Properties

Label 8325.2.a.ch
Level $8325$
Weight $2$
Character orbit 8325.a
Self dual yes
Analytic conductor $66.475$
Analytic rank $0$
Dimension $5$
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] = [8325,2,Mod(1,8325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8325, 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("8325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.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: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{3} + 2) q^{4} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{4} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{3} + 2) q^{4} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + 1) q^{11} + (\beta_{3} + \beta_{2} - 1) q^{13} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 4) q^{14} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{17} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{19} + (2 \beta_{4} + 2 \beta_1) q^{22} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{26} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 5) q^{28} + 2 \beta_1 q^{29} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{31} + (\beta_{4} + 4 \beta_{2} + 4 \beta_1) q^{32} + (2 \beta_{4} - \beta_{3} - \beta_{2} - 7) q^{34} - q^{37} + (\beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 1) q^{38} + (\beta_{3} + 1) q^{41} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{44} + (4 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{47} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5) q^{49} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 5) q^{52} + (4 \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{53} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} - 9 \beta_1 + 4) q^{56} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{58} + (\beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 + 7) q^{59} + (2 \beta_{4} + 2 \beta_{2} - 4) q^{61} + (3 \beta_{3} - 3 \beta_1 + 6) q^{62} + ( - \beta_{3} + 4 \beta_{2} + 8 \beta_1) q^{64} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{67} + ( - 4 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 + 7) q^{68} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{71} + (4 \beta_{4} + \beta_{3} - 4 \beta_1 - 1) q^{73} - \beta_{4} q^{74} + (2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 4) q^{76} + (2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{77} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 7) q^{79} + (2 \beta_{4} + 2 \beta_1) q^{82} + (2 \beta_{4} + 2 \beta_{3} - \beta_1 + 5) q^{83} + (2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 8) q^{86} + (6 \beta_{4} + 4 \beta_{2} + 6 \beta_1) q^{88} + ( - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{89} + ( - \beta_{4} - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{91} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 14) q^{92} + ( - \beta_{3} + 6 \beta_{2} + 9 \beta_1 - 4) q^{94} + ( - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{97} + (7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 10 q^{4} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 10 q^{4} - 11 q^{7} + 6 q^{8} + 5 q^{11} - 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 8 q^{22} + 4 q^{23} + 4 q^{26} - 28 q^{28} + 4 q^{29} + 8 q^{31} + 14 q^{32} - 32 q^{34} - 5 q^{37} - 2 q^{38} + 5 q^{41} - 10 q^{43} + 46 q^{44} + 7 q^{47} + 22 q^{49} + 32 q^{52} - q^{53} - 8 q^{56} + 16 q^{58} + 30 q^{59} - 14 q^{61} + 24 q^{62} + 20 q^{64} - 24 q^{67} + 20 q^{68} + 7 q^{71} - 5 q^{73} - 2 q^{74} + 12 q^{76} - 17 q^{77} + 28 q^{79} + 8 q^{82} + 27 q^{83} - 44 q^{86} + 28 q^{88} - 6 q^{89} - 14 q^{91} + 56 q^{92} + 4 q^{94} + 26 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 3\beta_{3} + 10\beta_{2} + 20\beta _1 + 18 \) 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
−1.38679
2.10563
−1.62871
−0.383115
3.29298
−2.47408 0 4.12105 0 0 −4.78404 −5.24765 0 0
1.2 −1.13359 0 −0.714970 0 0 −2.46164 3.07767 0 0
1.3 0.728950 0 −1.46863 0 0 −2.55244 −2.52846 0 0
1.4 2.15510 0 2.64446 0 0 2.62521 1.38887 0 0
1.5 2.72362 0 5.41809 0 0 −3.82710 9.30957 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 8325.2.a.ch 5
3.b odd 2 1 925.2.a.f 5
5.b even 2 1 1665.2.a.p 5
15.d odd 2 1 185.2.a.e 5
15.e even 4 2 925.2.b.f 10
60.h even 2 1 2960.2.a.w 5
105.g even 2 1 9065.2.a.k 5
555.b odd 2 1 6845.2.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.e 5 15.d odd 2 1
925.2.a.f 5 3.b odd 2 1
925.2.b.f 10 15.e even 4 2
1665.2.a.p 5 5.b even 2 1
2960.2.a.w 5 60.h even 2 1
6845.2.a.f 5 555.b odd 2 1
8325.2.a.ch 5 1.a even 1 1 trivial
9065.2.a.k 5 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{5} - 2T_{2}^{4} - 8T_{2}^{3} + 14T_{2}^{2} + 11T_{2} - 12 \) Copy content Toggle raw display
\( T_{7}^{5} + 11T_{7}^{4} + 32T_{7}^{3} - 32T_{7}^{2} - 268T_{7} - 302 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 8T_{11}^{3} + 48T_{11}^{2} + 16T_{11} - 96 \) Copy content Toggle raw display
\( T_{13}^{5} + 4T_{13}^{4} - 28T_{13}^{3} - 60T_{13}^{2} + 148T_{13} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 8 T^{3} + 14 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 11 T^{4} + 32 T^{3} + \cdots - 302 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} - 8 T^{3} + 48 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 28 T^{3} - 60 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{5} - 52 T^{3} + 12 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$19$ \( T^{5} + 4 T^{4} - 38 T^{3} - 66 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} - 72 T^{3} + 208 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 32 T^{3} + 48 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$31$ \( T^{5} - 8 T^{4} - 30 T^{3} + 342 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$37$ \( (T + 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} - 5 T^{4} - 8 T^{3} + 48 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$43$ \( T^{5} + 10 T^{4} - 80 T^{3} + \cdots + 2528 \) Copy content Toggle raw display
$47$ \( T^{5} - 7 T^{4} - 92 T^{3} + 592 T^{2} + \cdots - 978 \) Copy content Toggle raw display
$53$ \( T^{5} + T^{4} - 144 T^{3} - 40 T^{2} + \cdots + 528 \) Copy content Toggle raw display
$59$ \( T^{5} - 30 T^{4} + 302 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{5} + 14 T^{4} - 8 T^{3} + \cdots + 3296 \) Copy content Toggle raw display
$67$ \( T^{5} + 24 T^{4} + 94 T^{3} + \cdots - 10952 \) Copy content Toggle raw display
$71$ \( T^{5} - 7 T^{4} - 132 T^{3} + \cdots + 7104 \) Copy content Toggle raw display
$73$ \( T^{5} + 5 T^{4} - 192 T^{3} + \cdots - 368 \) Copy content Toggle raw display
$79$ \( T^{5} - 28 T^{4} + 134 T^{3} + \cdots - 19508 \) Copy content Toggle raw display
$83$ \( T^{5} - 27 T^{4} + 178 T^{3} + \cdots + 4818 \) Copy content Toggle raw display
$89$ \( T^{5} + 6 T^{4} - 248 T^{3} + \cdots - 22944 \) Copy content Toggle raw display
$97$ \( T^{5} - 26 T^{4} - 88 T^{3} + \cdots + 166976 \) Copy content Toggle raw display
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