Properties

Label 9075.2.a.cp
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 6x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + q^{3} + ( - \beta_{3} - \beta_{2} + 3) q^{4} + ( - \beta_{2} + \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 4 \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + q^{3} + ( - \beta_{3} - \beta_{2} + 3) q^{4} + ( - \beta_{2} + \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 4 \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{3} - \beta_{2} + 3) q^{12} + ( - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{13} + (2 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{14} + ( - 2 \beta_{3} + \beta_1 + 2) q^{16} + (\beta_{3} - 2 \beta_1) q^{17} + ( - \beta_{2} + \beta_1) q^{18} + ( - \beta_{3} - \beta_1 + 3) q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{23} + ( - 4 \beta_{2} + \beta_1 + 1) q^{24} + ( - 7 \beta_{2} + 3 \beta_1 + 5) q^{26} + q^{27} + ( - \beta_{3} + 10 \beta_{2} - 3 \beta_1 - 1) q^{28} + ( - \beta_{3} + \beta_1 + 4) q^{29} + ( - \beta_1 - 1) q^{31} + (2 \beta_{3} - \beta_{2} + 2) q^{32} + ( - \beta_{3} + 5 \beta_{2} - 8) q^{34} + ( - \beta_{3} - \beta_{2} + 3) q^{36} + ( - \beta_{3} - 5 \beta_{2} + 1) q^{37} + (\beta_{3} - 5 \beta_{2} + 3 \beta_1 - 4) q^{38} + ( - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{39} + ( - \beta_{2} - 2 \beta_1 + 5) q^{41} + (2 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{42} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{43} + ( - 8 \beta_{2} + \beta_1 + 6) q^{46} + (4 \beta_{2} - \beta_1 - 2) q^{47} + ( - 2 \beta_{3} + \beta_1 + 2) q^{48} + ( - \beta_{3} + 6 \beta_{2} + \beta_1 + 3) q^{49} + (\beta_{3} - 2 \beta_1) q^{51} + ( - 5 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 13) q^{52} + (\beta_{3} - 5 \beta_{2} + \beta_1 - 1) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + (7 \beta_{3} + 3 \beta_{2} + \beta_1 - 12) q^{56} + ( - \beta_{3} - \beta_1 + 3) q^{57} + (\beta_{3} - 8 \beta_{2} + 4 \beta_1 + 4) q^{58} + ( - 2 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{59} + (2 \beta_1 + 8) q^{61} + (2 \beta_{2} - \beta_1 - 4) q^{62} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{63} + (\beta_{3} + 4 \beta_{2} - 3) q^{64} + ( - 4 \beta_{3} - 4 \beta_{2} + \beta_1 - 2) q^{67} + (4 \beta_{3} + 5 \beta_{2} - 4 \beta_1 - 5) q^{68} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{69} + (\beta_{3} + 3 \beta_{2} - 6 \beta_1 + 1) q^{71} + ( - 4 \beta_{2} + \beta_1 + 1) q^{72} + (4 \beta_{3} - 2 \beta_{2} + \beta_1) q^{73} + ( - 4 \beta_{3} - 4 \beta_{2} + \beta_1 + 5) q^{74} + ( - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 11) q^{76} + ( - 7 \beta_{2} + 3 \beta_1 + 5) q^{78} + (3 \beta_{3} + 7 \beta_{2} - 6 \beta_1) q^{79} + q^{81} + ( - \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 7) q^{82} + (3 \beta_{3} + 9 \beta_{2} - 10) q^{83} + ( - \beta_{3} + 10 \beta_{2} - 3 \beta_1 - 1) q^{84} + (\beta_{3} - 15 \beta_{2} + 7 \beta_1 + 9) q^{86} + ( - \beta_{3} + \beta_1 + 4) q^{87} + (2 \beta_{3} + 3 \beta_1 + 3) q^{89} + ( - \beta_{3} + 8 \beta_{2} - 5 \beta_1 - 4) q^{91} + ( - 4 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 10) q^{92} + ( - \beta_1 - 1) q^{93} + (4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 8) q^{94} + (2 \beta_{3} - \beta_{2} + 2) q^{96} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{97} + (7 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} + 9 q^{12} + 10 q^{13} - 21 q^{14} + 7 q^{16} - q^{17} - q^{18} + 10 q^{19} - 4 q^{21} - q^{23} - 3 q^{24} + 9 q^{26} + 4 q^{27} + 12 q^{28} + 16 q^{29} - 5 q^{31} + 8 q^{32} - 23 q^{34} + 9 q^{36} - 7 q^{37} - 22 q^{38} + 10 q^{39} + 16 q^{41} - 21 q^{42} + 26 q^{43} + 9 q^{46} - q^{47} + 7 q^{48} + 24 q^{49} - q^{51} + 38 q^{52} - 12 q^{53} - q^{54} - 34 q^{56} + 10 q^{57} + 5 q^{58} - 11 q^{59} + 34 q^{61} - 13 q^{62} - 4 q^{63} - 3 q^{64} - 19 q^{67} - 10 q^{68} - q^{69} + 5 q^{71} - 3 q^{72} + q^{73} + 9 q^{74} + 46 q^{76} + 9 q^{78} + 11 q^{79} + 4 q^{81} - 30 q^{82} - 19 q^{83} + 12 q^{84} + 14 q^{86} + 16 q^{87} + 17 q^{89} - 6 q^{91} + 34 q^{92} - 5 q^{93} - 24 q^{94} + 8 q^{96} + 12 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 6x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 5\beta _1 - 1 \) 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
−0.933531
−2.48008
2.55157
1.86205
−2.55157 1.00000 4.51049 0 −2.55157 4.68008 −6.40567 1.00000 0
1.2 −1.86205 1.00000 1.46722 0 −1.86205 −1.28876 0.992053 1.00000 0
1.3 0.933531 1.00000 −1.12852 0 0.933531 −4.44402 −2.92057 1.00000 0
1.4 2.48008 1.00000 4.15081 0 2.48008 −2.94731 5.33418 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 9075.2.a.cp 4
5.b even 2 1 9075.2.a.de 4
11.b odd 2 1 9075.2.a.dh 4
11.c even 5 2 825.2.n.l yes 8
55.d odd 2 1 9075.2.a.cn 4
55.j even 10 2 825.2.n.h 8
55.k odd 20 4 825.2.bx.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.h 8 55.j even 10 2
825.2.n.l yes 8 11.c even 5 2
825.2.bx.i 16 55.k odd 20 4
9075.2.a.cn 4 55.d odd 2 1
9075.2.a.cp 4 1.a even 1 1 trivial
9075.2.a.de 4 5.b even 2 1
9075.2.a.dh 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{4} + T_{2}^{3} - 8T_{2}^{2} - 6T_{2} + 11 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} - 18T_{7}^{2} - 89T_{7} - 79 \) Copy content Toggle raw display
\( T_{13}^{4} - 10T_{13}^{3} + 24T_{13}^{2} + 5T_{13} - 31 \) Copy content Toggle raw display
\( T_{17}^{4} + T_{17}^{3} - 38T_{17}^{2} - 126T_{17} - 99 \) Copy content Toggle raw display
\( T_{19}^{4} - 10T_{19}^{3} + 16T_{19}^{2} + 45T_{19} - 81 \) Copy content Toggle raw display
\( T_{23}^{4} + T_{23}^{3} - 39T_{23}^{2} - 29T_{23} + 341 \) Copy content Toggle raw display
\( T_{37}^{4} + 7T_{37}^{3} - 42T_{37}^{2} - 128T_{37} + 341 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 8 T^{2} - 6 T + 11 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} - 18 T^{2} - 89 T - 79 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + 24 T^{2} + 5 T - 31 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} - 38 T^{2} - 126 T - 99 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 16 T^{2} + 45 T - 81 \) Copy content Toggle raw display
$23$ \( T^{4} + T^{3} - 39 T^{2} - 29 T + 341 \) Copy content Toggle raw display
$29$ \( T^{4} - 16 T^{3} + 80 T^{2} + \cdots - 101 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + T^{2} - 15 T - 1 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} - 42 T^{2} - 128 T + 341 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + 55 T^{2} + \cdots - 181 \) Copy content Toggle raw display
$43$ \( T^{4} - 26 T^{3} + 197 T^{2} + \cdots - 1089 \) Copy content Toggle raw display
$47$ \( T^{4} + T^{3} - 38 T^{2} - 6 T + 131 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} - 30 T^{2} + \cdots + 319 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + T^{2} - 189 T - 99 \) Copy content Toggle raw display
$61$ \( T^{4} - 34 T^{3} + 400 T^{2} + \cdots + 2864 \) Copy content Toggle raw display
$67$ \( T^{4} + 19 T^{3} - 18 T^{2} + \cdots + 2351 \) Copy content Toggle raw display
$71$ \( T^{4} - 5 T^{3} - 256 T^{2} + \cdots + 11749 \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} - 210 T^{2} + 1352 T - 1111 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} - 265 T^{2} + \cdots - 7781 \) Copy content Toggle raw display
$83$ \( T^{4} + 19 T^{3} - 93 T^{2} + \cdots - 9119 \) Copy content Toggle raw display
$89$ \( T^{4} - 17 T^{3} - 25 T^{2} + 711 T + 99 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + 38 T^{2} + \cdots - 139 \) Copy content Toggle raw display
show more
show less