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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1521,2,Mod(1351,1521)] 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("1521.1351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,0,0,0,0,28,0,0,0,0,0,-4,0,0,0,0,0,12,0,0,-52,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,-84,0,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1452461474\)
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: \( x^{12} - 2 x^{11} + 2 x^{10} - 4 x^{9} + 42 x^{8} - 98 x^{7} + 120 x^{6} - 70 x^{5} + 21 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + (2 \beta_{3} - \beta_{2} - 2) q^{4} + ( - \beta_{9} + \beta_{6} - \beta_1) q^{5} - \beta_{8} q^{7} + ( - \beta_{6} - 2 \beta_1) q^{8} + ( - 4 \beta_{3} + 4 \beta_{2} + 5) q^{10} + ( - \beta_{9} + 2 \beta_{6}) q^{11}+ \cdots + ( - 2 \beta_{9} - 5 \beta_{6} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} + 28 q^{10} - 4 q^{16} + 12 q^{22} - 52 q^{25} - 84 q^{40} + 44 q^{43} - 20 q^{49} - 84 q^{55} + 76 q^{61} - 40 q^{64} - 20 q^{79} - 4 q^{82} + 12 q^{88} + 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} - 4 x^{9} + 42 x^{8} - 98 x^{7} + 120 x^{6} - 70 x^{5} + 21 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4279 \nu^{11} + 11534 \nu^{10} + 9623 \nu^{9} - 5572 \nu^{8} + 103505 \nu^{7} + 298538 \nu^{6} + \cdots + 13971 ) / 100163 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5582 \nu^{11} + 32024 \nu^{10} - 46194 \nu^{9} + 57382 \nu^{8} - 303802 \nu^{7} + 1397798 \nu^{6} + \cdots + 112221 ) / 100163 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38434 \nu^{11} - 19996 \nu^{10} - 6262 \nu^{9} - 99743 \nu^{8} + 1429210 \nu^{7} - 1484178 \nu^{6} + \cdots + 172093 ) / 100163 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65218 \nu^{11} + 32253 \nu^{10} - 120013 \nu^{9} - 98562 \nu^{8} + 2189308 \nu^{7} + 171737 \nu^{6} + \cdots + 317458 ) / 100163 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 72427 \nu^{11} - 58333 \nu^{10} + 20516 \nu^{9} - 231915 \nu^{8} + 2753491 \nu^{7} - 3633798 \nu^{6} + \cdots - 316329 ) / 100163 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73721 \nu^{11} - 133272 \nu^{10} + 119078 \nu^{9} - 273912 \nu^{8} + 3052282 \nu^{7} + \cdots - 193850 ) / 100163 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 79691 \nu^{11} - 117725 \nu^{10} + 112908 \nu^{9} - 286916 \nu^{8} + 3234070 \nu^{7} + \cdots + 204421 ) / 100163 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 101689 \nu^{11} + 189732 \nu^{10} - 157236 \nu^{9} + 371269 \nu^{8} - 4221143 \nu^{7} + \cdots + 246129 ) / 100163 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 110364 \nu^{11} + 202290 \nu^{10} - 157594 \nu^{9} + 381201 \nu^{8} - 4550013 \nu^{7} + \cdots + 202767 ) / 100163 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 178831 \nu^{11} - 331623 \nu^{10} + 252741 \nu^{9} - 616714 \nu^{8} + 7384757 \nu^{7} + \cdots - 324453 ) / 100163 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 201202 \nu^{11} + 346355 \nu^{10} - 309866 \nu^{9} + 711879 \nu^{8} - 8237495 \nu^{7} + \cdots + 427189 ) / 100163 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} + 5\beta_{10} + 13\beta_{9} - 3\beta_{8} - 5\beta_{7} - \beta_{5} - 3\beta_{4} + 13\beta_{3} ) / 26 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -11\beta_{11} - 5\beta_{10} + 3\beta_{8} - 13\beta_{6} - 13\beta_1 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} + 34 \beta_{10} + 65 \beta_{9} - 23 \beta_{8} + 22 \beta_{7} - 13 \beta_{6} + 7 \beta_{5} + \cdots + 52 ) / 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{7} - 50\beta_{5} + 6\beta_{4} + 104\beta_{3} - 91\beta_{2} - 221 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 51 \beta_{11} - 217 \beta_{10} - 377 \beta_{9} + 151 \beta_{8} + 129 \beta_{7} + 39 \beta_{6} + \cdots + 455 ) / 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29\beta_{11} + 25\beta_{10} + 21\beta_{9} - 10\beta_{8} + 34\beta_{6} + 37\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 524 \beta_{11} - 1433 \beta_{10} - 2314 \beta_{9} + 956 \beta_{8} - 815 \beta_{7} - 39 \beta_{6} + \cdots - 3484 ) / 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1036\beta_{7} + 1944\beta_{5} + 216\beta_{4} - 5655\beta_{3} + 3601\beta_{2} + 9022 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4435 \beta_{11} + 9615 \beta_{10} + 14638 \beta_{9} - 6120 \beta_{8} - 5295 \beta_{7} + 1794 \beta_{6} + \cdots - 25545 ) / 26 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -14643\beta_{11} - 16218\beta_{10} - 17641\beta_{9} + 7786\beta_{8} - 15132\beta_{6} - 19994\beta_1 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 34457 \beta_{11} + 65028 \beta_{10} + 94627 \beta_{9} - 39841 \beta_{8} + 34968 \beta_{7} + \cdots + 183053 ) / 26 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
−1.85322 − 1.85322i
0.606238 − 0.606238i
1.54189 − 1.54189i
0.260051 + 0.260051i
−0.349314 − 0.349314i
0.794355 − 0.794355i
0.794355 + 0.794355i
−0.349314 + 0.349314i
0.260051 − 0.260051i
1.54189 + 1.54189i
0.606238 + 0.606238i
−1.85322 + 1.85322i
− 2.45945i 0 −4.04892 3.33722i 0 − 3.69202i 5.03922i 0 8.20775
1351.2 − 2.45945i 0 −4.04892 3.33722i 0 3.69202i 5.03922i 0 8.20775
1351.3 − 1.28184i 0 0.356896 2.16889i 0 − 1.04892i − 3.02115i 0 2.78017
1351.4 − 1.28184i 0 0.356896 2.16889i 0 1.04892i − 3.02115i 0 2.78017
1351.5 − 1.14367i 0 0.692021 − 3.48695i 0 − 3.35690i − 3.07878i 0 −3.98792
1351.6 − 1.14367i 0 0.692021 − 3.48695i 0 3.35690i − 3.07878i 0 −3.98792
1351.7 1.14367i 0 0.692021 3.48695i 0 − 3.35690i 3.07878i 0 −3.98792
1351.8 1.14367i 0 0.692021 3.48695i 0 3.35690i 3.07878i 0 −3.98792
1351.9 1.28184i 0 0.356896 − 2.16889i 0 − 1.04892i 3.02115i 0 2.78017
1351.10 1.28184i 0 0.356896 − 2.16889i 0 1.04892i 3.02115i 0 2.78017
1351.11 2.45945i 0 −4.04892 − 3.33722i 0 − 3.69202i − 5.03922i 0 8.20775
1351.12 2.45945i 0 −4.04892 − 3.33722i 0 3.69202i − 5.03922i 0 8.20775
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1351.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.n 12
3.b odd 2 1 inner 1521.2.b.n 12
13.b even 2 1 inner 1521.2.b.n 12
13.d odd 4 1 1521.2.a.v ✓ 6
13.d odd 4 1 1521.2.a.w yes 6
39.d odd 2 1 inner 1521.2.b.n 12
39.f even 4 1 1521.2.a.v ✓ 6
39.f even 4 1 1521.2.a.w yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1521.2.a.v ✓ 6 13.d odd 4 1
1521.2.a.v ✓ 6 39.f even 4 1
1521.2.a.w yes 6 13.d odd 4 1
1521.2.a.w yes 6 39.f even 4 1
1521.2.b.n 12 1.a even 1 1 trivial
1521.2.b.n 12 3.b odd 2 1 inner
1521.2.b.n 12 13.b even 2 1 inner
1521.2.b.n 12 39.d 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 \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 13 \) Copy content Toggle raw display
\( T_{5}^{6} + 28T_{5}^{4} + 245T_{5}^{2} + 637 \) Copy content Toggle raw display
\( T_{7}^{6} + 26T_{7}^{4} + 181T_{7}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 9 T^{4} + 20 T^{2} + 13)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 28 T^{4} + \cdots + 637)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 26 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 29 T^{4} + \cdots + 13)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 67 T^{4} + \cdots - 2197)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 89 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 60 T^{4} + \cdots - 2197)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 158 T^{4} + \cdots - 2197)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 166 T^{4} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 194 T^{4} + \cdots + 142129)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 165 T^{4} + \cdots + 165997)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 11 T^{2} + \cdots - 13)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 23 T^{4} + \cdots + 13)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 161 T^{4} + \cdots - 107653)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 120 T^{4} + \cdots + 832)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 19 T^{2} + \cdots - 169)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 133 T^{4} + \cdots + 8281)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 162 T^{4} + \cdots + 10933)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 90 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} + \cdots + 169)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 212 T^{4} + \cdots + 165997)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 263 T^{4} + \cdots + 251173)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 307 T^{4} + \cdots + 142129)^{2} \) Copy content Toggle raw display
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