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Properties

Label 2380.2.cb.b
Level $2380$
Weight $2$
Character orbit 2380.cb
Analytic conductor $19.004$
Analytic rank $0$
Dimension $92$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2380,2,Mod(781,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.781"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2380.cb (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [92] 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: no
Analytic conductor: \(19.0043956811\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 56 q^{9} - 24 q^{13} + 12 q^{15} + 4 q^{17} - 24 q^{21} + 46 q^{25} - 24 q^{33} + 14 q^{35} - 64 q^{43} + 16 q^{47} + 38 q^{49} - 22 q^{51} - 8 q^{53} + 18 q^{59} + 16 q^{67} - 36 q^{69} - 12 q^{77}+ \cdots + 44 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
781.1 0 −2.92080 − 1.68632i 0 0.866025 − 0.500000i 0 2.60434 + 0.466295i 0 4.18738 + 7.25276i 0
781.2 0 −2.90618 − 1.67788i 0 −0.866025 + 0.500000i 0 −2.41676 + 1.07668i 0 4.13058 + 7.15438i 0
781.3 0 −2.70301 − 1.56058i 0 −0.866025 + 0.500000i 0 −0.660525 − 2.56197i 0 3.37085 + 5.83848i 0
781.4 0 −2.56753 − 1.48237i 0 0.866025 − 0.500000i 0 −2.05447 + 1.66708i 0 2.89482 + 5.01398i 0
781.5 0 −2.33315 − 1.34704i 0 0.866025 − 0.500000i 0 1.17935 − 2.36836i 0 2.12905 + 3.68763i 0
781.6 0 −2.29941 − 1.32756i 0 −0.866025 + 0.500000i 0 2.54913 − 0.708459i 0 2.02485 + 3.50715i 0
781.7 0 −2.28469 − 1.31906i 0 0.866025 − 0.500000i 0 −2.35903 − 1.19790i 0 1.97986 + 3.42921i 0
781.8 0 −2.14890 − 1.24067i 0 −0.866025 + 0.500000i 0 1.32551 + 2.28977i 0 1.57851 + 2.73406i 0
781.9 0 −2.08418 − 1.20330i 0 0.866025 − 0.500000i 0 1.23531 + 2.33966i 0 1.39587 + 2.41772i 0
781.10 0 −1.74748 − 1.00891i 0 −0.866025 + 0.500000i 0 −2.38330 − 1.14887i 0 0.535791 + 0.928016i 0
781.11 0 −1.57801 − 0.911062i 0 0.866025 − 0.500000i 0 1.58859 − 2.11575i 0 0.160068 + 0.277246i 0
781.12 0 −1.44456 − 0.834019i 0 −0.866025 + 0.500000i 0 1.42675 − 2.22809i 0 −0.108826 − 0.188492i 0
781.13 0 −1.43555 − 0.828817i 0 −0.866025 + 0.500000i 0 −0.635268 + 2.56835i 0 −0.126124 − 0.218454i 0
781.14 0 −1.30936 − 0.755959i 0 −0.866025 + 0.500000i 0 2.47675 + 0.930443i 0 −0.357052 − 0.618432i 0
781.15 0 −1.18535 − 0.684359i 0 −0.866025 + 0.500000i 0 −1.83366 + 1.90727i 0 −0.563304 − 0.975672i 0
781.16 0 −1.10643 − 0.638799i 0 0.866025 − 0.500000i 0 2.57601 − 0.603461i 0 −0.683871 − 1.18450i 0
781.17 0 −1.04428 − 0.602915i 0 −0.866025 + 0.500000i 0 −2.63869 − 0.193203i 0 −0.772986 − 1.33885i 0
781.18 0 −0.890631 − 0.514206i 0 −0.866025 + 0.500000i 0 1.93180 − 1.80780i 0 −0.971184 − 1.68214i 0
781.19 0 −0.789610 − 0.455882i 0 0.866025 − 0.500000i 0 −2.01881 − 1.71008i 0 −1.08434 − 1.87814i 0
781.20 0 −0.484641 − 0.279807i 0 0.866025 − 0.500000i 0 2.00662 + 1.72438i 0 −1.34342 − 2.32686i 0
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 781.46
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.b even 2 1 inner
119.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2380.2.cb.b ✓ 92
7.c even 3 1 inner 2380.2.cb.b ✓ 92
17.b even 2 1 inner 2380.2.cb.b ✓ 92
119.j even 6 1 inner 2380.2.cb.b ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2380.2.cb.b ✓ 92 1.a even 1 1 trivial
2380.2.cb.b ✓ 92 7.c even 3 1 inner
2380.2.cb.b ✓ 92 17.b even 2 1 inner
2380.2.cb.b ✓ 92 119.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{92} - 97 T_{3}^{90} + 5059 T_{3}^{88} - 182382 T_{3}^{86} + 5034613 T_{3}^{84} - 112289343 T_{3}^{82} + \cdots + 43046721 \) acting on \(S_{2}^{\mathrm{new}}(2380, [\chi])\). Copy content Toggle raw display