Properties

Label 6007.2.a.b
Level 6007
Weight 2
Character orbit 6007.a
Self dual Yes
Analytic conductor 47.966
Analytic rank 1
Dimension 237
CM No

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Newspace parameters

Level: \( N \) = \( 6007 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9661364942\)
Analytic rank: \(1\)
Dimension: \(237\)
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.

\(\operatorname{Tr}(f)(q) = \) \(237q \) \(\mathstrut -\mathstrut 26q^{2} \) \(\mathstrut -\mathstrut 24q^{3} \) \(\mathstrut +\mathstrut 226q^{4} \) \(\mathstrut -\mathstrut 67q^{5} \) \(\mathstrut -\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 37q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 189q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(237q \) \(\mathstrut -\mathstrut 26q^{2} \) \(\mathstrut -\mathstrut 24q^{3} \) \(\mathstrut +\mathstrut 226q^{4} \) \(\mathstrut -\mathstrut 67q^{5} \) \(\mathstrut -\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 37q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 189q^{9} \) \(\mathstrut -\mathstrut 39q^{10} \) \(\mathstrut -\mathstrut 38q^{11} \) \(\mathstrut -\mathstrut 67q^{12} \) \(\mathstrut -\mathstrut 52q^{13} \) \(\mathstrut -\mathstrut 54q^{14} \) \(\mathstrut -\mathstrut 24q^{15} \) \(\mathstrut +\mathstrut 208q^{16} \) \(\mathstrut -\mathstrut 255q^{17} \) \(\mathstrut -\mathstrut 71q^{18} \) \(\mathstrut -\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut 154q^{20} \) \(\mathstrut -\mathstrut 60q^{21} \) \(\mathstrut -\mathstrut 39q^{22} \) \(\mathstrut -\mathstrut 118q^{23} \) \(\mathstrut -\mathstrut 85q^{24} \) \(\mathstrut +\mathstrut 170q^{25} \) \(\mathstrut -\mathstrut 61q^{26} \) \(\mathstrut -\mathstrut 87q^{27} \) \(\mathstrut -\mathstrut 99q^{28} \) \(\mathstrut -\mathstrut 87q^{29} \) \(\mathstrut -\mathstrut 30q^{30} \) \(\mathstrut -\mathstrut 28q^{31} \) \(\mathstrut -\mathstrut 156q^{32} \) \(\mathstrut -\mathstrut 173q^{33} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 113q^{35} \) \(\mathstrut +\mathstrut 152q^{36} \) \(\mathstrut -\mathstrut 49q^{37} \) \(\mathstrut -\mathstrut 145q^{38} \) \(\mathstrut -\mathstrut 49q^{39} \) \(\mathstrut -\mathstrut 91q^{40} \) \(\mathstrut -\mathstrut 197q^{41} \) \(\mathstrut -\mathstrut 61q^{42} \) \(\mathstrut -\mathstrut 63q^{43} \) \(\mathstrut -\mathstrut 106q^{44} \) \(\mathstrut -\mathstrut 181q^{45} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 119q^{47} \) \(\mathstrut -\mathstrut 142q^{48} \) \(\mathstrut +\mathstrut 150q^{49} \) \(\mathstrut -\mathstrut 89q^{50} \) \(\mathstrut -\mathstrut 40q^{51} \) \(\mathstrut -\mathstrut 97q^{52} \) \(\mathstrut -\mathstrut 190q^{53} \) \(\mathstrut -\mathstrut 97q^{54} \) \(\mathstrut -\mathstrut 55q^{55} \) \(\mathstrut -\mathstrut 154q^{56} \) \(\mathstrut -\mathstrut 202q^{57} \) \(\mathstrut -\mathstrut 27q^{58} \) \(\mathstrut -\mathstrut 86q^{59} \) \(\mathstrut -\mathstrut 48q^{60} \) \(\mathstrut -\mathstrut 96q^{61} \) \(\mathstrut -\mathstrut 239q^{62} \) \(\mathstrut -\mathstrut 149q^{63} \) \(\mathstrut +\mathstrut 183q^{64} \) \(\mathstrut -\mathstrut 259q^{65} \) \(\mathstrut -\mathstrut 72q^{66} \) \(\mathstrut -\mathstrut 28q^{67} \) \(\mathstrut -\mathstrut 482q^{68} \) \(\mathstrut -\mathstrut 83q^{69} \) \(\mathstrut +\mathstrut 20q^{70} \) \(\mathstrut -\mathstrut 63q^{71} \) \(\mathstrut -\mathstrut 193q^{72} \) \(\mathstrut -\mathstrut 206q^{73} \) \(\mathstrut -\mathstrut 132q^{74} \) \(\mathstrut -\mathstrut 89q^{75} \) \(\mathstrut -\mathstrut 11q^{76} \) \(\mathstrut -\mathstrut 179q^{77} \) \(\mathstrut -\mathstrut 58q^{78} \) \(\mathstrut -\mathstrut 32q^{79} \) \(\mathstrut -\mathstrut 320q^{80} \) \(\mathstrut +\mathstrut 57q^{81} \) \(\mathstrut -\mathstrut 77q^{82} \) \(\mathstrut -\mathstrut 245q^{83} \) \(\mathstrut -\mathstrut 133q^{84} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut -\mathstrut 39q^{86} \) \(\mathstrut -\mathstrut 179q^{87} \) \(\mathstrut -\mathstrut 104q^{88} \) \(\mathstrut -\mathstrut 227q^{89} \) \(\mathstrut -\mathstrut 146q^{90} \) \(\mathstrut -\mathstrut 36q^{91} \) \(\mathstrut -\mathstrut 315q^{92} \) \(\mathstrut -\mathstrut 87q^{93} \) \(\mathstrut -\mathstrut 48q^{94} \) \(\mathstrut -\mathstrut 111q^{95} \) \(\mathstrut -\mathstrut 134q^{96} \) \(\mathstrut -\mathstrut 221q^{97} \) \(\mathstrut -\mathstrut 161q^{98} \) \(\mathstrut -\mathstrut 68q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.79278 2.70470 5.79960 2.04099 −7.55362 −2.59733 −10.6114 4.31540 −5.70003
1.2 −2.78154 3.16840 5.73694 −2.96782 −8.81302 0.536102 −10.3944 7.03875 8.25511
1.3 −2.77308 −0.650314 5.68998 0.444469 1.80337 −4.52369 −10.2326 −2.57709 −1.23255
1.4 −2.73398 −2.84716 5.47464 −1.36635 7.78406 −2.19683 −9.49960 5.10630 3.73558
1.5 −2.72886 −1.67732 5.44669 −2.00878 4.57718 3.72534 −9.40555 −0.186597 5.48170
1.6 −2.72815 1.45530 5.44281 2.25925 −3.97027 3.02620 −9.39250 −0.882113 −6.16358
1.7 −2.72222 −1.56731 5.41047 −3.83172 4.26655 −2.76166 −9.28405 −0.543552 10.4308
1.8 −2.71930 −0.422286 5.39458 −1.71313 1.14832 0.813338 −9.23087 −2.82167 4.65850
1.9 −2.71184 0.480935 5.35409 −1.73647 −1.30422 3.65301 −9.09575 −2.76870 4.70903
1.10 −2.70797 −0.668026 5.33310 2.30781 1.80899 −3.86069 −9.02593 −2.55374 −6.24947
1.11 −2.69134 −2.16445 5.24332 1.36308 5.82528 5.10442 −8.72889 1.68484 −3.66852
1.12 −2.68501 −2.77509 5.20930 3.15364 7.45116 −1.25017 −8.61702 4.70113 −8.46756
1.13 −2.66512 −3.23472 5.10287 −3.53010 8.62093 1.66538 −8.26953 7.46344 9.40814
1.14 −2.63584 2.53685 4.94764 0.682671 −6.68672 1.00934 −7.76949 3.43560 −1.79941
1.15 −2.62005 0.737084 4.86469 −2.98180 −1.93120 −0.252259 −7.50563 −2.45671 7.81248
1.16 −2.61224 1.23083 4.82380 3.92824 −3.21523 −2.33975 −7.37643 −1.48505 −10.2615
1.17 −2.54661 1.71939 4.48524 −4.31092 −4.37861 4.46436 −6.32895 −0.0437112 10.9783
1.18 −2.54016 0.887378 4.45240 −3.77266 −2.25408 −4.31374 −6.22948 −2.21256 9.58314
1.19 −2.52945 −1.63373 4.39812 1.74511 4.13244 0.157946 −6.06591 −0.330925 −4.41416
1.20 −2.52132 1.70278 4.35706 −2.82870 −4.29327 −1.18756 −5.94292 −0.100525 7.13206
See next 80 embeddings (of 237 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.237
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(6007\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{237} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6007))\).