Properties

Label 1950.2.z.a
Level $1950$
Weight $2$
Character orbit 1950.z
Analytic conductor $15.571$
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] = [1950,2,Mod(1699,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1950.1699");
 
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.z (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: \(15.5708283941\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} - \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} - \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 3 \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{12} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}) q^{13} - \zeta_{12}^{2} q^{16} - 6 \zeta_{12} q^{17} - \zeta_{12}^{3} q^{18} + (6 \zeta_{12}^{2} - 6) q^{19} - 3 \zeta_{12} q^{22} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{23} + \zeta_{12}^{2} q^{24} + ( - 4 \zeta_{12}^{2} + 3) q^{26} + \zeta_{12}^{3} q^{27} + 2 \zeta_{12}^{2} q^{29} - 6 q^{31} - \zeta_{12} q^{32} + 3 \zeta_{12} q^{33} - 6 q^{34} - \zeta_{12}^{2} q^{36} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{37} + 6 \zeta_{12}^{3} q^{38} + (4 \zeta_{12}^{2} - 3) q^{39} + 10 \zeta_{12}^{2} q^{41} - 8 \zeta_{12} q^{43} - 3 q^{44} + ( - 5 \zeta_{12}^{2} + 5) q^{46} - 12 \zeta_{12}^{3} q^{47} + \zeta_{12} q^{48} - 7 \zeta_{12}^{2} q^{49} + 6 q^{51} + ( - 3 \zeta_{12}^{3} - \zeta_{12}) q^{52} - 12 \zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} - 6 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} + ( - 15 \zeta_{12}^{2} + 15) q^{61} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{62} - q^{64} + 3 q^{66} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{67} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{68} + (5 \zeta_{12}^{2} - 5) q^{69} + (7 \zeta_{12}^{2} - 7) q^{71} - \zeta_{12} q^{72} + \zeta_{12}^{3} q^{73} + (9 \zeta_{12}^{2} - 9) q^{74} + 6 \zeta_{12}^{2} q^{76} + (3 \zeta_{12}^{3} + \zeta_{12}) q^{78} - 6 q^{79} - \zeta_{12}^{2} q^{81} + 10 \zeta_{12} q^{82} - 5 \zeta_{12}^{3} q^{83} - 8 q^{86} - 2 \zeta_{12} q^{87} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{88} - 6 \zeta_{12}^{2} q^{89} - 5 \zeta_{12}^{3} q^{92} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{93} - 12 \zeta_{12}^{2} q^{94} + q^{96} + 5 \zeta_{12} q^{97} - 7 \zeta_{12} q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9} - 6 q^{11} - 2 q^{16} - 12 q^{19} + 2 q^{24} + 4 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 2 q^{36} - 4 q^{39} + 20 q^{41} - 12 q^{44} + 10 q^{46} - 14 q^{49} + 24 q^{51} + 2 q^{54} + 30 q^{61} - 4 q^{64} + 12 q^{66} - 10 q^{69} - 14 q^{71} - 18 q^{74} + 12 q^{76} - 24 q^{79} - 2 q^{81} - 32 q^{86} - 12 q^{89} - 24 q^{94} + 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(-1\)

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
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
1849.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 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
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.z.a 4
5.b even 2 1 inner 1950.2.z.a 4
5.c odd 4 1 1950.2.i.e 2
5.c odd 4 1 1950.2.i.v yes 2
13.c even 3 1 inner 1950.2.z.a 4
65.n even 6 1 inner 1950.2.z.a 4
65.q odd 12 1 1950.2.i.e 2
65.q odd 12 1 1950.2.i.v yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.e 2 5.c odd 4 1
1950.2.i.e 2 65.q odd 12 1
1950.2.i.v yes 2 5.c odd 4 1
1950.2.i.v yes 2 65.q odd 12 1
1950.2.z.a 4 1.a even 1 1 trivial
1950.2.z.a 4 5.b even 2 1 inner
1950.2.z.a 4 13.c even 3 1 inner
1950.2.z.a 4 65.n even 6 1 inner

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
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17}^{4} - 36T_{17}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$41$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 15 T + 225)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
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