LMFDB - Embedded newform 6001.2.a.c.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6001,2,Mod(1,6001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6001, 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("6001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6001 = 17 \cdot 353 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6001.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:	$47.9182262530$
Analytic rank:	$0$
Dimension:	$121$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24876 q^{2} -0.983386 q^{3} +3.05693 q^{4} -2.62112 q^{5} +2.21140 q^{6} +0.682123 q^{7} -2.37679 q^{8} -2.03295 q^{9} +O(q^{10})$	\(q-2.24876 q^{2} -0.983386 q^{3} +3.05693 q^{4} -2.62112 q^{5} +2.21140 q^{6} +0.682123 q^{7} -2.37679 q^{8} -2.03295 q^{9} +5.89428 q^{10} +2.84623 q^{11} -3.00614 q^{12} -4.17539 q^{13} -1.53393 q^{14} +2.57757 q^{15} -0.769031 q^{16} -1.00000 q^{17} +4.57163 q^{18} +8.35656 q^{19} -8.01259 q^{20} -0.670790 q^{21} -6.40049 q^{22} +7.80502 q^{23} +2.33730 q^{24} +1.87028 q^{25} +9.38946 q^{26} +4.94933 q^{27} +2.08520 q^{28} -3.71581 q^{29} -5.79635 q^{30} -0.0649985 q^{31} +6.48295 q^{32} -2.79894 q^{33} +2.24876 q^{34} -1.78793 q^{35} -6.21460 q^{36} +0.303231 q^{37} -18.7919 q^{38} +4.10602 q^{39} +6.22986 q^{40} +10.8872 q^{41} +1.50845 q^{42} -1.73818 q^{43} +8.70073 q^{44} +5.32862 q^{45} -17.5516 q^{46} -4.52680 q^{47} +0.756254 q^{48} -6.53471 q^{49} -4.20583 q^{50} +0.983386 q^{51} -12.7639 q^{52} +1.04525 q^{53} -11.1299 q^{54} -7.46031 q^{55} -1.62126 q^{56} -8.21772 q^{57} +8.35597 q^{58} -1.26157 q^{59} +7.87947 q^{60} +6.82594 q^{61} +0.146166 q^{62} -1.38672 q^{63} -13.0405 q^{64} +10.9442 q^{65} +6.29415 q^{66} -7.40662 q^{67} -3.05693 q^{68} -7.67534 q^{69} +4.02063 q^{70} +2.53678 q^{71} +4.83190 q^{72} -9.54883 q^{73} -0.681894 q^{74} -1.83921 q^{75} +25.5454 q^{76} +1.94148 q^{77} -9.23346 q^{78} +12.7063 q^{79} +2.01572 q^{80} +1.23175 q^{81} -24.4827 q^{82} -12.1730 q^{83} -2.05056 q^{84} +2.62112 q^{85} +3.90876 q^{86} +3.65407 q^{87} -6.76488 q^{88} -0.559578 q^{89} -11.9828 q^{90} -2.84813 q^{91} +23.8594 q^{92} +0.0639186 q^{93} +10.1797 q^{94} -21.9036 q^{95} -6.37524 q^{96} +5.88278 q^{97} +14.6950 q^{98} -5.78625 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9}+O(q^{10}) $ 121 * q + 9 * q^2 + 13 * q^3 + 127 * q^4 + 21 * q^5 + 19 * q^6 - 13 * q^7 + 24 * q^8 + 134 * q^9	\( 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9} - q^{10} + 40 q^{11} + 41 q^{12} + 14 q^{13} + 32 q^{14} + 49 q^{15} + 135 q^{16} - 121 q^{17} + 28 q^{18} + 34 q^{19} + 64 q^{20} + 34 q^{21} - 18 q^{22} + 37 q^{23} + 54 q^{24} + 128 q^{25} + 91 q^{26} + 55 q^{27} - 28 q^{28} + 45 q^{29} + 30 q^{30} + 67 q^{31} + 47 q^{32} + 40 q^{33} - 9 q^{34} + 59 q^{35} + 138 q^{36} - 16 q^{37} + 30 q^{38} + 37 q^{39} + 14 q^{40} + 89 q^{41} + 33 q^{42} + 16 q^{43} + 90 q^{44} + 83 q^{45} - 9 q^{46} + 135 q^{47} + 96 q^{48} + 128 q^{49} + 71 q^{50} - 13 q^{51} + 47 q^{52} + 52 q^{53} + 90 q^{54} + 93 q^{55} + 69 q^{56} - 4 q^{57} + 5 q^{58} + 170 q^{59} + 78 q^{60} - 2 q^{61} + 46 q^{62} - 10 q^{63} + 182 q^{64} + 50 q^{65} + 68 q^{66} + 46 q^{67} - 127 q^{68} + 97 q^{69} + 46 q^{70} + 191 q^{71} + 57 q^{72} - 12 q^{73} + 68 q^{74} + 86 q^{75} + 108 q^{76} + 62 q^{77} - 10 q^{78} + 130 q^{80} + 149 q^{81} + 14 q^{82} + 83 q^{83} + 126 q^{84} - 21 q^{85} + 132 q^{86} + 50 q^{87} - 42 q^{88} + 144 q^{89} + 9 q^{90} + 13 q^{91} + 50 q^{92} + 43 q^{93} + 41 q^{94} + 82 q^{95} + 110 q^{96} - 3 q^{97} + 36 q^{98} + 89 q^{99}+O(q^{100}) \) 121 * q + 9 * q^2 + 13 * q^3 + 127 * q^4 + 21 * q^5 + 19 * q^6 - 13 * q^7 + 24 * q^8 + 134 * q^9 - q^10 + 40 * q^11 + 41 * q^12 + 14 * q^13 + 32 * q^14 + 49 * q^15 + 135 * q^16 - 121 * q^17 + 28 * q^18 + 34 * q^19 + 64 * q^20 + 34 * q^21 - 18 * q^22 + 37 * q^23 + 54 * q^24 + 128 * q^25 + 91 * q^26 + 55 * q^27 - 28 * q^28 + 45 * q^29 + 30 * q^30 + 67 * q^31 + 47 * q^32 + 40 * q^33 - 9 * q^34 + 59 * q^35 + 138 * q^36 - 16 * q^37 + 30 * q^38 + 37 * q^39 + 14 * q^40 + 89 * q^41 + 33 * q^42 + 16 * q^43 + 90 * q^44 + 83 * q^45 - 9 * q^46 + 135 * q^47 + 96 * q^48 + 128 * q^49 + 71 * q^50 - 13 * q^51 + 47 * q^52 + 52 * q^53 + 90 * q^54 + 93 * q^55 + 69 * q^56 - 4 * q^57 + 5 * q^58 + 170 * q^59 + 78 * q^60 - 2 * q^61 + 46 * q^62 - 10 * q^63 + 182 * q^64 + 50 * q^65 + 68 * q^66 + 46 * q^67 - 127 * q^68 + 97 * q^69 + 46 * q^70 + 191 * q^71 + 57 * q^72 - 12 * q^73 + 68 * q^74 + 86 * q^75 + 108 * q^76 + 62 * q^77 - 10 * q^78 + 130 * q^80 + 149 * q^81 + 14 * q^82 + 83 * q^83 + 126 * q^84 - 21 * q^85 + 132 * q^86 + 50 * q^87 - 42 * q^88 + 144 * q^89 + 9 * q^90 + 13 * q^91 + 50 * q^92 + 43 * q^93 + 41 * q^94 + 82 * q^95 + 110 * q^96 - 3 * q^97 + 36 * q^98 + 89 * 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 $

$2$	−2.24876	−1.59012	−0.795058	−	0.606534i	$-0.792559\pi$
$2$	−2.24876	−1.59012	−0.795058	+	0.606534i	$0.792559\pi$
$3$	−0.983386	−0.567758	−0.283879	−	0.958860i	$-0.591621\pi$
$3$	−0.983386	−0.567758	−0.283879	+	0.958860i	$0.591621\pi$
$4$	3.05693	1.52847
$5$	−2.62112	−1.17220	−0.586101	−	0.810238i	$-0.699338\pi$
$5$	−2.62112	−1.17220	−0.586101	+	0.810238i	$0.699338\pi$
$6$	2.21140	0.902801
$7$	0.682123	0.257818	0.128909	−	0.991656i	$-0.458852\pi$
$7$	0.682123	0.257818	0.128909	+	0.991656i	$0.458852\pi$
$8$	−2.37679	−0.840322
$9$	−2.03295	−0.677651
$10$	5.89428	1.86394
$11$	2.84623	0.858170	0.429085	−	0.903264i	$-0.358836\pi$
$11$	2.84623	0.858170	0.429085	+	0.903264i	$0.358836\pi$
$12$	−3.00614	−0.867799
$13$	−4.17539	−1.15805	−0.579023	−	0.815311i	$-0.696566\pi$
$13$	−4.17539	−1.15805	−0.579023	+	0.815311i	$0.696566\pi$
$14$	−1.53393	−0.409961
$15$	2.57757	0.665527
$16$	−0.769031	−0.192258
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	4.57163	1.07754
$19$	8.35656	1.91713	0.958563	−	0.284880i	$-0.0919538\pi$
$19$	8.35656	1.91713	0.958563	+	0.284880i	$0.0919538\pi$
$20$	−8.01259	−1.79167
$21$	−0.670790	−0.146378
$22$	−6.40049	−1.36459
$23$	7.80502	1.62746	0.813729	−	0.581244i	$-0.197434\pi$
$23$	7.80502	1.62746	0.813729	+	0.581244i	$0.197434\pi$
$24$	2.33730	0.477099
$25$	1.87028	0.374057
$26$	9.38946	1.84143
$27$	4.94933	0.952500
$28$	2.08520	0.394067
$29$	−3.71581	−0.690008	−0.345004	−	0.938601i	$-0.612123\pi$
$29$	−3.71581	−0.690008	−0.345004	+	0.938601i	$0.612123\pi$
$30$	−5.79635	−1.05826
$31$	−0.0649985	−0.0116741	−0.00583704	−	0.999983i	$-0.501858\pi$
$31$	−0.0649985	−0.0116741	−0.00583704	+	0.999983i	$0.501858\pi$
$32$	6.48295	1.14603
$33$	−2.79894	−0.487233
$34$	2.24876	0.385660
$35$	−1.78793	−0.302215
$36$	−6.21460	−1.03577
$37$	0.303231	0.0498509	0.0249254	−	0.999689i	$-0.492065\pi$
$37$	0.303231	0.0498509	0.0249254	+	0.999689i	$0.492065\pi$
$38$	−18.7919	−3.04845
$39$	4.10602	0.657489
$40$	6.22986	0.985027
$41$	10.8872	1.70029	0.850146	−	0.526548i	$-0.176514\pi$
$41$	10.8872	1.70029	0.850146	+	0.526548i	$0.176514\pi$
$42$	1.50845	0.232759
$43$	−1.73818	−0.265070	−0.132535	−	0.991178i	$-0.542312\pi$
$43$	−1.73818	−0.265070	−0.132535	+	0.991178i	$0.542312\pi$
$44$	8.70073	1.31168
$45$	5.32862	0.794343
$46$	−17.5516	−2.58785
$47$	−4.52680	−0.660301	−0.330151	−	0.943928i	$-0.607100\pi$
$47$	−4.52680	−0.660301	−0.330151	+	0.943928i	$0.607100\pi$
$48$	0.756254	0.109156
$49$	−6.53471	−0.933530
$50$	−4.20583	−0.594794
$51$	0.983386	0.137702
$52$	−12.7639	−1.77003
$53$	1.04525	0.143576	0.0717881	−	0.997420i	$-0.477129\pi$
$53$	1.04525	0.143576	0.0717881	+	0.997420i	$0.477129\pi$
$54$	−11.1299	−1.51458
$55$	−7.46031	−1.00595
$56$	−1.62126	−0.216650
$57$	−8.21772	−1.08846
$58$	8.35597	1.09719
$59$	−1.26157	−0.164243	−0.0821215	−	0.996622i	$-0.526170\pi$
$59$	−1.26157	−0.164243	−0.0821215	+	0.996622i	$0.526170\pi$
$60$	7.87947	1.01724
$61$	6.82594	0.873972	0.436986	−	0.899468i	$-0.356046\pi$
$61$	6.82594	0.873972	0.436986	+	0.899468i	$0.356046\pi$
$62$	0.146166	0.0185631
$63$	−1.38672	−0.174711
$64$	−13.0405	−1.63007
$65$	10.9442	1.35746
$66$	6.29415	0.774757
$67$	−7.40662	−0.904862	−0.452431	−	0.891799i	$-0.649443\pi$
$67$	−7.40662	−0.904862	−0.452431	+	0.891799i	$0.649443\pi$
$68$	−3.05693	−0.370707
$69$	−7.67534	−0.924002
$70$	4.02063	0.480557
$71$	2.53678	0.301060	0.150530	−	0.988605i	$-0.451902\pi$
$71$	2.53678	0.301060	0.150530	+	0.988605i	$0.451902\pi$
$72$	4.83190	0.569445
$73$	−9.54883	−1.11761	−0.558803	−	0.829301i	$-0.688739\pi$
$73$	−9.54883	−1.11761	−0.558803	+	0.829301i	$0.688739\pi$
$74$	−0.681894	−0.0792686
$75$	−1.83921	−0.212374
$76$	25.5454	2.93026
$77$	1.94148	0.221252
$78$	−9.23346	−1.04548
$79$	12.7063	1.42957	0.714787	−	0.699342i	$-0.246524\pi$
$79$	12.7063	1.42957	0.714787	+	0.699342i	$0.246524\pi$
$80$	2.01572	0.225365
$81$	1.23175	0.136861
$82$	−24.4827	−2.70366
$83$	−12.1730	−1.33616	−0.668081	−	0.744088i	$-0.732884\pi$
$83$	−12.1730	−1.33616	−0.668081	+	0.744088i	$0.732884\pi$
$84$	−2.05056	−0.223735
$85$	2.62112	0.284301
$86$	3.90876	0.421492
$87$	3.65407	0.391758
$88$	−6.76488	−0.721139
$89$	−0.559578	−0.0593151	−0.0296576	−	0.999560i	$-0.509442\pi$
$89$	−0.559578	−0.0593151	−0.0296576	+	0.999560i	$0.509442\pi$
$90$	−11.9828	−1.26310
$91$	−2.84813	−0.298565
$92$	23.8594	2.48751
$93$	0.0639186	0.00662805
$94$	10.1797	1.04996
$95$	−21.9036	−2.24726
$96$	−6.37524	−0.650670
$97$	5.88278	0.597306	0.298653	−	0.954362i	$-0.403463\pi$
$97$	5.88278	0.597306	0.298653	+	0.954362i	$0.403463\pi$
$98$	14.6950	1.48442
$99$	−5.78625	−0.581540
$100$	5.71733	0.571733
$101$	1.10144	0.109598	0.0547988	−	0.998497i	$-0.482548\pi$
$101$	1.10144	0.109598	0.0547988	+	0.998497i	$0.482548\pi$
$102$	−2.21140	−0.218961
$103$	15.8028	1.55710	0.778549	−	0.627584i	$-0.215956\pi$
$103$	15.8028	1.55710	0.778549	+	0.627584i	$0.215956\pi$
$104$	9.92402	0.973131
$105$	1.75822	0.171585
$106$	−2.35052	−0.228303
$107$	−18.1468	−1.75431	−0.877157	−	0.480204i	$-0.840563\pi$
$107$	−18.1468	−1.75431	−0.877157	+	0.480204i	$0.840563\pi$
$108$	15.1298	1.45586
$109$	0.839324	0.0803927	0.0401963	−	0.999192i	$-0.487202\pi$
$109$	0.839324	0.0803927	0.0401963	+	0.999192i	$0.487202\pi$
$110$	16.7765	1.59957
$111$	−0.298193	−0.0283032
$112$	−0.524574	−0.0495676
$113$	12.3824	1.16484	0.582419	−	0.812889i	$-0.302106\pi$
$113$	12.3824	1.16484	0.582419	+	0.812889i	$0.302106\pi$
$114$	18.4797	1.73078
$115$	−20.4579	−1.90771
$116$	−11.3590	−1.05465
$117$	8.48837	0.784750
$118$	2.83698	0.261165
$119$	−0.682123	−0.0625302
$120$	−6.12635	−0.559257
$121$	−2.89898	−0.263544
$122$	−15.3499	−1.38972
$123$	−10.7063	−0.965354
$124$	−0.198696	−0.0178434
$125$	8.20337	0.733732
$126$	3.11841	0.277810
$127$	−7.03664	−0.624400	−0.312200	−	0.950016i	$-0.601066\pi$
$127$	−7.03664	−0.624400	−0.312200	+	0.950016i	$0.601066\pi$
$128$	16.3592	1.44596
$129$	1.70930	0.150496
$130$	−24.6109	−2.15852
$131$	13.4100	1.17164	0.585819	−	0.810442i	$-0.300773\pi$
$131$	13.4100	1.17164	0.585819	+	0.810442i	$0.300773\pi$
$132$	−8.55617	−0.744719
$133$	5.70021	0.494270
$134$	16.6557	1.43884
$135$	−12.9728	−1.11652
$136$	2.37679	0.203808
$137$	17.3369	1.48119	0.740594	−	0.671953i	$-0.234544\pi$
$137$	17.3369	1.48119	0.740594	+	0.671953i	$0.234544\pi$
$138$	17.2600	1.46927
$139$	−15.0215	−1.27411	−0.637055	−	0.770818i	$-0.719848\pi$
$139$	−15.0215	−1.27411	−0.637055	+	0.770818i	$0.719848\pi$
$140$	−5.46558	−0.461926
$141$	4.45159	0.374891
$142$	−5.70461	−0.478720
$143$	−11.8841	−0.993800
$144$	1.56340	0.130284
$145$	9.73959	0.808829
$146$	21.4730	1.77712
$147$	6.42614	0.530019
$148$	0.926957	0.0761954
$149$	−8.73061	−0.715240	−0.357620	−	0.933867i	$-0.616412\pi$
$149$	−8.73061	−0.715240	−0.357620	+	0.933867i	$0.616412\pi$
$150$	4.13595	0.337699
$151$	−9.73036	−0.791846	−0.395923	−	0.918284i	$-0.629575\pi$
$151$	−9.73036	−0.791846	−0.395923	+	0.918284i	$0.629575\pi$
$152$	−19.8618	−1.61100
$153$	2.03295	0.164354
$154$	−4.36593	−0.351816
$155$	0.170369	0.0136844
$156$	12.5518	1.00495
$157$	−18.6749	−1.49042	−0.745210	−	0.666830i	$-0.767651\pi$
$157$	−18.6749	−1.49042	−0.745210	+	0.666830i	$0.767651\pi$
$158$	−28.5735	−2.27319
$159$	−1.02788	−0.0815165
$160$	−16.9926	−1.34338
$161$	5.32398	0.419589
$162$	−2.76992	−0.217626
$163$	−4.56875	−0.357852	−0.178926	−	0.983863i	$-0.557262\pi$
$163$	−4.56875	−0.357852	−0.178926	+	0.983863i	$0.557262\pi$
$164$	33.2814	2.59884
$165$	7.33637	0.571135
$166$	27.3742	2.12465
$167$	5.23398	0.405017	0.202509	−	0.979280i	$-0.435091\pi$
$167$	5.23398	0.405017	0.202509	+	0.979280i	$0.435091\pi$
$168$	1.59433	0.123005
$169$	4.43389	0.341069
$170$	−5.89428	−0.452071
$171$	−16.9885	−1.29914
$172$	−5.31350	−0.405151
$173$	2.55660	0.194375	0.0971875	−	0.995266i	$-0.469015\pi$
$173$	2.55660	0.194375	0.0971875	+	0.995266i	$0.469015\pi$
$174$	−8.21714	−0.622940
$175$	1.27577	0.0964388
$176$	−2.18884	−0.164990
$177$	1.24061	0.0932503
$178$	1.25836	0.0943179
$179$	3.75635	0.280763	0.140382	−	0.990097i	$-0.455167\pi$
$179$	3.75635	0.280763	0.140382	+	0.990097i	$0.455167\pi$
$180$	16.2892	1.21413
$181$	−13.4163	−0.997229	−0.498614	−	0.866824i	$-0.666158\pi$
$181$	−13.4163	−0.997229	−0.498614	+	0.866824i	$0.666158\pi$
$182$	6.40477	0.474753
$183$	−6.71254	−0.496205
$184$	−18.5509	−1.36759
$185$	−0.794806	−0.0584353
$186$	−0.143738	−0.0105394
$187$	−2.84623	−0.208137
$188$	−13.8381	−1.00925
$189$	3.37606	0.245572
$190$	49.2559	3.57340
$191$	−23.2037	−1.67896	−0.839480	−	0.543390i	$-0.817140\pi$
$191$	−23.2037	−1.67896	−0.839480	+	0.543390i	$0.817140\pi$
$192$	12.8239	0.925484
$193$	−15.4844	−1.11459	−0.557295	−	0.830315i	$-0.688161\pi$
$193$	−15.4844	−1.11459	−0.557295	+	0.830315i	$0.688161\pi$
$194$	−13.2290	−0.949785
$195$	−10.7624	−0.770710
$196$	−19.9762	−1.42687
$197$	26.2169	1.86788	0.933939	−	0.357433i	$-0.116348\pi$
$197$	26.2169	1.86788	0.933939	+	0.357433i	$0.116348\pi$
$198$	13.0119	0.924715
$199$	−12.2579	−0.868938	−0.434469	−	0.900687i	$-0.643064\pi$
$199$	−12.2579	−0.868938	−0.434469	+	0.900687i	$0.643064\pi$
$200$	−4.44527	−0.314328
$201$	7.28356	0.513743
$202$	−2.47688	−0.174273
$203$	−2.53464	−0.177897
$204$	3.00614	0.210472
$205$	−28.5366	−1.99308
$206$	−35.5368	−2.47597
$207$	−15.8672	−1.10285
$208$	3.21100	0.222643
$209$	23.7847	1.64522
$210$	−3.95383	−0.272840
$211$	−2.37458	−0.163473	−0.0817364	−	0.996654i	$-0.526047\pi$
$211$	−2.37458	−0.163473	−0.0817364	+	0.996654i	$0.526047\pi$
$212$	3.19526	0.219451
$213$	−2.49463	−0.170929
$214$	40.8078	2.78956
$215$	4.55598	0.310716
$216$	−11.7635	−0.800406
$217$	−0.0443370	−0.00300979
$218$	−1.88744	−0.127834
$219$	9.39018	0.634530
$220$	−22.8057	−1.53756
$221$	4.17539	0.280867
$222$	0.670565	0.0450054
$223$	−21.8248	−1.46150	−0.730748	−	0.682647i	$-0.760828\pi$
$223$	−21.8248	−1.46150	−0.730748	+	0.682647i	$0.760828\pi$
$224$	4.42217	0.295469
$225$	−3.80220	−0.253480
$226$	−27.8451	−1.85223
$227$	−13.7261	−0.911033	−0.455516	−	0.890227i	$-0.650545\pi$
$227$	−13.7261	−0.911033	−0.455516	+	0.890227i	$0.650545\pi$
$228$	−25.1210	−1.66368
$229$	9.57309	0.632607	0.316304	−	0.948658i	$-0.397558\pi$
$229$	9.57309	0.632607	0.316304	+	0.948658i	$0.397558\pi$
$230$	46.0050	3.03348
$231$	−1.90922	−0.125618
$232$	8.83169	0.579829
$233$	10.0496	0.658369	0.329185	−	0.944266i	$-0.393226\pi$
$233$	10.0496	0.658369	0.329185	+	0.944266i	$0.393226\pi$
$234$	−19.0883	−1.24784
$235$	11.8653	0.774006
$236$	−3.85655	−0.251040
$237$	−12.4952	−0.811652
$238$	1.53393	0.0994301
$239$	10.7695	0.696621	0.348310	−	0.937379i	$-0.386755\pi$
$239$	10.7695	0.696621	0.348310	+	0.937379i	$0.386755\pi$
$240$	−1.98223	−0.127953
$241$	10.1673	0.654933	0.327466	−	0.944863i	$-0.393805\pi$
$241$	10.1673	0.654933	0.327466	+	0.944863i	$0.393805\pi$
$242$	6.51913	0.419065
$243$	−16.0593	−1.03020
$244$	20.8664	1.33584
$245$	17.1283	1.09429
$246$	24.0759	1.53502
$247$	−34.8919	−2.22012
$248$	0.154488	0.00980999
$249$	11.9708	0.758617
$250$	−18.4474	−1.16672
$251$	6.27774	0.396247	0.198124	−	0.980177i	$-0.436515\pi$
$251$	6.27774	0.396247	0.198124	+	0.980177i	$0.436515\pi$
$252$	−4.23912	−0.267040
$253$	22.2149	1.39664
$254$	15.8237	0.992869
$255$	−2.57757	−0.161414
$256$	−10.7068	−0.669178
$257$	−18.9263	−1.18059	−0.590294	−	0.807188i	$-0.700988\pi$
$257$	−18.9263	−1.18059	−0.590294	+	0.807188i	$0.700988\pi$
$258$	−3.84381	−0.239305
$259$	0.206841	0.0128525
$260$	33.4557	2.07484
$261$	7.55406	0.467585
$262$	−30.1559	−1.86304
$263$	20.5165	1.26510	0.632550	−	0.774520i	$-0.282008\pi$
$263$	20.5165	1.26510	0.632550	+	0.774520i	$0.282008\pi$
$264$	6.65249	0.409432
$265$	−2.73973	−0.168300
$266$	−12.8184	−0.785947
$267$	0.550281	0.0336766
$268$	−22.6415	−1.38305
$269$	10.3559	0.631408	0.315704	−	0.948858i	$-0.397759\pi$
$269$	10.3559	0.631408	0.315704	+	0.948858i	$0.397759\pi$
$270$	29.1728	1.77540
$271$	8.85436	0.537864	0.268932	−	0.963159i	$-0.413329\pi$
$271$	8.85436	0.537864	0.268932	+	0.963159i	$0.413329\pi$
$272$	0.769031	0.0466293
$273$	2.80081	0.169513
$274$	−38.9865	−2.35526
$275$	5.32326	0.321005
$276$	−23.4630	−1.41231
$277$	13.6110	0.817804	0.408902	−	0.912578i	$-0.365912\pi$
$277$	13.6110	0.817804	0.408902	+	0.912578i	$0.365912\pi$
$278$	33.7799	2.02598
$279$	0.132139	0.00791095
$280$	4.24953	0.253958
$281$	−2.22224	−0.132568	−0.0662838	−	0.997801i	$-0.521114\pi$
$281$	−2.22224	−0.132568	−0.0662838	+	0.997801i	$0.521114\pi$
$282$	−10.0106	−0.596120
$283$	22.3444	1.32824	0.664118	−	0.747627i	$-0.268807\pi$
$283$	22.3444	1.32824	0.664118	+	0.747627i	$0.268807\pi$
$284$	7.75476	0.460160
$285$	21.5397	1.27590
$286$	26.7246	1.58026
$287$	7.42640	0.438366
$288$	−13.1795	−0.776611
$289$	1.00000	0.0588235
$290$	−21.9020	−1.28613
$291$	−5.78504	−0.339125
$292$	−29.1901	−1.70822
$293$	−15.2619	−0.891608	−0.445804	−	0.895131i	$-0.647082\pi$
$293$	−15.2619	−0.891608	−0.445804	+	0.895131i	$0.647082\pi$
$294$	−14.4509	−0.842791
$295$	3.30674	0.192526
$296$	−0.720716	−0.0418908
$297$	14.0869	0.817407
$298$	19.6331	1.13731
$299$	−32.5890	−1.88467
$300$	−5.62234	−0.324606
$301$	−1.18565	−0.0683399
$302$	21.8813	1.25913
$303$	−1.08314	−0.0622249
$304$	−6.42645	−0.368582
$305$	−17.8916	−1.02447
$306$	−4.57163	−0.261343
$307$	11.4841	0.655430	0.327715	−	0.944777i	$-0.393721\pi$
$307$	11.4841	0.655430	0.327715	+	0.944777i	$0.393721\pi$
$308$	5.93497	0.338176
$309$	−15.5403	−0.884055
$310$	−0.383120	−0.0217597
$311$	32.4238	1.83859	0.919293	−	0.393575i	$-0.128762\pi$
$311$	32.4238	1.83859	0.919293	+	0.393575i	$0.128762\pi$
$312$	−9.75914	−0.552503
$313$	1.56960	0.0887192	0.0443596	−	0.999016i	$-0.485875\pi$
$313$	1.56960	0.0887192	0.0443596	+	0.999016i	$0.485875\pi$
$314$	41.9954	2.36994
$315$	3.63478	0.204796
$316$	38.8424	2.18505
$317$	−32.3893	−1.81916	−0.909582	−	0.415524i	$-0.863598\pi$
$317$	−32.3893	−1.81916	−0.909582	+	0.415524i	$0.863598\pi$
$318$	2.31147	0.129621
$319$	−10.5760	−0.592145
$320$	34.1809	1.91077
$321$	17.8453	0.996026
$322$	−11.9724	−0.667194
$323$	−8.35656	−0.464971
$324$	3.76539	0.209188
$325$	−7.80917	−0.433175
$326$	10.2740	0.569026
$327$	−0.825379	−0.0456436
$328$	−25.8765	−1.42879
$329$	−3.08784	−0.170238
$330$	−16.4977	−0.908171
$331$	−8.37551	−0.460360	−0.230180	−	0.973148i	$-0.573931\pi$
$331$	−8.37551	−0.460360	−0.230180	+	0.973148i	$0.573931\pi$
$332$	−37.2121	−2.04228
$333$	−0.616454	−0.0337815
$334$	−11.7700	−0.644024
$335$	19.4137	1.06068
$336$	0.515858	0.0281424
$337$	−14.9674	−0.815324	−0.407662	−	0.913133i	$-0.633656\pi$
$337$	−14.9674	−0.815324	−0.407662	+	0.913133i	$0.633656\pi$
$338$	−9.97077	−0.542338
$339$	−12.1767	−0.661346
$340$	8.01259	0.434544
$341$	−0.185001	−0.0100183
$342$	38.2031	2.06579
$343$	−9.23234	−0.498500
$344$	4.13129	0.222744
$345$	20.1180	1.08312
$346$	−5.74919	−0.309079
$347$	32.0979	1.72311	0.861554	−	0.507667i	$-0.169492\pi$
$347$	32.0979	1.72311	0.861554	+	0.507667i	$0.169492\pi$
$348$	11.1703	0.598788
$349$	−34.4946	−1.84645	−0.923226	−	0.384258i	$-0.874457\pi$
$349$	−34.4946	−1.84645	−0.923226	+	0.384258i	$0.874457\pi$
$350$	−2.86889	−0.153349
$351$	−20.6654	−1.10304
$352$	18.4519	0.983492
$353$	1.00000	0.0532246
$353$	1.00000	0.0532246
$354$	−2.78985	−0.148279
$355$	−6.64920	−0.352903
$356$	−1.71059	−0.0906612
$357$	0.670790	0.0355020
$358$	−8.44715	−0.446446
$359$	−8.94160	−0.471920	−0.235960	−	0.971763i	$-0.575823\pi$
$359$	−8.94160	−0.471920	−0.235960	+	0.971763i	$0.575823\pi$
$360$	−12.6650	−0.667504
$361$	50.8321	2.67537
$362$	30.1702	1.58571
$363$	2.85082	0.149629
$364$	−8.70655	−0.456347
$365$	25.0286	1.31006
$366$	15.0949	0.789023
$367$	−16.4060	−0.856384	−0.428192	−	0.903688i	$-0.640849\pi$
$367$	−16.4060	−0.856384	−0.428192	+	0.903688i	$0.640849\pi$
$368$	−6.00230	−0.312891
$369$	−22.1331	−1.15220
$370$	1.78733	0.0929188
$371$	0.712989	0.0370166
$372$	0.195395	0.0101308
$373$	−15.5727	−0.806324	−0.403162	−	0.915129i	$-0.632089\pi$
$373$	−15.5727	−0.806324	−0.403162	+	0.915129i	$0.632089\pi$
$374$	6.40049	0.330962
$375$	−8.06708	−0.416582
$376$	10.7592	0.554866
$377$	15.5150	0.799061
$378$	−7.59195	−0.390488
$379$	11.3314	0.582054	0.291027	−	0.956715i	$-0.406003\pi$
$379$	11.3314	0.582054	0.291027	+	0.956715i	$0.406003\pi$
$380$	−66.9577	−3.43486
$381$	6.91973	0.354508
$382$	52.1796	2.66974
$383$	2.19998	0.112414	0.0562069	−	0.998419i	$-0.482099\pi$
$383$	2.19998	0.112414	0.0562069	+	0.998419i	$0.482099\pi$
$384$	−16.0874	−0.820956
$385$	−5.08886	−0.259352
$386$	34.8207	1.77232
$387$	3.53364	0.179625
$388$	17.9833	0.912961
$389$	30.9067	1.56703	0.783515	−	0.621373i	$-0.213425\pi$
$389$	30.9067	1.56703	0.783515	+	0.621373i	$0.213425\pi$
$390$	24.2020	1.22552
$391$	−7.80502	−0.394717
$392$	15.5316	0.784465
$393$	−13.1872	−0.665207
$394$	−58.9556	−2.97014
$395$	−33.3048	−1.67575
$396$	−17.6882	−0.888864
$397$	−6.47622	−0.325032	−0.162516	−	0.986706i	$-0.551961\pi$
$397$	−6.47622	−0.325032	−0.162516	+	0.986706i	$0.551961\pi$
$398$	27.5651	1.38171
$399$	−5.60550	−0.280626
$400$	−1.43831	−0.0719153
$401$	8.98001	0.448440	0.224220	−	0.974539i	$-0.428017\pi$
$401$	8.98001	0.448440	0.224220	+	0.974539i	$0.428017\pi$
$402$	−16.3790	−0.816910
$403$	0.271394	0.0135191
$404$	3.36703	0.167516
$405$	−3.22858	−0.160429
$406$	5.69980	0.282876
$407$	0.863065	0.0427805
$408$	−2.33730	−0.115714
$409$	32.8442	1.62404	0.812020	−	0.583630i	$-0.198368\pi$
$409$	32.8442	1.62404	0.812020	+	0.583630i	$0.198368\pi$
$410$	64.1721	3.16923
$411$	−17.0488	−0.840956
$412$	48.3082	2.37997
$413$	−0.860549	−0.0423449
$414$	35.6816	1.75366
$415$	31.9070	1.56625
$416$	−27.0688	−1.32716
$417$	14.7720	0.723387
$418$	−53.4861	−2.61609
$419$	26.7347	1.30608	0.653038	−	0.757325i	$-0.273494\pi$
$419$	26.7347	1.30608	0.653038	+	0.757325i	$0.273494\pi$
$420$	5.37477	0.262262
$421$	−13.1321	−0.640018	−0.320009	−	0.947414i	$-0.603686\pi$
$421$	−13.1321	−0.640018	−0.320009	+	0.947414i	$0.603686\pi$
$422$	5.33987	0.259941
$423$	9.20277	0.447454
$424$	−2.48434	−0.120650
$425$	−1.87028	−0.0907221
$426$	5.60983	0.271797
$427$	4.65614	0.225326
$428$	−55.4734	−2.68141
$429$	11.6867	0.564238
$430$	−10.2453	−0.494074
$431$	6.57277	0.316599	0.158300	−	0.987391i	$-0.449399\pi$
$431$	6.57277	0.316599	0.158300	+	0.987391i	$0.449399\pi$
$432$	−3.80619	−0.183125
$433$	7.05699	0.339137	0.169569	−	0.985518i	$-0.445763\pi$
$433$	7.05699	0.339137	0.169569	+	0.985518i	$0.445763\pi$
$434$	0.0997034	0.00478592
$435$	−9.57777	−0.459219
$436$	2.56576	0.122877
$437$	65.2231	3.12004
$438$	−21.1163	−1.00898
$439$	4.76096	0.227228	0.113614	−	0.993525i	$-0.463757\pi$
$439$	4.76096	0.227228	0.113614	+	0.993525i	$0.463757\pi$
$440$	17.7316	0.845320
$441$	13.2848	0.632607
$442$	−9.38946	−0.446611
$443$	17.7774	0.844631	0.422315	−	0.906449i	$-0.361217\pi$
$443$	17.7774	0.844631	0.422315	+	0.906449i	$0.361217\pi$
$444$	−0.911556	−0.0432605
$445$	1.46672	0.0695293
$446$	49.0788	2.32395
$447$	8.58556	0.406083
$448$	−8.89526	−0.420261
$449$	−11.9480	−0.563863	−0.281931	−	0.959435i	$-0.590975\pi$
$449$	−11.9480	−0.563863	−0.281931	+	0.959435i	$0.590975\pi$
$450$	8.55024	0.403062
$451$	30.9874	1.45914
$452$	37.8522	1.78042
$453$	9.56870	0.449577
$454$	30.8667	1.44865
$455$	7.46530	0.349979
$456$	19.5318	0.914660
$457$	−32.1619	−1.50447	−0.752235	−	0.658895i	$-0.771024\pi$
$457$	−32.1619	−1.50447	−0.752235	+	0.658895i	$0.771024\pi$
$458$	−21.5276	−1.00592
$459$	−4.94933	−0.231015
$460$	−62.5384	−2.91587
$461$	38.5875	1.79720	0.898600	−	0.438769i	$-0.144585\pi$
$461$	38.5875	1.79720	0.898600	+	0.438769i	$0.144585\pi$
$462$	4.29339	0.199747
$463$	8.44551	0.392496	0.196248	−	0.980554i	$-0.437124\pi$
$463$	8.44551	0.392496	0.196248	+	0.980554i	$0.437124\pi$
$464$	2.85757	0.132659
$465$	−0.167539	−0.00776942
$466$	−22.5991	−1.04688
$467$	−23.6559	−1.09467	−0.547333	−	0.836915i	$-0.684357\pi$
$467$	−23.6559	−1.09467	−0.547333	+	0.836915i	$0.684357\pi$
$468$	25.9484	1.19946
$469$	−5.05223	−0.233290
$470$	−26.6822	−1.23076
$471$	18.3646	0.846198
$472$	2.99850	0.138017
$473$	−4.94726	−0.227475
$474$	28.0988	1.29062
$475$	15.6291	0.717114
$476$	−2.08520	−0.0955752
$477$	−2.12494	−0.0972945
$478$	−24.2180	−1.10771
$479$	−2.69722	−0.123239	−0.0616197	−	0.998100i	$-0.519627\pi$
$479$	−2.69722	−0.123239	−0.0616197	+	0.998100i	$0.519627\pi$
$480$	16.7103	0.762716
$481$	−1.26611	−0.0577296
$482$	−22.8638	−1.04142
$483$	−5.23553	−0.238225
$484$	−8.86200	−0.402818
$485$	−15.4195	−0.700163
$486$	36.1135	1.63814
$487$	−31.2447	−1.41583	−0.707915	−	0.706298i	$-0.750364\pi$
$487$	−31.2447	−1.41583	−0.707915	+	0.706298i	$0.750364\pi$
$488$	−16.2238	−0.734418
$489$	4.49284	0.203173
$490$	−38.5174	−1.74004
$491$	30.7707	1.38866	0.694330	−	0.719657i	$-0.255701\pi$
$491$	30.7707	1.38866	0.694330	+	0.719657i	$0.255701\pi$
$492$	−32.7284	−1.47551
$493$	3.71581	0.167352
$494$	78.4636	3.53024
$495$	15.1665	0.681682
$496$	0.0499859	0.00224443
$497$	1.73040	0.0776188
$498$	−26.9194	−1.20629
$499$	−32.8184	−1.46915	−0.734576	−	0.678527i	$-0.762619\pi$
$499$	−32.8184	−1.46915	−0.734576	+	0.678527i	$0.762619\pi$
$500$	25.0771	1.12148
$501$	−5.14702	−0.229952
$502$	−14.1171	−0.630079
$503$	−25.1372	−1.12081	−0.560407	−	0.828217i	$-0.689355\pi$
$503$	−25.1372	−1.12081	−0.560407	+	0.828217i	$0.689355\pi$
$504$	3.29595	0.146813
$505$	−2.88701	−0.128470
$506$	−49.9559	−2.22081
$507$	−4.36023	−0.193644
$508$	−21.5105	−0.954375
$509$	2.68100	0.118833	0.0594166	−	0.998233i	$-0.481076\pi$
$509$	2.68100	0.118833	0.0594166	+	0.998233i	$0.481076\pi$
$510$	5.79635	0.256667
$511$	−6.51348	−0.288139
$512$	−8.64123	−0.381892
$513$	41.3594	1.82606
$514$	42.5607	1.87727
$515$	−41.4211	−1.82523
$516$	5.22522	0.230027
$517$	−12.8843	−0.566651
$518$	−0.465136	−0.0204369
$519$	−2.51413	−0.110358
$520$	−26.0121	−1.14071
$521$	1.46899	0.0643575	0.0321788	−	0.999482i	$-0.489755\pi$
$521$	1.46899	0.0643575	0.0321788	+	0.999482i	$0.489755\pi$
$522$	−16.9873	−0.743514
$523$	2.39820	0.104866	0.0524330	−	0.998624i	$-0.483302\pi$
$523$	2.39820	0.104866	0.0524330	+	0.998624i	$0.483302\pi$
$524$	40.9935	1.79081
$525$	−1.25457	−0.0547539
$526$	−46.1366	−2.01165
$527$	0.0649985	0.00283138
$528$	2.15247	0.0936743
$529$	37.9183	1.64862
$530$	6.16100	0.267617
$531$	2.56472	0.111299
$532$	17.4251	0.755476
$533$	−45.4582	−1.96901
$534$	−1.23745	−0.0535497
$535$	47.5649	2.05641
$536$	17.6040	0.760376
$537$	−3.69394	−0.159405
$538$	−23.2879	−1.00401
$539$	−18.5993	−0.801127
$540$	−39.6570	−1.70657
$541$	9.56244	0.411121	0.205561	−	0.978644i	$-0.434098\pi$
$541$	9.56244	0.411121	0.205561	+	0.978644i	$0.434098\pi$
$542$	−19.9113	−0.855266
$543$	13.1934	0.566185
$544$	−6.48295	−0.277954
$545$	−2.19997	−0.0942364
$546$	−6.29836	−0.269545
$547$	33.2422	1.42133	0.710667	−	0.703529i	$-0.248393\pi$
$547$	33.2422	1.42133	0.710667	+	0.703529i	$0.248393\pi$
$548$	52.9976	2.26395
$549$	−13.8768	−0.592248
$550$	−11.9707	−0.510434
$551$	−31.0514	−1.32283
$552$	18.2427	0.776459
$553$	8.66728	0.368570
$554$	−30.6078	−1.30040
$555$	0.781601	0.0331771
$556$	−45.9198	−1.94743
$557$	25.0916	1.06317	0.531583	−	0.847006i	$-0.321597\pi$
$557$	25.0916	1.06317	0.531583	+	0.847006i	$0.321597\pi$
$558$	−0.297149	−0.0125793
$559$	7.25758	0.306963
$560$	1.37497	0.0581032
$561$	2.79894	0.118171
$562$	4.99728	0.210798
$563$	−30.3777	−1.28027	−0.640135	−	0.768263i	$-0.721121\pi$
$563$	−30.3777	−1.28027	−0.640135	+	0.768263i	$0.721121\pi$
$564$	13.6082	0.573009
$565$	−32.4558	−1.36543
$566$	−50.2472	−2.11205
$567$	0.840208	0.0352854
$568$	−6.02938	−0.252987
$569$	12.7099	0.532826	0.266413	−	0.963859i	$-0.414161\pi$
$569$	12.7099	0.532826	0.266413	+	0.963859i	$0.414161\pi$
$570$	−48.4376	−2.02883
$571$	8.12677	0.340095	0.170047	−	0.985436i	$-0.445608\pi$
$571$	8.12677	0.340095	0.170047	+	0.985436i	$0.445608\pi$
$572$	−36.3289	−1.51899
$573$	22.8182	0.953243
$574$	−16.7002	−0.697053
$575$	14.5976	0.608762
$576$	26.5108	1.10462
$577$	22.4564	0.934873	0.467437	−	0.884027i	$-0.345178\pi$
$577$	22.4564	0.934873	0.467437	+	0.884027i	$0.345178\pi$
$578$	−2.24876	−0.0935362
$579$	15.2271	0.632817
$580$	29.7733	1.23627
$581$	−8.30350	−0.344487
$582$	13.0092	0.539248
$583$	2.97502	0.123213
$584$	22.6955	0.939148
$585$	−22.2491	−0.919886
$586$	34.3203	1.41776
$587$	−3.66975	−0.151467	−0.0757334	−	0.997128i	$-0.524130\pi$
$587$	−3.66975	−0.151467	−0.0757334	+	0.997128i	$0.524130\pi$
$588$	19.6443	0.810116
$589$	−0.543164	−0.0223807
$590$	−7.43607	−0.306138
$591$	−25.7813	−1.06050
$592$	−0.233194	−0.00958421
$593$	2.35212	0.0965898	0.0482949	−	0.998833i	$-0.484621\pi$
$593$	2.35212	0.0965898	0.0482949	+	0.998833i	$0.484621\pi$
$594$	−31.6782	−1.29977
$595$	1.78793	0.0732980
$596$	−26.6889	−1.09322
$597$	12.0542	0.493347
$598$	73.2849	2.99684
$599$	26.9185	1.09986	0.549929	−	0.835211i	$-0.314655\pi$
$599$	26.9185	1.09986	0.549929	+	0.835211i	$0.314655\pi$
$600$	4.37142	0.178462
$601$	−6.15987	−0.251266	−0.125633	−	0.992077i	$-0.540096\pi$
$601$	−6.15987	−0.251266	−0.125633	+	0.992077i	$0.540096\pi$
$602$	2.66625	0.108668
$603$	15.0573	0.613181
$604$	−29.7451	−1.21031
$605$	7.59859	0.308927
$606$	2.43573	0.0989448
$607$	48.4157	1.96513	0.982565	−	0.185917i	$-0.0595255\pi$
$607$	48.4157	1.96513	0.982565	+	0.185917i	$0.0595255\pi$
$608$	54.1751	2.19709
$609$	2.49253	0.101002
$610$	40.2340	1.62903
$611$	18.9012	0.764659
$612$	6.21460	0.251210
$613$	−8.73637	−0.352859	−0.176429	−	0.984313i	$-0.556455\pi$
$613$	−8.73637	−0.352859	−0.176429	+	0.984313i	$0.556455\pi$
$614$	−25.8249	−1.04221
$615$	28.0625	1.13159
$616$	−4.61449	−0.185923
$617$	−15.9311	−0.641364	−0.320682	−	0.947187i	$-0.603912\pi$
$617$	−15.9311	−0.641364	−0.320682	+	0.947187i	$0.603912\pi$
$618$	34.9464	1.40575
$619$	21.8745	0.879210	0.439605	−	0.898191i	$-0.355118\pi$
$619$	21.8745	0.879210	0.439605	+	0.898191i	$0.355118\pi$
$620$	0.520807	0.0209161
$621$	38.6296	1.55015
$622$	−72.9134	−2.92356
$623$	−0.381701	−0.0152925
$624$	−3.15766	−0.126407
$625$	−30.8535	−1.23414
$626$	−3.52966	−0.141074
$627$	−23.3895	−0.934087
$628$	−57.0879	−2.27806
$629$	−0.303231	−0.0120906
$630$	−8.17375	−0.325650
$631$	36.5043	1.45321	0.726606	−	0.687055i	$-0.241097\pi$
$631$	36.5043	1.45321	0.726606	+	0.687055i	$0.241097\pi$
$632$	−30.2003	−1.20130
$633$	2.33513	0.0928130
$634$	72.8359	2.89268
$635$	18.4439	0.731923
$636$	−3.14217	−0.124595
$637$	27.2850	1.08107
$638$	23.7830	0.941578
$639$	−5.15715	−0.204014
$640$	−42.8794	−1.69496
$641$	−32.7194	−1.29234	−0.646169	−	0.763194i	$-0.723630\pi$
$641$	−32.7194	−1.29234	−0.646169	+	0.763194i	$0.723630\pi$
$642$	−40.1298	−1.58380
$643$	46.7364	1.84310	0.921552	−	0.388256i	$-0.126922\pi$
$643$	46.7364	1.84310	0.921552	+	0.388256i	$0.126922\pi$
$644$	16.2751	0.641327
$645$	−4.48029	−0.176411
$646$	18.7919	0.739358
$647$	−21.7376	−0.854593	−0.427297	−	0.904111i	$-0.640534\pi$
$647$	−21.7376	−0.854593	−0.427297	+	0.904111i	$0.640534\pi$
$648$	−2.92762	−0.115008
$649$	−3.59073	−0.140948
$650$	17.5610	0.688798
$651$	0.0436004	0.00170883
$652$	−13.9664	−0.546965
$653$	42.7190	1.67172	0.835862	−	0.548939i	$-0.184968\pi$
$653$	42.7190	1.67172	0.835862	+	0.548939i	$0.184968\pi$
$654$	1.85608	0.0725786
$655$	−35.1493	−1.37340
$656$	−8.37257	−0.326894
$657$	19.4123	0.757346
$658$	6.94381	0.270698
$659$	−8.05658	−0.313840	−0.156920	−	0.987611i	$-0.550156\pi$
$659$	−8.05658	−0.313840	−0.156920	+	0.987611i	$0.550156\pi$
$660$	22.4268	0.872961
$661$	49.1594	1.91208	0.956040	−	0.293237i	$-0.0947326\pi$
$661$	49.1594	1.91208	0.956040	+	0.293237i	$0.0947326\pi$
$662$	18.8345	0.732025
$663$	−4.10602	−0.159465
$664$	28.9327	1.12281
$665$	−14.9409	−0.579385
$666$	1.38626	0.0537164
$667$	−29.0019	−1.12296
$668$	15.9999	0.619055
$669$	21.4622	0.829776
$670$	−43.6567	−1.68661
$671$	19.4282	0.750017
$672$	−4.34870	−0.167755
$673$	10.5216	0.405580	0.202790	−	0.979222i	$-0.434999\pi$
$673$	10.5216	0.405580	0.202790	+	0.979222i	$0.434999\pi$
$674$	33.6580	1.29646
$675$	9.25666	0.356289
$676$	13.5541	0.521312
$677$	16.2710	0.625347	0.312673	−	0.949861i	$-0.398775\pi$
$677$	16.2710	0.625347	0.312673	+	0.949861i	$0.398775\pi$
$678$	27.3825	1.05162
$679$	4.01278	0.153996
$680$	−6.22986	−0.238904
$681$	13.4980	0.517246
$682$	0.416023	0.0159303
$683$	−0.774592	−0.0296389	−0.0148195	−	0.999890i	$-0.504717\pi$
$683$	−0.774592	−0.0296389	−0.0148195	+	0.999890i	$0.504717\pi$
$684$	−51.9327	−1.98569
$685$	−45.4420	−1.73625
$686$	20.7613	0.792672
$687$	−9.41404	−0.359168
$688$	1.33671	0.0509618
$689$	−4.36433	−0.166268
$690$	−45.2406	−1.72228
$691$	−4.74756	−0.180606	−0.0903029	−	0.995914i	$-0.528784\pi$
$691$	−4.74756	−0.180606	−0.0903029	+	0.995914i	$0.528784\pi$
$692$	7.81536	0.297096
$693$	−3.94693	−0.149932
$694$	−72.1806	−2.73994
$695$	39.3733	1.49351
$696$	−8.68496	−0.329203
$697$	−10.8872	−0.412381
$698$	77.5701	2.93607
$699$	−9.88260	−0.373794
$700$	3.89993	0.147403
$701$	26.4618	0.999447	0.499723	−	0.866185i	$-0.333435\pi$
$701$	26.4618	0.999447	0.499723	+	0.866185i	$0.333435\pi$
$702$	46.4716	1.75396
$703$	2.53397	0.0955704
$704$	−37.1164	−1.39888
$705$	−11.6682	−0.439448
$706$	−2.24876	−0.0846333
$707$	0.751319	0.0282563
$708$	3.79247	0.142530
$709$	1.07964	0.0405468	0.0202734	−	0.999794i	$-0.493546\pi$
$709$	1.07964	0.0405468	0.0202734	+	0.999794i	$0.493546\pi$
$710$	14.9525	0.561157
$711$	−25.8314	−0.968752
$712$	1.33000	0.0498438
$713$	−0.507315	−0.0189991
$714$	−1.50845	−0.0564523
$715$	31.1497	1.16493
$716$	11.4829	0.429137
$717$	−10.5906	−0.395512
$718$	20.1075	0.750406
$719$	45.3551	1.69146	0.845729	−	0.533613i	$-0.179166\pi$
$719$	45.3551	1.69146	0.845729	+	0.533613i	$0.179166\pi$
$720$	−4.09787	−0.152719
$721$	10.7795	0.401449
$722$	−114.309	−4.25415
$723$	−9.99837	−0.371843
$724$	−41.0128	−1.52423
$725$	−6.94962	−0.258102
$726$	−6.41081	−0.237928
$727$	48.4450	1.79673	0.898363	−	0.439254i	$-0.144757\pi$
$727$	48.4450	1.79673	0.898363	+	0.439254i	$0.144757\pi$
$728$	6.76941	0.250891
$729$	12.0972	0.448045
$730$	−56.2835	−2.08314
$731$	1.73818	0.0642889
$732$	−20.5198	−0.758432
$733$	9.32878	0.344567	0.172283	−	0.985047i	$-0.444886\pi$
$733$	9.32878	0.344567	0.172283	+	0.985047i	$0.444886\pi$
$734$	36.8931	1.36175
$735$	−16.8437	−0.621289
$736$	50.5995	1.86512
$737$	−21.0809	−0.776526
$738$	49.7721	1.83214
$739$	2.75593	0.101378	0.0506892	−	0.998714i	$-0.483858\pi$
$739$	2.75593	0.101378	0.0506892	+	0.998714i	$0.483858\pi$
$740$	−2.42967	−0.0893163
$741$	34.3122	1.26049
$742$	−1.60334	−0.0588606
$743$	8.59199	0.315209	0.157605	−	0.987502i	$-0.449623\pi$
$743$	8.59199	0.315209	0.157605	+	0.987502i	$0.449623\pi$
$744$	−0.151921	−0.00556970
$745$	22.8840	0.838405
$746$	35.0193	1.28215
$747$	24.7472	0.905451
$748$	−8.70073	−0.318130
$749$	−12.3783	−0.452294
$750$	18.1409	0.662413
$751$	42.2283	1.54093	0.770466	−	0.637481i	$-0.220023\pi$
$751$	42.2283	1.54093	0.770466	+	0.637481i	$0.220023\pi$
$752$	3.48125	0.126948
$753$	−6.17344	−0.224973
$754$	−34.8894	−1.27060
$755$	25.5045	0.928203
$756$	10.3204	0.375348
$757$	43.7933	1.59169	0.795847	−	0.605498i	$-0.207026\pi$
$757$	43.7933	1.59169	0.795847	+	0.605498i	$0.207026\pi$
$758$	−25.4816	−0.925533
$759$	−21.8458	−0.792951
$760$	52.0602	1.88842
$761$	−2.48522	−0.0900893	−0.0450446	−	0.998985i	$-0.514343\pi$
$761$	−2.48522	−0.0900893	−0.0450446	+	0.998985i	$0.514343\pi$
$762$	−15.5608	−0.563709
$763$	0.572523	0.0207267
$764$	−70.9321	−2.56623
$765$	−5.32862	−0.192657
$766$	−4.94723	−0.178751
$767$	5.26757	0.190201
$768$	10.5290	0.379931
$769$	9.77089	0.352347	0.176174	−	0.984359i	$-0.443628\pi$
$769$	9.77089	0.352347	0.176174	+	0.984359i	$0.443628\pi$
$770$	11.4436	0.412400
$771$	18.6118	0.670288
$772$	−47.3347	−1.70361
$773$	19.2627	0.692831	0.346415	−	0.938081i	$-0.387399\pi$
$773$	19.2627	0.692831	0.346415	+	0.938081i	$0.387399\pi$
$774$	−7.94631	−0.285624
$775$	−0.121566	−0.00436677
$776$	−13.9821	−0.501929
$777$	−0.203404	−0.00729709
$778$	−69.5017	−2.49176
$779$	90.9793	3.25967
$780$	−32.8999	−1.17800
$781$	7.22025	0.258361
$782$	17.5516	0.627645
$783$	−18.3908	−0.657233
$784$	5.02539	0.179478
$785$	48.9492	1.74707
$786$	29.6549	1.05775
$787$	−47.4237	−1.69047	−0.845237	−	0.534392i	$-0.820541\pi$
$787$	−47.4237	−1.69047	−0.845237	+	0.534392i	$0.820541\pi$
$788$	80.1433	2.85499
$789$	−20.1756	−0.718270
$790$	74.8947	2.66463
$791$	8.44633	0.300317
$792$	13.7527	0.488680
$793$	−28.5010	−1.01210
$794$	14.5635	0.516838
$795$	2.69421	0.0955538
$796$	−37.4715	−1.32814
$797$	−9.16274	−0.324561	−0.162280	−	0.986745i	$-0.551885\pi$
$797$	−9.16274	−0.324561	−0.162280	+	0.986745i	$0.551885\pi$
$798$	12.6054	0.446228
$799$	4.52680	0.160147
$800$	12.1250	0.428682
$801$	1.13760	0.0401949
$802$	−20.1939	−0.713072
$803$	−27.1781	−0.959096
$804$	22.2654	0.785238
$805$	−13.9548	−0.491843
$806$	−0.610301	−0.0214970
$807$	−10.1838	−0.358487
$808$	−2.61790	−0.0920972
$809$	−11.7066	−0.411581	−0.205791	−	0.978596i	$-0.565977\pi$
$809$	−11.7066	−0.411581	−0.205791	+	0.978596i	$0.565977\pi$
$810$	7.26030	0.255101
$811$	26.8683	0.943475	0.471737	−	0.881739i	$-0.343627\pi$
$811$	26.8683	0.943475	0.471737	+	0.881739i	$0.343627\pi$
$812$	−7.74822	−0.271909
$813$	−8.70725	−0.305377
$814$	−1.94083	−0.0680260
$815$	11.9753	0.419475
$816$	−0.756254	−0.0264742
$817$	−14.5252	−0.508173
$818$	−73.8587	−2.58241
$819$	5.79012	0.202323
$820$	−87.2345	−3.04636
$821$	12.2860	0.428785	0.214392	−	0.976748i	$-0.431223\pi$
$821$	12.2860	0.428785	0.214392	+	0.976748i	$0.431223\pi$
$822$	38.3388	1.33722
$823$	−37.6599	−1.31274	−0.656371	−	0.754438i	$-0.727909\pi$
$823$	−37.6599	−1.31274	−0.656371	+	0.754438i	$0.727909\pi$
$824$	−37.5600	−1.30846
$825$	−5.23482	−0.182253
$826$	1.93517	0.0673332
$827$	22.2689	0.774365	0.387182	−	0.922003i	$-0.373448\pi$
$827$	22.2689	0.774365	0.387182	+	0.922003i	$0.373448\pi$
$828$	−48.5050	−1.68567
$829$	47.3863	1.64580	0.822898	−	0.568189i	$-0.192356\pi$
$829$	47.3863	1.64580	0.822898	+	0.568189i	$0.192356\pi$
$830$	−71.7512	−2.49052
$831$	−13.3848	−0.464315
$832$	54.4494	1.88769
$833$	6.53471	0.226414
$834$	−33.2187	−1.15027
$835$	−13.7189	−0.474762
$836$	72.7081	2.51466
$837$	−0.321699	−0.0111196
$838$	−60.1200	−2.07681
$839$	40.5398	1.39959	0.699795	−	0.714344i	$-0.253275\pi$
$839$	40.5398	1.39959	0.699795	+	0.714344i	$0.253275\pi$
$840$	−4.17893	−0.144187
$841$	−15.1928	−0.523888
$842$	29.5309	1.01770
$843$	2.18532	0.0752663
$844$	−7.25893	−0.249863
$845$	−11.6218	−0.399801
$846$	−20.6948	−0.711503
$847$	−1.97746	−0.0679465
$848$	−0.803829	−0.0276036
$849$	−21.9732	−0.754117
$850$	4.20583	0.144259
$851$	2.36672	0.0811302
$852$	−7.62592	−0.261260
$853$	39.9148	1.36666	0.683329	−	0.730111i	$-0.260531\pi$
$853$	39.9148	1.36666	0.683329	+	0.730111i	$0.260531\pi$
$854$	−10.4705	−0.358295
$855$	44.5289	1.52286
$856$	43.1310	1.47419
$857$	30.3192	1.03568	0.517841	−	0.855477i	$-0.326736\pi$
$857$	30.3192	1.03568	0.517841	+	0.855477i	$0.326736\pi$
$858$	−26.2805	−0.897203
$859$	−56.1950	−1.91735	−0.958674	−	0.284508i	$-0.908170\pi$
$859$	−56.1950	−1.91735	−0.958674	+	0.284508i	$0.908170\pi$
$860$	13.9273	0.474918
$861$	−7.30301	−0.248886
$862$	−14.7806	−0.503429
$863$	−44.7551	−1.52348	−0.761741	−	0.647881i	$-0.775655\pi$
$863$	−44.7551	−1.52348	−0.761741	+	0.647881i	$0.775655\pi$
$864$	32.0863	1.09160
$865$	−6.70117	−0.227847
$866$	−15.8695	−0.539267
$867$	−0.983386	−0.0333975
$868$	−0.135535	−0.00460037
$869$	36.1651	1.22682
$870$	21.5381	0.730211
$871$	30.9255	1.04787
$872$	−1.99490	−0.0675557
$873$	−11.9594	−0.404765
$874$	−146.671	−4.96123
$875$	5.59571	0.189170
$876$	28.7051	0.969857
$877$	−3.70861	−0.125231	−0.0626154	−	0.998038i	$-0.519944\pi$
$877$	−3.70861	−0.125231	−0.0626154	+	0.998038i	$0.519944\pi$
$878$	−10.7063	−0.361319
$879$	15.0083	0.506218
$880$	5.73721	0.193401
$881$	−14.3599	−0.483798	−0.241899	−	0.970301i	$-0.577770\pi$
$881$	−14.3599	−0.483798	−0.241899	+	0.970301i	$0.577770\pi$
$882$	−29.8742	−1.00592
$883$	26.1897	0.881354	0.440677	−	0.897666i	$-0.354738\pi$
$883$	26.1897	0.881354	0.440677	+	0.897666i	$0.354738\pi$
$884$	12.7639	0.429296
$885$	−3.25180	−0.109308
$886$	−39.9772	−1.34306
$887$	−11.8835	−0.399010	−0.199505	−	0.979897i	$-0.563933\pi$
$887$	−11.8835	−0.399010	−0.199505	+	0.979897i	$0.563933\pi$
$888$	0.708742	0.0237838
$889$	−4.79985	−0.160982
$890$	−3.29831	−0.110560
$891$	3.50585	0.117450
$892$	−66.7169	−2.23385
$893$	−37.8285	−1.26588
$894$	−19.3069	−0.645719
$895$	−9.84587	−0.329111
$896$	11.1590	0.372796
$897$	32.0475	1.07004
$898$	26.8683	0.896607
$899$	0.241522	0.00805521
$900$	−11.6231	−0.387436
$901$	−1.04525	−0.0348223
$902$	−69.6833	−2.32020
$903$	1.16595	0.0388006
$904$	−29.4304	−0.978839
$905$	35.1659	1.16895
$906$	−21.5177	−0.714879
$907$	43.4886	1.44402	0.722008	−	0.691885i	$-0.243219\pi$
$907$	43.4886	1.44402	0.722008	+	0.691885i	$0.243219\pi$
$908$	−41.9597	−1.39248
$909$	−2.23918	−0.0742689
$910$	−16.7877	−0.556507
$911$	−41.9563	−1.39007	−0.695037	−	0.718974i	$-0.744612\pi$
$911$	−41.9563	−1.39007	−0.695037	+	0.718974i	$0.744612\pi$
$912$	6.31968	0.209266
$913$	−34.6472	−1.14665
$914$	72.3244	2.39228
$915$	17.5944	0.581652
$916$	29.2643	0.966919
$917$	9.14728	0.302070
$918$	11.1299	0.367341
$919$	−18.6353	−0.614722	−0.307361	−	0.951593i	$-0.599446\pi$
$919$	−18.6353	−0.614722	−0.307361	+	0.951593i	$0.599446\pi$
$920$	48.6241	1.60309
$921$	−11.2933	−0.372125
$922$	−86.7742	−2.85775
$923$	−10.5920	−0.348641
$924$	−5.83636	−0.192002
$925$	0.567128	0.0186471
$926$	−18.9919	−0.624114
$927$	−32.1264	−1.05517
$928$	−24.0894	−0.790773
$929$	3.87399	0.127102	0.0635508	−	0.997979i	$-0.479758\pi$
$929$	3.87399	0.127102	0.0635508	+	0.997979i	$0.479758\pi$
$930$	0.376755	0.0123543
$931$	−54.6077	−1.78969
$932$	30.7208	1.00629
$933$	−31.8851	−1.04387
$934$	53.1965	1.74064
$935$	7.46031	0.243978
$936$	−20.1751	−0.659443
$937$	−21.6701	−0.707930	−0.353965	−	0.935259i	$-0.615167\pi$
$937$	−21.6701	−0.707930	−0.353965	+	0.935259i	$0.615167\pi$
$938$	11.3613	0.370958
$939$	−1.54352	−0.0503710
$940$	36.2714	1.18304
$941$	36.2826	1.18278	0.591389	−	0.806387i	$-0.298580\pi$
$941$	36.2826	1.18278	0.591389	+	0.806387i	$0.298580\pi$
$942$	−41.2977	−1.34555
$943$	84.9746	2.76715
$944$	0.970189	0.0315770
$945$	−8.84906	−0.287860
$946$	11.1252	0.361712
$947$	−41.0951	−1.33541	−0.667706	−	0.744425i	$-0.732724\pi$
$947$	−41.0951	−1.33541	−0.667706	+	0.744425i	$0.732724\pi$
$948$	−38.1970	−1.24058
$949$	39.8701	1.29424
$950$	−35.1462	−1.14029
$951$	31.8512	1.03285
$952$	1.62126	0.0525454
$953$	26.6883	0.864518	0.432259	−	0.901749i	$-0.357717\pi$
$953$	26.6883	0.864518	0.432259	+	0.901749i	$0.357717\pi$
$954$	4.77849	0.154709
$955$	60.8197	1.96808
$956$	32.9216	1.06476
$957$	10.4003	0.336195
$958$	6.06542	0.195965
$959$	11.8259	0.381878
$960$	−33.6130	−1.08485
$961$	−30.9958	−0.999864
$962$	2.84718	0.0917966
$963$	36.8915	1.18881
$964$	31.0807	1.00104
$965$	40.5864	1.30652
$966$	11.7735	0.378805
$967$	12.5530	0.403678	0.201839	−	0.979419i	$-0.435308\pi$
$967$	12.5530	0.403678	0.201839	+	0.979419i	$0.435308\pi$
$968$	6.89027	0.221462
$969$	8.21772	0.263991
$970$	34.6748	1.11334
$971$	−39.3073	−1.26143	−0.630716	−	0.776014i	$-0.717239\pi$
$971$	−39.3073	−1.26143	−0.630716	+	0.776014i	$0.717239\pi$
$972$	−49.0922	−1.57463
$973$	−10.2465	−0.328489
$974$	70.2618	2.25133
$975$	7.67943	0.245938
$976$	−5.24936	−0.168028
$977$	44.5699	1.42592	0.712958	−	0.701207i	$-0.247355\pi$
$977$	44.5699	1.42592	0.712958	+	0.701207i	$0.247355\pi$
$978$	−10.1033	−0.323069
$979$	−1.59269	−0.0509025
$980$	52.3600	1.67258
$981$	−1.70631	−0.0544782
$982$	−69.1959	−2.20813
$983$	50.2056	1.60131	0.800655	−	0.599125i	$-0.204485\pi$
$983$	50.2056	1.60131	0.800655	+	0.599125i	$0.204485\pi$
$984$	25.4466	0.811208
$985$	−68.7177	−2.18953
$986$	−8.35597	−0.266108
$987$	3.03653	0.0966539
$988$	−106.662	−3.39338
$989$	−13.5665	−0.431390
$990$	−34.1058	−1.08395
$991$	−41.5443	−1.31970	−0.659850	−	0.751398i	$-0.729380\pi$
$991$	−41.5443	−1.31970	−0.659850	+	0.751398i	$0.729380\pi$
$992$	−0.421382	−0.0133789
$993$	8.23636	0.261373
$994$	−3.89125	−0.123423
$995$	32.1294	1.01857
$996$	36.5938	1.15952
$997$	−2.30330	−0.0729461	−0.0364731	−	0.999335i	$-0.511612\pi$
$997$	−2.30330	−0.0729461	−0.0364731	+	0.999335i	$0.511612\pi$
$998$	73.8007	2.33612
$999$	1.50079	0.0474829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6001.2.a.c.1.14	✓	121

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6001.2.a.c.1.14	✓	121	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:	\( N \)	\(=\)	\( 6001 = 17 \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24876 q^{2} -0.983386 q^{3} +3.05693 q^{4} -2.62112 q^{5} +2.21140 q^{6} +0.682123 q^{7} -2.37679 q^{8} -2.03295 q^{9} +O(q^{10})\)	\(q-2.24876 q^{2} -0.983386 q^{3} +3.05693 q^{4} -2.62112 q^{5} +2.21140 q^{6} +0.682123 q^{7} -2.37679 q^{8} -2.03295 q^{9} +5.89428 q^{10} +2.84623 q^{11} -3.00614 q^{12} -4.17539 q^{13} -1.53393 q^{14} +2.57757 q^{15} -0.769031 q^{16} -1.00000 q^{17} +4.57163 q^{18} +8.35656 q^{19} -8.01259 q^{20} -0.670790 q^{21} -6.40049 q^{22} +7.80502 q^{23} +2.33730 q^{24} +1.87028 q^{25} +9.38946 q^{26} +4.94933 q^{27} +2.08520 q^{28} -3.71581 q^{29} -5.79635 q^{30} -0.0649985 q^{31} +6.48295 q^{32} -2.79894 q^{33} +2.24876 q^{34} -1.78793 q^{35} -6.21460 q^{36} +0.303231 q^{37} -18.7919 q^{38} +4.10602 q^{39} +6.22986 q^{40} +10.8872 q^{41} +1.50845 q^{42} -1.73818 q^{43} +8.70073 q^{44} +5.32862 q^{45} -17.5516 q^{46} -4.52680 q^{47} +0.756254 q^{48} -6.53471 q^{49} -4.20583 q^{50} +0.983386 q^{51} -12.7639 q^{52} +1.04525 q^{53} -11.1299 q^{54} -7.46031 q^{55} -1.62126 q^{56} -8.21772 q^{57} +8.35597 q^{58} -1.26157 q^{59} +7.87947 q^{60} +6.82594 q^{61} +0.146166 q^{62} -1.38672 q^{63} -13.0405 q^{64} +10.9442 q^{65} +6.29415 q^{66} -7.40662 q^{67} -3.05693 q^{68} -7.67534 q^{69} +4.02063 q^{70} +2.53678 q^{71} +4.83190 q^{72} -9.54883 q^{73} -0.681894 q^{74} -1.83921 q^{75} +25.5454 q^{76} +1.94148 q^{77} -9.23346 q^{78} +12.7063 q^{79} +2.01572 q^{80} +1.23175 q^{81} -24.4827 q^{82} -12.1730 q^{83} -2.05056 q^{84} +2.62112 q^{85} +3.90876 q^{86} +3.65407 q^{87} -6.76488 q^{88} -0.559578 q^{89} -11.9828 q^{90} -2.84813 q^{91} +23.8594 q^{92} +0.0639186 q^{93} +10.1797 q^{94} -21.9036 q^{95} -6.37524 q^{96} +5.88278 q^{97} +14.6950 q^{98} -5.78625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9}+O(q^{10}) \) 121 * q + 9 * q^2 + 13 * q^3 + 127 * q^4 + 21 * q^5 + 19 * q^6 - 13 * q^7 + 24 * q^8 + 134 * q^9	\( 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9} - q^{10} + 40 q^{11} + 41 q^{12} + 14 q^{13} + 32 q^{14} + 49 q^{15} + 135 q^{16} - 121 q^{17} + 28 q^{18} + 34 q^{19} + 64 q^{20} + 34 q^{21} - 18 q^{22} + 37 q^{23} + 54 q^{24} + 128 q^{25} + 91 q^{26} + 55 q^{27} - 28 q^{28} + 45 q^{29} + 30 q^{30} + 67 q^{31} + 47 q^{32} + 40 q^{33} - 9 q^{34} + 59 q^{35} + 138 q^{36} - 16 q^{37} + 30 q^{38} + 37 q^{39} + 14 q^{40} + 89 q^{41} + 33 q^{42} + 16 q^{43} + 90 q^{44} + 83 q^{45} - 9 q^{46} + 135 q^{47} + 96 q^{48} + 128 q^{49} + 71 q^{50} - 13 q^{51} + 47 q^{52} + 52 q^{53} + 90 q^{54} + 93 q^{55} + 69 q^{56} - 4 q^{57} + 5 q^{58} + 170 q^{59} + 78 q^{60} - 2 q^{61} + 46 q^{62} - 10 q^{63} + 182 q^{64} + 50 q^{65} + 68 q^{66} + 46 q^{67} - 127 q^{68} + 97 q^{69} + 46 q^{70} + 191 q^{71} + 57 q^{72} - 12 q^{73} + 68 q^{74} + 86 q^{75} + 108 q^{76} + 62 q^{77} - 10 q^{78} + 130 q^{80} + 149 q^{81} + 14 q^{82} + 83 q^{83} + 126 q^{84} - 21 q^{85} + 132 q^{86} + 50 q^{87} - 42 q^{88} + 144 q^{89} + 9 q^{90} + 13 q^{91} + 50 q^{92} + 43 q^{93} + 41 q^{94} + 82 q^{95} + 110 q^{96} - 3 q^{97} + 36 q^{98} + 89 q^{99}+O(q^{100}) \) 121 * q + 9 * q^2 + 13 * q^3 + 127 * q^4 + 21 * q^5 + 19 * q^6 - 13 * q^7 + 24 * q^8 + 134 * q^9 - q^10 + 40 * q^11 + 41 * q^12 + 14 * q^13 + 32 * q^14 + 49 * q^15 + 135 * q^16 - 121 * q^17 + 28 * q^18 + 34 * q^19 + 64 * q^20 + 34 * q^21 - 18 * q^22 + 37 * q^23 + 54 * q^24 + 128 * q^25 + 91 * q^26 + 55 * q^27 - 28 * q^28 + 45 * q^29 + 30 * q^30 + 67 * q^31 + 47 * q^32 + 40 * q^33 - 9 * q^34 + 59 * q^35 + 138 * q^36 - 16 * q^37 + 30 * q^38 + 37 * q^39 + 14 * q^40 + 89 * q^41 + 33 * q^42 + 16 * q^43 + 90 * q^44 + 83 * q^45 - 9 * q^46 + 135 * q^47 + 96 * q^48 + 128 * q^49 + 71 * q^50 - 13 * q^51 + 47 * q^52 + 52 * q^53 + 90 * q^54 + 93 * q^55 + 69 * q^56 - 4 * q^57 + 5 * q^58 + 170 * q^59 + 78 * q^60 - 2 * q^61 + 46 * q^62 - 10 * q^63 + 182 * q^64 + 50 * q^65 + 68 * q^66 + 46 * q^67 - 127 * q^68 + 97 * q^69 + 46 * q^70 + 191 * q^71 + 57 * q^72 - 12 * q^73 + 68 * q^74 + 86 * q^75 + 108 * q^76 + 62 * q^77 - 10 * q^78 + 130 * q^80 + 149 * q^81 + 14 * q^82 + 83 * q^83 + 126 * q^84 - 21 * q^85 + 132 * q^86 + 50 * q^87 - 42 * q^88 + 144 * q^89 + 9 * q^90 + 13 * q^91 + 50 * q^92 + 43 * q^93 + 41 * q^94 + 82 * q^95 + 110 * q^96 - 3 * q^97 + 36 * q^98 + 89 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more