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Properties

Label 1242.2.e.b
Level $1242$
Weight $2$
Character orbit 1242.e
Analytic conductor $9.917$
Analytic rank $0$
Dimension $10$
CM no
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: 10.0.288778218147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} + \beta_{5} q^{4} + \beta_{2} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 1) q^{2} + \beta_{5} q^{4} + \beta_{2} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{3} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - 5 q^{4} + q^{5} + 5 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} - 5 q^{4} + q^{5} + 5 q^{7} + 10 q^{8} - 2 q^{10} + 11 q^{11} + 6 q^{13} + 5 q^{14} - 5 q^{16} - 2 q^{17} - 6 q^{19} + q^{20} + 11 q^{22} + 5 q^{23} - 12 q^{26} - 10 q^{28} + 8 q^{29} + 4 q^{31} - 5 q^{32} + q^{34} - 46 q^{35} - 28 q^{37} + 3 q^{38} + q^{40} + 24 q^{41} + 27 q^{43} - 22 q^{44} - 10 q^{46} + 9 q^{47} - 12 q^{49} + 6 q^{52} + 26 q^{53} + 16 q^{55} + 5 q^{56} + 8 q^{58} + 9 q^{59} + 3 q^{61} - 8 q^{62} + 10 q^{64} - 5 q^{65} + 5 q^{67} + q^{68} + 23 q^{70} - 54 q^{71} + 34 q^{73} + 14 q^{74} + 3 q^{76} + 13 q^{77} - 11 q^{79} - 2 q^{80} - 48 q^{82} + 23 q^{83} + 23 q^{85} + 27 q^{86} + 11 q^{88} - 78 q^{89} - 30 q^{91} + 5 q^{92} + 9 q^{94} + 37 q^{95} + 28 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 134 \nu^{9} + 132 \nu^{8} - 429 \nu^{7} + 95 \nu^{6} - 1947 \nu^{5} + 3531 \nu^{4} + 6253 \nu^{3} + \cdots + 37125 ) / 14559 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 999 \nu^{9} - 537 \nu^{8} - 4321 \nu^{7} - 4688 \nu^{6} - 34543 \nu^{5} - 20431 \nu^{4} + \cdots - 9081 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} + \cdots + 29709 ) / 24265 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2704 \nu^{9} + 8748 \nu^{8} - 28431 \nu^{7} + 47767 \nu^{6} - 129033 \nu^{5} + 234009 \nu^{4} + \cdots + 291084 ) / 72795 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3301 \nu^{9} + 2962 \nu^{8} - 21759 \nu^{7} + 8823 \nu^{6} - 104352 \nu^{5} + 42836 \nu^{4} + \cdots - 18639 ) / 72795 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3509 \nu^{9} + 7368 \nu^{8} - 23946 \nu^{7} + 42362 \nu^{6} - 108678 \nu^{5} + 197094 \nu^{4} + \cdots + 194139 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4858 \nu^{9} - 1154 \nu^{8} - 32647 \nu^{7} - 15316 \nu^{6} - 164966 \nu^{5} - 67267 \nu^{4} + \cdots - 40662 ) / 72795 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6592 \nu^{9} - 3379 \nu^{8} + 41313 \nu^{7} - 4601 \nu^{6} + 213629 \nu^{5} - 11527 \nu^{4} + \cdots + 43308 ) / 72795 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9564 \nu^{9} - 7538 \nu^{8} + 60896 \nu^{7} - 18587 \nu^{6} + 293173 \nu^{5} - 92449 \nu^{4} + \cdots - 132759 ) / 72795 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{5} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} - \beta_{8} - 8\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 2\beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{9} + 5\beta_{8} + 3\beta_{7} + 31\beta_{5} + 5\beta_{2} + 10\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{9} - 11\beta_{8} + 3\beta_{7} + 3\beta_{6} + 11\beta_{5} + 11\beta_{4} - 29\beta_{3} - 29\beta_{2} + 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} - 8\beta_{4} + 7\beta_{3} - 15\beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -46\beta_{9} + 43\beta_{8} - 24\beta_{7} - 43\beta_{5} + 133\beta_{2} - 46\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 200 \beta_{9} - 112 \beta_{8} - 114 \beta_{7} - 114 \beta_{6} - 554 \beta_{5} + 112 \beta_{4} + \cdots - 554 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{6} - 58\beta_{4} + 200\beta_{3} + 68\beta _1 - 61 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(461\) \(649\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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
−1.04536 + 1.81062i
0.187540 − 0.324828i
0.827154 − 1.43267i
−0.539982 + 0.935277i
1.07065 − 1.85442i
−1.04536 − 1.81062i
0.187540 + 0.324828i
0.827154 + 1.43267i
−0.539982 − 0.935277i
1.07065 + 1.85442i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.43330 + 2.48254i 0 1.80778 + 3.13117i 1.00000 0 2.86660
415.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.536235 + 0.928786i 0 −0.121951 − 0.211225i 1.00000 0 1.07247
415.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.217761 + 0.377174i 0 2.31474 + 4.00925i 1.00000 0 0.435523
415.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.990153 − 1.71499i 0 0.245502 + 0.425221i 1.00000 0 −1.98031
415.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.69714 − 2.93953i 0 −1.74607 − 3.02428i 1.00000 0 −3.39428
829.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.43330 − 2.48254i 0 1.80778 − 3.13117i 1.00000 0 2.86660
829.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.536235 − 0.928786i 0 −0.121951 + 0.211225i 1.00000 0 1.07247
829.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.217761 − 0.377174i 0 2.31474 − 4.00925i 1.00000 0 0.435523
829.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.990153 + 1.71499i 0 0.245502 − 0.425221i 1.00000 0 −1.98031
829.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.69714 + 2.93953i 0 −1.74607 + 3.02428i 1.00000 0 −3.39428
\(n\): 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.b 10
3.b odd 2 1 414.2.e.d ✓ 10
9.c even 3 1 inner 1242.2.e.b 10
9.c even 3 1 3726.2.a.u 5
9.d odd 6 1 414.2.e.d ✓ 10
9.d odd 6 1 3726.2.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.d ✓ 10 3.b odd 2 1
414.2.e.d ✓ 10 9.d odd 6 1
1242.2.e.b 10 1.a even 1 1 trivial
1242.2.e.b 10 9.c even 3 1 inner
3726.2.a.r 5 9.d odd 6 1
3726.2.a.u 5 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - T_{5}^{9} + 13T_{5}^{8} + 2T_{5}^{7} + 124T_{5}^{6} - T_{5}^{5} + 334T_{5}^{4} + 341T_{5}^{3} + 580T_{5}^{2} + 225T_{5} + 81 \) 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
$5$ \( T^{10} - T^{9} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + \cdots + 84681 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots + 1234321 \) Copy content Toggle raw display
$17$ \( (T^{5} + T^{4} - 75 T^{3} + \cdots + 2037)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} + \cdots - 857)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{10} - 8 T^{9} + \cdots + 8590761 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + \cdots + 3207681 \) Copy content Toggle raw display
$37$ \( (T^{5} + 14 T^{4} + \cdots + 19189)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 24 T^{9} + \cdots + 38900169 \) Copy content Toggle raw display
$43$ \( T^{10} - 27 T^{9} + \cdots + 458329 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 1378265625 \) Copy content Toggle raw display
$53$ \( (T^{5} - 13 T^{4} + \cdots - 213)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 9 T^{9} + \cdots + 26594649 \) Copy content Toggle raw display
$61$ \( T^{10} - 3 T^{9} + \cdots + 3374569 \) Copy content Toggle raw display
$67$ \( T^{10} - 5 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$71$ \( (T^{5} + 27 T^{4} + \cdots - 21357)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 17 T^{4} + \cdots - 297)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 119968209 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 107557641 \) Copy content Toggle raw display
$89$ \( (T^{5} + 39 T^{4} + \cdots - 62667)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 28 T^{9} + \cdots + 20007729 \) Copy content Toggle raw display
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