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Properties

Label 9075.2.a.df
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{2} + q^{3} + \beta_1 q^{4} + ( - \beta_{3} - \beta_{2}) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{2} + q^{3} + \beta_1 q^{4} + ( - \beta_{3} - \beta_{2}) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - 1) q^{8} + q^{9} + \beta_1 q^{12} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{13} + ( - 3 \beta_{3} + \beta_1 - 3) q^{14} + (\beta_{2} - 2 \beta_1 - 2) q^{16} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{17}+ \cdots + (7 \beta_{3} + \beta_{2} - 2 \beta_1 + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + q^{12} + 7 q^{13} - 5 q^{14} - 9 q^{16} + 8 q^{17} + q^{18} - 11 q^{19} - 5 q^{23} - 3 q^{24} - 12 q^{26} + 4 q^{27} + 5 q^{28} - 17 q^{29} - 5 q^{31} + 17 q^{34} + q^{36} - 15 q^{37} + q^{38} + 7 q^{39} - 10 q^{41} - 5 q^{42} - 4 q^{43} - 15 q^{46} + 8 q^{47} - 9 q^{48} - 8 q^{49} + 8 q^{51} - 7 q^{52} - 10 q^{53} + q^{54} + 10 q^{56} - 11 q^{57} + 7 q^{58} + 9 q^{59} - 37 q^{61} - 20 q^{62} - 7 q^{64} - 3 q^{67} + 17 q^{68} - 5 q^{69} + 13 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} - 29 q^{76} - 12 q^{78} - 20 q^{79} + 4 q^{81} - 5 q^{82} - 17 q^{83} + 5 q^{84} + 34 q^{86} - 17 q^{87} - 24 q^{89} - 20 q^{91} - 10 q^{92} - 5 q^{93} - 23 q^{94} + 32 q^{97} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.82709
−1.95630
−0.209057
1.33826
−1.95630 1.00000 1.82709 0 −1.95630 1.54732 0.338261 1.00000 0
1.2 −0.209057 1.00000 −1.95630 0 −0.209057 0.488830 0.827091 1.00000 0
1.3 1.33826 1.00000 −0.209057 0 1.33826 −3.78339 −2.95630 1.00000 0
1.4 1.82709 1.00000 1.33826 0 1.82709 1.74724 −1.20906 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.df 4
5.b even 2 1 1815.2.a.q 4
11.b odd 2 1 9075.2.a.co 4
11.d odd 10 2 825.2.n.j 8
15.d odd 2 1 5445.2.a.bq 4
55.d odd 2 1 1815.2.a.u 4
55.h odd 10 2 165.2.m.c ✓ 8
55.l even 20 4 825.2.bx.e 16
165.d even 2 1 5445.2.a.bj 4
165.r even 10 2 495.2.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.c ✓ 8 55.h odd 10 2
495.2.n.c 8 165.r even 10 2
825.2.n.j 8 11.d odd 10 2
825.2.bx.e 16 55.l even 20 4
1815.2.a.q 4 5.b even 2 1
1815.2.a.u 4 55.d odd 2 1
5445.2.a.bj 4 165.d even 2 1
5445.2.a.bq 4 15.d odd 2 1
9075.2.a.co 4 11.b odd 2 1
9075.2.a.df 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{4} - T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 10T_{7}^{2} + 15T_{7} - 5 \) Copy content Toggle raw display
\( T_{13}^{4} - 7T_{13}^{3} - 16T_{13}^{2} + 202T_{13} - 359 \) Copy content Toggle raw display
\( T_{17}^{4} - 8T_{17}^{3} - 6T_{17}^{2} + 88T_{17} - 59 \) Copy content Toggle raw display
\( T_{19}^{4} + 11T_{19}^{3} + 6T_{19}^{2} - 184T_{19} - 239 \) Copy content Toggle raw display
\( T_{23}^{4} + 5T_{23}^{3} - 25T_{23}^{2} + 25T_{23} - 5 \) Copy content Toggle raw display
\( T_{37}^{4} + 15T_{37}^{3} + 65T_{37}^{2} + 45T_{37} - 155 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 10 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} + \cdots - 359 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots - 59 \) Copy content Toggle raw display
$19$ \( T^{4} + 11 T^{3} + \cdots - 239 \) Copy content Toggle raw display
$23$ \( T^{4} + 5 T^{3} + \cdots - 5 \) Copy content Toggle raw display
$29$ \( T^{4} + 17 T^{3} + \cdots - 419 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots - 5 \) Copy content Toggle raw display
$37$ \( T^{4} + 15 T^{3} + \cdots - 155 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} + \cdots - 5 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots - 569 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{4} - 9 T^{3} + \cdots + 1341 \) Copy content Toggle raw display
$61$ \( T^{4} + 37 T^{3} + \cdots + 6571 \) Copy content Toggle raw display
$67$ \( T^{4} + 3 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$71$ \( T^{4} - 13 T^{3} + \cdots - 389 \) Copy content Toggle raw display
$73$ \( T^{4} - 15 T^{3} + \cdots - 4205 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} + \cdots - 305 \) Copy content Toggle raw display
$83$ \( T^{4} + 17 T^{3} + \cdots - 14669 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots - 31869 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + \cdots + 61 \) Copy content Toggle raw display
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