Properties

Label 3626.2.a.bg
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3626,2,Mod(1,3626)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3626, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3626.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.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: \(28.9537557729\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 21x^{6} + 151x^{4} - 406x^{2} + 242 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{4} q^{6} + q^{8} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{4} q^{6} + q^{8} + ( - \beta_{6} + 2) q^{9} - \beta_{2} q^{10} - \beta_{7} q^{11} + \beta_{4} q^{12} + (\beta_{4} + \beta_{3}) q^{13} + ( - \beta_{7} + \beta_1 - 1) q^{15} + q^{16} + \beta_{3} q^{17} + ( - \beta_{6} + 2) q^{18} + ( - \beta_{5} + \beta_{4}) q^{19} - \beta_{2} q^{20} - \beta_{7} q^{22} + ( - \beta_1 + 1) q^{23} + \beta_{4} q^{24} + (2 \beta_{7} + \beta_1 + 6) q^{25} + (\beta_{4} + \beta_{3}) q^{26} + (3 \beta_{4} - \beta_{3} + \beta_{2}) q^{27} + (\beta_{7} + \beta_{6} - \beta_1 + 2) q^{29} + ( - \beta_{7} + \beta_1 - 1) q^{30} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{31}+ \cdots + ( - 5 \beta_{7} - 2 \beta_{6} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 18 q^{9} + 2 q^{11} - 8 q^{15} + 8 q^{16} + 18 q^{18} + 2 q^{22} + 10 q^{23} + 42 q^{25} + 14 q^{29} - 8 q^{30} + 8 q^{32} + 18 q^{36} + 8 q^{37} + 30 q^{39} - 20 q^{43} + 2 q^{44} + 10 q^{46} + 42 q^{50} - 12 q^{51} + 40 q^{53} + 28 q^{57} + 14 q^{58} - 8 q^{60} + 8 q^{64} + 10 q^{67} + 40 q^{71} + 18 q^{72} + 8 q^{74} + 30 q^{78} + 22 q^{79} + 92 q^{81} + 8 q^{85} - 20 q^{86} + 2 q^{88} + 10 q^{92} - 36 q^{93} + 24 q^{95} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 21x^{6} + 151x^{4} - 406x^{2} + 242 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 8\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 31\nu^{5} - 126\nu^{3} + 64\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 41\nu^{5} - 145\nu^{3} + 52\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 31\nu^{5} - 137\nu^{3} + 130\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 31\nu^{5} - 137\nu^{3} + 152\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 15\nu^{4} - 60\nu^{2} + 35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{6} - 15\nu^{4} + 62\nu^{2} - 46 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} - 4\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} + 4\beta_{6} + \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{5} - 28\beta_{4} - 2\beta_{3} + 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{7} + 29\beta_{6} + 15\beta _1 + 260 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 137\beta_{5} - 198\beta_{4} - 31\beta_{3} + 102\beta_{2} \) 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
2.38022
−0.904109
2.81330
−2.56953
2.56953
−2.81330
0.904109
−2.38022
1.00000 −3.21136 1.00000 4.00765 −3.21136 0 1.00000 7.31284 4.00765
1.2 1.00000 −3.09330 1.00000 −1.59233 −3.09330 0 1.00000 6.56849 −1.59233
1.3 1.00000 −1.02018 1.00000 −4.36631 −1.02018 0 1.00000 −1.95923 −4.36631
1.4 1.00000 −0.279098 1.00000 1.82718 −0.279098 0 1.00000 −2.92210 1.82718
1.5 1.00000 0.279098 1.00000 −1.82718 0.279098 0 1.00000 −2.92210 −1.82718
1.6 1.00000 1.02018 1.00000 4.36631 1.02018 0 1.00000 −1.95923 4.36631
1.7 1.00000 3.09330 1.00000 1.59233 3.09330 0 1.00000 6.56849 1.59233
1.8 1.00000 3.21136 1.00000 −4.00765 3.21136 0 1.00000 7.31284 −4.00765
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(37\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bg 8
7.b odd 2 1 inner 3626.2.a.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bg 8 1.a even 1 1 trivial
3626.2.a.bg 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{8} - 21T_{3}^{6} + 121T_{3}^{4} - 112T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{8} - 41T_{5}^{6} + 521T_{5}^{4} - 2096T_{5}^{2} + 2592 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 31T_{11}^{2} + 48T_{11} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 21 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{8} - 41 T^{6} + \cdots + 2592 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 31 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 53 T^{6} + \cdots + 288 \) Copy content Toggle raw display
$17$ \( T^{8} - 44 T^{6} + \cdots + 128 \) Copy content Toggle raw display
$19$ \( T^{8} - 84 T^{6} + \cdots + 61952 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} - 33 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 7 T^{3} + \cdots - 236)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 133 T^{6} + \cdots + 349448 \) Copy content Toggle raw display
$37$ \( (T - 1)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 141 T^{6} + \cdots + 10368 \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + \cdots - 864)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 212 T^{6} + \cdots + 73728 \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + \cdots - 8448)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 224 T^{6} + \cdots + 2957312 \) Copy content Toggle raw display
$61$ \( T^{8} - 233 T^{6} + \cdots + 3338528 \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} + \cdots + 3504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 20 T^{3} + \cdots - 384)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 625 T^{6} + \cdots + 538970112 \) Copy content Toggle raw display
$79$ \( (T^{4} - 11 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 256 T^{6} + \cdots + 80000 \) Copy content Toggle raw display
$89$ \( T^{8} - 268 T^{6} + \cdots + 839808 \) Copy content Toggle raw display
$97$ \( T^{8} - 512 T^{6} + \cdots + 16866432 \) Copy content Toggle raw display
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