Properties

Label 2450.2.c.p
Level $2450$
Weight $2$
Character orbit 2450.c
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2450,2,Mod(99,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, 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("2450.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (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: \(19.5633484952\)
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 70)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

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

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 2.00000i −1.00000 0 2.00000 0 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 2.00000 0 1.00000i −1.00000 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
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 2450.2.c.p 2
5.b even 2 1 inner 2450.2.c.p 2
5.c odd 4 1 490.2.a.j 1
5.c odd 4 1 2450.2.a.f 1
7.b odd 2 1 2450.2.c.f 2
7.d odd 6 2 350.2.j.a 4
15.e even 4 1 4410.2.a.c 1
20.e even 4 1 3920.2.a.g 1
35.c odd 2 1 2450.2.c.f 2
35.f even 4 1 490.2.a.g 1
35.f even 4 1 2450.2.a.p 1
35.i odd 6 2 350.2.j.a 4
35.k even 12 2 70.2.e.b 2
35.k even 12 2 350.2.e.h 2
35.l odd 12 2 490.2.e.a 2
105.k odd 4 1 4410.2.a.m 1
105.w odd 12 2 630.2.k.e 2
140.j odd 4 1 3920.2.a.be 1
140.x odd 12 2 560.2.q.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 35.k even 12 2
350.2.e.h 2 35.k even 12 2
350.2.j.a 4 7.d odd 6 2
350.2.j.a 4 35.i odd 6 2
490.2.a.g 1 35.f even 4 1
490.2.a.j 1 5.c odd 4 1
490.2.e.a 2 35.l odd 12 2
560.2.q.d 2 140.x odd 12 2
630.2.k.e 2 105.w odd 12 2
2450.2.a.f 1 5.c odd 4 1
2450.2.a.p 1 35.f even 4 1
2450.2.c.f 2 7.b odd 2 1
2450.2.c.f 2 35.c odd 2 1
2450.2.c.p 2 1.a even 1 1 trivial
2450.2.c.p 2 5.b even 2 1 inner
3920.2.a.g 1 20.e even 4 1
3920.2.a.be 1 140.j odd 4 1
4410.2.a.c 1 15.e even 4 1
4410.2.a.m 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2450, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display
\( T_{19} + 1 \) Copy content Toggle raw display
\( T_{31} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 121 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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