Properties

Label 6041.2.a.e
Level 6041
Weight 2
Character orbit 6041.a
Self dual Yes
Analytic conductor 48.238
Analytic rank 0
Dimension 112
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: \(0\)
Dimension: \(112\)
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.

\(\operatorname{Tr}(f)(q) = \) \(112q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 131q^{4} \) \(\mathstrut +\mathstrut 13q^{5} \) \(\mathstrut +\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 112q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 116q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(112q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 131q^{4} \) \(\mathstrut +\mathstrut 13q^{5} \) \(\mathstrut +\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 112q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 116q^{9} \) \(\mathstrut +\mathstrut 32q^{10} \) \(\mathstrut +\mathstrut 14q^{11} \) \(\mathstrut +\mathstrut 36q^{12} \) \(\mathstrut +\mathstrut 22q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut +\mathstrut 19q^{15} \) \(\mathstrut +\mathstrut 169q^{16} \) \(\mathstrut +\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 18q^{18} \) \(\mathstrut +\mathstrut 52q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut -\mathstrut 14q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 38q^{23} \) \(\mathstrut +\mathstrut 64q^{24} \) \(\mathstrut +\mathstrut 99q^{25} \) \(\mathstrut +\mathstrut 45q^{26} \) \(\mathstrut +\mathstrut 65q^{27} \) \(\mathstrut -\mathstrut 131q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut 133q^{31} \) \(\mathstrut -\mathstrut 26q^{32} \) \(\mathstrut +\mathstrut 27q^{33} \) \(\mathstrut +\mathstrut 52q^{34} \) \(\mathstrut -\mathstrut 13q^{35} \) \(\mathstrut +\mathstrut 183q^{36} \) \(\mathstrut -\mathstrut 13q^{37} \) \(\mathstrut +\mathstrut 20q^{38} \) \(\mathstrut +\mathstrut 74q^{39} \) \(\mathstrut +\mathstrut 92q^{40} \) \(\mathstrut +\mathstrut 25q^{41} \) \(\mathstrut -\mathstrut 18q^{42} \) \(\mathstrut -\mathstrut 11q^{43} \) \(\mathstrut +\mathstrut 16q^{44} \) \(\mathstrut +\mathstrut 63q^{45} \) \(\mathstrut +\mathstrut 28q^{46} \) \(\mathstrut +\mathstrut 71q^{47} \) \(\mathstrut +\mathstrut 70q^{48} \) \(\mathstrut +\mathstrut 112q^{49} \) \(\mathstrut +\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 57q^{51} \) \(\mathstrut +\mathstrut 79q^{52} \) \(\mathstrut -\mathstrut 10q^{53} \) \(\mathstrut +\mathstrut 75q^{54} \) \(\mathstrut +\mathstrut 146q^{55} \) \(\mathstrut +\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 83q^{57} \) \(\mathstrut -\mathstrut 19q^{58} \) \(\mathstrut +\mathstrut 56q^{59} \) \(\mathstrut -\mathstrut 3q^{60} \) \(\mathstrut +\mathstrut 80q^{61} \) \(\mathstrut +\mathstrut 42q^{62} \) \(\mathstrut -\mathstrut 116q^{63} \) \(\mathstrut +\mathstrut 263q^{64} \) \(\mathstrut -\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 48q^{66} \) \(\mathstrut +\mathstrut 29q^{67} \) \(\mathstrut +\mathstrut 57q^{68} \) \(\mathstrut +\mathstrut 56q^{69} \) \(\mathstrut -\mathstrut 32q^{70} \) \(\mathstrut +\mathstrut 100q^{71} \) \(\mathstrut -\mathstrut 62q^{72} \) \(\mathstrut +\mathstrut 73q^{73} \) \(\mathstrut +\mathstrut 24q^{74} \) \(\mathstrut +\mathstrut 89q^{75} \) \(\mathstrut +\mathstrut 155q^{76} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut +\mathstrut 33q^{78} \) \(\mathstrut +\mathstrut 140q^{79} \) \(\mathstrut +\mathstrut 80q^{80} \) \(\mathstrut +\mathstrut 120q^{81} \) \(\mathstrut +\mathstrut 114q^{82} \) \(\mathstrut +\mathstrut 36q^{83} \) \(\mathstrut -\mathstrut 36q^{84} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 12q^{86} \) \(\mathstrut +\mathstrut 96q^{87} \) \(\mathstrut +\mathstrut 29q^{88} \) \(\mathstrut +\mathstrut 47q^{89} \) \(\mathstrut +\mathstrut 52q^{90} \) \(\mathstrut -\mathstrut 22q^{91} \) \(\mathstrut +\mathstrut 81q^{92} \) \(\mathstrut -\mathstrut 10q^{93} \) \(\mathstrut +\mathstrut 127q^{94} \) \(\mathstrut +\mathstrut 96q^{95} \) \(\mathstrut +\mathstrut 175q^{96} \) \(\mathstrut +\mathstrut 80q^{97} \) \(\mathstrut -\mathstrut 3q^{98} \) \(\mathstrut +\mathstrut 74q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81373 −3.44120 5.91707 1.30649 9.68259 −1.00000 −11.0216 8.84183 −3.67611
1.2 −2.80869 2.13107 5.88877 −3.52544 −5.98552 −1.00000 −10.9224 1.54145 9.90189
1.3 −2.77254 −2.14289 5.68701 −0.156494 5.94125 −1.00000 −10.2224 1.59196 0.433887
1.4 −2.76309 2.60057 5.63466 −1.29315 −7.18561 −1.00000 −10.0429 3.76297 3.57310
1.5 −2.71246 −0.244067 5.35744 −2.10613 0.662022 −1.00000 −9.10692 −2.94043 5.71281
1.6 −2.71151 1.27956 5.35230 1.05589 −3.46954 −1.00000 −9.08979 −1.36273 −2.86306
1.7 −2.67889 −2.12545 5.17645 −3.74850 5.69384 −1.00000 −8.50937 1.51753 10.0418
1.8 −2.57911 −0.702301 4.65183 2.85376 1.81131 −1.00000 −6.83937 −2.50677 −7.36016
1.9 −2.52776 3.15958 4.38957 3.22941 −7.98667 −1.00000 −6.04026 6.98297 −8.16316
1.10 −2.46248 −0.0194781 4.06380 −1.44141 0.0479644 −1.00000 −5.08207 −2.99962 3.54944
1.11 −2.45673 3.01946 4.03553 −1.39061 −7.41801 −1.00000 −5.00075 6.11715 3.41635
1.12 −2.43414 −1.84866 3.92503 3.83434 4.49989 −1.00000 −4.68578 0.417546 −9.33330
1.13 −2.41077 2.26439 3.81183 4.38346 −5.45893 −1.00000 −4.36791 2.12745 −10.5675
1.14 −2.37296 0.810716 3.63095 0.528258 −1.92380 −1.00000 −3.87017 −2.34274 −1.25354
1.15 −2.27713 −2.93009 3.18534 −2.32769 6.67221 −1.00000 −2.69918 5.58544 5.30045
1.16 −2.25493 0.275101 3.08473 2.79647 −0.620335 −1.00000 −2.44598 −2.92432 −6.30585
1.17 −2.22590 −2.92856 2.95465 1.01796 6.51870 −1.00000 −2.12496 5.57647 −2.26588
1.18 −2.13001 −1.53283 2.53693 −2.08639 3.26494 −1.00000 −1.14367 −0.650428 4.44403
1.19 −2.10602 −0.719393 2.43533 1.34496 1.51506 −1.00000 −0.916817 −2.48247 −2.83252
1.20 −2.06364 −0.217047 2.25861 3.48472 0.447908 −1.00000 −0.533675 −2.95289 −7.19121
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.112
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 \(T_{2}^{112} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).