Properties

Label 6525.2.a.bc
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} + (2 \beta - 2) q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} + (2 \beta - 2) q^{7} + (\beta + 4) q^{8} + (\beta + 3) q^{11} + 2 q^{13} + 8 q^{14} + 3 \beta q^{16} + (2 \beta - 4) q^{17} + ( - 2 \beta + 2) q^{19} + (4 \beta + 4) q^{22} + ( - \beta + 5) q^{23} + 2 \beta q^{26} + (4 \beta + 4) q^{28} - q^{29} + 4 q^{31} + (\beta + 4) q^{32} + ( - 2 \beta + 8) q^{34} + ( - 3 \beta - 3) q^{37} - 8 q^{38} + ( - 3 \beta - 3) q^{41} + (3 \beta - 3) q^{43} + (6 \beta + 10) q^{44} + (4 \beta - 4) q^{46} + (2 \beta - 10) q^{47} + ( - 4 \beta + 13) q^{49} + (2 \beta + 4) q^{52} + ( - \beta - 5) q^{53} + 8 \beta q^{56} - \beta q^{58} - 12 q^{59} + (2 \beta + 4) q^{61} + 4 \beta q^{62} + ( - \beta + 4) q^{64} + ( - 6 \beta + 2) q^{67} + 2 \beta q^{68} + (2 \beta + 6) q^{71} + (3 \beta + 3) q^{73} + ( - 6 \beta - 12) q^{74} + ( - 4 \beta - 4) q^{76} + (6 \beta + 2) q^{77} - 12 q^{79} + ( - 6 \beta - 12) q^{82} + (\beta - 1) q^{83} + 12 q^{86} + (8 \beta + 16) q^{88} + (4 \beta - 6) q^{89} + (4 \beta - 4) q^{91} + (2 \beta + 6) q^{92} + ( - 8 \beta + 8) q^{94} + ( - 3 \beta + 1) q^{97} + (9 \beta - 16) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 7 q^{11} + 4 q^{13} + 16 q^{14} + 3 q^{16} - 6 q^{17} + 2 q^{19} + 12 q^{22} + 9 q^{23} + 2 q^{26} + 12 q^{28} - 2 q^{29} + 8 q^{31} + 9 q^{32} + 14 q^{34} - 9 q^{37} - 16 q^{38} - 9 q^{41} - 3 q^{43} + 26 q^{44} - 4 q^{46} - 18 q^{47} + 22 q^{49} + 10 q^{52} - 11 q^{53} + 8 q^{56} - q^{58} - 24 q^{59} + 10 q^{61} + 4 q^{62} + 7 q^{64} - 2 q^{67} + 2 q^{68} + 14 q^{71} + 9 q^{73} - 30 q^{74} - 12 q^{76} + 10 q^{77} - 24 q^{79} - 30 q^{82} - q^{83} + 24 q^{86} + 40 q^{88} - 8 q^{89} - 4 q^{91} + 14 q^{92} + 8 q^{94} - q^{97} - 23 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 0 0 −5.12311 2.43845 0 0
1.2 2.56155 0 4.56155 0 0 3.12311 6.56155 0 0
\(n\): e.g. 2-40 or 990-1000
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.bc 2
3.b odd 2 1 2175.2.a.m 2
5.b even 2 1 1305.2.a.i 2
15.d odd 2 1 435.2.a.h 2
15.e even 4 2 2175.2.c.h 4
60.h even 2 1 6960.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.h 2 15.d odd 2 1
1305.2.a.i 2 5.b even 2 1
2175.2.a.m 2 3.b odd 2 1
2175.2.c.h 4 15.e even 4 2
6525.2.a.bc 2 1.a even 1 1 trivial
6960.2.a.bx 2 60.h even 2 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 152 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$97$ \( T^{2} + T - 38 \) Copy content Toggle raw display
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