Properties

Label 261.2.i.a
Level $261$
Weight $2$
Character orbit 261.i
Analytic conductor $2.084$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [261,2,Mod(115,261)] 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("261.115"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 24 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} - 16 q^{9} - 2 q^{13} - 20 q^{16} + 20 q^{20} - 10 q^{22} - 8 q^{23} - 14 q^{24} - 22 q^{25} + 30 q^{30} + 2 q^{33} + 2 q^{34} - 44 q^{35} - 34 q^{36} - 8 q^{38}+ \cdots - 18 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −2.30002 − 1.32792i −0.409872 − 1.68286i 2.52674 + 4.37645i −1.47546 − 2.55557i −1.29198 + 4.41489i 1.33159 − 2.30638i − 8.10957i −2.66401 + 1.37951i 7.83718i
115.2 −2.26306 − 1.30658i 1.66679 + 0.470956i 2.41430 + 4.18170i 0.170896 + 0.296000i −3.15672 − 3.24360i −0.576467 + 0.998470i − 7.39161i 2.55640 + 1.56997i − 0.893156i
115.3 −2.06455 − 1.19197i −1.45470 + 0.940137i 1.84159 + 3.18973i −0.878765 − 1.52207i 4.12392 − 0.207008i −1.89717 + 3.28600i − 4.01260i 1.23228 − 2.73523i 4.18985i
115.4 −1.81340 − 1.04696i 0.858622 − 1.50425i 1.19227 + 2.06507i 1.51290 + 2.62042i −3.13192 + 1.82885i 1.24575 − 2.15770i − 0.805192i −1.52554 − 2.58316i − 6.33580i
115.5 −1.73523 − 1.00183i 0.462463 + 1.66917i 1.00734 + 1.74477i −1.07929 − 1.86938i 0.869753 − 3.35970i 1.45265 − 2.51607i − 0.0294314i −2.57226 + 1.54386i 4.32506i
115.6 −1.66971 − 0.964006i −0.0168015 + 1.73197i 0.858615 + 1.48716i 1.99062 + 3.44786i 1.69768 − 2.87568i −0.902398 + 1.56300i 0.545186i −2.99944 − 0.0581992i − 7.67588i
115.7 −1.41480 − 0.816835i −0.201963 − 1.72024i 0.334440 + 0.579267i 0.0190953 + 0.0330740i −1.11941 + 2.59876i −1.71429 + 2.96923i 2.17461i −2.91842 + 0.694848i − 0.0623908i
115.8 −1.33932 − 0.773255i −1.46098 − 0.930347i 0.195847 + 0.339217i 0.856420 + 1.48336i 1.23732 + 2.37574i −0.149308 + 0.258609i 2.48726i 1.26891 + 2.71843i − 2.64892i
115.9 −1.06011 − 0.612057i 1.70377 − 0.311731i −0.250773 − 0.434351i −0.514238 − 0.890686i −1.99698 − 0.712333i 1.16581 − 2.01925i 3.06218i 2.80565 − 1.06223i 1.25897i
115.10 −0.732333 − 0.422813i −1.63408 − 0.574260i −0.642459 − 1.11277i −2.08988 − 3.61977i 0.953889 + 1.11146i 0.362528 − 0.627917i 2.77781i 2.34045 + 1.87678i 3.53451i
115.11 −0.691757 − 0.399386i −1.14970 + 1.29545i −0.680982 − 1.17949i 0.238375 + 0.412877i 1.31270 − 0.436964i 0.272609 − 0.472172i 2.68544i −0.356388 − 2.97876i − 0.380814i
115.12 −0.541770 − 0.312791i 1.71659 + 0.230880i −0.804323 − 1.39313i 1.42008 + 2.45966i −0.857782 − 0.662020i −2.18288 + 3.78086i 2.25751i 2.89339 + 0.792655i − 1.77676i
115.13 −0.0951021 − 0.0549072i 0.575869 − 1.63352i −0.993970 − 1.72161i −0.922069 − 1.59707i −0.144458 + 0.123731i −1.06391 + 1.84275i 0.437933i −2.33675 − 1.88138i 0.202513i
115.14 −0.0450906 − 0.0260330i 0.798660 + 1.53693i −0.998645 − 1.72970i 1.25131 + 2.16733i 0.00399887 − 0.0900924i 2.15547 − 3.73339i 0.208123i −1.72429 + 2.45496i − 0.130301i
115.15 0.0450906 + 0.0260330i −0.798660 − 1.53693i −0.998645 − 1.72970i 1.25131 + 2.16733i 0.00399887 − 0.0900924i 2.15547 − 3.73339i − 0.208123i −1.72429 + 2.45496i 0.130301i
115.16 0.0951021 + 0.0549072i −0.575869 + 1.63352i −0.993970 − 1.72161i −0.922069 − 1.59707i −0.144458 + 0.123731i −1.06391 + 1.84275i − 0.437933i −2.33675 − 1.88138i − 0.202513i
115.17 0.541770 + 0.312791i −1.71659 − 0.230880i −0.804323 − 1.39313i 1.42008 + 2.45966i −0.857782 − 0.662020i −2.18288 + 3.78086i − 2.25751i 2.89339 + 0.792655i 1.77676i
115.18 0.691757 + 0.399386i 1.14970 − 1.29545i −0.680982 − 1.17949i 0.238375 + 0.412877i 1.31270 − 0.436964i 0.272609 − 0.472172i − 2.68544i −0.356388 − 2.97876i 0.380814i
115.19 0.732333 + 0.422813i 1.63408 + 0.574260i −0.642459 − 1.11277i −2.08988 − 3.61977i 0.953889 + 1.11146i 0.362528 − 0.627917i − 2.77781i 2.34045 + 1.87678i − 3.53451i
115.20 1.06011 + 0.612057i −1.70377 + 0.311731i −0.250773 − 0.434351i −0.514238 − 0.890686i −1.99698 − 0.712333i 1.16581 − 2.01925i − 3.06218i 2.80565 − 1.06223i − 1.25897i
See all 56 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 115.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.b even 2 1 inner
261.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.i.a ✓ 56
3.b odd 2 1 783.2.i.a 56
9.c even 3 1 inner 261.2.i.a ✓ 56
9.d odd 6 1 783.2.i.a 56
29.b even 2 1 inner 261.2.i.a ✓ 56
87.d odd 2 1 783.2.i.a 56
261.h odd 6 1 783.2.i.a 56
261.i even 6 1 inner 261.2.i.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.i.a ✓ 56 1.a even 1 1 trivial
261.2.i.a ✓ 56 9.c even 3 1 inner
261.2.i.a ✓ 56 29.b even 2 1 inner
261.2.i.a ✓ 56 261.i even 6 1 inner
783.2.i.a 56 3.b odd 2 1
783.2.i.a 56 9.d odd 6 1
783.2.i.a 56 87.d odd 2 1
783.2.i.a 56 261.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(261, [\chi])\).