Properties

Label 8047.2.a.d
Level 8047
Weight 2
Character orbit 8047.a
Self dual Yes
Analytic conductor 64.256
Analytic rank 0
Dimension 156
CM No

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Newspace parameters

Level: \( N \) = \( 8047 = 13 \cdot 619 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8047.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(156\)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(156q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 23q^{3} \) \(\mathstrut +\mathstrut 161q^{4} \) \(\mathstrut +\mathstrut 39q^{5} \) \(\mathstrut +\mathstrut 25q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 169q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(156q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 23q^{3} \) \(\mathstrut +\mathstrut 161q^{4} \) \(\mathstrut +\mathstrut 39q^{5} \) \(\mathstrut +\mathstrut 25q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 169q^{9} \) \(\mathstrut +\mathstrut 11q^{10} \) \(\mathstrut +\mathstrut 23q^{11} \) \(\mathstrut +\mathstrut 57q^{12} \) \(\mathstrut -\mathstrut 156q^{13} \) \(\mathstrut +\mathstrut 18q^{14} \) \(\mathstrut +\mathstrut 32q^{15} \) \(\mathstrut +\mathstrut 159q^{16} \) \(\mathstrut +\mathstrut 119q^{17} \) \(\mathstrut +\mathstrut 36q^{18} \) \(\mathstrut +\mathstrut 35q^{19} \) \(\mathstrut +\mathstrut 109q^{20} \) \(\mathstrut +\mathstrut 33q^{21} \) \(\mathstrut +\mathstrut 11q^{22} \) \(\mathstrut +\mathstrut 55q^{23} \) \(\mathstrut +\mathstrut 63q^{24} \) \(\mathstrut +\mathstrut 189q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut +\mathstrut 89q^{27} \) \(\mathstrut +\mathstrut 54q^{28} \) \(\mathstrut -\mathstrut 55q^{29} \) \(\mathstrut +\mathstrut 47q^{31} \) \(\mathstrut +\mathstrut 112q^{32} \) \(\mathstrut +\mathstrut 109q^{33} \) \(\mathstrut +\mathstrut 51q^{34} \) \(\mathstrut +\mathstrut 25q^{35} \) \(\mathstrut +\mathstrut 162q^{36} \) \(\mathstrut +\mathstrut 53q^{37} \) \(\mathstrut +\mathstrut 37q^{38} \) \(\mathstrut -\mathstrut 23q^{39} \) \(\mathstrut +\mathstrut 25q^{40} \) \(\mathstrut +\mathstrut 113q^{41} \) \(\mathstrut +\mathstrut 26q^{42} \) \(\mathstrut +\mathstrut 31q^{43} \) \(\mathstrut +\mathstrut 86q^{44} \) \(\mathstrut +\mathstrut 144q^{45} \) \(\mathstrut +\mathstrut 37q^{46} \) \(\mathstrut +\mathstrut 115q^{47} \) \(\mathstrut +\mathstrut 129q^{48} \) \(\mathstrut +\mathstrut 189q^{49} \) \(\mathstrut +\mathstrut 72q^{50} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 161q^{52} \) \(\mathstrut +\mathstrut 51q^{53} \) \(\mathstrut +\mathstrut 108q^{54} \) \(\mathstrut +\mathstrut 22q^{55} \) \(\mathstrut +\mathstrut 39q^{56} \) \(\mathstrut +\mathstrut 102q^{57} \) \(\mathstrut +\mathstrut 31q^{58} \) \(\mathstrut +\mathstrut 75q^{59} \) \(\mathstrut +\mathstrut 97q^{60} \) \(\mathstrut +\mathstrut 7q^{61} \) \(\mathstrut +\mathstrut 77q^{62} \) \(\mathstrut +\mathstrut 94q^{63} \) \(\mathstrut +\mathstrut 158q^{64} \) \(\mathstrut -\mathstrut 39q^{65} \) \(\mathstrut +\mathstrut 48q^{66} \) \(\mathstrut +\mathstrut 37q^{67} \) \(\mathstrut +\mathstrut 235q^{68} \) \(\mathstrut +\mathstrut 27q^{69} \) \(\mathstrut +\mathstrut 38q^{70} \) \(\mathstrut +\mathstrut 70q^{71} \) \(\mathstrut +\mathstrut 152q^{72} \) \(\mathstrut +\mathstrut 155q^{73} \) \(\mathstrut -\mathstrut 18q^{74} \) \(\mathstrut +\mathstrut 80q^{75} \) \(\mathstrut +\mathstrut 21q^{76} \) \(\mathstrut +\mathstrut 101q^{77} \) \(\mathstrut -\mathstrut 25q^{78} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 211q^{80} \) \(\mathstrut +\mathstrut 220q^{81} \) \(\mathstrut +\mathstrut 45q^{82} \) \(\mathstrut +\mathstrut 132q^{83} \) \(\mathstrut +\mathstrut 86q^{84} \) \(\mathstrut +\mathstrut 74q^{85} \) \(\mathstrut +\mathstrut 35q^{86} \) \(\mathstrut +\mathstrut 53q^{87} \) \(\mathstrut +\mathstrut 51q^{88} \) \(\mathstrut +\mathstrut 190q^{89} \) \(\mathstrut -\mathstrut 27q^{90} \) \(\mathstrut -\mathstrut 19q^{91} \) \(\mathstrut +\mathstrut 125q^{92} \) \(\mathstrut +\mathstrut 96q^{93} \) \(\mathstrut -\mathstrut 19q^{94} \) \(\mathstrut +\mathstrut 72q^{95} \) \(\mathstrut +\mathstrut 146q^{96} \) \(\mathstrut +\mathstrut 155q^{97} \) \(\mathstrut +\mathstrut 135q^{98} \) \(\mathstrut +\mathstrut 89q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78003 2.35788 5.72856 3.19176 −6.55498 −2.60821 −10.3655 2.55961 −8.87319
1.2 −2.75185 0.388525 5.57269 0.235747 −1.06916 −0.913788 −9.83151 −2.84905 −0.648741
1.3 −2.71116 −0.430887 5.35037 1.49097 1.16820 4.38752 −9.08338 −2.81434 −4.04226
1.4 −2.66092 −2.09208 5.08047 4.25784 5.56685 −2.19372 −8.19688 1.37680 −11.3297
1.5 −2.65345 0.515937 5.04077 −0.00500829 −1.36901 0.475978 −8.06853 −2.73381 0.0132892
1.6 −2.61991 −3.07843 4.86391 −2.25474 8.06519 3.02795 −7.50318 6.47672 5.90721
1.7 −2.53163 −1.76740 4.40915 3.70059 4.47441 1.97275 −6.09908 0.123713 −9.36852
1.8 −2.51520 3.16825 4.32624 1.81160 −7.96880 1.29182 −5.85097 7.03782 −4.55654
1.9 −2.51030 2.30230 4.30163 −2.11540 −5.77947 3.11614 −5.77778 2.30059 5.31029
1.10 −2.48921 1.66409 4.19617 0.253243 −4.14227 −2.12862 −5.46674 −0.230804 −0.630376
1.11 −2.43558 −1.59767 3.93207 −2.50698 3.89125 −1.32806 −4.70573 −0.447465 6.10595
1.12 −2.42534 −2.30091 3.88226 −2.29855 5.58047 −0.451041 −4.56512 2.29416 5.57476
1.13 −2.42132 −1.26535 3.86281 0.549340 3.06382 1.75655 −4.51045 −1.39889 −1.33013
1.14 −2.40103 1.80838 3.76492 2.80011 −4.34197 −2.97647 −4.23763 0.270237 −6.72314
1.15 −2.37112 0.716095 3.62222 −3.83331 −1.69795 −1.71381 −3.84647 −2.48721 9.08923
1.16 −2.35417 1.10999 3.54214 −1.30122 −2.61310 −1.47122 −3.63046 −1.76793 3.06330
1.17 −2.22460 3.13495 2.94885 −1.36986 −6.97401 0.348323 −2.11081 6.82790 3.04739
1.18 −2.19459 1.61581 2.81621 −2.64713 −3.54605 3.12381 −1.79125 −0.389142 5.80935
1.19 −2.16081 −2.05453 2.66911 −1.91838 4.43946 −4.69995 −1.44582 1.22110 4.14525
1.20 −2.15517 0.169019 2.64475 4.16569 −0.364265 2.92802 −1.38954 −2.97143 −8.97775
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.156
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(619\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{156} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).