LMFDB - Embedded newform 3040.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1,3040)] 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("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-2,0,5,0,3,0,-8] 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:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} -4.00000 q^{11} -1.56155 q^{13} -1.56155 q^{15} +3.56155 q^{17} -1.00000 q^{19} +0.684658 q^{21} +6.68466 q^{23} +1.00000 q^{25} -5.56155 q^{27} -7.56155 q^{29} +3.12311 q^{31} -6.24621 q^{33} -0.438447 q^{35} +7.12311 q^{37} -2.43845 q^{39} -8.24621 q^{41} -2.00000 q^{43} +0.561553 q^{45} -5.12311 q^{47} -6.80776 q^{49} +5.56155 q^{51} +2.43845 q^{53} +4.00000 q^{55} -1.56155 q^{57} -7.80776 q^{59} -13.1231 q^{61} -0.246211 q^{63} +1.56155 q^{65} -6.43845 q^{67} +10.4384 q^{69} -8.00000 q^{71} -5.80776 q^{73} +1.56155 q^{75} -1.75379 q^{77} -3.12311 q^{79} -7.00000 q^{81} -14.0000 q^{83} -3.56155 q^{85} -11.8078 q^{87} -10.0000 q^{89} -0.684658 q^{91} +4.87689 q^{93} +1.00000 q^{95} +7.12311 q^{97} +2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{11} + q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 11 q^{21} + q^{23} + 2 q^{25} - 7 q^{27} - 11 q^{29} - 2 q^{31} + 4 q^{33} - 5 q^{35} + 6 q^{37} - 9 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^11 + q^13 + q^15 + 3 * q^17 - 2 * q^19 - 11 * q^21 + q^23 + 2 * q^25 - 7 * q^27 - 11 * q^29 - 2 * q^31 + 4 * q^33 - 5 * q^35 + 6 * q^37 - 9 * q^39 - 4 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 + 7 * q^51 + 9 * q^53 + 8 * q^55 + q^57 + 5 * q^59 - 18 * q^61 + 16 * q^63 - q^65 - 17 * q^67 + 25 * q^69 - 16 * q^71 + 9 * q^73 - q^75 - 20 * q^77 + 2 * q^79 - 14 * q^81 - 28 * q^83 - 3 * q^85 - 3 * q^87 - 20 * q^89 + 11 * q^91 + 18 * q^93 + 2 * q^95 + 6 * q^97 - 12 * 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	0
$2$	0	0
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−1.56155	−0.433097	−0.216548	−	0.976272i	$-0.569480\pi$
$13$	−1.56155	−0.433097	−0.216548	+	0.976272i	$0.569480\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.684658	0.149405
$22$	0	0
$23$	6.68466	1.39385	0.696924	−	0.717145i	$-0.254552\pi$
$23$	6.68466	1.39385	0.696924	+	0.717145i	$0.254552\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−7.56155	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$29$	−7.56155	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	0	0
$33$	−6.24621	−1.08733
$34$	0	0
$35$	−0.438447	−0.0741111
$36$	0	0
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	0	0
$39$	−2.43845	−0.390464
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	−5.12311	−0.747282	−0.373641	−	0.927573i	$-0.621891\pi$
$47$	−5.12311	−0.747282	−0.373641	+	0.927573i	$0.621891\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	2.43845	0.334946	0.167473	−	0.985877i	$-0.446439\pi$
$53$	2.43845	0.334946	0.167473	+	0.985877i	$0.446439\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−1.56155	−0.206833
$58$	0	0
$59$	−7.80776	−1.01648	−0.508242	−	0.861214i	$-0.669705\pi$
$59$	−7.80776	−1.01648	−0.508242	+	0.861214i	$0.669705\pi$
$60$	0	0
$61$	−13.1231	−1.68024	−0.840121	−	0.542399i	$-0.817516\pi$
$61$	−13.1231	−1.68024	−0.840121	+	0.542399i	$0.817516\pi$
$62$	0	0
$63$	−0.246211	−0.0310197
$64$	0	0
$65$	1.56155	0.193687
$66$	0	0
$67$	−6.43845	−0.786582	−0.393291	−	0.919414i	$-0.628663\pi$
$67$	−6.43845	−0.786582	−0.393291	+	0.919414i	$0.628663\pi$
$68$	0	0
$69$	10.4384	1.25664
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−5.80776	−0.679747	−0.339874	−	0.940471i	$-0.610384\pi$
$73$	−5.80776	−0.679747	−0.339874	+	0.940471i	$0.610384\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	−1.75379	−0.199863
$78$	0	0
$79$	−3.12311	−0.351377	−0.175688	−	0.984446i	$-0.556215\pi$
$79$	−3.12311	−0.351377	−0.175688	+	0.984446i	$0.556215\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	−3.56155	−0.386305
$86$	0	0
$87$	−11.8078	−1.26593
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−0.684658	−0.0717717
$92$	0	0
$93$	4.87689	0.505710
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	7.12311	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$97$	7.12311	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$98$	0	0
$99$	2.24621	0.225753
$100$	0	0
$101$	5.12311	0.509768	0.254884	−	0.966972i	$-0.417963\pi$
$101$	5.12311	0.509768	0.254884	+	0.966972i	$0.417963\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−0.684658	−0.0668158
$106$	0	0
$107$	0.684658	0.0661884	0.0330942	−	0.999452i	$-0.489464\pi$
$107$	0.684658	0.0661884	0.0330942	+	0.999452i	$0.489464\pi$
$108$	0	0
$109$	15.5616	1.49053	0.745263	−	0.666770i	$-0.232324\pi$
$109$	15.5616	1.49053	0.745263	+	0.666770i	$0.232324\pi$
$110$	0	0
$111$	11.1231	1.05576
$112$	0	0
$113$	10.2462	0.963882	0.481941	−	0.876204i	$-0.339932\pi$
$113$	10.2462	0.963882	0.481941	+	0.876204i	$0.339932\pi$
$114$	0	0
$115$	−6.68466	−0.623348
$116$	0	0
$117$	0.876894	0.0810689
$118$	0	0
$119$	1.56155	0.143147
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−12.8769	−1.16107
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.36932	0.831392	0.415696	−	0.909504i	$-0.363538\pi$
$127$	9.36932	0.831392	0.415696	+	0.909504i	$0.363538\pi$
$128$	0	0
$129$	−3.12311	−0.274974
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	−0.438447	−0.0380182
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	17.8078	1.52142	0.760710	−	0.649092i	$-0.224851\pi$
$137$	17.8078	1.52142	0.760710	+	0.649092i	$0.224851\pi$
$138$	0	0
$139$	−14.2462	−1.20835	−0.604174	−	0.796852i	$-0.706497\pi$
$139$	−14.2462	−1.20835	−0.604174	+	0.796852i	$0.706497\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	6.24621	0.522334
$144$	0	0
$145$	7.56155	0.627953
$146$	0	0
$147$	−10.6307	−0.876804
$148$	0	0
$149$	−5.12311	−0.419701	−0.209851	−	0.977733i	$-0.567298\pi$
$149$	−5.12311	−0.419701	−0.209851	+	0.977733i	$0.567298\pi$
$150$	0	0
$151$	4.87689	0.396876	0.198438	−	0.980113i	$-0.436413\pi$
$151$	4.87689	0.396876	0.198438	+	0.980113i	$0.436413\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−3.12311	−0.250854
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	3.80776	0.301975
$160$	0	0
$161$	2.93087	0.230985
$162$	0	0
$163$	20.2462	1.58581	0.792903	−	0.609348i	$-0.208569\pi$
$163$	20.2462	1.58581	0.792903	+	0.609348i	$0.208569\pi$
$164$	0	0
$165$	6.24621	0.486267
$166$	0	0
$167$	5.36932	0.415490	0.207745	−	0.978183i	$-0.433388\pi$
$167$	5.36932	0.415490	0.207745	+	0.978183i	$0.433388\pi$
$168$	0	0
$169$	−10.5616	−0.812427
$170$	0	0
$171$	0.561553	0.0429430
$172$	0	0
$173$	11.1231	0.845674	0.422837	−	0.906206i	$-0.361034\pi$
$173$	11.1231	0.845674	0.422837	+	0.906206i	$0.361034\pi$
$174$	0	0
$175$	0.438447	0.0331435
$176$	0	0
$177$	−12.1922	−0.916425
$178$	0	0
$179$	−2.24621	−0.167890	−0.0839449	−	0.996470i	$-0.526752\pi$
$179$	−2.24621	−0.167890	−0.0839449	+	0.996470i	$0.526752\pi$
$180$	0	0
$181$	24.2462	1.80221	0.901103	−	0.433604i	$-0.142758\pi$
$181$	24.2462	1.80221	0.901103	+	0.433604i	$0.142758\pi$
$182$	0	0
$183$	−20.4924	−1.51484
$184$	0	0
$185$	−7.12311	−0.523701
$186$	0	0
$187$	−14.2462	−1.04179
$188$	0	0
$189$	−2.43845	−0.177371
$190$	0	0
$191$	16.6847	1.20726	0.603630	−	0.797265i	$-0.293721\pi$
$191$	16.6847	1.20726	0.603630	+	0.797265i	$0.293721\pi$
$192$	0	0
$193$	0.876894	0.0631202	0.0315601	−	0.999502i	$-0.489952\pi$
$193$	0.876894	0.0631202	0.0315601	+	0.999502i	$0.489952\pi$
$194$	0	0
$195$	2.43845	0.174621
$196$	0	0
$197$	−7.75379	−0.552435	−0.276217	−	0.961095i	$-0.589081\pi$
$197$	−7.75379	−0.552435	−0.276217	+	0.961095i	$0.589081\pi$
$198$	0	0
$199$	−9.56155	−0.677801	−0.338900	−	0.940822i	$-0.610055\pi$
$199$	−9.56155	−0.677801	−0.338900	+	0.940822i	$0.610055\pi$
$200$	0	0
$201$	−10.0540	−0.709153
$202$	0	0
$203$	−3.31534	−0.232691
$204$	0	0
$205$	8.24621	0.575940
$206$	0	0
$207$	−3.75379	−0.260906
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−14.0540	−0.967516	−0.483758	−	0.875202i	$-0.660728\pi$
$211$	−14.0540	−0.967516	−0.483758	+	0.875202i	$0.660728\pi$
$212$	0	0
$213$	−12.4924	−0.855967
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	1.36932	0.0929553
$218$	0	0
$219$	−9.06913	−0.612835
$220$	0	0
$221$	−5.56155	−0.374111
$222$	0	0
$223$	−9.36932	−0.627416	−0.313708	−	0.949520i	$-0.601571\pi$
$223$	−9.36932	−0.627416	−0.313708	+	0.949520i	$0.601571\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−16.6847	−1.10740	−0.553700	−	0.832716i	$-0.686784\pi$
$227$	−16.6847	−1.10740	−0.553700	+	0.832716i	$0.686784\pi$
$228$	0	0
$229$	20.7386	1.37045	0.685224	−	0.728333i	$-0.259704\pi$
$229$	20.7386	1.37045	0.685224	+	0.728333i	$0.259704\pi$
$230$	0	0
$231$	−2.73863	−0.180189
$232$	0	0
$233$	6.49242	0.425333	0.212666	−	0.977125i	$-0.431785\pi$
$233$	6.49242	0.425333	0.212666	+	0.977125i	$0.431785\pi$
$234$	0	0
$235$	5.12311	0.334195
$236$	0	0
$237$	−4.87689	−0.316788
$238$	0	0
$239$	−23.8078	−1.54000	−0.769998	−	0.638046i	$-0.779743\pi$
$239$	−23.8078	−1.54000	−0.769998	+	0.638046i	$0.779743\pi$
$240$	0	0
$241$	27.8617	1.79473	0.897366	−	0.441287i	$-0.145478\pi$
$241$	27.8617	1.79473	0.897366	+	0.441287i	$0.145478\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	6.80776	0.434932
$246$	0	0
$247$	1.56155	0.0993592
$248$	0	0
$249$	−21.8617	−1.38543
$250$	0	0
$251$	22.2462	1.40417	0.702084	−	0.712094i	$-0.252253\pi$
$251$	22.2462	1.40417	0.702084	+	0.712094i	$0.252253\pi$
$252$	0	0
$253$	−26.7386	−1.68104
$254$	0	0
$255$	−5.56155	−0.348278
$256$	0	0
$257$	−6.63068	−0.413611	−0.206805	−	0.978382i	$-0.566307\pi$
$257$	−6.63068	−0.413611	−0.206805	+	0.978382i	$0.566307\pi$
$258$	0	0
$259$	3.12311	0.194060
$260$	0	0
$261$	4.24621	0.262834
$262$	0	0
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	−2.43845	−0.149793
$266$	0	0
$267$	−15.6155	−0.955655
$268$	0	0
$269$	−7.75379	−0.472757	−0.236378	−	0.971661i	$-0.575960\pi$
$269$	−7.75379	−0.472757	−0.236378	+	0.971661i	$0.575960\pi$
$270$	0	0
$271$	25.1771	1.52940	0.764699	−	0.644387i	$-0.222887\pi$
$271$	25.1771	1.52940	0.764699	+	0.644387i	$0.222887\pi$
$272$	0	0
$273$	−1.06913	−0.0647067
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	21.6155	1.29875	0.649376	−	0.760468i	$-0.275030\pi$
$277$	21.6155	1.29875	0.649376	+	0.760468i	$0.275030\pi$
$278$	0	0
$279$	−1.75379	−0.104997
$280$	0	0
$281$	7.75379	0.462552	0.231276	−	0.972888i	$-0.425710\pi$
$281$	7.75379	0.462552	0.231276	+	0.972888i	$0.425710\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	1.56155	0.0924984
$286$	0	0
$287$	−3.61553	−0.213418
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	11.1231	0.652048
$292$	0	0
$293$	−22.9309	−1.33964	−0.669818	−	0.742525i	$-0.733628\pi$
$293$	−22.9309	−1.33964	−0.669818	+	0.742525i	$0.733628\pi$
$294$	0	0
$295$	7.80776	0.454586
$296$	0	0
$297$	22.2462	1.29086
$298$	0	0
$299$	−10.4384	−0.603671
$300$	0	0
$301$	−0.876894	−0.0505434
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	13.1231	0.751427
$306$	0	0
$307$	−22.2462	−1.26966	−0.634829	−	0.772653i	$-0.718930\pi$
$307$	−22.2462	−1.26966	−0.634829	+	0.772653i	$0.718930\pi$
$308$	0	0
$309$	−24.9848	−1.42134
$310$	0	0
$311$	−9.56155	−0.542186	−0.271093	−	0.962553i	$-0.587385\pi$
$311$	−9.56155	−0.542186	−0.271093	+	0.962553i	$0.587385\pi$
$312$	0	0
$313$	15.1771	0.857859	0.428930	−	0.903338i	$-0.358891\pi$
$313$	15.1771	0.857859	0.428930	+	0.903338i	$0.358891\pi$
$314$	0	0
$315$	0.246211	0.0138724
$316$	0	0
$317$	9.56155	0.537030	0.268515	−	0.963275i	$-0.413467\pi$
$317$	9.56155	0.537030	0.268515	+	0.963275i	$0.413467\pi$
$318$	0	0
$319$	30.2462	1.69346
$320$	0	0
$321$	1.06913	0.0596730
$322$	0	0
$323$	−3.56155	−0.198170
$324$	0	0
$325$	−1.56155	−0.0866194
$326$	0	0
$327$	24.3002	1.34380
$328$	0	0
$329$	−2.24621	−0.123838
$330$	0	0
$331$	7.80776	0.429154	0.214577	−	0.976707i	$-0.431163\pi$
$331$	7.80776	0.429154	0.214577	+	0.976707i	$0.431163\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	6.43845	0.351770
$336$	0	0
$337$	23.1231	1.25960	0.629798	−	0.776759i	$-0.283138\pi$
$337$	23.1231	1.25960	0.629798	+	0.776759i	$0.283138\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	0	0
$341$	−12.4924	−0.676503
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	−10.4384	−0.561987
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	8.68466	0.463553
$352$	0	0
$353$	25.4233	1.35315	0.676573	−	0.736376i	$-0.263464\pi$
$353$	25.4233	1.35315	0.676573	+	0.736376i	$0.263464\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	2.43845	0.129056
$358$	0	0
$359$	−22.9309	−1.21025	−0.605123	−	0.796132i	$-0.706876\pi$
$359$	−22.9309	−1.21025	−0.605123	+	0.796132i	$0.706876\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.80776	0.409801
$364$	0	0
$365$	5.80776	0.303992
$366$	0	0
$367$	26.8769	1.40296	0.701481	−	0.712688i	$-0.252522\pi$
$367$	26.8769	1.40296	0.701481	+	0.712688i	$0.252522\pi$
$368$	0	0
$369$	4.63068	0.241064
$370$	0	0
$371$	1.06913	0.0555065
$372$	0	0
$373$	−4.19224	−0.217066	−0.108533	−	0.994093i	$-0.534615\pi$
$373$	−4.19224	−0.217066	−0.108533	+	0.994093i	$0.534615\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	11.8078	0.608131
$378$	0	0
$379$	20.6847	1.06250	0.531250	−	0.847215i	$-0.321723\pi$
$379$	20.6847	1.06250	0.531250	+	0.847215i	$0.321723\pi$
$380$	0	0
$381$	14.6307	0.749553
$382$	0	0
$383$	−22.7386	−1.16189	−0.580945	−	0.813943i	$-0.697317\pi$
$383$	−22.7386	−1.16189	−0.580945	+	0.813943i	$0.697317\pi$
$384$	0	0
$385$	1.75379	0.0893814
$386$	0	0
$387$	1.12311	0.0570907
$388$	0	0
$389$	−1.12311	−0.0569437	−0.0284719	−	0.999595i	$-0.509064\pi$
$389$	−1.12311	−0.0569437	−0.0284719	+	0.999595i	$0.509064\pi$
$390$	0	0
$391$	23.8078	1.20401
$392$	0	0
$393$	−24.9848	−1.26032
$394$	0	0
$395$	3.12311	0.157140
$396$	0	0
$397$	−31.8617	−1.59909	−0.799547	−	0.600603i	$-0.794927\pi$
$397$	−31.8617	−1.59909	−0.799547	+	0.600603i	$0.794927\pi$
$398$	0	0
$399$	−0.684658	−0.0342758
$400$	0	0
$401$	28.7386	1.43514	0.717569	−	0.696487i	$-0.245255\pi$
$401$	28.7386	1.43514	0.717569	+	0.696487i	$0.245255\pi$
$402$	0	0
$403$	−4.87689	−0.242935
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	−28.4924	−1.41232
$408$	0	0
$409$	−24.7386	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	−24.7386	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$410$	0	0
$411$	27.8078	1.37166
$412$	0	0
$413$	−3.42329	−0.168449
$414$	0	0
$415$	14.0000	0.687233
$416$	0	0
$417$	−22.2462	−1.08940
$418$	0	0
$419$	−31.1231	−1.52046	−0.760232	−	0.649652i	$-0.774915\pi$
$419$	−31.1231	−1.52046	−0.760232	+	0.649652i	$0.774915\pi$
$420$	0	0
$421$	−34.3002	−1.67169	−0.835844	−	0.548966i	$-0.815021\pi$
$421$	−34.3002	−1.67169	−0.835844	+	0.548966i	$0.815021\pi$
$422$	0	0
$423$	2.87689	0.139879
$424$	0	0
$425$	3.56155	0.172761
$426$	0	0
$427$	−5.75379	−0.278445
$428$	0	0
$429$	9.75379	0.470917
$430$	0	0
$431$	−12.4924	−0.601739	−0.300869	−	0.953665i	$-0.597277\pi$
$431$	−12.4924	−0.601739	−0.300869	+	0.953665i	$0.597277\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	11.8078	0.566139
$436$	0	0
$437$	−6.68466	−0.319771
$438$	0	0
$439$	6.63068	0.316465	0.158233	−	0.987402i	$-0.449420\pi$
$439$	6.63068	0.316465	0.158233	+	0.987402i	$0.449420\pi$
$440$	0	0
$441$	3.82292	0.182044
$442$	0	0
$443$	−29.6155	−1.40708	−0.703538	−	0.710658i	$-0.748398\pi$
$443$	−29.6155	−1.40708	−0.703538	+	0.710658i	$0.748398\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	−32.7386	−1.54503	−0.772516	−	0.634996i	$-0.781002\pi$
$449$	−32.7386	−1.54503	−0.772516	+	0.634996i	$0.781002\pi$
$450$	0	0
$451$	32.9848	1.55320
$452$	0	0
$453$	7.61553	0.357809
$454$	0	0
$455$	0.684658	0.0320973
$456$	0	0
$457$	12.4384	0.581846	0.290923	−	0.956746i	$-0.406038\pi$
$457$	12.4384	0.581846	0.290923	+	0.956746i	$0.406038\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	7.75379	0.361130	0.180565	−	0.983563i	$-0.442207\pi$
$461$	7.75379	0.361130	0.180565	+	0.983563i	$0.442207\pi$
$462$	0	0
$463$	35.8617	1.66664	0.833318	−	0.552794i	$-0.186438\pi$
$463$	35.8617	1.66664	0.833318	+	0.552794i	$0.186438\pi$
$464$	0	0
$465$	−4.87689	−0.226161
$466$	0	0
$467$	27.8617	1.28929	0.644644	−	0.764483i	$-0.277006\pi$
$467$	27.8617	1.28929	0.644644	+	0.764483i	$0.277006\pi$
$468$	0	0
$469$	−2.82292	−0.130350
$470$	0	0
$471$	9.36932	0.431715
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−1.36932	−0.0626967
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−11.1231	−0.507170
$482$	0	0
$483$	4.57671	0.208247
$484$	0	0
$485$	−7.12311	−0.323444
$486$	0	0
$487$	2.63068	0.119208	0.0596038	−	0.998222i	$-0.481016\pi$
$487$	2.63068	0.119208	0.0596038	+	0.998222i	$0.481016\pi$
$488$	0	0
$489$	31.6155	1.42970
$490$	0	0
$491$	−11.1231	−0.501979	−0.250989	−	0.967990i	$-0.580756\pi$
$491$	−11.1231	−0.501979	−0.250989	+	0.967990i	$0.580756\pi$
$492$	0	0
$493$	−26.9309	−1.21291
$494$	0	0
$495$	−2.24621	−0.100960
$496$	0	0
$497$	−3.50758	−0.157336
$498$	0	0
$499$	20.8769	0.934578	0.467289	−	0.884105i	$-0.345231\pi$
$499$	20.8769	0.934578	0.467289	+	0.884105i	$0.345231\pi$
$500$	0	0
$501$	8.38447	0.374591
$502$	0	0
$503$	25.4233	1.13357	0.566784	−	0.823866i	$-0.308187\pi$
$503$	25.4233	1.13357	0.566784	+	0.823866i	$0.308187\pi$
$504$	0	0
$505$	−5.12311	−0.227975
$506$	0	0
$507$	−16.4924	−0.732454
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−2.54640	−0.112646
$512$	0	0
$513$	5.56155	0.245549
$514$	0	0
$515$	16.0000	0.705044
$516$	0	0
$517$	20.4924	0.901256
$518$	0	0
$519$	17.3693	0.762428
$520$	0	0
$521$	−1.12311	−0.0492042	−0.0246021	−	0.999697i	$-0.507832\pi$
$521$	−1.12311	−0.0492042	−0.0246021	+	0.999697i	$0.507832\pi$
$522$	0	0
$523$	14.9309	0.652881	0.326441	−	0.945218i	$-0.394151\pi$
$523$	14.9309	0.652881	0.326441	+	0.945218i	$0.394151\pi$
$524$	0	0
$525$	0.684658	0.0298809
$526$	0	0
$527$	11.1231	0.484530
$528$	0	0
$529$	21.6847	0.942811
$530$	0	0
$531$	4.38447	0.190270
$532$	0	0
$533$	12.8769	0.557760
$534$	0	0
$535$	−0.684658	−0.0296004
$536$	0	0
$537$	−3.50758	−0.151363
$538$	0	0
$539$	27.2311	1.17292
$540$	0	0
$541$	−25.1231	−1.08013	−0.540063	−	0.841624i	$-0.681600\pi$
$541$	−25.1231	−1.08013	−0.540063	+	0.841624i	$0.681600\pi$
$542$	0	0
$543$	37.8617	1.62480
$544$	0	0
$545$	−15.5616	−0.666584
$546$	0	0
$547$	−38.2462	−1.63529	−0.817645	−	0.575723i	$-0.804721\pi$
$547$	−38.2462	−1.63529	−0.817645	+	0.575723i	$0.804721\pi$
$548$	0	0
$549$	7.36932	0.314515
$550$	0	0
$551$	7.56155	0.322133
$552$	0	0
$553$	−1.36932	−0.0582293
$554$	0	0
$555$	−11.1231	−0.472150
$556$	0	0
$557$	4.63068	0.196208	0.0981042	−	0.995176i	$-0.468722\pi$
$557$	4.63068	0.196208	0.0981042	+	0.995176i	$0.468722\pi$
$558$	0	0
$559$	3.12311	0.132093
$560$	0	0
$561$	−22.2462	−0.939236
$562$	0	0
$563$	−18.7386	−0.789739	−0.394870	−	0.918737i	$-0.629210\pi$
$563$	−18.7386	−0.789739	−0.394870	+	0.918737i	$0.629210\pi$
$564$	0	0
$565$	−10.2462	−0.431061
$566$	0	0
$567$	−3.06913	−0.128891
$568$	0	0
$569$	−9.12311	−0.382460	−0.191230	−	0.981545i	$-0.561248\pi$
$569$	−9.12311	−0.382460	−0.191230	+	0.981545i	$0.561248\pi$
$570$	0	0
$571$	2.63068	0.110091	0.0550453	−	0.998484i	$-0.482470\pi$
$571$	2.63068	0.110091	0.0550453	+	0.998484i	$0.482470\pi$
$572$	0	0
$573$	26.0540	1.08842
$574$	0	0
$575$	6.68466	0.278770
$576$	0	0
$577$	−33.4233	−1.39143	−0.695715	−	0.718318i	$-0.744912\pi$
$577$	−33.4233	−1.39143	−0.695715	+	0.718318i	$0.744912\pi$
$578$	0	0
$579$	1.36932	0.0569069
$580$	0	0
$581$	−6.13826	−0.254658
$582$	0	0
$583$	−9.75379	−0.403961
$584$	0	0
$585$	−0.876894	−0.0362551
$586$	0	0
$587$	−32.2462	−1.33094	−0.665472	−	0.746423i	$-0.731770\pi$
$587$	−32.2462	−1.33094	−0.665472	+	0.746423i	$0.731770\pi$
$588$	0	0
$589$	−3.12311	−0.128685
$590$	0	0
$591$	−12.1080	−0.498055
$592$	0	0
$593$	8.24621	0.338631	0.169316	−	0.985562i	$-0.445844\pi$
$593$	8.24621	0.338631	0.169316	+	0.985562i	$0.445844\pi$
$594$	0	0
$595$	−1.56155	−0.0640174
$596$	0	0
$597$	−14.9309	−0.611080
$598$	0	0
$599$	−31.6155	−1.29178	−0.645888	−	0.763432i	$-0.723513\pi$
$599$	−31.6155	−1.29178	−0.645888	+	0.763432i	$0.723513\pi$
$600$	0	0
$601$	0.738634	0.0301295	0.0150647	−	0.999887i	$-0.495205\pi$
$601$	0.738634	0.0301295	0.0150647	+	0.999887i	$0.495205\pi$
$602$	0	0
$603$	3.61553	0.147236
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−0.876894	−0.0355920	−0.0177960	−	0.999842i	$-0.505665\pi$
$607$	−0.876894	−0.0355920	−0.0177960	+	0.999842i	$0.505665\pi$
$608$	0	0
$609$	−5.17708	−0.209786
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−16.6307	−0.671707	−0.335853	−	0.941914i	$-0.609025\pi$
$613$	−16.6307	−0.671707	−0.335853	+	0.941914i	$0.609025\pi$
$614$	0	0
$615$	12.8769	0.519246
$616$	0	0
$617$	−42.9848	−1.73050	−0.865252	−	0.501337i	$-0.832842\pi$
$617$	−42.9848	−1.73050	−0.865252	+	0.501337i	$0.832842\pi$
$618$	0	0
$619$	14.2462	0.572604	0.286302	−	0.958139i	$-0.407574\pi$
$619$	14.2462	0.572604	0.286302	+	0.958139i	$0.407574\pi$
$620$	0	0
$621$	−37.1771	−1.49186
$622$	0	0
$623$	−4.38447	−0.175660
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.24621	0.249450
$628$	0	0
$629$	25.3693	1.01154
$630$	0	0
$631$	22.7386	0.905211	0.452605	−	0.891711i	$-0.350495\pi$
$631$	22.7386	0.905211	0.452605	+	0.891711i	$0.350495\pi$
$632$	0	0
$633$	−21.9460	−0.872276
$634$	0	0
$635$	−9.36932	−0.371810
$636$	0	0
$637$	10.6307	0.421203
$638$	0	0
$639$	4.49242	0.177717
$640$	0	0
$641$	1.12311	0.0443600	0.0221800	−	0.999754i	$-0.492939\pi$
$641$	1.12311	0.0443600	0.0221800	+	0.999754i	$0.492939\pi$
$642$	0	0
$643$	−18.4924	−0.729270	−0.364635	−	0.931151i	$-0.618806\pi$
$643$	−18.4924	−0.729270	−0.364635	+	0.931151i	$0.618806\pi$
$644$	0	0
$645$	3.12311	0.122972
$646$	0	0
$647$	−5.31534	−0.208968	−0.104484	−	0.994527i	$-0.533319\pi$
$647$	−5.31534	−0.208968	−0.104484	+	0.994527i	$0.533319\pi$
$648$	0	0
$649$	31.2311	1.22593
$650$	0	0
$651$	2.13826	0.0838050
$652$	0	0
$653$	20.6307	0.807341	0.403671	−	0.914904i	$-0.367734\pi$
$653$	20.6307	0.807341	0.403671	+	0.914904i	$0.367734\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	3.26137	0.127238
$658$	0	0
$659$	47.8078	1.86233	0.931163	−	0.364603i	$-0.118795\pi$
$659$	47.8078	1.86233	0.931163	+	0.364603i	$0.118795\pi$
$660$	0	0
$661$	−8.93087	−0.347371	−0.173685	−	0.984801i	$-0.555568\pi$
$661$	−8.93087	−0.347371	−0.173685	+	0.984801i	$0.555568\pi$
$662$	0	0
$663$	−8.68466	−0.337284
$664$	0	0
$665$	0.438447	0.0170023
$666$	0	0
$667$	−50.5464	−1.95716
$668$	0	0
$669$	−14.6307	−0.565655
$670$	0	0
$671$	52.4924	2.02645
$672$	0	0
$673$	−3.50758	−0.135207	−0.0676036	−	0.997712i	$-0.521535\pi$
$673$	−3.50758	−0.135207	−0.0676036	+	0.997712i	$0.521535\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	35.8078	1.37620	0.688102	−	0.725614i	$-0.258444\pi$
$677$	35.8078	1.37620	0.688102	+	0.725614i	$0.258444\pi$
$678$	0	0
$679$	3.12311	0.119854
$680$	0	0
$681$	−26.0540	−0.998391
$682$	0	0
$683$	17.7538	0.679330	0.339665	−	0.940547i	$-0.389686\pi$
$683$	17.7538	0.679330	0.339665	+	0.940547i	$0.389686\pi$
$684$	0	0
$685$	−17.8078	−0.680400
$686$	0	0
$687$	32.3845	1.23554
$688$	0	0
$689$	−3.80776	−0.145064
$690$	0	0
$691$	40.1080	1.52578	0.762889	−	0.646529i	$-0.223780\pi$
$691$	40.1080	1.52578	0.762889	+	0.646529i	$0.223780\pi$
$692$	0	0
$693$	0.984845	0.0374112
$694$	0	0
$695$	14.2462	0.540390
$696$	0	0
$697$	−29.3693	−1.11244
$698$	0	0
$699$	10.1383	0.383464
$700$	0	0
$701$	2.87689	0.108659	0.0543294	−	0.998523i	$-0.482698\pi$
$701$	2.87689	0.108659	0.0543294	+	0.998523i	$0.482698\pi$
$702$	0	0
$703$	−7.12311	−0.268653
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	2.24621	0.0844775
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	1.75379	0.0657722
$712$	0	0
$713$	20.8769	0.781846
$714$	0	0
$715$	−6.24621	−0.233595
$716$	0	0
$717$	−37.1771	−1.38840
$718$	0	0
$719$	27.4233	1.02272	0.511358	−	0.859368i	$-0.329143\pi$
$719$	27.4233	1.02272	0.511358	+	0.859368i	$0.329143\pi$
$720$	0	0
$721$	−7.01515	−0.261258
$722$	0	0
$723$	43.5076	1.61806
$724$	0	0
$725$	−7.56155	−0.280829
$726$	0	0
$727$	−1.31534	−0.0487833	−0.0243917	−	0.999702i	$-0.507765\pi$
$727$	−1.31534	−0.0487833	−0.0243917	+	0.999702i	$0.507765\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−7.12311	−0.263458
$732$	0	0
$733$	−45.6155	−1.68485	−0.842424	−	0.538815i	$-0.818872\pi$
$733$	−45.6155	−1.68485	−0.842424	+	0.538815i	$0.818872\pi$
$734$	0	0
$735$	10.6307	0.392119
$736$	0	0
$737$	25.7538	0.948653
$738$	0	0
$739$	8.87689	0.326542	0.163271	−	0.986581i	$-0.447796\pi$
$739$	8.87689	0.326542	0.163271	+	0.986581i	$0.447796\pi$
$740$	0	0
$741$	2.43845	0.0895786
$742$	0	0
$743$	−10.6307	−0.390002	−0.195001	−	0.980803i	$-0.562471\pi$
$743$	−10.6307	−0.390002	−0.195001	+	0.980803i	$0.562471\pi$
$744$	0	0
$745$	5.12311	0.187696
$746$	0	0
$747$	7.86174	0.287646
$748$	0	0
$749$	0.300187	0.0109686
$750$	0	0
$751$	6.63068	0.241957	0.120979	−	0.992655i	$-0.461397\pi$
$751$	6.63068	0.241957	0.120979	+	0.992655i	$0.461397\pi$
$752$	0	0
$753$	34.7386	1.26595
$754$	0	0
$755$	−4.87689	−0.177488
$756$	0	0
$757$	36.7386	1.33529	0.667644	−	0.744481i	$-0.267303\pi$
$757$	36.7386	1.33529	0.667644	+	0.744481i	$0.267303\pi$
$758$	0	0
$759$	−41.7538	−1.51557
$760$	0	0
$761$	30.6847	1.11232	0.556159	−	0.831076i	$-0.312275\pi$
$761$	30.6847	1.11232	0.556159	+	0.831076i	$0.312275\pi$
$762$	0	0
$763$	6.82292	0.247006
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	12.1922	0.440236
$768$	0	0
$769$	−7.17708	−0.258812	−0.129406	−	0.991592i	$-0.541307\pi$
$769$	−7.17708	−0.258812	−0.129406	+	0.991592i	$0.541307\pi$
$770$	0	0
$771$	−10.3542	−0.372896
$772$	0	0
$773$	−25.1771	−0.905557	−0.452778	−	0.891623i	$-0.649567\pi$
$773$	−25.1771	−0.905557	−0.452778	+	0.891623i	$0.649567\pi$
$774$	0	0
$775$	3.12311	0.112185
$776$	0	0
$777$	4.87689	0.174958
$778$	0	0
$779$	8.24621	0.295451
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	42.0540	1.50289
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	2.05398	0.0732163	0.0366082	−	0.999330i	$-0.488345\pi$
$787$	2.05398	0.0732163	0.0366082	+	0.999330i	$0.488345\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	4.49242	0.159732
$792$	0	0
$793$	20.4924	0.727707
$794$	0	0
$795$	−3.80776	−0.135047
$796$	0	0
$797$	4.19224	0.148497	0.0742483	−	0.997240i	$-0.476344\pi$
$797$	4.19224	0.148497	0.0742483	+	0.997240i	$0.476344\pi$
$798$	0	0
$799$	−18.2462	−0.645505
$800$	0	0
$801$	5.61553	0.198415
$802$	0	0
$803$	23.2311	0.819806
$804$	0	0
$805$	−2.93087	−0.103300
$806$	0	0
$807$	−12.1080	−0.426220
$808$	0	0
$809$	11.0691	0.389170	0.194585	−	0.980886i	$-0.437664\pi$
$809$	11.0691	0.389170	0.194585	+	0.980886i	$0.437664\pi$
$810$	0	0
$811$	55.0388	1.93267	0.966337	−	0.257279i	$-0.0828259\pi$
$811$	55.0388	1.93267	0.966337	+	0.257279i	$0.0828259\pi$
$812$	0	0
$813$	39.3153	1.37885
$814$	0	0
$815$	−20.2462	−0.709194
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	0.384472	0.0134345
$820$	0	0
$821$	13.1231	0.458000	0.229000	−	0.973426i	$-0.426454\pi$
$821$	13.1231	0.458000	0.229000	+	0.973426i	$0.426454\pi$
$822$	0	0
$823$	−41.4233	−1.44393	−0.721963	−	0.691932i	$-0.756760\pi$
$823$	−41.4233	−1.44393	−0.721963	+	0.691932i	$0.756760\pi$
$824$	0	0
$825$	−6.24621	−0.217465
$826$	0	0
$827$	−1.94602	−0.0676699	−0.0338350	−	0.999427i	$-0.510772\pi$
$827$	−1.94602	−0.0676699	−0.0338350	+	0.999427i	$0.510772\pi$
$828$	0	0
$829$	47.5616	1.65188	0.825941	−	0.563757i	$-0.190645\pi$
$829$	47.5616	1.65188	0.825941	+	0.563757i	$0.190645\pi$
$830$	0	0
$831$	33.7538	1.17091
$832$	0	0
$833$	−24.2462	−0.840081
$834$	0	0
$835$	−5.36932	−0.185813
$836$	0	0
$837$	−17.3693	−0.600371
$838$	0	0
$839$	−47.2311	−1.63060	−0.815299	−	0.579041i	$-0.803427\pi$
$839$	−47.2311	−1.63060	−0.815299	+	0.579041i	$0.803427\pi$
$840$	0	0
$841$	28.1771	0.971623
$842$	0	0
$843$	12.1080	0.417020
$844$	0	0
$845$	10.5616	0.363328
$846$	0	0
$847$	2.19224	0.0753261
$848$	0	0
$849$	−9.36932	−0.321554
$850$	0	0
$851$	47.6155	1.63224
$852$	0	0
$853$	4.73863	0.162248	0.0811239	−	0.996704i	$-0.474149\pi$
$853$	4.73863	0.162248	0.0811239	+	0.996704i	$0.474149\pi$
$854$	0	0
$855$	−0.561553	−0.0192047
$856$	0	0
$857$	40.9848	1.40002	0.700008	−	0.714135i	$-0.253180\pi$
$857$	40.9848	1.40002	0.700008	+	0.714135i	$0.253180\pi$
$858$	0	0
$859$	−54.7386	−1.86766	−0.933829	−	0.357720i	$-0.883554\pi$
$859$	−54.7386	−1.86766	−0.933829	+	0.357720i	$0.883554\pi$
$860$	0	0
$861$	−5.64584	−0.192410
$862$	0	0
$863$	−49.4773	−1.68423	−0.842113	−	0.539301i	$-0.818688\pi$
$863$	−49.4773	−1.68423	−0.842113	+	0.539301i	$0.818688\pi$
$864$	0	0
$865$	−11.1231	−0.378197
$866$	0	0
$867$	−6.73863	−0.228856
$868$	0	0
$869$	12.4924	0.423776
$870$	0	0
$871$	10.0540	0.340666
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	−0.438447	−0.0148222
$876$	0	0
$877$	−4.19224	−0.141562	−0.0707809	−	0.997492i	$-0.522549\pi$
$877$	−4.19224	−0.141562	−0.0707809	+	0.997492i	$0.522549\pi$
$878$	0	0
$879$	−35.8078	−1.20777
$880$	0	0
$881$	6.49242	0.218735	0.109368	−	0.994001i	$-0.465117\pi$
$881$	6.49242	0.218735	0.109368	+	0.994001i	$0.465117\pi$
$882$	0	0
$883$	36.2462	1.21978	0.609891	−	0.792485i	$-0.291213\pi$
$883$	36.2462	1.21978	0.609891	+	0.792485i	$0.291213\pi$
$884$	0	0
$885$	12.1922	0.409838
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	4.10795	0.137776
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	5.12311	0.171438
$894$	0	0
$895$	2.24621	0.0750826
$896$	0	0
$897$	−16.3002	−0.544247
$898$	0	0
$899$	−23.6155	−0.787622
$900$	0	0
$901$	8.68466	0.289328
$902$	0	0
$903$	−1.36932	−0.0455680
$904$	0	0
$905$	−24.2462	−0.805971
$906$	0	0
$907$	19.4233	0.644940	0.322470	−	0.946580i	$-0.395487\pi$
$907$	19.4233	0.644940	0.322470	+	0.946580i	$0.395487\pi$
$908$	0	0
$909$	−2.87689	−0.0954206
$910$	0	0
$911$	26.7386	0.885890	0.442945	−	0.896549i	$-0.353934\pi$
$911$	26.7386	0.885890	0.442945	+	0.896549i	$0.353934\pi$
$912$	0	0
$913$	56.0000	1.85333
$914$	0	0
$915$	20.4924	0.677459
$916$	0	0
$917$	−7.01515	−0.231661
$918$	0	0
$919$	−0.192236	−0.00634128	−0.00317064	−	0.999995i	$-0.501009\pi$
$919$	−0.192236	−0.00634128	−0.00317064	+	0.999995i	$0.501009\pi$
$920$	0	0
$921$	−34.7386	−1.14468
$922$	0	0
$923$	12.4924	0.411193
$924$	0	0
$925$	7.12311	0.234206
$926$	0	0
$927$	8.98485	0.295101
$928$	0	0
$929$	32.0540	1.05166	0.525828	−	0.850591i	$-0.323755\pi$
$929$	32.0540	1.05166	0.525828	+	0.850591i	$0.323755\pi$
$930$	0	0
$931$	6.80776	0.223115
$932$	0	0
$933$	−14.9309	−0.488815
$934$	0	0
$935$	14.2462	0.465901
$936$	0	0
$937$	−20.0540	−0.655135	−0.327567	−	0.944828i	$-0.606229\pi$
$937$	−20.0540	−0.655135	−0.327567	+	0.944828i	$0.606229\pi$
$938$	0	0
$939$	23.6998	0.773414
$940$	0	0
$941$	26.6847	0.869895	0.434948	−	0.900456i	$-0.356767\pi$
$941$	26.6847	0.869895	0.434948	+	0.900456i	$0.356767\pi$
$942$	0	0
$943$	−55.1231	−1.79506
$944$	0	0
$945$	2.43845	0.0793227
$946$	0	0
$947$	−8.63068	−0.280460	−0.140230	−	0.990119i	$-0.544784\pi$
$947$	−8.63068	−0.280460	−0.140230	+	0.990119i	$0.544784\pi$
$948$	0	0
$949$	9.06913	0.294396
$950$	0	0
$951$	14.9309	0.484167
$952$	0	0
$953$	−31.2311	−1.01167	−0.505837	−	0.862629i	$-0.668816\pi$
$953$	−31.2311	−1.01167	−0.505837	+	0.862629i	$0.668816\pi$
$954$	0	0
$955$	−16.6847	−0.539903
$956$	0	0
$957$	47.2311	1.52676
$958$	0	0
$959$	7.80776	0.252126
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	−0.384472	−0.0123894
$964$	0	0
$965$	−0.876894	−0.0282282
$966$	0	0
$967$	30.8769	0.992934	0.496467	−	0.868056i	$-0.334630\pi$
$967$	30.8769	0.992934	0.496467	+	0.868056i	$0.334630\pi$
$968$	0	0
$969$	−5.56155	−0.178663
$970$	0	0
$971$	−52.9848	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$971$	−52.9848	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$972$	0	0
$973$	−6.24621	−0.200244
$974$	0	0
$975$	−2.43845	−0.0780928
$976$	0	0
$977$	−3.12311	−0.0999170	−0.0499585	−	0.998751i	$-0.515909\pi$
$977$	−3.12311	−0.0999170	−0.0499585	+	0.998751i	$0.515909\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	−8.73863	−0.279003
$982$	0	0
$983$	−11.6155	−0.370478	−0.185239	−	0.982694i	$-0.559306\pi$
$983$	−11.6155	−0.370478	−0.185239	+	0.982694i	$0.559306\pi$
$984$	0	0
$985$	7.75379	0.247056
$986$	0	0
$987$	−3.50758	−0.111647
$988$	0	0
$989$	−13.3693	−0.425120
$990$	0	0
$991$	12.8769	0.409048	0.204524	−	0.978862i	$-0.434435\pi$
$991$	12.8769	0.409048	0.204524	+	0.978862i	$0.434435\pi$
$992$	0	0
$993$	12.1922	0.386909
$994$	0	0
$995$	9.56155	0.303122
$996$	0	0
$997$	−8.24621	−0.261160	−0.130580	−	0.991438i	$-0.541684\pi$
$997$	−8.24621	−0.261160	−0.130580	+	0.991438i	$0.541684\pi$
$998$	0	0
$999$	−39.6155	−1.25338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.e.1.2	✓	2
4.3	odd	2		3040.2.a.h.1.1	yes	2
8.3	odd	2		6080.2.a.bc.1.2		2
8.5	even	2		6080.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.e.1.2	✓	2	1.1	even	1	trivial
3040.2.a.h.1.1	yes	2	4.3	odd	2
6080.2.a.bc.1.2		2	8.3	odd	2
6080.2.a.bi.1.1		2	8.5	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:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} -4.00000 q^{11} -1.56155 q^{13} -1.56155 q^{15} +3.56155 q^{17} -1.00000 q^{19} +0.684658 q^{21} +6.68466 q^{23} +1.00000 q^{25} -5.56155 q^{27} -7.56155 q^{29} +3.12311 q^{31} -6.24621 q^{33} -0.438447 q^{35} +7.12311 q^{37} -2.43845 q^{39} -8.24621 q^{41} -2.00000 q^{43} +0.561553 q^{45} -5.12311 q^{47} -6.80776 q^{49} +5.56155 q^{51} +2.43845 q^{53} +4.00000 q^{55} -1.56155 q^{57} -7.80776 q^{59} -13.1231 q^{61} -0.246211 q^{63} +1.56155 q^{65} -6.43845 q^{67} +10.4384 q^{69} -8.00000 q^{71} -5.80776 q^{73} +1.56155 q^{75} -1.75379 q^{77} -3.12311 q^{79} -7.00000 q^{81} -14.0000 q^{83} -3.56155 q^{85} -11.8078 q^{87} -10.0000 q^{89} -0.684658 q^{91} +4.87689 q^{93} +1.00000 q^{95} +7.12311 q^{97} +2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{11} + q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 11 q^{21} + q^{23} + 2 q^{25} - 7 q^{27} - 11 q^{29} - 2 q^{31} + 4 q^{33} - 5 q^{35} + 6 q^{37} - 9 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^11 + q^13 + q^15 + 3 * q^17 - 2 * q^19 - 11 * q^21 + q^23 + 2 * q^25 - 7 * q^27 - 11 * q^29 - 2 * q^31 + 4 * q^33 - 5 * q^35 + 6 * q^37 - 9 * q^39 - 4 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 + 7 * q^51 + 9 * q^53 + 8 * q^55 + q^57 + 5 * q^59 - 18 * q^61 + 16 * q^63 - q^65 - 17 * q^67 + 25 * q^69 - 16 * q^71 + 9 * q^73 - q^75 - 20 * q^77 + 2 * q^79 - 14 * q^81 - 28 * q^83 - 3 * q^85 - 3 * q^87 - 20 * q^89 + 11 * q^91 + 18 * q^93 + 2 * q^95 + 6 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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