Properties

Label 1225.2.b.c
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} + 4 q^{11} - q^{16} + 3 i q^{18} + 4 i q^{22} - 8 i q^{23} - 2 q^{29} + 5 i q^{32} + 3 q^{36} - 6 i q^{37} + 12 i q^{43} + 4 q^{44} + 8 q^{46} + 10 i q^{53} - 2 i q^{58} - 7 q^{64} + 4 i q^{67} + 16 q^{71} + 9 i q^{72} + 6 q^{74} - 8 q^{79} + 9 q^{81} - 12 q^{86} + 12 i q^{88} - 8 i q^{92} + 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 0
99.2 1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
35.c odd 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.c 2
5.b even 2 1 inner 1225.2.b.c 2
5.c odd 4 1 49.2.a.a 1
5.c odd 4 1 1225.2.a.c 1
7.b odd 2 1 CM 1225.2.b.c 2
15.e even 4 1 441.2.a.c 1
20.e even 4 1 784.2.a.f 1
35.c odd 2 1 inner 1225.2.b.c 2
35.f even 4 1 49.2.a.a 1
35.f even 4 1 1225.2.a.c 1
35.k even 12 2 49.2.c.a 2
35.l odd 12 2 49.2.c.a 2
40.i odd 4 1 3136.2.a.n 1
40.k even 4 1 3136.2.a.o 1
55.e even 4 1 5929.2.a.c 1
60.l odd 4 1 7056.2.a.bg 1
65.h odd 4 1 8281.2.a.d 1
105.k odd 4 1 441.2.a.c 1
105.w odd 12 2 441.2.e.d 2
105.x even 12 2 441.2.e.d 2
140.j odd 4 1 784.2.a.f 1
140.w even 12 2 784.2.i.f 2
140.x odd 12 2 784.2.i.f 2
280.s even 4 1 3136.2.a.n 1
280.y odd 4 1 3136.2.a.o 1
385.l odd 4 1 5929.2.a.c 1
420.w even 4 1 7056.2.a.bg 1
455.s even 4 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 5.c odd 4 1
49.2.a.a 1 35.f even 4 1
49.2.c.a 2 35.k even 12 2
49.2.c.a 2 35.l odd 12 2
441.2.a.c 1 15.e even 4 1
441.2.a.c 1 105.k odd 4 1
441.2.e.d 2 105.w odd 12 2
441.2.e.d 2 105.x even 12 2
784.2.a.f 1 20.e even 4 1
784.2.a.f 1 140.j odd 4 1
784.2.i.f 2 140.w even 12 2
784.2.i.f 2 140.x odd 12 2
1225.2.a.c 1 5.c odd 4 1
1225.2.a.c 1 35.f even 4 1
1225.2.b.c 2 1.a even 1 1 trivial
1225.2.b.c 2 5.b even 2 1 inner
1225.2.b.c 2 7.b odd 2 1 CM
1225.2.b.c 2 35.c odd 2 1 inner
3136.2.a.n 1 40.i odd 4 1
3136.2.a.n 1 280.s even 4 1
3136.2.a.o 1 40.k even 4 1
3136.2.a.o 1 280.y odd 4 1
5929.2.a.c 1 55.e even 4 1
5929.2.a.c 1 385.l odd 4 1
7056.2.a.bg 1 60.l odd 4 1
7056.2.a.bg 1 420.w even 4 1
8281.2.a.d 1 65.h odd 4 1
8281.2.a.d 1 455.s even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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