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Properties

Label 1372.2.e.a
Level $1372$
Weight $2$
Character orbit 1372.e
Analytic conductor $10.955$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [1372,2,Mod(361,1372)] 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("1372.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1372, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1372 = 2^{2} \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1372.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-7,0,-7,0,0,0,-9,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9554751573\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - x^{11} + 10 x^{10} - 7 x^{9} + 68 x^{8} - 44 x^{7} + 225 x^{6} - 77 x^{5} + 490 x^{4} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{8} - \beta_{6} - \beta_{5} + \cdots - 2) q^{3} + (\beta_{11} + \beta_{8}) q^{5} + (\beta_{10} + 2 \beta_{8} + \cdots + 2 \beta_1) q^{9} + ( - \beta_{11} + 2 \beta_{10} + \cdots - \beta_{4}) q^{11}+ \cdots + (\beta_{9} - 3 \beta_{8} + 2 \beta_{7} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 7 q^{3} - 7 q^{5} - 9 q^{9} - 3 q^{11} + 14 q^{13} + 22 q^{15} - 7 q^{19} - 11 q^{23} - 3 q^{25} + 56 q^{27} + 12 q^{29} - 14 q^{31} + 6 q^{37} - 5 q^{39} - 6 q^{43} - 21 q^{45} - 42 q^{47} + 17 q^{51}+ \cdots - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 10 x^{10} - 7 x^{9} + 68 x^{8} - 44 x^{7} + 225 x^{6} - 77 x^{5} + 490 x^{4} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12913 \nu^{11} - 425609 \nu^{10} + 3061232 \nu^{9} - 1591856 \nu^{8} + 20536204 \nu^{7} + \cdots + 22385944 ) / 127211644 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 63723 \nu^{11} + 1725936 \nu^{10} - 3313883 \nu^{9} + 11185944 \nu^{8} - 29290964 \nu^{7} + \cdots + 60776905 ) / 127211644 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5084 \nu^{11} + 33382 \nu^{10} + 3870 \nu^{9} + 222904 \nu^{8} - 252592 \nu^{7} + \cdots + 2346491 ) / 4543273 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 182621 \nu^{11} + 233529 \nu^{10} + 7594049 \nu^{9} + 6905192 \nu^{8} + 53967792 \nu^{7} + \cdots + 137695537 ) / 127211644 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42735 \nu^{11} + 8172 \nu^{10} + 367055 \nu^{9} + 157692 \nu^{8} + 2601204 \nu^{7} + \cdots + 6456919 ) / 18173092 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50907 \nu^{11} + 60295 \nu^{10} - 456837 \nu^{9} + 304776 \nu^{8} - 2982708 \nu^{7} + \cdots - 52425261 ) / 18173092 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 922417 \nu^{11} - 1221562 \nu^{10} + 9166966 \nu^{9} - 9026304 \nu^{8} + 61620512 \nu^{7} + \cdots - 56802662 ) / 127211644 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 133889 \nu^{11} + 649589 \nu^{10} - 681678 \nu^{9} + 4255576 \nu^{8} - 7032028 \nu^{7} + \cdots - 42948906 ) / 18173092 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2584630 \nu^{11} - 3898215 \nu^{10} + 19906849 \nu^{9} - 33984104 \nu^{8} + 130893744 \nu^{7} + \cdots + 73531409 ) / 127211644 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3203246 \nu^{11} - 1710741 \nu^{10} + 29547887 \nu^{9} - 22338120 \nu^{8} + 187549592 \nu^{7} + \cdots - 148774633 ) / 127211644 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 3\beta_{8} + \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 4\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 6\beta_{10} + 12\beta_{8} - 6\beta_{7} - 6\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + 6\beta_{8} - 18\beta_{6} + 6\beta_{5} - \beta_{4} - 13\beta_{2} - 18\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{9} + 6\beta_{8} + 32\beta_{7} + 6\beta_{5} + 17\beta_{4} + 6\beta_{3} - 6\beta_{2} + 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{11} - 11 \beta_{10} - 8 \beta_{8} - 11 \beta_{7} - 11 \beta_{5} - 50 \beta_{4} + \cdots + 85 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 50 \beta_{11} + 166 \beta_{10} + 50 \beta_{9} - 275 \beta_{8} - 3 \beta_{6} + 137 \beta_{5} + \cdots - 275 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 60 \beta_{9} - 95 \beta_{8} + 84 \beta_{7} + 412 \beta_{6} - 95 \beta_{5} + 347 \beta_{4} + \cdots - 136 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 287 \beta_{11} - 854 \beta_{10} + 1176 \beta_{8} - 854 \beta_{7} - 854 \beta_{5} - 359 \beta_{4} + \cdots + 43 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 359 \beta_{11} + 558 \beta_{10} + 359 \beta_{9} + 569 \beta_{8} - 2030 \beta_{6} + 969 \beta_{5} + \cdots + 569 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1372\mathbb{Z}\right)^\times\).

\(n\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.869996 − 1.50688i
−0.174968 − 0.303054i
−1.15344 − 1.99781i
1.07594 + 1.86358i
1.09252 + 1.89229i
0.529947 + 0.917895i
−0.869996 + 1.50688i
−0.174968 + 0.303054i
−1.15344 + 1.99781i
1.07594 − 1.86358i
1.09252 − 1.89229i
0.529947 − 0.917895i
0 −1.59252 + 2.75832i 0 0.598651 + 1.03689i 0 0 0 −3.57222 − 6.18726i 0
361.2 0 −1.57594 + 2.72960i 0 −2.01664 − 3.49292i 0 0 0 −3.46716 − 6.00529i 0
361.3 0 −1.02995 + 1.78392i 0 −1.41759 − 2.45533i 0 0 0 −0.621582 − 1.07661i 0
361.4 0 −0.325032 + 0.562971i 0 0.794119 + 1.37546i 0 0 0 1.28871 + 2.23211i 0
361.5 0 0.369996 − 0.640851i 0 −0.975161 − 1.68903i 0 0 0 1.22621 + 2.12385i 0
361.6 0 0.653437 − 1.13179i 0 −0.483381 − 0.837240i 0 0 0 0.646041 + 1.11898i 0
1353.1 0 −1.59252 − 2.75832i 0 0.598651 − 1.03689i 0 0 0 −3.57222 + 6.18726i 0
1353.2 0 −1.57594 − 2.72960i 0 −2.01664 + 3.49292i 0 0 0 −3.46716 + 6.00529i 0
1353.3 0 −1.02995 − 1.78392i 0 −1.41759 + 2.45533i 0 0 0 −0.621582 + 1.07661i 0
1353.4 0 −0.325032 − 0.562971i 0 0.794119 − 1.37546i 0 0 0 1.28871 − 2.23211i 0
1353.5 0 0.369996 + 0.640851i 0 −0.975161 + 1.68903i 0 0 0 1.22621 − 2.12385i 0
1353.6 0 0.653437 + 1.13179i 0 −0.483381 + 0.837240i 0 0 0 0.646041 − 1.11898i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1372.2.e.a 12
7.b odd 2 1 1372.2.e.d 12
7.c even 3 1 1372.2.a.d yes 6
7.c even 3 1 inner 1372.2.e.a 12
7.d odd 6 1 1372.2.a.a ✓ 6
7.d odd 6 1 1372.2.e.d 12
28.f even 6 1 5488.2.a.q 6
28.g odd 6 1 5488.2.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1372.2.a.a ✓ 6 7.d odd 6 1
1372.2.a.d yes 6 7.c even 3 1
1372.2.e.a 12 1.a even 1 1 trivial
1372.2.e.a 12 7.c even 3 1 inner
1372.2.e.d 12 7.b odd 2 1
1372.2.e.d 12 7.d odd 6 1
5488.2.a.g 6 28.g odd 6 1
5488.2.a.q 6 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1372, [\chi])\):

\( T_{3}^{12} + 7 T_{3}^{11} + 38 T_{3}^{10} + 105 T_{3}^{9} + 251 T_{3}^{8} + 301 T_{3}^{7} + 525 T_{3}^{6} + \cdots + 169 \) Copy content Toggle raw display
\( T_{11}^{12} + 3 T_{11}^{11} + 49 T_{11}^{10} - 48 T_{11}^{9} + 1249 T_{11}^{8} - 1645 T_{11}^{7} + \cdots + 187489 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 187489 \) Copy content Toggle raw display
$13$ \( (T^{6} - 7 T^{5} + \cdots + 349)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 34 T^{10} + \cdots + 121801 \) Copy content Toggle raw display
$19$ \( T^{12} + 7 T^{11} + \cdots + 175561 \) Copy content Toggle raw display
$23$ \( T^{12} + 11 T^{11} + \cdots + 2550409 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 351)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 992691049 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 986022801 \) Copy content Toggle raw display
$41$ \( (T^{6} - 136 T^{4} + \cdots + 2113)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + \cdots + 7631)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1423477441 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 129208689 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 393506569 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20810082049 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 421521961 \) Copy content Toggle raw display
$71$ \( (T^{3} + 14 T^{2} + 49 T + 7)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 19217999641 \) Copy content Toggle raw display
$79$ \( T^{12} + 6 T^{11} + \cdots + 241081 \) Copy content Toggle raw display
$83$ \( (T^{6} - 49 T^{5} + \cdots + 20327)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 115259571001 \) Copy content Toggle raw display
$97$ \( (T^{6} + 21 T^{5} + \cdots + 297317)^{2} \) Copy content Toggle raw display
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