Properties

Label 1225.2.b.k
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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (2 \beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - 2 q^{6} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 4 \beta_{2} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (2 \beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - 2 q^{6} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 4 \beta_{2} - 5) q^{9} + ( - 2 \beta_{2} + 1) q^{11} + (4 \beta_{3} + 2 \beta_1) q^{12} - 2 \beta_1 q^{13} + 3 \beta_{2} q^{16} + 4 \beta_1 q^{17} + ( - 4 \beta_{3} - \beta_1) q^{18} + ( - 4 \beta_{2} - 2) q^{19} + ( - 2 \beta_{3} + 3 \beta_1) q^{22} + (5 \beta_{3} + 2 \beta_1) q^{23} + ( - 2 \beta_{2} - 6) q^{24} + ( - 2 \beta_{2} + 2) q^{26} + ( - 12 \beta_{3} - 4 \beta_1) q^{27} - 5 q^{29} - 6 \beta_{2} q^{31} + (5 \beta_{3} + \beta_1) q^{32} + ( - 2 \beta_{3} + 2 \beta_1) q^{33} + (4 \beta_{2} - 4) q^{34} + ( - 5 \beta_{2} - 9) q^{36} - 3 \beta_{3} q^{37} + ( - 4 \beta_{3} + 2 \beta_1) q^{38} + 4 q^{39} + (2 \beta_{2} - 6) q^{41} + (3 \beta_{3} - 2 \beta_1) q^{43} + (\beta_{2} - 1) q^{44} + ( - 3 \beta_{2} - 2) q^{46} - 2 \beta_{3} q^{47} + 6 \beta_{3} q^{48} - 8 q^{51} - 2 \beta_{3} q^{52} + (6 \beta_{3} + 4 \beta_1) q^{53} + (8 \beta_{2} + 4) q^{54} + ( - 12 \beta_{3} - 4 \beta_1) q^{57} - 5 \beta_1 q^{58} + (6 \beta_{2} + 8) q^{59} + (6 \beta_{2} + 6) q^{61} + ( - 6 \beta_{3} + 6 \beta_1) q^{62} + (2 \beta_{2} - 1) q^{64} + (4 \beta_{2} - 2) q^{66} + (3 \beta_{3} + 2 \beta_1) q^{67} + 4 \beta_{3} q^{68} + ( - 10 \beta_{2} - 14) q^{69} + (6 \beta_{2} + 5) q^{71} + ( - 13 \beta_{3} - 6 \beta_1) q^{72} + (10 \beta_{3} - 2 \beta_1) q^{73} + 3 \beta_{2} q^{74} + ( - 2 \beta_{2} - 6) q^{76} + 4 \beta_1 q^{78} + (10 \beta_{2} + 5) q^{79} + (12 \beta_{2} + 17) q^{81} + (2 \beta_{3} - 8 \beta_1) q^{82} + (4 \beta_{3} + 6 \beta_1) q^{83} + ( - 5 \beta_{2} + 2) q^{86} + ( - 10 \beta_{3} - 10 \beta_1) q^{87} + ( - 3 \beta_{3} + 4 \beta_1) q^{88} + (2 \beta_{2} + 16) q^{89} + (7 \beta_{3} + 5 \beta_1) q^{92} - 12 \beta_{3} q^{93} + 2 \beta_{2} q^{94} + ( - 10 \beta_{2} - 12) q^{96} + (4 \beta_{3} + 2 \beta_1) q^{97} + ( - 2 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{16} - 20 q^{24} + 12 q^{26} - 20 q^{29} + 12 q^{31} - 24 q^{34} - 26 q^{36} + 16 q^{39} - 28 q^{41} - 6 q^{44} - 2 q^{46} - 32 q^{51} + 20 q^{59} + 12 q^{61} - 8 q^{64} - 16 q^{66} - 36 q^{69} + 8 q^{71} - 6 q^{74} - 20 q^{76} + 44 q^{81} + 18 q^{86} + 60 q^{89} - 4 q^{94} - 28 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 1.23607i −0.618034 0 −2.00000 0 2.23607i 1.47214 0
99.2 0.618034i 3.23607i 1.61803 0 −2.00000 0 2.23607i −7.47214 0
99.3 0.618034i 3.23607i 1.61803 0 −2.00000 0 2.23607i −7.47214 0
99.4 1.61803i 1.23607i −0.618034 0 −2.00000 0 2.23607i 1.47214 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.k 4
5.b even 2 1 inner 1225.2.b.k 4
5.c odd 4 1 1225.2.a.n 2
5.c odd 4 1 1225.2.a.u 2
7.b odd 2 1 175.2.b.c 4
21.c even 2 1 1575.2.d.k 4
28.d even 2 1 2800.2.g.s 4
35.c odd 2 1 175.2.b.c 4
35.f even 4 1 175.2.a.d 2
35.f even 4 1 175.2.a.e yes 2
105.g even 2 1 1575.2.d.k 4
105.k odd 4 1 1575.2.a.n 2
105.k odd 4 1 1575.2.a.s 2
140.c even 2 1 2800.2.g.s 4
140.j odd 4 1 2800.2.a.bh 2
140.j odd 4 1 2800.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 35.f even 4 1
175.2.a.e yes 2 35.f even 4 1
175.2.b.c 4 7.b odd 2 1
175.2.b.c 4 35.c odd 2 1
1225.2.a.n 2 5.c odd 4 1
1225.2.a.u 2 5.c odd 4 1
1225.2.b.k 4 1.a even 1 1 trivial
1225.2.b.k 4 5.b even 2 1 inner
1575.2.a.n 2 105.k odd 4 1
1575.2.a.s 2 105.k odd 4 1
1575.2.d.k 4 21.c even 2 1
1575.2.d.k 4 105.g even 2 1
2800.2.a.bh 2 140.j odd 4 1
2800.2.a.bp 2 140.j odd 4 1
2800.2.g.s 4 28.d even 2 1
2800.2.g.s 4 140.c even 2 1

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 20 \) Copy content Toggle raw display
\( T_{31}^{2} - 6T_{31} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 42T^{2} + 121 \) Copy content Toggle raw display
$29$ \( (T + 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 14 T + 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 42T^{2} + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 252 T^{2} + 13456 \) Copy content Toggle raw display
$79$ \( (T^{2} - 125)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} - 30 T + 220)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
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