LMFDB - Embedded newform 9075.2.a.cc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
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			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} -1.21432 q^{6} -4.90321 q^{7} -3.06668 q^{8} +1.00000 q^{9} +0.525428 q^{12} -4.14764 q^{13} -5.95407 q^{14} -2.67307 q^{16} +5.33185 q^{17} +1.21432 q^{18} +5.18421 q^{19} +4.90321 q^{21} +4.00000 q^{23} +3.06668 q^{24} -5.03657 q^{26} -1.00000 q^{27} +2.57628 q^{28} -1.80642 q^{29} +2.62222 q^{31} +2.88739 q^{32} +6.47457 q^{34} -0.525428 q^{36} +5.80642 q^{37} +6.29529 q^{38} +4.14764 q^{39} -1.80642 q^{41} +5.95407 q^{42} -4.90321 q^{43} +4.85728 q^{46} +7.05086 q^{47} +2.67307 q^{48} +17.0415 q^{49} -5.33185 q^{51} +2.17929 q^{52} -7.18421 q^{53} -1.21432 q^{54} +15.0366 q^{56} -5.18421 q^{57} -2.19358 q^{58} +1.67307 q^{59} -0.755569 q^{61} +3.18421 q^{62} -4.90321 q^{63} +8.85236 q^{64} -4.85728 q^{67} -2.80150 q^{68} -4.00000 q^{69} +0.428639 q^{71} -3.06668 q^{72} -12.7096 q^{73} +7.05086 q^{74} -2.72393 q^{76} +5.03657 q^{78} +6.42864 q^{79} +1.00000 q^{81} -2.19358 q^{82} +2.90321 q^{83} -2.57628 q^{84} -5.95407 q^{86} +1.80642 q^{87} +0.622216 q^{89} +20.3368 q^{91} -2.10171 q^{92} -2.62222 q^{93} +8.56199 q^{94} -2.88739 q^{96} -2.75557 q^{97} +20.6938 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9} - 5 q^{12} - 6 q^{13} + 2 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} + 2 q^{19} + 8 q^{21} + 12 q^{23} + 9 q^{24} - 8 q^{26} - 3 q^{27}+ \cdots + 29 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 5 * q^4 + 3 * q^6 - 8 * q^7 - 9 * q^8 + 3 * q^9 - 5 * q^12 - 6 * q^13 + 2 * q^14 + 5 * q^16 - 4 * q^17 - 3 * q^18 + 2 * q^19 + 8 * q^21 + 12 * q^23 + 9 * q^24 - 8 * q^26 - 3 * q^27 - 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 5 * q^36 + 4 * q^37 + 6 * q^38 + 6 * q^39 + 8 * q^41 - 2 * q^42 - 8 * q^43 - 12 * q^46 + 8 * q^47 - 5 * q^48 + 11 * q^49 + 4 * q^51 + 26 * q^52 - 8 * q^53 + 3 * q^54 + 38 * q^56 - 2 * q^57 - 20 * q^58 - 8 * q^59 - 2 * q^61 - 4 * q^62 - 8 * q^63 + 33 * q^64 + 12 * q^67 - 28 * q^68 - 12 * q^69 - 12 * q^71 - 9 * q^72 - 18 * q^73 + 8 * q^74 + 18 * q^76 + 8 * q^78 + 6 * q^79 + 3 * q^81 - 20 * q^82 + 2 * q^83 + 12 * q^84 + 2 * q^86 - 8 * q^87 + 2 * q^89 + 8 * q^91 + 20 * q^92 - 8 * q^93 + 12 * q^94 + 11 * q^96 - 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	$0.358751\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.525428	−0.262714
$5$	0	0
$5$	0	0
$6$	−1.21432	−0.495744
$7$	−4.90321	−1.85324	−0.926620	−	0.375999i	$-0.877300\pi$
$7$	−4.90321	−1.85324	−0.926620	+	0.375999i	$0.877300\pi$
$8$	−3.06668	−1.08423
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.525428	0.151678
$13$	−4.14764	−1.15035	−0.575175	−	0.818031i	$-0.695066\pi$
$13$	−4.14764	−1.15035	−0.575175	+	0.818031i	$0.695066\pi$
$14$	−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	$0.276198\pi$
$18$	1.21432	0.286218
$19$	5.18421	1.18934	0.594669	−	0.803970i	$-0.297283\pi$
$19$	5.18421	1.18934	0.594669	+	0.803970i	$0.297283\pi$
$20$	0	0
$21$	4.90321	1.06997
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	3.06668	0.625983
$25$	0	0
$26$	−5.03657	−0.987752
$27$	−1.00000	−0.192450
$28$	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
$36$	−0.525428	−0.0875713
$37$	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
$39$	4.14764	0.664154
$40$	0	0
$41$	−1.80642	−0.282116	−0.141058	−	0.990001i	$-0.545050\pi$
$41$	−1.80642	−0.282116	−0.141058	+	0.990001i	$0.545050\pi$
$42$	5.95407	0.918732
$43$	−4.90321	−0.747733	−0.373866	−	0.927483i	$-0.621968\pi$
$43$	−4.90321	−0.747733	−0.373866	+	0.927483i	$0.621968\pi$
$44$	0	0
$45$	0	0
$46$	4.85728	0.716167
$47$	7.05086	1.02847	0.514236	−	0.857648i	$-0.328075\pi$
$47$	7.05086	1.02847	0.514236	+	0.857648i	$0.328075\pi$
$48$	2.67307	0.385825
$49$	17.0415	2.43450
$50$	0	0
$51$	−5.33185	−0.746609
$52$	2.17929	0.302213
$53$	−7.18421	−0.986827	−0.493413	−	0.869795i	$-0.664251\pi$
$53$	−7.18421	−0.986827	−0.493413	+	0.869795i	$0.664251\pi$
$54$	−1.21432	−0.165248
$55$	0	0
$56$	15.0366	2.00935
$57$	−5.18421	−0.686665
$58$	−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
$61$	−0.755569	−0.0967407	−0.0483703	−	0.998829i	$-0.515403\pi$
$61$	−0.755569	−0.0967407	−0.0483703	+	0.998829i	$0.515403\pi$
$62$	3.18421	0.404395
$63$	−4.90321	−0.617747
$64$	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
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0.428639	0.0508701	0.0254351	−	0.999676i	$-0.491903\pi$
$71$	0.428639	0.0508701	0.0254351	+	0.999676i	$0.491903\pi$
$72$	−3.06668	−0.361411
$73$	−12.7096	−1.48755	−0.743775	−	0.668430i	$-0.766967\pi$
$73$	−12.7096	−1.48755	−0.743775	+	0.668430i	$0.766967\pi$
$74$	7.05086	0.819645
$75$	0	0
$76$	−2.72393	−0.312456
$77$	0	0
$78$	5.03657	0.570279
$79$	6.42864	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$79$	6.42864	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−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	$0.449065\pi$
$84$	−2.57628	−0.281095
$85$	0	0
$86$	−5.95407	−0.642044
$87$	1.80642	0.193669
$88$	0	0
$89$	0.622216	0.0659547	0.0329774	−	0.999456i	$-0.489501\pi$
$89$	0.622216	0.0659547	0.0329774	+	0.999456i	$0.489501\pi$
$90$	0	0
$91$	20.3368	2.13187
$92$	−2.10171	−0.219118
$93$	−2.62222	−0.271911
$94$	8.56199	0.883102
$95$	0	0
$96$	−2.88739	−0.294693
$97$	−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	$0.153540\pi$
$102$	−6.47457	−0.641078
$103$	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	$0.682068\pi$
$108$	0.525428	0.0505593
$109$	−15.7146	−1.50518	−0.752591	−	0.658488i	$-0.771196\pi$
$109$	−15.7146	−1.50518	−0.752591	+	0.658488i	$0.771196\pi$
$110$	0	0
$111$	−5.80642	−0.551121
$112$	13.1066	1.23846
$113$	−1.76494	−0.166031	−0.0830156	−	0.996548i	$-0.526455\pi$
$113$	−1.76494	−0.166031	−0.0830156	+	0.996548i	$0.526455\pi$
$114$	−6.29529	−0.589608
$115$	0	0
$116$	0.949145	0.0881259
$117$	−4.14764	−0.383450
$118$	2.03164	0.187028
$119$	−26.1432	−2.39654
$120$	0	0
$121$	0	0
$122$	−0.917502	−0.0830667
$123$	1.80642	0.162880
$124$	−1.37778	−0.123729
$125$	0	0
$126$	−5.95407	−0.530430
$127$	−18.7096	−1.66021	−0.830106	−	0.557606i	$-0.811720\pi$
$127$	−18.7096	−1.66021	−0.830106	+	0.557606i	$0.811720\pi$
$128$	4.97481	0.439715
$129$	4.90321	0.431704
$130$	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	$-0.203252\pi$
$137$	18.7971	1.60594	0.802970	+	0.596019i	$0.203252\pi$
$138$	−4.85728	−0.413479
$139$	−14.0415	−1.19098	−0.595492	−	0.803361i	$-0.703043\pi$
$139$	−14.0415	−1.19098	−0.595492	+	0.803361i	$0.703043\pi$
$140$	0	0
$141$	−7.05086	−0.593789
$142$	0.520505	0.0436798
$143$	0	0
$144$	−2.67307	−0.222756
$145$	0	0
$146$	−15.4336	−1.27729
$147$	−17.0415	−1.40556
$148$	−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	$-0.495766\pi$
$151$	0.326929	0.0266051	0.0133026	+	0.999912i	$0.495766\pi$
$152$	−15.8983	−1.28952
$153$	5.33185	0.431055
$154$	0	0
$155$	0	0
$156$	−2.17929	−0.174483
$157$	−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
$159$	7.18421	0.569745
$160$	0	0
$161$	−19.6128	−1.54571
$162$	1.21432	0.0954060
$163$	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	$0.667835\pi$
$168$	−15.0366	−1.16010
$169$	4.20294	0.323303
$170$	0	0
$171$	5.18421	0.396446
$172$	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	$0.677120\pi$
$174$	2.19358	0.166295
$175$	0	0
$176$	0	0
$177$	−1.67307	−0.125756
$178$	0.755569	0.0566323
$179$	12.8573	0.960998	0.480499	−	0.876995i	$-0.340456\pi$
$179$	12.8573	0.960998	0.480499	+	0.876995i	$0.340456\pi$
$180$	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
$183$	0.755569	0.0558532
$184$	−12.2667	−0.904314
$185$	0	0
$186$	−3.18421	−0.233477
$187$	0	0
$188$	−3.70471	−0.270194
$189$	4.90321	0.356656
$190$	0	0
$191$	14.3684	1.03966	0.519831	−	0.854269i	$-0.325995\pi$
$191$	14.3684	1.03966	0.519831	+	0.854269i	$0.325995\pi$
$192$	−8.85236	−0.638864
$193$	11.7605	0.846539	0.423269	−	0.906004i	$-0.360882\pi$
$193$	11.7605	0.846539	0.423269	+	0.906004i	$0.360882\pi$
$194$	−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
$201$	4.85728	0.342606
$202$	21.6227	1.52137
$203$	8.85728	0.621659
$204$	2.80150	0.196144
$205$	0	0
$206$	6.00984	0.418726
$207$	4.00000	0.278019
$208$	11.0869	0.768741
$209$	0	0
$210$	0	0
$211$	−1.95851	−0.134830	−0.0674148	−	0.997725i	$-0.521475\pi$
$211$	−1.95851	−0.134830	−0.0674148	+	0.997725i	$0.521475\pi$
$212$	3.77478	0.259253
$213$	−0.428639	−0.0293699
$214$	−13.5986	−0.929578
$215$	0	0
$216$	3.06668	0.208661
$217$	−12.8573	−0.872809
$218$	−19.0825	−1.29243
$219$	12.7096	0.858838
$220$	0	0
$221$	−22.1146	−1.48759
$222$	−7.05086	−0.473222
$223$	−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	$0.432510\pi$
$228$	2.72393	0.180396
$229$	−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	$0.485150\pi$
$234$	−5.03657	−0.329251
$235$	0	0
$236$	−0.879077	−0.0572231
$237$	−6.42864	−0.417585
$238$	−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	$-0.486195\pi$
$241$	1.34614	0.0867126	0.0433563	+	0.999060i	$0.486195\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0.396997	0.0254151
$245$	0	0
$246$	2.19358	0.139857
$247$	−21.5022	−1.36816
$248$	−8.04149	−0.510635
$249$	−2.90321	−0.183984
$250$	0	0
$251$	1.08250	0.0683267	0.0341633	−	0.999416i	$-0.489123\pi$
$251$	1.08250	0.0683267	0.0341633	+	0.999416i	$0.489123\pi$
$252$	2.57628	0.162291
$253$	0	0
$254$	−22.7195	−1.42555
$255$	0	0
$256$	−11.6637	−0.728981
$257$	−0.133353	−0.00831834	−0.00415917	−	0.999991i	$-0.501324\pi$
$257$	−0.133353	−0.00831834	−0.00415917	+	0.999991i	$0.501324\pi$
$258$	5.95407	0.370684
$259$	−28.4701	−1.76905
$260$	0	0
$261$	−1.80642	−0.111815
$262$	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
$267$	−0.622216	−0.0380790
$268$	2.55215	0.155897
$269$	−26.8573	−1.63752	−0.818759	−	0.574138i	$-0.805337\pi$
$269$	−26.8573	−1.63752	−0.818759	+	0.574138i	$0.805337\pi$
$270$	0	0
$271$	−3.08250	−0.187248	−0.0936242	−	0.995608i	$-0.529845\pi$
$271$	−3.08250	−0.187248	−0.0936242	+	0.995608i	$0.529845\pi$
$272$	−14.2524	−0.864180
$273$	−20.3368	−1.23084
$274$	22.8256	1.37895
$275$	0	0
$276$	2.10171	0.126508
$277$	8.70964	0.523311	0.261656	−	0.965161i	$-0.415732\pi$
$277$	8.70964	0.523311	0.261656	+	0.965161i	$0.415732\pi$
$278$	−17.0509	−1.02264
$279$	2.62222	0.156988
$280$	0	0
$281$	20.3783	1.21567	0.607833	−	0.794065i	$-0.292039\pi$
$281$	20.3783	1.21567	0.607833	+	0.794065i	$0.292039\pi$
$282$	−8.56199	−0.509859
$283$	6.32248	0.375833	0.187916	−	0.982185i	$-0.439827\pi$
$283$	6.32248	0.375833	0.187916	+	0.982185i	$0.439827\pi$
$284$	−0.225219	−0.0133643
$285$	0	0
$286$	0	0
$287$	8.85728	0.522829
$288$	2.88739	0.170141
$289$	11.4286	0.672273
$290$	0	0
$291$	2.75557	0.161534
$292$	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	$0.661971\pi$
$294$	−20.6938	−1.20689
$295$	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
$303$	−17.8064	−1.02295
$304$	−13.8578	−0.794797
$305$	0	0
$306$	6.47457	0.370127
$307$	9.58565	0.547082	0.273541	−	0.961860i	$-0.411805\pi$
$307$	9.58565	0.547082	0.273541	+	0.961860i	$0.411805\pi$
$308$	0	0
$309$	−4.94914	−0.281547
$310$	0	0
$311$	14.5303	0.823941	0.411970	−	0.911197i	$-0.364841\pi$
$311$	14.5303	0.823941	0.411970	+	0.911197i	$0.364841\pi$
$312$	−12.7195	−0.720099
$313$	−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	$-0.508856\pi$
$317$	−0.990632	−0.0556394	−0.0278197	+	0.999613i	$0.508856\pi$
$318$	8.72393	0.489213
$319$	0	0
$320$	0	0
$321$	11.1985	0.625039
$322$	−23.8163	−1.32723
$323$	27.6414	1.53801
$324$	−0.525428	−0.0291904
$325$	0	0
$326$	14.7841	0.818818
$327$	15.7146	0.869017
$328$	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
$333$	5.80642	0.318190
$334$	−15.7921	−0.864107
$335$	0	0
$336$	−13.1066	−0.715025
$337$	3.16992	0.172676	0.0863382	−	0.996266i	$-0.472483\pi$
$337$	3.16992	0.172676	0.0863382	+	0.996266i	$0.472483\pi$
$338$	5.10372	0.277606
$339$	1.76494	0.0958582
$340$	0	0
$341$	0	0
$342$	6.29529	0.340410
$343$	−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	$0.542645\pi$
$348$	−0.949145	−0.0508795
$349$	−18.2034	−0.974407	−0.487203	−	0.873289i	$-0.661983\pi$
$349$	−18.2034	−0.974407	−0.487203	+	0.873289i	$0.661983\pi$
$350$	0	0
$351$	4.14764	0.221385
$352$	0	0
$353$	22.4099	1.19276	0.596379	−	0.802703i	$-0.296605\pi$
$353$	22.4099	1.19276	0.596379	+	0.802703i	$0.296605\pi$
$354$	−2.03164	−0.107981
$355$	0	0
$356$	−0.326929	−0.0173272
$357$	26.1432	1.38364
$358$	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
$366$	0.917502	0.0479586
$367$	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
$369$	−1.80642	−0.0940387
$370$	0	0
$371$	35.2257	1.82883
$372$	1.37778	0.0714348
$373$	17.0049	0.880481	0.440241	−	0.897880i	$-0.354893\pi$
$373$	17.0049	0.880481	0.440241	+	0.897880i	$0.354893\pi$
$374$	0	0
$375$	0	0
$376$	−21.6227	−1.11511
$377$	7.49240	0.385878
$378$	5.95407	0.306244
$379$	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
$381$	18.7096	0.958524
$382$	17.4479	0.892710
$383$	1.21585	0.0621271	0.0310635	−	0.999517i	$-0.490111\pi$
$383$	1.21585	0.0621271	0.0310635	+	0.999517i	$0.490111\pi$
$384$	−4.97481	−0.253870
$385$	0	0
$386$	14.2810	0.726884
$387$	−4.90321	−0.249244
$388$	1.44785	0.0735035
$389$	2.26671	0.114927	0.0574633	−	0.998348i	$-0.481699\pi$
$389$	2.26671	0.114927	0.0574633	+	0.998348i	$0.481699\pi$
$390$	0	0
$391$	21.3274	1.07857
$392$	−52.2607	−2.63957
$393$	−1.24443	−0.0627733
$394$	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
$399$	25.4193	1.27256
$400$	0	0
$401$	17.5625	0.877028	0.438514	−	0.898724i	$-0.355505\pi$
$401$	17.5625	0.877028	0.438514	+	0.898724i	$0.355505\pi$
$402$	5.89829	0.294180
$403$	−10.8760	−0.541773
$404$	−9.35599	−0.465478
$405$	0	0
$406$	10.7556	0.533790
$407$	0	0
$408$	16.3511	0.809498
$409$	21.3461	1.05550	0.527749	−	0.849400i	$-0.323036\pi$
$409$	21.3461	1.05550	0.527749	+	0.849400i	$0.323036\pi$
$410$	0	0
$411$	−18.7971	−0.927190
$412$	−2.60042	−0.128113
$413$	−8.20342	−0.403664
$414$	4.85728	0.238722
$415$	0	0
$416$	−11.9759	−0.587165
$417$	14.0415	0.687615
$418$	0	0
$419$	28.8573	1.40977	0.704885	−	0.709321i	$-0.250999\pi$
$419$	28.8573	1.40977	0.704885	+	0.709321i	$0.250999\pi$
$420$	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
$423$	7.05086	0.342824
$424$	22.0316	1.06995
$425$	0	0
$426$	−0.520505	−0.0252186
$427$	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	$0.571469\pi$
$432$	2.67307	0.128608
$433$	−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
$438$	15.4336	0.737444
$439$	36.5303	1.74350	0.871749	−	0.489952i	$-0.162986\pi$
$439$	36.5303	1.74350	0.871749	+	0.489952i	$0.162986\pi$
$440$	0	0
$441$	17.0415	0.811499
$442$	−26.8542	−1.27732
$443$	−38.2766	−1.81857	−0.909287	−	0.416170i	$-0.863372\pi$
$443$	−38.2766	−1.81857	−0.909287	+	0.416170i	$0.863372\pi$
$444$	3.05086	0.144787
$445$	0	0
$446$	−31.5843	−1.49556
$447$	−3.05086	−0.144300
$448$	−43.4050	−2.05069
$449$	−31.8479	−1.50300	−0.751498	−	0.659735i	$-0.770668\pi$
$449$	−31.8479	−1.50300	−0.751498	+	0.659735i	$0.770668\pi$
$450$	0	0
$451$	0	0
$452$	0.927346	0.0436187
$453$	−0.326929	−0.0153605
$454$	7.70027	0.361391
$455$	0	0
$456$	15.8983	0.744506
$457$	−1.39207	−0.0651185	−0.0325592	−	0.999470i	$-0.510366\pi$
$457$	−1.39207	−0.0651185	−0.0325592	+	0.999470i	$0.510366\pi$
$458$	−28.3269	−1.32363
$459$	−5.33185	−0.248870
$460$	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	$-0.403853\pi$
$467$	12.8573	0.594964	0.297482	+	0.954727i	$0.403853\pi$
$468$	2.17929	0.100738
$469$	23.8163	1.09973
$470$	0	0
$471$	19.9081	0.917318
$472$	−5.13077	−0.236163
$473$	0	0
$474$	−7.80642	−0.358561
$475$	0	0
$476$	13.7364	0.629605
$477$	−7.18421	−0.328942
$478$	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
$483$	19.6128	0.892415
$484$	0	0
$485$	0	0
$486$	−1.21432	−0.0550827
$487$	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
$489$	−12.1748	−0.550565
$490$	0	0
$491$	32.9403	1.48657	0.743286	−	0.668973i	$-0.233266\pi$
$491$	32.9403	1.48657	0.743286	+	0.668973i	$0.233266\pi$
$492$	−0.949145	−0.0427908
$493$	−9.63158	−0.433785
$494$	−26.1106	−1.17477
$495$	0	0
$496$	−7.00937	−0.314730
$497$	−2.10171	−0.0942746
$498$	−3.52543	−0.157978
$499$	−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
$501$	13.0049	0.581017
$502$	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	$0.883059\pi$
$504$	15.0366	0.669782
$505$	0	0
$506$	0	0
$507$	−4.20294	−0.186659
$508$	9.83056	0.436160
$509$	−38.8573	−1.72232	−0.861159	−	0.508335i	$-0.830261\pi$
$509$	−38.8573	−1.72232	−0.861159	+	0.508335i	$0.830261\pi$
$510$	0	0
$511$	62.3180	2.75679
$512$	−24.1131	−1.06566
$513$	−5.18421	−0.228888
$514$	−0.161933	−0.00714257
$515$	0	0
$516$	−2.57628	−0.113415
$517$	0	0
$518$	−34.5718	−1.51900
$519$	13.8938	0.609872
$520$	0	0
$521$	11.1111	0.486785	0.243393	−	0.969928i	$-0.421740\pi$
$521$	11.1111	0.486785	0.243393	+	0.969928i	$0.421740\pi$
$522$	−2.19358	−0.0960102
$523$	−27.3002	−1.19375	−0.596877	−	0.802332i	$-0.703592\pi$
$523$	−27.3002	−1.19375	−0.596877	+	0.802332i	$0.703592\pi$
$524$	−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
$531$	1.67307	0.0726051
$532$	13.3560	0.579055
$533$	7.49240	0.324532
$534$	−0.755569	−0.0326967
$535$	0	0
$536$	14.8957	0.643396
$537$	−12.8573	−0.554833
$538$	−32.6133	−1.40606
$539$	0	0
$540$	0	0
$541$	16.1017	0.692267	0.346133	−	0.938185i	$-0.387494\pi$
$541$	16.1017	0.692267	0.346133	+	0.938185i	$0.387494\pi$
$542$	−3.74314	−0.160782
$543$	−0.917502	−0.0393738
$544$	15.3951	0.660061
$545$	0	0
$546$	−24.6953	−1.05686
$547$	−40.0370	−1.71186	−0.855930	−	0.517091i	$-0.827015\pi$
$547$	−40.0370	−1.71186	−0.855930	+	0.517091i	$0.827015\pi$
$548$	−9.87649	−0.421903
$549$	−0.755569	−0.0322469
$550$	0	0
$551$	−9.36488	−0.398957
$552$	12.2667	0.522106
$553$	−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	$0.295422\pi$
$558$	3.18421	0.134798
$559$	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	$0.257161\pi$
$564$	3.70471	0.155997
$565$	0	0
$566$	7.67752	0.322710
$567$	−4.90321	−0.205916
$568$	−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	$-0.428156\pi$
$571$	10.6953	0.447586	0.223793	+	0.974637i	$0.428156\pi$
$572$	0	0
$573$	−14.3684	−0.600249
$574$	10.7556	0.448929
$575$	0	0
$576$	8.85236	0.368848
$577$	−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
$579$	−11.7605	−0.488749
$580$	0	0
$581$	−14.2351	−0.590570
$582$	3.34614	0.138702
$583$	0	0
$584$	38.9763	1.61285
$585$	0	0
$586$	−20.2524	−0.836620
$587$	−10.9590	−0.452326	−0.226163	−	0.974089i	$-0.572618\pi$
$587$	−10.9590	−0.452326	−0.226163	+	0.974089i	$0.572618\pi$
$588$	8.95407	0.369260
$589$	13.5941	0.560136
$590$	0	0
$591$	−3.82071	−0.157163
$592$	−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
$597$	13.7146	0.561299
$598$	−20.1463	−0.823842
$599$	41.7146	1.70441	0.852205	−	0.523208i	$-0.175265\pi$
$599$	41.7146	1.70441	0.852205	+	0.523208i	$0.175265\pi$
$600$	0	0
$601$	−14.5906	−0.595162	−0.297581	−	0.954697i	$-0.596180\pi$
$601$	−14.5906	−0.595162	−0.297581	+	0.954697i	$0.596180\pi$
$602$	29.1941	1.18986
$603$	−4.85728	−0.197804
$604$	−0.171778	−0.00698953
$605$	0	0
$606$	−21.6227	−0.878362
$607$	19.9826	0.811071	0.405535	−	0.914079i	$-0.367085\pi$
$607$	19.9826	0.811071	0.405535	+	0.914079i	$0.367085\pi$
$608$	14.9688	0.607066
$609$	−8.85728	−0.358915
$610$	0	0
$611$	−29.2444	−1.18310
$612$	−2.80150	−0.113244
$613$	−19.0781	−0.770555	−0.385278	−	0.922801i	$-0.625894\pi$
$613$	−19.0781	−0.770555	−0.385278	+	0.922801i	$0.625894\pi$
$614$	11.6400	0.469754
$615$	0	0
$616$	0	0
$617$	−39.3590	−1.58454	−0.792268	−	0.610174i	$-0.791100\pi$
$617$	−39.3590	−1.58454	−0.792268	+	0.610174i	$0.791100\pi$
$618$	−6.00984	−0.241751
$619$	−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
$621$	−4.00000	−0.160514
$622$	17.6445	0.707480
$623$	−3.05086	−0.122230
$624$	−11.0869	−0.443833
$625$	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
$633$	1.95851	0.0778439
$634$	−1.20294	−0.0477750
$635$	0	0
$636$	−3.77478	−0.149680
$637$	−70.6820	−2.80052
$638$	0	0
$639$	0.428639	0.0169567
$640$	0	0
$641$	−6.25380	−0.247010	−0.123505	−	0.992344i	$-0.539414\pi$
$641$	−6.25380	−0.247010	−0.123505	+	0.992344i	$0.539414\pi$
$642$	13.5986	0.536692
$643$	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	$-0.369376\pi$
$647$	20.2953	0.797890	0.398945	+	0.916975i	$0.369376\pi$
$648$	−3.06668	−0.120470
$649$	0	0
$650$	0	0
$651$	12.8573	0.503916
$652$	−6.39700	−0.250526
$653$	−10.6222	−0.415679	−0.207840	−	0.978163i	$-0.566643\pi$
$653$	−10.6222	−0.415679	−0.207840	+	0.978163i	$0.566643\pi$
$654$	19.0825	0.746185
$655$	0	0
$656$	4.82870	0.188529
$657$	−12.7096	−0.495850
$658$	−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
$663$	22.1146	0.858861
$664$	−8.90321	−0.345512
$665$	0	0
$666$	7.05086	0.273215
$667$	−7.22570	−0.279780
$668$	6.83314	0.264382
$669$	26.0098	1.00560
$670$	0	0
$671$	0	0
$672$	14.1575	0.546137
$673$	−10.2208	−0.393982	−0.196991	−	0.980405i	$-0.563117\pi$
$673$	−10.2208	−0.393982	−0.196991	+	0.980405i	$0.563117\pi$
$674$	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	$0.586211\pi$
$678$	2.14320	0.0823090
$679$	13.5111	0.518510
$680$	0	0
$681$	−6.34122	−0.242996
$682$	0	0
$683$	−10.3970	−0.397830	−0.198915	−	0.980017i	$-0.563742\pi$
$683$	−10.3970	−0.397830	−0.198915	+	0.980017i	$0.563742\pi$
$684$	−2.72393	−0.104152
$685$	0	0
$686$	−59.7877	−2.28270
$687$	23.3274	0.889996
$688$	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
$696$	−5.53972	−0.209982
$697$	−9.63158	−0.364822
$698$	−22.1048	−0.836678
$699$	−1.42372	−0.0538499
$700$	0	0
$701$	−48.9688	−1.84953	−0.924764	−	0.380542i	$-0.875737\pi$
$701$	−48.9688	−1.84953	−0.924764	+	0.380542i	$0.875737\pi$
$702$	5.03657	0.190093
$703$	30.1017	1.13531
$704$	0	0
$705$	0	0
$706$	27.2128	1.02417
$707$	−87.3087	−3.28358
$708$	0.879077	0.0330378
$709$	−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
$711$	6.42864	0.241093
$712$	−1.90813	−0.0715103
$713$	10.4889	0.392811
$714$	31.7462	1.18807
$715$	0	0
$716$	−6.75557	−0.252467
$717$	−18.9590	−0.708036
$718$	−25.8983	−0.966516
$719$	5.83500	0.217609	0.108804	−	0.994063i	$-0.465298\pi$
$719$	5.83500	0.217609	0.108804	+	0.994063i	$0.465298\pi$
$720$	0	0
$721$	−24.2667	−0.903739
$722$	9.56400	0.355935
$723$	−1.34614	−0.0500635
$724$	−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
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.1432	−0.966941
$732$	−0.396997	−0.0146734
$733$	45.2083	1.66981	0.834904	−	0.550395i	$-0.185523\pi$
$733$	45.2083	1.66981	0.834904	+	0.550395i	$0.185523\pi$
$734$	42.6637	1.57475
$735$	0	0
$736$	11.5496	0.425723
$737$	0	0
$738$	−2.19358	−0.0807467
$739$	−5.65433	−0.207998	−0.103999	−	0.994577i	$-0.533164\pi$
$739$	−5.65433	−0.207998	−0.103999	+	0.994577i	$0.533164\pi$
$740$	0	0
$741$	21.5022	0.789905
$742$	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	$0.526341\pi$
$744$	8.04149	0.294815
$745$	0	0
$746$	20.6494	0.756029
$747$	2.90321	0.106223
$748$	0	0
$749$	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
$753$	−1.08250	−0.0394484
$754$	9.09817	0.331336
$755$	0	0
$756$	−2.57628	−0.0936985
$757$	−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	$0.587227\pi$
$762$	22.7195	0.823040
$763$	77.0518	2.78946
$764$	−7.54956	−0.273134
$765$	0	0
$766$	1.47643	0.0533457
$767$	−6.93930	−0.250564
$768$	11.6637	0.420878
$769$	−38.8573	−1.40123	−0.700615	−	0.713540i	$-0.747091\pi$
$769$	−38.8573	−1.40123	−0.700615	+	0.713540i	$0.747091\pi$
$770$	0	0
$771$	0.133353	0.00480259
$772$	−6.17929	−0.222397
$773$	−36.3368	−1.30694	−0.653471	−	0.756951i	$-0.726688\pi$
$773$	−36.3368	−1.30694	−0.653471	+	0.756951i	$0.726688\pi$
$774$	−5.95407	−0.214015
$775$	0	0
$776$	8.45044	0.303353
$777$	28.4701	1.02136
$778$	2.75251	0.0986821
$779$	−9.36488	−0.335532
$780$	0	0
$781$	0	0
$782$	25.8983	0.926121
$783$	1.80642	0.0645563
$784$	−45.5531	−1.62690
$785$	0	0
$786$	−1.51114	−0.0539005
$787$	33.5482	1.19586	0.597932	−	0.801547i	$-0.295989\pi$
$787$	33.5482	1.19586	0.597932	+	0.801547i	$0.295989\pi$
$788$	−2.00751	−0.0715145
$789$	−0.147643	−0.00525624
$790$	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	$-0.592238\pi$
$797$	−16.1334	−0.571473	−0.285736	+	0.958308i	$0.592238\pi$
$798$	30.8671	1.09268
$799$	37.5941	1.32998
$800$	0	0
$801$	0.622216	0.0219849
$802$	21.3265	0.753063
$803$	0	0
$804$	−2.55215	−0.0900073
$805$	0	0
$806$	−13.2070	−0.465195
$807$	26.8573	0.945421
$808$	−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	$-0.575890\pi$
$811$	−13.4509	−0.472325	−0.236163	+	0.971714i	$0.575890\pi$
$812$	−4.65386	−0.163318
$813$	3.08250	0.108108
$814$	0	0
$815$	0	0
$816$	14.2524	0.498934
$817$	−25.4193	−0.889308
$818$	25.9210	0.906308
$819$	20.3368	0.710624
$820$	0	0
$821$	−24.1748	−0.843708	−0.421854	−	0.906664i	$-0.638620\pi$
$821$	−24.1748	−0.843708	−0.421854	+	0.906664i	$0.638620\pi$
$822$	−22.8256	−0.796135
$823$	−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	$0.386079\pi$
$828$	−2.10171	−0.0730395
$829$	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
$831$	−8.70964	−0.302134
$832$	−36.7164	−1.27291
$833$	90.8627	3.14821
$834$	17.0509	0.590423
$835$	0	0
$836$	0	0
$837$	−2.62222	−0.0906370
$838$	35.0420	1.21050
$839$	−52.8988	−1.82627	−0.913134	−	0.407659i	$-0.866345\pi$
$839$	−52.8988	−1.82627	−0.913134	+	0.407659i	$0.866345\pi$
$840$	0	0
$841$	−25.7368	−0.887477
$842$	−43.0954	−1.48517
$843$	−20.3783	−0.701865
$844$	1.02906	0.0354216
$845$	0	0
$846$	8.56199	0.294367
$847$	0	0
$848$	19.2039	0.659465
$849$	−6.32248	−0.216987
$850$	0	0
$851$	23.2257	0.796167
$852$	0.225219	0.00771588
$853$	46.9229	1.60661	0.803305	−	0.595568i	$-0.203073\pi$
$853$	46.9229	1.60661	0.803305	+	0.595568i	$0.203073\pi$
$854$	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
$861$	−8.85728	−0.301855
$862$	−11.2257	−0.382349
$863$	32.6824	1.11252	0.556262	−	0.831007i	$-0.312235\pi$
$863$	32.6824	1.11252	0.556262	+	0.831007i	$0.312235\pi$
$864$	−2.88739	−0.0982310
$865$	0	0
$866$	−7.63254	−0.259364
$867$	−11.4286	−0.388137
$868$	6.75557	0.229299
$869$	0	0
$870$	0	0
$871$	20.1463	0.682630
$872$	48.1915	1.63197
$873$	−2.75557	−0.0932619
$874$	25.1811	0.851765
$875$	0	0
$876$	−6.67799	−0.225628
$877$	−49.1798	−1.66068	−0.830341	−	0.557255i	$-0.811855\pi$
$877$	−49.1798	−1.66068	−0.830341	+	0.557255i	$0.811855\pi$
$878$	44.3595	1.49706
$879$	16.6780	0.562535
$880$	0	0
$881$	33.8163	1.13930	0.569650	−	0.821888i	$-0.307079\pi$
$881$	33.8163	1.13930	0.569650	+	0.821888i	$0.307079\pi$
$882$	20.6938	0.696797
$883$	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	$0.540940\pi$
$888$	17.8064	0.597544
$889$	91.7373	3.07677
$890$	0	0
$891$	0	0
$892$	13.6663	0.457581
$893$	36.5531	1.22320
$894$	−3.70471	−0.123904
$895$	0	0
$896$	−24.3926	−0.814898
$897$	16.5906	0.553943
$898$	−38.6735	−1.29055
$899$	−4.73683	−0.157982
$900$	0	0
$901$	−38.3051	−1.27613
$902$	0	0
$903$	−24.0415	−0.800051
$904$	5.41249	0.180017
$905$	0	0
$906$	−0.396997	−0.0131893
$907$	−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
$909$	17.8064	0.590602
$910$	0	0
$911$	−45.3274	−1.50176	−0.750882	−	0.660436i	$-0.770371\pi$
$911$	−45.3274	−1.50176	−0.750882	+	0.660436i	$0.770371\pi$
$912$	13.8578	0.458876
$913$	0	0
$914$	−1.69042	−0.0559142
$915$	0	0
$916$	12.2569	0.404978
$917$	−6.10171	−0.201496
$918$	−6.47457	−0.213693
$919$	16.3269	0.538576	0.269288	−	0.963060i	$-0.413212\pi$
$919$	16.3269	0.538576	0.269288	+	0.963060i	$0.413212\pi$
$920$	0	0
$921$	−9.58565	−0.315858
$922$	9.35599	0.308123
$923$	−1.77784	−0.0585184
$924$	0	0
$925$	0	0
$926$	5.68598	0.186853
$927$	4.94914	0.162551
$928$	−5.21585	−0.171219
$929$	−29.6128	−0.971566	−0.485783	−	0.874079i	$-0.661465\pi$
$929$	−29.6128	−0.971566	−0.485783	+	0.874079i	$0.661465\pi$
$930$	0	0
$931$	88.3466	2.89544
$932$	−0.748060	−0.0245035
$933$	−14.5303	−0.475702
$934$	15.6128	0.510868
$935$	0	0
$936$	12.7195	0.415749
$937$	−8.92195	−0.291467	−0.145734	−	0.989324i	$-0.546554\pi$
$937$	−8.92195	−0.291467	−0.145734	+	0.989324i	$0.546554\pi$
$938$	28.9206	0.944290
$939$	21.0321	0.686357
$940$	0	0
$941$	22.2766	0.726195	0.363097	−	0.931751i	$-0.381719\pi$
$941$	22.2766	0.726195	0.363097	+	0.931751i	$0.381719\pi$
$942$	24.1748	0.787659
$943$	−7.22570	−0.235301
$944$	−4.47224	−0.145559
$945$	0	0
$946$	0	0
$947$	44.4612	1.44480	0.722398	−	0.691477i	$-0.243040\pi$
$947$	44.4612	1.44480	0.722398	+	0.691477i	$0.243040\pi$
$948$	3.37778	0.109705
$949$	52.7150	1.71120
$950$	0	0
$951$	0.990632	0.0321234
$952$	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	$0.608048\pi$
$954$	−8.72393	−0.282448
$955$	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
$963$	−11.1985	−0.360867
$964$	−0.707300	−0.0227806
$965$	0	0
$966$	23.8163	0.766276
$967$	−27.4839	−0.883824	−0.441912	−	0.897058i	$-0.645700\pi$
$967$	−27.4839	−0.883824	−0.441912	+	0.897058i	$0.645700\pi$
$968$	0	0
$969$	−27.6414	−0.887971
$970$	0	0
$971$	−5.81532	−0.186622	−0.0933112	−	0.995637i	$-0.529745\pi$
$971$	−5.81532	−0.186622	−0.0933112	+	0.995637i	$0.529745\pi$
$972$	0.525428	0.0168531
$973$	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	$-0.218210\pi$
$977$	48.3912	1.54817	0.774085	+	0.633081i	$0.218210\pi$
$978$	−14.7841	−0.472745
$979$	0	0
$980$	0	0
$981$	−15.7146	−0.501727
$982$	40.0000	1.27645
$983$	−49.9724	−1.59387	−0.796936	−	0.604064i	$-0.793547\pi$
$983$	−49.9724	−1.59387	−0.796936	+	0.604064i	$0.793547\pi$
$984$	−5.53972	−0.176600
$985$	0	0
$986$	−11.6958	−0.372471
$987$	34.5718	1.10043
$988$	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
$993$	17.5812	0.557923
$994$	−2.55215	−0.0809492
$995$	0	0
$996$	1.52543	0.0483350
$997$	4.33138	0.137176	0.0685880	−	0.997645i	$-0.478151\pi$
$997$	4.33138	0.137176	0.0685880	+	0.997645i	$0.478151\pi$
$998$	−1.98126	−0.0627158
$999$	−5.80642	−0.183707

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.3	✓	6	55.32	even	4
165.2.c.a.34.4	yes	6	55.43	even	4
495.2.c.d.199.3		6	165.98	odd	4
495.2.c.d.199.4		6	165.32	odd	4
825.2.a.h.1.3		3	55.54	odd	2
825.2.a.n.1.1		3	11.10	odd	2
1815.2.c.d.364.3		6	5.3	odd	4
1815.2.c.d.364.4		6	5.2	odd	4
2475.2.a.y.1.3		3	33.32	even	2
2475.2.a.be.1.1		3	165.164	even	2
2640.2.d.i.529.2		6	220.87	odd	4
2640.2.d.i.529.5		6	220.43	odd	4
9075.2.a.cc.1.3		3	1.1	even	1	trivial
9075.2.a.ck.1.1		3	5.4	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} -1.21432 q^{6} -4.90321 q^{7} -3.06668 q^{8} +1.00000 q^{9} +0.525428 q^{12} -4.14764 q^{13} -5.95407 q^{14} -2.67307 q^{16} +5.33185 q^{17} +1.21432 q^{18} +5.18421 q^{19} +4.90321 q^{21} +4.00000 q^{23} +3.06668 q^{24} -5.03657 q^{26} -1.00000 q^{27} +2.57628 q^{28} -1.80642 q^{29} +2.62222 q^{31} +2.88739 q^{32} +6.47457 q^{34} -0.525428 q^{36} +5.80642 q^{37} +6.29529 q^{38} +4.14764 q^{39} -1.80642 q^{41} +5.95407 q^{42} -4.90321 q^{43} +4.85728 q^{46} +7.05086 q^{47} +2.67307 q^{48} +17.0415 q^{49} -5.33185 q^{51} +2.17929 q^{52} -7.18421 q^{53} -1.21432 q^{54} +15.0366 q^{56} -5.18421 q^{57} -2.19358 q^{58} +1.67307 q^{59} -0.755569 q^{61} +3.18421 q^{62} -4.90321 q^{63} +8.85236 q^{64} -4.85728 q^{67} -2.80150 q^{68} -4.00000 q^{69} +0.428639 q^{71} -3.06668 q^{72} -12.7096 q^{73} +7.05086 q^{74} -2.72393 q^{76} +5.03657 q^{78} +6.42864 q^{79} +1.00000 q^{81} -2.19358 q^{82} +2.90321 q^{83} -2.57628 q^{84} -5.95407 q^{86} +1.80642 q^{87} +0.622216 q^{89} +20.3368 q^{91} -2.10171 q^{92} -2.62222 q^{93} +8.56199 q^{94} -2.88739 q^{96} -2.75557 q^{97} +20.6938 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9} - 5 q^{12} - 6 q^{13} + 2 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} + 2 q^{19} + 8 q^{21} + 12 q^{23} + 9 q^{24} - 8 q^{26} - 3 q^{27}+ \cdots + 29 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 5 * q^4 + 3 * q^6 - 8 * q^7 - 9 * q^8 + 3 * q^9 - 5 * q^12 - 6 * q^13 + 2 * q^14 + 5 * q^16 - 4 * q^17 - 3 * q^18 + 2 * q^19 + 8 * q^21 + 12 * q^23 + 9 * q^24 - 8 * q^26 - 3 * q^27 - 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 5 * q^36 + 4 * q^37 + 6 * q^38 + 6 * q^39 + 8 * q^41 - 2 * q^42 - 8 * q^43 - 12 * q^46 + 8 * q^47 - 5 * q^48 + 11 * q^49 + 4 * q^51 + 26 * q^52 - 8 * q^53 + 3 * q^54 + 38 * q^56 - 2 * q^57 - 20 * q^58 - 8 * q^59 - 2 * q^61 - 4 * q^62 - 8 * q^63 + 33 * q^64 + 12 * q^67 - 28 * q^68 - 12 * q^69 - 12 * q^71 - 9 * q^72 - 18 * q^73 + 8 * q^74 + 18 * q^76 + 8 * q^78 + 6 * q^79 + 3 * q^81 - 20 * q^82 + 2 * q^83 + 12 * q^84 + 2 * q^86 - 8 * q^87 + 2 * q^89 + 8 * q^91 + 20 * q^92 - 8 * q^93 + 12 * q^94 + 11 * q^96 - 8 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more