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Properties

Label 8330.2.a.cv
Level $8330$
Weight $2$
Character orbit 8330.a
Self dual yes
Analytic conductor $66.515$
Analytic rank $0$
Dimension $10$
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] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) 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("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8330.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,-2,10,-10,-2,0,10,12,-10,-2,-2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.5153848837\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{3} + \beta_{2} + 1) q^{9} - q^{10} + \beta_{5} q^{11} - \beta_1 q^{12} + (\beta_{8} + \beta_{6} - \beta_1) q^{13} + \beta_1 q^{15}+ \cdots + ( - 3 \beta_{9} - \beta_{8} - 3 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{9} + 10 \nu^{8} + 15 \nu^{7} - 196 \nu^{6} - 14 \nu^{5} + 1104 \nu^{4} - 321 \nu^{3} + \cdots + 374 ) / 152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 10\nu^{8} - 15\nu^{7} + 196\nu^{6} + 14\nu^{5} - 1104\nu^{4} + 321\nu^{3} + 1932\nu^{2} - 741\nu - 982 ) / 152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6\nu^{9} - 3\nu^{8} - 109\nu^{7} + 36\nu^{6} + 502\nu^{5} + 102\nu^{4} - 316\nu^{3} - 1043\nu^{2} - 209\nu + 682 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - \nu^{7} - 19\nu^{6} + 18\nu^{5} + 99\nu^{4} - 75\nu^{3} - 129\nu^{2} + 58\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{9} - 18 \nu^{8} - 331 \nu^{7} + 330 \nu^{6} + 1834 \nu^{5} - 1478 \nu^{4} - 2941 \nu^{3} + \cdots - 50 ) / 152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31 \nu^{9} + 44 \nu^{8} + 579 \nu^{7} - 794 \nu^{6} - 2904 \nu^{5} + 3520 \nu^{4} + 3463 \nu^{3} + \cdots + 726 ) / 152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18 \nu^{9} + 9 \nu^{8} + 365 \nu^{7} - 146 \nu^{6} - 2209 \nu^{5} + 359 \nu^{4} + 4368 \nu^{3} + \cdots - 944 ) / 76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21 \nu^{9} - 20 \nu^{8} - 410 \nu^{7} + 354 \nu^{6} + 2289 \nu^{5} - 1429 \nu^{4} - 3823 \nu^{3} + \cdots + 202 ) / 76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 8\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + 11\beta_{3} + 12\beta_{2} - 2\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{9} - 14 \beta_{8} - 16 \beta_{7} + 13 \beta_{6} - \beta_{5} - 3 \beta_{4} - 6 \beta_{3} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{9} + 7 \beta_{8} + 37 \beta_{7} + 58 \beta_{6} - 15 \beta_{5} + 15 \beta_{4} + 115 \beta_{3} + \cdots + 312 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 332 \beta_{9} - 160 \beta_{8} - 204 \beta_{7} + 156 \beta_{6} - 16 \beta_{5} - 52 \beta_{4} + \cdots - 386 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 136 \beta_{9} + 150 \beta_{8} + 514 \beta_{7} + 802 \beta_{6} - 180 \beta_{5} + 188 \beta_{4} + \cdots + 3248 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3762 \beta_{9} - 1755 \beta_{8} - 2419 \beta_{7} + 1801 \beta_{6} - 190 \beta_{5} - 694 \beta_{4} + \cdots - 4818 \) 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
3.18918
2.79976
1.56890
1.35219
0.924404
−0.255150
−0.664810
−1.18925
−2.34515
−3.38008
1.00000 −3.18918 1.00000 −1.00000 −3.18918 0 1.00000 7.17088 −1.00000
1.2 1.00000 −2.79976 1.00000 −1.00000 −2.79976 0 1.00000 4.83868 −1.00000
1.3 1.00000 −1.56890 1.00000 −1.00000 −1.56890 0 1.00000 −0.538549 −1.00000
1.4 1.00000 −1.35219 1.00000 −1.00000 −1.35219 0 1.00000 −1.17159 −1.00000
1.5 1.00000 −0.924404 1.00000 −1.00000 −0.924404 0 1.00000 −2.14548 −1.00000
1.6 1.00000 0.255150 1.00000 −1.00000 0.255150 0 1.00000 −2.93490 −1.00000
1.7 1.00000 0.664810 1.00000 −1.00000 0.664810 0 1.00000 −2.55803 −1.00000
1.8 1.00000 1.18925 1.00000 −1.00000 1.18925 0 1.00000 −1.58568 −1.00000
1.9 1.00000 2.34515 1.00000 −1.00000 2.34515 0 1.00000 2.49972 −1.00000
1.10 1.00000 3.38008 1.00000 −1.00000 3.38008 0 1.00000 8.42495 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

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 8330.2.a.cv ✓ 10
7.b odd 2 1 8330.2.a.cw yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8330.2.a.cv ✓ 10 1.a even 1 1 trivial
8330.2.a.cw yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8330))\):

\( T_{3}^{10} + 2 T_{3}^{9} - 19 T_{3}^{8} - 38 T_{3}^{7} + 100 T_{3}^{6} + 194 T_{3}^{5} - 151 T_{3}^{4} + \cdots - 28 \) Copy content Toggle raw display
\( T_{11}^{10} + 2 T_{11}^{9} - 67 T_{11}^{8} - 74 T_{11}^{7} + 1595 T_{11}^{6} + 964 T_{11}^{5} + \cdots - 120896 \) Copy content Toggle raw display
\( T_{13}^{10} + 4 T_{13}^{9} - 88 T_{13}^{8} - 216 T_{13}^{7} + 3009 T_{13}^{6} + 2572 T_{13}^{5} + \cdots + 109256 \) Copy content Toggle raw display
\( T_{19}^{10} + 14 T_{19}^{9} - 19 T_{19}^{8} - 934 T_{19}^{7} - 2248 T_{19}^{6} + 12842 T_{19}^{5} + \cdots - 25794 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots - 28 \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots - 120896 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 109256 \) Copy content Toggle raw display
$17$ \( (T + 1)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + 14 T^{9} + \cdots - 25794 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots - 43904 \) Copy content Toggle raw display
$29$ \( T^{10} - 10 T^{9} + \cdots + 68642 \) Copy content Toggle raw display
$31$ \( T^{10} + 2 T^{9} + \cdots + 1282624 \) Copy content Toggle raw display
$37$ \( T^{10} - 196 T^{8} + \cdots - 520704 \) Copy content Toggle raw display
$41$ \( T^{10} - 22 T^{9} + \cdots + 5923904 \) Copy content Toggle raw display
$43$ \( T^{10} - 14 T^{9} + \cdots - 12836864 \) Copy content Toggle raw display
$47$ \( T^{10} + 18 T^{9} + \cdots - 49213454 \) Copy content Toggle raw display
$53$ \( T^{10} - 22 T^{9} + \cdots - 103868 \) Copy content Toggle raw display
$59$ \( T^{10} - 6 T^{9} + \cdots + 491054 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 117761056 \) Copy content Toggle raw display
$67$ \( T^{10} - 14 T^{9} + \cdots + 37484288 \) Copy content Toggle raw display
$71$ \( T^{10} - 20 T^{9} + \cdots - 686152 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 693266368 \) Copy content Toggle raw display
$79$ \( T^{10} - 44 T^{9} + \cdots - 12533024 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 521531136 \) Copy content Toggle raw display
$89$ \( T^{10} - 190 T^{8} + \cdots + 735104 \) Copy content Toggle raw display
$97$ \( T^{10} - 40 T^{9} + \cdots + 165312 \) Copy content Toggle raw display
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