Properties

Label 6041.2.a.d
Level 6041
Weight 2
Character orbit 6041.a
Self dual Yes
Analytic conductor 48.238
Analytic rank 1
Dimension 101
CM No

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Newspace parameters

Level: \( N \) = \( 6041 = 7 \cdot 863 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6041.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2376278611\)
Analytic rank: \(1\)
Dimension: \(101\)
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.

\(\operatorname{Tr}(f)(q) = \) \(101q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 17q^{3} \) \(\mathstrut +\mathstrut 85q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 101q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 88q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(101q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 17q^{3} \) \(\mathstrut +\mathstrut 85q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 101q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 88q^{9} \) \(\mathstrut -\mathstrut 23q^{10} \) \(\mathstrut -\mathstrut 13q^{11} \) \(\mathstrut -\mathstrut 31q^{12} \) \(\mathstrut -\mathstrut 35q^{13} \) \(\mathstrut -\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 45q^{16} \) \(\mathstrut -\mathstrut 19q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 59q^{19} \) \(\mathstrut -\mathstrut 31q^{20} \) \(\mathstrut +\mathstrut 17q^{21} \) \(\mathstrut -\mathstrut 13q^{22} \) \(\mathstrut -\mathstrut 29q^{23} \) \(\mathstrut -\mathstrut 59q^{24} \) \(\mathstrut +\mathstrut 103q^{25} \) \(\mathstrut -\mathstrut 18q^{26} \) \(\mathstrut -\mathstrut 47q^{27} \) \(\mathstrut -\mathstrut 85q^{28} \) \(\mathstrut -\mathstrut 26q^{29} \) \(\mathstrut -\mathstrut 8q^{30} \) \(\mathstrut -\mathstrut 125q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 18q^{33} \) \(\mathstrut -\mathstrut 66q^{34} \) \(\mathstrut +\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 40q^{36} \) \(\mathstrut +\mathstrut 22q^{37} \) \(\mathstrut -\mathstrut 31q^{38} \) \(\mathstrut -\mathstrut 94q^{39} \) \(\mathstrut -\mathstrut 79q^{40} \) \(\mathstrut -\mathstrut 39q^{41} \) \(\mathstrut +\mathstrut 17q^{42} \) \(\mathstrut -\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 53q^{44} \) \(\mathstrut -\mathstrut 50q^{45} \) \(\mathstrut -\mathstrut 37q^{46} \) \(\mathstrut -\mathstrut 47q^{47} \) \(\mathstrut -\mathstrut 81q^{48} \) \(\mathstrut +\mathstrut 101q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 23q^{51} \) \(\mathstrut -\mathstrut 56q^{52} \) \(\mathstrut -\mathstrut 5q^{53} \) \(\mathstrut -\mathstrut 77q^{54} \) \(\mathstrut -\mathstrut 155q^{55} \) \(\mathstrut +\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 61q^{57} \) \(\mathstrut -\mathstrut 31q^{58} \) \(\mathstrut -\mathstrut 33q^{59} \) \(\mathstrut -\mathstrut 48q^{60} \) \(\mathstrut -\mathstrut 96q^{61} \) \(\mathstrut -\mathstrut 38q^{62} \) \(\mathstrut -\mathstrut 88q^{63} \) \(\mathstrut -\mathstrut 33q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 91q^{66} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 41q^{68} \) \(\mathstrut -\mathstrut 91q^{69} \) \(\mathstrut +\mathstrut 23q^{70} \) \(\mathstrut -\mathstrut 116q^{71} \) \(\mathstrut -\mathstrut 5q^{72} \) \(\mathstrut -\mathstrut 62q^{73} \) \(\mathstrut -\mathstrut 23q^{74} \) \(\mathstrut -\mathstrut 94q^{75} \) \(\mathstrut -\mathstrut 112q^{76} \) \(\mathstrut +\mathstrut 13q^{77} \) \(\mathstrut +\mathstrut 17q^{78} \) \(\mathstrut -\mathstrut 127q^{79} \) \(\mathstrut -\mathstrut 87q^{80} \) \(\mathstrut +\mathstrut 37q^{81} \) \(\mathstrut -\mathstrut 118q^{82} \) \(\mathstrut -\mathstrut 58q^{83} \) \(\mathstrut +\mathstrut 31q^{84} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut -\mathstrut 26q^{86} \) \(\mathstrut -\mathstrut 82q^{87} \) \(\mathstrut -\mathstrut 40q^{88} \) \(\mathstrut -\mathstrut 57q^{89} \) \(\mathstrut -\mathstrut 123q^{90} \) \(\mathstrut +\mathstrut 35q^{91} \) \(\mathstrut -\mathstrut 28q^{92} \) \(\mathstrut -\mathstrut 10q^{93} \) \(\mathstrut -\mathstrut 107q^{94} \) \(\mathstrut -\mathstrut 70q^{95} \) \(\mathstrut -\mathstrut 76q^{96} \) \(\mathstrut -\mathstrut 69q^{97} \) \(\mathstrut +\mathstrut 3q^{98} \) \(\mathstrut -\mathstrut 67q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.71484 1.06768 5.37033 2.50113 −2.89858 −1.00000 −9.14989 −1.86006 −6.79016
1.2 −2.68338 −0.794633 5.20055 −0.431207 2.13231 −1.00000 −8.58829 −2.36856 1.15709
1.3 −2.67103 −2.02602 5.13442 3.05924 5.41156 −1.00000 −8.37215 1.10475 −8.17132
1.4 −2.59007 −1.31514 4.70847 −2.80668 3.40631 −1.00000 −7.01513 −1.27040 7.26950
1.5 −2.44864 −2.39074 3.99581 −0.961081 5.85406 −1.00000 −4.88702 2.71565 2.35334
1.6 −2.44717 0.719096 3.98863 −3.39767 −1.75975 −1.00000 −4.86651 −2.48290 8.31466
1.7 −2.41856 2.09832 3.84945 −1.02982 −5.07493 −1.00000 −4.47302 1.40297 2.49069
1.8 −2.41637 −2.07802 3.83883 1.82876 5.02127 −1.00000 −4.44330 1.31818 −4.41896
1.9 −2.41440 0.0560859 3.82935 0.0819711 −0.135414 −1.00000 −4.41678 −2.99685 −0.197911
1.10 −2.41346 2.83250 3.82477 1.74211 −6.83611 −1.00000 −4.40400 5.02306 −4.20452
1.11 −2.37665 1.96222 3.64847 −3.56190 −4.66352 −1.00000 −3.91784 0.850318 8.46540
1.12 −2.34195 3.16467 3.48474 −1.79579 −7.41150 −1.00000 −3.47719 7.01511 4.20566
1.13 −2.15616 −2.99968 2.64901 −1.14985 6.46779 −1.00000 −1.39937 5.99811 2.47926
1.14 −2.14147 −2.96762 2.58591 4.08817 6.35507 −1.00000 −1.25470 5.80675 −8.75470
1.15 −2.08521 1.06538 2.34812 1.41345 −2.22154 −1.00000 −0.725898 −1.86497 −2.94734
1.16 −2.00569 2.11059 2.02278 3.12491 −4.23318 −1.00000 −0.0456843 1.45460 −6.26760
1.17 −1.98854 −0.0463309 1.95428 −0.0496669 0.0921307 −1.00000 0.0909182 −2.99785 0.0987645
1.18 −1.87092 −1.43026 1.50033 1.61115 2.67590 −1.00000 0.934843 −0.954356 −3.01432
1.19 −1.83237 1.94542 1.35759 1.21630 −3.56474 −1.00000 1.17714 0.784666 −2.22871
1.20 −1.79653 2.70133 1.22753 2.96874 −4.85303 −1.00000 1.38776 4.29718 −5.33345
See next 80 embeddings (of 101 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.101
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(863\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{101} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).