LMFDB - Embedded newform 1035.2.a.p.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1035,2,Mod(1,1035)] 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("1035.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,-6,0,6,0,0,0,4] 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:	$8.26451660920$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.98838128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 $ x^6 - x^5 - 10x^4 - x^3 + 16x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.493507$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.810417 q^{2} -1.34322 q^{4} -1.00000 q^{5} -4.60617 q^{7} -2.70941 q^{8} -0.810417 q^{10} +5.35375 q^{11} +5.51982 q^{13} -3.73292 q^{14} +0.490703 q^{16} +1.17660 q^{17} -1.73292 q^{19} +1.34322 q^{20} +4.33877 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.47336 q^{26} +6.18712 q^{28} -3.08635 q^{29} +4.34322 q^{31} +5.81648 q^{32} +0.953534 q^{34} +4.60617 q^{35} +8.81926 q^{37} -1.40438 q^{38} +2.70941 q^{40} -7.27347 q^{41} +8.52038 q^{43} -7.19129 q^{44} +0.810417 q^{46} +9.69252 q^{47} +14.2168 q^{49} +0.810417 q^{50} -7.41436 q^{52} +10.2018 q^{53} -5.35375 q^{55} +12.4800 q^{56} -2.50123 q^{58} -11.7468 q^{59} +1.33270 q^{61} +3.51982 q^{62} +3.73237 q^{64} -5.51982 q^{65} -6.01391 q^{67} -1.58043 q^{68} +3.73292 q^{70} -11.3991 q^{71} -1.51982 q^{73} +7.14728 q^{74} +2.32770 q^{76} -24.6603 q^{77} +1.26764 q^{79} -0.490703 q^{80} -5.89454 q^{82} +1.15555 q^{83} -1.17660 q^{85} +6.90506 q^{86} -14.5055 q^{88} -5.40223 q^{89} -25.4252 q^{91} -1.34322 q^{92} +7.85498 q^{94} +1.73292 q^{95} +16.4440 q^{97} +11.5215 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 22 * q^37 + 36 * q^38 - 18 * q^41 + 8 * q^43 + 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 12 * q^97 - 4 * 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 $

$2$	0.810417	0.573051	0.286526	−	0.958073i	$-0.407500\pi$
$2$	0.810417	0.573051	0.286526	+	0.958073i	$0.407500\pi$
$3$	0	0
$3$	0	0
$4$	−1.34322	−0.671612
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.60617	−1.74097	−0.870484	−	0.492196i	$-0.836194\pi$
$7$	−4.60617	−1.74097	−0.870484	+	0.492196i	$0.836194\pi$
$8$	−2.70941	−0.957919
$9$	0	0
$10$	−0.810417	−0.256276
$11$	5.35375	1.61422	0.807108	−	0.590404i	$-0.201031\pi$
$11$	5.35375	1.61422	0.807108	+	0.590404i	$0.201031\pi$
$12$	0	0
$13$	5.51982	1.53092	0.765462	−	0.643482i	$-0.222511\pi$
$13$	5.51982	1.53092	0.765462	+	0.643482i	$0.222511\pi$
$14$	−3.73292	−0.997664
$15$	0	0
$16$	0.490703	0.122676
$17$	1.17660	0.285367	0.142683	−	0.989768i	$-0.454427\pi$
$17$	1.17660	0.285367	0.142683	+	0.989768i	$0.454427\pi$
$18$	0	0
$19$	−1.73292	−0.397558	−0.198779	−	0.980044i	$-0.563698\pi$
$19$	−1.73292	−0.397558	−0.198779	+	0.980044i	$0.563698\pi$
$20$	1.34322	0.300354
$21$	0	0
$22$	4.33877	0.925028
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	4.47336	0.877297
$27$	0	0
$28$	6.18712	1.16926
$29$	−3.08635	−0.573120	−0.286560	−	0.958062i	$-0.592512\pi$
$29$	−3.08635	−0.573120	−0.286560	+	0.958062i	$0.592512\pi$
$30$	0	0
$31$	4.34322	0.780066	0.390033	−	0.920801i	$-0.372464\pi$
$31$	4.34322	0.780066	0.390033	+	0.920801i	$0.372464\pi$
$32$	5.81648	1.02822
$33$	0	0
$34$	0.953534	0.163530
$35$	4.60617	0.778585
$36$	0	0
$37$	8.81926	1.44988	0.724939	−	0.688813i	$-0.241868\pi$
$37$	8.81926	1.44988	0.724939	+	0.688813i	$0.241868\pi$
$38$	−1.40438	−0.227821
$39$	0	0
$40$	2.70941	0.428395
$41$	−7.27347	−1.13593	−0.567963	−	0.823054i	$-0.692268\pi$
$41$	−7.27347	−1.13593	−0.567963	+	0.823054i	$0.692268\pi$
$42$	0	0
$43$	8.52038	1.29935	0.649673	−	0.760214i	$-0.274906\pi$
$43$	8.52038	1.29935	0.649673	+	0.760214i	$0.274906\pi$
$44$	−7.19129	−1.08413
$45$	0	0
$46$	0.810417	0.119489
$47$	9.69252	1.41380	0.706899	−	0.707314i	$-0.250093\pi$
$47$	9.69252	1.41380	0.706899	+	0.707314i	$0.250093\pi$
$48$	0	0
$49$	14.2168	2.03097
$50$	0.810417	0.114610
$51$	0	0
$52$	−7.41436	−1.02819
$53$	10.2018	1.40133	0.700663	−	0.713492i	$-0.252887\pi$
$53$	10.2018	1.40133	0.700663	+	0.713492i	$0.252887\pi$
$54$	0	0
$55$	−5.35375	−0.721899
$56$	12.4800	1.66771
$57$	0	0
$58$	−2.50123	−0.328427
$59$	−11.7468	−1.52931	−0.764653	−	0.644442i	$-0.777090\pi$
$59$	−11.7468	−1.52931	−0.764653	+	0.644442i	$0.777090\pi$
$60$	0	0
$61$	1.33270	0.170635	0.0853174	−	0.996354i	$-0.472810\pi$
$61$	1.33270	0.170635	0.0853174	+	0.996354i	$0.472810\pi$
$62$	3.51982	0.447018
$63$	0	0
$64$	3.73237	0.466546
$65$	−5.51982	−0.684650
$66$	0	0
$67$	−6.01391	−0.734716	−0.367358	−	0.930080i	$-0.619738\pi$
$67$	−6.01391	−0.734716	−0.367358	+	0.930080i	$0.619738\pi$
$68$	−1.58043	−0.191656
$69$	0	0
$70$	3.73292	0.446169
$71$	−11.3991	−1.35283	−0.676415	−	0.736521i	$-0.736467\pi$
$71$	−11.3991	−1.35283	−0.676415	+	0.736521i	$0.736467\pi$
$72$	0	0
$73$	−1.51982	−0.177882	−0.0889408	−	0.996037i	$-0.528348\pi$
$73$	−1.51982	−0.177882	−0.0889408	+	0.996037i	$0.528348\pi$
$74$	7.14728	0.830854
$75$	0	0
$76$	2.32770	0.267005
$77$	−24.6603	−2.81030
$78$	0	0
$79$	1.26764	0.142621	0.0713103	−	0.997454i	$-0.477282\pi$
$79$	1.26764	0.142621	0.0713103	+	0.997454i	$0.477282\pi$
$80$	−0.490703	−0.0548623
$81$	0	0
$82$	−5.89454	−0.650943
$83$	1.15555	0.126838	0.0634189	−	0.997987i	$-0.479800\pi$
$83$	1.15555	0.126838	0.0634189	+	0.997987i	$0.479800\pi$
$84$	0	0
$85$	−1.17660	−0.127620
$86$	6.90506	0.744591
$87$	0	0
$88$	−14.5055	−1.54629
$89$	−5.40223	−0.572635	−0.286317	−	0.958135i	$-0.592431\pi$
$89$	−5.40223	−0.572635	−0.286317	+	0.958135i	$0.592431\pi$
$90$	0	0
$91$	−25.4252	−2.66529
$92$	−1.34322	−0.140041
$93$	0	0
$94$	7.85498	0.810179
$95$	1.73292	0.177793
$96$	0	0
$97$	16.4440	1.66964	0.834819	−	0.550524i	$-0.185572\pi$
$97$	16.4440	1.66964	0.834819	+	0.550524i	$0.185572\pi$
$98$	11.5215	1.16385
$99$	0	0
$100$	−1.34322	−0.134322
$101$	−3.43403	−0.341699	−0.170849	−	0.985297i	$-0.554651\pi$
$101$	−3.43403	−0.341699	−0.170849	+	0.985297i	$0.554651\pi$
$102$	0	0
$103$	9.42879	0.929046	0.464523	−	0.885561i	$-0.346226\pi$
$103$	9.42879	0.929046	0.464523	+	0.885561i	$0.346226\pi$
$104$	−14.9554	−1.46650
$105$	0	0
$106$	8.26772	0.803032
$107$	−14.9093	−1.44134	−0.720669	−	0.693279i	$-0.756165\pi$
$107$	−14.9093	−1.44134	−0.720669	+	0.693279i	$0.756165\pi$
$108$	0	0
$109$	8.26216	0.791371	0.395686	−	0.918386i	$-0.370507\pi$
$109$	8.26216	0.791371	0.395686	+	0.918386i	$0.370507\pi$
$110$	−4.33877	−0.413685
$111$	0	0
$112$	−2.26026	−0.213575
$113$	10.6280	0.999798	0.499899	−	0.866084i	$-0.333370\pi$
$113$	10.6280	0.999798	0.499899	+	0.866084i	$0.333370\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	4.14566	0.384915
$117$	0	0
$118$	−9.51982	−0.876371
$119$	−5.41960	−0.496814
$120$	0	0
$121$	17.6626	1.60569
$122$	1.08004	0.0977825
$123$	0	0
$124$	−5.83393	−0.523902
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.23560	−0.198377	−0.0991887	−	0.995069i	$-0.531625\pi$
$127$	−2.23560	−0.198377	−0.0991887	+	0.995069i	$0.531625\pi$
$128$	−8.60819	−0.760864
$129$	0	0
$130$	−4.47336	−0.392339
$131$	10.6831	0.933386	0.466693	−	0.884419i	$-0.345445\pi$
$131$	10.6831	0.933386	0.466693	+	0.884419i	$0.345445\pi$
$132$	0	0
$133$	7.98211	0.692136
$134$	−4.87377	−0.421030
$135$	0	0
$136$	−3.18788	−0.273358
$137$	0.326633	0.0279061	0.0139531	−	0.999903i	$-0.495558\pi$
$137$	0.326633	0.0279061	0.0139531	+	0.999903i	$0.495558\pi$
$138$	0	0
$139$	9.54894	0.809931	0.404965	−	0.914332i	$-0.367284\pi$
$139$	9.54894	0.809931	0.404965	+	0.914332i	$0.367284\pi$
$140$	−6.18712	−0.522907
$141$	0	0
$142$	−9.23805	−0.775240
$143$	29.5517	2.47124
$144$	0	0
$145$	3.08635	0.256307
$146$	−1.23169	−0.101935
$147$	0	0
$148$	−11.8463	−0.973756
$149$	22.5368	1.84628	0.923142	−	0.384460i	$-0.125612\pi$
$149$	22.5368	1.84628	0.923142	+	0.384460i	$0.125612\pi$
$150$	0	0
$151$	16.3305	1.32896	0.664478	−	0.747308i	$-0.268654\pi$
$151$	16.3305	1.32896	0.664478	+	0.747308i	$0.268654\pi$
$152$	4.69517	0.380829
$153$	0	0
$154$	−19.9851	−1.61045
$155$	−4.34322	−0.348856
$156$	0	0
$157$	−13.2666	−1.05879	−0.529397	−	0.848374i	$-0.677582\pi$
$157$	−13.2666	−1.05879	−0.529397	+	0.848374i	$0.677582\pi$
$158$	1.02732	0.0817289
$159$	0	0
$160$	−5.81648	−0.459833
$161$	−4.60617	−0.363017
$162$	0	0
$163$	−15.7261	−1.23176	−0.615881	−	0.787839i	$-0.711200\pi$
$163$	−15.7261	−1.23176	−0.615881	+	0.787839i	$0.711200\pi$
$164$	9.76990	0.762901
$165$	0	0
$166$	0.936475	0.0726846
$167$	1.59282	0.123256	0.0616279	−	0.998099i	$-0.480371\pi$
$167$	1.59282	0.123256	0.0616279	+	0.998099i	$0.480371\pi$
$168$	0	0
$169$	17.4684	1.34373
$170$	−0.953534	−0.0731327
$171$	0	0
$172$	−11.4448	−0.872657
$173$	−6.90506	−0.524982	−0.262491	−	0.964934i	$-0.584544\pi$
$173$	−6.90506	−0.524982	−0.262491	+	0.964934i	$0.584544\pi$
$174$	0	0
$175$	−4.60617	−0.348194
$176$	2.62710	0.198025
$177$	0	0
$178$	−4.37805	−0.328149
$179$	−25.1515	−1.87991	−0.939957	−	0.341294i	$-0.889135\pi$
$179$	−25.1515	−1.87991	−0.939957	+	0.341294i	$0.889135\pi$
$180$	0	0
$181$	−12.9678	−0.963886	−0.481943	−	0.876203i	$-0.660069\pi$
$181$	−12.9678	−0.963886	−0.481943	+	0.876203i	$0.660069\pi$
$182$	−20.6050	−1.52735
$183$	0	0
$184$	−2.70941	−0.199740
$185$	−8.81926	−0.648405
$186$	0	0
$187$	6.29920	0.460643
$188$	−13.0192	−0.949525
$189$	0	0
$190$	1.40438	0.101885
$191$	3.90897	0.282843	0.141421	−	0.989949i	$-0.454833\pi$
$191$	3.90897	0.282843	0.141421	+	0.989949i	$0.454833\pi$
$192$	0	0
$193$	−2.68645	−0.193375	−0.0966874	−	0.995315i	$-0.530825\pi$
$193$	−2.68645	−0.193375	−0.0966874	+	0.995315i	$0.530825\pi$
$194$	13.3265	0.956788
$195$	0	0
$196$	−19.0964	−1.36403
$197$	9.78139	0.696895	0.348448	−	0.937328i	$-0.386709\pi$
$197$	9.78139	0.696895	0.348448	+	0.937328i	$0.386709\pi$
$198$	0	0
$199$	−19.0867	−1.35302	−0.676509	−	0.736434i	$-0.736508\pi$
$199$	−19.0867	−1.35302	−0.676509	+	0.736434i	$0.736508\pi$
$200$	−2.70941	−0.191584
$201$	0	0
$202$	−2.78299	−0.195811
$203$	14.2162	0.997784
$204$	0	0
$205$	7.27347	0.508001
$206$	7.64125	0.532391
$207$	0	0
$208$	2.70859	0.187807
$209$	−9.27760	−0.641745
$210$	0	0
$211$	−8.09524	−0.557299	−0.278650	−	0.960393i	$-0.589887\pi$
$211$	−8.09524	−0.557299	−0.278650	+	0.960393i	$0.589887\pi$
$212$	−13.7033	−0.941149
$213$	0	0
$214$	−12.0828	−0.825960
$215$	−8.52038	−0.581085
$216$	0	0
$217$	−20.0056	−1.35807
$218$	6.69579	0.453496
$219$	0	0
$220$	7.19129	0.484837
$221$	6.49460	0.436874
$222$	0	0
$223$	14.2094	0.951534	0.475767	−	0.879571i	$-0.342171\pi$
$223$	14.2094	0.951534	0.475767	+	0.879571i	$0.342171\pi$
$224$	−26.7917	−1.79010
$225$	0	0
$226$	8.61311	0.572936
$227$	16.6256	1.10348	0.551740	−	0.834016i	$-0.313964\pi$
$227$	16.6256	1.10348	0.551740	+	0.834016i	$0.313964\pi$
$228$	0	0
$229$	15.3640	1.01528	0.507640	−	0.861569i	$-0.330518\pi$
$229$	15.3640	1.01528	0.507640	+	0.861569i	$0.330518\pi$
$230$	−0.810417	−0.0534373
$231$	0	0
$232$	8.36217	0.549003
$233$	15.0878	0.988433	0.494217	−	0.869339i	$-0.335455\pi$
$233$	15.0878	0.988433	0.494217	+	0.869339i	$0.335455\pi$
$234$	0	0
$235$	−9.69252	−0.632270
$236$	15.7786	1.02710
$237$	0	0
$238$	−4.39214	−0.284700
$239$	1.42847	0.0924000	0.0462000	−	0.998932i	$-0.485289\pi$
$239$	1.42847	0.0924000	0.0462000	+	0.998932i	$0.485289\pi$
$240$	0	0
$241$	10.5954	0.682511	0.341255	−	0.939971i	$-0.389148\pi$
$241$	10.5954	0.682511	0.341255	+	0.939971i	$0.389148\pi$
$242$	14.3141	0.920145
$243$	0	0
$244$	−1.79012	−0.114600
$245$	−14.2168	−0.908278
$246$	0	0
$247$	−9.56539	−0.608631
$248$	−11.7676	−0.747241
$249$	0	0
$250$	−0.810417	−0.0512552
$251$	−15.3939	−0.971657	−0.485829	−	0.874054i	$-0.661482\pi$
$251$	−15.3939	−0.971657	−0.485829	+	0.874054i	$0.661482\pi$
$252$	0	0
$253$	5.35375	0.336587
$254$	−1.81177	−0.113680
$255$	0	0
$256$	−14.4410	−0.902560
$257$	7.92004	0.494038	0.247019	−	0.969011i	$-0.420549\pi$
$257$	7.92004	0.494038	0.247019	+	0.969011i	$0.420549\pi$
$258$	0	0
$259$	−40.6230	−2.52419
$260$	7.41436	0.459819
$261$	0	0
$262$	8.65776	0.534878
$263$	17.7857	1.09671	0.548355	−	0.836246i	$-0.315254\pi$
$263$	17.7857	1.09671	0.548355	+	0.836246i	$0.315254\pi$
$264$	0	0
$265$	−10.2018	−0.626692
$266$	6.46883	0.396629
$267$	0	0
$268$	8.07803	0.493444
$269$	0.155319	0.00946996	0.00473498	−	0.999989i	$-0.498493\pi$
$269$	0.155319	0.00946996	0.00473498	+	0.999989i	$0.498493\pi$
$270$	0	0
$271$	26.1967	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$271$	26.1967	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$272$	0.577359	0.0350076
$273$	0	0
$274$	0.264709	0.0159916
$275$	5.35375	0.322843
$276$	0	0
$277$	−17.6542	−1.06074	−0.530369	−	0.847767i	$-0.677947\pi$
$277$	−17.6542	−1.06074	−0.530369	+	0.847767i	$0.677947\pi$
$278$	7.73862	0.464132
$279$	0	0
$280$	−12.4800	−0.745821
$281$	18.7534	1.11873	0.559366	−	0.828921i	$-0.311045\pi$
$281$	18.7534	1.11873	0.559366	+	0.828921i	$0.311045\pi$
$282$	0	0
$283$	16.3826	0.973847	0.486923	−	0.873445i	$-0.338119\pi$
$283$	16.3826	0.973847	0.486923	+	0.873445i	$0.338119\pi$
$284$	15.3116	0.908577
$285$	0	0
$286$	23.9492	1.41615
$287$	33.5028	1.97761
$288$	0	0
$289$	−15.6156	−0.918566
$290$	2.50123	0.146877
$291$	0	0
$292$	2.04146	0.119468
$293$	−16.5506	−0.966897	−0.483448	−	0.875373i	$-0.660616\pi$
$293$	−16.5506	−0.966897	−0.483448	+	0.875373i	$0.660616\pi$
$294$	0	0
$295$	11.7468	0.683927
$296$	−23.8950	−1.38887
$297$	0	0
$298$	18.2642	1.05801
$299$	5.51982	0.319220
$300$	0	0
$301$	−39.2463	−2.26212
$302$	13.2345	0.761560
$303$	0	0
$304$	−0.850347	−0.0487707
$305$	−1.33270	−0.0763102
$306$	0	0
$307$	17.3084	0.987844	0.493922	−	0.869506i	$-0.335563\pi$
$307$	17.3084	0.987844	0.493922	+	0.869506i	$0.335563\pi$
$308$	33.1243	1.88743
$309$	0	0
$310$	−3.51982	−0.199912
$311$	−16.8210	−0.953834	−0.476917	−	0.878948i	$-0.658246\pi$
$311$	−16.8210	−0.953834	−0.476917	+	0.878948i	$0.658246\pi$
$312$	0	0
$313$	−8.44566	−0.477377	−0.238688	−	0.971096i	$-0.576717\pi$
$313$	−8.44566	−0.477377	−0.238688	+	0.971096i	$0.576717\pi$
$314$	−10.7515	−0.606743
$315$	0	0
$316$	−1.70273	−0.0957858
$317$	32.7283	1.83821	0.919104	−	0.394016i	$-0.128915\pi$
$317$	32.7283	1.83821	0.919104	+	0.394016i	$0.128915\pi$
$318$	0	0
$319$	−16.5235	−0.925140
$320$	−3.73237	−0.208646
$321$	0	0
$322$	−3.73292	−0.208027
$323$	−2.03894	−0.113450
$324$	0	0
$325$	5.51982	0.306185
$326$	−12.7447	−0.705863
$327$	0	0
$328$	19.7068	1.08812
$329$	−44.6454	−2.46138
$330$	0	0
$331$	−15.2008	−0.835512	−0.417756	−	0.908559i	$-0.637183\pi$
$331$	−15.2008	−0.835512	−0.417756	+	0.908559i	$0.637183\pi$
$332$	−1.55216	−0.0851859
$333$	0	0
$334$	1.29085	0.0706319
$335$	6.01391	0.328575
$336$	0	0
$337$	11.9751	0.652327	0.326164	−	0.945313i	$-0.394244\pi$
$337$	11.9751	0.652327	0.326164	+	0.945313i	$0.394244\pi$
$338$	14.1567	0.770023
$339$	0	0
$340$	1.58043	0.0857111
$341$	23.2525	1.25920
$342$	0	0
$343$	−33.2418	−1.79489
$344$	−23.0852	−1.24467
$345$	0	0
$346$	−5.59597	−0.300841
$347$	−15.6679	−0.841095	−0.420547	−	0.907271i	$-0.638162\pi$
$347$	−15.6679	−0.841095	−0.420547	+	0.907271i	$0.638162\pi$
$348$	0	0
$349$	24.0603	1.28792	0.643958	−	0.765061i	$-0.277291\pi$
$349$	24.0603	1.28792	0.643958	+	0.765061i	$0.277291\pi$
$350$	−3.73292	−0.199533
$351$	0	0
$352$	31.1400	1.65977
$353$	−22.5772	−1.20166	−0.600830	−	0.799376i	$-0.705163\pi$
$353$	−22.5772	−1.20166	−0.600830	+	0.799376i	$0.705163\pi$
$354$	0	0
$355$	11.3991	0.605004
$356$	7.25641	0.384589
$357$	0	0
$358$	−20.3832	−1.07729
$359$	−24.3132	−1.28320	−0.641602	−	0.767038i	$-0.721730\pi$
$359$	−24.3132	−1.28320	−0.641602	+	0.767038i	$0.721730\pi$
$360$	0	0
$361$	−15.9970	−0.841947
$362$	−10.5093	−0.552356
$363$	0	0
$364$	34.1518	1.79004
$365$	1.51982	0.0795511
$366$	0	0
$367$	16.0482	0.837708	0.418854	−	0.908053i	$-0.362432\pi$
$367$	16.0482	0.837708	0.418854	+	0.908053i	$0.362432\pi$
$368$	0.490703	0.0255797
$369$	0	0
$370$	−7.14728	−0.371569
$371$	−46.9913	−2.43967
$372$	0	0
$373$	14.8470	0.768747	0.384373	−	0.923178i	$-0.374418\pi$
$373$	14.8470	0.768747	0.384373	+	0.923178i	$0.374418\pi$
$374$	5.10498	0.263972
$375$	0	0
$376$	−26.2610	−1.35431
$377$	−17.0361	−0.877403
$378$	0	0
$379$	21.8303	1.12135	0.560673	−	0.828038i	$-0.310543\pi$
$379$	21.8303	1.12135	0.560673	+	0.828038i	$0.310543\pi$
$380$	−2.32770	−0.119408
$381$	0	0
$382$	3.16789	0.162083
$383$	−11.4590	−0.585528	−0.292764	−	0.956185i	$-0.594575\pi$
$383$	−11.4590	−0.585528	−0.292764	+	0.956185i	$0.594575\pi$
$384$	0	0
$385$	24.6603	1.25680
$386$	−2.17714	−0.110814
$387$	0	0
$388$	−22.0880	−1.12135
$389$	30.7060	1.55686	0.778428	−	0.627734i	$-0.216018\pi$
$389$	30.7060	1.55686	0.778428	+	0.627734i	$0.216018\pi$
$390$	0	0
$391$	1.17660	0.0595031
$392$	−38.5191	−1.94551
$393$	0	0
$394$	7.92700	0.399357
$395$	−1.26764	−0.0637819
$396$	0	0
$397$	−36.3445	−1.82408	−0.912039	−	0.410102i	$-0.865493\pi$
$397$	−36.3445	−1.82408	−0.912039	+	0.410102i	$0.865493\pi$
$398$	−15.4682	−0.775348
$399$	0	0
$400$	0.490703	0.0245351
$401$	−7.23228	−0.361163	−0.180581	−	0.983560i	$-0.557798\pi$
$401$	−7.23228	−0.361163	−0.180581	+	0.983560i	$0.557798\pi$
$402$	0	0
$403$	23.9738	1.19422
$404$	4.61267	0.229489
$405$	0	0
$406$	11.5211	0.571781
$407$	47.2161	2.34042
$408$	0	0
$409$	−13.2289	−0.654129	−0.327064	−	0.945002i	$-0.606059\pi$
$409$	−13.2289	−0.654129	−0.327064	+	0.945002i	$0.606059\pi$
$410$	5.89454	0.291111
$411$	0	0
$412$	−12.6650	−0.623959
$413$	54.1079	2.66247
$414$	0	0
$415$	−1.15555	−0.0567236
$416$	32.1060	1.57412
$417$	0	0
$418$	−7.51872	−0.367753
$419$	−18.3115	−0.894574	−0.447287	−	0.894390i	$-0.647610\pi$
$419$	−18.3115	−0.894574	−0.447287	+	0.894390i	$0.647610\pi$
$420$	0	0
$421$	14.1996	0.692049	0.346024	−	0.938226i	$-0.387531\pi$
$421$	14.1996	0.692049	0.346024	+	0.938226i	$0.387531\pi$
$422$	−6.56052	−0.319361
$423$	0	0
$424$	−27.6408	−1.34236
$425$	1.17660	0.0570733
$426$	0	0
$427$	−6.13864	−0.297070
$428$	20.0266	0.968020
$429$	0	0
$430$	−6.90506	−0.332991
$431$	−35.0475	−1.68818	−0.844090	−	0.536202i	$-0.819859\pi$
$431$	−35.0475	−1.68818	−0.844090	+	0.536202i	$0.819859\pi$
$432$	0	0
$433$	3.14590	0.151182	0.0755911	−	0.997139i	$-0.475916\pi$
$433$	3.14590	0.151182	0.0755911	+	0.997139i	$0.475916\pi$
$434$	−16.2129	−0.778244
$435$	0	0
$436$	−11.0979	−0.531495
$437$	−1.73292	−0.0828966
$438$	0	0
$439$	1.95513	0.0933135	0.0466567	−	0.998911i	$-0.485143\pi$
$439$	1.95513	0.0933135	0.0466567	+	0.998911i	$0.485143\pi$
$440$	14.5055	0.691521
$441$	0	0
$442$	5.26334	0.250351
$443$	−27.4890	−1.30604	−0.653020	−	0.757340i	$-0.726498\pi$
$443$	−27.4890	−1.30604	−0.653020	+	0.757340i	$0.726498\pi$
$444$	0	0
$445$	5.40223	0.256090
$446$	11.5156	0.545277
$447$	0	0
$448$	−17.1919	−0.812242
$449$	2.83238	0.133668	0.0668341	−	0.997764i	$-0.478710\pi$
$449$	2.83238	0.133668	0.0668341	+	0.997764i	$0.478710\pi$
$450$	0	0
$451$	−38.9403	−1.83363
$452$	−14.2758	−0.671477
$453$	0	0
$454$	13.4737	0.632350
$455$	25.4252	1.19195
$456$	0	0
$457$	−2.97045	−0.138952	−0.0694760	−	0.997584i	$-0.522133\pi$
$457$	−2.97045	−0.138952	−0.0694760	+	0.997584i	$0.522133\pi$
$458$	12.4512	0.581808
$459$	0	0
$460$	1.34322	0.0626282
$461$	16.1855	0.753832	0.376916	−	0.926247i	$-0.376984\pi$
$461$	16.1855	0.753832	0.376916	+	0.926247i	$0.376984\pi$
$462$	0	0
$463$	−2.40718	−0.111871	−0.0559356	−	0.998434i	$-0.517814\pi$
$463$	−2.40718	−0.111871	−0.0559356	+	0.998434i	$0.517814\pi$
$464$	−1.51448	−0.0703080
$465$	0	0
$466$	12.2274	0.566423
$467$	10.0639	0.465700	0.232850	−	0.972513i	$-0.425195\pi$
$467$	10.0639	0.465700	0.232850	+	0.972513i	$0.425195\pi$
$468$	0	0
$469$	27.7011	1.27912
$470$	−7.85498	−0.362323
$471$	0	0
$472$	31.8269	1.46495
$473$	45.6160	2.09742
$474$	0	0
$475$	−1.73292	−0.0795116
$476$	7.27975	0.333667
$477$	0	0
$478$	1.15766	0.0529499
$479$	−26.0812	−1.19168	−0.595841	−	0.803102i	$-0.703181\pi$
$479$	−26.0812	−1.19168	−0.595841	+	0.803102i	$0.703181\pi$
$480$	0	0
$481$	48.6808	2.21965
$482$	8.58670	0.391114
$483$	0	0
$484$	−23.7249	−1.07840
$485$	−16.4440	−0.746685
$486$	0	0
$487$	23.7436	1.07592	0.537962	−	0.842969i	$-0.319194\pi$
$487$	23.7436	1.07592	0.537962	+	0.842969i	$0.319194\pi$
$488$	−3.61082	−0.163454
$489$	0	0
$490$	−11.5215	−0.520490
$491$	−2.79506	−0.126139	−0.0630696	−	0.998009i	$-0.520089\pi$
$491$	−2.79506	−0.126139	−0.0630696	+	0.998009i	$0.520089\pi$
$492$	0	0
$493$	−3.63139	−0.163549
$494$	−7.75195	−0.348777
$495$	0	0
$496$	2.13123	0.0956952
$497$	52.5064	2.35523
$498$	0	0
$499$	−8.35214	−0.373893	−0.186947	−	0.982370i	$-0.559859\pi$
$499$	−8.35214	−0.373893	−0.186947	+	0.982370i	$0.559859\pi$
$500$	1.34322	0.0600708
$501$	0	0
$502$	−12.4755	−0.556809
$503$	−19.8111	−0.883333	−0.441667	−	0.897179i	$-0.645613\pi$
$503$	−19.8111	−0.883333	−0.441667	+	0.897179i	$0.645613\pi$
$504$	0	0
$505$	3.43403	0.152812
$506$	4.33877	0.192882
$507$	0	0
$508$	3.00291	0.133233
$509$	−29.1471	−1.29192	−0.645961	−	0.763371i	$-0.723543\pi$
$509$	−29.1471	−1.29192	−0.645961	+	0.763371i	$0.723543\pi$
$510$	0	0
$511$	7.00056	0.309686
$512$	5.51319	0.243651
$513$	0	0
$514$	6.41853	0.283109
$515$	−9.42879	−0.415482
$516$	0	0
$517$	51.8913	2.28218
$518$	−32.9216	−1.44649
$519$	0	0
$520$	14.9554	0.655839
$521$	−3.73563	−0.163661	−0.0818306	−	0.996646i	$-0.526077\pi$
$521$	−3.73563	−0.163661	−0.0818306	+	0.996646i	$0.526077\pi$
$522$	0	0
$523$	−28.4551	−1.24425	−0.622127	−	0.782916i	$-0.713731\pi$
$523$	−28.4551	−1.24425	−0.622127	+	0.782916i	$0.713731\pi$
$524$	−14.3498	−0.626874
$525$	0	0
$526$	14.4138	0.628471
$527$	5.11022	0.222605
$528$	0	0
$529$	1.00000	0.0434783
$530$	−8.26772	−0.359127
$531$	0	0
$532$	−10.7218	−0.464847
$533$	−40.1483	−1.73901
$534$	0	0
$535$	14.9093	0.644586
$536$	16.2941	0.703798
$537$	0	0
$538$	0.125873	0.00542677
$539$	76.1132	3.27843
$540$	0	0
$541$	37.3591	1.60619	0.803097	−	0.595849i	$-0.203184\pi$
$541$	37.3591	1.60619	0.803097	+	0.595849i	$0.203184\pi$
$542$	21.2302	0.911917
$543$	0	0
$544$	6.84366	0.293419
$545$	−8.26216	−0.353912
$546$	0	0
$547$	−38.4683	−1.64479	−0.822394	−	0.568919i	$-0.807362\pi$
$547$	−38.4683	−1.64479	−0.822394	+	0.568919i	$0.807362\pi$
$548$	−0.438742	−0.0187421
$549$	0	0
$550$	4.33877	0.185006
$551$	5.34838	0.227849
$552$	0	0
$553$	−5.83896	−0.248298
$554$	−14.3073	−0.607857
$555$	0	0
$556$	−12.8264	−0.543959
$557$	−29.8981	−1.26682	−0.633412	−	0.773815i	$-0.718346\pi$
$557$	−29.8981	−1.26682	−0.633412	+	0.773815i	$0.718346\pi$
$558$	0	0
$559$	47.0310	1.98920
$560$	2.26026	0.0955134
$561$	0	0
$562$	15.1980	0.641090
$563$	−36.7777	−1.54999	−0.774997	−	0.631965i	$-0.782249\pi$
$563$	−36.7777	−1.54999	−0.774997	+	0.631965i	$0.782249\pi$
$564$	0	0
$565$	−10.6280	−0.447123
$566$	13.2768	0.558064
$567$	0	0
$568$	30.8849	1.29590
$569$	−19.0192	−0.797328	−0.398664	−	0.917097i	$-0.630526\pi$
$569$	−19.0192	−0.797328	−0.398664	+	0.917097i	$0.630526\pi$
$570$	0	0
$571$	41.8743	1.75238	0.876192	−	0.481962i	$-0.160076\pi$
$571$	41.8743	1.75238	0.876192	+	0.481962i	$0.160076\pi$
$572$	−39.6946	−1.65972
$573$	0	0
$574$	27.1513	1.13327
$575$	1.00000	0.0417029
$576$	0	0
$577$	30.6755	1.27704	0.638520	−	0.769605i	$-0.279547\pi$
$577$	30.6755	1.27704	0.638520	+	0.769605i	$0.279547\pi$
$578$	−12.6552	−0.526385
$579$	0	0
$580$	−4.14566	−0.172139
$581$	−5.32265	−0.220821
$582$	0	0
$583$	54.6180	2.26204
$584$	4.11781	0.170396
$585$	0	0
$586$	−13.4129	−0.554081
$587$	−1.52558	−0.0629675	−0.0314837	−	0.999504i	$-0.510023\pi$
$587$	−1.52558	−0.0629675	−0.0314837	+	0.999504i	$0.510023\pi$
$588$	0	0
$589$	−7.52644	−0.310122
$590$	9.51982	0.391925
$591$	0	0
$592$	4.32764	0.177865
$593$	−5.12072	−0.210283	−0.105141	−	0.994457i	$-0.533529\pi$
$593$	−5.12072	−0.210283	−0.105141	+	0.994457i	$0.533529\pi$
$594$	0	0
$595$	5.41960	0.222182
$596$	−30.2719	−1.23999
$597$	0	0
$598$	4.47336	0.182929
$599$	18.5010	0.755931	0.377966	−	0.925820i	$-0.376624\pi$
$599$	18.5010	0.755931	0.377966	+	0.925820i	$0.376624\pi$
$600$	0	0
$601$	−30.5128	−1.24464	−0.622322	−	0.782761i	$-0.713811\pi$
$601$	−30.5128	−1.24464	−0.622322	+	0.782761i	$0.713811\pi$
$602$	−31.8059	−1.29631
$603$	0	0
$604$	−21.9355	−0.892544
$605$	−17.6626	−0.718088
$606$	0	0
$607$	−20.6414	−0.837808	−0.418904	−	0.908031i	$-0.637586\pi$
$607$	−20.6414	−0.837808	−0.418904	+	0.908031i	$0.637586\pi$
$608$	−10.0795	−0.408777
$609$	0	0
$610$	−1.08004	−0.0437296
$611$	53.5010	2.16442
$612$	0	0
$613$	−11.4793	−0.463643	−0.231822	−	0.972758i	$-0.574469\pi$
$613$	−11.4793	−0.463643	−0.231822	+	0.972758i	$0.574469\pi$
$614$	14.0270	0.566085
$615$	0	0
$616$	66.8147	2.69204
$617$	−19.8055	−0.797340	−0.398670	−	0.917095i	$-0.630528\pi$
$617$	−19.8055	−0.797340	−0.398670	+	0.917095i	$0.630528\pi$
$618$	0	0
$619$	−24.4491	−0.982692	−0.491346	−	0.870965i	$-0.663495\pi$
$619$	−24.4491	−0.982692	−0.491346	+	0.870965i	$0.663495\pi$
$620$	5.83393	0.234296
$621$	0	0
$622$	−13.6320	−0.546595
$623$	24.8836	0.996939
$624$	0	0
$625$	1.00000	0.0400000
$626$	−6.84450	−0.273561
$627$	0	0
$628$	17.8201	0.711099
$629$	10.3767	0.413747
$630$	0	0
$631$	46.5829	1.85444	0.927218	−	0.374522i	$-0.122193\pi$
$631$	46.5829	1.85444	0.927218	+	0.374522i	$0.122193\pi$
$632$	−3.43455	−0.136619
$633$	0	0
$634$	26.5236	1.05339
$635$	2.23560	0.0887171
$636$	0	0
$637$	78.4742	3.10926
$638$	−13.3909	−0.530153
$639$	0	0
$640$	8.60819	0.340269
$641$	18.8170	0.743226	0.371613	−	0.928388i	$-0.378805\pi$
$641$	18.8170	0.743226	0.371613	+	0.928388i	$0.378805\pi$
$642$	0	0
$643$	−17.1868	−0.677783	−0.338891	−	0.940826i	$-0.610052\pi$
$643$	−17.1868	−0.677783	−0.338891	+	0.940826i	$0.610052\pi$
$644$	6.18712	0.243807
$645$	0	0
$646$	−1.65239	−0.0650126
$647$	−16.6153	−0.653214	−0.326607	−	0.945160i	$-0.605905\pi$
$647$	−16.6153	−0.653214	−0.326607	+	0.945160i	$0.605905\pi$
$648$	0	0
$649$	−62.8896	−2.46863
$650$	4.47336	0.175459
$651$	0	0
$652$	21.1237	0.827267
$653$	−8.71155	−0.340909	−0.170455	−	0.985366i	$-0.554524\pi$
$653$	−8.71155	−0.340909	−0.170455	+	0.985366i	$0.554524\pi$
$654$	0	0
$655$	−10.6831	−0.417423
$656$	−3.56911	−0.139350
$657$	0	0
$658$	−36.1814	−1.41050
$659$	24.1116	0.939256	0.469628	−	0.882865i	$-0.344388\pi$
$659$	24.1116	0.939256	0.469628	+	0.882865i	$0.344388\pi$
$660$	0	0
$661$	39.6144	1.54082	0.770411	−	0.637548i	$-0.220051\pi$
$661$	39.6144	1.54082	0.770411	+	0.637548i	$0.220051\pi$
$662$	−12.3190	−0.478791
$663$	0	0
$664$	−3.13085	−0.121500
$665$	−7.98211	−0.309533
$666$	0	0
$667$	−3.08635	−0.119504
$668$	−2.13951	−0.0827802
$669$	0	0
$670$	4.87377	0.188290
$671$	7.13494	0.275441
$672$	0	0
$673$	20.0269	0.771980	0.385990	−	0.922503i	$-0.373860\pi$
$673$	20.0269	0.771980	0.385990	+	0.922503i	$0.373860\pi$
$674$	9.70485	0.373817
$675$	0	0
$676$	−23.4640	−0.902463
$677$	−51.3624	−1.97402	−0.987010	−	0.160662i	$-0.948637\pi$
$677$	−51.3624	−1.97402	−0.987010	+	0.160662i	$0.948637\pi$
$678$	0	0
$679$	−75.7440	−2.90679
$680$	3.18788	0.122250
$681$	0	0
$682$	18.8442	0.721583
$683$	−1.78656	−0.0683609	−0.0341804	−	0.999416i	$-0.510882\pi$
$683$	−1.78656	−0.0683609	−0.0341804	+	0.999416i	$0.510882\pi$
$684$	0	0
$685$	−0.326633	−0.0124800
$686$	−26.9397	−1.02856
$687$	0	0
$688$	4.18097	0.159398
$689$	56.3122	2.14532
$690$	0	0
$691$	−51.7197	−1.96751	−0.983755	−	0.179515i	$-0.942547\pi$
$691$	−51.7197	−1.96751	−0.983755	+	0.179515i	$0.942547\pi$
$692$	9.27504	0.352584
$693$	0	0
$694$	−12.6975	−0.481990
$695$	−9.54894	−0.362212
$696$	0	0
$697$	−8.55794	−0.324155
$698$	19.4988	0.738042
$699$	0	0
$700$	6.18712	0.233851
$701$	33.4929	1.26501	0.632504	−	0.774557i	$-0.282027\pi$
$701$	33.4929	1.26501	0.632504	+	0.774557i	$0.282027\pi$
$702$	0	0
$703$	−15.2830	−0.576411
$704$	19.9822	0.753106
$705$	0	0
$706$	−18.2969	−0.688613
$707$	15.8177	0.594887
$708$	0	0
$709$	−15.8720	−0.596085	−0.298043	−	0.954553i	$-0.596334\pi$
$709$	−15.8720	−0.596085	−0.298043	+	0.954553i	$0.596334\pi$
$710$	9.23805	0.346698
$711$	0	0
$712$	14.6368	0.548538
$713$	4.34322	0.162655
$714$	0	0
$715$	−29.5517	−1.10517
$716$	33.7842	1.26257
$717$	0	0
$718$	−19.7038	−0.735341
$719$	−15.3345	−0.571881	−0.285941	−	0.958247i	$-0.592306\pi$
$719$	−15.3345	−0.571881	−0.285941	+	0.958247i	$0.592306\pi$
$720$	0	0
$721$	−43.4306	−1.61744
$722$	−12.9642	−0.482479
$723$	0	0
$724$	17.4186	0.647358
$725$	−3.08635	−0.114624
$726$	0	0
$727$	27.1804	1.00806	0.504032	−	0.863685i	$-0.331849\pi$
$727$	27.1804	1.00806	0.504032	+	0.863685i	$0.331849\pi$
$728$	68.8873	2.55313
$729$	0	0
$730$	1.23169	0.0455868
$731$	10.0250	0.370790
$732$	0	0
$733$	−4.70308	−0.173712	−0.0868560	−	0.996221i	$-0.527682\pi$
$733$	−4.70308	−0.173712	−0.0868560	+	0.996221i	$0.527682\pi$
$734$	13.0057	0.480050
$735$	0	0
$736$	5.81648	0.214398
$737$	−32.1970	−1.18599
$738$	0	0
$739$	1.19298	0.0438845	0.0219422	−	0.999759i	$-0.493015\pi$
$739$	1.19298	0.0438845	0.0219422	+	0.999759i	$0.493015\pi$
$740$	11.8463	0.435477
$741$	0	0
$742$	−38.0825	−1.39805
$743$	10.0273	0.367867	0.183934	−	0.982939i	$-0.441117\pi$
$743$	10.0273	0.367867	0.183934	+	0.982939i	$0.441117\pi$
$744$	0	0
$745$	−22.5368	−0.825683
$746$	12.0322	0.440531
$747$	0	0
$748$	−8.46125	−0.309374
$749$	68.6748	2.50932
$750$	0	0
$751$	−19.2248	−0.701524	−0.350762	−	0.936465i	$-0.614077\pi$
$751$	−19.2248	−0.701524	−0.350762	+	0.936465i	$0.614077\pi$
$752$	4.75615	0.173439
$753$	0	0
$754$	−13.8063	−0.502797
$755$	−16.3305	−0.594327
$756$	0	0
$757$	−29.0674	−1.05647	−0.528236	−	0.849098i	$-0.677146\pi$
$757$	−29.0674	−1.05647	−0.528236	+	0.849098i	$0.677146\pi$
$758$	17.6916	0.642588
$759$	0	0
$760$	−4.69517	−0.170312
$761$	−34.9604	−1.26731	−0.633657	−	0.773614i	$-0.718447\pi$
$761$	−34.9604	−1.26731	−0.633657	+	0.773614i	$0.718447\pi$
$762$	0	0
$763$	−38.0569	−1.37775
$764$	−5.25062	−0.189961
$765$	0	0
$766$	−9.28658	−0.335538
$767$	−64.8404	−2.34125
$768$	0	0
$769$	8.00084	0.288518	0.144259	−	0.989540i	$-0.453920\pi$
$769$	8.00084	0.288518	0.144259	+	0.989540i	$0.453920\pi$
$770$	19.9851	0.720213
$771$	0	0
$772$	3.60851	0.129873
$773$	−25.3463	−0.911644	−0.455822	−	0.890071i	$-0.650655\pi$
$773$	−25.3463	−0.911644	−0.455822	+	0.890071i	$0.650655\pi$
$774$	0	0
$775$	4.34322	0.156013
$776$	−44.5535	−1.59938
$777$	0	0
$778$	24.8846	0.892158
$779$	12.6043	0.451596
$780$	0	0
$781$	−61.0282	−2.18376
$782$	0.953534	0.0340983
$783$	0	0
$784$	6.97622	0.249151
$785$	13.2666	0.473507
$786$	0	0
$787$	−5.10486	−0.181969	−0.0909843	−	0.995852i	$-0.529001\pi$
$787$	−5.10486	−0.181969	−0.0909843	+	0.995852i	$0.529001\pi$
$788$	−13.1386	−0.468044
$789$	0	0
$790$	−1.02732	−0.0365503
$791$	−48.9544	−1.74062
$792$	0	0
$793$	7.35627	0.261229
$794$	−29.4542	−1.04529
$795$	0	0
$796$	25.6377	0.908704
$797$	44.0182	1.55921	0.779603	−	0.626274i	$-0.215421\pi$
$797$	44.0182	1.55921	0.779603	+	0.626274i	$0.215421\pi$
$798$	0	0
$799$	11.4042	0.403451
$800$	5.81648	0.205644
$801$	0	0
$802$	−5.86116	−0.206965
$803$	−8.13674	−0.287139
$804$	0	0
$805$	4.60617	0.162346
$806$	19.4288	0.684350
$807$	0	0
$808$	9.30418	0.327320
$809$	6.15978	0.216566	0.108283	−	0.994120i	$-0.465465\pi$
$809$	6.15978	0.216566	0.108283	+	0.994120i	$0.465465\pi$
$810$	0	0
$811$	−2.23801	−0.0785873	−0.0392936	−	0.999228i	$-0.512511\pi$
$811$	−2.23801	−0.0785873	−0.0392936	+	0.999228i	$0.512511\pi$
$812$	−19.0956	−0.670124
$813$	0	0
$814$	38.2647	1.34118
$815$	15.7261	0.550861
$816$	0	0
$817$	−14.7651	−0.516565
$818$	−10.7209	−0.374849
$819$	0	0
$820$	−9.76990	−0.341180
$821$	−39.9914	−1.39571	−0.697855	−	0.716239i	$-0.745862\pi$
$821$	−39.9914	−1.39571	−0.697855	+	0.716239i	$0.745862\pi$
$822$	0	0
$823$	23.3116	0.812590	0.406295	−	0.913742i	$-0.366821\pi$
$823$	23.3116	0.812590	0.406295	+	0.913742i	$0.366821\pi$
$824$	−25.5464	−0.889951
$825$	0	0
$826$	43.8499	1.52573
$827$	43.4988	1.51260	0.756301	−	0.654224i	$-0.227005\pi$
$827$	43.4988	1.51260	0.756301	+	0.654224i	$0.227005\pi$
$828$	0	0
$829$	33.1090	1.14992	0.574961	−	0.818181i	$-0.305017\pi$
$829$	33.1090	1.14992	0.574961	+	0.818181i	$0.305017\pi$
$830$	−0.936475	−0.0325055
$831$	0	0
$832$	20.6020	0.714246
$833$	16.7274	0.579571
$834$	0	0
$835$	−1.59282	−0.0551217
$836$	12.4619	0.431004
$837$	0	0
$838$	−14.8399	−0.512637
$839$	52.5227	1.81329	0.906643	−	0.421899i	$-0.138636\pi$
$839$	52.5227	1.81329	0.906643	+	0.421899i	$0.138636\pi$
$840$	0	0
$841$	−19.4745	−0.671533
$842$	11.5076	0.396579
$843$	0	0
$844$	10.8737	0.374289
$845$	−17.4684	−0.600932
$846$	0	0
$847$	−81.3571	−2.79546
$848$	5.00606	0.171909
$849$	0	0
$850$	0.953534	0.0327059
$851$	8.81926	0.302320
$852$	0	0
$853$	22.9673	0.786385	0.393192	−	0.919456i	$-0.371371\pi$
$853$	22.9673	0.786385	0.393192	+	0.919456i	$0.371371\pi$
$854$	−4.97486	−0.170236
$855$	0	0
$856$	40.3954	1.38069
$857$	55.1228	1.88296	0.941480	−	0.337069i	$-0.109436\pi$
$857$	55.1228	1.88296	0.941480	+	0.337069i	$0.109436\pi$
$858$	0	0
$859$	−49.9526	−1.70436	−0.852180	−	0.523249i	$-0.824720\pi$
$859$	−49.9526	−1.70436	−0.852180	+	0.523249i	$0.824720\pi$
$860$	11.4448	0.390264
$861$	0	0
$862$	−28.4031	−0.967413
$863$	−12.0802	−0.411214	−0.205607	−	0.978635i	$-0.565917\pi$
$863$	−12.0802	−0.411214	−0.205607	+	0.978635i	$0.565917\pi$
$864$	0	0
$865$	6.90506	0.234779
$866$	2.54949	0.0866351
$867$	0	0
$868$	26.8721	0.912097
$869$	6.78663	0.230221
$870$	0	0
$871$	−33.1957	−1.12479
$872$	−22.3855	−0.758070
$873$	0	0
$874$	−1.40438	−0.0475040
$875$	4.60617	0.155717
$876$	0	0
$877$	45.5573	1.53836	0.769180	−	0.639033i	$-0.220665\pi$
$877$	45.5573	1.53836	0.769180	+	0.639033i	$0.220665\pi$
$878$	1.58447	0.0534734
$879$	0	0
$880$	−2.62710	−0.0885595
$881$	6.57232	0.221427	0.110714	−	0.993852i	$-0.464686\pi$
$881$	6.57232	0.221427	0.110714	+	0.993852i	$0.464686\pi$
$882$	0	0
$883$	5.51200	0.185494	0.0927468	−	0.995690i	$-0.470435\pi$
$883$	5.51200	0.185494	0.0927468	+	0.995690i	$0.470435\pi$
$884$	−8.72371	−0.293410
$885$	0	0
$886$	−22.2775	−0.748428
$887$	31.2681	1.04988	0.524941	−	0.851139i	$-0.324087\pi$
$887$	31.2681	1.04988	0.524941	+	0.851139i	$0.324087\pi$
$888$	0	0
$889$	10.2975	0.345369
$890$	4.37805	0.146753
$891$	0	0
$892$	−19.0865	−0.639062
$893$	−16.7963	−0.562067
$894$	0	0
$895$	25.1515	0.840723
$896$	39.6508	1.32464
$897$	0	0
$898$	2.29541	0.0765987
$899$	−13.4047	−0.447072
$900$	0	0
$901$	12.0034	0.399892
$902$	−31.5579	−1.05076
$903$	0	0
$904$	−28.7956	−0.957726
$905$	12.9678	0.431063
$906$	0	0
$907$	−12.7989	−0.424982	−0.212491	−	0.977163i	$-0.568158\pi$
$907$	−12.7989	−0.424982	−0.212491	+	0.977163i	$0.568158\pi$
$908$	−22.3319	−0.741110
$909$	0	0
$910$	20.6050	0.683050
$911$	31.6963	1.05015	0.525073	−	0.851057i	$-0.324038\pi$
$911$	31.6963	1.05015	0.525073	+	0.851057i	$0.324038\pi$
$912$	0	0
$913$	6.18651	0.204744
$914$	−2.40730	−0.0796265
$915$	0	0
$916$	−20.6373	−0.681875
$917$	−49.2081	−1.62500
$918$	0	0
$919$	7.63249	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$919$	7.63249	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$920$	2.70941	0.0893264
$921$	0	0
$922$	13.1170	0.431984
$923$	−62.9212	−2.07108
$924$	0	0
$925$	8.81926	0.289976
$926$	−1.95082	−0.0641080
$927$	0	0
$928$	−17.9517	−0.589293
$929$	−17.2979	−0.567525	−0.283763	−	0.958895i	$-0.591583\pi$
$929$	−17.2979	−0.567525	−0.283763	+	0.958895i	$0.591583\pi$
$930$	0	0
$931$	−24.6365	−0.807429
$932$	−20.2663	−0.663844
$933$	0	0
$934$	8.15593	0.266870
$935$	−6.29920	−0.206006
$936$	0	0
$937$	12.5967	0.411517	0.205759	−	0.978603i	$-0.434034\pi$
$937$	12.5967	0.411517	0.205759	+	0.978603i	$0.434034\pi$
$938$	22.4494	0.732999
$939$	0	0
$940$	13.0192	0.424640
$941$	19.7213	0.642895	0.321448	−	0.946927i	$-0.395831\pi$
$941$	19.7213	0.642895	0.321448	+	0.946927i	$0.395831\pi$
$942$	0	0
$943$	−7.27347	−0.236857
$944$	−5.76420	−0.187609
$945$	0	0
$946$	36.9679	1.20193
$947$	24.8824	0.808569	0.404285	−	0.914633i	$-0.367521\pi$
$947$	24.8824	0.808569	0.404285	+	0.914633i	$0.367521\pi$
$948$	0	0
$949$	−8.38914	−0.272323
$950$	−1.40438	−0.0455642
$951$	0	0
$952$	14.6839	0.475908
$953$	24.9095	0.806899	0.403449	−	0.915002i	$-0.367811\pi$
$953$	24.9095	0.806899	0.403449	+	0.915002i	$0.367811\pi$
$954$	0	0
$955$	−3.90897	−0.126491
$956$	−1.91876	−0.0620570
$957$	0	0
$958$	−21.1367	−0.682895
$959$	−1.50453	−0.0485837
$960$	0	0
$961$	−12.1364	−0.391497
$962$	39.4517	1.27197
$963$	0	0
$964$	−14.2320	−0.458383
$965$	2.68645	0.0864799
$966$	0	0
$967$	−29.4706	−0.947709	−0.473855	−	0.880603i	$-0.657138\pi$
$967$	−29.4706	−0.947709	−0.473855	+	0.880603i	$0.657138\pi$
$968$	−47.8552	−1.53813
$969$	0	0
$970$	−13.3265	−0.427889
$971$	−36.7868	−1.18054	−0.590272	−	0.807204i	$-0.700980\pi$
$971$	−36.7868	−1.18054	−0.590272	+	0.807204i	$0.700980\pi$
$972$	0	0
$973$	−43.9840	−1.41006
$974$	19.2422	0.616560
$975$	0	0
$976$	0.653960	0.0209327
$977$	−14.6435	−0.468488	−0.234244	−	0.972178i	$-0.575262\pi$
$977$	−14.6435	−0.468488	−0.234244	+	0.972178i	$0.575262\pi$
$978$	0	0
$979$	−28.9222	−0.924357
$980$	19.0964	0.610011
$981$	0	0
$982$	−2.26516	−0.0722843
$983$	−1.10622	−0.0352831	−0.0176415	−	0.999844i	$-0.505616\pi$
$983$	−1.10622	−0.0352831	−0.0176415	+	0.999844i	$0.505616\pi$
$984$	0	0
$985$	−9.78139	−0.311661
$986$	−2.94294	−0.0937222
$987$	0	0
$988$	12.8485	0.408764
$989$	8.52038	0.270932
$990$	0	0
$991$	−42.7058	−1.35659	−0.678297	−	0.734788i	$-0.737282\pi$
$991$	−42.7058	−1.35659	−0.678297	+	0.734788i	$0.737282\pi$
$992$	25.2623	0.802079
$993$	0	0
$994$	42.5520	1.34967
$995$	19.0867	0.605088
$996$	0	0
$997$	30.0771	0.952553	0.476276	−	0.879296i	$-0.341986\pi$
$997$	30.0771	0.952553	0.476276	+	0.879296i	$0.341986\pi$
$998$	−6.76871	−0.214260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.p.1.4	✓	6
3.2	odd	2		1035.2.a.q.1.3	yes	6
5.4	even	2		5175.2.a.bz.1.3		6
15.14	odd	2		5175.2.a.by.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.4	✓	6	1.1	even	1	trivial
1035.2.a.q.1.3	yes	6	3.2	odd	2
5175.2.a.by.1.4		6	15.14	odd	2
5175.2.a.bz.1.3		6	5.4	even	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+0.810417 q^{2} -1.34322 q^{4} -1.00000 q^{5} -4.60617 q^{7} -2.70941 q^{8} -0.810417 q^{10} +5.35375 q^{11} +5.51982 q^{13} -3.73292 q^{14} +0.490703 q^{16} +1.17660 q^{17} -1.73292 q^{19} +1.34322 q^{20} +4.33877 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.47336 q^{26} +6.18712 q^{28} -3.08635 q^{29} +4.34322 q^{31} +5.81648 q^{32} +0.953534 q^{34} +4.60617 q^{35} +8.81926 q^{37} -1.40438 q^{38} +2.70941 q^{40} -7.27347 q^{41} +8.52038 q^{43} -7.19129 q^{44} +0.810417 q^{46} +9.69252 q^{47} +14.2168 q^{49} +0.810417 q^{50} -7.41436 q^{52} +10.2018 q^{53} -5.35375 q^{55} +12.4800 q^{56} -2.50123 q^{58} -11.7468 q^{59} +1.33270 q^{61} +3.51982 q^{62} +3.73237 q^{64} -5.51982 q^{65} -6.01391 q^{67} -1.58043 q^{68} +3.73292 q^{70} -11.3991 q^{71} -1.51982 q^{73} +7.14728 q^{74} +2.32770 q^{76} -24.6603 q^{77} +1.26764 q^{79} -0.490703 q^{80} -5.89454 q^{82} +1.15555 q^{83} -1.17660 q^{85} +6.90506 q^{86} -14.5055 q^{88} -5.40223 q^{89} -25.4252 q^{91} -1.34322 q^{92} +7.85498 q^{94} +1.73292 q^{95} +16.4440 q^{97} +11.5215 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 22 * q^37 + 36 * q^38 - 18 * q^41 + 8 * q^43 + 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 12 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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Database

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