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Properties

Label 1950.2.i.ba
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: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{2} + 1) q^{6} + \beta_1 q^{7} + q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{2} + 1) q^{6} + \beta_1 q^{7} + q^{8} + ( - \beta_{2} - 1) q^{9} + 3 \beta_{2} q^{11} - q^{12} + (\beta_{2} - \beta_1 - 1) q^{13} + \beta_{3} q^{14} + \beta_{2} q^{16} + ( - 4 \beta_{2} - \beta_1 - 4) q^{17} + q^{18} - \beta_1 q^{19} - \beta_{3} q^{21} + ( - 3 \beta_{2} - 3) q^{22} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{23} - \beta_{2} q^{24} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{26} - q^{27} + ( - \beta_{3} - \beta_1) q^{28} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + ( - \beta_{2} - 1) q^{32} + (3 \beta_{2} + 3) q^{33} + ( - \beta_{3} + 4) q^{34} + \beta_{2} q^{36} + (3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{37} - \beta_{3} q^{38} + (\beta_{3} + 2 \beta_{2} + 1) q^{39} + ( - 3 \beta_{3} - 3 \beta_1) q^{41} + (\beta_{3} + \beta_1) q^{42} + ( - 2 \beta_{2} - \beta_1 - 2) q^{43} + 3 q^{44} + (\beta_{2} + \beta_1 + 1) q^{46} + 6 q^{47} + (\beta_{2} + 1) q^{48} + 3 \beta_{2} q^{49} + (\beta_{3} - 4) q^{51} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{52} + ( - \beta_{3} + 4) q^{53} - \beta_{2} q^{54} + \beta_1 q^{56} + \beta_{3} q^{57} + (2 \beta_{2} + 2 \beta_1 + 2) q^{58} + (6 \beta_{2} + 6) q^{59} + ( - 5 \beta_{2} - 3 \beta_1 - 5) q^{61} + (\beta_{3} + 6 \beta_{2} + \beta_1) q^{62} + ( - \beta_{3} - \beta_1) q^{63} + q^{64} - 3 q^{66} + 4 \beta_{2} q^{67} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{68} + ( - \beta_{2} - \beta_1 - 1) q^{69} + (11 \beta_{2} - \beta_1 + 11) q^{71} + ( - \beta_{2} - 1) q^{72} + (2 \beta_{3} + 5) q^{73} + ( - \beta_{2} - 3 \beta_1 - 1) q^{74} + (\beta_{3} + \beta_1) q^{76} + 3 \beta_{3} q^{77} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{78} + ( - \beta_{3} + 6) q^{79} + \beta_{2} q^{81} + 3 \beta_1 q^{82} + (4 \beta_{3} - 1) q^{83} - \beta_1 q^{84} + ( - \beta_{3} + 2) q^{86} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{87} + 3 \beta_{2} q^{88} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{89} + (\beta_{3} - 10 \beta_{2} - \beta_1) q^{91} + (\beta_{3} - 1) q^{92} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{93} + 6 \beta_{2} q^{94} - q^{96} + ( - 11 \beta_{2} + 2 \beta_1 - 11) q^{97} + ( - 3 \beta_{2} - 3) q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{8} - 2 q^{9} - 6 q^{11} - 4 q^{12} - 6 q^{13} - 2 q^{16} - 8 q^{17} + 4 q^{18} - 6 q^{22} + 2 q^{23} + 2 q^{24} - 4 q^{27} + 4 q^{29} + 24 q^{31} - 2 q^{32} + 6 q^{33} + 16 q^{34} - 2 q^{36} - 2 q^{37} - 4 q^{43} + 12 q^{44} + 2 q^{46} + 24 q^{47} + 2 q^{48} - 6 q^{49} - 16 q^{51} + 6 q^{52} + 16 q^{53} + 2 q^{54} + 4 q^{58} + 12 q^{59} - 10 q^{61} - 12 q^{62} + 4 q^{64} - 12 q^{66} - 8 q^{67} - 8 q^{68} - 2 q^{69} + 22 q^{71} - 2 q^{72} + 20 q^{73} - 2 q^{74} - 6 q^{78} + 24 q^{79} - 2 q^{81} - 4 q^{83} + 8 q^{86} - 4 q^{87} - 6 q^{88} + 8 q^{89} + 20 q^{91} - 4 q^{92} + 12 q^{93} - 12 q^{94} - 4 q^{96} - 22 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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
−1.58114 − 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 − 2.73861i
−0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i 0 0.500000 + 0.866025i −1.58114 − 2.73861i 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 1.58114 + 2.73861i 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 −1.58114 + 2.73861i 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 1.58114 − 2.73861i 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.ba ✓ 4
5.b even 2 1 1950.2.i.bf yes 4
5.c odd 4 2 1950.2.z.o 8
13.c even 3 1 inner 1950.2.i.ba ✓ 4
65.n even 6 1 1950.2.i.bf yes 4
65.q odd 12 2 1950.2.z.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.ba ✓ 4 1.a even 1 1 trivial
1950.2.i.ba ✓ 4 13.c even 3 1 inner
1950.2.i.bf yes 4 5.b even 2 1
1950.2.i.bf yes 4 65.n even 6 1
1950.2.z.o 8 5.c odd 4 2
1950.2.z.o 8 65.q odd 12 2

Hecke kernels

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

\( T_{7}^{4} + 10T_{7}^{2} + 100 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17}^{4} + 8T_{17}^{3} + 58T_{17}^{2} + 48T_{17} + 36 \) Copy content Toggle raw display
\( T_{19}^{4} + 10T_{19}^{2} + 100 \) 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
$7$ \( T^{4} + 10T^{2} + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} + 10T^{2} + 100 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 7921 \) Copy content Toggle raw display
$41$ \( T^{4} + 90T^{2} + 8100 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$47$ \( (T - 6)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + \cdots + 4225 \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 22 T^{3} + \cdots + 12321 \) Copy content Toggle raw display
$73$ \( (T^{2} - 10 T - 15)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 159)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$97$ \( T^{4} + 22 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
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