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Properties

Label 7865.2.a.bj
Level $7865$
Weight $2$
Character orbit 7865.a
Self dual yes
Analytic conductor $62.802$
Analytic rank $1$
Dimension $22$
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] = [7865,2,Mod(1,7865)] 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("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7865.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,-1,-7,19,-22,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.8023411897\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 715)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{2} - 7 q^{3} + 19 q^{4} - 22 q^{5} + 11 q^{6} - 5 q^{7} - 3 q^{8} + 25 q^{9} + q^{10} - 24 q^{12} - 22 q^{13} - q^{14} + 7 q^{15} + 17 q^{16} + 8 q^{17} - 15 q^{18} + q^{19} - 19 q^{20} - 7 q^{21}+ \cdots - 52 q^{98}+O(q^{100}) \) 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 −2.68191 −0.422348 5.19266 −1.00000 1.13270 1.63689 −8.56243 −2.82162 2.68191
1.2 −2.63724 −3.36726 4.95504 −1.00000 8.88027 −1.51491 −7.79316 8.33842 2.63724
1.3 −2.13085 −1.26779 2.54054 −1.00000 2.70147 0.140787 −1.15181 −1.39272 2.13085
1.4 −2.00968 −2.70847 2.03881 −1.00000 5.44317 3.78340 −0.0780047 4.33583 2.00968
1.5 −1.84716 0.0768496 1.41198 −1.00000 −0.141953 −4.65953 1.08616 −2.99409 1.84716
1.6 −1.69631 2.64900 0.877468 −1.00000 −4.49353 2.38264 1.90416 4.01722 1.69631
1.7 −1.38153 2.18096 −0.0913787 −1.00000 −3.01305 −2.98037 2.88930 1.75657 1.38153
1.8 −1.17175 −1.54348 −0.627009 −1.00000 1.80857 0.207051 3.07819 −0.617665 1.17175
1.9 −0.702555 −0.777773 −1.50642 −1.00000 0.546429 3.98440 2.46345 −2.39507 0.702555
1.10 −0.172693 1.88595 −1.97018 −1.00000 −0.325689 −0.0945814 0.685620 0.556798 0.172693
1.11 −0.140969 −3.38989 −1.98013 −1.00000 0.477871 −3.17246 0.561076 8.49138 0.140969
1.12 −0.0974215 0.448159 −1.99051 −1.00000 −0.0436603 0.106320 0.388761 −2.79915 0.0974215
1.13 0.452627 3.11105 −1.79513 −1.00000 1.40815 −1.70036 −1.71778 6.67864 −0.452627
1.14 0.582746 −0.306220 −1.66041 −1.00000 −0.178448 −3.02370 −2.13309 −2.90623 −0.582746
1.15 0.760118 −2.68706 −1.42222 −1.00000 −2.04248 −3.45711 −2.60129 4.22030 −0.760118
1.16 1.23026 −2.30841 −0.486459 −1.00000 −2.83994 2.54300 −3.05899 2.32875 −1.23026
1.17 1.57868 −1.44217 0.492223 −1.00000 −2.27673 −0.654083 −2.38029 −0.920132 −1.57868
1.18 1.71202 2.30206 0.931001 −1.00000 3.94117 −1.16533 −1.83014 2.29950 −1.71202
1.19 1.99666 1.89326 1.98664 −1.00000 3.78020 0.233793 −0.0266834 0.584444 −1.99666
1.20 2.33964 −0.564896 3.47392 −1.00000 −1.32165 3.23954 3.44845 −2.68089 −2.33964
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(11\) \( +1 \)
\(13\) \( +1 \)

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 7865.2.a.bj 22
11.b odd 2 1 7865.2.a.bk 22
11.c even 5 2 715.2.v.c ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.v.c ✓ 44 11.c even 5 2
7865.2.a.bj 22 1.a even 1 1 trivial
7865.2.a.bk 22 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):

\( T_{2}^{22} + T_{2}^{21} - 31 T_{2}^{20} - 29 T_{2}^{19} + 406 T_{2}^{18} + 352 T_{2}^{17} - 2933 T_{2}^{16} + \cdots + 5 \) Copy content Toggle raw display
\( T_{3}^{22} + 7 T_{3}^{21} - 21 T_{3}^{20} - 241 T_{3}^{19} - T_{3}^{18} + 3354 T_{3}^{17} + 3660 T_{3}^{16} + \cdots - 496 \) Copy content Toggle raw display