Properties

Label 538.2.a.d
Level $538$
Weight $2$
Character orbit 538.a
Self dual yes
Analytic conductor $4.296$
Analytic rank $1$
Dimension $7$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{6} + \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{6} - \beta_1 + 1) q^{6} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} - \beta_{5} - \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{6} + \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{6} - \beta_1 + 1) q^{6} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} + \beta_{5} - 5 \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20} - 5 q^{21} + 12 q^{22} - 22 q^{23} + 4 q^{24} + 9 q^{25} - 3 q^{26} - 22 q^{27} - 3 q^{28} - 7 q^{29} + 6 q^{30} - 3 q^{31} - 7 q^{32} - 19 q^{33} + 8 q^{34} - 25 q^{35} + 7 q^{36} - 13 q^{37} + 7 q^{38} + q^{39} + 6 q^{40} - 11 q^{41} + 5 q^{42} - 14 q^{43} - 12 q^{44} - 17 q^{45} + 22 q^{46} - 23 q^{47} - 4 q^{48} + 6 q^{49} - 9 q^{50} - 21 q^{51} + 3 q^{52} - 21 q^{53} + 22 q^{54} + 15 q^{55} + 3 q^{56} - 14 q^{57} + 7 q^{58} - 20 q^{59} - 6 q^{60} + 13 q^{61} + 3 q^{62} - 12 q^{63} + 7 q^{64} - 29 q^{65} + 19 q^{66} - 30 q^{67} - 8 q^{68} + 13 q^{69} + 25 q^{70} - 23 q^{71} - 7 q^{72} + 9 q^{73} + 13 q^{74} + 3 q^{75} - 7 q^{76} + 8 q^{77} - q^{78} + 11 q^{79} - 6 q^{80} + 43 q^{81} + 11 q^{82} - 3 q^{83} - 5 q^{84} + 21 q^{85} + 14 q^{86} + 28 q^{87} + 12 q^{88} + 16 q^{89} + 17 q^{90} - 10 q^{91} - 22 q^{92} - 13 q^{93} + 23 q^{94} + 4 q^{96} + 21 q^{97} - 6 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 8\nu^{4} - 5\nu^{3} - 11\nu^{2} + 5\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 9\nu^{3} + 37\nu^{2} + 11\nu - 30 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - \nu^{5} + 12\nu^{4} + 11\nu^{3} - 37\nu^{2} - 23\nu + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 11\nu^{3} + 39\nu^{2} + 21\nu - 32 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} + 3\nu^{5} + 48\nu^{4} - 11\nu^{3} - 113\nu^{2} - 3\nu + 58 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 8\beta_{5} + 8\beta_{4} + \beta_{3} - 2\beta_{2} + 11\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 3\beta_{5} + 10\beta_{4} + 18\beta_{3} - 3\beta_{2} + 43\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{6} + 56\beta_{5} + 58\beta_{4} + 16\beta_{3} - 21\beta_{2} + 95\beta _1 + 38 \) Copy content Toggle raw display

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
−1.23849
0.870832
−2.30699
−1.49844
1.71952
2.78033
0.673230
−1.00000 −3.38066 1.00000 −2.51475 3.38066 2.26176 −1.00000 8.42889 2.51475
1.2 −1.00000 −2.79016 1.00000 2.36234 2.79016 −3.88429 −1.00000 4.78499 −2.36234
1.3 −1.00000 −1.20937 1.00000 −2.40321 1.20937 1.52017 −1.00000 −1.53742 2.40321
1.4 −1.00000 −0.370477 1.00000 2.98038 0.370477 −2.56529 −1.00000 −2.86275 −2.98038
1.5 −1.00000 0.292058 1.00000 −2.36976 −0.292058 3.96531 −1.00000 −2.91470 2.36976
1.6 −1.00000 0.980985 1.00000 −0.685853 −0.980985 −1.46090 −1.00000 −2.03767 0.685853
1.7 −1.00000 2.47763 1.00000 −3.36915 −2.47763 −2.83676 −1.00000 3.13865 3.36915
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(269\) \( +1 \)

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 538.2.a.d 7
3.b odd 2 1 4842.2.a.o 7
4.b odd 2 1 4304.2.a.i 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.d 7 1.a even 1 1 trivial
4304.2.a.i 7 4.b odd 2 1
4842.2.a.o 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 4T_{3}^{6} - 6T_{3}^{5} - 30T_{3}^{4} + 31T_{3}^{2} + 2T_{3} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 4 T^{6} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{7} + 6 T^{6} + \cdots + 233 \) Copy content Toggle raw display
$7$ \( T^{7} + 3 T^{6} + \cdots - 563 \) Copy content Toggle raw display
$11$ \( T^{7} + 12 T^{6} + \cdots - 480 \) Copy content Toggle raw display
$13$ \( T^{7} - 3 T^{6} + \cdots - 89 \) Copy content Toggle raw display
$17$ \( T^{7} + 8 T^{6} + \cdots - 32 \) Copy content Toggle raw display
$19$ \( T^{7} + 7 T^{6} + \cdots + 6788 \) Copy content Toggle raw display
$23$ \( T^{7} + 22 T^{6} + \cdots + 47584 \) Copy content Toggle raw display
$29$ \( T^{7} + 7 T^{6} + \cdots + 35296 \) Copy content Toggle raw display
$31$ \( T^{7} + 3 T^{6} + \cdots + 44411 \) Copy content Toggle raw display
$37$ \( T^{7} + 13 T^{6} + \cdots - 73351 \) Copy content Toggle raw display
$41$ \( T^{7} + 11 T^{6} + \cdots + 32644 \) Copy content Toggle raw display
$43$ \( T^{7} + 14 T^{6} + \cdots + 55392 \) Copy content Toggle raw display
$47$ \( T^{7} + 23 T^{6} + \cdots - 149280 \) Copy content Toggle raw display
$53$ \( T^{7} + 21 T^{6} + \cdots + 1074432 \) Copy content Toggle raw display
$59$ \( T^{7} + 20 T^{6} + \cdots + 149669 \) Copy content Toggle raw display
$61$ \( T^{7} - 13 T^{6} + \cdots + 495031 \) Copy content Toggle raw display
$67$ \( T^{7} + 30 T^{6} + \cdots + 1051936 \) Copy content Toggle raw display
$71$ \( T^{7} + 23 T^{6} + \cdots - 67860 \) Copy content Toggle raw display
$73$ \( T^{7} - 9 T^{6} + \cdots + 378085 \) Copy content Toggle raw display
$79$ \( T^{7} - 11 T^{6} + \cdots - 76640 \) Copy content Toggle raw display
$83$ \( T^{7} + 3 T^{6} + \cdots - 3711332 \) Copy content Toggle raw display
$89$ \( T^{7} - 16 T^{6} + \cdots - 818177 \) Copy content Toggle raw display
$97$ \( T^{7} - 21 T^{6} + \cdots + 5604 \) Copy content Toggle raw display
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