Properties

Label 825.2.a.k
Level $825$
Weight $2$
Character orbit 825.a
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
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] = [825,2,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
0.311108
−1.48119
2.17009
−1.90321 −1.00000 1.62222 0 1.90321 4.42864 0.719004 1.00000 0
1.2 0.193937 −1.00000 −1.96239 0 −0.193937 −3.35026 −0.768452 1.00000 0
1.3 2.70928 −1.00000 5.34017 0 −2.70928 −1.07838 9.04945 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 825.2.a.k 3
3.b odd 2 1 2475.2.a.bb 3
5.b even 2 1 165.2.a.c 3
5.c odd 4 2 825.2.c.g 6
11.b odd 2 1 9075.2.a.cf 3
15.d odd 2 1 495.2.a.e 3
15.e even 4 2 2475.2.c.r 6
20.d odd 2 1 2640.2.a.be 3
35.c odd 2 1 8085.2.a.bk 3
55.d odd 2 1 1815.2.a.m 3
60.h even 2 1 7920.2.a.cj 3
165.d even 2 1 5445.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 5.b even 2 1
495.2.a.e 3 15.d odd 2 1
825.2.a.k 3 1.a even 1 1 trivial
825.2.c.g 6 5.c odd 4 2
1815.2.a.m 3 55.d odd 2 1
2475.2.a.bb 3 3.b odd 2 1
2475.2.c.r 6 15.e even 4 2
2640.2.a.be 3 20.d odd 2 1
5445.2.a.z 3 165.d even 2 1
7920.2.a.cj 3 60.h even 2 1
8085.2.a.bk 3 35.c odd 2 1
9075.2.a.cf 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 16T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$23$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$31$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$37$ \( (T - 2)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 14 T^{2} + 44 T + 8 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 80 T - 400 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 52 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} - 16 T + 320 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 52 T - 248 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$73$ \( T^{3} - 14 T^{2} + 4 T + 344 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 64 T + 800 \) Copy content Toggle raw display
$83$ \( T^{3} - 120T - 16 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 52 T - 200 \) Copy content Toggle raw display
$97$ \( T^{3} + 22 T^{2} + 108 T + 8 \) Copy content Toggle raw display
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