LMFDB - Embedded newform 9075.2.a.bu.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} -1.85410 q^{13} +4.85410 q^{14} -4.85410 q^{16} +7.47214 q^{17} +1.61803 q^{18} +2.76393 q^{19} -3.00000 q^{21} -7.61803 q^{23} +2.23607 q^{24} -3.00000 q^{26} -1.00000 q^{27} +1.85410 q^{28} -3.61803 q^{29} -8.85410 q^{31} -3.38197 q^{32} +12.0902 q^{34} +0.618034 q^{36} -8.85410 q^{37} +4.47214 q^{38} +1.85410 q^{39} +0.236068 q^{41} -4.85410 q^{42} +1.76393 q^{43} -12.3262 q^{46} -9.38197 q^{47} +4.85410 q^{48} +2.00000 q^{49} -7.47214 q^{51} -1.14590 q^{52} +0.472136 q^{53} -1.61803 q^{54} -6.70820 q^{56} -2.76393 q^{57} -5.85410 q^{58} -4.14590 q^{59} +5.76393 q^{61} -14.3262 q^{62} +3.00000 q^{63} +4.23607 q^{64} +8.70820 q^{67} +4.61803 q^{68} +7.61803 q^{69} +1.47214 q^{71} -2.23607 q^{72} -6.85410 q^{73} -14.3262 q^{74} +1.70820 q^{76} +3.00000 q^{78} +7.56231 q^{79} +1.00000 q^{81} +0.381966 q^{82} +10.3820 q^{83} -1.85410 q^{84} +2.85410 q^{86} +3.61803 q^{87} +2.56231 q^{89} -5.56231 q^{91} -4.70820 q^{92} +8.85410 q^{93} -15.1803 q^{94} +3.38197 q^{96} +10.4164 q^{97} +3.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} + q^{18} + 10 q^{19} - 6 q^{21} - 13 q^{23} - 6 q^{26} - 2 q^{27} - 3 q^{28} - 5 q^{29} - 11 q^{31} - 9 q^{32} + 13 q^{34} - q^{36} - 11 q^{37} - 3 q^{39} - 4 q^{41} - 3 q^{42} + 8 q^{43} - 9 q^{46} - 21 q^{47} + 3 q^{48} + 4 q^{49} - 6 q^{51} - 9 q^{52} - 8 q^{53} - q^{54} - 10 q^{57} - 5 q^{58} - 15 q^{59} + 16 q^{61} - 13 q^{62} + 6 q^{63} + 4 q^{64} + 4 q^{67} + 7 q^{68} + 13 q^{69} - 6 q^{71} - 7 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} + 23 q^{83} + 3 q^{84} - q^{86} + 5 q^{87} - 15 q^{89} + 9 q^{91} + 4 q^{92} + 11 q^{93} - 8 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 6 * q^17 + q^18 + 10 * q^19 - 6 * q^21 - 13 * q^23 - 6 * q^26 - 2 * q^27 - 3 * q^28 - 5 * q^29 - 11 * q^31 - 9 * q^32 + 13 * q^34 - q^36 - 11 * q^37 - 3 * q^39 - 4 * q^41 - 3 * q^42 + 8 * q^43 - 9 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 - 6 * q^51 - 9 * q^52 - 8 * q^53 - q^54 - 10 * q^57 - 5 * q^58 - 15 * q^59 + 16 * q^61 - 13 * q^62 + 6 * q^63 + 4 * q^64 + 4 * q^67 + 7 * q^68 + 13 * q^69 - 6 * q^71 - 7 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 + 23 * q^83 + 3 * q^84 - q^86 + 5 * q^87 - 15 * q^89 + 9 * q^91 + 4 * q^92 + 11 * q^93 - 8 * q^94 + 9 * q^96 - 6 * q^97 + 2 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.618034	−0.178411
$13$	−1.85410	−0.514235	−0.257118	−	0.966380i	$-0.582773\pi$
$13$	−1.85410	−0.514235	−0.257118	+	0.966380i	$0.582773\pi$
$14$	4.85410	1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	7.47214	1.81226	0.906130	−	0.423000i	$-0.139023\pi$
$17$	7.47214	1.81226	0.906130	+	0.423000i	$0.139023\pi$
$18$	1.61803	0.381374
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−7.61803	−1.58847	−0.794235	−	0.607611i	$-0.792128\pi$
$23$	−7.61803	−1.58847	−0.794235	+	0.607611i	$0.792128\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	1.85410	0.350392
$29$	−3.61803	−0.671852	−0.335926	−	0.941888i	$-0.609049\pi$
$29$	−3.61803	−0.671852	−0.335926	+	0.941888i	$0.609049\pi$
$30$	0	0
$31$	−8.85410	−1.59024	−0.795122	−	0.606450i	$-0.792593\pi$
$31$	−8.85410	−1.59024	−0.795122	+	0.606450i	$0.792593\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	12.0902	2.07345
$35$	0	0
$36$	0.618034	0.103006
$37$	−8.85410	−1.45561	−0.727803	−	0.685787i	$-0.759458\pi$
$37$	−8.85410	−1.45561	−0.727803	+	0.685787i	$0.759458\pi$
$38$	4.47214	0.725476
$39$	1.85410	0.296894
$40$	0	0
$41$	0.236068	0.0368676	0.0184338	−	0.999830i	$-0.494132\pi$
$41$	0.236068	0.0368676	0.0184338	+	0.999830i	$0.494132\pi$
$42$	−4.85410	−0.749004
$43$	1.76393	0.268997	0.134499	−	0.990914i	$-0.457058\pi$
$43$	1.76393	0.268997	0.134499	+	0.990914i	$0.457058\pi$
$44$	0	0
$45$	0	0
$46$	−12.3262	−1.81740
$47$	−9.38197	−1.36850	−0.684250	−	0.729247i	$-0.739870\pi$
$47$	−9.38197	−1.36850	−0.684250	+	0.729247i	$0.739870\pi$
$48$	4.85410	0.700629
$49$	2.00000	0.285714
$50$	0	0
$51$	−7.47214	−1.04631
$52$	−1.14590	−0.158907
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	−6.70820	−0.896421
$57$	−2.76393	−0.366092
$58$	−5.85410	−0.768681
$59$	−4.14590	−0.539750	−0.269875	−	0.962895i	$-0.586982\pi$
$59$	−4.14590	−0.539750	−0.269875	+	0.962895i	$0.586982\pi$
$60$	0	0
$61$	5.76393	0.737996	0.368998	−	0.929430i	$-0.379701\pi$
$61$	5.76393	0.737996	0.368998	+	0.929430i	$0.379701\pi$
$62$	−14.3262	−1.81943
$63$	3.00000	0.377964
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	8.70820	1.06388	0.531938	−	0.846783i	$-0.321464\pi$
$67$	8.70820	1.06388	0.531938	+	0.846783i	$0.321464\pi$
$68$	4.61803	0.560019
$69$	7.61803	0.917104
$70$	0	0
$71$	1.47214	0.174710	0.0873552	−	0.996177i	$-0.472158\pi$
$71$	1.47214	0.174710	0.0873552	+	0.996177i	$0.472158\pi$
$72$	−2.23607	−0.263523
$73$	−6.85410	−0.802212	−0.401106	−	0.916032i	$-0.631374\pi$
$73$	−6.85410	−0.802212	−0.401106	+	0.916032i	$0.631374\pi$
$74$	−14.3262	−1.66539
$75$	0	0
$76$	1.70820	0.195944
$77$	0	0
$78$	3.00000	0.339683
$79$	7.56231	0.850826	0.425413	−	0.904999i	$-0.360129\pi$
$79$	7.56231	0.850826	0.425413	+	0.904999i	$0.360129\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.381966	0.0421811
$83$	10.3820	1.13957	0.569784	−	0.821794i	$-0.307027\pi$
$83$	10.3820	1.13957	0.569784	+	0.821794i	$0.307027\pi$
$84$	−1.85410	−0.202299
$85$	0	0
$86$	2.85410	0.307766
$87$	3.61803	0.387894
$88$	0	0
$89$	2.56231	0.271604	0.135802	−	0.990736i	$-0.456639\pi$
$89$	2.56231	0.271604	0.135802	+	0.990736i	$0.456639\pi$
$90$	0	0
$91$	−5.56231	−0.583088
$92$	−4.70820	−0.490864
$93$	8.85410	0.918128
$94$	−15.1803	−1.56573
$95$	0	0
$96$	3.38197	0.345170
$97$	10.4164	1.05763	0.528813	−	0.848738i	$-0.322637\pi$
$97$	10.4164	1.05763	0.528813	+	0.848738i	$0.322637\pi$
$98$	3.23607	0.326892
$99$	0	0
$100$	0	0
$101$	1.09017	0.108476	0.0542380	−	0.998528i	$-0.482727\pi$
$101$	1.09017	0.108476	0.0542380	+	0.998528i	$0.482727\pi$
$102$	−12.0902	−1.19711
$103$	−15.1803	−1.49576	−0.747882	−	0.663832i	$-0.768929\pi$
$103$	−15.1803	−1.49576	−0.747882	+	0.663832i	$0.768929\pi$
$104$	4.14590	0.406539
$105$	0	0
$106$	0.763932	0.0741996
$107$	16.4164	1.58703	0.793517	−	0.608548i	$-0.208248\pi$
$107$	16.4164	1.58703	0.793517	+	0.608548i	$0.208248\pi$
$108$	−0.618034	−0.0594703
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	8.85410	0.840394
$112$	−14.5623	−1.37601
$113$	−8.47214	−0.796992	−0.398496	−	0.917170i	$-0.630468\pi$
$113$	−8.47214	−0.796992	−0.398496	+	0.917170i	$0.630468\pi$
$114$	−4.47214	−0.418854
$115$	0	0
$116$	−2.23607	−0.207614
$117$	−1.85410	−0.171412
$118$	−6.70820	−0.617540
$119$	22.4164	2.05491
$120$	0	0
$121$	0	0
$122$	9.32624	0.844358
$123$	−0.236068	−0.0212855
$124$	−5.47214	−0.491412
$125$	0	0
$126$	4.85410	0.432438
$127$	6.61803	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$127$	6.61803	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$128$	13.6180	1.20368
$129$	−1.76393	−0.155306
$130$	0	0
$131$	−7.32624	−0.640096	−0.320048	−	0.947401i	$-0.603699\pi$
$131$	−7.32624	−0.640096	−0.320048	+	0.947401i	$0.603699\pi$
$132$	0	0
$133$	8.29180	0.718990
$134$	14.0902	1.21721
$135$	0	0
$136$	−16.7082	−1.43272
$137$	−12.7984	−1.09344	−0.546719	−	0.837316i	$-0.684124\pi$
$137$	−12.7984	−1.09344	−0.546719	+	0.837316i	$0.684124\pi$
$138$	12.3262	1.04928
$139$	−5.85410	−0.496538	−0.248269	−	0.968691i	$-0.579862\pi$
$139$	−5.85410	−0.496538	−0.248269	+	0.968691i	$0.579862\pi$
$140$	0	0
$141$	9.38197	0.790104
$142$	2.38197	0.199890
$143$	0	0
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−11.0902	−0.917829
$147$	−2.00000	−0.164957
$148$	−5.47214	−0.449807
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	9.18034	0.747085	0.373543	−	0.927613i	$-0.378143\pi$
$151$	9.18034	0.747085	0.373543	+	0.927613i	$0.378143\pi$
$152$	−6.18034	−0.501292
$153$	7.47214	0.604086
$154$	0	0
$155$	0	0
$156$	1.14590	0.0917453
$157$	−17.7984	−1.42046	−0.710232	−	0.703967i	$-0.751410\pi$
$157$	−17.7984	−1.42046	−0.710232	+	0.703967i	$0.751410\pi$
$158$	12.2361	0.973449
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	−22.8541	−1.80116
$162$	1.61803	0.127125
$163$	−17.0902	−1.33861	−0.669303	−	0.742990i	$-0.733407\pi$
$163$	−17.0902	−1.33861	−0.669303	+	0.742990i	$0.733407\pi$
$164$	0.145898	0.0113927
$165$	0	0
$166$	16.7984	1.30381
$167$	−20.9443	−1.62072	−0.810358	−	0.585935i	$-0.800727\pi$
$167$	−20.9443	−1.62072	−0.810358	+	0.585935i	$0.800727\pi$
$168$	6.70820	0.517549
$169$	−9.56231	−0.735562
$170$	0	0
$171$	2.76393	0.211363
$172$	1.09017	0.0831247
$173$	−5.47214	−0.416039	−0.208019	−	0.978125i	$-0.566702\pi$
$173$	−5.47214	−0.416039	−0.208019	+	0.978125i	$0.566702\pi$
$174$	5.85410	0.443798
$175$	0	0
$176$	0	0
$177$	4.14590	0.311625
$178$	4.14590	0.310748
$179$	−20.1246	−1.50418	−0.752092	−	0.659058i	$-0.770955\pi$
$179$	−20.1246	−1.50418	−0.752092	+	0.659058i	$0.770955\pi$
$180$	0	0
$181$	−12.7984	−0.951296	−0.475648	−	0.879636i	$-0.657786\pi$
$181$	−12.7984	−0.951296	−0.475648	+	0.879636i	$0.657786\pi$
$182$	−9.00000	−0.667124
$183$	−5.76393	−0.426082
$184$	17.0344	1.25580
$185$	0	0
$186$	14.3262	1.05045
$187$	0	0
$188$	−5.79837	−0.422890
$189$	−3.00000	−0.218218
$190$	0	0
$191$	15.7426	1.13910	0.569549	−	0.821957i	$-0.307118\pi$
$191$	15.7426	1.13910	0.569549	+	0.821957i	$0.307118\pi$
$192$	−4.23607	−0.305712
$193$	18.4721	1.32965	0.664827	−	0.746998i	$-0.268505\pi$
$193$	18.4721	1.32965	0.664827	+	0.746998i	$0.268505\pi$
$194$	16.8541	1.21005
$195$	0	0
$196$	1.23607	0.0882906
$197$	9.70820	0.691681	0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	0	0
$199$	−8.29180	−0.587790	−0.293895	−	0.955838i	$-0.594952\pi$
$199$	−8.29180	−0.587790	−0.293895	+	0.955838i	$0.594952\pi$
$200$	0	0
$201$	−8.70820	−0.614229
$202$	1.76393	0.124110
$203$	−10.8541	−0.761809
$204$	−4.61803	−0.323327
$205$	0	0
$206$	−24.5623	−1.71134
$207$	−7.61803	−0.529490
$208$	9.00000	0.624038
$209$	0	0
$210$	0	0
$211$	−13.1803	−0.907372	−0.453686	−	0.891162i	$-0.649891\pi$
$211$	−13.1803	−0.907372	−0.453686	+	0.891162i	$0.649891\pi$
$212$	0.291796	0.0200406
$213$	−1.47214	−0.100869
$214$	26.5623	1.81576
$215$	0	0
$216$	2.23607	0.152145
$217$	−26.5623	−1.80317
$218$	−24.2705	−1.64381
$219$	6.85410	0.463157
$220$	0	0
$221$	−13.8541	−0.931928
$222$	14.3262	0.961514
$223$	12.7082	0.851004	0.425502	−	0.904957i	$-0.360097\pi$
$223$	12.7082	0.851004	0.425502	+	0.904957i	$0.360097\pi$
$224$	−10.1459	−0.677901
$225$	0	0
$226$	−13.7082	−0.911856
$227$	−14.5623	−0.966534	−0.483267	−	0.875473i	$-0.660550\pi$
$227$	−14.5623	−0.966534	−0.483267	+	0.875473i	$0.660550\pi$
$228$	−1.70820	−0.113129
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	8.09017	0.531146
$233$	−6.32624	−0.414446	−0.207223	−	0.978294i	$-0.566443\pi$
$233$	−6.32624	−0.414446	−0.207223	+	0.978294i	$0.566443\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	−2.56231	−0.166792
$237$	−7.56231	−0.491225
$238$	36.2705	2.35107
$239$	20.6525	1.33590	0.667949	−	0.744207i	$-0.267173\pi$
$239$	20.6525	1.33590	0.667949	+	0.744207i	$0.267173\pi$
$240$	0	0
$241$	−7.85410	−0.505927	−0.252964	−	0.967476i	$-0.581405\pi$
$241$	−7.85410	−0.505927	−0.252964	+	0.967476i	$0.581405\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	3.56231	0.228053
$245$	0	0
$246$	−0.381966	−0.0243533
$247$	−5.12461	−0.326071
$248$	19.7984	1.25720
$249$	−10.3820	−0.657930
$250$	0	0
$251$	−20.5623	−1.29788	−0.648941	−	0.760839i	$-0.724788\pi$
$251$	−20.5623	−1.29788	−0.648941	+	0.760839i	$0.724788\pi$
$252$	1.85410	0.116797
$253$	0	0
$254$	10.7082	0.671892
$255$	0	0
$256$	13.5623	0.847644
$257$	−10.0344	−0.625931	−0.312966	−	0.949764i	$-0.601322\pi$
$257$	−10.0344	−0.625931	−0.312966	+	0.949764i	$0.601322\pi$
$258$	−2.85410	−0.177689
$259$	−26.5623	−1.65050
$260$	0	0
$261$	−3.61803	−0.223951
$262$	−11.8541	−0.732349
$263$	−11.1246	−0.685973	−0.342986	−	0.939340i	$-0.611439\pi$
$263$	−11.1246	−0.685973	−0.342986	+	0.939340i	$0.611439\pi$
$264$	0	0
$265$	0	0
$266$	13.4164	0.822613
$267$	−2.56231	−0.156811
$268$	5.38197	0.328756
$269$	−0.527864	−0.0321844	−0.0160922	−	0.999871i	$-0.505123\pi$
$269$	−0.527864	−0.0321844	−0.0160922	+	0.999871i	$0.505123\pi$
$270$	0	0
$271$	22.2705	1.35284	0.676419	−	0.736517i	$-0.263531\pi$
$271$	22.2705	1.35284	0.676419	+	0.736517i	$0.263531\pi$
$272$	−36.2705	−2.19922
$273$	5.56231	0.336646
$274$	−20.7082	−1.25103
$275$	0	0
$276$	4.70820	0.283401
$277$	23.9787	1.44074	0.720371	−	0.693589i	$-0.243972\pi$
$277$	23.9787	1.44074	0.720371	+	0.693589i	$0.243972\pi$
$278$	−9.47214	−0.568101
$279$	−8.85410	−0.530081
$280$	0	0
$281$	3.32624	0.198427	0.0992134	−	0.995066i	$-0.468367\pi$
$281$	3.32624	0.198427	0.0992134	+	0.995066i	$0.468367\pi$
$282$	15.1803	0.903976
$283$	27.0902	1.61034	0.805172	−	0.593042i	$-0.202073\pi$
$283$	27.0902	1.61034	0.805172	+	0.593042i	$0.202073\pi$
$284$	0.909830	0.0539885
$285$	0	0
$286$	0	0
$287$	0.708204	0.0418040
$288$	−3.38197	−0.199284
$289$	38.8328	2.28428
$290$	0	0
$291$	−10.4164	−0.610621
$292$	−4.23607	−0.247897
$293$	−19.4164	−1.13432	−0.567159	−	0.823608i	$-0.691958\pi$
$293$	−19.4164	−1.13432	−0.567159	+	0.823608i	$0.691958\pi$
$294$	−3.23607	−0.188731
$295$	0	0
$296$	19.7984	1.15076
$297$	0	0
$298$	−26.1803	−1.51659
$299$	14.1246	0.816847
$300$	0	0
$301$	5.29180	0.305014
$302$	14.8541	0.854758
$303$	−1.09017	−0.0626286
$304$	−13.4164	−0.769484
$305$	0	0
$306$	12.0902	0.691149
$307$	15.8885	0.906807	0.453404	−	0.891305i	$-0.350210\pi$
$307$	15.8885	0.906807	0.453404	+	0.891305i	$0.350210\pi$
$308$	0	0
$309$	15.1803	0.863579
$310$	0	0
$311$	−18.5279	−1.05062	−0.525309	−	0.850911i	$-0.676050\pi$
$311$	−18.5279	−1.05062	−0.525309	+	0.850911i	$0.676050\pi$
$312$	−4.14590	−0.234715
$313$	−30.7082	−1.73573	−0.867865	−	0.496800i	$-0.834508\pi$
$313$	−30.7082	−1.73573	−0.867865	+	0.496800i	$0.834508\pi$
$314$	−28.7984	−1.62519
$315$	0	0
$316$	4.67376	0.262920
$317$	−9.18034	−0.515619	−0.257810	−	0.966196i	$-0.583001\pi$
$317$	−9.18034	−0.515619	−0.257810	+	0.966196i	$0.583001\pi$
$318$	−0.763932	−0.0428392
$319$	0	0
$320$	0	0
$321$	−16.4164	−0.916275
$322$	−36.9787	−2.06074
$323$	20.6525	1.14913
$324$	0.618034	0.0343352
$325$	0	0
$326$	−27.6525	−1.53153
$327$	15.0000	0.829502
$328$	−0.527864	−0.0291464
$329$	−28.1459	−1.55173
$330$	0	0
$331$	−16.4164	−0.902327	−0.451164	−	0.892441i	$-0.648991\pi$
$331$	−16.4164	−0.902327	−0.451164	+	0.892441i	$0.648991\pi$
$332$	6.41641	0.352146
$333$	−8.85410	−0.485202
$334$	−33.8885	−1.85430
$335$	0	0
$336$	14.5623	0.794439
$337$	−23.7082	−1.29147	−0.645734	−	0.763562i	$-0.723449\pi$
$337$	−23.7082	−1.29147	−0.645734	+	0.763562i	$0.723449\pi$
$338$	−15.4721	−0.841573
$339$	8.47214	0.460143
$340$	0	0
$341$	0	0
$342$	4.47214	0.241825
$343$	−15.0000	−0.809924
$344$	−3.94427	−0.212661
$345$	0	0
$346$	−8.85410	−0.475999
$347$	−12.3262	−0.661707	−0.330854	−	0.943682i	$-0.607337\pi$
$347$	−12.3262	−0.661707	−0.330854	+	0.943682i	$0.607337\pi$
$348$	2.23607	0.119866
$349$	−28.9443	−1.54935	−0.774676	−	0.632359i	$-0.782087\pi$
$349$	−28.9443	−1.54935	−0.774676	+	0.632359i	$0.782087\pi$
$350$	0	0
$351$	1.85410	0.0989646
$352$	0	0
$353$	8.88854	0.473089	0.236545	−	0.971621i	$-0.423985\pi$
$353$	8.88854	0.473089	0.236545	+	0.971621i	$0.423985\pi$
$354$	6.70820	0.356537
$355$	0	0
$356$	1.58359	0.0839302
$357$	−22.4164	−1.18640
$358$	−32.5623	−1.72097
$359$	−18.2148	−0.961339	−0.480670	−	0.876902i	$-0.659606\pi$
$359$	−18.2148	−0.961339	−0.480670	+	0.876902i	$0.659606\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	−20.7082	−1.08840
$363$	0	0
$364$	−3.43769	−0.180184
$365$	0	0
$366$	−9.32624	−0.487490
$367$	18.8328	0.983065	0.491532	−	0.870859i	$-0.336437\pi$
$367$	18.8328	0.983065	0.491532	+	0.870859i	$0.336437\pi$
$368$	36.9787	1.92765
$369$	0.236068	0.0122892
$370$	0	0
$371$	1.41641	0.0735362
$372$	5.47214	0.283717
$373$	17.4164	0.901787	0.450894	−	0.892578i	$-0.351105\pi$
$373$	17.4164	0.901787	0.450894	+	0.892578i	$0.351105\pi$
$374$	0	0
$375$	0	0
$376$	20.9787	1.08189
$377$	6.70820	0.345490
$378$	−4.85410	−0.249668
$379$	−23.2148	−1.19246	−0.596232	−	0.802812i	$-0.703336\pi$
$379$	−23.2148	−1.19246	−0.596232	+	0.802812i	$0.703336\pi$
$380$	0	0
$381$	−6.61803	−0.339052
$382$	25.4721	1.30327
$383$	21.5279	1.10002	0.550011	−	0.835157i	$-0.314623\pi$
$383$	21.5279	1.10002	0.550011	+	0.835157i	$0.314623\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	29.8885	1.52129
$387$	1.76393	0.0896657
$388$	6.43769	0.326824
$389$	−18.0902	−0.917208	−0.458604	−	0.888641i	$-0.651650\pi$
$389$	−18.0902	−0.917208	−0.458604	+	0.888641i	$0.651650\pi$
$390$	0	0
$391$	−56.9230	−2.87872
$392$	−4.47214	−0.225877
$393$	7.32624	0.369560
$394$	15.7082	0.791368
$395$	0	0
$396$	0	0
$397$	19.5623	0.981804	0.490902	−	0.871215i	$-0.336667\pi$
$397$	19.5623	0.981804	0.490902	+	0.871215i	$0.336667\pi$
$398$	−13.4164	−0.672504
$399$	−8.29180	−0.415109
$400$	0	0
$401$	0.819660	0.0409319	0.0204659	−	0.999791i	$-0.493485\pi$
$401$	0.819660	0.0409319	0.0204659	+	0.999791i	$0.493485\pi$
$402$	−14.0902	−0.702754
$403$	16.4164	0.817760
$404$	0.673762	0.0335209
$405$	0	0
$406$	−17.5623	−0.871603
$407$	0	0
$408$	16.7082	0.827179
$409$	10.1246	0.500630	0.250315	−	0.968164i	$-0.419466\pi$
$409$	10.1246	0.500630	0.250315	+	0.968164i	$0.419466\pi$
$410$	0	0
$411$	12.7984	0.631297
$412$	−9.38197	−0.462216
$413$	−12.4377	−0.612019
$414$	−12.3262	−0.605802
$415$	0	0
$416$	6.27051	0.307437
$417$	5.85410	0.286677
$418$	0	0
$419$	18.9443	0.925488	0.462744	−	0.886492i	$-0.346865\pi$
$419$	18.9443	0.925488	0.462744	+	0.886492i	$0.346865\pi$
$420$	0	0
$421$	−7.27051	−0.354343	−0.177171	−	0.984180i	$-0.556695\pi$
$421$	−7.27051	−0.354343	−0.177171	+	0.984180i	$0.556695\pi$
$422$	−21.3262	−1.03815
$423$	−9.38197	−0.456167
$424$	−1.05573	−0.0512707
$425$	0	0
$426$	−2.38197	−0.115407
$427$	17.2918	0.836809
$428$	10.1459	0.490420
$429$	0	0
$430$	0	0
$431$	−16.1459	−0.777721	−0.388860	−	0.921297i	$-0.627131\pi$
$431$	−16.1459	−0.777721	−0.388860	+	0.921297i	$0.627131\pi$
$432$	4.85410	0.233543
$433$	27.1803	1.30620	0.653102	−	0.757270i	$-0.273467\pi$
$433$	27.1803	1.30620	0.653102	+	0.757270i	$0.273467\pi$
$434$	−42.9787	−2.06304
$435$	0	0
$436$	−9.27051	−0.443977
$437$	−21.0557	−1.00723
$438$	11.0902	0.529909
$439$	−7.43769	−0.354982	−0.177491	−	0.984122i	$-0.556798\pi$
$439$	−7.43769	−0.354982	−0.177491	+	0.984122i	$0.556798\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−22.4164	−1.06624
$443$	27.5066	1.30688	0.653438	−	0.756980i	$-0.273326\pi$
$443$	27.5066	1.30688	0.653438	+	0.756980i	$0.273326\pi$
$444$	5.47214	0.259696
$445$	0	0
$446$	20.5623	0.973653
$447$	16.1803	0.765304
$448$	12.7082	0.600406
$449$	14.4721	0.682982	0.341491	−	0.939885i	$-0.389068\pi$
$449$	14.4721	0.682982	0.341491	+	0.939885i	$0.389068\pi$
$450$	0	0
$451$	0	0
$452$	−5.23607	−0.246284
$453$	−9.18034	−0.431330
$454$	−23.5623	−1.10583
$455$	0	0
$456$	6.18034	0.289421
$457$	23.5279	1.10059	0.550294	−	0.834971i	$-0.314516\pi$
$457$	23.5279	1.10059	0.550294	+	0.834971i	$0.314516\pi$
$458$	−16.1803	−0.756058
$459$	−7.47214	−0.348769
$460$	0	0
$461$	9.18034	0.427571	0.213786	−	0.976881i	$-0.431421\pi$
$461$	9.18034	0.427571	0.213786	+	0.976881i	$0.431421\pi$
$462$	0	0
$463$	9.29180	0.431826	0.215913	−	0.976413i	$-0.430727\pi$
$463$	9.29180	0.431826	0.215913	+	0.976413i	$0.430727\pi$
$464$	17.5623	0.815310
$465$	0	0
$466$	−10.2361	−0.474177
$467$	4.96556	0.229779	0.114889	−	0.993378i	$-0.463349\pi$
$467$	4.96556	0.229779	0.114889	+	0.993378i	$0.463349\pi$
$468$	−1.14590	−0.0529692
$469$	26.1246	1.20632
$470$	0	0
$471$	17.7984	0.820106
$472$	9.27051	0.426710
$473$	0	0
$474$	−12.2361	−0.562021
$475$	0	0
$476$	13.8541	0.635002
$477$	0.472136	0.0216176
$478$	33.4164	1.52843
$479$	−7.36068	−0.336318	−0.168159	−	0.985760i	$-0.553782\pi$
$479$	−7.36068	−0.336318	−0.168159	+	0.985760i	$0.553782\pi$
$480$	0	0
$481$	16.4164	0.748524
$482$	−12.7082	−0.578843
$483$	22.8541	1.03990
$484$	0	0
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	1.27051	0.0575723	0.0287861	−	0.999586i	$-0.490836\pi$
$487$	1.27051	0.0575723	0.0287861	+	0.999586i	$0.490836\pi$
$488$	−12.8885	−0.583437
$489$	17.0902	0.772844
$490$	0	0
$491$	13.8541	0.625227	0.312613	−	0.949880i	$-0.398796\pi$
$491$	13.8541	0.625227	0.312613	+	0.949880i	$0.398796\pi$
$492$	−0.145898	−0.00657759
$493$	−27.0344	−1.21757
$494$	−8.29180	−0.373066
$495$	0	0
$496$	42.9787	1.92980
$497$	4.41641	0.198103
$498$	−16.7984	−0.752753
$499$	11.8328	0.529710	0.264855	−	0.964288i	$-0.414676\pi$
$499$	11.8328	0.529710	0.264855	+	0.964288i	$0.414676\pi$
$500$	0	0
$501$	20.9443	0.935721
$502$	−33.2705	−1.48494
$503$	−3.43769	−0.153279	−0.0766396	−	0.997059i	$-0.524419\pi$
$503$	−3.43769	−0.153279	−0.0766396	+	0.997059i	$0.524419\pi$
$504$	−6.70820	−0.298807
$505$	0	0
$506$	0	0
$507$	9.56231	0.424677
$508$	4.09017	0.181472
$509$	−14.7984	−0.655926	−0.327963	−	0.944691i	$-0.606362\pi$
$509$	−14.7984	−0.655926	−0.327963	+	0.944691i	$0.606362\pi$
$510$	0	0
$511$	−20.5623	−0.909623
$512$	−5.29180	−0.233867
$513$	−2.76393	−0.122031
$514$	−16.2361	−0.716142
$515$	0	0
$516$	−1.09017	−0.0479921
$517$	0	0
$518$	−42.9787	−1.88838
$519$	5.47214	0.240200
$520$	0	0
$521$	−43.2492	−1.89478	−0.947391	−	0.320077i	$-0.896291\pi$
$521$	−43.2492	−1.89478	−0.947391	+	0.320077i	$0.896291\pi$
$522$	−5.85410	−0.256227
$523$	38.7984	1.69653	0.848267	−	0.529568i	$-0.177646\pi$
$523$	38.7984	1.69653	0.848267	+	0.529568i	$0.177646\pi$
$524$	−4.52786	−0.197801
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−66.1591	−2.88193
$528$	0	0
$529$	35.0344	1.52324
$530$	0	0
$531$	−4.14590	−0.179917
$532$	5.12461	0.222180
$533$	−0.437694	−0.0189586
$534$	−4.14590	−0.179411
$535$	0	0
$536$	−19.4721	−0.841068
$537$	20.1246	0.868441
$538$	−0.854102	−0.0368230
$539$	0	0
$540$	0	0
$541$	24.8328	1.06765	0.533823	−	0.845596i	$-0.320755\pi$
$541$	24.8328	1.06765	0.533823	+	0.845596i	$0.320755\pi$
$542$	36.0344	1.54781
$543$	12.7984	0.549231
$544$	−25.2705	−1.08346
$545$	0	0
$546$	9.00000	0.385164
$547$	21.2918	0.910371	0.455186	−	0.890397i	$-0.349573\pi$
$547$	21.2918	0.910371	0.455186	+	0.890397i	$0.349573\pi$
$548$	−7.90983	−0.337891
$549$	5.76393	0.245999
$550$	0	0
$551$	−10.0000	−0.426014
$552$	−17.0344	−0.725034
$553$	22.6869	0.964746
$554$	38.7984	1.64838
$555$	0	0
$556$	−3.61803	−0.153439
$557$	−30.5410	−1.29406	−0.647032	−	0.762463i	$-0.723990\pi$
$557$	−30.5410	−1.29406	−0.647032	+	0.762463i	$0.723990\pi$
$558$	−14.3262	−0.606478
$559$	−3.27051	−0.138328
$560$	0	0
$561$	0	0
$562$	5.38197	0.227025
$563$	−17.8328	−0.751564	−0.375782	−	0.926708i	$-0.622626\pi$
$563$	−17.8328	−0.751564	−0.375782	+	0.926708i	$0.622626\pi$
$564$	5.79837	0.244156
$565$	0	0
$566$	43.8328	1.84243
$567$	3.00000	0.125988
$568$	−3.29180	−0.138121
$569$	−15.5279	−0.650962	−0.325481	−	0.945549i	$-0.605526\pi$
$569$	−15.5279	−0.650962	−0.325481	+	0.945549i	$0.605526\pi$
$570$	0	0
$571$	−13.5836	−0.568456	−0.284228	−	0.958757i	$-0.591737\pi$
$571$	−13.5836	−0.568456	−0.284228	+	0.958757i	$0.591737\pi$
$572$	0	0
$573$	−15.7426	−0.657658
$574$	1.14590	0.0478289
$575$	0	0
$576$	4.23607	0.176503
$577$	20.2148	0.841552	0.420776	−	0.907164i	$-0.361758\pi$
$577$	20.2148	0.841552	0.420776	+	0.907164i	$0.361758\pi$
$578$	62.8328	2.61350
$579$	−18.4721	−0.767676
$580$	0	0
$581$	31.1459	1.29215
$582$	−16.8541	−0.698625
$583$	0	0
$584$	15.3262	0.634204
$585$	0	0
$586$	−31.4164	−1.29780
$587$	−34.3050	−1.41592	−0.707958	−	0.706254i	$-0.750384\pi$
$587$	−34.3050	−1.41592	−0.707958	+	0.706254i	$0.750384\pi$
$588$	−1.23607	−0.0509746
$589$	−24.4721	−1.00836
$590$	0	0
$591$	−9.70820	−0.399342
$592$	42.9787	1.76641
$593$	25.5066	1.04743	0.523715	−	0.851894i	$-0.324546\pi$
$593$	25.5066	1.04743	0.523715	+	0.851894i	$0.324546\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	8.29180	0.339361
$598$	22.8541	0.934574
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	29.3820	1.19852	0.599258	−	0.800556i	$-0.295462\pi$
$601$	29.3820	1.19852	0.599258	+	0.800556i	$0.295462\pi$
$602$	8.56231	0.348974
$603$	8.70820	0.354625
$604$	5.67376	0.230862
$605$	0	0
$606$	−1.76393	−0.0716548
$607$	28.8541	1.17115	0.585576	−	0.810618i	$-0.300868\pi$
$607$	28.8541	1.17115	0.585576	+	0.810618i	$0.300868\pi$
$608$	−9.34752	−0.379092
$609$	10.8541	0.439830
$610$	0	0
$611$	17.3951	0.703731
$612$	4.61803	0.186673
$613$	24.9787	1.00888	0.504440	−	0.863447i	$-0.331699\pi$
$613$	24.9787	1.00888	0.504440	+	0.863447i	$0.331699\pi$
$614$	25.7082	1.03750
$615$	0	0
$616$	0	0
$617$	−9.05573	−0.364570	−0.182285	−	0.983246i	$-0.558349\pi$
$617$	−9.05573	−0.364570	−0.182285	+	0.983246i	$0.558349\pi$
$618$	24.5623	0.988041
$619$	2.63932	0.106083	0.0530416	−	0.998592i	$-0.483108\pi$
$619$	2.63932	0.106083	0.0530416	+	0.998592i	$0.483108\pi$
$620$	0	0
$621$	7.61803	0.305701
$622$	−29.9787	−1.20204
$623$	7.68692	0.307970
$624$	−9.00000	−0.360288
$625$	0	0
$626$	−49.6869	−1.98589
$627$	0	0
$628$	−11.0000	−0.438948
$629$	−66.1591	−2.63793
$630$	0	0
$631$	−49.1803	−1.95784	−0.978919	−	0.204248i	$-0.934525\pi$
$631$	−49.1803	−1.95784	−0.978919	+	0.204248i	$0.934525\pi$
$632$	−16.9098	−0.672637
$633$	13.1803	0.523871
$634$	−14.8541	−0.589932
$635$	0	0
$636$	−0.291796	−0.0115705
$637$	−3.70820	−0.146924
$638$	0	0
$639$	1.47214	0.0582368
$640$	0	0
$641$	39.1591	1.54669	0.773345	−	0.633986i	$-0.218582\pi$
$641$	39.1591	1.54669	0.773345	+	0.633986i	$0.218582\pi$
$642$	−26.5623	−1.04833
$643$	−20.3820	−0.803786	−0.401893	−	0.915687i	$-0.631648\pi$
$643$	−20.3820	−0.803786	−0.401893	+	0.915687i	$0.631648\pi$
$644$	−14.1246	−0.556588
$645$	0	0
$646$	33.4164	1.31475
$647$	−13.2016	−0.519009	−0.259505	−	0.965742i	$-0.583559\pi$
$647$	−13.2016	−0.519009	−0.259505	+	0.965742i	$0.583559\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	26.5623	1.04106
$652$	−10.5623	−0.413652
$653$	4.94427	0.193484	0.0967422	−	0.995309i	$-0.469158\pi$
$653$	4.94427	0.193484	0.0967422	+	0.995309i	$0.469158\pi$
$654$	24.2705	0.949052
$655$	0	0
$656$	−1.14590	−0.0447398
$657$	−6.85410	−0.267404
$658$	−45.5410	−1.77537
$659$	44.0689	1.71668	0.858340	−	0.513081i	$-0.171496\pi$
$659$	44.0689	1.71668	0.858340	+	0.513081i	$0.171496\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	−26.5623	−1.03237
$663$	13.8541	0.538049
$664$	−23.2148	−0.900908
$665$	0	0
$666$	−14.3262	−0.555130
$667$	27.5623	1.06722
$668$	−12.9443	−0.500829
$669$	−12.7082	−0.491328
$670$	0	0
$671$	0	0
$672$	10.1459	0.391387
$673$	−22.5066	−0.867565	−0.433782	−	0.901018i	$-0.642821\pi$
$673$	−22.5066	−0.867565	−0.433782	+	0.901018i	$0.642821\pi$
$674$	−38.3607	−1.47760
$675$	0	0
$676$	−5.90983	−0.227301
$677$	−7.52786	−0.289319	−0.144660	−	0.989481i	$-0.546209\pi$
$677$	−7.52786	−0.289319	−0.144660	+	0.989481i	$0.546209\pi$
$678$	13.7082	0.526460
$679$	31.2492	1.19924
$680$	0	0
$681$	14.5623	0.558029
$682$	0	0
$683$	12.7082	0.486266	0.243133	−	0.969993i	$-0.421825\pi$
$683$	12.7082	0.486266	0.243133	+	0.969993i	$0.421825\pi$
$684$	1.70820	0.0653148
$685$	0	0
$686$	−24.2705	−0.926652
$687$	10.0000	0.381524
$688$	−8.56231	−0.326435
$689$	−0.875388	−0.0333496
$690$	0	0
$691$	−24.7082	−0.939944	−0.469972	−	0.882681i	$-0.655736\pi$
$691$	−24.7082	−0.939944	−0.469972	+	0.882681i	$0.655736\pi$
$692$	−3.38197	−0.128563
$693$	0	0
$694$	−19.9443	−0.757074
$695$	0	0
$696$	−8.09017	−0.306657
$697$	1.76393	0.0668137
$698$	−46.8328	−1.77265
$699$	6.32624	0.239280
$700$	0	0
$701$	−14.2361	−0.537689	−0.268844	−	0.963184i	$-0.586642\pi$
$701$	−14.2361	−0.537689	−0.268844	+	0.963184i	$0.586642\pi$
$702$	3.00000	0.113228
$703$	−24.4721	−0.922984
$704$	0	0
$705$	0	0
$706$	14.3820	0.541272
$707$	3.27051	0.123000
$708$	2.56231	0.0962974
$709$	−9.47214	−0.355734	−0.177867	−	0.984055i	$-0.556920\pi$
$709$	−9.47214	−0.355734	−0.177867	+	0.984055i	$0.556920\pi$
$710$	0	0
$711$	7.56231	0.283609
$712$	−5.72949	−0.214722
$713$	67.4508	2.52605
$714$	−36.2705	−1.35739
$715$	0	0
$716$	−12.4377	−0.464818
$717$	−20.6525	−0.771281
$718$	−29.4721	−1.09989
$719$	−26.0557	−0.971715	−0.485857	−	0.874038i	$-0.661493\pi$
$719$	−26.0557	−0.971715	−0.485857	+	0.874038i	$0.661493\pi$
$720$	0	0
$721$	−45.5410	−1.69604
$722$	−18.3820	−0.684106
$723$	7.85410	0.292097
$724$	−7.90983	−0.293967
$725$	0	0
$726$	0	0
$727$	−0.236068	−0.00875528	−0.00437764	−	0.999990i	$-0.501393\pi$
$727$	−0.236068	−0.00875528	−0.00437764	+	0.999990i	$0.501393\pi$
$728$	12.4377	0.460972
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.1803	0.487492
$732$	−3.56231	−0.131667
$733$	6.43769	0.237782	0.118891	−	0.992907i	$-0.462066\pi$
$733$	6.43769	0.237782	0.118891	+	0.992907i	$0.462066\pi$
$734$	30.4721	1.12475
$735$	0	0
$736$	25.7639	0.949671
$737$	0	0
$738$	0.381966	0.0140604
$739$	−44.2705	−1.62852	−0.814259	−	0.580502i	$-0.802856\pi$
$739$	−44.2705	−1.62852	−0.814259	+	0.580502i	$0.802856\pi$
$740$	0	0
$741$	5.12461	0.188257
$742$	2.29180	0.0841345
$743$	−37.1803	−1.36401	−0.682007	−	0.731345i	$-0.738893\pi$
$743$	−37.1803	−1.36401	−0.682007	+	0.731345i	$0.738893\pi$
$744$	−19.7984	−0.725844
$745$	0	0
$746$	28.1803	1.03176
$747$	10.3820	0.379856
$748$	0	0
$749$	49.2492	1.79953
$750$	0	0
$751$	−17.4721	−0.637567	−0.318784	−	0.947828i	$-0.603274\pi$
$751$	−17.4721	−0.637567	−0.318784	+	0.947828i	$0.603274\pi$
$752$	45.5410	1.66071
$753$	20.5623	0.749332
$754$	10.8541	0.395283
$755$	0	0
$756$	−1.85410	−0.0674330
$757$	39.0344	1.41873	0.709365	−	0.704841i	$-0.248982\pi$
$757$	39.0344	1.41873	0.709365	+	0.704841i	$0.248982\pi$
$758$	−37.5623	−1.36432
$759$	0	0
$760$	0	0
$761$	0.111456	0.00404028	0.00202014	−	0.999998i	$-0.499357\pi$
$761$	0.111456	0.00404028	0.00202014	+	0.999998i	$0.499357\pi$
$762$	−10.7082	−0.387917
$763$	−45.0000	−1.62911
$764$	9.72949	0.352001
$765$	0	0
$766$	34.8328	1.25856
$767$	7.68692	0.277559
$768$	−13.5623	−0.489388
$769$	24.1459	0.870723	0.435362	−	0.900256i	$-0.356620\pi$
$769$	24.1459	0.870723	0.435362	+	0.900256i	$0.356620\pi$
$770$	0	0
$771$	10.0344	0.361382
$772$	11.4164	0.410886
$773$	21.6525	0.778785	0.389393	−	0.921072i	$-0.372685\pi$
$773$	21.6525	0.778785	0.389393	+	0.921072i	$0.372685\pi$
$774$	2.85410	0.102589
$775$	0	0
$776$	−23.2918	−0.836127
$777$	26.5623	0.952917
$778$	−29.2705	−1.04940
$779$	0.652476	0.0233774
$780$	0	0
$781$	0	0
$782$	−92.1033	−3.29361
$783$	3.61803	0.129298
$784$	−9.70820	−0.346722
$785$	0	0
$786$	11.8541	0.422822
$787$	−25.6180	−0.913184	−0.456592	−	0.889676i	$-0.650930\pi$
$787$	−25.6180	−0.913184	−0.456592	+	0.889676i	$0.650930\pi$
$788$	6.00000	0.213741
$789$	11.1246	0.396047
$790$	0	0
$791$	−25.4164	−0.903703
$792$	0	0
$793$	−10.6869	−0.379504
$794$	31.6525	1.12330
$795$	0	0
$796$	−5.12461	−0.181637
$797$	41.3951	1.46629	0.733145	−	0.680072i	$-0.238052\pi$
$797$	41.3951	1.46629	0.733145	+	0.680072i	$0.238052\pi$
$798$	−13.4164	−0.474936
$799$	−70.1033	−2.48008
$800$	0	0
$801$	2.56231	0.0905346
$802$	1.32624	0.0468311
$803$	0	0
$804$	−5.38197	−0.189807
$805$	0	0
$806$	26.5623	0.935617
$807$	0.527864	0.0185817
$808$	−2.43769	−0.0857578
$809$	−2.23607	−0.0786160	−0.0393080	−	0.999227i	$-0.512515\pi$
$809$	−2.23607	−0.0786160	−0.0393080	+	0.999227i	$0.512515\pi$
$810$	0	0
$811$	19.7082	0.692049	0.346024	−	0.938226i	$-0.387531\pi$
$811$	19.7082	0.692049	0.346024	+	0.938226i	$0.387531\pi$
$812$	−6.70820	−0.235412
$813$	−22.2705	−0.781061
$814$	0	0
$815$	0	0
$816$	36.2705	1.26972
$817$	4.87539	0.170568
$818$	16.3820	0.572782
$819$	−5.56231	−0.194363
$820$	0	0
$821$	−56.3951	−1.96820	−0.984102	−	0.177606i	$-0.943165\pi$
$821$	−56.3951	−1.96820	−0.984102	+	0.177606i	$0.943165\pi$
$822$	20.7082	0.722282
$823$	−3.47214	−0.121031	−0.0605155	−	0.998167i	$-0.519274\pi$
$823$	−3.47214	−0.121031	−0.0605155	+	0.998167i	$0.519274\pi$
$824$	33.9443	1.18250
$825$	0	0
$826$	−20.1246	−0.700225
$827$	46.8673	1.62973	0.814867	−	0.579648i	$-0.196810\pi$
$827$	46.8673	1.62973	0.814867	+	0.579648i	$0.196810\pi$
$828$	−4.70820	−0.163621
$829$	−25.3262	−0.879617	−0.439808	−	0.898092i	$-0.644954\pi$
$829$	−25.3262	−0.879617	−0.439808	+	0.898092i	$0.644954\pi$
$830$	0	0
$831$	−23.9787	−0.831812
$832$	−7.85410	−0.272292
$833$	14.9443	0.517788
$834$	9.47214	0.327993
$835$	0	0
$836$	0	0
$837$	8.85410	0.306043
$838$	30.6525	1.05887
$839$	24.7984	0.856135	0.428067	−	0.903747i	$-0.359195\pi$
$839$	24.7984	0.856135	0.428067	+	0.903747i	$0.359195\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	−11.7639	−0.405412
$843$	−3.32624	−0.114562
$844$	−8.14590	−0.280393
$845$	0	0
$846$	−15.1803	−0.521911
$847$	0	0
$848$	−2.29180	−0.0787006
$849$	−27.0902	−0.929732
$850$	0	0
$851$	67.4508	2.31219
$852$	−0.909830	−0.0311703
$853$	−53.6869	−1.83821	−0.919103	−	0.394018i	$-0.871085\pi$
$853$	−53.6869	−1.83821	−0.919103	+	0.394018i	$0.871085\pi$
$854$	27.9787	0.957412
$855$	0	0
$856$	−36.7082	−1.25466
$857$	−10.0902	−0.344674	−0.172337	−	0.985038i	$-0.555132\pi$
$857$	−10.0902	−0.344674	−0.172337	+	0.985038i	$0.555132\pi$
$858$	0	0
$859$	−2.56231	−0.0874247	−0.0437124	−	0.999044i	$-0.513919\pi$
$859$	−2.56231	−0.0874247	−0.0437124	+	0.999044i	$0.513919\pi$
$860$	0	0
$861$	−0.708204	−0.0241355
$862$	−26.1246	−0.889808
$863$	34.6180	1.17841	0.589206	−	0.807983i	$-0.299441\pi$
$863$	34.6180	1.17841	0.589206	+	0.807983i	$0.299441\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	43.9787	1.49446
$867$	−38.8328	−1.31883
$868$	−16.4164	−0.557209
$869$	0	0
$870$	0	0
$871$	−16.1459	−0.547083
$872$	33.5410	1.13584
$873$	10.4164	0.352542
$874$	−34.0689	−1.15240
$875$	0	0
$876$	4.23607	0.143123
$877$	29.5836	0.998967	0.499483	−	0.866323i	$-0.333523\pi$
$877$	29.5836	0.998967	0.499483	+	0.866323i	$0.333523\pi$
$878$	−12.0344	−0.406143
$879$	19.4164	0.654899
$880$	0	0
$881$	−34.1803	−1.15156	−0.575782	−	0.817603i	$-0.695302\pi$
$881$	−34.1803	−1.15156	−0.575782	+	0.817603i	$0.695302\pi$
$882$	3.23607	0.108964
$883$	−16.6869	−0.561559	−0.280780	−	0.959772i	$-0.590593\pi$
$883$	−16.6869	−0.561559	−0.280780	+	0.959772i	$0.590593\pi$
$884$	−8.56231	−0.287982
$885$	0	0
$886$	44.5066	1.49523
$887$	−18.1803	−0.610436	−0.305218	−	0.952282i	$-0.598729\pi$
$887$	−18.1803	−0.610436	−0.305218	+	0.952282i	$0.598729\pi$
$888$	−19.7984	−0.664390
$889$	19.8541	0.665885
$890$	0	0
$891$	0	0
$892$	7.85410	0.262975
$893$	−25.9311	−0.867752
$894$	26.1803	0.875602
$895$	0	0
$896$	40.8541	1.36484
$897$	−14.1246	−0.471607
$898$	23.4164	0.781416
$899$	32.0344	1.06841
$900$	0	0
$901$	3.52786	0.117530
$902$	0	0
$903$	−5.29180	−0.176100
$904$	18.9443	0.630077
$905$	0	0
$906$	−14.8541	−0.493494
$907$	53.8328	1.78749	0.893745	−	0.448576i	$-0.148069\pi$
$907$	53.8328	1.78749	0.893745	+	0.448576i	$0.148069\pi$
$908$	−9.00000	−0.298675
$909$	1.09017	0.0361587
$910$	0	0
$911$	54.0344	1.79024	0.895120	−	0.445824i	$-0.147089\pi$
$911$	54.0344	1.79024	0.895120	+	0.445824i	$0.147089\pi$
$912$	13.4164	0.444262
$913$	0	0
$914$	38.0689	1.25921
$915$	0	0
$916$	−6.18034	−0.204204
$917$	−21.9787	−0.725801
$918$	−12.0902	−0.399035
$919$	−15.8541	−0.522979	−0.261489	−	0.965206i	$-0.584214\pi$
$919$	−15.8541	−0.522979	−0.261489	+	0.965206i	$0.584214\pi$
$920$	0	0
$921$	−15.8885	−0.523545
$922$	14.8541	0.489194
$923$	−2.72949	−0.0898423
$924$	0	0
$925$	0	0
$926$	15.0344	0.494062
$927$	−15.1803	−0.498588
$928$	12.2361	0.401669
$929$	−54.4721	−1.78717	−0.893586	−	0.448891i	$-0.851819\pi$
$929$	−54.4721	−1.78717	−0.893586	+	0.448891i	$0.851819\pi$
$930$	0	0
$931$	5.52786	0.181168
$932$	−3.90983	−0.128071
$933$	18.5279	0.606575
$934$	8.03444	0.262895
$935$	0	0
$936$	4.14590	0.135513
$937$	36.4164	1.18967	0.594836	−	0.803847i	$-0.297217\pi$
$937$	36.4164	1.18967	0.594836	+	0.803847i	$0.297217\pi$
$938$	42.2705	1.38018
$939$	30.7082	1.00212
$940$	0	0
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	28.7984	0.938302
$943$	−1.79837	−0.0585631
$944$	20.1246	0.655000
$945$	0	0
$946$	0	0
$947$	−21.4934	−0.698442	−0.349221	−	0.937040i	$-0.613554\pi$
$947$	−21.4934	−0.698442	−0.349221	+	0.937040i	$0.613554\pi$
$948$	−4.67376	−0.151797
$949$	12.7082	0.412526
$950$	0	0
$951$	9.18034	0.297693
$952$	−50.1246	−1.62455
$953$	−55.7214	−1.80499	−0.902496	−	0.430698i	$-0.858267\pi$
$953$	−55.7214	−1.80499	−0.902496	+	0.430698i	$0.858267\pi$
$954$	0.763932	0.0247332
$955$	0	0
$956$	12.7639	0.412815
$957$	0	0
$958$	−11.9098	−0.384789
$959$	−38.3951	−1.23984
$960$	0	0
$961$	47.3951	1.52887
$962$	26.5623	0.856403
$963$	16.4164	0.529011
$964$	−4.85410	−0.156340
$965$	0	0
$966$	36.9787	1.18977
$967$	−5.21478	−0.167696	−0.0838480	−	0.996479i	$-0.526721\pi$
$967$	−5.21478	−0.167696	−0.0838480	+	0.996479i	$0.526721\pi$
$968$	0	0
$969$	−20.6525	−0.663453
$970$	0	0
$971$	59.4853	1.90897	0.954487	−	0.298253i	$-0.0964038\pi$
$971$	59.4853	1.90897	0.954487	+	0.298253i	$0.0964038\pi$
$972$	−0.618034	−0.0198234
$973$	−17.5623	−0.563022
$974$	2.05573	0.0658698
$975$	0	0
$976$	−27.9787	−0.895577
$977$	36.2705	1.16040	0.580198	−	0.814475i	$-0.302975\pi$
$977$	36.2705	1.16040	0.580198	+	0.814475i	$0.302975\pi$
$978$	27.6525	0.884229
$979$	0	0
$980$	0	0
$981$	−15.0000	−0.478913
$982$	22.4164	0.715336
$983$	0.875388	0.0279205	0.0139603	−	0.999903i	$-0.495556\pi$
$983$	0.875388	0.0279205	0.0139603	+	0.999903i	$0.495556\pi$
$984$	0.527864	0.0168277
$985$	0	0
$986$	−43.7426	−1.39305
$987$	28.1459	0.895894
$988$	−3.16718	−0.100762
$989$	−13.4377	−0.427294
$990$	0	0
$991$	−48.4508	−1.53909	−0.769546	−	0.638591i	$-0.779517\pi$
$991$	−48.4508	−1.53909	−0.769546	+	0.638591i	$0.779517\pi$
$992$	29.9443	0.950732
$993$	16.4164	0.520959
$994$	7.14590	0.226654
$995$	0	0
$996$	−6.41641	−0.203312
$997$	14.7082	0.465813	0.232907	−	0.972499i	$-0.425176\pi$
$997$	14.7082	0.465813	0.232907	+	0.972499i	$0.425176\pi$
$998$	19.1459	0.606053
$999$	8.85410	0.280131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bu.1.2		2
5.4	even	2		9075.2.a.bb.1.1		2
11.2	odd	10		825.2.n.e.301.1	yes	4
11.6	odd	10		825.2.n.e.751.1	yes	4
11.10	odd	2		9075.2.a.y.1.1		2
55.2	even	20		825.2.bx.a.499.1		8
55.13	even	20		825.2.bx.a.499.2		8
55.17	even	20		825.2.bx.a.124.2		8
55.24	odd	10		825.2.n.a.301.1	✓	4
55.28	even	20		825.2.bx.a.124.1		8
55.39	odd	10		825.2.n.a.751.1	yes	4
55.54	odd	2		9075.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.a.301.1	✓	4	55.24	odd	10
825.2.n.a.751.1	yes	4	55.39	odd	10
825.2.n.e.301.1	yes	4	11.2	odd	10
825.2.n.e.751.1	yes	4	11.6	odd	10
825.2.bx.a.124.1		8	55.28	even	20
825.2.bx.a.124.2		8	55.17	even	20
825.2.bx.a.499.1		8	55.2	even	20
825.2.bx.a.499.2		8	55.13	even	20
9075.2.a.y.1.1		2	11.10	odd	2
9075.2.a.bb.1.1		2	5.4	even	2
9075.2.a.bu.1.2		2	1.1	even	1	trivial
9075.2.a.bz.1.2		2	55.54	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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} -1.85410 q^{13} +4.85410 q^{14} -4.85410 q^{16} +7.47214 q^{17} +1.61803 q^{18} +2.76393 q^{19} -3.00000 q^{21} -7.61803 q^{23} +2.23607 q^{24} -3.00000 q^{26} -1.00000 q^{27} +1.85410 q^{28} -3.61803 q^{29} -8.85410 q^{31} -3.38197 q^{32} +12.0902 q^{34} +0.618034 q^{36} -8.85410 q^{37} +4.47214 q^{38} +1.85410 q^{39} +0.236068 q^{41} -4.85410 q^{42} +1.76393 q^{43} -12.3262 q^{46} -9.38197 q^{47} +4.85410 q^{48} +2.00000 q^{49} -7.47214 q^{51} -1.14590 q^{52} +0.472136 q^{53} -1.61803 q^{54} -6.70820 q^{56} -2.76393 q^{57} -5.85410 q^{58} -4.14590 q^{59} +5.76393 q^{61} -14.3262 q^{62} +3.00000 q^{63} +4.23607 q^{64} +8.70820 q^{67} +4.61803 q^{68} +7.61803 q^{69} +1.47214 q^{71} -2.23607 q^{72} -6.85410 q^{73} -14.3262 q^{74} +1.70820 q^{76} +3.00000 q^{78} +7.56231 q^{79} +1.00000 q^{81} +0.381966 q^{82} +10.3820 q^{83} -1.85410 q^{84} +2.85410 q^{86} +3.61803 q^{87} +2.56231 q^{89} -5.56231 q^{91} -4.70820 q^{92} +8.85410 q^{93} -15.1803 q^{94} +3.38197 q^{96} +10.4164 q^{97} +3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} + q^{18} + 10 q^{19} - 6 q^{21} - 13 q^{23} - 6 q^{26} - 2 q^{27} - 3 q^{28} - 5 q^{29} - 11 q^{31} - 9 q^{32} + 13 q^{34} - q^{36} - 11 q^{37} - 3 q^{39} - 4 q^{41} - 3 q^{42} + 8 q^{43} - 9 q^{46} - 21 q^{47} + 3 q^{48} + 4 q^{49} - 6 q^{51} - 9 q^{52} - 8 q^{53} - q^{54} - 10 q^{57} - 5 q^{58} - 15 q^{59} + 16 q^{61} - 13 q^{62} + 6 q^{63} + 4 q^{64} + 4 q^{67} + 7 q^{68} + 13 q^{69} - 6 q^{71} - 7 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} + 23 q^{83} + 3 q^{84} - q^{86} + 5 q^{87} - 15 q^{89} + 9 q^{91} + 4 q^{92} + 11 q^{93} - 8 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 6 * q^17 + q^18 + 10 * q^19 - 6 * q^21 - 13 * q^23 - 6 * q^26 - 2 * q^27 - 3 * q^28 - 5 * q^29 - 11 * q^31 - 9 * q^32 + 13 * q^34 - q^36 - 11 * q^37 - 3 * q^39 - 4 * q^41 - 3 * q^42 + 8 * q^43 - 9 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 - 6 * q^51 - 9 * q^52 - 8 * q^53 - q^54 - 10 * q^57 - 5 * q^58 - 15 * q^59 + 16 * q^61 - 13 * q^62 + 6 * q^63 + 4 * q^64 + 4 * q^67 + 7 * q^68 + 13 * q^69 - 6 * q^71 - 7 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 + 23 * q^83 + 3 * q^84 - q^86 + 5 * q^87 - 15 * q^89 + 9 * q^91 + 4 * q^92 + 11 * q^93 - 8 * q^94 + 9 * q^96 - 6 * q^97 + 2 * q^98

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