Properties

Label 538.2.a.e
Level $538$
Weight $2$
Character orbit 538.a
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
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: \(0\)
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: \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{2} q^{3} + q^{4} + (\beta_{4} + 1) q^{5} + \beta_{2} q^{6} + (\beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{2} q^{3} + q^{4} + (\beta_{4} + 1) q^{5} + \beta_{2} q^{6} + (\beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{9} + (\beta_{4} + 1) q^{10} + (\beta_{5} - \beta_{4}) q^{11} + \beta_{2} q^{12} + (\beta_{3} + \beta_1 - 1) q^{13} + (\beta_{6} - \beta_{3} - \beta_{2} + 1) q^{14} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 1) q^{15}+ \cdots + (\beta_{5} - 3 \beta_{4} + 2 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20} - 6 q^{21} - 3 q^{22} + 12 q^{23} + q^{24} + 22 q^{25} - 9 q^{26} - 14 q^{27} + 6 q^{28} - 5 q^{29} + 8 q^{30} + 14 q^{31} + 7 q^{32} - 4 q^{33} + 8 q^{34} - 4 q^{35} + 12 q^{36} + 13 q^{37} - 11 q^{38} - 18 q^{39} + 7 q^{40} + 12 q^{41} - 6 q^{42} - 11 q^{43} - 3 q^{44} + 3 q^{45} + 12 q^{46} + 2 q^{47} + q^{48} + 15 q^{49} + 22 q^{50} - 26 q^{51} - 9 q^{52} + 19 q^{53} - 14 q^{54} - 40 q^{55} + 6 q^{56} - 12 q^{57} - 5 q^{58} - 9 q^{59} + 8 q^{60} - 3 q^{61} + 14 q^{62} - 26 q^{63} + 7 q^{64} - 10 q^{65} - 4 q^{66} - 33 q^{67} + 8 q^{68} - 64 q^{69} - 4 q^{70} + 28 q^{71} + 12 q^{72} - 14 q^{73} + 13 q^{74} - 45 q^{75} - 11 q^{76} + 10 q^{77} - 18 q^{78} + 2 q^{79} + 7 q^{80} + 15 q^{81} + 12 q^{82} - 7 q^{83} - 6 q^{84} - 16 q^{85} - 11 q^{86} + 16 q^{87} - 3 q^{88} + 18 q^{89} + 3 q^{90} - 26 q^{91} + 12 q^{92} + 6 q^{93} + 2 q^{94} - 34 q^{95} + q^{96} - 4 q^{97} + 15 q^{98} - 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} - 15\nu^{4} - 12\nu^{3} + 51\nu^{2} + 21\nu - 48 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - \nu^{5} + 45\nu^{4} + 10\nu^{3} - 151\nu^{2} - 17\nu + 138 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} + \nu^{5} - 73\nu^{4} - 8\nu^{3} + 225\nu^{2} + 13\nu - 186 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{6} + 3\nu^{5} - 251\nu^{4} - 24\nu^{3} + 801\nu^{2} + 47\nu - 696 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -21\nu^{6} - 3\nu^{5} + 311\nu^{4} + 20\nu^{3} - 997\nu^{2} - 39\nu + 856 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 3\beta_{3} - 2\beta_{2} + 6\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{6} + 13\beta_{5} + \beta_{4} - 11\beta_{3} - 12\beta_{2} + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} - 27\beta_{5} + 26\beta_{4} - 37\beta_{3} - 22\beta_{2} + 55\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 145\beta_{6} + 147\beta_{5} + 13\beta_{4} - 113\beta_{3} - 129\beta_{2} - 4\beta _1 + 458 \) 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
−1.26381
1.61949
3.22542
−3.33118
1.14366
1.19866
−1.59223
1.00000 −3.13760 1.00000 3.00565 −3.13760 −0.324393 1.00000 6.84453 3.00565
1.2 1.00000 −2.60058 1.00000 −3.80694 −2.60058 0.487379 1.00000 3.76302 −3.80694
1.3 1.00000 −0.383827 1.00000 4.15424 −0.383827 4.87818 1.00000 −2.85268 4.15424
1.4 1.00000 0.366193 1.00000 −0.953965 0.366193 3.52894 1.00000 −2.86590 −0.953965
1.5 1.00000 1.65255 1.00000 0.725043 1.65255 1.29978 1.00000 −0.269065 0.725043
1.6 1.00000 2.12827 1.00000 3.84433 2.12827 −4.95403 1.00000 1.52953 3.84433
1.7 1.00000 2.97499 1.00000 0.0316474 2.97499 1.08414 1.00000 5.85056 0.0316474
\(n\): 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.e 7
3.b odd 2 1 4842.2.a.n 7
4.b odd 2 1 4304.2.a.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.e 7 1.a even 1 1 trivial
4304.2.a.h 7 4.b odd 2 1
4842.2.a.n 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - T_{3}^{6} - 16T_{3}^{5} + 19T_{3}^{4} + 63T_{3}^{3} - 87T_{3}^{2} - 10T_{3} + 12 \) 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
$3$ \( T^{7} - T^{6} + \cdots + 12 \) Copy content Toggle raw display
$5$ \( T^{7} - 7 T^{6} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{7} - 6 T^{6} + \cdots - 19 \) Copy content Toggle raw display
$11$ \( T^{7} + 3 T^{6} + \cdots + 288 \) Copy content Toggle raw display
$13$ \( T^{7} + 9 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{7} - 8 T^{6} + \cdots - 22016 \) Copy content Toggle raw display
$19$ \( T^{7} + 11 T^{6} + \cdots - 20 \) Copy content Toggle raw display
$23$ \( T^{7} - 12 T^{6} + \cdots + 2432 \) Copy content Toggle raw display
$29$ \( T^{7} + 5 T^{6} + \cdots - 27040 \) Copy content Toggle raw display
$31$ \( T^{7} - 14 T^{6} + \cdots + 250907 \) Copy content Toggle raw display
$37$ \( T^{7} - 13 T^{6} + \cdots + 2236 \) Copy content Toggle raw display
$41$ \( T^{7} - 12 T^{6} + \cdots - 288917 \) Copy content Toggle raw display
$43$ \( T^{7} + 11 T^{6} + \cdots - 227616 \) Copy content Toggle raw display
$47$ \( T^{7} - 2 T^{6} + \cdots + 64896 \) Copy content Toggle raw display
$53$ \( T^{7} - 19 T^{6} + \cdots - 26988 \) Copy content Toggle raw display
$59$ \( T^{7} + 9 T^{6} + \cdots - 146660 \) Copy content Toggle raw display
$61$ \( T^{7} + 3 T^{6} + \cdots + 210068 \) Copy content Toggle raw display
$67$ \( T^{7} + 33 T^{6} + \cdots - 251936 \) Copy content Toggle raw display
$71$ \( T^{7} - 28 T^{6} + \cdots + 28293 \) Copy content Toggle raw display
$73$ \( T^{7} + 14 T^{6} + \cdots - 382603 \) Copy content Toggle raw display
$79$ \( T^{7} - 2 T^{6} + \cdots + 14720 \) Copy content Toggle raw display
$83$ \( T^{7} + 7 T^{6} + \cdots - 441152 \) Copy content Toggle raw display
$89$ \( T^{7} - 18 T^{6} + \cdots + 1143155 \) Copy content Toggle raw display
$97$ \( T^{7} + 4 T^{6} + \cdots + 15927 \) Copy content Toggle raw display
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