LMFDB - Embedded newform 9075.2.a.dr.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9075,2,Mod(1,9075)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,1,-6,9,0,-1,-4,0,6,0,0,-9,2,-10,0,11,2,1,2,0,4,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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			1.5
Root			$1.53672$ 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)

$f(q)$	$=$	\(q+1.53672 q^{2} -1.00000 q^{3} +0.361523 q^{4} -1.53672 q^{6} +2.86503 q^{7} -2.51789 q^{8} +1.00000 q^{9} -0.361523 q^{12} -2.43598 q^{13} +4.40277 q^{14} -4.59235 q^{16} +3.88488 q^{17} +1.53672 q^{18} -5.93848 q^{19} -2.86503 q^{21} +6.02732 q^{23} +2.51789 q^{24} -3.74343 q^{26} -1.00000 q^{27} +1.03578 q^{28} -4.03698 q^{29} -0.783128 q^{31} -2.02140 q^{32} +5.96999 q^{34} +0.361523 q^{36} -8.02406 q^{37} -9.12581 q^{38} +2.43598 q^{39} +7.54195 q^{41} -4.40277 q^{42} +9.90572 q^{43} +9.26233 q^{46} -0.551103 q^{47} +4.59235 q^{48} +1.20842 q^{49} -3.88488 q^{51} -0.880664 q^{52} -2.75907 q^{53} -1.53672 q^{54} -7.21383 q^{56} +5.93848 q^{57} -6.20372 q^{58} -14.5497 q^{59} -13.7901 q^{61} -1.20345 q^{62} +2.86503 q^{63} +6.07836 q^{64} -3.96724 q^{67} +1.40447 q^{68} -6.02732 q^{69} +9.27271 q^{71} -2.51789 q^{72} +0.615424 q^{73} -12.3308 q^{74} -2.14690 q^{76} +3.74343 q^{78} -4.53993 q^{79} +1.00000 q^{81} +11.5899 q^{82} -11.6797 q^{83} -1.03578 q^{84} +15.2224 q^{86} +4.03698 q^{87} +16.9894 q^{89} -6.97917 q^{91} +2.17902 q^{92} +0.783128 q^{93} -0.846893 q^{94} +2.02140 q^{96} -8.95511 q^{97} +1.85700 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 6 q^{3} + 9 q^{4} - q^{6} - 4 q^{7} + 6 q^{9} - 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}+ \cdots + 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

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 $

$2$	1.53672	1.08663	0.543314	−	0.839529i	$-0.317169\pi$
$2$	1.53672	1.08663	0.543314	+	0.839529i	$0.317169\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.361523	0.180762
$5$	0	0
$5$	0	0
$6$	−1.53672	−0.627365
$7$	2.86503	1.08288	0.541440	−	0.840739i	$-0.317879\pi$
$7$	2.86503	1.08288	0.541440	+	0.840739i	$0.317879\pi$
$8$	−2.51789	−0.890208
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.361523	−0.104363
$13$	−2.43598	−0.675620	−0.337810	−	0.941214i	$-0.609686\pi$
$13$	−2.43598	−0.675620	−0.337810	+	0.941214i	$0.609686\pi$
$14$	4.40277	1.17669
$15$	0	0
$16$	−4.59235	−1.14809
$17$	3.88488	0.942222	0.471111	−	0.882074i	$-0.343853\pi$
$17$	3.88488	0.942222	0.471111	+	0.882074i	$0.343853\pi$
$18$	1.53672	0.362210
$19$	−5.93848	−1.36238	−0.681191	−	0.732106i	$-0.738538\pi$
$19$	−5.93848	−1.36238	−0.681191	+	0.732106i	$0.738538\pi$
$20$	0	0
$21$	−2.86503	−0.625202
$22$	0	0
$23$	6.02732	1.25678	0.628392	−	0.777897i	$-0.283714\pi$
$23$	6.02732	1.25678	0.628392	+	0.777897i	$0.283714\pi$
$24$	2.51789	0.513962
$25$	0	0
$26$	−3.74343	−0.734148
$27$	−1.00000	−0.192450
$28$	1.03578	0.195743
$29$	−4.03698	−0.749648	−0.374824	−	0.927096i	$-0.622297\pi$
$29$	−4.03698	−0.749648	−0.374824	+	0.927096i	$0.622297\pi$
$30$	0	0
$31$	−0.783128	−0.140654	−0.0703270	−	0.997524i	$-0.522404\pi$
$31$	−0.783128	−0.140654	−0.0703270	+	0.997524i	$0.522404\pi$
$32$	−2.02140	−0.357336
$33$	0	0
$34$	5.96999	1.02385
$35$	0	0
$36$	0.361523	0.0602539
$37$	−8.02406	−1.31915	−0.659574	−	0.751640i	$-0.729263\pi$
$37$	−8.02406	−1.31915	−0.659574	+	0.751640i	$0.729263\pi$
$38$	−9.12581	−1.48040
$39$	2.43598	0.390069
$40$	0	0
$41$	7.54195	1.17785	0.588927	−	0.808186i	$-0.299550\pi$
$41$	7.54195	1.17785	0.588927	+	0.808186i	$0.299550\pi$
$42$	−4.40277	−0.679362
$43$	9.90572	1.51061	0.755304	−	0.655374i	$-0.227489\pi$
$43$	9.90572	1.51061	0.755304	+	0.655374i	$0.227489\pi$
$44$	0	0
$45$	0	0
$46$	9.26233	1.36566
$47$	−0.551103	−0.0803866	−0.0401933	−	0.999192i	$-0.512797\pi$
$47$	−0.551103	−0.0803866	−0.0401933	+	0.999192i	$0.512797\pi$
$48$	4.59235	0.662848
$49$	1.20842	0.172631
$50$	0	0
$51$	−3.88488	−0.543992
$52$	−0.880664	−0.122126
$53$	−2.75907	−0.378987	−0.189493	−	0.981882i	$-0.560685\pi$
$53$	−2.75907	−0.378987	−0.189493	+	0.981882i	$0.560685\pi$
$54$	−1.53672	−0.209122
$55$	0	0
$56$	−7.21383	−0.963989
$57$	5.93848	0.786571
$58$	−6.20372	−0.814589
$59$	−14.5497	−1.89421	−0.947103	−	0.320931i	$-0.896004\pi$
$59$	−14.5497	−1.89421	−0.947103	+	0.320931i	$0.896004\pi$
$60$	0	0
$61$	−13.7901	−1.76565	−0.882824	−	0.469704i	$-0.844361\pi$
$61$	−13.7901	−1.76565	−0.882824	+	0.469704i	$0.844361\pi$
$62$	−1.20345	−0.152839
$63$	2.86503	0.360960
$64$	6.07836	0.759795
$65$	0	0
$66$	0	0
$67$	−3.96724	−0.484675	−0.242338	−	0.970192i	$-0.577914\pi$
$67$	−3.96724	−0.484675	−0.242338	+	0.970192i	$0.577914\pi$
$68$	1.40447	0.170318
$69$	−6.02732	−0.725604
$70$	0	0
$71$	9.27271	1.10047	0.550234	−	0.835010i	$-0.314538\pi$
$71$	9.27271	1.10047	0.550234	+	0.835010i	$0.314538\pi$
$72$	−2.51789	−0.296736
$73$	0.615424	0.0720300	0.0360150	−	0.999351i	$-0.488534\pi$
$73$	0.615424	0.0720300	0.0360150	+	0.999351i	$0.488534\pi$
$74$	−12.3308	−1.43342
$75$	0	0
$76$	−2.14690	−0.246266
$77$	0	0
$78$	3.74343	0.423861
$79$	−4.53993	−0.510782	−0.255391	−	0.966838i	$-0.582204\pi$
$79$	−4.53993	−0.510782	−0.255391	+	0.966838i	$0.582204\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.5899	1.27989
$83$	−11.6797	−1.28202	−0.641008	−	0.767534i	$-0.721483\pi$
$83$	−11.6797	−1.28202	−0.641008	+	0.767534i	$0.721483\pi$
$84$	−1.03578	−0.113012
$85$	0	0
$86$	15.2224	1.64147
$87$	4.03698	0.432809
$88$	0	0
$89$	16.9894	1.80087	0.900437	−	0.434986i	$-0.143247\pi$
$89$	16.9894	1.80087	0.900437	+	0.434986i	$0.143247\pi$
$90$	0	0
$91$	−6.97917	−0.731616
$92$	2.17902	0.227178
$93$	0.783128	0.0812066
$94$	−0.846893	−0.0873504
$95$	0	0
$96$	2.02140	0.206308
$97$	−8.95511	−0.909254	−0.454627	−	0.890682i	$-0.650227\pi$
$97$	−8.95511	−0.909254	−0.454627	+	0.890682i	$0.650227\pi$
$98$	1.85700	0.187586
$99$	0	0
$100$	0	0
$101$	4.05035	0.403024	0.201512	−	0.979486i	$-0.435414\pi$
$101$	4.05035	0.403024	0.201512	+	0.979486i	$0.435414\pi$
$102$	−5.96999	−0.591117
$103$	−3.09729	−0.305185	−0.152593	−	0.988289i	$-0.548762\pi$
$103$	−3.09729	−0.305185	−0.152593	+	0.988289i	$0.548762\pi$
$104$	6.13353	0.601442
$105$	0	0
$106$	−4.23992	−0.411818
$107$	5.42307	0.524268	0.262134	−	0.965032i	$-0.415574\pi$
$107$	5.42307	0.524268	0.262134	+	0.965032i	$0.415574\pi$
$108$	−0.361523	−0.0347876
$109$	2.29776	0.220085	0.110043	−	0.993927i	$-0.464901\pi$
$109$	2.29776	0.220085	0.110043	+	0.993927i	$0.464901\pi$
$110$	0	0
$111$	8.02406	0.761610
$112$	−13.1572	−1.24324
$113$	9.61495	0.904498	0.452249	−	0.891892i	$-0.350622\pi$
$113$	9.61495	0.904498	0.452249	+	0.891892i	$0.350622\pi$
$114$	9.12581	0.854711
$115$	0	0
$116$	−1.45946	−0.135508
$117$	−2.43598	−0.225207
$118$	−22.3588	−2.05830
$119$	11.1303	1.02031
$120$	0	0
$121$	0	0
$122$	−21.1917	−1.91860
$123$	−7.54195	−0.680035
$124$	−0.283119	−0.0254248
$125$	0	0
$126$	4.40277	0.392230
$127$	−19.6140	−1.74046	−0.870229	−	0.492648i	$-0.836029\pi$
$127$	−19.6140	−1.74046	−0.870229	+	0.492648i	$0.836029\pi$
$128$	13.3836	1.18295
$129$	−9.90572	−0.872150
$130$	0	0
$131$	−4.00843	−0.350218	−0.175109	−	0.984549i	$-0.556028\pi$
$131$	−4.00843	−0.350218	−0.175109	+	0.984549i	$0.556028\pi$
$132$	0	0
$133$	−17.0140	−1.47530
$134$	−6.09655	−0.526662
$135$	0	0
$136$	−9.78169	−0.838773
$137$	−23.1747	−1.97995	−0.989975	−	0.141242i	$-0.954891\pi$
$137$	−23.1747	−1.97995	−0.989975	+	0.141242i	$0.954891\pi$
$138$	−9.26233	−0.788462
$139$	13.8238	1.17252	0.586260	−	0.810123i	$-0.300600\pi$
$139$	13.8238	1.17252	0.586260	+	0.810123i	$0.300600\pi$
$140$	0	0
$141$	0.551103	0.0464112
$142$	14.2496	1.19580
$143$	0	0
$144$	−4.59235	−0.382696
$145$	0	0
$146$	0.945738	0.0782698
$147$	−1.20842	−0.0996685
$148$	−2.90088	−0.238451
$149$	−13.5293	−1.10836	−0.554181	−	0.832396i	$-0.686968\pi$
$149$	−13.5293	−1.10836	−0.554181	+	0.832396i	$0.686968\pi$
$150$	0	0
$151$	−21.7316	−1.76849	−0.884245	−	0.467024i	$-0.845326\pi$
$151$	−21.7316	−1.76849	−0.884245	+	0.467024i	$0.845326\pi$
$152$	14.9524	1.21280
$153$	3.88488	0.314074
$154$	0	0
$155$	0	0
$156$	0.880664	0.0705096
$157$	10.0477	0.801894	0.400947	−	0.916101i	$-0.368681\pi$
$157$	10.0477	0.801894	0.400947	+	0.916101i	$0.368681\pi$
$158$	−6.97662	−0.555030
$159$	2.75907	0.218808
$160$	0	0
$161$	17.2685	1.36095
$162$	1.53672	0.120737
$163$	−17.0119	−1.33248	−0.666239	−	0.745739i	$-0.732097\pi$
$163$	−17.0119	−1.33248	−0.666239	+	0.745739i	$0.732097\pi$
$164$	2.72659	0.212911
$165$	0	0
$166$	−17.9485	−1.39308
$167$	8.17422	0.632540	0.316270	−	0.948669i	$-0.397569\pi$
$167$	8.17422	0.632540	0.316270	+	0.948669i	$0.397569\pi$
$168$	7.21383	0.556559
$169$	−7.06599	−0.543538
$170$	0	0
$171$	−5.93848	−0.454127
$172$	3.58115	0.273060
$173$	−0.0419137	−0.00318664	−0.00159332	−	0.999999i	$-0.500507\pi$
$173$	−0.0419137	−0.00318664	−0.00159332	+	0.999999i	$0.500507\pi$
$174$	6.20372	0.470303
$175$	0	0
$176$	0	0
$177$	14.5497	1.09362
$178$	26.1081	1.95688
$179$	−21.5288	−1.60914	−0.804570	−	0.593857i	$-0.797604\pi$
$179$	−21.5288	−1.60914	−0.804570	+	0.593857i	$0.797604\pi$
$180$	0	0
$181$	10.9246	0.812020	0.406010	−	0.913869i	$-0.366920\pi$
$181$	10.9246	0.812020	0.406010	+	0.913869i	$0.366920\pi$
$182$	−10.7251	−0.794995
$183$	13.7901	1.01940
$184$	−15.1761	−1.11880
$185$	0	0
$186$	1.20345	0.0882414
$187$	0	0
$188$	−0.199236	−0.0145308
$189$	−2.86503	−0.208401
$190$	0	0
$191$	−0.485281	−0.0351137	−0.0175569	−	0.999846i	$-0.505589\pi$
$191$	−0.485281	−0.0351137	−0.0175569	+	0.999846i	$0.505589\pi$
$192$	−6.07836	−0.438668
$193$	12.5864	0.905989	0.452994	−	0.891513i	$-0.350356\pi$
$193$	12.5864	0.905989	0.452994	+	0.891513i	$0.350356\pi$
$194$	−13.7615	−0.988021
$195$	0	0
$196$	0.436870	0.0312050
$197$	−15.8941	−1.13241	−0.566204	−	0.824265i	$-0.691588\pi$
$197$	−15.8941	−1.13241	−0.566204	+	0.824265i	$0.691588\pi$
$198$	0	0
$199$	−5.02089	−0.355922	−0.177961	−	0.984038i	$-0.556950\pi$
$199$	−5.02089	−0.355922	−0.177961	+	0.984038i	$0.556950\pi$
$200$	0	0
$201$	3.96724	0.279827
$202$	6.22427	0.437938
$203$	−11.5661	−0.811779
$204$	−1.40447	−0.0983329
$205$	0	0
$206$	−4.75969	−0.331623
$207$	6.02732	0.418928
$208$	11.1869	0.775670
$209$	0	0
$210$	0	0
$211$	−0.800040	−0.0550770	−0.0275385	−	0.999621i	$-0.508767\pi$
$211$	−0.800040	−0.0550770	−0.0275385	+	0.999621i	$0.508767\pi$
$212$	−0.997466	−0.0685063
$213$	−9.27271	−0.635356
$214$	8.33376	0.569684
$215$	0	0
$216$	2.51789	0.171321
$217$	−2.24369	−0.152311
$218$	3.53102	0.239151
$219$	−0.615424	−0.0415865
$220$	0	0
$221$	−9.46350	−0.636584
$222$	12.3308	0.827587
$223$	8.10773	0.542933	0.271467	−	0.962448i	$-0.412491\pi$
$223$	8.10773	0.542933	0.271467	+	0.962448i	$0.412491\pi$
$224$	−5.79137	−0.386952
$225$	0	0
$226$	14.7755	0.982853
$227$	8.70426	0.577722	0.288861	−	0.957371i	$-0.406723\pi$
$227$	8.70426	0.577722	0.288861	+	0.957371i	$0.406723\pi$
$228$	2.14690	0.142182
$229$	29.6210	1.95741	0.978705	−	0.205272i	$-0.0658080\pi$
$229$	29.6210	1.95741	0.978705	+	0.205272i	$0.0658080\pi$
$230$	0	0
$231$	0	0
$232$	10.1647	0.667342
$233$	−6.71715	−0.440055	−0.220028	−	0.975494i	$-0.570615\pi$
$233$	−6.71715	−0.440055	−0.220028	+	0.975494i	$0.570615\pi$
$234$	−3.74343	−0.244716
$235$	0	0
$236$	−5.26004	−0.342400
$237$	4.53993	0.294900
$238$	17.1042	1.10870
$239$	−2.23252	−0.144409	−0.0722047	−	0.997390i	$-0.523003\pi$
$239$	−2.23252	−0.144409	−0.0722047	+	0.997390i	$0.523003\pi$
$240$	0	0
$241$	8.20298	0.528400	0.264200	−	0.964468i	$-0.414892\pi$
$241$	8.20298	0.528400	0.264200	+	0.964468i	$0.414892\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−4.98546	−0.319161
$245$	0	0
$246$	−11.5899	−0.738945
$247$	14.4660	0.920452
$248$	1.97183	0.125211
$249$	11.6797	0.740172
$250$	0	0
$251$	−0.900721	−0.0568530	−0.0284265	−	0.999596i	$-0.509050\pi$
$251$	−0.900721	−0.0568530	−0.0284265	+	0.999596i	$0.509050\pi$
$252$	1.03578	0.0652478
$253$	0	0
$254$	−30.1413	−1.89123
$255$	0	0
$256$	8.41013	0.525633
$257$	−4.27720	−0.266804	−0.133402	−	0.991062i	$-0.542590\pi$
$257$	−4.27720	−0.266804	−0.133402	+	0.991062i	$0.542590\pi$
$258$	−15.2224	−0.947703
$259$	−22.9892	−1.42848
$260$	0	0
$261$	−4.03698	−0.249883
$262$	−6.15986	−0.380557
$263$	−3.64614	−0.224831	−0.112415	−	0.993661i	$-0.535859\pi$
$263$	−3.64614	−0.224831	−0.112415	+	0.993661i	$0.535859\pi$
$264$	0	0
$265$	0	0
$266$	−26.1458	−1.60310
$267$	−16.9894	−1.03974
$268$	−1.43425	−0.0876107
$269$	−11.2872	−0.688192	−0.344096	−	0.938934i	$-0.611815\pi$
$269$	−11.2872	−0.688192	−0.344096	+	0.938934i	$0.611815\pi$
$270$	0	0
$271$	−9.58466	−0.582227	−0.291113	−	0.956689i	$-0.594026\pi$
$271$	−9.58466	−0.582227	−0.291113	+	0.956689i	$0.594026\pi$
$272$	−17.8407	−1.08175
$273$	6.97917	0.422399
$274$	−35.6132	−2.15147
$275$	0	0
$276$	−2.17902	−0.131161
$277$	−26.9810	−1.62113	−0.810565	−	0.585648i	$-0.800840\pi$
$277$	−26.9810	−1.62113	−0.810565	+	0.585648i	$0.800840\pi$
$278$	21.2434	1.27409
$279$	−0.783128	−0.0468846
$280$	0	0
$281$	−18.9918	−1.13296	−0.566479	−	0.824076i	$-0.691695\pi$
$281$	−18.9918	−1.13296	−0.566479	+	0.824076i	$0.691695\pi$
$282$	0.846893	0.0504317
$283$	−27.0944	−1.61060	−0.805298	−	0.592870i	$-0.797995\pi$
$283$	−27.0944	−1.61060	−0.805298	+	0.592870i	$0.797995\pi$
$284$	3.35230	0.198922
$285$	0	0
$286$	0	0
$287$	21.6079	1.27548
$288$	−2.02140	−0.119112
$289$	−1.90771	−0.112218
$290$	0	0
$291$	8.95511	0.524958
$292$	0.222490	0.0130202
$293$	−9.07599	−0.530225	−0.265112	−	0.964218i	$-0.585409\pi$
$293$	−9.07599	−0.530225	−0.265112	+	0.964218i	$0.585409\pi$
$294$	−1.85700	−0.108303
$295$	0	0
$296$	20.2037	1.17432
$297$	0	0
$298$	−20.7908	−1.20438
$299$	−14.6824	−0.849108
$300$	0	0
$301$	28.3802	1.63581
$302$	−33.3954	−1.92169
$303$	−4.05035	−0.232686
$304$	27.2716	1.56413
$305$	0	0
$306$	5.96999	0.341282
$307$	−6.10964	−0.348696	−0.174348	−	0.984684i	$-0.555782\pi$
$307$	−6.10964	−0.348696	−0.174348	+	0.984684i	$0.555782\pi$
$308$	0	0
$309$	3.09729	0.176199
$310$	0	0
$311$	−14.8604	−0.842655	−0.421327	−	0.906909i	$-0.638436\pi$
$311$	−14.8604	−0.842655	−0.421327	+	0.906909i	$0.638436\pi$
$312$	−6.13353	−0.347243
$313$	−27.4142	−1.54954	−0.774772	−	0.632240i	$-0.782136\pi$
$313$	−27.4142	−1.54954	−0.774772	+	0.632240i	$0.782136\pi$
$314$	15.4406	0.871361
$315$	0	0
$316$	−1.64129	−0.0923298
$317$	−13.9328	−0.782543	−0.391272	−	0.920275i	$-0.627965\pi$
$317$	−13.9328	−0.782543	−0.391272	+	0.920275i	$0.627965\pi$
$318$	4.23992	0.237763
$319$	0	0
$320$	0	0
$321$	−5.42307	−0.302686
$322$	26.5369	1.47884
$323$	−23.0703	−1.28367
$324$	0.361523	0.0200846
$325$	0	0
$326$	−26.1427	−1.44791
$327$	−2.29776	−0.127066
$328$	−18.9898	−1.04854
$329$	−1.57893	−0.0870491
$330$	0	0
$331$	24.9385	1.37074	0.685371	−	0.728194i	$-0.259640\pi$
$331$	24.9385	1.37074	0.685371	+	0.728194i	$0.259640\pi$
$332$	−4.22249	−0.231739
$333$	−8.02406	−0.439716
$334$	12.5615	0.687336
$335$	0	0
$336$	13.1572	0.717786
$337$	16.2380	0.884541	0.442271	−	0.896882i	$-0.354173\pi$
$337$	16.2380	0.884541	0.442271	+	0.896882i	$0.354173\pi$
$338$	−10.8585	−0.590623
$339$	−9.61495	−0.522212
$340$	0	0
$341$	0	0
$342$	−9.12581	−0.493468
$343$	−16.5931	−0.895942
$344$	−24.9415	−1.34476
$345$	0	0
$346$	−0.0644098	−0.00346269
$347$	27.4616	1.47422	0.737109	−	0.675774i	$-0.236191\pi$
$347$	27.4616	1.47422	0.737109	+	0.675774i	$0.236191\pi$
$348$	1.45946	0.0782353
$349$	25.1370	1.34555	0.672777	−	0.739845i	$-0.265101\pi$
$349$	25.1370	1.34555	0.672777	+	0.739845i	$0.265101\pi$
$350$	0	0
$351$	2.43598	0.130023
$352$	0	0
$353$	−12.2362	−0.651267	−0.325633	−	0.945496i	$-0.605577\pi$
$353$	−12.2362	−0.651267	−0.325633	+	0.945496i	$0.605577\pi$
$354$	22.3588	1.18836
$355$	0	0
$356$	6.14207	0.325529
$357$	−11.1303	−0.589079
$358$	−33.0839	−1.74854
$359$	−3.88413	−0.204997	−0.102498	−	0.994733i	$-0.532684\pi$
$359$	−3.88413	−0.204997	−0.102498	+	0.994733i	$0.532684\pi$
$360$	0	0
$361$	16.2656	0.856083
$362$	16.7881	0.882364
$363$	0	0
$364$	−2.52313	−0.132248
$365$	0	0
$366$	21.1917	1.10771
$367$	19.5703	1.02156	0.510780	−	0.859712i	$-0.329357\pi$
$367$	19.5703	1.02156	0.510780	+	0.859712i	$0.329357\pi$
$368$	−27.6795	−1.44290
$369$	7.54195	0.392618
$370$	0	0
$371$	−7.90482	−0.410398
$372$	0.283119	0.0146790
$373$	−0.729978	−0.0377968	−0.0188984	−	0.999821i	$-0.506016\pi$
$373$	−0.729978	−0.0377968	−0.0188984	+	0.999821i	$0.506016\pi$
$374$	0	0
$375$	0	0
$376$	1.38761	0.0715608
$377$	9.83401	0.506477
$378$	−4.40277	−0.226454
$379$	36.0155	1.84999	0.924997	−	0.379975i	$-0.124067\pi$
$379$	36.0155	1.84999	0.924997	+	0.379975i	$0.124067\pi$
$380$	0	0
$381$	19.6140	1.00485
$382$	−0.745744	−0.0381556
$383$	−12.4551	−0.636428	−0.318214	−	0.948019i	$-0.603083\pi$
$383$	−12.4551	−0.636428	−0.318214	+	0.948019i	$0.603083\pi$
$384$	−13.3836	−0.682977
$385$	0	0
$386$	19.3418	0.984473
$387$	9.90572	0.503536
$388$	−3.23748	−0.164358
$389$	−20.4612	−1.03742	−0.518711	−	0.854950i	$-0.673588\pi$
$389$	−20.4612	−1.03742	−0.518711	+	0.854950i	$0.673588\pi$
$390$	0	0
$391$	23.4154	1.18417
$392$	−3.04266	−0.153677
$393$	4.00843	0.202199
$394$	−24.4249	−1.23051
$395$	0	0
$396$	0	0
$397$	−23.9046	−1.19974	−0.599869	−	0.800098i	$-0.704781\pi$
$397$	−23.9046	−1.19974	−0.599869	+	0.800098i	$0.704781\pi$
$398$	−7.71573	−0.386755
$399$	17.0140	0.851763
$400$	0	0
$401$	−9.45101	−0.471961	−0.235980	−	0.971758i	$-0.575830\pi$
$401$	−9.45101	−0.471961	−0.235980	+	0.971758i	$0.575830\pi$
$402$	6.09655	0.304068
$403$	1.90769	0.0950286
$404$	1.46429	0.0728513
$405$	0	0
$406$	−17.7739	−0.882103
$407$	0	0
$408$	9.78169	0.484266
$409$	−7.69520	−0.380503	−0.190252	−	0.981735i	$-0.560930\pi$
$409$	−7.69520	−0.380503	−0.190252	+	0.981735i	$0.560930\pi$
$410$	0	0
$411$	23.1747	1.14312
$412$	−1.11974	−0.0551658
$413$	−41.6853	−2.05120
$414$	9.26233	0.455219
$415$	0	0
$416$	4.92409	0.241423
$417$	−13.8238	−0.676955
$418$	0	0
$419$	−30.0599	−1.46852	−0.734262	−	0.678866i	$-0.762472\pi$
$419$	−30.0599	−1.46852	−0.734262	+	0.678866i	$0.762472\pi$
$420$	0	0
$421$	−23.8194	−1.16088	−0.580442	−	0.814301i	$-0.697120\pi$
$421$	−23.8194	−1.16088	−0.580442	+	0.814301i	$0.697120\pi$
$422$	−1.22944	−0.0598482
$423$	−0.551103	−0.0267955
$424$	6.94702	0.337377
$425$	0	0
$426$	−14.2496	−0.690396
$427$	−39.5092	−1.91199
$428$	1.96056	0.0947675
$429$	0	0
$430$	0	0
$431$	−0.608493	−0.0293101	−0.0146550	−	0.999893i	$-0.504665\pi$
$431$	−0.608493	−0.0293101	−0.0146550	+	0.999893i	$0.504665\pi$
$432$	4.59235	0.220949
$433$	17.1030	0.821919	0.410959	−	0.911654i	$-0.365194\pi$
$433$	17.1030	0.821919	0.410959	+	0.911654i	$0.365194\pi$
$434$	−3.44793	−0.165506
$435$	0	0
$436$	0.830693	0.0397830
$437$	−35.7931	−1.71222
$438$	−0.945738	−0.0451891
$439$	11.5579	0.551628	0.275814	−	0.961211i	$-0.411053\pi$
$439$	11.5579	0.551628	0.275814	+	0.961211i	$0.411053\pi$
$440$	0	0
$441$	1.20842	0.0575436
$442$	−14.5428	−0.691730
$443$	−17.7741	−0.844471	−0.422235	−	0.906486i	$-0.638754\pi$
$443$	−17.7741	−0.844471	−0.422235	+	0.906486i	$0.638754\pi$
$444$	2.90088	0.137670
$445$	0	0
$446$	12.4593	0.589967
$447$	13.5293	0.639913
$448$	17.4147	0.822768
$449$	21.7125	1.02467	0.512337	−	0.858784i	$-0.328780\pi$
$449$	21.7125	1.02467	0.512337	+	0.858784i	$0.328780\pi$
$450$	0	0
$451$	0	0
$452$	3.47603	0.163499
$453$	21.7316	1.02104
$454$	13.3761	0.627769
$455$	0	0
$456$	−14.9524	−0.700212
$457$	0.874148	0.0408909	0.0204455	−	0.999791i	$-0.493492\pi$
$457$	0.874148	0.0408909	0.0204455	+	0.999791i	$0.493492\pi$
$458$	45.5193	2.12698
$459$	−3.88488	−0.181331
$460$	0	0
$461$	6.51959	0.303648	0.151824	−	0.988408i	$-0.451485\pi$
$461$	6.51959	0.303648	0.151824	+	0.988408i	$0.451485\pi$
$462$	0	0
$463$	−12.5689	−0.584124	−0.292062	−	0.956399i	$-0.594341\pi$
$463$	−12.5689	−0.584124	−0.292062	+	0.956399i	$0.594341\pi$
$464$	18.5392	0.860661
$465$	0	0
$466$	−10.3224	−0.478177
$467$	26.3110	1.21753	0.608764	−	0.793352i	$-0.291666\pi$
$467$	26.3110	1.21753	0.608764	+	0.793352i	$0.291666\pi$
$468$	−0.880664	−0.0407087
$469$	−11.3663	−0.524846
$470$	0	0
$471$	−10.0477	−0.462974
$472$	36.6344	1.68624
$473$	0	0
$474$	6.97662	0.320447
$475$	0	0
$476$	4.02387	0.184434
$477$	−2.75907	−0.126329
$478$	−3.43076	−0.156919
$479$	−6.99425	−0.319575	−0.159788	−	0.987151i	$-0.551081\pi$
$479$	−6.99425	−0.319575	−0.159788	+	0.987151i	$0.551081\pi$
$480$	0	0
$481$	19.5465	0.891243
$482$	12.6057	0.574175
$483$	−17.2685	−0.785743
$484$	0	0
$485$	0	0
$486$	−1.53672	−0.0697073
$487$	6.08580	0.275774	0.137887	−	0.990448i	$-0.455969\pi$
$487$	6.08580	0.275774	0.137887	+	0.990448i	$0.455969\pi$
$488$	34.7220	1.57179
$489$	17.0119	0.769306
$490$	0	0
$491$	4.58064	0.206722	0.103361	−	0.994644i	$-0.467040\pi$
$491$	4.58064	0.206722	0.103361	+	0.994644i	$0.467040\pi$
$492$	−2.72659	−0.122924
$493$	−15.6832	−0.706335
$494$	22.2303	1.00019
$495$	0	0
$496$	3.59640	0.161483
$497$	26.5666	1.19168
$498$	17.9485	0.804292
$499$	−21.2025	−0.949152	−0.474576	−	0.880214i	$-0.657399\pi$
$499$	−21.2025	−0.949152	−0.474576	+	0.880214i	$0.657399\pi$
$500$	0	0
$501$	−8.17422	−0.365197
$502$	−1.38416	−0.0617781
$503$	−14.8190	−0.660746	−0.330373	−	0.943850i	$-0.607175\pi$
$503$	−14.8190	−0.660746	−0.330373	+	0.943850i	$0.607175\pi$
$504$	−7.21383	−0.321330
$505$	0	0
$506$	0	0
$507$	7.06599	0.313812
$508$	−7.09090	−0.314608
$509$	−23.6991	−1.05044	−0.525222	−	0.850965i	$-0.676018\pi$
$509$	−23.6991	−1.05044	−0.525222	+	0.850965i	$0.676018\pi$
$510$	0	0
$511$	1.76321	0.0779999
$512$	−13.8431	−0.611783
$513$	5.93848	0.262190
$514$	−6.57287	−0.289917
$515$	0	0
$516$	−3.58115	−0.157651
$517$	0	0
$518$	−35.3281	−1.55223
$519$	0.0419137	0.00183981
$520$	0	0
$521$	−34.8182	−1.52541	−0.762707	−	0.646745i	$-0.776130\pi$
$521$	−34.8182	−1.52541	−0.762707	+	0.646745i	$0.776130\pi$
$522$	−6.20372	−0.271530
$523$	−16.6591	−0.728453	−0.364226	−	0.931310i	$-0.618667\pi$
$523$	−16.6591	−0.728453	−0.364226	+	0.931310i	$0.618667\pi$
$524$	−1.44914	−0.0633060
$525$	0	0
$526$	−5.60312	−0.244307
$527$	−3.04236	−0.132527
$528$	0	0
$529$	13.3286	0.579504
$530$	0	0
$531$	−14.5497	−0.631402
$532$	−6.15094	−0.266677
$533$	−18.3721	−0.795782
$534$	−26.1081	−1.12981
$535$	0	0
$536$	9.98906	0.431462
$537$	21.5288	0.929038
$538$	−17.3453	−0.747810
$539$	0	0
$540$	0	0
$541$	4.99608	0.214798	0.107399	−	0.994216i	$-0.465748\pi$
$541$	4.99608	0.214798	0.107399	+	0.994216i	$0.465748\pi$
$542$	−14.7290	−0.632664
$543$	−10.9246	−0.468820
$544$	−7.85289	−0.336690
$545$	0	0
$546$	10.7251	0.458990
$547$	−27.0209	−1.15533	−0.577665	−	0.816274i	$-0.696036\pi$
$547$	−27.0209	−1.15533	−0.577665	+	0.816274i	$0.696036\pi$
$548$	−8.37820	−0.357899
$549$	−13.7901	−0.588549
$550$	0	0
$551$	23.9735	1.02131
$552$	15.1761	0.645938
$553$	−13.0071	−0.553116
$554$	−41.4624	−1.76157
$555$	0	0
$556$	4.99763	0.211947
$557$	−8.26537	−0.350215	−0.175107	−	0.984549i	$-0.556027\pi$
$557$	−8.26537	−0.350215	−0.175107	+	0.984549i	$0.556027\pi$
$558$	−1.20345	−0.0509462
$559$	−24.1302	−1.02060
$560$	0	0
$561$	0	0
$562$	−29.1852	−1.23110
$563$	18.9867	0.800194	0.400097	−	0.916473i	$-0.368976\pi$
$563$	18.9867	0.800194	0.400097	+	0.916473i	$0.368976\pi$
$564$	0.199236	0.00838936
$565$	0	0
$566$	−41.6367	−1.75012
$567$	2.86503	0.120320
$568$	−23.3477	−0.979646
$569$	43.0309	1.80395	0.901974	−	0.431790i	$-0.142118\pi$
$569$	43.0309	1.80395	0.901974	+	0.431790i	$0.142118\pi$
$570$	0	0
$571$	8.68683	0.363533	0.181766	−	0.983342i	$-0.441819\pi$
$571$	8.68683	0.363533	0.181766	+	0.983342i	$0.441819\pi$
$572$	0	0
$573$	0.485281	0.0202729
$574$	33.2055	1.38597
$575$	0	0
$576$	6.07836	0.253265
$577$	11.0409	0.459639	0.229819	−	0.973233i	$-0.426186\pi$
$577$	11.0409	0.459639	0.229819	+	0.973233i	$0.426186\pi$
$578$	−2.93162	−0.121939
$579$	−12.5864	−0.523073
$580$	0	0
$581$	−33.4628	−1.38827
$582$	13.7615	0.570434
$583$	0	0
$584$	−1.54957	−0.0641216
$585$	0	0
$586$	−13.9473	−0.576157
$587$	−26.1401	−1.07892	−0.539459	−	0.842012i	$-0.681371\pi$
$587$	−26.1401	−1.07892	−0.539459	+	0.842012i	$0.681371\pi$
$588$	−0.436870	−0.0180162
$589$	4.65059	0.191624
$590$	0	0
$591$	15.8941	0.653796
$592$	36.8493	1.51450
$593$	12.0171	0.493484	0.246742	−	0.969081i	$-0.420640\pi$
$593$	12.0171	0.493484	0.246742	+	0.969081i	$0.420640\pi$
$594$	0	0
$595$	0	0
$596$	−4.89115	−0.200349
$597$	5.02089	0.205491
$598$	−22.5629	−0.922665
$599$	−41.8072	−1.70820	−0.854098	−	0.520112i	$-0.825890\pi$
$599$	−41.8072	−1.70820	−0.854098	+	0.520112i	$0.825890\pi$
$600$	0	0
$601$	40.5539	1.65423	0.827115	−	0.562033i	$-0.189981\pi$
$601$	40.5539	1.65423	0.827115	+	0.562033i	$0.189981\pi$
$602$	43.6126	1.77752
$603$	−3.96724	−0.161558
$604$	−7.85646	−0.319675
$605$	0	0
$606$	−6.22427	−0.252844
$607$	−29.2269	−1.18628	−0.593141	−	0.805099i	$-0.702112\pi$
$607$	−29.2269	−1.18628	−0.593141	+	0.805099i	$0.702112\pi$
$608$	12.0040	0.486828
$609$	11.5661	0.468681
$610$	0	0
$611$	1.34248	0.0543108
$612$	1.40447	0.0567725
$613$	−20.1259	−0.812876	−0.406438	−	0.913678i	$-0.633229\pi$
$613$	−20.1259	−0.812876	−0.406438	+	0.913678i	$0.633229\pi$
$614$	−9.38884	−0.378903
$615$	0	0
$616$	0	0
$617$	8.98070	0.361550	0.180775	−	0.983525i	$-0.442139\pi$
$617$	8.98070	0.361550	0.180775	+	0.983525i	$0.442139\pi$
$618$	4.75969	0.191463
$619$	24.0303	0.965860	0.482930	−	0.875659i	$-0.339573\pi$
$619$	24.0303	0.965860	0.482930	+	0.875659i	$0.339573\pi$
$620$	0	0
$621$	−6.02732	−0.241868
$622$	−22.8363	−0.915653
$623$	48.6752	1.95013
$624$	−11.1869	−0.447834
$625$	0	0
$626$	−42.1282	−1.68378
$627$	0	0
$628$	3.63248	0.144952
$629$	−31.1725	−1.24293
$630$	0	0
$631$	23.0323	0.916902	0.458451	−	0.888720i	$-0.348404\pi$
$631$	23.0323	0.916902	0.458451	+	0.888720i	$0.348404\pi$
$632$	11.4310	0.454702
$633$	0.800040	0.0317987
$634$	−21.4109	−0.850334
$635$	0	0
$636$	0.997466	0.0395521
$637$	−2.94368	−0.116633
$638$	0	0
$639$	9.27271	0.366823
$640$	0	0
$641$	17.8410	0.704677	0.352339	−	0.935873i	$-0.385387\pi$
$641$	17.8410	0.704677	0.352339	+	0.935873i	$0.385387\pi$
$642$	−8.33376	−0.328907
$643$	−21.4561	−0.846147	−0.423074	−	0.906095i	$-0.639049\pi$
$643$	−21.4561	−0.846147	−0.423074	+	0.906095i	$0.639049\pi$
$644$	6.24295	0.246007
$645$	0	0
$646$	−35.4527	−1.39487
$647$	−7.91168	−0.311040	−0.155520	−	0.987833i	$-0.549705\pi$
$647$	−7.91168	−0.311040	−0.155520	+	0.987833i	$0.549705\pi$
$648$	−2.51789	−0.0989120
$649$	0	0
$650$	0	0
$651$	2.24369	0.0879371
$652$	−6.15021	−0.240861
$653$	−20.7286	−0.811171	−0.405585	−	0.914057i	$-0.632932\pi$
$653$	−20.7286	−0.811171	−0.405585	+	0.914057i	$0.632932\pi$
$654$	−3.53102	−0.138074
$655$	0	0
$656$	−34.6353	−1.35228
$657$	0.615424	0.0240100
$658$	−2.42638	−0.0945900
$659$	44.9515	1.75106	0.875531	−	0.483162i	$-0.160512\pi$
$659$	44.9515	1.75106	0.875531	+	0.483162i	$0.160512\pi$
$660$	0	0
$661$	32.0760	1.24761	0.623806	−	0.781579i	$-0.285585\pi$
$661$	32.0760	1.24761	0.623806	+	0.781579i	$0.285585\pi$
$662$	38.3235	1.48949
$663$	9.46350	0.367532
$664$	29.4082	1.14126
$665$	0	0
$666$	−12.3308	−0.477808
$667$	−24.3322	−0.942145
$668$	2.95517	0.114339
$669$	−8.10773	−0.313463
$670$	0	0
$671$	0	0
$672$	5.79137	0.223407
$673$	11.0442	0.425723	0.212861	−	0.977082i	$-0.431722\pi$
$673$	11.0442	0.425723	0.212861	+	0.977082i	$0.431722\pi$
$674$	24.9534	0.961168
$675$	0	0
$676$	−2.55452	−0.0982507
$677$	−7.99883	−0.307420	−0.153710	−	0.988116i	$-0.549122\pi$
$677$	−7.99883	−0.307420	−0.153710	+	0.988116i	$0.549122\pi$
$678$	−14.7755	−0.567451
$679$	−25.6567	−0.984613
$680$	0	0
$681$	−8.70426	−0.333548
$682$	0	0
$683$	16.6442	0.636873	0.318436	−	0.947944i	$-0.396842\pi$
$683$	16.6442	0.636873	0.318436	+	0.947944i	$0.396842\pi$
$684$	−2.14690	−0.0820887
$685$	0	0
$686$	−25.4990	−0.973556
$687$	−29.6210	−1.13011
$688$	−45.4905	−1.73431
$689$	6.72104	0.256051
$690$	0	0
$691$	−12.4814	−0.474815	−0.237408	−	0.971410i	$-0.576298\pi$
$691$	−12.4814	−0.474815	−0.237408	+	0.971410i	$0.576298\pi$
$692$	−0.0151528	−0.000576022 0
$693$	0	0
$694$	42.2010	1.60193
$695$	0	0
$696$	−10.1647	−0.385290
$697$	29.2996	1.10980
$698$	38.6287	1.46212
$699$	6.71715	0.254066
$700$	0	0
$701$	−3.13045	−0.118235	−0.0591177	−	0.998251i	$-0.518829\pi$
$701$	−3.13045	−0.118235	−0.0591177	+	0.998251i	$0.518829\pi$
$702$	3.74343	0.141287
$703$	47.6508	1.79718
$704$	0	0
$705$	0	0
$706$	−18.8037	−0.707685
$707$	11.6044	0.436427
$708$	5.26004	0.197684
$709$	−8.71471	−0.327288	−0.163644	−	0.986519i	$-0.552325\pi$
$709$	−8.71471	−0.327288	−0.163644	+	0.986519i	$0.552325\pi$
$710$	0	0
$711$	−4.53993	−0.170261
$712$	−42.7774	−1.60315
$713$	−4.72016	−0.176771
$714$	−17.1042	−0.640110
$715$	0	0
$716$	−7.78317	−0.290871
$717$	2.23252	0.0833748
$718$	−5.96884	−0.222755
$719$	10.3566	0.386237	0.193118	−	0.981175i	$-0.438140\pi$
$719$	10.3566	0.386237	0.193118	+	0.981175i	$0.438140\pi$
$720$	0	0
$721$	−8.87385	−0.330479
$722$	24.9957	0.930244
$723$	−8.20298	−0.305072
$724$	3.94950	0.146782
$725$	0	0
$726$	0	0
$727$	−18.2456	−0.676693	−0.338346	−	0.941022i	$-0.609868\pi$
$727$	−18.2456	−0.676693	−0.338346	+	0.941022i	$0.609868\pi$
$728$	17.5728	0.651290
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.4825	1.42333
$732$	4.98546	0.184268
$733$	3.14121	0.116023	0.0580116	−	0.998316i	$-0.481524\pi$
$733$	3.14121	0.116023	0.0580116	+	0.998316i	$0.481524\pi$
$734$	30.0741	1.11006
$735$	0	0
$736$	−12.1836	−0.449094
$737$	0	0
$738$	11.5899	0.426630
$739$	33.9751	1.24980	0.624898	−	0.780706i	$-0.285141\pi$
$739$	33.9751	1.24980	0.624898	+	0.780706i	$0.285141\pi$
$740$	0	0
$741$	−14.4660	−0.531423
$742$	−12.1475	−0.445950
$743$	40.2606	1.47702	0.738510	−	0.674243i	$-0.235530\pi$
$743$	40.2606	1.47702	0.738510	+	0.674243i	$0.235530\pi$
$744$	−1.97183	−0.0722907
$745$	0	0
$746$	−1.12178	−0.0410711
$747$	−11.6797	−0.427339
$748$	0	0
$749$	15.5373	0.567719
$750$	0	0
$751$	39.7342	1.44992	0.724960	−	0.688791i	$-0.241858\pi$
$751$	39.7342	1.44992	0.724960	+	0.688791i	$0.241858\pi$
$752$	2.53085	0.0922908
$753$	0.900721	0.0328241
$754$	15.1122	0.550352
$755$	0	0
$756$	−1.03578	−0.0376708
$757$	−51.6449	−1.87707	−0.938533	−	0.345188i	$-0.887815\pi$
$757$	−51.6449	−1.87707	−0.938533	+	0.345188i	$0.887815\pi$
$758$	55.3460	2.01026
$759$	0	0
$760$	0	0
$761$	39.8286	1.44379	0.721893	−	0.692004i	$-0.243272\pi$
$761$	39.8286	1.44379	0.721893	+	0.692004i	$0.243272\pi$
$762$	30.1413	1.09190
$763$	6.58315	0.238326
$764$	−0.175440	−0.00634721
$765$	0	0
$766$	−19.1401	−0.691560
$767$	35.4427	1.27976
$768$	−8.41013	−0.303475
$769$	−46.2949	−1.66944	−0.834719	−	0.550676i	$-0.814370\pi$
$769$	−46.2949	−1.66944	−0.834719	+	0.550676i	$0.814370\pi$
$770$	0	0
$771$	4.27720	0.154039
$772$	4.55028	0.163768
$773$	−7.01535	−0.252325	−0.126162	−	0.992010i	$-0.540266\pi$
$773$	−7.01535	−0.252325	−0.126162	+	0.992010i	$0.540266\pi$
$774$	15.2224	0.547157
$775$	0	0
$776$	22.5480	0.809425
$777$	22.9892	0.824733
$778$	−31.4432	−1.12729
$779$	−44.7877	−1.60469
$780$	0	0
$781$	0	0
$782$	35.9830	1.28675
$783$	4.03698	0.144270
$784$	−5.54947	−0.198195
$785$	0	0
$786$	6.15986	0.219715
$787$	−2.77476	−0.0989095	−0.0494548	−	0.998776i	$-0.515748\pi$
$787$	−2.77476	−0.0989095	−0.0494548	+	0.998776i	$0.515748\pi$
$788$	−5.74608	−0.204696
$789$	3.64614	0.129806
$790$	0	0
$791$	27.5471	0.979464
$792$	0	0
$793$	33.5926	1.19291
$794$	−36.7348	−1.30367
$795$	0	0
$796$	−1.81517	−0.0643370
$797$	−42.9669	−1.52197	−0.760983	−	0.648772i	$-0.775283\pi$
$797$	−42.9669	−1.52197	−0.760983	+	0.648772i	$0.775283\pi$
$798$	26.1458	0.925550
$799$	−2.14097	−0.0757420
$800$	0	0
$801$	16.9894	0.600291
$802$	−14.5236	−0.512846
$803$	0	0
$804$	1.43425	0.0505821
$805$	0	0
$806$	2.93159	0.103261
$807$	11.2872	0.397328
$808$	−10.1983	−0.358775
$809$	33.2867	1.17030	0.585149	−	0.810926i	$-0.301036\pi$
$809$	33.2867	1.17030	0.585149	+	0.810926i	$0.301036\pi$
$810$	0	0
$811$	37.8675	1.32971	0.664855	−	0.746973i	$-0.268493\pi$
$811$	37.8675	1.32971	0.664855	+	0.746973i	$0.268493\pi$
$812$	−4.18140	−0.146739
$813$	9.58466	0.336149
$814$	0	0
$815$	0	0
$816$	17.8407	0.624550
$817$	−58.8250	−2.05802
$818$	−11.8254	−0.413465
$819$	−6.97917	−0.243872
$820$	0	0
$821$	15.5524	0.542782	0.271391	−	0.962469i	$-0.412516\pi$
$821$	15.5524	0.542782	0.271391	+	0.962469i	$0.412516\pi$
$822$	35.6132	1.24215
$823$	19.7207	0.687421	0.343710	−	0.939076i	$-0.388316\pi$
$823$	19.7207	0.687421	0.343710	+	0.939076i	$0.388316\pi$
$824$	7.79864	0.271678
$825$	0	0
$826$	−64.0588	−2.22889
$827$	−5.77933	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$827$	−5.77933	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$828$	2.17902	0.0757260
$829$	34.0085	1.18116	0.590581	−	0.806978i	$-0.298898\pi$
$829$	34.0085	1.18116	0.590581	+	0.806978i	$0.298898\pi$
$830$	0	0
$831$	26.9810	0.935960
$832$	−14.8068	−0.513333
$833$	4.69455	0.162657
$834$	−21.2434	−0.735599
$835$	0	0
$836$	0	0
$837$	0.783128	0.0270689
$838$	−46.1939	−1.59574
$839$	38.0383	1.31323	0.656614	−	0.754226i	$-0.271988\pi$
$839$	38.0383	1.31323	0.656614	+	0.754226i	$0.271988\pi$
$840$	0	0
$841$	−12.7028	−0.438028
$842$	−36.6038	−1.26145
$843$	18.9918	0.654114
$844$	−0.289233	−0.00995580
$845$	0	0
$846$	−0.846893	−0.0291168
$847$	0	0
$848$	12.6706	0.435110
$849$	27.0944	0.929878
$850$	0	0
$851$	−48.3636	−1.65788
$852$	−3.35230	−0.114848
$853$	48.9458	1.67587	0.837936	−	0.545769i	$-0.183762\pi$
$853$	48.9458	1.67587	0.837936	+	0.545769i	$0.183762\pi$
$854$	−60.7148	−2.07762
$855$	0	0
$856$	−13.6547	−0.466707
$857$	27.4604	0.938030	0.469015	−	0.883190i	$-0.344609\pi$
$857$	27.4604	0.938030	0.469015	+	0.883190i	$0.344609\pi$
$858$	0	0
$859$	−1.25914	−0.0429613	−0.0214806	−	0.999769i	$-0.506838\pi$
$859$	−1.25914	−0.0429613	−0.0214806	+	0.999769i	$0.506838\pi$
$860$	0	0
$861$	−21.6079	−0.736397
$862$	−0.935086	−0.0318491
$863$	−26.7154	−0.909404	−0.454702	−	0.890644i	$-0.650254\pi$
$863$	−26.7154	−0.909404	−0.454702	+	0.890644i	$0.650254\pi$
$864$	2.02140	0.0687694
$865$	0	0
$866$	26.2826	0.893120
$867$	1.90771	0.0647892
$868$	−0.811145	−0.0275321
$869$	0	0
$870$	0	0
$871$	9.66412	0.327456
$872$	−5.78550	−0.195922
$873$	−8.95511	−0.303085
$874$	−55.0042	−1.86054
$875$	0	0
$876$	−0.222490	−0.00751724
$877$	26.7109	0.901962	0.450981	−	0.892533i	$-0.351074\pi$
$877$	26.7109	0.901962	0.450981	+	0.892533i	$0.351074\pi$
$878$	17.7613	0.599415
$879$	9.07599	0.306125
$880$	0	0
$881$	−31.7144	−1.06849	−0.534243	−	0.845331i	$-0.679403\pi$
$881$	−31.7144	−1.06849	−0.534243	+	0.845331i	$0.679403\pi$
$882$	1.85700	0.0625286
$883$	47.8895	1.61161	0.805804	−	0.592182i	$-0.201733\pi$
$883$	47.8895	1.61161	0.805804	+	0.592182i	$0.201733\pi$
$884$	−3.42127	−0.115070
$885$	0	0
$886$	−27.3138	−0.917626
$887$	34.0800	1.14429	0.572147	−	0.820151i	$-0.306111\pi$
$887$	34.0800	1.14429	0.572147	+	0.820151i	$0.306111\pi$
$888$	−20.2037	−0.677991
$889$	−56.1947	−1.88471
$890$	0	0
$891$	0	0
$892$	2.93113	0.0981415
$893$	3.27271	0.109517
$894$	20.7908	0.695348
$895$	0	0
$896$	38.3444	1.28100
$897$	14.6824	0.490233
$898$	33.3661	1.11344
$899$	3.16147	0.105441
$900$	0	0
$901$	−10.7186	−0.357090
$902$	0	0
$903$	−28.3802	−0.944435
$904$	−24.2094	−0.805191
$905$	0	0
$906$	33.3954	1.10949
$907$	22.5941	0.750226	0.375113	−	0.926979i	$-0.377604\pi$
$907$	22.5941	0.750226	0.375113	+	0.926979i	$0.377604\pi$
$908$	3.14679	0.104430
$909$	4.05035	0.134341
$910$	0	0
$911$	−15.8393	−0.524779	−0.262390	−	0.964962i	$-0.584511\pi$
$911$	−15.8393	−0.524779	−0.262390	+	0.964962i	$0.584511\pi$
$912$	−27.2716	−0.903052
$913$	0	0
$914$	1.34333	0.0444333
$915$	0	0
$916$	10.7087	0.353825
$917$	−11.4843	−0.379245
$918$	−5.96999	−0.197039
$919$	0.360925	0.0119058	0.00595291	−	0.999982i	$-0.498105\pi$
$919$	0.360925	0.0119058	0.00595291	+	0.999982i	$0.498105\pi$
$920$	0	0
$921$	6.10964	0.201320
$922$	10.0188	0.329952
$923$	−22.5882	−0.743499
$924$	0	0
$925$	0	0
$926$	−19.3149	−0.634726
$927$	−3.09729	−0.101728
$928$	8.16034	0.267876
$929$	0.480345	0.0157596	0.00787980	−	0.999969i	$-0.497492\pi$
$929$	0.480345	0.0157596	0.00787980	+	0.999969i	$0.497492\pi$
$930$	0	0
$931$	−7.17616	−0.235189
$932$	−2.42841	−0.0795451
$933$	14.8604	0.486507
$934$	40.4328	1.32300
$935$	0	0
$936$	6.13353	0.200481
$937$	−52.1442	−1.70348	−0.851739	−	0.523967i	$-0.824452\pi$
$937$	−52.1442	−1.70348	−0.851739	+	0.523967i	$0.824452\pi$
$938$	−17.4668	−0.570312
$939$	27.4142	0.894630
$940$	0	0
$941$	14.9475	0.487275	0.243637	−	0.969866i	$-0.421659\pi$
$941$	14.9475	0.487275	0.243637	+	0.969866i	$0.421659\pi$
$942$	−15.4406	−0.503081
$943$	45.4577	1.48031
$944$	66.8171	2.17471
$945$	0	0
$946$	0	0
$947$	27.2440	0.885311	0.442656	−	0.896692i	$-0.354036\pi$
$947$	27.2440	0.885311	0.442656	+	0.896692i	$0.354036\pi$
$948$	1.64129	0.0533066
$949$	−1.49916	−0.0486649
$950$	0	0
$951$	13.9328	0.451802
$952$	−28.0249	−0.908291
$953$	−37.8748	−1.22688	−0.613442	−	0.789740i	$-0.710216\pi$
$953$	−37.8748	−1.22688	−0.613442	+	0.789740i	$0.710216\pi$
$954$	−4.23992	−0.137273
$955$	0	0
$956$	−0.807106	−0.0261037
$957$	0	0
$958$	−10.7482	−0.347260
$959$	−66.3964	−2.14405
$960$	0	0
$961$	−30.3867	−0.980216
$962$	30.0376	0.968450
$963$	5.42307	0.174756
$964$	2.96557	0.0955144
$965$	0	0
$966$	−26.5369	−0.853811
$967$	−43.6845	−1.40480	−0.702400	−	0.711783i	$-0.747888\pi$
$967$	−43.6845	−1.40480	−0.702400	+	0.711783i	$0.747888\pi$
$968$	0	0
$969$	23.0703	0.741125
$970$	0	0
$971$	−2.73545	−0.0877848	−0.0438924	−	0.999036i	$-0.513976\pi$
$971$	−2.73545	−0.0877848	−0.0438924	+	0.999036i	$0.513976\pi$
$972$	−0.361523	−0.0115959
$973$	39.6057	1.26970
$974$	9.35220	0.299664
$975$	0	0
$976$	63.3292	2.02712
$977$	−36.5804	−1.17031	−0.585156	−	0.810921i	$-0.698967\pi$
$977$	−36.5804	−1.17031	−0.585156	+	0.810921i	$0.698967\pi$
$978$	26.1427	0.835950
$979$	0	0
$980$	0	0
$981$	2.29776	0.0733618
$982$	7.03919	0.224630
$983$	−6.44168	−0.205458	−0.102729	−	0.994709i	$-0.532757\pi$
$983$	−6.44168	−0.205458	−0.102729	+	0.994709i	$0.532757\pi$
$984$	18.9898	0.605372
$985$	0	0
$986$	−24.1007	−0.767523
$987$	1.57893	0.0502578
$988$	5.22981	0.166382
$989$	59.7050	1.89851
$990$	0	0
$991$	−13.5718	−0.431123	−0.215561	−	0.976490i	$-0.569158\pi$
$991$	−13.5718	−0.431123	−0.215561	+	0.976490i	$0.569158\pi$
$992$	1.58301	0.0502607
$993$	−24.9385	−0.791398
$994$	40.8256	1.29491
$995$	0	0
$996$	4.22249	0.133795
$997$	1.84851	0.0585429	0.0292715	−	0.999571i	$-0.490681\pi$
$997$	1.84851	0.0585429	0.0292715	+	0.999571i	$0.490681\pi$
$998$	−32.5823	−1.03138
$999$	8.02406	0.253870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dr.1.5		6
5.2	odd	4		1815.2.c.h.364.9	yes	12
5.3	odd	4		1815.2.c.h.364.4	✓	12
5.4	even	2		9075.2.a.dp.1.2		6
11.10	odd	2		9075.2.a.do.1.2		6
55.32	even	4		1815.2.c.i.364.4	yes	12
55.43	even	4		1815.2.c.i.364.9	yes	12
55.54	odd	2		9075.2.a.ds.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.c.h.364.4	✓	12	5.3	odd	4
1815.2.c.h.364.9	yes	12	5.2	odd	4
1815.2.c.i.364.4	yes	12	55.32	even	4
1815.2.c.i.364.9	yes	12	55.43	even	4
9075.2.a.do.1.2		6	11.10	odd	2
9075.2.a.dp.1.2		6	5.4	even	2
9075.2.a.dr.1.5		6	1.1	even	1	trivial
9075.2.a.ds.1.5		6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.53672 q^{2} -1.00000 q^{3} +0.361523 q^{4} -1.53672 q^{6} +2.86503 q^{7} -2.51789 q^{8} +1.00000 q^{9} -0.361523 q^{12} -2.43598 q^{13} +4.40277 q^{14} -4.59235 q^{16} +3.88488 q^{17} +1.53672 q^{18} -5.93848 q^{19} -2.86503 q^{21} +6.02732 q^{23} +2.51789 q^{24} -3.74343 q^{26} -1.00000 q^{27} +1.03578 q^{28} -4.03698 q^{29} -0.783128 q^{31} -2.02140 q^{32} +5.96999 q^{34} +0.361523 q^{36} -8.02406 q^{37} -9.12581 q^{38} +2.43598 q^{39} +7.54195 q^{41} -4.40277 q^{42} +9.90572 q^{43} +9.26233 q^{46} -0.551103 q^{47} +4.59235 q^{48} +1.20842 q^{49} -3.88488 q^{51} -0.880664 q^{52} -2.75907 q^{53} -1.53672 q^{54} -7.21383 q^{56} +5.93848 q^{57} -6.20372 q^{58} -14.5497 q^{59} -13.7901 q^{61} -1.20345 q^{62} +2.86503 q^{63} +6.07836 q^{64} -3.96724 q^{67} +1.40447 q^{68} -6.02732 q^{69} +9.27271 q^{71} -2.51789 q^{72} +0.615424 q^{73} -12.3308 q^{74} -2.14690 q^{76} +3.74343 q^{78} -4.53993 q^{79} +1.00000 q^{81} +11.5899 q^{82} -11.6797 q^{83} -1.03578 q^{84} +15.2224 q^{86} +4.03698 q^{87} +16.9894 q^{89} -6.97917 q^{91} +2.17902 q^{92} +0.783128 q^{93} -0.846893 q^{94} +2.02140 q^{96} -8.95511 q^{97} +1.85700 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 6 q^{3} + 9 q^{4} - q^{6} - 4 q^{7} + 6 q^{9} - 9 q^{12} + 2 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} + q^{18} + 2 q^{19} + 4 q^{21} - 12 q^{23} - 6 q^{27} - 24 q^{28} - 12 q^{29} + 18 q^{31}+ \cdots + 53 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

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Representations

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Database

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