Properties

Label 2550.2.f.e
Level $2550$
Weight $2$
Character orbit 2550.f
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
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] = [2550,2,Mod(1699,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2550.1699");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.f (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: \(20.3618525154\)
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 102)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} - q^{4} - i q^{6} + 2 q^{7} - i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{3} - q^{4} - i q^{6} + 2 q^{7} - i q^{8} + q^{9} + q^{12} + 6 i q^{13} + 2 i q^{14} + q^{16} + ( - i - 4) q^{17} + i q^{18} - 2 q^{21} + 6 q^{23} + i q^{24} - 6 q^{26} - q^{27} - 2 q^{28} + 6 i q^{29} - 10 i q^{31} + i q^{32} + ( - 4 i + 1) q^{34} - q^{36} + 2 q^{37} - 6 i q^{39} - 2 i q^{42} - 4 i q^{43} + 6 i q^{46} + 8 i q^{47} - q^{48} - 3 q^{49} + (i + 4) q^{51} - 6 i q^{52} + 6 i q^{53} - i q^{54} - 2 i q^{56} - 6 q^{58} + 10 i q^{61} + 10 q^{62} + 2 q^{63} - q^{64} + 8 i q^{67} + (i + 4) q^{68} - 6 q^{69} - 10 i q^{71} - i q^{72} + 16 q^{73} + 2 i q^{74} + 6 q^{78} + 6 i q^{79} + q^{81} + 16 i q^{83} + 2 q^{84} + 4 q^{86} - 6 i q^{87} - 10 q^{89} + 12 i q^{91} - 6 q^{92} + 10 i q^{93} - 8 q^{94} - i q^{96} + 12 q^{97} - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{7} + 2 q^{9} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 4 q^{21} + 12 q^{23} - 12 q^{26} - 2 q^{27} - 4 q^{28} + 2 q^{34} - 2 q^{36} + 4 q^{37} - 2 q^{48} - 6 q^{49} + 8 q^{51} - 12 q^{58} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 8 q^{68} - 12 q^{69} + 32 q^{73} + 12 q^{78} + 2 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} - 12 q^{92} - 16 q^{94} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 0 1.00000i 2.00000 1.00000i 1.00000 0
1699.2 1.00000i −1.00000 −1.00000 0 1.00000i 2.00000 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
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2550.2.f.e 2
5.b even 2 1 2550.2.f.j 2
5.c odd 4 1 102.2.b.a 2
5.c odd 4 1 2550.2.c.f 2
15.e even 4 1 306.2.b.a 2
17.b even 2 1 2550.2.f.j 2
20.e even 4 1 816.2.c.a 2
40.i odd 4 1 3264.2.c.j 2
40.k even 4 1 3264.2.c.i 2
60.l odd 4 1 2448.2.c.a 2
85.c even 2 1 inner 2550.2.f.e 2
85.f odd 4 1 1734.2.a.e 1
85.g odd 4 1 102.2.b.a 2
85.g odd 4 1 2550.2.c.f 2
85.i odd 4 1 1734.2.a.d 1
85.k odd 8 2 1734.2.f.h 4
85.n odd 8 2 1734.2.f.h 4
255.k even 4 1 5202.2.a.n 1
255.o even 4 1 306.2.b.a 2
255.r even 4 1 5202.2.a.h 1
340.r even 4 1 816.2.c.a 2
680.u even 4 1 3264.2.c.i 2
680.bi odd 4 1 3264.2.c.j 2
1020.x odd 4 1 2448.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.b.a 2 5.c odd 4 1
102.2.b.a 2 85.g odd 4 1
306.2.b.a 2 15.e even 4 1
306.2.b.a 2 255.o even 4 1
816.2.c.a 2 20.e even 4 1
816.2.c.a 2 340.r even 4 1
1734.2.a.d 1 85.i odd 4 1
1734.2.a.e 1 85.f odd 4 1
1734.2.f.h 4 85.k odd 8 2
1734.2.f.h 4 85.n odd 8 2
2448.2.c.a 2 60.l odd 4 1
2448.2.c.a 2 1020.x odd 4 1
2550.2.c.f 2 5.c odd 4 1
2550.2.c.f 2 85.g odd 4 1
2550.2.f.e 2 1.a even 1 1 trivial
2550.2.f.e 2 85.c even 2 1 inner
2550.2.f.j 2 5.b even 2 1
2550.2.f.j 2 17.b even 2 1
3264.2.c.i 2 40.k even 4 1
3264.2.c.i 2 680.u even 4 1
3264.2.c.j 2 40.i odd 4 1
3264.2.c.j 2 680.bi odd 4 1
5202.2.a.h 1 255.r even 4 1
5202.2.a.n 1 255.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2550, [\chi])\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{23} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 100 \) Copy content Toggle raw display
$73$ \( (T - 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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