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Properties

Label 1950.2.i.p
Level $1950$
Weight $2$
Character orbit 1950.i
Analytic conductor $15.571$
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] = [1950,2,Mod(451,1950)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.451");
 
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.i (of order \(3\), degree \(2\), 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: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + q^{12} + (4 \zeta_{6} - 3) q^{13} + (\zeta_{6} - 1) q^{16} - 6 \zeta_{6} q^{17} - q^{18} + 3 \zeta_{6} q^{22} + ( - 9 \zeta_{6} + 9) q^{23} + ( - \zeta_{6} + 1) q^{24} + (3 \zeta_{6} + 1) q^{26} + q^{27} + ( - 7 \zeta_{6} + 7) q^{29} + q^{31} + \zeta_{6} q^{32} - 3 \zeta_{6} q^{33} - 6 q^{34} + (\zeta_{6} - 1) q^{36} + ( - \zeta_{6} + 1) q^{37} + ( - 3 \zeta_{6} - 1) q^{39} - 13 \zeta_{6} q^{43} + 3 q^{44} - 9 \zeta_{6} q^{46} - 11 q^{47} - \zeta_{6} q^{48} + ( - 7 \zeta_{6} + 7) q^{49} + 6 q^{51} + ( - \zeta_{6} + 4) q^{52} + 6 q^{53} + ( - \zeta_{6} + 1) q^{54} - 7 \zeta_{6} q^{58} + 11 \zeta_{6} q^{59} + ( - \zeta_{6} + 1) q^{62} + q^{64} - 3 q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + (6 \zeta_{6} - 6) q^{68} + 9 \zeta_{6} q^{69} - 4 \zeta_{6} q^{71} + \zeta_{6} q^{72} - \zeta_{6} q^{74} + (\zeta_{6} - 4) q^{78} + q^{79} + (\zeta_{6} - 1) q^{81} - 4 q^{83} - 13 q^{86} + 7 \zeta_{6} q^{87} + ( - 3 \zeta_{6} + 3) q^{88} + ( - 4 \zeta_{6} + 4) q^{89} - 9 q^{92} + (\zeta_{6} - 1) q^{93} + (11 \zeta_{6} - 11) q^{94} - q^{96} - 4 \zeta_{6} q^{97} - 7 \zeta_{6} q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{8} - q^{9} - 3 q^{11} + 2 q^{12} - 2 q^{13} - q^{16} - 6 q^{17} - 2 q^{18} + 3 q^{22} + 9 q^{23} + q^{24} + 5 q^{26} + 2 q^{27} + 7 q^{29} + 2 q^{31} + q^{32} - 3 q^{33} - 12 q^{34} - q^{36} + q^{37} - 5 q^{39} - 13 q^{43} + 6 q^{44} - 9 q^{46} - 22 q^{47} - q^{48} + 7 q^{49} + 12 q^{51} + 7 q^{52} + 12 q^{53} + q^{54} - 7 q^{58} + 11 q^{59} + q^{62} + 2 q^{64} - 6 q^{66} + 4 q^{67} - 6 q^{68} + 9 q^{69} - 4 q^{71} + q^{72} - q^{74} - 7 q^{78} + 2 q^{79} - q^{81} - 8 q^{83} - 26 q^{86} + 7 q^{87} + 3 q^{88} + 4 q^{89} - 18 q^{92} - q^{93} - 11 q^{94} - 2 q^{96} - 4 q^{97} - 7 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 −1.00000 −0.500000 − 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 −1.00000 −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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.p 2
5.b even 2 1 1950.2.i.l 2
5.c odd 4 2 390.2.y.d ✓ 4
13.c even 3 1 inner 1950.2.i.p 2
15.e even 4 2 1170.2.bp.c 4
65.n even 6 1 1950.2.i.l 2
65.q odd 12 2 390.2.y.d ✓ 4
195.bl even 12 2 1170.2.bp.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.d ✓ 4 5.c odd 4 2
390.2.y.d ✓ 4 65.q odd 12 2
1170.2.bp.c 4 15.e even 4 2
1170.2.bp.c 4 195.bl even 12 2
1950.2.i.l 2 5.b even 2 1
1950.2.i.l 2 65.n even 6 1
1950.2.i.p 2 1.a even 1 1 trivial
1950.2.i.p 2 13.c even 3 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
\( T_{17}^{2} + 6T_{17} + 36 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$47$ \( (T + 11)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
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