LMFDB - Embedded newform 3042.2.a.bh.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1,3042)] 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("3042.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$3042 = 2 \cdot 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3042.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,1,0,-9,3,0,1,5,0,0,-9,0,3,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1014)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.15883 q^{5} -4.69202 q^{7} +1.00000 q^{8} +3.15883 q^{10} -0.137063 q^{11} -4.69202 q^{14} +1.00000 q^{16} +5.60388 q^{17} -4.98792 q^{19} +3.15883 q^{20} -0.137063 q^{22} +6.09783 q^{23} +4.97823 q^{25} -4.69202 q^{28} +0.850855 q^{29} +6.23490 q^{31} +1.00000 q^{32} +5.60388 q^{34} -14.8213 q^{35} +11.7017 q^{37} -4.98792 q^{38} +3.15883 q^{40} +4.27413 q^{41} -2.09783 q^{43} -0.137063 q^{44} +6.09783 q^{46} -4.98792 q^{47} +15.0151 q^{49} +4.97823 q^{50} +1.82908 q^{53} -0.432960 q^{55} -4.69202 q^{56} +0.850855 q^{58} +5.89977 q^{59} +4.39612 q^{61} +6.23490 q^{62} +1.00000 q^{64} -4.71379 q^{67} +5.60388 q^{68} -14.8213 q^{70} +0.0978347 q^{71} -2.32304 q^{73} +11.7017 q^{74} -4.98792 q^{76} +0.643104 q^{77} +14.5157 q^{79} +3.15883 q^{80} +4.27413 q^{82} +9.85623 q^{83} +17.7017 q^{85} -2.09783 q^{86} -0.137063 q^{88} -17.0858 q^{89} +6.09783 q^{92} -4.98792 q^{94} -15.7560 q^{95} +2.12737 q^{97} +15.0151 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + 3 q^{4} + q^{5} - 9 q^{7} + 3 q^{8} + q^{10} + 5 q^{11} - 9 q^{14} + 3 q^{16} + 8 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} + 18 q^{25} - 9 q^{28} - 11 q^{29} - 5 q^{31} + 3 q^{32} + 8 q^{34}+ \cdots + 20 q^{98}+O(q^{100})$ 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 9 * q^7 + 3 * q^8 + q^10 + 5 * q^11 - 9 * q^14 + 3 * q^16 + 8 * q^17 + 4 * q^19 + q^20 + 5 * q^22 + 18 * q^25 - 9 * q^28 - 11 * q^29 - 5 * q^31 + 3 * q^32 + 8 * q^34 + 4 * q^35 + 8 * q^37 + 4 * q^38 + q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 4 * q^47 + 20 * q^49 + 18 * q^50 - 5 * q^53 + 18 * q^55 - 9 * q^56 - 11 * q^58 - 5 * q^59 + 22 * q^61 - 5 * q^62 + 3 * q^64 - 6 * q^67 + 8 * q^68 + 4 * q^70 - 18 * q^71 + 13 * q^73 + 8 * q^74 + 4 * q^76 + 6 * q^77 + 31 * q^79 + q^80 + 2 * q^82 + 13 * q^83 + 26 * q^85 + 12 * q^86 + 5 * q^88 - 14 * q^89 + 4 * q^94 - 8 * q^95 + 23 * q^97 + 20 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.15883	1.41267	0.706337	−	0.707876i	$-0.250346\pi$
$5$	3.15883	1.41267	0.706337	+	0.707876i	$0.250346\pi$
$6$	0	0
$7$	−4.69202	−1.77342	−0.886709	−	0.462329i	$-0.847014\pi$
$7$	−4.69202	−1.77342	−0.886709	+	0.462329i	$0.847014\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.15883	0.998911
$11$	−0.137063	−0.0413262	−0.0206631	−	0.999786i	$-0.506578\pi$
$11$	−0.137063	−0.0413262	−0.0206631	+	0.999786i	$0.506578\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.69202	−1.25400
$15$	0	0
$16$	1.00000	0.250000
$17$	5.60388	1.35914	0.679570	−	0.733611i	$-0.262167\pi$
$17$	5.60388	1.35914	0.679570	+	0.733611i	$0.262167\pi$
$18$	0	0
$19$	−4.98792	−1.14431	−0.572153	−	0.820147i	$-0.693892\pi$
$19$	−4.98792	−1.14431	−0.572153	+	0.820147i	$0.693892\pi$
$20$	3.15883	0.706337
$21$	0	0
$22$	−0.137063	−0.0292220
$23$	6.09783	1.27149	0.635743	−	0.771901i	$-0.280694\pi$
$23$	6.09783	1.27149	0.635743	+	0.771901i	$0.280694\pi$
$24$	0	0
$25$	4.97823	0.995646
$26$	0	0
$27$	0	0
$28$	−4.69202	−0.886709
$29$	0.850855	0.158000	0.0789999	−	0.996875i	$-0.474827\pi$
$29$	0.850855	0.158000	0.0789999	+	0.996875i	$0.474827\pi$
$30$	0	0
$31$	6.23490	1.11982	0.559910	−	0.828553i	$-0.310836\pi$
$31$	6.23490	1.11982	0.559910	+	0.828553i	$0.310836\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.60388	0.961057
$35$	−14.8213	−2.50526
$36$	0	0
$37$	11.7017	1.92375	0.961875	−	0.273491i	$-0.0881783\pi$
$37$	11.7017	1.92375	0.961875	+	0.273491i	$0.0881783\pi$
$38$	−4.98792	−0.809147
$39$	0	0
$40$	3.15883	0.499455
$41$	4.27413	0.667506	0.333753	−	0.942660i	$-0.391685\pi$
$41$	4.27413	0.667506	0.333753	+	0.942660i	$0.391685\pi$
$42$	0	0
$43$	−2.09783	−0.319917	−0.159958	−	0.987124i	$-0.551136\pi$
$43$	−2.09783	−0.319917	−0.159958	+	0.987124i	$0.551136\pi$
$44$	−0.137063	−0.0206631
$45$	0	0
$46$	6.09783	0.899077
$47$	−4.98792	−0.727563	−0.363781	−	0.931484i	$-0.618514\pi$
$47$	−4.98792	−0.727563	−0.363781	+	0.931484i	$0.618514\pi$
$48$	0	0
$49$	15.0151	2.14501
$50$	4.97823	0.704028
$51$	0	0
$52$	0	0
$53$	1.82908	0.251244	0.125622	−	0.992078i	$-0.459907\pi$
$53$	1.82908	0.251244	0.125622	+	0.992078i	$0.459907\pi$
$54$	0	0
$55$	−0.432960	−0.0583804
$56$	−4.69202	−0.626998
$57$	0	0
$58$	0.850855	0.111723
$59$	5.89977	0.768085	0.384042	−	0.923315i	$-0.374532\pi$
$59$	5.89977	0.768085	0.384042	+	0.923315i	$0.374532\pi$
$60$	0	0
$61$	4.39612	0.562866	0.281433	−	0.959581i	$-0.409190\pi$
$61$	4.39612	0.562866	0.281433	+	0.959581i	$0.409190\pi$
$62$	6.23490	0.791833
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.71379	−0.575881	−0.287941	−	0.957648i	$-0.592971\pi$
$67$	−4.71379	−0.575881	−0.287941	+	0.957648i	$0.592971\pi$
$68$	5.60388	0.679570
$69$	0	0
$70$	−14.8213	−1.77149
$71$	0.0978347	0.0116108	0.00580542	−	0.999983i	$-0.498152\pi$
$71$	0.0978347	0.0116108	0.00580542	+	0.999983i	$0.498152\pi$
$72$	0	0
$73$	−2.32304	−0.271892	−0.135946	−	0.990716i	$-0.543407\pi$
$73$	−2.32304	−0.271892	−0.135946	+	0.990716i	$0.543407\pi$
$74$	11.7017	1.36030
$75$	0	0
$76$	−4.98792	−0.572153
$77$	0.643104	0.0732885
$78$	0	0
$79$	14.5157	1.63315	0.816574	−	0.577241i	$-0.195871\pi$
$79$	14.5157	1.63315	0.816574	+	0.577241i	$0.195871\pi$
$80$	3.15883	0.353168
$81$	0	0
$82$	4.27413	0.471998
$83$	9.85623	1.08186	0.540931	−	0.841067i	$-0.318072\pi$
$83$	9.85623	1.08186	0.540931	+	0.841067i	$0.318072\pi$
$84$	0	0
$85$	17.7017	1.92002
$86$	−2.09783	−0.226215
$87$	0	0
$88$	−0.137063	−0.0146110
$89$	−17.0858	−1.81109	−0.905543	−	0.424254i	$-0.860536\pi$
$89$	−17.0858	−1.81109	−0.905543	+	0.424254i	$0.860536\pi$
$90$	0	0
$91$	0	0
$92$	6.09783	0.635743
$93$	0	0
$94$	−4.98792	−0.514465
$95$	−15.7560	−1.61653
$96$	0	0
$97$	2.12737	0.216002	0.108001	−	0.994151i	$-0.465555\pi$
$97$	2.12737	0.216002	0.108001	+	0.994151i	$0.465555\pi$
$98$	15.0151	1.51675
$99$	0	0
$100$	4.97823	0.497823
$101$	9.18598	0.914039	0.457020	−	0.889457i	$-0.348917\pi$
$101$	9.18598	0.914039	0.457020	+	0.889457i	$0.348917\pi$
$102$	0	0
$103$	0.225209	0.0221905	0.0110953	−	0.999938i	$-0.496468\pi$
$103$	0.225209	0.0221905	0.0110953	+	0.999938i	$0.496468\pi$
$104$	0	0
$105$	0	0
$106$	1.82908	0.177656
$107$	−11.2838	−1.09085	−0.545424	−	0.838160i	$-0.683631\pi$
$107$	−11.2838	−1.09085	−0.545424	+	0.838160i	$0.683631\pi$
$108$	0	0
$109$	−0.195669	−0.0187417	−0.00937086	−	0.999956i	$-0.502983\pi$
$109$	−0.195669	−0.0187417	−0.00937086	+	0.999956i	$0.502983\pi$
$110$	−0.432960	−0.0412811
$111$	0	0
$112$	−4.69202	−0.443354
$113$	0.439665	0.0413602	0.0206801	−	0.999786i	$-0.493417\pi$
$113$	0.439665	0.0413602	0.0206801	+	0.999786i	$0.493417\pi$
$114$	0	0
$115$	19.2620	1.79619
$116$	0.850855	0.0789999
$117$	0	0
$118$	5.89977	0.543118
$119$	−26.2935	−2.41032
$120$	0	0
$121$	−10.9812	−0.998292
$122$	4.39612	0.398006
$123$	0	0
$124$	6.23490	0.559910
$125$	−0.0687686	−0.00615085
$126$	0	0
$127$	−7.87263	−0.698583	−0.349291	−	0.937014i	$-0.613578\pi$
$127$	−7.87263	−0.698583	−0.349291	+	0.937014i	$0.613578\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	0.621334	0.0542862	0.0271431	−	0.999632i	$-0.491359\pi$
$131$	0.621334	0.0542862	0.0271431	+	0.999632i	$0.491359\pi$
$132$	0	0
$133$	23.4034	2.02933
$134$	−4.71379	−0.407210
$135$	0	0
$136$	5.60388	0.480528
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	−13.6582	−1.15847	−0.579235	−	0.815160i	$-0.696649\pi$
$139$	−13.6582	−1.15847	−0.579235	+	0.815160i	$0.696649\pi$
$140$	−14.8213	−1.25263
$141$	0	0
$142$	0.0978347	0.00821010
$143$	0	0
$144$	0	0
$145$	2.68771	0.223202
$146$	−2.32304	−0.192256
$147$	0	0
$148$	11.7017	0.961875
$149$	−16.0586	−1.31557	−0.657786	−	0.753205i	$-0.728507\pi$
$149$	−16.0586	−1.31557	−0.657786	+	0.753205i	$0.728507\pi$
$150$	0	0
$151$	−21.8823	−1.78076	−0.890379	−	0.455221i	$-0.849560\pi$
$151$	−21.8823	−1.78076	−0.890379	+	0.455221i	$0.849560\pi$
$152$	−4.98792	−0.404574
$153$	0	0
$154$	0.643104	0.0518228
$155$	19.6950	1.58194
$156$	0	0
$157$	−7.90217	−0.630661	−0.315331	−	0.948982i	$-0.602115\pi$
$157$	−7.90217	−0.630661	−0.315331	+	0.948982i	$0.602115\pi$
$158$	14.5157	1.15481
$159$	0	0
$160$	3.15883	0.249728
$161$	−28.6112	−2.25488
$162$	0	0
$163$	8.01938	0.628126	0.314063	−	0.949402i	$-0.398310\pi$
$163$	8.01938	0.628126	0.314063	+	0.949402i	$0.398310\pi$
$164$	4.27413	0.333753
$165$	0	0
$166$	9.85623	0.764992
$167$	17.0858	1.32214	0.661068	−	0.750326i	$-0.270104\pi$
$167$	17.0858	1.32214	0.661068	+	0.750326i	$0.270104\pi$
$168$	0	0
$169$	0	0
$170$	17.7017	1.35766
$171$	0	0
$172$	−2.09783	−0.159958
$173$	15.3448	1.16664	0.583322	−	0.812241i	$-0.301752\pi$
$173$	15.3448	1.16664	0.583322	+	0.812241i	$0.301752\pi$
$174$	0	0
$175$	−23.3580	−1.76570
$176$	−0.137063	−0.0103315
$177$	0	0
$178$	−17.0858	−1.28063
$179$	−0.523499	−0.0391282	−0.0195641	−	0.999809i	$-0.506228\pi$
$179$	−0.523499	−0.0391282	−0.0195641	+	0.999809i	$0.506228\pi$
$180$	0	0
$181$	8.89008	0.660795	0.330397	−	0.943842i	$-0.392817\pi$
$181$	8.89008	0.660795	0.330397	+	0.943842i	$0.392817\pi$
$182$	0	0
$183$	0	0
$184$	6.09783	0.449538
$185$	36.9638	2.71763
$186$	0	0
$187$	−0.768086	−0.0561680
$188$	−4.98792	−0.363781
$189$	0	0
$190$	−15.7560	−1.14306
$191$	7.03146	0.508779	0.254389	−	0.967102i	$-0.418126\pi$
$191$	7.03146	0.508779	0.254389	+	0.967102i	$0.418126\pi$
$192$	0	0
$193$	−17.7560	−1.27811	−0.639053	−	0.769163i	$-0.720673\pi$
$193$	−17.7560	−1.27811	−0.639053	+	0.769163i	$0.720673\pi$
$194$	2.12737	0.152737
$195$	0	0
$196$	15.0151	1.07250
$197$	−18.6571	−1.32926	−0.664632	−	0.747171i	$-0.731412\pi$
$197$	−18.6571	−1.32926	−0.664632	+	0.747171i	$0.731412\pi$
$198$	0	0
$199$	7.66248	0.543179	0.271589	−	0.962413i	$-0.412451\pi$
$199$	7.66248	0.543179	0.271589	+	0.962413i	$0.412451\pi$
$200$	4.97823	0.352014
$201$	0	0
$202$	9.18598	0.646323
$203$	−3.99223	−0.280200
$204$	0	0
$205$	13.5013	0.942969
$206$	0.225209	0.0156911
$207$	0	0
$208$	0	0
$209$	0.683661	0.0472898
$210$	0	0
$211$	11.1642	0.768576	0.384288	−	0.923213i	$-0.374447\pi$
$211$	11.1642	0.768576	0.384288	+	0.923213i	$0.374447\pi$
$212$	1.82908	0.125622
$213$	0	0
$214$	−11.2838	−0.771346
$215$	−6.62671	−0.451938
$216$	0	0
$217$	−29.2543	−1.98591
$218$	−0.195669	−0.0132524
$219$	0	0
$220$	−0.432960	−0.0291902
$221$	0	0
$222$	0	0
$223$	−24.6353	−1.64970	−0.824852	−	0.565349i	$-0.808742\pi$
$223$	−24.6353	−1.64970	−0.824852	+	0.565349i	$0.808742\pi$
$224$	−4.69202	−0.313499
$225$	0	0
$226$	0.439665	0.0292461
$227$	−7.47650	−0.496233	−0.248116	−	0.968730i	$-0.579812\pi$
$227$	−7.47650	−0.496233	−0.248116	+	0.968730i	$0.579812\pi$
$228$	0	0
$229$	19.2271	1.27056	0.635282	−	0.772280i	$-0.280884\pi$
$229$	19.2271	1.27056	0.635282	+	0.772280i	$0.280884\pi$
$230$	19.2620	1.27010
$231$	0	0
$232$	0.850855	0.0558614
$233$	−3.70171	−0.242507	−0.121254	−	0.992622i	$-0.538691\pi$
$233$	−3.70171	−0.242507	−0.121254	+	0.992622i	$0.538691\pi$
$234$	0	0
$235$	−15.7560	−1.02781
$236$	5.89977	0.384042
$237$	0	0
$238$	−26.2935	−1.70435
$239$	8.51334	0.550682	0.275341	−	0.961347i	$-0.411209\pi$
$239$	8.51334	0.550682	0.275341	+	0.961347i	$0.411209\pi$
$240$	0	0
$241$	−17.4330	−1.12296	−0.561478	−	0.827492i	$-0.689767\pi$
$241$	−17.4330	−1.12296	−0.561478	+	0.827492i	$0.689767\pi$
$242$	−10.9812	−0.705899
$243$	0	0
$244$	4.39612	0.281433
$245$	47.4301	3.03020
$246$	0	0
$247$	0	0
$248$	6.23490	0.395916
$249$	0	0
$250$	−0.0687686	−0.00434931
$251$	3.48427	0.219925	0.109963	−	0.993936i	$-0.464927\pi$
$251$	3.48427	0.219925	0.109963	+	0.993936i	$0.464927\pi$
$252$	0	0
$253$	−0.835790	−0.0525456
$254$	−7.87263	−0.493972
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6039	0.848586	0.424293	−	0.905525i	$-0.360523\pi$
$257$	13.6039	0.848586	0.424293	+	0.905525i	$0.360523\pi$
$258$	0	0
$259$	−54.9047	−3.41161
$260$	0	0
$261$	0	0
$262$	0.621334	0.0383861
$263$	−11.4577	−0.706513	−0.353256	−	0.935527i	$-0.614926\pi$
$263$	−11.4577	−0.706513	−0.353256	+	0.935527i	$0.614926\pi$
$264$	0	0
$265$	5.77777	0.354926
$266$	23.4034	1.43496
$267$	0	0
$268$	−4.71379	−0.287941
$269$	−22.3666	−1.36371	−0.681857	−	0.731485i	$-0.738828\pi$
$269$	−22.3666	−1.36371	−0.681857	+	0.731485i	$0.738828\pi$
$270$	0	0
$271$	3.87263	0.235245	0.117623	−	0.993058i	$-0.462473\pi$
$271$	3.87263	0.235245	0.117623	+	0.993058i	$0.462473\pi$
$272$	5.60388	0.339785
$273$	0	0
$274$	−4.00000	−0.241649
$275$	−0.682333	−0.0411462
$276$	0	0
$277$	28.7090	1.72496	0.862478	−	0.506094i	$-0.168911\pi$
$277$	28.7090	1.72496	0.862478	+	0.506094i	$0.168911\pi$
$278$	−13.6582	−0.819163
$279$	0	0
$280$	−14.8213	−0.885743
$281$	29.0858	1.73511	0.867555	−	0.497341i	$-0.165690\pi$
$281$	29.0858	1.73511	0.867555	+	0.497341i	$0.165690\pi$
$282$	0	0
$283$	13.7560	0.817710	0.408855	−	0.912599i	$-0.365928\pi$
$283$	13.7560	0.817710	0.408855	+	0.912599i	$0.365928\pi$
$284$	0.0978347	0.00580542
$285$	0	0
$286$	0	0
$287$	−20.0543	−1.18377
$288$	0	0
$289$	14.4034	0.847260
$290$	2.68771	0.157828
$291$	0	0
$292$	−2.32304	−0.135946
$293$	−27.7362	−1.62036	−0.810182	−	0.586179i	$-0.800632\pi$
$293$	−27.7362	−1.62036	−0.810182	+	0.586179i	$0.800632\pi$
$294$	0	0
$295$	18.6364	1.08505
$296$	11.7017	0.680148
$297$	0	0
$298$	−16.0586	−0.930250
$299$	0	0
$300$	0	0
$301$	9.84309	0.567346
$302$	−21.8823	−1.25919
$303$	0	0
$304$	−4.98792	−0.286077
$305$	13.8866	0.795146
$306$	0	0
$307$	−12.4590	−0.711075	−0.355538	−	0.934662i	$-0.615702\pi$
$307$	−12.4590	−0.711075	−0.355538	+	0.934662i	$0.615702\pi$
$308$	0.643104	0.0366443
$309$	0	0
$310$	19.6950	1.11860
$311$	6.09783	0.345776	0.172888	−	0.984941i	$-0.444690\pi$
$311$	6.09783	0.345776	0.172888	+	0.984941i	$0.444690\pi$
$312$	0	0
$313$	−12.7385	−0.720025	−0.360013	−	0.932947i	$-0.617228\pi$
$313$	−12.7385	−0.720025	−0.360013	+	0.932947i	$0.617228\pi$
$314$	−7.90217	−0.445945
$315$	0	0
$316$	14.5157	0.816574
$317$	−14.8140	−0.832038	−0.416019	−	0.909356i	$-0.636575\pi$
$317$	−14.8140	−0.832038	−0.416019	+	0.909356i	$0.636575\pi$
$318$	0	0
$319$	−0.116621	−0.00652952
$320$	3.15883	0.176584
$321$	0	0
$322$	−28.6112	−1.59444
$323$	−27.9517	−1.55527
$324$	0	0
$325$	0	0
$326$	8.01938	0.444152
$327$	0	0
$328$	4.27413	0.235999
$329$	23.4034	1.29027
$330$	0	0
$331$	7.70171	0.423324	0.211662	−	0.977343i	$-0.432112\pi$
$331$	7.70171	0.423324	0.211662	+	0.977343i	$0.432112\pi$
$332$	9.85623	0.540931
$333$	0	0
$334$	17.0858	0.934891
$335$	−14.8901	−0.813532
$336$	0	0
$337$	26.5961	1.44878	0.724391	−	0.689389i	$-0.242121\pi$
$337$	26.5961	1.44878	0.724391	+	0.689389i	$0.242121\pi$
$338$	0	0
$339$	0	0
$340$	17.7017	0.960010
$341$	−0.854576	−0.0462779
$342$	0	0
$343$	−37.6069	−2.03058
$344$	−2.09783	−0.113108
$345$	0	0
$346$	15.3448	0.824942
$347$	−0.911854	−0.0489509	−0.0244754	−	0.999700i	$-0.507792\pi$
$347$	−0.911854	−0.0489509	−0.0244754	+	0.999700i	$0.507792\pi$
$348$	0	0
$349$	−17.7211	−0.948588	−0.474294	−	0.880366i	$-0.657297\pi$
$349$	−17.7211	−0.948588	−0.474294	+	0.880366i	$0.657297\pi$
$350$	−23.3580	−1.24854
$351$	0	0
$352$	−0.137063	−0.00730550
$353$	26.4349	1.40699	0.703493	−	0.710702i	$-0.251622\pi$
$353$	26.4349	1.40699	0.703493	+	0.710702i	$0.251622\pi$
$354$	0	0
$355$	0.309043	0.0164023
$356$	−17.0858	−0.905543
$357$	0	0
$358$	−0.523499	−0.0276678
$359$	−7.76941	−0.410054	−0.205027	−	0.978756i	$-0.565728\pi$
$359$	−7.76941	−0.410054	−0.205027	+	0.978756i	$0.565728\pi$
$360$	0	0
$361$	5.87933	0.309438
$362$	8.89008	0.467252
$363$	0	0
$364$	0	0
$365$	−7.33811	−0.384094
$366$	0	0
$367$	−13.3274	−0.695682	−0.347841	−	0.937553i	$-0.613085\pi$
$367$	−13.3274	−0.695682	−0.347841	+	0.937553i	$0.613085\pi$
$368$	6.09783	0.317872
$369$	0	0
$370$	36.9638	1.92165
$371$	−8.58211	−0.445561
$372$	0	0
$373$	6.70304	0.347070	0.173535	−	0.984828i	$-0.444481\pi$
$373$	6.70304	0.347070	0.173535	+	0.984828i	$0.444481\pi$
$374$	−0.768086	−0.0397168
$375$	0	0
$376$	−4.98792	−0.257232
$377$	0	0
$378$	0	0
$379$	2.41550	0.124076	0.0620380	−	0.998074i	$-0.480240\pi$
$379$	2.41550	0.124076	0.0620380	+	0.998074i	$0.480240\pi$
$380$	−15.7560	−0.808266
$381$	0	0
$382$	7.03146	0.359761
$383$	−10.0978	−0.515975	−0.257988	−	0.966148i	$-0.583059\pi$
$383$	−10.0978	−0.515975	−0.257988	+	0.966148i	$0.583059\pi$
$384$	0	0
$385$	2.03146	0.103533
$386$	−17.7560	−0.903757
$387$	0	0
$388$	2.12737	0.108001
$389$	−25.1336	−1.27432	−0.637162	−	0.770730i	$-0.719892\pi$
$389$	−25.1336	−1.27432	−0.637162	+	0.770730i	$0.719892\pi$
$390$	0	0
$391$	34.1715	1.72813
$392$	15.0151	0.758375
$393$	0	0
$394$	−18.6571	−0.939931
$395$	45.8528	2.30710
$396$	0	0
$397$	20.8358	1.04572	0.522859	−	0.852419i	$-0.324865\pi$
$397$	20.8358	1.04572	0.522859	+	0.852419i	$0.324865\pi$
$398$	7.66248	0.384085
$399$	0	0
$400$	4.97823	0.248911
$401$	5.95646	0.297451	0.148726	−	0.988878i	$-0.452483\pi$
$401$	5.95646	0.297451	0.148726	+	0.988878i	$0.452483\pi$
$402$	0	0
$403$	0	0
$404$	9.18598	0.457020
$405$	0	0
$406$	−3.99223	−0.198131
$407$	−1.60388	−0.0795012
$408$	0	0
$409$	−1.80194	−0.0891001	−0.0445500	−	0.999007i	$-0.514185\pi$
$409$	−1.80194	−0.0891001	−0.0445500	+	0.999007i	$0.514185\pi$
$410$	13.5013	0.666779
$411$	0	0
$412$	0.225209	0.0110953
$413$	−27.6819	−1.36214
$414$	0	0
$415$	31.1342	1.52832
$416$	0	0
$417$	0	0
$418$	0.683661	0.0334389
$419$	28.4499	1.38987	0.694935	−	0.719072i	$-0.255433\pi$
$419$	28.4499	1.38987	0.694935	+	0.719072i	$0.255433\pi$
$420$	0	0
$421$	13.9323	0.679019	0.339509	−	0.940603i	$-0.389739\pi$
$421$	13.9323	0.679019	0.339509	+	0.940603i	$0.389739\pi$
$422$	11.1642	0.543465
$423$	0	0
$424$	1.82908	0.0888282
$425$	27.8974	1.35322
$426$	0	0
$427$	−20.6267	−0.998196
$428$	−11.2838	−0.545424
$429$	0	0
$430$	−6.62671	−0.319568
$431$	15.9022	0.765980	0.382990	−	0.923752i	$-0.374894\pi$
$431$	15.9022	0.765980	0.382990	+	0.923752i	$0.374894\pi$
$432$	0	0
$433$	4.77718	0.229577	0.114788	−	0.993390i	$-0.463381\pi$
$433$	4.77718	0.229577	0.114788	+	0.993390i	$0.463381\pi$
$434$	−29.2543	−1.40425
$435$	0	0
$436$	−0.195669	−0.00937086
$437$	−30.4155	−1.45497
$438$	0	0
$439$	−33.6316	−1.60515	−0.802575	−	0.596552i	$-0.796537\pi$
$439$	−33.6316	−1.60515	−0.802575	+	0.596552i	$0.796537\pi$
$440$	−0.432960	−0.0206406
$441$	0	0
$442$	0	0
$443$	35.3749	1.68071	0.840357	−	0.542033i	$-0.182345\pi$
$443$	35.3749	1.68071	0.840357	+	0.542033i	$0.182345\pi$
$444$	0	0
$445$	−53.9711	−2.55847
$446$	−24.6353	−1.16652
$447$	0	0
$448$	−4.69202	−0.221677
$449$	−18.0629	−0.852442	−0.426221	−	0.904619i	$-0.640155\pi$
$449$	−18.0629	−0.852442	−0.426221	+	0.904619i	$0.640155\pi$
$450$	0	0
$451$	−0.585826	−0.0275855
$452$	0.439665	0.0206801
$453$	0	0
$454$	−7.47650	−0.350890
$455$	0	0
$456$	0	0
$457$	15.4668	0.723507	0.361753	−	0.932274i	$-0.382178\pi$
$457$	15.4668	0.723507	0.361753	+	0.932274i	$0.382178\pi$
$458$	19.2271	0.898425
$459$	0	0
$460$	19.2620	0.898097
$461$	−18.8092	−0.876033	−0.438017	−	0.898967i	$-0.644319\pi$
$461$	−18.8092	−0.876033	−0.438017	+	0.898967i	$0.644319\pi$
$462$	0	0
$463$	15.8431	0.736291	0.368145	−	0.929768i	$-0.379993\pi$
$463$	15.8431	0.736291	0.368145	+	0.929768i	$0.379993\pi$
$464$	0.850855	0.0395000
$465$	0	0
$466$	−3.70171	−0.171478
$467$	−22.0006	−1.01807	−0.509033	−	0.860747i	$-0.669997\pi$
$467$	−22.0006	−1.01807	−0.509033	+	0.860747i	$0.669997\pi$
$468$	0	0
$469$	22.1172	1.02128
$470$	−15.7560	−0.726770
$471$	0	0
$472$	5.89977	0.271559
$473$	0.287536	0.0132209
$474$	0	0
$475$	−24.8310	−1.13932
$476$	−26.2935	−1.20516
$477$	0	0
$478$	8.51334	0.389391
$479$	−21.3491	−0.975466	−0.487733	−	0.872993i	$-0.662176\pi$
$479$	−21.3491	−0.975466	−0.487733	+	0.872993i	$0.662176\pi$
$480$	0	0
$481$	0	0
$482$	−17.4330	−0.794050
$483$	0	0
$484$	−10.9812	−0.499146
$485$	6.72002	0.305141
$486$	0	0
$487$	31.6394	1.43372	0.716859	−	0.697219i	$-0.245579\pi$
$487$	31.6394	1.43372	0.716859	+	0.697219i	$0.245579\pi$
$488$	4.39612	0.199003
$489$	0	0
$490$	47.4301	2.14267
$491$	1.39911	0.0631409	0.0315704	−	0.999502i	$-0.489949\pi$
$491$	1.39911	0.0631409	0.0315704	+	0.999502i	$0.489949\pi$
$492$	0	0
$493$	4.76809	0.214744
$494$	0	0
$495$	0	0
$496$	6.23490	0.279955
$497$	−0.459042	−0.0205909
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−0.0687686	−0.00307543
$501$	0	0
$502$	3.48427	0.155511
$503$	18.3827	0.819645	0.409822	−	0.912165i	$-0.365591\pi$
$503$	18.3827	0.819645	0.409822	+	0.912165i	$0.365591\pi$
$504$	0	0
$505$	29.0170	1.29124
$506$	−0.835790	−0.0371554
$507$	0	0
$508$	−7.87263	−0.349291
$509$	0.132751	0.00588411	0.00294205	−	0.999996i	$-0.499064\pi$
$509$	0.132751	0.00588411	0.00294205	+	0.999996i	$0.499064\pi$
$510$	0	0
$511$	10.8998	0.482178
$512$	1.00000	0.0441942
$513$	0	0
$514$	13.6039	0.600041
$515$	0.711399	0.0313480
$516$	0	0
$517$	0.683661	0.0300674
$518$	−54.9047	−2.41237
$519$	0	0
$520$	0	0
$521$	−37.0508	−1.62323	−0.811613	−	0.584195i	$-0.801410\pi$
$521$	−37.0508	−1.62323	−0.811613	+	0.584195i	$0.801410\pi$
$522$	0	0
$523$	−3.15346	−0.137891	−0.0689455	−	0.997620i	$-0.521963\pi$
$523$	−3.15346	−0.137891	−0.0689455	+	0.997620i	$0.521963\pi$
$524$	0.621334	0.0271431
$525$	0	0
$526$	−11.4577	−0.499580
$527$	34.9396	1.52199
$528$	0	0
$529$	14.1836	0.616678
$530$	5.77777	0.250970
$531$	0	0
$532$	23.4034	1.01467
$533$	0	0
$534$	0	0
$535$	−35.6437	−1.54101
$536$	−4.71379	−0.203605
$537$	0	0
$538$	−22.3666	−0.964292
$539$	−2.05802	−0.0886450
$540$	0	0
$541$	4.07846	0.175347	0.0876733	−	0.996149i	$-0.472057\pi$
$541$	4.07846	0.175347	0.0876733	+	0.996149i	$0.472057\pi$
$542$	3.87263	0.166344
$543$	0	0
$544$	5.60388	0.240264
$545$	−0.618087	−0.0264759
$546$	0	0
$547$	−23.0508	−0.985583	−0.492791	−	0.870148i	$-0.664023\pi$
$547$	−23.0508	−0.985583	−0.492791	+	0.870148i	$0.664023\pi$
$548$	−4.00000	−0.170872
$549$	0	0
$550$	−0.682333	−0.0290948
$551$	−4.24400	−0.180800
$552$	0	0
$553$	−68.1081	−2.89625
$554$	28.7090	1.21973
$555$	0	0
$556$	−13.6582	−0.579235
$557$	−20.4155	−0.865033	−0.432516	−	0.901626i	$-0.642374\pi$
$557$	−20.4155	−0.865033	−0.432516	+	0.901626i	$0.642374\pi$
$558$	0	0
$559$	0	0
$560$	−14.8213	−0.626315
$561$	0	0
$562$	29.0858	1.22691
$563$	−21.5609	−0.908685	−0.454342	−	0.890827i	$-0.650126\pi$
$563$	−21.5609	−0.908685	−0.454342	+	0.890827i	$0.650126\pi$
$564$	0	0
$565$	1.38883	0.0584285
$566$	13.7560	0.578208
$567$	0	0
$568$	0.0978347	0.00410505
$569$	−8.98792	−0.376793	−0.188397	−	0.982093i	$-0.560329\pi$
$569$	−8.98792	−0.376793	−0.188397	+	0.982093i	$0.560329\pi$
$570$	0	0
$571$	13.5603	0.567482	0.283741	−	0.958901i	$-0.408424\pi$
$571$	13.5603	0.567482	0.283741	+	0.958901i	$0.408424\pi$
$572$	0	0
$573$	0	0
$574$	−20.0543	−0.837050
$575$	30.3564	1.26595
$576$	0	0
$577$	16.2825	0.677849	0.338924	−	0.940814i	$-0.389937\pi$
$577$	16.2825	0.677849	0.338924	+	0.940814i	$0.389937\pi$
$578$	14.4034	0.599103
$579$	0	0
$580$	2.68771	0.111601
$581$	−46.2457	−1.91859
$582$	0	0
$583$	−0.250700	−0.0103830
$584$	−2.32304	−0.0961282
$585$	0	0
$586$	−27.7362	−1.14577
$587$	−47.5706	−1.96345	−0.981725	−	0.190307i	$-0.939052\pi$
$587$	−47.5706	−1.96345	−0.981725	+	0.190307i	$0.939052\pi$
$588$	0	0
$589$	−31.0992	−1.28142
$590$	18.6364	0.767248
$591$	0	0
$592$	11.7017	0.480937
$593$	−31.0267	−1.27411	−0.637056	−	0.770817i	$-0.719848\pi$
$593$	−31.0267	−1.27411	−0.637056	+	0.770817i	$0.719848\pi$
$594$	0	0
$595$	−83.0568	−3.40500
$596$	−16.0586	−0.657786
$597$	0	0
$598$	0	0
$599$	−22.3263	−0.912228	−0.456114	−	0.889921i	$-0.650759\pi$
$599$	−22.3263	−0.912228	−0.456114	+	0.889921i	$0.650759\pi$
$600$	0	0
$601$	−8.18060	−0.333694	−0.166847	−	0.985983i	$-0.553359\pi$
$601$	−8.18060	−0.333694	−0.166847	+	0.985983i	$0.553359\pi$
$602$	9.84309	0.401174
$603$	0	0
$604$	−21.8823	−0.890379
$605$	−34.6878	−1.41026
$606$	0	0
$607$	3.30798	0.134267	0.0671334	−	0.997744i	$-0.478615\pi$
$607$	3.30798	0.134267	0.0671334	+	0.997744i	$0.478615\pi$
$608$	−4.98792	−0.202287
$609$	0	0
$610$	13.8866	0.562253
$611$	0	0
$612$	0	0
$613$	−10.8853	−0.439653	−0.219827	−	0.975539i	$-0.570549\pi$
$613$	−10.8853	−0.439653	−0.219827	+	0.975539i	$0.570549\pi$
$614$	−12.4590	−0.502806
$615$	0	0
$616$	0.643104	0.0259114
$617$	34.5676	1.39164	0.695820	−	0.718216i	$-0.255041\pi$
$617$	34.5676	1.39164	0.695820	+	0.718216i	$0.255041\pi$
$618$	0	0
$619$	2.86592	0.115191	0.0575955	−	0.998340i	$-0.481657\pi$
$619$	2.86592	0.115191	0.0575955	+	0.998340i	$0.481657\pi$
$620$	19.6950	0.790970
$621$	0	0
$622$	6.09783	0.244501
$623$	80.1667	3.21181
$624$	0	0
$625$	−25.1084	−1.00434
$626$	−12.7385	−0.509135
$627$	0	0
$628$	−7.90217	−0.315331
$629$	65.5749	2.61464
$630$	0	0
$631$	−42.6631	−1.69839	−0.849195	−	0.528079i	$-0.822912\pi$
$631$	−42.6631	−1.69839	−0.849195	+	0.528079i	$0.822912\pi$
$632$	14.5157	0.577405
$633$	0	0
$634$	−14.8140	−0.588340
$635$	−24.8683	−0.986869
$636$	0	0
$637$	0	0
$638$	−0.116621	−0.00461707
$639$	0	0
$640$	3.15883	0.124864
$641$	−41.3927	−1.63491	−0.817456	−	0.575991i	$-0.804616\pi$
$641$	−41.3927	−1.63491	−0.817456	+	0.575991i	$0.804616\pi$
$642$	0	0
$643$	13.7125	0.540767	0.270383	−	0.962753i	$-0.412850\pi$
$643$	13.7125	0.540767	0.270383	+	0.962753i	$0.412850\pi$
$644$	−28.6112	−1.12744
$645$	0	0
$646$	−27.9517	−1.09974
$647$	−18.0086	−0.707992	−0.353996	−	0.935247i	$-0.615177\pi$
$647$	−18.0086	−0.707992	−0.353996	+	0.935247i	$0.615177\pi$
$648$	0	0
$649$	−0.808643	−0.0317420
$650$	0	0
$651$	0	0
$652$	8.01938	0.314063
$653$	−46.5652	−1.82224	−0.911119	−	0.412143i	$-0.864780\pi$
$653$	−46.5652	−1.82224	−0.911119	+	0.412143i	$0.864780\pi$
$654$	0	0
$655$	1.96269	0.0766887
$656$	4.27413	0.166877
$657$	0	0
$658$	23.4034	0.912360
$659$	−13.8562	−0.539762	−0.269881	−	0.962894i	$-0.586984\pi$
$659$	−13.8562	−0.539762	−0.269881	+	0.962894i	$0.586984\pi$
$660$	0	0
$661$	−43.1051	−1.67660	−0.838298	−	0.545213i	$-0.816449\pi$
$661$	−43.1051	−1.67660	−0.838298	+	0.545213i	$0.816449\pi$
$662$	7.70171	0.299335
$663$	0	0
$664$	9.85623	0.382496
$665$	73.9275	2.86679
$666$	0	0
$667$	5.18837	0.200895
$668$	17.0858	0.661068
$669$	0	0
$670$	−14.8901	−0.575254
$671$	−0.602548	−0.0232611
$672$	0	0
$673$	30.7415	1.18500	0.592499	−	0.805571i	$-0.298141\pi$
$673$	30.7415	1.18500	0.592499	+	0.805571i	$0.298141\pi$
$674$	26.5961	1.02444
$675$	0	0
$676$	0	0
$677$	−16.5894	−0.637582	−0.318791	−	0.947825i	$-0.603277\pi$
$677$	−16.5894	−0.637582	−0.318791	+	0.947825i	$0.603277\pi$
$678$	0	0
$679$	−9.98169	−0.383062
$680$	17.7017	0.678830
$681$	0	0
$682$	−0.854576	−0.0327234
$683$	34.9885	1.33880	0.669399	−	0.742903i	$-0.266552\pi$
$683$	34.9885	1.33880	0.669399	+	0.742903i	$0.266552\pi$
$684$	0	0
$685$	−12.6353	−0.482771
$686$	−37.6069	−1.43584
$687$	0	0
$688$	−2.09783	−0.0799792
$689$	0	0
$690$	0	0
$691$	14.0871	0.535898	0.267949	−	0.963433i	$-0.413654\pi$
$691$	14.0871	0.535898	0.267949	+	0.963433i	$0.413654\pi$
$692$	15.3448	0.583322
$693$	0	0
$694$	−0.911854	−0.0346135
$695$	−43.1439	−1.63654
$696$	0	0
$697$	23.9517	0.907234
$698$	−17.7211	−0.670753
$699$	0	0
$700$	−23.3580	−0.882848
$701$	−48.6112	−1.83602	−0.918009	−	0.396559i	$-0.870204\pi$
$701$	−48.6112	−1.83602	−0.918009	+	0.396559i	$0.870204\pi$
$702$	0	0
$703$	−58.3672	−2.20136
$704$	−0.137063	−0.00516577
$705$	0	0
$706$	26.4349	0.994890
$707$	−43.1008	−1.62097
$708$	0	0
$709$	17.2862	0.649197	0.324599	−	0.945852i	$-0.394771\pi$
$709$	17.2862	0.649197	0.324599	+	0.945852i	$0.394771\pi$
$710$	0.309043	0.0115982
$711$	0	0
$712$	−17.0858	−0.640316
$713$	38.0194	1.42384
$714$	0	0
$715$	0	0
$716$	−0.523499	−0.0195641
$717$	0	0
$718$	−7.76941	−0.289952
$719$	−29.1207	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$719$	−29.1207	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$720$	0	0
$721$	−1.05669	−0.0393531
$722$	5.87933	0.218806
$723$	0	0
$724$	8.89008	0.330397
$725$	4.23575	0.157312
$726$	0	0
$727$	45.5666	1.68997	0.844985	−	0.534790i	$-0.179609\pi$
$727$	45.5666	1.68997	0.844985	+	0.534790i	$0.179609\pi$
$728$	0	0
$729$	0	0
$730$	−7.33811	−0.271596
$731$	−11.7560	−0.434812
$732$	0	0
$733$	−21.7995	−0.805185	−0.402592	−	0.915379i	$-0.631891\pi$
$733$	−21.7995	−0.805185	−0.402592	+	0.915379i	$0.631891\pi$
$734$	−13.3274	−0.491922
$735$	0	0
$736$	6.09783	0.224769
$737$	0.646088	0.0237990
$738$	0	0
$739$	−41.5663	−1.52904	−0.764521	−	0.644599i	$-0.777024\pi$
$739$	−41.5663	−1.52904	−0.764521	+	0.644599i	$0.777024\pi$
$740$	36.9638	1.35881
$741$	0	0
$742$	−8.58211	−0.315059
$743$	−28.8224	−1.05739	−0.528695	−	0.848812i	$-0.677319\pi$
$743$	−28.8224	−1.05739	−0.528695	+	0.848812i	$0.677319\pi$
$744$	0	0
$745$	−50.7265	−1.85847
$746$	6.70304	0.245416
$747$	0	0
$748$	−0.768086	−0.0280840
$749$	52.9439	1.93453
$750$	0	0
$751$	16.6203	0.606482	0.303241	−	0.952914i	$-0.401931\pi$
$751$	16.6203	0.606482	0.303241	+	0.952914i	$0.401931\pi$
$752$	−4.98792	−0.181891
$753$	0	0
$754$	0	0
$755$	−69.1226	−2.51563
$756$	0	0
$757$	−40.3913	−1.46805	−0.734024	−	0.679123i	$-0.762360\pi$
$757$	−40.3913	−1.46805	−0.734024	+	0.679123i	$0.762360\pi$
$758$	2.41550	0.0877350
$759$	0	0
$760$	−15.7560	−0.571530
$761$	3.29483	0.119438	0.0597188	−	0.998215i	$-0.480980\pi$
$761$	3.29483	0.119438	0.0597188	+	0.998215i	$0.480980\pi$
$762$	0	0
$763$	0.918085	0.0332369
$764$	7.03146	0.254389
$765$	0	0
$766$	−10.0978	−0.364850
$767$	0	0
$768$	0	0
$769$	−2.35258	−0.0848363	−0.0424182	−	0.999100i	$-0.513506\pi$
$769$	−2.35258	−0.0848363	−0.0424182	+	0.999100i	$0.513506\pi$
$770$	2.03146	0.0732087
$771$	0	0
$772$	−17.7560	−0.639053
$773$	20.3937	0.733512	0.366756	−	0.930317i	$-0.380468\pi$
$773$	20.3937	0.733512	0.366756	+	0.930317i	$0.380468\pi$
$774$	0	0
$775$	31.0388	1.11494
$776$	2.12737	0.0763683
$777$	0	0
$778$	−25.1336	−0.901083
$779$	−21.3190	−0.763832
$780$	0	0
$781$	−0.0134095	−0.000479831 0
$782$	34.1715	1.22197
$783$	0	0
$784$	15.0151	0.536252
$785$	−24.9616	−0.890919
$786$	0	0
$787$	19.6775	0.701429	0.350714	−	0.936482i	$-0.385939\pi$
$787$	19.6775	0.701429	0.350714	+	0.936482i	$0.385939\pi$
$788$	−18.6571	−0.664632
$789$	0	0
$790$	45.8528	1.63137
$791$	−2.06292	−0.0733489
$792$	0	0
$793$	0	0
$794$	20.8358	0.739435
$795$	0	0
$796$	7.66248	0.271589
$797$	45.8689	1.62476	0.812380	−	0.583128i	$-0.198172\pi$
$797$	45.8689	1.62476	0.812380	+	0.583128i	$0.198172\pi$
$798$	0	0
$799$	−27.9517	−0.988859
$800$	4.97823	0.176007
$801$	0	0
$802$	5.95646	0.210330
$803$	0.318404	0.0112362
$804$	0	0
$805$	−90.3779	−3.18540
$806$	0	0
$807$	0	0
$808$	9.18598	0.323162
$809$	−38.1414	−1.34098	−0.670490	−	0.741919i	$-0.733916\pi$
$809$	−38.1414	−1.34098	−0.670490	+	0.741919i	$0.733916\pi$
$810$	0	0
$811$	46.6983	1.63980	0.819899	−	0.572509i	$-0.194030\pi$
$811$	46.6983	1.63980	0.819899	+	0.572509i	$0.194030\pi$
$812$	−3.99223	−0.140100
$813$	0	0
$814$	−1.60388	−0.0562158
$815$	25.3319	0.887337
$816$	0	0
$817$	10.4638	0.366083
$818$	−1.80194	−0.0630033
$819$	0	0
$820$	13.5013	0.471484
$821$	−32.6950	−1.14106	−0.570532	−	0.821276i	$-0.693263\pi$
$821$	−32.6950	−1.14106	−0.570532	+	0.821276i	$0.693263\pi$
$822$	0	0
$823$	−21.6799	−0.755715	−0.377858	−	0.925864i	$-0.623339\pi$
$823$	−21.6799	−0.755715	−0.377858	+	0.925864i	$0.623339\pi$
$824$	0.225209	0.00784554
$825$	0	0
$826$	−27.6819	−0.963175
$827$	13.9172	0.483950	0.241975	−	0.970283i	$-0.422205\pi$
$827$	13.9172	0.483950	0.241975	+	0.970283i	$0.422205\pi$
$828$	0	0
$829$	−16.4047	−0.569760	−0.284880	−	0.958563i	$-0.591954\pi$
$829$	−16.4047	−0.569760	−0.284880	+	0.958563i	$0.591954\pi$
$830$	31.1342	1.08068
$831$	0	0
$832$	0	0
$833$	84.1426	2.91537
$834$	0	0
$835$	53.9711	1.86775
$836$	0.683661	0.0236449
$837$	0	0
$838$	28.4499	0.982787
$839$	−11.6146	−0.400982	−0.200491	−	0.979696i	$-0.564254\pi$
$839$	−11.6146	−0.400982	−0.200491	+	0.979696i	$0.564254\pi$
$840$	0	0
$841$	−28.2760	−0.975036
$842$	13.9323	0.480139
$843$	0	0
$844$	11.1642	0.384288
$845$	0	0
$846$	0	0
$847$	51.5241	1.77039
$848$	1.82908	0.0628110
$849$	0	0
$850$	27.8974	0.956872
$851$	71.3551	2.44602
$852$	0	0
$853$	26.2983	0.900436	0.450218	−	0.892919i	$-0.351346\pi$
$853$	26.2983	0.900436	0.450218	+	0.892919i	$0.351346\pi$
$854$	−20.6267	−0.705832
$855$	0	0
$856$	−11.2838	−0.385673
$857$	48.6305	1.66119	0.830594	−	0.556879i	$-0.188001\pi$
$857$	48.6305	1.66119	0.830594	+	0.556879i	$0.188001\pi$
$858$	0	0
$859$	33.6185	1.14705	0.573524	−	0.819189i	$-0.305576\pi$
$859$	33.6185	1.14705	0.573524	+	0.819189i	$0.305576\pi$
$860$	−6.62671	−0.225969
$861$	0	0
$862$	15.9022	0.541630
$863$	5.78879	0.197053	0.0985264	−	0.995134i	$-0.468587\pi$
$863$	5.78879	0.197053	0.0985264	+	0.995134i	$0.468587\pi$
$864$	0	0
$865$	48.4717	1.64809
$866$	4.77718	0.162335
$867$	0	0
$868$	−29.2543	−0.992955
$869$	−1.98957	−0.0674917
$870$	0	0
$871$	0	0
$872$	−0.195669	−0.00662620
$873$	0	0
$874$	−30.4155	−1.02882
$875$	0.322664	0.0109080
$876$	0	0
$877$	9.50604	0.320996	0.160498	−	0.987036i	$-0.448690\pi$
$877$	9.50604	0.320996	0.160498	+	0.987036i	$0.448690\pi$
$878$	−33.6316	−1.13501
$879$	0	0
$880$	−0.432960	−0.0145951
$881$	46.7875	1.57631	0.788155	−	0.615477i	$-0.211037\pi$
$881$	46.7875	1.57631	0.788155	+	0.615477i	$0.211037\pi$
$882$	0	0
$883$	3.03146	0.102017	0.0510084	−	0.998698i	$-0.483756\pi$
$883$	3.03146	0.102017	0.0510084	+	0.998698i	$0.483756\pi$
$884$	0	0
$885$	0	0
$886$	35.3749	1.18844
$887$	−37.0180	−1.24294	−0.621472	−	0.783436i	$-0.713465\pi$
$887$	−37.0180	−1.24294	−0.621472	+	0.783436i	$0.713465\pi$
$888$	0	0
$889$	36.9385	1.23888
$890$	−53.9711	−1.80911
$891$	0	0
$892$	−24.6353	−0.824852
$893$	24.8793	0.832555
$894$	0	0
$895$	−1.65365	−0.0552753
$896$	−4.69202	−0.156749
$897$	0	0
$898$	−18.0629	−0.602767
$899$	5.30499	0.176931
$900$	0	0
$901$	10.2500	0.341476
$902$	−0.585826	−0.0195059
$903$	0	0
$904$	0.439665	0.0146230
$905$	28.0823	0.933487
$906$	0	0
$907$	19.0965	0.634089	0.317045	−	0.948411i	$-0.397310\pi$
$907$	19.0965	0.634089	0.317045	+	0.948411i	$0.397310\pi$
$908$	−7.47650	−0.248116
$909$	0	0
$910$	0	0
$911$	31.3142	1.03749	0.518743	−	0.854930i	$-0.326400\pi$
$911$	31.3142	1.03749	0.518743	+	0.854930i	$0.326400\pi$
$912$	0	0
$913$	−1.35093	−0.0447092
$914$	15.4668	0.511597
$915$	0	0
$916$	19.2271	0.635282
$917$	−2.91531	−0.0962721
$918$	0	0
$919$	30.3967	1.00270	0.501348	−	0.865246i	$-0.332838\pi$
$919$	30.3967	1.00270	0.501348	+	0.865246i	$0.332838\pi$
$920$	19.2620	0.635051
$921$	0	0
$922$	−18.8092	−0.619449
$923$	0	0
$924$	0	0
$925$	58.2538	1.91537
$926$	15.8431	0.520636
$927$	0	0
$928$	0.850855	0.0279307
$929$	40.5810	1.33142	0.665710	−	0.746210i	$-0.268129\pi$
$929$	40.5810	1.33142	0.665710	+	0.746210i	$0.268129\pi$
$930$	0	0
$931$	−74.8939	−2.45455
$932$	−3.70171	−0.121254
$933$	0	0
$934$	−22.0006	−0.719881
$935$	−2.42626	−0.0793470
$936$	0	0
$937$	−18.7047	−0.611056	−0.305528	−	0.952183i	$-0.598833\pi$
$937$	−18.7047	−0.611056	−0.305528	+	0.952183i	$0.598833\pi$
$938$	22.1172	0.722153
$939$	0	0
$940$	−15.7560	−0.513904
$941$	4.04998	0.132026	0.0660128	−	0.997819i	$-0.478972\pi$
$941$	4.04998	0.132026	0.0660128	+	0.997819i	$0.478972\pi$
$942$	0	0
$943$	26.0629	0.848725
$944$	5.89977	0.192021
$945$	0	0
$946$	0.287536	0.00934861
$947$	−11.5356	−0.374856	−0.187428	−	0.982278i	$-0.560015\pi$
$947$	−11.5356	−0.374856	−0.187428	+	0.982278i	$0.560015\pi$
$948$	0	0
$949$	0	0
$950$	−24.8310	−0.805624
$951$	0	0
$952$	−26.2935	−0.852177
$953$	9.57109	0.310038	0.155019	−	0.987911i	$-0.450456\pi$
$953$	9.57109	0.310038	0.155019	+	0.987911i	$0.450456\pi$
$954$	0	0
$955$	22.2112	0.718738
$956$	8.51334	0.275341
$957$	0	0
$958$	−21.3491	−0.689759
$959$	18.7681	0.606053
$960$	0	0
$961$	7.87395	0.253998
$962$	0	0
$963$	0	0
$964$	−17.4330	−0.561478
$965$	−56.0883	−1.80555
$966$	0	0
$967$	−61.2073	−1.96829	−0.984147	−	0.177357i	$-0.943245\pi$
$967$	−61.2073	−1.96829	−0.984147	+	0.177357i	$0.943245\pi$
$968$	−10.9812	−0.352950
$969$	0	0
$970$	6.72002	0.215767
$971$	28.8595	0.926145	0.463072	−	0.886320i	$-0.346747\pi$
$971$	28.8595	0.926145	0.463072	+	0.886320i	$0.346747\pi$
$972$	0	0
$973$	64.0844	2.05445
$974$	31.6394	1.01379
$975$	0	0
$976$	4.39612	0.140717
$977$	46.6305	1.49184	0.745922	−	0.666034i	$-0.232009\pi$
$977$	46.6305	1.49184	0.745922	+	0.666034i	$0.232009\pi$
$978$	0	0
$979$	2.34183	0.0748452
$980$	47.4301	1.51510
$981$	0	0
$982$	1.39911	0.0446473
$983$	55.6883	1.77618	0.888090	−	0.459669i	$-0.152032\pi$
$983$	55.6883	1.77618	0.888090	+	0.459669i	$0.152032\pi$
$984$	0	0
$985$	−58.9347	−1.87782
$986$	4.76809	0.151847
$987$	0	0
$988$	0	0
$989$	−12.7922	−0.406770
$990$	0	0
$991$	−9.32172	−0.296114	−0.148057	−	0.988979i	$-0.547302\pi$
$991$	−9.32172	−0.296114	−0.148057	+	0.988979i	$0.547302\pi$
$992$	6.23490	0.197958
$993$	0	0
$994$	−0.459042	−0.0145599
$995$	24.2045	0.767334
$996$	0	0
$997$	46.0253	1.45764	0.728819	−	0.684707i	$-0.240070\pi$
$997$	46.0253	1.45764	0.728819	+	0.684707i	$0.240070\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.bh.1.3		3
3.2	odd	2		1014.2.a.l.1.1	✓	3
12.11	even	2		8112.2.a.cj.1.1		3
13.5	odd	4		3042.2.b.o.1351.1		6
13.8	odd	4		3042.2.b.o.1351.6		6
13.12	even	2		3042.2.a.ba.1.1		3
39.2	even	12		1014.2.i.h.823.3		12
39.5	even	4		1014.2.b.f.337.6		6
39.8	even	4		1014.2.b.f.337.1		6
39.11	even	12		1014.2.i.h.823.4		12
39.17	odd	6		1014.2.e.l.991.3		6
39.20	even	12		1014.2.i.h.361.1		12
39.23	odd	6		1014.2.e.l.529.3		6
39.29	odd	6		1014.2.e.n.529.1		6
39.32	even	12		1014.2.i.h.361.6		12
39.35	odd	6		1014.2.e.n.991.1		6
39.38	odd	2		1014.2.a.n.1.3	yes	3
156.155	even	2		8112.2.a.cm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.l.1.1	✓	3	3.2	odd	2
1014.2.a.n.1.3	yes	3	39.38	odd	2
1014.2.b.f.337.1		6	39.8	even	4
1014.2.b.f.337.6		6	39.5	even	4
1014.2.e.l.529.3		6	39.23	odd	6
1014.2.e.l.991.3		6	39.17	odd	6
1014.2.e.n.529.1		6	39.29	odd	6
1014.2.e.n.991.1		6	39.35	odd	6
1014.2.i.h.361.1		12	39.20	even	12
1014.2.i.h.361.6		12	39.32	even	12
1014.2.i.h.823.3		12	39.2	even	12
1014.2.i.h.823.4		12	39.11	even	12
3042.2.a.ba.1.1		3	13.12	even	2
3042.2.a.bh.1.3		3	1.1	even	1	trivial
3042.2.b.o.1351.1		6	13.5	odd	4
3042.2.b.o.1351.6		6	13.8	odd	4
8112.2.a.cj.1.1		3	12.11	even	2
8112.2.a.cm.1.3		3	156.155	even	2

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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

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3042.2.a.bh.1.3

Level	$3042$
Weight	$2$
Character	3042.1
Self dual	yes
Analytic conductor	$24.290$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$