Properties

Label 225.3.g.f
Level $225$
Weight $3$
Character orbit 225.g
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,3,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{4} - 6 \beta_1 q^{7} - 5 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{4} - 6 \beta_1 q^{7} - 5 \beta_{3} q^{8} + 18 q^{11} - 6 \beta_{3} q^{13} - 18 \beta_{2} q^{14} + 11 q^{16} - 4 \beta_1 q^{17} - 10 \beta_{2} q^{19} + 18 \beta_1 q^{22} + 16 \beta_{3} q^{23} + 18 q^{26} + 6 \beta_{3} q^{28} + 22 q^{31} - 9 \beta_1 q^{32} - 12 \beta_{2} q^{34} - 6 \beta_1 q^{37} - 10 \beta_{3} q^{38} + 18 q^{41} + 24 \beta_{3} q^{43} - 18 \beta_{2} q^{44} - 48 q^{46} + 36 \beta_1 q^{47} + 59 \beta_{2} q^{49} - 6 \beta_1 q^{52} - 4 \beta_{3} q^{53} - 90 q^{56} + 90 \beta_{2} q^{59} + 2 q^{61} + 22 \beta_1 q^{62} - 71 \beta_{2} q^{64} - 36 \beta_1 q^{67} + 4 \beta_{3} q^{68} - 72 q^{71} - 36 \beta_{3} q^{73} - 18 \beta_{2} q^{74} - 10 q^{76} - 108 \beta_1 q^{77} + 70 \beta_{2} q^{79} + 18 \beta_1 q^{82} - 44 \beta_{3} q^{83} - 72 q^{86} - 90 \beta_{3} q^{88} + 90 \beta_{2} q^{89} - 108 q^{91} + 16 \beta_1 q^{92} + 108 \beta_{2} q^{94} + 84 \beta_1 q^{97} + 59 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 72 q^{11} + 44 q^{16} + 72 q^{26} + 88 q^{31} + 72 q^{41} - 192 q^{46} - 360 q^{56} + 8 q^{61} - 288 q^{71} - 40 q^{76} - 288 q^{86} - 432 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i 0 1.00000i 0 0 7.34847 + 7.34847i −6.12372 + 6.12372i 0 0
82.2 1.22474 + 1.22474i 0 1.00000i 0 0 −7.34847 7.34847i 6.12372 6.12372i 0 0
118.1 −1.22474 + 1.22474i 0 1.00000i 0 0 7.34847 7.34847i −6.12372 6.12372i 0 0
118.2 1.22474 1.22474i 0 1.00000i 0 0 −7.34847 + 7.34847i 6.12372 + 6.12372i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.g.f 4
3.b odd 2 1 75.3.f.a 4
5.b even 2 1 inner 225.3.g.f 4
5.c odd 4 2 inner 225.3.g.f 4
12.b even 2 1 1200.3.bg.j 4
15.d odd 2 1 75.3.f.a 4
15.e even 4 2 75.3.f.a 4
60.h even 2 1 1200.3.bg.j 4
60.l odd 4 2 1200.3.bg.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.f.a 4 3.b odd 2 1
75.3.f.a 4 15.d odd 2 1
75.3.f.a 4 15.e even 4 2
225.3.g.f 4 1.a even 1 1 trivial
225.3.g.f 4 5.b even 2 1 inner
225.3.g.f 4 5.c odd 4 2 inner
1200.3.bg.j 4 12.b even 2 1
1200.3.bg.j 4 60.h even 2 1
1200.3.bg.j 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 11664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11664 \) Copy content Toggle raw display
$11$ \( (T - 18)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 11664 \) Copy content Toggle raw display
$17$ \( T^{4} + 2304 \) Copy content Toggle raw display
$19$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 589824 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 22)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 11664 \) Copy content Toggle raw display
$41$ \( (T - 18)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 2985984 \) Copy content Toggle raw display
$47$ \( T^{4} + 15116544 \) Copy content Toggle raw display
$53$ \( T^{4} + 2304 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 15116544 \) Copy content Toggle raw display
$71$ \( (T + 72)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 15116544 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 33732864 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 448084224 \) Copy content Toggle raw display
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