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Properties

Label 6790.2.a.u
Level $6790$
Weight $2$
Character orbit 6790.a
Self dual yes
Analytic conductor $54.218$
Analytic rank $1$
Dimension $10$
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] = [6790,2,Mod(1,6790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6790, 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("6790.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6790 = 2 \cdot 5 \cdot 7 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6790.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: \(54.2184229724\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 11x^{8} + 34x^{7} + 38x^{6} - 126x^{5} - 32x^{4} + 160x^{3} - 36x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{7} - q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{7} - q^{8} + \beta_{2} q^{9} - q^{10} + ( - \beta_{8} + \beta_{6}) q^{11} + \beta_1 q^{12} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + ( - 4 \beta_{9} - 3 \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 3 q^{3} + 10 q^{4} + 10 q^{5} - 3 q^{6} - 10 q^{7} - 10 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 3 q^{3} + 10 q^{4} + 10 q^{5} - 3 q^{6} - 10 q^{7} - 10 q^{8} + q^{9} - 10 q^{10} - q^{11} + 3 q^{12} + q^{13} + 10 q^{14} + 3 q^{15} + 10 q^{16} - 10 q^{17} - q^{18} - 11 q^{19} + 10 q^{20} - 3 q^{21} + q^{22} + 8 q^{23} - 3 q^{24} + 10 q^{25} - q^{26} + 6 q^{27} - 10 q^{28} - 23 q^{29} - 3 q^{30} - 4 q^{31} - 10 q^{32} + q^{33} + 10 q^{34} - 10 q^{35} + q^{36} - 5 q^{37} + 11 q^{38} - 12 q^{39} - 10 q^{40} - 30 q^{41} + 3 q^{42} + 5 q^{43} - q^{44} + q^{45} - 8 q^{46} + 4 q^{47} + 3 q^{48} + 10 q^{49} - 10 q^{50} - 18 q^{51} + q^{52} - q^{53} - 6 q^{54} - q^{55} + 10 q^{56} - 26 q^{57} + 23 q^{58} - 23 q^{59} + 3 q^{60} - 33 q^{61} + 4 q^{62} - q^{63} + 10 q^{64} + q^{65} - q^{66} + 13 q^{67} - 10 q^{68} + 6 q^{69} + 10 q^{70} + 10 q^{71} - q^{72} + 10 q^{73} + 5 q^{74} + 3 q^{75} - 11 q^{76} + q^{77} + 12 q^{78} - 4 q^{79} + 10 q^{80} - 30 q^{81} + 30 q^{82} + q^{83} - 3 q^{84} - 10 q^{85} - 5 q^{86} - 18 q^{87} + q^{88} - 32 q^{89} - q^{90} - q^{91} + 8 q^{92} + 30 q^{93} - 4 q^{94} - 11 q^{95} - 3 q^{96} + 10 q^{97} - 10 q^{98} - 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 11x^{8} + 34x^{7} + 38x^{6} - 126x^{5} - 32x^{4} + 160x^{3} - 36x^{2} - 32x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 13\nu^{3} + 16\nu^{2} - 18\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 3\nu^{7} - 9\nu^{6} + 26\nu^{5} + 26\nu^{4} - 60\nu^{3} - 28\nu^{2} + 20\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 2\nu^{8} - 12\nu^{7} + 19\nu^{6} + 50\nu^{5} - 52\nu^{4} - 80\nu^{3} + 36\nu^{2} + 32\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 3\nu^{8} + 11\nu^{7} - 32\nu^{6} - 40\nu^{5} + 104\nu^{4} + 48\nu^{3} - 96\nu^{2} + 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + 3\nu^{8} + 11\nu^{7} - 32\nu^{6} - 42\nu^{5} + 110\nu^{4} + 58\nu^{3} - 124\nu^{2} - 4\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 2\nu^{8} - 14\nu^{7} + 25\nu^{6} + 66\nu^{5} - 102\nu^{4} - 110\nu^{3} + 140\nu^{2} + 28\nu - 24 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{9} + 4\beta_{8} - \beta_{7} - 3\beta_{5} - 3\beta_{4} + 11\beta_{3} + 9\beta_{2} + 27\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{9} + 24\beta_{8} - 10\beta_{7} - 14\beta_{5} - 12\beta_{4} + 25\beta_{3} + 37\beta_{2} + 23\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 39 \beta_{9} + 54 \beta_{8} - 13 \beta_{7} + 2 \beta_{6} - 41 \beta_{5} - 35 \beta_{4} + 98 \beta_{3} + \cdots + 94 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 139 \beta_{9} + 222 \beta_{8} - 77 \beta_{7} + 6 \beta_{6} - 141 \beta_{5} - 109 \beta_{4} + 241 \beta_{3} + \cdots + 443 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 382 \beta_{9} + 540 \beta_{8} - 122 \beta_{7} + 40 \beta_{6} - 410 \beta_{5} - 312 \beta_{4} + \cdots + 743 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.17505
−1.94553
−1.65156
−0.470218
0.250543
0.666879
1.04024
2.07189
2.42437
2.78845
−1.00000 −2.17505 1.00000 1.00000 2.17505 −1.00000 −1.00000 1.73085 −1.00000
1.2 −1.00000 −1.94553 1.00000 1.00000 1.94553 −1.00000 −1.00000 0.785105 −1.00000
1.3 −1.00000 −1.65156 1.00000 1.00000 1.65156 −1.00000 −1.00000 −0.272365 −1.00000
1.4 −1.00000 −0.470218 1.00000 1.00000 0.470218 −1.00000 −1.00000 −2.77890 −1.00000
1.5 −1.00000 0.250543 1.00000 1.00000 −0.250543 −1.00000 −1.00000 −2.93723 −1.00000
1.6 −1.00000 0.666879 1.00000 1.00000 −0.666879 −1.00000 −1.00000 −2.55527 −1.00000
1.7 −1.00000 1.04024 1.00000 1.00000 −1.04024 −1.00000 −1.00000 −1.91791 −1.00000
1.8 −1.00000 2.07189 1.00000 1.00000 −2.07189 −1.00000 −1.00000 1.29272 −1.00000
1.9 −1.00000 2.42437 1.00000 1.00000 −2.42437 −1.00000 −1.00000 2.87757 −1.00000
1.10 −1.00000 2.78845 1.00000 1.00000 −2.78845 −1.00000 −1.00000 4.77543 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)
\(97\) \( -1 \)

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 6790.2.a.u ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6790.2.a.u ✓ 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6790))\):

\( T_{3}^{10} - 3T_{3}^{9} - 11T_{3}^{8} + 34T_{3}^{7} + 38T_{3}^{6} - 126T_{3}^{5} - 32T_{3}^{4} + 160T_{3}^{3} - 36T_{3}^{2} - 32T_{3} + 8 \) Copy content Toggle raw display
\( T_{11}^{10} + T_{11}^{9} - 46 T_{11}^{8} - 51 T_{11}^{7} + 723 T_{11}^{6} + 870 T_{11}^{5} - 4380 T_{11}^{4} + \cdots - 3620 \) Copy content Toggle raw display
\( T_{23}^{10} - 8 T_{23}^{9} - 84 T_{23}^{8} + 614 T_{23}^{7} + 2441 T_{23}^{6} - 14350 T_{23}^{5} + \cdots - 118430 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( (T + 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots - 3620 \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots + 512 \) Copy content Toggle raw display
$17$ \( T^{10} + 10 T^{9} + \cdots + 65600 \) Copy content Toggle raw display
$19$ \( T^{10} + 11 T^{9} + \cdots + 1568 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots - 118430 \) Copy content Toggle raw display
$29$ \( T^{10} + 23 T^{9} + \cdots - 45268 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} + \cdots - 33639680 \) Copy content Toggle raw display
$37$ \( T^{10} + 5 T^{9} + \cdots - 1451840 \) Copy content Toggle raw display
$41$ \( T^{10} + 30 T^{9} + \cdots - 5859964 \) Copy content Toggle raw display
$43$ \( T^{10} - 5 T^{9} + \cdots + 5793880 \) Copy content Toggle raw display
$47$ \( T^{10} - 4 T^{9} + \cdots + 1296197 \) Copy content Toggle raw display
$53$ \( T^{10} + T^{9} + \cdots - 37776932 \) Copy content Toggle raw display
$59$ \( T^{10} + 23 T^{9} + \cdots + 10676608 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 110774120 \) Copy content Toggle raw display
$67$ \( T^{10} - 13 T^{9} + \cdots + 1096468 \) Copy content Toggle raw display
$71$ \( T^{10} - 10 T^{9} + \cdots - 8952064 \) Copy content Toggle raw display
$73$ \( T^{10} - 10 T^{9} + \cdots + 171952 \) Copy content Toggle raw display
$79$ \( T^{10} + 4 T^{9} + \cdots + 33043018 \) Copy content Toggle raw display
$83$ \( T^{10} - T^{9} + \cdots + 82067324 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 1208480704 \) Copy content Toggle raw display
$97$ \( (T - 1)^{10} \) Copy content Toggle raw display
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