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Properties

Label 6040.2.a.n
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} - \beta_{7} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} - \beta_{7} q^{7} + \beta_{2} q^{9} + ( - \beta_{9} + \beta_1 - 1) q^{11} - \beta_{11} q^{13} + \beta_1 q^{15} + (\beta_{3} - 1) q^{17} + (\beta_{12} + \beta_{11} - \beta_{10} + \cdots + 2) q^{19}+ \cdots + (\beta_{12} + \beta_{11} - \beta_{10} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 49 \nu^{12} + 146 \nu^{11} + 769 \nu^{10} - 2522 \nu^{9} - 3344 \nu^{8} + 14165 \nu^{7} + \cdots + 4040 ) / 212 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 178 \nu^{12} - 711 \nu^{11} - 2588 \nu^{10} + 12639 \nu^{9} + 9026 \nu^{8} - 74902 \nu^{7} + \cdots - 8423 ) / 212 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 486 \nu^{12} - 1765 \nu^{11} - 7452 \nu^{10} + 31293 \nu^{9} + 31378 \nu^{8} - 184594 \nu^{7} + \cdots - 15237 ) / 212 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 741 \nu^{12} - 2766 \nu^{11} - 11097 \nu^{10} + 48834 \nu^{9} + 43236 \nu^{8} - 285965 \nu^{7} + \cdots - 28888 ) / 212 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 379 \nu^{12} + 1404 \nu^{11} + 5723 \nu^{10} - 24848 \nu^{9} - 22886 \nu^{8} + 146093 \nu^{7} + \cdots + 14604 ) / 106 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 779 \nu^{12} - 2916 \nu^{11} - 11715 \nu^{10} + 51640 \nu^{9} + 46236 \nu^{8} - 304011 \nu^{7} + \cdots - 31106 ) / 212 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 785 \nu^{12} + 2962 \nu^{11} + 11701 \nu^{10} - 52362 \nu^{9} - 44824 \nu^{8} + 307329 \nu^{7} + \cdots + 32784 ) / 212 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19 \nu^{12} - 70 \nu^{11} - 287 \nu^{10} + 1238 \nu^{9} + 1148 \nu^{8} - 7271 \nu^{7} - 71 \nu^{6} + \cdots - 756 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1092 \nu^{12} - 4079 \nu^{11} - 16426 \nu^{10} + 72203 \nu^{9} + 64938 \nu^{8} - 424736 \nu^{7} + \cdots - 42039 ) / 212 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1991 \nu^{12} + 7385 \nu^{11} + 29981 \nu^{10} - 130545 \nu^{9} - 118974 \nu^{8} + 766063 \nu^{7} + \cdots + 75647 ) / 212 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{12} + 8 \beta_{11} - 10 \beta_{10} - 20 \beta_{9} - 9 \beta_{8} - 10 \beta_{7} + \beta_{6} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{12} + \beta_{10} - 15 \beta_{9} - 13 \beta_{8} - 13 \beta_{7} + 9 \beta_{6} + 11 \beta_{5} + \cdots + 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 82 \beta_{12} + 57 \beta_{11} - 86 \beta_{10} - 166 \beta_{9} - 71 \beta_{8} - 86 \beta_{7} + 12 \beta_{6} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 108 \beta_{12} - 3 \beta_{11} + 20 \beta_{10} - 154 \beta_{9} - 128 \beta_{8} - 125 \beta_{7} + \cdots + 829 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 634 \beta_{12} + 398 \beta_{11} - 703 \beta_{10} - 1310 \beta_{9} - 543 \beta_{8} - 699 \beta_{7} + \cdots + 432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 883 \beta_{12} - 52 \beta_{11} + 258 \beta_{10} - 1386 \beta_{9} - 1148 \beta_{8} - 1080 \beta_{7} + \cdots + 5848 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4798 \beta_{12} + 2783 \beta_{11} - 5615 \beta_{10} - 10164 \beta_{9} - 4133 \beta_{8} - 5532 \beta_{7} + \cdots + 3311 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6937 \beta_{12} - 610 \beta_{11} + 2764 \beta_{10} - 11772 \beta_{9} - 9867 \beta_{8} - 8889 \beta_{7} + \cdots + 41881 \) 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
2.79963
2.64979
2.12021
1.97615
1.74594
0.429184
0.291870
0.278392
−0.565329
−1.14309
−1.28532
−2.51561
−2.78181
0 −2.79963 0 −1.00000 0 2.90399 0 4.83794 0
1.2 0 −2.64979 0 −1.00000 0 1.08783 0 4.02136 0
1.3 0 −2.12021 0 −1.00000 0 −4.61476 0 1.49528 0
1.4 0 −1.97615 0 −1.00000 0 −2.05862 0 0.905164 0
1.5 0 −1.74594 0 −1.00000 0 3.52041 0 0.0483066 0
1.6 0 −0.429184 0 −1.00000 0 −1.03124 0 −2.81580 0
1.7 0 −0.291870 0 −1.00000 0 0.544006 0 −2.91481 0
1.8 0 −0.278392 0 −1.00000 0 −2.48850 0 −2.92250 0
1.9 0 0.565329 0 −1.00000 0 3.63919 0 −2.68040 0
1.10 0 1.14309 0 −1.00000 0 3.09474 0 −1.69334 0
1.11 0 1.28532 0 −1.00000 0 −3.04407 0 −1.34796 0
1.12 0 2.51561 0 −1.00000 0 0.120976 0 3.32830 0
1.13 0 2.78181 0 −1.00000 0 −1.67394 0 4.73847 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.n ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.n ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{13} + 4 T_{3}^{12} - 14 T_{3}^{11} - 70 T_{3}^{10} + 41 T_{3}^{9} + 403 T_{3}^{8} + 109 T_{3}^{7} + \cdots - 11 \) Copy content Toggle raw display
\( T_{7}^{13} - 45 T_{7}^{11} + 3 T_{7}^{10} + 738 T_{7}^{9} + 31 T_{7}^{8} - 5580 T_{7}^{7} - 1233 T_{7}^{6} + \cdots - 1024 \) Copy content Toggle raw display
\( T_{11}^{13} + 14 T_{11}^{12} + 51 T_{11}^{11} - 124 T_{11}^{10} - 1140 T_{11}^{9} - 1089 T_{11}^{8} + \cdots + 20408 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} \) Copy content Toggle raw display
$3$ \( T^{13} + 4 T^{12} + \cdots - 11 \) Copy content Toggle raw display
$5$ \( (T + 1)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} - 45 T^{11} + \cdots - 1024 \) Copy content Toggle raw display
$11$ \( T^{13} + 14 T^{12} + \cdots + 20408 \) Copy content Toggle raw display
$13$ \( T^{13} - 5 T^{12} + \cdots - 4184 \) Copy content Toggle raw display
$17$ \( T^{13} + 8 T^{12} + \cdots - 4400 \) Copy content Toggle raw display
$19$ \( T^{13} - 16 T^{12} + \cdots + 257836 \) Copy content Toggle raw display
$23$ \( T^{13} + 4 T^{12} + \cdots - 24 \) Copy content Toggle raw display
$29$ \( T^{13} + 6 T^{12} + \cdots - 832368 \) Copy content Toggle raw display
$31$ \( T^{13} - 11 T^{12} + \cdots - 8678271 \) Copy content Toggle raw display
$37$ \( T^{13} - 6 T^{12} + \cdots - 13574336 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 119172736 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 1062812352 \) Copy content Toggle raw display
$47$ \( T^{13} + 22 T^{12} + \cdots + 12599664 \) Copy content Toggle raw display
$53$ \( T^{13} + 17 T^{12} + \cdots - 48858416 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 194640048 \) Copy content Toggle raw display
$61$ \( T^{13} - 10 T^{12} + \cdots + 44283344 \) Copy content Toggle raw display
$67$ \( T^{13} - 12 T^{12} + \cdots - 10077 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 7381389376 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 8989505596 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 48857386432 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 7802154023809 \) Copy content Toggle raw display
$89$ \( T^{13} + 53 T^{12} + \cdots + 48844096 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 813884668368 \) Copy content Toggle raw display
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