Properties

Label 950.2.j.d
Level $950$
Weight $2$
Character orbit 950.j
Analytic conductor $7.586$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(49,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.j (of order \(6\), degree \(2\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{2} q^{9} + 5 q^{11} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{13} + ( - \zeta_{12}^{2} + 1) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + 3 \zeta_{12}^{3} q^{18} + ( - 5 \zeta_{12}^{2} + 3) q^{19} - 5 \zeta_{12} q^{22} + (\zeta_{12}^{3} - \zeta_{12}) q^{23} + 2 q^{26} + (\zeta_{12}^{3} - \zeta_{12}) q^{28} + 6 \zeta_{12}^{2} q^{29} + 4 q^{31} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{32} + ( - 3 \zeta_{12}^{2} + 3) q^{36} - 11 \zeta_{12}^{3} q^{37} + (5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{38} + ( - 9 \zeta_{12}^{2} + 9) q^{41} + 6 \zeta_{12} q^{43} + 5 \zeta_{12}^{2} q^{44} + q^{46} + 6 q^{49} - 2 \zeta_{12} q^{52} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{53} + q^{56} - 6 \zeta_{12}^{3} q^{58} - 4 \zeta_{12} q^{62} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} - q^{64} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{67} + (6 \zeta_{12}^{2} - 6) q^{71} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{72} + 14 \zeta_{12} q^{73} + (11 \zeta_{12}^{2} - 11) q^{74} + ( - 2 \zeta_{12}^{2} + 5) q^{76} + 5 \zeta_{12}^{3} q^{77} + ( - 10 \zeta_{12}^{2} + 10) q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{82} - 14 \zeta_{12}^{3} q^{83} - 6 \zeta_{12}^{2} q^{86} - 5 \zeta_{12}^{3} q^{88} - 7 \zeta_{12}^{2} q^{89} - 2 \zeta_{12}^{2} q^{91} - \zeta_{12} q^{92} + 2 \zeta_{12} q^{97} - 6 \zeta_{12} q^{98} - 15 \zeta_{12}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 6 q^{9} + 20 q^{11} + 2 q^{14} - 2 q^{16} + 2 q^{19} + 8 q^{26} + 12 q^{29} + 16 q^{31} + 6 q^{36} + 18 q^{41} + 10 q^{44} + 4 q^{46} + 24 q^{49} + 4 q^{56} - 4 q^{64} - 12 q^{71} - 22 q^{74} + 16 q^{76} + 20 q^{79} - 18 q^{81} - 12 q^{86} - 14 q^{89} - 4 q^{91} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
49.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
349.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 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
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.d 4
5.b even 2 1 inner 950.2.j.d 4
5.c odd 4 1 190.2.e.a 2
5.c odd 4 1 950.2.e.f 2
15.e even 4 1 1710.2.l.h 2
19.c even 3 1 inner 950.2.j.d 4
20.e even 4 1 1520.2.q.f 2
95.i even 6 1 inner 950.2.j.d 4
95.l even 12 1 3610.2.a.c 1
95.m odd 12 1 190.2.e.a 2
95.m odd 12 1 950.2.e.f 2
95.m odd 12 1 3610.2.a.g 1
285.v even 12 1 1710.2.l.h 2
380.v even 12 1 1520.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.a 2 5.c odd 4 1
190.2.e.a 2 95.m odd 12 1
950.2.e.f 2 5.c odd 4 1
950.2.e.f 2 95.m odd 12 1
950.2.j.d 4 1.a even 1 1 trivial
950.2.j.d 4 5.b even 2 1 inner
950.2.j.d 4 19.c even 3 1 inner
950.2.j.d 4 95.i even 6 1 inner
1520.2.q.f 2 20.e even 4 1
1520.2.q.f 2 380.v even 12 1
1710.2.l.h 2 15.e even 4 1
1710.2.l.h 2 285.v even 12 1
3610.2.a.c 1 95.l even 12 1
3610.2.a.g 1 95.m odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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