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Properties

Label 6040.2.a.l
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.a (trivial)

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: yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + q^{5} + (\beta_{4} + \beta_1) q^{7} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + q^{5} + (\beta_{4} + \beta_1) q^{7} - \beta_{5} q^{9} + (\beta_{7} - \beta_{3} - \beta_1 - 1) q^{11} + ( - \beta_{8} - \beta_{7} + 2 \beta_{5} + \cdots - 1) q^{13}+ \cdots + (\beta_{8} + 2 \beta_{7} + \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{8} + 4\nu^{7} - 41\nu^{6} - 47\nu^{5} + 164\nu^{4} + 141\nu^{3} - 181\nu^{2} - 89\nu + 31 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{8} - 8\nu^{7} - 74\nu^{6} + 68\nu^{5} + 192\nu^{4} - 100\nu^{3} - 106\nu^{2} + 22\nu - 23 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{8} + \nu^{7} + 11\nu^{6} - 9\nu^{5} - 32\nu^{4} + 17\nu^{3} + 27\nu^{2} - 10\nu - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{8} - 9\nu^{7} - 184\nu^{6} + 70\nu^{5} + 580\nu^{4} - 67\nu^{3} - 532\nu^{2} - 24\nu + 70 ) / 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -25\nu^{8} + 23\nu^{7} + 268\nu^{6} - 189\nu^{5} - 734\nu^{4} + 242\nu^{3} + 555\nu^{2} - 34\nu - 59 ) / 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -27\nu^{8} + 16\nu^{7} + 304\nu^{6} - 123\nu^{5} - 917\nu^{4} + 122\nu^{3} + 771\nu^{2} - 18\nu - 71 ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57\nu^{8} - 41\nu^{7} - 636\nu^{6} + 329\nu^{5} + 1894\nu^{4} - 389\nu^{3} - 1606\nu^{2} + 38\nu + 160 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{8} + \beta_{7} + 7\beta_{6} - 8\beta_{5} + 7\beta_{4} - 2\beta_{3} - 7\beta_{2} + 7\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{8} - 2\beta_{7} + 9\beta_{5} + 8\beta_{4} - \beta_{3} - 10\beta_{2} + 30\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 56\beta_{8} + 6\beta_{7} + 47\beta_{6} - 57\beta_{5} + 50\beta_{4} - 20\beta_{3} - 51\beta_{2} + 47\beta _1 + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{8} - 24\beta_{7} + 66\beta_{5} + 61\beta_{4} - 14\beta_{3} - 81\beta_{2} + 197\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 386 \beta_{8} + 28 \beta_{7} + 320 \beta_{6} - 396 \beta_{5} + 358 \beta_{4} - 161 \beta_{3} + \cdots + 663 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.390645
2.02556
−1.10984
−0.357360
−1.41589
2.68790
0.289144
1.10659
−2.61675
0 −1.96251 0 1.00000 0 −2.16922 0 0.851433 0
1.2 0 −1.75822 0 1.00000 0 1.53187 0 0.0913304 0
1.3 0 −1.57672 0 1.00000 0 −0.208807 0 −0.513952 0
1.4 0 −1.13091 0 1.00000 0 2.44094 0 −1.72105 0
1.5 0 0.0722232 0 1.00000 0 −0.709626 0 −2.99478 0
1.6 0 0.325171 0 1.00000 0 2.31586 0 −2.89426 0
1.7 0 1.17504 0 1.00000 0 −3.16934 0 −1.61928 0
1.8 0 2.35518 0 1.00000 0 0.202920 0 2.54689 0
1.9 0 2.50073 0 1.00000 0 −2.23459 0 3.25367 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(151\) \(1\)

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 6040.2.a.l ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.l ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{9} - 12T_{3}^{7} - 4T_{3}^{6} + 46T_{3}^{5} + 25T_{3}^{4} - 55T_{3}^{3} - 26T_{3}^{2} + 16T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{9} + 2T_{7}^{8} - 15T_{7}^{7} - 25T_{7}^{6} + 72T_{7}^{5} + 95T_{7}^{4} - 110T_{7}^{3} - 99T_{7}^{2} + 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{9} + 6T_{11}^{8} - 20T_{11}^{7} - 116T_{11}^{6} + 98T_{11}^{5} + 429T_{11}^{4} + 361T_{11}^{3} + 116T_{11}^{2} + 10T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 12 T^{7} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 2 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{9} + 6 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{9} + 9 T^{8} + \cdots - 233 \) Copy content Toggle raw display
$17$ \( T^{9} + 2 T^{8} + \cdots + 932 \) Copy content Toggle raw display
$19$ \( T^{9} + 10 T^{8} + \cdots - 7985 \) Copy content Toggle raw display
$23$ \( T^{9} + 6 T^{8} + \cdots + 142909 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots - 16883 \) Copy content Toggle raw display
$31$ \( T^{9} - 9 T^{8} + \cdots - 59293 \) Copy content Toggle raw display
$37$ \( T^{9} + 12 T^{8} + \cdots + 231608 \) Copy content Toggle raw display
$41$ \( T^{9} + 20 T^{8} + \cdots - 4780 \) Copy content Toggle raw display
$43$ \( T^{9} - T^{8} + \cdots + 1306532 \) Copy content Toggle raw display
$47$ \( T^{9} - 22 T^{8} + \cdots + 1103020 \) Copy content Toggle raw display
$53$ \( T^{9} + 35 T^{8} + \cdots + 6006232 \) Copy content Toggle raw display
$59$ \( T^{9} - 14 T^{8} + \cdots - 7313 \) Copy content Toggle raw display
$61$ \( T^{9} + 22 T^{8} + \cdots - 2080 \) Copy content Toggle raw display
$67$ \( T^{9} - 4 T^{8} + \cdots - 772507 \) Copy content Toggle raw display
$71$ \( T^{9} + 22 T^{8} + \cdots - 24347200 \) Copy content Toggle raw display
$73$ \( T^{9} + 34 T^{8} + \cdots + 437831 \) Copy content Toggle raw display
$79$ \( T^{9} - 8 T^{8} + \cdots - 28424708 \) Copy content Toggle raw display
$83$ \( T^{9} + 3 T^{8} + \cdots + 3251687 \) Copy content Toggle raw display
$89$ \( T^{9} + 5 T^{8} + \cdots + 846388 \) Copy content Toggle raw display
$97$ \( T^{9} + 33 T^{8} + \cdots + 57618964 \) Copy content Toggle raw display
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