LMFDB - Embedded newform 8048.2.a.v.1.19

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:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.893789 q^{3} -3.85908 q^{5} -3.77625 q^{7} -2.20114 q^{9} +O(q^{10})$	$q+0.893789 q^{3} -3.85908 q^{5} -3.77625 q^{7} -2.20114 q^{9} +4.71894 q^{11} -0.511554 q^{13} -3.44921 q^{15} +5.63108 q^{17} -0.0695678 q^{19} -3.37517 q^{21} -5.20252 q^{23} +9.89252 q^{25} -4.64872 q^{27} -1.66971 q^{29} +0.370767 q^{31} +4.21774 q^{33} +14.5729 q^{35} +0.932872 q^{37} -0.457221 q^{39} +5.06469 q^{41} +6.38770 q^{43} +8.49438 q^{45} +4.19872 q^{47} +7.26006 q^{49} +5.03300 q^{51} +1.31018 q^{53} -18.2108 q^{55} -0.0621790 q^{57} +13.0216 q^{59} +10.4141 q^{61} +8.31206 q^{63} +1.97413 q^{65} -2.09964 q^{67} -4.64996 q^{69} -3.30951 q^{71} +1.07644 q^{73} +8.84182 q^{75} -17.8199 q^{77} -10.1610 q^{79} +2.44844 q^{81} -11.7519 q^{83} -21.7308 q^{85} -1.49237 q^{87} -16.5010 q^{89} +1.93176 q^{91} +0.331388 q^{93} +0.268468 q^{95} -12.5981 q^{97} -10.3871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	0.893789	0.516029	0.258015	−	0.966141i	$-0.416932\pi$
$3$	0.893789	0.516029	0.258015	+	0.966141i	$0.416932\pi$
$4$	0	0
$5$	−3.85908	−1.72583	−0.862917	−	0.505346i	$-0.831365\pi$
$5$	−3.85908	−1.72583	−0.862917	+	0.505346i	$0.831365\pi$
$6$	0	0
$7$	−3.77625	−1.42729	−0.713644	−	0.700509i	$-0.752957\pi$
$7$	−3.77625	−1.42729	−0.713644	+	0.700509i	$0.752957\pi$
$8$	0	0
$9$	−2.20114	−0.733714
$10$	0	0
$11$	4.71894	1.42281	0.711407	−	0.702780i	$-0.248058\pi$
$11$	4.71894	1.42281	0.711407	+	0.702780i	$0.248058\pi$
$12$	0	0
$13$	−0.511554	−0.141880	−0.0709398	−	0.997481i	$-0.522600\pi$
$13$	−0.511554	−0.141880	−0.0709398	+	0.997481i	$0.522600\pi$
$14$	0	0
$15$	−3.44921	−0.890581
$16$	0	0
$17$	5.63108	1.36574	0.682869	−	0.730541i	$-0.260732\pi$
$17$	5.63108	1.36574	0.682869	+	0.730541i	$0.260732\pi$
$18$	0	0
$19$	−0.0695678	−0.0159600	−0.00797998	−	0.999968i	$-0.502540\pi$
$19$	−0.0695678	−0.0159600	−0.00797998	+	0.999968i	$0.502540\pi$
$20$	0	0
$21$	−3.37517	−0.736523
$22$	0	0
$23$	−5.20252	−1.08480	−0.542401	−	0.840120i	$-0.682485\pi$
$23$	−5.20252	−1.08480	−0.542401	+	0.840120i	$0.682485\pi$
$24$	0	0
$25$	9.89252	1.97850
$26$	0	0
$27$	−4.64872	−0.894647
$28$	0	0
$29$	−1.66971	−0.310057	−0.155028	−	0.987910i	$-0.549547\pi$
$29$	−1.66971	−0.310057	−0.155028	+	0.987910i	$0.549547\pi$
$30$	0	0
$31$	0.370767	0.0665918	0.0332959	−	0.999446i	$-0.489400\pi$
$31$	0.370767	0.0665918	0.0332959	+	0.999446i	$0.489400\pi$
$32$	0	0
$33$	4.21774	0.734214
$34$	0	0
$35$	14.5729	2.46326
$36$	0	0
$37$	0.932872	0.153363	0.0766816	−	0.997056i	$-0.475568\pi$
$37$	0.932872	0.153363	0.0766816	+	0.997056i	$0.475568\pi$
$38$	0	0
$39$	−0.457221	−0.0732140
$40$	0	0
$41$	5.06469	0.790972	0.395486	−	0.918472i	$-0.370576\pi$
$41$	5.06469	0.790972	0.395486	+	0.918472i	$0.370576\pi$
$42$	0	0
$43$	6.38770	0.974115	0.487058	−	0.873370i	$-0.338070\pi$
$43$	6.38770	0.974115	0.487058	+	0.873370i	$0.338070\pi$
$44$	0	0
$45$	8.49438	1.26627
$46$	0	0
$47$	4.19872	0.612447	0.306223	−	0.951960i	$-0.400935\pi$
$47$	4.19872	0.612447	0.306223	+	0.951960i	$0.400935\pi$
$48$	0	0
$49$	7.26006	1.03715
$50$	0	0
$51$	5.03300	0.704761
$52$	0	0
$53$	1.31018	0.179968	0.0899839	−	0.995943i	$-0.471318\pi$
$53$	1.31018	0.179968	0.0899839	+	0.995943i	$0.471318\pi$
$54$	0	0
$55$	−18.2108	−2.45554
$56$	0	0
$57$	−0.0621790	−0.00823581
$58$	0	0
$59$	13.0216	1.69527	0.847635	−	0.530579i	$-0.178026\pi$
$59$	13.0216	1.69527	0.847635	+	0.530579i	$0.178026\pi$
$60$	0	0
$61$	10.4141	1.33339	0.666696	−	0.745330i	$-0.267708\pi$
$61$	10.4141	1.33339	0.666696	+	0.745330i	$0.267708\pi$
$62$	0	0
$63$	8.31206	1.04722
$64$	0	0
$65$	1.97413	0.244861
$66$	0	0
$67$	−2.09964	−0.256512	−0.128256	−	0.991741i	$-0.540938\pi$
$67$	−2.09964	−0.256512	−0.128256	+	0.991741i	$0.540938\pi$
$68$	0	0
$69$	−4.64996	−0.559789
$70$	0	0
$71$	−3.30951	−0.392767	−0.196383	−	0.980527i	$-0.562920\pi$
$71$	−3.30951	−0.392767	−0.196383	+	0.980527i	$0.562920\pi$
$72$	0	0
$73$	1.07644	0.125987	0.0629937	−	0.998014i	$-0.479935\pi$
$73$	1.07644	0.125987	0.0629937	+	0.998014i	$0.479935\pi$
$74$	0	0
$75$	8.84182	1.02097
$76$	0	0
$77$	−17.8199	−2.03077
$78$	0	0
$79$	−10.1610	−1.14320	−0.571598	−	0.820534i	$-0.693676\pi$
$79$	−10.1610	−1.14320	−0.571598	+	0.820534i	$0.693676\pi$
$80$	0	0
$81$	2.44844	0.272049
$82$	0	0
$83$	−11.7519	−1.28993	−0.644967	−	0.764210i	$-0.723129\pi$
$83$	−11.7519	−1.28993	−0.644967	+	0.764210i	$0.723129\pi$
$84$	0	0
$85$	−21.7308	−2.35704
$86$	0	0
$87$	−1.49237	−0.159999
$88$	0	0
$89$	−16.5010	−1.74910	−0.874551	−	0.484933i	$-0.838844\pi$
$89$	−16.5010	−1.74910	−0.874551	+	0.484933i	$0.838844\pi$
$90$	0	0
$91$	1.93176	0.202503
$92$	0	0
$93$	0.331388	0.0343633
$94$	0	0
$95$	0.268468	0.0275442
$96$	0	0
$97$	−12.5981	−1.27914	−0.639572	−	0.768731i	$-0.720889\pi$
$97$	−12.5981	−1.27914	−0.639572	+	0.768731i	$0.720889\pi$
$98$	0	0
$99$	−10.3871	−1.04394
$100$	0	0
$101$	−16.9935	−1.69091	−0.845457	−	0.534044i	$-0.820672\pi$
$101$	−16.9935	−1.69091	−0.845457	+	0.534044i	$0.820672\pi$
$102$	0	0
$103$	15.7173	1.54867	0.774336	−	0.632775i	$-0.218084\pi$
$103$	15.7173	1.54867	0.774336	+	0.632775i	$0.218084\pi$
$104$	0	0
$105$	13.0251	1.27112
$106$	0	0
$107$	7.81103	0.755121	0.377561	−	0.925985i	$-0.376763\pi$
$107$	7.81103	0.755121	0.377561	+	0.925985i	$0.376763\pi$
$108$	0	0
$109$	−1.47716	−0.141486	−0.0707430	−	0.997495i	$-0.522537\pi$
$109$	−1.47716	−0.141486	−0.0707430	+	0.997495i	$0.522537\pi$
$110$	0	0
$111$	0.833791	0.0791399
$112$	0	0
$113$	−4.93874	−0.464598	−0.232299	−	0.972644i	$-0.574625\pi$
$113$	−4.93874	−0.464598	−0.232299	+	0.972644i	$0.574625\pi$
$114$	0	0
$115$	20.0770	1.87219
$116$	0	0
$117$	1.12600	0.104099
$118$	0	0
$119$	−21.2644	−1.94930
$120$	0	0
$121$	11.2684	1.02440
$122$	0	0
$123$	4.52677	0.408165
$124$	0	0
$125$	−18.8806	−1.68873
$126$	0	0
$127$	12.0431	1.06865	0.534325	−	0.845279i	$-0.320566\pi$
$127$	12.0431	1.06865	0.534325	+	0.845279i	$0.320566\pi$
$128$	0	0
$129$	5.70926	0.502672
$130$	0	0
$131$	−0.173395	−0.0151496	−0.00757479	−	0.999971i	$-0.502411\pi$
$131$	−0.173395	−0.0151496	−0.00757479	+	0.999971i	$0.502411\pi$
$132$	0	0
$133$	0.262705	0.0227795
$134$	0	0
$135$	17.9398	1.54401
$136$	0	0
$137$	−10.3811	−0.886914	−0.443457	−	0.896296i	$-0.646248\pi$
$137$	−10.3811	−0.886914	−0.443457	+	0.896296i	$0.646248\pi$
$138$	0	0
$139$	−14.1517	−1.20033	−0.600166	−	0.799875i	$-0.704899\pi$
$139$	−14.1517	−1.20033	−0.600166	+	0.799875i	$0.704899\pi$
$140$	0	0
$141$	3.75277	0.316041
$142$	0	0
$143$	−2.41399	−0.201868
$144$	0	0
$145$	6.44354	0.535107
$146$	0	0
$147$	6.48896	0.535201
$148$	0	0
$149$	19.1922	1.57228	0.786142	−	0.618046i	$-0.212076\pi$
$149$	19.1922	1.57228	0.786142	+	0.618046i	$0.212076\pi$
$150$	0	0
$151$	7.33354	0.596795	0.298397	−	0.954442i	$-0.403548\pi$
$151$	7.33354	0.596795	0.298397	+	0.954442i	$0.403548\pi$
$152$	0	0
$153$	−12.3948	−1.00206
$154$	0	0
$155$	−1.43082	−0.114926
$156$	0	0
$157$	20.3956	1.62775	0.813873	−	0.581043i	$-0.197355\pi$
$157$	20.3956	1.62775	0.813873	+	0.581043i	$0.197355\pi$
$158$	0	0
$159$	1.17103	0.0928687
$160$	0	0
$161$	19.6460	1.54832
$162$	0	0
$163$	−1.52868	−0.119735	−0.0598677	−	0.998206i	$-0.519068\pi$
$163$	−1.52868	−0.119735	−0.0598677	+	0.998206i	$0.519068\pi$
$164$	0	0
$165$	−16.2766	−1.26713
$166$	0	0
$167$	−6.89472	−0.533530	−0.266765	−	0.963762i	$-0.585955\pi$
$167$	−6.89472	−0.533530	−0.266765	+	0.963762i	$0.585955\pi$
$168$	0	0
$169$	−12.7383	−0.979870
$170$	0	0
$171$	0.153129	0.0117100
$172$	0	0
$173$	−24.2254	−1.84182	−0.920912	−	0.389770i	$-0.872555\pi$
$173$	−24.2254	−1.84182	−0.920912	+	0.389770i	$0.872555\pi$
$174$	0	0
$175$	−37.3566	−2.82389
$176$	0	0
$177$	11.6386	0.874810
$178$	0	0
$179$	−17.5123	−1.30893	−0.654467	−	0.756091i	$-0.727107\pi$
$179$	−17.5123	−1.30893	−0.654467	+	0.756091i	$0.727107\pi$
$180$	0	0
$181$	−14.0756	−1.04623	−0.523115	−	0.852262i	$-0.675230\pi$
$181$	−14.0756	−1.04623	−0.523115	+	0.852262i	$0.675230\pi$
$182$	0	0
$183$	9.30803	0.688069
$184$	0	0
$185$	−3.60003	−0.264679
$186$	0	0
$187$	26.5728	1.94319
$188$	0	0
$189$	17.5547	1.27692
$190$	0	0
$191$	18.0871	1.30874	0.654370	−	0.756175i	$-0.272934\pi$
$191$	18.0871	1.30874	0.654370	+	0.756175i	$0.272934\pi$
$192$	0	0
$193$	−4.16569	−0.299853	−0.149926	−	0.988697i	$-0.547904\pi$
$193$	−4.16569	−0.299853	−0.149926	+	0.988697i	$0.547904\pi$
$194$	0	0
$195$	1.76446	0.126355
$196$	0	0
$197$	−17.9008	−1.27538	−0.637689	−	0.770294i	$-0.720109\pi$
$197$	−17.9008	−1.27538	−0.637689	+	0.770294i	$0.720109\pi$
$198$	0	0
$199$	−18.8700	−1.33766	−0.668830	−	0.743415i	$-0.733205\pi$
$199$	−18.8700	−1.33766	−0.668830	+	0.743415i	$0.733205\pi$
$200$	0	0
$201$	−1.87664	−0.132368
$202$	0	0
$203$	6.30523	0.442541
$204$	0	0
$205$	−19.5451	−1.36509
$206$	0	0
$207$	11.4515	0.795933
$208$	0	0
$209$	−0.328287	−0.0227081
$210$	0	0
$211$	−18.2052	−1.25330	−0.626649	−	0.779302i	$-0.715574\pi$
$211$	−18.2052	−1.25330	−0.626649	+	0.779302i	$0.715574\pi$
$212$	0	0
$213$	−2.95801	−0.202679
$214$	0	0
$215$	−24.6507	−1.68116
$216$	0	0
$217$	−1.40011	−0.0950457
$218$	0	0
$219$	0.962107	0.0650132
$220$	0	0
$221$	−2.88060	−0.193770
$222$	0	0
$223$	−3.04587	−0.203967	−0.101983	−	0.994786i	$-0.532519\pi$
$223$	−3.04587	−0.203967	−0.101983	+	0.994786i	$0.532519\pi$
$224$	0	0
$225$	−21.7748	−1.45165
$226$	0	0
$227$	7.06816	0.469130	0.234565	−	0.972100i	$-0.424633\pi$
$227$	7.06816	0.469130	0.234565	+	0.972100i	$0.424633\pi$
$228$	0	0
$229$	3.43621	0.227071	0.113536	−	0.993534i	$-0.463782\pi$
$229$	3.43621	0.227071	0.113536	+	0.993534i	$0.463782\pi$
$230$	0	0
$231$	−15.9272	−1.04794
$232$	0	0
$233$	−19.0437	−1.24759	−0.623797	−	0.781586i	$-0.714411\pi$
$233$	−19.0437	−1.24759	−0.623797	+	0.781586i	$0.714411\pi$
$234$	0	0
$235$	−16.2032	−1.05698
$236$	0	0
$237$	−9.08175	−0.589923
$238$	0	0
$239$	20.8015	1.34553	0.672767	−	0.739854i	$-0.265106\pi$
$239$	20.8015	1.34553	0.672767	+	0.739854i	$0.265106\pi$
$240$	0	0
$241$	4.43572	0.285730	0.142865	−	0.989742i	$-0.454369\pi$
$241$	4.43572	0.285730	0.142865	+	0.989742i	$0.454369\pi$
$242$	0	0
$243$	16.1346	1.03503
$244$	0	0
$245$	−28.0172	−1.78995
$246$	0	0
$247$	0.0355877	0.00226439
$248$	0	0
$249$	−10.5037	−0.665644
$250$	0	0
$251$	22.4720	1.41842	0.709210	−	0.704997i	$-0.249052\pi$
$251$	22.4720	1.41842	0.709210	+	0.704997i	$0.249052\pi$
$252$	0	0
$253$	−24.5504	−1.54347
$254$	0	0
$255$	−19.4228	−1.21630
$256$	0	0
$257$	0.369671	0.0230595	0.0115297	−	0.999934i	$-0.496330\pi$
$257$	0.369671	0.0230595	0.0115297	+	0.999934i	$0.496330\pi$
$258$	0	0
$259$	−3.52276	−0.218893
$260$	0	0
$261$	3.67526	0.227493
$262$	0	0
$263$	−2.44589	−0.150820	−0.0754099	−	0.997153i	$-0.524027\pi$
$263$	−2.44589	−0.150820	−0.0754099	+	0.997153i	$0.524027\pi$
$264$	0	0
$265$	−5.05611	−0.310594
$266$	0	0
$267$	−14.7484	−0.902588
$268$	0	0
$269$	−3.85067	−0.234780	−0.117390	−	0.993086i	$-0.537453\pi$
$269$	−3.85067	−0.234780	−0.117390	+	0.993086i	$0.537453\pi$
$270$	0	0
$271$	29.7078	1.80462	0.902310	−	0.431087i	$-0.141870\pi$
$271$	29.7078	1.80462	0.902310	+	0.431087i	$0.141870\pi$
$272$	0	0
$273$	1.72658	0.104498
$274$	0	0
$275$	46.6822	2.81504
$276$	0	0
$277$	21.6836	1.30284	0.651420	−	0.758717i	$-0.274173\pi$
$277$	21.6836	1.30284	0.651420	+	0.758717i	$0.274173\pi$
$278$	0	0
$279$	−0.816111	−0.0488593
$280$	0	0
$281$	−2.96764	−0.177035	−0.0885174	−	0.996075i	$-0.528213\pi$
$281$	−2.96764	−0.177035	−0.0885174	+	0.996075i	$0.528213\pi$
$282$	0	0
$283$	0.541058	0.0321626	0.0160813	−	0.999871i	$-0.494881\pi$
$283$	0.541058	0.0321626	0.0160813	+	0.999871i	$0.494881\pi$
$284$	0	0
$285$	0.239954	0.0142136
$286$	0	0
$287$	−19.1255	−1.12894
$288$	0	0
$289$	14.7091	0.865241
$290$	0	0
$291$	−11.2601	−0.660076
$292$	0	0
$293$	−11.3653	−0.663966	−0.331983	−	0.943285i	$-0.607718\pi$
$293$	−11.3653	−0.663966	−0.331983	+	0.943285i	$0.607718\pi$
$294$	0	0
$295$	−50.2515	−2.92576
$296$	0	0
$297$	−21.9371	−1.27292
$298$	0	0
$299$	2.66137	0.153911
$300$	0	0
$301$	−24.1216	−1.39034
$302$	0	0
$303$	−15.1886	−0.872561
$304$	0	0
$305$	−40.1889	−2.30121
$306$	0	0
$307$	−8.95952	−0.511347	−0.255674	−	0.966763i	$-0.582297\pi$
$307$	−8.95952	−0.511347	−0.255674	+	0.966763i	$0.582297\pi$
$308$	0	0
$309$	14.0480	0.799160
$310$	0	0
$311$	22.9435	1.30101	0.650503	−	0.759504i	$-0.274558\pi$
$311$	22.9435	1.30101	0.650503	+	0.759504i	$0.274558\pi$
$312$	0	0
$313$	−29.0611	−1.64263	−0.821314	−	0.570477i	$-0.806759\pi$
$313$	−29.0611	−1.64263	−0.821314	+	0.570477i	$0.806759\pi$
$314$	0	0
$315$	−32.0769	−1.80733
$316$	0	0
$317$	22.3051	1.25278	0.626390	−	0.779510i	$-0.284532\pi$
$317$	22.3051	1.25278	0.626390	+	0.779510i	$0.284532\pi$
$318$	0	0
$319$	−7.87926	−0.441154
$320$	0	0
$321$	6.98142	0.389665
$322$	0	0
$323$	−0.391742	−0.0217971
$324$	0	0
$325$	−5.06056	−0.280709
$326$	0	0
$327$	−1.32027	−0.0730109
$328$	0	0
$329$	−15.8554	−0.874138
$330$	0	0
$331$	−16.7757	−0.922076	−0.461038	−	0.887380i	$-0.652523\pi$
$331$	−16.7757	−0.922076	−0.461038	+	0.887380i	$0.652523\pi$
$332$	0	0
$333$	−2.05338	−0.112525
$334$	0	0
$335$	8.10269	0.442697
$336$	0	0
$337$	−27.7893	−1.51378	−0.756890	−	0.653542i	$-0.773282\pi$
$337$	−27.7893	−1.51378	−0.756890	+	0.653542i	$0.773282\pi$
$338$	0	0
$339$	−4.41420	−0.239746
$340$	0	0
$341$	1.74963	0.0947478
$342$	0	0
$343$	−0.982048	−0.0530256
$344$	0	0
$345$	17.9446	0.966104
$346$	0	0
$347$	−1.80385	−0.0968359	−0.0484179	−	0.998827i	$-0.515418\pi$
$347$	−1.80385	−0.0968359	−0.0484179	+	0.998827i	$0.515418\pi$
$348$	0	0
$349$	24.4308	1.30775	0.653876	−	0.756602i	$-0.273142\pi$
$349$	24.4308	1.30775	0.653876	+	0.756602i	$0.273142\pi$
$350$	0	0
$351$	2.37807	0.126932
$352$	0	0
$353$	−9.08469	−0.483530	−0.241765	−	0.970335i	$-0.577726\pi$
$353$	−9.08469	−0.483530	−0.241765	+	0.970335i	$0.577726\pi$
$354$	0	0
$355$	12.7717	0.677850
$356$	0	0
$357$	−19.0059	−1.00590
$358$	0	0
$359$	24.4405	1.28992	0.644961	−	0.764215i	$-0.276873\pi$
$359$	24.4405	1.28992	0.644961	+	0.764215i	$0.276873\pi$
$360$	0	0
$361$	−18.9952	−0.999745
$362$	0	0
$363$	10.0716	0.528622
$364$	0	0
$365$	−4.15406	−0.217433
$366$	0	0
$367$	−34.6446	−1.80843	−0.904216	−	0.427075i	$-0.859544\pi$
$367$	−34.6446	−1.80843	−0.904216	+	0.427075i	$0.859544\pi$
$368$	0	0
$369$	−11.1481	−0.580347
$370$	0	0
$371$	−4.94758	−0.256866
$372$	0	0
$373$	−28.5395	−1.47772	−0.738859	−	0.673860i	$-0.764635\pi$
$373$	−28.5395	−1.47772	−0.738859	+	0.673860i	$0.764635\pi$
$374$	0	0
$375$	−16.8753	−0.871437
$376$	0	0
$377$	0.854146	0.0439907
$378$	0	0
$379$	−5.70976	−0.293291	−0.146645	−	0.989189i	$-0.546848\pi$
$379$	−5.70976	−0.293291	−0.146645	+	0.989189i	$0.546848\pi$
$380$	0	0
$381$	10.7640	0.551455
$382$	0	0
$383$	−20.8571	−1.06575	−0.532873	−	0.846195i	$-0.678888\pi$
$383$	−20.8571	−1.06575	−0.532873	+	0.846195i	$0.678888\pi$
$384$	0	0
$385$	68.7685	3.50477
$386$	0	0
$387$	−14.0602	−0.714722
$388$	0	0
$389$	−23.3177	−1.18225	−0.591126	−	0.806579i	$-0.701316\pi$
$389$	−23.3177	−1.18225	−0.591126	+	0.806579i	$0.701316\pi$
$390$	0	0
$391$	−29.2958	−1.48155
$392$	0	0
$393$	−0.154979	−0.00781763
$394$	0	0
$395$	39.2120	1.97297
$396$	0	0
$397$	−33.4612	−1.67937	−0.839684	−	0.543076i	$-0.817260\pi$
$397$	−33.4612	−1.67937	−0.839684	+	0.543076i	$0.817260\pi$
$398$	0	0
$399$	0.234803	0.0117549
$400$	0	0
$401$	−27.3510	−1.36584	−0.682922	−	0.730491i	$-0.739291\pi$
$401$	−27.3510	−1.36584	−0.682922	+	0.730491i	$0.739291\pi$
$402$	0	0
$403$	−0.189668	−0.00944801
$404$	0	0
$405$	−9.44874	−0.469512
$406$	0	0
$407$	4.40217	0.218207
$408$	0	0
$409$	−2.71339	−0.134169	−0.0670843	−	0.997747i	$-0.521370\pi$
$409$	−2.71339	−0.134169	−0.0670843	+	0.997747i	$0.521370\pi$
$410$	0	0
$411$	−9.27848	−0.457674
$412$	0	0
$413$	−49.1729	−2.41964
$414$	0	0
$415$	45.3514	2.22621
$416$	0	0
$417$	−12.6487	−0.619407
$418$	0	0
$419$	4.86004	0.237428	0.118714	−	0.992928i	$-0.462123\pi$
$419$	4.86004	0.237428	0.118714	+	0.992928i	$0.462123\pi$
$420$	0	0
$421$	13.2128	0.643952	0.321976	−	0.946748i	$-0.395653\pi$
$421$	13.2128	0.643952	0.321976	+	0.946748i	$0.395653\pi$
$422$	0	0
$423$	−9.24198	−0.449360
$424$	0	0
$425$	55.7056	2.70212
$426$	0	0
$427$	−39.3263	−1.90313
$428$	0	0
$429$	−2.15760	−0.104170
$430$	0	0
$431$	−23.1046	−1.11291	−0.556455	−	0.830878i	$-0.687839\pi$
$431$	−23.1046	−1.11291	−0.556455	+	0.830878i	$0.687839\pi$
$432$	0	0
$433$	2.97652	0.143043	0.0715213	−	0.997439i	$-0.477215\pi$
$433$	2.97652	0.143043	0.0715213	+	0.997439i	$0.477215\pi$
$434$	0	0
$435$	5.75917	0.276131
$436$	0	0
$437$	0.361928	0.0173134
$438$	0	0
$439$	−26.2388	−1.25231	−0.626155	−	0.779699i	$-0.715372\pi$
$439$	−26.2388	−1.25231	−0.626155	+	0.779699i	$0.715372\pi$
$440$	0	0
$441$	−15.9804	−0.760972
$442$	0	0
$443$	28.1621	1.33802	0.669012	−	0.743252i	$-0.266718\pi$
$443$	28.1621	1.33802	0.669012	+	0.743252i	$0.266718\pi$
$444$	0	0
$445$	63.6787	3.01866
$446$	0	0
$447$	17.1537	0.811344
$448$	0	0
$449$	14.4204	0.680541	0.340271	−	0.940328i	$-0.389481\pi$
$449$	14.4204	0.680541	0.340271	+	0.940328i	$0.389481\pi$
$450$	0	0
$451$	23.9000	1.12541
$452$	0	0
$453$	6.55464	0.307964
$454$	0	0
$455$	−7.45480	−0.349487
$456$	0	0
$457$	3.73893	0.174900	0.0874499	−	0.996169i	$-0.472128\pi$
$457$	3.73893	0.174900	0.0874499	+	0.996169i	$0.472128\pi$
$458$	0	0
$459$	−26.1774	−1.22185
$460$	0	0
$461$	−12.1666	−0.566654	−0.283327	−	0.959023i	$-0.591438\pi$
$461$	−12.1666	−0.566654	−0.283327	+	0.959023i	$0.591438\pi$
$462$	0	0
$463$	25.0242	1.16298	0.581488	−	0.813555i	$-0.302471\pi$
$463$	25.0242	1.16298	0.581488	+	0.813555i	$0.302471\pi$
$464$	0	0
$465$	−1.27885	−0.0593054
$466$	0	0
$467$	2.41158	0.111594	0.0557972	−	0.998442i	$-0.482230\pi$
$467$	2.41158	0.111594	0.0557972	+	0.998442i	$0.482230\pi$
$468$	0	0
$469$	7.92877	0.366117
$470$	0	0
$471$	18.2294	0.839965
$472$	0	0
$473$	30.1432	1.38599
$474$	0	0
$475$	−0.688201	−0.0315768
$476$	0	0
$477$	−2.88390	−0.132045
$478$	0	0
$479$	12.3217	0.562992	0.281496	−	0.959562i	$-0.409169\pi$
$479$	12.3217	0.562992	0.281496	+	0.959562i	$0.409169\pi$
$480$	0	0
$481$	−0.477214	−0.0217591
$482$	0	0
$483$	17.5594	0.798981
$484$	0	0
$485$	48.6172	2.20759
$486$	0	0
$487$	34.7800	1.57603	0.788016	−	0.615655i	$-0.211109\pi$
$487$	34.7800	1.57603	0.788016	+	0.615655i	$0.211109\pi$
$488$	0	0
$489$	−1.36632	−0.0617870
$490$	0	0
$491$	6.56397	0.296228	0.148114	−	0.988970i	$-0.452680\pi$
$491$	6.56397	0.296228	0.148114	+	0.988970i	$0.452680\pi$
$492$	0	0
$493$	−9.40226	−0.423457
$494$	0	0
$495$	40.0845	1.80166
$496$	0	0
$497$	12.4975	0.560591
$498$	0	0
$499$	−27.2924	−1.22178	−0.610888	−	0.791717i	$-0.709187\pi$
$499$	−27.2924	−1.22178	−0.610888	+	0.791717i	$0.709187\pi$
$500$	0	0
$501$	−6.16243	−0.275317
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	65.5792	2.91824
$506$	0	0
$507$	−11.3854	−0.505642
$508$	0	0
$509$	3.54941	0.157325	0.0786625	−	0.996901i	$-0.474935\pi$
$509$	3.54941	0.157325	0.0786625	+	0.996901i	$0.474935\pi$
$510$	0	0
$511$	−4.06489	−0.179820
$512$	0	0
$513$	0.323402	0.0142785
$514$	0	0
$515$	−60.6544	−2.67275
$516$	0	0
$517$	19.8135	0.871398
$518$	0	0
$519$	−21.6524	−0.950436
$520$	0	0
$521$	10.8364	0.474751	0.237376	−	0.971418i	$-0.423713\pi$
$521$	10.8364	0.474751	0.237376	+	0.971418i	$0.423713\pi$
$522$	0	0
$523$	−11.3289	−0.495377	−0.247689	−	0.968840i	$-0.579671\pi$
$523$	−11.3289	−0.495377	−0.247689	+	0.968840i	$0.579671\pi$
$524$	0	0
$525$	−33.3889	−1.45721
$526$	0	0
$527$	2.08782	0.0909470
$528$	0	0
$529$	4.06626	0.176794
$530$	0	0
$531$	−28.6624	−1.24384
$532$	0	0
$533$	−2.59086	−0.112223
$534$	0	0
$535$	−30.1434	−1.30321
$536$	0	0
$537$	−15.6523	−0.675449
$538$	0	0
$539$	34.2598	1.47567
$540$	0	0
$541$	−20.4018	−0.877140	−0.438570	−	0.898697i	$-0.644515\pi$
$541$	−20.4018	−0.877140	−0.438570	+	0.898697i	$0.644515\pi$
$542$	0	0
$543$	−12.5806	−0.539886
$544$	0	0
$545$	5.70047	0.244181
$546$	0	0
$547$	−24.3885	−1.04278	−0.521388	−	0.853319i	$-0.674586\pi$
$547$	−24.3885	−1.04278	−0.521388	+	0.853319i	$0.674586\pi$
$548$	0	0
$549$	−22.9229	−0.978327
$550$	0	0
$551$	0.116158	0.00494849
$552$	0	0
$553$	38.3703	1.63167
$554$	0	0
$555$	−3.21767	−0.136582
$556$	0	0
$557$	−12.3148	−0.521793	−0.260897	−	0.965367i	$-0.584018\pi$
$557$	−12.3148	−0.521793	−0.260897	+	0.965367i	$0.584018\pi$
$558$	0	0
$559$	−3.26765	−0.138207
$560$	0	0
$561$	23.7505	1.00274
$562$	0	0
$563$	−29.9321	−1.26149	−0.630745	−	0.775990i	$-0.717250\pi$
$563$	−29.9321	−1.26149	−0.630745	+	0.775990i	$0.717250\pi$
$564$	0	0
$565$	19.0590	0.801819
$566$	0	0
$567$	−9.24593	−0.388293
$568$	0	0
$569$	19.1154	0.801360	0.400680	−	0.916218i	$-0.368774\pi$
$569$	19.1154	0.801360	0.400680	+	0.916218i	$0.368774\pi$
$570$	0	0
$571$	−15.4172	−0.645191	−0.322595	−	0.946537i	$-0.604555\pi$
$571$	−15.4172	−0.645191	−0.322595	+	0.946537i	$0.604555\pi$
$572$	0	0
$573$	16.1661	0.675348
$574$	0	0
$575$	−51.4661	−2.14628
$576$	0	0
$577$	4.37226	0.182019	0.0910097	−	0.995850i	$-0.470991\pi$
$577$	4.37226	0.182019	0.0910097	+	0.995850i	$0.470991\pi$
$578$	0	0
$579$	−3.72325	−0.154733
$580$	0	0
$581$	44.3780	1.84111
$582$	0	0
$583$	6.18269	0.256061
$584$	0	0
$585$	−4.34534	−0.179657
$586$	0	0
$587$	−16.8749	−0.696500	−0.348250	−	0.937402i	$-0.613224\pi$
$587$	−16.8749	−0.696500	−0.348250	+	0.937402i	$0.613224\pi$
$588$	0	0
$589$	−0.0257935	−0.00106280
$590$	0	0
$591$	−15.9995	−0.658133
$592$	0	0
$593$	25.1993	1.03481	0.517405	−	0.855741i	$-0.326898\pi$
$593$	25.1993	1.03481	0.517405	+	0.855741i	$0.326898\pi$
$594$	0	0
$595$	82.0610	3.36417
$596$	0	0
$597$	−16.8658	−0.690272
$598$	0	0
$599$	−33.5870	−1.37233	−0.686165	−	0.727446i	$-0.740707\pi$
$599$	−33.5870	−1.37233	−0.686165	+	0.727446i	$0.740707\pi$
$600$	0	0
$601$	19.6769	0.802636	0.401318	−	0.915939i	$-0.368552\pi$
$601$	19.6769	0.802636	0.401318	+	0.915939i	$0.368552\pi$
$602$	0	0
$603$	4.62161	0.188206
$604$	0	0
$605$	−43.4858	−1.76795
$606$	0	0
$607$	0.481239	0.0195329	0.00976644	−	0.999952i	$-0.496891\pi$
$607$	0.481239	0.0195329	0.00976644	+	0.999952i	$0.496891\pi$
$608$	0	0
$609$	5.63555	0.228364
$610$	0	0
$611$	−2.14787	−0.0868937
$612$	0	0
$613$	−7.08677	−0.286232	−0.143116	−	0.989706i	$-0.545712\pi$
$613$	−7.08677	−0.286232	−0.143116	+	0.989706i	$0.545712\pi$
$614$	0	0
$615$	−17.4692	−0.704425
$616$	0	0
$617$	5.30571	0.213600	0.106800	−	0.994281i	$-0.465940\pi$
$617$	5.30571	0.213600	0.106800	+	0.994281i	$0.465940\pi$
$618$	0	0
$619$	−30.7918	−1.23763	−0.618813	−	0.785539i	$-0.712386\pi$
$619$	−30.7918	−1.23763	−0.618813	+	0.785539i	$0.712386\pi$
$620$	0	0
$621$	24.1851	0.970515
$622$	0	0
$623$	62.3119	2.49647
$624$	0	0
$625$	23.3993	0.935972
$626$	0	0
$627$	−0.293419	−0.0117180
$628$	0	0
$629$	5.25308	0.209454
$630$	0	0
$631$	35.2732	1.40421	0.702103	−	0.712076i	$-0.252245\pi$
$631$	35.2732	1.40421	0.702103	+	0.712076i	$0.252245\pi$
$632$	0	0
$633$	−16.2716	−0.646739
$634$	0	0
$635$	−46.4752	−1.84431
$636$	0	0
$637$	−3.71391	−0.147151
$638$	0	0
$639$	7.28470	0.288178
$640$	0	0
$641$	17.8655	0.705644	0.352822	−	0.935690i	$-0.385222\pi$
$641$	17.8655	0.705644	0.352822	+	0.935690i	$0.385222\pi$
$642$	0	0
$643$	8.61818	0.339868	0.169934	−	0.985455i	$-0.445645\pi$
$643$	8.61818	0.339868	0.169934	+	0.985455i	$0.445645\pi$
$644$	0	0
$645$	−22.0325	−0.867529
$646$	0	0
$647$	−15.4910	−0.609016	−0.304508	−	0.952510i	$-0.598492\pi$
$647$	−15.4910	−0.609016	−0.304508	+	0.952510i	$0.598492\pi$
$648$	0	0
$649$	61.4483	2.41206
$650$	0	0
$651$	−1.25140	−0.0490464
$652$	0	0
$653$	21.5939	0.845035	0.422517	−	0.906355i	$-0.361147\pi$
$653$	21.5939	0.845035	0.422517	+	0.906355i	$0.361147\pi$
$654$	0	0
$655$	0.669145	0.0261457
$656$	0	0
$657$	−2.36939	−0.0924386
$658$	0	0
$659$	−25.2479	−0.983517	−0.491759	−	0.870731i	$-0.663646\pi$
$659$	−25.2479	−0.983517	−0.491759	+	0.870731i	$0.663646\pi$
$660$	0	0
$661$	−26.7554	−1.04067	−0.520333	−	0.853964i	$-0.674192\pi$
$661$	−26.7554	−1.04067	−0.520333	+	0.853964i	$0.674192\pi$
$662$	0	0
$663$	−2.57465	−0.0999912
$664$	0	0
$665$	−1.01380	−0.0393136
$666$	0	0
$667$	8.68670	0.336350
$668$	0	0
$669$	−2.72237	−0.105253
$670$	0	0
$671$	49.1436	1.89717
$672$	0	0
$673$	44.1952	1.70360	0.851801	−	0.523866i	$-0.175511\pi$
$673$	44.1952	1.70360	0.851801	+	0.523866i	$0.175511\pi$
$674$	0	0
$675$	−45.9876	−1.77006
$676$	0	0
$677$	−25.5496	−0.981952	−0.490976	−	0.871173i	$-0.663360\pi$
$677$	−25.5496	−0.981952	−0.490976	+	0.871173i	$0.663360\pi$
$678$	0	0
$679$	47.5736	1.82571
$680$	0	0
$681$	6.31745	0.242085
$682$	0	0
$683$	4.16513	0.159374	0.0796872	−	0.996820i	$-0.474608\pi$
$683$	4.16513	0.159374	0.0796872	+	0.996820i	$0.474608\pi$
$684$	0	0
$685$	40.0614	1.53067
$686$	0	0
$687$	3.07125	0.117175
$688$	0	0
$689$	−0.670230	−0.0255337
$690$	0	0
$691$	−34.4452	−1.31036	−0.655179	−	0.755474i	$-0.727407\pi$
$691$	−34.4452	−1.31036	−0.655179	+	0.755474i	$0.727407\pi$
$692$	0	0
$693$	39.2241	1.49000
$694$	0	0
$695$	54.6126	2.07158
$696$	0	0
$697$	28.5197	1.08026
$698$	0	0
$699$	−17.0210	−0.643795
$700$	0	0
$701$	−19.9463	−0.753363	−0.376682	−	0.926343i	$-0.622935\pi$
$701$	−19.9463	−0.753363	−0.376682	+	0.926343i	$0.622935\pi$
$702$	0	0
$703$	−0.0648979	−0.00244767
$704$	0	0
$705$	−14.4823	−0.545434
$706$	0	0
$707$	64.1716	2.41342
$708$	0	0
$709$	−42.3849	−1.59180	−0.795900	−	0.605429i	$-0.793002\pi$
$709$	−42.3849	−1.59180	−0.795900	+	0.605429i	$0.793002\pi$
$710$	0	0
$711$	22.3657	0.838779
$712$	0	0
$713$	−1.92893	−0.0722389
$714$	0	0
$715$	9.31580	0.348391
$716$	0	0
$717$	18.5921	0.694336
$718$	0	0
$719$	29.6547	1.10593	0.552967	−	0.833203i	$-0.313495\pi$
$719$	29.6547	1.10593	0.552967	+	0.833203i	$0.313495\pi$
$720$	0	0
$721$	−59.3525	−2.21040
$722$	0	0
$723$	3.96460	0.147445
$724$	0	0
$725$	−16.5176	−0.613449
$726$	0	0
$727$	−9.18661	−0.340713	−0.170356	−	0.985383i	$-0.554492\pi$
$727$	−9.18661	−0.340713	−0.170356	+	0.985383i	$0.554492\pi$
$728$	0	0
$729$	7.07557	0.262058
$730$	0	0
$731$	35.9697	1.33039
$732$	0	0
$733$	−3.92172	−0.144852	−0.0724260	−	0.997374i	$-0.523074\pi$
$733$	−3.92172	−0.144852	−0.0724260	+	0.997374i	$0.523074\pi$
$734$	0	0
$735$	−25.0414	−0.923668
$736$	0	0
$737$	−9.90809	−0.364969
$738$	0	0
$739$	40.3182	1.48313	0.741564	−	0.670882i	$-0.234084\pi$
$739$	40.3182	1.48313	0.741564	+	0.670882i	$0.234084\pi$
$740$	0	0
$741$	0.0318079	0.00116849
$742$	0	0
$743$	−12.1172	−0.444535	−0.222268	−	0.974986i	$-0.571346\pi$
$743$	−12.1172	−0.444535	−0.222268	+	0.974986i	$0.571346\pi$
$744$	0	0
$745$	−74.0641	−2.71350
$746$	0	0
$747$	25.8675	0.946442
$748$	0	0
$749$	−29.4964	−1.07778
$750$	0	0
$751$	15.8609	0.578771	0.289386	−	0.957213i	$-0.406549\pi$
$751$	15.8609	0.578771	0.289386	+	0.957213i	$0.406549\pi$
$752$	0	0
$753$	20.0852	0.731947
$754$	0	0
$755$	−28.3007	−1.02997
$756$	0	0
$757$	24.4267	0.887805	0.443902	−	0.896075i	$-0.353594\pi$
$757$	24.4267	0.887805	0.443902	+	0.896075i	$0.353594\pi$
$758$	0	0
$759$	−21.9429	−0.796477
$760$	0	0
$761$	20.2267	0.733216	0.366608	−	0.930375i	$-0.380519\pi$
$761$	20.2267	0.733216	0.366608	+	0.930375i	$0.380519\pi$
$762$	0	0
$763$	5.57811	0.201941
$764$	0	0
$765$	47.8326	1.72939
$766$	0	0
$767$	−6.66126	−0.240524
$768$	0	0
$769$	−2.17563	−0.0784551	−0.0392275	−	0.999230i	$-0.512490\pi$
$769$	−2.17563	−0.0784551	−0.0392275	+	0.999230i	$0.512490\pi$
$770$	0	0
$771$	0.330408	0.0118994
$772$	0	0
$773$	3.49777	0.125806	0.0629030	−	0.998020i	$-0.479964\pi$
$773$	3.49777	0.125806	0.0629030	+	0.998020i	$0.479964\pi$
$774$	0	0
$775$	3.66782	0.131752
$776$	0	0
$777$	−3.14860	−0.112955
$778$	0	0
$779$	−0.352339	−0.0126239
$780$	0	0
$781$	−15.6174	−0.558834
$782$	0	0
$783$	7.76201	0.277392
$784$	0	0
$785$	−78.7083	−2.80922
$786$	0	0
$787$	25.5395	0.910384	0.455192	−	0.890393i	$-0.349571\pi$
$787$	25.5395	0.910384	0.455192	+	0.890393i	$0.349571\pi$
$788$	0	0
$789$	−2.18611	−0.0778275
$790$	0	0
$791$	18.6499	0.663115
$792$	0	0
$793$	−5.32738	−0.189181
$794$	0	0
$795$	−4.51910	−0.160276
$796$	0	0
$797$	−31.0283	−1.09908	−0.549540	−	0.835467i	$-0.685197\pi$
$797$	−31.0283	−1.09908	−0.549540	+	0.835467i	$0.685197\pi$
$798$	0	0
$799$	23.6434	0.836442
$800$	0	0
$801$	36.3210	1.28334
$802$	0	0
$803$	5.07964	0.179257
$804$	0	0
$805$	−75.8156	−2.67215
$806$	0	0
$807$	−3.44169	−0.121153
$808$	0	0
$809$	33.7935	1.18812	0.594059	−	0.804422i	$-0.297525\pi$
$809$	33.7935	1.18812	0.594059	+	0.804422i	$0.297525\pi$
$810$	0	0
$811$	14.3158	0.502697	0.251348	−	0.967897i	$-0.419126\pi$
$811$	14.3158	0.502697	0.251348	+	0.967897i	$0.419126\pi$
$812$	0	0
$813$	26.5525	0.931237
$814$	0	0
$815$	5.89930	0.206643
$816$	0	0
$817$	−0.444378	−0.0155468
$818$	0	0
$819$	−4.25207	−0.148579
$820$	0	0
$821$	−46.7093	−1.63017	−0.815083	−	0.579344i	$-0.803309\pi$
$821$	−46.7093	−1.63017	−0.815083	+	0.579344i	$0.803309\pi$
$822$	0	0
$823$	−8.77876	−0.306008	−0.153004	−	0.988226i	$-0.548895\pi$
$823$	−8.77876	−0.306008	−0.153004	+	0.988226i	$0.548895\pi$
$824$	0	0
$825$	41.7241	1.45265
$826$	0	0
$827$	40.5870	1.41135	0.705674	−	0.708537i	$-0.250644\pi$
$827$	40.5870	1.41135	0.705674	+	0.708537i	$0.250644\pi$
$828$	0	0
$829$	36.2857	1.26025	0.630127	−	0.776492i	$-0.283003\pi$
$829$	36.2857	1.26025	0.630127	+	0.776492i	$0.283003\pi$
$830$	0	0
$831$	19.3806	0.672304
$832$	0	0
$833$	40.8820	1.41648
$834$	0	0
$835$	26.6073	0.920784
$836$	0	0
$837$	−1.72360	−0.0595762
$838$	0	0
$839$	18.1603	0.626963	0.313481	−	0.949594i	$-0.398505\pi$
$839$	18.1603	0.626963	0.313481	+	0.949594i	$0.398505\pi$
$840$	0	0
$841$	−26.2121	−0.903865
$842$	0	0
$843$	−2.65245	−0.0913551
$844$	0	0
$845$	49.1582	1.69109
$846$	0	0
$847$	−42.5524	−1.46212
$848$	0	0
$849$	0.483592	0.0165968
$850$	0	0
$851$	−4.85329	−0.166369
$852$	0	0
$853$	−42.2922	−1.44806	−0.724028	−	0.689771i	$-0.757711\pi$
$853$	−42.2922	−1.44806	−0.724028	+	0.689771i	$0.757711\pi$
$854$	0	0
$855$	−0.590936	−0.0202096
$856$	0	0
$857$	39.9946	1.36619	0.683095	−	0.730330i	$-0.260633\pi$
$857$	39.9946	1.36619	0.683095	+	0.730330i	$0.260633\pi$
$858$	0	0
$859$	−55.2653	−1.88563	−0.942813	−	0.333322i	$-0.891830\pi$
$859$	−55.2653	−1.88563	−0.942813	+	0.333322i	$0.891830\pi$
$860$	0	0
$861$	−17.0942	−0.582569
$862$	0	0
$863$	−14.9582	−0.509185	−0.254592	−	0.967048i	$-0.581941\pi$
$863$	−14.9582	−0.509185	−0.254592	+	0.967048i	$0.581941\pi$
$864$	0	0
$865$	93.4879	3.17868
$866$	0	0
$867$	13.1468	0.446490
$868$	0	0
$869$	−47.9490	−1.62656
$870$	0	0
$871$	1.07408	0.0363938
$872$	0	0
$873$	27.7302	0.938526
$874$	0	0
$875$	71.2979	2.41031
$876$	0	0
$877$	−33.7083	−1.13825	−0.569124	−	0.822252i	$-0.692717\pi$
$877$	−33.7083	−1.13825	−0.569124	+	0.822252i	$0.692717\pi$
$878$	0	0
$879$	−10.1582	−0.342626
$880$	0	0
$881$	24.4774	0.824666	0.412333	−	0.911033i	$-0.364714\pi$
$881$	24.4774	0.824666	0.412333	+	0.911033i	$0.364714\pi$
$882$	0	0
$883$	−22.1930	−0.746853	−0.373426	−	0.927660i	$-0.621817\pi$
$883$	−22.1930	−0.746853	−0.373426	+	0.927660i	$0.621817\pi$
$884$	0	0
$885$	−44.9143	−1.50978
$886$	0	0
$887$	−8.44417	−0.283528	−0.141764	−	0.989901i	$-0.545277\pi$
$887$	−8.44417	−0.283528	−0.141764	+	0.989901i	$0.545277\pi$
$888$	0	0
$889$	−45.4777	−1.52527
$890$	0	0
$891$	11.5541	0.387076
$892$	0	0
$893$	−0.292096	−0.00977462
$894$	0	0
$895$	67.5816	2.25900
$896$	0	0
$897$	2.37871	0.0794227
$898$	0	0
$899$	−0.619073	−0.0206472
$900$	0	0
$901$	7.37776	0.245789
$902$	0	0
$903$	−21.5596	−0.717458
$904$	0	0
$905$	54.3188	1.80562
$906$	0	0
$907$	−16.7572	−0.556413	−0.278207	−	0.960521i	$-0.589740\pi$
$907$	−16.7572	−0.556413	−0.278207	+	0.960521i	$0.589740\pi$
$908$	0	0
$909$	37.4050	1.24065
$910$	0	0
$911$	48.2461	1.59847	0.799233	−	0.601021i	$-0.205239\pi$
$911$	48.2461	1.59847	0.799233	+	0.601021i	$0.205239\pi$
$912$	0	0
$913$	−55.4564	−1.83534
$914$	0	0
$915$	−35.9204	−1.18749
$916$	0	0
$917$	0.654782	0.0216228
$918$	0	0
$919$	−18.0249	−0.594587	−0.297294	−	0.954786i	$-0.596084\pi$
$919$	−18.0249	−0.594587	−0.297294	+	0.954786i	$0.596084\pi$
$920$	0	0
$921$	−8.00793	−0.263870
$922$	0	0
$923$	1.69299	0.0557256
$924$	0	0
$925$	9.22845	0.303430
$926$	0	0
$927$	−34.5960	−1.13628
$928$	0	0
$929$	9.46901	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$929$	9.46901	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$930$	0	0
$931$	−0.505066	−0.0165529
$932$	0	0
$933$	20.5067	0.671358
$934$	0	0
$935$	−102.546	−3.35363
$936$	0	0
$937$	−29.3194	−0.957824	−0.478912	−	0.877863i	$-0.658969\pi$
$937$	−29.3194	−0.957824	−0.478912	+	0.877863i	$0.658969\pi$
$938$	0	0
$939$	−25.9745	−0.847644
$940$	0	0
$941$	54.4765	1.77588	0.887942	−	0.459956i	$-0.152135\pi$
$941$	54.4765	1.77588	0.887942	+	0.459956i	$0.152135\pi$
$942$	0	0
$943$	−26.3492	−0.858047
$944$	0	0
$945$	−67.7452	−2.20375
$946$	0	0
$947$	0.898972	0.0292127	0.0146063	−	0.999893i	$-0.495350\pi$
$947$	0.898972	0.0292127	0.0146063	+	0.999893i	$0.495350\pi$
$948$	0	0
$949$	−0.550655	−0.0178750
$950$	0	0
$951$	19.9361	0.646472
$952$	0	0
$953$	−47.1222	−1.52644	−0.763218	−	0.646141i	$-0.776382\pi$
$953$	−47.1222	−1.52644	−0.763218	+	0.646141i	$0.776382\pi$
$954$	0	0
$955$	−69.7998	−2.25867
$956$	0	0
$957$	−7.04239	−0.227648
$958$	0	0
$959$	39.2015	1.26588
$960$	0	0
$961$	−30.8625	−0.995566
$962$	0	0
$963$	−17.1932	−0.554043
$964$	0	0
$965$	16.0757	0.517496
$966$	0	0
$967$	−24.3871	−0.784235	−0.392118	−	0.919915i	$-0.628257\pi$
$967$	−24.3871	−0.784235	−0.392118	+	0.919915i	$0.628257\pi$
$968$	0	0
$969$	−0.350135	−0.0112480
$970$	0	0
$971$	36.8714	1.18326	0.591630	−	0.806209i	$-0.298485\pi$
$971$	36.8714	1.18326	0.591630	+	0.806209i	$0.298485\pi$
$972$	0	0
$973$	53.4404	1.71322
$974$	0	0
$975$	−4.52307	−0.144854
$976$	0	0
$977$	10.0886	0.322763	0.161382	−	0.986892i	$-0.448405\pi$
$977$	10.0886	0.322763	0.161382	+	0.986892i	$0.448405\pi$
$978$	0	0
$979$	−77.8673	−2.48865
$980$	0	0
$981$	3.25143	0.103810
$982$	0	0
$983$	−29.7977	−0.950399	−0.475200	−	0.879878i	$-0.657624\pi$
$983$	−29.7977	−0.950399	−0.475200	+	0.879878i	$0.657624\pi$
$984$	0	0
$985$	69.0806	2.20109
$986$	0	0
$987$	−14.1714	−0.451081
$988$	0	0
$989$	−33.2322	−1.05672
$990$	0	0
$991$	14.3363	0.455408	0.227704	−	0.973730i	$-0.426878\pi$
$991$	14.3363	0.455408	0.227704	+	0.973730i	$0.426878\pi$
$992$	0	0
$993$	−14.9939	−0.475818
$994$	0	0
$995$	72.8209	2.30858
$996$	0	0
$997$	−5.98970	−0.189696	−0.0948478	−	0.995492i	$-0.530236\pi$
$997$	−5.98970	−0.189696	−0.0948478	+	0.995492i	$0.530236\pi$
$998$	0	0
$999$	−4.33666	−0.137206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.19		28
4.3	odd	2		4024.2.a.d.1.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.10	✓	28	4.3	odd	2
8048.2.a.v.1.19		28	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)

\(f(q)\)	\(=\)	\(q+0.893789 q^{3} -3.85908 q^{5} -3.77625 q^{7} -2.20114 q^{9} +O(q^{10})\)	\(q+0.893789 q^{3} -3.85908 q^{5} -3.77625 q^{7} -2.20114 q^{9} +4.71894 q^{11} -0.511554 q^{13} -3.44921 q^{15} +5.63108 q^{17} -0.0695678 q^{19} -3.37517 q^{21} -5.20252 q^{23} +9.89252 q^{25} -4.64872 q^{27} -1.66971 q^{29} +0.370767 q^{31} +4.21774 q^{33} +14.5729 q^{35} +0.932872 q^{37} -0.457221 q^{39} +5.06469 q^{41} +6.38770 q^{43} +8.49438 q^{45} +4.19872 q^{47} +7.26006 q^{49} +5.03300 q^{51} +1.31018 q^{53} -18.2108 q^{55} -0.0621790 q^{57} +13.0216 q^{59} +10.4141 q^{61} +8.31206 q^{63} +1.97413 q^{65} -2.09964 q^{67} -4.64996 q^{69} -3.30951 q^{71} +1.07644 q^{73} +8.84182 q^{75} -17.8199 q^{77} -10.1610 q^{79} +2.44844 q^{81} -11.7519 q^{83} -21.7308 q^{85} -1.49237 q^{87} -16.5010 q^{89} +1.93176 q^{91} +0.331388 q^{93} +0.268468 q^{95} -12.5981 q^{97} -10.3871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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