LMFDB - Embedded newform 1339.2.a.g.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13041 q^{2} +1.24412 q^{3} -0.722174 q^{4} -3.15829 q^{5} -1.40637 q^{6} -4.57377 q^{7} +3.07717 q^{8} -1.45216 q^{9} +O(q^{10})$	\(q-1.13041 q^{2} +1.24412 q^{3} -0.722174 q^{4} -3.15829 q^{5} -1.40637 q^{6} -4.57377 q^{7} +3.07717 q^{8} -1.45216 q^{9} +3.57016 q^{10} -3.66167 q^{11} -0.898472 q^{12} -1.00000 q^{13} +5.17023 q^{14} -3.92929 q^{15} -2.03412 q^{16} -4.09670 q^{17} +1.64154 q^{18} -0.591761 q^{19} +2.28083 q^{20} -5.69032 q^{21} +4.13919 q^{22} +6.91797 q^{23} +3.82838 q^{24} +4.97478 q^{25} +1.13041 q^{26} -5.53903 q^{27} +3.30306 q^{28} +1.63183 q^{29} +4.44171 q^{30} +3.68143 q^{31} -3.85496 q^{32} -4.55556 q^{33} +4.63095 q^{34} +14.4453 q^{35} +1.04871 q^{36} -6.63658 q^{37} +0.668932 q^{38} -1.24412 q^{39} -9.71859 q^{40} -6.90245 q^{41} +6.43240 q^{42} +4.18717 q^{43} +2.64436 q^{44} +4.58634 q^{45} -7.82014 q^{46} +0.196825 q^{47} -2.53069 q^{48} +13.9194 q^{49} -5.62354 q^{50} -5.09679 q^{51} +0.722174 q^{52} +3.72381 q^{53} +6.26137 q^{54} +11.5646 q^{55} -14.0743 q^{56} -0.736223 q^{57} -1.84464 q^{58} +9.88610 q^{59} +2.83763 q^{60} -6.09653 q^{61} -4.16153 q^{62} +6.64185 q^{63} +8.42592 q^{64} +3.15829 q^{65} +5.14965 q^{66} +11.3448 q^{67} +2.95853 q^{68} +8.60680 q^{69} -16.3291 q^{70} -4.91414 q^{71} -4.46855 q^{72} +1.46415 q^{73} +7.50206 q^{74} +6.18923 q^{75} +0.427354 q^{76} +16.7476 q^{77} +1.40637 q^{78} -13.6566 q^{79} +6.42432 q^{80} -2.53474 q^{81} +7.80259 q^{82} -12.7760 q^{83} +4.10940 q^{84} +12.9386 q^{85} -4.73321 q^{86} +2.03020 q^{87} -11.2676 q^{88} +8.83350 q^{89} -5.18445 q^{90} +4.57377 q^{91} -4.99598 q^{92} +4.58015 q^{93} -0.222492 q^{94} +1.86895 q^{95} -4.79604 q^{96} -16.5779 q^{97} -15.7346 q^{98} +5.31733 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9}+O(q^{10}) $ 30 * q + 5 * q^3 + 34 * q^4 + 17 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 49 * q^9	\( 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9} + 5 q^{10} + q^{11} + 15 q^{12} - 30 q^{13} + 24 q^{14} + 6 q^{15} + 38 q^{16} + 17 q^{17} + 8 q^{18} + 9 q^{19} + 31 q^{20} + 27 q^{21} + 2 q^{22} + 14 q^{23} + 26 q^{24} + 47 q^{25} + 14 q^{27} - 6 q^{28} + 53 q^{29} + 25 q^{30} + 19 q^{31} - 4 q^{32} + q^{33} - 22 q^{34} + 9 q^{35} + 61 q^{36} + 46 q^{38} - 5 q^{39} - 35 q^{40} + 28 q^{41} - 7 q^{42} + 6 q^{43} - 12 q^{44} + 68 q^{45} + 7 q^{46} + 12 q^{47} + 13 q^{48} + 54 q^{49} - 18 q^{50} + 10 q^{51} - 34 q^{52} + 37 q^{53} + 22 q^{54} + 11 q^{55} + 67 q^{56} - 57 q^{57} - 5 q^{58} + 61 q^{59} - 102 q^{60} + 16 q^{61} - 2 q^{62} - 7 q^{63} + 29 q^{64} - 17 q^{65} - 83 q^{66} - 2 q^{67} + 57 q^{68} + 98 q^{69} - 10 q^{70} + 50 q^{71} - 8 q^{72} - 10 q^{73} - 13 q^{74} + 5 q^{75} - 19 q^{76} + 54 q^{77} - 3 q^{78} + 3 q^{79} + 76 q^{80} + 118 q^{81} + 7 q^{82} + 6 q^{83} + 44 q^{84} + 33 q^{85} - 29 q^{86} - 10 q^{87} - 13 q^{88} + 77 q^{89} + 38 q^{90} - 2 q^{91} - 3 q^{92} + 34 q^{93} - 25 q^{94} + 24 q^{95} - 28 q^{96} + 12 q^{97} - 14 q^{98} - 58 q^{99}+O(q^{100}) \) 30 * q + 5 * q^3 + 34 * q^4 + 17 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 49 * q^9 + 5 * q^10 + q^11 + 15 * q^12 - 30 * q^13 + 24 * q^14 + 6 * q^15 + 38 * q^16 + 17 * q^17 + 8 * q^18 + 9 * q^19 + 31 * q^20 + 27 * q^21 + 2 * q^22 + 14 * q^23 + 26 * q^24 + 47 * q^25 + 14 * q^27 - 6 * q^28 + 53 * q^29 + 25 * q^30 + 19 * q^31 - 4 * q^32 + q^33 - 22 * q^34 + 9 * q^35 + 61 * q^36 + 46 * q^38 - 5 * q^39 - 35 * q^40 + 28 * q^41 - 7 * q^42 + 6 * q^43 - 12 * q^44 + 68 * q^45 + 7 * q^46 + 12 * q^47 + 13 * q^48 + 54 * q^49 - 18 * q^50 + 10 * q^51 - 34 * q^52 + 37 * q^53 + 22 * q^54 + 11 * q^55 + 67 * q^56 - 57 * q^57 - 5 * q^58 + 61 * q^59 - 102 * q^60 + 16 * q^61 - 2 * q^62 - 7 * q^63 + 29 * q^64 - 17 * q^65 - 83 * q^66 - 2 * q^67 + 57 * q^68 + 98 * q^69 - 10 * q^70 + 50 * q^71 - 8 * q^72 - 10 * q^73 - 13 * q^74 + 5 * q^75 - 19 * q^76 + 54 * q^77 - 3 * q^78 + 3 * q^79 + 76 * q^80 + 118 * q^81 + 7 * q^82 + 6 * q^83 + 44 * q^84 + 33 * q^85 - 29 * q^86 - 10 * q^87 - 13 * q^88 + 77 * q^89 + 38 * q^90 - 2 * q^91 - 3 * q^92 + 34 * q^93 - 25 * q^94 + 24 * q^95 - 28 * q^96 + 12 * q^97 - 14 * q^98 - 58 * 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 $

$2$	−1.13041	−0.799320	−0.399660	−	0.916663i	$-0.630872\pi$
$2$	−1.13041	−0.799320	−0.399660	+	0.916663i	$0.630872\pi$
$3$	1.24412	0.718294	0.359147	−	0.933281i	$-0.383068\pi$
$3$	1.24412	0.718294	0.359147	+	0.933281i	$0.383068\pi$
$4$	−0.722174	−0.361087
$5$	−3.15829	−1.41243	−0.706215	−	0.707998i	$-0.749599\pi$
$5$	−3.15829	−1.41243	−0.706215	+	0.707998i	$0.749599\pi$
$6$	−1.40637	−0.574147
$7$	−4.57377	−1.72872	−0.864361	−	0.502872i	$-0.832277\pi$
$7$	−4.57377	−1.72872	−0.864361	+	0.502872i	$0.832277\pi$
$8$	3.07717	1.08794
$9$	−1.45216	−0.484054
$10$	3.57016	1.12898
$11$	−3.66167	−1.10403	−0.552017	−	0.833833i	$-0.686142\pi$
$11$	−3.66167	−1.10403	−0.552017	+	0.833833i	$0.686142\pi$
$12$	−0.898472	−0.259367
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	5.17023	1.38180
$15$	−3.92929	−1.01454
$16$	−2.03412	−0.508529
$17$	−4.09670	−0.993596	−0.496798	−	0.867866i	$-0.665491\pi$
$17$	−4.09670	−0.993596	−0.496798	+	0.867866i	$0.665491\pi$
$18$	1.64154	0.386914
$19$	−0.591761	−0.135759	−0.0678796	−	0.997694i	$-0.521623\pi$
$19$	−0.591761	−0.135759	−0.0678796	+	0.997694i	$0.521623\pi$
$20$	2.28083	0.510010
$21$	−5.69032	−1.24173
$22$	4.13919	0.882478
$23$	6.91797	1.44250	0.721248	−	0.692677i	$-0.243569\pi$
$23$	6.91797	1.44250	0.721248	+	0.692677i	$0.243569\pi$
$24$	3.82838	0.781464
$25$	4.97478	0.994956
$26$	1.13041	0.221692
$27$	−5.53903	−1.06599
$28$	3.30306	0.624219
$29$	1.63183	0.303024	0.151512	−	0.988455i	$-0.451586\pi$
$29$	1.63183	0.303024	0.151512	+	0.988455i	$0.451586\pi$
$30$	4.44171	0.810942
$31$	3.68143	0.661205	0.330603	−	0.943770i	$-0.392748\pi$
$31$	3.68143	0.661205	0.330603	+	0.943770i	$0.392748\pi$
$32$	−3.85496	−0.681467
$33$	−4.55556	−0.793022
$34$	4.63095	0.794201
$35$	14.4453	2.44170
$36$	1.04871	0.174786
$37$	−6.63658	−1.09105	−0.545524	−	0.838095i	$-0.683669\pi$
$37$	−6.63658	−1.09105	−0.545524	+	0.838095i	$0.683669\pi$
$38$	0.668932	0.108515
$39$	−1.24412	−0.199219
$40$	−9.71859	−1.53664
$41$	−6.90245	−1.07798	−0.538991	−	0.842312i	$-0.681194\pi$
$41$	−6.90245	−1.07798	−0.538991	+	0.842312i	$0.681194\pi$
$42$	6.43240	0.992540
$43$	4.18717	0.638537	0.319268	−	0.947664i	$-0.396563\pi$
$43$	4.18717	0.638537	0.319268	+	0.947664i	$0.396563\pi$
$44$	2.64436	0.398653
$45$	4.58634	0.683692
$46$	−7.82014	−1.15302
$47$	0.196825	0.0287098	0.0143549	−	0.999897i	$-0.495431\pi$
$47$	0.196825	0.0287098	0.0143549	+	0.999897i	$0.495431\pi$
$48$	−2.53069	−0.365273
$49$	13.9194	1.98848
$50$	−5.62354	−0.795289
$51$	−5.09679	−0.713694
$52$	0.722174	0.100148
$53$	3.72381	0.511504	0.255752	−	0.966742i	$-0.417677\pi$
$53$	3.72381	0.511504	0.255752	+	0.966742i	$0.417677\pi$
$54$	6.26137	0.852065
$55$	11.5646	1.55937
$56$	−14.0743	−1.88075
$57$	−0.736223	−0.0975151
$58$	−1.84464	−0.242213
$59$	9.88610	1.28706	0.643530	−	0.765421i	$-0.277469\pi$
$59$	9.88610	1.28706	0.643530	+	0.765421i	$0.277469\pi$
$60$	2.83763	0.366337
$61$	−6.09653	−0.780580	−0.390290	−	0.920692i	$-0.627625\pi$
$61$	−6.09653	−0.780580	−0.390290	+	0.920692i	$0.627625\pi$
$62$	−4.16153	−0.528515
$63$	6.64185	0.836794
$64$	8.42592	1.05324
$65$	3.15829	0.391737
$66$	5.14965	0.633878
$67$	11.3448	1.38599	0.692993	−	0.720945i	$-0.256292\pi$
$67$	11.3448	1.38599	0.692993	+	0.720945i	$0.256292\pi$
$68$	2.95853	0.358775
$69$	8.60680	1.03614
$70$	−16.3291	−1.95170
$71$	−4.91414	−0.583202	−0.291601	−	0.956540i	$-0.594188\pi$
$71$	−4.91414	−0.583202	−0.291601	+	0.956540i	$0.594188\pi$
$72$	−4.46855	−0.526624
$73$	1.46415	0.171366	0.0856830	−	0.996322i	$-0.472693\pi$
$73$	1.46415	0.171366	0.0856830	+	0.996322i	$0.472693\pi$
$74$	7.50206	0.872097
$75$	6.18923	0.714671
$76$	0.427354	0.0490209
$77$	16.7476	1.90857
$78$	1.40637	0.159240
$79$	−13.6566	−1.53649	−0.768247	−	0.640154i	$-0.778871\pi$
$79$	−13.6566	−1.53649	−0.768247	+	0.640154i	$0.778871\pi$
$80$	6.42432	0.718261
$81$	−2.53474	−0.281638
$82$	7.80259	0.861652
$83$	−12.7760	−1.40235	−0.701174	−	0.712990i	$-0.747341\pi$
$83$	−12.7760	−1.40235	−0.701174	+	0.712990i	$0.747341\pi$
$84$	4.10940	0.448373
$85$	12.9386	1.40338
$86$	−4.73321	−0.510396
$87$	2.03020	0.217660
$88$	−11.2676	−1.20113
$89$	8.83350	0.936350	0.468175	−	0.883636i	$-0.344912\pi$
$89$	8.83350	0.936350	0.468175	+	0.883636i	$0.344912\pi$
$90$	−5.18445	−0.546489
$91$	4.57377	0.479461
$92$	−4.99598	−0.520867
$93$	4.58015	0.474940
$94$	−0.222492	−0.0229483
$95$	1.86895	0.191750
$96$	−4.79604	−0.489494
$97$	−16.5779	−1.68323	−0.841617	−	0.540075i	$-0.818396\pi$
$97$	−16.5779	−1.68323	−0.841617	+	0.540075i	$0.818396\pi$
$98$	−15.7346	−1.58943
$99$	5.31733	0.534412
$100$	−3.59266	−0.359266
$101$	15.5348	1.54577	0.772887	−	0.634544i	$-0.218812\pi$
$101$	15.5348	1.54577	0.772887	+	0.634544i	$0.218812\pi$
$102$	5.76147	0.570470
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−3.07717	−0.301742
$105$	17.9717	1.75386
$106$	−4.20943	−0.408856
$107$	−9.78683	−0.946128	−0.473064	−	0.881028i	$-0.656852\pi$
$107$	−9.78683	−0.946128	−0.473064	+	0.881028i	$0.656852\pi$
$108$	4.00014	0.384914
$109$	−18.6166	−1.78314	−0.891572	−	0.452880i	$-0.850397\pi$
$109$	−18.6166	−1.78314	−0.891572	+	0.452880i	$0.850397\pi$
$110$	−13.0727	−1.24644
$111$	−8.25672	−0.783693
$112$	9.30358	0.879105
$113$	4.14105	0.389557	0.194779	−	0.980847i	$-0.437601\pi$
$113$	4.14105	0.389557	0.194779	+	0.980847i	$0.437601\pi$
$114$	0.832233	0.0779458
$115$	−21.8489	−2.03742
$116$	−1.17847	−0.109418
$117$	1.45216	0.134252
$118$	−11.1753	−1.02877
$119$	18.7374	1.71765
$120$	−12.0911	−1.10376
$121$	2.40782	0.218893
$122$	6.89157	0.623934
$123$	−8.58749	−0.774308
$124$	−2.65864	−0.238753
$125$	0.0796448	0.00712365
$126$	−7.50801	−0.668867
$127$	6.23426	0.553202	0.276601	−	0.960985i	$-0.410792\pi$
$127$	6.23426	0.553202	0.276601	+	0.960985i	$0.410792\pi$
$128$	−1.81482	−0.160409
$129$	5.20935	0.458657
$130$	−3.57016	−0.313124
$131$	4.99361	0.436293	0.218147	−	0.975916i	$-0.429999\pi$
$131$	4.99361	0.436293	0.218147	+	0.975916i	$0.429999\pi$
$132$	3.28991	0.286350
$133$	2.70658	0.234690
$134$	−12.8242	−1.10785
$135$	17.4939	1.50563
$136$	−12.6063	−1.08098
$137$	7.36450	0.629192	0.314596	−	0.949226i	$-0.398131\pi$
$137$	7.36450	0.629192	0.314596	+	0.949226i	$0.398131\pi$
$138$	−9.72921	−0.828205
$139$	−18.9709	−1.60909	−0.804545	−	0.593891i	$-0.797591\pi$
$139$	−18.9709	−1.60909	−0.804545	+	0.593891i	$0.797591\pi$
$140$	−10.4320	−0.881665
$141$	0.244874	0.0206221
$142$	5.55500	0.466165
$143$	3.66167	0.306204
$144$	2.95387	0.246155
$145$	−5.15380	−0.428000
$146$	−1.65509	−0.136976
$147$	17.3174	1.42831
$148$	4.79277	0.393963
$149$	−3.79184	−0.310640	−0.155320	−	0.987864i	$-0.549641\pi$
$149$	−3.79184	−0.310640	−0.155320	+	0.987864i	$0.549641\pi$
$150$	−6.99637	−0.571251
$151$	17.8729	1.45448	0.727238	−	0.686385i	$-0.240804\pi$
$151$	17.8729	1.45448	0.727238	+	0.686385i	$0.240804\pi$
$152$	−1.82095	−0.147699
$153$	5.94907	0.480954
$154$	−18.9317	−1.52556
$155$	−11.6270	−0.933905
$156$	0.898472	0.0719354
$157$	10.2596	0.818804	0.409402	−	0.912354i	$-0.365737\pi$
$157$	10.2596	0.818804	0.409402	+	0.912354i	$0.365737\pi$
$158$	15.4376	1.22815
$159$	4.63287	0.367410
$160$	12.1751	0.962524
$161$	−31.6412	−2.49368
$162$	2.86530	0.225119
$163$	−8.55010	−0.669696	−0.334848	−	0.942272i	$-0.608685\pi$
$163$	−8.55010	−0.669696	−0.334848	+	0.942272i	$0.608685\pi$
$164$	4.98477	0.389245
$165$	14.3878	1.12009
$166$	14.4421	1.12093
$167$	11.0291	0.853461	0.426731	−	0.904379i	$-0.359665\pi$
$167$	11.0291	0.853461	0.426731	+	0.904379i	$0.359665\pi$
$168$	−17.5101	−1.35093
$169$	1.00000	0.0769231
$170$	−14.6259	−1.12175
$171$	0.859332	0.0657148
$172$	−3.02386	−0.230567
$173$	22.8628	1.73822	0.869112	−	0.494615i	$-0.164691\pi$
$173$	22.8628	1.73822	0.869112	+	0.494615i	$0.164691\pi$
$174$	−2.29496	−0.173980
$175$	−22.7535	−1.72000
$176$	7.44826	0.561434
$177$	12.2995	0.924488
$178$	−9.98548	−0.748443
$179$	−7.29720	−0.545419	−0.272709	−	0.962096i	$-0.587920\pi$
$179$	−7.29720	−0.545419	−0.272709	+	0.962096i	$0.587920\pi$
$180$	−3.31214	−0.246872
$181$	6.77213	0.503369	0.251684	−	0.967809i	$-0.419016\pi$
$181$	6.77213	0.503369	0.251684	+	0.967809i	$0.419016\pi$
$182$	−5.17023	−0.383243
$183$	−7.58482	−0.560686
$184$	21.2878	1.56936
$185$	20.9602	1.54103
$186$	−5.17745	−0.379629
$187$	15.0008	1.09696
$188$	−0.142142	−0.0103667
$189$	25.3342	1.84279
$190$	−2.11268	−0.153270
$191$	−13.0071	−0.941161	−0.470581	−	0.882357i	$-0.655955\pi$
$191$	−13.0071	−0.941161	−0.470581	+	0.882357i	$0.655955\pi$
$192$	10.4829	0.756536
$193$	−19.3132	−1.39019	−0.695097	−	0.718916i	$-0.744639\pi$
$193$	−19.3132	−1.39019	−0.695097	+	0.718916i	$0.744639\pi$
$194$	18.7398	1.34544
$195$	3.92929	0.281383
$196$	−10.0522	−0.718014
$197$	−2.25716	−0.160816	−0.0804079	−	0.996762i	$-0.525622\pi$
$197$	−2.25716	−0.160816	−0.0804079	+	0.996762i	$0.525622\pi$
$198$	−6.01077	−0.427167
$199$	−1.74598	−0.123769	−0.0618846	−	0.998083i	$-0.519711\pi$
$199$	−1.74598	−0.123769	−0.0618846	+	0.998083i	$0.519711\pi$
$200$	15.3083	1.08246
$201$	14.1143	0.995545
$202$	−17.5607	−1.23557
$203$	−7.46363	−0.523844
$204$	3.68077	0.257706
$205$	21.7999	1.52257
$206$	−1.13041	−0.0787594
$207$	−10.0460	−0.698246
$208$	2.03412	0.141041
$209$	2.16683	0.149883
$210$	−20.3154	−1.40189
$211$	11.8012	0.812431	0.406216	−	0.913777i	$-0.366848\pi$
$211$	11.8012	0.812431	0.406216	+	0.913777i	$0.366848\pi$
$212$	−2.68924	−0.184698
$213$	−6.11379	−0.418910
$214$	11.0631	0.756259
$215$	−13.2243	−0.901888
$216$	−17.0445	−1.15973
$217$	−16.8380	−1.14304
$218$	21.0443	1.42530
$219$	1.82158	0.123091
$220$	−8.35166	−0.563069
$221$	4.09670	0.275574
$222$	9.33347	0.626422
$223$	16.6628	1.11582	0.557912	−	0.829900i	$-0.311603\pi$
$223$	16.6628	1.11582	0.557912	+	0.829900i	$0.311603\pi$
$224$	17.6317	1.17807
$225$	−7.22419	−0.481612
$226$	−4.68108	−0.311381
$227$	−5.65583	−0.375391	−0.187695	−	0.982227i	$-0.560102\pi$
$227$	−5.65583	−0.375391	−0.187695	+	0.982227i	$0.560102\pi$
$228$	0.531681	0.0352114
$229$	−12.1067	−0.800032	−0.400016	−	0.916508i	$-0.630995\pi$
$229$	−12.1067	−0.800032	−0.400016	+	0.916508i	$0.630995\pi$
$230$	24.6983	1.62855
$231$	20.8361	1.37091
$232$	5.02143	0.329673
$233$	−16.9513	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$233$	−16.9513	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$234$	−1.64154	−0.107311
$235$	−0.621629	−0.0405506
$236$	−7.13949	−0.464741
$237$	−16.9905	−1.10365
$238$	−21.1809	−1.37295
$239$	10.3913	0.672157	0.336079	−	0.941834i	$-0.390899\pi$
$239$	10.3913	0.672157	0.336079	+	0.941834i	$0.390899\pi$
$240$	7.99264	0.515923
$241$	−0.463491	−0.0298561	−0.0149280	−	0.999889i	$-0.504752\pi$
$241$	−0.463491	−0.0298561	−0.0149280	+	0.999889i	$0.504752\pi$
$242$	−2.72183	−0.174966
$243$	13.4636	0.863688
$244$	4.40275	0.281857
$245$	−43.9613	−2.80859
$246$	9.70738	0.618920
$247$	0.591761	0.0376528
$248$	11.3284	0.719354
$249$	−15.8949	−1.00730
$250$	−0.0900313	−0.00569408
$251$	29.6848	1.87369	0.936843	−	0.349750i	$-0.113734\pi$
$251$	29.6848	1.87369	0.936843	+	0.349750i	$0.113734\pi$
$252$	−4.79657	−0.302156
$253$	−25.3313	−1.59257
$254$	−7.04727	−0.442185
$255$	16.0971	1.00804
$256$	−14.8003	−0.925021
$257$	−15.4462	−0.963505	−0.481752	−	0.876307i	$-0.660000\pi$
$257$	−15.4462	−0.963505	−0.481752	+	0.876307i	$0.660000\pi$
$258$	−5.88869	−0.366614
$259$	30.3542	1.88612
$260$	−2.28083	−0.141451
$261$	−2.36969	−0.146680
$262$	−5.64482	−0.348738
$263$	−1.04868	−0.0646644	−0.0323322	−	0.999477i	$-0.510293\pi$
$263$	−1.04868	−0.0646644	−0.0323322	+	0.999477i	$0.510293\pi$
$264$	−14.0182	−0.862764
$265$	−11.7609	−0.722464
$266$	−3.05954	−0.187592
$267$	10.9900	0.672574
$268$	−8.19290	−0.500461
$269$	18.7313	1.14207	0.571035	−	0.820926i	$-0.306542\pi$
$269$	18.7313	1.14207	0.571035	+	0.820926i	$0.306542\pi$
$270$	−19.7752	−1.20348
$271$	−6.05339	−0.367717	−0.183859	−	0.982953i	$-0.558859\pi$
$271$	−6.05339	−0.367717	−0.183859	+	0.982953i	$0.558859\pi$
$272$	8.33317	0.505272
$273$	5.69032	0.344394
$274$	−8.32490	−0.502926
$275$	−18.2160	−1.09847
$276$	−6.21561	−0.374136
$277$	−14.1703	−0.851410	−0.425705	−	0.904862i	$-0.639974\pi$
$277$	−14.1703	−0.851410	−0.425705	+	0.904862i	$0.639974\pi$
$278$	21.4449	1.28618
$279$	−5.34604	−0.320059
$280$	44.4506	2.65643
$281$	21.0989	1.25865	0.629327	−	0.777141i	$-0.283331\pi$
$281$	21.0989	1.25865	0.629327	+	0.777141i	$0.283331\pi$
$282$	−0.276808	−0.0164837
$283$	15.4092	0.915981	0.457990	−	0.888957i	$-0.348569\pi$
$283$	15.4092	0.915981	0.457990	+	0.888957i	$0.348569\pi$
$284$	3.54887	0.210587
$285$	2.32520	0.137733
$286$	−4.13919	−0.244755
$287$	31.5702	1.86353
$288$	5.59802	0.329867
$289$	−0.217039	−0.0127670
$290$	5.82590	0.342109
$291$	−20.6250	−1.20906
$292$	−1.05737	−0.0618781
$293$	28.1102	1.64222	0.821108	−	0.570772i	$-0.193356\pi$
$293$	28.1102	1.64222	0.821108	+	0.570772i	$0.193356\pi$
$294$	−19.5757	−1.14168
$295$	−31.2232	−1.81788
$296$	−20.4219	−1.18700
$297$	20.2821	1.17689
$298$	4.28633	0.248301
$299$	−6.91797	−0.400077
$300$	−4.46970	−0.258059
$301$	−19.1511	−1.10385
$302$	−20.2037	−1.16259
$303$	19.3272	1.11032
$304$	1.20371	0.0690375
$305$	19.2546	1.10251
$306$	−6.72489	−0.384436
$307$	−19.2107	−1.09641	−0.548207	−	0.836343i	$-0.684689\pi$
$307$	−19.2107	−1.09641	−0.548207	+	0.836343i	$0.684689\pi$
$308$	−12.0947	−0.689160
$309$	1.24412	0.0707756
$310$	13.1433	0.746490
$311$	−9.32268	−0.528641	−0.264320	−	0.964435i	$-0.585148\pi$
$311$	−9.32268	−0.528641	−0.264320	+	0.964435i	$0.585148\pi$
$312$	−3.82838	−0.216739
$313$	9.79610	0.553708	0.276854	−	0.960912i	$-0.410708\pi$
$313$	9.79610	0.553708	0.276854	+	0.960912i	$0.410708\pi$
$314$	−11.5975	−0.654487
$315$	−20.9769	−1.18191
$316$	9.86248	0.554808
$317$	10.3820	0.583114	0.291557	−	0.956554i	$-0.405827\pi$
$317$	10.3820	0.583114	0.291557	+	0.956554i	$0.405827\pi$
$318$	−5.23704	−0.293679
$319$	−5.97523	−0.334549
$320$	−26.6115	−1.48763
$321$	−12.1760	−0.679598
$322$	35.7675	1.99325
$323$	2.42427	0.134890
$324$	1.83053	0.101696
$325$	−4.97478	−0.275951
$326$	9.66512	0.535301
$327$	−23.1613	−1.28082
$328$	−21.2400	−1.17278
$329$	−0.900230	−0.0496313
$330$	−16.2641	−0.895308
$331$	−29.5052	−1.62175	−0.810876	−	0.585218i	$-0.801009\pi$
$331$	−29.5052	−1.62175	−0.810876	+	0.585218i	$0.801009\pi$
$332$	9.22650	0.506370
$333$	9.63739	0.528126
$334$	−12.4675	−0.682189
$335$	−35.8301	−1.95761
$336$	11.5748	0.631456
$337$	−27.3158	−1.48799	−0.743993	−	0.668188i	$-0.767070\pi$
$337$	−27.3158	−1.48799	−0.743993	+	0.668188i	$0.767070\pi$
$338$	−1.13041	−0.0614862
$339$	5.15197	0.279817
$340$	−9.34389	−0.506744
$341$	−13.4802	−0.729993
$342$	−0.971398	−0.0525272
$343$	−31.6475	−1.70880
$344$	12.8846	0.694693
$345$	−27.1827	−1.46347
$346$	−25.8443	−1.38940
$347$	7.25345	0.389385	0.194693	−	0.980864i	$-0.437629\pi$
$347$	7.25345	0.389385	0.194693	+	0.980864i	$0.437629\pi$
$348$	−1.46616	−0.0785943
$349$	−9.65565	−0.516855	−0.258428	−	0.966031i	$-0.583204\pi$
$349$	−9.65565	−0.516855	−0.258428	+	0.966031i	$0.583204\pi$
$350$	25.7208	1.37483
$351$	5.53903	0.295652
$352$	14.1156	0.752363
$353$	−4.25344	−0.226388	−0.113194	−	0.993573i	$-0.536108\pi$
$353$	−4.25344	−0.226388	−0.113194	+	0.993573i	$0.536108\pi$
$354$	−13.9035	−0.738962
$355$	15.5203	0.823731
$356$	−6.37933	−0.338104
$357$	23.3116	1.23378
$358$	8.24883	0.435964
$359$	−0.313234	−0.0165318	−0.00826592	−	0.999966i	$-0.502631\pi$
$359$	−0.313234	−0.0165318	−0.00826592	+	0.999966i	$0.502631\pi$
$360$	14.1130	0.743819
$361$	−18.6498	−0.981569
$362$	−7.65528	−0.402353
$363$	2.99563	0.157230
$364$	−3.30306	−0.173127
$365$	−4.62421	−0.242042
$366$	8.57395	0.448168
$367$	−6.75041	−0.352368	−0.176184	−	0.984357i	$-0.556375\pi$
$367$	−6.75041	−0.352368	−0.176184	+	0.984357i	$0.556375\pi$
$368$	−14.0720	−0.733551
$369$	10.0235	0.521801
$370$	−23.6937	−1.23177
$371$	−17.0318	−0.884249
$372$	−3.30767	−0.171495
$373$	−1.00544	−0.0520600	−0.0260300	−	0.999661i	$-0.508287\pi$
$373$	−1.00544	−0.0520600	−0.0260300	+	0.999661i	$0.508287\pi$
$374$	−16.9570	−0.876826
$375$	0.0990878	0.00511687
$376$	0.605663	0.0312347
$377$	−1.63183	−0.0840437
$378$	−28.6381	−1.47298
$379$	−23.8288	−1.22400	−0.612002	−	0.790857i	$-0.709635\pi$
$379$	−23.8288	−1.22400	−0.612002	+	0.790857i	$0.709635\pi$
$380$	−1.34971	−0.0692386
$381$	7.75618	0.397361
$382$	14.7034	0.752289
$383$	−27.5134	−1.40587	−0.702935	−	0.711254i	$-0.748128\pi$
$383$	−27.5134	−1.40587	−0.702935	+	0.711254i	$0.748128\pi$
$384$	−2.25785	−0.115221
$385$	−52.8938	−2.69572
$386$	21.8318	1.11121
$387$	−6.08044	−0.309086
$388$	11.9721	0.607794
$389$	32.8736	1.66676	0.833380	−	0.552701i	$-0.186403\pi$
$389$	32.8736	1.66676	0.833380	+	0.552701i	$0.186403\pi$
$390$	−4.44171	−0.224915
$391$	−28.3409	−1.43326
$392$	42.8322	2.16335
$393$	6.21265	0.313387
$394$	2.55151	0.128543
$395$	43.1316	2.17019
$396$	−3.84004	−0.192969
$397$	−24.5767	−1.23347	−0.616734	−	0.787171i	$-0.711545\pi$
$397$	−24.5767	−1.23347	−0.616734	+	0.787171i	$0.711545\pi$
$398$	1.97367	0.0989313
$399$	3.36731	0.168576
$400$	−10.1193	−0.505964
$401$	29.5856	1.47744	0.738718	−	0.674015i	$-0.235432\pi$
$401$	29.5856	1.47744	0.738718	+	0.674015i	$0.235432\pi$
$402$	−15.9549	−0.795759
$403$	−3.68143	−0.183385
$404$	−11.2189	−0.558159
$405$	8.00545	0.397794
$406$	8.43696	0.418719
$407$	24.3010	1.20455
$408$	−15.6837	−0.776459
$409$	15.9596	0.789154	0.394577	−	0.918863i	$-0.370891\pi$
$409$	15.9596	0.789154	0.394577	+	0.918863i	$0.370891\pi$
$410$	−24.6428	−1.21702
$411$	9.16233	0.451945
$412$	−0.722174	−0.0355790
$413$	−45.2167	−2.22497
$414$	11.3561	0.558122
$415$	40.3503	1.98072
$416$	3.85496	0.189005
$417$	−23.6021	−1.15580
$418$	−2.44941	−0.119805
$419$	−4.29439	−0.209795	−0.104897	−	0.994483i	$-0.533451\pi$
$419$	−4.29439	−0.209795	−0.104897	+	0.994483i	$0.533451\pi$
$420$	−12.9787	−0.633295
$421$	9.54506	0.465198	0.232599	−	0.972573i	$-0.425277\pi$
$421$	9.54506	0.465198	0.232599	+	0.972573i	$0.425277\pi$
$422$	−13.3402	−0.649393
$423$	−0.285821	−0.0138971
$424$	11.4588	0.556488
$425$	−20.3802	−0.988585
$426$	6.91109	0.334843
$427$	27.8841	1.34941
$428$	7.06779	0.341635
$429$	4.55556	0.219945
$430$	14.9489	0.720898
$431$	16.4941	0.794493	0.397247	−	0.917712i	$-0.369966\pi$
$431$	16.4941	0.794493	0.397247	+	0.917712i	$0.369966\pi$
$432$	11.2670	0.542085
$433$	−2.51258	−0.120747	−0.0603736	−	0.998176i	$-0.519229\pi$
$433$	−2.51258	−0.120747	−0.0603736	+	0.998176i	$0.519229\pi$
$434$	19.0339	0.913655
$435$	−6.41195	−0.307430
$436$	13.4444	0.643870
$437$	−4.09379	−0.195832
$438$	−2.05914	−0.0983893
$439$	19.5323	0.932224	0.466112	−	0.884726i	$-0.345654\pi$
$439$	19.5323	0.932224	0.466112	+	0.884726i	$0.345654\pi$
$440$	35.5863	1.69651
$441$	−20.2131	−0.962531
$442$	−4.63095	−0.220272
$443$	−13.5751	−0.644974	−0.322487	−	0.946574i	$-0.604519\pi$
$443$	−13.5751	−0.644974	−0.322487	+	0.946574i	$0.604519\pi$
$444$	5.96279	0.282981
$445$	−27.8988	−1.32253
$446$	−18.8358	−0.891900
$447$	−4.71751	−0.223131
$448$	−38.5382	−1.82076
$449$	−26.5747	−1.25414	−0.627068	−	0.778965i	$-0.715745\pi$
$449$	−26.5747	−1.25414	−0.627068	+	0.778965i	$0.715745\pi$
$450$	8.16629	0.384963
$451$	25.2745	1.19013
$452$	−2.99056	−0.140664
$453$	22.2361	1.04474
$454$	6.39341	0.300058
$455$	−14.4453	−0.677205
$456$	−2.26548	−0.106091
$457$	18.2678	0.854530	0.427265	−	0.904127i	$-0.359477\pi$
$457$	18.2678	0.854530	0.427265	+	0.904127i	$0.359477\pi$
$458$	13.6855	0.639482
$459$	22.6918	1.05916
$460$	15.7787	0.735688
$461$	13.4382	0.625881	0.312940	−	0.949773i	$-0.398686\pi$
$461$	13.4382	0.625881	0.312940	+	0.949773i	$0.398686\pi$
$462$	−23.5533	−1.09580
$463$	22.7421	1.05692	0.528458	−	0.848959i	$-0.322770\pi$
$463$	22.7421	1.05692	0.528458	+	0.848959i	$0.322770\pi$
$464$	−3.31934	−0.154096
$465$	−14.4654	−0.670819
$466$	19.1619	0.887656
$467$	−20.3573	−0.942022	−0.471011	−	0.882127i	$-0.656111\pi$
$467$	−20.3573	−0.942022	−0.471011	+	0.882127i	$0.656111\pi$
$468$	−1.04871	−0.0484768
$469$	−51.8884	−2.39598
$470$	0.702695	0.0324129
$471$	12.7642	0.588142
$472$	30.4212	1.40025
$473$	−15.3320	−0.704967
$474$	19.2063	0.882173
$475$	−2.94388	−0.135075
$476$	−13.5316	−0.620222
$477$	−5.40757	−0.247596
$478$	−11.7464	−0.537269
$479$	−0.407995	−0.0186418	−0.00932088	−	0.999957i	$-0.502967\pi$
$479$	−0.407995	−0.0186418	−0.00932088	+	0.999957i	$0.502967\pi$
$480$	15.1473	0.691375
$481$	6.63658	0.302602
$482$	0.523934	0.0238645
$483$	−39.3655	−1.79119
$484$	−1.73887	−0.0790395
$485$	52.3579	2.37745
$486$	−15.2193	−0.690363
$487$	27.1135	1.22863	0.614315	−	0.789061i	$-0.289432\pi$
$487$	27.1135	1.22863	0.614315	+	0.789061i	$0.289432\pi$
$488$	−18.7601	−0.849228
$489$	−10.6374	−0.481038
$490$	49.6943	2.24496
$491$	−8.49499	−0.383373	−0.191687	−	0.981456i	$-0.561396\pi$
$491$	−8.49499	−0.383373	−0.191687	+	0.981456i	$0.561396\pi$
$492$	6.20166	0.279592
$493$	−6.68513	−0.301083
$494$	−0.668932	−0.0300967
$495$	−16.7937	−0.754820
$496$	−7.48847	−0.336242
$497$	22.4762	1.00819
$498$	17.9677	0.805154
$499$	26.2862	1.17673	0.588366	−	0.808595i	$-0.299772\pi$
$499$	26.2862	1.17673	0.588366	+	0.808595i	$0.299772\pi$
$500$	−0.0575174	−0.00257226
$501$	13.7216	0.613036
$502$	−33.5559	−1.49768
$503$	−35.0578	−1.56315	−0.781575	−	0.623812i	$-0.785583\pi$
$503$	−35.0578	−1.56315	−0.781575	+	0.623812i	$0.785583\pi$
$504$	20.4381	0.910386
$505$	−49.0635	−2.18330
$506$	28.6348	1.27297
$507$	1.24412	0.0552534
$508$	−4.50222	−0.199754
$509$	38.5309	1.70785	0.853925	−	0.520396i	$-0.174216\pi$
$509$	38.5309	1.70785	0.853925	+	0.520396i	$0.174216\pi$
$510$	−18.1964	−0.805749
$511$	−6.69669	−0.296244
$512$	20.3601	0.899797
$513$	3.27778	0.144718
$514$	17.4605	0.770149
$515$	−3.15829	−0.139171
$516$	−3.76205	−0.165615
$517$	−0.720707	−0.0316966
$518$	−34.3127	−1.50761
$519$	28.4441	1.24856
$520$	9.71859	0.426189
$521$	−20.5333	−0.899580	−0.449790	−	0.893134i	$-0.648501\pi$
$521$	−20.5333	−0.899580	−0.449790	+	0.893134i	$0.648501\pi$
$522$	2.67872	0.117244
$523$	−12.8176	−0.560474	−0.280237	−	0.959931i	$-0.590413\pi$
$523$	−12.8176	−0.560474	−0.280237	+	0.959931i	$0.590413\pi$
$524$	−3.60625	−0.157540
$525$	−28.3081	−1.23547
$526$	1.18544	0.0516875
$527$	−15.0817	−0.656971
$528$	9.26654	0.403275
$529$	24.8583	1.08080
$530$	13.2946	0.577480
$531$	−14.3562	−0.623007
$532$	−1.95462	−0.0847435
$533$	6.90245	0.298978
$534$	−12.4231	−0.537602
$535$	30.9096	1.33634
$536$	34.9098	1.50787
$537$	−9.07861	−0.391771
$538$	−21.1741	−0.912880
$539$	−50.9681	−2.19535
$540$	−12.6336	−0.543664
$541$	33.1693	1.42606	0.713029	−	0.701134i	$-0.247323\pi$
$541$	33.1693	1.42606	0.713029	+	0.701134i	$0.247323\pi$
$542$	6.84281	0.293924
$543$	8.42536	0.361567
$544$	15.7926	0.677103
$545$	58.7965	2.51856
$546$	−6.43240	−0.275281
$547$	−44.5047	−1.90288	−0.951442	−	0.307828i	$-0.900398\pi$
$547$	−44.5047	−1.90288	−0.951442	+	0.307828i	$0.900398\pi$
$548$	−5.31845	−0.227193
$549$	8.85314	0.377843
$550$	20.5916	0.878027
$551$	−0.965655	−0.0411383
$552$	26.4846	1.12726
$553$	62.4623	2.65617
$554$	16.0182	0.680549
$555$	26.0771	1.10691
$556$	13.7003	0.581022
$557$	12.2218	0.517856	0.258928	−	0.965897i	$-0.416631\pi$
$557$	12.2218	0.517856	0.258928	+	0.965897i	$0.416631\pi$
$558$	6.04321	0.255830
$559$	−4.18717	−0.177098
$560$	−29.3834	−1.24167
$561$	18.6628	0.787943
$562$	−23.8504	−1.00607
$563$	10.2170	0.430596	0.215298	−	0.976548i	$-0.430928\pi$
$563$	10.2170	0.430596	0.215298	+	0.976548i	$0.430928\pi$
$564$	−0.176842	−0.00744637
$565$	−13.0786	−0.550222
$566$	−17.4187	−0.732162
$567$	11.5933	0.486874
$568$	−15.1217	−0.634491
$569$	−31.5775	−1.32380	−0.661899	−	0.749593i	$-0.730249\pi$
$569$	−31.5775	−1.32380	−0.661899	+	0.749593i	$0.730249\pi$
$570$	−2.62843	−0.110093
$571$	18.0191	0.754077	0.377038	−	0.926198i	$-0.376942\pi$
$571$	18.0191	0.754077	0.377038	+	0.926198i	$0.376942\pi$
$572$	−2.64436	−0.110566
$573$	−16.1824	−0.676030
$574$	−35.6873	−1.48956
$575$	34.4154	1.43522
$576$	−12.2358	−0.509825
$577$	32.3880	1.34833	0.674164	−	0.738581i	$-0.264504\pi$
$577$	32.3880	1.34833	0.674164	+	0.738581i	$0.264504\pi$
$578$	0.245343	0.0102049
$579$	−24.0280	−0.998568
$580$	3.72194	0.154545
$581$	58.4345	2.42427
$582$	23.3146	0.966423
$583$	−13.6354	−0.564719
$584$	4.50545	0.186437
$585$	−4.58634	−0.189622
$586$	−31.7761	−1.31266
$587$	39.6783	1.63770	0.818850	−	0.574008i	$-0.194612\pi$
$587$	39.6783	1.63770	0.818850	+	0.574008i	$0.194612\pi$
$588$	−12.5062	−0.515745
$589$	−2.17853	−0.0897647
$590$	35.2949	1.45307
$591$	−2.80818	−0.115513
$592$	13.4996	0.554829
$593$	−23.6662	−0.971854	−0.485927	−	0.873999i	$-0.661518\pi$
$593$	−23.6662	−0.971854	−0.485927	+	0.873999i	$0.661518\pi$
$594$	−22.9271	−0.940709
$595$	−59.1780	−2.42606
$596$	2.73837	0.112168
$597$	−2.17221	−0.0889027
$598$	7.82014	0.319789
$599$	19.5343	0.798151	0.399076	−	0.916918i	$-0.369331\pi$
$599$	19.5343	0.798151	0.399076	+	0.916918i	$0.369331\pi$
$600$	19.0453	0.777523
$601$	21.5158	0.877648	0.438824	−	0.898573i	$-0.355395\pi$
$601$	21.5158	0.877648	0.438824	+	0.898573i	$0.355395\pi$
$602$	21.6486	0.882332
$603$	−16.4744	−0.670891
$604$	−12.9073	−0.525192
$605$	−7.60460	−0.309171
$606$	−21.8477	−0.887501
$607$	41.0132	1.66468	0.832338	−	0.554268i	$-0.187002\pi$
$607$	41.0132	1.66468	0.832338	+	0.554268i	$0.187002\pi$
$608$	2.28121	0.0925155
$609$	−9.28566	−0.376274
$610$	−21.7656	−0.881262
$611$	−0.196825	−0.00796267
$612$	−4.29627	−0.173666
$613$	−37.5110	−1.51505	−0.757527	−	0.652804i	$-0.773592\pi$
$613$	−37.5110	−1.51505	−0.757527	+	0.652804i	$0.773592\pi$
$614$	21.7160	0.876386
$615$	27.1218	1.09365
$616$	51.5353	2.07642
$617$	7.05138	0.283878	0.141939	−	0.989875i	$-0.454666\pi$
$617$	7.05138	0.283878	0.141939	+	0.989875i	$0.454666\pi$
$618$	−1.40637	−0.0565724
$619$	−37.0930	−1.49089	−0.745446	−	0.666566i	$-0.767764\pi$
$619$	−37.0930	−1.49089	−0.745446	+	0.666566i	$0.767764\pi$
$620$	8.39674	0.337221
$621$	−38.3189	−1.53768
$622$	10.5384	0.422553
$623$	−40.4024	−1.61869
$624$	2.53069	0.101309
$625$	−25.1255	−1.00502
$626$	−11.0736	−0.442590
$627$	2.69580	0.107660
$628$	−7.40921	−0.295660
$629$	27.1881	1.08406
$630$	23.7125	0.944727
$631$	−1.79108	−0.0713019	−0.0356509	−	0.999364i	$-0.511350\pi$
$631$	−1.79108	−0.0713019	−0.0356509	+	0.999364i	$0.511350\pi$
$632$	−42.0239	−1.67162
$633$	14.6822	0.583564
$634$	−11.7360	−0.466095
$635$	−19.6896	−0.781358
$636$	−3.34574	−0.132667
$637$	−13.9194	−0.551505
$638$	6.75446	0.267412
$639$	7.13613	0.282301
$640$	5.73172	0.226566
$641$	20.7286	0.818730	0.409365	−	0.912371i	$-0.365750\pi$
$641$	20.7286	0.818730	0.409365	+	0.912371i	$0.365750\pi$
$642$	13.7639	0.543217
$643$	10.0473	0.396226	0.198113	−	0.980179i	$-0.436519\pi$
$643$	10.0473	0.396226	0.198113	+	0.980179i	$0.436519\pi$
$644$	22.8505	0.900434
$645$	−16.4526	−0.647821
$646$	−2.74042	−0.107820
$647$	38.1212	1.49870	0.749349	−	0.662175i	$-0.230366\pi$
$647$	38.1212	1.49870	0.749349	+	0.662175i	$0.230366\pi$
$648$	−7.79984	−0.306407
$649$	−36.1996	−1.42096
$650$	5.62354	0.220573
$651$	−20.9486	−0.821038
$652$	6.17466	0.241818
$653$	−15.4723	−0.605477	−0.302738	−	0.953074i	$-0.597901\pi$
$653$	−15.4723	−0.605477	−0.302738	+	0.953074i	$0.597901\pi$
$654$	26.1817	1.02379
$655$	−15.7712	−0.616234
$656$	14.0404	0.548185
$657$	−2.12618	−0.0829504
$658$	1.01763	0.0396713
$659$	19.9511	0.777183	0.388592	−	0.921410i	$-0.372962\pi$
$659$	19.9511	0.777183	0.388592	+	0.921410i	$0.372962\pi$
$660$	−10.3905	−0.404449
$661$	36.4240	1.41673	0.708364	−	0.705847i	$-0.249434\pi$
$661$	36.4240	1.41673	0.708364	+	0.705847i	$0.249434\pi$
$662$	33.3529	1.29630
$663$	5.09679	0.197943
$664$	−39.3140	−1.52568
$665$	−8.54815	−0.331483
$666$	−10.8942	−0.422142
$667$	11.2890	0.437111
$668$	−7.96497	−0.308174
$669$	20.7305	0.801489
$670$	40.5027	1.56475
$671$	22.3235	0.861788
$672$	21.9360	0.846198
$673$	40.8124	1.57320	0.786601	−	0.617462i	$-0.211839\pi$
$673$	40.8124	1.57320	0.786601	+	0.617462i	$0.211839\pi$
$674$	30.8780	1.18938
$675$	−27.5555	−1.06061
$676$	−0.722174	−0.0277759
$677$	18.3543	0.705412	0.352706	−	0.935734i	$-0.385262\pi$
$677$	18.3543	0.705412	0.352706	+	0.935734i	$0.385262\pi$
$678$	−5.82383	−0.223663
$679$	75.8236	2.90984
$680$	39.8142	1.52680
$681$	−7.03655	−0.269641
$682$	15.2381	0.583499
$683$	27.5489	1.05413	0.527065	−	0.849825i	$-0.323293\pi$
$683$	27.5489	1.05413	0.527065	+	0.849825i	$0.323293\pi$
$684$	−0.620588	−0.0237288
$685$	−23.2592	−0.888689
$686$	35.7746	1.36588
$687$	−15.0622	−0.574658
$688$	−8.51718	−0.324715
$689$	−3.72381	−0.141866
$690$	30.7276	1.16978
$691$	40.9528	1.55792	0.778959	−	0.627075i	$-0.215748\pi$
$691$	40.9528	1.55792	0.778959	+	0.627075i	$0.215748\pi$
$692$	−16.5109	−0.627650
$693$	−24.3203	−0.923850
$694$	−8.19937	−0.311244
$695$	59.9156	2.27273
$696$	6.24727	0.236802
$697$	28.2773	1.07108
$698$	10.9148	0.413133
$699$	−21.0894	−0.797675
$700$	16.4320	0.621071
$701$	10.5084	0.396897	0.198448	−	0.980111i	$-0.436410\pi$
$701$	10.5084	0.396897	0.198448	+	0.980111i	$0.436410\pi$
$702$	−6.26137	−0.236320
$703$	3.92727	0.148120
$704$	−30.8529	−1.16281
$705$	−0.773382	−0.0291272
$706$	4.80813	0.180956
$707$	−71.0527	−2.67221
$708$	−8.88239	−0.333821
$709$	−15.3438	−0.576248	−0.288124	−	0.957593i	$-0.593031\pi$
$709$	−15.3438	−0.576248	−0.288124	+	0.957593i	$0.593031\pi$
$710$	−17.5443	−0.658425
$711$	19.8317	0.743745
$712$	27.1822	1.01870
$713$	25.4681	0.953786
$714$	−26.3516	−0.986184
$715$	−11.5646	−0.432492
$716$	5.26985	0.196944
$717$	12.9280	0.482806
$718$	0.354083	0.0132142
$719$	13.3754	0.498818	0.249409	−	0.968398i	$-0.419764\pi$
$719$	13.3754	0.498818	0.249409	+	0.968398i	$0.419764\pi$
$720$	−9.32916	−0.347677
$721$	−4.57377	−0.170336
$722$	21.0819	0.784588
$723$	−0.576639	−0.0214454
$724$	−4.89066	−0.181760
$725$	8.11802	0.301496
$726$	−3.38628	−0.125677
$727$	−31.5010	−1.16831	−0.584154	−	0.811643i	$-0.698573\pi$
$727$	−31.5010	−1.16831	−0.584154	+	0.811643i	$0.698573\pi$
$728$	14.0743	0.521627
$729$	24.3545	0.902020
$730$	5.22726	0.193469
$731$	−17.1536	−0.634448
$732$	5.47756	0.202457
$733$	−37.8216	−1.39697	−0.698485	−	0.715624i	$-0.746142\pi$
$733$	−37.8216	−1.39697	−0.698485	+	0.715624i	$0.746142\pi$
$734$	7.63072	0.281655
$735$	−54.6932	−2.01739
$736$	−26.6685	−0.983014
$737$	−41.5408	−1.53018
$738$	−11.3306	−0.417086
$739$	−33.4839	−1.23172	−0.615862	−	0.787854i	$-0.711192\pi$
$739$	−33.4839	−1.23172	−0.615862	+	0.787854i	$0.711192\pi$
$740$	−15.1369	−0.556445
$741$	0.736223	0.0270458
$742$	19.2529	0.706798
$743$	−4.93159	−0.180923	−0.0904613	−	0.995900i	$-0.528834\pi$
$743$	−4.93159	−0.180923	−0.0904613	+	0.995900i	$0.528834\pi$
$744$	14.0939	0.516708
$745$	11.9757	0.438757
$746$	1.13656	0.0416126
$747$	18.5528	0.678812
$748$	−10.8332	−0.396100
$749$	44.7627	1.63559
$750$	−0.112010	−0.00409002
$751$	12.4381	0.453872	0.226936	−	0.973910i	$-0.427129\pi$
$751$	12.4381	0.453872	0.226936	+	0.973910i	$0.427129\pi$
$752$	−0.400364	−0.0145998
$753$	36.9315	1.34586
$754$	1.84464	0.0671778
$755$	−56.4478	−2.05434
$756$	−18.2957	−0.665409
$757$	35.7706	1.30010	0.650052	−	0.759890i	$-0.274747\pi$
$757$	35.7706	1.30010	0.650052	+	0.759890i	$0.274747\pi$
$758$	26.9363	0.978370
$759$	−31.5152	−1.14393
$760$	5.75108	0.208614
$761$	38.8502	1.40832	0.704160	−	0.710042i	$-0.251324\pi$
$761$	38.8502	1.40832	0.704160	+	0.710042i	$0.251324\pi$
$762$	−8.76766	−0.317619
$763$	85.1478	3.08256
$764$	9.39339	0.339841
$765$	−18.7889	−0.679313
$766$	31.1014	1.12374
$767$	−9.88610	−0.356966
$768$	−18.4134	−0.664437
$769$	32.9070	1.18666	0.593328	−	0.804961i	$-0.297814\pi$
$769$	32.9070	1.18666	0.593328	+	0.804961i	$0.297814\pi$
$770$	59.7917	2.15474
$771$	−19.2169	−0.692080
$772$	13.9475	0.501981
$773$	43.8814	1.57830	0.789152	−	0.614197i	$-0.210520\pi$
$773$	43.8814	1.57830	0.789152	+	0.614197i	$0.210520\pi$
$774$	6.87339	0.247059
$775$	18.3143	0.657870
$776$	−51.0131	−1.83126
$777$	37.7643	1.35479
$778$	−37.1607	−1.33227
$779$	4.08460	0.146346
$780$	−2.83763	−0.101604
$781$	17.9940	0.643875
$782$	32.0368	1.14563
$783$	−9.03877	−0.323019
$784$	−28.3136	−1.01120
$785$	−32.4027	−1.15650
$786$	−7.02284	−0.250497
$787$	30.8993	1.10144	0.550721	−	0.834690i	$-0.314353\pi$
$787$	30.8993	1.10144	0.550721	+	0.834690i	$0.314353\pi$
$788$	1.63006	0.0580685
$789$	−1.30468	−0.0464480
$790$	−48.7564	−1.73468
$791$	−18.9402	−0.673436
$792$	16.3624	0.581411
$793$	6.09653	0.216494
$794$	27.7817	0.985936
$795$	−14.6319	−0.518941
$796$	1.26090	0.0446915
$797$	−14.1296	−0.500495	−0.250248	−	0.968182i	$-0.580512\pi$
$797$	−14.1296	−0.500495	−0.250248	+	0.968182i	$0.580512\pi$
$798$	−3.80644	−0.134747
$799$	−0.806332	−0.0285260
$800$	−19.1776	−0.678030
$801$	−12.8277	−0.453244
$802$	−33.4439	−1.18094
$803$	−5.36124	−0.189194
$804$	−10.1930	−0.359478
$805$	99.9320	3.52214
$806$	4.16153	0.146584
$807$	23.3041	0.820342
$808$	47.8033	1.68172
$809$	44.0813	1.54982	0.774908	−	0.632074i	$-0.217796\pi$
$809$	44.0813	1.54982	0.774908	+	0.632074i	$0.217796\pi$
$810$	−9.04944	−0.317965
$811$	−17.9881	−0.631647	−0.315823	−	0.948818i	$-0.602281\pi$
$811$	−17.9881	−0.631647	−0.315823	+	0.948818i	$0.602281\pi$
$812$	5.39004	0.189153
$813$	−7.53115	−0.264129
$814$	−27.4701	−0.962825
$815$	27.0037	0.945898
$816$	10.3675	0.362934
$817$	−2.47780	−0.0866873
$818$	−18.0409	−0.630787
$819$	−6.64185	−0.232085
$820$	−15.7433	−0.549781
$821$	−12.3980	−0.432693	−0.216347	−	0.976317i	$-0.569414\pi$
$821$	−12.3980	−0.432693	−0.216347	+	0.976317i	$0.569414\pi$
$822$	−10.3572	−0.361248
$823$	28.9785	1.01013	0.505063	−	0.863082i	$-0.331469\pi$
$823$	28.9785	1.01013	0.505063	+	0.863082i	$0.331469\pi$
$824$	3.07717	0.107198
$825$	−22.6629	−0.789022
$826$	51.1134	1.77846
$827$	−3.73872	−0.130008	−0.0650040	−	0.997885i	$-0.520706\pi$
$827$	−3.73872	−0.130008	−0.0650040	+	0.997885i	$0.520706\pi$
$828$	7.25497	0.252128
$829$	18.1467	0.630260	0.315130	−	0.949048i	$-0.397952\pi$
$829$	18.1467	0.630260	0.315130	+	0.949048i	$0.397952\pi$
$830$	−45.6124	−1.58323
$831$	−17.6296	−0.611562
$832$	−8.42592	−0.292116
$833$	−57.0234	−1.97574
$834$	26.6800	0.923854
$835$	−34.8332	−1.20545
$836$	−1.56483	−0.0541208
$837$	−20.3916	−0.704836
$838$	4.85442	0.167693
$839$	−30.1431	−1.04066	−0.520328	−	0.853967i	$-0.674190\pi$
$839$	−30.1431	−1.04066	−0.520328	+	0.853967i	$0.674190\pi$
$840$	55.3019	1.90810
$841$	−26.3371	−0.908177
$842$	−10.7898	−0.371842
$843$	26.2496	0.904083
$844$	−8.52255	−0.293358
$845$	−3.15829	−0.108648
$846$	0.323095	0.0111082
$847$	−11.0128	−0.378405
$848$	−7.57466	−0.260115
$849$	19.1709	0.657943
$850$	23.0380	0.790196
$851$	−45.9117	−1.57383
$852$	4.41522	0.151263
$853$	−14.8838	−0.509613	−0.254806	−	0.966992i	$-0.582012\pi$
$853$	−14.8838	−0.509613	−0.254806	+	0.966992i	$0.582012\pi$
$854$	−31.5205	−1.07861
$855$	−2.71402	−0.0928175
$856$	−30.1157	−1.02934
$857$	−14.5825	−0.498130	−0.249065	−	0.968487i	$-0.580123\pi$
$857$	−14.5825	−0.498130	−0.249065	+	0.968487i	$0.580123\pi$
$858$	−5.14965	−0.175806
$859$	13.8676	0.473155	0.236578	−	0.971613i	$-0.423974\pi$
$859$	13.8676	0.473155	0.236578	+	0.971613i	$0.423974\pi$
$860$	9.55023	0.325660
$861$	39.2772	1.33856
$862$	−18.6451	−0.635055
$863$	−32.8056	−1.11672	−0.558358	−	0.829600i	$-0.688569\pi$
$863$	−32.8056	−1.11672	−0.558358	+	0.829600i	$0.688569\pi$
$864$	21.3527	0.726435
$865$	−72.2072	−2.45512
$866$	2.84025	0.0965156
$867$	−0.270023	−0.00917046
$868$	12.1600	0.412737
$869$	50.0061	1.69634
$870$	7.24813	0.245735
$871$	−11.3448	−0.384403
$872$	−57.2864	−1.93996
$873$	24.0738	0.814775
$874$	4.62765	0.156533
$875$	−0.364277	−0.0123148
$876$	−1.31550	−0.0444466
$877$	−44.3679	−1.49820	−0.749099	−	0.662458i	$-0.769513\pi$
$877$	−44.3679	−1.49820	−0.749099	+	0.662458i	$0.769513\pi$
$878$	−22.0795	−0.745146
$879$	34.9725	1.17959
$880$	−23.5238	−0.792986
$881$	33.7952	1.13859	0.569295	−	0.822133i	$-0.307216\pi$
$881$	33.7952	1.13859	0.569295	+	0.822133i	$0.307216\pi$
$882$	22.8491	0.769370
$883$	−29.8466	−1.00442	−0.502209	−	0.864747i	$-0.667479\pi$
$883$	−29.8466	−1.00442	−0.502209	+	0.864747i	$0.667479\pi$
$884$	−2.95853	−0.0995062
$885$	−38.8454	−1.30577
$886$	15.3454	0.515540
$887$	−37.5890	−1.26211	−0.631057	−	0.775736i	$-0.717379\pi$
$887$	−37.5890	−1.26211	−0.631057	+	0.775736i	$0.717379\pi$
$888$	−25.4073	−0.852614
$889$	−28.5141	−0.956331
$890$	31.5370	1.05712
$891$	9.28139	0.310938
$892$	−12.0334	−0.402909
$893$	−0.116473	−0.00389763
$894$	5.33272	0.178353
$895$	23.0467	0.770365
$896$	8.30055	0.277302
$897$	−8.60680	−0.287373
$898$	30.0403	1.00246
$899$	6.00749	0.200361
$900$	5.21712	0.173904
$901$	−15.2553	−0.508229
$902$	−28.5705	−0.951294
$903$	−23.8263	−0.792891
$904$	12.7427	0.423816
$905$	−21.3883	−0.710973
$906$	−25.1359	−0.835083
$907$	−23.6710	−0.785984	−0.392992	−	0.919542i	$-0.628560\pi$
$907$	−23.6710	−0.785984	−0.392992	+	0.919542i	$0.628560\pi$
$908$	4.08450	0.135549
$909$	−22.5591	−0.748237
$910$	16.3291	0.541304
$911$	−51.6621	−1.71164	−0.855820	−	0.517273i	$-0.826947\pi$
$911$	−51.6621	−1.71164	−0.855820	+	0.517273i	$0.826947\pi$
$912$	1.49756	0.0495892
$913$	46.7815	1.54824
$914$	−20.6500	−0.683043
$915$	23.9550	0.791929
$916$	8.74313	0.288881
$917$	−22.8396	−0.754230
$918$	−25.6510	−0.846608
$919$	27.5707	0.909474	0.454737	−	0.890626i	$-0.349733\pi$
$919$	27.5707	0.909474	0.454737	+	0.890626i	$0.349733\pi$
$920$	−67.2330	−2.21660
$921$	−23.9005	−0.787548
$922$	−15.1907	−0.500279
$923$	4.91414	0.161751
$924$	−15.0473	−0.495019
$925$	−33.0156	−1.08554
$926$	−25.7079	−0.844815
$927$	−1.45216	−0.0476952
$928$	−6.29065	−0.206501
$929$	−48.7509	−1.59947	−0.799733	−	0.600356i	$-0.795025\pi$
$929$	−48.7509	−1.59947	−0.799733	+	0.600356i	$0.795025\pi$
$930$	16.3519	0.536199
$931$	−8.23693	−0.269954
$932$	12.2418	0.400992
$933$	−11.5985	−0.379719
$934$	23.0121	0.752977
$935$	−47.3767	−1.54938
$936$	4.46855	0.146059
$937$	2.54261	0.0830635	0.0415318	−	0.999137i	$-0.486776\pi$
$937$	2.54261	0.0830635	0.0415318	+	0.999137i	$0.486776\pi$
$938$	58.6551	1.91516
$939$	12.1875	0.397725
$940$	0.448924	0.0146423
$941$	37.5066	1.22268	0.611340	−	0.791368i	$-0.290631\pi$
$941$	37.5066	1.22268	0.611340	+	0.791368i	$0.290631\pi$
$942$	−14.4287	−0.470114
$943$	−47.7509	−1.55498
$944$	−20.1095	−0.654508
$945$	−80.0128	−2.60282
$946$	17.3315	0.563495
$947$	−25.1527	−0.817353	−0.408677	−	0.912679i	$-0.634010\pi$
$947$	−25.1527	−0.817353	−0.408677	+	0.912679i	$0.634010\pi$
$948$	12.2701	0.398515
$949$	−1.46415	−0.0475284
$950$	3.32779	0.107968
$951$	12.9165	0.418847
$952$	57.6581	1.86871
$953$	−1.40237	−0.0454271	−0.0227136	−	0.999742i	$-0.507231\pi$
$953$	−1.40237	−0.0454271	−0.0227136	+	0.999742i	$0.507231\pi$
$954$	6.11277	0.197908
$955$	41.0802	1.32932
$956$	−7.50433	−0.242707
$957$	−7.43392	−0.240304
$958$	0.461201	0.0149007
$959$	−33.6835	−1.08770
$960$	−33.1079	−1.06855
$961$	−17.4470	−0.562808
$962$	−7.50206	−0.241876
$963$	14.2121	0.457977
$964$	0.334721	0.0107806
$965$	60.9966	1.96355
$966$	44.4991	1.43174
$967$	14.6590	0.471400	0.235700	−	0.971826i	$-0.424262\pi$
$967$	14.6590	0.471400	0.235700	+	0.971826i	$0.424262\pi$
$968$	7.40929	0.238144
$969$	3.01608	0.0968906
$970$	−59.1858	−1.90034
$971$	−9.12260	−0.292758	−0.146379	−	0.989229i	$-0.546762\pi$
$971$	−9.12260	−0.292758	−0.146379	+	0.989229i	$0.546762\pi$
$972$	−9.72304	−0.311867
$973$	86.7685	2.78167
$974$	−30.6494	−0.982069
$975$	−6.18923	−0.198214
$976$	12.4010	0.396948
$977$	−22.4205	−0.717295	−0.358647	−	0.933473i	$-0.616762\pi$
$977$	−22.4205	−0.717295	−0.358647	+	0.933473i	$0.616762\pi$
$978$	12.0246	0.384504
$979$	−32.3454	−1.03376
$980$	31.7477	1.01414
$981$	27.0342	0.863137
$982$	9.60282	0.306438
$983$	36.0339	1.14930	0.574652	−	0.818398i	$-0.305137\pi$
$983$	36.0339	1.14930	0.574652	+	0.818398i	$0.305137\pi$
$984$	−26.4252	−0.842404
$985$	7.12875	0.227141
$986$	7.55694	0.240662
$987$	−1.12000	−0.0356499
$988$	−0.427354	−0.0135960
$989$	28.9667	0.921087
$990$	18.9837	0.603343
$991$	−24.3374	−0.773104	−0.386552	−	0.922268i	$-0.626334\pi$
$991$	−24.3374	−0.773104	−0.386552	+	0.922268i	$0.626334\pi$
$992$	−14.1918	−0.450589
$993$	−36.7080	−1.16489
$994$	−25.4073	−0.805869
$995$	5.51431	0.174815
$996$	11.4789	0.363722
$997$	49.5368	1.56885	0.784424	−	0.620225i	$-0.212959\pi$
$997$	49.5368	1.56885	0.784424	+	0.620225i	$0.212959\pi$
$998$	−29.7142	−0.940585
$999$	36.7602	1.16304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.g.1.11	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.g.1.11	✓	30	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13041 q^{2} +1.24412 q^{3} -0.722174 q^{4} -3.15829 q^{5} -1.40637 q^{6} -4.57377 q^{7} +3.07717 q^{8} -1.45216 q^{9} +O(q^{10})\)	\(q-1.13041 q^{2} +1.24412 q^{3} -0.722174 q^{4} -3.15829 q^{5} -1.40637 q^{6} -4.57377 q^{7} +3.07717 q^{8} -1.45216 q^{9} +3.57016 q^{10} -3.66167 q^{11} -0.898472 q^{12} -1.00000 q^{13} +5.17023 q^{14} -3.92929 q^{15} -2.03412 q^{16} -4.09670 q^{17} +1.64154 q^{18} -0.591761 q^{19} +2.28083 q^{20} -5.69032 q^{21} +4.13919 q^{22} +6.91797 q^{23} +3.82838 q^{24} +4.97478 q^{25} +1.13041 q^{26} -5.53903 q^{27} +3.30306 q^{28} +1.63183 q^{29} +4.44171 q^{30} +3.68143 q^{31} -3.85496 q^{32} -4.55556 q^{33} +4.63095 q^{34} +14.4453 q^{35} +1.04871 q^{36} -6.63658 q^{37} +0.668932 q^{38} -1.24412 q^{39} -9.71859 q^{40} -6.90245 q^{41} +6.43240 q^{42} +4.18717 q^{43} +2.64436 q^{44} +4.58634 q^{45} -7.82014 q^{46} +0.196825 q^{47} -2.53069 q^{48} +13.9194 q^{49} -5.62354 q^{50} -5.09679 q^{51} +0.722174 q^{52} +3.72381 q^{53} +6.26137 q^{54} +11.5646 q^{55} -14.0743 q^{56} -0.736223 q^{57} -1.84464 q^{58} +9.88610 q^{59} +2.83763 q^{60} -6.09653 q^{61} -4.16153 q^{62} +6.64185 q^{63} +8.42592 q^{64} +3.15829 q^{65} +5.14965 q^{66} +11.3448 q^{67} +2.95853 q^{68} +8.60680 q^{69} -16.3291 q^{70} -4.91414 q^{71} -4.46855 q^{72} +1.46415 q^{73} +7.50206 q^{74} +6.18923 q^{75} +0.427354 q^{76} +16.7476 q^{77} +1.40637 q^{78} -13.6566 q^{79} +6.42432 q^{80} -2.53474 q^{81} +7.80259 q^{82} -12.7760 q^{83} +4.10940 q^{84} +12.9386 q^{85} -4.73321 q^{86} +2.03020 q^{87} -11.2676 q^{88} +8.83350 q^{89} -5.18445 q^{90} +4.57377 q^{91} -4.99598 q^{92} +4.58015 q^{93} -0.222492 q^{94} +1.86895 q^{95} -4.79604 q^{96} -16.5779 q^{97} -15.7346 q^{98} +5.31733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9}+O(q^{10}) \) 30 * q + 5 * q^3 + 34 * q^4 + 17 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 49 * q^9	\( 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9} + 5 q^{10} + q^{11} + 15 q^{12} - 30 q^{13} + 24 q^{14} + 6 q^{15} + 38 q^{16} + 17 q^{17} + 8 q^{18} + 9 q^{19} + 31 q^{20} + 27 q^{21} + 2 q^{22} + 14 q^{23} + 26 q^{24} + 47 q^{25} + 14 q^{27} - 6 q^{28} + 53 q^{29} + 25 q^{30} + 19 q^{31} - 4 q^{32} + q^{33} - 22 q^{34} + 9 q^{35} + 61 q^{36} + 46 q^{38} - 5 q^{39} - 35 q^{40} + 28 q^{41} - 7 q^{42} + 6 q^{43} - 12 q^{44} + 68 q^{45} + 7 q^{46} + 12 q^{47} + 13 q^{48} + 54 q^{49} - 18 q^{50} + 10 q^{51} - 34 q^{52} + 37 q^{53} + 22 q^{54} + 11 q^{55} + 67 q^{56} - 57 q^{57} - 5 q^{58} + 61 q^{59} - 102 q^{60} + 16 q^{61} - 2 q^{62} - 7 q^{63} + 29 q^{64} - 17 q^{65} - 83 q^{66} - 2 q^{67} + 57 q^{68} + 98 q^{69} - 10 q^{70} + 50 q^{71} - 8 q^{72} - 10 q^{73} - 13 q^{74} + 5 q^{75} - 19 q^{76} + 54 q^{77} - 3 q^{78} + 3 q^{79} + 76 q^{80} + 118 q^{81} + 7 q^{82} + 6 q^{83} + 44 q^{84} + 33 q^{85} - 29 q^{86} - 10 q^{87} - 13 q^{88} + 77 q^{89} + 38 q^{90} - 2 q^{91} - 3 q^{92} + 34 q^{93} - 25 q^{94} + 24 q^{95} - 28 q^{96} + 12 q^{97} - 14 q^{98} - 58 q^{99}+O(q^{100}) \) 30 * q + 5 * q^3 + 34 * q^4 + 17 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 49 * q^9 + 5 * q^10 + q^11 + 15 * q^12 - 30 * q^13 + 24 * q^14 + 6 * q^15 + 38 * q^16 + 17 * q^17 + 8 * q^18 + 9 * q^19 + 31 * q^20 + 27 * q^21 + 2 * q^22 + 14 * q^23 + 26 * q^24 + 47 * q^25 + 14 * q^27 - 6 * q^28 + 53 * q^29 + 25 * q^30 + 19 * q^31 - 4 * q^32 + q^33 - 22 * q^34 + 9 * q^35 + 61 * q^36 + 46 * q^38 - 5 * q^39 - 35 * q^40 + 28 * q^41 - 7 * q^42 + 6 * q^43 - 12 * q^44 + 68 * q^45 + 7 * q^46 + 12 * q^47 + 13 * q^48 + 54 * q^49 - 18 * q^50 + 10 * q^51 - 34 * q^52 + 37 * q^53 + 22 * q^54 + 11 * q^55 + 67 * q^56 - 57 * q^57 - 5 * q^58 + 61 * q^59 - 102 * q^60 + 16 * q^61 - 2 * q^62 - 7 * q^63 + 29 * q^64 - 17 * q^65 - 83 * q^66 - 2 * q^67 + 57 * q^68 + 98 * q^69 - 10 * q^70 + 50 * q^71 - 8 * q^72 - 10 * q^73 - 13 * q^74 + 5 * q^75 - 19 * q^76 + 54 * q^77 - 3 * q^78 + 3 * q^79 + 76 * q^80 + 118 * q^81 + 7 * q^82 + 6 * q^83 + 44 * q^84 + 33 * q^85 - 29 * q^86 - 10 * q^87 - 13 * q^88 + 77 * q^89 + 38 * q^90 - 2 * q^91 - 3 * q^92 + 34 * q^93 - 25 * q^94 + 24 * q^95 - 28 * q^96 + 12 * q^97 - 14 * q^98 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more