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Properties

Label 351.2.g.d
Level $351$
Weight $2$
Character orbit 351.g
Analytic conductor $2.803$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(55,351)] 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("351.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-6,-2] 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: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
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: \( x^{10} + 8x^{8} - 4x^{7} + 52x^{6} - 19x^{5} + 100x^{4} - 24x^{3} + 150x^{2} - 36x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} + \beta_{6} - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{9} + \beta_{6} + \beta_{5} - 1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{8} + (\beta_{8} - \beta_{7} + \cdots - 2 \beta_1) q^{10}+ \cdots + ( - \beta_{9} - 3 \beta_{8} + \cdots + 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 2 q^{5} - 3 q^{7} + 12 q^{8} + 6 q^{10} - 2 q^{11} + 5 q^{13} - 6 q^{14} - 4 q^{16} + 8 q^{17} - q^{19} + 30 q^{20} + 7 q^{22} + 13 q^{23} + 8 q^{25} - 26 q^{26} - 7 q^{28} + 17 q^{29}+ \cdots + 70 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 8x^{8} - 4x^{7} + 52x^{6} - 19x^{5} + 100x^{4} - 24x^{3} + 150x^{2} - 36x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6136 \nu^{9} + 9552 \nu^{8} - 10348 \nu^{7} + 39845 \nu^{6} - 101888 \nu^{5} + 517400 \nu^{4} + \cdots + 1205568 ) / 4749375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3184 \nu^{9} + 19812 \nu^{8} - 21463 \nu^{7} + 140320 \nu^{6} - 211328 \nu^{5} + 1073150 \nu^{4} + \cdots + 4767783 ) / 1583125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21128 \nu^{9} - 107196 \nu^{8} + 116129 \nu^{7} - 620185 \nu^{6} + 1143424 \nu^{5} + \cdots - 15581814 ) / 4749375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21231 \nu^{9} - 78408 \nu^{8} + 84942 \nu^{7} - 784505 \nu^{6} + 836352 \nu^{5} - 4247100 \nu^{4} + \cdots - 5059072 ) / 1583125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 133952 \nu^{9} - 6136 \nu^{8} + 1062064 \nu^{7} - 525460 \nu^{6} + 6925659 \nu^{5} - 2443200 \nu^{4} + \cdots + 156351 ) / 4749375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59332 \nu^{9} + 9601 \nu^{8} - 511724 \nu^{7} + 277485 \nu^{6} - 3479744 \nu^{5} + \cdots + 3025584 ) / 949875 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 133952 \nu^{9} + 6136 \nu^{8} - 1062064 \nu^{7} + 525460 \nu^{6} - 6925659 \nu^{5} + \cdots + 4593024 ) / 1583125 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 83812 \nu^{9} - 64009 \nu^{8} - 695834 \nu^{7} - 112240 \nu^{6} - 4172154 \nu^{5} + \cdots - 174156 ) / 949875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - \beta_{3} - 5\beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 7\beta_{8} + \beta_{7} - 15\beta_{6} + \beta_{4} + 7\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{8} - 8\beta_{7} + 11\beta_{6} + 28\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{5} - 10\beta_{4} - 46\beta_{3} - 23\beta_{2} - 23\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{9} - 79\beta_{8} + 54\beta_{7} - 97\beta_{6} + 54\beta_{4} + 79\beta_{3} + 168\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -52\beta_{9} + 301\beta_{8} - 81\beta_{7} + 527\beta_{6} + 52\beta_{5} + 201\beta_{2} - 527 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -29\beta_{5} - 353\beta_{4} - 583\beta_{3} - 1048\beta_{2} - 1048\beta _1 + 771 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−1.30800 − 2.26552i
−0.683601 − 1.18403i
0.124958 + 0.216433i
0.754386 + 1.30663i
1.11226 + 1.92649i
−1.30800 + 2.26552i
−0.683601 + 1.18403i
0.124958 − 0.216433i
0.754386 − 1.30663i
1.11226 − 1.92649i
−1.30800 − 2.26552i 0 −2.42173 + 4.19455i −3.84345 0 −0.358824 + 0.621501i 7.43846 0 5.02723 + 8.70742i
55.2 −0.683601 − 1.18403i 0 0.0653801 − 0.113242i 1.13076 0 0.587702 − 1.01793i −2.91318 0 −0.772989 − 1.33886i
55.3 0.124958 + 0.216433i 0 0.968771 − 1.67796i 2.93754 0 −1.94751 + 3.37319i 0.984052 0 0.367068 + 0.635781i
55.4 0.754386 + 1.30663i 0 −0.138195 + 0.239361i 0.723610 0 2.28408 − 3.95613i 2.60053 0 0.545881 + 0.945493i
55.5 1.11226 + 1.92649i 0 −1.47423 + 2.55344i −1.94846 0 −2.06544 + 3.57745i −2.10987 0 −2.16719 − 3.75369i
217.1 −1.30800 + 2.26552i 0 −2.42173 − 4.19455i −3.84345 0 −0.358824 − 0.621501i 7.43846 0 5.02723 − 8.70742i
217.2 −0.683601 + 1.18403i 0 0.0653801 + 0.113242i 1.13076 0 0.587702 + 1.01793i −2.91318 0 −0.772989 + 1.33886i
217.3 0.124958 − 0.216433i 0 0.968771 + 1.67796i 2.93754 0 −1.94751 − 3.37319i 0.984052 0 0.367068 − 0.635781i
217.4 0.754386 − 1.30663i 0 −0.138195 − 0.239361i 0.723610 0 2.28408 + 3.95613i 2.60053 0 0.545881 − 0.945493i
217.5 1.11226 − 1.92649i 0 −1.47423 − 2.55344i −1.94846 0 −2.06544 − 3.57745i −2.10987 0 −2.16719 + 3.75369i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 55.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.g.d ✓ 10
3.b odd 2 1 351.2.g.e yes 10
13.c even 3 1 inner 351.2.g.d ✓ 10
13.c even 3 1 4563.2.a.bd 5
13.e even 6 1 4563.2.a.be 5
39.h odd 6 1 4563.2.a.bc 5
39.i odd 6 1 351.2.g.e yes 10
39.i odd 6 1 4563.2.a.bf 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.g.d ✓ 10 1.a even 1 1 trivial
351.2.g.d ✓ 10 13.c even 3 1 inner
351.2.g.e yes 10 3.b odd 2 1
351.2.g.e yes 10 39.i odd 6 1
4563.2.a.bc 5 39.h odd 6 1
4563.2.a.bd 5 13.c even 3 1
4563.2.a.be 5 13.e even 6 1
4563.2.a.bf 5 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8T_{2}^{8} - 4T_{2}^{7} + 52T_{2}^{6} - 19T_{2}^{5} + 100T_{2}^{4} - 24T_{2}^{3} + 150T_{2}^{2} - 36T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} + T^{4} - 14 T^{3} + \cdots - 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 3844 \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots + 173889 \) Copy content Toggle raw display
$13$ \( T^{10} - 5 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 8 T^{9} + \cdots + 125316 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 576 \) Copy content Toggle raw display
$23$ \( T^{10} - 13 T^{9} + \cdots + 6561 \) Copy content Toggle raw display
$29$ \( T^{10} - 17 T^{9} + \cdots + 138384 \) Copy content Toggle raw display
$31$ \( (T^{5} + 3 T^{4} + \cdots + 1042)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 11 T^{9} + \cdots + 669124 \) Copy content Toggle raw display
$41$ \( T^{10} - 16 T^{9} + \cdots + 56070144 \) Copy content Toggle raw display
$43$ \( T^{10} - 12 T^{9} + \cdots + 22500 \) Copy content Toggle raw display
$47$ \( (T^{5} + 3 T^{4} + \cdots + 1593)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 20 T^{4} + \cdots - 486)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 19 T^{9} + \cdots + 4782969 \) Copy content Toggle raw display
$61$ \( T^{10} - 6 T^{9} + \cdots + 487204 \) Copy content Toggle raw display
$67$ \( T^{10} - 26 T^{9} + \cdots + 58033924 \) Copy content Toggle raw display
$71$ \( T^{10} + 20 T^{9} + \cdots + 9585216 \) Copy content Toggle raw display
$73$ \( (T^{5} - 12 T^{4} + \cdots + 2029)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 12 T^{4} + \cdots - 8618)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 20 T^{4} + \cdots + 36027)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 28 T^{9} + \cdots + 360000 \) Copy content Toggle raw display
$97$ \( T^{10} - 5 T^{9} + \cdots + 69169 \) Copy content Toggle raw display
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