Properties

Label 137.14.a.b
Level $137$
Weight $14$
Character orbit 137.a
Self dual yes
Analytic conductor $146.906$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,14,Mod(1,137)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("137.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 137.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(146.906372664\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 192 q^{2} + 5103 q^{3} + 331776 q^{4} + 62500 q^{5} + 222425 q^{6} + 1997511 q^{7} + 2334726 q^{8} + 41009413 q^{9} + 13137365 q^{10} + 7820697 q^{11} + 42745784 q^{12} + 66267378 q^{13} + 36345348 q^{14}+ \cdots - 7632076398172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −176.599 2101.11 22995.1 30122.1 −371053. 6899.61 −2.61420e6 2.82035e6 −5.31953e6
1.2 −174.361 430.472 22209.7 −10966.2 −75057.4 458433. −2.44415e6 −1.40902e6 1.91208e6
1.3 −169.735 1771.08 20617.8 −15582.9 −300614. −412907. −2.10909e6 1.54241e6 2.64496e6
1.4 −167.481 459.119 19858.0 −40210.0 −76893.8 −495349. −1.95384e6 −1.38353e6 6.73442e6
1.5 −161.398 −1562.94 17857.2 −39179.2 252254. −67632.8 −1.55995e6 848449. 6.32344e6
1.6 −159.066 −1941.36 17109.9 48917.9 308804. 406620. −1.41854e6 2.17455e6 −7.78116e6
1.7 −158.011 −2326.30 16775.5 12974.5 367581. −127352. −1.35628e6 3.81736e6 −2.05012e6
1.8 −152.124 706.902 14949.8 −49661.5 −107537. 337884. −1.02802e6 −1.09461e6 7.55472e6
1.9 −145.399 −796.895 12948.8 3191.23 115868. −271948. −691639. −959281. −464001.
1.10 −144.768 1452.52 12765.9 38936.1 −210279. −275181. −662158. 515494. −5.63672e6
1.11 −136.560 999.507 10456.5 −5911.73 −136492. −349994. −309244. −595310. 807304.
1.12 −135.596 −1087.26 10194.4 26538.1 147429. 431959. −271517. −412182. −3.59848e6
1.13 −133.940 −738.808 9747.79 9283.76 98955.6 246144. −208382. −1.04849e6 −1.24346e6
1.14 −133.260 −1119.69 9566.31 −37065.1 149210. 577387. −183142. −340624. 4.93930e6
1.15 −128.799 2347.34 8397.10 47986.7 −302335. 546065. −26416.7 3.91570e6 −6.18062e6
1.16 −126.636 −1923.59 7844.59 −50613.5 243595. −594202. 43994.6 2.10586e6 6.40947e6
1.17 −108.666 1340.42 3616.32 18488.0 −145658. 536536. 497221. 202405. −2.00902e6
1.18 −105.594 −587.209 2958.09 62024.0 62005.8 287492. 552669. −1.24951e6 −6.54936e6
1.19 −101.070 −1676.86 2023.20 52809.5 169481. −165775. 623482. 1.21753e6 −5.33747e6
1.20 −96.9118 708.896 1199.89 −24990.2 −68700.4 −24881.0 677618. −1.09179e6 2.42184e6
See all 76 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.76
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(137\) \( -1 \)

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 137.14.a.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.14.a.b ✓ 76 1.a even 1 1 trivial