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Properties

Label 4641.2.a.t
Level $4641$
Weight $2$
Character orbit 4641.a
Self dual yes
Analytic conductor $37.059$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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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] = [4641,2,Mod(1,4641)] 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("4641.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4641, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4641.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,1,-12,15,-3] 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: \(37.0585715781\)
Analytic rank: \(0\)
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: \( x^{12} - x^{11} - 19 x^{10} + 16 x^{9} + 136 x^{8} - 91 x^{7} - 456 x^{6} + 220 x^{5} + 722 x^{4} + \cdots + 85 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} - \beta_1 q^{6} - q^{7} + (\beta_{10} - \beta_{9} + \beta_1) q^{8} + q^{9} + ( - \beta_{11} - \beta_{3}) q^{10} + (\beta_{10} + 1) q^{11}+ \cdots + (\beta_{10} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 12 q^{3} + 15 q^{4} - 3 q^{5} - q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9} + q^{10} + 14 q^{11} - 15 q^{12} - 12 q^{13} - q^{14} + 3 q^{15} + 5 q^{16} + 12 q^{17} + q^{18} + 6 q^{19} - q^{20}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 19 x^{10} + 16 x^{9} + 136 x^{8} - 91 x^{7} - 456 x^{6} + 220 x^{5} + 722 x^{4} + \cdots + 85 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} - 18 \nu^{9} - 19 \nu^{8} + 114 \nu^{7} + 122 \nu^{6} - 298 \nu^{5} - 302 \nu^{4} + \cdots - 44 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2 \nu^{11} + \nu^{10} - 35 \nu^{9} - 21 \nu^{8} + 215 \nu^{7} + 145 \nu^{6} - 543 \nu^{5} - 379 \nu^{4} + \cdots - 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5 \nu^{11} + 4 \nu^{10} - 87 \nu^{9} - 79 \nu^{8} + 527 \nu^{7} + 528 \nu^{6} - 1288 \nu^{5} + \cdots - 253 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6 \nu^{11} + 5 \nu^{10} - 105 \nu^{9} - 96 \nu^{8} + 643 \nu^{7} + 626 \nu^{6} - 1606 \nu^{5} + \cdots - 275 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11 \nu^{11} + 9 \nu^{10} - 192 \nu^{9} - 175 \nu^{8} + 1170 \nu^{7} + 1154 \nu^{6} - 2894 \nu^{5} + \cdots - 542 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14 \nu^{11} + 11 \nu^{10} - 245 \nu^{9} - 214 \nu^{8} + 1499 \nu^{7} + 1408 \nu^{6} - 3734 \nu^{5} + \cdots - 627 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15 \nu^{11} + 12 \nu^{10} - 263 \nu^{9} - 233 \nu^{8} + 1613 \nu^{7} + 1532 \nu^{6} - 4032 \nu^{5} + \cdots - 693 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} + 12 \nu^{10} - 263 \nu^{9} - 233 \nu^{8} + 1613 \nu^{7} + 1532 \nu^{6} - 4032 \nu^{5} + \cdots - 693 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21 \nu^{11} - 18 \nu^{10} + 369 \nu^{9} + 345 \nu^{8} - 2269 \nu^{7} - 2244 \nu^{6} + 5692 \nu^{5} + \cdots + 979 ) / 2 \) 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
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 8\beta_{10} - 7\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} - \beta_{8} - 10\beta_{7} + 10\beta_{6} + 10\beta_{5} - \beta_{3} + 45\beta_{2} + 10\beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{11} + 54 \beta_{10} - 44 \beta_{9} + 10 \beta_{8} - 9 \beta_{7} + 12 \beta_{6} - \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{11} + 14 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} - 78 \beta_{7} + 76 \beta_{6} + 77 \beta_{5} + \cdots + 444 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 90 \beta_{11} + 347 \beta_{10} - 274 \beta_{9} + 78 \beta_{8} - 61 \beta_{7} + 105 \beta_{6} - 14 \beta_{5} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17 \beta_{11} + 135 \beta_{10} - 34 \beta_{9} - 104 \beta_{8} - 558 \beta_{7} + 529 \beta_{6} + \cdots + 2700 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 662 \beta_{11} + 2193 \beta_{10} - 1716 \beta_{9} + 560 \beta_{8} - 376 \beta_{7} + 817 \beta_{6} + \cdots + 364 \) 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
−2.51924
−2.26441
−1.69095
−1.59535
−0.729657
−0.596335
0.525406
1.11728
1.81545
1.81703
2.54577
2.57500
−2.51924 −1.00000 4.34657 −1.26360 2.51924 −1.00000 −5.91159 1.00000 3.18331
1.2 −2.26441 −1.00000 3.12755 1.99264 2.26441 −1.00000 −2.55325 1.00000 −4.51216
1.3 −1.69095 −1.00000 0.859298 −3.05176 1.69095 −1.00000 1.92887 1.00000 5.16036
1.4 −1.59535 −1.00000 0.545142 2.04940 1.59535 −1.00000 2.32101 1.00000 −3.26951
1.5 −0.729657 −1.00000 −1.46760 1.79318 0.729657 −1.00000 2.53016 1.00000 −1.30841
1.6 −0.596335 −1.00000 −1.64438 −4.30184 0.596335 −1.00000 2.17327 1.00000 2.56534
1.7 0.525406 −1.00000 −1.72395 −1.05112 −0.525406 −1.00000 −1.95659 1.00000 −0.552267
1.8 1.11728 −1.00000 −0.751696 0.177320 −1.11728 −1.00000 −3.07440 1.00000 0.198115
1.9 1.81545 −1.00000 1.29586 −1.34587 −1.81545 −1.00000 −1.27833 1.00000 −2.44337
1.10 1.81703 −1.00000 1.30160 4.31133 −1.81703 −1.00000 −1.26901 1.00000 7.83383
1.11 2.54577 −1.00000 4.48097 −3.15427 −2.54577 −1.00000 6.31598 1.00000 −8.03006
1.12 2.57500 −1.00000 4.63063 0.844591 −2.57500 −1.00000 6.77388 1.00000 2.17482
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( +1 \)
\(17\) \( -1 \)

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 4641.2.a.t ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4641.2.a.t ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4641))\):

\( T_{2}^{12} - T_{2}^{11} - 19 T_{2}^{10} + 16 T_{2}^{9} + 136 T_{2}^{8} - 91 T_{2}^{7} - 456 T_{2}^{6} + \cdots + 85 \) Copy content Toggle raw display
\( T_{5}^{12} + 3 T_{5}^{11} - 32 T_{5}^{10} - 91 T_{5}^{9} + 316 T_{5}^{8} + 791 T_{5}^{7} - 1367 T_{5}^{6} + \cdots + 350 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 85 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 350 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{11} + \cdots + 150272 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( (T - 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 15545872 \) Copy content Toggle raw display
$23$ \( T^{12} - 11 T^{11} + \cdots - 1023520 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 2718238 \) Copy content Toggle raw display
$31$ \( T^{12} - 23 T^{11} + \cdots + 40000 \) Copy content Toggle raw display
$37$ \( T^{12} - 22 T^{11} + \cdots - 57746600 \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} + \cdots + 43775450 \) Copy content Toggle raw display
$43$ \( T^{12} + 14 T^{11} + \cdots - 22290560 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 711474688 \) Copy content Toggle raw display
$53$ \( T^{12} + 11 T^{11} + \cdots + 20008960 \) Copy content Toggle raw display
$59$ \( T^{12} - 15 T^{11} + \cdots + 3116920 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 3632542150 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 2787244000 \) Copy content Toggle raw display
$71$ \( T^{12} - 54 T^{11} + \cdots + 12251392 \) Copy content Toggle raw display
$73$ \( T^{12} - 10 T^{11} + \cdots + 398080 \) Copy content Toggle raw display
$79$ \( T^{12} - 44 T^{11} + \cdots + 45749696 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 82962476056 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 662262272 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 15053281024 \) Copy content Toggle raw display
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