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Properties

Label 684.2.r.b
Level $684$
Weight $2$
Character orbit 684.r
Analytic conductor $5.462$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [684,2,Mod(487,684)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(684, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("684.487"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{17} + x^{16} + 3 x^{14} - 12 x^{13} + 28 x^{12} - 24 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{9} - \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{14} + \beta_{8} + \beta_{2}) q^{7} + (\beta_{13} - \beta_{12}) q^{8} + (\beta_{18} + \beta_{17} - \beta_{16} + \cdots - 1) q^{10}+ \cdots + ( - 3 \beta_{19} - 2 \beta_{18} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + q^{4} - 4 q^{8} + 6 q^{10} + 6 q^{13} - 9 q^{14} - 11 q^{16} + 12 q^{19} + 14 q^{20} + 8 q^{22} - 10 q^{25} + 7 q^{28} + 12 q^{31} + 29 q^{32} - 6 q^{34} - 25 q^{38} - 46 q^{40} - 12 q^{41}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{17} + x^{16} + 3 x^{14} - 12 x^{13} + 28 x^{12} - 24 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{18} - 6 \nu^{14} - 6 \nu^{13} - 10 \nu^{12} - 16 \nu^{11} - 7 \nu^{10} + 10 \nu^{9} + \cdots - 448 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{19} + 6 \nu^{18} - 6 \nu^{17} - 2 \nu^{16} + 4 \nu^{13} + 66 \nu^{12} - 19 \nu^{11} + \cdots + 5120 ) / 1792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{19} - 60 \nu^{18} - 3 \nu^{17} - 120 \nu^{16} - 161 \nu^{15} - 182 \nu^{14} - 215 \nu^{13} + \cdots - 18944 ) / 7168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{19} - 9 \nu^{17} - 4 \nu^{16} + 13 \nu^{15} + 10 \nu^{14} + 3 \nu^{13} + 34 \nu^{12} + \cdots - 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{19} + 10 \nu^{18} - 10 \nu^{17} + 20 \nu^{16} - 42 \nu^{14} - 12 \nu^{13} + 12 \nu^{12} + \cdots + 6144 ) / 1792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{19} + 4 \nu^{18} - 4 \nu^{17} + 15 \nu^{16} - 21 \nu^{14} + 12 \nu^{13} + 37 \nu^{12} + \cdots + 4608 ) / 1792 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4 \nu^{19} - 3 \nu^{18} - 4 \nu^{17} + \nu^{16} + 7 \nu^{14} + 54 \nu^{13} + 9 \nu^{12} + 118 \nu^{11} + \cdots + 1024 ) / 1792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{19} + 5 \nu^{18} - 5 \nu^{17} + 10 \nu^{16} + 21 \nu^{15} + 29 \nu^{13} + 62 \nu^{12} + \cdots + 3968 ) / 896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{19} - 13 \nu^{17} - 4 \nu^{16} + \nu^{15} - 14 \nu^{14} - 17 \nu^{13} + 26 \nu^{12} + \cdots + 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{19} + 9 \nu^{18} - 9 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 15 \nu^{13} + 64 \nu^{12} + \cdots + 5888 ) / 1792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5 \nu^{19} + 9 \nu^{18} - 9 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 15 \nu^{13} + 64 \nu^{12} + \cdots + 5888 ) / 1792 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6 \nu^{19} + 13 \nu^{18} - 6 \nu^{17} + 12 \nu^{16} + 14 \nu^{15} - 14 \nu^{14} - 52 \nu^{13} + \cdots + 1536 ) / 1792 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11 \nu^{19} + 38 \nu^{18} + 11 \nu^{17} - 22 \nu^{16} + 49 \nu^{15} - 28 \nu^{14} - 61 \nu^{13} + \cdots + 6144 ) / 3584 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13 \nu^{19} - 34 \nu^{18} - 15 \nu^{17} - 26 \nu^{16} - 21 \nu^{15} - 28 \nu^{14} - 39 \nu^{13} + \cdots - 8704 ) / 3584 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( \nu^{19} + \nu^{18} - \nu^{17} + 3 \nu^{16} + \nu^{15} - \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + \cdots + 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( \nu^{19} - 2 \nu^{18} + 3 \nu^{17} - 2 \nu^{16} + \nu^{15} + 3 \nu^{13} - 12 \nu^{12} + 28 \nu^{11} + \cdots - 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 8 \nu^{19} - \nu^{18} + \nu^{17} - 9 \nu^{16} + 7 \nu^{15} + 21 \nu^{14} - 17 \nu^{13} + \cdots - 2048 ) / 896 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - \beta_{17} + \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} - \beta_{9} + \cdots - \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{12} + \beta_{11} + \beta_{8} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{19} + 2 \beta_{17} + 2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{11} + 6 \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{18} - \beta_{17} - 4 \beta_{16} - 2 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} + 4 \beta_{12} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{19} + 6 \beta_{18} - 2 \beta_{17} - \beta_{16} - 5 \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{19} - \beta_{18} - 9 \beta_{17} + \beta_{16} - \beta_{15} + 4 \beta_{14} - 5 \beta_{12} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6 \beta_{19} + 2 \beta_{18} - 4 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2 \beta_{19} + 5 \beta_{18} + 21 \beta_{17} + 14 \beta_{16} + 10 \beta_{15} + 9 \beta_{14} + 4 \beta_{13} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - \beta_{19} + 20 \beta_{17} - 11 \beta_{16} + \beta_{15} - 33 \beta_{14} - 23 \beta_{13} + 12 \beta_{12} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13 \beta_{19} + 15 \beta_{18} + 9 \beta_{17} - 13 \beta_{16} - 21 \beta_{15} - 20 \beta_{14} - 4 \beta_{13} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4 \beta_{19} - 8 \beta_{18} - 50 \beta_{17} - 8 \beta_{16} - 4 \beta_{15} + 38 \beta_{14} - 39 \beta_{13} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 56 \beta_{19} + 37 \beta_{18} - 21 \beta_{17} + 44 \beta_{16} + 6 \beta_{15} + 9 \beta_{14} + \cdots - 52 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7 \beta_{19} + 14 \beta_{18} + 42 \beta_{17} + 15 \beta_{16} + 23 \beta_{15} + 11 \beta_{14} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 5 \beta_{19} - 69 \beta_{18} + 75 \beta_{17} + 17 \beta_{16} + 103 \beta_{15} + 72 \beta_{14} + \cdots - 66 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 182 \beta_{19} - 150 \beta_{18} + 24 \beta_{17} + 70 \beta_{16} + 150 \beta_{15} - 54 \beta_{14} + \cdots + 176 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(\beta_{12}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
0.894531 − 1.09536i
−0.141961 − 1.40707i
−0.237943 − 1.39405i
1.34699 − 0.430844i
1.32127 + 0.504239i
−1.34963 − 0.422503i
0.818463 + 1.15331i
0.478868 + 1.33067i
−1.25575 + 0.650459i
−0.874835 + 1.11115i
0.894531 + 1.09536i
−0.141961 + 1.40707i
−0.237943 + 1.39405i
1.34699 + 0.430844i
1.32127 − 0.504239i
−1.34963 + 0.422503i
0.818463 − 1.15331i
0.478868 − 1.33067i
−1.25575 − 0.650459i
−0.874835 − 1.11115i
−1.39588 − 0.227006i 0 1.89694 + 0.633745i 0.720511 + 1.24796i 0 1.30023i −2.50402 − 1.31525i 0 −0.722448 − 1.90556i
487.2 −1.14758 + 0.826477i 0 0.633871 − 1.89689i 1.44325 + 2.49978i 0 − 2.96861i 0.840324 + 2.70071i 0 −3.72225 − 1.67588i
487.3 −1.08831 + 0.903091i 0 0.368852 − 1.96569i −1.43271 − 2.48152i 0 2.64551i 1.37377 + 2.47240i 0 3.80027 + 1.40681i
487.4 −1.04662 − 0.951102i 0 0.190808 + 1.99088i −1.45389 − 2.51821i 0 − 5.22116i 1.69383 − 2.26516i 0 −0.873411 + 4.01839i
487.5 −0.223949 − 1.39637i 0 −1.89969 + 0.625432i −0.127074 − 0.220099i 0 1.43470i 1.29877 + 2.51261i 0 −0.278881 + 0.226733i
487.6 0.308915 + 1.38006i 0 −1.80914 + 0.852645i −0.984694 − 1.70554i 0 − 0.355075i −1.73557 − 2.23333i 0 2.04956 − 1.88581i
487.7 0.589563 − 1.28546i 0 −1.30483 − 1.51572i 2.18826 + 3.79018i 0 − 0.239504i −2.71769 + 0.783701i 0 6.16226 − 0.578382i
487.8 0.912961 − 1.08005i 0 −0.333006 − 1.97208i −1.38652 − 2.40152i 0 0.149639i −2.43396 − 1.44077i 0 −3.85959 − 0.694987i
487.9 1.19119 + 0.762280i 0 0.837858 + 1.81604i 0.542680 + 0.939950i 0 4.07547i −0.386284 + 2.80193i 0 −0.0700709 + 1.53333i
487.10 1.39971 + 0.202053i 0 1.91835 + 0.565628i 0.490176 + 0.849009i 0 − 4.28530i 2.57084 + 1.17932i 0 0.514557 + 1.28740i
559.1 −1.39588 + 0.227006i 0 1.89694 − 0.633745i 0.720511 − 1.24796i 0 − 1.30023i −2.50402 + 1.31525i 0 −0.722448 + 1.90556i
559.2 −1.14758 − 0.826477i 0 0.633871 + 1.89689i 1.44325 − 2.49978i 0 2.96861i 0.840324 − 2.70071i 0 −3.72225 + 1.67588i
559.3 −1.08831 − 0.903091i 0 0.368852 + 1.96569i −1.43271 + 2.48152i 0 − 2.64551i 1.37377 − 2.47240i 0 3.80027 − 1.40681i
559.4 −1.04662 + 0.951102i 0 0.190808 − 1.99088i −1.45389 + 2.51821i 0 5.22116i 1.69383 + 2.26516i 0 −0.873411 − 4.01839i
559.5 −0.223949 + 1.39637i 0 −1.89969 − 0.625432i −0.127074 + 0.220099i 0 − 1.43470i 1.29877 − 2.51261i 0 −0.278881 − 0.226733i
559.6 0.308915 − 1.38006i 0 −1.80914 − 0.852645i −0.984694 + 1.70554i 0 0.355075i −1.73557 + 2.23333i 0 2.04956 + 1.88581i
559.7 0.589563 + 1.28546i 0 −1.30483 + 1.51572i 2.18826 − 3.79018i 0 0.239504i −2.71769 − 0.783701i 0 6.16226 + 0.578382i
559.8 0.912961 + 1.08005i 0 −0.333006 + 1.97208i −1.38652 + 2.40152i 0 − 0.149639i −2.43396 + 1.44077i 0 −3.85959 + 0.694987i
559.9 1.19119 − 0.762280i 0 0.837858 − 1.81604i 0.542680 − 0.939950i 0 − 4.07547i −0.386284 − 2.80193i 0 −0.0700709 − 1.53333i
559.10 1.39971 − 0.202053i 0 1.91835 − 0.565628i 0.490176 − 0.849009i 0 4.28530i 2.57084 − 1.17932i 0 0.514557 − 1.28740i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.r.b 20
3.b odd 2 1 228.2.k.b yes 20
4.b odd 2 1 684.2.r.c 20
12.b even 2 1 228.2.k.a ✓ 20
19.d odd 6 1 684.2.r.c 20
57.f even 6 1 228.2.k.a ✓ 20
76.f even 6 1 inner 684.2.r.b 20
228.n odd 6 1 228.2.k.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.k.a ✓ 20 12.b even 2 1
228.2.k.a ✓ 20 57.f even 6 1
228.2.k.b yes 20 3.b odd 2 1
228.2.k.b yes 20 228.n odd 6 1
684.2.r.b 20 1.a even 1 1 trivial
684.2.r.b 20 76.f even 6 1 inner
684.2.r.c 20 4.b odd 2 1
684.2.r.c 20 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{20} + 30 T_{5}^{18} + 24 T_{5}^{17} + 614 T_{5}^{16} + 476 T_{5}^{15} + 6612 T_{5}^{14} + \cdots + 50176 \) Copy content Toggle raw display
\( T_{23}^{20} - 78 T_{23}^{18} + 4714 T_{23}^{16} - 16284 T_{23}^{15} - 74860 T_{23}^{14} + 467976 T_{23}^{13} + \cdots + 295936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 30 T^{18} + \cdots + 50176 \) Copy content Toggle raw display
$7$ \( T^{20} + 82 T^{18} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{20} + 116 T^{18} + \cdots + 10445824 \) Copy content Toggle raw display
$13$ \( T^{20} - 6 T^{19} + \cdots + 117649 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1107558400 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} - 78 T^{18} + \cdots + 295936 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 708837376 \) Copy content Toggle raw display
$31$ \( (T^{10} - 6 T^{9} + \cdots + 133909)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22559739601 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 124084289536 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 11511112196809 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 94686674944 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 876665877799936 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 437545206784 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 55822580045401 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
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