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Properties

Label 3627.2.a.u
Level $3627$
Weight $2$
Character orbit 3627.a
Self dual yes
Analytic conductor $28.962$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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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] = [3627,2,Mod(1,3627)] 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("3627.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3627, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3627 = 3^{2} \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3627.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.9617408131\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 17x^{10} + 103x^{8} - 275x^{6} + 316x^{4} - 128x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + \beta_{3} + 1) q^{4} + ( - \beta_{9} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} - 1) q^{7} + ( - \beta_{11} + \beta_{8} + \cdots + 2 \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} - 1) q^{10}+ \cdots + ( - 4 \beta_{11} + 3 \beta_{10} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{4} - 10 q^{7} - 6 q^{10} - 12 q^{13} + 10 q^{16} - 22 q^{19} - 10 q^{22} + 10 q^{25} - 38 q^{28} - 12 q^{31} - 54 q^{34} - 76 q^{37} - 34 q^{40} + 22 q^{43} - 24 q^{46} + 42 q^{49} - 10 q^{52}+ \cdots - 100 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 17x^{10} + 103x^{8} - 275x^{6} + 316x^{4} - 128x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} - 12\nu^{6} + 39\nu^{4} - 32\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 14\nu^{6} + 61\nu^{4} - 88\nu^{2} + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 14\nu^{6} - 61\nu^{4} + 92\nu^{2} - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 17\nu^{9} + 101\nu^{7} - 251\nu^{5} + 238\nu^{3} - 64\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 17\nu^{8} + 103\nu^{6} - 271\nu^{4} + 280\nu^{2} - 60 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{10} + 33\nu^{8} - 190\nu^{6} + 463\nu^{4} - 440\nu^{2} + 96 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} - 49\nu^{9} + 279\nu^{7} - 675\nu^{5} + 650\nu^{3} - 176\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{11} - 51\nu^{9} + 307\nu^{7} - 797\nu^{5} + 826\nu^{3} - 208\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} - 33\nu^{9} + 190\nu^{7} - 461\nu^{5} + 422\nu^{3} - 68\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2\nu^{11} - 33\nu^{9} + 190\nu^{7} - 463\nu^{5} + 440\nu^{3} - 96\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{8} + \beta_{5} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{6} + 6\beta_{4} + 8\beta_{3} - \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{11} + 2\beta_{10} + 9\beta_{8} + 9\beta_{5} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} + 22\beta_{6} + 38\beta_{4} + 58\beta_{3} - 9\beta_{2} + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -93\beta_{11} + 22\beta_{10} + 2\beta_{9} + 71\beta_{8} + 65\beta_{5} + 276\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93\beta_{7} + 186\beta_{6} + 254\beta_{4} + 416\beta_{3} - 65\beta_{2} + 575 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -719\beta_{11} + 186\beta_{10} + 24\beta_{9} + 537\beta_{8} + 449\beta_{5} + 1936\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 719\beta_{7} + 1442\beta_{6} + 1750\beta_{4} + 2986\beta_{3} - 449\beta_{2} + 3893 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -5353\beta_{11} + 1442\beta_{10} + 206\beta_{9} + 3979\beta_{8} + 3097\beta_{5} + 13712\beta_1 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.68259
−2.05034
−1.78373
−1.33861
−0.621125
−0.490364
0.490364
0.621125
1.33861
1.78373
2.05034
2.68259
−2.68259 0 5.19630 0.692545 0 −3.93814 −8.57436 0 −1.85782
1.2 −2.05034 0 2.20388 3.53724 0 0.521315 −0.418033 0 −7.25254
1.3 −1.78373 0 1.18170 −2.42678 0 −1.54220 1.45963 0 4.32872
1.4 −1.33861 0 −0.208134 −0.358509 0 0.940311 2.95582 0 0.479902
1.5 −0.621125 0 −1.61420 0.967413 0 4.17172 2.24487 0 −0.600885
1.6 −0.490364 0 −1.75954 −3.88003 0 −5.15301 1.84354 0 1.90263
1.7 0.490364 0 −1.75954 3.88003 0 −5.15301 −1.84354 0 1.90263
1.8 0.621125 0 −1.61420 −0.967413 0 4.17172 −2.24487 0 −0.600885
1.9 1.33861 0 −0.208134 0.358509 0 0.940311 −2.95582 0 0.479902
1.10 1.78373 0 1.18170 2.42678 0 −1.54220 −1.45963 0 4.32872
1.11 2.05034 0 2.20388 −3.53724 0 0.521315 0.418033 0 −7.25254
1.12 2.68259 0 5.19630 −0.692545 0 −3.93814 8.57436 0 −1.85782
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(13\) \( +1 \)
\(31\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3627.2.a.u ✓ 12
3.b odd 2 1 inner 3627.2.a.u ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3627.2.a.u ✓ 12 1.a even 1 1 trivial
3627.2.a.u ✓ 12 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3627))\):

\( T_{2}^{12} - 17T_{2}^{10} + 103T_{2}^{8} - 275T_{2}^{6} + 316T_{2}^{4} - 128T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{12} - 35T_{5}^{10} + 403T_{5}^{8} - 1672T_{5}^{6} + 1936T_{5}^{4} - 720T_{5}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 17 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 35 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{6} + 5 T^{5} - 19 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 98 T^{10} + \cdots + 40804 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 145 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( (T^{6} + 11 T^{5} + \cdots + 13232)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 140 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 281971264 \) Copy content Toggle raw display
$31$ \( (T + 1)^{12} \) Copy content Toggle raw display
$37$ \( (T^{6} + 38 T^{5} + \cdots - 4502)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 249 T^{10} + \cdots + 719104 \) Copy content Toggle raw display
$43$ \( (T^{6} - 11 T^{5} + \cdots + 14336)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1073741824 \) Copy content Toggle raw display
$53$ \( T^{12} - 312 T^{10} + \cdots + 51380224 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 209207296 \) Copy content Toggle raw display
$61$ \( (T^{6} + 2 T^{5} + \cdots - 39392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 23 T^{5} + \cdots - 191744)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 814759936 \) Copy content Toggle raw display
$73$ \( (T^{6} + 18 T^{5} + \cdots + 1454)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + T^{5} + \cdots - 2205536)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 303 T^{10} + \cdots + 29073664 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 619810816 \) Copy content Toggle raw display
$97$ \( (T^{6} + 50 T^{5} + \cdots + 1568)^{2} \) Copy content Toggle raw display
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