Properties

Label 3626.2.a.bh
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 32x^{8} + 323x^{6} - 1412x^{4} + 2721x^{2} - 1800 \) 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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{8} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{8} q^{6} + q^{8} + ( - \beta_{3} - \beta_{2} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{8} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{8} q^{6} + q^{8} + ( - \beta_{3} - \beta_{2} + 4) q^{9} - \beta_1 q^{10} + ( - \beta_{5} + 1) q^{11} + \beta_{8} q^{12} + ( - \beta_{9} - \beta_1) q^{13} + (\beta_{6} + \beta_{5} - \beta_{3} + 1) q^{15} + q^{16} + ( - \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{3} - \beta_{2} + 4) q^{18} + (\beta_{7} - 2 \beta_{4}) q^{19} - \beta_1 q^{20} + ( - \beta_{5} + 1) q^{22} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{23} + \beta_{8} q^{24} + (\beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{25} + ( - \beta_{9} - \beta_1) q^{26} + (3 \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{27}+ \cdots + ( - 6 \beta_{5} - 3 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 38 q^{9} + 8 q^{11} + 12 q^{15} + 10 q^{16} + 38 q^{18} + 8 q^{22} - 4 q^{23} + 14 q^{25} + 36 q^{29} + 12 q^{30} + 10 q^{32} + 38 q^{36} - 10 q^{37} + 4 q^{39} + 4 q^{43} + 8 q^{44} - 4 q^{46} + 14 q^{50} + 20 q^{51} - 4 q^{53} - 20 q^{57} + 36 q^{58} + 12 q^{60} + 10 q^{64} + 52 q^{65} - 12 q^{67} + 40 q^{71} + 38 q^{72} - 10 q^{74} + 4 q^{78} + 82 q^{81} - 8 q^{85} + 4 q^{86} + 8 q^{88} - 4 q^{92} + 52 q^{93} + 8 q^{95} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 32x^{8} + 323x^{6} - 1412x^{4} + 2721x^{2} - 1800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{8} - 76\nu^{6} + 925\nu^{4} - 3949\nu^{2} + 4887 ) / 177 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\nu^{8} - 431\nu^{6} + 2975\nu^{4} - 7166\nu^{2} + 4758 ) / 177 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97\nu^{9} - 2624\nu^{7} + 18401\nu^{5} - 47714\nu^{3} + 43647\nu ) / 5310 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -34\nu^{8} + 938\nu^{6} - 6875\nu^{4} + 18458\nu^{2} - 15465 ) / 177 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -37\nu^{8} + 1052\nu^{6} - 8174\nu^{4} + 22523\nu^{2} - 17574 ) / 177 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 157\nu^{9} - 4904\nu^{7} + 46151\nu^{5} - 166184\nu^{3} + 190257\nu ) / 5310 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 209\nu^{9} - 5818\nu^{7} + 43297\nu^{5} - 114868\nu^{3} + 87219\nu ) / 1770 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 341\nu^{9} - 9772\nu^{7} + 77443\nu^{5} - 222397\nu^{3} + 190281\nu ) / 2655 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{9} + 4\beta_{8} + 2\beta_{7} - 8\beta_{4} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 19\beta_{5} + 42\beta_{3} + 24\beta_{2} + 67 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -69\beta_{9} + 88\beta_{8} + 49\beta_{7} - 163\beta_{4} + 193\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50\beta_{6} + 337\beta_{5} + 775\beta_{3} + 454\beta_{2} + 1041 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1313\beta_{9} + 1651\beta_{8} + 942\beta_{7} - 2965\beta_{4} + 3366\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 975\beta_{6} + 5993\beta_{5} + 13974\beta_{3} + 8215\beta_{2} + 17974 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -23905\beta_{9} + 29936\beta_{8} + 17171\beta_{7} - 53167\beta_{4} + 59896\beta_1 \) 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
2.25483
1.85747
−2.04532
1.17019
−4.23236
4.23236
−1.17019
2.04532
−1.85747
−2.25483
1.00000 −3.27400 1.00000 −2.25483 −3.27400 0 1.00000 7.71911 −2.25483
1.2 1.00000 −3.18017 1.00000 −1.85747 −3.18017 0 1.00000 7.11351 −1.85747
1.3 1.00000 −2.85745 1.00000 2.04532 −2.85745 0 1.00000 5.16501 2.04532
1.4 1.00000 −2.04457 1.00000 −1.17019 −2.04457 0 1.00000 1.18025 −1.17019
1.5 1.00000 −0.906713 1.00000 4.23236 −0.906713 0 1.00000 −2.17787 4.23236
1.6 1.00000 0.906713 1.00000 −4.23236 0.906713 0 1.00000 −2.17787 −4.23236
1.7 1.00000 2.04457 1.00000 1.17019 2.04457 0 1.00000 1.18025 1.17019
1.8 1.00000 2.85745 1.00000 −2.04532 2.85745 0 1.00000 5.16501 −2.04532
1.9 1.00000 3.18017 1.00000 1.85747 3.18017 0 1.00000 7.11351 1.85747
1.10 1.00000 3.27400 1.00000 2.25483 3.27400 0 1.00000 7.71911 2.25483
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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.bh 10
7.b odd 2 1 inner 3626.2.a.bh 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bh 10 1.a even 1 1 trivial
3626.2.a.bh 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{10} - 34T_{3}^{8} + 427T_{3}^{6} - 2378T_{3}^{4} + 5385T_{3}^{2} - 3042 \) Copy content Toggle raw display
\( T_{5}^{10} - 32T_{5}^{8} + 323T_{5}^{6} - 1412T_{5}^{4} + 2721T_{5}^{2} - 1800 \) Copy content Toggle raw display
\( T_{11}^{5} - 4T_{11}^{4} - 29T_{11}^{3} + 144T_{11}^{2} - 171T_{11} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 34 T^{8} + \cdots - 3042 \) Copy content Toggle raw display
$5$ \( T^{10} - 32 T^{8} + \cdots - 1800 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} - 29 T^{3} + \cdots + 54)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 104 T^{8} + \cdots - 131072 \) Copy content Toggle raw display
$17$ \( T^{10} - 130 T^{8} + \cdots - 1782272 \) Copy content Toggle raw display
$19$ \( T^{10} - 130 T^{8} + \cdots - 1782272 \) Copy content Toggle raw display
$23$ \( (T^{5} + 2 T^{4} + \cdots - 216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 18 T^{4} + \cdots + 1858)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 224 T^{8} + \cdots - 32902272 \) Copy content Toggle raw display
$37$ \( (T + 1)^{10} \) Copy content Toggle raw display
$41$ \( T^{10} - 74 T^{8} + \cdots - 2 \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} + \cdots + 7520)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 296 T^{8} + \cdots - 346112 \) Copy content Toggle raw display
$53$ \( (T^{5} + 2 T^{4} + \cdots + 46208)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 194 T^{8} + \cdots - 1384448 \) Copy content Toggle raw display
$61$ \( T^{10} - 256 T^{8} + \cdots - 109512 \) Copy content Toggle raw display
$67$ \( (T^{5} + 6 T^{4} + \cdots + 3812)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 20 T^{4} + \cdots - 17920)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 615233042 \) Copy content Toggle raw display
$79$ \( (T^{5} - 81 T^{3} + \cdots - 56)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 314 T^{8} + \cdots - 913952 \) Copy content Toggle raw display
$89$ \( T^{10} - 242 T^{8} + \cdots - 61649408 \) Copy content Toggle raw display
$97$ \( T^{10} - 466 T^{8} + \cdots - 54080000 \) Copy content Toggle raw display
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