Properties

Label 6014.2.a.j
Level 6014
Weight 2
Character orbit 6014.a
Self dual Yes
Analytic conductor 48.022
Analytic rank 0
Dimension 32
CM No

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Newspace parameters

Level: \( N \) = \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0220317756\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

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

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 −1.00000 −3.39762 1.00000 2.43962 3.39762 3.88412 −1.00000 8.54381 −2.43962
1.2 −1.00000 −3.38837 1.00000 −3.95025 3.38837 −2.93892 −1.00000 8.48103 3.95025
1.3 −1.00000 −2.82503 1.00000 −0.629658 2.82503 −0.109398 −1.00000 4.98077 0.629658
1.4 −1.00000 −2.75899 1.00000 2.79935 2.75899 −3.68185 −1.00000 4.61205 −2.79935
1.5 −1.00000 −2.60322 1.00000 0.943880 2.60322 3.39783 −1.00000 3.77677 −0.943880
1.6 −1.00000 −2.29039 1.00000 −1.87953 2.29039 0.391521 −1.00000 2.24587 1.87953
1.7 −1.00000 −2.05301 1.00000 −2.11820 2.05301 −1.54348 −1.00000 1.21485 2.11820
1.8 −1.00000 −1.88640 1.00000 4.44357 1.88640 −1.18572 −1.00000 0.558507 −4.44357
1.9 −1.00000 −1.64406 1.00000 −4.02843 1.64406 2.75493 −1.00000 −0.297060 4.02843
1.10 −1.00000 −1.52309 1.00000 −0.0721797 1.52309 3.89998 −1.00000 −0.680188 0.0721797
1.11 −1.00000 −1.01996 1.00000 −1.74888 1.01996 2.14281 −1.00000 −1.95968 1.74888
1.12 −1.00000 −0.848161 1.00000 4.13342 0.848161 1.58594 −1.00000 −2.28062 −4.13342
1.13 −1.00000 −0.676893 1.00000 −1.03316 0.676893 −4.68414 −1.00000 −2.54182 1.03316
1.14 −1.00000 −0.475406 1.00000 1.98672 0.475406 3.40549 −1.00000 −2.77399 −1.98672
1.15 −1.00000 −0.416033 1.00000 0.917555 0.416033 −0.743221 −1.00000 −2.82692 −0.917555
1.16 −1.00000 −0.0623199 1.00000 1.97455 0.0623199 −0.185636 −1.00000 −2.99612 −1.97455
1.17 −1.00000 0.00816480 1.00000 4.13053 −0.00816480 −5.05598 −1.00000 −2.99993 −4.13053
1.18 −1.00000 0.284664 1.00000 −2.60924 −0.284664 −1.90185 −1.00000 −2.91897 2.60924
1.19 −1.00000 0.502069 1.00000 0.295460 −0.502069 −2.57922 −1.00000 −2.74793 −0.295460
1.20 −1.00000 0.519470 1.00000 −3.34361 −0.519470 4.82750 −1.00000 −2.73015 3.34361
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(31\) \(1\)
\(97\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\).