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Properties

Label 2550.2.f.p
Level $2550$
Weight $2$
Character orbit 2550.f
Analytic conductor $20.362$
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] = [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: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 510)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} - q^{4} - \beta_1 q^{6} + (\beta_{3} + 1) q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} - q^{4} - \beta_1 q^{6} + (\beta_{3} + 1) q^{7} - \beta_1 q^{8} + q^{9} + q^{12} + ( - \beta_{2} + 3 \beta_1) q^{13} + (\beta_{2} + \beta_1) q^{14} + q^{16} + (\beta_{2} - 2) q^{17} + \beta_1 q^{18} + ( - \beta_{3} - 1) q^{21} - 2 q^{23} + \beta_1 q^{24} + (\beta_{3} - 3) q^{26} - q^{27} + ( - \beta_{3} - 1) q^{28} + 2 \beta_{2} q^{29} + 2 \beta_1 q^{31} + \beta_1 q^{32} + ( - \beta_{3} - 2 \beta_1) q^{34} - q^{36} + (2 \beta_{3} + 4) q^{37} + (\beta_{2} - 3 \beta_1) q^{39} + ( - \beta_{2} + 5 \beta_1) q^{41} + ( - \beta_{2} - \beta_1) q^{42} + ( - \beta_{2} + \beta_1) q^{43} - 2 \beta_1 q^{46} - 4 \beta_1 q^{47} - q^{48} + (2 \beta_{3} + 7) q^{49} + ( - \beta_{2} + 2) q^{51} + (\beta_{2} - 3 \beta_1) q^{52} - 2 \beta_{2} q^{53} - \beta_1 q^{54} + ( - \beta_{2} - \beta_1) q^{56} - 2 \beta_{3} q^{58} + (\beta_{3} + 7) q^{59} + (2 \beta_{2} - 4 \beta_1) q^{61} - 2 q^{62} + (\beta_{3} + 1) q^{63} - q^{64} + ( - \beta_{2} - 11 \beta_1) q^{67} + ( - \beta_{2} + 2) q^{68} + 2 q^{69} + (3 \beta_{2} + 3 \beta_1) q^{71} - \beta_1 q^{72} + ( - \beta_{3} - 3) q^{73} + (2 \beta_{2} + 4 \beta_1) q^{74} + ( - \beta_{3} + 3) q^{78} + 2 \beta_{2} q^{79} + q^{81} + (\beta_{3} - 5) q^{82} + (2 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{3} + 1) q^{84} + (\beta_{3} - 1) q^{86} - 2 \beta_{2} q^{87} + ( - 2 \beta_{3} + 8) q^{89} + (2 \beta_{2} - 10 \beta_1) q^{91} + 2 q^{92} - 2 \beta_1 q^{93} + 4 q^{94} - \beta_1 q^{96} + ( - \beta_{3} + 1) q^{97} + (2 \beta_{2} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{7} + 4 q^{9} + 4 q^{12} + 4 q^{16} - 8 q^{17} - 4 q^{21} - 8 q^{23} - 12 q^{26} - 4 q^{27} - 4 q^{28} - 4 q^{36} + 16 q^{37} - 4 q^{48} + 28 q^{49} + 8 q^{51} + 28 q^{59} - 8 q^{62} + 4 q^{63} - 4 q^{64} + 8 q^{68} + 8 q^{69} - 12 q^{73} + 12 q^{78} + 4 q^{81} - 20 q^{82} + 4 q^{84} - 4 q^{86} + 32 q^{89} + 8 q^{92} + 16 q^{94} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.30278i
− 1.30278i
− 2.30278i
1.30278i
− 1.00000i −1.00000 −1.00000 0 1.00000i −2.60555 1.00000i 1.00000 0
1699.2 − 1.00000i −1.00000 −1.00000 0 1.00000i 4.60555 1.00000i 1.00000 0
1699.3 1.00000i −1.00000 −1.00000 0 − 1.00000i −2.60555 − 1.00000i 1.00000 0
1699.4 1.00000i −1.00000 −1.00000 0 − 1.00000i 4.60555 − 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.p 4
5.b even 2 1 2550.2.f.s 4
5.c odd 4 1 510.2.c.c ✓ 4
5.c odd 4 1 2550.2.c.o 4
15.e even 4 1 1530.2.c.g 4
17.b even 2 1 2550.2.f.s 4
20.e even 4 1 4080.2.h.q 4
85.c even 2 1 inner 2550.2.f.p 4
85.f odd 4 1 8670.2.a.bk 2
85.g odd 4 1 510.2.c.c ✓ 4
85.g odd 4 1 2550.2.c.o 4
85.i odd 4 1 8670.2.a.bh 2
255.o even 4 1 1530.2.c.g 4
340.r even 4 1 4080.2.h.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.c.c ✓ 4 5.c odd 4 1
510.2.c.c ✓ 4 85.g odd 4 1
1530.2.c.g 4 15.e even 4 1
1530.2.c.g 4 255.o even 4 1
2550.2.c.o 4 5.c odd 4 1
2550.2.c.o 4 85.g odd 4 1
2550.2.f.p 4 1.a even 1 1 trivial
2550.2.f.p 4 85.c even 2 1 inner
2550.2.f.s 4 5.b even 2 1
2550.2.f.s 4 17.b even 2 1
4080.2.h.q 4 20.e even 4 1
4080.2.h.q 4 340.r even 4 1
8670.2.a.bh 2 85.i odd 4 1
8670.2.a.bk 2 85.f odd 4 1

Hecke kernels

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

\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{4} + 44T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T + 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
$43$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 14 T + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} + 268 T^{2} + 11664 \) Copy content Toggle raw display
$71$ \( T^{4} + 252 T^{2} + 11664 \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
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