Properties

Label 8026.2.a.b
Level 8026
Weight 2
Character orbit 8026.a
Self dual Yes
Analytic conductor 64.088
Analytic rank 1
Dimension 81
CM No

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Newspace parameters

Level: \( N \) = \( 8026 = 2 \cdot 4013 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8026.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0879326623\)
Analytic rank: \(1\)
Dimension: \(81\)
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.

\(\operatorname{Tr}(f)(q) = \) \(81q \) \(\mathstrut -\mathstrut 81q^{2} \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 81q^{4} \) \(\mathstrut -\mathstrut 26q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 81q^{8} \) \(\mathstrut +\mathstrut 59q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(81q \) \(\mathstrut -\mathstrut 81q^{2} \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 81q^{4} \) \(\mathstrut -\mathstrut 26q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 81q^{8} \) \(\mathstrut +\mathstrut 59q^{9} \) \(\mathstrut +\mathstrut 26q^{10} \) \(\mathstrut -\mathstrut 41q^{11} \) \(\mathstrut -\mathstrut 10q^{12} \) \(\mathstrut +\mathstrut 33q^{13} \) \(\mathstrut -\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 81q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 59q^{18} \) \(\mathstrut -\mathstrut 32q^{19} \) \(\mathstrut -\mathstrut 26q^{20} \) \(\mathstrut -\mathstrut 23q^{21} \) \(\mathstrut +\mathstrut 41q^{22} \) \(\mathstrut -\mathstrut 28q^{23} \) \(\mathstrut +\mathstrut 10q^{24} \) \(\mathstrut +\mathstrut 81q^{25} \) \(\mathstrut -\mathstrut 33q^{26} \) \(\mathstrut -\mathstrut 37q^{27} \) \(\mathstrut +\mathstrut 3q^{28} \) \(\mathstrut -\mathstrut 35q^{29} \) \(\mathstrut +\mathstrut 7q^{30} \) \(\mathstrut -\mathstrut 29q^{31} \) \(\mathstrut -\mathstrut 81q^{32} \) \(\mathstrut -\mathstrut 7q^{33} \) \(\mathstrut +\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 67q^{35} \) \(\mathstrut +\mathstrut 59q^{36} \) \(\mathstrut +\mathstrut 13q^{37} \) \(\mathstrut +\mathstrut 32q^{38} \) \(\mathstrut -\mathstrut 42q^{39} \) \(\mathstrut +\mathstrut 26q^{40} \) \(\mathstrut -\mathstrut 66q^{41} \) \(\mathstrut +\mathstrut 23q^{42} \) \(\mathstrut -\mathstrut 22q^{43} \) \(\mathstrut -\mathstrut 41q^{44} \) \(\mathstrut -\mathstrut 65q^{45} \) \(\mathstrut +\mathstrut 28q^{46} \) \(\mathstrut -\mathstrut 71q^{47} \) \(\mathstrut -\mathstrut 10q^{48} \) \(\mathstrut +\mathstrut 64q^{49} \) \(\mathstrut -\mathstrut 81q^{50} \) \(\mathstrut -\mathstrut 43q^{51} \) \(\mathstrut +\mathstrut 33q^{52} \) \(\mathstrut -\mathstrut 37q^{53} \) \(\mathstrut +\mathstrut 37q^{54} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 3q^{56} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut +\mathstrut 35q^{58} \) \(\mathstrut -\mathstrut 162q^{59} \) \(\mathstrut -\mathstrut 7q^{60} \) \(\mathstrut +\mathstrut 19q^{61} \) \(\mathstrut +\mathstrut 29q^{62} \) \(\mathstrut -\mathstrut 16q^{63} \) \(\mathstrut +\mathstrut 81q^{64} \) \(\mathstrut -\mathstrut 45q^{65} \) \(\mathstrut +\mathstrut 7q^{66} \) \(\mathstrut -\mathstrut 43q^{67} \) \(\mathstrut -\mathstrut 9q^{68} \) \(\mathstrut -\mathstrut 21q^{69} \) \(\mathstrut +\mathstrut 67q^{70} \) \(\mathstrut -\mathstrut 99q^{71} \) \(\mathstrut -\mathstrut 59q^{72} \) \(\mathstrut +\mathstrut 53q^{73} \) \(\mathstrut -\mathstrut 13q^{74} \) \(\mathstrut -\mathstrut 61q^{75} \) \(\mathstrut -\mathstrut 32q^{76} \) \(\mathstrut -\mathstrut 31q^{77} \) \(\mathstrut +\mathstrut 42q^{78} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 26q^{80} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut +\mathstrut 66q^{82} \) \(\mathstrut -\mathstrut 112q^{83} \) \(\mathstrut -\mathstrut 23q^{84} \) \(\mathstrut +\mathstrut 17q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut -\mathstrut 15q^{87} \) \(\mathstrut +\mathstrut 41q^{88} \) \(\mathstrut -\mathstrut 111q^{89} \) \(\mathstrut +\mathstrut 65q^{90} \) \(\mathstrut -\mathstrut 49q^{91} \) \(\mathstrut -\mathstrut 28q^{92} \) \(\mathstrut -\mathstrut 19q^{93} \) \(\mathstrut +\mathstrut 71q^{94} \) \(\mathstrut -\mathstrut 53q^{95} \) \(\mathstrut +\mathstrut 10q^{96} \) \(\mathstrut +\mathstrut 50q^{97} \) \(\mathstrut -\mathstrut 64q^{98} \) \(\mathstrut -\mathstrut 97q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.26929 1.00000 −1.66866 3.26929 −1.36572 −1.00000 7.68829 1.66866
1.2 −1.00000 −3.25240 1.00000 −3.99675 3.25240 3.80728 −1.00000 7.57814 3.99675
1.3 −1.00000 −3.23458 1.00000 −3.62121 3.23458 −0.933365 −1.00000 7.46248 3.62121
1.4 −1.00000 −3.21582 1.00000 3.05469 3.21582 −0.0968570 −1.00000 7.34152 −3.05469
1.5 −1.00000 −3.01169 1.00000 2.74703 3.01169 −2.66211 −1.00000 6.07025 −2.74703
1.6 −1.00000 −2.83489 1.00000 −3.31707 2.83489 3.86348 −1.00000 5.03658 3.31707
1.7 −1.00000 −2.80063 1.00000 −1.23643 2.80063 −3.84630 −1.00000 4.84350 1.23643
1.8 −1.00000 −2.77693 1.00000 2.15661 2.77693 3.32117 −1.00000 4.71136 −2.15661
1.9 −1.00000 −2.70930 1.00000 1.99036 2.70930 −3.73074 −1.00000 4.34031 −1.99036
1.10 −1.00000 −2.56900 1.00000 −2.27157 2.56900 3.80635 −1.00000 3.59977 2.27157
1.11 −1.00000 −2.53693 1.00000 0.430224 2.53693 2.70951 −1.00000 3.43604 −0.430224
1.12 −1.00000 −2.51028 1.00000 0.572444 2.51028 −2.83976 −1.00000 3.30151 −0.572444
1.13 −1.00000 −2.45434 1.00000 −0.137052 2.45434 1.42878 −1.00000 3.02378 0.137052
1.14 −1.00000 −2.44759 1.00000 0.910070 2.44759 −1.13462 −1.00000 2.99071 −0.910070
1.15 −1.00000 −2.39342 1.00000 1.94313 2.39342 4.16913 −1.00000 2.72848 −1.94313
1.16 −1.00000 −2.34885 1.00000 3.93410 2.34885 −0.279699 −1.00000 2.51708 −3.93410
1.17 −1.00000 −2.22289 1.00000 −1.72959 2.22289 1.27645 −1.00000 1.94125 1.72959
1.18 −1.00000 −2.20617 1.00000 −2.45356 2.20617 0.571999 −1.00000 1.86717 2.45356
1.19 −1.00000 −1.96246 1.00000 −3.42476 1.96246 −2.06086 −1.00000 0.851259 3.42476
1.20 −1.00000 −1.92493 1.00000 −0.857375 1.92493 1.26508 −1.00000 0.705338 0.857375
See all 81 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.81
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4013\) \(1\)