Properties

Label 1950.2.e.l
Level $1950$
Weight $2$
Character orbit 1950.e
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1950,2,Mod(1249,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1950.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (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: \(15.5708283941\)
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 390)
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.

\(f(q)\) \(=\) \( q + i q^{2} - i q^{3} - q^{4} + q^{6} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - i q^{3} - q^{4} + q^{6} - i q^{8} - q^{9} + i q^{12} - i q^{13} + q^{16} + 6 i q^{17} - i q^{18} - 4 i q^{23} - q^{24} + q^{26} + i q^{27} + 10 q^{29} + i q^{32} - 6 q^{34} + q^{36} + 6 i q^{37} - q^{39} + 2 q^{41} - 4 i q^{43} + 4 q^{46} - i q^{48} + 7 q^{49} + 6 q^{51} + i q^{52} - 6 i q^{53} - q^{54} + 10 i q^{58} + 6 q^{61} - q^{64} - 4 i q^{67} - 6 i q^{68} - 4 q^{69} + 16 q^{71} + i q^{72} - 2 i q^{73} - 6 q^{74} - i q^{78} + q^{81} + 2 i q^{82} + 4 i q^{83} + 4 q^{86} - 10 i q^{87} + 6 q^{89} + 4 i q^{92} + q^{96} - 14 i q^{97} + 7 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{16} - 2 q^{24} + 2 q^{26} + 20 q^{29} - 12 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 8 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{69} + 32 q^{71} - 12 q^{74} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 0 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 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 1950.2.e.l 2
3.b odd 2 1 5850.2.e.p 2
5.b even 2 1 inner 1950.2.e.l 2
5.c odd 4 1 390.2.a.a 1
5.c odd 4 1 1950.2.a.y 1
15.d odd 2 1 5850.2.e.p 2
15.e even 4 1 1170.2.a.m 1
15.e even 4 1 5850.2.a.m 1
20.e even 4 1 3120.2.a.q 1
60.l odd 4 1 9360.2.a.bn 1
65.f even 4 1 5070.2.b.c 2
65.h odd 4 1 5070.2.a.s 1
65.k even 4 1 5070.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.a 1 5.c odd 4 1
1170.2.a.m 1 15.e even 4 1
1950.2.a.y 1 5.c odd 4 1
1950.2.e.l 2 1.a even 1 1 trivial
1950.2.e.l 2 5.b even 2 1 inner
3120.2.a.q 1 20.e even 4 1
5070.2.a.s 1 65.h odd 4 1
5070.2.b.c 2 65.f even 4 1
5070.2.b.c 2 65.k even 4 1
5850.2.a.m 1 15.e even 4 1
5850.2.e.p 2 3.b odd 2 1
5850.2.e.p 2 15.d odd 2 1
9360.2.a.bn 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) 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
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) 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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