LMFDB - Embedded newform 9075.2.a.du.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 38x^4 - 25x^3 - 41x^2 + 20x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.40736$ 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)

$f(q)$	$=$	$q-1.40736 q^{2} +1.00000 q^{3} -0.0193259 q^{4} -1.40736 q^{6} +2.16377 q^{7} +2.84193 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.40736 q^{2} +1.00000 q^{3} -0.0193259 q^{4} -1.40736 q^{6} +2.16377 q^{7} +2.84193 q^{8} +1.00000 q^{9} -0.0193259 q^{12} -2.44925 q^{13} -3.04521 q^{14} -3.96097 q^{16} -2.42748 q^{17} -1.40736 q^{18} -4.99266 q^{19} +2.16377 q^{21} -2.41686 q^{23} +2.84193 q^{24} +3.44699 q^{26} +1.00000 q^{27} -0.0418169 q^{28} +5.42701 q^{29} +2.30813 q^{31} -0.109320 q^{32} +3.41635 q^{34} -0.0193259 q^{36} +9.35888 q^{37} +7.02649 q^{38} -2.44925 q^{39} +5.90849 q^{41} -3.04521 q^{42} -3.11043 q^{43} +3.40141 q^{46} -11.9915 q^{47} -3.96097 q^{48} -2.31809 q^{49} -2.42748 q^{51} +0.0473341 q^{52} -7.05531 q^{53} -1.40736 q^{54} +6.14928 q^{56} -4.99266 q^{57} -7.63778 q^{58} +4.27958 q^{59} -5.16368 q^{61} -3.24838 q^{62} +2.16377 q^{63} +8.07580 q^{64} +14.0857 q^{67} +0.0469133 q^{68} -2.41686 q^{69} +6.82043 q^{71} +2.84193 q^{72} +3.60194 q^{73} -13.1714 q^{74} +0.0964877 q^{76} +3.44699 q^{78} -10.7029 q^{79} +1.00000 q^{81} -8.31540 q^{82} -2.38378 q^{83} -0.0418169 q^{84} +4.37751 q^{86} +5.42701 q^{87} -10.3078 q^{89} -5.29963 q^{91} +0.0467081 q^{92} +2.30813 q^{93} +16.8765 q^{94} -0.109320 q^{96} +5.27674 q^{97} +3.26240 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 8 q^{3} + 7 q^{4} - q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - q^2 + 8 * q^3 + 7 * q^4 - q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9	$ 8 q - q^{2} + 8 q^{3} + 7 q^{4} - q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} + 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} - 6 q^{19} - 8 q^{21} + 7 q^{23} - 3 q^{24} - q^{26} + 8 q^{27} - 11 q^{28} - 28 q^{29} + 10 q^{31} - 12 q^{32} - 8 q^{34} + 7 q^{36} + 14 q^{37} + 19 q^{38} - 7 q^{39} - 14 q^{41} - 8 q^{42} - 11 q^{43} + 5 q^{47} - 7 q^{48} + 2 q^{49} + q^{51} - q^{52} - 5 q^{53} - q^{54} - 4 q^{56} - 6 q^{57} - 3 q^{58} - 2 q^{59} - 36 q^{61} + 17 q^{62} - 8 q^{63} - 23 q^{64} + 6 q^{67} + 3 q^{68} + 7 q^{69} - 17 q^{71} - 3 q^{72} - 32 q^{73} - 47 q^{74} - 5 q^{76} - q^{78} - 41 q^{79} + 8 q^{81} + 6 q^{83} - 11 q^{84} - 9 q^{86} - 28 q^{87} + 21 q^{89} - 9 q^{91} + 69 q^{92} + 10 q^{93} - 51 q^{94} - 12 q^{96} - 4 q^{97} + 27 q^{98}+O(q^{100}) $ 8 * q - q^2 + 8 * q^3 + 7 * q^4 - q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 + 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 - 6 * q^19 - 8 * q^21 + 7 * q^23 - 3 * q^24 - q^26 + 8 * q^27 - 11 * q^28 - 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 + 14 * q^37 + 19 * q^38 - 7 * q^39 - 14 * q^41 - 8 * q^42 - 11 * q^43 + 5 * q^47 - 7 * q^48 + 2 * q^49 + q^51 - q^52 - 5 * q^53 - q^54 - 4 * q^56 - 6 * q^57 - 3 * q^58 - 2 * q^59 - 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 + 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 - 47 * q^74 - 5 * q^76 - q^78 - 41 * q^79 + 8 * q^81 + 6 * q^83 - 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 + 69 * q^92 + 10 * q^93 - 51 * q^94 - 12 * q^96 - 4 * q^97 + 27 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.40736	−0.995157	−0.497578	−	0.867419i	$-0.665777\pi$
$2$	−1.40736	−0.995157	−0.497578	+	0.867419i	$0.665777\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.0193259	−0.00966296
$5$	0	0
$5$	0	0
$6$	−1.40736	−0.574554
$7$	2.16377	0.817829	0.408914	−	0.912573i	$-0.365907\pi$
$7$	2.16377	0.817829	0.408914	+	0.912573i	$0.365907\pi$
$8$	2.84193	1.00477
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.0193259	−0.00557891
$13$	−2.44925	−0.679301	−0.339650	−	0.940552i	$-0.610309\pi$
$13$	−2.44925	−0.679301	−0.339650	+	0.940552i	$0.610309\pi$
$14$	−3.04521	−0.813868
$15$	0	0
$16$	−3.96097	−0.990244
$17$	−2.42748	−0.588751	−0.294375	−	0.955690i	$-0.595112\pi$
$17$	−2.42748	−0.588751	−0.294375	+	0.955690i	$0.595112\pi$
$18$	−1.40736	−0.331719
$19$	−4.99266	−1.14539	−0.572697	−	0.819767i	$-0.694103\pi$
$19$	−4.99266	−1.14539	−0.572697	+	0.819767i	$0.694103\pi$
$20$	0	0
$21$	2.16377	0.472174
$22$	0	0
$23$	−2.41686	−0.503951	−0.251976	−	0.967734i	$-0.581080\pi$
$23$	−2.41686	−0.503951	−0.251976	+	0.967734i	$0.581080\pi$
$24$	2.84193	0.580106
$25$	0	0
$26$	3.44699	0.676011
$27$	1.00000	0.192450
$28$	−0.0418169	−0.00790264
$29$	5.42701	1.00777	0.503885	−	0.863771i	$-0.331904\pi$
$29$	5.42701	1.00777	0.503885	+	0.863771i	$0.331904\pi$
$30$	0	0
$31$	2.30813	0.414553	0.207276	−	0.978282i	$-0.433540\pi$
$31$	2.30813	0.414553	0.207276	+	0.978282i	$0.433540\pi$
$32$	−0.109320	−0.0193252
$33$	0	0
$34$	3.41635	0.585899
$35$	0	0
$36$	−0.0193259	−0.00322099
$37$	9.35888	1.53859	0.769295	−	0.638893i	$-0.220608\pi$
$37$	9.35888	1.53859	0.769295	+	0.638893i	$0.220608\pi$
$38$	7.02649	1.13985
$39$	−2.44925	−0.392194
$40$	0	0
$41$	5.90849	0.922752	0.461376	−	0.887205i	$-0.347356\pi$
$41$	5.90849	0.922752	0.461376	+	0.887205i	$0.347356\pi$
$42$	−3.04521	−0.469887
$43$	−3.11043	−0.474336	−0.237168	−	0.971469i	$-0.576219\pi$
$43$	−3.11043	−0.474336	−0.237168	+	0.971469i	$0.576219\pi$
$44$	0	0
$45$	0	0
$46$	3.40141	0.501510
$47$	−11.9915	−1.74915	−0.874573	−	0.484894i	$-0.838858\pi$
$47$	−11.9915	−1.74915	−0.874573	+	0.484894i	$0.838858\pi$
$48$	−3.96097	−0.571717
$49$	−2.31809	−0.331156
$50$	0	0
$51$	−2.42748	−0.339915
$52$	0.0473341	0.00656405
$53$	−7.05531	−0.969122	−0.484561	−	0.874758i	$-0.661021\pi$
$53$	−7.05531	−0.969122	−0.484561	+	0.874758i	$0.661021\pi$
$54$	−1.40736	−0.191518
$55$	0	0
$56$	6.14928	0.821732
$57$	−4.99266	−0.661294
$58$	−7.63778	−1.00289
$59$	4.27958	0.557154	0.278577	−	0.960414i	$-0.410137\pi$
$59$	4.27958	0.557154	0.278577	+	0.960414i	$0.410137\pi$
$60$	0	0
$61$	−5.16368	−0.661142	−0.330571	−	0.943781i	$-0.607241\pi$
$61$	−5.16368	−0.661142	−0.330571	+	0.943781i	$0.607241\pi$
$62$	−3.24838	−0.412545
$63$	2.16377	0.272610
$64$	8.07580	1.00948
$65$	0	0
$66$	0	0
$67$	14.0857	1.72084	0.860422	−	0.509582i	$-0.170200\pi$
$67$	14.0857	1.72084	0.860422	+	0.509582i	$0.170200\pi$
$68$	0.0469133	0.00568907
$69$	−2.41686	−0.290956
$70$	0	0
$71$	6.82043	0.809436	0.404718	−	0.914442i	$-0.367370\pi$
$71$	6.82043	0.809436	0.404718	+	0.914442i	$0.367370\pi$
$72$	2.84193	0.334924
$73$	3.60194	0.421576	0.210788	−	0.977532i	$-0.432397\pi$
$73$	3.60194	0.421576	0.210788	+	0.977532i	$0.432397\pi$
$74$	−13.1714	−1.53114
$75$	0	0
$76$	0.0964877	0.0110679
$77$	0	0
$78$	3.44699	0.390295
$79$	−10.7029	−1.20417	−0.602086	−	0.798431i	$-0.705664\pi$
$79$	−10.7029	−1.20417	−0.602086	+	0.798431i	$0.705664\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.31540	−0.918283
$83$	−2.38378	−0.261653	−0.130827	−	0.991405i	$-0.541763\pi$
$83$	−2.38378	−0.261653	−0.130827	+	0.991405i	$0.541763\pi$
$84$	−0.0418169	−0.00456259
$85$	0	0
$86$	4.37751	0.472039
$87$	5.42701	0.581836
$88$	0	0
$89$	−10.3078	−1.09262	−0.546310	−	0.837583i	$-0.683968\pi$
$89$	−10.3078	−1.09262	−0.546310	+	0.837583i	$0.683968\pi$
$90$	0	0
$91$	−5.29963	−0.555552
$92$	0.0467081	0.00486966
$93$	2.30813	0.239342
$94$	16.8765	1.74067
$95$	0	0
$96$	−0.109320	−0.0111574
$97$	5.27674	0.535772	0.267886	−	0.963451i	$-0.413675\pi$
$97$	5.27674	0.535772	0.267886	+	0.963451i	$0.413675\pi$
$98$	3.26240	0.329552
$99$	0	0
$100$	0	0
$101$	−15.8254	−1.57469	−0.787343	−	0.616515i	$-0.788544\pi$
$101$	−15.8254	−1.57469	−0.787343	+	0.616515i	$0.788544\pi$
$102$	3.41635	0.338269
$103$	−1.74784	−0.172220	−0.0861098	−	0.996286i	$-0.527444\pi$
$103$	−1.74784	−0.172220	−0.0861098	+	0.996286i	$0.527444\pi$
$104$	−6.96060	−0.682543
$105$	0	0
$106$	9.92940	0.964428
$107$	8.92746	0.863050	0.431525	−	0.902101i	$-0.357976\pi$
$107$	8.92746	0.863050	0.431525	+	0.902101i	$0.357976\pi$
$108$	−0.0193259	−0.00185964
$109$	−18.8496	−1.80546	−0.902730	−	0.430207i	$-0.858441\pi$
$109$	−18.8496	−1.80546	−0.902730	+	0.430207i	$0.858441\pi$
$110$	0	0
$111$	9.35888	0.888306
$112$	−8.57065	−0.809850
$113$	−0.702486	−0.0660843	−0.0330422	−	0.999454i	$-0.510520\pi$
$113$	−0.702486	−0.0660843	−0.0330422	+	0.999454i	$0.510520\pi$
$114$	7.02649	0.658091
$115$	0	0
$116$	−0.104882	−0.00973804
$117$	−2.44925	−0.226434
$118$	−6.02293	−0.554456
$119$	−5.25251	−0.481497
$120$	0	0
$121$	0	0
$122$	7.26718	0.657940
$123$	5.90849	0.532751
$124$	−0.0446068	−0.00400581
$125$	0	0
$126$	−3.04521	−0.271289
$127$	−19.2895	−1.71166	−0.855832	−	0.517254i	$-0.826954\pi$
$127$	−19.2895	−1.71166	−0.855832	+	0.517254i	$0.826954\pi$
$128$	−11.1470	−0.985261
$129$	−3.11043	−0.273858
$130$	0	0
$131$	−0.731549	−0.0639157	−0.0319579	−	0.999489i	$-0.510174\pi$
$131$	−0.731549	−0.0639157	−0.0319579	+	0.999489i	$0.510174\pi$
$132$	0	0
$133$	−10.8030	−0.936737
$134$	−19.8237	−1.71251
$135$	0	0
$136$	−6.89872	−0.591561
$137$	13.4988	1.15328	0.576639	−	0.816999i	$-0.304364\pi$
$137$	13.4988	1.15328	0.576639	+	0.816999i	$0.304364\pi$
$138$	3.40141	0.289547
$139$	−16.2925	−1.38191	−0.690955	−	0.722898i	$-0.742810\pi$
$139$	−16.2925	−1.38191	−0.690955	+	0.722898i	$0.742810\pi$
$140$	0	0
$141$	−11.9915	−1.00987
$142$	−9.59883	−0.805516
$143$	0	0
$144$	−3.96097	−0.330081
$145$	0	0
$146$	−5.06925	−0.419534
$147$	−2.31809	−0.191193
$148$	−0.180869	−0.0148673
$149$	1.96471	0.160956	0.0804778	−	0.996756i	$-0.474355\pi$
$149$	1.96471	0.160956	0.0804778	+	0.996756i	$0.474355\pi$
$150$	0	0
$151$	6.41511	0.522055	0.261027	−	0.965331i	$-0.415939\pi$
$151$	6.41511	0.522055	0.261027	+	0.965331i	$0.415939\pi$
$152$	−14.1888	−1.15086
$153$	−2.42748	−0.196250
$154$	0	0
$155$	0	0
$156$	0.0473341	0.00378976
$157$	10.1011	0.806156	0.403078	−	0.915166i	$-0.367940\pi$
$157$	10.1011	0.806156	0.403078	+	0.915166i	$0.367940\pi$
$158$	15.0629	1.19834
$159$	−7.05531	−0.559523
$160$	0	0
$161$	−5.22954	−0.412146
$162$	−1.40736	−0.110573
$163$	−17.1635	−1.34435	−0.672176	−	0.740391i	$-0.734640\pi$
$163$	−17.1635	−1.34435	−0.672176	+	0.740391i	$0.734640\pi$
$164$	−0.114187	−0.00891651
$165$	0	0
$166$	3.35484	0.260386
$167$	−12.9371	−1.00110	−0.500550	−	0.865708i	$-0.666869\pi$
$167$	−12.9371	−1.00110	−0.500550	+	0.865708i	$0.666869\pi$
$168$	6.14928	0.474427
$169$	−7.00116	−0.538551
$170$	0	0
$171$	−4.99266	−0.381798
$172$	0.0601119	0.00458349
$173$	−0.186614	−0.0141880	−0.00709402	−	0.999975i	$-0.502258\pi$
$173$	−0.186614	−0.0141880	−0.00709402	+	0.999975i	$0.502258\pi$
$174$	−7.63778	−0.579018
$175$	0	0
$176$	0	0
$177$	4.27958	0.321673
$178$	14.5068	1.08733
$179$	7.93874	0.593369	0.296685	−	0.954975i	$-0.404119\pi$
$179$	7.93874	0.593369	0.296685	+	0.954975i	$0.404119\pi$
$180$	0	0
$181$	3.35297	0.249224	0.124612	−	0.992206i	$-0.460231\pi$
$181$	3.35297	0.249224	0.124612	+	0.992206i	$0.460231\pi$
$182$	7.45850	0.552861
$183$	−5.16368	−0.381711
$184$	−6.86855	−0.506357
$185$	0	0
$186$	−3.24838	−0.238183
$187$	0	0
$188$	0.231747	0.0169019
$189$	2.16377	0.157391
$190$	0	0
$191$	23.6161	1.70880	0.854399	−	0.519617i	$-0.173926\pi$
$191$	23.6161	1.70880	0.854399	+	0.519617i	$0.173926\pi$
$192$	8.07580	0.582821
$193$	−13.4928	−0.971234	−0.485617	−	0.874172i	$-0.661405\pi$
$193$	−13.4928	−0.971234	−0.485617	+	0.874172i	$0.661405\pi$
$194$	−7.42629	−0.533177
$195$	0	0
$196$	0.0447992	0.00319995
$197$	−19.9986	−1.42484	−0.712422	−	0.701751i	$-0.752402\pi$
$197$	−19.9986	−1.42484	−0.712422	+	0.701751i	$0.752402\pi$
$198$	0	0
$199$	9.08289	0.643869	0.321934	−	0.946762i	$-0.395667\pi$
$199$	9.08289	0.643869	0.321934	+	0.946762i	$0.395667\pi$
$200$	0	0
$201$	14.0857	0.993530
$202$	22.2721	1.56706
$203$	11.7428	0.824183
$204$	0.0469133	0.00328459
$205$	0	0
$206$	2.45984	0.171385
$207$	−2.41686	−0.167984
$208$	9.70143	0.672673
$209$	0	0
$210$	0	0
$211$	−26.8389	−1.84767	−0.923835	−	0.382792i	$-0.874963\pi$
$211$	−26.8389	−1.84767	−0.923835	+	0.382792i	$0.874963\pi$
$212$	0.136350	0.00936458
$213$	6.82043	0.467328
$214$	−12.5642	−0.858870
$215$	0	0
$216$	2.84193	0.193369
$217$	4.99427	0.339033
$218$	26.5282	1.79672
$219$	3.60194	0.243397
$220$	0	0
$221$	5.94552	0.399939
$222$	−13.1714	−0.884003
$223$	5.82507	0.390075	0.195038	−	0.980796i	$-0.437517\pi$
$223$	5.82507	0.390075	0.195038	+	0.980796i	$0.437517\pi$
$224$	−0.236544	−0.0158047
$225$	0	0
$226$	0.988654	0.0657643
$227$	18.2902	1.21396	0.606982	−	0.794715i	$-0.292380\pi$
$227$	18.2902	1.21396	0.606982	+	0.794715i	$0.292380\pi$
$228$	0.0964877	0.00639006
$229$	−23.5038	−1.55318	−0.776588	−	0.630008i	$-0.783051\pi$
$229$	−23.5038	−1.55318	−0.776588	+	0.630008i	$0.783051\pi$
$230$	0	0
$231$	0	0
$232$	15.4232	1.01258
$233$	27.3946	1.79468	0.897341	−	0.441338i	$-0.145496\pi$
$233$	27.3946	1.79468	0.897341	+	0.441338i	$0.145496\pi$
$234$	3.44699	0.225337
$235$	0	0
$236$	−0.0827068	−0.00538376
$237$	−10.7029	−0.695230
$238$	7.39220	0.479165
$239$	−21.9138	−1.41749	−0.708743	−	0.705467i	$-0.750737\pi$
$239$	−21.9138	−1.41749	−0.708743	+	0.705467i	$0.750737\pi$
$240$	0	0
$241$	−22.7034	−1.46245	−0.731227	−	0.682134i	$-0.761052\pi$
$241$	−22.7034	−1.46245	−0.731227	+	0.682134i	$0.761052\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0.0997929	0.00638859
$245$	0	0
$246$	−8.31540	−0.530171
$247$	12.2283	0.778068
$248$	6.55955	0.416532
$249$	−2.38378	−0.151066
$250$	0	0
$251$	2.60228	0.164254	0.0821271	−	0.996622i	$-0.473829\pi$
$251$	2.60228	0.164254	0.0821271	+	0.996622i	$0.473829\pi$
$252$	−0.0418169	−0.00263421
$253$	0	0
$254$	27.1473	1.70337
$255$	0	0
$256$	−0.463778	−0.0289862
$257$	16.9205	1.05547	0.527736	−	0.849408i	$-0.323041\pi$
$257$	16.9205	1.05547	0.527736	+	0.849408i	$0.323041\pi$
$258$	4.37751	0.272532
$259$	20.2505	1.25830
$260$	0	0
$261$	5.42701	0.335923
$262$	1.02956	0.0636062
$263$	17.8845	1.10281	0.551403	−	0.834239i	$-0.314093\pi$
$263$	17.8845	1.10281	0.551403	+	0.834239i	$0.314093\pi$
$264$	0	0
$265$	0	0
$266$	15.2037	0.932200
$267$	−10.3078	−0.630824
$268$	−0.272219	−0.0166284
$269$	−28.7834	−1.75496	−0.877478	−	0.479616i	$-0.840776\pi$
$269$	−28.7834	−1.75496	−0.877478	+	0.479616i	$0.840776\pi$
$270$	0	0
$271$	22.3952	1.36041	0.680205	−	0.733022i	$-0.261891\pi$
$271$	22.3952	1.36041	0.680205	+	0.733022i	$0.261891\pi$
$272$	9.61519	0.583007
$273$	−5.29963	−0.320748
$274$	−18.9977	−1.14769
$275$	0	0
$276$	0.0467081	0.00281150
$277$	2.73513	0.164338	0.0821690	−	0.996618i	$-0.473815\pi$
$277$	2.73513	0.164338	0.0821690	+	0.996618i	$0.473815\pi$
$278$	22.9294	1.37522
$279$	2.30813	0.138184
$280$	0	0
$281$	4.51488	0.269335	0.134667	−	0.990891i	$-0.457003\pi$
$281$	4.51488	0.269335	0.134667	+	0.990891i	$0.457003\pi$
$282$	16.8765	1.00498
$283$	13.6296	0.810195	0.405098	−	0.914273i	$-0.367238\pi$
$283$	13.6296	0.810195	0.405098	+	0.914273i	$0.367238\pi$
$284$	−0.131811	−0.00782154
$285$	0	0
$286$	0	0
$287$	12.7846	0.754653
$288$	−0.109320	−0.00644174
$289$	−11.1073	−0.653373
$290$	0	0
$291$	5.27674	0.309328
$292$	−0.0696109	−0.00407367
$293$	−10.7326	−0.627004	−0.313502	−	0.949588i	$-0.601502\pi$
$293$	−10.7326	−0.627004	−0.313502	+	0.949588i	$0.601502\pi$
$294$	3.26240	0.190267
$295$	0	0
$296$	26.5973	1.54593
$297$	0	0
$298$	−2.76507	−0.160176
$299$	5.91952	0.342334
$300$	0	0
$301$	−6.73026	−0.387926
$302$	−9.02840	−0.519526
$303$	−15.8254	−0.909146
$304$	19.7758	1.13422
$305$	0	0
$306$	3.41635	0.195300
$307$	−5.07712	−0.289766	−0.144883	−	0.989449i	$-0.546281\pi$
$307$	−5.07712	−0.289766	−0.144883	+	0.989449i	$0.546281\pi$
$308$	0	0
$309$	−1.74784	−0.0994310
$310$	0	0
$311$	−4.07719	−0.231196	−0.115598	−	0.993296i	$-0.536878\pi$
$311$	−4.07719	−0.231196	−0.115598	+	0.993296i	$0.536878\pi$
$312$	−6.96060	−0.394066
$313$	29.3267	1.65764	0.828820	−	0.559515i	$-0.189013\pi$
$313$	29.3267	1.65764	0.828820	+	0.559515i	$0.189013\pi$
$314$	−14.2159	−0.802251
$315$	0	0
$316$	0.206844	0.0116359
$317$	1.16203	0.0652662	0.0326331	−	0.999467i	$-0.489611\pi$
$317$	1.16203	0.0652662	0.0326331	+	0.999467i	$0.489611\pi$
$318$	9.92940	0.556813
$319$	0	0
$320$	0	0
$321$	8.92746	0.498282
$322$	7.35987	0.410150
$323$	12.1196	0.674352
$324$	−0.0193259	−0.00107366
$325$	0	0
$326$	24.1554	1.33784
$327$	−18.8496	−1.04238
$328$	16.7915	0.927156
$329$	−25.9470	−1.43050
$330$	0	0
$331$	14.8212	0.814644	0.407322	−	0.913285i	$-0.366463\pi$
$331$	14.8212	0.814644	0.407322	+	0.913285i	$0.366463\pi$
$332$	0.0460686	0.00252835
$333$	9.35888	0.512864
$334$	18.2071	0.996250
$335$	0	0
$336$	−8.57065	−0.467567
$337$	−11.3574	−0.618674	−0.309337	−	0.950952i	$-0.600107\pi$
$337$	−11.3574	−0.618674	−0.309337	+	0.950952i	$0.600107\pi$
$338$	9.85318	0.535942
$339$	−0.702486	−0.0381538
$340$	0	0
$341$	0	0
$342$	7.02649	0.379949
$343$	−20.1622	−1.08866
$344$	−8.83961	−0.476600
$345$	0	0
$346$	0.262635	0.0141193
$347$	7.01735	0.376711	0.188356	−	0.982101i	$-0.439684\pi$
$347$	7.01735	0.376711	0.188356	+	0.982101i	$0.439684\pi$
$348$	−0.104882	−0.00562226
$349$	5.51667	0.295301	0.147650	−	0.989040i	$-0.452829\pi$
$349$	5.51667	0.295301	0.147650	+	0.989040i	$0.452829\pi$
$350$	0	0
$351$	−2.44925	−0.130731
$352$	0	0
$353$	14.6513	0.779810	0.389905	−	0.920855i	$-0.372508\pi$
$353$	14.6513	0.779810	0.389905	+	0.920855i	$0.372508\pi$
$354$	−6.02293	−0.320115
$355$	0	0
$356$	0.199207	0.0105579
$357$	−5.25251	−0.277993
$358$	−11.1727	−0.590496
$359$	−8.85323	−0.467256	−0.233628	−	0.972326i	$-0.575060\pi$
$359$	−8.85323	−0.467256	−0.233628	+	0.972326i	$0.575060\pi$
$360$	0	0
$361$	5.92666	0.311930
$362$	−4.71885	−0.248017
$363$	0	0
$364$	0.102420	0.00536827
$365$	0	0
$366$	7.26718	0.379862
$367$	−21.5484	−1.12482	−0.562409	−	0.826859i	$-0.690125\pi$
$367$	−21.5484	−1.12482	−0.562409	+	0.826859i	$0.690125\pi$
$368$	9.57314	0.499034
$369$	5.90849	0.307584
$370$	0	0
$371$	−15.2661	−0.792576
$372$	−0.0446068	−0.00231275
$373$	4.26418	0.220791	0.110395	−	0.993888i	$-0.464788\pi$
$373$	4.26418	0.220791	0.110395	+	0.993888i	$0.464788\pi$
$374$	0	0
$375$	0	0
$376$	−34.0791	−1.75749
$377$	−13.2921	−0.684579
$378$	−3.04521	−0.156629
$379$	−37.0106	−1.90110	−0.950552	−	0.310564i	$-0.899482\pi$
$379$	−37.0106	−1.90110	−0.950552	+	0.310564i	$0.899482\pi$
$380$	0	0
$381$	−19.2895	−0.988230
$382$	−33.2364	−1.70052
$383$	−9.84449	−0.503030	−0.251515	−	0.967853i	$-0.580929\pi$
$383$	−9.84449	−0.503030	−0.251515	+	0.967853i	$0.580929\pi$
$384$	−11.1470	−0.568841
$385$	0	0
$386$	18.9893	0.966530
$387$	−3.11043	−0.158112
$388$	−0.101978	−0.00517714
$389$	−2.94018	−0.149073	−0.0745365	−	0.997218i	$-0.523748\pi$
$389$	−2.94018	−0.149073	−0.0745365	+	0.997218i	$0.523748\pi$
$390$	0	0
$391$	5.86689	0.296702
$392$	−6.58785	−0.332737
$393$	−0.731549	−0.0369018
$394$	28.1454	1.41794
$395$	0	0
$396$	0	0
$397$	34.7013	1.74161	0.870805	−	0.491629i	$-0.163598\pi$
$397$	34.7013	1.74161	0.870805	+	0.491629i	$0.163598\pi$
$398$	−12.7829	−0.640750
$399$	−10.8030	−0.540825
$400$	0	0
$401$	−10.7539	−0.537026	−0.268513	−	0.963276i	$-0.586532\pi$
$401$	−10.7539	−0.537026	−0.268513	+	0.963276i	$0.586532\pi$
$402$	−19.8237	−0.988718
$403$	−5.65320	−0.281606
$404$	0.305840	0.0152161
$405$	0	0
$406$	−16.5264	−0.820192
$407$	0	0
$408$	−6.89872	−0.341538
$409$	−19.8402	−0.981033	−0.490516	−	0.871432i	$-0.663192\pi$
$409$	−19.8402	−0.981033	−0.490516	+	0.871432i	$0.663192\pi$
$410$	0	0
$411$	13.4988	0.665845
$412$	0.0337786	0.00166415
$413$	9.26004	0.455657
$414$	3.40141	0.167170
$415$	0	0
$416$	0.267752	0.0131276
$417$	−16.2925	−0.797846
$418$	0	0
$419$	26.5771	1.29838	0.649189	−	0.760628i	$-0.275109\pi$
$419$	26.5771	1.29838	0.649189	+	0.760628i	$0.275109\pi$
$420$	0	0
$421$	−40.0227	−1.95058	−0.975292	−	0.220918i	$-0.929095\pi$
$421$	−40.0227	−1.95058	−0.975292	+	0.220918i	$0.929095\pi$
$422$	37.7722	1.83872
$423$	−11.9915	−0.583049
$424$	−20.0507	−0.973747
$425$	0	0
$426$	−9.59883	−0.465065
$427$	−11.1730	−0.540701
$428$	−0.172531	−0.00833961
$429$	0	0
$430$	0	0
$431$	19.6309	0.945589	0.472794	−	0.881173i	$-0.343245\pi$
$431$	19.6309	0.945589	0.472794	+	0.881173i	$0.343245\pi$
$432$	−3.96097	−0.190572
$433$	−40.0733	−1.92580	−0.962900	−	0.269857i	$-0.913023\pi$
$433$	−40.0733	−1.92580	−0.962900	+	0.269857i	$0.913023\pi$
$434$	−7.02876	−0.337391
$435$	0	0
$436$	0.364285	0.0174461
$437$	12.0666	0.577223
$438$	−5.06925	−0.242218
$439$	7.34629	0.350619	0.175310	−	0.984513i	$-0.443907\pi$
$439$	7.34629	0.350619	0.175310	+	0.984513i	$0.443907\pi$
$440$	0	0
$441$	−2.31809	−0.110385
$442$	−8.36751	−0.398002
$443$	−27.7088	−1.31649	−0.658243	−	0.752806i	$-0.728700\pi$
$443$	−27.7088	−1.31649	−0.658243	+	0.752806i	$0.728700\pi$
$444$	−0.180869	−0.00858366
$445$	0	0
$446$	−8.19799	−0.388186
$447$	1.96471	0.0929277
$448$	17.4742	0.825578
$449$	−3.32449	−0.156893	−0.0784463	−	0.996918i	$-0.524996\pi$
$449$	−3.32449	−0.156893	−0.0784463	+	0.996918i	$0.524996\pi$
$450$	0	0
$451$	0	0
$452$	0.0135762	0.000638570 0
$453$	6.41511	0.301408
$454$	−25.7410	−1.20808
$455$	0	0
$456$	−14.1888	−0.664450
$457$	−17.7293	−0.829341	−0.414670	−	0.909972i	$-0.636103\pi$
$457$	−17.7293	−0.829341	−0.414670	+	0.909972i	$0.636103\pi$
$458$	33.0784	1.54565
$459$	−2.42748	−0.113305
$460$	0	0
$461$	−24.4362	−1.13811	−0.569055	−	0.822300i	$-0.692691\pi$
$461$	−24.4362	−1.13811	−0.569055	+	0.822300i	$0.692691\pi$
$462$	0	0
$463$	−23.1516	−1.07594	−0.537972	−	0.842963i	$-0.680809\pi$
$463$	−23.1516	−1.07594	−0.537972	+	0.842963i	$0.680809\pi$
$464$	−21.4962	−0.997938
$465$	0	0
$466$	−38.5542	−1.78599
$467$	−23.1400	−1.07079	−0.535397	−	0.844601i	$-0.679838\pi$
$467$	−23.1400	−1.07079	−0.535397	+	0.844601i	$0.679838\pi$
$468$	0.0473341	0.00218802
$469$	30.4783	1.40736
$470$	0	0
$471$	10.1011	0.465434
$472$	12.1623	0.559813
$473$	0	0
$474$	15.0629	0.691862
$475$	0	0
$476$	0.101510	0.00465269
$477$	−7.05531	−0.323041
$478$	30.8407	1.41062
$479$	−10.7694	−0.492067	−0.246034	−	0.969261i	$-0.579127\pi$
$479$	−10.7694	−0.492067	−0.246034	+	0.969261i	$0.579127\pi$
$480$	0	0
$481$	−22.9223	−1.04517
$482$	31.9520	1.45537
$483$	−5.22954	−0.237952
$484$	0	0
$485$	0	0
$486$	−1.40736	−0.0638393
$487$	18.4916	0.837932	0.418966	−	0.908002i	$-0.362393\pi$
$487$	18.4916	0.837932	0.418966	+	0.908002i	$0.362393\pi$
$488$	−14.6748	−0.664298
$489$	−17.1635	−0.776162
$490$	0	0
$491$	−35.3882	−1.59705	−0.798523	−	0.601965i	$-0.794385\pi$
$491$	−35.3882	−1.59705	−0.798523	+	0.601965i	$0.794385\pi$
$492$	−0.114187	−0.00514795
$493$	−13.1740	−0.593325
$494$	−17.2097	−0.774299
$495$	0	0
$496$	−9.14246	−0.410508
$497$	14.7578	0.661980
$498$	3.35484	0.150334
$499$	−4.03340	−0.180560	−0.0902799	−	0.995916i	$-0.528776\pi$
$499$	−4.03340	−0.180560	−0.0902799	+	0.995916i	$0.528776\pi$
$500$	0	0
$501$	−12.9371	−0.577985
$502$	−3.66235	−0.163459
$503$	34.2194	1.52577	0.762885	−	0.646534i	$-0.223782\pi$
$503$	34.2194	1.52577	0.762885	+	0.646534i	$0.223782\pi$
$504$	6.14928	0.273911
$505$	0	0
$506$	0	0
$507$	−7.00116	−0.310932
$508$	0.372787	0.0165397
$509$	24.2833	1.07634	0.538169	−	0.842837i	$-0.319116\pi$
$509$	24.2833	1.07634	0.538169	+	0.842837i	$0.319116\pi$
$510$	0	0
$511$	7.79379	0.344777
$512$	22.9466	1.01411
$513$	−4.99266	−0.220431
$514$	−23.8133	−1.05036
$515$	0	0
$516$	0.0601119	0.00264628
$517$	0	0
$518$	−28.4998	−1.25221
$519$	−0.186614	−0.00819147
$520$	0	0
$521$	43.0776	1.88727	0.943633	−	0.330994i	$-0.107384\pi$
$521$	43.0776	1.88727	0.943633	+	0.330994i	$0.107384\pi$
$522$	−7.63778	−0.334296
$523$	33.7657	1.47647	0.738235	−	0.674544i	$-0.235660\pi$
$523$	33.7657	1.47647	0.738235	+	0.674544i	$0.235660\pi$
$524$	0.0141379	0.000617615 0
$525$	0	0
$526$	−25.1700	−1.09746
$527$	−5.60295	−0.244068
$528$	0	0
$529$	−17.1588	−0.746033
$530$	0	0
$531$	4.27958	0.185718
$532$	0.208777	0.00905165
$533$	−14.4714	−0.626826
$534$	14.5068	0.627769
$535$	0	0
$536$	40.0306	1.72906
$537$	7.93874	0.342582
$538$	40.5088	1.74646
$539$	0	0
$540$	0	0
$541$	−4.80869	−0.206742	−0.103371	−	0.994643i	$-0.532963\pi$
$541$	−4.80869	−0.206742	−0.103371	+	0.994643i	$0.532963\pi$
$542$	−31.5182	−1.35382
$543$	3.35297	0.143890
$544$	0.265372	0.0113777
$545$	0	0
$546$	7.45850	0.319194
$547$	21.9395	0.938065	0.469033	−	0.883181i	$-0.344603\pi$
$547$	21.9395	0.938065	0.469033	+	0.883181i	$0.344603\pi$
$548$	−0.260876	−0.0111441
$549$	−5.16368	−0.220381
$550$	0	0
$551$	−27.0952	−1.15429
$552$	−6.86855	−0.292345
$553$	−23.1587	−0.984807
$554$	−3.84933	−0.163542
$555$	0	0
$556$	0.314867	0.0133533
$557$	23.6616	1.00257	0.501286	−	0.865281i	$-0.332860\pi$
$557$	23.6616	1.00257	0.501286	+	0.865281i	$0.332860\pi$
$558$	−3.24838	−0.137515
$559$	7.61823	0.322217
$560$	0	0
$561$	0	0
$562$	−6.35408	−0.268031
$563$	34.3960	1.44962	0.724810	−	0.688949i	$-0.241927\pi$
$563$	34.3960	1.44962	0.724810	+	0.688949i	$0.241927\pi$
$564$	0.231747	0.00975833
$565$	0	0
$566$	−19.1818	−0.806271
$567$	2.16377	0.0908699
$568$	19.3832	0.813299
$569$	−38.2009	−1.60147	−0.800733	−	0.599021i	$-0.795557\pi$
$569$	−38.2009	−1.60147	−0.800733	+	0.599021i	$0.795557\pi$
$570$	0	0
$571$	18.0690	0.756165	0.378083	−	0.925772i	$-0.376584\pi$
$571$	18.0690	0.756165	0.378083	+	0.925772i	$0.376584\pi$
$572$	0	0
$573$	23.6161	0.986575
$574$	−17.9926	−0.750998
$575$	0	0
$576$	8.07580	0.336492
$577$	−30.8655	−1.28495	−0.642474	−	0.766308i	$-0.722092\pi$
$577$	−30.8655	−1.28495	−0.642474	+	0.766308i	$0.722092\pi$
$578$	15.6321	0.650208
$579$	−13.4928	−0.560742
$580$	0	0
$581$	−5.15795	−0.213988
$582$	−7.42629	−0.307830
$583$	0	0
$584$	10.2365	0.423588
$585$	0	0
$586$	15.1046	0.623967
$587$	−22.5809	−0.932014	−0.466007	−	0.884781i	$-0.654308\pi$
$587$	−22.5809	−0.932014	−0.466007	+	0.884781i	$0.654308\pi$
$588$	0.0447992	0.00184749
$589$	−11.5237	−0.474827
$590$	0	0
$591$	−19.9986	−0.822634
$592$	−37.0703	−1.52358
$593$	−20.9313	−0.859544	−0.429772	−	0.902937i	$-0.641406\pi$
$593$	−20.9313	−0.859544	−0.429772	+	0.902937i	$0.641406\pi$
$594$	0	0
$595$	0	0
$596$	−0.0379699	−0.00155531
$597$	9.08289	0.371738
$598$	−8.33091	−0.340676
$599$	−2.55498	−0.104394	−0.0521969	−	0.998637i	$-0.516622\pi$
$599$	−2.55498	−0.104394	−0.0521969	+	0.998637i	$0.516622\pi$
$600$	0	0
$601$	−6.47777	−0.264234	−0.132117	−	0.991234i	$-0.542177\pi$
$601$	−6.47777	−0.264234	−0.132117	+	0.991234i	$0.542177\pi$
$602$	9.47193	0.386047
$603$	14.0857	0.573615
$604$	−0.123978	−0.00504459
$605$	0	0
$606$	22.2721	0.904743
$607$	−35.5206	−1.44174	−0.720869	−	0.693071i	$-0.756257\pi$
$607$	−35.5206	−1.44174	−0.720869	+	0.693071i	$0.756257\pi$
$608$	0.545798	0.0221350
$609$	11.7428	0.475843
$610$	0	0
$611$	29.3703	1.18820
$612$	0.0469133	0.00189636
$613$	−13.1667	−0.531800	−0.265900	−	0.964001i	$-0.585669\pi$
$613$	−13.1667	−0.531800	−0.265900	+	0.964001i	$0.585669\pi$
$614$	7.14535	0.288363
$615$	0	0
$616$	0	0
$617$	−30.6703	−1.23474	−0.617370	−	0.786673i	$-0.711802\pi$
$617$	−30.6703	−1.23474	−0.617370	+	0.786673i	$0.711802\pi$
$618$	2.45984	0.0989495
$619$	−35.7516	−1.43698	−0.718489	−	0.695539i	$-0.755166\pi$
$619$	−35.7516	−1.43698	−0.718489	+	0.695539i	$0.755166\pi$
$620$	0	0
$621$	−2.41686	−0.0969854
$622$	5.73809	0.230077
$623$	−22.3036	−0.893576
$624$	9.70143	0.388368
$625$	0	0
$626$	−41.2733	−1.64961
$627$	0	0
$628$	−0.195213	−0.00778985
$629$	−22.7185	−0.905846
$630$	0	0
$631$	3.92017	0.156060	0.0780298	−	0.996951i	$-0.475137\pi$
$631$	3.92017	0.156060	0.0780298	+	0.996951i	$0.475137\pi$
$632$	−30.4169	−1.20992
$633$	−26.8389	−1.06675
$634$	−1.63540	−0.0649501
$635$	0	0
$636$	0.136350	0.00540664
$637$	5.67759	0.224954
$638$	0	0
$639$	6.82043	0.269812
$640$	0	0
$641$	22.2700	0.879614	0.439807	−	0.898092i	$-0.355047\pi$
$641$	22.2700	0.879614	0.439807	+	0.898092i	$0.355047\pi$
$642$	−12.5642	−0.495869
$643$	−14.7424	−0.581383	−0.290692	−	0.956817i	$-0.593885\pi$
$643$	−14.7424	−0.581383	−0.290692	+	0.956817i	$0.593885\pi$
$644$	0.101066	0.00398255
$645$	0	0
$646$	−17.0567	−0.671086
$647$	18.7654	0.737743	0.368872	−	0.929480i	$-0.379744\pi$
$647$	18.7654	0.737743	0.368872	+	0.929480i	$0.379744\pi$
$648$	2.84193	0.111641
$649$	0	0
$650$	0	0
$651$	4.99427	0.195741
$652$	0.331701	0.0129904
$653$	−22.6349	−0.885773	−0.442886	−	0.896578i	$-0.646045\pi$
$653$	−22.6349	−0.885773	−0.442886	+	0.896578i	$0.646045\pi$
$654$	26.5282	1.03733
$655$	0	0
$656$	−23.4034	−0.913749
$657$	3.60194	0.140525
$658$	36.5168	1.42357
$659$	20.4233	0.795580	0.397790	−	0.917476i	$-0.369777\pi$
$659$	20.4233	0.795580	0.397790	+	0.917476i	$0.369777\pi$
$660$	0	0
$661$	−12.4374	−0.483758	−0.241879	−	0.970306i	$-0.577764\pi$
$661$	−12.4374	−0.483758	−0.241879	+	0.970306i	$0.577764\pi$
$662$	−20.8588	−0.810699
$663$	5.94552	0.230905
$664$	−6.77452	−0.262902
$665$	0	0
$666$	−13.1714	−0.510380
$667$	−13.1163	−0.507867
$668$	0.250020	0.00967358
$669$	5.82507	0.225210
$670$	0	0
$671$	0	0
$672$	−0.236544	−0.00912487
$673$	23.1015	0.890498	0.445249	−	0.895407i	$-0.353115\pi$
$673$	23.1015	0.890498	0.445249	+	0.895407i	$0.353115\pi$
$674$	15.9839	0.615678
$675$	0	0
$676$	0.135304	0.00520399
$677$	29.6960	1.14131	0.570656	−	0.821189i	$-0.306689\pi$
$677$	29.6960	1.14131	0.570656	+	0.821189i	$0.306689\pi$
$678$	0.988654	0.0379690
$679$	11.4177	0.438170
$680$	0	0
$681$	18.2902	0.700883
$682$	0	0
$683$	−50.1762	−1.91994	−0.959969	−	0.280105i	$-0.909631\pi$
$683$	−50.1762	−1.91994	−0.959969	+	0.280105i	$0.909631\pi$
$684$	0.0964877	0.00368930
$685$	0	0
$686$	28.3756	1.08339
$687$	−23.5038	−0.896727
$688$	12.3203	0.469708
$689$	17.2803	0.658325
$690$	0	0
$691$	−23.8224	−0.906246	−0.453123	−	0.891448i	$-0.649690\pi$
$691$	−23.8224	−0.906246	−0.453123	+	0.891448i	$0.649690\pi$
$692$	0.00360649	0.000137098 0
$693$	0	0
$694$	−9.87597	−0.374887
$695$	0	0
$696$	15.4232	0.584613
$697$	−14.3428	−0.543271
$698$	−7.76397	−0.293871
$699$	27.3946	1.03616
$700$	0	0
$701$	−11.3927	−0.430296	−0.215148	−	0.976581i	$-0.569023\pi$
$701$	−11.3927	−0.430296	−0.215148	+	0.976581i	$0.569023\pi$
$702$	3.44699	0.130098
$703$	−46.7257	−1.76229
$704$	0	0
$705$	0	0
$706$	−20.6197	−0.776033
$707$	−34.2426	−1.28782
$708$	−0.0827068	−0.00310831
$709$	38.6380	1.45108	0.725541	−	0.688179i	$-0.241590\pi$
$709$	38.6380	1.45108	0.725541	+	0.688179i	$0.241590\pi$
$710$	0	0
$711$	−10.7029	−0.401391
$712$	−29.2939	−1.09783
$713$	−5.57845	−0.208914
$714$	7.39220	0.276646
$715$	0	0
$716$	−0.153423	−0.00573370
$717$	−21.9138	−0.818386
$718$	12.4597	0.464993
$719$	−10.9187	−0.407197	−0.203599	−	0.979054i	$-0.565264\pi$
$719$	−10.9187	−0.407197	−0.203599	+	0.979054i	$0.565264\pi$
$720$	0	0
$721$	−3.78192	−0.140846
$722$	−8.34097	−0.310419
$723$	−22.7034	−0.844349
$724$	−0.0647991	−0.00240824
$725$	0	0
$726$	0	0
$727$	−9.96114	−0.369438	−0.184719	−	0.982791i	$-0.559138\pi$
$727$	−9.96114	−0.369438	−0.184719	+	0.982791i	$0.559138\pi$
$728$	−15.0612	−0.558203
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.55051	0.279266
$732$	0.0997929	0.00368845
$733$	5.29250	0.195483	0.0977415	−	0.995212i	$-0.468838\pi$
$733$	5.29250	0.195483	0.0977415	+	0.995212i	$0.468838\pi$
$734$	30.3265	1.11937
$735$	0	0
$736$	0.264212	0.00973897
$737$	0	0
$738$	−8.31540	−0.306094
$739$	−13.0032	−0.478332	−0.239166	−	0.970979i	$-0.576874\pi$
$739$	−13.0032	−0.478332	−0.239166	+	0.970979i	$0.576874\pi$
$740$	0	0
$741$	12.2283	0.449218
$742$	21.4849	0.788737
$743$	−54.0230	−1.98191	−0.990955	−	0.134191i	$-0.957156\pi$
$743$	−54.0230	−1.98191	−0.990955	+	0.134191i	$0.957156\pi$
$744$	6.55955	0.240485
$745$	0	0
$746$	−6.00126	−0.219722
$747$	−2.38378	−0.0872178
$748$	0	0
$749$	19.3170	0.705827
$750$	0	0
$751$	−43.9454	−1.60359	−0.801795	−	0.597599i	$-0.796122\pi$
$751$	−43.9454	−1.60359	−0.801795	+	0.597599i	$0.796122\pi$
$752$	47.4982	1.73208
$753$	2.60228	0.0948322
$754$	18.7069	0.681263
$755$	0	0
$756$	−0.0418169	−0.00152086
$757$	−20.0109	−0.727309	−0.363654	−	0.931534i	$-0.618471\pi$
$757$	−20.0109	−0.727309	−0.363654	+	0.931534i	$0.618471\pi$
$758$	52.0873	1.89190
$759$	0	0
$760$	0	0
$761$	17.6343	0.639241	0.319621	−	0.947546i	$-0.396445\pi$
$761$	17.6343	0.639241	0.319621	+	0.947546i	$0.396445\pi$
$762$	27.1473	0.983443
$763$	−40.7861	−1.47656
$764$	−0.456402	−0.0165120
$765$	0	0
$766$	13.8548	0.500594
$767$	−10.4818	−0.378475
$768$	−0.463778	−0.0167352
$769$	−37.2865	−1.34458	−0.672292	−	0.740286i	$-0.734690\pi$
$769$	−37.2865	−1.34458	−0.672292	+	0.740286i	$0.734690\pi$
$770$	0	0
$771$	16.9205	0.609377
$772$	0.260761	0.00938499
$773$	20.4462	0.735397	0.367699	−	0.929945i	$-0.380146\pi$
$773$	20.4462	0.735397	0.367699	+	0.929945i	$0.380146\pi$
$774$	4.37751	0.157346
$775$	0	0
$776$	14.9961	0.538329
$777$	20.2505	0.726482
$778$	4.13790	0.148351
$779$	−29.4991	−1.05692
$780$	0	0
$781$	0	0
$782$	−8.25686	−0.295265
$783$	5.42701	0.193945
$784$	9.18190	0.327925
$785$	0	0
$786$	1.02956	0.0367230
$787$	4.23306	0.150892	0.0754461	−	0.997150i	$-0.475962\pi$
$787$	4.23306	0.150892	0.0754461	+	0.997150i	$0.475962\pi$
$788$	0.386492	0.0137682
$789$	17.8845	0.636705
$790$	0	0
$791$	−1.52002	−0.0540457
$792$	0	0
$793$	12.6472	0.449114
$794$	−48.8374	−1.73317
$795$	0	0
$796$	−0.175535	−0.00622168
$797$	16.9793	0.601437	0.300718	−	0.953713i	$-0.402774\pi$
$797$	16.9793	0.601437	0.300718	+	0.953713i	$0.402774\pi$
$798$	15.2037	0.538206
$799$	29.1092	1.02981
$800$	0	0
$801$	−10.3078	−0.364207
$802$	15.1347	0.534425
$803$	0	0
$804$	−0.272219	−0.00960044
$805$	0	0
$806$	7.95612	0.280242
$807$	−28.7834	−1.01322
$808$	−44.9746	−1.58220
$809$	−13.9967	−0.492099	−0.246050	−	0.969257i	$-0.579133\pi$
$809$	−13.9967	−0.492099	−0.246050	+	0.969257i	$0.579133\pi$
$810$	0	0
$811$	−48.8158	−1.71415	−0.857077	−	0.515188i	$-0.827722\pi$
$811$	−48.8158	−1.71415	−0.857077	+	0.515188i	$0.827722\pi$
$812$	−0.226940	−0.00796405
$813$	22.3952	0.785434
$814$	0	0
$815$	0	0
$816$	9.61519	0.336599
$817$	15.5293	0.543302
$818$	27.9223	0.976281
$819$	−5.29963	−0.185184
$820$	0	0
$821$	−28.2697	−0.986619	−0.493310	−	0.869854i	$-0.664213\pi$
$821$	−28.2697	−0.986619	−0.493310	+	0.869854i	$0.664213\pi$
$822$	−18.9977	−0.662620
$823$	16.6345	0.579842	0.289921	−	0.957051i	$-0.406371\pi$
$823$	16.6345	0.579842	0.289921	+	0.957051i	$0.406371\pi$
$824$	−4.96723	−0.173042
$825$	0	0
$826$	−13.0322	−0.453450
$827$	31.9893	1.11238	0.556188	−	0.831057i	$-0.312263\pi$
$827$	31.9893	1.11238	0.556188	+	0.831057i	$0.312263\pi$
$828$	0.0467081	0.00162322
$829$	53.6294	1.86263	0.931314	−	0.364218i	$-0.118664\pi$
$829$	53.6294	1.86263	0.931314	+	0.364218i	$0.118664\pi$
$830$	0	0
$831$	2.73513	0.0948806
$832$	−19.7797	−0.685737
$833$	5.62712	0.194968
$834$	22.9294	0.793982
$835$	0	0
$836$	0	0
$837$	2.30813	0.0797808
$838$	−37.4037	−1.29209
$839$	37.8292	1.30601	0.653004	−	0.757355i	$-0.273509\pi$
$839$	37.8292	1.30601	0.653004	+	0.757355i	$0.273509\pi$
$840$	0	0
$841$	0.452416	0.0156006
$842$	56.3264	1.94114
$843$	4.51488	0.155501
$844$	0.518687	0.0178539
$845$	0	0
$846$	16.8765	0.580225
$847$	0	0
$848$	27.9459	0.959667
$849$	13.6296	0.467766
$850$	0	0
$851$	−22.6192	−0.775375
$852$	−0.131811	−0.00451577
$853$	22.1875	0.759687	0.379843	−	0.925051i	$-0.375978\pi$
$853$	22.1875	0.759687	0.379843	+	0.925051i	$0.375978\pi$
$854$	15.7245	0.538082
$855$	0	0
$856$	25.3712	0.867169
$857$	−13.7471	−0.469592	−0.234796	−	0.972045i	$-0.575442\pi$
$857$	−13.7471	−0.469592	−0.234796	+	0.972045i	$0.575442\pi$
$858$	0	0
$859$	43.9972	1.50117	0.750583	−	0.660776i	$-0.229773\pi$
$859$	43.9972	1.50117	0.750583	+	0.660776i	$0.229773\pi$
$860$	0	0
$861$	12.7846	0.435699
$862$	−27.6279	−0.941009
$863$	5.26659	0.179277	0.0896385	−	0.995974i	$-0.471429\pi$
$863$	5.26659	0.179277	0.0896385	+	0.995974i	$0.471429\pi$
$864$	−0.109320	−0.00371914
$865$	0	0
$866$	56.3978	1.91647
$867$	−11.1073	−0.377225
$868$	−0.0965189	−0.00327606
$869$	0	0
$870$	0	0
$871$	−34.4995	−1.16897
$872$	−53.5691	−1.81408
$873$	5.27674	0.178591
$874$	−16.9821	−0.574428
$875$	0	0
$876$	−0.0696109	−0.00235193
$877$	−0.301130	−0.0101684	−0.00508421	−	0.999987i	$-0.501618\pi$
$877$	−0.301130	−0.0101684	−0.00508421	+	0.999987i	$0.501618\pi$
$878$	−10.3389	−0.348921
$879$	−10.7326	−0.362001
$880$	0	0
$881$	24.5711	0.827821	0.413911	−	0.910318i	$-0.364163\pi$
$881$	24.5711	0.827821	0.413911	+	0.910318i	$0.364163\pi$
$882$	3.26240	0.109851
$883$	−2.07849	−0.0699467	−0.0349733	−	0.999388i	$-0.511135\pi$
$883$	−2.07849	−0.0699467	−0.0349733	+	0.999388i	$0.511135\pi$
$884$	−0.114903	−0.00386459
$885$	0	0
$886$	38.9964	1.31011
$887$	−30.5436	−1.02555	−0.512776	−	0.858522i	$-0.671383\pi$
$887$	−30.5436	−1.02555	−0.512776	+	0.858522i	$0.671383\pi$
$888$	26.5973	0.892545
$889$	−41.7380	−1.39985
$890$	0	0
$891$	0	0
$892$	−0.112575	−0.00376928
$893$	59.8697	2.00346
$894$	−2.76507	−0.0924777
$895$	0	0
$896$	−24.1195	−0.805775
$897$	5.91952	0.197647
$898$	4.67877	0.156133
$899$	12.5263	0.417774
$900$	0	0
$901$	17.1266	0.570571
$902$	0	0
$903$	−6.73026	−0.223969
$904$	−1.99641	−0.0663997
$905$	0	0
$906$	−9.02840	−0.299949
$907$	−35.8505	−1.19039	−0.595197	−	0.803579i	$-0.702926\pi$
$907$	−35.8505	−1.19039	−0.595197	+	0.803579i	$0.702926\pi$
$908$	−0.353475	−0.0117305
$909$	−15.8254	−0.524896
$910$	0	0
$911$	9.39731	0.311347	0.155673	−	0.987809i	$-0.450245\pi$
$911$	9.39731	0.311347	0.155673	+	0.987809i	$0.450245\pi$
$912$	19.7758	0.654842
$913$	0	0
$914$	24.9516	0.825324
$915$	0	0
$916$	0.454233	0.0150083
$917$	−1.58291	−0.0522721
$918$	3.41635	0.112756
$919$	−0.858655	−0.0283244	−0.0141622	−	0.999900i	$-0.504508\pi$
$919$	−0.858655	−0.0283244	−0.0141622	+	0.999900i	$0.504508\pi$
$920$	0	0
$921$	−5.07712	−0.167297
$922$	34.3907	1.13260
$923$	−16.7050	−0.549850
$924$	0	0
$925$	0	0
$926$	32.5827	1.07073
$927$	−1.74784	−0.0574065
$928$	−0.593281	−0.0194754
$929$	40.1611	1.31764	0.658822	−	0.752299i	$-0.271055\pi$
$929$	40.1611	1.31764	0.658822	+	0.752299i	$0.271055\pi$
$930$	0	0
$931$	11.5734	0.379304
$932$	−0.529426	−0.0173419
$933$	−4.07719	−0.133481
$934$	32.5665	1.06561
$935$	0	0
$936$	−6.96060	−0.227514
$937$	31.5591	1.03099	0.515495	−	0.856892i	$-0.327608\pi$
$937$	31.5591	1.03099	0.515495	+	0.856892i	$0.327608\pi$
$938$	−42.8940	−1.40054
$939$	29.3267	0.957039
$940$	0	0
$941$	10.9750	0.357774	0.178887	−	0.983870i	$-0.442750\pi$
$941$	10.9750	0.357774	0.178887	+	0.983870i	$0.442750\pi$
$942$	−14.2159	−0.463180
$943$	−14.2800	−0.465022
$944$	−16.9513	−0.551718
$945$	0	0
$946$	0	0
$947$	−32.3272	−1.05049	−0.525246	−	0.850951i	$-0.676027\pi$
$947$	−32.3272	−1.05049	−0.525246	+	0.850951i	$0.676027\pi$
$948$	0.206844	0.00671797
$949$	−8.82208	−0.286377
$950$	0	0
$951$	1.16203	0.0376815
$952$	−14.9273	−0.483795
$953$	−4.55744	−0.147630	−0.0738149	−	0.997272i	$-0.523517\pi$
$953$	−4.55744	−0.147630	−0.0738149	+	0.997272i	$0.523517\pi$
$954$	9.92940	0.321476
$955$	0	0
$956$	0.423504	0.0136971
$957$	0	0
$958$	15.1565	0.489684
$959$	29.2082	0.943183
$960$	0	0
$961$	−25.6725	−0.828146
$962$	32.2600	1.04010
$963$	8.92746	0.287683
$964$	0.438764	0.0141316
$965$	0	0
$966$	7.35987	0.236800
$967$	−3.69371	−0.118782	−0.0593909	−	0.998235i	$-0.518916\pi$
$967$	−3.69371	−0.118782	−0.0593909	+	0.998235i	$0.518916\pi$
$968$	0	0
$969$	12.1196	0.389337
$970$	0	0
$971$	−53.3607	−1.71242	−0.856212	−	0.516624i	$-0.827189\pi$
$971$	−53.3607	−1.71242	−0.856212	+	0.516624i	$0.827189\pi$
$972$	−0.0193259	−0.000619879 0
$973$	−35.2532	−1.13017
$974$	−26.0243	−0.833874
$975$	0	0
$976$	20.4532	0.654692
$977$	36.4623	1.16653	0.583267	−	0.812281i	$-0.301774\pi$
$977$	36.4623	1.16653	0.583267	+	0.812281i	$0.301774\pi$
$978$	24.1554	0.772403
$979$	0	0
$980$	0	0
$981$	−18.8496	−0.601820
$982$	49.8040	1.58931
$983$	5.13985	0.163936	0.0819678	−	0.996635i	$-0.473880\pi$
$983$	5.13985	0.163936	0.0819678	+	0.996635i	$0.473880\pi$
$984$	16.7915	0.535294
$985$	0	0
$986$	18.5406	0.590452
$987$	−25.9470	−0.825901
$988$	−0.236323	−0.00751843
$989$	7.51749	0.239042
$990$	0	0
$991$	−57.4924	−1.82631	−0.913154	−	0.407616i	$-0.866360\pi$
$991$	−57.4924	−1.82631	−0.913154	+	0.407616i	$0.866360\pi$
$992$	−0.252325	−0.00801133
$993$	14.8212	0.470335
$994$	−20.7697	−0.658774
$995$	0	0
$996$	0.0460686	0.00145974
$997$	6.68078	0.211583	0.105791	−	0.994388i	$-0.466262\pi$
$997$	6.68078	0.211583	0.105791	+	0.994388i	$0.466262\pi$
$998$	5.67646	0.179685
$999$	9.35888	0.296102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.du.1.3		8
5.4	even	2		9075.2.a.dv.1.6		8
11.3	even	5		825.2.n.m.526.2	✓	16
11.4	even	5		825.2.n.m.676.2	yes	16
11.10	odd	2		9075.2.a.dw.1.6		8
55.3	odd	20		825.2.bx.j.724.6		32
55.4	even	10		825.2.n.n.676.3	yes	16
55.14	even	10		825.2.n.n.526.3	yes	16
55.37	odd	20		825.2.bx.j.49.6		32
55.47	odd	20		825.2.bx.j.724.3		32
55.48	odd	20		825.2.bx.j.49.3		32
55.54	odd	2		9075.2.a.dt.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.m.526.2	✓	16	11.3	even	5
825.2.n.m.676.2	yes	16	11.4	even	5
825.2.n.n.526.3	yes	16	55.14	even	10
825.2.n.n.676.3	yes	16	55.4	even	10
825.2.bx.j.49.3		32	55.48	odd	20
825.2.bx.j.49.6		32	55.37	odd	20
825.2.bx.j.724.3		32	55.47	odd	20
825.2.bx.j.724.6		32	55.3	odd	20
9075.2.a.dt.1.3		8	55.54	odd	2
9075.2.a.du.1.3		8	1.1	even	1	trivial
9075.2.a.dv.1.6		8	5.4	even	2
9075.2.a.dw.1.6		8	11.10	odd	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)

\(f(q)\)	\(=\)	\(q-1.40736 q^{2} +1.00000 q^{3} -0.0193259 q^{4} -1.40736 q^{6} +2.16377 q^{7} +2.84193 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.40736 q^{2} +1.00000 q^{3} -0.0193259 q^{4} -1.40736 q^{6} +2.16377 q^{7} +2.84193 q^{8} +1.00000 q^{9} -0.0193259 q^{12} -2.44925 q^{13} -3.04521 q^{14} -3.96097 q^{16} -2.42748 q^{17} -1.40736 q^{18} -4.99266 q^{19} +2.16377 q^{21} -2.41686 q^{23} +2.84193 q^{24} +3.44699 q^{26} +1.00000 q^{27} -0.0418169 q^{28} +5.42701 q^{29} +2.30813 q^{31} -0.109320 q^{32} +3.41635 q^{34} -0.0193259 q^{36} +9.35888 q^{37} +7.02649 q^{38} -2.44925 q^{39} +5.90849 q^{41} -3.04521 q^{42} -3.11043 q^{43} +3.40141 q^{46} -11.9915 q^{47} -3.96097 q^{48} -2.31809 q^{49} -2.42748 q^{51} +0.0473341 q^{52} -7.05531 q^{53} -1.40736 q^{54} +6.14928 q^{56} -4.99266 q^{57} -7.63778 q^{58} +4.27958 q^{59} -5.16368 q^{61} -3.24838 q^{62} +2.16377 q^{63} +8.07580 q^{64} +14.0857 q^{67} +0.0469133 q^{68} -2.41686 q^{69} +6.82043 q^{71} +2.84193 q^{72} +3.60194 q^{73} -13.1714 q^{74} +0.0964877 q^{76} +3.44699 q^{78} -10.7029 q^{79} +1.00000 q^{81} -8.31540 q^{82} -2.38378 q^{83} -0.0418169 q^{84} +4.37751 q^{86} +5.42701 q^{87} -10.3078 q^{89} -5.29963 q^{91} +0.0467081 q^{92} +2.30813 q^{93} +16.8765 q^{94} -0.109320 q^{96} +5.27674 q^{97} +3.26240 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 8 q^{3} + 7 q^{4} - q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 + 8 * q^3 + 7 * q^4 - q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 7 q^{4} - q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} + 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} - 6 q^{19} - 8 q^{21} + 7 q^{23} - 3 q^{24} - q^{26} + 8 q^{27} - 11 q^{28} - 28 q^{29} + 10 q^{31} - 12 q^{32} - 8 q^{34} + 7 q^{36} + 14 q^{37} + 19 q^{38} - 7 q^{39} - 14 q^{41} - 8 q^{42} - 11 q^{43} + 5 q^{47} - 7 q^{48} + 2 q^{49} + q^{51} - q^{52} - 5 q^{53} - q^{54} - 4 q^{56} - 6 q^{57} - 3 q^{58} - 2 q^{59} - 36 q^{61} + 17 q^{62} - 8 q^{63} - 23 q^{64} + 6 q^{67} + 3 q^{68} + 7 q^{69} - 17 q^{71} - 3 q^{72} - 32 q^{73} - 47 q^{74} - 5 q^{76} - q^{78} - 41 q^{79} + 8 q^{81} + 6 q^{83} - 11 q^{84} - 9 q^{86} - 28 q^{87} + 21 q^{89} - 9 q^{91} + 69 q^{92} + 10 q^{93} - 51 q^{94} - 12 q^{96} - 4 q^{97} + 27 q^{98}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 7 * q^4 - q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 + 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 - 6 * q^19 - 8 * q^21 + 7 * q^23 - 3 * q^24 - q^26 + 8 * q^27 - 11 * q^28 - 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 + 14 * q^37 + 19 * q^38 - 7 * q^39 - 14 * q^41 - 8 * q^42 - 11 * q^43 + 5 * q^47 - 7 * q^48 + 2 * q^49 + q^51 - q^52 - 5 * q^53 - q^54 - 4 * q^56 - 6 * q^57 - 3 * q^58 - 2 * q^59 - 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 + 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 - 47 * q^74 - 5 * q^76 - q^78 - 41 * q^79 + 8 * q^81 + 6 * q^83 - 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 + 69 * q^92 + 10 * q^93 - 51 * q^94 - 12 * q^96 - 4 * q^97 + 27 * q^98

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