LMFDB - Embedded newform 9065.2.a.bc.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] 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("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [34,-2,10,30,-34,8,0,-6,28,2,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.559083 q^{2} +0.530934 q^{3} -1.68743 q^{4} -1.00000 q^{5} +0.296836 q^{6} -2.06158 q^{8} -2.71811 q^{9} -0.559083 q^{10} -2.04956 q^{11} -0.895912 q^{12} -1.94110 q^{13} -0.530934 q^{15} +2.22226 q^{16} -6.52022 q^{17} -1.51965 q^{18} +3.41391 q^{19} +1.68743 q^{20} -1.14588 q^{22} -1.41393 q^{23} -1.09456 q^{24} +1.00000 q^{25} -1.08524 q^{26} -3.03594 q^{27} -4.24571 q^{29} -0.296836 q^{30} +0.765169 q^{31} +5.36558 q^{32} -1.08818 q^{33} -3.64534 q^{34} +4.58661 q^{36} +1.00000 q^{37} +1.90866 q^{38} -1.03060 q^{39} +2.06158 q^{40} -8.48866 q^{41} -11.3123 q^{43} +3.45849 q^{44} +2.71811 q^{45} -0.790503 q^{46} -7.94442 q^{47} +1.17987 q^{48} +0.559083 q^{50} -3.46181 q^{51} +3.27547 q^{52} +1.16492 q^{53} -1.69734 q^{54} +2.04956 q^{55} +1.81256 q^{57} -2.37371 q^{58} +1.16578 q^{59} +0.895912 q^{60} +0.714569 q^{61} +0.427793 q^{62} -1.44471 q^{64} +1.94110 q^{65} -0.608385 q^{66} +7.56750 q^{67} +11.0024 q^{68} -0.750702 q^{69} -12.5740 q^{71} +5.60359 q^{72} -3.08360 q^{73} +0.559083 q^{74} +0.530934 q^{75} -5.76072 q^{76} -0.576190 q^{78} +9.66550 q^{79} -2.22226 q^{80} +6.54244 q^{81} -4.74587 q^{82} +14.6622 q^{83} +6.52022 q^{85} -6.32450 q^{86} -2.25419 q^{87} +4.22534 q^{88} -12.7892 q^{89} +1.51965 q^{90} +2.38590 q^{92} +0.406254 q^{93} -4.44159 q^{94} -3.41391 q^{95} +2.84877 q^{96} +13.0641 q^{97} +5.57094 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) $ 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * q^99

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 $

$2$	0.559083	0.395332	0.197666	−	0.980269i	$-0.436664\pi$
$2$	0.559083	0.395332	0.197666	+	0.980269i	$0.436664\pi$
$3$	0.530934	0.306535	0.153267	−	0.988185i	$-0.451020\pi$
$3$	0.530934	0.306535	0.153267	+	0.988185i	$0.451020\pi$
$4$	−1.68743	−0.843713
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.296836	0.121183
$7$	0	0
$7$	0	0
$8$	−2.06158	−0.728878
$9$	−2.71811	−0.906036
$10$	−0.559083	−0.176798
$11$	−2.04956	−0.617967	−0.308983	−	0.951067i	$-0.599989\pi$
$11$	−2.04956	−0.617967	−0.308983	+	0.951067i	$0.599989\pi$
$12$	−0.895912	−0.258627
$13$	−1.94110	−0.538365	−0.269183	−	0.963089i	$-0.586754\pi$
$13$	−1.94110	−0.538365	−0.269183	+	0.963089i	$0.586754\pi$
$14$	0	0
$15$	−0.530934	−0.137087
$16$	2.22226	0.555564
$17$	−6.52022	−1.58138	−0.790692	−	0.612214i	$-0.790279\pi$
$17$	−6.52022	−1.58138	−0.790692	+	0.612214i	$0.790279\pi$
$18$	−1.51965	−0.358185
$19$	3.41391	0.783204	0.391602	−	0.920135i	$-0.371921\pi$
$19$	3.41391	0.783204	0.391602	+	0.920135i	$0.371921\pi$
$20$	1.68743	0.377320
$21$	0	0
$22$	−1.14588	−0.244302
$23$	−1.41393	−0.294824	−0.147412	−	0.989075i	$-0.547094\pi$
$23$	−1.41393	−0.294824	−0.147412	+	0.989075i	$0.547094\pi$
$24$	−1.09456	−0.223427
$25$	1.00000	0.200000
$26$	−1.08524	−0.212833
$27$	−3.03594	−0.584267
$28$	0	0
$29$	−4.24571	−0.788409	−0.394204	−	0.919023i	$-0.628980\pi$
$29$	−4.24571	−0.788409	−0.394204	+	0.919023i	$0.628980\pi$
$30$	−0.296836	−0.0541947
$31$	0.765169	0.137428	0.0687142	−	0.997636i	$-0.478110\pi$
$31$	0.765169	0.137428	0.0687142	+	0.997636i	$0.478110\pi$
$32$	5.36558	0.948510
$33$	−1.08818	−0.189428
$34$	−3.64534	−0.625171
$35$	0	0
$36$	4.58661	0.764435
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	1.90866	0.309625
$39$	−1.03060	−0.165028
$40$	2.06158	0.325964
$41$	−8.48866	−1.32571	−0.662853	−	0.748750i	$-0.730655\pi$
$41$	−8.48866	−1.32571	−0.662853	+	0.748750i	$0.730655\pi$
$42$	0	0
$43$	−11.3123	−1.72510	−0.862552	−	0.505969i	$-0.831135\pi$
$43$	−11.3123	−1.72510	−0.862552	+	0.505969i	$0.831135\pi$
$44$	3.45849	0.521386
$45$	2.71811	0.405192
$46$	−0.790503	−0.116553
$47$	−7.94442	−1.15881	−0.579406	−	0.815039i	$-0.696715\pi$
$47$	−7.94442	−1.15881	−0.579406	+	0.815039i	$0.696715\pi$
$48$	1.17987	0.170300
$49$	0	0
$50$	0.559083	0.0790663
$51$	−3.46181	−0.484750
$52$	3.27547	0.454226
$53$	1.16492	0.160014	0.0800069	−	0.996794i	$-0.474506\pi$
$53$	1.16492	0.160014	0.0800069	+	0.996794i	$0.474506\pi$
$54$	−1.69734	−0.230979
$55$	2.04956	0.276363
$56$	0	0
$57$	1.81256	0.240080
$58$	−2.37371	−0.311683
$59$	1.16578	0.151772	0.0758859	−	0.997117i	$-0.475822\pi$
$59$	1.16578	0.151772	0.0758859	+	0.997117i	$0.475822\pi$
$60$	0.895912	0.115662
$61$	0.714569	0.0914912	0.0457456	−	0.998953i	$-0.485434\pi$
$61$	0.714569	0.0914912	0.0457456	+	0.998953i	$0.485434\pi$
$62$	0.427793	0.0543298
$63$	0	0
$64$	−1.44471	−0.180588
$65$	1.94110	0.240764
$66$	−0.608385	−0.0748870
$67$	7.56750	0.924517	0.462259	−	0.886745i	$-0.347039\pi$
$67$	7.56750	0.924517	0.462259	+	0.886745i	$0.347039\pi$
$68$	11.0024	1.33423
$69$	−0.750702	−0.0903739
$70$	0	0
$71$	−12.5740	−1.49226	−0.746130	−	0.665800i	$-0.768091\pi$
$71$	−12.5740	−1.49226	−0.746130	+	0.665800i	$0.768091\pi$
$72$	5.60359	0.660390
$73$	−3.08360	−0.360908	−0.180454	−	0.983583i	$-0.557757\pi$
$73$	−3.08360	−0.360908	−0.180454	+	0.983583i	$0.557757\pi$
$74$	0.559083	0.0649921
$75$	0.530934	0.0613070
$76$	−5.76072	−0.660800
$77$	0	0
$78$	−0.576190	−0.0652407
$79$	9.66550	1.08745	0.543727	−	0.839262i	$-0.317013\pi$
$79$	9.66550	1.08745	0.543727	+	0.839262i	$0.317013\pi$
$80$	−2.22226	−0.248456
$81$	6.54244	0.726938
$82$	−4.74587	−0.524093
$83$	14.6622	1.60939	0.804695	−	0.593688i	$-0.202329\pi$
$83$	14.6622	1.60939	0.804695	+	0.593688i	$0.202329\pi$
$84$	0	0
$85$	6.52022	0.707217
$86$	−6.32450	−0.681988
$87$	−2.25419	−0.241675
$88$	4.22534	0.450422
$89$	−12.7892	−1.35565	−0.677826	−	0.735222i	$-0.737078\pi$
$89$	−12.7892	−1.35565	−0.677826	+	0.735222i	$0.737078\pi$
$90$	1.51965	0.160185
$91$	0	0
$92$	2.38590	0.248747
$93$	0.406254	0.0421266
$94$	−4.44159	−0.458115
$95$	−3.41391	−0.350260
$96$	2.84877	0.290751
$97$	13.0641	1.32646	0.663229	−	0.748416i	$-0.269185\pi$
$97$	13.0641	1.32646	0.663229	+	0.748416i	$0.269185\pi$
$98$	0	0
$99$	5.57094	0.559900
$100$	−1.68743	−0.168743
$101$	−14.0771	−1.40072	−0.700362	−	0.713788i	$-0.746978\pi$
$101$	−14.0771	−1.40072	−0.700362	+	0.713788i	$0.746978\pi$
$102$	−1.93544	−0.191637
$103$	−8.28546	−0.816390	−0.408195	−	0.912895i	$-0.633842\pi$
$103$	−8.28546	−0.816390	−0.408195	+	0.912895i	$0.633842\pi$
$104$	4.00174	0.392403
$105$	0	0
$106$	0.651286	0.0632585
$107$	−7.63947	−0.738535	−0.369268	−	0.929323i	$-0.620391\pi$
$107$	−7.63947	−0.738535	−0.369268	+	0.929323i	$0.620391\pi$
$108$	5.12292	0.492953
$109$	18.8621	1.80667	0.903333	−	0.428941i	$-0.141113\pi$
$109$	18.8621	1.80667	0.903333	+	0.428941i	$0.141113\pi$
$110$	1.14588	0.109255
$111$	0.530934	0.0503940
$112$	0	0
$113$	−18.1624	−1.70857	−0.854287	−	0.519802i	$-0.826006\pi$
$113$	−18.1624	−1.70857	−0.854287	+	0.519802i	$0.826006\pi$
$114$	1.01337	0.0949110
$115$	1.41393	0.131849
$116$	7.16432	0.665191
$117$	5.27613	0.487779
$118$	0.651769	0.0600002
$119$	0	0
$120$	1.09456	0.0999194
$121$	−6.79929	−0.618117
$122$	0.399504	0.0361694
$123$	−4.50692	−0.406375
$124$	−1.29117	−0.115950
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.206562	−0.0183294	−0.00916471	−	0.999958i	$-0.502917\pi$
$127$	−0.206562	−0.0183294	−0.00916471	+	0.999958i	$0.502917\pi$
$128$	−11.5389	−1.01990
$129$	−6.00607	−0.528805
$130$	1.08524	0.0951817
$131$	8.13609	0.710853	0.355427	−	0.934704i	$-0.384336\pi$
$131$	8.13609	0.710853	0.355427	+	0.934704i	$0.384336\pi$
$132$	1.83623	0.159823
$133$	0	0
$134$	4.23086	0.365491
$135$	3.03594	0.261292
$136$	13.4419	1.15264
$137$	7.70137	0.657972	0.328986	−	0.944335i	$-0.393293\pi$
$137$	7.70137	0.657972	0.328986	+	0.944335i	$0.393293\pi$
$138$	−0.419705	−0.0357276
$139$	2.06381	0.175050	0.0875249	−	0.996162i	$-0.472104\pi$
$139$	2.06381	0.175050	0.0875249	+	0.996162i	$0.472104\pi$
$140$	0	0
$141$	−4.21796	−0.355216
$142$	−7.02992	−0.589938
$143$	3.97842	0.332692
$144$	−6.04034	−0.503362
$145$	4.24571	0.352587
$146$	−1.72399	−0.142678
$147$	0	0
$148$	−1.68743	−0.138706
$149$	12.8881	1.05584	0.527919	−	0.849295i	$-0.322973\pi$
$149$	12.8881	1.05584	0.527919	+	0.849295i	$0.322973\pi$
$150$	0.296836	0.0242366
$151$	−8.28453	−0.674185	−0.337093	−	0.941471i	$-0.609444\pi$
$151$	−8.28453	−0.674185	−0.337093	+	0.941471i	$0.609444\pi$
$152$	−7.03804	−0.570860
$153$	17.7227	1.43279
$154$	0	0
$155$	−0.765169	−0.0614598
$156$	1.73906	0.139236
$157$	14.7065	1.17371	0.586855	−	0.809692i	$-0.300366\pi$
$157$	14.7065	1.17371	0.586855	+	0.809692i	$0.300366\pi$
$158$	5.40382	0.429905
$159$	0.618494	0.0490498
$160$	−5.36558	−0.424187
$161$	0	0
$162$	3.65777	0.287382
$163$	9.40725	0.736833	0.368416	−	0.929661i	$-0.379900\pi$
$163$	9.40725	0.736833	0.368416	+	0.929661i	$0.379900\pi$
$164$	14.3240	1.11852
$165$	1.08818	0.0847149
$166$	8.19742	0.636243
$167$	14.4956	1.12170	0.560851	−	0.827917i	$-0.310474\pi$
$167$	14.4956	1.12170	0.560851	+	0.827917i	$0.310474\pi$
$168$	0	0
$169$	−9.23212	−0.710163
$170$	3.64534	0.279585
$171$	−9.27938	−0.709612
$172$	19.0886	1.45549
$173$	20.6766	1.57201	0.786006	−	0.618219i	$-0.212146\pi$
$173$	20.6766	1.57201	0.786006	+	0.618219i	$0.212146\pi$
$174$	−1.26028	−0.0955417
$175$	0	0
$176$	−4.55466	−0.343320
$177$	0.618953	0.0465234
$178$	−7.15023	−0.535932
$179$	−1.73706	−0.129834	−0.0649169	−	0.997891i	$-0.520678\pi$
$179$	−1.73706	−0.129834	−0.0649169	+	0.997891i	$0.520678\pi$
$180$	−4.58661	−0.341866
$181$	−7.03225	−0.522703	−0.261352	−	0.965244i	$-0.584168\pi$
$181$	−7.03225	−0.522703	−0.261352	+	0.965244i	$0.584168\pi$
$182$	0	0
$183$	0.379389	0.0280452
$184$	2.91492	0.214891
$185$	−1.00000	−0.0735215
$186$	0.227130	0.0166540
$187$	13.3636	0.977243
$188$	13.4056	0.977705
$189$	0	0
$190$	−1.90866	−0.138469
$191$	−8.42607	−0.609689	−0.304844	−	0.952402i	$-0.598604\pi$
$191$	−8.42607	−0.609689	−0.304844	+	0.952402i	$0.598604\pi$
$192$	−0.767044	−0.0553566
$193$	−0.773093	−0.0556485	−0.0278242	−	0.999613i	$-0.508858\pi$
$193$	−0.773093	−0.0556485	−0.0278242	+	0.999613i	$0.508858\pi$
$194$	7.30392	0.524391
$195$	1.03060	0.0738027
$196$	0	0
$197$	4.93747	0.351780	0.175890	−	0.984410i	$-0.443720\pi$
$197$	4.93747	0.351780	0.175890	+	0.984410i	$0.443720\pi$
$198$	3.11462	0.221346
$199$	14.0397	0.995250	0.497625	−	0.867392i	$-0.334206\pi$
$199$	14.0397	0.995250	0.497625	+	0.867392i	$0.334206\pi$
$200$	−2.06158	−0.145776
$201$	4.01784	0.283397
$202$	−7.87027	−0.553750
$203$	0	0
$204$	5.84154	0.408990
$205$	8.48866	0.592874
$206$	−4.63226	−0.322745
$207$	3.84321	0.267121
$208$	−4.31363	−0.299097
$209$	−6.99702	−0.483994
$210$	0	0
$211$	2.70448	0.186184	0.0930920	−	0.995658i	$-0.470325\pi$
$211$	2.70448	0.186184	0.0930920	+	0.995658i	$0.470325\pi$
$212$	−1.96571	−0.135006
$213$	−6.67597	−0.457430
$214$	−4.27110	−0.291966
$215$	11.3123	0.771490
$216$	6.25883	0.425859
$217$	0	0
$218$	10.5455	0.714232
$219$	−1.63719	−0.110631
$220$	−3.45849	−0.233171
$221$	12.6564	0.851363
$222$	0.296836	0.0199224
$223$	−16.8820	−1.13050	−0.565252	−	0.824918i	$-0.691221\pi$
$223$	−16.8820	−1.13050	−0.565252	+	0.824918i	$0.691221\pi$
$224$	0	0
$225$	−2.71811	−0.181207
$226$	−10.1543	−0.675453
$227$	14.7448	0.978645	0.489323	−	0.872103i	$-0.337244\pi$
$227$	14.7448	0.978645	0.489323	+	0.872103i	$0.337244\pi$
$228$	−3.05856	−0.202558
$229$	19.1552	1.26581	0.632905	−	0.774229i	$-0.281862\pi$
$229$	19.1552	1.26581	0.632905	+	0.774229i	$0.281862\pi$
$230$	0.790503	0.0521242
$231$	0	0
$232$	8.75287	0.574654
$233$	15.2833	1.00125	0.500623	−	0.865666i	$-0.333104\pi$
$233$	15.2833	1.00125	0.500623	+	0.865666i	$0.333104\pi$
$234$	2.94980	0.192834
$235$	7.94442	0.518237
$236$	−1.96717	−0.128052
$237$	5.13174	0.333342
$238$	0	0
$239$	−26.8358	−1.73586	−0.867931	−	0.496684i	$-0.834551\pi$
$239$	−26.8358	−1.73586	−0.867931	+	0.496684i	$0.834551\pi$
$240$	−1.17987	−0.0761604
$241$	13.9409	0.898013	0.449007	−	0.893528i	$-0.351778\pi$
$241$	13.9409	0.898013	0.449007	+	0.893528i	$0.351778\pi$
$242$	−3.80137	−0.244361
$243$	12.5814	0.807099
$244$	−1.20578	−0.0771923
$245$	0	0
$246$	−2.51974	−0.160653
$247$	−6.62675	−0.421650
$248$	−1.57746	−0.100169
$249$	7.78469	0.493335
$250$	−0.559083	−0.0353595
$251$	4.63233	0.292390	0.146195	−	0.989256i	$-0.453297\pi$
$251$	4.63233	0.292390	0.146195	+	0.989256i	$0.453297\pi$
$252$	0	0
$253$	2.89793	0.182191
$254$	−0.115485	−0.00724620
$255$	3.46181	0.216787
$256$	−3.56178	−0.222611
$257$	−0.286330	−0.0178608	−0.00893039	−	0.999960i	$-0.502843\pi$
$257$	−0.286330	−0.0178608	−0.00893039	+	0.999960i	$0.502843\pi$
$258$	−3.35789	−0.209053
$259$	0	0
$260$	−3.27547	−0.203136
$261$	11.5403	0.714327
$262$	4.54875	0.281023
$263$	−0.925895	−0.0570931	−0.0285466	−	0.999592i	$-0.509088\pi$
$263$	−0.925895	−0.0570931	−0.0285466	+	0.999592i	$0.509088\pi$
$264$	2.24337	0.138070
$265$	−1.16492	−0.0715603
$266$	0	0
$267$	−6.79022	−0.415555
$268$	−12.7696	−0.780027
$269$	2.43167	0.148261	0.0741307	−	0.997249i	$-0.476382\pi$
$269$	2.43167	0.148261	0.0741307	+	0.997249i	$0.476382\pi$
$270$	1.69734	0.103297
$271$	−5.01816	−0.304832	−0.152416	−	0.988316i	$-0.548705\pi$
$271$	−5.01816	−0.304832	−0.152416	+	0.988316i	$0.548705\pi$
$272$	−14.4896	−0.878561
$273$	0	0
$274$	4.30571	0.260117
$275$	−2.04956	−0.123593
$276$	1.26675	0.0762496
$277$	−2.35639	−0.141582	−0.0707908	−	0.997491i	$-0.522552\pi$
$277$	−2.35639	−0.141582	−0.0707908	+	0.997491i	$0.522552\pi$
$278$	1.15384	0.0692027
$279$	−2.07981	−0.124515
$280$	0	0
$281$	−15.5879	−0.929896	−0.464948	−	0.885338i	$-0.653927\pi$
$281$	−15.5879	−0.929896	−0.464948	+	0.885338i	$0.653927\pi$
$282$	−2.35819	−0.140428
$283$	6.56274	0.390114	0.195057	−	0.980792i	$-0.437511\pi$
$283$	6.56274	0.390114	0.195057	+	0.980792i	$0.437511\pi$
$284$	21.2177	1.25904
$285$	−1.81256	−0.107367
$286$	2.22427	0.131524
$287$	0	0
$288$	−14.5842	−0.859385
$289$	25.5132	1.50078
$290$	2.37371	0.139389
$291$	6.93618	0.406606
$292$	5.20335	0.304503
$293$	7.67703	0.448497	0.224248	−	0.974532i	$-0.428007\pi$
$293$	7.67703	0.448497	0.224248	+	0.974532i	$0.428007\pi$
$294$	0	0
$295$	−1.16578	−0.0678744
$296$	−2.06158	−0.119827
$297$	6.22235	0.361057
$298$	7.20554	0.417406
$299$	2.74458	0.158723
$300$	−0.895912	−0.0517255
$301$	0	0
$302$	−4.63174	−0.266527
$303$	−7.47401	−0.429371
$304$	7.58659	0.435121
$305$	−0.714569	−0.0409161
$306$	9.90844	0.566428
$307$	24.3428	1.38932	0.694659	−	0.719339i	$-0.255555\pi$
$307$	24.3428	1.38932	0.694659	+	0.719339i	$0.255555\pi$
$308$	0	0
$309$	−4.39903	−0.250252
$310$	−0.427793	−0.0242970
$311$	11.1874	0.634381	0.317191	−	0.948362i	$-0.397261\pi$
$311$	11.1874	0.634381	0.317191	+	0.948362i	$0.397261\pi$
$312$	2.12466	0.120285
$313$	16.0894	0.909425	0.454713	−	0.890638i	$-0.349742\pi$
$313$	16.0894	0.909425	0.454713	+	0.890638i	$0.349742\pi$
$314$	8.22218	0.464004
$315$	0	0
$316$	−16.3098	−0.917498
$317$	−4.90702	−0.275605	−0.137803	−	0.990460i	$-0.544004\pi$
$317$	−4.90702	−0.275605	−0.137803	+	0.990460i	$0.544004\pi$
$318$	0.345790	0.0193909
$319$	8.70186	0.487210
$320$	1.44471	0.0807616
$321$	−4.05605	−0.226387
$322$	0	0
$323$	−22.2594	−1.23855
$324$	−11.0399	−0.613327
$325$	−1.94110	−0.107673
$326$	5.25944	0.291293
$327$	10.0146	0.553806
$328$	17.5000	0.966278
$329$	0	0
$330$	0.608385	0.0334905
$331$	19.4934	1.07146	0.535728	−	0.844391i	$-0.320037\pi$
$331$	19.4934	1.07146	0.535728	+	0.844391i	$0.320037\pi$
$332$	−24.7415	−1.35786
$333$	−2.71811	−0.148951
$334$	8.10424	0.443444
$335$	−7.56750	−0.413457
$336$	0	0
$337$	−8.12725	−0.442720	−0.221360	−	0.975192i	$-0.571050\pi$
$337$	−8.12725	−0.442720	−0.221360	+	0.975192i	$0.571050\pi$
$338$	−5.16152	−0.280750
$339$	−9.64303	−0.523737
$340$	−11.0024	−0.596688
$341$	−1.56826	−0.0849262
$342$	−5.18794	−0.280532
$343$	0	0
$344$	23.3211	1.25739
$345$	0.750702	0.0404164
$346$	11.5599	0.621466
$347$	−13.2178	−0.709568	−0.354784	−	0.934948i	$-0.615446\pi$
$347$	−13.2178	−0.709568	−0.354784	+	0.934948i	$0.615446\pi$
$348$	3.80378	0.203904
$349$	−21.6956	−1.16134	−0.580670	−	0.814139i	$-0.697210\pi$
$349$	−21.6956	−1.16134	−0.580670	+	0.814139i	$0.697210\pi$
$350$	0	0
$351$	5.89307	0.314549
$352$	−10.9971	−0.586148
$353$	−4.42366	−0.235448	−0.117724	−	0.993046i	$-0.537560\pi$
$353$	−4.42366	−0.235448	−0.117724	+	0.993046i	$0.537560\pi$
$354$	0.346046	0.0183922
$355$	12.5740	0.667359
$356$	21.5808	1.14378
$357$	0	0
$358$	−0.971160	−0.0513274
$359$	−23.7797	−1.25504	−0.627521	−	0.778599i	$-0.715930\pi$
$359$	−23.7797	−1.25504	−0.627521	+	0.778599i	$0.715930\pi$
$360$	−5.60359	−0.295335
$361$	−7.34522	−0.386591
$362$	−3.93161	−0.206641
$363$	−3.60997	−0.189475
$364$	0	0
$365$	3.08360	0.161403
$366$	0.212110	0.0110872
$367$	0.945229	0.0493406	0.0246703	−	0.999696i	$-0.492146\pi$
$367$	0.945229	0.0493406	0.0246703	+	0.999696i	$0.492146\pi$
$368$	−3.14211	−0.163794
$369$	23.0731	1.20114
$370$	−0.559083	−0.0290654
$371$	0	0
$372$	−0.685524	−0.0355428
$373$	−35.2595	−1.82567	−0.912834	−	0.408330i	$-0.866111\pi$
$373$	−35.2595	−1.82567	−0.912834	+	0.408330i	$0.866111\pi$
$374$	7.47137	0.386335
$375$	−0.530934	−0.0274173
$376$	16.3780	0.844633
$377$	8.24137	0.424452
$378$	0	0
$379$	25.6281	1.31643	0.658215	−	0.752830i	$-0.271312\pi$
$379$	25.6281	1.31643	0.658215	+	0.752830i	$0.271312\pi$
$380$	5.76072	0.295519
$381$	−0.109671	−0.00561861
$382$	−4.71087	−0.241029
$383$	−5.73686	−0.293140	−0.146570	−	0.989200i	$-0.546823\pi$
$383$	−5.73686	−0.293140	−0.146570	+	0.989200i	$0.546823\pi$
$384$	−6.12638	−0.312636
$385$	0	0
$386$	−0.432224	−0.0219996
$387$	30.7480	1.56301
$388$	−22.0447	−1.11915
$389$	5.46951	0.277315	0.138658	−	0.990340i	$-0.455721\pi$
$389$	5.46951	0.277315	0.138658	+	0.990340i	$0.455721\pi$
$390$	0.576190	0.0291765
$391$	9.21911	0.466230
$392$	0	0
$393$	4.31973	0.217901
$394$	2.76046	0.139070
$395$	−9.66550	−0.486324
$396$	−9.40054	−0.472395
$397$	−2.39397	−0.120150	−0.0600751	−	0.998194i	$-0.519134\pi$
$397$	−2.39397	−0.120150	−0.0600751	+	0.998194i	$0.519134\pi$
$398$	7.84938	0.393454
$399$	0	0
$400$	2.22226	0.111113
$401$	15.7007	0.784053	0.392027	−	0.919954i	$-0.371774\pi$
$401$	15.7007	0.784053	0.392027	+	0.919954i	$0.371774\pi$
$402$	2.24631	0.112036
$403$	−1.48527	−0.0739867
$404$	23.7541	1.18181
$405$	−6.54244	−0.325097
$406$	0	0
$407$	−2.04956	−0.101593
$408$	7.13678	0.353323
$409$	24.5582	1.21432	0.607162	−	0.794578i	$-0.292308\pi$
$409$	24.5582	1.21432	0.607162	+	0.794578i	$0.292308\pi$
$410$	4.74587	0.234382
$411$	4.08892	0.201692
$412$	13.9811	0.688799
$413$	0	0
$414$	2.14867	0.105601
$415$	−14.6622	−0.719741
$416$	−10.4152	−0.510645
$417$	1.09574	0.0536589
$418$	−3.91192	−0.191338
$419$	1.58546	0.0774549	0.0387275	−	0.999250i	$-0.487670\pi$
$419$	1.58546	0.0774549	0.0387275	+	0.999250i	$0.487670\pi$
$420$	0	0
$421$	40.0569	1.95225	0.976127	−	0.217202i	$-0.0696931\pi$
$421$	40.0569	1.95225	0.976127	+	0.217202i	$0.0696931\pi$
$422$	1.51203	0.0736044
$423$	21.5938	1.04993
$424$	−2.40157	−0.116631
$425$	−6.52022	−0.316277
$426$	−3.73242	−0.180837
$427$	0	0
$428$	12.8910	0.623112
$429$	2.11228	0.101982
$430$	6.32450	0.304994
$431$	−11.7106	−0.564080	−0.282040	−	0.959403i	$-0.591011\pi$
$431$	−11.7106	−0.564080	−0.282040	+	0.959403i	$0.591011\pi$
$432$	−6.74664	−0.324598
$433$	11.6998	0.562257	0.281128	−	0.959670i	$-0.409291\pi$
$433$	11.6998	0.562257	0.281128	+	0.959670i	$0.409291\pi$
$434$	0	0
$435$	2.25419	0.108080
$436$	−31.8285	−1.52431
$437$	−4.82702	−0.230907
$438$	−0.915325	−0.0437359
$439$	26.1184	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$439$	26.1184	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$440$	−4.22534	−0.201435
$441$	0	0
$442$	7.07599	0.336571
$443$	−24.6031	−1.16893	−0.584465	−	0.811419i	$-0.698695\pi$
$443$	−24.6031	−1.16893	−0.584465	+	0.811419i	$0.698695\pi$
$444$	−0.895912	−0.0425181
$445$	12.7892	0.606266
$446$	−9.43846	−0.446924
$447$	6.84275	0.323651
$448$	0	0
$449$	−36.4551	−1.72042	−0.860212	−	0.509937i	$-0.829669\pi$
$449$	−36.4551	−1.72042	−0.860212	+	0.509937i	$0.829669\pi$
$450$	−1.51965	−0.0716370
$451$	17.3980	0.819242
$452$	30.6477	1.44155
$453$	−4.39854	−0.206661
$454$	8.24356	0.386889
$455$	0	0
$456$	−3.73674	−0.174989
$457$	−31.0886	−1.45426	−0.727132	−	0.686497i	$-0.759147\pi$
$457$	−31.0886	−1.45426	−0.727132	+	0.686497i	$0.759147\pi$
$458$	10.7093	0.500415
$459$	19.7950	0.923951
$460$	−2.38590	−0.111243
$461$	−28.0059	−1.30437	−0.652183	−	0.758061i	$-0.726147\pi$
$461$	−28.0059	−1.30437	−0.652183	+	0.758061i	$0.726147\pi$
$462$	0	0
$463$	−29.1126	−1.35298	−0.676488	−	0.736454i	$-0.736499\pi$
$463$	−29.1126	−1.35298	−0.676488	+	0.736454i	$0.736499\pi$
$464$	−9.43507	−0.438012
$465$	−0.406254	−0.0188396
$466$	8.54466	0.395824
$467$	−27.0864	−1.25341	−0.626704	−	0.779257i	$-0.715597\pi$
$467$	−27.0864	−1.25341	−0.626704	+	0.779257i	$0.715597\pi$
$468$	−8.90308	−0.411545
$469$	0	0
$470$	4.44159	0.204875
$471$	7.80820	0.359783
$472$	−2.40335	−0.110623
$473$	23.1852	1.06606
$474$	2.86907	0.131781
$475$	3.41391	0.156641
$476$	0	0
$477$	−3.16637	−0.144978
$478$	−15.0034	−0.686241
$479$	−21.3239	−0.974312	−0.487156	−	0.873315i	$-0.661966\pi$
$479$	−21.3239	−0.974312	−0.487156	+	0.873315i	$0.661966\pi$
$480$	−2.84877	−0.130028
$481$	−1.94110	−0.0885067
$482$	7.79413	0.355013
$483$	0	0
$484$	11.4733	0.521513
$485$	−13.0641	−0.593210
$486$	7.03406	0.319072
$487$	16.7544	0.759213	0.379606	−	0.925148i	$-0.376059\pi$
$487$	16.7544	0.759213	0.379606	+	0.925148i	$0.376059\pi$
$488$	−1.47314	−0.0666859
$489$	4.99463	0.225865
$490$	0	0
$491$	−32.2228	−1.45419	−0.727096	−	0.686535i	$-0.759131\pi$
$491$	−32.2228	−1.45419	−0.727096	+	0.686535i	$0.759131\pi$
$492$	7.60509	0.342864
$493$	27.6830	1.24678
$494$	−3.70491	−0.166692
$495$	−5.57094	−0.250395
$496$	1.70040	0.0763503
$497$	0	0
$498$	4.35229	0.195031
$499$	12.9362	0.579103	0.289552	−	0.957162i	$-0.406494\pi$
$499$	12.9362	0.579103	0.289552	+	0.957162i	$0.406494\pi$
$500$	1.68743	0.0754640
$501$	7.69620	0.343841
$502$	2.58986	0.115591
$503$	−34.9710	−1.55928	−0.779641	−	0.626227i	$-0.784598\pi$
$503$	−34.9710	−1.55928	−0.779641	+	0.626227i	$0.784598\pi$
$504$	0	0
$505$	14.0771	0.626423
$506$	1.62019	0.0720260
$507$	−4.90164	−0.217690
$508$	0.348558	0.0154648
$509$	0.726009	0.0321798	0.0160899	−	0.999871i	$-0.494878\pi$
$509$	0.726009	0.0321798	0.0160899	+	0.999871i	$0.494878\pi$
$510$	1.93544	0.0857026
$511$	0	0
$512$	21.0864	0.931897
$513$	−10.3644	−0.457600
$514$	−0.160082	−0.00706093
$515$	8.28546	0.365101
$516$	10.1348	0.446159
$517$	16.2826	0.716107
$518$	0	0
$519$	10.9779	0.481876
$520$	−4.00174	−0.175488
$521$	42.6678	1.86931	0.934656	−	0.355554i	$-0.115708\pi$
$521$	42.6678	1.86931	0.934656	+	0.355554i	$0.115708\pi$
$522$	6.45199	0.282396
$523$	−25.3202	−1.10718	−0.553588	−	0.832791i	$-0.686742\pi$
$523$	−25.3202	−1.10718	−0.553588	+	0.832791i	$0.686742\pi$
$524$	−13.7290	−0.599756
$525$	0	0
$526$	−0.517652	−0.0225707
$527$	−4.98907	−0.217327
$528$	−2.41822	−0.105240
$529$	−21.0008	−0.913079
$530$	−0.651286	−0.0282901
$531$	−3.16872	−0.137511
$532$	0	0
$533$	16.4774	0.713714
$534$	−3.79630	−0.164282
$535$	7.63947	0.330283
$536$	−15.6010	−0.673860
$537$	−0.922263	−0.0397986
$538$	1.35951	0.0586124
$539$	0	0
$540$	−5.12292	−0.220455
$541$	−1.31457	−0.0565178	−0.0282589	−	0.999601i	$-0.508996\pi$
$541$	−1.31457	−0.0565178	−0.0282589	+	0.999601i	$0.508996\pi$
$542$	−2.80557	−0.120510
$543$	−3.73366	−0.160227
$544$	−34.9848	−1.49996
$545$	−18.8621	−0.807965
$546$	0	0
$547$	8.18693	0.350048	0.175024	−	0.984564i	$-0.444000\pi$
$547$	8.18693	0.350048	0.175024	+	0.984564i	$0.444000\pi$
$548$	−12.9955	−0.555140
$549$	−1.94228	−0.0828943
$550$	−1.14588	−0.0488603
$551$	−14.4945	−0.617485
$552$	1.54763	0.0658715
$553$	0	0
$554$	−1.31742	−0.0559717
$555$	−0.530934	−0.0225369
$556$	−3.48252	−0.147692
$557$	39.5746	1.67683	0.838415	−	0.545033i	$-0.183483\pi$
$557$	39.5746	1.67683	0.838415	+	0.545033i	$0.183483\pi$
$558$	−1.16279	−0.0492248
$559$	21.9583	0.928736
$560$	0	0
$561$	7.09519	0.299559
$562$	−8.71494	−0.367617
$563$	−39.3820	−1.65975	−0.829877	−	0.557946i	$-0.811590\pi$
$563$	−39.3820	−1.65975	−0.829877	+	0.557946i	$0.811590\pi$
$564$	7.11750	0.299701
$565$	18.1624	0.764097
$566$	3.66912	0.154224
$567$	0	0
$568$	25.9223	1.08768
$569$	−1.07337	−0.0449979	−0.0224989	−	0.999747i	$-0.507162\pi$
$569$	−1.07337	−0.0449979	−0.0224989	+	0.999747i	$0.507162\pi$
$570$	−1.01337	−0.0424455
$571$	−44.0451	−1.84323	−0.921615	−	0.388106i	$-0.873130\pi$
$571$	−44.0451	−1.84323	−0.921615	+	0.388106i	$0.873130\pi$
$572$	−6.71328	−0.280696
$573$	−4.47369	−0.186891
$574$	0	0
$575$	−1.41393	−0.0589648
$576$	3.92687	0.163620
$577$	15.7670	0.656391	0.328195	−	0.944610i	$-0.393560\pi$
$577$	15.7670	0.656391	0.328195	+	0.944610i	$0.393560\pi$
$578$	14.2640	0.593305
$579$	−0.410462	−0.0170582
$580$	−7.16432	−0.297482
$581$	0	0
$582$	3.87790	0.160744
$583$	−2.38757	−0.0988832
$584$	6.35709	0.263058
$585$	−5.27613	−0.218141
$586$	4.29210	0.177305
$587$	37.6543	1.55416	0.777080	−	0.629402i	$-0.216700\pi$
$587$	37.6543	1.55416	0.777080	+	0.629402i	$0.216700\pi$
$588$	0	0
$589$	2.61222	0.107635
$590$	−0.651769	−0.0268329
$591$	2.62147	0.107833
$592$	2.22226	0.0913342
$593$	15.1328	0.621428	0.310714	−	0.950503i	$-0.399432\pi$
$593$	15.1328	0.621428	0.310714	+	0.950503i	$0.399432\pi$
$594$	3.47881	0.142737
$595$	0	0
$596$	−21.7478	−0.890823
$597$	7.45417	0.305079
$598$	1.53445	0.0627482
$599$	−6.72505	−0.274778	−0.137389	−	0.990517i	$-0.543871\pi$
$599$	−6.72505	−0.274778	−0.137389	+	0.990517i	$0.543871\pi$
$600$	−1.09456	−0.0446853
$601$	−43.5725	−1.77736	−0.888680	−	0.458529i	$-0.848377\pi$
$601$	−43.5725	−1.77736	−0.888680	+	0.458529i	$0.848377\pi$
$602$	0	0
$603$	−20.5693	−0.837646
$604$	13.9795	0.568819
$605$	6.79929	0.276430
$606$	−4.17860	−0.169744
$607$	0.539111	0.0218819	0.0109409	−	0.999940i	$-0.496517\pi$
$607$	0.539111	0.0218819	0.0109409	+	0.999940i	$0.496517\pi$
$608$	18.3176	0.742877
$609$	0	0
$610$	−0.399504	−0.0161754
$611$	15.4209	0.623864
$612$	−29.9057	−1.20887
$613$	−33.4197	−1.34981	−0.674904	−	0.737906i	$-0.735815\pi$
$613$	−33.4197	−1.34981	−0.674904	+	0.737906i	$0.735815\pi$
$614$	13.6097	0.549241
$615$	4.50692	0.181736
$616$	0	0
$617$	39.7593	1.60065	0.800325	−	0.599566i	$-0.204660\pi$
$617$	39.7593	1.60065	0.800325	+	0.599566i	$0.204660\pi$
$618$	−2.45943	−0.0989326
$619$	−10.6841	−0.429431	−0.214715	−	0.976677i	$-0.568882\pi$
$619$	−10.6841	−0.429431	−0.214715	+	0.976677i	$0.568882\pi$
$620$	1.29117	0.0518545
$621$	4.29259	0.172256
$622$	6.25471	0.250791
$623$	0	0
$624$	−2.29025	−0.0916836
$625$	1.00000	0.0400000
$626$	8.99530	0.359524
$627$	−3.71496	−0.148361
$628$	−24.8162	−0.990274
$629$	−6.52022	−0.259978
$630$	0	0
$631$	−28.6655	−1.14116	−0.570578	−	0.821243i	$-0.693281\pi$
$631$	−28.6655	−1.14116	−0.570578	+	0.821243i	$0.693281\pi$
$632$	−19.9262	−0.792621
$633$	1.43590	0.0570719
$634$	−2.74343	−0.108956
$635$	0.206562	0.00819716
$636$	−1.04366	−0.0413840
$637$	0	0
$638$	4.86506	0.192610
$639$	34.1775	1.35204
$640$	11.5389	0.456114
$641$	2.21155	0.0873511	0.0436756	−	0.999046i	$-0.486093\pi$
$641$	2.21155	0.0873511	0.0436756	+	0.999046i	$0.486093\pi$
$642$	−2.26767	−0.0894978
$643$	−24.7172	−0.974750	−0.487375	−	0.873193i	$-0.662046\pi$
$643$	−24.7172	−0.974750	−0.487375	+	0.873193i	$0.662046\pi$
$644$	0	0
$645$	6.00607	0.236489
$646$	−12.4449	−0.489637
$647$	−4.09476	−0.160981	−0.0804907	−	0.996755i	$-0.525649\pi$
$647$	−4.09476	−0.160981	−0.0804907	+	0.996755i	$0.525649\pi$
$648$	−13.4878	−0.529849
$649$	−2.38934	−0.0937899
$650$	−1.08524	−0.0425666
$651$	0	0
$652$	−15.8740	−0.621675
$653$	34.8286	1.36295	0.681474	−	0.731842i	$-0.261339\pi$
$653$	34.8286	1.36295	0.681474	+	0.731842i	$0.261339\pi$
$654$	5.59897	0.218937
$655$	−8.13609	−0.317903
$656$	−18.8640	−0.736515
$657$	8.38157	0.326996
$658$	0	0
$659$	43.6998	1.70230	0.851151	−	0.524922i	$-0.175905\pi$
$659$	43.6998	1.70230	0.851151	+	0.524922i	$0.175905\pi$
$660$	−1.83623	−0.0714751
$661$	9.77749	0.380300	0.190150	−	0.981755i	$-0.439103\pi$
$661$	9.77749	0.380300	0.190150	+	0.981755i	$0.439103\pi$
$662$	10.8984	0.423580
$663$	6.71972	0.260972
$664$	−30.2274	−1.17305
$665$	0	0
$666$	−1.51965	−0.0588852
$667$	6.00312	0.232442
$668$	−24.4602	−0.946394
$669$	−8.96324	−0.346539
$670$	−4.23086	−0.163453
$671$	−1.46455	−0.0565385
$672$	0	0
$673$	23.1685	0.893080	0.446540	−	0.894764i	$-0.352656\pi$
$673$	23.1685	0.893080	0.446540	+	0.894764i	$0.352656\pi$
$674$	−4.54381	−0.175021
$675$	−3.03594	−0.116853
$676$	15.5785	0.599174
$677$	30.3950	1.16817	0.584087	−	0.811691i	$-0.301453\pi$
$677$	30.3950	1.16817	0.584087	+	0.811691i	$0.301453\pi$
$678$	−5.39126	−0.207050
$679$	0	0
$680$	−13.4419	−0.515475
$681$	7.82850	0.299989
$682$	−0.876789	−0.0335740
$683$	16.7020	0.639083	0.319542	−	0.947572i	$-0.396471\pi$
$683$	16.7020	0.639083	0.319542	+	0.947572i	$0.396471\pi$
$684$	15.6583	0.598709
$685$	−7.70137	−0.294254
$686$	0	0
$687$	10.1701	0.388015
$688$	−25.1388	−0.958406
$689$	−2.26123	−0.0861459
$690$	0.419705	0.0159779
$691$	29.6033	1.12616	0.563081	−	0.826402i	$-0.309616\pi$
$691$	29.6033	1.12616	0.563081	+	0.826402i	$0.309616\pi$
$692$	−34.8902	−1.32633
$693$	0	0
$694$	−7.38985	−0.280515
$695$	−2.06381	−0.0782846
$696$	4.64720	0.176151
$697$	55.3479	2.09645
$698$	−12.1297	−0.459115
$699$	8.11445	0.306917
$700$	0	0
$701$	6.80969	0.257199	0.128599	−	0.991697i	$-0.458952\pi$
$701$	6.80969	0.257199	0.128599	+	0.991697i	$0.458952\pi$
$702$	3.29472	0.124351
$703$	3.41391	0.128758
$704$	2.96102	0.111598
$705$	4.21796	0.158858
$706$	−2.47319	−0.0930799
$707$	0	0
$708$	−1.04444	−0.0392524
$709$	−17.6447	−0.662662	−0.331331	−	0.943515i	$-0.607498\pi$
$709$	−17.6447	−0.662662	−0.331331	+	0.943515i	$0.607498\pi$
$710$	7.02992	0.263828
$711$	−26.2719	−0.985272
$712$	26.3659	0.988106
$713$	−1.08189	−0.0405172
$714$	0	0
$715$	−3.97842	−0.148784
$716$	2.93116	0.109542
$717$	−14.2480	−0.532103
$718$	−13.2948	−0.496158
$719$	44.4726	1.65855	0.829275	−	0.558841i	$-0.188754\pi$
$719$	44.4726	1.65855	0.829275	+	0.558841i	$0.188754\pi$
$720$	6.04034	0.225110
$721$	0	0
$722$	−4.10659	−0.152832
$723$	7.40171	0.275272
$724$	11.8664	0.441011
$725$	−4.24571	−0.157682
$726$	−2.01828	−0.0749053
$727$	−20.7667	−0.770194	−0.385097	−	0.922876i	$-0.625832\pi$
$727$	−20.7667	−0.770194	−0.385097	+	0.922876i	$0.625832\pi$
$728$	0	0
$729$	−12.9474	−0.479534
$730$	1.72399	0.0638077
$731$	73.7584	2.72805
$732$	−0.640191	−0.0236621
$733$	−1.16963	−0.0432012	−0.0216006	−	0.999767i	$-0.506876\pi$
$733$	−1.16963	−0.0432012	−0.0216006	+	0.999767i	$0.506876\pi$
$734$	0.528462	0.0195059
$735$	0	0
$736$	−7.58654	−0.279644
$737$	−15.5101	−0.571321
$738$	12.8998	0.474848
$739$	6.51308	0.239588	0.119794	−	0.992799i	$-0.461777\pi$
$739$	6.51308	0.239588	0.119794	+	0.992799i	$0.461777\pi$
$740$	1.68743	0.0620310
$741$	−3.51837	−0.129250
$742$	0	0
$743$	−25.0553	−0.919189	−0.459594	−	0.888129i	$-0.652005\pi$
$743$	−25.0553	−0.919189	−0.459594	+	0.888129i	$0.652005\pi$
$744$	−0.837525	−0.0307052
$745$	−12.8881	−0.472185
$746$	−19.7130	−0.721744
$747$	−39.8536	−1.45817
$748$	−22.5501	−0.824513
$749$	0	0
$750$	−0.296836	−0.0108389
$751$	−8.79249	−0.320842	−0.160421	−	0.987049i	$-0.551285\pi$
$751$	−8.79249	−0.320842	−0.160421	+	0.987049i	$0.551285\pi$
$752$	−17.6545	−0.643795
$753$	2.45946	0.0896278
$754$	4.60761	0.167799
$755$	8.28453	0.301505
$756$	0	0
$757$	−11.1501	−0.405256	−0.202628	−	0.979256i	$-0.564948\pi$
$757$	−11.1501	−0.405256	−0.202628	+	0.979256i	$0.564948\pi$
$758$	14.3283	0.520426
$759$	1.53861	0.0558480
$760$	7.03804	0.255297
$761$	50.6613	1.83647	0.918236	−	0.396034i	$-0.129614\pi$
$761$	50.6613	1.83647	0.918236	+	0.396034i	$0.129614\pi$
$762$	−0.0613151	−0.00222121
$763$	0	0
$764$	14.2184	0.514402
$765$	−17.7227	−0.640764
$766$	−3.20739	−0.115888
$767$	−2.26290	−0.0817087
$768$	−1.89107	−0.0682381
$769$	48.5250	1.74986	0.874928	−	0.484253i	$-0.160909\pi$
$769$	48.5250	1.74986	0.874928	+	0.484253i	$0.160909\pi$
$770$	0	0
$771$	−0.152022	−0.00547496
$772$	1.30454	0.0469513
$773$	29.4931	1.06079	0.530396	−	0.847750i	$-0.322043\pi$
$773$	29.4931	1.06079	0.530396	+	0.847750i	$0.322043\pi$
$774$	17.1907	0.617906
$775$	0.765169	0.0274857
$776$	−26.9327	−0.966827
$777$	0	0
$778$	3.05791	0.109631
$779$	−28.9795	−1.03830
$780$	−1.73906	−0.0622683
$781$	25.7712	0.922168
$782$	5.15425	0.184316
$783$	12.8897	0.460641
$784$	0	0
$785$	−14.7065	−0.524899
$786$	2.41509	0.0861433
$787$	−3.05645	−0.108951	−0.0544753	−	0.998515i	$-0.517349\pi$
$787$	−3.05645	−0.108951	−0.0544753	+	0.998515i	$0.517349\pi$
$788$	−8.33161	−0.296801
$789$	−0.491589	−0.0175010
$790$	−5.40382	−0.192259
$791$	0	0
$792$	−11.4849	−0.408099
$793$	−1.38705	−0.0492557
$794$	−1.33843	−0.0474991
$795$	−0.618494	−0.0219357
$796$	−23.6910	−0.839705
$797$	−32.0649	−1.13580	−0.567899	−	0.823098i	$-0.692244\pi$
$797$	−32.0649	−1.13580	−0.567899	+	0.823098i	$0.692244\pi$
$798$	0	0
$799$	51.7993	1.83253
$800$	5.36558	0.189702
$801$	34.7624	1.22827
$802$	8.77798	0.309961
$803$	6.32004	0.223029
$804$	−6.77981	−0.239106
$805$	0	0
$806$	−0.830391	−0.0292493
$807$	1.29106	0.0454473
$808$	29.0210	1.02096
$809$	42.7337	1.50244	0.751219	−	0.660054i	$-0.229466\pi$
$809$	42.7337	1.50244	0.751219	+	0.660054i	$0.229466\pi$
$810$	−3.65777	−0.128521
$811$	19.1945	0.674009	0.337004	−	0.941503i	$-0.390586\pi$
$811$	19.1945	0.674009	0.337004	+	0.941503i	$0.390586\pi$
$812$	0	0
$813$	−2.66431	−0.0934415
$814$	−1.14588	−0.0401630
$815$	−9.40725	−0.329522
$816$	−7.69302	−0.269310
$817$	−38.6190	−1.35111
$818$	13.7301	0.480061
$819$	0	0
$820$	−14.3240	−0.500215
$821$	29.1456	1.01719	0.508594	−	0.861007i	$-0.330166\pi$
$821$	29.1456	1.01719	0.508594	+	0.861007i	$0.330166\pi$
$822$	2.28605	0.0797350
$823$	27.1799	0.947431	0.473715	−	0.880678i	$-0.342913\pi$
$823$	27.1799	0.947431	0.473715	+	0.880678i	$0.342913\pi$
$824$	17.0811	0.595049
$825$	−1.08818	−0.0378857
$826$	0	0
$827$	−32.1445	−1.11777	−0.558887	−	0.829244i	$-0.688771\pi$
$827$	−32.1445	−1.11777	−0.558887	+	0.829244i	$0.688771\pi$
$828$	−6.48512	−0.225374
$829$	−5.82592	−0.202343	−0.101171	−	0.994869i	$-0.532259\pi$
$829$	−5.82592	−0.202343	−0.101171	+	0.994869i	$0.532259\pi$
$830$	−8.19742	−0.284537
$831$	−1.25109	−0.0433997
$832$	2.80433	0.0972225
$833$	0	0
$834$	0.612613	0.0212130
$835$	−14.4956	−0.501640
$836$	11.8070	0.408352
$837$	−2.32301	−0.0802948
$838$	0.886406	0.0306204
$839$	34.2430	1.18220	0.591101	−	0.806598i	$-0.298694\pi$
$839$	34.2430	1.18220	0.591101	+	0.806598i	$0.298694\pi$
$840$	0	0
$841$	−10.9739	−0.378411
$842$	22.3951	0.771787
$843$	−8.27615	−0.285046
$844$	−4.56361	−0.157086
$845$	9.23212	0.317594
$846$	12.0727	0.415069
$847$	0	0
$848$	2.58875	0.0888979
$849$	3.48438	0.119584
$850$	−3.64534	−0.125034
$851$	−1.41393	−0.0484688
$852$	11.2652	0.385940
$853$	34.5143	1.18175	0.590874	−	0.806764i	$-0.298783\pi$
$853$	34.5143	1.18175	0.590874	+	0.806764i	$0.298783\pi$
$854$	0	0
$855$	9.27938	0.317348
$856$	15.7494	0.538302
$857$	21.4428	0.732471	0.366236	−	0.930522i	$-0.380646\pi$
$857$	21.4428	0.732471	0.366236	+	0.930522i	$0.380646\pi$
$858$	1.18094	0.0403166
$859$	−36.9248	−1.25986	−0.629930	−	0.776652i	$-0.716916\pi$
$859$	−36.9248	−1.25986	−0.629930	+	0.776652i	$0.716916\pi$
$860$	−19.0886	−0.650916
$861$	0	0
$862$	−6.54721	−0.222999
$863$	−35.4093	−1.20535	−0.602673	−	0.797988i	$-0.705898\pi$
$863$	−35.4093	−1.20535	−0.602673	+	0.797988i	$0.705898\pi$
$864$	−16.2896	−0.554183
$865$	−20.6766	−0.703025
$866$	6.54116	0.222278
$867$	13.5458	0.460041
$868$	0	0
$869$	−19.8100	−0.672010
$870$	1.26028	0.0427276
$871$	−14.6893	−0.497728
$872$	−38.8858	−1.31684
$873$	−35.5097	−1.20182
$874$	−2.69870	−0.0912850
$875$	0	0
$876$	2.76264	0.0933408
$877$	26.2487	0.886357	0.443178	−	0.896433i	$-0.353851\pi$
$877$	26.2487	0.886357	0.443178	+	0.896433i	$0.353851\pi$
$878$	14.6024	0.492806
$879$	4.07599	0.137480
$880$	4.55466	0.153537
$881$	−38.2803	−1.28969	−0.644847	−	0.764311i	$-0.723079\pi$
$881$	−38.2803	−1.28969	−0.644847	+	0.764311i	$0.723079\pi$
$882$	0	0
$883$	−16.2982	−0.548477	−0.274239	−	0.961662i	$-0.588426\pi$
$883$	−16.2982	−0.548477	−0.274239	+	0.961662i	$0.588426\pi$
$884$	−21.3568	−0.718306
$885$	−0.618953	−0.0208059
$886$	−13.7552	−0.462115
$887$	44.6702	1.49988	0.749938	−	0.661508i	$-0.230083\pi$
$887$	44.6702	1.49988	0.749938	+	0.661508i	$0.230083\pi$
$888$	−1.09456	−0.0367311
$889$	0	0
$890$	7.15023	0.239676
$891$	−13.4092	−0.449224
$892$	28.4872	0.953821
$893$	−27.1215	−0.907587
$894$	3.82567	0.127949
$895$	1.73706	0.0580634
$896$	0	0
$897$	1.45719	0.0486541
$898$	−20.3814	−0.680138
$899$	−3.24869	−0.108350
$900$	4.58661	0.152887
$901$	−7.59552	−0.253043
$902$	9.72696	0.323872
$903$	0	0
$904$	37.4432	1.24534
$905$	7.03225	0.233760
$906$	−2.45915	−0.0816998
$907$	−31.8439	−1.05736	−0.528680	−	0.848821i	$-0.677313\pi$
$907$	−31.8439	−1.05736	−0.528680	+	0.848821i	$0.677313\pi$
$908$	−24.8807	−0.825696
$909$	38.2631	1.26911
$910$	0	0
$911$	−46.2352	−1.53184	−0.765920	−	0.642936i	$-0.777716\pi$
$911$	−46.2352	−1.53184	−0.765920	+	0.642936i	$0.777716\pi$
$912$	4.02798	0.133380
$913$	−30.0512	−0.994550
$914$	−17.3811	−0.574917
$915$	−0.379389	−0.0125422
$916$	−32.3230	−1.06798
$917$	0	0
$918$	11.0670	0.365267
$919$	41.0722	1.35485	0.677424	−	0.735593i	$-0.263096\pi$
$919$	41.0722	1.35485	0.677424	+	0.735593i	$0.263096\pi$
$920$	−2.91492	−0.0961020
$921$	12.9244	0.425875
$922$	−15.6577	−0.515657
$923$	24.4075	0.803382
$924$	0	0
$925$	1.00000	0.0328798
$926$	−16.2763	−0.534874
$927$	22.5208	0.739679
$928$	−22.7807	−0.747814
$929$	50.9719	1.67233	0.836167	−	0.548475i	$-0.184792\pi$
$929$	50.9719	1.67233	0.836167	+	0.548475i	$0.184792\pi$
$930$	−0.227130	−0.00744788
$931$	0	0
$932$	−25.7895	−0.844764
$933$	5.93979	0.194460
$934$	−15.1435	−0.495512
$935$	−13.3636	−0.437036
$936$	−10.8772	−0.355531
$937$	−36.8950	−1.20531	−0.602654	−	0.798003i	$-0.705890\pi$
$937$	−36.8950	−1.20531	−0.602654	+	0.798003i	$0.705890\pi$
$938$	0	0
$939$	8.54240	0.278771
$940$	−13.4056	−0.437243
$941$	−28.8694	−0.941116	−0.470558	−	0.882369i	$-0.655947\pi$
$941$	−28.8694	−0.941116	−0.470558	+	0.882369i	$0.655947\pi$
$942$	4.36544	0.142234
$943$	12.0023	0.390850
$944$	2.59067	0.0843190
$945$	0	0
$946$	12.9625	0.421446
$947$	12.0082	0.390213	0.195106	−	0.980782i	$-0.437495\pi$
$947$	12.0082	0.390213	0.195106	+	0.980782i	$0.437495\pi$
$948$	−8.65943	−0.281245
$949$	5.98559	0.194300
$950$	1.90866	0.0619251
$951$	−2.60530	−0.0844827
$952$	0	0
$953$	−35.5463	−1.15146	−0.575728	−	0.817641i	$-0.695281\pi$
$953$	−35.5463	−1.15146	−0.575728	+	0.817641i	$0.695281\pi$
$954$	−1.77027	−0.0573145
$955$	8.42607	0.272661
$956$	45.2834	1.46457
$957$	4.62011	0.149347
$958$	−11.9218	−0.385177
$959$	0	0
$960$	0.767044	0.0247562
$961$	−30.4145	−0.981113
$962$	−1.08524	−0.0349895
$963$	20.7649	0.669140
$964$	−23.5243	−0.757666
$965$	0.773093	0.0248868
$966$	0	0
$967$	27.9449	0.898648	0.449324	−	0.893369i	$-0.351665\pi$
$967$	27.9449	0.898648	0.449324	+	0.893369i	$0.351665\pi$
$968$	14.0173	0.450532
$969$	−11.8183	−0.379658
$970$	−7.30392	−0.234515
$971$	−29.9589	−0.961425	−0.480713	−	0.876878i	$-0.659622\pi$
$971$	−29.9589	−0.961425	−0.480713	+	0.876878i	$0.659622\pi$
$972$	−21.2302	−0.680960
$973$	0	0
$974$	9.36709	0.300141
$975$	−1.03060	−0.0330056
$976$	1.58796	0.0508292
$977$	−11.8853	−0.380246	−0.190123	−	0.981760i	$-0.560889\pi$
$977$	−11.8853	−0.380246	−0.190123	+	0.981760i	$0.560889\pi$
$978$	2.79242	0.0892916
$979$	26.2123	0.837748
$980$	0	0
$981$	−51.2694	−1.63690
$982$	−18.0152	−0.574888
$983$	19.9876	0.637507	0.318753	−	0.947838i	$-0.396736\pi$
$983$	19.9876	0.637507	0.318753	+	0.947838i	$0.396736\pi$
$984$	9.29136	0.296198
$985$	−4.93747	−0.157321
$986$	15.4771	0.492891
$987$	0	0
$988$	11.1822	0.355752
$989$	15.9947	0.508602
$990$	−3.11462	−0.0989891
$991$	−17.7861	−0.564994	−0.282497	−	0.959268i	$-0.591163\pi$
$991$	−17.7861	−0.564994	−0.282497	+	0.959268i	$0.591163\pi$
$992$	4.10558	0.130352
$993$	10.3497	0.328439
$994$	0	0
$995$	−14.0397	−0.445089
$996$	−13.1361	−0.416233
$997$	−38.4135	−1.21657	−0.608284	−	0.793719i	$-0.708142\pi$
$997$	−38.4135	−1.21657	−0.608284	+	0.793719i	$0.708142\pi$
$998$	7.23241	0.228938
$999$	−3.03594	−0.0960529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bc.1.20	yes	34
7.6	odd	2		9065.2.a.bb.1.20	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bb.1.20	✓	34	7.6	odd	2
9065.2.a.bc.1.20	yes	34	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q+0.559083 q^{2} +0.530934 q^{3} -1.68743 q^{4} -1.00000 q^{5} +0.296836 q^{6} -2.06158 q^{8} -2.71811 q^{9} -0.559083 q^{10} -2.04956 q^{11} -0.895912 q^{12} -1.94110 q^{13} -0.530934 q^{15} +2.22226 q^{16} -6.52022 q^{17} -1.51965 q^{18} +3.41391 q^{19} +1.68743 q^{20} -1.14588 q^{22} -1.41393 q^{23} -1.09456 q^{24} +1.00000 q^{25} -1.08524 q^{26} -3.03594 q^{27} -4.24571 q^{29} -0.296836 q^{30} +0.765169 q^{31} +5.36558 q^{32} -1.08818 q^{33} -3.64534 q^{34} +4.58661 q^{36} +1.00000 q^{37} +1.90866 q^{38} -1.03060 q^{39} +2.06158 q^{40} -8.48866 q^{41} -11.3123 q^{43} +3.45849 q^{44} +2.71811 q^{45} -0.790503 q^{46} -7.94442 q^{47} +1.17987 q^{48} +0.559083 q^{50} -3.46181 q^{51} +3.27547 q^{52} +1.16492 q^{53} -1.69734 q^{54} +2.04956 q^{55} +1.81256 q^{57} -2.37371 q^{58} +1.16578 q^{59} +0.895912 q^{60} +0.714569 q^{61} +0.427793 q^{62} -1.44471 q^{64} +1.94110 q^{65} -0.608385 q^{66} +7.56750 q^{67} +11.0024 q^{68} -0.750702 q^{69} -12.5740 q^{71} +5.60359 q^{72} -3.08360 q^{73} +0.559083 q^{74} +0.530934 q^{75} -5.76072 q^{76} -0.576190 q^{78} +9.66550 q^{79} -2.22226 q^{80} +6.54244 q^{81} -4.74587 q^{82} +14.6622 q^{83} +6.52022 q^{85} -6.32450 q^{86} -2.25419 q^{87} +4.22534 q^{88} -12.7892 q^{89} +1.51965 q^{90} +2.38590 q^{92} +0.406254 q^{93} -4.44159 q^{94} -3.41391 q^{95} +2.84877 q^{96} +13.0641 q^{97} +5.57094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) \) 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more