LMFDB - Embedded newform 6013.2.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$109$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31924 q^{2} +3.03113 q^{3} +3.37885 q^{4} +1.64684 q^{5} -7.02991 q^{6} +1.00000 q^{7} -3.19788 q^{8} +6.18776 q^{9} +O(q^{10})$	\(q-2.31924 q^{2} +3.03113 q^{3} +3.37885 q^{4} +1.64684 q^{5} -7.02991 q^{6} +1.00000 q^{7} -3.19788 q^{8} +6.18776 q^{9} -3.81941 q^{10} +1.18478 q^{11} +10.2417 q^{12} +5.12534 q^{13} -2.31924 q^{14} +4.99179 q^{15} +0.658941 q^{16} -6.39433 q^{17} -14.3509 q^{18} +6.08738 q^{19} +5.56443 q^{20} +3.03113 q^{21} -2.74778 q^{22} +5.34244 q^{23} -9.69321 q^{24} -2.28792 q^{25} -11.8869 q^{26} +9.66251 q^{27} +3.37885 q^{28} +7.62059 q^{29} -11.5771 q^{30} -10.7372 q^{31} +4.86753 q^{32} +3.59122 q^{33} +14.8300 q^{34} +1.64684 q^{35} +20.9075 q^{36} +7.75870 q^{37} -14.1181 q^{38} +15.5356 q^{39} -5.26640 q^{40} -2.27736 q^{41} -7.02991 q^{42} -10.2084 q^{43} +4.00320 q^{44} +10.1902 q^{45} -12.3904 q^{46} -5.91309 q^{47} +1.99734 q^{48} +1.00000 q^{49} +5.30622 q^{50} -19.3821 q^{51} +17.3178 q^{52} +5.89523 q^{53} -22.4096 q^{54} +1.95114 q^{55} -3.19788 q^{56} +18.4516 q^{57} -17.6739 q^{58} +6.73270 q^{59} +16.8665 q^{60} +4.52365 q^{61} +24.9022 q^{62} +6.18776 q^{63} -12.6068 q^{64} +8.44062 q^{65} -8.32889 q^{66} -0.200977 q^{67} -21.6055 q^{68} +16.1936 q^{69} -3.81941 q^{70} +7.70302 q^{71} -19.7877 q^{72} -2.54847 q^{73} -17.9942 q^{74} -6.93498 q^{75} +20.5683 q^{76} +1.18478 q^{77} -36.0307 q^{78} +5.43372 q^{79} +1.08517 q^{80} +10.7251 q^{81} +5.28172 q^{82} +4.01327 q^{83} +10.2417 q^{84} -10.5304 q^{85} +23.6758 q^{86} +23.0990 q^{87} -3.78879 q^{88} -10.7483 q^{89} -23.6336 q^{90} +5.12534 q^{91} +18.0513 q^{92} -32.5460 q^{93} +13.7139 q^{94} +10.0249 q^{95} +14.7541 q^{96} -6.33525 q^{97} -2.31924 q^{98} +7.33113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) $ 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * 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 $

$2$	−2.31924	−1.63995	−0.819974	−	0.572401i	$-0.806012\pi$
$2$	−2.31924	−1.63995	−0.819974	+	0.572401i	$0.806012\pi$
$3$	3.03113	1.75002	0.875012	−	0.484101i	$-0.160853\pi$
$3$	3.03113	1.75002	0.875012	+	0.484101i	$0.160853\pi$
$4$	3.37885	1.68943
$5$	1.64684	0.736489	0.368245	−	0.929729i	$-0.379959\pi$
$5$	1.64684	0.736489	0.368245	+	0.929729i	$0.379959\pi$
$6$	−7.02991	−2.86995
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.19788	−1.13062
$9$	6.18776	2.06259
$10$	−3.81941	−1.20780
$11$	1.18478	0.357225	0.178612	−	0.983920i	$-0.442839\pi$
$11$	1.18478	0.357225	0.178612	+	0.983920i	$0.442839\pi$
$12$	10.2417	2.95654
$13$	5.12534	1.42151	0.710757	−	0.703438i	$-0.248353\pi$
$13$	5.12534	1.42151	0.710757	+	0.703438i	$0.248353\pi$
$14$	−2.31924	−0.619842
$15$	4.99179	1.28887
$16$	0.658941	0.164735
$17$	−6.39433	−1.55085	−0.775426	−	0.631438i	$-0.782465\pi$
$17$	−6.39433	−1.55085	−0.775426	+	0.631438i	$0.782465\pi$
$18$	−14.3509	−3.38253
$19$	6.08738	1.39654	0.698270	−	0.715835i	$-0.253954\pi$
$19$	6.08738	1.39654	0.698270	+	0.715835i	$0.253954\pi$
$20$	5.56443	1.24424
$21$	3.03113	0.661447
$22$	−2.74778	−0.585829
$23$	5.34244	1.11398	0.556988	−	0.830521i	$-0.311957\pi$
$23$	5.34244	1.11398	0.556988	+	0.830521i	$0.311957\pi$
$24$	−9.69321	−1.97862
$25$	−2.28792	−0.457584
$26$	−11.8869	−2.33121
$27$	9.66251	1.85955
$28$	3.37885	0.638543
$29$	7.62059	1.41511	0.707554	−	0.706659i	$-0.249798\pi$
$29$	7.62059	1.41511	0.707554	+	0.706659i	$0.249798\pi$
$30$	−11.5771	−2.11369
$31$	−10.7372	−1.92847	−0.964233	−	0.265057i	$-0.914609\pi$
$31$	−10.7372	−1.92847	−0.964233	+	0.265057i	$0.914609\pi$
$32$	4.86753	0.860466
$33$	3.59122	0.625152
$34$	14.8300	2.54332
$35$	1.64684	0.278367
$36$	20.9075	3.48459
$37$	7.75870	1.27552	0.637761	−	0.770234i	$-0.279861\pi$
$37$	7.75870	1.27552	0.637761	+	0.770234i	$0.279861\pi$
$38$	−14.1181	−2.29025
$39$	15.5356	2.48768
$40$	−5.26640	−0.832692
$41$	−2.27736	−0.355663	−0.177832	−	0.984061i	$-0.556908\pi$
$41$	−2.27736	−0.355663	−0.177832	+	0.984061i	$0.556908\pi$
$42$	−7.02991	−1.08474
$43$	−10.2084	−1.55677	−0.778386	−	0.627786i	$-0.783961\pi$
$43$	−10.2084	−1.55677	−0.778386	+	0.627786i	$0.783961\pi$
$44$	4.00320	0.603505
$45$	10.1902	1.51907
$46$	−12.3904	−1.82686
$47$	−5.91309	−0.862513	−0.431257	−	0.902229i	$-0.641930\pi$
$47$	−5.91309	−0.862513	−0.431257	+	0.902229i	$0.641930\pi$
$48$	1.99734	0.288291
$49$	1.00000	0.142857
$50$	5.30622	0.750413
$51$	−19.3821	−2.71403
$52$	17.3178	2.40154
$53$	5.89523	0.809773	0.404886	−	0.914367i	$-0.367311\pi$
$53$	5.89523	0.809773	0.404886	+	0.914367i	$0.367311\pi$
$54$	−22.4096	−3.04957
$55$	1.95114	0.263092
$56$	−3.19788	−0.427335
$57$	18.4516	2.44398
$58$	−17.6739	−2.32070
$59$	6.73270	0.876523	0.438261	−	0.898848i	$-0.355594\pi$
$59$	6.73270	0.876523	0.438261	+	0.898848i	$0.355594\pi$
$60$	16.8665	2.17746
$61$	4.52365	0.579195	0.289597	−	0.957149i	$-0.406479\pi$
$61$	4.52365	0.579195	0.289597	+	0.957149i	$0.406479\pi$
$62$	24.9022	3.16258
$63$	6.18776	0.779584
$64$	−12.6068	−1.57585
$65$	8.44062	1.04693
$66$	−8.32889	−1.02522
$67$	−0.200977	−0.0245532	−0.0122766	−	0.999925i	$-0.503908\pi$
$67$	−0.200977	−0.0245532	−0.0122766	+	0.999925i	$0.503908\pi$
$68$	−21.6055	−2.62005
$69$	16.1936	1.94949
$70$	−3.81941	−0.456507
$71$	7.70302	0.914181	0.457090	−	0.889420i	$-0.348892\pi$
$71$	7.70302	0.914181	0.457090	+	0.889420i	$0.348892\pi$
$72$	−19.7877	−2.33201
$73$	−2.54847	−0.298276	−0.149138	−	0.988816i	$-0.547650\pi$
$73$	−2.54847	−0.298276	−0.149138	+	0.988816i	$0.547650\pi$
$74$	−17.9942	−2.09179
$75$	−6.93498	−0.800782
$76$	20.5683	2.35935
$77$	1.18478	0.135018
$78$	−36.0307	−4.07967
$79$	5.43372	0.611341	0.305670	−	0.952137i	$-0.401119\pi$
$79$	5.43372	0.611341	0.305670	+	0.952137i	$0.401119\pi$
$80$	1.08517	0.121326
$81$	10.7251	1.19168
$82$	5.28172	0.583269
$83$	4.01327	0.440513	0.220257	−	0.975442i	$-0.429311\pi$
$83$	4.01327	0.440513	0.220257	+	0.975442i	$0.429311\pi$
$84$	10.2417	1.11747
$85$	−10.5304	−1.14219
$86$	23.6758	2.55302
$87$	23.0990	2.47647
$88$	−3.78879	−0.403886
$89$	−10.7483	−1.13932	−0.569661	−	0.821880i	$-0.692925\pi$
$89$	−10.7483	−1.13932	−0.569661	+	0.821880i	$0.692925\pi$
$90$	−23.6336	−2.49120
$91$	5.12534	0.537282
$92$	18.0513	1.88198
$93$	−32.5460	−3.37486
$94$	13.7139	1.41448
$95$	10.0249	1.02854
$96$	14.7541	1.50584
$97$	−6.33525	−0.643247	−0.321624	−	0.946868i	$-0.604229\pi$
$97$	−6.33525	−0.643247	−0.321624	+	0.946868i	$0.604229\pi$
$98$	−2.31924	−0.234278
$99$	7.33113	0.736807
$100$	−7.73054	−0.773054
$101$	5.01109	0.498622	0.249311	−	0.968424i	$-0.419796\pi$
$101$	5.01109	0.498622	0.249311	+	0.968424i	$0.419796\pi$
$102$	44.9515	4.45087
$103$	−6.24112	−0.614955	−0.307478	−	0.951555i	$-0.599485\pi$
$103$	−6.24112	−0.614955	−0.307478	+	0.951555i	$0.599485\pi$
$104$	−16.3902	−1.60720
$105$	4.99179	0.487149
$106$	−13.6724	−1.32798
$107$	11.2229	1.08496	0.542478	−	0.840070i	$-0.317486\pi$
$107$	11.2229	1.08496	0.542478	+	0.840070i	$0.317486\pi$
$108$	32.6482	3.14158
$109$	−13.0611	−1.25103	−0.625515	−	0.780212i	$-0.715111\pi$
$109$	−13.0611	−1.25103	−0.625515	+	0.780212i	$0.715111\pi$
$110$	−4.52516	−0.431457
$111$	23.5176	2.23219
$112$	0.658941	0.0622640
$113$	2.34478	0.220578	0.110289	−	0.993900i	$-0.464822\pi$
$113$	2.34478	0.220578	0.110289	+	0.993900i	$0.464822\pi$
$114$	−42.7937	−4.00800
$115$	8.79814	0.820431
$116$	25.7489	2.39072
$117$	31.7144	2.93200
$118$	−15.6147	−1.43745
$119$	−6.39433	−0.586167
$120$	−15.9632	−1.45723
$121$	−9.59630	−0.872391
$122$	−10.4914	−0.949848
$123$	−6.90296	−0.622419
$124$	−36.2796	−3.25800
$125$	−12.0020	−1.07349
$126$	−14.3509	−1.27848
$127$	−14.0360	−1.24549	−0.622747	−	0.782423i	$-0.713983\pi$
$127$	−14.0360	−1.24549	−0.622747	+	0.782423i	$0.713983\pi$
$128$	19.5031	1.72385
$129$	−30.9431	−2.72439
$130$	−19.5758	−1.71691
$131$	3.60801	0.315234	0.157617	−	0.987500i	$-0.449619\pi$
$131$	3.60801	0.315234	0.157617	+	0.987500i	$0.449619\pi$
$132$	12.1342	1.05615
$133$	6.08738	0.527842
$134$	0.466112	0.0402659
$135$	15.9126	1.36954
$136$	20.4483	1.75343
$137$	9.12559	0.779651	0.389826	−	0.920889i	$-0.372535\pi$
$137$	9.12559	0.779651	0.389826	+	0.920889i	$0.372535\pi$
$138$	−37.5569	−3.19705
$139$	−14.4230	−1.22334	−0.611670	−	0.791113i	$-0.709502\pi$
$139$	−14.4230	−1.22334	−0.611670	+	0.791113i	$0.709502\pi$
$140$	5.56443	0.470280
$141$	−17.9234	−1.50942
$142$	−17.8651	−1.49921
$143$	6.07240	0.507800
$144$	4.07736	0.339780
$145$	12.5499	1.04221
$146$	5.91051	0.489157
$147$	3.03113	0.250004
$148$	26.2155	2.15490
$149$	−7.11121	−0.582573	−0.291286	−	0.956636i	$-0.594083\pi$
$149$	−7.11121	−0.582573	−0.291286	+	0.956636i	$0.594083\pi$
$150$	16.0838	1.31324
$151$	−17.0462	−1.38720	−0.693601	−	0.720360i	$-0.743977\pi$
$151$	−17.0462	−1.38720	−0.693601	+	0.720360i	$0.743977\pi$
$152$	−19.4667	−1.57896
$153$	−39.5666	−3.19877
$154$	−2.74778	−0.221423
$155$	−17.6825	−1.42029
$156$	52.4925	4.20276
$157$	19.5705	1.56190	0.780948	−	0.624596i	$-0.214736\pi$
$157$	19.5705	1.56190	0.780948	+	0.624596i	$0.214736\pi$
$158$	−12.6021	−1.00257
$159$	17.8692	1.41712
$160$	8.01604	0.633724
$161$	5.34244	0.421043
$162$	−24.8740	−1.95428
$163$	14.7845	1.15801	0.579006	−	0.815323i	$-0.303441\pi$
$163$	14.7845	1.15801	0.579006	+	0.815323i	$0.303441\pi$
$164$	−7.69485	−0.600867
$165$	5.91417	0.460418
$166$	−9.30771	−0.722418
$167$	7.23587	0.559928	0.279964	−	0.960010i	$-0.409677\pi$
$167$	7.23587	0.559928	0.279964	+	0.960010i	$0.409677\pi$
$168$	−9.69321	−0.747847
$169$	13.2691	1.02070
$170$	24.4226	1.87313
$171$	37.6672	2.88048
$172$	−34.4928	−2.63005
$173$	−17.1092	−1.30079	−0.650394	−	0.759597i	$-0.725396\pi$
$173$	−17.1092	−1.30079	−0.650394	+	0.759597i	$0.725396\pi$
$174$	−53.5721	−4.06129
$175$	−2.28792	−0.172950
$176$	0.780700	0.0588474
$177$	20.4077	1.53394
$178$	24.9279	1.86843
$179$	1.24969	0.0934065	0.0467032	−	0.998909i	$-0.485128\pi$
$179$	1.24969	0.0934065	0.0467032	+	0.998909i	$0.485128\pi$
$180$	34.4313	2.56636
$181$	17.9073	1.33104	0.665521	−	0.746379i	$-0.268209\pi$
$181$	17.9073	1.33104	0.665521	+	0.746379i	$0.268209\pi$
$182$	−11.8869	−0.881114
$183$	13.7118	1.01360
$184$	−17.0845	−1.25949
$185$	12.7773	0.939408
$186$	75.4818	5.53460
$187$	−7.57587	−0.554003
$188$	−19.9795	−1.45715
$189$	9.66251	0.702845
$190$	−23.2502	−1.68675
$191$	13.4239	0.971323	0.485661	−	0.874147i	$-0.338579\pi$
$191$	13.4239	0.971323	0.485661	+	0.874147i	$0.338579\pi$
$192$	−38.2130	−2.75778
$193$	−11.6318	−0.837278	−0.418639	−	0.908153i	$-0.637493\pi$
$193$	−11.6318	−0.837278	−0.418639	+	0.908153i	$0.637493\pi$
$194$	14.6929	1.05489
$195$	25.5846	1.83215
$196$	3.37885	0.241347
$197$	−3.38305	−0.241032	−0.120516	−	0.992711i	$-0.538455\pi$
$197$	−3.38305	−0.241032	−0.120516	+	0.992711i	$0.538455\pi$
$198$	−17.0026	−1.20832
$199$	27.2690	1.93305	0.966526	−	0.256570i	$-0.0825925\pi$
$199$	27.2690	1.93305	0.966526	+	0.256570i	$0.0825925\pi$
$200$	7.31650	0.517354
$201$	−0.609186	−0.0429687
$202$	−11.6219	−0.817713
$203$	7.62059	0.534861
$204$	−65.4891	−4.58515
$205$	−3.75044	−0.261942
$206$	14.4746	1.00849
$207$	33.0577	2.29767
$208$	3.37730	0.234173
$209$	7.21220	0.498878
$210$	−11.5771	−0.798898
$211$	7.66551	0.527716	0.263858	−	0.964562i	$-0.415005\pi$
$211$	7.66551	0.527716	0.263858	+	0.964562i	$0.415005\pi$
$212$	19.9191	1.36805
$213$	23.3489	1.59984
$214$	−26.0285	−1.77927
$215$	−16.8117	−1.14655
$216$	−30.8996	−2.10245
$217$	−10.7372	−0.728891
$218$	30.2918	2.05162
$219$	−7.72476	−0.521991
$220$	6.59263	0.444475
$221$	−32.7731	−2.20456
$222$	−54.5429	−3.66068
$223$	−15.3162	−1.02565	−0.512825	−	0.858493i	$-0.671401\pi$
$223$	−15.3162	−1.02565	−0.512825	+	0.858493i	$0.671401\pi$
$224$	4.86753	0.325226
$225$	−14.1571	−0.943805
$226$	−5.43809	−0.361736
$227$	20.9724	1.39199	0.695994	−	0.718047i	$-0.254964\pi$
$227$	20.9724	1.39199	0.695994	+	0.718047i	$0.254964\pi$
$228$	62.3454	4.12892
$229$	26.3555	1.74162	0.870810	−	0.491620i	$-0.163595\pi$
$229$	26.3555	1.74162	0.870810	+	0.491620i	$0.163595\pi$
$230$	−20.4050	−1.34546
$231$	3.59122	0.236285
$232$	−24.3698	−1.59995
$233$	0.159976	0.0104804	0.00524019	−	0.999986i	$-0.498332\pi$
$233$	0.159976	0.0104804	0.00524019	+	0.999986i	$0.498332\pi$
$234$	−73.5531	−4.80832
$235$	−9.73792	−0.635232
$236$	22.7488	1.48082
$237$	16.4703	1.06986
$238$	14.8300	0.961283
$239$	−5.14720	−0.332945	−0.166472	−	0.986046i	$-0.553238\pi$
$239$	−5.14720	−0.332945	−0.166472	+	0.986046i	$0.553238\pi$
$240$	3.28929	0.212323
$241$	−15.1274	−0.974444	−0.487222	−	0.873278i	$-0.661990\pi$
$241$	−15.1274	−0.974444	−0.487222	+	0.873278i	$0.661990\pi$
$242$	22.2561	1.43067
$243$	3.52158	0.225909
$244$	15.2848	0.978507
$245$	1.64684	0.105213
$246$	16.0096	1.02073
$247$	31.1999	1.98520
$248$	34.3365	2.18037
$249$	12.1647	0.770909
$250$	27.8355	1.76047
$251$	−20.8228	−1.31433	−0.657163	−	0.753749i	$-0.728244\pi$
$251$	−20.8228	−1.31433	−0.657163	+	0.753749i	$0.728244\pi$
$252$	20.9075	1.31705
$253$	6.32962	0.397940
$254$	32.5528	2.04254
$255$	−31.9191	−1.99885
$256$	−20.0187	−1.25117
$257$	−27.7150	−1.72882	−0.864408	−	0.502791i	$-0.832306\pi$
$257$	−27.7150	−1.72882	−0.864408	+	0.502791i	$0.832306\pi$
$258$	71.7644	4.46785
$259$	7.75870	0.482102
$260$	28.5196	1.76871
$261$	47.1544	2.91878
$262$	−8.36783	−0.516967
$263$	20.3314	1.25368	0.626842	−	0.779146i	$-0.284347\pi$
$263$	20.3314	1.25368	0.626842	+	0.779146i	$0.284347\pi$
$264$	−11.4843	−0.706811
$265$	9.70851	0.596389
$266$	−14.1181	−0.865634
$267$	−32.5796	−1.99384
$268$	−0.679070	−0.0414808
$269$	−2.93400	−0.178889	−0.0894447	−	0.995992i	$-0.528509\pi$
$269$	−2.93400	−0.178889	−0.0894447	+	0.995992i	$0.528509\pi$
$270$	−36.9051	−2.24597
$271$	−14.1314	−0.858419	−0.429209	−	0.903205i	$-0.641208\pi$
$271$	−14.1314	−0.858419	−0.429209	+	0.903205i	$0.641208\pi$
$272$	−4.21348	−0.255480
$273$	15.5356	0.940256
$274$	−21.1644	−1.27859
$275$	−2.71068	−0.163460
$276$	54.7159	3.29351
$277$	−19.4136	−1.16645	−0.583226	−	0.812310i	$-0.698210\pi$
$277$	−19.4136	−1.16645	−0.583226	+	0.812310i	$0.698210\pi$
$278$	33.4502	2.00621
$279$	−66.4395	−3.97763
$280$	−5.26640	−0.314728
$281$	−25.2475	−1.50614	−0.753069	−	0.657941i	$-0.771428\pi$
$281$	−25.2475	−1.50614	−0.753069	+	0.657941i	$0.771428\pi$
$282$	41.5685	2.47537
$283$	−22.5169	−1.33849	−0.669244	−	0.743042i	$-0.733382\pi$
$283$	−22.5169	−1.33849	−0.669244	+	0.743042i	$0.733382\pi$
$284$	26.0274	1.54444
$285$	30.3869	1.79996
$286$	−14.0833	−0.832765
$287$	−2.27736	−0.134428
$288$	30.1191	1.77478
$289$	23.8875	1.40514
$290$	−29.1062	−1.70917
$291$	−19.2030	−1.12570
$292$	−8.61092	−0.503916
$293$	30.0825	1.75744	0.878718	−	0.477341i	$-0.158399\pi$
$293$	30.0825	1.75744	0.878718	+	0.477341i	$0.158399\pi$
$294$	−7.02991	−0.409993
$295$	11.0877	0.645550
$296$	−24.8114	−1.44213
$297$	11.4480	0.664278
$298$	16.4926	0.955389
$299$	27.3818	1.58353
$300$	−23.4323	−1.35286
$301$	−10.2084	−0.588404
$302$	39.5342	2.27494
$303$	15.1893	0.872600
$304$	4.01122	0.230059
$305$	7.44973	0.426571
$306$	91.7642	5.24581
$307$	5.68328	0.324362	0.162181	−	0.986761i	$-0.448147\pi$
$307$	5.68328	0.324362	0.162181	+	0.986761i	$0.448147\pi$
$308$	4.00320	0.228103
$309$	−18.9176	−1.07619
$310$	41.0099	2.32921
$311$	−2.28904	−0.129799	−0.0648997	−	0.997892i	$-0.520673\pi$
$311$	−2.28904	−0.129799	−0.0648997	+	0.997892i	$0.520673\pi$
$312$	−49.6810	−2.81263
$313$	4.65374	0.263045	0.131522	−	0.991313i	$-0.458013\pi$
$313$	4.65374	0.263045	0.131522	+	0.991313i	$0.458013\pi$
$314$	−45.3886	−2.56143
$315$	10.1902	0.574155
$316$	18.3597	1.03282
$317$	17.2308	0.967777	0.483889	−	0.875130i	$-0.339224\pi$
$317$	17.2308	0.967777	0.483889	+	0.875130i	$0.339224\pi$
$318$	−41.4430	−2.32401
$319$	9.02873	0.505512
$320$	−20.7614	−1.16060
$321$	34.0180	1.89870
$322$	−12.3904	−0.690489
$323$	−38.9247	−2.16583
$324$	36.2385	2.01325
$325$	−11.7264	−0.650461
$326$	−34.2887	−1.89908
$327$	−39.5900	−2.18933
$328$	7.28272	0.402121
$329$	−5.91309	−0.325999
$330$	−13.7164	−0.755061
$331$	22.4834	1.23580	0.617900	−	0.786257i	$-0.287984\pi$
$331$	22.4834	1.23580	0.617900	+	0.786257i	$0.287984\pi$
$332$	13.5602	0.744215
$333$	48.0089	2.63087
$334$	−16.7817	−0.918253
$335$	−0.330976	−0.0180832
$336$	1.99734	0.108964
$337$	−28.8038	−1.56904	−0.784521	−	0.620102i	$-0.787091\pi$
$337$	−28.8038	−1.56904	−0.784521	+	0.620102i	$0.787091\pi$
$338$	−30.7742	−1.67390
$339$	7.10732	0.386017
$340$	−35.5808	−1.92964
$341$	−12.7213	−0.688895
$342$	−87.3591	−4.72384
$343$	1.00000	0.0539949
$344$	32.6454	1.76012
$345$	26.6683	1.43577
$346$	39.6802	2.13322
$347$	−10.8197	−0.580834	−0.290417	−	0.956900i	$-0.593794\pi$
$347$	−10.8197	−0.580834	−0.290417	+	0.956900i	$0.593794\pi$
$348$	78.0482	4.18382
$349$	17.8549	0.955752	0.477876	−	0.878427i	$-0.341407\pi$
$349$	17.8549	0.955752	0.477876	+	0.878427i	$0.341407\pi$
$350$	5.30622	0.283629
$351$	49.5237	2.64338
$352$	5.76695	0.307380
$353$	−12.8811	−0.685594	−0.342797	−	0.939410i	$-0.611374\pi$
$353$	−12.8811	−0.685594	−0.342797	+	0.939410i	$0.611374\pi$
$354$	−47.3302	−2.51557
$355$	12.6856	0.673284
$356$	−36.3171	−1.92480
$357$	−19.3821	−1.02581
$358$	−2.89833	−0.153182
$359$	−30.1383	−1.59064	−0.795319	−	0.606191i	$-0.792697\pi$
$359$	−30.1383	−1.59064	−0.795319	+	0.606191i	$0.792697\pi$
$360$	−32.5872	−1.71750
$361$	18.0561	0.950323
$362$	−41.5313	−2.18284
$363$	−29.0876	−1.52670
$364$	17.3178	0.907698
$365$	−4.19693	−0.219677
$366$	−31.8009	−1.66226
$367$	−19.1210	−0.998110	−0.499055	−	0.866570i	$-0.666319\pi$
$367$	−19.1210	−0.998110	−0.499055	+	0.866570i	$0.666319\pi$
$368$	3.52035	0.183511
$369$	−14.0917	−0.733586
$370$	−29.6336	−1.54058
$371$	5.89523	0.306065
$372$	−109.968	−5.70158
$373$	−0.983259	−0.0509112	−0.0254556	−	0.999676i	$-0.508104\pi$
$373$	−0.983259	−0.0509112	−0.0254556	+	0.999676i	$0.508104\pi$
$374$	17.5702	0.908535
$375$	−36.3797	−1.87864
$376$	18.9094	0.975177
$377$	39.0581	2.01160
$378$	−22.4096	−1.15263
$379$	−31.5450	−1.62036	−0.810178	−	0.586184i	$-0.800629\pi$
$379$	−31.5450	−1.62036	−0.810178	+	0.586184i	$0.800629\pi$
$380$	33.8728	1.73764
$381$	−42.5450	−2.17965
$382$	−31.1333	−1.59292
$383$	15.8326	0.809007	0.404504	−	0.914536i	$-0.367444\pi$
$383$	15.8326	0.809007	0.404504	+	0.914536i	$0.367444\pi$
$384$	59.1166	3.01678
$385$	1.95114	0.0994395
$386$	26.9770	1.37309
$387$	−63.1673	−3.21098
$388$	−21.4059	−1.08672
$389$	−1.70292	−0.0863416	−0.0431708	−	0.999068i	$-0.513746\pi$
$389$	−1.70292	−0.0863416	−0.0431708	+	0.999068i	$0.513746\pi$
$390$	−59.3368	−3.00463
$391$	−34.1613	−1.72761
$392$	−3.19788	−0.161518
$393$	10.9364	0.551667
$394$	7.84608	0.395280
$395$	8.94846	0.450246
$396$	24.7708	1.24478
$397$	−4.08442	−0.204991	−0.102496	−	0.994733i	$-0.532683\pi$
$397$	−4.08442	−0.204991	−0.102496	+	0.994733i	$0.532683\pi$
$398$	−63.2433	−3.17010
$399$	18.4516	0.923737
$400$	−1.50760	−0.0753801
$401$	−31.7083	−1.58344	−0.791719	−	0.610885i	$-0.790814\pi$
$401$	−31.7083	−1.58344	−0.791719	+	0.610885i	$0.790814\pi$
$402$	1.41285	0.0704664
$403$	−55.0320	−2.74134
$404$	16.9317	0.842385
$405$	17.6625	0.877656
$406$	−17.6739	−0.877143
$407$	9.19235	0.455648
$408$	61.9816	3.06854
$409$	−0.419241	−0.0207301	−0.0103651	−	0.999946i	$-0.503299\pi$
$409$	−0.419241	−0.0207301	−0.0103651	+	0.999946i	$0.503299\pi$
$410$	8.69815	0.429571
$411$	27.6609	1.36441
$412$	−21.0878	−1.03892
$413$	6.73270	0.331294
$414$	−76.6687	−3.76806
$415$	6.60921	0.324433
$416$	24.9478	1.22316
$417$	−43.7179	−2.14087
$418$	−16.7268	−0.818134
$419$	7.30320	0.356785	0.178392	−	0.983959i	$-0.442910\pi$
$419$	7.30320	0.356785	0.178392	+	0.983959i	$0.442910\pi$
$420$	16.8665	0.823002
$421$	11.3649	0.553893	0.276947	−	0.960885i	$-0.410678\pi$
$421$	11.3649	0.553893	0.276947	+	0.960885i	$0.410678\pi$
$422$	−17.7781	−0.865426
$423$	−36.5888	−1.77901
$424$	−18.8523	−0.915548
$425$	14.6297	0.709645
$426$	−54.1515	−2.62365
$427$	4.52365	0.218915
$428$	37.9204	1.83295
$429$	18.4063	0.888662
$430$	38.9902	1.88027
$431$	−10.1198	−0.487454	−0.243727	−	0.969844i	$-0.578370\pi$
$431$	−10.1198	−0.487454	−0.243727	+	0.969844i	$0.578370\pi$
$432$	6.36702	0.306334
$433$	3.84773	0.184910	0.0924550	−	0.995717i	$-0.470529\pi$
$433$	3.84773	0.184910	0.0924550	+	0.995717i	$0.470529\pi$
$434$	24.9022	1.19534
$435$	38.0404	1.82390
$436$	−44.1316	−2.11352
$437$	32.5214	1.55571
$438$	17.9155	0.856037
$439$	20.1952	0.963864	0.481932	−	0.876209i	$-0.339935\pi$
$439$	20.1952	0.963864	0.481932	+	0.876209i	$0.339935\pi$
$440$	−6.23953	−0.297458
$441$	6.18776	0.294655
$442$	76.0086	3.61536
$443$	−6.50705	−0.309159	−0.154580	−	0.987980i	$-0.549402\pi$
$443$	−6.50705	−0.309159	−0.154580	+	0.987980i	$0.549402\pi$
$444$	79.4626	3.77113
$445$	−17.7008	−0.839098
$446$	35.5219	1.68201
$447$	−21.5550	−1.01952
$448$	−12.6068	−0.595617
$449$	1.79501	0.0847117	0.0423558	−	0.999103i	$-0.486514\pi$
$449$	1.79501	0.0847117	0.0423558	+	0.999103i	$0.486514\pi$
$450$	32.8336	1.54779
$451$	−2.69817	−0.127052
$452$	7.92265	0.372650
$453$	−51.6693	−2.42764
$454$	−48.6400	−2.28279
$455$	8.44062	0.395702
$456$	−59.0062	−2.76322
$457$	−36.3229	−1.69911	−0.849557	−	0.527498i	$-0.823130\pi$
$457$	−36.3229	−1.69911	−0.849557	+	0.527498i	$0.823130\pi$
$458$	−61.1246	−2.85616
$459$	−61.7853	−2.88389
$460$	29.7276	1.38606
$461$	6.84375	0.318745	0.159373	−	0.987218i	$-0.449053\pi$
$461$	6.84375	0.318745	0.159373	+	0.987218i	$0.449053\pi$
$462$	−8.32889	−0.387495
$463$	−33.6672	−1.56465	−0.782324	−	0.622872i	$-0.785966\pi$
$463$	−33.6672	−1.56465	−0.782324	+	0.622872i	$0.785966\pi$
$464$	5.02152	0.233118
$465$	−53.5980	−2.48555
$466$	−0.371022	−0.0171873
$467$	35.8138	1.65727	0.828633	−	0.559792i	$-0.189119\pi$
$467$	35.8138	1.65727	0.828633	+	0.559792i	$0.189119\pi$
$468$	107.158	4.95339
$469$	−0.200977	−0.00928023
$470$	22.5845	1.04175
$471$	59.3208	2.73336
$472$	−21.5304	−0.991017
$473$	−12.0948	−0.556117
$474$	−38.1985	−1.75452
$475$	−13.9274	−0.639034
$476$	−21.6055	−0.990286
$477$	36.4783	1.67023
$478$	11.9376	0.546012
$479$	0.588134	0.0268725	0.0134363	−	0.999910i	$-0.495723\pi$
$479$	0.588134	0.0268725	0.0134363	+	0.999910i	$0.495723\pi$
$480$	24.2977	1.10903
$481$	39.7660	1.81317
$482$	35.0841	1.59804
$483$	16.1936	0.736836
$484$	−32.4245	−1.47384
$485$	−10.4331	−0.473745
$486$	−8.16737	−0.370479
$487$	−3.94727	−0.178868	−0.0894340	−	0.995993i	$-0.528506\pi$
$487$	−3.94727	−0.178868	−0.0894340	+	0.995993i	$0.528506\pi$
$488$	−14.4661	−0.654851
$489$	44.8138	2.02655
$490$	−3.81941	−0.172543
$491$	−13.3365	−0.601868	−0.300934	−	0.953645i	$-0.597298\pi$
$491$	−13.3365	−0.601868	−0.300934	+	0.953645i	$0.597298\pi$
$492$	−23.3241	−1.05153
$493$	−48.7286	−2.19463
$494$	−72.3599	−3.25562
$495$	12.0732	0.542650
$496$	−7.07520	−0.317686
$497$	7.70302	0.345528
$498$	−28.2129	−1.26425
$499$	21.3624	0.956312	0.478156	−	0.878275i	$-0.341305\pi$
$499$	21.3624	0.956312	0.478156	+	0.878275i	$0.341305\pi$
$500$	−40.5531	−1.81359
$501$	21.9329	0.979888
$502$	48.2930	2.15542
$503$	0.837181	0.0373280	0.0186640	−	0.999826i	$-0.494059\pi$
$503$	0.837181	0.0373280	0.0186640	+	0.999826i	$0.494059\pi$
$504$	−19.7877	−0.881416
$505$	8.25246	0.367230
$506$	−14.6799	−0.652600
$507$	40.2205	1.78625
$508$	−47.4256	−2.10417
$509$	7.41482	0.328656	0.164328	−	0.986406i	$-0.447454\pi$
$509$	7.41482	0.328656	0.164328	+	0.986406i	$0.447454\pi$
$510$	74.0280	3.27802
$511$	−2.54847	−0.112738
$512$	7.42184	0.328002
$513$	58.8194	2.59694
$514$	64.2777	2.83517
$515$	−10.2781	−0.452908
$516$	−104.552	−4.60265
$517$	−7.00571	−0.308111
$518$	−17.9942	−0.790622
$519$	−51.8602	−2.27641
$520$	−26.9921	−1.18368
$521$	14.7867	0.647815	0.323908	−	0.946089i	$-0.395003\pi$
$521$	14.7867	0.647815	0.323908	+	0.946089i	$0.395003\pi$
$522$	−109.362	−4.78665
$523$	6.56218	0.286944	0.143472	−	0.989654i	$-0.454173\pi$
$523$	6.56218	0.286944	0.143472	+	0.989654i	$0.454173\pi$
$524$	12.1909	0.532564
$525$	−6.93498	−0.302667
$526$	−47.1532	−2.05598
$527$	68.6575	2.99077
$528$	2.36640	0.102984
$529$	5.54167	0.240942
$530$	−22.5163	−0.978046
$531$	41.6603	1.80790
$532$	20.5683	0.891751
$533$	−11.6722	−0.505580
$534$	75.5598	3.26979
$535$	18.4823	0.799059
$536$	0.642700	0.0277604
$537$	3.78799	0.163464
$538$	6.80465	0.293369
$539$	1.18478	0.0510321
$540$	53.7664	2.31374
$541$	14.9973	0.644784	0.322392	−	0.946606i	$-0.395513\pi$
$541$	14.9973	0.644784	0.322392	+	0.946606i	$0.395513\pi$
$542$	32.7739	1.40776
$543$	54.2795	2.32936
$544$	−31.1246	−1.33446
$545$	−21.5096	−0.921370
$546$	−36.0307	−1.54197
$547$	−31.7740	−1.35856	−0.679279	−	0.733880i	$-0.737707\pi$
$547$	−31.7740	−1.35856	−0.679279	+	0.733880i	$0.737707\pi$
$548$	30.8340	1.31716
$549$	27.9913	1.19464
$550$	6.28670	0.268066
$551$	46.3894	1.97626
$552$	−51.7854	−2.20413
$553$	5.43372	0.231065
$554$	45.0248	1.91292
$555$	38.7298	1.64399
$556$	−48.7331	−2.06674
$557$	−3.74638	−0.158739	−0.0793696	−	0.996845i	$-0.525291\pi$
$557$	−3.74638	−0.158739	−0.0793696	+	0.996845i	$0.525291\pi$
$558$	154.089	6.52310
$559$	−52.3217	−2.21297
$560$	1.08517	0.0458568
$561$	−22.9635	−0.969519
$562$	58.5549	2.46999
$563$	−5.16990	−0.217885	−0.108943	−	0.994048i	$-0.534747\pi$
$563$	−5.16990	−0.217885	−0.108943	+	0.994048i	$0.534747\pi$
$564$	−60.5604	−2.55005
$565$	3.86147	0.162453
$566$	52.2219	2.19505
$567$	10.7251	0.450411
$568$	−24.6334	−1.03359
$569$	10.9253	0.458013	0.229007	−	0.973425i	$-0.426452\pi$
$569$	10.9253	0.458013	0.229007	+	0.973425i	$0.426452\pi$
$570$	−70.4744	−2.95185
$571$	−4.07348	−0.170470	−0.0852350	−	0.996361i	$-0.527164\pi$
$571$	−4.07348	−0.170470	−0.0852350	+	0.996361i	$0.527164\pi$
$572$	20.5178	0.857890
$573$	40.6897	1.69984
$574$	5.28172	0.220455
$575$	−12.2231	−0.509737
$576$	−78.0080	−3.25033
$577$	17.5048	0.728733	0.364366	−	0.931256i	$-0.381286\pi$
$577$	17.5048	0.728733	0.364366	+	0.931256i	$0.381286\pi$
$578$	−55.4006	−2.30436
$579$	−35.2576	−1.46526
$580$	42.4043	1.76074
$581$	4.01327	0.166498
$582$	44.5362	1.84609
$583$	6.98456	0.289271
$584$	8.14972	0.337238
$585$	52.2285	2.15938
$586$	−69.7683	−2.88210
$587$	−7.45946	−0.307885	−0.153943	−	0.988080i	$-0.549197\pi$
$587$	−7.45946	−0.307885	−0.153943	+	0.988080i	$0.549197\pi$
$588$	10.2417	0.422363
$589$	−65.3616	−2.69318
$590$	−25.7149	−1.05867
$591$	−10.2545	−0.421812
$592$	5.11252	0.210123
$593$	35.2783	1.44871	0.724353	−	0.689429i	$-0.242139\pi$
$593$	35.2783	1.44871	0.724353	+	0.689429i	$0.242139\pi$
$594$	−26.5505	−1.08938
$595$	−10.5304	−0.431706
$596$	−24.0277	−0.984214
$597$	82.6561	3.38289
$598$	−63.5049	−2.59691
$599$	−29.7386	−1.21509	−0.607543	−	0.794287i	$-0.707845\pi$
$599$	−29.7386	−1.21509	−0.607543	+	0.794287i	$0.707845\pi$
$600$	22.1773	0.905383
$601$	−15.0481	−0.613825	−0.306913	−	0.951738i	$-0.599296\pi$
$601$	−15.0481	−0.613825	−0.306913	+	0.951738i	$0.599296\pi$
$602$	23.6758	0.964952
$603$	−1.24359	−0.0506431
$604$	−57.5966	−2.34357
$605$	−15.8036	−0.642506
$606$	−35.2275	−1.43102
$607$	5.36717	0.217847	0.108923	−	0.994050i	$-0.465260\pi$
$607$	5.36717	0.217847	0.108923	+	0.994050i	$0.465260\pi$
$608$	29.6305	1.20167
$609$	23.0990	0.936020
$610$	−17.2777	−0.699553
$611$	−30.3066	−1.22607
$612$	−133.690	−5.40408
$613$	24.4031	0.985631	0.492815	−	0.870134i	$-0.335968\pi$
$613$	24.4031	0.985631	0.492815	+	0.870134i	$0.335968\pi$
$614$	−13.1809	−0.531936
$615$	−11.3681	−0.458405
$616$	−3.78879	−0.152655
$617$	35.1737	1.41604	0.708019	−	0.706193i	$-0.249589\pi$
$617$	35.1737	1.41604	0.708019	+	0.706193i	$0.249589\pi$
$618$	43.8745	1.76489
$619$	−6.19250	−0.248898	−0.124449	−	0.992226i	$-0.539716\pi$
$619$	−6.19250	−0.248898	−0.124449	+	0.992226i	$0.539716\pi$
$620$	−59.7466	−2.39948
$621$	51.6214	2.07150
$622$	5.30882	0.212864
$623$	−10.7483	−0.430623
$624$	10.2370	0.409809
$625$	−8.32584	−0.333034
$626$	−10.7931	−0.431380
$627$	21.8611	0.873049
$628$	66.1259	2.63871
$629$	−49.6117	−1.97815
$630$	−23.6336	−0.941585
$631$	−0.275267	−0.0109582	−0.00547911	−	0.999985i	$-0.501744\pi$
$631$	−0.275267	−0.0109582	−0.00547911	+	0.999985i	$0.501744\pi$
$632$	−17.3764	−0.691196
$633$	23.2352	0.923516
$634$	−39.9623	−1.58710
$635$	−23.1151	−0.917293
$636$	60.3775	2.39412
$637$	5.12534	0.203073
$638$	−20.9397	−0.829012
$639$	47.6644	1.88558
$640$	32.1186	1.26960
$641$	−21.9951	−0.868752	−0.434376	−	0.900732i	$-0.643031\pi$
$641$	−21.9951	−0.868752	−0.434376	+	0.900732i	$0.643031\pi$
$642$	−78.8958	−3.11377
$643$	24.9184	0.982684	0.491342	−	0.870967i	$-0.336507\pi$
$643$	24.9184	0.982684	0.491342	+	0.870967i	$0.336507\pi$
$644$	18.0513	0.711322
$645$	−50.9584	−2.00648
$646$	90.2755	3.55184
$647$	6.26837	0.246435	0.123217	−	0.992380i	$-0.460679\pi$
$647$	6.26837	0.246435	0.123217	+	0.992380i	$0.460679\pi$
$648$	−34.2976	−1.34734
$649$	7.97677	0.313115
$650$	27.1962	1.06672
$651$	−32.5460	−1.27558
$652$	49.9547	1.95638
$653$	−2.31739	−0.0906863	−0.0453431	−	0.998971i	$-0.514438\pi$
$653$	−2.31739	−0.0906863	−0.0453431	+	0.998971i	$0.514438\pi$
$654$	91.8185	3.59039
$655$	5.94182	0.232166
$656$	−1.50064	−0.0585902
$657$	−15.7693	−0.615221
$658$	13.7139	0.534622
$659$	−23.0499	−0.897896	−0.448948	−	0.893558i	$-0.648201\pi$
$659$	−23.0499	−0.897896	−0.448948	+	0.893558i	$0.648201\pi$
$660$	19.9831	0.777842
$661$	34.4082	1.33832	0.669162	−	0.743116i	$-0.266653\pi$
$661$	34.4082	1.33832	0.669162	+	0.743116i	$0.266653\pi$
$662$	−52.1443	−2.02665
$663$	−99.3397	−3.85803
$664$	−12.8340	−0.498054
$665$	10.0249	0.388750
$666$	−111.344	−4.31449
$667$	40.7126	1.57640
$668$	24.4489	0.945958
$669$	−46.4255	−1.79491
$670$	0.767612	0.0296554
$671$	5.35953	0.206903
$672$	14.7541	0.569153
$673$	−28.2861	−1.09035	−0.545175	−	0.838322i	$-0.683537\pi$
$673$	−28.2861	−1.09035	−0.545175	+	0.838322i	$0.683537\pi$
$674$	66.8027	2.57315
$675$	−22.1070	−0.850900
$676$	44.8344	1.72440
$677$	−7.83501	−0.301124	−0.150562	−	0.988601i	$-0.548108\pi$
$677$	−7.83501	−0.301124	−0.150562	+	0.988601i	$0.548108\pi$
$678$	−16.4836	−0.633047
$679$	−6.33525	−0.243125
$680$	33.6751	1.29138
$681$	63.5702	2.43601
$682$	29.5036	1.12975
$683$	49.3746	1.88927	0.944634	−	0.328126i	$-0.106417\pi$
$683$	49.3746	1.88927	0.944634	+	0.328126i	$0.106417\pi$
$684$	127.272	4.86636
$685$	15.0284	0.574205
$686$	−2.31924	−0.0885488
$687$	79.8870	3.04788
$688$	−6.72675	−0.256455
$689$	30.2151	1.15110
$690$	−61.8501	−2.35459
$691$	−32.1810	−1.22422	−0.612111	−	0.790772i	$-0.709679\pi$
$691$	−32.1810	−1.22422	−0.612111	+	0.790772i	$0.709679\pi$
$692$	−57.8094	−2.19758
$693$	7.33113	0.278487
$694$	25.0935	0.952538
$695$	−23.7523	−0.900976
$696$	−73.8680	−2.79996
$697$	14.5622	0.551581
$698$	−41.4098	−1.56738
$699$	0.484908	0.0183409
$700$	−7.73054	−0.292187
$701$	−14.1053	−0.532748	−0.266374	−	0.963870i	$-0.585826\pi$
$701$	−14.1053	−0.532748	−0.266374	+	0.963870i	$0.585826\pi$
$702$	−114.857	−4.33500
$703$	47.2301	1.78132
$704$	−14.9363	−0.562934
$705$	−29.5169	−1.11167
$706$	29.8744	1.12434
$707$	5.01109	0.188461
$708$	68.9546	2.59147
$709$	−43.3902	−1.62955	−0.814776	−	0.579775i	$-0.803140\pi$
$709$	−43.3902	−1.62955	−0.814776	+	0.579775i	$0.803140\pi$
$710$	−29.4210	−1.10415
$711$	33.6225	1.26094
$712$	34.3719	1.28814
$713$	−57.3631	−2.14826
$714$	44.9515	1.68227
$715$	10.0003	0.373989
$716$	4.22253	0.157803
$717$	−15.6019	−0.582662
$718$	69.8978	2.60856
$719$	7.94878	0.296439	0.148220	−	0.988954i	$-0.452646\pi$
$719$	7.94878	0.296439	0.148220	+	0.988954i	$0.452646\pi$
$720$	6.71477	0.250245
$721$	−6.24112	−0.232431
$722$	−41.8764	−1.55848
$723$	−45.8533	−1.70530
$724$	60.5063	2.24870
$725$	−17.4353	−0.647530
$726$	67.4611	2.50372
$727$	−41.6980	−1.54649	−0.773247	−	0.634106i	$-0.781368\pi$
$727$	−41.6980	−1.54649	−0.773247	+	0.634106i	$0.781368\pi$
$728$	−16.3902	−0.607463
$729$	−21.5009	−0.796328
$730$	9.73367	0.360259
$731$	65.2761	2.41432
$732$	46.3301	1.71241
$733$	5.45241	0.201389	0.100695	−	0.994917i	$-0.467894\pi$
$733$	5.45241	0.201389	0.100695	+	0.994917i	$0.467894\pi$
$734$	44.3462	1.63685
$735$	4.99179	0.184125
$736$	26.0045	0.958538
$737$	−0.238113	−0.00877100
$738$	32.6820	1.20304
$739$	−24.5090	−0.901577	−0.450789	−	0.892631i	$-0.648857\pi$
$739$	−24.5090	−0.901577	−0.450789	+	0.892631i	$0.648857\pi$
$740$	43.1727	1.58706
$741$	94.5709	3.47415
$742$	−13.6724	−0.501931
$743$	48.5534	1.78125	0.890626	−	0.454737i	$-0.150267\pi$
$743$	48.5534	1.78125	0.890626	+	0.454737i	$0.150267\pi$
$744$	104.078	3.81570
$745$	−11.7110	−0.429059
$746$	2.28041	0.0834917
$747$	24.8331	0.908597
$748$	−25.5978	−0.935947
$749$	11.2229	0.410075
$750$	84.3732	3.08087
$751$	44.2012	1.61292	0.806462	−	0.591286i	$-0.201380\pi$
$751$	44.2012	1.61292	0.806462	+	0.591286i	$0.201380\pi$
$752$	−3.89638	−0.142086
$753$	−63.1167	−2.30010
$754$	−90.5850	−3.29891
$755$	−28.0724	−1.02166
$756$	32.6482	1.18740
$757$	10.4023	0.378077	0.189038	−	0.981970i	$-0.439463\pi$
$757$	10.4023	0.378077	0.189038	+	0.981970i	$0.439463\pi$
$758$	73.1602	2.65730
$759$	19.1859	0.696404
$760$	−32.0586	−1.16289
$761$	−0.343218	−0.0124417	−0.00622083	−	0.999981i	$-0.501980\pi$
$761$	−0.343218	−0.0124417	−0.00622083	+	0.999981i	$0.501980\pi$
$762$	98.6718	3.57450
$763$	−13.0611	−0.472845
$764$	45.3575	1.64098
$765$	−65.1598	−2.35586
$766$	−36.7195	−1.32673
$767$	34.5074	1.24599
$768$	−60.6794	−2.18958
$769$	44.3665	1.59990	0.799948	−	0.600069i	$-0.204860\pi$
$769$	44.3665	1.59990	0.799948	+	0.600069i	$0.204860\pi$
$770$	−4.52516	−0.163075
$771$	−84.0079	−3.02547
$772$	−39.3023	−1.41452
$773$	12.9482	0.465713	0.232857	−	0.972511i	$-0.425193\pi$
$773$	12.9482	0.465713	0.232857	+	0.972511i	$0.425193\pi$
$774$	146.500	5.26583
$775$	24.5659	0.882434
$776$	20.2594	0.727270
$777$	23.5176	0.843690
$778$	3.94948	0.141596
$779$	−13.8631	−0.496698
$780$	86.4467	3.09529
$781$	9.12639	0.326568
$782$	79.2282	2.83319
$783$	73.6341	2.63147
$784$	0.658941	0.0235336
$785$	32.2295	1.15032
$786$	−25.3640	−0.904704
$787$	−13.2914	−0.473789	−0.236894	−	0.971535i	$-0.576129\pi$
$787$	−13.2914	−0.473789	−0.236894	+	0.971535i	$0.576129\pi$
$788$	−11.4308	−0.407206
$789$	61.6270	2.19398
$790$	−20.7536	−0.738380
$791$	2.34478	0.0833706
$792$	−23.4441	−0.833050
$793$	23.1853	0.823333
$794$	9.47274	0.336175
$795$	29.4278	1.04370
$796$	92.1381	3.26575
$797$	6.29706	0.223053	0.111527	−	0.993761i	$-0.464426\pi$
$797$	6.29706	0.223053	0.111527	+	0.993761i	$0.464426\pi$
$798$	−42.7937	−1.51488
$799$	37.8103	1.33763
$800$	−11.1365	−0.393735
$801$	−66.5081	−2.34995
$802$	73.5391	2.59675
$803$	−3.01938	−0.106552
$804$	−2.05835	−0.0725924
$805$	8.79814	0.310094
$806$	127.632	4.49565
$807$	−8.89335	−0.313061
$808$	−16.0249	−0.563753
$809$	−4.36705	−0.153537	−0.0767687	−	0.997049i	$-0.524460\pi$
$809$	−4.36705	−0.153537	−0.0767687	+	0.997049i	$0.524460\pi$
$810$	−40.9635	−1.43931
$811$	29.7660	1.04523	0.522613	−	0.852570i	$-0.324957\pi$
$811$	29.7660	1.04523	0.522613	+	0.852570i	$0.324957\pi$
$812$	25.7489	0.903608
$813$	−42.8340	−1.50225
$814$	−21.3192	−0.747238
$815$	24.3477	0.852863
$816$	−12.7716	−0.447096
$817$	−62.1426	−2.17409
$818$	0.972318	0.0339963
$819$	31.7144	1.10819
$820$	−12.6722	−0.442532
$821$	−4.31901	−0.150735	−0.0753673	−	0.997156i	$-0.524013\pi$
$821$	−4.31901	−0.150735	−0.0753673	+	0.997156i	$0.524013\pi$
$822$	−64.1520	−2.23756
$823$	5.15417	0.179663	0.0898315	−	0.995957i	$-0.471367\pi$
$823$	5.15417	0.179663	0.0898315	+	0.995957i	$0.471367\pi$
$824$	19.9584	0.695283
$825$	−8.21643	−0.286059
$826$	−15.6147	−0.543305
$827$	−7.72863	−0.268751	−0.134375	−	0.990931i	$-0.542903\pi$
$827$	−7.72863	−0.268751	−0.134375	+	0.990931i	$0.542903\pi$
$828$	111.697	3.88175
$829$	1.29865	0.0451040	0.0225520	−	0.999746i	$-0.492821\pi$
$829$	1.29865	0.0451040	0.0225520	+	0.999746i	$0.492821\pi$
$830$	−15.3283	−0.532053
$831$	−58.8452	−2.04132
$832$	−64.6143	−2.24010
$833$	−6.39433	−0.221550
$834$	101.392	3.51092
$835$	11.9163	0.412381
$836$	24.3690	0.842818
$837$	−103.749	−3.58608
$838$	−16.9378	−0.585108
$839$	57.2421	1.97622	0.988108	−	0.153761i	$-0.0491386\pi$
$839$	57.2421	1.97622	0.988108	+	0.153761i	$0.0491386\pi$
$840$	−15.9632	−0.550781
$841$	29.0734	1.00253
$842$	−26.3580	−0.908355
$843$	−76.5284	−2.63578
$844$	25.9006	0.891537
$845$	21.8521	0.751736
$846$	84.8580	2.91748
$847$	−9.59630	−0.329733
$848$	3.88461	0.133398
$849$	−68.2516	−2.34239
$850$	−33.9297	−1.16378
$851$	41.4504	1.42090
$852$	78.8924	2.70281
$853$	−37.9841	−1.30055	−0.650275	−	0.759699i	$-0.725346\pi$
$853$	−37.9841	−1.30055	−0.650275	+	0.759699i	$0.725346\pi$
$854$	−10.4914	−0.359009
$855$	62.0319	2.12145
$856$	−35.8894	−1.22668
$857$	42.2812	1.44430	0.722149	−	0.691738i	$-0.243155\pi$
$857$	42.2812	1.44430	0.722149	+	0.691738i	$0.243155\pi$
$858$	−42.6884	−1.45736
$859$	−1.00000	−0.0341196
$859$	−1.00000	−0.0341196
$860$	−56.8041	−1.93700
$861$	−6.90296	−0.235252
$862$	23.4702	0.799399
$863$	−15.7211	−0.535151	−0.267576	−	0.963537i	$-0.586223\pi$
$863$	−15.7211	−0.535151	−0.267576	+	0.963537i	$0.586223\pi$
$864$	47.0326	1.60008
$865$	−28.1761	−0.958016
$866$	−8.92378	−0.303242
$867$	72.4060	2.45904
$868$	−36.2796	−1.23141
$869$	6.43776	0.218386
$870$	−88.2246	−2.99109
$871$	−1.03007	−0.0349027
$872$	41.7680	1.41444
$873$	−39.2010	−1.32675
$874$	−75.4249	−2.55128
$875$	−12.0020	−0.405743
$876$	−26.1008	−0.881865
$877$	13.1111	0.442731	0.221365	−	0.975191i	$-0.428949\pi$
$877$	13.1111	0.442731	0.221365	+	0.975191i	$0.428949\pi$
$878$	−46.8374	−1.58069
$879$	91.1839	3.07556
$880$	1.28569	0.0433405
$881$	−32.5518	−1.09670	−0.548349	−	0.836249i	$-0.684744\pi$
$881$	−32.5518	−1.09670	−0.548349	+	0.836249i	$0.684744\pi$
$882$	−14.3509	−0.483219
$883$	5.76480	0.194001	0.0970005	−	0.995284i	$-0.469075\pi$
$883$	5.76480	0.194001	0.0970005	+	0.995284i	$0.469075\pi$
$884$	−110.736	−3.72444
$885$	33.6082	1.12973
$886$	15.0914	0.507005
$887$	15.1647	0.509181	0.254591	−	0.967049i	$-0.418059\pi$
$887$	15.1647	0.509181	0.254591	+	0.967049i	$0.418059\pi$
$888$	−75.2067	−2.52377
$889$	−14.0360	−0.470753
$890$	41.0523	1.37608
$891$	12.7069	0.425696
$892$	−51.7513	−1.73276
$893$	−35.9952	−1.20453
$894$	49.9911	1.67195
$895$	2.05805	0.0687929
$896$	19.5031	0.651554
$897$	82.9979	2.77122
$898$	−4.16305	−0.138923
$899$	−81.8242	−2.72899
$900$	−47.8347	−1.59449
$901$	−37.6961	−1.25584
$902$	6.25768	0.208358
$903$	−30.9431	−1.02972
$904$	−7.49832	−0.249390
$905$	29.4905	0.980298
$906$	119.833	3.98120
$907$	7.77814	0.258269	0.129134	−	0.991627i	$-0.458780\pi$
$907$	7.77814	0.258269	0.129134	+	0.991627i	$0.458780\pi$
$908$	70.8627	2.35166
$909$	31.0074	1.02845
$910$	−19.5758	−0.648931
$911$	−6.93044	−0.229616	−0.114808	−	0.993388i	$-0.536625\pi$
$911$	−6.93044	−0.229616	−0.114808	+	0.993388i	$0.536625\pi$
$912$	12.1585	0.402609
$913$	4.75484	0.157362
$914$	84.2413	2.78646
$915$	22.5811	0.746509
$916$	89.0513	2.94234
$917$	3.60801	0.119147
$918$	143.295	4.72943
$919$	−41.7806	−1.37822	−0.689108	−	0.724659i	$-0.741998\pi$
$919$	−41.7806	−1.37822	−0.689108	+	0.724659i	$0.741998\pi$
$920$	−28.1354	−0.927598
$921$	17.2268	0.567641
$922$	−15.8723	−0.522725
$923$	39.4806	1.29952
$924$	12.1342	0.399186
$925$	−17.7513	−0.583658
$926$	78.0822	2.56594
$927$	−38.6185	−1.26840
$928$	37.0935	1.21765
$929$	−16.8225	−0.551928	−0.275964	−	0.961168i	$-0.588997\pi$
$929$	−16.8225	−0.551928	−0.275964	+	0.961168i	$0.588997\pi$
$930$	124.306	4.07617
$931$	6.08738	0.199506
$932$	0.540535	0.0177058
$933$	−6.93838	−0.227152
$934$	−83.0607	−2.71783
$935$	−12.4763	−0.408017
$936$	−101.419	−3.31498
$937$	39.5364	1.29160	0.645799	−	0.763508i	$-0.276524\pi$
$937$	39.5364	1.29160	0.645799	+	0.763508i	$0.276524\pi$
$938$	0.466112	0.0152191
$939$	14.1061	0.460335
$940$	−32.9030	−1.07318
$941$	−3.46219	−0.112864	−0.0564320	−	0.998406i	$-0.517972\pi$
$941$	−3.46219	−0.112864	−0.0564320	+	0.998406i	$0.517972\pi$
$942$	−137.579	−4.48256
$943$	−12.1666	−0.396200
$944$	4.43645	0.144394
$945$	15.9126	0.517637
$946$	28.0506	0.912003
$947$	39.0736	1.26972	0.634861	−	0.772626i	$-0.281057\pi$
$947$	39.0736	1.26972	0.634861	+	0.772626i	$0.281057\pi$
$948$	55.6508	1.80745
$949$	−13.0618	−0.424004
$950$	32.3010	1.04798
$951$	52.2288	1.69363
$952$	20.4483	0.662734
$953$	−23.2234	−0.752278	−0.376139	−	0.926563i	$-0.622749\pi$
$953$	−23.2234	−0.752278	−0.376139	+	0.926563i	$0.622749\pi$
$954$	−84.6017	−2.73908
$955$	22.1071	0.715369
$956$	−17.3916	−0.562486
$957$	27.3673	0.884658
$958$	−1.36402	−0.0440695
$959$	9.12559	0.294681
$960$	−62.9306	−2.03108
$961$	84.2884	2.71898
$962$	−92.2267	−2.97351
$963$	69.4444	2.23782
$964$	−51.1134	−1.64625
$965$	−19.1558	−0.616646
$966$	−37.5569	−1.20837
$967$	23.8746	0.767755	0.383877	−	0.923384i	$-0.374589\pi$
$967$	23.8746	0.767755	0.383877	+	0.923384i	$0.374589\pi$
$968$	30.6878	0.986345
$969$	−117.986	−3.79025
$970$	24.1969	0.776916
$971$	30.0242	0.963523	0.481762	−	0.876302i	$-0.339997\pi$
$971$	30.0242	0.963523	0.481762	+	0.876302i	$0.339997\pi$
$972$	11.8989	0.381657
$973$	−14.4230	−0.462379
$974$	9.15465	0.293334
$975$	−35.5441	−1.13832
$976$	2.98082	0.0954137
$977$	44.4253	1.42129	0.710646	−	0.703550i	$-0.248403\pi$
$977$	44.4253	1.42129	0.710646	+	0.703550i	$0.248403\pi$
$978$	−103.934	−3.32343
$979$	−12.7344	−0.406994
$980$	5.56443	0.177749
$981$	−80.8191	−2.58036
$982$	30.9305	0.987031
$983$	−28.4577	−0.907659	−0.453830	−	0.891089i	$-0.649942\pi$
$983$	−28.4577	−0.907659	−0.453830	+	0.891089i	$0.649942\pi$
$984$	22.0749	0.703721
$985$	−5.57134	−0.177518
$986$	113.013	3.59907
$987$	−17.9234	−0.570507
$988$	105.420	3.35385
$989$	−54.5380	−1.73421
$990$	−28.0006	−0.889917
$991$	−29.1700	−0.926617	−0.463308	−	0.886197i	$-0.653338\pi$
$991$	−29.1700	−0.926617	−0.463308	+	0.886197i	$0.653338\pi$
$992$	−52.2638	−1.65938
$993$	68.1502	2.16268
$994$	−17.8651	−0.566647
$995$	44.9078	1.42367
$996$	41.1029	1.30239
$997$	23.5922	0.747173	0.373587	−	0.927595i	$-0.378128\pi$
$997$	23.5922	0.747173	0.373587	+	0.927595i	$0.378128\pi$
$998$	−49.5444	−1.56830
$999$	74.9685	2.37190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.e.1.10	✓	109

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.e.1.10	✓	109	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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31924 q^{2} +3.03113 q^{3} +3.37885 q^{4} +1.64684 q^{5} -7.02991 q^{6} +1.00000 q^{7} -3.19788 q^{8} +6.18776 q^{9} +O(q^{10})\)	\(q-2.31924 q^{2} +3.03113 q^{3} +3.37885 q^{4} +1.64684 q^{5} -7.02991 q^{6} +1.00000 q^{7} -3.19788 q^{8} +6.18776 q^{9} -3.81941 q^{10} +1.18478 q^{11} +10.2417 q^{12} +5.12534 q^{13} -2.31924 q^{14} +4.99179 q^{15} +0.658941 q^{16} -6.39433 q^{17} -14.3509 q^{18} +6.08738 q^{19} +5.56443 q^{20} +3.03113 q^{21} -2.74778 q^{22} +5.34244 q^{23} -9.69321 q^{24} -2.28792 q^{25} -11.8869 q^{26} +9.66251 q^{27} +3.37885 q^{28} +7.62059 q^{29} -11.5771 q^{30} -10.7372 q^{31} +4.86753 q^{32} +3.59122 q^{33} +14.8300 q^{34} +1.64684 q^{35} +20.9075 q^{36} +7.75870 q^{37} -14.1181 q^{38} +15.5356 q^{39} -5.26640 q^{40} -2.27736 q^{41} -7.02991 q^{42} -10.2084 q^{43} +4.00320 q^{44} +10.1902 q^{45} -12.3904 q^{46} -5.91309 q^{47} +1.99734 q^{48} +1.00000 q^{49} +5.30622 q^{50} -19.3821 q^{51} +17.3178 q^{52} +5.89523 q^{53} -22.4096 q^{54} +1.95114 q^{55} -3.19788 q^{56} +18.4516 q^{57} -17.6739 q^{58} +6.73270 q^{59} +16.8665 q^{60} +4.52365 q^{61} +24.9022 q^{62} +6.18776 q^{63} -12.6068 q^{64} +8.44062 q^{65} -8.32889 q^{66} -0.200977 q^{67} -21.6055 q^{68} +16.1936 q^{69} -3.81941 q^{70} +7.70302 q^{71} -19.7877 q^{72} -2.54847 q^{73} -17.9942 q^{74} -6.93498 q^{75} +20.5683 q^{76} +1.18478 q^{77} -36.0307 q^{78} +5.43372 q^{79} +1.08517 q^{80} +10.7251 q^{81} +5.28172 q^{82} +4.01327 q^{83} +10.2417 q^{84} -10.5304 q^{85} +23.6758 q^{86} +23.0990 q^{87} -3.78879 q^{88} -10.7483 q^{89} -23.6336 q^{90} +5.12534 q^{91} +18.0513 q^{92} -32.5460 q^{93} +13.7139 q^{94} +10.0249 q^{95} +14.7541 q^{96} -6.33525 q^{97} -2.31924 q^{98} +7.33113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more