.519831	−	0.854269i	$-0.674005\pi$
$191$	−14.3684	−1.03966	−0.519831	+	0.854269i	$0.674005\pi$
$192$	0	0
$193$	−11.7605	−0.846539	−0.423269	−	0.906004i	$-0.639118\pi$
$193$	−11.7605	−0.846539	−0.423269	+	0.906004i	$0.639118\pi$
$194$	−3.34614	−0.240239
$195$	0	0
$196$	−8.95407	−0.639576
$197$	3.82071	0.272215	0.136107	−	0.990694i	$-0.456541\pi$
$197$	3.82071	0.272215	0.136107	+	0.990694i	$0.456541\pi$
$198$	0	0
$199$	−13.7146	−0.972199	−0.486100	−	0.873903i	$-0.661581\pi$
$199$	−13.7146	−0.972199	−0.486100	+	0.873903i	$0.661581\pi$
$200$	0	0
$201$	0	0
$202$	21.6227	1.52137
$203$	−8.85728	−0.621659
$204$	0	0
$205$	0	0
$206$	6.00984	0.418726
$207$	0	0
$208$	−11.0869	−0.768741
$209$	−5.18421	−0.358599
$210$	0	0
$211$	1.95851	0.134830	0.0674148	−	0.997725i	$-0.478525\pi$
$211$	1.95851	0.134830	0.0674148	+	0.997725i	$0.478525\pi$
$212$	−3.77478	−0.259253
$213$	0	0
$214$	−13.5986	−0.929578
$215$	0	0
$216$	0	0
$217$	12.8573	0.872809
$218$	19.0825	1.29243
$219$	0	0
$220$	0	0
$221$	22.1146	1.48759
$222$	0	0
$223$	−26.0098	−1.74175	−0.870874	−	0.491506i	$-0.836446\pi$
$223$	−26.0098	−1.74175	−0.870874	+	0.491506i	$0.836446\pi$
$224$	14.1575	0.945937
$225$	0	0
$226$	2.14320	0.142563
$227$	6.34122	0.420882	0.210441	−	0.977607i	$-0.432510\pi$
$227$	6.34122	0.420882	0.210441	+	0.977607i	$0.432510\pi$
$228$	0	0
$229$	−23.3274	−1.54152	−0.770759	−	0.637127i	$-0.780123\pi$
$229$	−23.3274	−1.54152	−0.770759	+	0.637127i	$0.780123\pi$
$230$	0	0
$231$	0	0
$232$	5.53972	0.363700
$233$	1.42372	0.0932708	0.0466354	−	0.998912i	$-0.485150\pi$
$233$	1.42372	0.0932708	0.0466354	+	0.998912i	$0.485150\pi$
$234$	0	0
$235$	0	0
$236$	0.879077	0.0572231
$237$	0	0
$238$	31.7462	2.05780
$239$	18.9590	1.22636	0.613178	−	0.789945i	$-0.289891\pi$
$239$	18.9590	1.22636	0.613178	+	0.789945i	$0.289891\pi$
$240$	0	0
$241$	−1.34614	−0.0867126	−0.0433563	−	0.999060i	$-0.513805\pi$
$241$	−1.34614	−0.0867126	−0.0433563	+	0.999060i	$0.513805\pi$
$242$	1.21432	0.0780594
$243$	0	0
$244$	−0.396997	−0.0254151
$245$	0	0
$246$	0	0
$247$	−21.5022	−1.36816
$248$	−8.04149	−0.510635
$249$	0	0
$250$	0	0
$251$	−1.08250	−0.0683267	−0.0341633	−	0.999416i	$-0.510877\pi$
$251$	−1.08250	−0.0683267	−0.0341633	+	0.999416i	$0.510877\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	22.7195	1.42555
$255$	0	0
$256$	−11.6637	−0.728981
$257$	0.133353	0.00831834	0.00415917	−	0.999991i	$-0.498676\pi$
$257$	0.133353	0.00831834	0.00415917	+	0.999991i	$0.498676\pi$
$258$	0	0
$259$	28.4701	1.76905
$260$	0	0
$261$	0	0
$262$	1.51114	0.0933584
$263$	0.147643	0.00910407	0.00455203	−	0.999990i	$-0.498551\pi$
$263$	0.147643	0.00910407	0.00455203	+	0.999990i	$0.498551\pi$
$264$	0	0
$265$	0	0
$266$	−30.8671	−1.89258
$267$	0	0
$268$	2.55215	0.155897
$269$	26.8573	1.63752	0.818759	−	0.574138i	$-0.194663\pi$
$269$	26.8573	1.63752	0.818759	+	0.574138i	$0.194663\pi$
$270$	0	0
$271$	3.08250	0.187248	0.0936242	−	0.995608i	$-0.470155\pi$
$271$	3.08250	0.187248	0.0936242	+	0.995608i	$0.470155\pi$
$272$	−14.2524	−0.864180
$273$	0	0
$274$	−22.8256	−1.37895
$275$	0	0
$276$	0	0
$277$	−8.70964	−0.523311	−0.261656	−	0.965161i	$-0.584268\pi$
$277$	−8.70964	−0.523311	−0.261656	+	0.965161i	$0.584268\pi$
$278$	17.0509	1.02264
$279$	0	0
$280$	0	0
$281$	20.3783	1.21567	0.607833	−	0.794065i	$-0.292039\pi$
$281$	20.3783	1.21567	0.607833	+	0.794065i	$0.292039\pi$
$282$	0	0
$283$	−6.32248	−0.375833	−0.187916	−	0.982185i	$-0.560173\pi$
$283$	−6.32248	−0.375833	−0.187916	+	0.982185i	$0.560173\pi$
$284$	0.225219	0.0133643
$285$	0	0
$286$	5.03657	0.297818
$287$	−8.85728	−0.522829
$288$	0	0
$289$	11.4286	0.672273
$290$	0	0
$291$	0	0
$292$	−6.67799	−0.390800
$293$	−16.6780	−0.974339	−0.487169	−	0.873308i	$-0.661971\pi$
$293$	−16.6780	−0.974339	−0.487169	+	0.873308i	$0.661971\pi$
$294$	0	0
$295$	0	0
$296$	−17.8064	−1.03498
$297$	0	0
$298$	3.70471	0.214608
$299$	−16.5906	−0.959458
$300$	0	0
$301$	24.0415	1.38573
$302$	−0.396997	−0.0228446
$303$	0	0
$304$	13.8578	0.794797
$305$	0	0
$306$	0	0
$307$	−9.58565	−0.547082	−0.273541	−	0.961860i	$-0.588195\pi$
$307$	−9.58565	−0.547082	−0.273541	+	0.961860i	$0.588195\pi$
$308$	−2.57628	−0.146797
$309$	0	0
$310$	0	0
$311$	−14.5303	−0.823941	−0.411970	−	0.911197i	$-0.635159\pi$
$311$	−14.5303	−0.823941	−0.411970	+	0.911197i	$0.635159\pi$
$312$	0	0
$313$	−21.0321	−1.18881	−0.594403	−	0.804167i	$-0.702612\pi$
$313$	−21.0321	−1.18881	−0.594403	+	0.804167i	$0.702612\pi$
$314$	−24.1748	−1.36427
$315$	0	0
$316$	3.37778	0.190015
$317$	0.990632	0.0556394	0.0278197	−	0.999613i	$-0.491144\pi$
$317$	0.990632	0.0556394	0.0278197	+	0.999613i	$0.491144\pi$
$318$	0	0
$319$	−1.80642	−0.101140
$320$	0	0
$321$	0	0
$322$	−23.8163	−1.32723
$323$	−27.6414	−1.53801
$324$	0	0
$325$	0	0
$326$	14.7841	0.818818
$327$	0	0
$328$	5.53972	0.305880
$329$	−34.5718	−1.90601
$330$	0	0
$331$	−17.5812	−0.966350	−0.483175	−	0.875524i	$-0.660517\pi$
$331$	−17.5812	−0.966350	−0.483175	+	0.875524i	$0.660517\pi$
$332$	−1.52543	−0.0837187
$333$	0	0
$334$	−15.7921	−0.864107
$335$	0	0
$336$	0	0
$337$	−3.16992	−0.172676	−0.0863382	−	0.996266i	$-0.527517\pi$
$337$	−3.16992	−0.172676	−0.0863382	+	0.996266i	$0.527517\pi$
$338$	5.10372	0.277606
$339$	0	0
$340$	0	0
$341$	2.62222	0.142001
$342$	0	0
$343$	49.2355	2.65847
$344$	−15.0366	−0.810717
$345$	0	0
$346$	−16.8716	−0.907021
$347$	−4.97634	−0.267144	−0.133572	−	0.991039i	$-0.542645\pi$
$347$	−4.97634	−0.267144	−0.133572	+	0.991039i	$0.542645\pi$
$348$	0	0
$349$	18.2034	0.974407	0.487203	−	0.873289i	$-0.338017\pi$
$349$	18.2034	0.974407	0.487203	+	0.873289i	$0.338017\pi$
$350$	0	0
$351$	0	0
$352$	2.88739	0.153898
$353$	−22.4099	−1.19276	−0.596379	−	0.802703i	$-0.703395\pi$
$353$	−22.4099	−1.19276	−0.596379	+	0.802703i	$0.703395\pi$
$354$	0	0
$355$	0	0
$356$	0.326929	0.0173272
$357$	0	0
$358$	−15.6128	−0.825165
$359$	−21.3274	−1.12562	−0.562809	−	0.826587i	$-0.690279\pi$
$359$	−21.3274	−1.12562	−0.562809	+	0.826587i	$0.690279\pi$
$360$	0	0
$361$	7.87601	0.414527
$362$	1.11414	0.0585579
$363$	0	0
$364$	−10.6855	−0.560072
$365$	0	0
$366$	0	0
$367$	35.1338	1.83397	0.916985	−	0.398921i	$-0.130615\pi$
$367$	35.1338	1.83397	0.916985	+	0.398921i	$0.130615\pi$
$368$	10.6923	0.557374
$369$	0	0
$370$	0	0
$371$	35.2257	1.82883
$372$	0	0
$373$	−17.0049	−0.880481	−0.440241	−	0.897880i	$-0.645107\pi$
$373$	−17.0049	−0.880481	−0.440241	+	0.897880i	$0.645107\pi$
$374$	6.47457	0.334792
$375$	0	0
$376$	21.6227	1.11511
$377$	−7.49240	−0.385878
$378$	0	0
$379$	2.36842	0.121657	0.0608287	−	0.998148i	$-0.480626\pi$
$379$	2.36842	0.121657	0.0608287	+	0.998148i	$0.480626\pi$
$380$	0	0
$381$	0	0
$382$	−17.4479	−0.892710
$383$	−1.21585	−0.0621271	−0.0310635	−	0.999517i	$-0.509889\pi$
$383$	−1.21585	−0.0621271	−0.0310635	+	0.999517i	$0.509889\pi$
$384$	0	0
$385$	0	0
$386$	−14.2810	−0.726884
$387$	0	0
$388$	1.44785	0.0735035
$389$	−2.26671	−0.114927	−0.0574633	−	0.998348i	$-0.518301\pi$
$389$	−2.26671	−0.114927	−0.0574633	+	0.998348i	$0.518301\pi$
$390$	0	0
$391$	−21.3274	−1.07857
$392$	−52.2607	−2.63957
$393$	0	0
$394$	4.63957	0.233738
$395$	0	0
$396$	0	0
$397$	−18.4889	−0.927929	−0.463965	−	0.885854i	$-0.653574\pi$
$397$	−18.4889	−0.927929	−0.463965	+	0.885854i	$0.653574\pi$
$398$	−16.6539	−0.834782
$399$	0	0
$400$	0	0
$401$	−17.5625	−0.877028	−0.438514	−	0.898724i	$-0.644495\pi$
$401$	−17.5625	−0.877028	−0.438514	+	0.898724i	$0.644495\pi$
$402$	0	0
$403$	10.8760	0.541773
$404$	−9.35599	−0.465478
$405$	0	0
$406$	−10.7556	−0.533790
$407$	5.80642	0.287814
$408$	0	0
$409$	−21.3461	−1.05550	−0.527749	−	0.849400i	$-0.676964\pi$
$409$	−21.3461	−1.05550	−0.527749	+	0.849400i	$0.676964\pi$
$410$	0	0
$411$	0	0
$412$	−2.60042	−0.128113
$413$	−8.20342	−0.403664
$414$	0	0
$415$	0	0
$416$	11.9759	0.587165
$417$	0	0
$418$	−6.29529	−0.307913
$419$	−28.8573	−1.40977	−0.704885	−	0.709321i	$-0.749001\pi$
$419$	−28.8573	−1.40977	−0.704885	+	0.709321i	$0.749001\pi$
$420$	0	0
$421$	−35.4893	−1.72964	−0.864822	−	0.502078i	$-0.832569\pi$
$421$	−35.4893	−1.72964	−0.864822	+	0.502078i	$0.832569\pi$
$422$	2.37826	0.115772
$423$	0	0
$424$	−22.0316	−1.06995
$425$	0	0
$426$	0	0
$427$	3.70471	0.179284
$428$	5.88400	0.284414
$429$	0	0
$430$	0	0
$431$	−9.24443	−0.445289	−0.222644	−	0.974900i	$-0.571469\pi$
$431$	−9.24443	−0.445289	−0.222644	+	0.974900i	$0.571469\pi$
$432$	0	0
$433$	−6.28544	−0.302059	−0.151030	−	0.988529i	$-0.548259\pi$
$433$	−6.28544	−0.302059	−0.151030	+	0.988529i	$0.548259\pi$
$434$	15.6128	0.749441
$435$	0	0
$436$	−8.25686	−0.395432
$437$	20.7368	0.991977
$438$	0	0
$439$	−36.5303	−1.74350	−0.871749	−	0.489952i	$-0.837014\pi$
$439$	−36.5303	−1.74350	−0.871749	+	0.489952i	$0.837014\pi$
$440$	0	0
$441$	0	0
$442$	26.8542	1.27732
$443$	38.2766	1.81857	0.909287	−	0.416170i	$-0.136628\pi$
$443$	38.2766	1.81857	0.909287	+	0.416170i	$0.136628\pi$
$444$	0	0
$445$	0	0
$446$	−31.5843	−1.49556
$447$	0	0
$448$	43.4050	2.05069
$449$	31.8479	1.50300	0.751498	−	0.659735i	$-0.229332\pi$
$449$	31.8479	1.50300	0.751498	+	0.659735i	$0.229332\pi$
$450$	0	0
$451$	−1.80642	−0.0850612
$452$	−0.927346	−0.0436187
$453$	0	0
$454$	7.70027	0.361391
$455$	0	0
$456$	0	0
$457$	1.39207	0.0651185	0.0325592	−	0.999470i	$-0.489634\pi$
$457$	1.39207	0.0651185	0.0325592	+	0.999470i	$0.489634\pi$
$458$	−28.3269	−1.32363
$459$	0	0
$460$	0	0
$461$	7.70471	0.358844	0.179422	−	0.983772i	$-0.442577\pi$
$461$	7.70471	0.358844	0.179422	+	0.983772i	$0.442577\pi$
$462$	0	0
$463$	4.68244	0.217611	0.108806	−	0.994063i	$-0.465297\pi$
$463$	4.68244	0.217611	0.108806	+	0.994063i	$0.465297\pi$
$464$	4.82870	0.224167
$465$	0	0
$466$	1.72885	0.0800873
$467$	−12.8573	−0.594964	−0.297482	−	0.954727i	$-0.596147\pi$
$467$	−12.8573	−0.594964	−0.297482	+	0.954727i	$0.596147\pi$
$468$	0	0
$469$	−23.8163	−1.09973
$470$	0	0
$471$	0	0
$472$	5.13077	0.236163
$473$	4.90321	0.225450
$474$	0	0
$475$	0	0
$476$	−13.7364	−0.629605
$477$	0	0
$478$	23.0223	1.05301
$479$	−8.38715	−0.383219	−0.191609	−	0.981471i	$-0.561371\pi$
$479$	−8.38715	−0.383219	−0.191609	+	0.981471i	$0.561371\pi$
$480$	0	0
$481$	24.0830	1.09809
$482$	−1.63465	−0.0744561
$483$	0	0
$484$	−0.525428	−0.0238831
$485$	0	0
$486$	0	0
$487$	9.83500	0.445667	0.222833	−	0.974857i	$-0.428469\pi$
$487$	9.83500	0.445667	0.222833	+	0.974857i	$0.428469\pi$
$488$	−2.31708	−0.104890
$489$	0	0
$490$	0	0
$491$	32.9403	1.48657	0.743286	−	0.668973i	$-0.233266\pi$
$491$	32.9403	1.48657	0.743286	+	0.668973i	$0.233266\pi$
$492$	0	0
$493$	−9.63158	−0.433785
$494$	−26.1106	−1.17477
$495$	0	0
$496$	−7.00937	−0.314730
$497$	−2.10171	−0.0942746
$498$	0	0
$499$	−1.63158	−0.0730397	−0.0365199	−	0.999333i	$-0.511627\pi$
$499$	−1.63158	−0.0730397	−0.0365199	+	0.999333i	$0.511627\pi$
$500$	0	0
$501$	0	0
$502$	−1.31450	−0.0586689
$503$	−41.8622	−1.86654	−0.933272	−	0.359171i	$-0.883059\pi$
$503$	−41.8622	−1.86654	−0.933272	+	0.359171i	$0.883059\pi$
$504$	0	0
$505$	0	0
$506$	−4.85728	−0.215932
$507$	0	0
$508$	−9.83056	−0.436160
$509$	38.8573	1.72232	0.861159	−	0.508335i	$-0.169739\pi$
$509$	38.8573	1.72232	0.861159	+	0.508335i	$0.169739\pi$
$510$	0	0
$511$	62.3180	2.75679
$512$	−24.1131	−1.06566
$513$	0	0
$514$	0.161933	0.00714257
$515$	0	0
$516$	0	0
$517$	−7.05086	−0.310096
$518$	34.5718	1.51900
$519$	0	0
$520$	0	0
$521$	−11.1111	−0.486785	−0.243393	−	0.969928i	$-0.578260\pi$
$521$	−11.1111	−0.486785	−0.243393	+	0.969928i	$0.578260\pi$
$522$	0	0
$523$	27.3002	1.19375	0.596877	−	0.802332i	$-0.296408\pi$
$523$	27.3002	1.19375	0.596877	+	0.802332i	$0.296408\pi$
$524$	−0.653858	−0.0285639
$525$	0	0
$526$	0.179286	0.00781724
$527$	13.9813	0.609033
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	13.3560	0.579055
$533$	−7.49240	−0.324532
$534$	0	0
$535$	0	0
$536$	14.8957	0.643396
$537$	0	0
$538$	32.6133	1.40606
$539$	17.0415	0.734029
$540$	0	0
$541$	−16.1017	−0.692267	−0.346133	−	0.938185i	$-0.612506\pi$
$541$	−16.1017	−0.692267	−0.346133	+	0.938185i	$0.612506\pi$
$542$	3.74314	0.160782
$543$	0	0
$544$	15.3951	0.660061
$545$	0	0
$546$	0	0
$547$	40.0370	1.71186	0.855930	−	0.517091i	$-0.172985\pi$
$547$	40.0370	1.71186	0.855930	+	0.517091i	$0.172985\pi$
$548$	9.87649	0.421903
$549$	0	0
$550$	0	0
$551$	9.36488	0.398957
$552$	0	0
$553$	−31.5210	−1.34041
$554$	−10.5763	−0.449343
$555$	0	0
$556$	−7.37778	−0.312888
$557$	28.2908	1.19872	0.599361	−	0.800479i	$-0.295422\pi$
$557$	28.2908	1.19872	0.599361	+	0.800479i	$0.295422\pi$
$558$	0	0
$559$	20.3368	0.860154
$560$	0	0
$561$	0	0
$562$	24.7457	1.04384
$563$	32.7926	1.38204	0.691022	−	0.722834i	$-0.257161\pi$
$563$	32.7926	1.38204	0.691022	+	0.722834i	$0.257161\pi$
$564$	0	0
$565$	0	0
$566$	−7.67752	−0.322710
$567$	0	0
$568$	1.31450	0.0551551
$569$	8.88586	0.372515	0.186257	−	0.982501i	$-0.440364\pi$
$569$	8.88586	0.372515	0.186257	+	0.982501i	$0.440364\pi$
$570$	0	0
$571$	−10.6953	−0.447586	−0.223793	−	0.974637i	$-0.571844\pi$
$571$	−10.6953	−0.447586	−0.223793	+	0.974637i	$0.571844\pi$
$572$	−2.17929	−0.0911205
$573$	0	0
$574$	−10.7556	−0.448929
$575$	0	0
$576$	0	0
$577$	−27.1338	−1.12960	−0.564798	−	0.825229i	$-0.691046\pi$
$577$	−27.1338	−1.12960	−0.564798	+	0.825229i	$0.691046\pi$
$578$	13.8780	0.577250
$579$	0	0
$580$	0	0
$581$	14.2351	0.590570
$582$	0	0
$583$	7.18421	0.297540
$584$	−38.9763	−1.61285
$585$	0	0
$586$	−20.2524	−0.836620
$587$	10.9590	0.452326	0.226163	−	0.974089i	$-0.427382\pi$
$587$	10.9590	0.452326	0.226163	+	0.974089i	$0.427382\pi$
$588$	0	0
$589$	−13.5941	−0.560136
$590$	0	0
$591$	0	0
$592$	−15.5210	−0.637908
$593$	23.7003	0.973253	0.486627	−	0.873610i	$-0.338227\pi$
$593$	23.7003	0.973253	0.486627	+	0.873610i	$0.338227\pi$
$594$	0	0
$595$	0	0
$596$	−1.60300	−0.0656616
$597$	0	0
$598$	−20.1463	−0.823842
$599$	−41.7146	−1.70441	−0.852205	−	0.523208i	$-0.824735\pi$
$599$	−41.7146	−1.70441	−0.852205	+	0.523208i	$0.824735\pi$
$600$	0	0
$601$	14.5906	0.595162	0.297581	−	0.954697i	$-0.403820\pi$
$601$	14.5906	0.595162	0.297581	+	0.954697i	$0.403820\pi$
$602$	29.1941	1.18986
$603$	0	0
$604$	0.171778	0.00698953
$605$	0	0
$606$	0	0
$607$	−19.9826	−0.811071	−0.405535	−	0.914079i	$-0.632915\pi$
$607$	−19.9826	−0.811071	−0.405535	+	0.914079i	$0.632915\pi$
$608$	−14.9688	−0.607066
$609$	0	0
$610$	0	0
$611$	−29.2444	−1.18310
$612$	0	0
$613$	19.0781	0.770555	0.385278	−	0.922801i	$-0.374106\pi$
$613$	19.0781	0.770555	0.385278	+	0.922801i	$0.374106\pi$
$614$	−11.6400	−0.469754
$615$	0	0
$616$	−15.0366	−0.605840
$617$	39.3590	1.58454	0.792268	−	0.610174i	$-0.208900\pi$
$617$	39.3590	1.58454	0.792268	+	0.610174i	$0.208900\pi$
$618$	0	0
$619$	−23.0923	−0.928160	−0.464080	−	0.885793i	$-0.653615\pi$
$619$	−23.0923	−0.928160	−0.464080	+	0.885793i	$0.653615\pi$
$620$	0	0
$621$	0	0
$622$	−17.6445	−0.707480
$623$	−3.05086	−0.122230
$624$	0	0
$625$	0	0
$626$	−25.5397	−1.02077
$627$	0	0
$628$	10.4603	0.417411
$629$	30.9590	1.23442
$630$	0	0
$631$	−25.5111	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$631$	−25.5111	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$632$	19.7146	0.784203
$633$	0	0
$634$	1.20294	0.0477750
$635$	0	0
$636$	0	0
$637$	70.6820	2.80052
$638$	−2.19358	−0.0868445
$639$	0	0
$640$	0	0
$641$	6.25380	0.247010	0.123505	−	0.992344i	$-0.460586\pi$
$641$	6.25380	0.247010	0.123505	+	0.992344i	$0.460586\pi$
$642$	0	0
$643$	6.84743	0.270036	0.135018	−	0.990843i	$-0.456891\pi$
$643$	6.84743	0.270036	0.135018	+	0.990843i	$0.456891\pi$
$644$	10.3051	0.406079
$645$	0	0
$646$	−33.5655	−1.32062
$647$	−20.2953	−0.797890	−0.398945	−	0.916975i	$-0.630624\pi$
$647$	−20.2953	−0.797890	−0.398945	+	0.916975i	$0.630624\pi$
$648$	0	0
$649$	−1.67307	−0.0656738
$650$	0	0
$651$	0	0
$652$	−6.39700	−0.250526
$653$	10.6222	0.415679	0.207840	−	0.978163i	$-0.433357\pi$
$653$	10.6222	0.415679	0.207840	+	0.978163i	$0.433357\pi$
$654$	0	0
$655$	0	0
$656$	4.82870	0.188529
$657$	0	0
$658$	−41.9813	−1.63660
$659$	−10.1017	−0.393507	−0.196753	−	0.980453i	$-0.563040\pi$
$659$	−10.1017	−0.393507	−0.196753	+	0.980453i	$0.563040\pi$
$660$	0	0
$661$	21.6128	0.840642	0.420321	−	0.907375i	$-0.361917\pi$
$661$	21.6128	0.840642	0.420321	+	0.907375i	$0.361917\pi$
$662$	−21.3492	−0.829760
$663$	0	0
$664$	−8.90321	−0.345512
$665$	0	0
$666$	0	0
$667$	7.22570	0.279780
$668$	6.83314	0.264382
$669$	0	0
$670$	0	0
$671$	0.755569	0.0291684
$672$	0	0
$673$	10.2208	0.393982	0.196991	−	0.980405i	$-0.436883\pi$
$673$	10.2208	0.393982	0.196991	+	0.980405i	$0.436883\pi$
$674$	−3.84929	−0.148269
$675$	0	0
$676$	−2.20834	−0.0849363
$677$	−13.9224	−0.535082	−0.267541	−	0.963546i	$-0.586211\pi$
$677$	−13.9224	−0.535082	−0.267541	+	0.963546i	$0.586211\pi$
$678$	0	0
$679$	−13.5111	−0.518510
$680$	0	0
$681$	0	0
$682$	3.18421	0.121930
$683$	10.3970	0.397830	0.198915	−	0.980017i	$-0.436258\pi$
$683$	10.3970	0.397830	0.198915	+	0.980017i	$0.436258\pi$
$684$	0	0
$685$	0	0
$686$	59.7877	2.28270
$687$	0	0
$688$	−13.1066	−0.499686
$689$	29.7975	1.13520
$690$	0	0
$691$	−0.977725	−0.0371944	−0.0185972	−	0.999827i	$-0.505920\pi$
$691$	−0.977725	−0.0371944	−0.0185972	+	0.999827i	$0.505920\pi$
$692$	7.30021	0.277512
$693$	0	0
$694$	−6.04287	−0.229384
$695$	0	0
$696$	0	0
$697$	−9.63158	−0.364822
$698$	22.1048	0.836678
$699$	0	0
$700$	0	0
$701$	−48.9688	−1.84953	−0.924764	−	0.380542i	$-0.875737\pi$
$701$	−48.9688	−1.84953	−0.924764	+	0.380542i	$0.875737\pi$
$702$	0	0
$703$	−30.1017	−1.13531
$704$	8.85236	0.333636
$705$	0	0
$706$	−27.2128	−1.02417
$707$	87.3087	3.28358
$708$	0	0
$709$	−37.2672	−1.39960	−0.699799	−	0.714340i	$-0.746727\pi$
$709$	−37.2672	−1.39960	−0.699799	+	0.714340i	$0.746727\pi$
$710$	0	0
$711$	0	0
$712$	1.90813	0.0715103
$713$	−10.4889	−0.392811
$714$	0	0
$715$	0	0
$716$	6.75557	0.252467
$717$	0	0
$718$	−25.8983	−0.966516
$719$	−5.83500	−0.217609	−0.108804	−	0.994063i	$-0.534702\pi$
$719$	−5.83500	−0.217609	−0.108804	+	0.994063i	$0.534702\pi$
$720$	0	0
$721$	24.2667	0.903739
$722$	9.56400	0.355935
$723$	0	0
$724$	−0.482081	−0.0179164
$725$	0	0
$726$	0	0
$727$	−46.8385	−1.73715	−0.868573	−	0.495562i	$-0.834962\pi$
$727$	−46.8385	−1.73715	−0.868573	+	0.495562i	$0.834962\pi$
$728$	−62.3663	−2.31145
$729$	0	0
$730$	0	0
$731$	26.1432	0.966941
$732$	0	0
$733$	−45.2083	−1.66981	−0.834904	−	0.550395i	$-0.814477\pi$
$733$	−45.2083	−1.66981	−0.834904	+	0.550395i	$0.814477\pi$
$734$	42.6637	1.57475
$735$	0	0
$736$	−11.5496	−0.425723
$737$	−4.85728	−0.178920
$738$	0	0
$739$	5.65433	0.207998	0.103999	−	0.994577i	$-0.466836\pi$
$739$	5.65433	0.207998	0.103999	+	0.994577i	$0.466836\pi$
$740$	0	0
$741$	0	0
$742$	42.7753	1.57033
$743$	−4.50622	−0.165317	−0.0826585	−	0.996578i	$-0.526341\pi$
$743$	−4.50622	−0.165317	−0.0826585	+	0.996578i	$0.526341\pi$
$744$	0	0
$745$	0	0
$746$	−20.6494	−0.756029
$747$	0	0
$748$	−2.80150	−0.102433
$749$	−54.9086	−2.00632
$750$	0	0
$751$	47.5121	1.73374	0.866870	−	0.498534i	$-0.166128\pi$
$751$	47.5121	1.73374	0.866870	+	0.498534i	$0.166128\pi$
$752$	18.8474	0.687295
$753$	0	0
$754$	−9.09817	−0.331336
$755$	0	0
$756$	0	0
$757$	−46.6637	−1.69602	−0.848011	−	0.529979i	$-0.822200\pi$
$757$	−46.6637	−1.69602	−0.848011	+	0.529979i	$0.822200\pi$
$758$	2.87601	0.104462
$759$	0	0
$760$	0	0
$761$	−14.9304	−0.541227	−0.270613	−	0.962688i	$-0.587227\pi$
$761$	−14.9304	−0.541227	−0.270613	+	0.962688i	$0.587227\pi$
$762$	0	0
$763$	77.0518	2.78946
$764$	7.54956	0.273134
$765$	0	0
$766$	−1.47643	−0.0533457
$767$	−6.93930	−0.250564
$768$	0	0
$769$	38.8573	1.40123	0.700615	−	0.713540i	$-0.252909\pi$
$769$	38.8573	1.40123	0.700615	+	0.713540i	$0.252909\pi$
$770$	0	0
$771$	0	0
$772$	6.17929	0.222397
$773$	36.3368	1.30694	0.653471	−	0.756951i	$-0.273312\pi$
$773$	36.3368	1.30694	0.653471	+	0.756951i	$0.273312\pi$
$774$	0	0
$775$	0	0
$776$	8.45044	0.303353
$777$	0	0
$778$	−2.75251	−0.0986821
$779$	9.36488	0.335532
$780$	0	0
$781$	−0.428639	−0.0153379
$782$	−25.8983	−0.926121
$783$	0	0
$784$	−45.5531	−1.62690
$785$	0	0
$786$	0	0
$787$	−33.5482	−1.19586	−0.597932	−	0.801547i	$-0.704011\pi$
$787$	−33.5482	−1.19586	−0.597932	+	0.801547i	$0.704011\pi$
$788$	−2.00751	−0.0715145
$789$	0	0
$790$	0	0
$791$	8.65386	0.307696
$792$	0	0
$793$	3.13383	0.111286
$794$	−22.4514	−0.796770
$795$	0	0
$796$	7.20601	0.255410
$797$	16.1334	0.571473	0.285736	−	0.958308i	$-0.407762\pi$
$797$	16.1334	0.571473	0.285736	+	0.958308i	$0.407762\pi$
$798$	0	0
$799$	−37.5941	−1.32998
$800$	0	0
$801$	0	0
$802$	−21.3265	−0.753063
$803$	12.7096	0.448513
$804$	0	0
$805$	0	0
$806$	13.2070	0.465195
$807$	0	0
$808$	−54.6065	−1.92105
$809$	−25.7431	−0.905081	−0.452540	−	0.891744i	$-0.649482\pi$
$809$	−25.7431	−0.905081	−0.452540	+	0.891744i	$0.649482\pi$
$810$	0	0
$811$	13.4509	0.472325	0.236163	−	0.971714i	$-0.424110\pi$
$811$	13.4509	0.472325	0.236163	+	0.971714i	$0.424110\pi$
$812$	4.65386	0.163318
$813$	0	0
$814$	7.05086	0.247132
$815$	0	0
$816$	0	0
$817$	−25.4193	−0.889308
$818$	−25.9210	−0.906308
$819$	0	0
$820$	0	0
$821$	−24.1748	−0.843708	−0.421854	−	0.906664i	$-0.638620\pi$
$821$	−24.1748	−0.843708	−0.421854	+	0.906664i	$0.638620\pi$
$822$	0	0
$823$	−40.9117	−1.42609	−0.713046	−	0.701118i	$-0.752685\pi$
$823$	−40.9117	−1.42609	−0.713046	+	0.701118i	$0.752685\pi$
$824$	−15.1774	−0.528731
$825$	0	0
$826$	−9.96158	−0.346608
$827$	20.1476	0.700602	0.350301	−	0.936637i	$-0.386079\pi$
$827$	20.1476	0.700602	0.350301	+	0.936637i	$0.386079\pi$
$828$	0	0
$829$	31.4322	1.09168	0.545842	−	0.837888i	$-0.316210\pi$
$829$	31.4322	1.09168	0.545842	+	0.837888i	$0.316210\pi$
$830$	0	0
$831$	0	0
$832$	36.7164	1.27291
$833$	90.8627	3.14821
$834$	0	0
$835$	0	0
$836$	2.72393	0.0942089
$837$	0	0
$838$	−35.0420	−1.21050
$839$	52.8988	1.82627	0.913134	−	0.407659i	$-0.133655\pi$
$839$	52.8988	1.82627	0.913134	+	0.407659i	$0.133655\pi$
$840$	0	0
$841$	−25.7368	−0.887477
$842$	−43.0954	−1.48517
$843$	0	0
$844$	−1.02906	−0.0354216
$845$	0	0
$846$	0	0
$847$	4.90321	0.168476
$848$	−19.2039	−0.659465
$849$	0	0
$850$	0	0
$851$	−23.2257	−0.796167
$852$	0	0
$853$	−46.9229	−1.60661	−0.803305	−	0.595568i	$-0.796927\pi$
$853$	−46.9229	−1.60661	−0.803305	+	0.595568i	$0.796927\pi$
$854$	4.49871	0.153943
$855$	0	0
$856$	34.3422	1.17379
$857$	25.1481	0.859043	0.429522	−	0.903057i	$-0.358682\pi$
$857$	25.1481	0.859043	0.429522	+	0.903057i	$0.358682\pi$
$858$	0	0
$859$	1.84791	0.0630499	0.0315250	−	0.999503i	$-0.489964\pi$
$859$	1.84791	0.0630499	0.0315250	+	0.999503i	$0.489964\pi$
$860$	0	0
$861$	0	0
$862$	−11.2257	−0.382349
$863$	−32.6824	−1.11252	−0.556262	−	0.831007i	$-0.687765\pi$
$863$	−32.6824	−1.11252	−0.556262	+	0.831007i	$0.687765\pi$
$864$	0	0
$865$	0	0
$866$	−7.63254	−0.259364
$867$	0	0
$868$	−6.75557	−0.229299
$869$	−6.42864	−0.218077
$870$	0	0
$871$	−20.1463	−0.682630
$872$	−48.1915	−1.63197
$873$	0	0
$874$	25.1811	0.851765
$875$	0	0
$876$	0	0
$877$	49.1798	1.66068	0.830341	−	0.557255i	$-0.188145\pi$
$877$	49.1798	1.66068	0.830341	+	0.557255i	$0.188145\pi$
$878$	−44.3595	−1.49706
$879$	0	0
$880$	0	0
$881$	−33.8163	−1.13930	−0.569650	−	0.821888i	$-0.692921\pi$
$881$	−33.8163	−1.13930	−0.569650	+	0.821888i	$0.692921\pi$
$882$	0	0
$883$	24.7368	0.832461	0.416230	−	0.909259i	$-0.363351\pi$
$883$	24.7368	0.832461	0.416230	+	0.909259i	$0.363351\pi$
$884$	−11.6196	−0.390810
$885$	0	0
$886$	46.4800	1.56153
$887$	−7.64004	−0.256528	−0.128264	−	0.991740i	$-0.540940\pi$
$887$	−7.64004	−0.256528	−0.128264	+	0.991740i	$0.540940\pi$
$888$	0	0
$889$	91.7373	3.07677
$890$	0	0
$891$	0	0
$892$	13.6663	0.457581
$893$	36.5531	1.22320
$894$	0	0
$895$	0	0
$896$	24.3926	0.814898
$897$	0	0
$898$	38.6735	1.29055
$899$	−4.73683	−0.157982
$900$	0	0
$901$	38.3051	1.27613
$902$	−2.19358	−0.0730381
$903$	0	0
$904$	−5.41249	−0.180017
$905$	0	0
$906$	0	0
$907$	−30.3970	−1.00932	−0.504658	−	0.863319i	$-0.668381\pi$
$907$	−30.3970	−1.00932	−0.504658	+	0.863319i	$0.668381\pi$
$908$	−3.33185	−0.110571
$909$	0	0
$910$	0	0
$911$	45.3274	1.50176	0.750882	−	0.660436i	$-0.229629\pi$
$911$	45.3274	1.50176	0.750882	+	0.660436i	$0.229629\pi$
$912$	0	0
$913$	2.90321	0.0960823
$914$	1.69042	0.0559142
$915$	0	0
$916$	12.2569	0.404978
$917$	6.10171	0.201496
$918$	0	0
$919$	−16.3269	−0.538576	−0.269288	−	0.963060i	$-0.586788\pi$
$919$	−16.3269	−0.538576	−0.269288	+	0.963060i	$0.586788\pi$
$920$	0	0
$921$	0	0
$922$	9.35599	0.308123
$923$	−1.77784	−0.0585184
$924$	0	0
$925$	0	0
$926$	5.68598	0.186853
$927$	0	0
$928$	−5.21585	−0.171219
$929$	29.6128	0.971566	0.485783	−	0.874079i	$-0.338535\pi$
$929$	29.6128	0.971566	0.485783	+	0.874079i	$0.338535\pi$
$930$	0	0
$931$	−88.3466	−2.89544
$932$	−0.748060	−0.0245035
$933$	0	0
$934$	−15.6128	−0.510868
$935$	0	0
$936$	0	0
$937$	8.92195	0.291467	0.145734	−	0.989324i	$-0.453446\pi$
$937$	8.92195	0.291467	0.145734	+	0.989324i	$0.453446\pi$
$938$	−28.9206	−0.944290
$939$	0	0
$940$	0	0
$941$	22.2766	0.726195	0.363097	−	0.931751i	$-0.381719\pi$
$941$	22.2766	0.726195	0.363097	+	0.931751i	$0.381719\pi$
$942$	0	0
$943$	7.22570	0.235301
$944$	4.47224	0.145559
$945$	0	0
$946$	5.95407	0.193583
$947$	−44.4612	−1.44480	−0.722398	−	0.691477i	$-0.756960\pi$
$947$	−44.4612	−1.44480	−0.722398	+	0.691477i	$0.756960\pi$
$948$	0	0
$949$	52.7150	1.71120
$950$	0	0
$951$	0	0
$952$	−80.1727	−2.59841
$953$	−20.5575	−0.665924	−0.332962	−	0.942940i	$-0.608048\pi$
$953$	−20.5575	−0.665924	−0.332962	+	0.942940i	$0.608048\pi$
$954$	0	0
$955$	0	0
$956$	−9.96158	−0.322180
$957$	0	0
$958$	−10.1847	−0.329052
$959$	−92.1659	−2.97619
$960$	0	0
$961$	−24.1240	−0.778193
$962$	29.2444	0.942878
$963$	0	0
$964$	0.707300	0.0227806
$965$	0	0
$966$	0	0
$967$	27.4839	0.883824	0.441912	−	0.897058i	$-0.354300\pi$
$967$	27.4839	0.883824	0.441912	+	0.897058i	$0.354300\pi$
$968$	−3.06668	−0.0985667
$969$	0	0
$970$	0	0
$971$	5.81532	0.186622	0.0933112	−	0.995637i	$-0.470255\pi$
$971$	5.81532	0.186622	0.0933112	+	0.995637i	$0.470255\pi$
$972$	0	0
$973$	68.8484	2.20718
$974$	11.9428	0.382673
$975$	0	0
$976$	−2.01969	−0.0646487
$977$	−48.3912	−1.54817	−0.774085	−	0.633081i	$-0.781790\pi$
$977$	−48.3912	−1.54817	−0.774085	+	0.633081i	$0.781790\pi$
$978$	0	0
$979$	−0.622216	−0.0198861
$980$	0	0
$981$	0	0
$982$	40.0000	1.27645
$983$	49.9724	1.59387	0.796936	−	0.604064i	$-0.206453\pi$
$983$	49.9724	1.59387	0.796936	+	0.604064i	$0.206453\pi$
$984$	0	0
$985$	0	0
$986$	−11.6958	−0.372471
$987$	0	0
$988$	11.2979	0.359433
$989$	−19.6128	−0.623652
$990$	0	0
$991$	−7.35905	−0.233768	−0.116884	−	0.993146i	$-0.537291\pi$
$991$	−7.35905	−0.233768	−0.116884	+	0.993146i	$0.537291\pi$
$992$	7.57136	0.240391
$993$	0	0
$994$	−2.55215	−0.0809492
$995$	0	0
$996$	0	0
$997$	−4.33138	−0.137176	−0.0685880	−	0.997645i	$-0.521849\pi$
$997$	−4.33138	−0.137176	−0.0685880	+	0.997645i	$0.521849\pi$
$998$	−1.98126	−0.0627158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.y.1.3		3
3.2	odd	2		825.2.a.n.1.1		3
5.2	odd	4		495.2.c.d.199.4		6
5.3	odd	4		495.2.c.d.199.3		6
5.4	even	2		2475.2.a.be.1.1		3
15.2	even	4		165.2.c.a.34.3	✓	6
15.8	even	4		165.2.c.a.34.4	yes	6
15.14	odd	2		825.2.a.h.1.3		3
33.32	even	2		9075.2.a.cc.1.3		3
60.23	odd	4		2640.2.d.i.529.5		6
60.47	odd	4		2640.2.d.i.529.2		6
165.32	odd	4		1815.2.c.d.364.4		6
165.98	odd	4		1815.2.c.d.364.3		6
165.164	even	2		9075.2.a.ck.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.3	✓	6	15.2	even	4
165.2.c.a.34.4	yes	6	15.8	even	4
495.2.c.d.199.3		6	5.3	odd	4
495.2.c.d.199.4		6	5.2	odd	4
825.2.a.h.1.3		3	15.14	odd	2
825.2.a.n.1.1		3	3.2	odd	2
1815.2.c.d.364.3		6	165.98	odd	4
1815.2.c.d.364.4		6	165.32	odd	4
2475.2.a.y.1.3		3	1.1	even	1	trivial
2475.2.a.be.1.1		3	5.4	even	2
2640.2.d.i.529.2		6	60.47	odd	4
2640.2.d.i.529.5		6	60.23	odd	4
9075.2.a.cc.1.3		3	33.32	even	2
9075.2.a.ck.1.1		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)

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} -0.525428 q^{4} +4.90321 q^{7} -3.06668 q^{8} +1.00000 q^{11} +4.14764 q^{13} +5.95407 q^{14} -2.67307 q^{16} +5.33185 q^{17} -5.18421 q^{19} +1.21432 q^{22} -4.00000 q^{23} +5.03657 q^{26} -2.57628 q^{28} -1.80642 q^{29} +2.62222 q^{31} +2.88739 q^{32} +6.47457 q^{34} +5.80642 q^{37} -6.29529 q^{38} -1.80642 q^{41} +4.90321 q^{43} -0.525428 q^{44} -4.85728 q^{46} -7.05086 q^{47} +17.0415 q^{49} -2.17929 q^{52} +7.18421 q^{53} -15.0366 q^{56} -2.19358 q^{58} -1.67307 q^{59} +0.755569 q^{61} +3.18421 q^{62} +8.85236 q^{64} -4.85728 q^{67} -2.80150 q^{68} -0.428639 q^{71} +12.7096 q^{73} +7.05086 q^{74} +2.72393 q^{76} +4.90321 q^{77} -6.42864 q^{79} -2.19358 q^{82} +2.90321 q^{83} +5.95407 q^{86} -3.06668 q^{88} -0.622216 q^{89} +20.3368 q^{91} +2.10171 q^{92} -8.56199 q^{94} -2.75557 q^{97} +20.6938 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} + 3 q^{11} + 6 q^{13} - 2 q^{14} + 5 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 8 q^{26} + 12 q^{28} + 8 q^{29} + 8 q^{31} - 11 q^{32} + 26 q^{34}+ \cdots + 29 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 + 3 * q^11 + 6 * q^13 - 2 * q^14 + 5 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 8 * q^26 + 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 4 * q^37 - 6 * q^38 + 8 * q^41 + 8 * q^43 + 5 * q^44 + 12 * q^46 - 8 * q^47 + 11 * q^49 - 26 * q^52 + 8 * q^53 - 38 * q^56 - 20 * q^58 + 8 * q^59 + 2 * q^61 - 4 * q^62 + 33 * q^64 + 12 * q^67 - 28 * q^68 + 12 * q^71 + 18 * q^73 + 8 * q^74 - 18 * q^76 + 8 * q^77 - 6 * q^79 - 20 * q^82 + 2 * q^83 - 2 * q^86 - 9 * q^88 - 2 * q^89 + 8 * q^91 - 20 * q^92 - 12 * q^94 - 8 * q^97 + 29 * q^98

Introduction

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