LMFDB - Embedded newform 151.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [151,2,Mod(1,151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	151.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:	$1.20574107052$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.4838537.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 $ x^6 - x^5 - 7x^4 + 3x^3 + 13x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.183668$ of defining polynomial
Character	$\chi$	$=$	151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.183668 q^{2} -3.29466 q^{3} -1.96627 q^{4} +3.71611 q^{5} -0.605125 q^{6} +3.65107 q^{7} -0.728478 q^{8} +7.85477 q^{9} +O(q^{10})$	\(q+0.183668 q^{2} -3.29466 q^{3} -1.96627 q^{4} +3.71611 q^{5} -0.605125 q^{6} +3.65107 q^{7} -0.728478 q^{8} +7.85477 q^{9} +0.682533 q^{10} -0.912147 q^{11} +6.47817 q^{12} -1.27865 q^{13} +0.670586 q^{14} -12.2433 q^{15} +3.79873 q^{16} +1.65822 q^{17} +1.44267 q^{18} +3.74742 q^{19} -7.30687 q^{20} -12.0290 q^{21} -0.167533 q^{22} -0.248713 q^{23} +2.40009 q^{24} +8.80951 q^{25} -0.234848 q^{26} -15.9948 q^{27} -7.17897 q^{28} +1.93860 q^{29} -2.24871 q^{30} -5.44267 q^{31} +2.15466 q^{32} +3.00521 q^{33} +0.304563 q^{34} +13.5678 q^{35} -15.4446 q^{36} -4.41526 q^{37} +0.688284 q^{38} +4.21272 q^{39} -2.70711 q^{40} +8.64880 q^{41} -2.20935 q^{42} +1.41210 q^{43} +1.79352 q^{44} +29.1892 q^{45} -0.0456807 q^{46} +2.19095 q^{47} -12.5155 q^{48} +6.33031 q^{49} +1.61803 q^{50} -5.46328 q^{51} +2.51417 q^{52} -10.4493 q^{53} -2.93774 q^{54} -3.38964 q^{55} -2.65972 q^{56} -12.3465 q^{57} +0.356059 q^{58} -0.862276 q^{59} +24.0736 q^{60} -2.48235 q^{61} -0.999648 q^{62} +28.6783 q^{63} -7.20172 q^{64} -4.75161 q^{65} +0.551962 q^{66} -13.4435 q^{67} -3.26051 q^{68} +0.819424 q^{69} +2.49198 q^{70} +8.78436 q^{71} -5.72203 q^{72} -14.3562 q^{73} -0.810944 q^{74} -29.0243 q^{75} -7.36843 q^{76} -3.33031 q^{77} +0.773744 q^{78} -10.1620 q^{79} +14.1165 q^{80} +29.1331 q^{81} +1.58851 q^{82} +7.06074 q^{83} +23.6523 q^{84} +6.16215 q^{85} +0.259358 q^{86} -6.38702 q^{87} +0.664479 q^{88} +11.7770 q^{89} +5.36114 q^{90} -4.66844 q^{91} +0.489036 q^{92} +17.9317 q^{93} +0.402409 q^{94} +13.9259 q^{95} -7.09888 q^{96} -0.860838 q^{97} +1.16268 q^{98} -7.16470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 5 q^{3} + 3 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) $ 6 * q + q^2 - 5 * q^3 + 3 * q^4 + 6 * q^5 - 2 * q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 6 q + q^{2} - 5 q^{3} + 3 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} + 8 q^{10} + 8 q^{11} - 11 q^{12} - q^{13} + 6 q^{14} - 18 q^{15} - 3 q^{16} + 9 q^{17} - 16 q^{18} - 6 q^{19} + 3 q^{20} + 13 q^{21} - 12 q^{22} - 4 q^{23} + q^{24} - 4 q^{25} - 7 q^{26} - 2 q^{27} - 24 q^{28} - 2 q^{29} - 16 q^{30} - 8 q^{31} - 11 q^{32} + 3 q^{33} - 9 q^{34} + 5 q^{35} - 25 q^{36} - 12 q^{37} - 3 q^{38} - 22 q^{39} + 9 q^{40} + 41 q^{41} - 24 q^{42} + q^{43} + 12 q^{45} - 17 q^{46} + 28 q^{47} + 9 q^{48} + 33 q^{49} + 15 q^{50} - 31 q^{51} + 15 q^{52} + 14 q^{53} + 27 q^{54} + q^{55} - 9 q^{56} - 28 q^{57} + q^{58} + 12 q^{59} + 17 q^{60} + 5 q^{61} - 9 q^{62} + 4 q^{63} - 27 q^{64} + 13 q^{65} + 8 q^{66} - 15 q^{67} + 11 q^{68} + 33 q^{69} - 33 q^{70} - 2 q^{71} + q^{72} - 7 q^{73} + 53 q^{74} - 32 q^{75} - 3 q^{76} - 15 q^{77} + 16 q^{78} - 9 q^{79} + 16 q^{80} + 66 q^{81} + 10 q^{82} - 11 q^{83} + 49 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 24 q^{88} + 36 q^{89} + 19 q^{90} - 35 q^{91} - 38 q^{92} - q^{93} + 42 q^{94} + 32 q^{95} - 34 q^{96} + 11 q^{97} + 15 q^{98} + 17 q^{99}+O(q^{100}) \) 6 * q + q^2 - 5 * q^3 + 3 * q^4 + 6 * q^5 - 2 * q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 + 8 * q^10 + 8 * q^11 - 11 * q^12 - q^13 + 6 * q^14 - 18 * q^15 - 3 * q^16 + 9 * q^17 - 16 * q^18 - 6 * q^19 + 3 * q^20 + 13 * q^21 - 12 * q^22 - 4 * q^23 + q^24 - 4 * q^25 - 7 * q^26 - 2 * q^27 - 24 * q^28 - 2 * q^29 - 16 * q^30 - 8 * q^31 - 11 * q^32 + 3 * q^33 - 9 * q^34 + 5 * q^35 - 25 * q^36 - 12 * q^37 - 3 * q^38 - 22 * q^39 + 9 * q^40 + 41 * q^41 - 24 * q^42 + q^43 + 12 * q^45 - 17 * q^46 + 28 * q^47 + 9 * q^48 + 33 * q^49 + 15 * q^50 - 31 * q^51 + 15 * q^52 + 14 * q^53 + 27 * q^54 + q^55 - 9 * q^56 - 28 * q^57 + q^58 + 12 * q^59 + 17 * q^60 + 5 * q^61 - 9 * q^62 + 4 * q^63 - 27 * q^64 + 13 * q^65 + 8 * q^66 - 15 * q^67 + 11 * q^68 + 33 * q^69 - 33 * q^70 - 2 * q^71 + q^72 - 7 * q^73 + 53 * q^74 - 32 * q^75 - 3 * q^76 - 15 * q^77 + 16 * q^78 - 9 * q^79 + 16 * q^80 + 66 * q^81 + 10 * q^82 - 11 * q^83 + 49 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 24 * q^88 + 36 * q^89 + 19 * q^90 - 35 * q^91 - 38 * q^92 - q^93 + 42 * q^94 + 32 * q^95 - 34 * q^96 + 11 * q^97 + 15 * q^98 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.183668	0.129873	0.0649366	−	0.997889i	$-0.479315\pi$
$2$	0.183668	0.129873	0.0649366	+	0.997889i	$0.479315\pi$
$3$	−3.29466	−1.90217	−0.951086	−	0.308927i	$-0.900030\pi$
$3$	−3.29466	−1.90217	−0.951086	+	0.308927i	$0.900030\pi$
$4$	−1.96627	−0.983133
$5$	3.71611	1.66190	0.830948	−	0.556349i	$-0.187798\pi$
$5$	3.71611	1.66190	0.830948	+	0.556349i	$0.187798\pi$
$6$	−0.605125	−0.247041
$7$	3.65107	1.37997	0.689987	−	0.723821i	$-0.257616\pi$
$7$	3.65107	1.37997	0.689987	+	0.723821i	$0.257616\pi$
$8$	−0.728478	−0.257556
$9$	7.85477	2.61826
$10$	0.682533	0.215836
$11$	−0.912147	−0.275023	−0.137511	−	0.990500i	$-0.543910\pi$
$11$	−0.912147	−0.275023	−0.137511	+	0.990500i	$0.543910\pi$
$12$	6.47817	1.87009
$13$	−1.27865	−0.354634	−0.177317	−	0.984154i	$-0.556742\pi$
$13$	−1.27865	−0.354634	−0.177317	+	0.984154i	$0.556742\pi$
$14$	0.670586	0.179222
$15$	−12.2433	−3.16121
$16$	3.79873	0.949683
$17$	1.65822	0.402178	0.201089	−	0.979573i	$-0.435552\pi$
$17$	1.65822	0.402178	0.201089	+	0.979573i	$0.435552\pi$
$18$	1.44267	0.340041
$19$	3.74742	0.859718	0.429859	−	0.902896i	$-0.358563\pi$
$19$	3.74742	0.859718	0.429859	+	0.902896i	$0.358563\pi$
$20$	−7.30687	−1.63387
$21$	−12.0290	−2.62495
$22$	−0.167533	−0.0357181
$23$	−0.248713	−0.0518602	−0.0259301	−	0.999664i	$-0.508255\pi$
$23$	−0.248713	−0.0518602	−0.0259301	+	0.999664i	$0.508255\pi$
$24$	2.40009	0.489915
$25$	8.80951	1.76190
$26$	−0.234848	−0.0460575
$27$	−15.9948	−3.07820
$28$	−7.17897	−1.35670
$29$	1.93860	0.359989	0.179994	−	0.983668i	$-0.442392\pi$
$29$	1.93860	0.359989	0.179994	+	0.983668i	$0.442392\pi$
$30$	−2.24871	−0.410557
$31$	−5.44267	−0.977533	−0.488767	−	0.872415i	$-0.662553\pi$
$31$	−5.44267	−0.977533	−0.488767	+	0.872415i	$0.662553\pi$
$32$	2.15466	0.380894
$33$	3.00521	0.523140
$34$	0.304563	0.0522322
$35$	13.5678	2.29338
$36$	−15.4446	−2.57409
$37$	−4.41526	−0.725864	−0.362932	−	0.931816i	$-0.618224\pi$
$37$	−4.41526	−0.725864	−0.362932	+	0.931816i	$0.618224\pi$
$38$	0.688284	0.111654
$39$	4.21272	0.674575
$40$	−2.70711	−0.428031
$41$	8.64880	1.35072	0.675358	−	0.737490i	$-0.263989\pi$
$41$	8.64880	1.35072	0.675358	+	0.737490i	$0.263989\pi$
$42$	−2.20935	−0.340911
$43$	1.41210	0.215343	0.107671	−	0.994187i	$-0.465661\pi$
$43$	1.41210	0.215343	0.107671	+	0.994187i	$0.465661\pi$
$44$	1.79352	0.270384
$45$	29.1892	4.35127
$46$	−0.0456807	−0.00673526
$47$	2.19095	0.319583	0.159791	−	0.987151i	$-0.448918\pi$
$47$	2.19095	0.319583	0.159791	+	0.987151i	$0.448918\pi$
$48$	−12.5155	−1.80646
$49$	6.33031	0.904330
$50$	1.61803	0.228824
$51$	−5.46328	−0.765012
$52$	2.51417	0.348652
$53$	−10.4493	−1.43532	−0.717661	−	0.696393i	$-0.754787\pi$
$53$	−10.4493	−1.43532	−0.717661	+	0.696393i	$0.754787\pi$
$54$	−2.93774	−0.399776
$55$	−3.38964	−0.457059
$56$	−2.65972	−0.355421
$57$	−12.3465	−1.63533
$58$	0.356059	0.0467529
$59$	−0.862276	−0.112259	−0.0561294	−	0.998424i	$-0.517876\pi$
$59$	−0.862276	−0.112259	−0.0561294	+	0.998424i	$0.517876\pi$
$60$	24.0736	3.10789
$61$	−2.48235	−0.317833	−0.158916	−	0.987292i	$-0.550800\pi$
$61$	−2.48235	−0.317833	−0.158916	+	0.987292i	$0.550800\pi$
$62$	−0.999648	−0.126955
$63$	28.6783	3.61313
$64$	−7.20172	−0.900215
$65$	−4.75161	−0.589365
$66$	0.551962	0.0679419
$67$	−13.4435	−1.64239	−0.821195	−	0.570647i	$-0.806692\pi$
$67$	−13.4435	−1.64239	−0.821195	+	0.570647i	$0.806692\pi$
$68$	−3.26051	−0.395395
$69$	0.819424	0.0986471
$70$	2.49198	0.297848
$71$	8.78436	1.04251	0.521256	−	0.853400i	$-0.325464\pi$
$71$	8.78436	1.04251	0.521256	+	0.853400i	$0.325464\pi$
$72$	−5.72203	−0.674347
$73$	−14.3562	−1.68026	−0.840131	−	0.542383i	$-0.817522\pi$
$73$	−14.3562	−1.68026	−0.840131	+	0.542383i	$0.817522\pi$
$74$	−0.810944	−0.0942703
$75$	−29.0243	−3.35144
$76$	−7.36843	−0.845217
$77$	−3.33031	−0.379524
$78$	0.773744	0.0876092
$79$	−10.1620	−1.14331	−0.571655	−	0.820494i	$-0.693698\pi$
$79$	−10.1620	−1.14331	−0.571655	+	0.820494i	$0.693698\pi$
$80$	14.1165	1.57828
$81$	29.1331	3.23701
$82$	1.58851	0.175422
$83$	7.06074	0.775017	0.387509	−	0.921866i	$-0.373336\pi$
$83$	7.06074	0.775017	0.387509	+	0.921866i	$0.373336\pi$
$84$	23.6523	2.58067
$85$	6.16215	0.668378
$86$	0.259358	0.0279673
$87$	−6.38702	−0.684760
$88$	0.664479	0.0708337
$89$	11.7770	1.24836	0.624180	−	0.781280i	$-0.285433\pi$
$89$	11.7770	1.24836	0.624180	+	0.781280i	$0.285433\pi$
$90$	5.36114	0.565114
$91$	−4.66844	−0.489386
$92$	0.489036	0.0509855
$93$	17.9317	1.85944
$94$	0.402409	0.0415053
$95$	13.9259	1.42876
$96$	−7.09888	−0.724526
$97$	−0.860838	−0.0874048	−0.0437024	−	0.999045i	$-0.513915\pi$
$97$	−0.860838	−0.0874048	−0.0437024	+	0.999045i	$0.513915\pi$
$98$	1.16268	0.117448
$99$	−7.16470	−0.720080
$100$	−17.3218	−1.73218
$101$	−7.72970	−0.769134	−0.384567	−	0.923097i	$-0.625649\pi$
$101$	−7.72970	−0.769134	−0.384567	+	0.923097i	$0.625649\pi$
$102$	−1.00343	−0.0993545
$103$	1.97534	0.194636	0.0973179	−	0.995253i	$-0.468974\pi$
$103$	1.97534	0.194636	0.0973179	+	0.995253i	$0.468974\pi$
$104$	0.931469	0.0913381
$105$	−44.7012	−4.36239
$106$	−1.91921	−0.186410
$107$	−0.319598	−0.0308967	−0.0154484	−	0.999881i	$-0.504918\pi$
$107$	−0.319598	−0.0308967	−0.0154484	+	0.999881i	$0.504918\pi$
$108$	31.4501	3.02628
$109$	12.6271	1.20945	0.604727	−	0.796433i	$-0.293282\pi$
$109$	12.6271	1.20945	0.604727	+	0.796433i	$0.293282\pi$
$110$	−0.622570	−0.0593597
$111$	14.5468	1.38072
$112$	13.8694	1.31054
$113$	−13.1810	−1.23997	−0.619984	−	0.784615i	$-0.712861\pi$
$113$	−13.1810	−1.23997	−0.619984	+	0.784615i	$0.712861\pi$
$114$	−2.26766	−0.212386
$115$	−0.924246	−0.0861864
$116$	−3.81180	−0.353917
$117$	−10.0435	−0.928523
$118$	−0.158373	−0.0145794
$119$	6.05429	0.554996
$120$	8.91899	0.814189
$121$	−10.1680	−0.924363
$122$	−0.455930	−0.0412780
$123$	−28.4948	−2.56929
$124$	10.7017	0.961045
$125$	14.1566	1.26620
$126$	5.26730	0.469249
$127$	0.830524	0.0736971	0.0368485	−	0.999321i	$-0.488268\pi$
$127$	0.830524	0.0736971	0.0368485	+	0.999321i	$0.488268\pi$
$128$	−5.63206	−0.497808
$129$	−4.65238	−0.409619
$130$	−0.872722	−0.0765428
$131$	9.31948	0.814247	0.407124	−	0.913373i	$-0.366532\pi$
$131$	9.31948	0.814247	0.407124	+	0.913373i	$0.366532\pi$
$132$	−5.90904	−0.514316
$133$	13.6821	1.18639
$134$	−2.46916	−0.213303
$135$	−59.4385	−5.11566
$136$	−1.20798	−0.103583
$137$	4.27610	0.365332	0.182666	−	0.983175i	$-0.441527\pi$
$137$	4.27610	0.365332	0.182666	+	0.983175i	$0.441527\pi$
$138$	0.150502	0.0128116
$139$	11.8522	1.00529	0.502645	−	0.864493i	$-0.332360\pi$
$139$	11.8522	1.00529	0.502645	+	0.864493i	$0.332360\pi$
$140$	−26.6779	−2.25469
$141$	−7.21843	−0.607902
$142$	1.61341	0.135394
$143$	1.16632	0.0975323
$144$	29.8382	2.48652
$145$	7.20405	0.598264
$146$	−2.63678	−0.218221
$147$	−20.8562	−1.72019
$148$	8.68158	0.713621
$149$	−21.2817	−1.74347	−0.871734	−	0.489979i	$-0.837004\pi$
$149$	−21.2817	−1.74347	−0.871734	+	0.489979i	$0.837004\pi$
$150$	−5.33085	−0.435262
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−2.72992	−0.221425
$153$	13.0250	1.05301
$154$	−0.611673	−0.0492900
$155$	−20.2256	−1.62456
$156$	−8.28332	−0.663197
$157$	−15.8157	−1.26223	−0.631115	−	0.775689i	$-0.717403\pi$
$157$	−15.8157	−1.26223	−0.631115	+	0.775689i	$0.717403\pi$
$158$	−1.86643	−0.148485
$159$	34.4269	2.73023
$160$	8.00698	0.633007
$161$	−0.908069	−0.0715658
$162$	5.35084	0.420401
$163$	−13.1212	−1.02773	−0.513866	−	0.857871i	$-0.671787\pi$
$163$	−13.1212	−1.02773	−0.513866	+	0.857871i	$0.671787\pi$
$164$	−17.0058	−1.32793
$165$	11.1677	0.869405
$166$	1.29684	0.100654
$167$	20.1890	1.56227	0.781134	−	0.624363i	$-0.214641\pi$
$167$	20.1890	1.56227	0.781134	+	0.624363i	$0.214641\pi$
$168$	8.76288	0.676071
$169$	−11.3651	−0.874235
$170$	1.13179	0.0868045
$171$	29.4352	2.25096
$172$	−2.77656	−0.211711
$173$	0.672028	0.0510933	0.0255467	−	0.999674i	$-0.491867\pi$
$173$	0.672028	0.0510933	0.0255467	+	0.999674i	$0.491867\pi$
$174$	−1.17309	−0.0889320
$175$	32.1641	2.43138
$176$	−3.46500	−0.261184
$177$	2.84091	0.213536
$178$	2.16307	0.162129
$179$	−2.07329	−0.154965	−0.0774823	−	0.996994i	$-0.524688\pi$
$179$	−2.07329	−0.154965	−0.0774823	+	0.996994i	$0.524688\pi$
$180$	−57.3938	−4.27788
$181$	2.78471	0.206985	0.103493	−	0.994630i	$-0.466998\pi$
$181$	2.78471	0.206985	0.103493	+	0.994630i	$0.466998\pi$
$182$	−0.857446	−0.0635581
$183$	8.17850	0.604572
$184$	0.181182	0.0133569
$185$	−16.4076	−1.20631
$186$	3.29350	0.241491
$187$	−1.51254	−0.110608
$188$	−4.30799	−0.314193
$189$	−58.3982	−4.24784
$190$	2.55774	0.185558
$191$	12.6722	0.916930	0.458465	−	0.888712i	$-0.348399\pi$
$191$	12.6722	0.916930	0.458465	+	0.888712i	$0.348399\pi$
$192$	23.7272	1.71236
$193$	−6.20943	−0.446964	−0.223482	−	0.974708i	$-0.571742\pi$
$193$	−6.20943	−0.446964	−0.223482	+	0.974708i	$0.571742\pi$
$194$	−0.158109	−0.0113515
$195$	15.6549	1.12107
$196$	−12.4471	−0.889077
$197$	−14.8505	−1.05805	−0.529027	−	0.848605i	$-0.677443\pi$
$197$	−14.8505	−1.05805	−0.529027	+	0.848605i	$0.677443\pi$
$198$	−1.31593	−0.0935191
$199$	15.0571	1.06737	0.533685	−	0.845683i	$-0.320807\pi$
$199$	15.0571	1.06737	0.533685	+	0.845683i	$0.320807\pi$
$200$	−6.41753	−0.453788
$201$	44.2919	3.12411
$202$	−1.41970	−0.0998899
$203$	7.07796	0.496775
$204$	10.7423	0.752108
$205$	32.1399	2.24475
$206$	0.362807	0.0252780
$207$	−1.95358	−0.135783
$208$	−4.85725	−0.336790
$209$	−3.41820	−0.236442
$210$	−8.21021	−0.566558
$211$	−14.1331	−0.972961	−0.486481	−	0.873691i	$-0.661720\pi$
$211$	−14.1331	−0.972961	−0.486481	+	0.873691i	$0.661720\pi$
$212$	20.5461	1.41111
$213$	−28.9415	−1.98304
$214$	−0.0587001	−0.00401266
$215$	5.24751	0.357878
$216$	11.6519	0.792809
$217$	−19.8716	−1.34897
$218$	2.31919	0.157076
$219$	47.2987	3.19615
$220$	6.66493	0.449350
$221$	−2.12029	−0.142626
$222$	2.67178	0.179318
$223$	−9.31277	−0.623629	−0.311815	−	0.950143i	$-0.600937\pi$
$223$	−9.31277	−0.623629	−0.311815	+	0.950143i	$0.600937\pi$
$224$	7.86683	0.525625
$225$	69.1967	4.61311
$226$	−2.42094	−0.161039
$227$	0.485548	0.0322270	0.0161135	−	0.999870i	$-0.494871\pi$
$227$	0.485548	0.0322270	0.0161135	+	0.999870i	$0.494871\pi$
$228$	24.2765	1.60775
$229$	15.3956	1.01737	0.508685	−	0.860953i	$-0.330132\pi$
$229$	15.3956	1.01737	0.508685	+	0.860953i	$0.330132\pi$
$230$	−0.169755	−0.0111933
$231$	10.9722	0.721920
$232$	−1.41223	−0.0927172
$233$	−20.5505	−1.34631	−0.673154	−	0.739502i	$-0.735061\pi$
$233$	−20.5505	−1.34631	−0.673154	+	0.739502i	$0.735061\pi$
$234$	−1.84468	−0.120590
$235$	8.14182	0.531114
$236$	1.69546	0.110365
$237$	33.4802	2.17477
$238$	1.11198	0.0720791
$239$	−2.85049	−0.184383	−0.0921915	−	0.995741i	$-0.529387\pi$
$239$	−2.85049	−0.184383	−0.0921915	+	0.995741i	$0.529387\pi$
$240$	−46.5091	−3.00215
$241$	11.0331	0.710707	0.355354	−	0.934732i	$-0.384360\pi$
$241$	11.0331	0.710707	0.355354	+	0.934732i	$0.384360\pi$
$242$	−1.86754	−0.120050
$243$	−47.9992	−3.07915
$244$	4.88097	0.312472
$245$	23.5242	1.50290
$246$	−5.23360	−0.333682
$247$	−4.79165	−0.304885
$248$	3.96487	0.251769
$249$	−23.2627	−1.47422
$250$	2.60011	0.164446
$251$	−14.4565	−0.912486	−0.456243	−	0.889855i	$-0.650805\pi$
$251$	−14.4565	−0.912486	−0.456243	+	0.889855i	$0.650805\pi$
$252$	−56.3892	−3.55219
$253$	0.226863	0.0142627
$254$	0.152541	0.00957127
$255$	−20.3022	−1.27137
$256$	13.3690	0.835563
$257$	28.0657	1.75069	0.875345	−	0.483500i	$-0.160635\pi$
$257$	28.0657	1.75069	0.875345	+	0.483500i	$0.160635\pi$
$258$	−0.854495	−0.0531985
$259$	−16.1204	−1.00167
$260$	9.34294	0.579424
$261$	15.2272	0.942543
$262$	1.71170	0.105749
$263$	−10.9878	−0.677537	−0.338768	−	0.940870i	$-0.610010\pi$
$263$	−10.9878	−0.677537	−0.338768	+	0.940870i	$0.610010\pi$
$264$	−2.18923	−0.134738
$265$	−38.8308	−2.38536
$266$	2.51297	0.154080
$267$	−38.8012	−2.37460
$268$	26.4336	1.61469
$269$	−14.6293	−0.891965	−0.445983	−	0.895042i	$-0.647146\pi$
$269$	−14.6293	−0.891965	−0.445983	+	0.895042i	$0.647146\pi$
$270$	−10.9170	−0.664387
$271$	−31.9043	−1.93805	−0.969025	−	0.246964i	$-0.920567\pi$
$271$	−31.9043	−1.93805	−0.969025	+	0.246964i	$0.920567\pi$
$272$	6.29915	0.381942
$273$	15.3809	0.930896
$274$	0.785384	0.0474468
$275$	−8.03556	−0.484562
$276$	−1.61121	−0.0969832
$277$	14.7225	0.884588	0.442294	−	0.896870i	$-0.354165\pi$
$277$	14.7225	0.884588	0.442294	+	0.896870i	$0.354165\pi$
$278$	2.17687	0.130560
$279$	−42.7510	−2.55943
$280$	−9.88384	−0.590672
$281$	27.3492	1.63152	0.815758	−	0.578394i	$-0.196320\pi$
$281$	27.3492	1.63152	0.815758	+	0.578394i	$0.196320\pi$
$282$	−1.32580	−0.0789502
$283$	26.2831	1.56237	0.781183	−	0.624302i	$-0.214616\pi$
$283$	26.2831	1.56237	0.781183	+	0.624302i	$0.214616\pi$
$284$	−17.2724	−1.02493
$285$	−45.8809	−2.71775
$286$	0.214216	0.0126668
$287$	31.5774	1.86395
$288$	16.9244	0.997279
$289$	−14.2503	−0.838253
$290$	1.32316	0.0776985
$291$	2.83617	0.166259
$292$	28.2280	1.65192
$293$	16.2252	0.947884	0.473942	−	0.880556i	$-0.342831\pi$
$293$	16.2252	0.947884	0.473942	+	0.880556i	$0.342831\pi$
$294$	−3.83063	−0.223407
$295$	−3.20432	−0.186563
$296$	3.21642	0.186951
$297$	14.5896	0.846575
$298$	−3.90879	−0.226430
$299$	0.318017	0.0183914
$300$	57.0695	3.29491
$301$	5.15567	0.297168
$302$	0.183668	0.0105689
$303$	25.4667	1.46302
$304$	14.2355	0.816460
$305$	−9.22471	−0.528205
$306$	2.39227	0.136757
$307$	0.0943867	0.00538693	0.00269347	−	0.999996i	$-0.499143\pi$
$307$	0.0943867	0.00538693	0.00269347	+	0.999996i	$0.499143\pi$
$308$	6.54828	0.373123
$309$	−6.50806	−0.370231
$310$	−3.71480	−0.210987
$311$	−13.8783	−0.786968	−0.393484	−	0.919332i	$-0.628730\pi$
$311$	−13.8783	−0.786968	−0.393484	+	0.919332i	$0.628730\pi$
$312$	−3.06887	−0.173741
$313$	−13.4879	−0.762379	−0.381190	−	0.924497i	$-0.624486\pi$
$313$	−13.4879	−0.762379	−0.381190	+	0.924497i	$0.624486\pi$
$314$	−2.90485	−0.163930
$315$	106.572	6.00465
$316$	19.9811	1.12403
$317$	−2.90664	−0.163253	−0.0816267	−	0.996663i	$-0.526012\pi$
$317$	−2.90664	−0.163253	−0.0816267	+	0.996663i	$0.526012\pi$
$318$	6.32313	0.354583
$319$	−1.76829	−0.0990050
$320$	−26.7624	−1.49607
$321$	1.05297	0.0587709
$322$	−0.166784	−0.00929449
$323$	6.21406	0.345760
$324$	−57.2835	−3.18241
$325$	−11.2643	−0.624830
$326$	−2.40995	−0.133475
$327$	−41.6019	−2.30059
$328$	−6.30046	−0.347885
$329$	7.99931	0.441016
$330$	2.05116	0.112912
$331$	−11.1032	−0.610288	−0.305144	−	0.952306i	$-0.598705\pi$
$331$	−11.1032	−0.610288	−0.305144	+	0.952306i	$0.598705\pi$
$332$	−13.8833	−0.761945
$333$	−34.6809	−1.90050
$334$	3.70808	0.202897
$335$	−49.9578	−2.72948
$336$	−45.6951	−2.49287
$337$	−0.880266	−0.0479512	−0.0239756	−	0.999713i	$-0.507632\pi$
$337$	−0.880266	−0.0479512	−0.0239756	+	0.999713i	$0.507632\pi$
$338$	−2.08740	−0.113540
$339$	43.4270	2.35863
$340$	−12.1164	−0.657105
$341$	4.96452	0.268844
$342$	5.40631	0.292340
$343$	−2.44508	−0.132022
$344$	−1.02868	−0.0554628
$345$	3.04507	0.163941
$346$	0.123430	0.00663565
$347$	34.5128	1.85275	0.926373	−	0.376608i	$-0.122910\pi$
$347$	34.5128	1.85275	0.926373	+	0.376608i	$0.122910\pi$
$348$	12.5586	0.673210
$349$	−28.8423	−1.54389	−0.771947	−	0.635687i	$-0.780717\pi$
$349$	−28.8423	−1.54389	−0.771947	+	0.635687i	$0.780717\pi$
$350$	5.90753	0.315771
$351$	20.4518	1.09164
$352$	−1.96537	−0.104755
$353$	23.9585	1.27518	0.637591	−	0.770375i	$-0.279931\pi$
$353$	23.9585	1.27518	0.637591	+	0.770375i	$0.279931\pi$
$354$	0.521785	0.0277326
$355$	32.6437	1.73255
$356$	−23.1567	−1.22730
$357$	−19.9468	−1.05570
$358$	−0.380797	−0.0201258
$359$	−1.51542	−0.0799807	−0.0399904	−	0.999200i	$-0.512733\pi$
$359$	−1.51542	−0.0799807	−0.0399904	+	0.999200i	$0.512733\pi$
$360$	−21.2637	−1.12070
$361$	−4.95681	−0.260885
$362$	0.511463	0.0268819
$363$	33.5000	1.75830
$364$	9.17940	0.481131
$365$	−53.3492	−2.79242
$366$	1.50213	0.0785178
$367$	−7.67846	−0.400812	−0.200406	−	0.979713i	$-0.564226\pi$
$367$	−7.67846	−0.400812	−0.200406	+	0.979713i	$0.564226\pi$
$368$	−0.944794	−0.0492508
$369$	67.9343	3.53652
$370$	−3.01356	−0.156668
$371$	−38.1511	−1.98071
$372$	−35.2586	−1.82807
$373$	23.5390	1.21880	0.609402	−	0.792861i	$-0.291409\pi$
$373$	23.5390	1.21880	0.609402	+	0.792861i	$0.291409\pi$
$374$	−0.277806	−0.0143650
$375$	−46.6410	−2.40853
$376$	−1.59606	−0.0823105
$377$	−2.47879	−0.127664
$378$	−10.7259	−0.551681
$379$	−23.0030	−1.18159	−0.590793	−	0.806823i	$-0.701185\pi$
$379$	−23.0030	−1.18159	−0.590793	+	0.806823i	$0.701185\pi$
$380$	−27.3819	−1.40466
$381$	−2.73629	−0.140184
$382$	2.32749	0.119085
$383$	20.8953	1.06770	0.533851	−	0.845579i	$-0.320744\pi$
$383$	20.8953	1.06770	0.533851	+	0.845579i	$0.320744\pi$
$384$	18.5557	0.946917
$385$	−12.3758	−0.630730
$386$	−1.14048	−0.0580487
$387$	11.0917	0.563823
$388$	1.69264	0.0859306
$389$	15.5215	0.786971	0.393486	−	0.919331i	$-0.371269\pi$
$389$	15.5215	0.786971	0.393486	+	0.919331i	$0.371269\pi$
$390$	2.87532	0.145597
$391$	−0.412422	−0.0208571
$392$	−4.61149	−0.232916
$393$	−30.7045	−1.54884
$394$	−2.72757	−0.137413
$395$	−37.7630	−1.90006
$396$	14.0877	0.707934
$397$	−32.1277	−1.61244	−0.806222	−	0.591614i	$-0.798491\pi$
$397$	−32.1277	−1.61244	−0.806222	+	0.591614i	$0.798491\pi$
$398$	2.76552	0.138623
$399$	−45.0779	−2.25672
$400$	33.4650	1.67325
$401$	1.10536	0.0551989	0.0275994	−	0.999619i	$-0.491214\pi$
$401$	1.10536	0.0551989	0.0275994	+	0.999619i	$0.491214\pi$
$402$	8.13502	0.405738
$403$	6.95928	0.346667
$404$	15.1986	0.756161
$405$	108.262	5.37958
$406$	1.30000	0.0645178
$407$	4.02736	0.199629
$408$	3.97988	0.197033
$409$	30.4605	1.50617	0.753086	−	0.657922i	$-0.228564\pi$
$409$	30.4605	1.50617	0.753086	+	0.657922i	$0.228564\pi$
$410$	5.90309	0.291533
$411$	−14.0883	−0.694924
$412$	−3.88404	−0.191353
$413$	−3.14823	−0.154914
$414$	−0.358812	−0.0176346
$415$	26.2385	1.28800
$416$	−2.75506	−0.135078
$417$	−39.0489	−1.91223
$418$	−0.627816	−0.0307075
$419$	12.4139	0.606459	0.303230	−	0.952917i	$-0.401935\pi$
$419$	12.4139	0.606459	0.303230	+	0.952917i	$0.401935\pi$
$420$	87.8945	4.28881
$421$	14.3449	0.699126	0.349563	−	0.936913i	$-0.386330\pi$
$421$	14.3449	0.699126	0.349563	+	0.936913i	$0.386330\pi$
$422$	−2.59580	−0.126362
$423$	17.2094	0.836750
$424$	7.61208	0.369675
$425$	14.6081	0.708598
$426$	−5.31564	−0.257543
$427$	−9.06324	−0.438601
$428$	0.628415	0.0303756
$429$	−3.84262	−0.185523
$430$	0.963803	0.0464787
$431$	−40.0541	−1.92934	−0.964668	−	0.263468i	$-0.915134\pi$
$431$	−40.0541	−1.92934	−0.964668	+	0.263468i	$0.915134\pi$
$432$	−60.7600	−2.92332
$433$	30.1463	1.44874	0.724369	−	0.689412i	$-0.242131\pi$
$433$	30.1463	1.44874	0.724369	+	0.689412i	$0.242131\pi$
$434$	−3.64978	−0.175195
$435$	−23.7349	−1.13800
$436$	−24.8282	−1.18905
$437$	−0.932033	−0.0445852
$438$	8.68727	0.415094
$439$	18.4456	0.880359	0.440179	−	0.897910i	$-0.354915\pi$
$439$	18.4456	0.880359	0.440179	+	0.897910i	$0.354915\pi$
$440$	2.46928	0.117718
$441$	49.7231	2.36777
$442$	−0.389430	−0.0185233
$443$	4.84442	0.230165	0.115083	−	0.993356i	$-0.463287\pi$
$443$	4.84442	0.230165	0.115083	+	0.993356i	$0.463287\pi$
$444$	−28.6028	−1.35743
$445$	43.7647	2.07465
$446$	−1.71046	−0.0809928
$447$	70.1161	3.31638
$448$	−26.2940	−1.24227
$449$	−21.8751	−1.03235	−0.516174	−	0.856484i	$-0.672644\pi$
$449$	−21.8751	−1.03235	−0.516174	+	0.856484i	$0.672644\pi$
$450$	12.7092	0.599119
$451$	−7.88897	−0.371477
$452$	25.9174	1.21905
$453$	−3.29466	−0.154797
$454$	0.0891800	0.00418542
$455$	−17.3485	−0.813309
$456$	8.99414	0.421189
$457$	0.383056	0.0179186	0.00895930	−	0.999960i	$-0.497148\pi$
$457$	0.383056	0.0179186	0.00895930	+	0.999960i	$0.497148\pi$
$458$	2.82768	0.132129
$459$	−26.5230	−1.23799
$460$	1.81731	0.0847327
$461$	34.9368	1.62717	0.813585	−	0.581447i	$-0.197513\pi$
$461$	34.9368	1.62717	0.813585	+	0.581447i	$0.197513\pi$
$462$	2.01525	0.0937581
$463$	13.6404	0.633925	0.316963	−	0.948438i	$-0.397337\pi$
$463$	13.6404	0.633925	0.316963	+	0.948438i	$0.397337\pi$
$464$	7.36422	0.341875
$465$	66.6364	3.09019
$466$	−3.77448	−0.174849
$467$	−30.3922	−1.40638	−0.703192	−	0.711000i	$-0.748243\pi$
$467$	−30.3922	−1.40638	−0.703192	+	0.711000i	$0.748243\pi$
$468$	19.7482	0.912862
$469$	−49.0833	−2.26646
$470$	1.49540	0.0689775
$471$	52.1073	2.40098
$472$	0.628149	0.0289129
$473$	−1.28804	−0.0592241
$474$	6.14925	0.282444
$475$	33.0130	1.51474
$476$	−11.9043	−0.545634
$477$	−82.0768	−3.75804
$478$	−0.523545	−0.0239464
$479$	−23.2203	−1.06096	−0.530482	−	0.847696i	$-0.677989\pi$
$479$	−23.2203	−1.06096	−0.530482	+	0.847696i	$0.677989\pi$
$480$	−26.3802	−1.20409
$481$	5.64558	0.257416
$482$	2.02644	0.0923019
$483$	2.99178	0.136130
$484$	19.9930	0.908771
$485$	−3.19897	−0.145258
$486$	−8.81594	−0.399899
$487$	23.9343	1.08457	0.542284	−	0.840195i	$-0.317560\pi$
$487$	23.9343	1.08457	0.542284	+	0.840195i	$0.317560\pi$
$488$	1.80834	0.0818597
$489$	43.2298	1.95492
$490$	4.32065	0.195187
$491$	14.5262	0.655557	0.327778	−	0.944755i	$-0.393700\pi$
$491$	14.5262	0.655557	0.327778	+	0.944755i	$0.393700\pi$
$492$	56.0284	2.52596
$493$	3.21463	0.144780
$494$	−0.880075	−0.0395964
$495$	−26.6249	−1.19670
$496$	−20.6753	−0.928347
$497$	32.0723	1.43864
$498$	−4.27263	−0.191461
$499$	19.6649	0.880322	0.440161	−	0.897919i	$-0.354921\pi$
$499$	19.6649	0.880322	0.440161	+	0.897919i	$0.354921\pi$
$500$	−27.8356	−1.24484
$501$	−66.5157	−2.97170
$502$	−2.65520	−0.118508
$503$	7.56135	0.337144	0.168572	−	0.985689i	$-0.446084\pi$
$503$	7.56135	0.337144	0.168572	+	0.985689i	$0.446084\pi$
$504$	−20.8915	−0.930582
$505$	−28.7244	−1.27822
$506$	0.0416675	0.00185235
$507$	37.4440	1.66294
$508$	−1.63303	−0.0724540
$509$	−4.07748	−0.180731	−0.0903656	−	0.995909i	$-0.528804\pi$
$509$	−4.07748	−0.180731	−0.0903656	+	0.995909i	$0.528804\pi$
$510$	−3.72887	−0.165117
$511$	−52.4154	−2.31872
$512$	13.7196	0.606326
$513$	−59.9393	−2.64639
$514$	5.15478	0.227368
$515$	7.34058	0.323465
$516$	9.14781	0.402710
$517$	−1.99847	−0.0878925
$518$	−2.96081	−0.130091
$519$	−2.21410	−0.0971883
$520$	3.46145	0.151794
$521$	−33.7144	−1.47706	−0.738528	−	0.674223i	$-0.764479\pi$
$521$	−33.7144	−1.47706	−0.738528	+	0.674223i	$0.764479\pi$
$522$	2.79677	0.122411
$523$	−36.7350	−1.60631	−0.803154	−	0.595772i	$-0.796846\pi$
$523$	−36.7350	−1.60631	−0.803154	+	0.595772i	$0.796846\pi$
$524$	−18.3246	−0.800513
$525$	−105.970	−4.62490
$526$	−2.01811	−0.0879939
$527$	−9.02517	−0.393142
$528$	11.4160	0.496817
$529$	−22.9381	−0.997311
$530$	−7.13199	−0.309794
$531$	−6.77298	−0.293922
$532$	−26.9027	−1.16638
$533$	−11.0588	−0.479010
$534$	−7.12656	−0.308396
$535$	−1.18766	−0.0513472
$536$	9.79333	0.423007
$537$	6.83077	0.294769
$538$	−2.68694	−0.115842
$539$	−5.77417	−0.248711
$540$	116.872	5.02937
$541$	17.3851	0.747442	0.373721	−	0.927541i	$-0.378082\pi$
$541$	17.3851	0.747442	0.373721	+	0.927541i	$0.378082\pi$
$542$	−5.85982	−0.251701
$543$	−9.17465	−0.393722
$544$	3.57291	0.153187
$545$	46.9236	2.00999
$546$	2.82499	0.120898
$547$	27.2857	1.16665	0.583326	−	0.812238i	$-0.301751\pi$
$547$	27.2857	1.16665	0.583326	+	0.812238i	$0.301751\pi$
$548$	−8.40795	−0.359170
$549$	−19.4983	−0.832168
$550$	−1.47588	−0.0629317
$551$	7.26475	0.309489
$552$	−0.596933	−0.0254071
$553$	−37.1020	−1.57774
$554$	2.70406	0.114884
$555$	54.0575	2.29461
$556$	−23.3045	−0.988333
$557$	22.0118	0.932669	0.466335	−	0.884608i	$-0.345574\pi$
$557$	22.0118	0.932669	0.466335	+	0.884608i	$0.345574\pi$
$558$	−7.85200	−0.332402
$559$	−1.80558	−0.0763679
$560$	51.5404	2.17798
$561$	4.98331	0.210395
$562$	5.02318	0.211890
$563$	−34.7867	−1.46609	−0.733043	−	0.680183i	$-0.761900\pi$
$563$	−34.7867	−1.46609	−0.733043	+	0.680183i	$0.761900\pi$
$564$	14.1934	0.597648
$565$	−48.9822	−2.06070
$566$	4.82737	0.202910
$567$	106.367	4.46700
$568$	−6.39922	−0.268505
$569$	4.70112	0.197081	0.0985407	−	0.995133i	$-0.468583\pi$
$569$	4.70112	0.197081	0.0985407	+	0.995133i	$0.468583\pi$
$570$	−8.42688	−0.352963
$571$	−10.1822	−0.426112	−0.213056	−	0.977040i	$-0.568342\pi$
$571$	−10.1822	−0.426112	−0.213056	+	0.977040i	$0.568342\pi$
$572$	−2.29329	−0.0958873
$573$	−41.7506	−1.74416
$574$	5.79977	0.242078
$575$	−2.19104	−0.0913726
$576$	−56.5679	−2.35700
$577$	20.5479	0.855422	0.427711	−	0.903916i	$-0.359320\pi$
$577$	20.5479	0.855422	0.427711	+	0.903916i	$0.359320\pi$
$578$	−2.61733	−0.108867
$579$	20.4579	0.850203
$580$	−14.1651	−0.588173
$581$	25.7793	1.06950
$582$	0.520914	0.0215926
$583$	9.53129	0.394746
$584$	10.4582	0.432761
$585$	−37.3228	−1.54311
$586$	2.98005	0.123105
$587$	8.40929	0.347088	0.173544	−	0.984826i	$-0.444478\pi$
$587$	8.40929	0.347088	0.173544	+	0.984826i	$0.444478\pi$
$588$	41.0089	1.69118
$589$	−20.3960	−0.840403
$590$	−0.588532	−0.0242295
$591$	48.9273	2.01260
$592$	−16.7724	−0.689341
$593$	−7.48556	−0.307395	−0.153697	−	0.988118i	$-0.549118\pi$
$593$	−7.48556	−0.307395	−0.153697	+	0.988118i	$0.549118\pi$
$594$	2.67965	0.109947
$595$	22.4984	0.922345
$596$	41.8456	1.71406
$597$	−49.6080	−2.03032
$598$	0.0584097	0.00238855
$599$	4.69324	0.191760	0.0958802	−	0.995393i	$-0.469433\pi$
$599$	4.69324	0.191760	0.0958802	+	0.995393i	$0.469433\pi$
$600$	21.1436	0.863183
$601$	−1.89605	−0.0773416	−0.0386708	−	0.999252i	$-0.512312\pi$
$601$	−1.89605	−0.0773416	−0.0386708	+	0.999252i	$0.512312\pi$
$602$	0.946933	0.0385941
$603$	−105.596	−4.30020
$604$	−1.96627	−0.0800062
$605$	−37.7854	−1.53620
$606$	4.67743	0.190008
$607$	−19.6483	−0.797501	−0.398751	−	0.917059i	$-0.630556\pi$
$607$	−19.6483	−0.797501	−0.398751	+	0.917059i	$0.630556\pi$
$608$	8.07444	0.327462
$609$	−23.3195	−0.944952
$610$	−1.69429	−0.0685997
$611$	−2.80146	−0.113335
$612$	−25.6105	−1.03524
$613$	−17.3399	−0.700352	−0.350176	−	0.936684i	$-0.613878\pi$
$613$	−17.3399	−0.700352	−0.350176	+	0.936684i	$0.613878\pi$
$614$	0.0173359	0.000699619 0
$615$	−105.890	−4.26990
$616$	2.42606	0.0977487
$617$	6.60055	0.265728	0.132864	−	0.991134i	$-0.457583\pi$
$617$	6.60055	0.265728	0.132864	+	0.991134i	$0.457583\pi$
$618$	−1.19533	−0.0480830
$619$	−6.67297	−0.268209	−0.134105	−	0.990967i	$-0.542816\pi$
$619$	−6.67297	−0.268209	−0.134105	+	0.990967i	$0.542816\pi$
$620$	39.7689	1.59716
$621$	3.97812	0.159636
$622$	−2.54901	−0.102206
$623$	42.9987	1.72271
$624$	16.0030	0.640632
$625$	8.55986	0.342394
$626$	−2.47730	−0.0990127
$627$	11.2618	0.449753
$628$	31.0979	1.24094
$629$	−7.32149	−0.291927
$630$	19.5739	0.779843
$631$	16.7963	0.668651	0.334325	−	0.942458i	$-0.391492\pi$
$631$	16.7963	0.668651	0.334325	+	0.942458i	$0.391492\pi$
$632$	7.40276	0.294466
$633$	46.5637	1.85074
$634$	−0.533859	−0.0212022
$635$	3.08632	0.122477
$636$	−67.6924	−2.68418
$637$	−8.09426	−0.320706
$638$	−0.324778	−0.0128581
$639$	68.9992	2.72956
$640$	−20.9294	−0.827306
$641$	20.5409	0.811316	0.405658	−	0.914025i	$-0.367042\pi$
$641$	20.5409	0.811316	0.405658	+	0.914025i	$0.367042\pi$
$642$	0.193397	0.00763277
$643$	1.06457	0.0419825	0.0209913	−	0.999780i	$-0.493318\pi$
$643$	1.06457	0.0419825	0.0209913	+	0.999780i	$0.493318\pi$
$644$	1.78550	0.0703587
$645$	−17.2888	−0.680745
$646$	1.14133	0.0449049
$647$	29.6687	1.16640	0.583198	−	0.812330i	$-0.301801\pi$
$647$	29.6687	1.16640	0.583198	+	0.812330i	$0.301801\pi$
$648$	−21.2228	−0.833712
$649$	0.786522	0.0308737
$650$	−2.06889	−0.0811487
$651$	65.4701	2.56597
$652$	25.7998	1.01040
$653$	−20.1399	−0.788137	−0.394069	−	0.919081i	$-0.628933\pi$
$653$	−20.1399	−0.788137	−0.394069	+	0.919081i	$0.628933\pi$
$654$	−7.64095	−0.298785
$655$	34.6323	1.35319
$656$	32.8545	1.28275
$657$	−112.764	−4.39936
$658$	1.46922	0.0572762
$659$	36.5416	1.42346	0.711730	−	0.702453i	$-0.247912\pi$
$659$	36.5416	1.42346	0.711730	+	0.702453i	$0.247912\pi$
$660$	−21.9587	−0.854740
$661$	22.1043	0.859758	0.429879	−	0.902886i	$-0.358556\pi$
$661$	22.1043	0.859758	0.429879	+	0.902886i	$0.358556\pi$
$662$	−2.03931	−0.0792600
$663$	6.98563	0.271299
$664$	−5.14360	−0.199610
$665$	50.8443	1.97166
$666$	−6.36978	−0.246824
$667$	−0.482155	−0.0186691
$668$	−39.6969	−1.53592
$669$	30.6824	1.18625
$670$	−9.17567	−0.354487
$671$	2.26427	0.0874111
$672$	−25.9185	−0.999828
$673$	30.5471	1.17750	0.588751	−	0.808314i	$-0.299620\pi$
$673$	30.5471	1.17750	0.588751	+	0.808314i	$0.299620\pi$
$674$	−0.161677	−0.00622757
$675$	−140.906	−5.42349
$676$	22.3467	0.859489
$677$	24.5063	0.941854	0.470927	−	0.882172i	$-0.343920\pi$
$677$	24.5063	0.941854	0.470927	+	0.882172i	$0.343920\pi$
$678$	7.97617	0.306323
$679$	−3.14298	−0.120616
$680$	−4.48899	−0.172145
$681$	−1.59972	−0.0613013
$682$	0.911825	0.0349156
$683$	−36.8637	−1.41055	−0.705275	−	0.708933i	$-0.749177\pi$
$683$	−36.8637	−1.41055	−0.705275	+	0.708933i	$0.749177\pi$
$684$	−57.8774	−2.21300
$685$	15.8905	0.607144
$686$	−0.449085	−0.0171461
$687$	−50.7232	−1.93521
$688$	5.36418	0.204507
$689$	13.3610	0.509014
$690$	0.559284	0.0212916
$691$	−12.4203	−0.472492	−0.236246	−	0.971693i	$-0.575917\pi$
$691$	−12.4203	−0.472492	−0.236246	+	0.971693i	$0.575917\pi$
$692$	−1.32139	−0.0502315
$693$	−26.1588	−0.993692
$694$	6.33892	0.240622
$695$	44.0441	1.67069
$696$	4.65280	0.176364
$697$	14.3416	0.543228
$698$	−5.29743	−0.200511
$699$	67.7069	2.56091
$700$	−63.2432	−2.39037
$701$	28.1266	1.06233	0.531164	−	0.847269i	$-0.321755\pi$
$701$	28.1266	1.06233	0.531164	+	0.847269i	$0.321755\pi$
$702$	3.75635	0.141774
$703$	−16.5459	−0.624039
$704$	6.56903	0.247579
$705$	−26.8245	−1.01027
$706$	4.40042	0.165612
$707$	−28.2217	−1.06138
$708$	−5.58598	−0.209934
$709$	1.43051	0.0537241	0.0268620	−	0.999639i	$-0.491449\pi$
$709$	1.43051	0.0537241	0.0268620	+	0.999639i	$0.491449\pi$
$710$	5.99562	0.225012
$711$	−79.8198	−2.99348
$712$	−8.57929	−0.321523
$713$	1.35366	0.0506951
$714$	−3.66360	−0.137107
$715$	4.33417	0.162089
$716$	4.07663	0.152351
$717$	9.39139	0.350728
$718$	−0.278335	−0.0103874
$719$	16.8995	0.630244	0.315122	−	0.949051i	$-0.397955\pi$
$719$	16.8995	0.630244	0.315122	+	0.949051i	$0.397955\pi$
$720$	110.882	4.13233
$721$	7.21209	0.268592
$722$	−0.910409	−0.0338819
$723$	−36.3504	−1.35189
$724$	−5.47547	−0.203494
$725$	17.0781	0.634265
$726$	6.15290	0.228356
$727$	26.9422	0.999233	0.499616	−	0.866247i	$-0.333474\pi$
$727$	26.9422	0.999233	0.499616	+	0.866247i	$0.333474\pi$
$728$	3.40086	0.126044
$729$	70.7417	2.62006
$730$	−9.79856	−0.362661
$731$	2.34157	0.0866062
$732$	−16.0811	−0.594375
$733$	−9.89979	−0.365657	−0.182829	−	0.983145i	$-0.558525\pi$
$733$	−9.89979	−0.365657	−0.182829	+	0.983145i	$0.558525\pi$
$734$	−1.41029	−0.0520548
$735$	−77.5041	−2.85878
$736$	−0.535893	−0.0197533
$737$	12.2625	0.451694
$738$	12.4774	0.459299
$739$	36.7085	1.35034	0.675171	−	0.737661i	$-0.264070\pi$
$739$	36.7085	1.35034	0.675171	+	0.737661i	$0.264070\pi$
$740$	32.2617	1.18596
$741$	15.7868	0.579944
$742$	−7.00716	−0.257241
$743$	−40.2612	−1.47704	−0.738521	−	0.674231i	$-0.764475\pi$
$743$	−40.2612	−1.47704	−0.738521	+	0.674231i	$0.764475\pi$
$744$	−13.0629	−0.478909
$745$	−79.0854	−2.89746
$746$	4.32338	0.158290
$747$	55.4605	2.02919
$748$	2.97406	0.108742
$749$	−1.16688	−0.0426367
$750$	−8.56648	−0.312804
$751$	20.0555	0.731836	0.365918	−	0.930647i	$-0.380755\pi$
$751$	20.0555	0.731836	0.365918	+	0.930647i	$0.380755\pi$
$752$	8.32284	0.303503
$753$	47.6292	1.73571
$754$	−0.455276	−0.0165802
$755$	3.71611	0.135243
$756$	114.826	4.17619
$757$	38.8419	1.41173	0.705867	−	0.708345i	$-0.250558\pi$
$757$	38.8419	1.41173	0.705867	+	0.708345i	$0.250558\pi$
$758$	−4.22493	−0.153456
$759$	−0.747435	−0.0271302
$760$	−10.1447	−0.367986
$761$	9.26836	0.335978	0.167989	−	0.985789i	$-0.446273\pi$
$761$	9.26836	0.335978	0.167989	+	0.985789i	$0.446273\pi$
$762$	−0.502570	−0.0182062
$763$	46.1023	1.66901
$764$	−24.9170	−0.901464
$765$	48.4022	1.74999
$766$	3.83781	0.138666
$767$	1.10255	0.0398108
$768$	−44.0463	−1.58939
$769$	−8.24686	−0.297389	−0.148695	−	0.988883i	$-0.547507\pi$
$769$	−8.24686	−0.297389	−0.148695	+	0.988883i	$0.547507\pi$
$770$	−2.27305	−0.0819149
$771$	−92.4668	−3.33011
$772$	12.2094	0.439425
$773$	−17.9148	−0.644352	−0.322176	−	0.946680i	$-0.604414\pi$
$773$	−17.9148	−0.644352	−0.322176	+	0.946680i	$0.604414\pi$
$774$	2.03720	0.0732255
$775$	−47.9473	−1.72232
$776$	0.627102	0.0225116
$777$	53.1113	1.90536
$778$	2.85081	0.102206
$779$	32.4107	1.16123
$780$	−30.7818	−1.10216
$781$	−8.01263	−0.286714
$782$	−0.0757488	−0.00270877
$783$	−31.0075	−1.10812
$784$	24.0472	0.858827
$785$	−58.7730	−2.09770
$786$	−5.63945	−0.201153
$787$	37.4489	1.33491	0.667454	−	0.744651i	$-0.267384\pi$
$787$	37.4489	1.33491	0.667454	+	0.744651i	$0.267384\pi$
$788$	29.2000	1.04021
$789$	36.2010	1.28879
$790$	−6.93587	−0.246767
$791$	−48.1249	−1.71112
$792$	5.21933	0.185461
$793$	3.17406	0.112714
$794$	−5.90085	−0.209413
$795$	127.934	4.53736
$796$	−29.6063	−1.04937
$797$	−6.56429	−0.232519	−0.116259	−	0.993219i	$-0.537090\pi$
$797$	−6.56429	−0.232519	−0.116259	+	0.993219i	$0.537090\pi$
$798$	−8.27938	−0.293087
$799$	3.63308	0.128529
$800$	18.9815	0.671098
$801$	92.5057	3.26853
$802$	0.203019	0.00716886
$803$	13.0949	0.462110
$804$	−87.0896	−3.07141
$805$	−3.37449	−0.118935
$806$	1.27820	0.0450227
$807$	48.1986	1.69667
$808$	5.63091	0.198095
$809$	12.8394	0.451409	0.225704	−	0.974196i	$-0.427532\pi$
$809$	12.8394	0.451409	0.225704	+	0.974196i	$0.427532\pi$
$810$	19.8843	0.698664
$811$	−1.31098	−0.0460347	−0.0230174	−	0.999735i	$-0.507327\pi$
$811$	−1.31098	−0.0460347	−0.0230174	+	0.999735i	$0.507327\pi$
$812$	−13.9171	−0.488396
$813$	105.114	3.68650
$814$	0.739700	0.0259265
$815$	−48.7598	−1.70798
$816$	−20.7535	−0.726519
$817$	5.29173	0.185134
$818$	5.59463	0.195612
$819$	−36.6696	−1.28134
$820$	−63.1956	−2.20689
$821$	−46.1086	−1.60920	−0.804600	−	0.593817i	$-0.797620\pi$
$821$	−46.1086	−1.60920	−0.804600	+	0.593817i	$0.797620\pi$
$822$	−2.58757	−0.0902520
$823$	6.36527	0.221879	0.110940	−	0.993827i	$-0.464614\pi$
$823$	6.36527	0.221879	0.110940	+	0.993827i	$0.464614\pi$
$824$	−1.43899	−0.0501296
$825$	26.4744	0.921721
$826$	−0.578231	−0.0201192
$827$	18.4051	0.640009	0.320004	−	0.947416i	$-0.396316\pi$
$827$	18.4051	0.640009	0.320004	+	0.947416i	$0.396316\pi$
$828$	3.84127	0.133493
$829$	0.214845	0.00746186	0.00373093	−	0.999993i	$-0.498812\pi$
$829$	0.214845	0.00746186	0.00373093	+	0.999993i	$0.498812\pi$
$830$	4.81919	0.167277
$831$	−48.5055	−1.68264
$832$	9.20849	0.319247
$833$	10.4971	0.363702
$834$	−7.17205	−0.248348
$835$	75.0245	2.59633
$836$	6.72109	0.232454
$837$	87.0545	3.00905
$838$	2.28004	0.0787628
$839$	8.66817	0.299259	0.149629	−	0.988742i	$-0.452192\pi$
$839$	8.66817	0.299259	0.149629	+	0.988742i	$0.452192\pi$
$840$	32.5639	1.12356
$841$	−25.2418	−0.870408
$842$	2.63470	0.0907977
$843$	−90.1062	−3.10342
$844$	27.7894	0.956550
$845$	−42.2338	−1.45289
$846$	3.16083	0.108671
$847$	−37.1240	−1.27560
$848$	−39.6941	−1.36310
$849$	−86.5938	−2.97189
$850$	2.68305	0.0920279
$851$	1.09813	0.0376435
$852$	56.9066	1.94959
$853$	−22.5241	−0.771211	−0.385605	−	0.922664i	$-0.626007\pi$
$853$	−22.5241	−0.771211	−0.385605	+	0.922664i	$0.626007\pi$
$854$	−1.66463	−0.0569625
$855$	109.384	3.74087
$856$	0.232820	0.00795764
$857$	−45.0950	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$857$	−45.0950	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$858$	−0.705768	−0.0240945
$859$	−21.0436	−0.718000	−0.359000	−	0.933338i	$-0.616882\pi$
$859$	−21.0436	−0.718000	−0.359000	+	0.933338i	$0.616882\pi$
$860$	−10.3180	−0.351841
$861$	−104.037	−3.54556
$862$	−7.35667	−0.250569
$863$	−2.01110	−0.0684587	−0.0342293	−	0.999414i	$-0.510898\pi$
$863$	−2.01110	−0.0684587	−0.0342293	+	0.999414i	$0.510898\pi$
$864$	−34.4634	−1.17247
$865$	2.49733	0.0849118
$866$	5.53693	0.188152
$867$	46.9499	1.59450
$868$	39.0728	1.32622
$869$	9.26919	0.314436
$870$	−4.35935	−0.147796
$871$	17.1896	0.582448
$872$	−9.19854	−0.311502
$873$	−6.76168	−0.228848
$874$	−0.171185	−0.00579042
$875$	51.6866	1.74733
$876$	−93.0018	−3.14224
$877$	0.542916	0.0183330	0.00916648	−	0.999958i	$-0.497082\pi$
$877$	0.542916	0.0183330	0.00916648	+	0.999958i	$0.497082\pi$
$878$	3.38787	0.114335
$879$	−53.4563	−1.80304
$880$	−12.8763	−0.434061
$881$	−8.69333	−0.292886	−0.146443	−	0.989219i	$-0.546782\pi$
$881$	−8.69333	−0.292886	−0.146443	+	0.989219i	$0.546782\pi$
$882$	9.13257	0.307510
$883$	35.6778	1.20065	0.600327	−	0.799755i	$-0.295037\pi$
$883$	35.6778	1.20065	0.600327	+	0.799755i	$0.295037\pi$
$884$	4.16905	0.140220
$885$	10.5571	0.354874
$886$	0.889767	0.0298923
$887$	13.8237	0.464154	0.232077	−	0.972697i	$-0.425448\pi$
$887$	13.8237	0.464154	0.232077	+	0.972697i	$0.425448\pi$
$888$	−10.5970	−0.355612
$889$	3.03230	0.101700
$890$	8.03820	0.269441
$891$	−26.5737	−0.890251
$892$	18.3114	0.613111
$893$	8.21042	0.274751
$894$	12.8781	0.430708
$895$	−7.70457	−0.257535
$896$	−20.5630	−0.686963
$897$	−1.04776	−0.0349836
$898$	−4.01776	−0.134074
$899$	−10.5512	−0.351901
$900$	−136.059	−4.53530
$901$	−17.3273	−0.577255
$902$	−1.44896	−0.0482449
$903$	−16.9862	−0.565264
$904$	9.60209	0.319361
$905$	10.3483	0.343989
$906$	−0.605125	−0.0201039
$907$	3.22925	0.107225	0.0536127	−	0.998562i	$-0.482926\pi$
$907$	3.22925	0.107225	0.0536127	+	0.998562i	$0.482926\pi$
$908$	−0.954717	−0.0316834
$909$	−60.7150	−2.01379
$910$	−3.18637	−0.105627
$911$	−36.7510	−1.21761	−0.608807	−	0.793318i	$-0.708352\pi$
$911$	−36.7510	−1.21761	−0.608807	+	0.793318i	$0.708352\pi$
$912$	−46.9010	−1.55305
$913$	−6.44043	−0.213147
$914$	0.0703553	0.00232715
$915$	30.3923	1.00474
$916$	−30.2718	−1.00021
$917$	34.0261	1.12364
$918$	−4.87143	−0.160781
$919$	−50.3548	−1.66105	−0.830526	−	0.556980i	$-0.811960\pi$
$919$	−50.3548	−1.66105	−0.830526	+	0.556980i	$0.811960\pi$
$920$	0.673293	0.0221978
$921$	−0.310972	−0.0102469
$922$	6.41679	0.211326
$923$	−11.2321	−0.369710
$924$	−21.5743	−0.709743
$925$	−38.8963	−1.27890
$926$	2.50532	0.0823299
$927$	15.5158	0.509606
$928$	4.17703	0.137118
$929$	49.9006	1.63719	0.818593	−	0.574374i	$-0.194755\pi$
$929$	49.9006	1.63719	0.818593	+	0.574374i	$0.194755\pi$
$930$	12.2390	0.401333
$931$	23.7224	0.777469
$932$	40.4078	1.32360
$933$	45.7243	1.49695
$934$	−5.58209	−0.182652
$935$	−5.62078	−0.183819
$936$	7.31648	0.239147
$937$	−39.6375	−1.29490	−0.647451	−	0.762107i	$-0.724165\pi$
$937$	−39.6375	−1.29490	−0.647451	+	0.762107i	$0.724165\pi$
$938$	−9.01506	−0.294352
$939$	44.4379	1.45018
$940$	−16.0090	−0.522156
$941$	25.3577	0.826637	0.413318	−	0.910587i	$-0.364370\pi$
$941$	25.3577	0.826637	0.413318	+	0.910587i	$0.364370\pi$
$942$	9.57048	0.311823
$943$	−2.15107	−0.0700484
$944$	−3.27556	−0.106610
$945$	−217.014	−7.05947
$946$	−0.236572	−0.00769163
$947$	5.89253	0.191481	0.0957407	−	0.995406i	$-0.469478\pi$
$947$	5.89253	0.191481	0.0957407	+	0.995406i	$0.469478\pi$
$948$	−65.8309	−2.13809
$949$	18.3565	0.595878
$950$	6.06344	0.196724
$951$	9.57639	0.310536
$952$	−4.41041	−0.142942
$953$	10.0351	0.325070	0.162535	−	0.986703i	$-0.448033\pi$
$953$	10.0351	0.325070	0.162535	+	0.986703i	$0.448033\pi$
$954$	−15.0749	−0.488069
$955$	47.0914	1.52384
$956$	5.60482	0.181273
$957$	5.82590	0.188325
$958$	−4.26484	−0.137791
$959$	15.6123	0.504149
$960$	88.1730	2.84577
$961$	−1.37730	−0.0444291
$962$	1.03691	0.0334315
$963$	−2.51037	−0.0808956
$964$	−21.6941	−0.698720
$965$	−23.0749	−0.742809
$966$	0.549495	0.0176797
$967$	−45.6262	−1.46724	−0.733620	−	0.679560i	$-0.762171\pi$
$967$	−45.6262	−1.46724	−0.733620	+	0.679560i	$0.762171\pi$
$968$	7.40716	0.238075
$969$	−20.4732	−0.657695
$970$	−0.587550	−0.0188651
$971$	50.0563	1.60638	0.803191	−	0.595721i	$-0.203134\pi$
$971$	50.0563	1.60638	0.803191	+	0.595721i	$0.203134\pi$
$972$	94.3792	3.02722
$973$	43.2731	1.38727
$974$	4.39598	0.140856
$975$	37.1120	1.18853
$976$	−9.42980	−0.301840
$977$	16.8418	0.538817	0.269409	−	0.963026i	$-0.413172\pi$
$977$	16.8418	0.538817	0.269409	+	0.963026i	$0.413172\pi$
$978$	7.93996	0.253892
$979$	−10.7424	−0.343327
$980$	−46.2547	−1.47755
$981$	99.1827	3.16666
$982$	2.66800	0.0851393
$983$	−54.2760	−1.73114	−0.865568	−	0.500792i	$-0.833042\pi$
$983$	−54.2760	−1.73114	−0.865568	+	0.500792i	$0.833042\pi$
$984$	20.7579	0.661736
$985$	−55.1862	−1.75838
$986$	0.590426	0.0188030
$987$	−26.3550	−0.838889
$988$	9.42166	0.299743
$989$	−0.351207	−0.0111677
$990$	−4.89015	−0.155419
$991$	−5.73279	−0.182108	−0.0910540	−	0.995846i	$-0.529024\pi$
$991$	−5.73279	−0.182108	−0.0910540	+	0.995846i	$0.529024\pi$
$992$	−11.7271	−0.372337
$993$	36.5813	1.16087
$994$	5.89067	0.186841
$995$	55.9539	1.77386
$996$	45.7407	1.44935
$997$	19.9095	0.630541	0.315270	−	0.949002i	$-0.397905\pi$
$997$	19.9095	0.630541	0.315270	+	0.949002i	$0.397905\pi$
$998$	3.61182	0.114330
$999$	70.6213	2.23436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.a.c.1.4	✓	6
3.2	odd	2		1359.2.a.i.1.3		6
4.3	odd	2		2416.2.a.o.1.6		6
5.4	even	2		3775.2.a.p.1.3		6
7.6	odd	2		7399.2.a.e.1.4		6
8.3	odd	2		9664.2.a.bc.1.1		6
8.5	even	2		9664.2.a.bh.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.a.c.1.4	✓	6	1.1	even	1	trivial
1359.2.a.i.1.3		6	3.2	odd	2
2416.2.a.o.1.6		6	4.3	odd	2
3775.2.a.p.1.3		6	5.4	even	2
7399.2.a.e.1.4		6	7.6	odd	2
9664.2.a.bc.1.1		6	8.3	odd	2
9664.2.a.bh.1.6		6	8.5	even	2

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:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.183668 q^{2} -3.29466 q^{3} -1.96627 q^{4} +3.71611 q^{5} -0.605125 q^{6} +3.65107 q^{7} -0.728478 q^{8} +7.85477 q^{9} +O(q^{10})\)	\(q+0.183668 q^{2} -3.29466 q^{3} -1.96627 q^{4} +3.71611 q^{5} -0.605125 q^{6} +3.65107 q^{7} -0.728478 q^{8} +7.85477 q^{9} +0.682533 q^{10} -0.912147 q^{11} +6.47817 q^{12} -1.27865 q^{13} +0.670586 q^{14} -12.2433 q^{15} +3.79873 q^{16} +1.65822 q^{17} +1.44267 q^{18} +3.74742 q^{19} -7.30687 q^{20} -12.0290 q^{21} -0.167533 q^{22} -0.248713 q^{23} +2.40009 q^{24} +8.80951 q^{25} -0.234848 q^{26} -15.9948 q^{27} -7.17897 q^{28} +1.93860 q^{29} -2.24871 q^{30} -5.44267 q^{31} +2.15466 q^{32} +3.00521 q^{33} +0.304563 q^{34} +13.5678 q^{35} -15.4446 q^{36} -4.41526 q^{37} +0.688284 q^{38} +4.21272 q^{39} -2.70711 q^{40} +8.64880 q^{41} -2.20935 q^{42} +1.41210 q^{43} +1.79352 q^{44} +29.1892 q^{45} -0.0456807 q^{46} +2.19095 q^{47} -12.5155 q^{48} +6.33031 q^{49} +1.61803 q^{50} -5.46328 q^{51} +2.51417 q^{52} -10.4493 q^{53} -2.93774 q^{54} -3.38964 q^{55} -2.65972 q^{56} -12.3465 q^{57} +0.356059 q^{58} -0.862276 q^{59} +24.0736 q^{60} -2.48235 q^{61} -0.999648 q^{62} +28.6783 q^{63} -7.20172 q^{64} -4.75161 q^{65} +0.551962 q^{66} -13.4435 q^{67} -3.26051 q^{68} +0.819424 q^{69} +2.49198 q^{70} +8.78436 q^{71} -5.72203 q^{72} -14.3562 q^{73} -0.810944 q^{74} -29.0243 q^{75} -7.36843 q^{76} -3.33031 q^{77} +0.773744 q^{78} -10.1620 q^{79} +14.1165 q^{80} +29.1331 q^{81} +1.58851 q^{82} +7.06074 q^{83} +23.6523 q^{84} +6.16215 q^{85} +0.259358 q^{86} -6.38702 q^{87} +0.664479 q^{88} +11.7770 q^{89} +5.36114 q^{90} -4.66844 q^{91} +0.489036 q^{92} +17.9317 q^{93} +0.402409 q^{94} +13.9259 q^{95} -7.09888 q^{96} -0.860838 q^{97} +1.16268 q^{98} -7.16470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 5 q^{3} + 3 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) 6 * q + q^2 - 5 * q^3 + 3 * q^4 + 6 * q^5 - 2 * q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 6 q + q^{2} - 5 q^{3} + 3 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} + 8 q^{10} + 8 q^{11} - 11 q^{12} - q^{13} + 6 q^{14} - 18 q^{15} - 3 q^{16} + 9 q^{17} - 16 q^{18} - 6 q^{19} + 3 q^{20} + 13 q^{21} - 12 q^{22} - 4 q^{23} + q^{24} - 4 q^{25} - 7 q^{26} - 2 q^{27} - 24 q^{28} - 2 q^{29} - 16 q^{30} - 8 q^{31} - 11 q^{32} + 3 q^{33} - 9 q^{34} + 5 q^{35} - 25 q^{36} - 12 q^{37} - 3 q^{38} - 22 q^{39} + 9 q^{40} + 41 q^{41} - 24 q^{42} + q^{43} + 12 q^{45} - 17 q^{46} + 28 q^{47} + 9 q^{48} + 33 q^{49} + 15 q^{50} - 31 q^{51} + 15 q^{52} + 14 q^{53} + 27 q^{54} + q^{55} - 9 q^{56} - 28 q^{57} + q^{58} + 12 q^{59} + 17 q^{60} + 5 q^{61} - 9 q^{62} + 4 q^{63} - 27 q^{64} + 13 q^{65} + 8 q^{66} - 15 q^{67} + 11 q^{68} + 33 q^{69} - 33 q^{70} - 2 q^{71} + q^{72} - 7 q^{73} + 53 q^{74} - 32 q^{75} - 3 q^{76} - 15 q^{77} + 16 q^{78} - 9 q^{79} + 16 q^{80} + 66 q^{81} + 10 q^{82} - 11 q^{83} + 49 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 24 q^{88} + 36 q^{89} + 19 q^{90} - 35 q^{91} - 38 q^{92} - q^{93} + 42 q^{94} + 32 q^{95} - 34 q^{96} + 11 q^{97} + 15 q^{98} + 17 q^{99}+O(q^{100}) \) 6 * q + q^2 - 5 * q^3 + 3 * q^4 + 6 * q^5 - 2 * q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 + 8 * q^10 + 8 * q^11 - 11 * q^12 - q^13 + 6 * q^14 - 18 * q^15 - 3 * q^16 + 9 * q^17 - 16 * q^18 - 6 * q^19 + 3 * q^20 + 13 * q^21 - 12 * q^22 - 4 * q^23 + q^24 - 4 * q^25 - 7 * q^26 - 2 * q^27 - 24 * q^28 - 2 * q^29 - 16 * q^30 - 8 * q^31 - 11 * q^32 + 3 * q^33 - 9 * q^34 + 5 * q^35 - 25 * q^36 - 12 * q^37 - 3 * q^38 - 22 * q^39 + 9 * q^40 + 41 * q^41 - 24 * q^42 + q^43 + 12 * q^45 - 17 * q^46 + 28 * q^47 + 9 * q^48 + 33 * q^49 + 15 * q^50 - 31 * q^51 + 15 * q^52 + 14 * q^53 + 27 * q^54 + q^55 - 9 * q^56 - 28 * q^57 + q^58 + 12 * q^59 + 17 * q^60 + 5 * q^61 - 9 * q^62 + 4 * q^63 - 27 * q^64 + 13 * q^65 + 8 * q^66 - 15 * q^67 + 11 * q^68 + 33 * q^69 - 33 * q^70 - 2 * q^71 + q^72 - 7 * q^73 + 53 * q^74 - 32 * q^75 - 3 * q^76 - 15 * q^77 + 16 * q^78 - 9 * q^79 + 16 * q^80 + 66 * q^81 + 10 * q^82 - 11 * q^83 + 49 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 24 * q^88 + 36 * q^89 + 19 * q^90 - 35 * q^91 - 38 * q^92 - q^93 + 42 * q^94 + 32 * q^95 - 34 * q^96 + 11 * q^97 + 15 * q^98 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more