LMFDB - Embedded newform 8048.2.a.w.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.137144 q^{3} +1.57958 q^{5} -4.44075 q^{7} -2.98119 q^{9} +O(q^{10})$	$q+0.137144 q^{3} +1.57958 q^{5} -4.44075 q^{7} -2.98119 q^{9} +3.56721 q^{11} -0.104051 q^{13} +0.216631 q^{15} -6.27710 q^{17} -3.22333 q^{19} -0.609023 q^{21} +0.996099 q^{23} -2.50491 q^{25} -0.820286 q^{27} -6.05098 q^{29} +0.827883 q^{31} +0.489222 q^{33} -7.01454 q^{35} +3.24249 q^{37} -0.0142700 q^{39} -10.9148 q^{41} +10.1934 q^{43} -4.70905 q^{45} +5.38152 q^{47} +12.7203 q^{49} -0.860868 q^{51} -2.80073 q^{53} +5.63471 q^{55} -0.442061 q^{57} +7.69410 q^{59} +0.864762 q^{61} +13.2387 q^{63} -0.164358 q^{65} -3.77761 q^{67} +0.136609 q^{69} +1.33473 q^{71} +10.1644 q^{73} -0.343534 q^{75} -15.8411 q^{77} +9.16131 q^{79} +8.83108 q^{81} +16.2039 q^{83} -9.91521 q^{85} -0.829857 q^{87} +17.9010 q^{89} +0.462066 q^{91} +0.113539 q^{93} -5.09152 q^{95} +6.97066 q^{97} -10.6345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.137144	0.0791803	0.0395901	−	0.999216i	$-0.487395\pi$
$3$	0.137144	0.0791803	0.0395901	+	0.999216i	$0.487395\pi$
$4$	0	0
$5$	1.57958	0.706412	0.353206	−	0.935546i	$-0.385091\pi$
$5$	1.57958	0.706412	0.353206	+	0.935546i	$0.385091\pi$
$6$	0	0
$7$	−4.44075	−1.67845	−0.839223	−	0.543788i	$-0.816990\pi$
$7$	−4.44075	−1.67845	−0.839223	+	0.543788i	$0.816990\pi$
$8$	0	0
$9$	−2.98119	−0.993730
$10$	0	0
$11$	3.56721	1.07555	0.537777	−	0.843087i	$-0.319264\pi$
$11$	3.56721	1.07555	0.537777	+	0.843087i	$0.319264\pi$
$12$	0	0
$13$	−0.104051	−0.0288586	−0.0144293	−	0.999896i	$-0.504593\pi$
$13$	−0.104051	−0.0288586	−0.0144293	+	0.999896i	$0.504593\pi$
$14$	0	0
$15$	0.216631	0.0559339
$16$	0	0
$17$	−6.27710	−1.52242	−0.761210	−	0.648505i	$-0.775394\pi$
$17$	−6.27710	−1.52242	−0.761210	+	0.648505i	$0.775394\pi$
$18$	0	0
$19$	−3.22333	−0.739482	−0.369741	−	0.929135i	$-0.620554\pi$
$19$	−3.22333	−0.739482	−0.369741	+	0.929135i	$0.620554\pi$
$20$	0	0
$21$	−0.609023	−0.132900
$22$	0	0
$23$	0.996099	0.207701	0.103851	−	0.994593i	$-0.466884\pi$
$23$	0.996099	0.207701	0.103851	+	0.994593i	$0.466884\pi$
$24$	0	0
$25$	−2.50491	−0.500982
$26$	0	0
$27$	−0.820286	−0.157864
$28$	0	0
$29$	−6.05098	−1.12364	−0.561820	−	0.827260i	$-0.689899\pi$
$29$	−6.05098	−1.12364	−0.561820	+	0.827260i	$0.689899\pi$
$30$	0	0
$31$	0.827883	0.148692	0.0743461	−	0.997233i	$-0.476313\pi$
$31$	0.827883	0.148692	0.0743461	+	0.997233i	$0.476313\pi$
$32$	0	0
$33$	0.489222	0.0851626
$34$	0	0
$35$	−7.01454	−1.18567
$36$	0	0
$37$	3.24249	0.533063	0.266531	−	0.963826i	$-0.414122\pi$
$37$	3.24249	0.533063	0.266531	+	0.963826i	$0.414122\pi$
$38$	0	0
$39$	−0.0142700	−0.00228503
$40$	0	0
$41$	−10.9148	−1.70461	−0.852303	−	0.523049i	$-0.824795\pi$
$41$	−10.9148	−1.70461	−0.852303	+	0.523049i	$0.824795\pi$
$42$	0	0
$43$	10.1934	1.55448	0.777241	−	0.629203i	$-0.216619\pi$
$43$	10.1934	1.55448	0.777241	+	0.629203i	$0.216619\pi$
$44$	0	0
$45$	−4.70905	−0.701983
$46$	0	0
$47$	5.38152	0.784976	0.392488	−	0.919757i	$-0.371615\pi$
$47$	5.38152	0.784976	0.392488	+	0.919757i	$0.371615\pi$
$48$	0	0
$49$	12.7203	1.81718
$50$	0	0
$51$	−0.860868	−0.120546
$52$	0	0
$53$	−2.80073	−0.384710	−0.192355	−	0.981325i	$-0.561612\pi$
$53$	−2.80073	−0.384710	−0.192355	+	0.981325i	$0.561612\pi$
$54$	0	0
$55$	5.63471	0.759784
$56$	0	0
$57$	−0.442061	−0.0585524
$58$	0	0
$59$	7.69410	1.00169	0.500843	−	0.865538i	$-0.333023\pi$
$59$	7.69410	1.00169	0.500843	+	0.865538i	$0.333023\pi$
$60$	0	0
$61$	0.864762	0.110721	0.0553607	−	0.998466i	$-0.482369\pi$
$61$	0.864762	0.110721	0.0553607	+	0.998466i	$0.482369\pi$
$62$	0	0
$63$	13.2387	1.66792
$64$	0	0
$65$	−0.164358	−0.0203861
$66$	0	0
$67$	−3.77761	−0.461508	−0.230754	−	0.973012i	$-0.574119\pi$
$67$	−3.77761	−0.461508	−0.230754	+	0.973012i	$0.574119\pi$
$68$	0	0
$69$	0.136609	0.0164458
$70$	0	0
$71$	1.33473	0.158403	0.0792014	−	0.996859i	$-0.474763\pi$
$71$	1.33473	0.158403	0.0792014	+	0.996859i	$0.474763\pi$
$72$	0	0
$73$	10.1644	1.18965	0.594824	−	0.803856i	$-0.297222\pi$
$73$	10.1644	1.18965	0.594824	+	0.803856i	$0.297222\pi$
$74$	0	0
$75$	−0.343534	−0.0396679
$76$	0	0
$77$	−15.8411	−1.80526
$78$	0	0
$79$	9.16131	1.03073	0.515364	−	0.856972i	$-0.327657\pi$
$79$	9.16131	1.03073	0.515364	+	0.856972i	$0.327657\pi$
$80$	0	0
$81$	8.83108	0.981231
$82$	0	0
$83$	16.2039	1.77861	0.889307	−	0.457311i	$-0.151187\pi$
$83$	16.2039	1.77861	0.889307	+	0.457311i	$0.151187\pi$
$84$	0	0
$85$	−9.91521	−1.07546
$86$	0	0
$87$	−0.829857	−0.0889701
$88$	0	0
$89$	17.9010	1.89750	0.948751	−	0.316024i	$-0.102348\pi$
$89$	17.9010	1.89750	0.948751	+	0.316024i	$0.102348\pi$
$90$	0	0
$91$	0.462066	0.0484376
$92$	0	0
$93$	0.113539	0.0117735
$94$	0	0
$95$	−5.09152	−0.522379
$96$	0	0
$97$	6.97066	0.707763	0.353882	−	0.935290i	$-0.384862\pi$
$97$	6.97066	0.707763	0.353882	+	0.935290i	$0.384862\pi$
$98$	0	0
$99$	−10.6345	−1.06881
$100$	0	0
$101$	−9.40013	−0.935348	−0.467674	−	0.883901i	$-0.654908\pi$
$101$	−9.40013	−0.935348	−0.467674	+	0.883901i	$0.654908\pi$
$102$	0	0
$103$	−0.271379	−0.0267397	−0.0133699	−	0.999911i	$-0.504256\pi$
$103$	−0.271379	−0.0267397	−0.0133699	+	0.999911i	$0.504256\pi$
$104$	0	0
$105$	−0.962004	−0.0938820
$106$	0	0
$107$	−4.92068	−0.475700	−0.237850	−	0.971302i	$-0.576443\pi$
$107$	−4.92068	−0.475700	−0.237850	+	0.971302i	$0.576443\pi$
$108$	0	0
$109$	−11.7504	−1.12548	−0.562742	−	0.826633i	$-0.690254\pi$
$109$	−11.7504	−1.12548	−0.562742	+	0.826633i	$0.690254\pi$
$110$	0	0
$111$	0.444689	0.0422080
$112$	0	0
$113$	13.5473	1.27442	0.637212	−	0.770688i	$-0.280087\pi$
$113$	13.5473	1.27442	0.637212	+	0.770688i	$0.280087\pi$
$114$	0	0
$115$	1.57342	0.146722
$116$	0	0
$117$	0.310197	0.0286777
$118$	0	0
$119$	27.8750	2.55530
$120$	0	0
$121$	1.72498	0.156816
$122$	0	0
$123$	−1.49690	−0.134971
$124$	0	0
$125$	−11.8546	−1.06031
$126$	0	0
$127$	−18.8678	−1.67425	−0.837125	−	0.547012i	$-0.815765\pi$
$127$	−18.8678	−1.67425	−0.837125	+	0.547012i	$0.815765\pi$
$128$	0	0
$129$	1.39797	0.123084
$130$	0	0
$131$	−18.3612	−1.60422	−0.802111	−	0.597175i	$-0.796290\pi$
$131$	−18.3612	−1.60422	−0.802111	+	0.597175i	$0.796290\pi$
$132$	0	0
$133$	14.3140	1.24118
$134$	0	0
$135$	−1.29571	−0.111517
$136$	0	0
$137$	−17.8295	−1.52328	−0.761640	−	0.648000i	$-0.775606\pi$
$137$	−17.8295	−1.52328	−0.761640	+	0.648000i	$0.775606\pi$
$138$	0	0
$139$	1.65275	0.140184	0.0700922	−	0.997541i	$-0.477671\pi$
$139$	1.65275	0.140184	0.0700922	+	0.997541i	$0.477671\pi$
$140$	0	0
$141$	0.738045	0.0621546
$142$	0	0
$143$	−0.371173	−0.0310390
$144$	0	0
$145$	−9.55804	−0.793752
$146$	0	0
$147$	1.74451	0.143885
$148$	0	0
$149$	23.1950	1.90020	0.950102	−	0.311938i	$-0.100978\pi$
$149$	23.1950	1.90020	0.950102	+	0.311938i	$0.100978\pi$
$150$	0	0
$151$	22.7186	1.84881	0.924406	−	0.381410i	$-0.124561\pi$
$151$	22.7186	1.84881	0.924406	+	0.381410i	$0.124561\pi$
$152$	0	0
$153$	18.7132	1.51288
$154$	0	0
$155$	1.30771	0.105038
$156$	0	0
$157$	20.3725	1.62590	0.812951	−	0.582331i	$-0.197859\pi$
$157$	20.3725	1.62590	0.812951	+	0.582331i	$0.197859\pi$
$158$	0	0
$159$	−0.384104	−0.0304614
$160$	0	0
$161$	−4.42343	−0.348615
$162$	0	0
$163$	−6.93616	−0.543282	−0.271641	−	0.962399i	$-0.587566\pi$
$163$	−6.93616	−0.543282	−0.271641	+	0.962399i	$0.587566\pi$
$164$	0	0
$165$	0.772768	0.0601599
$166$	0	0
$167$	−0.925307	−0.0716024	−0.0358012	−	0.999359i	$-0.511398\pi$
$167$	−0.925307	−0.0716024	−0.0358012	+	0.999359i	$0.511398\pi$
$168$	0	0
$169$	−12.9892	−0.999167
$170$	0	0
$171$	9.60935	0.734845
$172$	0	0
$173$	−14.7523	−1.12159	−0.560797	−	0.827953i	$-0.689505\pi$
$173$	−14.7523	−1.12159	−0.560797	+	0.827953i	$0.689505\pi$
$174$	0	0
$175$	11.1237	0.840872
$176$	0	0
$177$	1.05520	0.0793138
$178$	0	0
$179$	−2.16479	−0.161804	−0.0809019	−	0.996722i	$-0.525780\pi$
$179$	−2.16479	−0.161804	−0.0809019	+	0.996722i	$0.525780\pi$
$180$	0	0
$181$	18.8184	1.39876	0.699381	−	0.714749i	$-0.253459\pi$
$181$	18.8184	1.39876	0.699381	+	0.714749i	$0.253459\pi$
$182$	0	0
$183$	0.118597	0.00876695
$184$	0	0
$185$	5.12179	0.376562
$186$	0	0
$187$	−22.3917	−1.63745
$188$	0	0
$189$	3.64269	0.264966
$190$	0	0
$191$	7.36926	0.533221	0.266610	−	0.963804i	$-0.414096\pi$
$191$	7.36926	0.533221	0.266610	+	0.963804i	$0.414096\pi$
$192$	0	0
$193$	−11.7609	−0.846571	−0.423286	−	0.905996i	$-0.639123\pi$
$193$	−11.7609	−0.846571	−0.423286	+	0.905996i	$0.639123\pi$
$194$	0	0
$195$	−0.0225407	−0.00161417
$196$	0	0
$197$	14.4379	1.02866	0.514329	−	0.857593i	$-0.328041\pi$
$197$	14.4379	1.02866	0.514329	+	0.857593i	$0.328041\pi$
$198$	0	0
$199$	−6.70039	−0.474978	−0.237489	−	0.971390i	$-0.576324\pi$
$199$	−6.70039	−0.474978	−0.237489	+	0.971390i	$0.576324\pi$
$200$	0	0
$201$	−0.518077	−0.0365423
$202$	0	0
$203$	26.8709	1.88597
$204$	0	0
$205$	−17.2409	−1.20415
$206$	0	0
$207$	−2.96956	−0.206399
$208$	0	0
$209$	−11.4983	−0.795352
$210$	0	0
$211$	−11.1352	−0.766579	−0.383289	−	0.923628i	$-0.625209\pi$
$211$	−11.1352	−0.766579	−0.383289	+	0.923628i	$0.625209\pi$
$212$	0	0
$213$	0.183050	0.0125424
$214$	0	0
$215$	16.1014	1.09810
$216$	0	0
$217$	−3.67642	−0.249572
$218$	0	0
$219$	1.39398	0.0941967
$220$	0	0
$221$	0.653140	0.0439350
$222$	0	0
$223$	18.7155	1.25328	0.626642	−	0.779308i	$-0.284429\pi$
$223$	18.7155	1.25328	0.626642	+	0.779308i	$0.284429\pi$
$224$	0	0
$225$	7.46762	0.497841
$226$	0	0
$227$	12.2064	0.810166	0.405083	−	0.914280i	$-0.367243\pi$
$227$	12.2064	0.810166	0.405083	+	0.914280i	$0.367243\pi$
$228$	0	0
$229$	4.20381	0.277796	0.138898	−	0.990307i	$-0.455644\pi$
$229$	4.20381	0.277796	0.138898	+	0.990307i	$0.455644\pi$
$230$	0	0
$231$	−2.17251	−0.142941
$232$	0	0
$233$	14.5967	0.956263	0.478132	−	0.878288i	$-0.341314\pi$
$233$	14.5967	0.956263	0.478132	+	0.878288i	$0.341314\pi$
$234$	0	0
$235$	8.50057	0.554516
$236$	0	0
$237$	1.25642	0.0816133
$238$	0	0
$239$	1.06640	0.0689800	0.0344900	−	0.999405i	$-0.489019\pi$
$239$	1.06640	0.0689800	0.0344900	+	0.999405i	$0.489019\pi$
$240$	0	0
$241$	−0.501466	−0.0323023	−0.0161511	−	0.999870i	$-0.505141\pi$
$241$	−0.501466	−0.0323023	−0.0161511	+	0.999870i	$0.505141\pi$
$242$	0	0
$243$	3.67199	0.235558
$244$	0	0
$245$	20.0927	1.28368
$246$	0	0
$247$	0.335391	0.0213404
$248$	0	0
$249$	2.22228	0.140831
$250$	0	0
$251$	−10.3338	−0.652262	−0.326131	−	0.945325i	$-0.605745\pi$
$251$	−10.3338	−0.652262	−0.326131	+	0.945325i	$0.605745\pi$
$252$	0	0
$253$	3.55329	0.223394
$254$	0	0
$255$	−1.35981	−0.0851549
$256$	0	0
$257$	−7.84115	−0.489117	−0.244559	−	0.969634i	$-0.578643\pi$
$257$	−7.84115	−0.489117	−0.244559	+	0.969634i	$0.578643\pi$
$258$	0	0
$259$	−14.3991	−0.894717
$260$	0	0
$261$	18.0391	1.11659
$262$	0	0
$263$	20.1755	1.24407	0.622036	−	0.782989i	$-0.286306\pi$
$263$	20.1755	1.24407	0.622036	+	0.782989i	$0.286306\pi$
$264$	0	0
$265$	−4.42399	−0.271763
$266$	0	0
$267$	2.45502	0.150245
$268$	0	0
$269$	−1.42468	−0.0868645	−0.0434323	−	0.999056i	$-0.513829\pi$
$269$	−1.42468	−0.0868645	−0.0434323	+	0.999056i	$0.513829\pi$
$270$	0	0
$271$	25.8290	1.56900	0.784500	−	0.620128i	$-0.212920\pi$
$271$	25.8290	1.56900	0.784500	+	0.620128i	$0.212920\pi$
$272$	0	0
$273$	0.0633696	0.00383531
$274$	0	0
$275$	−8.93554	−0.538833
$276$	0	0
$277$	−11.0093	−0.661487	−0.330744	−	0.943721i	$-0.607300\pi$
$277$	−11.0093	−0.661487	−0.330744	+	0.943721i	$0.607300\pi$
$278$	0	0
$279$	−2.46808	−0.147760
$280$	0	0
$281$	7.50370	0.447633	0.223817	−	0.974631i	$-0.428148\pi$
$281$	7.50370	0.447633	0.223817	+	0.974631i	$0.428148\pi$
$282$	0	0
$283$	−14.6706	−0.872079	−0.436039	−	0.899928i	$-0.643619\pi$
$283$	−14.6706	−0.872079	−0.436039	+	0.899928i	$0.643619\pi$
$284$	0	0
$285$	−0.698272	−0.0413621
$286$	0	0
$287$	48.4699	2.86109
$288$	0	0
$289$	22.4020	1.31776
$290$	0	0
$291$	0.955986	0.0560409
$292$	0	0
$293$	−24.3818	−1.42440	−0.712200	−	0.701977i	$-0.752301\pi$
$293$	−24.3818	−1.42440	−0.712200	+	0.701977i	$0.752301\pi$
$294$	0	0
$295$	12.1535	0.707603
$296$	0	0
$297$	−2.92613	−0.169791
$298$	0	0
$299$	−0.103645	−0.00599397
$300$	0	0
$301$	−45.2664	−2.60911
$302$	0	0
$303$	−1.28917	−0.0740611
$304$	0	0
$305$	1.36597	0.0782149
$306$	0	0
$307$	16.4237	0.937349	0.468675	−	0.883371i	$-0.344732\pi$
$307$	16.4237	0.937349	0.468675	+	0.883371i	$0.344732\pi$
$308$	0	0
$309$	−0.0372180	−0.00211726
$310$	0	0
$311$	−4.26482	−0.241836	−0.120918	−	0.992663i	$-0.538584\pi$
$311$	−4.26482	−0.241836	−0.120918	+	0.992663i	$0.538584\pi$
$312$	0	0
$313$	14.1420	0.799354	0.399677	−	0.916656i	$-0.369122\pi$
$313$	14.1420	0.799354	0.399677	+	0.916656i	$0.369122\pi$
$314$	0	0
$315$	20.9117	1.17824
$316$	0	0
$317$	−0.151631	−0.00851646	−0.00425823	−	0.999991i	$-0.501355\pi$
$317$	−0.151631	−0.00851646	−0.00425823	+	0.999991i	$0.501355\pi$
$318$	0	0
$319$	−21.5851	−1.20853
$320$	0	0
$321$	−0.674843	−0.0376660
$322$	0	0
$323$	20.2331	1.12580
$324$	0	0
$325$	0.260639	0.0144577
$326$	0	0
$327$	−1.61150	−0.0891161
$328$	0	0
$329$	−23.8980	−1.31754
$330$	0	0
$331$	−7.49245	−0.411822	−0.205911	−	0.978571i	$-0.566016\pi$
$331$	−7.49245	−0.411822	−0.205911	+	0.978571i	$0.566016\pi$
$332$	0	0
$333$	−9.66649	−0.529721
$334$	0	0
$335$	−5.96705	−0.326015
$336$	0	0
$337$	−16.5644	−0.902322	−0.451161	−	0.892443i	$-0.648990\pi$
$337$	−16.5644	−0.902322	−0.451161	+	0.892443i	$0.648990\pi$
$338$	0	0
$339$	1.85794	0.100909
$340$	0	0
$341$	2.95323	0.159926
$342$	0	0
$343$	−25.4022	−1.37159
$344$	0	0
$345$	0.215786	0.0116175
$346$	0	0
$347$	32.0620	1.72118	0.860589	−	0.509299i	$-0.170095\pi$
$347$	32.0620	1.72118	0.860589	+	0.509299i	$0.170095\pi$
$348$	0	0
$349$	18.1181	0.969842	0.484921	−	0.874558i	$-0.338848\pi$
$349$	18.1181	0.969842	0.484921	+	0.874558i	$0.338848\pi$
$350$	0	0
$351$	0.0853518	0.00455574
$352$	0	0
$353$	−15.2174	−0.809940	−0.404970	−	0.914330i	$-0.632718\pi$
$353$	−15.2174	−0.809940	−0.404970	+	0.914330i	$0.632718\pi$
$354$	0	0
$355$	2.10831	0.111898
$356$	0	0
$357$	3.82290	0.202329
$358$	0	0
$359$	26.1169	1.37840	0.689199	−	0.724572i	$-0.257962\pi$
$359$	26.1169	1.37840	0.689199	+	0.724572i	$0.257962\pi$
$360$	0	0
$361$	−8.61017	−0.453167
$362$	0	0
$363$	0.236571	0.0124167
$364$	0	0
$365$	16.0555	0.840382
$366$	0	0
$367$	12.8002	0.668165	0.334082	−	0.942544i	$-0.391574\pi$
$367$	12.8002	0.668165	0.334082	+	0.942544i	$0.391574\pi$
$368$	0	0
$369$	32.5391	1.69392
$370$	0	0
$371$	12.4373	0.645714
$372$	0	0
$373$	−26.2157	−1.35740	−0.678700	−	0.734416i	$-0.737456\pi$
$373$	−26.2157	−1.35740	−0.678700	+	0.734416i	$0.737456\pi$
$374$	0	0
$375$	−1.62580	−0.0839558
$376$	0	0
$377$	0.629612	0.0324267
$378$	0	0
$379$	16.0306	0.823438	0.411719	−	0.911311i	$-0.364929\pi$
$379$	16.0306	0.823438	0.411719	+	0.911311i	$0.364929\pi$
$380$	0	0
$381$	−2.58762	−0.132568
$382$	0	0
$383$	7.07663	0.361599	0.180799	−	0.983520i	$-0.442131\pi$
$383$	7.07663	0.361599	0.180799	+	0.983520i	$0.442131\pi$
$384$	0	0
$385$	−25.0223	−1.27526
$386$	0	0
$387$	−30.3885	−1.54474
$388$	0	0
$389$	20.1616	1.02223	0.511116	−	0.859512i	$-0.329232\pi$
$389$	20.1616	1.02223	0.511116	+	0.859512i	$0.329232\pi$
$390$	0	0
$391$	−6.25261	−0.316208
$392$	0	0
$393$	−2.51813	−0.127023
$394$	0	0
$395$	14.4711	0.728118
$396$	0	0
$397$	−6.58287	−0.330385	−0.165192	−	0.986261i	$-0.552824\pi$
$397$	−6.58287	−0.330385	−0.165192	+	0.986261i	$0.552824\pi$
$398$	0	0
$399$	1.96308	0.0982769
$400$	0	0
$401$	27.0532	1.35097	0.675486	−	0.737373i	$-0.263934\pi$
$401$	27.0532	1.35097	0.675486	+	0.737373i	$0.263934\pi$
$402$	0	0
$403$	−0.0861423	−0.00429105
$404$	0	0
$405$	13.9494	0.693153
$406$	0	0
$407$	11.5667	0.573338
$408$	0	0
$409$	−5.81189	−0.287379	−0.143690	−	0.989623i	$-0.545897\pi$
$409$	−5.81189	−0.287379	−0.143690	+	0.989623i	$0.545897\pi$
$410$	0	0
$411$	−2.44522	−0.120614
$412$	0	0
$413$	−34.1676	−1.68128
$414$	0	0
$415$	25.5955	1.25643
$416$	0	0
$417$	0.226665	0.0110998
$418$	0	0
$419$	18.6706	0.912118	0.456059	−	0.889949i	$-0.349261\pi$
$419$	18.6706	0.912118	0.456059	+	0.889949i	$0.349261\pi$
$420$	0	0
$421$	6.85783	0.334230	0.167115	−	0.985937i	$-0.446555\pi$
$421$	6.85783	0.334230	0.167115	+	0.985937i	$0.446555\pi$
$422$	0	0
$423$	−16.0433	−0.780054
$424$	0	0
$425$	15.7236	0.762706
$426$	0	0
$427$	−3.84019	−0.185840
$428$	0	0
$429$	−0.0509042	−0.00245768
$430$	0	0
$431$	−28.2264	−1.35962	−0.679809	−	0.733389i	$-0.737937\pi$
$431$	−28.2264	−1.35962	−0.679809	+	0.733389i	$0.737937\pi$
$432$	0	0
$433$	−20.5692	−0.988491	−0.494246	−	0.869322i	$-0.664556\pi$
$433$	−20.5692	−0.988491	−0.494246	+	0.869322i	$0.664556\pi$
$434$	0	0
$435$	−1.31083	−0.0628495
$436$	0	0
$437$	−3.21075	−0.153591
$438$	0	0
$439$	14.4953	0.691822	0.345911	−	0.938267i	$-0.387570\pi$
$439$	14.4953	0.691822	0.345911	+	0.938267i	$0.387570\pi$
$440$	0	0
$441$	−37.9215	−1.80579
$442$	0	0
$443$	−39.9584	−1.89848	−0.949240	−	0.314553i	$-0.898145\pi$
$443$	−39.9584	−1.89848	−0.949240	+	0.314553i	$0.898145\pi$
$444$	0	0
$445$	28.2762	1.34042
$446$	0	0
$447$	3.18105	0.150459
$448$	0	0
$449$	35.6375	1.68184	0.840918	−	0.541162i	$-0.182016\pi$
$449$	35.6375	1.68184	0.840918	+	0.541162i	$0.182016\pi$
$450$	0	0
$451$	−38.9354	−1.83340
$452$	0	0
$453$	3.11572	0.146389
$454$	0	0
$455$	0.729872	0.0342169
$456$	0	0
$457$	−38.2192	−1.78782	−0.893910	−	0.448247i	$-0.852049\pi$
$457$	−38.2192	−1.78782	−0.893910	+	0.448247i	$0.852049\pi$
$458$	0	0
$459$	5.14902	0.240336
$460$	0	0
$461$	7.84710	0.365476	0.182738	−	0.983162i	$-0.441504\pi$
$461$	7.84710	0.365476	0.182738	+	0.983162i	$0.441504\pi$
$462$	0	0
$463$	15.3588	0.713784	0.356892	−	0.934146i	$-0.383836\pi$
$463$	15.3588	0.713784	0.356892	+	0.934146i	$0.383836\pi$
$464$	0	0
$465$	0.179345	0.00831693
$466$	0	0
$467$	29.2438	1.35324	0.676620	−	0.736332i	$-0.263444\pi$
$467$	29.2438	1.35324	0.676620	+	0.736332i	$0.263444\pi$
$468$	0	0
$469$	16.7754	0.774616
$470$	0	0
$471$	2.79397	0.128739
$472$	0	0
$473$	36.3621	1.67193
$474$	0	0
$475$	8.07414	0.370467
$476$	0	0
$477$	8.34951	0.382298
$478$	0	0
$479$	30.2804	1.38355	0.691774	−	0.722114i	$-0.256830\pi$
$479$	30.2804	1.38355	0.691774	+	0.722114i	$0.256830\pi$
$480$	0	0
$481$	−0.337386	−0.0153835
$482$	0	0
$483$	−0.606648	−0.0276034
$484$	0	0
$485$	11.0107	0.499972
$486$	0	0
$487$	−18.1071	−0.820510	−0.410255	−	0.911971i	$-0.634560\pi$
$487$	−18.1071	−0.820510	−0.410255	+	0.911971i	$0.634560\pi$
$488$	0	0
$489$	−0.951255	−0.0430172
$490$	0	0
$491$	−26.5911	−1.20004	−0.600019	−	0.799986i	$-0.704840\pi$
$491$	−26.5911	−1.20004	−0.600019	+	0.799986i	$0.704840\pi$
$492$	0	0
$493$	37.9826	1.71065
$494$	0	0
$495$	−16.7981	−0.755021
$496$	0	0
$497$	−5.92719	−0.265871
$498$	0	0
$499$	−37.2818	−1.66896	−0.834480	−	0.551038i	$-0.814232\pi$
$499$	−37.2818	−1.66896	−0.834480	+	0.551038i	$0.814232\pi$
$500$	0	0
$501$	−0.126901	−0.00566950
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−14.8483	−0.660741
$506$	0	0
$507$	−1.78139	−0.0791143
$508$	0	0
$509$	41.5372	1.84111	0.920553	−	0.390617i	$-0.127738\pi$
$509$	41.5372	1.84111	0.920553	+	0.390617i	$0.127738\pi$
$510$	0	0
$511$	−45.1374	−1.99676
$512$	0	0
$513$	2.64405	0.116738
$514$	0	0
$515$	−0.428666	−0.0188893
$516$	0	0
$517$	19.1970	0.844283
$518$	0	0
$519$	−2.02319	−0.0888081
$520$	0	0
$521$	27.4398	1.20216	0.601079	−	0.799189i	$-0.294738\pi$
$521$	27.4398	1.20216	0.601079	+	0.799189i	$0.294738\pi$
$522$	0	0
$523$	5.86143	0.256302	0.128151	−	0.991755i	$-0.459096\pi$
$523$	5.86143	0.256302	0.128151	+	0.991755i	$0.459096\pi$
$524$	0	0
$525$	1.52555	0.0665804
$526$	0	0
$527$	−5.19670	−0.226372
$528$	0	0
$529$	−22.0078	−0.956860
$530$	0	0
$531$	−22.9376	−0.995406
$532$	0	0
$533$	1.13570	0.0491926
$534$	0	0
$535$	−7.77263	−0.336040
$536$	0	0
$537$	−0.296888	−0.0128117
$538$	0	0
$539$	45.3758	1.95448
$540$	0	0
$541$	−1.80838	−0.0777482	−0.0388741	−	0.999244i	$-0.512377\pi$
$541$	−1.80838	−0.0777482	−0.0388741	+	0.999244i	$0.512377\pi$
$542$	0	0
$543$	2.58084	0.110754
$544$	0	0
$545$	−18.5607	−0.795055
$546$	0	0
$547$	34.8669	1.49080	0.745401	−	0.666616i	$-0.232258\pi$
$547$	34.8669	1.49080	0.745401	+	0.666616i	$0.232258\pi$
$548$	0	0
$549$	−2.57802	−0.110027
$550$	0	0
$551$	19.5043	0.830911
$552$	0	0
$553$	−40.6831	−1.73002
$554$	0	0
$555$	0.702425	0.0298163
$556$	0	0
$557$	12.6406	0.535601	0.267801	−	0.963474i	$-0.413703\pi$
$557$	12.6406	0.535601	0.267801	+	0.963474i	$0.413703\pi$
$558$	0	0
$559$	−1.06064	−0.0448602
$560$	0	0
$561$	−3.07090	−0.129653
$562$	0	0
$563$	−11.8213	−0.498209	−0.249104	−	0.968477i	$-0.580136\pi$
$563$	−11.8213	−0.498209	−0.249104	+	0.968477i	$0.580136\pi$
$564$	0	0
$565$	21.3991	0.900269
$566$	0	0
$567$	−39.2166	−1.64694
$568$	0	0
$569$	−7.65733	−0.321012	−0.160506	−	0.987035i	$-0.551313\pi$
$569$	−7.65733	−0.321012	−0.160506	+	0.987035i	$0.551313\pi$
$570$	0	0
$571$	−1.37037	−0.0573483	−0.0286741	−	0.999589i	$-0.509129\pi$
$571$	−1.37037	−0.0573483	−0.0286741	+	0.999589i	$0.509129\pi$
$572$	0	0
$573$	1.01065	0.0422206
$574$	0	0
$575$	−2.49514	−0.104055
$576$	0	0
$577$	−5.50469	−0.229163	−0.114582	−	0.993414i	$-0.536553\pi$
$577$	−5.50469	−0.229163	−0.114582	+	0.993414i	$0.536553\pi$
$578$	0	0
$579$	−1.61295	−0.0670318
$580$	0	0
$581$	−71.9577	−2.98531
$582$	0	0
$583$	−9.99078	−0.413776
$584$	0	0
$585$	0.489982	0.0202583
$586$	0	0
$587$	−9.86409	−0.407135	−0.203567	−	0.979061i	$-0.565254\pi$
$587$	−9.86409	−0.407135	−0.203567	+	0.979061i	$0.565254\pi$
$588$	0	0
$589$	−2.66854	−0.109955
$590$	0	0
$591$	1.98008	0.0814494
$592$	0	0
$593$	−29.6737	−1.21855	−0.609277	−	0.792958i	$-0.708540\pi$
$593$	−29.6737	−1.21855	−0.609277	+	0.792958i	$0.708540\pi$
$594$	0	0
$595$	44.0310	1.80509
$596$	0	0
$597$	−0.918920	−0.0376089
$598$	0	0
$599$	−28.6620	−1.17110	−0.585548	−	0.810638i	$-0.699121\pi$
$599$	−28.6620	−1.17110	−0.585548	+	0.810638i	$0.699121\pi$
$600$	0	0
$601$	−41.9195	−1.70993	−0.854966	−	0.518685i	$-0.826422\pi$
$601$	−41.9195	−1.70993	−0.854966	+	0.518685i	$0.826422\pi$
$602$	0	0
$603$	11.2618	0.458615
$604$	0	0
$605$	2.72475	0.110777
$606$	0	0
$607$	19.7054	0.799820	0.399910	−	0.916555i	$-0.369041\pi$
$607$	19.7054	0.799820	0.399910	+	0.916555i	$0.369041\pi$
$608$	0	0
$609$	3.68519	0.149331
$610$	0	0
$611$	−0.559954	−0.0226533
$612$	0	0
$613$	34.3426	1.38708	0.693542	−	0.720416i	$-0.256049\pi$
$613$	34.3426	1.38708	0.693542	+	0.720416i	$0.256049\pi$
$614$	0	0
$615$	−2.36448	−0.0953452
$616$	0	0
$617$	34.8171	1.40169	0.700843	−	0.713316i	$-0.252808\pi$
$617$	34.8171	1.40169	0.700843	+	0.713316i	$0.252808\pi$
$618$	0	0
$619$	−21.3748	−0.859126	−0.429563	−	0.903037i	$-0.641332\pi$
$619$	−21.3748	−0.859126	−0.429563	+	0.903037i	$0.641332\pi$
$620$	0	0
$621$	−0.817086	−0.0327885
$622$	0	0
$623$	−79.4939	−3.18486
$624$	0	0
$625$	−6.20086	−0.248035
$626$	0	0
$627$	−1.57692	−0.0629762
$628$	0	0
$629$	−20.3535	−0.811545
$630$	0	0
$631$	13.7806	0.548598	0.274299	−	0.961644i	$-0.411554\pi$
$631$	13.7806	0.548598	0.274299	+	0.961644i	$0.411554\pi$
$632$	0	0
$633$	−1.52713	−0.0606979
$634$	0	0
$635$	−29.8034	−1.18271
$636$	0	0
$637$	−1.32356	−0.0524413
$638$	0	0
$639$	−3.97907	−0.157410
$640$	0	0
$641$	27.3790	1.08141	0.540703	−	0.841214i	$-0.318158\pi$
$641$	27.3790	1.08141	0.540703	+	0.841214i	$0.318158\pi$
$642$	0	0
$643$	−26.5203	−1.04586	−0.522929	−	0.852376i	$-0.675161\pi$
$643$	−26.5203	−1.04586	−0.522929	+	0.852376i	$0.675161\pi$
$644$	0	0
$645$	2.20821	0.0869482
$646$	0	0
$647$	−13.6398	−0.536238	−0.268119	−	0.963386i	$-0.586402\pi$
$647$	−13.6398	−0.536238	−0.268119	+	0.963386i	$0.586402\pi$
$648$	0	0
$649$	27.4465	1.07737
$650$	0	0
$651$	−0.504200	−0.0197612
$652$	0	0
$653$	47.8602	1.87292	0.936458	−	0.350779i	$-0.114083\pi$
$653$	47.8602	1.87292	0.936458	+	0.350779i	$0.114083\pi$
$654$	0	0
$655$	−29.0030	−1.13324
$656$	0	0
$657$	−30.3019	−1.18219
$658$	0	0
$659$	−43.7693	−1.70501	−0.852505	−	0.522718i	$-0.824918\pi$
$659$	−43.7693	−1.70501	−0.852505	+	0.522718i	$0.824918\pi$
$660$	0	0
$661$	−39.6164	−1.54090	−0.770449	−	0.637502i	$-0.779968\pi$
$661$	−39.6164	−1.54090	−0.770449	+	0.637502i	$0.779968\pi$
$662$	0	0
$663$	0.0895744	0.00347878
$664$	0	0
$665$	22.6102	0.876784
$666$	0	0
$667$	−6.02738	−0.233381
$668$	0	0
$669$	2.56672	0.0992353
$670$	0	0
$671$	3.08479	0.119087
$672$	0	0
$673$	24.1058	0.929212	0.464606	−	0.885518i	$-0.346196\pi$
$673$	24.1058	0.929212	0.464606	+	0.885518i	$0.346196\pi$
$674$	0	0
$675$	2.05474	0.0790871
$676$	0	0
$677$	26.9028	1.03396	0.516978	−	0.855998i	$-0.327057\pi$
$677$	26.9028	1.03396	0.516978	+	0.855998i	$0.327057\pi$
$678$	0	0
$679$	−30.9550	−1.18794
$680$	0	0
$681$	1.67403	0.0641491
$682$	0	0
$683$	4.13391	0.158180	0.0790898	−	0.996867i	$-0.474799\pi$
$683$	4.13391	0.158180	0.0790898	+	0.996867i	$0.474799\pi$
$684$	0	0
$685$	−28.1633	−1.07606
$686$	0	0
$687$	0.576529	0.0219960
$688$	0	0
$689$	0.291419	0.0111022
$690$	0	0
$691$	18.2722	0.695107	0.347554	−	0.937660i	$-0.387012\pi$
$691$	18.2722	0.695107	0.347554	+	0.937660i	$0.387012\pi$
$692$	0	0
$693$	47.2253	1.79394
$694$	0	0
$695$	2.61066	0.0990280
$696$	0	0
$697$	68.5133	2.59513
$698$	0	0
$699$	2.00186	0.0757172
$700$	0	0
$701$	−32.9166	−1.24324	−0.621621	−	0.783318i	$-0.713525\pi$
$701$	−32.9166	−1.24324	−0.621621	+	0.783318i	$0.713525\pi$
$702$	0	0
$703$	−10.4516	−0.394190
$704$	0	0
$705$	1.16580	0.0439067
$706$	0	0
$707$	41.7436	1.56993
$708$	0	0
$709$	37.6661	1.41458	0.707289	−	0.706924i	$-0.249918\pi$
$709$	37.6661	1.41458	0.707289	+	0.706924i	$0.249918\pi$
$710$	0	0
$711$	−27.3116	−1.02427
$712$	0	0
$713$	0.824653	0.0308835
$714$	0	0
$715$	−0.586299	−0.0219263
$716$	0	0
$717$	0.146251	0.00546185
$718$	0	0
$719$	−39.6610	−1.47910	−0.739552	−	0.673099i	$-0.764963\pi$
$719$	−39.6610	−1.47910	−0.739552	+	0.673099i	$0.764963\pi$
$720$	0	0
$721$	1.20512	0.0448812
$722$	0	0
$723$	−0.0687732	−0.00255770
$724$	0	0
$725$	15.1572	0.562923
$726$	0	0
$727$	−0.326534	−0.0121105	−0.00605524	−	0.999982i	$-0.501927\pi$
$727$	−0.326534	−0.0121105	−0.00605524	+	0.999982i	$0.501927\pi$
$728$	0	0
$729$	−25.9896	−0.962579
$730$	0	0
$731$	−63.9851	−2.36658
$732$	0	0
$733$	15.2221	0.562239	0.281120	−	0.959673i	$-0.409294\pi$
$733$	15.2221	0.562239	0.281120	+	0.959673i	$0.409294\pi$
$734$	0	0
$735$	2.75560	0.101642
$736$	0	0
$737$	−13.4755	−0.496377
$738$	0	0
$739$	−1.80611	−0.0664389	−0.0332195	−	0.999448i	$-0.510576\pi$
$739$	−1.80611	−0.0664389	−0.0332195	+	0.999448i	$0.510576\pi$
$740$	0	0
$741$	0.0459970	0.00168974
$742$	0	0
$743$	39.4995	1.44910	0.724548	−	0.689224i	$-0.242049\pi$
$743$	39.4995	1.44910	0.724548	+	0.689224i	$0.242049\pi$
$744$	0	0
$745$	36.6384	1.34233
$746$	0	0
$747$	−48.3071	−1.76746
$748$	0	0
$749$	21.8515	0.798436
$750$	0	0
$751$	36.2290	1.32202	0.661008	−	0.750379i	$-0.270129\pi$
$751$	36.2290	1.32202	0.661008	+	0.750379i	$0.270129\pi$
$752$	0	0
$753$	−1.41722	−0.0516463
$754$	0	0
$755$	35.8859	1.30602
$756$	0	0
$757$	20.6460	0.750393	0.375197	−	0.926945i	$-0.377575\pi$
$757$	20.6460	0.750393	0.375197	+	0.926945i	$0.377575\pi$
$758$	0	0
$759$	0.487314	0.0176884
$760$	0	0
$761$	−14.9927	−0.543487	−0.271743	−	0.962370i	$-0.587600\pi$
$761$	−14.9927	−0.543487	−0.271743	+	0.962370i	$0.587600\pi$
$762$	0	0
$763$	52.1806	1.88906
$764$	0	0
$765$	29.5591	1.06871
$766$	0	0
$767$	−0.800581	−0.0289073
$768$	0	0
$769$	−8.34663	−0.300987	−0.150494	−	0.988611i	$-0.548086\pi$
$769$	−8.34663	−0.300987	−0.150494	+	0.988611i	$0.548086\pi$
$770$	0	0
$771$	−1.07537	−0.0387284
$772$	0	0
$773$	18.0952	0.650838	0.325419	−	0.945570i	$-0.394495\pi$
$773$	18.0952	0.650838	0.325419	+	0.945570i	$0.394495\pi$
$774$	0	0
$775$	−2.07377	−0.0744921
$776$	0	0
$777$	−1.97475	−0.0708439
$778$	0	0
$779$	35.1820	1.26052
$780$	0	0
$781$	4.76125	0.170371
$782$	0	0
$783$	4.96354	0.177382
$784$	0	0
$785$	32.1801	1.14856
$786$	0	0
$787$	−7.53615	−0.268635	−0.134317	−	0.990938i	$-0.542884\pi$
$787$	−7.53615	−0.268635	−0.134317	+	0.990938i	$0.542884\pi$
$788$	0	0
$789$	2.76695	0.0985059
$790$	0	0
$791$	−60.1603	−2.13905
$792$	0	0
$793$	−0.0899796	−0.00319527
$794$	0	0
$795$	−0.606724	−0.0215183
$796$	0	0
$797$	−35.8685	−1.27053	−0.635263	−	0.772296i	$-0.719108\pi$
$797$	−35.8685	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$798$	0	0
$799$	−33.7803	−1.19506
$800$	0	0
$801$	−53.3663	−1.88561
$802$	0	0
$803$	36.2584	1.27953
$804$	0	0
$805$	−6.98718	−0.246266
$806$	0	0
$807$	−0.195387	−0.00687796
$808$	0	0
$809$	−35.8340	−1.25986	−0.629928	−	0.776654i	$-0.716916\pi$
$809$	−35.8340	−1.25986	−0.629928	+	0.776654i	$0.716916\pi$
$810$	0	0
$811$	39.5546	1.38895	0.694475	−	0.719516i	$-0.255636\pi$
$811$	39.5546	1.38895	0.694475	+	0.719516i	$0.255636\pi$
$812$	0	0
$813$	3.54230	0.124234
$814$	0	0
$815$	−10.9563	−0.383781
$816$	0	0
$817$	−32.8567	−1.14951
$818$	0	0
$819$	−1.37751	−0.0481340
$820$	0	0
$821$	−42.2012	−1.47283	−0.736416	−	0.676529i	$-0.763483\pi$
$821$	−42.2012	−1.47283	−0.736416	+	0.676529i	$0.763483\pi$
$822$	0	0
$823$	29.8169	1.03935	0.519676	−	0.854363i	$-0.326053\pi$
$823$	29.8169	1.03935	0.519676	+	0.854363i	$0.326053\pi$
$824$	0	0
$825$	−1.22546	−0.0426650
$826$	0	0
$827$	5.40176	0.187838	0.0939188	−	0.995580i	$-0.470061\pi$
$827$	5.40176	0.187838	0.0939188	+	0.995580i	$0.470061\pi$
$828$	0	0
$829$	−5.14687	−0.178758	−0.0893790	−	0.995998i	$-0.528488\pi$
$829$	−5.14687	−0.178758	−0.0893790	+	0.995998i	$0.528488\pi$
$830$	0	0
$831$	−1.50987	−0.0523767
$832$	0	0
$833$	−79.8463	−2.76651
$834$	0	0
$835$	−1.46160	−0.0505808
$836$	0	0
$837$	−0.679101	−0.0234732
$838$	0	0
$839$	10.4855	0.362000	0.181000	−	0.983483i	$-0.442067\pi$
$839$	10.4855	0.362000	0.181000	+	0.983483i	$0.442067\pi$
$840$	0	0
$841$	7.61439	0.262565
$842$	0	0
$843$	1.02909	0.0354437
$844$	0	0
$845$	−20.5175	−0.705824
$846$	0	0
$847$	−7.66019	−0.263207
$848$	0	0
$849$	−2.01199	−0.0690514
$850$	0	0
$851$	3.22984	0.110718
$852$	0	0
$853$	−18.4657	−0.632252	−0.316126	−	0.948717i	$-0.602382\pi$
$853$	−18.4657	−0.632252	−0.316126	+	0.948717i	$0.602382\pi$
$854$	0	0
$855$	15.1788	0.519104
$856$	0	0
$857$	−21.1416	−0.722184	−0.361092	−	0.932530i	$-0.617596\pi$
$857$	−21.1416	−0.722184	−0.361092	+	0.932530i	$0.617596\pi$
$858$	0	0
$859$	2.62460	0.0895503	0.0447752	−	0.998997i	$-0.485743\pi$
$859$	2.62460	0.0895503	0.0447752	+	0.998997i	$0.485743\pi$
$860$	0	0
$861$	6.64737	0.226542
$862$	0	0
$863$	−26.5965	−0.905356	−0.452678	−	0.891674i	$-0.649531\pi$
$863$	−26.5965	−0.905356	−0.452678	+	0.891674i	$0.649531\pi$
$864$	0	0
$865$	−23.3025	−0.792307
$866$	0	0
$867$	3.07230	0.104341
$868$	0	0
$869$	32.6803	1.10860
$870$	0	0
$871$	0.393065	0.0133185
$872$	0	0
$873$	−20.7809	−0.703326
$874$	0	0
$875$	52.6435	1.77968
$876$	0	0
$877$	51.7946	1.74898	0.874490	−	0.485043i	$-0.161196\pi$
$877$	51.7946	1.74898	0.874490	+	0.485043i	$0.161196\pi$
$878$	0	0
$879$	−3.34382	−0.112784
$880$	0	0
$881$	−33.0169	−1.11237	−0.556183	−	0.831060i	$-0.687735\pi$
$881$	−33.0169	−1.11237	−0.556183	+	0.831060i	$0.687735\pi$
$882$	0	0
$883$	24.4735	0.823597	0.411799	−	0.911275i	$-0.364901\pi$
$883$	24.4735	0.823597	0.411799	+	0.911275i	$0.364901\pi$
$884$	0	0
$885$	1.66678	0.0560282
$886$	0	0
$887$	−18.1946	−0.610916	−0.305458	−	0.952206i	$-0.598809\pi$
$887$	−18.1946	−0.610916	−0.305458	+	0.952206i	$0.598809\pi$
$888$	0	0
$889$	83.7874	2.81014
$890$	0	0
$891$	31.5023	1.05537
$892$	0	0
$893$	−17.3464	−0.580475
$894$	0	0
$895$	−3.41946	−0.114300
$896$	0	0
$897$	−0.0142144	−0.000474604 0
$898$	0	0
$899$	−5.00951	−0.167076
$900$	0	0
$901$	17.5804	0.585690
$902$	0	0
$903$	−6.20803	−0.206590
$904$	0	0
$905$	29.7253	0.988102
$906$	0	0
$907$	23.9916	0.796628	0.398314	−	0.917249i	$-0.369595\pi$
$907$	23.9916	0.796628	0.398314	+	0.917249i	$0.369595\pi$
$908$	0	0
$909$	28.0236	0.929483
$910$	0	0
$911$	35.6670	1.18170	0.590850	−	0.806782i	$-0.298793\pi$
$911$	35.6670	1.18170	0.590850	+	0.806782i	$0.298793\pi$
$912$	0	0
$913$	57.8028	1.91300
$914$	0	0
$915$	0.187334	0.00619308
$916$	0	0
$917$	81.5373	2.69260
$918$	0	0
$919$	−42.5231	−1.40271	−0.701354	−	0.712813i	$-0.747421\pi$
$919$	−42.5231	−1.40271	−0.701354	+	0.712813i	$0.747421\pi$
$920$	0	0
$921$	2.25241	0.0742196
$922$	0	0
$923$	−0.138880	−0.00457129
$924$	0	0
$925$	−8.12216	−0.267055
$926$	0	0
$927$	0.809032	0.0265721
$928$	0	0
$929$	3.09898	0.101674	0.0508371	−	0.998707i	$-0.483811\pi$
$929$	3.09898	0.101674	0.0508371	+	0.998707i	$0.483811\pi$
$930$	0	0
$931$	−41.0015	−1.34377
$932$	0	0
$933$	−0.584896	−0.0191486
$934$	0	0
$935$	−35.3696	−1.15671
$936$	0	0
$937$	22.4502	0.733414	0.366707	−	0.930336i	$-0.380485\pi$
$937$	22.4502	0.733414	0.366707	+	0.930336i	$0.380485\pi$
$938$	0	0
$939$	1.93950	0.0632931
$940$	0	0
$941$	−26.3639	−0.859440	−0.429720	−	0.902962i	$-0.641388\pi$
$941$	−26.3639	−0.859440	−0.429720	+	0.902962i	$0.641388\pi$
$942$	0	0
$943$	−10.8722	−0.354048
$944$	0	0
$945$	5.75393	0.187175
$946$	0	0
$947$	35.5597	1.15554	0.577768	−	0.816201i	$-0.303924\pi$
$947$	35.5597	1.15554	0.577768	+	0.816201i	$0.303924\pi$
$948$	0	0
$949$	−1.05761	−0.0343316
$950$	0	0
$951$	−0.0207954	−0.000674335 0
$952$	0	0
$953$	24.1635	0.782732	0.391366	−	0.920235i	$-0.372003\pi$
$953$	24.1635	0.782732	0.391366	+	0.920235i	$0.372003\pi$
$954$	0	0
$955$	11.6404	0.376674
$956$	0	0
$957$	−2.96027	−0.0956921
$958$	0	0
$959$	79.1765	2.55674
$960$	0	0
$961$	−30.3146	−0.977891
$962$	0	0
$963$	14.6695	0.472718
$964$	0	0
$965$	−18.5774	−0.598028
$966$	0	0
$967$	25.2563	0.812188	0.406094	−	0.913831i	$-0.366891\pi$
$967$	25.2563	0.812188	0.406094	+	0.913831i	$0.366891\pi$
$968$	0	0
$969$	2.77486	0.0891413
$970$	0	0
$971$	−8.28223	−0.265789	−0.132895	−	0.991130i	$-0.542427\pi$
$971$	−8.28223	−0.265789	−0.132895	+	0.991130i	$0.542427\pi$
$972$	0	0
$973$	−7.33945	−0.235292
$974$	0	0
$975$	0.0357452	0.00114476
$976$	0	0
$977$	32.2547	1.03192	0.515959	−	0.856613i	$-0.327436\pi$
$977$	32.2547	1.03192	0.515959	+	0.856613i	$0.327436\pi$
$978$	0	0
$979$	63.8566	2.04087
$980$	0	0
$981$	35.0302	1.11843
$982$	0	0
$983$	40.3667	1.28750	0.643749	−	0.765237i	$-0.277378\pi$
$983$	40.3667	1.28750	0.643749	+	0.765237i	$0.277378\pi$
$984$	0	0
$985$	22.8059	0.726656
$986$	0	0
$987$	−3.27747	−0.104323
$988$	0	0
$989$	10.1537	0.322867
$990$	0	0
$991$	37.9851	1.20664	0.603319	−	0.797500i	$-0.293845\pi$
$991$	37.9851	1.20664	0.603319	+	0.797500i	$0.293845\pi$
$992$	0	0
$993$	−1.02755	−0.0326082
$994$	0	0
$995$	−10.5838	−0.335530
$996$	0	0
$997$	19.5045	0.617712	0.308856	−	0.951109i	$-0.400054\pi$
$997$	19.5045	0.617712	0.308856	+	0.951109i	$0.400054\pi$
$998$	0	0
$999$	−2.65977	−0.0841515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.13		29
4.3	odd	2		4024.2.a.e.1.17	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.17	✓	29	4.3	odd	2
8048.2.a.w.1.13		29	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.137144 q^{3} +1.57958 q^{5} -4.44075 q^{7} -2.98119 q^{9} +O(q^{10})\)	\(q+0.137144 q^{3} +1.57958 q^{5} -4.44075 q^{7} -2.98119 q^{9} +3.56721 q^{11} -0.104051 q^{13} +0.216631 q^{15} -6.27710 q^{17} -3.22333 q^{19} -0.609023 q^{21} +0.996099 q^{23} -2.50491 q^{25} -0.820286 q^{27} -6.05098 q^{29} +0.827883 q^{31} +0.489222 q^{33} -7.01454 q^{35} +3.24249 q^{37} -0.0142700 q^{39} -10.9148 q^{41} +10.1934 q^{43} -4.70905 q^{45} +5.38152 q^{47} +12.7203 q^{49} -0.860868 q^{51} -2.80073 q^{53} +5.63471 q^{55} -0.442061 q^{57} +7.69410 q^{59} +0.864762 q^{61} +13.2387 q^{63} -0.164358 q^{65} -3.77761 q^{67} +0.136609 q^{69} +1.33473 q^{71} +10.1644 q^{73} -0.343534 q^{75} -15.8411 q^{77} +9.16131 q^{79} +8.83108 q^{81} +16.2039 q^{83} -9.91521 q^{85} -0.829857 q^{87} +17.9010 q^{89} +0.462066 q^{91} +0.113539 q^{93} -5.09152 q^{95} +6.97066 q^{97} -10.6345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

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