Properties

Label 2151.4.a.f
Level $2151$
Weight $4$
Character orbit 2151.a
Self dual yes
Analytic conductor $126.913$
Analytic rank $0$
Dimension $37$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(126.913108422\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: no (minimal twist has level 239)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37q - 4q^{2} + 170q^{4} - 43q^{5} + 60q^{7} - 27q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 37q - 4q^{2} + 170q^{4} - 43q^{5} + 60q^{7} - 27q^{8} + 147q^{10} - 55q^{11} + 250q^{13} - 169q^{14} + 918q^{16} - 189q^{17} + 550q^{19} - 486q^{20} + 226q^{22} - 74q^{23} + 1604q^{25} - 560q^{26} + 829q^{28} - 389q^{29} + 1107q^{31} - 125q^{32} + 1423q^{34} - 270q^{35} + 1002q^{37} - 1037q^{38} + 1536q^{40} - 1518q^{41} + 1098q^{43} - 1037q^{44} + 1030q^{46} - 1214q^{47} + 4663q^{49} - 929q^{50} + 2895q^{52} - 904q^{53} + 1350q^{55} - 2556q^{56} + 1396q^{58} - 1658q^{59} + 2313q^{61} + 4519q^{62} + 3807q^{64} + 56q^{65} + 1535q^{67} + 6526q^{68} - 4099q^{70} + 3255q^{71} + 3154q^{73} + 2629q^{74} + 1981q^{76} + 3734q^{77} + 2260q^{79} + 8242q^{80} - 9898q^{82} + 939q^{83} + 1272q^{85} + 3457q^{86} - 1808q^{88} - 1486q^{89} + 174q^{91} + 14076q^{92} - 984q^{94} + 1828q^{95} + 6148q^{97} + 6243q^{98} + O(q^{100}) \)

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 −5.58089 0 23.1463 10.0408 0 25.4163 −84.5299 0 −56.0363
1.2 −5.29599 0 20.0475 −6.13814 0 −8.89768 −63.8037 0 32.5075
1.3 −5.11871 0 18.2012 −0.668880 0 7.37421 −52.2167 0 3.42380
1.4 −4.90101 0 16.0199 −18.8248 0 −10.8883 −39.3055 0 92.2607
1.5 −4.79841 0 15.0248 14.4400 0 1.92684 −33.7078 0 −69.2892
1.6 −4.53815 0 12.5948 −19.7043 0 34.6627 −20.8518 0 89.4211
1.7 −4.41344 0 11.4784 −16.6195 0 −34.6158 −15.3518 0 73.3492
1.8 −3.91247 0 7.30741 2.39092 0 22.0359 2.70973 0 −9.35442
1.9 −3.84652 0 6.79575 −10.3358 0 2.00272 4.63218 0 39.7568
1.10 −3.37030 0 3.35895 1.43026 0 27.2799 15.6418 0 −4.82040
1.11 −2.83758 0 0.0518438 9.05330 0 −30.7893 22.5535 0 −25.6894
1.12 −2.73040 0 −0.544930 −10.5995 0 15.6616 23.3311 0 28.9408
1.13 −2.40786 0 −2.20221 19.1943 0 10.9735 24.5655 0 −46.2171
1.14 −1.50491 0 −5.73525 −18.0796 0 −23.9080 20.6703 0 27.2082
1.15 −1.28174 0 −6.35714 −12.7359 0 11.6522 18.4021 0 16.3241
1.16 −1.09413 0 −6.80288 10.8819 0 4.64141 16.1963 0 −11.9062
1.17 −0.918355 0 −7.15662 −19.7054 0 10.7153 13.9192 0 18.0965
1.18 −0.412743 0 −7.82964 15.2997 0 4.25811 6.53358 0 −6.31484
1.19 −0.271729 0 −7.92616 11.3325 0 −27.2237 4.32760 0 −3.07937
1.20 −0.123948 0 −7.98464 −3.85596 0 −18.4037 1.98126 0 0.477937
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(239\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.4.a.f 37
3.b odd 2 1 239.4.a.b 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.4.a.b 37 3.b odd 2 1
2151.4.a.f 37 1.a even 1 1 trivial