LMFDB - Embedded newform 3042.2.a.bi.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 = [3,3,0,3,2,0,4,3,0,2,3,0,0,4,0,3,-5] 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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.60388 q^{5} +1.10992 q^{7} +1.00000 q^{8} +3.60388 q^{10} +2.35690 q^{11} +1.10992 q^{14} +1.00000 q^{16} -5.96077 q^{17} +0.911854 q^{19} +3.60388 q^{20} +2.35690 q^{22} -3.38404 q^{23} +7.98792 q^{25} +1.10992 q^{28} +3.78017 q^{29} +8.49396 q^{31} +1.00000 q^{32} -5.96077 q^{34} +4.00000 q^{35} +4.89008 q^{37} +0.911854 q^{38} +3.60388 q^{40} +7.18598 q^{41} -0.515729 q^{43} +2.35690 q^{44} -3.38404 q^{46} -6.98792 q^{47} -5.76809 q^{49} +7.98792 q^{50} +3.38404 q^{53} +8.49396 q^{55} +1.10992 q^{56} +3.78017 q^{58} -10.1468 q^{59} -0.439665 q^{61} +8.49396 q^{62} +1.00000 q^{64} +2.14675 q^{67} -5.96077 q^{68} +4.00000 q^{70} -0.615957 q^{71} +6.32304 q^{73} +4.89008 q^{74} +0.911854 q^{76} +2.61596 q^{77} -15.4819 q^{79} +3.60388 q^{80} +7.18598 q^{82} +0.911854 q^{83} -21.4819 q^{85} -0.515729 q^{86} +2.35690 q^{88} -3.75063 q^{89} -3.38404 q^{92} -6.98792 q^{94} +3.28621 q^{95} -14.6746 q^{97} -5.76809 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} + 2 q^{10} + 3 q^{11} + 4 q^{14} + 3 q^{16} - 5 q^{17} - q^{19} + 2 q^{20} + 3 q^{22} + 5 q^{25} + 4 q^{28} + 10 q^{29} + 16 q^{31} + 3 q^{32} - 5 q^{34}+ \cdots + 3 q^{98}+O(q^{100})$ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 + 2 * q^10 + 3 * q^11 + 4 * q^14 + 3 * q^16 - 5 * q^17 - q^19 + 2 * q^20 + 3 * q^22 + 5 * q^25 + 4 * q^28 + 10 * q^29 + 16 * q^31 + 3 * q^32 - 5 * q^34 + 12 * q^35 + 14 * q^37 - q^38 + 2 * q^40 + 7 * q^41 + 11 * q^43 + 3 * q^44 - 2 * q^47 + 3 * q^49 + 5 * q^50 + 16 * q^55 + 4 * q^56 + 10 * q^58 - 3 * q^59 - 4 * q^61 + 16 * q^62 + 3 * q^64 - 21 * q^67 - 5 * q^68 + 12 * q^70 - 12 * q^71 - q^73 + 14 * q^74 - q^76 + 18 * q^77 - 18 * q^79 + 2 * q^80 + 7 * q^82 - q^83 - 36 * q^85 + 11 * q^86 + 3 * q^88 + 25 * q^89 - 2 * q^94 + 18 * q^95 - 23 * q^97 + 3 * 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
$5$	3.60388	1.61170	0.805851	−	0.592118i	$-0.201708\pi$
$5$	3.60388	1.61170	0.805851	+	0.592118i	$0.201708\pi$
$6$	0	0
$7$	1.10992	0.419509	0.209754	−	0.977754i	$-0.432734\pi$
$7$	1.10992	0.419509	0.209754	+	0.977754i	$0.432734\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.60388	1.13965
$11$	2.35690	0.710631	0.355315	−	0.934746i	$-0.384373\pi$
$11$	2.35690	0.710631	0.355315	+	0.934746i	$0.384373\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.10992	0.296638
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	0	0
$19$	0.911854	0.209194	0.104597	−	0.994515i	$-0.466645\pi$
$19$	0.911854	0.209194	0.104597	+	0.994515i	$0.466645\pi$
$20$	3.60388	0.805851
$21$	0	0
$22$	2.35690	0.502492
$23$	−3.38404	−0.705622	−0.352811	−	0.935695i	$-0.614774\pi$
$23$	−3.38404	−0.705622	−0.352811	+	0.935695i	$0.614774\pi$
$24$	0	0
$25$	7.98792	1.59758
$26$	0	0
$27$	0	0
$28$	1.10992	0.209754
$29$	3.78017	0.701959	0.350980	−	0.936383i	$-0.385849\pi$
$29$	3.78017	0.701959	0.350980	+	0.936383i	$0.385849\pi$
$30$	0	0
$31$	8.49396	1.52556	0.762780	−	0.646658i	$-0.223834\pi$
$31$	8.49396	1.52556	0.762780	+	0.646658i	$0.223834\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.96077	−1.02226
$35$	4.00000	0.676123
$36$	0	0
$37$	4.89008	0.803925	0.401962	−	0.915656i	$-0.368328\pi$
$37$	4.89008	0.803925	0.401962	+	0.915656i	$0.368328\pi$
$38$	0.911854	0.147922
$39$	0	0
$40$	3.60388	0.569823
$41$	7.18598	1.12226	0.561131	−	0.827727i	$-0.310366\pi$
$41$	7.18598	1.12226	0.561131	+	0.827727i	$0.310366\pi$
$42$	0	0
$43$	−0.515729	−0.0786480	−0.0393240	−	0.999227i	$-0.512520\pi$
$43$	−0.515729	−0.0786480	−0.0393240	+	0.999227i	$0.512520\pi$
$44$	2.35690	0.355315
$45$	0	0
$46$	−3.38404	−0.498950
$47$	−6.98792	−1.01929	−0.509646	−	0.860384i	$-0.670224\pi$
$47$	−6.98792	−1.01929	−0.509646	+	0.860384i	$0.670224\pi$
$48$	0	0
$49$	−5.76809	−0.824012
$50$	7.98792	1.12966
$51$	0	0
$52$	0	0
$53$	3.38404	0.464834	0.232417	−	0.972616i	$-0.425337\pi$
$53$	3.38404	0.464834	0.232417	+	0.972616i	$0.425337\pi$
$54$	0	0
$55$	8.49396	1.14533
$56$	1.10992	0.148319
$57$	0	0
$58$	3.78017	0.496360
$59$	−10.1468	−1.32099	−0.660497	−	0.750828i	$-0.729655\pi$
$59$	−10.1468	−1.32099	−0.660497	+	0.750828i	$0.729655\pi$
$60$	0	0
$61$	−0.439665	−0.0562933	−0.0281467	−	0.999604i	$-0.508961\pi$
$61$	−0.439665	−0.0562933	−0.0281467	+	0.999604i	$0.508961\pi$
$62$	8.49396	1.07873
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.14675	0.262268	0.131134	−	0.991365i	$-0.458138\pi$
$67$	2.14675	0.262268	0.131134	+	0.991365i	$0.458138\pi$
$68$	−5.96077	−0.722850
$69$	0	0
$70$	4.00000	0.478091
$71$	−0.615957	−0.0731007	−0.0365503	−	0.999332i	$-0.511637\pi$
$71$	−0.615957	−0.0731007	−0.0365503	+	0.999332i	$0.511637\pi$
$72$	0	0
$73$	6.32304	0.740056	0.370028	−	0.929021i	$-0.379348\pi$
$73$	6.32304	0.740056	0.370028	+	0.929021i	$0.379348\pi$
$74$	4.89008	0.568461
$75$	0	0
$76$	0.911854	0.104597
$77$	2.61596	0.298116
$78$	0	0
$79$	−15.4819	−1.74185	−0.870924	−	0.491418i	$-0.836479\pi$
$79$	−15.4819	−1.74185	−0.870924	+	0.491418i	$0.836479\pi$
$80$	3.60388	0.402926
$81$	0	0
$82$	7.18598	0.793559
$83$	0.911854	0.100089	0.0500445	−	0.998747i	$-0.484064\pi$
$83$	0.911854	0.100089	0.0500445	+	0.998747i	$0.484064\pi$
$84$	0	0
$85$	−21.4819	−2.33004
$86$	−0.515729	−0.0556125
$87$	0	0
$88$	2.35690	0.251246
$89$	−3.75063	−0.397566	−0.198783	−	0.980044i	$-0.563699\pi$
$89$	−3.75063	−0.397566	−0.198783	+	0.980044i	$0.563699\pi$
$90$	0	0
$91$	0	0
$92$	−3.38404	−0.352811
$93$	0	0
$94$	−6.98792	−0.720749
$95$	3.28621	0.337158
$96$	0	0
$97$	−14.6746	−1.48998	−0.744988	−	0.667078i	$-0.767545\pi$
$97$	−14.6746	−1.48998	−0.744988	+	0.667078i	$0.767545\pi$
$98$	−5.76809	−0.582665
$99$	0	0
$100$	7.98792	0.798792
$101$	−8.76809	−0.872457	−0.436229	−	0.899836i	$-0.643686\pi$
$101$	−8.76809	−0.872457	−0.436229	+	0.899836i	$0.643686\pi$
$102$	0	0
$103$	18.8116	1.85356	0.926782	−	0.375599i	$-0.122563\pi$
$103$	18.8116	1.85356	0.926782	+	0.375599i	$0.122563\pi$
$104$	0	0
$105$	0	0
$106$	3.38404	0.328687
$107$	−18.0519	−1.74514	−0.872572	−	0.488486i	$-0.837549\pi$
$107$	−18.0519	−1.74514	−0.872572	+	0.488486i	$0.837549\pi$
$108$	0	0
$109$	6.09783	0.584067	0.292033	−	0.956408i	$-0.405668\pi$
$109$	6.09783	0.584067	0.292033	+	0.956408i	$0.405668\pi$
$110$	8.49396	0.809867
$111$	0	0
$112$	1.10992	0.104877
$113$	12.2010	1.14778	0.573889	−	0.818933i	$-0.305434\pi$
$113$	12.2010	1.14778	0.573889	+	0.818933i	$0.305434\pi$
$114$	0	0
$115$	−12.1957	−1.13725
$116$	3.78017	0.350980
$117$	0	0
$118$	−10.1468	−0.934084
$119$	−6.61596	−0.606484
$120$	0	0
$121$	−5.44504	−0.495004
$122$	−0.439665	−0.0398054
$123$	0	0
$124$	8.49396	0.762780
$125$	10.7681	0.963127
$126$	0	0
$127$	11.4276	1.01403	0.507017	−	0.861936i	$-0.330748\pi$
$127$	11.4276	1.01403	0.507017	+	0.861936i	$0.330748\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−2.29590	−0.200593	−0.100297	−	0.994958i	$-0.531979\pi$
$131$	−2.29590	−0.200593	−0.100297	+	0.994958i	$0.531979\pi$
$132$	0	0
$133$	1.01208	0.0877586
$134$	2.14675	0.185451
$135$	0	0
$136$	−5.96077	−0.511132
$137$	−9.08038	−0.775789	−0.387894	−	0.921704i	$-0.626797\pi$
$137$	−9.08038	−0.775789	−0.387894	+	0.921704i	$0.626797\pi$
$138$	0	0
$139$	18.9051	1.60351	0.801757	−	0.597650i	$-0.203899\pi$
$139$	18.9051	1.60351	0.801757	+	0.597650i	$0.203899\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−0.615957	−0.0516900
$143$	0	0
$144$	0	0
$145$	13.6233	1.13135
$146$	6.32304	0.523299
$147$	0	0
$148$	4.89008	0.401962
$149$	−18.6896	−1.53111	−0.765557	−	0.643368i	$-0.777537\pi$
$149$	−18.6896	−1.53111	−0.765557	+	0.643368i	$0.777537\pi$
$150$	0	0
$151$	−0.317667	−0.0258514	−0.0129257	−	0.999916i	$-0.504114\pi$
$151$	−0.317667	−0.0258514	−0.0129257	+	0.999916i	$0.504114\pi$
$152$	0.911854	0.0739611
$153$	0	0
$154$	2.61596	0.210800
$155$	30.6112	2.45875
$156$	0	0
$157$	18.8901	1.50759	0.753796	−	0.657108i	$-0.228220\pi$
$157$	18.8901	1.50759	0.753796	+	0.657108i	$0.228220\pi$
$158$	−15.4819	−1.23167
$159$	0	0
$160$	3.60388	0.284911
$161$	−3.75600	−0.296015
$162$	0	0
$163$	−4.33273	−0.339366	−0.169683	−	0.985499i	$-0.554274\pi$
$163$	−4.33273	−0.339366	−0.169683	+	0.985499i	$0.554274\pi$
$164$	7.18598	0.561131
$165$	0	0
$166$	0.911854	0.0707736
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	0	0
$170$	−21.4819	−1.64758
$171$	0	0
$172$	−0.515729	−0.0393240
$173$	10.9879	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$173$	10.9879	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$174$	0	0
$175$	8.86592	0.670201
$176$	2.35690	0.177658
$177$	0	0
$178$	−3.75063	−0.281121
$179$	−4.65519	−0.347945	−0.173972	−	0.984751i	$-0.555660\pi$
$179$	−4.65519	−0.347945	−0.173972	+	0.984751i	$0.555660\pi$
$180$	0	0
$181$	−1.06638	−0.0792631	−0.0396315	−	0.999214i	$-0.512618\pi$
$181$	−1.06638	−0.0792631	−0.0396315	+	0.999214i	$0.512618\pi$
$182$	0	0
$183$	0	0
$184$	−3.38404	−0.249475
$185$	17.6233	1.29569
$186$	0	0
$187$	−14.0489	−1.02736
$188$	−6.98792	−0.509646
$189$	0	0
$190$	3.28621	0.238407
$191$	−0.890084	−0.0644042	−0.0322021	−	0.999481i	$-0.510252\pi$
$191$	−0.890084	−0.0644042	−0.0322021	+	0.999481i	$0.510252\pi$
$192$	0	0
$193$	16.2174	1.16736	0.583678	−	0.811985i	$-0.301613\pi$
$193$	16.2174	1.16736	0.583678	+	0.811985i	$0.301613\pi$
$194$	−14.6746	−1.05357
$195$	0	0
$196$	−5.76809	−0.412006
$197$	11.4711	0.817284	0.408642	−	0.912695i	$-0.366003\pi$
$197$	11.4711	0.817284	0.408642	+	0.912695i	$0.366003\pi$
$198$	0	0
$199$	−3.79954	−0.269343	−0.134671	−	0.990890i	$-0.542998\pi$
$199$	−3.79954	−0.269343	−0.134671	+	0.990890i	$0.542998\pi$
$200$	7.98792	0.564831
$201$	0	0
$202$	−8.76809	−0.616920
$203$	4.19567	0.294478
$204$	0	0
$205$	25.8974	1.80875
$206$	18.8116	1.31067
$207$	0	0
$208$	0	0
$209$	2.14914	0.148659
$210$	0	0
$211$	−25.0465	−1.72427	−0.862137	−	0.506675i	$-0.830874\pi$
$211$	−25.0465	−1.72427	−0.862137	+	0.506675i	$0.830874\pi$
$212$	3.38404	0.232417
$213$	0	0
$214$	−18.0519	−1.23400
$215$	−1.85862	−0.126757
$216$	0	0
$217$	9.42758	0.639986
$218$	6.09783	0.412997
$219$	0	0
$220$	8.49396	0.572663
$221$	0	0
$222$	0	0
$223$	12.9879	0.869735	0.434868	−	0.900494i	$-0.356795\pi$
$223$	12.9879	0.869735	0.434868	+	0.900494i	$0.356795\pi$
$224$	1.10992	0.0741594
$225$	0	0
$226$	12.2010	0.811602
$227$	13.8049	0.916265	0.458132	−	0.888884i	$-0.348519\pi$
$227$	13.8049	0.916265	0.458132	+	0.888884i	$0.348519\pi$
$228$	0	0
$229$	−11.5603	−0.763928	−0.381964	−	0.924177i	$-0.624752\pi$
$229$	−11.5603	−0.763928	−0.381964	+	0.924177i	$0.624752\pi$
$230$	−12.1957	−0.804159
$231$	0	0
$232$	3.78017	0.248180
$233$	−9.77479	−0.640368	−0.320184	−	0.947355i	$-0.603745\pi$
$233$	−9.77479	−0.640368	−0.320184	+	0.947355i	$0.603745\pi$
$234$	0	0
$235$	−25.1836	−1.64280
$236$	−10.1468	−0.660497
$237$	0	0
$238$	−6.61596	−0.428849
$239$	0.944378	0.0610867	0.0305434	−	0.999533i	$-0.490276\pi$
$239$	0.944378	0.0610867	0.0305434	+	0.999533i	$0.490276\pi$
$240$	0	0
$241$	−0.219833	−0.0141607	−0.00708033	−	0.999975i	$-0.502254\pi$
$241$	−0.219833	−0.0141607	−0.00708033	+	0.999975i	$0.502254\pi$
$242$	−5.44504	−0.350021
$243$	0	0
$244$	−0.439665	−0.0281467
$245$	−20.7875	−1.32806
$246$	0	0
$247$	0	0
$248$	8.49396	0.539367
$249$	0	0
$250$	10.7681	0.681034
$251$	−16.2543	−1.02596	−0.512980	−	0.858400i	$-0.671459\pi$
$251$	−16.2543	−1.02596	−0.512980	+	0.858400i	$0.671459\pi$
$252$	0	0
$253$	−7.97584	−0.501437
$254$	11.4276	0.717030
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.4373	1.39960	0.699799	−	0.714340i	$-0.253273\pi$
$257$	22.4373	1.39960	0.699799	+	0.714340i	$0.253273\pi$
$258$	0	0
$259$	5.42758	0.337254
$260$	0	0
$261$	0	0
$262$	−2.29590	−0.141841
$263$	10.4940	0.647085	0.323543	−	0.946214i	$-0.395126\pi$
$263$	10.4940	0.647085	0.323543	+	0.946214i	$0.395126\pi$
$264$	0	0
$265$	12.1957	0.749174
$266$	1.01208	0.0620547
$267$	0	0
$268$	2.14675	0.131134
$269$	−26.4155	−1.61058	−0.805291	−	0.592880i	$-0.797991\pi$
$269$	−26.4155	−1.61058	−0.805291	+	0.592880i	$0.797991\pi$
$270$	0	0
$271$	−22.0301	−1.33824	−0.669118	−	0.743157i	$-0.733328\pi$
$271$	−22.0301	−1.33824	−0.669118	+	0.743157i	$0.733328\pi$
$272$	−5.96077	−0.361425
$273$	0	0
$274$	−9.08038	−0.548566
$275$	18.8267	1.13529
$276$	0	0
$277$	2.17629	0.130761	0.0653804	−	0.997860i	$-0.479174\pi$
$277$	2.17629	0.130761	0.0653804	+	0.997860i	$0.479174\pi$
$278$	18.9051	1.13386
$279$	0	0
$280$	4.00000	0.239046
$281$	−25.0030	−1.49155	−0.745776	−	0.666196i	$-0.767921\pi$
$281$	−25.0030	−1.49155	−0.745776	+	0.666196i	$0.767921\pi$
$282$	0	0
$283$	−16.3153	−0.969842	−0.484921	−	0.874558i	$-0.661152\pi$
$283$	−16.3153	−0.969842	−0.484921	+	0.874558i	$0.661152\pi$
$284$	−0.615957	−0.0365503
$285$	0	0
$286$	0	0
$287$	7.97584	0.470799
$288$	0	0
$289$	18.5308	1.09005
$290$	13.6233	0.799985
$291$	0	0
$292$	6.32304	0.370028
$293$	1.87800	0.109714	0.0548570	−	0.998494i	$-0.482530\pi$
$293$	1.87800	0.109714	0.0548570	+	0.998494i	$0.482530\pi$
$294$	0	0
$295$	−36.5676	−2.12905
$296$	4.89008	0.284230
$297$	0	0
$298$	−18.6896	−1.08266
$299$	0	0
$300$	0	0
$301$	−0.572417	−0.0329935
$302$	−0.317667	−0.0182797
$303$	0	0
$304$	0.911854	0.0522984
$305$	−1.58450	−0.0907281
$306$	0	0
$307$	−23.9801	−1.36862	−0.684310	−	0.729192i	$-0.739896\pi$
$307$	−23.9801	−1.36862	−0.684310	+	0.729192i	$0.739896\pi$
$308$	2.61596	0.149058
$309$	0	0
$310$	30.6112	1.73860
$311$	5.38404	0.305301	0.152651	−	0.988280i	$-0.451219\pi$
$311$	5.38404	0.305301	0.152651	+	0.988280i	$0.451219\pi$
$312$	0	0
$313$	18.9487	1.07104	0.535522	−	0.844522i	$-0.320115\pi$
$313$	18.9487	1.07104	0.535522	+	0.844522i	$0.320115\pi$
$314$	18.8901	1.06603
$315$	0	0
$316$	−15.4819	−0.870924
$317$	11.5013	0.645975	0.322987	−	0.946403i	$-0.395313\pi$
$317$	11.5013	0.645975	0.322987	+	0.946403i	$0.395313\pi$
$318$	0	0
$319$	8.90946	0.498834
$320$	3.60388	0.201463
$321$	0	0
$322$	−3.75600	−0.209314
$323$	−5.43535	−0.302431
$324$	0	0
$325$	0	0
$326$	−4.33273	−0.239968
$327$	0	0
$328$	7.18598	0.396779
$329$	−7.75600	−0.427602
$330$	0	0
$331$	−34.6112	−1.90240	−0.951201	−	0.308572i	$-0.900149\pi$
$331$	−34.6112	−1.90240	−0.951201	+	0.308572i	$0.900149\pi$
$332$	0.911854	0.0500445
$333$	0	0
$334$	−14.0000	−0.766046
$335$	7.73663	0.422697
$336$	0	0
$337$	1.95407	0.106445	0.0532224	−	0.998583i	$-0.483051\pi$
$337$	1.95407	0.106445	0.0532224	+	0.998583i	$0.483051\pi$
$338$	0	0
$339$	0	0
$340$	−21.4819	−1.16502
$341$	20.0194	1.08411
$342$	0	0
$343$	−14.1715	−0.765189
$344$	−0.515729	−0.0278063
$345$	0	0
$346$	10.9879	0.590714
$347$	6.41550	0.344402	0.172201	−	0.985062i	$-0.444912\pi$
$347$	6.41550	0.344402	0.172201	+	0.985062i	$0.444912\pi$
$348$	0	0
$349$	−1.08575	−0.0581190	−0.0290595	−	0.999578i	$-0.509251\pi$
$349$	−1.08575	−0.0581190	−0.0290595	+	0.999578i	$0.509251\pi$
$350$	8.86592	0.473903
$351$	0	0
$352$	2.35690	0.125623
$353$	4.28919	0.228291	0.114145	−	0.993464i	$-0.463587\pi$
$353$	4.28919	0.228291	0.114145	+	0.993464i	$0.463587\pi$
$354$	0	0
$355$	−2.21983	−0.117816
$356$	−3.75063	−0.198783
$357$	0	0
$358$	−4.65519	−0.246034
$359$	−15.5060	−0.818378	−0.409189	−	0.912450i	$-0.634188\pi$
$359$	−15.5060	−0.818378	−0.409189	+	0.912450i	$0.634188\pi$
$360$	0	0
$361$	−18.1685	−0.956238
$362$	−1.06638	−0.0560475
$363$	0	0
$364$	0	0
$365$	22.7875	1.19275
$366$	0	0
$367$	17.4276	0.909712	0.454856	−	0.890565i	$-0.349691\pi$
$367$	17.4276	0.909712	0.454856	+	0.890565i	$0.349691\pi$
$368$	−3.38404	−0.176405
$369$	0	0
$370$	17.6233	0.916189
$371$	3.75600	0.195002
$372$	0	0
$373$	8.19567	0.424356	0.212178	−	0.977231i	$-0.431944\pi$
$373$	8.19567	0.424356	0.212178	+	0.977231i	$0.431944\pi$
$374$	−14.0489	−0.726452
$375$	0	0
$376$	−6.98792	−0.360374
$377$	0	0
$378$	0	0
$379$	15.0476	0.772943	0.386471	−	0.922301i	$-0.373694\pi$
$379$	15.0476	0.772943	0.386471	+	0.922301i	$0.373694\pi$
$380$	3.28621	0.168579
$381$	0	0
$382$	−0.890084	−0.0455406
$383$	−11.1207	−0.568240	−0.284120	−	0.958789i	$-0.591701\pi$
$383$	−11.1207	−0.568240	−0.284120	+	0.958789i	$0.591701\pi$
$384$	0	0
$385$	9.42758	0.480474
$386$	16.2174	0.825446
$387$	0	0
$388$	−14.6746	−0.744988
$389$	8.04354	0.407824	0.203912	−	0.978989i	$-0.434634\pi$
$389$	8.04354	0.407824	0.203912	+	0.978989i	$0.434634\pi$
$390$	0	0
$391$	20.1715	1.02012
$392$	−5.76809	−0.291332
$393$	0	0
$394$	11.4711	0.577907
$395$	−55.7948	−2.80734
$396$	0	0
$397$	21.9081	1.09954	0.549769	−	0.835317i	$-0.314716\pi$
$397$	21.9081	1.09954	0.549769	+	0.835317i	$0.314716\pi$
$398$	−3.79954	−0.190454
$399$	0	0
$400$	7.98792	0.399396
$401$	17.4426	0.871044	0.435522	−	0.900178i	$-0.356564\pi$
$401$	17.4426	0.871044	0.435522	+	0.900178i	$0.356564\pi$
$402$	0	0
$403$	0	0
$404$	−8.76809	−0.436229
$405$	0	0
$406$	4.19567	0.208228
$407$	11.5254	0.571294
$408$	0	0
$409$	17.4330	0.862004	0.431002	−	0.902351i	$-0.358160\pi$
$409$	17.4330	0.862004	0.431002	+	0.902351i	$0.358160\pi$
$410$	25.8974	1.27898
$411$	0	0
$412$	18.8116	0.926782
$413$	−11.2620	−0.554169
$414$	0	0
$415$	3.28621	0.161314
$416$	0	0
$417$	0	0
$418$	2.14914	0.105118
$419$	−9.97584	−0.487352	−0.243676	−	0.969857i	$-0.578353\pi$
$419$	−9.97584	−0.487352	−0.243676	+	0.969857i	$0.578353\pi$
$420$	0	0
$421$	−0.615957	−0.0300199	−0.0150100	−	0.999887i	$-0.504778\pi$
$421$	−0.615957	−0.0300199	−0.0150100	+	0.999887i	$0.504778\pi$
$422$	−25.0465	−1.21925
$423$	0	0
$424$	3.38404	0.164344
$425$	−47.6142	−2.30963
$426$	0	0
$427$	−0.487991	−0.0236156
$428$	−18.0519	−0.872572
$429$	0	0
$430$	−1.85862	−0.0896308
$431$	−14.7922	−0.712518	−0.356259	−	0.934387i	$-0.615948\pi$
$431$	−14.7922	−0.712518	−0.356259	+	0.934387i	$0.615948\pi$
$432$	0	0
$433$	16.5321	0.794483	0.397242	−	0.917714i	$-0.369968\pi$
$433$	16.5321	0.794483	0.397242	+	0.917714i	$0.369968\pi$
$434$	9.42758	0.452538
$435$	0	0
$436$	6.09783	0.292033
$437$	−3.08575	−0.147612
$438$	0	0
$439$	3.50125	0.167106	0.0835529	−	0.996503i	$-0.473373\pi$
$439$	3.50125	0.167106	0.0835529	+	0.996503i	$0.473373\pi$
$440$	8.49396	0.404934
$441$	0	0
$442$	0	0
$443$	−17.4077	−0.827066	−0.413533	−	0.910489i	$-0.635705\pi$
$443$	−17.4077	−0.827066	−0.413533	+	0.910489i	$0.635705\pi$
$444$	0	0
$445$	−13.5168	−0.640758
$446$	12.9879	0.614996
$447$	0	0
$448$	1.10992	0.0524386
$449$	34.1497	1.61163	0.805813	−	0.592170i	$-0.201729\pi$
$449$	34.1497	1.61163	0.805813	+	0.592170i	$0.201729\pi$
$450$	0	0
$451$	16.9366	0.797514
$452$	12.2010	0.573889
$453$	0	0
$454$	13.8049	0.647897
$455$	0	0
$456$	0	0
$457$	−9.40342	−0.439873	−0.219937	−	0.975514i	$-0.570585\pi$
$457$	−9.40342	−0.439873	−0.219937	+	0.975514i	$0.570585\pi$
$458$	−11.5603	−0.540179
$459$	0	0
$460$	−12.1957	−0.568626
$461$	−0.733169	−0.0341471	−0.0170735	−	0.999854i	$-0.505435\pi$
$461$	−0.733169	−0.0341471	−0.0170735	+	0.999854i	$0.505435\pi$
$462$	0	0
$463$	−7.24267	−0.336595	−0.168298	−	0.985736i	$-0.553827\pi$
$463$	−7.24267	−0.336595	−0.168298	+	0.985736i	$0.553827\pi$
$464$	3.78017	0.175490
$465$	0	0
$466$	−9.77479	−0.452808
$467$	−30.2446	−1.39955	−0.699776	−	0.714362i	$-0.746717\pi$
$467$	−30.2446	−1.39955	−0.699776	+	0.714362i	$0.746717\pi$
$468$	0	0
$469$	2.38271	0.110024
$470$	−25.1836	−1.16163
$471$	0	0
$472$	−10.1468	−0.467042
$473$	−1.21552	−0.0558897
$474$	0	0
$475$	7.28382	0.334204
$476$	−6.61596	−0.303242
$477$	0	0
$478$	0.944378	0.0431948
$479$	−36.7198	−1.67777	−0.838884	−	0.544310i	$-0.816792\pi$
$479$	−36.7198	−1.67777	−0.838884	+	0.544310i	$0.816792\pi$
$480$	0	0
$481$	0	0
$482$	−0.219833	−0.0100131
$483$	0	0
$484$	−5.44504	−0.247502
$485$	−52.8853	−2.40140
$486$	0	0
$487$	28.6547	1.29847	0.649234	−	0.760588i	$-0.275089\pi$
$487$	28.6547	1.29847	0.649234	+	0.760588i	$0.275089\pi$
$488$	−0.439665	−0.0199027
$489$	0	0
$490$	−20.7875	−0.939082
$491$	−30.4295	−1.37326	−0.686632	−	0.727005i	$-0.740912\pi$
$491$	−30.4295	−1.37326	−0.686632	+	0.727005i	$0.740912\pi$
$492$	0	0
$493$	−22.5327	−1.01482
$494$	0	0
$495$	0	0
$496$	8.49396	0.381390
$497$	−0.683661	−0.0306664
$498$	0	0
$499$	−15.9715	−0.714984	−0.357492	−	0.933916i	$-0.616368\pi$
$499$	−15.9715	−0.714984	−0.357492	+	0.933916i	$0.616368\pi$
$500$	10.7681	0.481563
$501$	0	0
$502$	−16.2543	−0.725464
$503$	41.9711	1.87140	0.935698	−	0.352801i	$-0.114771\pi$
$503$	41.9711	1.87140	0.935698	+	0.352801i	$0.114771\pi$
$504$	0	0
$505$	−31.5991	−1.40614
$506$	−7.97584	−0.354569
$507$	0	0
$508$	11.4276	0.507017
$509$	0.914247	0.0405233	0.0202616	−	0.999795i	$-0.493550\pi$
$509$	0.914247	0.0405233	0.0202616	+	0.999795i	$0.493550\pi$
$510$	0	0
$511$	7.01805	0.310460
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.4373	0.989666
$515$	67.7948	2.98739
$516$	0	0
$517$	−16.4698	−0.724341
$518$	5.42758	0.238474
$519$	0	0
$520$	0	0
$521$	3.31096	0.145056	0.0725279	−	0.997366i	$-0.476893\pi$
$521$	3.31096	0.145056	0.0725279	+	0.997366i	$0.476893\pi$
$522$	0	0
$523$	−0.850855	−0.0372053	−0.0186026	−	0.999827i	$-0.505922\pi$
$523$	−0.850855	−0.0372053	−0.0186026	+	0.999827i	$0.505922\pi$
$524$	−2.29590	−0.100297
$525$	0	0
$526$	10.4940	0.457558
$527$	−50.6305	−2.20550
$528$	0	0
$529$	−11.5483	−0.502098
$530$	12.1957	0.529746
$531$	0	0
$532$	1.01208	0.0438793
$533$	0	0
$534$	0	0
$535$	−65.0568	−2.81265
$536$	2.14675	0.0927256
$537$	0	0
$538$	−26.4155	−1.13885
$539$	−13.5948	−0.585569
$540$	0	0
$541$	40.8853	1.75780	0.878898	−	0.477010i	$-0.158279\pi$
$541$	40.8853	1.75780	0.878898	+	0.477010i	$0.158279\pi$
$542$	−22.0301	−0.946275
$543$	0	0
$544$	−5.96077	−0.255566
$545$	21.9758	0.941341
$546$	0	0
$547$	−2.39075	−0.102221	−0.0511105	−	0.998693i	$-0.516276\pi$
$547$	−2.39075	−0.102221	−0.0511105	+	0.998693i	$0.516276\pi$
$548$	−9.08038	−0.387894
$549$	0	0
$550$	18.8267	0.802773
$551$	3.44696	0.146845
$552$	0	0
$553$	−17.1836	−0.730720
$554$	2.17629	0.0924618
$555$	0	0
$556$	18.9051	0.801757
$557$	27.1508	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$557$	27.1508	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−25.0030	−1.05469
$563$	6.52409	0.274958	0.137479	−	0.990505i	$-0.456100\pi$
$563$	6.52409	0.274958	0.137479	+	0.990505i	$0.456100\pi$
$564$	0	0
$565$	43.9711	1.84988
$566$	−16.3153	−0.685782
$567$	0	0
$568$	−0.615957	−0.0258450
$569$	−7.30021	−0.306041	−0.153020	−	0.988223i	$-0.548900\pi$
$569$	−7.30021	−0.306041	−0.153020	+	0.988223i	$0.548900\pi$
$570$	0	0
$571$	−43.6722	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$571$	−43.6722	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$572$	0	0
$573$	0	0
$574$	7.97584	0.332905
$575$	−27.0315	−1.12729
$576$	0	0
$577$	−16.8528	−0.701590	−0.350795	−	0.936452i	$-0.614089\pi$
$577$	−16.8528	−0.701590	−0.350795	+	0.936452i	$0.614089\pi$
$578$	18.5308	0.770779
$579$	0	0
$580$	13.6233	0.565675
$581$	1.01208	0.0419882
$582$	0	0
$583$	7.97584	0.330325
$584$	6.32304	0.261649
$585$	0	0
$586$	1.87800	0.0775796
$587$	−22.1825	−0.915571	−0.457785	−	0.889063i	$-0.651357\pi$
$587$	−22.1825	−0.915571	−0.457785	+	0.889063i	$0.651357\pi$
$588$	0	0
$589$	7.74525	0.319137
$590$	−36.5676	−1.50547
$591$	0	0
$592$	4.89008	0.200981
$593$	3.98493	0.163642	0.0818208	−	0.996647i	$-0.473926\pi$
$593$	3.98493	0.163642	0.0818208	+	0.996647i	$0.473926\pi$
$594$	0	0
$595$	−23.8431	−0.977471
$596$	−18.6896	−0.765557
$597$	0	0
$598$	0	0
$599$	−33.2379	−1.35806	−0.679032	−	0.734109i	$-0.737600\pi$
$599$	−33.2379	−1.35806	−0.679032	+	0.734109i	$0.737600\pi$
$600$	0	0
$601$	9.79715	0.399634	0.199817	−	0.979833i	$-0.435965\pi$
$601$	9.79715	0.399634	0.199817	+	0.979833i	$0.435965\pi$
$602$	−0.572417	−0.0233300
$603$	0	0
$604$	−0.317667	−0.0129257
$605$	−19.6233	−0.797799
$606$	0	0
$607$	−24.2258	−0.983295	−0.491647	−	0.870794i	$-0.663605\pi$
$607$	−24.2258	−0.983295	−0.491647	+	0.870794i	$0.663605\pi$
$608$	0.911854	0.0369806
$609$	0	0
$610$	−1.58450	−0.0641545
$611$	0	0
$612$	0	0
$613$	15.0556	0.608091	0.304045	−	0.952658i	$-0.401663\pi$
$613$	15.0556	0.608091	0.304045	+	0.952658i	$0.401663\pi$
$614$	−23.9801	−0.967760
$615$	0	0
$616$	2.61596	0.105400
$617$	−2.01879	−0.0812733	−0.0406366	−	0.999174i	$-0.512939\pi$
$617$	−2.01879	−0.0812733	−0.0406366	+	0.999174i	$0.512939\pi$
$618$	0	0
$619$	7.84309	0.315240	0.157620	−	0.987500i	$-0.449618\pi$
$619$	7.84309	0.315240	0.157620	+	0.987500i	$0.449618\pi$
$620$	30.6112	1.22937
$621$	0	0
$622$	5.38404	0.215880
$623$	−4.16288	−0.166782
$624$	0	0
$625$	−1.13275	−0.0453101
$626$	18.9487	0.757342
$627$	0	0
$628$	18.8901	0.753796
$629$	−29.1487	−1.16223
$630$	0	0
$631$	−24.5327	−0.976632	−0.488316	−	0.872667i	$-0.662389\pi$
$631$	−24.5327	−0.976632	−0.488316	+	0.872667i	$0.662389\pi$
$632$	−15.4819	−0.615836
$633$	0	0
$634$	11.5013	0.456773
$635$	41.1836	1.63432
$636$	0	0
$637$	0	0
$638$	8.90946	0.352729
$639$	0	0
$640$	3.60388	0.142456
$641$	41.6015	1.64316	0.821580	−	0.570093i	$-0.193093\pi$
$641$	41.6015	1.64316	0.821580	+	0.570093i	$0.193093\pi$
$642$	0	0
$643$	−45.4118	−1.79087	−0.895433	−	0.445196i	$-0.853134\pi$
$643$	−45.4118	−1.79087	−0.895433	+	0.445196i	$0.853134\pi$
$644$	−3.75600	−0.148007
$645$	0	0
$646$	−5.43535	−0.213851
$647$	−35.8345	−1.40880	−0.704399	−	0.709804i	$-0.748783\pi$
$647$	−35.8345	−1.40880	−0.704399	+	0.709804i	$0.748783\pi$
$648$	0	0
$649$	−23.9148	−0.938740
$650$	0	0
$651$	0	0
$652$	−4.33273	−0.169683
$653$	−18.5590	−0.726270	−0.363135	−	0.931737i	$-0.618294\pi$
$653$	−18.5590	−0.726270	−0.363135	+	0.931737i	$0.618294\pi$
$654$	0	0
$655$	−8.27413	−0.323297
$656$	7.18598	0.280565
$657$	0	0
$658$	−7.75600	−0.302361
$659$	3.97525	0.154854	0.0774268	−	0.996998i	$-0.475330\pi$
$659$	3.97525	0.154854	0.0774268	+	0.996998i	$0.475330\pi$
$660$	0	0
$661$	−1.23191	−0.0479159	−0.0239580	−	0.999713i	$-0.507627\pi$
$661$	−1.23191	−0.0479159	−0.0239580	+	0.999713i	$0.507627\pi$
$662$	−34.6112	−1.34520
$663$	0	0
$664$	0.911854	0.0353868
$665$	3.64742	0.141441
$666$	0	0
$667$	−12.7922	−0.495318
$668$	−14.0000	−0.541676
$669$	0	0
$670$	7.73663	0.298892
$671$	−1.03624	−0.0400038
$672$	0	0
$673$	−36.8256	−1.41952	−0.709762	−	0.704442i	$-0.751197\pi$
$673$	−36.8256	−1.41952	−0.709762	+	0.704442i	$0.751197\pi$
$674$	1.95407	0.0752678
$675$	0	0
$676$	0	0
$677$	25.9215	0.996246	0.498123	−	0.867106i	$-0.334023\pi$
$677$	25.9215	0.996246	0.498123	+	0.867106i	$0.334023\pi$
$678$	0	0
$679$	−16.2875	−0.625058
$680$	−21.4819	−0.823792
$681$	0	0
$682$	20.0194	0.766582
$683$	37.5472	1.43670	0.718352	−	0.695680i	$-0.244897\pi$
$683$	37.5472	1.43670	0.718352	+	0.695680i	$0.244897\pi$
$684$	0	0
$685$	−32.7245	−1.25034
$686$	−14.1715	−0.541071
$687$	0	0
$688$	−0.515729	−0.0196620
$689$	0	0
$690$	0	0
$691$	−45.2549	−1.72158	−0.860788	−	0.508963i	$-0.830029\pi$
$691$	−45.2549	−1.72158	−0.860788	+	0.508963i	$0.830029\pi$
$692$	10.9879	0.417698
$693$	0	0
$694$	6.41550	0.243529
$695$	68.1318	2.58439
$696$	0	0
$697$	−42.8340	−1.62245
$698$	−1.08575	−0.0410964
$699$	0	0
$700$	8.86592	0.335100
$701$	−36.0823	−1.36281	−0.681405	−	0.731907i	$-0.738631\pi$
$701$	−36.0823	−1.36281	−0.681405	+	0.731907i	$0.738631\pi$
$702$	0	0
$703$	4.45904	0.168176
$704$	2.35690	0.0888289
$705$	0	0
$706$	4.28919	0.161426
$707$	−9.73184	−0.366004
$708$	0	0
$709$	19.0664	0.716053	0.358026	−	0.933711i	$-0.383450\pi$
$709$	19.0664	0.716053	0.358026	+	0.933711i	$0.383450\pi$
$710$	−2.21983	−0.0833088
$711$	0	0
$712$	−3.75063	−0.140561
$713$	−28.7439	−1.07647
$714$	0	0
$715$	0	0
$716$	−4.65519	−0.173972
$717$	0	0
$718$	−15.5060	−0.578680
$719$	15.3056	0.570802	0.285401	−	0.958408i	$-0.407873\pi$
$719$	15.3056	0.570802	0.285401	+	0.958408i	$0.407873\pi$
$720$	0	0
$721$	20.8793	0.777587
$722$	−18.1685	−0.676162
$723$	0	0
$724$	−1.06638	−0.0396315
$725$	30.1957	1.12144
$726$	0	0
$727$	3.46250	0.128417	0.0642085	−	0.997937i	$-0.479548\pi$
$727$	3.46250	0.128417	0.0642085	+	0.997937i	$0.479548\pi$
$728$	0	0
$729$	0	0
$730$	22.7875	0.843402
$731$	3.07415	0.113701
$732$	0	0
$733$	26.0930	0.963769	0.481884	−	0.876235i	$-0.339953\pi$
$733$	26.0930	0.963769	0.481884	+	0.876235i	$0.339953\pi$
$734$	17.4276	0.643264
$735$	0	0
$736$	−3.38404	−0.124737
$737$	5.05967	0.186375
$738$	0	0
$739$	26.4993	0.974794	0.487397	−	0.873181i	$-0.337946\pi$
$739$	26.4993	0.974794	0.487397	+	0.873181i	$0.337946\pi$
$740$	17.6233	0.647844
$741$	0	0
$742$	3.75600	0.137887
$743$	−0.415502	−0.0152433	−0.00762164	−	0.999971i	$-0.502426\pi$
$743$	−0.415502	−0.0152433	−0.00762164	+	0.999971i	$0.502426\pi$
$744$	0	0
$745$	−67.3551	−2.46770
$746$	8.19567	0.300065
$747$	0	0
$748$	−14.0489	−0.513679
$749$	−20.0361	−0.732103
$750$	0	0
$751$	2.90946	0.106168	0.0530839	−	0.998590i	$-0.483095\pi$
$751$	2.90946	0.106168	0.0530839	+	0.998590i	$0.483095\pi$
$752$	−6.98792	−0.254823
$753$	0	0
$754$	0	0
$755$	−1.14483	−0.0416647
$756$	0	0
$757$	12.3720	0.449667	0.224833	−	0.974397i	$-0.427816\pi$
$757$	12.3720	0.449667	0.224833	+	0.974397i	$0.427816\pi$
$758$	15.0476	0.546553
$759$	0	0
$760$	3.28621	0.119203
$761$	42.4306	1.53811	0.769053	−	0.639184i	$-0.220728\pi$
$761$	42.4306	1.53811	0.769053	+	0.639184i	$0.220728\pi$
$762$	0	0
$763$	6.76809	0.245021
$764$	−0.890084	−0.0322021
$765$	0	0
$766$	−11.1207	−0.401806
$767$	0	0
$768$	0	0
$769$	−13.3341	−0.480839	−0.240419	−	0.970669i	$-0.577285\pi$
$769$	−13.3341	−0.480839	−0.240419	+	0.970669i	$0.577285\pi$
$770$	9.42758	0.339747
$771$	0	0
$772$	16.2174	0.583678
$773$	−5.85384	−0.210548	−0.105274	−	0.994443i	$-0.533572\pi$
$773$	−5.85384	−0.210548	−0.105274	+	0.994443i	$0.533572\pi$
$774$	0	0
$775$	67.8491	2.43721
$776$	−14.6746	−0.526786
$777$	0	0
$778$	8.04354	0.288375
$779$	6.55257	0.234770
$780$	0	0
$781$	−1.45175	−0.0519476
$782$	20.1715	0.721332
$783$	0	0
$784$	−5.76809	−0.206003
$785$	68.0775	2.42979
$786$	0	0
$787$	23.2965	0.830430	0.415215	−	0.909723i	$-0.363706\pi$
$787$	23.2965	0.830430	0.415215	+	0.909723i	$0.363706\pi$
$788$	11.4711	0.408642
$789$	0	0
$790$	−55.7948	−1.98509
$791$	13.5421	0.481503
$792$	0	0
$793$	0	0
$794$	21.9081	0.777491
$795$	0	0
$796$	−3.79954	−0.134671
$797$	−35.8103	−1.26847	−0.634233	−	0.773142i	$-0.718684\pi$
$797$	−35.8103	−1.26847	−0.634233	+	0.773142i	$0.718684\pi$
$798$	0	0
$799$	41.6534	1.47359
$800$	7.98792	0.282416
$801$	0	0
$802$	17.4426	0.615921
$803$	14.9028	0.525907
$804$	0	0
$805$	−13.5362	−0.477087
$806$	0	0
$807$	0	0
$808$	−8.76809	−0.308460
$809$	−28.3744	−0.997589	−0.498795	−	0.866720i	$-0.666224\pi$
$809$	−28.3744	−0.997589	−0.498795	+	0.866720i	$0.666224\pi$
$810$	0	0
$811$	−5.20344	−0.182717	−0.0913587	−	0.995818i	$-0.529121\pi$
$811$	−5.20344	−0.182717	−0.0913587	+	0.995818i	$0.529121\pi$
$812$	4.19567	0.147239
$813$	0	0
$814$	11.5254	0.403966
$815$	−15.6146	−0.546957
$816$	0	0
$817$	−0.470270	−0.0164527
$818$	17.4330	0.609529
$819$	0	0
$820$	25.8974	0.904376
$821$	−5.65338	−0.197304	−0.0986522	−	0.995122i	$-0.531453\pi$
$821$	−5.65338	−0.197304	−0.0986522	+	0.995122i	$0.531453\pi$
$822$	0	0
$823$	39.0616	1.36160	0.680801	−	0.732469i	$-0.261632\pi$
$823$	39.0616	1.36160	0.680801	+	0.732469i	$0.261632\pi$
$824$	18.8116	0.655334
$825$	0	0
$826$	−11.2620	−0.391857
$827$	−5.40283	−0.187875	−0.0939374	−	0.995578i	$-0.529945\pi$
$827$	−5.40283	−0.187875	−0.0939374	+	0.995578i	$0.529945\pi$
$828$	0	0
$829$	−8.38537	−0.291236	−0.145618	−	0.989341i	$-0.546517\pi$
$829$	−8.38537	−0.291236	−0.145618	+	0.989341i	$0.546517\pi$
$830$	3.28621	0.114066
$831$	0	0
$832$	0	0
$833$	34.3822	1.19127
$834$	0	0
$835$	−50.4543	−1.74604
$836$	2.14914	0.0743297
$837$	0	0
$838$	−9.97584	−0.344610
$839$	−3.98062	−0.137426	−0.0687132	−	0.997636i	$-0.521889\pi$
$839$	−3.98062	−0.137426	−0.0687132	+	0.997636i	$0.521889\pi$
$840$	0	0
$841$	−14.7103	−0.507253
$842$	−0.615957	−0.0212273
$843$	0	0
$844$	−25.0465	−0.862137
$845$	0	0
$846$	0	0
$847$	−6.04354	−0.207659
$848$	3.38404	0.116209
$849$	0	0
$850$	−47.6142	−1.63315
$851$	−16.5483	−0.567267
$852$	0	0
$853$	−6.29350	−0.215485	−0.107743	−	0.994179i	$-0.534362\pi$
$853$	−6.29350	−0.215485	−0.107743	+	0.994179i	$0.534362\pi$
$854$	−0.487991	−0.0166987
$855$	0	0
$856$	−18.0519	−0.617001
$857$	−4.37627	−0.149491	−0.0747453	−	0.997203i	$-0.523814\pi$
$857$	−4.37627	−0.149491	−0.0747453	+	0.997203i	$0.523814\pi$
$858$	0	0
$859$	15.0261	0.512683	0.256342	−	0.966586i	$-0.417483\pi$
$859$	15.0261	0.512683	0.256342	+	0.966586i	$0.417483\pi$
$860$	−1.85862	−0.0633786
$861$	0	0
$862$	−14.7922	−0.503826
$863$	6.21121	0.211432	0.105716	−	0.994396i	$-0.466287\pi$
$863$	6.21121	0.211432	0.105716	+	0.994396i	$0.466287\pi$
$864$	0	0
$865$	39.5991	1.34641
$866$	16.5321	0.561784
$867$	0	0
$868$	9.42758	0.319993
$869$	−36.4892	−1.23781
$870$	0	0
$871$	0	0
$872$	6.09783	0.206499
$873$	0	0
$874$	−3.08575	−0.104377
$875$	11.9517	0.404040
$876$	0	0
$877$	−38.2198	−1.29059	−0.645296	−	0.763933i	$-0.723266\pi$
$877$	−38.2198	−1.29059	−0.645296	+	0.763933i	$0.723266\pi$
$878$	3.50125	0.118162
$879$	0	0
$880$	8.49396	0.286331
$881$	−26.7832	−0.902347	−0.451174	−	0.892436i	$-0.648994\pi$
$881$	−26.7832	−0.902347	−0.451174	+	0.892436i	$0.648994\pi$
$882$	0	0
$883$	−34.4956	−1.16087	−0.580435	−	0.814307i	$-0.697117\pi$
$883$	−34.4956	−1.16087	−0.580435	+	0.814307i	$0.697117\pi$
$884$	0	0
$885$	0	0
$886$	−17.4077	−0.584824
$887$	23.9866	0.805391	0.402695	−	0.915334i	$-0.368073\pi$
$887$	23.9866	0.805391	0.402695	+	0.915334i	$0.368073\pi$
$888$	0	0
$889$	12.6837	0.425396
$890$	−13.5168	−0.453084
$891$	0	0
$892$	12.9879	0.434868
$893$	−6.37196	−0.213230
$894$	0	0
$895$	−16.7767	−0.560784
$896$	1.10992	0.0370797
$897$	0	0
$898$	34.1497	1.13959
$899$	32.1086	1.07088
$900$	0	0
$901$	−20.1715	−0.672010
$902$	16.9366	0.563927
$903$	0	0
$904$	12.2010	0.405801
$905$	−3.84309	−0.127748
$906$	0	0
$907$	23.9269	0.794480	0.397240	−	0.917715i	$-0.369968\pi$
$907$	23.9269	0.794480	0.397240	+	0.917715i	$0.369968\pi$
$908$	13.8049	0.458132
$909$	0	0
$910$	0	0
$911$	35.8866	1.18898	0.594488	−	0.804104i	$-0.297355\pi$
$911$	35.8866	1.18898	0.594488	+	0.804104i	$0.297355\pi$
$912$	0	0
$913$	2.14914	0.0711263
$914$	−9.40342	−0.311037
$915$	0	0
$916$	−11.5603	−0.381964
$917$	−2.54825	−0.0841507
$918$	0	0
$919$	33.2465	1.09670	0.548351	−	0.836249i	$-0.315256\pi$
$919$	33.2465	1.09670	0.548351	+	0.836249i	$0.315256\pi$
$920$	−12.1957	−0.402079
$921$	0	0
$922$	−0.733169	−0.0241456
$923$	0	0
$924$	0	0
$925$	39.0616	1.28434
$926$	−7.24267	−0.238009
$927$	0	0
$928$	3.78017	0.124090
$929$	54.2583	1.78016	0.890079	−	0.455806i	$-0.150649\pi$
$929$	54.2583	1.78016	0.890079	+	0.455806i	$0.150649\pi$
$930$	0	0
$931$	−5.25965	−0.172378
$932$	−9.77479	−0.320184
$933$	0	0
$934$	−30.2446	−0.989633
$935$	−50.6305	−1.65580
$936$	0	0
$937$	16.5265	0.539897	0.269948	−	0.962875i	$-0.412993\pi$
$937$	16.5265	0.539897	0.269948	+	0.962875i	$0.412993\pi$
$938$	2.38271	0.0777984
$939$	0	0
$940$	−25.1836	−0.821398
$941$	41.7017	1.35944	0.679718	−	0.733473i	$-0.262102\pi$
$941$	41.7017	1.35944	0.679718	+	0.733473i	$0.262102\pi$
$942$	0	0
$943$	−24.3177	−0.791892
$944$	−10.1468	−0.330249
$945$	0	0
$946$	−1.21552	−0.0395200
$947$	−3.00106	−0.0975215	−0.0487608	−	0.998810i	$-0.515527\pi$
$947$	−3.00106	−0.0975215	−0.0487608	+	0.998810i	$0.515527\pi$
$948$	0	0
$949$	0	0
$950$	7.28382	0.236318
$951$	0	0
$952$	−6.61596	−0.214424
$953$	−38.1450	−1.23564	−0.617818	−	0.786321i	$-0.711983\pi$
$953$	−38.1450	−1.23564	−0.617818	+	0.786321i	$0.711983\pi$
$954$	0	0
$955$	−3.20775	−0.103800
$956$	0.944378	0.0305434
$957$	0	0
$958$	−36.7198	−1.18636
$959$	−10.0785	−0.325450
$960$	0	0
$961$	41.1473	1.32733
$962$	0	0
$963$	0	0
$964$	−0.219833	−0.00708033
$965$	58.4456	1.88143
$966$	0	0
$967$	−26.8793	−0.864381	−0.432190	−	0.901782i	$-0.642259\pi$
$967$	−26.8793	−0.864381	−0.432190	+	0.901782i	$0.642259\pi$
$968$	−5.44504	−0.175010
$969$	0	0
$970$	−52.8853	−1.69804
$971$	−3.13647	−0.100654	−0.0503271	−	0.998733i	$-0.516026\pi$
$971$	−3.13647	−0.100654	−0.0503271	+	0.998733i	$0.516026\pi$
$972$	0	0
$973$	20.9831	0.672688
$974$	28.6547	0.918156
$975$	0	0
$976$	−0.439665	−0.0140733
$977$	−35.8864	−1.14811	−0.574053	−	0.818818i	$-0.694630\pi$
$977$	−35.8864	−1.14811	−0.574053	+	0.818818i	$0.694630\pi$
$978$	0	0
$979$	−8.83984	−0.282522
$980$	−20.7875	−0.664031
$981$	0	0
$982$	−30.4295	−0.971044
$983$	30.4370	0.970790	0.485395	−	0.874295i	$-0.338676\pi$
$983$	30.4370	0.970790	0.485395	+	0.874295i	$0.338676\pi$
$984$	0	0
$985$	41.3405	1.31722
$986$	−22.5327	−0.717588
$987$	0	0
$988$	0	0
$989$	1.74525	0.0554957
$990$	0	0
$991$	31.4470	0.998946	0.499473	−	0.866330i	$-0.333527\pi$
$991$	31.4470	0.998946	0.499473	+	0.866330i	$0.333527\pi$
$992$	8.49396	0.269683
$993$	0	0
$994$	−0.683661	−0.0216844
$995$	−13.6931	−0.434100
$996$	0	0
$997$	19.1099	0.605217	0.302609	−	0.953115i	$-0.402143\pi$
$997$	19.1099	0.605217	0.302609	+	0.953115i	$0.402143\pi$
$998$	−15.9715	−0.505570
$999$	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.bi.1.3		3
3.2	odd	2		338.2.a.g.1.1	✓	3
12.11	even	2		2704.2.a.v.1.3		3
13.5	odd	4		3042.2.b.n.1351.1		6
13.8	odd	4		3042.2.b.n.1351.6		6
13.12	even	2		3042.2.a.z.1.1		3
15.14	odd	2		8450.2.a.bx.1.3		3
39.2	even	12		338.2.e.e.147.3		12
39.5	even	4		338.2.b.d.337.4		6
39.8	even	4		338.2.b.d.337.1		6
39.11	even	12		338.2.e.e.147.6		12
39.17	odd	6		338.2.c.h.315.3		6
39.20	even	12		338.2.e.e.23.3		12
39.23	odd	6		338.2.c.h.191.3		6
39.29	odd	6		338.2.c.i.191.3		6
39.32	even	12		338.2.e.e.23.6		12
39.35	odd	6		338.2.c.i.315.3		6
39.38	odd	2		338.2.a.h.1.1	yes	3
156.47	odd	4		2704.2.f.m.337.5		6
156.83	odd	4		2704.2.f.m.337.6		6
156.155	even	2		2704.2.a.w.1.3		3
195.194	odd	2		8450.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.1	✓	3	3.2	odd	2
338.2.a.h.1.1	yes	3	39.38	odd	2
338.2.b.d.337.1		6	39.8	even	4
338.2.b.d.337.4		6	39.5	even	4
338.2.c.h.191.3		6	39.23	odd	6
338.2.c.h.315.3		6	39.17	odd	6
338.2.c.i.191.3		6	39.29	odd	6
338.2.c.i.315.3		6	39.35	odd	6
338.2.e.e.23.3		12	39.20	even	12
338.2.e.e.23.6		12	39.32	even	12
338.2.e.e.147.3		12	39.2	even	12
338.2.e.e.147.6		12	39.11	even	12
2704.2.a.v.1.3		3	12.11	even	2
2704.2.a.w.1.3		3	156.155	even	2
2704.2.f.m.337.5		6	156.47	odd	4
2704.2.f.m.337.6		6	156.83	odd	4
3042.2.a.z.1.1		3	13.12	even	2
3042.2.a.bi.1.3		3	1.1	even	1	trivial
3042.2.b.n.1351.1		6	13.5	odd	4
3042.2.b.n.1351.6		6	13.8	odd	4
8450.2.a.bn.1.3		3	195.194	odd	2
8450.2.a.bx.1.3		3	15.14	odd	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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Varieties

Fields

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Newspace parameters

Newform invariants

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.bi.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	$1$