LMFDB - Embedded newform 6039.2.a.p.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18061 q^{2} -0.606158 q^{4} -0.892999 q^{5} -4.86985 q^{7} -3.07686 q^{8} +O(q^{10})$	\(q+1.18061 q^{2} -0.606158 q^{4} -0.892999 q^{5} -4.86985 q^{7} -3.07686 q^{8} -1.05428 q^{10} -1.00000 q^{11} -2.04408 q^{13} -5.74940 q^{14} -2.42026 q^{16} +0.0469432 q^{17} -5.61003 q^{19} +0.541298 q^{20} -1.18061 q^{22} -5.07449 q^{23} -4.20255 q^{25} -2.41326 q^{26} +2.95190 q^{28} -8.04512 q^{29} +6.36277 q^{31} +3.29633 q^{32} +0.0554216 q^{34} +4.34878 q^{35} +6.41813 q^{37} -6.62326 q^{38} +2.74763 q^{40} -0.619902 q^{41} -8.82600 q^{43} +0.606158 q^{44} -5.99099 q^{46} -5.24012 q^{47} +16.7155 q^{49} -4.96158 q^{50} +1.23903 q^{52} +10.2366 q^{53} +0.892999 q^{55} +14.9839 q^{56} -9.49815 q^{58} -8.39059 q^{59} +1.00000 q^{61} +7.51195 q^{62} +8.73220 q^{64} +1.82536 q^{65} +1.06778 q^{67} -0.0284549 q^{68} +5.13421 q^{70} -4.12438 q^{71} +7.37393 q^{73} +7.57731 q^{74} +3.40056 q^{76} +4.86985 q^{77} -12.2271 q^{79} +2.16129 q^{80} -0.731863 q^{82} -3.44275 q^{83} -0.0419202 q^{85} -10.4201 q^{86} +3.07686 q^{88} +15.1497 q^{89} +9.95436 q^{91} +3.07594 q^{92} -6.18655 q^{94} +5.00975 q^{95} +7.25811 q^{97} +19.7345 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

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 $

$2$	1.18061	0.834818	0.417409	−	0.908719i	$-0.362938\pi$
$2$	1.18061	0.834818	0.417409	+	0.908719i	$0.362938\pi$
$3$	0	0
$3$	0	0
$4$	−0.606158	−0.303079
$5$	−0.892999	−0.399361	−0.199681	−	0.979861i	$-0.563990\pi$
$5$	−0.892999	−0.399361	−0.199681	+	0.979861i	$0.563990\pi$
$6$	0	0
$7$	−4.86985	−1.84063	−0.920316	−	0.391176i	$-0.872069\pi$
$7$	−4.86985	−1.84063	−0.920316	+	0.391176i	$0.872069\pi$
$8$	−3.07686	−1.08783
$9$	0	0
$10$	−1.05428	−0.333394
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.04408	−0.566925	−0.283463	−	0.958983i	$-0.591483\pi$
$13$	−2.04408	−0.566925	−0.283463	+	0.958983i	$0.591483\pi$
$14$	−5.74940	−1.53659
$15$	0	0
$16$	−2.42026	−0.605065
$17$	0.0469432	0.0113854	0.00569269	−	0.999984i	$-0.498188\pi$
$17$	0.0469432	0.0113854	0.00569269	+	0.999984i	$0.498188\pi$
$18$	0	0
$19$	−5.61003	−1.28703	−0.643514	−	0.765434i	$-0.722524\pi$
$19$	−5.61003	−1.28703	−0.643514	+	0.765434i	$0.722524\pi$
$20$	0.541298	0.121038
$21$	0	0
$22$	−1.18061	−0.251707
$23$	−5.07449	−1.05810	−0.529052	−	0.848590i	$-0.677452\pi$
$23$	−5.07449	−1.05810	−0.529052	+	0.848590i	$0.677452\pi$
$24$	0	0
$25$	−4.20255	−0.840510
$26$	−2.41326	−0.473279
$27$	0	0
$28$	2.95190	0.557857
$29$	−8.04512	−1.49394	−0.746970	−	0.664857i	$-0.768492\pi$
$29$	−8.04512	−1.49394	−0.746970	+	0.664857i	$0.768492\pi$
$30$	0	0
$31$	6.36277	1.14279	0.571393	−	0.820676i	$-0.306403\pi$
$31$	6.36277	1.14279	0.571393	+	0.820676i	$0.306403\pi$
$32$	3.29633	0.582715
$33$	0	0
$34$	0.0554216	0.00950473
$35$	4.34878	0.735077
$36$	0	0
$37$	6.41813	1.05513	0.527567	−	0.849514i	$-0.323104\pi$
$37$	6.41813	1.05513	0.527567	+	0.849514i	$0.323104\pi$
$38$	−6.62326	−1.07443
$39$	0	0
$40$	2.74763	0.434439
$41$	−0.619902	−0.0968125	−0.0484062	−	0.998828i	$-0.515414\pi$
$41$	−0.619902	−0.0968125	−0.0484062	+	0.998828i	$0.515414\pi$
$42$	0	0
$43$	−8.82600	−1.34595	−0.672976	−	0.739664i	$-0.734984\pi$
$43$	−8.82600	−1.34595	−0.672976	+	0.739664i	$0.734984\pi$
$44$	0.606158	0.0913817
$45$	0	0
$46$	−5.99099	−0.883324
$47$	−5.24012	−0.764350	−0.382175	−	0.924090i	$-0.624825\pi$
$47$	−5.24012	−0.764350	−0.382175	+	0.924090i	$0.624825\pi$
$48$	0	0
$49$	16.7155	2.38793
$50$	−4.96158	−0.701673
$51$	0	0
$52$	1.23903	0.171823
$53$	10.2366	1.40611	0.703053	−	0.711137i	$-0.251819\pi$
$53$	10.2366	1.40611	0.703053	+	0.711137i	$0.251819\pi$
$54$	0	0
$55$	0.892999	0.120412
$56$	14.9839	2.00230
$57$	0	0
$58$	−9.49815	−1.24717
$59$	−8.39059	−1.09236	−0.546181	−	0.837667i	$-0.683919\pi$
$59$	−8.39059	−1.09236	−0.546181	+	0.837667i	$0.683919\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	7.51195	0.954019
$63$	0	0
$64$	8.73220	1.09153
$65$	1.82536	0.226408
$66$	0	0
$67$	1.06778	0.130450	0.0652250	−	0.997871i	$-0.479223\pi$
$67$	1.06778	0.130450	0.0652250	+	0.997871i	$0.479223\pi$
$68$	−0.0284549	−0.00345067
$69$	0	0
$70$	5.13421	0.613656
$71$	−4.12438	−0.489474	−0.244737	−	0.969589i	$-0.578702\pi$
$71$	−4.12438	−0.489474	−0.244737	+	0.969589i	$0.578702\pi$
$72$	0	0
$73$	7.37393	0.863054	0.431527	−	0.902100i	$-0.357975\pi$
$73$	7.37393	0.863054	0.431527	+	0.902100i	$0.357975\pi$
$74$	7.57731	0.880845
$75$	0	0
$76$	3.40056	0.390071
$77$	4.86985	0.554971
$78$	0	0
$79$	−12.2271	−1.37566	−0.687831	−	0.725871i	$-0.741437\pi$
$79$	−12.2271	−1.37566	−0.687831	+	0.725871i	$0.741437\pi$
$80$	2.16129	0.241639
$81$	0	0
$82$	−0.731863	−0.0808208
$83$	−3.44275	−0.377891	−0.188945	−	0.981988i	$-0.560507\pi$
$83$	−3.44275	−0.377891	−0.188945	+	0.981988i	$0.560507\pi$
$84$	0	0
$85$	−0.0419202	−0.00454688
$86$	−10.4201	−1.12363
$87$	0	0
$88$	3.07686	0.327994
$89$	15.1497	1.60586	0.802931	−	0.596072i	$-0.203273\pi$
$89$	15.1497	1.60586	0.802931	+	0.596072i	$0.203273\pi$
$90$	0	0
$91$	9.95436	1.04350
$92$	3.07594	0.320689
$93$	0	0
$94$	−6.18655	−0.638094
$95$	5.00975	0.513990
$96$	0	0
$97$	7.25811	0.736950	0.368475	−	0.929638i	$-0.379880\pi$
$97$	7.25811	0.736950	0.368475	+	0.929638i	$0.379880\pi$
$98$	19.7345	1.99348
$99$	0	0
$100$	2.54741	0.254741
$101$	9.91171	0.986252	0.493126	−	0.869958i	$-0.335854\pi$
$101$	9.91171	0.986252	0.493126	+	0.869958i	$0.335854\pi$
$102$	0	0
$103$	1.40216	0.138159	0.0690796	−	0.997611i	$-0.477994\pi$
$103$	1.40216	0.138159	0.0690796	+	0.997611i	$0.477994\pi$
$104$	6.28934	0.616720
$105$	0	0
$106$	12.0855	1.17384
$107$	−15.8138	−1.52878	−0.764388	−	0.644757i	$-0.776959\pi$
$107$	−15.8138	−1.52878	−0.764388	+	0.644757i	$0.776959\pi$
$108$	0	0
$109$	−20.0582	−1.92123	−0.960616	−	0.277880i	$-0.910368\pi$
$109$	−20.0582	−1.92123	−0.960616	+	0.277880i	$0.910368\pi$
$110$	1.05428	0.100522
$111$	0	0
$112$	11.7863	1.11370
$113$	−0.838803	−0.0789079	−0.0394540	−	0.999221i	$-0.512562\pi$
$113$	−0.838803	−0.0789079	−0.0394540	+	0.999221i	$0.512562\pi$
$114$	0	0
$115$	4.53151	0.422566
$116$	4.87661	0.452782
$117$	0	0
$118$	−9.90603	−0.911924
$119$	−0.228606	−0.0209563
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.18061	0.106888
$123$	0	0
$124$	−3.85684	−0.346354
$125$	8.21787	0.735029
$126$	0	0
$127$	4.84557	0.429975	0.214988	−	0.976617i	$-0.431029\pi$
$127$	4.84557	0.429975	0.214988	+	0.976617i	$0.431029\pi$
$128$	3.71667	0.328510
$129$	0	0
$130$	2.15504	0.189009
$131$	−3.13067	−0.273528	−0.136764	−	0.990604i	$-0.543670\pi$
$131$	−3.13067	−0.273528	−0.136764	+	0.990604i	$0.543670\pi$
$132$	0	0
$133$	27.3200	2.36895
$134$	1.26063	0.108902
$135$	0	0
$136$	−0.144437	−0.0123854
$137$	−6.40919	−0.547574	−0.273787	−	0.961790i	$-0.588276\pi$
$137$	−6.40919	−0.547574	−0.273787	+	0.961790i	$0.588276\pi$
$138$	0	0
$139$	−2.30859	−0.195812	−0.0979058	−	0.995196i	$-0.531214\pi$
$139$	−2.30859	−0.195812	−0.0979058	+	0.995196i	$0.531214\pi$
$140$	−2.63604	−0.222786
$141$	0	0
$142$	−4.86929	−0.408622
$143$	2.04408	0.170934
$144$	0	0
$145$	7.18428	0.596622
$146$	8.70575	0.720493
$147$	0	0
$148$	−3.89040	−0.319789
$149$	20.7506	1.69996	0.849979	−	0.526817i	$-0.176615\pi$
$149$	20.7506	1.69996	0.849979	+	0.526817i	$0.176615\pi$
$150$	0	0
$151$	3.92540	0.319445	0.159722	−	0.987162i	$-0.448940\pi$
$151$	3.92540	0.319445	0.159722	+	0.987162i	$0.448940\pi$
$152$	17.2613	1.40007
$153$	0	0
$154$	5.74940	0.463300
$155$	−5.68194	−0.456385
$156$	0	0
$157$	−0.487164	−0.0388799	−0.0194400	−	0.999811i	$-0.506188\pi$
$157$	−0.487164	−0.0388799	−0.0194400	+	0.999811i	$0.506188\pi$
$158$	−14.4355	−1.14843
$159$	0	0
$160$	−2.94362	−0.232714
$161$	24.7120	1.94758
$162$	0	0
$163$	2.64283	0.207003	0.103501	−	0.994629i	$-0.466995\pi$
$163$	2.64283	0.207003	0.103501	+	0.994629i	$0.466995\pi$
$164$	0.375758	0.0293418
$165$	0	0
$166$	−4.06455	−0.315470
$167$	−18.1527	−1.40469	−0.702347	−	0.711834i	$-0.747865\pi$
$167$	−18.1527	−1.40469	−0.702347	+	0.711834i	$0.747865\pi$
$168$	0	0
$169$	−8.82175	−0.678596
$170$	−0.0494915	−0.00379582
$171$	0	0
$172$	5.34995	0.407930
$173$	−16.4825	−1.25314	−0.626569	−	0.779366i	$-0.715541\pi$
$173$	−16.4825	−1.25314	−0.626569	+	0.779366i	$0.715541\pi$
$174$	0	0
$175$	20.4658	1.54707
$176$	2.42026	0.182434
$177$	0	0
$178$	17.8859	1.34060
$179$	−23.4879	−1.75557	−0.877784	−	0.479057i	$-0.840979\pi$
$179$	−23.4879	−1.75557	−0.877784	+	0.479057i	$0.840979\pi$
$180$	0	0
$181$	−11.8902	−0.883795	−0.441898	−	0.897066i	$-0.645695\pi$
$181$	−11.8902	−0.883795	−0.441898	+	0.897066i	$0.645695\pi$
$182$	11.7522	0.871133
$183$	0	0
$184$	15.6135	1.15104
$185$	−5.73138	−0.421380
$186$	0	0
$187$	−0.0469432	−0.00343282
$188$	3.17634	0.231658
$189$	0	0
$190$	5.91457	0.429088
$191$	17.8085	1.28858	0.644290	−	0.764782i	$-0.277153\pi$
$191$	17.8085	1.28858	0.644290	+	0.764782i	$0.277153\pi$
$192$	0	0
$193$	−19.3340	−1.39169	−0.695847	−	0.718190i	$-0.744971\pi$
$193$	−19.3340	−1.39169	−0.695847	+	0.718190i	$0.744971\pi$
$194$	8.56901	0.615219
$195$	0	0
$196$	−10.1322	−0.723730
$197$	−8.78179	−0.625676	−0.312838	−	0.949806i	$-0.601280\pi$
$197$	−8.78179	−0.625676	−0.312838	+	0.949806i	$0.601280\pi$
$198$	0	0
$199$	13.3247	0.944561	0.472280	−	0.881448i	$-0.343431\pi$
$199$	13.3247	0.944561	0.472280	+	0.881448i	$0.343431\pi$
$200$	12.9307	0.914336
$201$	0	0
$202$	11.7019	0.823341
$203$	39.1786	2.74980
$204$	0	0
$205$	0.553572	0.0386632
$206$	1.65541	0.115338
$207$	0	0
$208$	4.94720	0.343026
$209$	5.61003	0.388054
$210$	0	0
$211$	25.8412	1.77898	0.889490	−	0.456954i	$-0.151060\pi$
$211$	25.8412	1.77898	0.889490	+	0.456954i	$0.151060\pi$
$212$	−6.20500	−0.426161
$213$	0	0
$214$	−18.6699	−1.27625
$215$	7.88161	0.537521
$216$	0	0
$217$	−30.9857	−2.10345
$218$	−23.6810	−1.60388
$219$	0	0
$220$	−0.541298	−0.0364943
$221$	−0.0959555	−0.00645466
$222$	0	0
$223$	0.872328	0.0584154	0.0292077	−	0.999573i	$-0.490702\pi$
$223$	0.872328	0.0584154	0.0292077	+	0.999573i	$0.490702\pi$
$224$	−16.0527	−1.07256
$225$	0	0
$226$	−0.990300	−0.0658738
$227$	11.0157	0.731137	0.365568	−	0.930785i	$-0.380875\pi$
$227$	11.0157	0.731137	0.365568	+	0.930785i	$0.380875\pi$
$228$	0	0
$229$	−17.0678	−1.12787	−0.563937	−	0.825818i	$-0.690714\pi$
$229$	−17.0678	−1.12787	−0.563937	+	0.825818i	$0.690714\pi$
$230$	5.34995	0.352765
$231$	0	0
$232$	24.7537	1.62516
$233$	14.8849	0.975139	0.487569	−	0.873084i	$-0.337884\pi$
$233$	14.8849	0.975139	0.487569	+	0.873084i	$0.337884\pi$
$234$	0	0
$235$	4.67942	0.305252
$236$	5.08602	0.331072
$237$	0	0
$238$	−0.269895	−0.0174947
$239$	15.5474	1.00568	0.502840	−	0.864379i	$-0.332288\pi$
$239$	15.5474	1.00568	0.502840	+	0.864379i	$0.332288\pi$
$240$	0	0
$241$	2.64687	0.170500	0.0852499	−	0.996360i	$-0.472831\pi$
$241$	2.64687	0.170500	0.0852499	+	0.996360i	$0.472831\pi$
$242$	1.18061	0.0758926
$243$	0	0
$244$	−0.606158	−0.0388053
$245$	−14.9269	−0.953646
$246$	0	0
$247$	11.4673	0.729649
$248$	−19.5773	−1.24316
$249$	0	0
$250$	9.70211	0.613615
$251$	12.5987	0.795225	0.397613	−	0.917553i	$-0.369839\pi$
$251$	12.5987	0.795225	0.397613	+	0.917553i	$0.369839\pi$
$252$	0	0
$253$	5.07449	0.319030
$254$	5.72074	0.358951
$255$	0	0
$256$	−13.0765	−0.817279
$257$	23.1534	1.44427	0.722136	−	0.691751i	$-0.243161\pi$
$257$	23.1534	1.44427	0.722136	+	0.691751i	$0.243161\pi$
$258$	0	0
$259$	−31.2553	−1.94211
$260$	−1.10646	−0.0686195
$261$	0	0
$262$	−3.69611	−0.228346
$263$	12.7818	0.788162	0.394081	−	0.919076i	$-0.371063\pi$
$263$	12.7818	0.788162	0.394081	+	0.919076i	$0.371063\pi$
$264$	0	0
$265$	−9.14129	−0.561545
$266$	32.2543	1.97764
$267$	0	0
$268$	−0.647242	−0.0395366
$269$	25.7058	1.56731	0.783657	−	0.621194i	$-0.213352\pi$
$269$	25.7058	1.56731	0.783657	+	0.621194i	$0.213352\pi$
$270$	0	0
$271$	−23.1665	−1.40727	−0.703633	−	0.710563i	$-0.748440\pi$
$271$	−23.1665	−1.40727	−0.703633	+	0.710563i	$0.748440\pi$
$272$	−0.113615	−0.00688889
$273$	0	0
$274$	−7.56677	−0.457125
$275$	4.20255	0.253423
$276$	0	0
$277$	6.71679	0.403573	0.201786	−	0.979430i	$-0.435325\pi$
$277$	6.71679	0.403573	0.201786	+	0.979430i	$0.435325\pi$
$278$	−2.72554	−0.163467
$279$	0	0
$280$	−13.3806	−0.799642
$281$	−4.91544	−0.293230	−0.146615	−	0.989194i	$-0.546838\pi$
$281$	−4.91544	−0.293230	−0.146615	+	0.989194i	$0.546838\pi$
$282$	0	0
$283$	28.7706	1.71024	0.855119	−	0.518433i	$-0.173484\pi$
$283$	28.7706	1.71024	0.855119	+	0.518433i	$0.173484\pi$
$284$	2.50003	0.148349
$285$	0	0
$286$	2.41326	0.142699
$287$	3.01883	0.178196
$288$	0	0
$289$	−16.9978	−0.999870
$290$	8.48184	0.498071
$291$	0	0
$292$	−4.46977	−0.261573
$293$	−1.04284	−0.0609233	−0.0304617	−	0.999536i	$-0.509698\pi$
$293$	−1.04284	−0.0609233	−0.0304617	+	0.999536i	$0.509698\pi$
$294$	0	0
$295$	7.49279	0.436247
$296$	−19.7477	−1.14781
$297$	0	0
$298$	24.4984	1.41916
$299$	10.3726	0.599865
$300$	0	0
$301$	42.9813	2.47740
$302$	4.63437	0.266678
$303$	0	0
$304$	13.5777	0.778735
$305$	−0.892999	−0.0511330
$306$	0	0
$307$	−19.4653	−1.11094	−0.555471	−	0.831536i	$-0.687462\pi$
$307$	−19.4653	−1.11094	−0.555471	+	0.831536i	$0.687462\pi$
$308$	−2.95190	−0.168200
$309$	0	0
$310$	−6.70817	−0.380998
$311$	11.4265	0.647936	0.323968	−	0.946068i	$-0.394983\pi$
$311$	11.4265	0.647936	0.323968	+	0.946068i	$0.394983\pi$
$312$	0	0
$313$	9.09920	0.514317	0.257159	−	0.966369i	$-0.417214\pi$
$313$	9.09920	0.514317	0.257159	+	0.966369i	$0.417214\pi$
$314$	−0.575151	−0.0324577
$315$	0	0
$316$	7.41158	0.416934
$317$	−14.0653	−0.789985	−0.394992	−	0.918684i	$-0.629253\pi$
$317$	−14.0653	−0.789985	−0.394992	+	0.918684i	$0.629253\pi$
$318$	0	0
$319$	8.04512	0.450440
$320$	−7.79785	−0.435913
$321$	0	0
$322$	29.1753	1.62587
$323$	−0.263352	−0.0146533
$324$	0	0
$325$	8.59034	0.476506
$326$	3.12016	0.172810
$327$	0	0
$328$	1.90735	0.105316
$329$	25.5186	1.40689
$330$	0	0
$331$	16.3873	0.900725	0.450363	−	0.892846i	$-0.351295\pi$
$331$	16.3873	0.900725	0.450363	+	0.892846i	$0.351295\pi$
$332$	2.08685	0.114531
$333$	0	0
$334$	−21.4312	−1.17266
$335$	−0.953526	−0.0520967
$336$	0	0
$337$	−6.92107	−0.377015	−0.188507	−	0.982072i	$-0.560365\pi$
$337$	−6.92107	−0.377015	−0.188507	+	0.982072i	$0.560365\pi$
$338$	−10.4151	−0.566504
$339$	0	0
$340$	0.0254102	0.00137806
$341$	−6.36277	−0.344563
$342$	0	0
$343$	−47.3130	−2.55466
$344$	27.1564	1.46417
$345$	0	0
$346$	−19.4594	−1.04614
$347$	23.8951	1.28276	0.641378	−	0.767225i	$-0.278363\pi$
$347$	23.8951	1.28276	0.641378	+	0.767225i	$0.278363\pi$
$348$	0	0
$349$	−25.9784	−1.39059	−0.695296	−	0.718723i	$-0.744727\pi$
$349$	−25.9784	−1.39059	−0.695296	+	0.718723i	$0.744727\pi$
$350$	24.1622	1.29152
$351$	0	0
$352$	−3.29633	−0.175695
$353$	11.3695	0.605136	0.302568	−	0.953128i	$-0.402156\pi$
$353$	11.3695	0.605136	0.302568	+	0.953128i	$0.402156\pi$
$354$	0	0
$355$	3.68307	0.195477
$356$	−9.18309	−0.486703
$357$	0	0
$358$	−27.7301	−1.46558
$359$	2.89034	0.152546	0.0762731	−	0.997087i	$-0.475698\pi$
$359$	2.89034	0.152546	0.0762731	+	0.997087i	$0.475698\pi$
$360$	0	0
$361$	12.4724	0.656443
$362$	−14.0378	−0.737808
$363$	0	0
$364$	−6.03391	−0.316263
$365$	−6.58492	−0.344670
$366$	0	0
$367$	−10.6261	−0.554677	−0.277338	−	0.960772i	$-0.589452\pi$
$367$	−10.6261	−0.554677	−0.277338	+	0.960772i	$0.589452\pi$
$368$	12.2816	0.640221
$369$	0	0
$370$	−6.76653	−0.351775
$371$	−49.8508	−2.58813
$372$	0	0
$373$	−36.1012	−1.86925	−0.934625	−	0.355635i	$-0.884265\pi$
$373$	−36.1012	−1.86925	−0.934625	+	0.355635i	$0.884265\pi$
$374$	−0.0554216	−0.00286578
$375$	0	0
$376$	16.1231	0.831486
$377$	16.4448	0.846952
$378$	0	0
$379$	−21.8570	−1.12272	−0.561358	−	0.827573i	$-0.689721\pi$
$379$	−21.8570	−1.12272	−0.561358	+	0.827573i	$0.689721\pi$
$380$	−3.03670	−0.155779
$381$	0	0
$382$	21.0249	1.07573
$383$	−19.2311	−0.982662	−0.491331	−	0.870973i	$-0.663490\pi$
$383$	−19.2311	−0.982662	−0.491331	+	0.870973i	$0.663490\pi$
$384$	0	0
$385$	−4.34878	−0.221634
$386$	−22.8260	−1.16181
$387$	0	0
$388$	−4.39956	−0.223354
$389$	−33.5284	−1.69996	−0.849978	−	0.526818i	$-0.823385\pi$
$389$	−33.5284	−1.69996	−0.849978	+	0.526818i	$0.823385\pi$
$390$	0	0
$391$	−0.238212	−0.0120469
$392$	−51.4312	−2.59767
$393$	0	0
$394$	−10.3679	−0.522326
$395$	10.9188	0.549386
$396$	0	0
$397$	−19.1981	−0.963527	−0.481763	−	0.876301i	$-0.660004\pi$
$397$	−19.1981	−0.963527	−0.481763	+	0.876301i	$0.660004\pi$
$398$	15.7313	0.788537
$399$	0	0
$400$	10.1713	0.508563
$401$	−30.0691	−1.50158	−0.750790	−	0.660541i	$-0.770327\pi$
$401$	−30.0691	−1.50158	−0.750790	+	0.660541i	$0.770327\pi$
$402$	0	0
$403$	−13.0060	−0.647874
$404$	−6.00806	−0.298912
$405$	0	0
$406$	46.2546	2.29558
$407$	−6.41813	−0.318135
$408$	0	0
$409$	34.3213	1.69708	0.848540	−	0.529132i	$-0.177482\pi$
$409$	34.3213	1.69708	0.848540	+	0.529132i	$0.177482\pi$
$410$	0.653553	0.0322767
$411$	0	0
$412$	−0.849931	−0.0418731
$413$	40.8610	2.01064
$414$	0	0
$415$	3.07437	0.150915
$416$	−6.73796	−0.330356
$417$	0	0
$418$	6.62326	0.323954
$419$	10.3143	0.503888	0.251944	−	0.967742i	$-0.418930\pi$
$419$	10.3143	0.503888	0.251944	+	0.967742i	$0.418930\pi$
$420$	0	0
$421$	22.2424	1.08403	0.542015	−	0.840369i	$-0.317662\pi$
$421$	22.2424	1.08403	0.542015	+	0.840369i	$0.317662\pi$
$422$	30.5084	1.48513
$423$	0	0
$424$	−31.4966	−1.52961
$425$	−0.197281	−0.00956954
$426$	0	0
$427$	−4.86985	−0.235669
$428$	9.58564	0.463339
$429$	0	0
$430$	9.30512	0.448733
$431$	4.84237	0.233249	0.116624	−	0.993176i	$-0.462793\pi$
$431$	4.84237	0.233249	0.116624	+	0.993176i	$0.462793\pi$
$432$	0	0
$433$	−22.1908	−1.06642	−0.533211	−	0.845982i	$-0.679015\pi$
$433$	−22.1908	−1.06642	−0.533211	+	0.845982i	$0.679015\pi$
$434$	−36.5821	−1.75600
$435$	0	0
$436$	12.1585	0.582284
$437$	28.4680	1.36181
$438$	0	0
$439$	−32.7712	−1.56408	−0.782042	−	0.623226i	$-0.785822\pi$
$439$	−32.7712	−1.56408	−0.782042	+	0.623226i	$0.785822\pi$
$440$	−2.74763	−0.130988
$441$	0	0
$442$	−0.113286	−0.00538847
$443$	−32.5209	−1.54511	−0.772557	−	0.634945i	$-0.781023\pi$
$443$	−32.5209	−1.54511	−0.772557	+	0.634945i	$0.781023\pi$
$444$	0	0
$445$	−13.5286	−0.641319
$446$	1.02988	0.0487662
$447$	0	0
$448$	−42.5246	−2.00910
$449$	22.9213	1.08172	0.540861	−	0.841112i	$-0.318098\pi$
$449$	22.9213	1.08172	0.540861	+	0.841112i	$0.318098\pi$
$450$	0	0
$451$	0.619902	0.0291901
$452$	0.508447	0.0239153
$453$	0	0
$454$	13.0052	0.610366
$455$	−8.88924	−0.416734
$456$	0	0
$457$	7.29516	0.341253	0.170627	−	0.985336i	$-0.445421\pi$
$457$	7.29516	0.341253	0.170627	+	0.985336i	$0.445421\pi$
$458$	−20.1505	−0.941570
$459$	0	0
$460$	−2.74681	−0.128071
$461$	−14.3569	−0.668666	−0.334333	−	0.942455i	$-0.608511\pi$
$461$	−14.3569	−0.668666	−0.334333	+	0.942455i	$0.608511\pi$
$462$	0	0
$463$	22.2605	1.03453	0.517265	−	0.855825i	$-0.326950\pi$
$463$	22.2605	1.03453	0.517265	+	0.855825i	$0.326950\pi$
$464$	19.4713	0.903930
$465$	0	0
$466$	17.5732	0.814064
$467$	−8.21946	−0.380351	−0.190176	−	0.981750i	$-0.560906\pi$
$467$	−8.21946	−0.380351	−0.190176	+	0.981750i	$0.560906\pi$
$468$	0	0
$469$	−5.19993	−0.240110
$470$	5.52458	0.254830
$471$	0	0
$472$	25.8167	1.18831
$473$	8.82600	0.405820
$474$	0	0
$475$	23.5764	1.08176
$476$	0.138571	0.00635141
$477$	0	0
$478$	18.3555	0.839561
$479$	0.741402	0.0338755	0.0169378	−	0.999857i	$-0.494608\pi$
$479$	0.741402	0.0338755	0.0169378	+	0.999857i	$0.494608\pi$
$480$	0	0
$481$	−13.1191	−0.598182
$482$	3.12492	0.142336
$483$	0	0
$484$	−0.606158	−0.0275526
$485$	−6.48149	−0.294309
$486$	0	0
$487$	−7.31688	−0.331559	−0.165780	−	0.986163i	$-0.553014\pi$
$487$	−7.31688	−0.331559	−0.165780	+	0.986163i	$0.553014\pi$
$488$	−3.07686	−0.139283
$489$	0	0
$490$	−17.6229	−0.796121
$491$	−9.44139	−0.426084	−0.213042	−	0.977043i	$-0.568337\pi$
$491$	−9.44139	−0.426084	−0.213042	+	0.977043i	$0.568337\pi$
$492$	0	0
$493$	−0.377663	−0.0170091
$494$	13.5385	0.609124
$495$	0	0
$496$	−15.3995	−0.691459
$497$	20.0851	0.900942
$498$	0	0
$499$	−6.78042	−0.303533	−0.151767	−	0.988416i	$-0.548496\pi$
$499$	−6.78042	−0.303533	−0.151767	+	0.988416i	$0.548496\pi$
$500$	−4.98132	−0.222772
$501$	0	0
$502$	14.8742	0.663868
$503$	−13.9283	−0.621032	−0.310516	−	0.950568i	$-0.600502\pi$
$503$	−13.9283	−0.621032	−0.310516	+	0.950568i	$0.600502\pi$
$504$	0	0
$505$	−8.85115	−0.393871
$506$	5.99099	0.266332
$507$	0	0
$508$	−2.93718	−0.130316
$509$	−10.3637	−0.459364	−0.229682	−	0.973266i	$-0.573769\pi$
$509$	−10.3637	−0.459364	−0.229682	+	0.973266i	$0.573769\pi$
$510$	0	0
$511$	−35.9100	−1.58856
$512$	−22.8716	−1.01079
$513$	0	0
$514$	27.3352	1.20570
$515$	−1.25213	−0.0551754
$516$	0	0
$517$	5.24012	0.230460
$518$	−36.9004	−1.62131
$519$	0	0
$520$	−5.61637	−0.246294
$521$	25.4751	1.11608	0.558042	−	0.829813i	$-0.311553\pi$
$521$	25.4751	1.11608	0.558042	+	0.829813i	$0.311553\pi$
$522$	0	0
$523$	−11.4023	−0.498589	−0.249295	−	0.968428i	$-0.580199\pi$
$523$	−11.4023	−0.498589	−0.249295	+	0.968428i	$0.580199\pi$
$524$	1.89768	0.0829005
$525$	0	0
$526$	15.0904	0.657972
$527$	0.298688	0.0130111
$528$	0	0
$529$	2.75040	0.119583
$530$	−10.7923	−0.468788
$531$	0	0
$532$	−16.5602	−0.717977
$533$	1.26713	0.0548854
$534$	0	0
$535$	14.1217	0.610534
$536$	−3.28540	−0.141908
$537$	0	0
$538$	30.3486	1.30842
$539$	−16.7155	−0.719987
$540$	0	0
$541$	−32.0169	−1.37651	−0.688256	−	0.725468i	$-0.741624\pi$
$541$	−32.0169	−1.37651	−0.688256	+	0.725468i	$0.741624\pi$
$542$	−27.3507	−1.17481
$543$	0	0
$544$	0.154740	0.00663444
$545$	17.9120	0.767266
$546$	0	0
$547$	30.7414	1.31441	0.657204	−	0.753713i	$-0.271739\pi$
$547$	30.7414	1.31441	0.657204	+	0.753713i	$0.271739\pi$
$548$	3.88498	0.165958
$549$	0	0
$550$	4.96158	0.211562
$551$	45.1333	1.92274
$552$	0	0
$553$	59.5444	2.53209
$554$	7.92992	0.336910
$555$	0	0
$556$	1.39937	0.0593464
$557$	−18.4676	−0.782498	−0.391249	−	0.920285i	$-0.627957\pi$
$557$	−18.4676	−0.782498	−0.391249	+	0.920285i	$0.627957\pi$
$558$	0	0
$559$	18.0410	0.763054
$560$	−10.5252	−0.444769
$561$	0	0
$562$	−5.80322	−0.244794
$563$	−35.9020	−1.51309	−0.756544	−	0.653942i	$-0.773114\pi$
$563$	−35.9020	−1.51309	−0.756544	+	0.653942i	$0.773114\pi$
$564$	0	0
$565$	0.749050	0.0315128
$566$	33.9669	1.42774
$567$	0	0
$568$	12.6901	0.532467
$569$	19.1639	0.803394	0.401697	−	0.915773i	$-0.368420\pi$
$569$	19.1639	0.803394	0.401697	+	0.915773i	$0.368420\pi$
$570$	0	0
$571$	−42.6413	−1.78448	−0.892242	−	0.451557i	$-0.850869\pi$
$571$	−42.6413	−1.78448	−0.892242	+	0.451557i	$0.850869\pi$
$572$	−1.23903	−0.0518066
$573$	0	0
$574$	3.56407	0.148761
$575$	21.3258	0.889347
$576$	0	0
$577$	−44.7459	−1.86280	−0.931399	−	0.364000i	$-0.881411\pi$
$577$	−44.7459	−1.86280	−0.931399	+	0.364000i	$0.881411\pi$
$578$	−20.0678	−0.834710
$579$	0	0
$580$	−4.35481	−0.180824
$581$	16.7657	0.695558
$582$	0	0
$583$	−10.2366	−0.423957
$584$	−22.6886	−0.938859
$585$	0	0
$586$	−1.23119	−0.0508599
$587$	36.1115	1.49048	0.745240	−	0.666797i	$-0.232335\pi$
$587$	36.1115	1.49048	0.745240	+	0.666797i	$0.232335\pi$
$588$	0	0
$589$	−35.6953	−1.47080
$590$	8.84607	0.364187
$591$	0	0
$592$	−15.5335	−0.638424
$593$	−6.16360	−0.253109	−0.126554	−	0.991960i	$-0.540392\pi$
$593$	−6.16360	−0.253109	−0.126554	+	0.991960i	$0.540392\pi$
$594$	0	0
$595$	0.204145	0.00836914
$596$	−12.5781	−0.515221
$597$	0	0
$598$	12.2461	0.500778
$599$	26.3147	1.07519	0.537595	−	0.843203i	$-0.319333\pi$
$599$	26.3147	1.07519	0.537595	+	0.843203i	$0.319333\pi$
$600$	0	0
$601$	−3.92916	−0.160274	−0.0801368	−	0.996784i	$-0.525536\pi$
$601$	−3.92916	−0.160274	−0.0801368	+	0.996784i	$0.525536\pi$
$602$	50.7442	2.06818
$603$	0	0
$604$	−2.37941	−0.0968169
$605$	−0.892999	−0.0363056
$606$	0	0
$607$	−40.6947	−1.65175	−0.825874	−	0.563855i	$-0.809318\pi$
$607$	−40.6947	−1.65175	−0.825874	+	0.563855i	$0.809318\pi$
$608$	−18.4925	−0.749971
$609$	0	0
$610$	−1.05428	−0.0426867
$611$	10.7112	0.433329
$612$	0	0
$613$	8.19146	0.330850	0.165425	−	0.986222i	$-0.447100\pi$
$613$	8.19146	0.330850	0.165425	+	0.986222i	$0.447100\pi$
$614$	−22.9809	−0.927435
$615$	0	0
$616$	−14.9839	−0.603717
$617$	−3.24947	−0.130819	−0.0654093	−	0.997859i	$-0.520835\pi$
$617$	−3.24947	−0.130819	−0.0654093	+	0.997859i	$0.520835\pi$
$618$	0	0
$619$	−29.4184	−1.18243	−0.591213	−	0.806516i	$-0.701351\pi$
$619$	−29.4184	−1.18243	−0.591213	+	0.806516i	$0.701351\pi$
$620$	3.44415	0.138321
$621$	0	0
$622$	13.4902	0.540908
$623$	−73.7767	−2.95580
$624$	0	0
$625$	13.6742	0.546968
$626$	10.7426	0.429361
$627$	0	0
$628$	0.295298	0.0117837
$629$	0.301287	0.0120131
$630$	0	0
$631$	9.91059	0.394534	0.197267	−	0.980350i	$-0.436793\pi$
$631$	9.91059	0.394534	0.197267	+	0.980350i	$0.436793\pi$
$632$	37.6212	1.49649
$633$	0	0
$634$	−16.6056	−0.659493
$635$	−4.32709	−0.171715
$636$	0	0
$637$	−34.1678	−1.35378
$638$	9.49815	0.376035
$639$	0	0
$640$	−3.31898	−0.131194
$641$	−13.6371	−0.538632	−0.269316	−	0.963052i	$-0.586798\pi$
$641$	−13.6371	−0.538632	−0.269316	+	0.963052i	$0.586798\pi$
$642$	0	0
$643$	−48.4940	−1.91242	−0.956209	−	0.292685i	$-0.905451\pi$
$643$	−48.4940	−1.91242	−0.956209	+	0.292685i	$0.905451\pi$
$644$	−14.9794	−0.590270
$645$	0	0
$646$	−0.310917	−0.0122329
$647$	35.0393	1.37754	0.688769	−	0.724981i	$-0.258151\pi$
$647$	35.0393	1.37754	0.688769	+	0.724981i	$0.258151\pi$
$648$	0	0
$649$	8.39059	0.329360
$650$	10.1419	0.397796
$651$	0	0
$652$	−1.60197	−0.0627382
$653$	26.0145	1.01803	0.509013	−	0.860759i	$-0.330010\pi$
$653$	26.0145	1.01803	0.509013	+	0.860759i	$0.330010\pi$
$654$	0	0
$655$	2.79569	0.109237
$656$	1.50032	0.0585778
$657$	0	0
$658$	30.1276	1.17450
$659$	44.0159	1.71462	0.857308	−	0.514804i	$-0.172135\pi$
$659$	44.0159	1.71462	0.857308	+	0.514804i	$0.172135\pi$
$660$	0	0
$661$	−6.27978	−0.244255	−0.122128	−	0.992514i	$-0.538972\pi$
$661$	−6.27978	−0.244255	−0.122128	+	0.992514i	$0.538972\pi$
$662$	19.3470	0.751942
$663$	0	0
$664$	10.5929	0.411083
$665$	−24.3968	−0.946066
$666$	0	0
$667$	40.8248	1.58074
$668$	11.0034	0.425733
$669$	0	0
$670$	−1.12574	−0.0434912
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−33.5651	−1.29384	−0.646920	−	0.762558i	$-0.723943\pi$
$673$	−33.5651	−1.29384	−0.646920	+	0.762558i	$0.723943\pi$
$674$	−8.17109	−0.314739
$675$	0	0
$676$	5.34737	0.205668
$677$	40.8492	1.56996	0.784981	−	0.619520i	$-0.212673\pi$
$677$	40.8492	1.56996	0.784981	+	0.619520i	$0.212673\pi$
$678$	0	0
$679$	−35.3460	−1.35645
$680$	0.128983	0.00494625
$681$	0	0
$682$	−7.51195	−0.287647
$683$	45.5785	1.74401	0.872007	−	0.489493i	$-0.162818\pi$
$683$	45.5785	1.74401	0.872007	+	0.489493i	$0.162818\pi$
$684$	0	0
$685$	5.72341	0.218680
$686$	−55.8583	−2.13268
$687$	0	0
$688$	21.3612	0.814388
$689$	−20.9244	−0.797157
$690$	0	0
$691$	−9.34416	−0.355469	−0.177734	−	0.984079i	$-0.556877\pi$
$691$	−9.34416	−0.355469	−0.177734	+	0.984079i	$0.556877\pi$
$692$	9.99097	0.379800
$693$	0	0
$694$	28.2108	1.07087
$695$	2.06157	0.0781996
$696$	0	0
$697$	−0.0291002	−0.00110225
$698$	−30.6704	−1.16089
$699$	0	0
$700$	−12.4055	−0.468884
$701$	−28.7536	−1.08601	−0.543003	−	0.839731i	$-0.682713\pi$
$701$	−28.7536	−1.08601	−0.543003	+	0.839731i	$0.682713\pi$
$702$	0	0
$703$	−36.0059	−1.35799
$704$	−8.73220	−0.329107
$705$	0	0
$706$	13.4229	0.505179
$707$	−48.2686	−1.81533
$708$	0	0
$709$	2.57308	0.0966339	0.0483170	−	0.998832i	$-0.484614\pi$
$709$	2.57308	0.0966339	0.0483170	+	0.998832i	$0.484614\pi$
$710$	4.34827	0.163188
$711$	0	0
$712$	−46.6134	−1.74691
$713$	−32.2878	−1.20919
$714$	0	0
$715$	−1.82536	−0.0682646
$716$	14.2374	0.532075
$717$	0	0
$718$	3.41237	0.127348
$719$	−34.1973	−1.27534	−0.637672	−	0.770308i	$-0.720103\pi$
$719$	−34.1973	−1.27534	−0.637672	+	0.770308i	$0.720103\pi$
$720$	0	0
$721$	−6.82833	−0.254300
$722$	14.7251	0.548010
$723$	0	0
$724$	7.20736	0.267859
$725$	33.8100	1.25567
$726$	0	0
$727$	−11.5995	−0.430201	−0.215100	−	0.976592i	$-0.569008\pi$
$727$	−11.5995	−0.430201	−0.215100	+	0.976592i	$0.569008\pi$
$728$	−30.6282	−1.13516
$729$	0	0
$730$	−7.77423	−0.287737
$731$	−0.414320	−0.0153242
$732$	0	0
$733$	6.21979	0.229733	0.114867	−	0.993381i	$-0.463356\pi$
$733$	6.21979	0.229733	0.114867	+	0.993381i	$0.463356\pi$
$734$	−12.5453	−0.463054
$735$	0	0
$736$	−16.7272	−0.616573
$737$	−1.06778	−0.0393321
$738$	0	0
$739$	−1.98442	−0.0729980	−0.0364990	−	0.999334i	$-0.511621\pi$
$739$	−1.98442	−0.0729980	−0.0364990	+	0.999334i	$0.511621\pi$
$740$	3.47412	0.127711
$741$	0	0
$742$	−58.8544	−2.16061
$743$	−45.2869	−1.66142	−0.830708	−	0.556709i	$-0.812064\pi$
$743$	−45.2869	−1.66142	−0.830708	+	0.556709i	$0.812064\pi$
$744$	0	0
$745$	−18.5303	−0.678897
$746$	−42.6215	−1.56048
$747$	0	0
$748$	0.0284549	0.00104042
$749$	77.0108	2.81391
$750$	0	0
$751$	15.6588	0.571398	0.285699	−	0.958319i	$-0.407774\pi$
$751$	15.6588	0.571398	0.285699	+	0.958319i	$0.407774\pi$
$752$	12.6824	0.462481
$753$	0	0
$754$	19.4150	0.707051
$755$	−3.50538	−0.127574
$756$	0	0
$757$	−15.0725	−0.547821	−0.273910	−	0.961755i	$-0.588317\pi$
$757$	−15.0725	−0.547821	−0.273910	+	0.961755i	$0.588317\pi$
$758$	−25.8046	−0.937264
$759$	0	0
$760$	−15.4143	−0.559135
$761$	29.0462	1.05292	0.526462	−	0.850199i	$-0.323518\pi$
$761$	29.0462	1.05292	0.526462	+	0.850199i	$0.323518\pi$
$762$	0	0
$763$	97.6808	3.53628
$764$	−10.7948	−0.390541
$765$	0	0
$766$	−22.7044	−0.820344
$767$	17.1510	0.619287
$768$	0	0
$769$	21.3781	0.770915	0.385457	−	0.922726i	$-0.374044\pi$
$769$	21.3781	0.770915	0.385457	+	0.922726i	$0.374044\pi$
$770$	−5.13421	−0.185024
$771$	0	0
$772$	11.7195	0.421793
$773$	14.1554	0.509134	0.254567	−	0.967055i	$-0.418067\pi$
$773$	14.1554	0.509134	0.254567	+	0.967055i	$0.418067\pi$
$774$	0	0
$775$	−26.7399	−0.960524
$776$	−22.3322	−0.801679
$777$	0	0
$778$	−39.5840	−1.41915
$779$	3.47767	0.124600
$780$	0	0
$781$	4.12438	0.147582
$782$	−0.281236	−0.0100570
$783$	0	0
$784$	−40.4558	−1.44485
$785$	0.435037	0.0155271
$786$	0	0
$787$	24.4555	0.871744	0.435872	−	0.900009i	$-0.356440\pi$
$787$	24.4555	0.871744	0.435872	+	0.900009i	$0.356440\pi$
$788$	5.32315	0.189629
$789$	0	0
$790$	12.8909	0.458637
$791$	4.08485	0.145240
$792$	0	0
$793$	−2.04408	−0.0725873
$794$	−22.6655	−0.804370
$795$	0	0
$796$	−8.07685	−0.286276
$797$	−0.994082	−0.0352122	−0.0176061	−	0.999845i	$-0.505604\pi$
$797$	−0.994082	−0.0352122	−0.0176061	+	0.999845i	$0.505604\pi$
$798$	0	0
$799$	−0.245988	−0.00870243
$800$	−13.8530	−0.489778
$801$	0	0
$802$	−35.4999	−1.25355
$803$	−7.37393	−0.260220
$804$	0	0
$805$	−22.0678	−0.777788
$806$	−15.3550	−0.540857
$807$	0	0
$808$	−30.4969	−1.07288
$809$	2.10756	0.0740977	0.0370489	−	0.999313i	$-0.488204\pi$
$809$	2.10756	0.0740977	0.0370489	+	0.999313i	$0.488204\pi$
$810$	0	0
$811$	32.0973	1.12709	0.563544	−	0.826086i	$-0.309438\pi$
$811$	32.0973	1.12709	0.563544	+	0.826086i	$0.309438\pi$
$812$	−23.7484	−0.833404
$813$	0	0
$814$	−7.57731	−0.265585
$815$	−2.36005	−0.0826689
$816$	0	0
$817$	49.5141	1.73228
$818$	40.5201	1.41675
$819$	0	0
$820$	−0.335552	−0.0117180
$821$	12.3051	0.429452	0.214726	−	0.976674i	$-0.431114\pi$
$821$	12.3051	0.429452	0.214726	+	0.976674i	$0.431114\pi$
$822$	0	0
$823$	−16.4630	−0.573864	−0.286932	−	0.957951i	$-0.592635\pi$
$823$	−16.4630	−0.573864	−0.286932	+	0.957951i	$0.592635\pi$
$824$	−4.31426	−0.150294
$825$	0	0
$826$	48.2409	1.67852
$827$	47.7080	1.65897	0.829485	−	0.558530i	$-0.188634\pi$
$827$	47.7080	1.65897	0.829485	+	0.558530i	$0.188634\pi$
$828$	0	0
$829$	39.8791	1.38506	0.692529	−	0.721390i	$-0.256497\pi$
$829$	39.8791	1.38506	0.692529	+	0.721390i	$0.256497\pi$
$830$	3.62964	0.125987
$831$	0	0
$832$	−17.8493	−0.618813
$833$	0.784678	0.0271875
$834$	0	0
$835$	16.2103	0.560981
$836$	−3.40056	−0.117611
$837$	0	0
$838$	12.1772	0.420655
$839$	34.7758	1.20059	0.600297	−	0.799777i	$-0.295049\pi$
$839$	34.7758	1.20059	0.600297	+	0.799777i	$0.295049\pi$
$840$	0	0
$841$	35.7239	1.23186
$842$	26.2597	0.904968
$843$	0	0
$844$	−15.6638	−0.539171
$845$	7.87781	0.271005
$846$	0	0
$847$	−4.86985	−0.167330
$848$	−24.7752	−0.850785
$849$	0	0
$850$	−0.232912	−0.00798882
$851$	−32.5687	−1.11644
$852$	0	0
$853$	36.7417	1.25801	0.629007	−	0.777400i	$-0.283462\pi$
$853$	36.7417	1.25801	0.629007	+	0.777400i	$0.283462\pi$
$854$	−5.74940	−0.196741
$855$	0	0
$856$	48.6567	1.66305
$857$	35.9515	1.22808	0.614039	−	0.789275i	$-0.289544\pi$
$857$	35.9515	1.22808	0.614039	+	0.789275i	$0.289544\pi$
$858$	0	0
$859$	37.6231	1.28368	0.641842	−	0.766837i	$-0.278171\pi$
$859$	37.6231	1.28368	0.641842	+	0.766837i	$0.278171\pi$
$860$	−4.77750	−0.162911
$861$	0	0
$862$	5.71695	0.194720
$863$	32.1729	1.09518	0.547589	−	0.836747i	$-0.315546\pi$
$863$	32.1729	1.09518	0.547589	+	0.836747i	$0.315546\pi$
$864$	0	0
$865$	14.7188	0.500455
$866$	−26.1987	−0.890269
$867$	0	0
$868$	18.7822	0.637511
$869$	12.2271	0.414778
$870$	0	0
$871$	−2.18262	−0.0739554
$872$	61.7164	2.08998
$873$	0	0
$874$	33.6096	1.13686
$875$	−40.0198	−1.35292
$876$	0	0
$877$	37.1819	1.25554	0.627771	−	0.778398i	$-0.283967\pi$
$877$	37.1819	1.25554	0.627771	+	0.778398i	$0.283967\pi$
$878$	−38.6900	−1.30572
$879$	0	0
$880$	−2.16129	−0.0728570
$881$	36.9036	1.24331	0.621657	−	0.783289i	$-0.286460\pi$
$881$	36.9036	1.24331	0.621657	+	0.783289i	$0.286460\pi$
$882$	0	0
$883$	15.1058	0.508351	0.254176	−	0.967158i	$-0.418196\pi$
$883$	15.1058	0.508351	0.254176	+	0.967158i	$0.418196\pi$
$884$	0.0581641	0.00195627
$885$	0	0
$886$	−38.3945	−1.28989
$887$	16.4143	0.551139	0.275569	−	0.961281i	$-0.411134\pi$
$887$	16.4143	0.551139	0.275569	+	0.961281i	$0.411134\pi$
$888$	0	0
$889$	−23.5972	−0.791426
$890$	−15.9721	−0.535385
$891$	0	0
$892$	−0.528768	−0.0177045
$893$	29.3972	0.983741
$894$	0	0
$895$	20.9747	0.701106
$896$	−18.0996	−0.604667
$897$	0	0
$898$	27.0611	0.903042
$899$	−51.1892	−1.70725
$900$	0	0
$901$	0.480539	0.0160091
$902$	0.731863	0.0243684
$903$	0	0
$904$	2.58088	0.0858387
$905$	10.6180	0.352954
$906$	0	0
$907$	−4.48810	−0.149025	−0.0745125	−	0.997220i	$-0.523740\pi$
$907$	−4.48810	−0.149025	−0.0745125	+	0.997220i	$0.523740\pi$
$908$	−6.67724	−0.221592
$909$	0	0
$910$	−10.4947	−0.347897
$911$	−27.3811	−0.907175	−0.453588	−	0.891212i	$-0.649856\pi$
$911$	−27.3811	−0.907175	−0.453588	+	0.891212i	$0.649856\pi$
$912$	0	0
$913$	3.44275	0.113938
$914$	8.61275	0.284884
$915$	0	0
$916$	10.3458	0.341835
$917$	15.2459	0.503464
$918$	0	0
$919$	9.01128	0.297255	0.148627	−	0.988893i	$-0.452515\pi$
$919$	9.01128	0.297255	0.148627	+	0.988893i	$0.452515\pi$
$920$	−13.9428	−0.459681
$921$	0	0
$922$	−16.9499	−0.558215
$923$	8.43056	0.277495
$924$	0	0
$925$	−26.9725	−0.886851
$926$	26.2809	0.863645
$927$	0	0
$928$	−26.5194	−0.870541
$929$	2.26479	0.0743054	0.0371527	−	0.999310i	$-0.488171\pi$
$929$	2.26479	0.0743054	0.0371527	+	0.999310i	$0.488171\pi$
$930$	0	0
$931$	−93.7744	−3.07333
$932$	−9.02256	−0.295544
$933$	0	0
$934$	−9.70398	−0.317524
$935$	0.0419202	0.00137094
$936$	0	0
$937$	31.1676	1.01820	0.509101	−	0.860707i	$-0.329978\pi$
$937$	31.1676	1.01820	0.509101	+	0.860707i	$0.329978\pi$
$938$	−6.13909	−0.200448
$939$	0	0
$940$	−2.83647	−0.0925154
$941$	27.3896	0.892876	0.446438	−	0.894815i	$-0.352692\pi$
$941$	27.3896	0.892876	0.446438	+	0.894815i	$0.352692\pi$
$942$	0	0
$943$	3.14568	0.102438
$944$	20.3074	0.660949
$945$	0	0
$946$	10.4201	0.338786
$947$	−40.8074	−1.32606	−0.663030	−	0.748592i	$-0.730730\pi$
$947$	−40.8074	−1.32606	−0.663030	+	0.748592i	$0.730730\pi$
$948$	0	0
$949$	−15.0729	−0.489287
$950$	27.8346	0.903074
$951$	0	0
$952$	0.703389	0.0227970
$953$	5.31907	0.172302	0.0861508	−	0.996282i	$-0.472543\pi$
$953$	5.31907	0.172302	0.0861508	+	0.996282i	$0.472543\pi$
$954$	0	0
$955$	−15.9030	−0.514609
$956$	−9.42420	−0.304801
$957$	0	0
$958$	0.875308	0.0282799
$959$	31.2118	1.00788
$960$	0	0
$961$	9.48478	0.305961
$962$	−15.4886	−0.499373
$963$	0	0
$964$	−1.60442	−0.0516749
$965$	17.2653	0.555789
$966$	0	0
$967$	0.551108	0.0177224	0.00886122	−	0.999961i	$-0.497179\pi$
$967$	0.551108	0.0177224	0.00886122	+	0.999961i	$0.497179\pi$
$968$	−3.07686	−0.0988940
$969$	0	0
$970$	−7.65212	−0.245695
$971$	1.65238	0.0530273	0.0265136	−	0.999648i	$-0.491559\pi$
$971$	1.65238	0.0530273	0.0265136	+	0.999648i	$0.491559\pi$
$972$	0	0
$973$	11.2425	0.360417
$974$	−8.63839	−0.276792
$975$	0	0
$976$	−2.42026	−0.0774706
$977$	−10.8937	−0.348520	−0.174260	−	0.984700i	$-0.555753\pi$
$977$	−10.8937	−0.348520	−0.174260	+	0.984700i	$0.555753\pi$
$978$	0	0
$979$	−15.1497	−0.484186
$980$	9.04806	0.289030
$981$	0	0
$982$	−11.1466	−0.355702
$983$	−19.5157	−0.622455	−0.311228	−	0.950335i	$-0.600740\pi$
$983$	−19.5157	−0.622455	−0.311228	+	0.950335i	$0.600740\pi$
$984$	0	0
$985$	7.84213	0.249871
$986$	−0.445873	−0.0141995
$987$	0	0
$988$	−6.95101	−0.221141
$989$	44.7874	1.42416
$990$	0	0
$991$	2.80366	0.0890612	0.0445306	−	0.999008i	$-0.485821\pi$
$991$	2.80366	0.0890612	0.0445306	+	0.999008i	$0.485821\pi$
$992$	20.9738	0.665919
$993$	0	0
$994$	23.7127	0.752123
$995$	−11.8989	−0.377221
$996$	0	0
$997$	−34.7140	−1.09940	−0.549702	−	0.835361i	$-0.685259\pi$
$997$	−34.7140	−1.09940	−0.549702	+	0.835361i	$0.685259\pi$
$998$	−8.00503	−0.253395
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.17	yes	25
3.2	odd	2		6039.2.a.m.1.9	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.9	✓	25	3.2	odd	2
6039.2.a.p.1.17	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18061 q^{2} -0.606158 q^{4} -0.892999 q^{5} -4.86985 q^{7} -3.07686 q^{8} +O(q^{10})\)	\(q+1.18061 q^{2} -0.606158 q^{4} -0.892999 q^{5} -4.86985 q^{7} -3.07686 q^{8} -1.05428 q^{10} -1.00000 q^{11} -2.04408 q^{13} -5.74940 q^{14} -2.42026 q^{16} +0.0469432 q^{17} -5.61003 q^{19} +0.541298 q^{20} -1.18061 q^{22} -5.07449 q^{23} -4.20255 q^{25} -2.41326 q^{26} +2.95190 q^{28} -8.04512 q^{29} +6.36277 q^{31} +3.29633 q^{32} +0.0554216 q^{34} +4.34878 q^{35} +6.41813 q^{37} -6.62326 q^{38} +2.74763 q^{40} -0.619902 q^{41} -8.82600 q^{43} +0.606158 q^{44} -5.99099 q^{46} -5.24012 q^{47} +16.7155 q^{49} -4.96158 q^{50} +1.23903 q^{52} +10.2366 q^{53} +0.892999 q^{55} +14.9839 q^{56} -9.49815 q^{58} -8.39059 q^{59} +1.00000 q^{61} +7.51195 q^{62} +8.73220 q^{64} +1.82536 q^{65} +1.06778 q^{67} -0.0284549 q^{68} +5.13421 q^{70} -4.12438 q^{71} +7.37393 q^{73} +7.57731 q^{74} +3.40056 q^{76} +4.86985 q^{77} -12.2271 q^{79} +2.16129 q^{80} -0.731863 q^{82} -3.44275 q^{83} -0.0419202 q^{85} -10.4201 q^{86} +3.07686 q^{88} +15.1497 q^{89} +9.95436 q^{91} +3.07594 q^{92} -6.18655 q^{94} +5.00975 q^{95} +7.25811 q^{97} +19.7345 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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