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Properties

Label 1242.2.e.d
Level $1242$
Weight $2$
Character orbit 1242.e
Analytic conductor $9.917$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1242,2,Mod(415,1242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1242, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1242.415");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1242 = 2 \cdot 3^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1242.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.91741993104\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.3184288458147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 2x^{8} + 9x^{7} + 6x^{6} - 45x^{5} + 18x^{4} + 81x^{3} - 54x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 414)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} - \beta_{2} q^{4} + (\beta_{7} + \beta_{4} + \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} - \beta_{2} q^{4} + (\beta_{7} + \beta_{4} + \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{9} + 2 \beta_{8} - 2 \beta_{6} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{2} - 5 q^{4} + 5 q^{5} - 3 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{2} - 5 q^{4} + 5 q^{5} - 3 q^{7} - 10 q^{8} + 10 q^{10} + 11 q^{11} + 3 q^{14} - 5 q^{16} - 22 q^{17} + 18 q^{19} + 5 q^{20} - 11 q^{22} - 5 q^{23} - 12 q^{25} + 6 q^{28} - 2 q^{29} - 4 q^{31} + 5 q^{32} - 11 q^{34} + 14 q^{35} + 8 q^{37} + 9 q^{38} - 5 q^{40} + 8 q^{41} - 5 q^{43} - 22 q^{44} - 10 q^{46} + 19 q^{47} - 4 q^{49} + 12 q^{50} + 6 q^{53} - 4 q^{55} + 3 q^{56} + 2 q^{58} + 19 q^{59} + 13 q^{61} - 8 q^{62} + 10 q^{64} - q^{65} - q^{67} + 11 q^{68} + 7 q^{70} - 54 q^{71} - 14 q^{73} + 4 q^{74} - 9 q^{76} + 19 q^{77} - 3 q^{79} - 10 q^{80} + 16 q^{82} + 9 q^{83} + 5 q^{85} + 5 q^{86} - 11 q^{88} - 18 q^{89} - 38 q^{91} - 5 q^{92} - 19 q^{94} + 17 q^{95} + 20 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 2x^{8} + 9x^{7} + 6x^{6} - 45x^{5} + 18x^{4} + 81x^{3} - 54x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} + \nu^{8} + \nu^{7} - 6\nu^{6} - 3\nu^{5} + 9\nu^{4} - 9\nu^{3} - 81\nu^{2} - 54\nu + 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{9} + \nu^{8} + 10\nu^{7} - 3\nu^{6} - 30\nu^{5} + 27\nu^{4} + 72\nu^{3} - 54\nu^{2} - 135\nu + 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 3\nu^{7} - 4\nu^{6} + 15\nu^{5} - 27\nu^{3} - 27\nu^{2} + 81\nu + 27 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{9} + \nu^{8} + 10\nu^{7} - 12\nu^{6} - 39\nu^{5} + 45\nu^{4} + 72\nu^{3} - 162\nu^{2} - 135\nu + 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 4\nu^{8} - 5\nu^{7} - 21\nu^{6} + 15\nu^{5} + 45\nu^{4} - 63\nu^{3} - 135\nu^{2} + 189\nu + 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} + 11\nu^{7} - 18\nu^{6} - 42\nu^{5} + 81\nu^{4} + 90\nu^{3} - 216\nu^{2} - 189\nu + 486 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 10\nu^{6} - 3\nu^{5} - 30\nu^{4} + 27\nu^{3} + 72\nu^{2} - 27\nu - 135 ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{9} + 10\nu^{8} + 10\nu^{7} - 39\nu^{6} - 48\nu^{5} + 153\nu^{4} + 45\nu^{3} - 378\nu^{2} - 216\nu + 729 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} + 4\nu^{8} + 4\nu^{7} - 16\nu^{6} - 18\nu^{5} + 66\nu^{4} + 9\nu^{3} - 153\nu^{2} - 27\nu + 297 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} - \beta_{8} - 2\beta_{7} + 3\beta_{6} + 4\beta_{5} - 3\beta_{4} - 3\beta_{3} + 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{9} + 5\beta_{8} + 4\beta_{7} + \beta_{5} + 3\beta_{3} - 6\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{9} - 4\beta_{8} - 2\beta_{7} + 6\beta_{6} - 5\beta_{5} - 6\beta_{4} - 3\beta_{2} - 3\beta _1 - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{9} + 8 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} - 14 \beta_{5} - 3 \beta_{4} + 15 \beta_{3} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 34\beta_{9} - 4\beta_{8} + 22\beta_{7} - 18\beta_{6} - 44\beta_{5} + 21\beta_{3} - 27\beta_{2} + 3\beta _1 + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{9} - 40\beta_{8} - 20\beta_{7} + 6\beta_{6} + 4\beta_{5} - 33\beta_{4} + 78\beta_{2} + 51\beta _1 + 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7 \beta_{9} + 53 \beta_{8} + 61 \beta_{7} - 60 \beta_{6} + 22 \beta_{5} - 30 \beta_{4} + 15 \beta_{3} + \cdots + 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16 \beta_{9} - 67 \beta_{8} + 31 \beta_{7} + 117 \beta_{6} + 100 \beta_{5} - 135 \beta_{4} - 60 \beta_{3} + \cdots - 120 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1242\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(649\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
−1.31829 + 1.12344i
1.70001 − 0.331620i
1.13442 + 1.30886i
−1.57184 − 0.727544i
1.05570 − 1.37313i
−1.31829 − 1.12344i
1.70001 + 0.331620i
1.13442 − 1.30886i
−1.57184 + 0.727544i
1.05570 + 1.37313i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.39019 + 2.40787i 0 0.0242331 + 0.0419729i −1.00000 0 −2.78037
415.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.332592 + 0.576066i 0 −2.28006 − 3.94917i −1.00000 0 −0.665183
415.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.559008 − 0.968230i 0 0.926205 + 1.60423i −1.00000 0 1.11802
415.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.52966 − 2.64946i 0 −1.44136 − 2.49651i −1.00000 0 3.05933
415.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.13411 − 3.69638i 0 1.27098 + 2.20140i −1.00000 0 4.26821
829.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.39019 − 2.40787i 0 0.0242331 − 0.0419729i −1.00000 0 −2.78037
829.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.332592 − 0.576066i 0 −2.28006 + 3.94917i −1.00000 0 −0.665183
829.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.559008 + 0.968230i 0 0.926205 − 1.60423i −1.00000 0 1.11802
829.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.52966 + 2.64946i 0 −1.44136 + 2.49651i −1.00000 0 3.05933
829.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.13411 + 3.69638i 0 1.27098 − 2.20140i −1.00000 0 4.26821
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1242.2.e.d 10
3.b odd 2 1 414.2.e.b ✓ 10
9.c even 3 1 inner 1242.2.e.d 10
9.c even 3 1 3726.2.a.q 5
9.d odd 6 1 414.2.e.b ✓ 10
9.d odd 6 1 3726.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.b ✓ 10 3.b odd 2 1
414.2.e.b ✓ 10 9.d odd 6 1
1242.2.e.d 10 1.a even 1 1 trivial
1242.2.e.d 10 9.c even 3 1 inner
3726.2.a.q 5 9.c even 3 1
3726.2.a.v 5 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 5 T_{5}^{9} + 31 T_{5}^{8} - 56 T_{5}^{7} + 262 T_{5}^{6} - 395 T_{5}^{5} + 1648 T_{5}^{4} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(1242, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{10} + 16 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{5} + 11 T^{4} + \cdots - 129)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 9 T^{4} + 17 T^{3} + \cdots + 37)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{10} + 2 T^{9} + \cdots + 328329 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} + \cdots + 65025 \) Copy content Toggle raw display
$37$ \( (T^{5} - 4 T^{4} - 65 T^{3} + \cdots - 83)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 8 T^{9} + \cdots + 3884841 \) Copy content Toggle raw display
$43$ \( T^{10} + 5 T^{9} + \cdots + 83521 \) Copy content Toggle raw display
$47$ \( T^{10} - 19 T^{9} + \cdots + 15896169 \) Copy content Toggle raw display
$53$ \( (T^{5} - 3 T^{4} + \cdots - 6585)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 19 T^{9} + \cdots + 14493249 \) Copy content Toggle raw display
$61$ \( T^{10} - 13 T^{9} + \cdots + 279841 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 146870161 \) Copy content Toggle raw display
$71$ \( (T^{5} + 27 T^{4} + \cdots + 3645)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 7 T^{4} + \cdots - 28053)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 10435235409 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 3697491249 \) Copy content Toggle raw display
$89$ \( (T^{5} + 9 T^{4} + \cdots + 220887)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 20 T^{9} + \cdots + 69705801 \) Copy content Toggle raw display
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