LMFDB - Embedded newform 9075.2.a.do.1.4

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:	$6$
Coefficient field:	6.6.860280160.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 9x^{3} + 23x^{2} - 22x - 2 $ x^6 - x^5 - 10x^4 + 9x^3 + 23x^2 - 22x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0838261$ 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)

$f(q)$	$=$	$q+0.0838261 q^{2} -1.00000 q^{3} -1.99297 q^{4} -0.0838261 q^{6} -2.34295 q^{7} -0.334715 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0838261 q^{2} -1.00000 q^{3} -1.99297 q^{4} -0.0838261 q^{6} -2.34295 q^{7} -0.334715 q^{8} +1.00000 q^{9} +1.99297 q^{12} -5.61615 q^{13} -0.196400 q^{14} +3.95789 q^{16} -2.13832 q^{17} +0.0838261 q^{18} +2.17529 q^{19} +2.34295 q^{21} -8.11851 q^{23} +0.334715 q^{24} -0.470780 q^{26} -1.00000 q^{27} +4.66943 q^{28} +5.11146 q^{29} +4.93848 q^{31} +1.00120 q^{32} -0.179247 q^{34} -1.99297 q^{36} +5.25915 q^{37} +0.182346 q^{38} +5.61615 q^{39} +8.59386 q^{41} +0.196400 q^{42} +6.99068 q^{43} -0.680543 q^{46} +5.75446 q^{47} -3.95789 q^{48} -1.51060 q^{49} +2.13832 q^{51} +11.1928 q^{52} -10.3207 q^{53} -0.0838261 q^{54} +0.784220 q^{56} -2.17529 q^{57} +0.428474 q^{58} -3.46830 q^{59} +11.7334 q^{61} +0.413974 q^{62} -2.34295 q^{63} -7.83185 q^{64} +9.16598 q^{67} +4.26160 q^{68} +8.11851 q^{69} -6.51765 q^{71} -0.334715 q^{72} -13.7966 q^{73} +0.440854 q^{74} -4.33530 q^{76} +0.470780 q^{78} -6.68282 q^{79} +1.00000 q^{81} +0.720390 q^{82} +11.7895 q^{83} -4.66943 q^{84} +0.586001 q^{86} -5.11146 q^{87} +2.38336 q^{89} +13.1583 q^{91} +16.1800 q^{92} -4.93848 q^{93} +0.482374 q^{94} -1.00120 q^{96} -2.10081 q^{97} -0.126628 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 6 q^{3} + 9 q^{4} + q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - q^2 - 6 * q^3 + 9 * q^4 + q^6 + 4 * q^7 + 6 * q^9	$ 6 q - q^{2} - 6 q^{3} + 9 q^{4} + q^{6} + 4 q^{7} + 6 q^{9} - 9 q^{12} - 2 q^{13} - 10 q^{14} + 11 q^{16} - 2 q^{17} - q^{18} - 2 q^{19} - 4 q^{21} - 12 q^{23} - 6 q^{27} + 24 q^{28} + 12 q^{29} + 18 q^{31} - 31 q^{32} - q^{34} + 9 q^{36} - 24 q^{37} - 32 q^{38} + 2 q^{39} - 6 q^{41} + 10 q^{42} - 10 q^{43} - 11 q^{46} - 8 q^{47} - 11 q^{48} + 12 q^{49} + 2 q^{51} + 8 q^{52} - 18 q^{53} + q^{54} - 42 q^{56} + 2 q^{57} + 8 q^{58} - 18 q^{59} + 4 q^{61} + 17 q^{62} + 4 q^{63} + 22 q^{64} - 12 q^{67} + 9 q^{68} + 12 q^{69} - 14 q^{73} - 4 q^{74} - 20 q^{76} + 2 q^{79} + 6 q^{81} - 36 q^{82} + 20 q^{83} - 24 q^{84} + 28 q^{86} - 12 q^{87} - 16 q^{89} + 24 q^{91} - 41 q^{92} - 18 q^{93} + 3 q^{94} + 31 q^{96} - 12 q^{97} - 53 q^{98}+O(q^{100}) $ 6 * q - q^2 - 6 * q^3 + 9 * q^4 + q^6 + 4 * q^7 + 6 * q^9 - 9 * q^12 - 2 * q^13 - 10 * q^14 + 11 * q^16 - 2 * q^17 - q^18 - 2 * q^19 - 4 * q^21 - 12 * q^23 - 6 * q^27 + 24 * q^28 + 12 * q^29 + 18 * q^31 - 31 * q^32 - q^34 + 9 * q^36 - 24 * q^37 - 32 * q^38 + 2 * q^39 - 6 * q^41 + 10 * q^42 - 10 * q^43 - 11 * q^46 - 8 * q^47 - 11 * q^48 + 12 * q^49 + 2 * q^51 + 8 * q^52 - 18 * q^53 + q^54 - 42 * q^56 + 2 * q^57 + 8 * q^58 - 18 * q^59 + 4 * q^61 + 17 * q^62 + 4 * q^63 + 22 * q^64 - 12 * q^67 + 9 * q^68 + 12 * q^69 - 14 * q^73 - 4 * q^74 - 20 * q^76 + 2 * q^79 + 6 * q^81 - 36 * q^82 + 20 * q^83 - 24 * q^84 + 28 * q^86 - 12 * q^87 - 16 * q^89 + 24 * q^91 - 41 * q^92 - 18 * q^93 + 3 * q^94 + 31 * q^96 - 12 * q^97 - 53 * 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 $

$2$	0.0838261	0.0592740	0.0296370	−	0.999561i	$-0.490565\pi$
$2$	0.0838261	0.0592740	0.0296370	+	0.999561i	$0.490565\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99297	−0.996487
$5$	0	0
$5$	0	0
$6$	−0.0838261	−0.0342219
$7$	−2.34295	−0.885551	−0.442775	−	0.896633i	$-0.646006\pi$
$7$	−2.34295	−0.885551	−0.442775	+	0.896633i	$0.646006\pi$
$8$	−0.334715	−0.118340
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.99297	0.575322
$13$	−5.61615	−1.55764	−0.778819	−	0.627248i	$-0.784181\pi$
$13$	−5.61615	−1.55764	−0.778819	+	0.627248i	$0.784181\pi$
$14$	−0.196400	−0.0524901
$15$	0	0
$16$	3.95789	0.989472
$17$	−2.13832	−0.518618	−0.259309	−	0.965794i	$-0.583495\pi$
$17$	−2.13832	−0.518618	−0.259309	+	0.965794i	$0.583495\pi$
$18$	0.0838261	0.0197580
$19$	2.17529	0.499047	0.249523	−	0.968369i	$-0.419726\pi$
$19$	2.17529	0.499047	0.249523	+	0.968369i	$0.419726\pi$
$20$	0	0
$21$	2.34295	0.511273
$22$	0	0
$23$	−8.11851	−1.69283	−0.846414	−	0.532526i	$-0.821243\pi$
$23$	−8.11851	−1.69283	−0.846414	+	0.532526i	$0.821243\pi$
$24$	0.334715	0.0683235
$25$	0	0
$26$	−0.470780	−0.0923275
$27$	−1.00000	−0.192450
$28$	4.66943	0.882439
$29$	5.11146	0.949175	0.474587	−	0.880208i	$-0.342597\pi$
$29$	5.11146	0.949175	0.474587	+	0.880208i	$0.342597\pi$
$30$	0	0
$31$	4.93848	0.886978	0.443489	−	0.896280i	$-0.353741\pi$
$31$	4.93848	0.886978	0.443489	+	0.896280i	$0.353741\pi$
$32$	1.00120	0.176990
$33$	0	0
$34$	−0.179247	−0.0307405
$35$	0	0
$36$	−1.99297	−0.332162
$37$	5.25915	0.864598	0.432299	−	0.901730i	$-0.357702\pi$
$37$	5.25915	0.864598	0.432299	+	0.901730i	$0.357702\pi$
$38$	0.182346	0.0295805
$39$	5.61615	0.899303
$40$	0	0
$41$	8.59386	1.34214	0.671068	−	0.741396i	$-0.265836\pi$
$41$	8.59386	1.34214	0.671068	+	0.741396i	$0.265836\pi$
$42$	0.196400	0.0303052
$43$	6.99068	1.06607	0.533034	−	0.846094i	$-0.321052\pi$
$43$	6.99068	1.06607	0.533034	+	0.846094i	$0.321052\pi$
$44$	0	0
$45$	0	0
$46$	−0.680543	−0.100341
$47$	5.75446	0.839374	0.419687	−	0.907669i	$-0.362140\pi$
$47$	5.75446	0.839374	0.419687	+	0.907669i	$0.362140\pi$
$48$	−3.95789	−0.571272
$49$	−1.51060	−0.215800
$50$	0	0
$51$	2.13832	0.299424
$52$	11.1928	1.55217
$53$	−10.3207	−1.41765	−0.708826	−	0.705383i	$-0.750775\pi$
$53$	−10.3207	−1.41765	−0.708826	+	0.705383i	$0.750775\pi$
$54$	−0.0838261	−0.0114073
$55$	0	0
$56$	0.784220	0.104796
$57$	−2.17529	−0.288125
$58$	0.428474	0.0562614
$59$	−3.46830	−0.451534	−0.225767	−	0.974181i	$-0.572489\pi$
$59$	−3.46830	−0.451534	−0.225767	+	0.974181i	$0.572489\pi$
$60$	0	0
$61$	11.7334	1.50230	0.751151	−	0.660130i	$-0.229499\pi$
$61$	11.7334	1.50230	0.751151	+	0.660130i	$0.229499\pi$
$62$	0.413974	0.0525747
$63$	−2.34295	−0.295184
$64$	−7.83185	−0.978981
$65$	0	0
$66$	0	0
$67$	9.16598	1.11980	0.559901	−	0.828559i	$-0.310839\pi$
$67$	9.16598	1.11980	0.559901	+	0.828559i	$0.310839\pi$
$68$	4.26160	0.516796
$69$	8.11851	0.977354
$70$	0	0
$71$	−6.51765	−0.773503	−0.386751	−	0.922184i	$-0.626403\pi$
$71$	−6.51765	−0.773503	−0.386751	+	0.922184i	$0.626403\pi$
$72$	−0.334715	−0.0394466
$73$	−13.7966	−1.61477	−0.807385	−	0.590025i	$-0.799118\pi$
$73$	−13.7966	−1.61477	−0.807385	+	0.590025i	$0.799118\pi$
$74$	0.440854	0.0512482
$75$	0	0
$76$	−4.33530	−0.497294
$77$	0	0
$78$	0.470780	0.0533053
$79$	−6.68282	−0.751876	−0.375938	−	0.926645i	$-0.622679\pi$
$79$	−6.68282	−0.751876	−0.375938	+	0.926645i	$0.622679\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.720390	0.0795537
$83$	11.7895	1.29407	0.647035	−	0.762460i	$-0.276009\pi$
$83$	11.7895	1.29407	0.647035	+	0.762460i	$0.276009\pi$
$84$	−4.66943	−0.509477
$85$	0	0
$86$	0.586001	0.0631902
$87$	−5.11146	−0.548006
$88$	0	0
$89$	2.38336	0.252635	0.126318	−	0.991990i	$-0.459684\pi$
$89$	2.38336	0.252635	0.126318	+	0.991990i	$0.459684\pi$
$90$	0	0
$91$	13.1583	1.37937
$92$	16.1800	1.68688
$93$	−4.93848	−0.512097
$94$	0.482374	0.0497531
$95$	0	0
$96$	−1.00120	−0.102185
$97$	−2.10081	−0.213305	−0.106653	−	0.994296i	$-0.534013\pi$
$97$	−2.10081	−0.213305	−0.106653	+	0.994296i	$0.534013\pi$
$98$	−0.126628	−0.0127913
$99$	0	0
$100$	0	0
$101$	−2.89635	−0.288197	−0.144099	−	0.989563i	$-0.546028\pi$
$101$	−2.89635	−0.288197	−0.144099	+	0.989563i	$0.546028\pi$
$102$	0.179247	0.0177481
$103$	−1.15527	−0.113833	−0.0569163	−	0.998379i	$-0.518127\pi$
$103$	−1.15527	−0.113833	−0.0569163	+	0.998379i	$0.518127\pi$
$104$	1.87981	0.184331
$105$	0	0
$106$	−0.865141	−0.0840299
$107$	16.9868	1.64217	0.821086	−	0.570805i	$-0.193369\pi$
$107$	16.9868	1.64217	0.821086	+	0.570805i	$0.193369\pi$
$108$	1.99297	0.191774
$109$	5.41383	0.518551	0.259275	−	0.965803i	$-0.416516\pi$
$109$	5.41383	0.518551	0.259275	+	0.965803i	$0.416516\pi$
$110$	0	0
$111$	−5.25915	−0.499176
$112$	−9.27312	−0.876228
$113$	6.82421	0.641968	0.320984	−	0.947085i	$-0.395986\pi$
$113$	6.82421	0.641968	0.320984	+	0.947085i	$0.395986\pi$
$114$	−0.182346	−0.0170783
$115$	0	0
$116$	−10.1870	−0.945840
$117$	−5.61615	−0.519213
$118$	−0.290734	−0.0267642
$119$	5.00996	0.459262
$120$	0	0
$121$	0	0
$122$	0.983561	0.0890475
$123$	−8.59386	−0.774882
$124$	−9.84227	−0.883862
$125$	0	0
$126$	−0.196400	−0.0174967
$127$	−4.53876	−0.402749	−0.201375	−	0.979514i	$-0.564541\pi$
$127$	−4.53876	−0.402749	−0.201375	+	0.979514i	$0.564541\pi$
$128$	−2.65892	−0.235018
$129$	−6.99068	−0.615495
$130$	0	0
$131$	−20.1090	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$131$	−20.1090	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$132$	0	0
$133$	−5.09660	−0.441931
$134$	0.768348	0.0663751
$135$	0	0
$136$	0.715727	0.0613731
$137$	−7.93676	−0.678083	−0.339042	−	0.940771i	$-0.610103\pi$
$137$	−7.93676	−0.678083	−0.339042	+	0.940771i	$0.610103\pi$
$138$	0.680543	0.0579317
$139$	8.27211	0.701631	0.350816	−	0.936445i	$-0.385904\pi$
$139$	8.27211	0.701631	0.350816	+	0.936445i	$0.385904\pi$
$140$	0	0
$141$	−5.75446	−0.484613
$142$	−0.546349	−0.0458486
$143$	0	0
$144$	3.95789	0.329824
$145$	0	0
$146$	−1.15651	−0.0957138
$147$	1.51060	0.124592
$148$	−10.4813	−0.861561
$149$	1.01157	0.0828710	0.0414355	−	0.999141i	$-0.486807\pi$
$149$	1.01157	0.0828710	0.0414355	+	0.999141i	$0.486807\pi$
$150$	0	0
$151$	−22.1738	−1.80448	−0.902241	−	0.431233i	$-0.858079\pi$
$151$	−22.1738	−1.80448	−0.902241	+	0.431233i	$0.858079\pi$
$152$	−0.728104	−0.0590571
$153$	−2.13832	−0.172873
$154$	0	0
$155$	0	0
$156$	−11.1928	−0.896144
$157$	−2.66179	−0.212434	−0.106217	−	0.994343i	$-0.533874\pi$
$157$	−2.66179	−0.212434	−0.106217	+	0.994343i	$0.533874\pi$
$158$	−0.560195	−0.0445667
$159$	10.3207	0.818482
$160$	0	0
$161$	19.0212	1.49908
$162$	0.0838261	0.00658600
$163$	−10.0076	−0.783859	−0.391930	−	0.919995i	$-0.628192\pi$
$163$	−10.0076	−0.783859	−0.391930	+	0.919995i	$0.628192\pi$
$164$	−17.1273	−1.33742
$165$	0	0
$166$	0.988272	0.0767047
$167$	12.4538	0.963705	0.481853	−	0.876252i	$-0.339964\pi$
$167$	12.4538	0.963705	0.481853	+	0.876252i	$0.339964\pi$
$168$	−0.784220	−0.0605039
$169$	18.5411	1.42624
$170$	0	0
$171$	2.17529	0.166349
$172$	−13.9322	−1.06232
$173$	23.0054	1.74907	0.874533	−	0.484966i	$-0.161168\pi$
$173$	23.0054	1.74907	0.874533	+	0.484966i	$0.161168\pi$
$174$	−0.428474	−0.0324825
$175$	0	0
$176$	0	0
$177$	3.46830	0.260693
$178$	0.199788	0.0149747
$179$	9.69004	0.724267	0.362134	−	0.932126i	$-0.382048\pi$
$179$	9.69004	0.724267	0.362134	+	0.932126i	$0.382048\pi$
$180$	0	0
$181$	3.99983	0.297305	0.148653	−	0.988889i	$-0.452506\pi$
$181$	3.99983	0.297305	0.148653	+	0.988889i	$0.452506\pi$
$182$	1.10301	0.0817607
$183$	−11.7334	−0.867355
$184$	2.71739	0.200329
$185$	0	0
$186$	−0.413974	−0.0303540
$187$	0	0
$188$	−11.4685	−0.836425
$189$	2.34295	0.170424
$190$	0	0
$191$	16.7717	1.21356	0.606780	−	0.794870i	$-0.292461\pi$
$191$	16.7717	1.21356	0.606780	+	0.794870i	$0.292461\pi$
$192$	7.83185	0.565215
$193$	15.3963	1.10825	0.554124	−	0.832434i	$-0.313053\pi$
$193$	15.3963	1.10825	0.554124	+	0.832434i	$0.313053\pi$
$194$	−0.176103	−0.0126434
$195$	0	0
$196$	3.01058	0.215042
$197$	10.6667	0.759970	0.379985	−	0.924993i	$-0.375929\pi$
$197$	10.6667	0.759970	0.379985	+	0.924993i	$0.375929\pi$
$198$	0	0
$199$	−11.8473	−0.839833	−0.419917	−	0.907563i	$-0.637941\pi$
$199$	−11.8473	−0.839833	−0.419917	+	0.907563i	$0.637941\pi$
$200$	0	0
$201$	−9.16598	−0.646518
$202$	−0.242789	−0.0170826
$203$	−11.9759	−0.840542
$204$	−4.26160	−0.298372
$205$	0	0
$206$	−0.0968421	−0.00674731
$207$	−8.11851	−0.564276
$208$	−22.2281	−1.54124
$209$	0	0
$210$	0	0
$211$	11.9597	0.823338	0.411669	−	0.911333i	$-0.364946\pi$
$211$	11.9597	0.823338	0.411669	+	0.911333i	$0.364946\pi$
$212$	20.5688	1.41267
$213$	6.51765	0.446582
$214$	1.42393	0.0973381
$215$	0	0
$216$	0.334715	0.0227745
$217$	−11.5706	−0.785464
$218$	0.453820	0.0307366
$219$	13.7966	0.932288
$220$	0	0
$221$	12.0091	0.807819
$222$	−0.440854	−0.0295882
$223$	7.69671	0.515410	0.257705	−	0.966224i	$-0.417034\pi$
$223$	7.69671	0.515410	0.257705	+	0.966224i	$0.417034\pi$
$224$	−2.34577	−0.156733
$225$	0	0
$226$	0.572047	0.0380520
$227$	−11.5028	−0.763467	−0.381733	−	0.924272i	$-0.624673\pi$
$227$	−11.5028	−0.763467	−0.381733	+	0.924272i	$0.624673\pi$
$228$	4.33530	0.287113
$229$	9.13308	0.603531	0.301766	−	0.953382i	$-0.402424\pi$
$229$	9.13308	0.603531	0.301766	+	0.953382i	$0.402424\pi$
$230$	0	0
$231$	0	0
$232$	−1.71088	−0.112325
$233$	−8.68471	−0.568955	−0.284477	−	0.958683i	$-0.591820\pi$
$233$	−8.68471	−0.568955	−0.284477	+	0.958683i	$0.591820\pi$
$234$	−0.470780	−0.0307758
$235$	0	0
$236$	6.91222	0.449947
$237$	6.68282	0.434096
$238$	0.419965	0.0272223
$239$	−16.4990	−1.06723	−0.533615	−	0.845728i	$-0.679167\pi$
$239$	−16.4990	−1.06723	−0.533615	+	0.845728i	$0.679167\pi$
$240$	0	0
$241$	21.7951	1.40395	0.701973	−	0.712204i	$-0.252303\pi$
$241$	21.7951	1.40395	0.701973	+	0.712204i	$0.252303\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−23.3843	−1.49702
$245$	0	0
$246$	−0.720390	−0.0459304
$247$	−12.2168	−0.777335
$248$	−1.65299	−0.104965
$249$	−11.7895	−0.747132
$250$	0	0
$251$	24.5736	1.55107	0.775535	−	0.631305i	$-0.217480\pi$
$251$	24.5736	1.55107	0.775535	+	0.631305i	$0.217480\pi$
$252$	4.66943	0.294146
$253$	0	0
$254$	−0.380466	−0.0238726
$255$	0	0
$256$	15.4408	0.965051
$257$	−26.9620	−1.68184	−0.840921	−	0.541158i	$-0.817986\pi$
$257$	−26.9620	−1.68184	−0.840921	+	0.541158i	$0.817986\pi$
$258$	−0.586001	−0.0364829
$259$	−12.3219	−0.765646
$260$	0	0
$261$	5.11146	0.316392
$262$	−1.68566	−0.104140
$263$	2.31551	0.142780	0.0713901	−	0.997448i	$-0.477256\pi$
$263$	2.31551	0.142780	0.0713901	+	0.997448i	$0.477256\pi$
$264$	0	0
$265$	0	0
$266$	−0.427228	−0.0261950
$267$	−2.38336	−0.145859
$268$	−18.2675	−1.11587
$269$	−21.5276	−1.31256	−0.656282	−	0.754516i	$-0.727872\pi$
$269$	−21.5276	−1.31256	−0.656282	+	0.754516i	$0.727872\pi$
$270$	0	0
$271$	−27.8385	−1.69107	−0.845535	−	0.533920i	$-0.820718\pi$
$271$	−27.8385	−1.69107	−0.845535	+	0.533920i	$0.820718\pi$
$272$	−8.46321	−0.513158
$273$	−13.1583	−0.796379
$274$	−0.665307	−0.0401927
$275$	0	0
$276$	−16.1800	−0.973920
$277$	1.20506	0.0724048	0.0362024	−	0.999344i	$-0.488474\pi$
$277$	1.20506	0.0724048	0.0362024	+	0.999344i	$0.488474\pi$
$278$	0.693419	0.0415885
$279$	4.93848	0.295659
$280$	0	0
$281$	−4.76376	−0.284182	−0.142091	−	0.989854i	$-0.545383\pi$
$281$	−4.76376	−0.284182	−0.142091	+	0.989854i	$0.545383\pi$
$282$	−0.482374	−0.0287250
$283$	15.6008	0.927371	0.463685	−	0.886000i	$-0.346527\pi$
$283$	15.6008	0.927371	0.463685	+	0.886000i	$0.346527\pi$
$284$	12.9895	0.770785
$285$	0	0
$286$	0	0
$287$	−20.1350	−1.18853
$288$	1.00120	0.0589966
$289$	−12.4276	−0.731036
$290$	0	0
$291$	2.10081	0.123152
$292$	27.4962	1.60910
$293$	−28.9754	−1.69276	−0.846382	−	0.532577i	$-0.821224\pi$
$293$	−28.9754	−1.69276	−0.846382	+	0.532577i	$0.821224\pi$
$294$	0.126628	0.00738507
$295$	0	0
$296$	−1.76032	−0.102316
$297$	0	0
$298$	0.0847959	0.00491210
$299$	45.5948	2.63681
$300$	0	0
$301$	−16.3788	−0.944058
$302$	−1.85875	−0.106959
$303$	2.89635	0.166391
$304$	8.60957	0.493793
$305$	0	0
$306$	−0.179247	−0.0102468
$307$	−16.6936	−0.952753	−0.476377	−	0.879241i	$-0.658050\pi$
$307$	−16.6936	−0.952753	−0.476377	+	0.879241i	$0.658050\pi$
$308$	0	0
$309$	1.15527	0.0657213
$310$	0	0
$311$	−7.69585	−0.436392	−0.218196	−	0.975905i	$-0.570017\pi$
$311$	−7.69585	−0.436392	−0.218196	+	0.975905i	$0.570017\pi$
$312$	−1.87981	−0.106423
$313$	−32.3970	−1.83118	−0.915592	−	0.402108i	$-0.868278\pi$
$313$	−32.3970	−1.83118	−0.915592	+	0.402108i	$0.868278\pi$
$314$	−0.223127	−0.0125918
$315$	0	0
$316$	13.3187	0.749234
$317$	−0.202096	−0.0113509	−0.00567543	−	0.999984i	$-0.501807\pi$
$317$	−0.202096	−0.0113509	−0.00567543	+	0.999984i	$0.501807\pi$
$318$	0.865141	0.0485147
$319$	0	0
$320$	0	0
$321$	−16.9868	−0.948108
$322$	1.59448	0.0888567
$323$	−4.65147	−0.258814
$324$	−1.99297	−0.110721
$325$	0	0
$326$	−0.838902	−0.0464625
$327$	−5.41383	−0.299386
$328$	−2.87650	−0.158828
$329$	−13.4824	−0.743309
$330$	0	0
$331$	−11.3828	−0.625656	−0.312828	−	0.949810i	$-0.601276\pi$
$331$	−11.3828	−0.625656	−0.312828	+	0.949810i	$0.601276\pi$
$332$	−23.4962	−1.28952
$333$	5.25915	0.288199
$334$	1.04395	0.0571227
$335$	0	0
$336$	9.27312	0.505890
$337$	18.1256	0.987365	0.493683	−	0.869642i	$-0.335650\pi$
$337$	18.1256	0.987365	0.493683	+	0.869642i	$0.335650\pi$
$338$	1.55423	0.0845388
$339$	−6.82421	−0.370640
$340$	0	0
$341$	0	0
$342$	0.182346	0.00986017
$343$	19.9399	1.07665
$344$	−2.33989	−0.126158
$345$	0	0
$346$	1.92845	0.103674
$347$	17.4879	0.938802	0.469401	−	0.882985i	$-0.344470\pi$
$347$	17.4879	0.938802	0.469401	+	0.882985i	$0.344470\pi$
$348$	10.1870	0.546081
$349$	25.1231	1.34481	0.672405	−	0.740183i	$-0.265261\pi$
$349$	25.1231	1.34481	0.672405	+	0.740183i	$0.265261\pi$
$350$	0	0
$351$	5.61615	0.299768
$352$	0	0
$353$	21.2148	1.12915	0.564576	−	0.825381i	$-0.309040\pi$
$353$	21.2148	1.12915	0.564576	+	0.825381i	$0.309040\pi$
$354$	0.290734	0.0154523
$355$	0	0
$356$	−4.74997	−0.251748
$357$	−5.00996	−0.265155
$358$	0.812278	0.0429302
$359$	−21.2283	−1.12039	−0.560193	−	0.828362i	$-0.689273\pi$
$359$	−21.2283	−1.12039	−0.560193	+	0.828362i	$0.689273\pi$
$360$	0	0
$361$	−14.2681	−0.750952
$362$	0.335290	0.0176225
$363$	0	0
$364$	−26.2242	−1.37452
$365$	0	0
$366$	−0.983561	−0.0514116
$367$	−24.6367	−1.28603	−0.643014	−	0.765855i	$-0.722316\pi$
$367$	−24.6367	−1.28603	−0.643014	+	0.765855i	$0.722316\pi$
$368$	−32.1322	−1.67501
$369$	8.59386	0.447379
$370$	0	0
$371$	24.1808	1.25540
$372$	9.84227	0.510298
$373$	−19.5612	−1.01284	−0.506420	−	0.862287i	$-0.669031\pi$
$373$	−19.5612	−1.01284	−0.506420	+	0.862287i	$0.669031\pi$
$374$	0	0
$375$	0	0
$376$	−1.92611	−0.0993314
$377$	−28.7067	−1.47847
$378$	0.196400	0.0101017
$379$	0.483951	0.0248589	0.0124294	−	0.999923i	$-0.496043\pi$
$379$	0.483951	0.0248589	0.0124294	+	0.999923i	$0.496043\pi$
$380$	0	0
$381$	4.53876	0.232528
$382$	1.40591	0.0719325
$383$	−1.78891	−0.0914088	−0.0457044	−	0.998955i	$-0.514553\pi$
$383$	−1.78891	−0.0914088	−0.0457044	+	0.998955i	$0.514553\pi$
$384$	2.65892	0.135688
$385$	0	0
$386$	1.29061	0.0656903
$387$	6.99068	0.355356
$388$	4.18686	0.212556
$389$	−37.0055	−1.87626	−0.938128	−	0.346290i	$-0.887441\pi$
$389$	−37.0055	−1.87626	−0.938128	+	0.346290i	$0.887441\pi$
$390$	0	0
$391$	17.3599	0.877930
$392$	0.505621	0.0255377
$393$	20.1090	1.01437
$394$	0.894146	0.0450464
$395$	0	0
$396$	0	0
$397$	−39.4552	−1.98020	−0.990100	−	0.140366i	$-0.955172\pi$
$397$	−39.4552	−1.98020	−0.990100	+	0.140366i	$0.955172\pi$
$398$	−0.993113	−0.0497803
$399$	5.09660	0.255149
$400$	0	0
$401$	−30.0905	−1.50265	−0.751324	−	0.659934i	$-0.770584\pi$
$401$	−30.0905	−1.50265	−0.751324	+	0.659934i	$0.770584\pi$
$402$	−0.768348	−0.0383217
$403$	−27.7352	−1.38159
$404$	5.77234	0.287185
$405$	0	0
$406$	−1.00389	−0.0498223
$407$	0	0
$408$	−0.715727	−0.0354338
$409$	−20.8161	−1.02929	−0.514645	−	0.857403i	$-0.672076\pi$
$409$	−20.8161	−1.02929	−0.514645	+	0.857403i	$0.672076\pi$
$410$	0	0
$411$	7.93676	0.391491
$412$	2.30243	0.113433
$413$	8.12603	0.399856
$414$	−0.680543	−0.0334469
$415$	0	0
$416$	−5.62291	−0.275686
$417$	−8.27211	−0.405087
$418$	0	0
$419$	14.5892	0.712731	0.356366	−	0.934347i	$-0.384016\pi$
$419$	14.5892	0.712731	0.356366	+	0.934347i	$0.384016\pi$
$420$	0	0
$421$	−15.7861	−0.769365	−0.384683	−	0.923049i	$-0.625689\pi$
$421$	−15.7861	−0.769365	−0.384683	+	0.923049i	$0.625689\pi$
$422$	1.00253	0.0488026
$423$	5.75446	0.279791
$424$	3.45448	0.167765
$425$	0	0
$426$	0.546349	0.0264707
$427$	−27.4906	−1.33036
$428$	−33.8541	−1.63640
$429$	0	0
$430$	0	0
$431$	29.3718	1.41479	0.707396	−	0.706818i	$-0.249870\pi$
$431$	29.3718	1.41479	0.707396	+	0.706818i	$0.249870\pi$
$432$	−3.95789	−0.190424
$433$	14.5467	0.699068	0.349534	−	0.936924i	$-0.386340\pi$
$433$	14.5467	0.699068	0.349534	+	0.936924i	$0.386340\pi$
$434$	−0.969919	−0.0465576
$435$	0	0
$436$	−10.7896	−0.516729
$437$	−17.6602	−0.844800
$438$	1.15651	0.0552604
$439$	−10.9210	−0.521232	−0.260616	−	0.965443i	$-0.583926\pi$
$439$	−10.9210	−0.521232	−0.260616	+	0.965443i	$0.583926\pi$
$440$	0	0
$441$	−1.51060	−0.0719333
$442$	1.00668	0.0478827
$443$	12.8567	0.610842	0.305421	−	0.952217i	$-0.401203\pi$
$443$	12.8567	0.610842	0.305421	+	0.952217i	$0.401203\pi$
$444$	10.4813	0.497422
$445$	0	0
$446$	0.645185	0.0305504
$447$	−1.01157	−0.0478456
$448$	18.3496	0.866938
$449$	2.39741	0.113141	0.0565704	−	0.998399i	$-0.481983\pi$
$449$	2.39741	0.113141	0.0565704	+	0.998399i	$0.481983\pi$
$450$	0	0
$451$	0	0
$452$	−13.6005	−0.639712
$453$	22.1738	1.04182
$454$	−0.964234	−0.0452537
$455$	0	0
$456$	0.728104	0.0340966
$457$	−12.9116	−0.603978	−0.301989	−	0.953311i	$-0.597651\pi$
$457$	−12.9116	−0.603978	−0.301989	+	0.953311i	$0.597651\pi$
$458$	0.765591	0.0357737
$459$	2.13832	0.0998080
$460$	0	0
$461$	−11.1109	−0.517487	−0.258744	−	0.965946i	$-0.583308\pi$
$461$	−11.1109	−0.517487	−0.258744	+	0.965946i	$0.583308\pi$
$462$	0	0
$463$	3.85587	0.179198	0.0895988	−	0.995978i	$-0.471442\pi$
$463$	3.85587	0.179198	0.0895988	+	0.995978i	$0.471442\pi$
$464$	20.2306	0.939182
$465$	0	0
$466$	−0.728006	−0.0337242
$467$	−27.2657	−1.26170	−0.630852	−	0.775903i	$-0.717295\pi$
$467$	−27.2657	−1.26170	−0.630852	+	0.775903i	$0.717295\pi$
$468$	11.1928	0.517389
$469$	−21.4754	−0.991642
$470$	0	0
$471$	2.66179	0.122649
$472$	1.16089	0.0534344
$473$	0	0
$474$	0.560195	0.0257306
$475$	0	0
$476$	−9.98471	−0.457649
$477$	−10.3207	−0.472551
$478$	−1.38304	−0.0632589
$479$	−3.20198	−0.146302	−0.0731511	−	0.997321i	$-0.523306\pi$
$479$	−3.20198	−0.146302	−0.0731511	+	0.997321i	$0.523306\pi$
$480$	0	0
$481$	−29.5361	−1.34673
$482$	1.82700	0.0832174
$483$	−19.0212	−0.865497
$484$	0	0
$485$	0	0
$486$	−0.0838261	−0.00380243
$487$	−18.0165	−0.816406	−0.408203	−	0.912891i	$-0.633844\pi$
$487$	−18.0165	−0.816406	−0.408203	+	0.912891i	$0.633844\pi$
$488$	−3.92733	−0.177782
$489$	10.0076	0.452561
$490$	0	0
$491$	−9.54166	−0.430609	−0.215305	−	0.976547i	$-0.569074\pi$
$491$	−9.54166	−0.430609	−0.215305	+	0.976547i	$0.569074\pi$
$492$	17.1273	0.772160
$493$	−10.9299	−0.492259
$494$	−1.02408	−0.0460757
$495$	0	0
$496$	19.5460	0.877640
$497$	15.2705	0.684976
$498$	−0.988272	−0.0442855
$499$	22.0383	0.986568	0.493284	−	0.869868i	$-0.335796\pi$
$499$	22.0383	0.986568	0.493284	+	0.869868i	$0.335796\pi$
$500$	0	0
$501$	−12.4538	−0.556396
$502$	2.05991	0.0919381
$503$	22.6392	1.00943	0.504715	−	0.863286i	$-0.331597\pi$
$503$	22.6392	1.00943	0.504715	+	0.863286i	$0.331597\pi$
$504$	0.784220	0.0349319
$505$	0	0
$506$	0	0
$507$	−18.5411	−0.823439
$508$	9.04562	0.401334
$509$	−6.61253	−0.293095	−0.146548	−	0.989204i	$-0.546816\pi$
$509$	−6.61253	−0.293095	−0.146548	+	0.989204i	$0.546816\pi$
$510$	0	0
$511$	32.3247	1.42996
$512$	6.61219	0.292220
$513$	−2.17529	−0.0960416
$514$	−2.26012	−0.0996895
$515$	0	0
$516$	13.9322	0.613333
$517$	0	0
$518$	−1.03290	−0.0453829
$519$	−23.0054	−1.00982
$520$	0	0
$521$	−22.0018	−0.963915	−0.481958	−	0.876195i	$-0.660074\pi$
$521$	−22.0018	−0.963915	−0.481958	+	0.876195i	$0.660074\pi$
$522$	0.428474	0.0187538
$523$	−5.37244	−0.234920	−0.117460	−	0.993078i	$-0.537475\pi$
$523$	−5.37244	−0.234920	−0.117460	+	0.993078i	$0.537475\pi$
$524$	40.0768	1.75076
$525$	0	0
$526$	0.194100	0.00846315
$527$	−10.5600	−0.460002
$528$	0	0
$529$	42.9103	1.86566
$530$	0	0
$531$	−3.46830	−0.150511
$532$	10.1574	0.440379
$533$	−48.2644	−2.09056
$534$	−0.199788	−0.00864565
$535$	0	0
$536$	−3.06799	−0.132517
$537$	−9.69004	−0.418156
$538$	−1.80458	−0.0778009
$539$	0	0
$540$	0	0
$541$	−1.73983	−0.0748013	−0.0374006	−	0.999300i	$-0.511908\pi$
$541$	−1.73983	−0.0748013	−0.0374006	+	0.999300i	$0.511908\pi$
$542$	−2.33360	−0.100236
$543$	−3.99983	−0.171649
$544$	−2.14089	−0.0917900
$545$	0	0
$546$	−1.10301	−0.0472045
$547$	−0.478639	−0.0204651	−0.0102326	−	0.999948i	$-0.503257\pi$
$547$	−0.478639	−0.0204651	−0.0102326	+	0.999948i	$0.503257\pi$
$548$	15.8177	0.675701
$549$	11.7334	0.500767
$550$	0	0
$551$	11.1189	0.473683
$552$	−2.71739	−0.115660
$553$	15.6575	0.665824
$554$	0.101015	0.00429172
$555$	0	0
$556$	−16.4861	−0.699166
$557$	12.2937	0.520899	0.260450	−	0.965487i	$-0.416129\pi$
$557$	12.2937	0.520899	0.260450	+	0.965487i	$0.416129\pi$
$558$	0.413974	0.0175249
$559$	−39.2607	−1.66055
$560$	0	0
$561$	0	0
$562$	−0.399328	−0.0168446
$563$	−7.90408	−0.333117	−0.166559	−	0.986032i	$-0.553266\pi$
$563$	−7.90408	−0.333117	−0.166559	+	0.986032i	$0.553266\pi$
$564$	11.4685	0.482910
$565$	0	0
$566$	1.30775	0.0549690
$567$	−2.34295	−0.0983945
$568$	2.18156	0.0915361
$569$	−32.6871	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$569$	−32.6871	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$570$	0	0
$571$	9.01812	0.377397	0.188698	−	0.982035i	$-0.439573\pi$
$571$	9.01812	0.377397	0.188698	+	0.982035i	$0.439573\pi$
$572$	0	0
$573$	−16.7717	−0.700649
$574$	−1.68783	−0.0704489
$575$	0	0
$576$	−7.83185	−0.326327
$577$	−19.9157	−0.829101	−0.414551	−	0.910026i	$-0.636061\pi$
$577$	−19.9157	−0.829101	−0.414551	+	0.910026i	$0.636061\pi$
$578$	−1.04176	−0.0433314
$579$	−15.3963	−0.639848
$580$	0	0
$581$	−27.6223	−1.14597
$582$	0.176103	0.00729970
$583$	0	0
$584$	4.61793	0.191091
$585$	0	0
$586$	−2.42890	−0.100337
$587$	−39.4082	−1.62655	−0.813274	−	0.581881i	$-0.802317\pi$
$587$	−39.4082	−1.62655	−0.813274	+	0.581881i	$0.802317\pi$
$588$	−3.01058	−0.124154
$589$	10.7427	0.442644
$590$	0	0
$591$	−10.6667	−0.438769
$592$	20.8151	0.855496
$593$	−14.3161	−0.587892	−0.293946	−	0.955822i	$-0.594969\pi$
$593$	−14.3161	−0.587892	−0.293946	+	0.955822i	$0.594969\pi$
$594$	0	0
$595$	0	0
$596$	−2.01603	−0.0825799
$597$	11.8473	0.484878
$598$	3.82203	0.156294
$599$	12.4850	0.510122	0.255061	−	0.966925i	$-0.417905\pi$
$599$	12.4850	0.510122	0.255061	+	0.966925i	$0.417905\pi$
$600$	0	0
$601$	30.2232	1.23283	0.616415	−	0.787422i	$-0.288584\pi$
$601$	30.2232	1.23283	0.616415	+	0.787422i	$0.288584\pi$
$602$	−1.37297	−0.0559581
$603$	9.16598	0.373267
$604$	44.1919	1.79814
$605$	0	0
$606$	0.242789	0.00986265
$607$	17.7783	0.721599	0.360800	−	0.932643i	$-0.382504\pi$
$607$	17.7783	0.721599	0.360800	+	0.932643i	$0.382504\pi$
$608$	2.17792	0.0883262
$609$	11.9759	0.485287
$610$	0	0
$611$	−32.3179	−1.30744
$612$	4.26160	0.172265
$613$	32.7940	1.32454	0.662268	−	0.749267i	$-0.269594\pi$
$613$	32.7940	1.32454	0.662268	+	0.749267i	$0.269594\pi$
$614$	−1.39936	−0.0564735
$615$	0	0
$616$	0	0
$617$	32.8457	1.32232	0.661159	−	0.750246i	$-0.270065\pi$
$617$	32.8457	1.32232	0.661159	+	0.750246i	$0.270065\pi$
$618$	0.0968421	0.00389556
$619$	13.7550	0.552860	0.276430	−	0.961034i	$-0.410849\pi$
$619$	13.7550	0.552860	0.276430	+	0.961034i	$0.410849\pi$
$620$	0	0
$621$	8.11851	0.325785
$622$	−0.645113	−0.0258667
$623$	−5.58408	−0.223721
$624$	22.2281	0.889835
$625$	0	0
$626$	−2.71571	−0.108542
$627$	0	0
$628$	5.30487	0.211687
$629$	−11.2457	−0.448396
$630$	0	0
$631$	17.4644	0.695246	0.347623	−	0.937634i	$-0.386989\pi$
$631$	17.4644	0.695246	0.347623	+	0.937634i	$0.386989\pi$
$632$	2.23684	0.0889768
$633$	−11.9597	−0.475355
$634$	−0.0169409	−0.000672810 0
$635$	0	0
$636$	−20.5688	−0.815606
$637$	8.48375	0.336138
$638$	0	0
$639$	−6.51765	−0.257834
$640$	0	0
$641$	20.0203	0.790754	0.395377	−	0.918519i	$-0.370614\pi$
$641$	20.0203	0.790754	0.395377	+	0.918519i	$0.370614\pi$
$642$	−1.42393	−0.0561982
$643$	3.13546	0.123650	0.0618252	−	0.998087i	$-0.480308\pi$
$643$	3.13546	0.123650	0.0618252	+	0.998087i	$0.480308\pi$
$644$	−37.9088	−1.49382
$645$	0	0
$646$	−0.389914	−0.0153410
$647$	−21.9166	−0.861629	−0.430814	−	0.902441i	$-0.641774\pi$
$647$	−21.9166	−0.861629	−0.430814	+	0.902441i	$0.641774\pi$
$648$	−0.334715	−0.0131489
$649$	0	0
$650$	0	0
$651$	11.5706	0.453488
$652$	19.9450	0.781105
$653$	−29.9875	−1.17350	−0.586751	−	0.809767i	$-0.699593\pi$
$653$	−29.9875	−1.17350	−0.586751	+	0.809767i	$0.699593\pi$
$654$	−0.453820	−0.0177458
$655$	0	0
$656$	34.0135	1.32801
$657$	−13.7966	−0.538257
$658$	−1.13018	−0.0440589
$659$	−39.5057	−1.53892	−0.769461	−	0.638694i	$-0.779475\pi$
$659$	−39.5057	−1.53892	−0.769461	+	0.638694i	$0.779475\pi$
$660$	0	0
$661$	−8.70467	−0.338572	−0.169286	−	0.985567i	$-0.554146\pi$
$661$	−8.70467	−0.338572	−0.169286	+	0.985567i	$0.554146\pi$
$662$	−0.954177	−0.0370851
$663$	−12.0091	−0.466394
$664$	−3.94614	−0.153140
$665$	0	0
$666$	0.440854	0.0170827
$667$	−41.4975	−1.60679
$668$	−24.8201	−0.960319
$669$	−7.69671	−0.297572
$670$	0	0
$671$	0	0
$672$	2.34577	0.0904900
$673$	−36.4886	−1.40653	−0.703265	−	0.710927i	$-0.748275\pi$
$673$	−36.4886	−1.40653	−0.703265	+	0.710927i	$0.748275\pi$
$674$	1.51940	0.0585251
$675$	0	0
$676$	−36.9519	−1.42123
$677$	−9.36270	−0.359838	−0.179919	−	0.983681i	$-0.557584\pi$
$677$	−9.36270	−0.359838	−0.179919	+	0.983681i	$0.557584\pi$
$678$	−0.572047	−0.0219693
$679$	4.92209	0.188893
$680$	0	0
$681$	11.5028	0.440788
$682$	0	0
$683$	5.14769	0.196971	0.0984854	−	0.995139i	$-0.468600\pi$
$683$	5.14769	0.196971	0.0984854	+	0.995139i	$0.468600\pi$
$684$	−4.33530	−0.165765
$685$	0	0
$686$	1.67148	0.0638175
$687$	−9.13308	−0.348449
$688$	27.6683	1.05485
$689$	57.9623	2.20819
$690$	0	0
$691$	−9.21857	−0.350691	−0.175345	−	0.984507i	$-0.556104\pi$
$691$	−9.21857	−0.350691	−0.175345	+	0.984507i	$0.556104\pi$
$692$	−45.8491	−1.74292
$693$	0	0
$694$	1.46595	0.0556465
$695$	0	0
$696$	1.71088	0.0648509
$697$	−18.3764	−0.696055
$698$	2.10597	0.0797123
$699$	8.68471	0.328486
$700$	0	0
$701$	−32.0934	−1.21215	−0.606075	−	0.795408i	$-0.707257\pi$
$701$	−32.0934	−1.21215	−0.606075	+	0.795408i	$0.707257\pi$
$702$	0.470780	0.0177684
$703$	11.4402	0.431475
$704$	0	0
$705$	0	0
$706$	1.77836	0.0669293
$707$	6.78599	0.255213
$708$	−6.91222	−0.259777
$709$	17.5173	0.657875	0.328938	−	0.944352i	$-0.393309\pi$
$709$	17.5173	0.657875	0.328938	+	0.944352i	$0.393309\pi$
$710$	0	0
$711$	−6.68282	−0.250625
$712$	−0.797746	−0.0298968
$713$	−40.0932	−1.50150
$714$	−0.419965	−0.0157168
$715$	0	0
$716$	−19.3120	−0.721723
$717$	16.4990	0.616165
$718$	−1.77949	−0.0664098
$719$	−37.7054	−1.40617	−0.703087	−	0.711104i	$-0.748195\pi$
$719$	−37.7054	−1.40617	−0.703087	+	0.711104i	$0.748195\pi$
$720$	0	0
$721$	2.70675	0.100805
$722$	−1.19604	−0.0445119
$723$	−21.7951	−0.810568
$724$	−7.97156	−0.296261
$725$	0	0
$726$	0	0
$727$	33.2685	1.23386	0.616931	−	0.787017i	$-0.288376\pi$
$727$	33.2685	1.23386	0.616931	+	0.787017i	$0.288376\pi$
$728$	−4.40430	−0.163234
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.9483	−0.552882
$732$	23.3843	0.864307
$733$	−6.67119	−0.246406	−0.123203	−	0.992381i	$-0.539317\pi$
$733$	−6.67119	−0.246406	−0.123203	+	0.992381i	$0.539317\pi$
$734$	−2.06520	−0.0762280
$735$	0	0
$736$	−8.12830	−0.299613
$737$	0	0
$738$	0.720390	0.0265179
$739$	41.2237	1.51644	0.758219	−	0.652000i	$-0.226070\pi$
$739$	41.2237	1.51644	0.758219	+	0.652000i	$0.226070\pi$
$740$	0	0
$741$	12.2168	0.448794
$742$	2.02698	0.0744127
$743$	12.1620	0.446181	0.223090	−	0.974798i	$-0.428386\pi$
$743$	12.1620	0.446181	0.223090	+	0.974798i	$0.428386\pi$
$744$	1.65299	0.0606014
$745$	0	0
$746$	−1.63974	−0.0600351
$747$	11.7895	0.431357
$748$	0	0
$749$	−39.7991	−1.45423
$750$	0	0
$751$	15.5528	0.567529	0.283764	−	0.958894i	$-0.408417\pi$
$751$	15.5528	0.567529	0.283764	+	0.958894i	$0.408417\pi$
$752$	22.7755	0.830538
$753$	−24.5736	−0.895510
$754$	−2.40637	−0.0876349
$755$	0	0
$756$	−4.66943	−0.169826
$757$	24.3108	0.883592	0.441796	−	0.897116i	$-0.354341\pi$
$757$	24.3108	0.883592	0.441796	+	0.897116i	$0.354341\pi$
$758$	0.0405677	0.00147349
$759$	0	0
$760$	0	0
$761$	19.3226	0.700442	0.350221	−	0.936667i	$-0.386106\pi$
$761$	19.3226	0.700442	0.350221	+	0.936667i	$0.386106\pi$
$762$	0.380466	0.0137828
$763$	−12.6843	−0.459203
$764$	−33.4256	−1.20930
$765$	0	0
$766$	−0.149957	−0.00541817
$767$	19.4785	0.703326
$768$	−15.4408	−0.557172
$769$	31.0283	1.11891	0.559454	−	0.828861i	$-0.311011\pi$
$769$	31.0283	1.11891	0.559454	+	0.828861i	$0.311011\pi$
$770$	0	0
$771$	26.9620	0.971012
$772$	−30.6844	−1.10435
$773$	−32.2032	−1.15827	−0.579134	−	0.815232i	$-0.696609\pi$
$773$	−32.2032	−1.15827	−0.579134	+	0.815232i	$0.696609\pi$
$774$	0.586001	0.0210634
$775$	0	0
$776$	0.703174	0.0252425
$777$	12.3219	0.442046
$778$	−3.10203	−0.111213
$779$	18.6942	0.669789
$780$	0	0
$781$	0	0
$782$	1.45522	0.0520384
$783$	−5.11146	−0.182669
$784$	−5.97878	−0.213528
$785$	0	0
$786$	1.68566	0.0601255
$787$	7.85951	0.280161	0.140081	−	0.990140i	$-0.455264\pi$
$787$	7.85951	0.280161	0.140081	+	0.990140i	$0.455264\pi$
$788$	−21.2584	−0.757300
$789$	−2.31551	−0.0824342
$790$	0	0
$791$	−15.9888	−0.568495
$792$	0	0
$793$	−65.8962	−2.34004
$794$	−3.30737	−0.117374
$795$	0	0
$796$	23.6114	0.836882
$797$	34.5079	1.22233	0.611167	−	0.791502i	$-0.290700\pi$
$797$	34.5079	1.22233	0.611167	+	0.791502i	$0.290700\pi$
$798$	0.427228	0.0151237
$799$	−12.3049	−0.435314
$800$	0	0
$801$	2.38336	0.0842118
$802$	−2.52237	−0.0890679
$803$	0	0
$804$	18.2675	0.644247
$805$	0	0
$806$	−2.32494	−0.0818924
$807$	21.5276	0.757809
$808$	0.969452	0.0341052
$809$	40.2902	1.41653	0.708264	−	0.705948i	$-0.249479\pi$
$809$	40.2902	1.41653	0.708264	+	0.705948i	$0.249479\pi$
$810$	0	0
$811$	34.7488	1.22020	0.610098	−	0.792326i	$-0.291130\pi$
$811$	34.7488	1.22020	0.610098	+	0.792326i	$0.291130\pi$
$812$	23.8676	0.837589
$813$	27.8385	0.976340
$814$	0	0
$815$	0	0
$816$	8.46321	0.296272
$817$	15.2068	0.532018
$818$	−1.74493	−0.0610102
$819$	13.1583	0.459789
$820$	0	0
$821$	−35.2735	−1.23105	−0.615527	−	0.788116i	$-0.711057\pi$
$821$	−35.2735	−1.23105	−0.615527	+	0.788116i	$0.711057\pi$
$822$	0.665307	0.0232053
$823$	−21.4792	−0.748717	−0.374359	−	0.927284i	$-0.622137\pi$
$823$	−21.4792	−0.748717	−0.374359	+	0.927284i	$0.622137\pi$
$824$	0.386688	0.0134709
$825$	0	0
$826$	0.681174	0.0237011
$827$	−1.74047	−0.0605222	−0.0302611	−	0.999542i	$-0.509634\pi$
$827$	−1.74047	−0.0605222	−0.0302611	+	0.999542i	$0.509634\pi$
$828$	16.1800	0.562293
$829$	−7.18450	−0.249528	−0.124764	−	0.992186i	$-0.539817\pi$
$829$	−7.18450	−0.249528	−0.124764	+	0.992186i	$0.539817\pi$
$830$	0	0
$831$	−1.20506	−0.0418029
$832$	43.9848	1.52490
$833$	3.23014	0.111918
$834$	−0.693419	−0.0240111
$835$	0	0
$836$	0	0
$837$	−4.93848	−0.170699
$838$	1.22296	0.0422464
$839$	−37.7646	−1.30378	−0.651889	−	0.758315i	$-0.726023\pi$
$839$	−37.7646	−1.30378	−0.651889	+	0.758315i	$0.726023\pi$
$840$	0	0
$841$	−2.87295	−0.0990674
$842$	−1.32328	−0.0456033
$843$	4.76376	0.164073
$844$	−23.8353	−0.820446
$845$	0	0
$846$	0.482374	0.0165844
$847$	0	0
$848$	−40.8480	−1.40273
$849$	−15.6008	−0.535418
$850$	0	0
$851$	−42.6964	−1.46362
$852$	−12.9895	−0.445013
$853$	13.3719	0.457845	0.228923	−	0.973445i	$-0.426480\pi$
$853$	13.3719	0.457845	0.228923	+	0.973445i	$0.426480\pi$
$854$	−2.30443	−0.0788560
$855$	0	0
$856$	−5.68573	−0.194334
$857$	−10.0571	−0.343544	−0.171772	−	0.985137i	$-0.554949\pi$
$857$	−10.0571	−0.343544	−0.171772	+	0.985137i	$0.554949\pi$
$858$	0	0
$859$	33.0078	1.12621	0.563106	−	0.826385i	$-0.309606\pi$
$859$	33.0078	1.12621	0.563106	+	0.826385i	$0.309606\pi$
$860$	0	0
$861$	20.1350	0.686198
$862$	2.46213	0.0838603
$863$	43.9961	1.49764	0.748822	−	0.662771i	$-0.230620\pi$
$863$	43.9961	1.49764	0.748822	+	0.662771i	$0.230620\pi$
$864$	−1.00120	−0.0340617
$865$	0	0
$866$	1.21939	0.0414365
$867$	12.4276	0.422064
$868$	23.0599	0.782704
$869$	0	0
$870$	0	0
$871$	−51.4775	−1.74425
$872$	−1.81209	−0.0613652
$873$	−2.10081	−0.0711017
$874$	−1.48038	−0.0500747
$875$	0	0
$876$	−27.4962	−0.929012
$877$	38.6032	1.30354	0.651769	−	0.758418i	$-0.274027\pi$
$877$	38.6032	1.30354	0.651769	+	0.758418i	$0.274027\pi$
$878$	−0.915467	−0.0308955
$879$	28.9754	0.977318
$880$	0	0
$881$	15.4513	0.520568	0.260284	−	0.965532i	$-0.416184\pi$
$881$	15.4513	0.520568	0.260284	+	0.965532i	$0.416184\pi$
$882$	−0.126628	−0.00426377
$883$	−8.88816	−0.299111	−0.149555	−	0.988753i	$-0.547784\pi$
$883$	−8.88816	−0.299111	−0.149555	+	0.988753i	$0.547784\pi$
$884$	−23.9338	−0.804981
$885$	0	0
$886$	1.07773	0.0362071
$887$	−38.5115	−1.29309	−0.646545	−	0.762876i	$-0.723787\pi$
$887$	−38.5115	−1.29309	−0.646545	+	0.762876i	$0.723787\pi$
$888$	1.76032	0.0590724
$889$	10.6341	0.356655
$890$	0	0
$891$	0	0
$892$	−15.3393	−0.513599
$893$	12.5177	0.418887
$894$	−0.0847959	−0.00283600
$895$	0	0
$896$	6.22972	0.208120
$897$	−45.5948	−1.52236
$898$	0.200966	0.00670631
$899$	25.2429	0.841897
$900$	0	0
$901$	22.0688	0.735219
$902$	0	0
$903$	16.3788	0.545052
$904$	−2.28417	−0.0759703
$905$	0	0
$906$	1.85875	0.0617527
$907$	−37.3893	−1.24149	−0.620745	−	0.784012i	$-0.713170\pi$
$907$	−37.3893	−1.24149	−0.620745	+	0.784012i	$0.713170\pi$
$908$	22.9248	0.760785
$909$	−2.89635	−0.0960658
$910$	0	0
$911$	−5.84881	−0.193780	−0.0968899	−	0.995295i	$-0.530889\pi$
$911$	−5.84881	−0.193780	−0.0968899	+	0.995295i	$0.530889\pi$
$912$	−8.60957	−0.285092
$913$	0	0
$914$	−1.08233	−0.0358002
$915$	0	0
$916$	−18.2020	−0.601411
$917$	47.1144	1.55585
$918$	0.179247	0.00591602
$919$	−22.6068	−0.745730	−0.372865	−	0.927886i	$-0.621624\pi$
$919$	−22.6068	−0.745730	−0.372865	+	0.927886i	$0.621624\pi$
$920$	0	0
$921$	16.6936	0.550072
$922$	−0.931385	−0.0306735
$923$	36.6041	1.20484
$924$	0	0
$925$	0	0
$926$	0.323223	0.0106218
$927$	−1.15527	−0.0379442
$928$	5.11762	0.167994
$929$	−38.6656	−1.26858	−0.634289	−	0.773096i	$-0.718707\pi$
$929$	−38.6656	−1.26858	−0.634289	+	0.773096i	$0.718707\pi$
$930$	0	0
$931$	−3.28600	−0.107694
$932$	17.3084	0.566956
$933$	7.69585	0.251951
$934$	−2.28557	−0.0747862
$935$	0	0
$936$	1.87981	0.0614435
$937$	36.8358	1.20337	0.601686	−	0.798733i	$-0.294496\pi$
$937$	36.8358	1.20337	0.601686	+	0.798733i	$0.294496\pi$
$938$	−1.80020	−0.0587786
$939$	32.3970	1.05724
$940$	0	0
$941$	22.6220	0.737457	0.368728	−	0.929537i	$-0.379793\pi$
$941$	22.6220	0.737457	0.368728	+	0.929537i	$0.379793\pi$
$942$	0.223127	0.00726988
$943$	−69.7694	−2.27200
$944$	−13.7271	−0.446780
$945$	0	0
$946$	0	0
$947$	22.8202	0.741558	0.370779	−	0.928721i	$-0.379091\pi$
$947$	22.8202	0.741558	0.370779	+	0.928721i	$0.379091\pi$
$948$	−13.3187	−0.432571
$949$	77.4837	2.51523
$950$	0	0
$951$	0.202096	0.00655342
$952$	−1.67691	−0.0543490
$953$	2.20673	0.0714830	0.0357415	−	0.999361i	$-0.488621\pi$
$953$	2.20673	0.0714830	0.0357415	+	0.999361i	$0.488621\pi$
$954$	−0.865141	−0.0280100
$955$	0	0
$956$	32.8820	1.06348
$957$	0	0
$958$	−0.268409	−0.00867191
$959$	18.5954	0.600477
$960$	0	0
$961$	−6.61138	−0.213270
$962$	−2.47590	−0.0798262
$963$	16.9868	0.547391
$964$	−43.4370	−1.39901
$965$	0	0
$966$	−1.59448	−0.0513014
$967$	23.8146	0.765825	0.382913	−	0.923785i	$-0.374921\pi$
$967$	23.8146	0.765825	0.382913	+	0.923785i	$0.374921\pi$
$968$	0	0
$969$	4.65147	0.149427
$970$	0	0
$971$	−28.7416	−0.922362	−0.461181	−	0.887306i	$-0.652574\pi$
$971$	−28.7416	−0.922362	−0.461181	+	0.887306i	$0.652574\pi$
$972$	1.99297	0.0639246
$973$	−19.3811	−0.621330
$974$	−1.51025	−0.0483917
$975$	0	0
$976$	46.4393	1.48649
$977$	−32.9635	−1.05460	−0.527298	−	0.849680i	$-0.676795\pi$
$977$	−32.9635	−1.05460	−0.527298	+	0.849680i	$0.676795\pi$
$978$	0.838902	0.0268251
$979$	0	0
$980$	0	0
$981$	5.41383	0.172850
$982$	−0.799840	−0.0255239
$983$	21.2431	0.677548	0.338774	−	0.940868i	$-0.389988\pi$
$983$	21.2431	0.677548	0.338774	+	0.940868i	$0.389988\pi$
$984$	2.87650	0.0916994
$985$	0	0
$986$	−0.916212	−0.0291781
$987$	13.4824	0.429149
$988$	24.3477	0.774604
$989$	−56.7539	−1.80467
$990$	0	0
$991$	−18.0931	−0.574746	−0.287373	−	0.957819i	$-0.592782\pi$
$991$	−18.0931	−0.574746	−0.287373	+	0.957819i	$0.592782\pi$
$992$	4.94443	0.156986
$993$	11.3828	0.361223
$994$	1.28007	0.0406013
$995$	0	0
$996$	23.4962	0.744507
$997$	−21.8967	−0.693475	−0.346737	−	0.937962i	$-0.612710\pi$
$997$	−21.8967	−0.693475	−0.346737	+	0.937962i	$0.612710\pi$
$998$	1.84738	0.0584778
$999$	−5.25915	−0.166392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.do.1.4		6
5.2	odd	4		1815.2.c.i.364.7	yes	12
5.3	odd	4		1815.2.c.i.364.6	yes	12
5.4	even	2		9075.2.a.ds.1.3		6
11.10	odd	2		9075.2.a.dr.1.3		6
55.32	even	4		1815.2.c.h.364.6	✓	12
55.43	even	4		1815.2.c.h.364.7	yes	12
55.54	odd	2		9075.2.a.dp.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.c.h.364.6	✓	12	55.32	even	4
1815.2.c.h.364.7	yes	12	55.43	even	4
1815.2.c.i.364.6	yes	12	5.3	odd	4
1815.2.c.i.364.7	yes	12	5.2	odd	4
9075.2.a.do.1.4		6	1.1	even	1	trivial
9075.2.a.dp.1.4		6	55.54	odd	2
9075.2.a.dr.1.3		6	11.10	odd	2
9075.2.a.ds.1.3		6	5.4	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0838261 q^{2} -1.00000 q^{3} -1.99297 q^{4} -0.0838261 q^{6} -2.34295 q^{7} -0.334715 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0838261 q^{2} -1.00000 q^{3} -1.99297 q^{4} -0.0838261 q^{6} -2.34295 q^{7} -0.334715 q^{8} +1.00000 q^{9} +1.99297 q^{12} -5.61615 q^{13} -0.196400 q^{14} +3.95789 q^{16} -2.13832 q^{17} +0.0838261 q^{18} +2.17529 q^{19} +2.34295 q^{21} -8.11851 q^{23} +0.334715 q^{24} -0.470780 q^{26} -1.00000 q^{27} +4.66943 q^{28} +5.11146 q^{29} +4.93848 q^{31} +1.00120 q^{32} -0.179247 q^{34} -1.99297 q^{36} +5.25915 q^{37} +0.182346 q^{38} +5.61615 q^{39} +8.59386 q^{41} +0.196400 q^{42} +6.99068 q^{43} -0.680543 q^{46} +5.75446 q^{47} -3.95789 q^{48} -1.51060 q^{49} +2.13832 q^{51} +11.1928 q^{52} -10.3207 q^{53} -0.0838261 q^{54} +0.784220 q^{56} -2.17529 q^{57} +0.428474 q^{58} -3.46830 q^{59} +11.7334 q^{61} +0.413974 q^{62} -2.34295 q^{63} -7.83185 q^{64} +9.16598 q^{67} +4.26160 q^{68} +8.11851 q^{69} -6.51765 q^{71} -0.334715 q^{72} -13.7966 q^{73} +0.440854 q^{74} -4.33530 q^{76} +0.470780 q^{78} -6.68282 q^{79} +1.00000 q^{81} +0.720390 q^{82} +11.7895 q^{83} -4.66943 q^{84} +0.586001 q^{86} -5.11146 q^{87} +2.38336 q^{89} +13.1583 q^{91} +16.1800 q^{92} -4.93848 q^{93} +0.482374 q^{94} -1.00120 q^{96} -2.10081 q^{97} -0.126628 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 6 q^{3} + 9 q^{4} + q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - q^2 - 6 * q^3 + 9 * q^4 + q^6 + 4 * q^7 + 6 * q^9	\( 6 q - q^{2} - 6 q^{3} + 9 q^{4} + q^{6} + 4 q^{7} + 6 q^{9} - 9 q^{12} - 2 q^{13} - 10 q^{14} + 11 q^{16} - 2 q^{17} - q^{18} - 2 q^{19} - 4 q^{21} - 12 q^{23} - 6 q^{27} + 24 q^{28} + 12 q^{29} + 18 q^{31} - 31 q^{32} - q^{34} + 9 q^{36} - 24 q^{37} - 32 q^{38} + 2 q^{39} - 6 q^{41} + 10 q^{42} - 10 q^{43} - 11 q^{46} - 8 q^{47} - 11 q^{48} + 12 q^{49} + 2 q^{51} + 8 q^{52} - 18 q^{53} + q^{54} - 42 q^{56} + 2 q^{57} + 8 q^{58} - 18 q^{59} + 4 q^{61} + 17 q^{62} + 4 q^{63} + 22 q^{64} - 12 q^{67} + 9 q^{68} + 12 q^{69} - 14 q^{73} - 4 q^{74} - 20 q^{76} + 2 q^{79} + 6 q^{81} - 36 q^{82} + 20 q^{83} - 24 q^{84} + 28 q^{86} - 12 q^{87} - 16 q^{89} + 24 q^{91} - 41 q^{92} - 18 q^{93} + 3 q^{94} + 31 q^{96} - 12 q^{97} - 53 q^{98}+O(q^{100}) \) 6 * q - q^2 - 6 * q^3 + 9 * q^4 + q^6 + 4 * q^7 + 6 * q^9 - 9 * q^12 - 2 * q^13 - 10 * q^14 + 11 * q^16 - 2 * q^17 - q^18 - 2 * q^19 - 4 * q^21 - 12 * q^23 - 6 * q^27 + 24 * q^28 + 12 * q^29 + 18 * q^31 - 31 * q^32 - q^34 + 9 * q^36 - 24 * q^37 - 32 * q^38 + 2 * q^39 - 6 * q^41 + 10 * q^42 - 10 * q^43 - 11 * q^46 - 8 * q^47 - 11 * q^48 + 12 * q^49 + 2 * q^51 + 8 * q^52 - 18 * q^53 + q^54 - 42 * q^56 + 2 * q^57 + 8 * q^58 - 18 * q^59 + 4 * q^61 + 17 * q^62 + 4 * q^63 + 22 * q^64 - 12 * q^67 + 9 * q^68 + 12 * q^69 - 14 * q^73 - 4 * q^74 - 20 * q^76 + 2 * q^79 + 6 * q^81 - 36 * q^82 + 20 * q^83 - 24 * q^84 + 28 * q^86 - 12 * q^87 - 16 * q^89 + 24 * q^91 - 41 * q^92 - 18 * q^93 + 3 * q^94 + 31 * q^96 - 12 * q^97 - 53 * q^98

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