Properties

Label 6525.2.a.by
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
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] = [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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2175)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} - 2 \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{11} + (\beta_{5} + \beta_{3} - 1) q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 2) q^{14}+ \cdots + ( - 3 \beta_{6} + 6 \beta_{5} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 11\nu^{4} + 2\nu^{3} + 28\nu^{2} - 10\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} + \nu^{5} - 10\nu^{4} - 8\nu^{3} + 22\nu^{2} + 11\nu - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 11\nu^{5} + 2\nu^{4} + 29\nu^{3} - 11\nu^{2} - 8\nu + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 11\nu^{5} + 2\nu^{4} + 30\nu^{3} - 11\nu^{2} - 14\nu + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + \nu^{6} + 11\nu^{5} - 12\nu^{4} - 28\nu^{3} + 32\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{3} + 7\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 9\beta_{6} - 10\beta_{5} + \beta_{4} - \beta_{2} + 39\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} + 9\beta_{6} + 2\beta_{5} - 10\beta_{3} + 49\beta_{2} - 2\beta _1 + 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{7} + 68\beta_{6} - 80\beta_{5} + 11\beta_{4} + 2\beta_{3} - 14\beta_{2} + 263\beta _1 - 17 \) 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
2.59587
2.57789
1.64893
0.510732
−0.135002
−0.485464
−1.98486
−2.72810
−2.59587 0 4.73855 0 0 −3.93826 −7.10893 0 0
1.2 −2.57789 0 4.64553 0 0 4.69867 −6.81989 0 0
1.3 −1.64893 0 0.718971 0 0 1.44112 2.11233 0 0
1.4 −0.510732 0 −1.73915 0 0 −4.82343 1.90970 0 0
1.5 0.135002 0 −1.98177 0 0 1.12340 −0.537546 0 0
1.6 0.485464 0 −1.76432 0 0 1.94912 −1.82745 0 0
1.7 1.98486 0 1.93969 0 0 2.16633 −0.119715 0 0
1.8 2.72810 0 5.44251 0 0 −4.61695 9.39150 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 6525.2.a.by 8
3.b odd 2 1 2175.2.a.bd yes 8
5.b even 2 1 6525.2.a.bz 8
15.d odd 2 1 2175.2.a.bc 8
15.e even 4 2 2175.2.c.p 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2175.2.a.bc 8 15.d odd 2 1
2175.2.a.bd yes 8 3.b odd 2 1
2175.2.c.p 16 15.e even 4 2
6525.2.a.by 8 1.a even 1 1 trivial
6525.2.a.bz 8 5.b even 2 1

Hecke kernels

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

\( T_{2}^{8} + 2T_{2}^{7} - 12T_{2}^{6} - 23T_{2}^{5} + 36T_{2}^{4} + 62T_{2}^{3} - 15T_{2}^{2} - 14T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 45T_{7}^{6} - 44T_{7}^{5} + 667T_{7}^{4} - 270T_{7}^{3} - 3428T_{7}^{2} + 5898T_{7} - 2817 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} - 36T_{11}^{6} - 238T_{11}^{5} + 265T_{11}^{4} + 2382T_{11}^{3} - 21T_{11}^{2} - 6052T_{11} - 1140 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots - 2817 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots - 1140 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots - 10503 \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots + 18954 \) Copy content Toggle raw display
$19$ \( T^{8} - 99 T^{6} + \cdots + 205760 \) Copy content Toggle raw display
$23$ \( T^{8} + 14 T^{7} + \cdots - 14976 \) Copy content Toggle raw display
$29$ \( (T + 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots + 1225792 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 20929536 \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots - 2160 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots + 486720 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 2390796 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 10368 \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 10368 \) Copy content Toggle raw display
$61$ \( T^{8} - 12 T^{7} + \cdots + 386112 \) Copy content Toggle raw display
$67$ \( T^{8} + 26 T^{7} + \cdots - 48194129 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots - 16256 \) Copy content Toggle raw display
$73$ \( T^{8} - 14 T^{7} + \cdots + 19650816 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + \cdots + 2032128 \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{7} + \cdots - 144320256 \) Copy content Toggle raw display
$89$ \( T^{8} + 34 T^{7} + \cdots - 12296640 \) Copy content Toggle raw display
$97$ \( T^{8} + 30 T^{7} + \cdots - 9904960 \) Copy content Toggle raw display
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