Properties

Label 1225.2.b.g
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,2,Mod(99,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not 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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{3} - 1) q^{4} - q^{6} + ( - \beta_{2} - 2 \beta_1) q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{3} - 1) q^{4} - q^{6} + ( - \beta_{2} - 2 \beta_1) q^{8} + \beta_{3} q^{9} + ( - \beta_{3} + 2) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{12} + 2 \beta_1 q^{13} + 3 q^{16} + 2 \beta_1 q^{17} + (\beta_{2} + 3 \beta_1) q^{18} - \beta_{3} q^{19} + ( - \beta_{2} - \beta_1) q^{22} + \beta_{2} q^{23} + (\beta_{3} - 1) q^{24} + ( - 2 \beta_{3} - 6) q^{26} + \beta_1 q^{27} + q^{29} - 6 q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - 5 \beta_{2} - \beta_1) q^{33} + ( - 2 \beta_{3} - 6) q^{34} + ( - \beta_{3} - 8) q^{36} + ( - \beta_{2} - 3 \beta_1) q^{38} - 2 q^{39} + ( - \beta_{3} - 5) q^{41} + ( - 3 \beta_{2} - 2 \beta_1) q^{43} + ( - \beta_{3} + 6) q^{44} + q^{46} + (\beta_{2} + \beta_1) q^{47} - 3 \beta_{2} q^{48} - 2 q^{51} + ( - 2 \beta_{2} - 8 \beta_1) q^{52} + (3 \beta_{2} + \beta_1) q^{53} + ( - \beta_{3} - 3) q^{54} + ( - 3 \beta_{2} - \beta_1) q^{57} + \beta_1 q^{58} + ( - 3 \beta_{3} + 4) q^{59} + (3 \beta_{3} - 3) q^{61} - 6 \beta_1 q^{62} + (\beta_{3} + 7) q^{64} + (\beta_{3} - 2) q^{66} + (6 \beta_{2} + 5 \beta_1) q^{67} + ( - 2 \beta_{2} - 8 \beta_1) q^{68} + ( - \beta_{3} + 3) q^{69} + (3 \beta_{3} - 4) q^{71} + (\beta_{2} - 5 \beta_1) q^{72} - 2 \beta_{2} q^{73} + (\beta_{3} + 8) q^{76} - 2 \beta_1 q^{78} + ( - \beta_{3} - 12) q^{79} + (3 \beta_{3} - 1) q^{81} + ( - \beta_{2} - 8 \beta_1) q^{82} + (4 \beta_{2} - 5 \beta_1) q^{83} + (2 \beta_{3} + 3) q^{86} - \beta_{2} q^{87} + ( - 3 \beta_{2} + \beta_1) q^{88} + (2 \beta_{3} + 3) q^{89} + (2 \beta_{2} + \beta_1) q^{92} + 6 \beta_{2} q^{93} + ( - \beta_{3} - 2) q^{94} + (2 \beta_{3} - 5) q^{96} + (\beta_{2} + 5 \beta_1) q^{97} + (2 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 0.414214i −3.82843 0 −1.00000 0 4.41421i 2.82843 0
99.2 0.414214i 2.41421i 1.82843 0 −1.00000 0 1.58579i −2.82843 0
99.3 0.414214i 2.41421i 1.82843 0 −1.00000 0 1.58579i −2.82843 0
99.4 2.41421i 0.414214i −3.82843 0 −1.00000 0 4.41421i 2.82843 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.2.b.g 4
5.b even 2 1 inner 1225.2.b.g 4
5.c odd 4 1 245.2.a.h 2
5.c odd 4 1 1225.2.a.k 2
7.b odd 2 1 1225.2.b.h 4
7.c even 3 2 175.2.k.a 8
15.e even 4 1 2205.2.a.n 2
20.e even 4 1 3920.2.a.bq 2
35.c odd 2 1 1225.2.b.h 4
35.f even 4 1 245.2.a.g 2
35.f even 4 1 1225.2.a.m 2
35.j even 6 2 175.2.k.a 8
35.k even 12 2 245.2.e.e 4
35.l odd 12 2 35.2.e.a 4
35.l odd 12 2 175.2.e.c 4
105.k odd 4 1 2205.2.a.q 2
105.x even 12 2 315.2.j.e 4
140.j odd 4 1 3920.2.a.bv 2
140.w even 12 2 560.2.q.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 35.l odd 12 2
175.2.e.c 4 35.l odd 12 2
175.2.k.a 8 7.c even 3 2
175.2.k.a 8 35.j even 6 2
245.2.a.g 2 35.f even 4 1
245.2.a.h 2 5.c odd 4 1
245.2.e.e 4 35.k even 12 2
315.2.j.e 4 105.x even 12 2
560.2.q.k 4 140.w even 12 2
1225.2.a.k 2 5.c odd 4 1
1225.2.a.m 2 35.f even 4 1
1225.2.b.g 4 1.a even 1 1 trivial
1225.2.b.g 4 5.b even 2 1 inner
1225.2.b.h 4 7.b odd 2 1
1225.2.b.h 4 35.c odd 2 1
2205.2.a.n 2 15.e even 4 1
2205.2.a.q 2 105.k odd 4 1
3920.2.a.bq 2 20.e even 4 1
3920.2.a.bv 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1225, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 8 \) Copy content Toggle raw display
\( T_{31} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T + 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 10 T + 17)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 63)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 246 T^{2} + 14161 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} + 24 T + 136)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 326 T^{2} + 25921 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 23)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
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