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Properties

Label 5265.2.a.be
Level $5265$
Weight $2$
Character orbit 5265.a
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
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] = [5265,2,Mod(1,5265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5265.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.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: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{6} - 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{6} - 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{8} - \beta_1 q^{10} + (\beta_{6} + \beta_{4} - \beta_{3} + 1) q^{11} + q^{13} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{14} + (\beta_{4} + \beta_{3} - 2) q^{16} + (\beta_{6} + \beta_{5} - \beta_{4} - 1) q^{17} + ( - \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{19} + ( - \beta_{2} - 1) q^{20} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{22}+ \cdots + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 5 q^{4} - 8 q^{5} - 6 q^{7} + 6 q^{8} - q^{10} + 9 q^{11} + 8 q^{13} - 3 q^{14} - 13 q^{16} - 6 q^{17} - 11 q^{19} - 5 q^{20} - 4 q^{22} + 3 q^{23} + 8 q^{25} + q^{26} - 13 q^{28} + 8 q^{29} - 18 q^{31} + 3 q^{32} - 9 q^{34} + 6 q^{35} - 18 q^{37} - 8 q^{38} - 6 q^{40} - 17 q^{41} - 17 q^{43} - 5 q^{44} + 3 q^{46} + 11 q^{47} - 16 q^{49} + q^{50} + 5 q^{52} - 10 q^{53} - 9 q^{55} - q^{56} - 10 q^{58} + 7 q^{59} - 21 q^{61} + 29 q^{62} - 10 q^{64} - 8 q^{65} - 13 q^{67} - 16 q^{68} + 3 q^{70} + 34 q^{71} - 16 q^{73} - 4 q^{74} - 2 q^{76} + 18 q^{77} - 37 q^{79} + 13 q^{80} + q^{82} + 3 q^{83} + 6 q^{85} - 2 q^{86} - 19 q^{88} - 14 q^{89} - 6 q^{91} - 14 q^{92} - 44 q^{94} + 11 q^{95} - 17 q^{97} - 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 10\nu^{4} + 10\nu^{3} - 11\nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + 3\nu^{6} + 4\nu^{5} - 16\nu^{4} + 20\nu^{2} - 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 3\nu^{6} + 5\nu^{5} - 17\nu^{4} - 6\nu^{3} + 24\nu^{2} + 2\nu - 6 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - \beta_{6} + \beta_{5} + 8\beta_{4} + 10\beta_{3} + 33\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{7} - 8\beta_{6} + 3\beta_{5} + 12\beta_{4} + 42\beta_{3} + 55\beta_{2} + 34\beta _1 + 63 \) 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
−1.89423
−1.84394
−1.24973
−0.475244
0.633210
1.43149
1.97477
2.42368
−1.89423 0 1.58811 −1.00000 0 1.67585 0.780216 0 1.89423
1.2 −1.84394 0 1.40013 −1.00000 0 −4.77020 1.10613 0 1.84394
1.3 −1.24973 0 −0.438165 −1.00000 0 −0.297448 3.04706 0 1.24973
1.4 −0.475244 0 −1.77414 −1.00000 0 2.49548 1.79364 0 0.475244
1.5 0.633210 0 −1.59905 −1.00000 0 −1.79265 −2.27895 0 −0.633210
1.6 1.43149 0 0.0491573 −1.00000 0 −1.25071 −2.79261 0 −1.43149
1.7 1.97477 0 1.89973 −1.00000 0 −0.248978 −0.198009 0 −1.97477
1.8 2.42368 0 3.87423 −1.00000 0 −1.81135 4.54253 0 −2.42368
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 5265.2.a.be 8
3.b odd 2 1 5265.2.a.bb 8
9.c even 3 2 1755.2.i.e 16
9.d odd 6 2 585.2.i.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.f ✓ 16 9.d odd 6 2
1755.2.i.e 16 9.c even 3 2
5265.2.a.bb 8 3.b odd 2 1
5265.2.a.be 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):

\( T_{2}^{8} - T_{2}^{7} - 10T_{2}^{6} + 7T_{2}^{5} + 33T_{2}^{4} - 14T_{2}^{3} - 38T_{2}^{2} + 7T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 2T_{7}^{6} - 50T_{7}^{5} - 49T_{7}^{4} + 75T_{7}^{3} + 129T_{7}^{2} + 51T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{8} - 9T_{11}^{7} - 2T_{11}^{6} + 158T_{11}^{5} - 181T_{11}^{4} - 503T_{11}^{3} + 316T_{11}^{2} + 497T_{11} + 81 \) Copy content Toggle raw display
\( T_{17}^{8} + 6T_{17}^{7} - 36T_{17}^{6} - 159T_{17}^{5} + 466T_{17}^{4} + 1014T_{17}^{3} - 1684T_{17}^{2} - 1441T_{17} + 822 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 10 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 6 \) Copy content Toggle raw display
$11$ \( T^{8} - 9 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T - 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 822 \) Copy content Toggle raw display
$19$ \( T^{8} + 11 T^{7} + \cdots + 51733 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots - 29976 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 557523 \) Copy content Toggle raw display
$31$ \( T^{8} + 18 T^{7} + \cdots + 4283 \) Copy content Toggle raw display
$37$ \( T^{8} + 18 T^{7} + \cdots + 236259 \) Copy content Toggle raw display
$41$ \( T^{8} + 17 T^{7} + \cdots - 55764 \) Copy content Toggle raw display
$43$ \( T^{8} + 17 T^{7} + \cdots - 129776 \) Copy content Toggle raw display
$47$ \( T^{8} - 11 T^{7} + \cdots - 3294 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots - 405738 \) Copy content Toggle raw display
$59$ \( T^{8} - 7 T^{7} + \cdots + 1281627 \) Copy content Toggle raw display
$61$ \( T^{8} + 21 T^{7} + \cdots + 5311639 \) Copy content Toggle raw display
$67$ \( T^{8} + 13 T^{7} + \cdots + 3125246 \) Copy content Toggle raw display
$71$ \( T^{8} - 34 T^{7} + \cdots + 1374696 \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 373998 \) Copy content Toggle raw display
$79$ \( T^{8} + 37 T^{7} + \cdots - 74164 \) Copy content Toggle raw display
$83$ \( T^{8} - 3 T^{7} + \cdots + 23847006 \) Copy content Toggle raw display
$89$ \( T^{8} + 14 T^{7} + \cdots + 1052154 \) Copy content Toggle raw display
$97$ \( T^{8} + 17 T^{7} + \cdots + 267489 \) Copy content Toggle raw display
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