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Properties

Label 6525.2.a.ce
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1305)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{7} + (\beta_{9} + \beta_{8} + \beta_1) q^{8} + (\beta_{5} - \beta_{2} - 1) q^{11} + ( - \beta_{7} - \beta_1) q^{13} + (\beta_{11} - 2 \beta_{5} + \cdots - \beta_{2}) q^{14}+ \cdots + (\beta_{8} - 2 \beta_{7} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} - 12 q^{11} - 16 q^{14} + 16 q^{16} - 20 q^{19} - 56 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 32 q^{41} - 68 q^{44} + 20 q^{46} - 4 q^{49} - 76 q^{56} - 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 24\nu^{2} - 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 17\nu^{9} + 97\nu^{7} - 205\nu^{5} + 105\nu^{3} + 59\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 19\nu^{8} + 127\nu^{6} - 349\nu^{4} + 343\nu^{2} - 43 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{11} + 37\nu^{9} - 239\nu^{7} + 632\nu^{5} - 621\nu^{3} + 131\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} - 37\nu^{9} + 242\nu^{7} - 662\nu^{5} + 693\nu^{3} - 164\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{11} + 91\nu^{9} - 575\nu^{7} + 1457\nu^{5} - 1239\nu^{3} + 11\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{11} - 91\nu^{9} + 575\nu^{7} - 1457\nu^{5} + 1251\nu^{3} - 71\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} - 18\nu^{8} + 113\nu^{6} - 287\nu^{4} + 250\nu^{2} - 15 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{10} - 55\nu^{8} + 351\nu^{6} - 907\nu^{4} + 827\nu^{2} - 91 ) / 4 \) 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
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{5} - \beta_{3} + 8\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 8\beta_{8} + \beta_{6} - \beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{11} + 10\beta_{10} + 10\beta_{5} - 8\beta_{3} + 56\beta_{2} + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66\beta_{9} + 56\beta_{8} + 2\beta_{7} + 12\beta_{6} - 10\beta_{4} + 181\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -78\beta_{11} + 80\beta_{10} + 74\beta_{5} - 50\beta_{3} + 381\beta_{2} + 489 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 461\beta_{9} + 385\beta_{8} + 30\beta_{7} + 108\beta_{6} - 68\beta_{4} + 1171\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -561\beta_{11} + 599\beta_{10} + 489\beta_{5} - 283\beta_{3} + 2576\beta_{2} + 3158 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3175\beta_{9} + 2648\beta_{8} + 316\beta_{7} + 877\beta_{6} - 379\beta_{4} + 7711\beta_1 \) 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
−2.62500
−2.51930
−1.78841
−1.35513
−1.27263
−0.328889
0.328889
1.27263
1.35513
1.78841
2.51930
2.62500
−2.62500 0 4.89062 0 0 1.33988 −7.58789 0 0
1.2 −2.51930 0 4.34685 0 0 4.08323 −5.91241 0 0
1.3 −1.78841 0 1.19839 0 0 −4.04635 1.43360 0 0
1.4 −1.35513 0 −0.163621 0 0 1.26969 2.93199 0 0
1.5 −1.27263 0 −0.380419 0 0 0.255813 3.02939 0 0
1.6 −0.328889 0 −1.89183 0 0 −1.86588 1.27998 0 0
1.7 0.328889 0 −1.89183 0 0 1.86588 −1.27998 0 0
1.8 1.27263 0 −0.380419 0 0 −0.255813 −3.02939 0 0
1.9 1.35513 0 −0.163621 0 0 −1.26969 −2.93199 0 0
1.10 1.78841 0 1.19839 0 0 4.04635 −1.43360 0 0
1.11 2.51930 0 4.34685 0 0 −4.08323 5.91241 0 0
1.12 2.62500 0 4.89062 0 0 −1.33988 7.58789 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6525.2.a.ce 12
3.b odd 2 1 6525.2.a.cf 12
5.b even 2 1 inner 6525.2.a.ce 12
5.c odd 4 2 1305.2.c.k ✓ 12
15.d odd 2 1 6525.2.a.cf 12
15.e even 4 2 1305.2.c.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.c.k ✓ 12 5.c odd 4 2
1305.2.c.l yes 12 15.e even 4 2
6525.2.a.ce 12 1.a even 1 1 trivial
6525.2.a.ce 12 5.b even 2 1 inner
6525.2.a.cf 12 3.b odd 2 1
6525.2.a.cf 12 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):

\( T_{2}^{12} - 20T_{2}^{10} + 148T_{2}^{8} - 502T_{2}^{6} + 792T_{2}^{4} - 496T_{2}^{2} + 45 \) Copy content Toggle raw display
\( T_{7}^{12} - 40T_{7}^{10} + 518T_{7}^{8} - 2412T_{7}^{6} + 4517T_{7}^{4} - 3036T_{7}^{2} + 180 \) Copy content Toggle raw display
\( T_{11}^{6} + 6T_{11}^{5} - 16T_{11}^{4} - 130T_{11}^{3} - 119T_{11}^{2} + 24T_{11} + 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 20 T^{10} + \cdots + 45 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 40 T^{10} + \cdots + 180 \) Copy content Toggle raw display
$11$ \( (T^{6} + 6 T^{5} - 16 T^{4} + \cdots + 30)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 100 T^{10} + \cdots + 2880 \) Copy content Toggle raw display
$17$ \( T^{12} - 144 T^{10} + \cdots + 8000 \) Copy content Toggle raw display
$19$ \( (T^{6} + 10 T^{5} + \cdots + 9832)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 96 T^{10} + \cdots + 2880 \) Copy content Toggle raw display
$29$ \( (T - 1)^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots + 33224)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16231322880 \) Copy content Toggle raw display
$41$ \( (T^{6} + 16 T^{5} + \cdots - 25920)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 588395520 \) Copy content Toggle raw display
$47$ \( T^{12} - 144 T^{10} + \cdots + 8000 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 550830080 \) Copy content Toggle raw display
$59$ \( (T^{6} + 22 T^{5} + \cdots + 204480)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 26 T^{5} + \cdots + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24224976180 \) Copy content Toggle raw display
$71$ \( (T^{6} + 10 T^{5} + \cdots - 539040)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 260 T^{10} + \cdots + 33592320 \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots - 17128)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 314265920 \) Copy content Toggle raw display
$89$ \( (T^{6} + 34 T^{5} + \cdots + 540)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 302330880 \) Copy content Toggle raw display
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