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Properties

Label 1950.2.i.y
Level $1950$
Weight $2$
Character orbit 1950.i
Analytic conductor $15.571$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{2} - 1) q^{2} + (\zeta_{12}^{2} - 1) q^{3} - \zeta_{12}^{2} q^{4} - \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{7} + q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{2} - 1) q^{2} + (\zeta_{12}^{2} - 1) q^{3} - \zeta_{12}^{2} q^{4} - \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{7} + q^{8} - \zeta_{12}^{2} q^{9} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots + 1) q^{11} + \cdots + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 2 q^{11} + 4 q^{12} - 6 q^{13} + 4 q^{14} - 2 q^{16} - 2 q^{17} + 4 q^{18} - 6 q^{19} + 4 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} - 6 q^{26} + 4 q^{27} - 2 q^{28} + 6 q^{29} + 16 q^{31} - 2 q^{32} + 2 q^{33} + 4 q^{34} - 2 q^{36} - 2 q^{37} + 12 q^{38} - 6 q^{39} + 8 q^{41} - 2 q^{42} + 10 q^{43} - 4 q^{44} - 2 q^{46} - 4 q^{47} - 2 q^{48} + 6 q^{49} + 4 q^{51} + 12 q^{52} + 24 q^{53} - 2 q^{54} - 2 q^{56} + 12 q^{57} + 6 q^{58} + 4 q^{59} - 4 q^{61} - 8 q^{62} - 2 q^{63} + 4 q^{64} - 4 q^{66} + 2 q^{67} - 2 q^{68} - 2 q^{69} - 6 q^{71} - 2 q^{72} + 16 q^{73} - 2 q^{74} - 6 q^{76} + 8 q^{77} + 12 q^{78} + 48 q^{79} - 2 q^{81} + 8 q^{82} + 28 q^{83} - 2 q^{84} - 20 q^{86} + 6 q^{87} + 2 q^{88} - 4 q^{89} + 12 q^{91} + 4 q^{92} - 8 q^{93} + 2 q^{94} + 4 q^{96} + 20 q^{97} + 6 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.36603 − 2.36603i 1.00000 −0.500000 − 0.866025i 0
451.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.366025 + 0.633975i 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 −1.36603 + 2.36603i 1.00000 −0.500000 + 0.866025i 0
601.2 −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.366025 − 0.633975i 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.y 4
5.b even 2 1 1950.2.i.bh 4
5.c odd 4 1 390.2.y.b ✓ 4
5.c odd 4 1 390.2.y.c yes 4
13.c even 3 1 inner 1950.2.i.y 4
15.e even 4 1 1170.2.bp.d 4
15.e even 4 1 1170.2.bp.e 4
65.n even 6 1 1950.2.i.bh 4
65.q odd 12 1 390.2.y.b ✓ 4
65.q odd 12 1 390.2.y.c yes 4
195.bl even 12 1 1170.2.bp.d 4
195.bl even 12 1 1170.2.bp.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.b ✓ 4 5.c odd 4 1
390.2.y.b ✓ 4 65.q odd 12 1
390.2.y.c yes 4 5.c odd 4 1
390.2.y.c yes 4 65.q odd 12 1
1170.2.bp.d 4 15.e even 4 1
1170.2.bp.d 4 195.bl even 12 1
1170.2.bp.e 4 15.e even 4 1
1170.2.bp.e 4 195.bl even 12 1
1950.2.i.y 4 1.a even 1 1 trivial
1950.2.i.y 4 13.c even 3 1 inner
1950.2.i.bh 4 5.b even 2 1
1950.2.i.bh 4 65.n even 6 1

Hecke kernels

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

\( T_{7}^{4} + 2T_{7}^{3} + 6T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} + 15T_{17}^{2} - 22T_{17} + 121 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} + 30T_{19}^{2} + 36T_{19} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T - 74)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 33)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 20449 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 59)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14 T + 46)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
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