Properties

Label 6033.2.a.e
Level 6033
Weight 2
Character orbit 6033.a
Self dual Yes
Analytic conductor 48.174
Analytic rank 0
Dimension 97
CM No

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Newspace parameters

Level: \( N \) = \( 6033 = 3 \cdot 2011 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6033.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1737475394\)
Analytic rank: \(0\)
Dimension: \(97\)
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.

\(\operatorname{Tr}(f)(q) = \) \(97q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 97q^{3} \) \(\mathstrut +\mathstrut 120q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 50q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 97q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(97q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 97q^{3} \) \(\mathstrut +\mathstrut 120q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 50q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 97q^{9} \) \(\mathstrut +\mathstrut 35q^{10} \) \(\mathstrut +\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 120q^{12} \) \(\mathstrut +\mathstrut 67q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 158q^{16} \) \(\mathstrut +\mathstrut 25q^{17} \) \(\mathstrut +\mathstrut 12q^{18} \) \(\mathstrut +\mathstrut 51q^{19} \) \(\mathstrut +\mathstrut 10q^{20} \) \(\mathstrut +\mathstrut 50q^{21} \) \(\mathstrut +\mathstrut 39q^{22} \) \(\mathstrut +\mathstrut 87q^{23} \) \(\mathstrut +\mathstrut 30q^{24} \) \(\mathstrut +\mathstrut 149q^{25} \) \(\mathstrut +\mathstrut 14q^{26} \) \(\mathstrut +\mathstrut 97q^{27} \) \(\mathstrut +\mathstrut 83q^{28} \) \(\mathstrut +\mathstrut 23q^{29} \) \(\mathstrut +\mathstrut 35q^{30} \) \(\mathstrut +\mathstrut 72q^{31} \) \(\mathstrut +\mathstrut 57q^{32} \) \(\mathstrut +\mathstrut 18q^{33} \) \(\mathstrut +\mathstrut 28q^{34} \) \(\mathstrut +\mathstrut 45q^{35} \) \(\mathstrut +\mathstrut 120q^{36} \) \(\mathstrut +\mathstrut 72q^{37} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut +\mathstrut 67q^{39} \) \(\mathstrut +\mathstrut 90q^{40} \) \(\mathstrut +\mathstrut 5q^{41} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut +\mathstrut 122q^{43} \) \(\mathstrut +\mathstrut 11q^{44} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 56q^{46} \) \(\mathstrut +\mathstrut 49q^{47} \) \(\mathstrut +\mathstrut 158q^{48} \) \(\mathstrut +\mathstrut 167q^{49} \) \(\mathstrut +\mathstrut 13q^{50} \) \(\mathstrut +\mathstrut 25q^{51} \) \(\mathstrut +\mathstrut 128q^{52} \) \(\mathstrut +\mathstrut 30q^{53} \) \(\mathstrut +\mathstrut 12q^{54} \) \(\mathstrut +\mathstrut 120q^{55} \) \(\mathstrut -\mathstrut 21q^{56} \) \(\mathstrut +\mathstrut 51q^{57} \) \(\mathstrut +\mathstrut 37q^{58} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 158q^{61} \) \(\mathstrut +\mathstrut 17q^{62} \) \(\mathstrut +\mathstrut 50q^{63} \) \(\mathstrut +\mathstrut 212q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 39q^{66} \) \(\mathstrut +\mathstrut 77q^{67} \) \(\mathstrut +\mathstrut 56q^{68} \) \(\mathstrut +\mathstrut 87q^{69} \) \(\mathstrut +\mathstrut 9q^{70} \) \(\mathstrut +\mathstrut 38q^{71} \) \(\mathstrut +\mathstrut 30q^{72} \) \(\mathstrut +\mathstrut 82q^{73} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 149q^{75} \) \(\mathstrut +\mathstrut 93q^{76} \) \(\mathstrut +\mathstrut 49q^{77} \) \(\mathstrut +\mathstrut 14q^{78} \) \(\mathstrut +\mathstrut 134q^{79} \) \(\mathstrut -\mathstrut 25q^{80} \) \(\mathstrut +\mathstrut 97q^{81} \) \(\mathstrut +\mathstrut 53q^{82} \) \(\mathstrut +\mathstrut 69q^{83} \) \(\mathstrut +\mathstrut 83q^{84} \) \(\mathstrut +\mathstrut 72q^{85} \) \(\mathstrut +\mathstrut 23q^{87} \) \(\mathstrut +\mathstrut 107q^{88} \) \(\mathstrut +\mathstrut 35q^{90} \) \(\mathstrut +\mathstrut 84q^{91} \) \(\mathstrut +\mathstrut 108q^{92} \) \(\mathstrut +\mathstrut 72q^{93} \) \(\mathstrut +\mathstrut 65q^{94} \) \(\mathstrut +\mathstrut 89q^{95} \) \(\mathstrut +\mathstrut 57q^{96} \) \(\mathstrut +\mathstrut 65q^{97} \) \(\mathstrut +\mathstrut 18q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.77569 1.00000 5.70444 −4.08383 −2.77569 2.32651 −10.2824 1.00000 11.3354
1.2 −2.72701 1.00000 5.43660 −1.98915 −2.72701 −1.33626 −9.37164 1.00000 5.42444
1.3 −2.72505 1.00000 5.42589 −0.505520 −2.72505 −2.64822 −9.33570 1.00000 1.37757
1.4 −2.69536 1.00000 5.26496 2.52315 −2.69536 0.986697 −8.80025 1.00000 −6.80079
1.5 −2.65791 1.00000 5.06448 3.14919 −2.65791 3.30721 −8.14510 1.00000 −8.37025
1.6 −2.63042 1.00000 4.91910 −0.782993 −2.63042 4.49130 −7.67844 1.00000 2.05960
1.7 −2.53542 1.00000 4.42834 −3.51748 −2.53542 −3.36027 −6.15684 1.00000 8.91826
1.8 −2.50001 1.00000 4.25007 −1.07876 −2.50001 3.84793 −5.62521 1.00000 2.69691
1.9 −2.41345 1.00000 3.82475 3.09815 −2.41345 4.57422 −4.40394 1.00000 −7.47724
1.10 −2.38207 1.00000 3.67426 −3.40207 −2.38207 2.20870 −3.98822 1.00000 8.10396
1.11 −2.37614 1.00000 3.64603 3.12222 −2.37614 −0.986181 −3.91118 1.00000 −7.41883
1.12 −2.30219 1.00000 3.30008 −3.13763 −2.30219 1.81298 −2.99304 1.00000 7.22342
1.13 −2.19685 1.00000 2.82615 3.79567 −2.19685 −1.66772 −1.81492 1.00000 −8.33851
1.14 −2.11811 1.00000 2.48641 0.996414 −2.11811 −4.64745 −1.03027 1.00000 −2.11052
1.15 −2.06099 1.00000 2.24767 −0.122069 −2.06099 −4.45814 −0.510454 1.00000 0.251582
1.16 −2.05972 1.00000 2.24246 −1.93322 −2.05972 2.16498 −0.499401 1.00000 3.98191
1.17 −1.96725 1.00000 1.87006 −1.28401 −1.96725 2.13094 0.255618 1.00000 2.52597
1.18 −1.95525 1.00000 1.82299 1.82748 −1.95525 −0.999221 0.346105 1.00000 −3.57318
1.19 −1.92742 1.00000 1.71496 −2.34894 −1.92742 −3.43132 0.549392 1.00000 4.52741
1.20 −1.88583 1.00000 1.55635 −4.37058 −1.88583 3.62055 0.836652 1.00000 8.24216
See all 97 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.97
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2011\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{97} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\).