Properties

Label 7935.2.a.be
Level $7935$
Weight $2$
Character orbit 7935.a
Self dual yes
Analytic conductor $63.361$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7935,2,Mod(1,7935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7935, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7935.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{8} + q^{9} - \beta_1 q^{10} + ( - \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{13} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{14} - q^{15} + (\beta_{4} - \beta_{2} + \beta_1 - 2) q^{16} + (\beta_{5} + \beta_{3} + 2 \beta_1 - 1) q^{17} - \beta_1 q^{18} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + 2) q^{19} + (\beta_{2} + 1) q^{20} + ( - \beta_{5} - \beta_1) q^{21} + ( - \beta_{5} - \beta_{2} - 1) q^{22} + (\beta_{3} + 1) q^{24} + q^{25} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_1 + 2) q^{26} - q^{27} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{28} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{29} + \beta_1 q^{30} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{32} + (\beta_{5} + \beta_{2}) q^{33} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{34} + (\beta_{5} + \beta_1) q^{35} + (\beta_{2} + 1) q^{36} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 1) q^{37} + (2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{38} + (\beta_{5} - \beta_{4} + \beta_1 - 1) q^{39} + ( - \beta_{3} - 1) q^{40} + ( - \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 - 3) q^{41} + ( - \beta_{5} - \beta_{3} - \beta_1 + 1) q^{42} + ( - 2 \beta_{5} - 3 \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{5} + \beta_{2} + \beta_1 - 1) q^{44} + q^{45} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{47} + ( - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{48} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 3) q^{49} - \beta_1 q^{50} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 + 1) q^{51} + (\beta_{5} + 2 \beta_{4} + 4 \beta_{2} - \beta_1 + 2) q^{52} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + 3) q^{53} + \beta_1 q^{54} + ( - \beta_{5} - \beta_{2}) q^{55} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{56} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} - 2) q^{57} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + 1) q^{58} + (\beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 3) q^{59} + ( - \beta_{2} - 1) q^{60} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{62} + (\beta_{5} + \beta_1) q^{63} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{64} + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{65} + (\beta_{5} + \beta_{2} + 1) q^{66} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{68} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{70} + ( - 3 \beta_{5} - 3 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{71} + ( - \beta_{3} - 1) q^{72} + (\beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{73} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{74} - q^{75} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 6) q^{76} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{77} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 2) q^{78} + (2 \beta_{5} + \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{79} + (\beta_{4} - \beta_{2} + \beta_1 - 2) q^{80} + q^{81} + (2 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + 6 \beta_1 - 1) q^{82} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 3) q^{83} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{84} + (\beta_{5} + \beta_{3} + 2 \beta_1 - 1) q^{85} + ( - 2 \beta_{5} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{86} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{87} + (3 \beta_{5} + 2 \beta_{2} + \beta_1) q^{88} + (2 \beta_{5} - \beta_{3} + \beta_{2} + 6 \beta_1) q^{89} - \beta_1 q^{90} + (4 \beta_{5} + \beta_{4} + \beta_{2} + 5 \beta_1 - 6) q^{91} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{93} + (2 \beta_{3} + 2 \beta_1 + 4) q^{94} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + 2) q^{95} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{96} + (3 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{97} + ( - 2 \beta_{5} + 3 \beta_1 + 4) q^{98} + ( - \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{9} + 4 q^{11} - 4 q^{12} + 10 q^{13} - 8 q^{14} - 6 q^{15} - 8 q^{16} - 8 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{22} + 6 q^{24} + 6 q^{25} + 14 q^{26} - 6 q^{27} - 4 q^{28} - 4 q^{29} - 6 q^{31} + 8 q^{32} - 4 q^{33} - 24 q^{34} - 2 q^{35} + 4 q^{36} - 6 q^{37} - 10 q^{38} - 10 q^{39} - 6 q^{40} - 20 q^{41} + 8 q^{42} + 4 q^{43} - 10 q^{44} + 6 q^{45} - 4 q^{47} + 8 q^{48} - 10 q^{49} + 8 q^{51} + 6 q^{52} + 22 q^{53} + 4 q^{55} + 8 q^{56} - 4 q^{57} + 4 q^{58} - 22 q^{59} - 4 q^{60} - 8 q^{61} + 16 q^{62} - 2 q^{63} - 24 q^{64} + 10 q^{65} + 2 q^{66} - 8 q^{67} - 2 q^{68} - 8 q^{70} - 2 q^{71} - 6 q^{72} - 18 q^{73} + 10 q^{74} - 6 q^{75} + 40 q^{76} - 22 q^{77} - 14 q^{78} - 20 q^{79} - 8 q^{80} + 6 q^{81} - 4 q^{82} + 18 q^{83} + 4 q^{84} - 8 q^{85} - 14 q^{86} + 4 q^{87} - 10 q^{88} - 6 q^{89} - 44 q^{91} + 6 q^{93} + 24 q^{94} + 4 q^{95} - 8 q^{96} - 12 q^{97} + 28 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 11\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 7\beta_{3} + \beta_{2} + 18\beta _1 + 7 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.27997
1.90131
0.184585
−0.821728
−1.45825
−2.08589
−2.27997 −1.00000 3.19828 1.00000 2.27997 −0.223188 −2.73205 1.00000 −2.27997
1.2 −1.90131 −1.00000 1.61498 1.00000 1.90131 0.941661 0.732051 1.00000 −1.90131
1.3 −0.184585 −1.00000 −1.96593 1.00000 0.184585 2.30634 0.732051 1.00000 −0.184585
1.4 0.821728 −1.00000 −1.32476 1.00000 −0.821728 −4.10637 −2.73205 1.00000 0.821728
1.5 1.45825 −1.00000 0.126482 1.00000 −1.45825 1.59751 −2.73205 1.00000 1.45825
1.6 2.08589 −1.00000 2.35095 1.00000 −2.08589 −2.51595 0.732051 1.00000 2.08589
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 7935.2.a.be yes 6
23.b odd 2 1 7935.2.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7935.2.a.bd 6 23.b odd 2 1
7935.2.a.be yes 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{6} - 8T_{2}^{4} + 2T_{2}^{3} + 16T_{2}^{2} - 8T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 14T_{7}^{4} - 8T_{7}^{3} + 52T_{7}^{2} - 24T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{6} - 4T_{11}^{5} - 11T_{11}^{4} + 10T_{11}^{3} + 7T_{11}^{2} - 6T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 8 T^{4} + 2 T^{3} + 16 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 14 T^{4} - 8 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} - 11 T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} - 10 T^{5} + 210 T^{3} + \cdots + 1198 \) Copy content Toggle raw display
$17$ \( T^{6} + 8 T^{5} - 8 T^{4} - 86 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} - 61 T^{4} + \cdots + 4021 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 4 T^{5} - 60 T^{4} - 24 T^{3} + \cdots + 208 \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} - 53 T^{4} + \cdots + 3421 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} - 152 T^{4} + \cdots - 61706 \) Copy content Toggle raw display
$41$ \( T^{6} + 20 T^{5} + 31 T^{4} + \cdots - 5819 \) Copy content Toggle raw display
$43$ \( T^{6} - 4 T^{5} - 106 T^{4} + \cdots - 506 \) Copy content Toggle raw display
$47$ \( T^{6} + 4 T^{5} - 52 T^{4} + \cdots + 2272 \) Copy content Toggle raw display
$53$ \( T^{6} - 22 T^{5} + 94 T^{4} + \cdots + 11902 \) Copy content Toggle raw display
$59$ \( T^{6} + 22 T^{5} + 34 T^{4} + \cdots + 68404 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} - 49 T^{4} - 300 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} - 60 T^{4} - 48 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{6} + 2 T^{5} - 211 T^{4} + \cdots - 2039 \) Copy content Toggle raw display
$73$ \( T^{6} + 18 T^{5} + 30 T^{4} + \cdots - 30888 \) Copy content Toggle raw display
$79$ \( T^{6} + 20 T^{5} - 85 T^{4} + \cdots + 65881 \) Copy content Toggle raw display
$83$ \( T^{6} - 18 T^{5} - 86 T^{4} + \cdots + 74038 \) Copy content Toggle raw display
$89$ \( T^{6} + 6 T^{5} - 290 T^{4} + \cdots - 373916 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} - 230 T^{4} + \cdots + 252982 \) Copy content Toggle raw display
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