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Properties

Label 1650.2.l.e
Level $1650$
Weight $2$
Character orbit 1650.l
Analytic conductor $13.175$
Analytic rank $0$
Dimension $8$
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] = [1650,2,Mod(43,1650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1650, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1650.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.l (of order \(4\), 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: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1871773696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 31x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} - \beta_{2} q^{6} + (\beta_{6} - \beta_{4} - \beta_1) q^{7} - \beta_{7} q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} - \beta_{2} q^{6} + (\beta_{6} - \beta_{4} - \beta_1) q^{7} - \beta_{7} q^{8} + \beta_{2} q^{9} + (\beta_{5} + 2 \beta_{2} - 2) q^{11} + \beta_{7} q^{12} - 2 \beta_{7} q^{13} + (\beta_{5} + \beta_{3} + 2 \beta_{2}) q^{14} - q^{16} - 2 \beta_1 q^{17} - \beta_{7} q^{18} + ( - \beta_{5} + \beta_{3} + 4) q^{19} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{21} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{22} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{23} + q^{24} - 2 q^{26} + \beta_{7} q^{27} + (\beta_{6} + \beta_{4} - \beta_1) q^{28} + ( - 2 \beta_{5} + 2 \beta_{3} + 4) q^{29} + ( - 2 \beta_{5} + 2 \beta_{3} + 4) q^{31} + \beta_1 q^{32} + (\beta_{7} - \beta_{6} - \beta_1) q^{33} + 2 \beta_{2} q^{34} - q^{36} + (6 \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{37}+ \cdots + (\beta_{3} - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{11} - 8 q^{16} + 24 q^{19} + 8 q^{24} - 16 q^{26} + 16 q^{29} + 16 q^{31} - 8 q^{36} + 16 q^{39} - 12 q^{44} + 8 q^{54} - 8 q^{56} + 12 q^{66} - 56 q^{71} + 56 q^{74} + 64 q^{79} - 8 q^{81} + 8 q^{84} - 8 q^{86} - 16 q^{91} + 8 q^{94} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 19\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 40\nu^{2} ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 9\nu^{4} - 97\nu^{2} + 108 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 40\nu^{3} - 63\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 9\nu^{4} - 97\nu^{2} - 108 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 40\nu^{3} + 120\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + 97\nu^{3} ) / 189 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} + 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 2\beta_{6} + 2\beta_{4} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 7\beta_{3} - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{6} + 19\beta_{4} + 61\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{5} - 20\beta_{3} - 97\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 240\beta_{7} - 97\beta_{6} - 97\beta_{4} + 97\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.62831 − 1.62831i
0.921201 + 0.921201i
−0.921201 − 0.921201i
1.62831 + 1.62831i
−1.62831 + 1.62831i
0.921201 − 0.921201i
−0.921201 + 0.921201i
1.62831 − 1.62831i
−0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 0 − 1.00000i −3.25662 − 3.25662i 0.707107 − 0.707107i 1.00000i 0
43.2 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 0 − 1.00000i 1.84240 + 1.84240i 0.707107 − 0.707107i 1.00000i 0
43.3 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 0 − 1.00000i −1.84240 − 1.84240i −0.707107 + 0.707107i 1.00000i 0
43.4 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 0 − 1.00000i 3.25662 + 3.25662i −0.707107 + 0.707107i 1.00000i 0
307.1 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 0 1.00000i −3.25662 + 3.25662i 0.707107 + 0.707107i − 1.00000i 0
307.2 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 0 1.00000i 1.84240 − 1.84240i 0.707107 + 0.707107i − 1.00000i 0
307.3 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 0 1.00000i −1.84240 + 1.84240i −0.707107 − 0.707107i − 1.00000i 0
307.4 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 0 1.00000i 3.25662 − 3.25662i −0.707107 − 0.707107i − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
55.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.l.e yes 8
5.b even 2 1 inner 1650.2.l.e yes 8
5.c odd 4 2 1650.2.l.d ✓ 8
11.b odd 2 1 1650.2.l.d ✓ 8
55.d odd 2 1 1650.2.l.d ✓ 8
55.e even 4 2 inner 1650.2.l.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1650.2.l.d ✓ 8 5.c odd 4 2
1650.2.l.d ✓ 8 11.b odd 2 1
1650.2.l.d ✓ 8 55.d odd 2 1
1650.2.l.e yes 8 1.a even 1 1 trivial
1650.2.l.e yes 8 5.b even 2 1 inner
1650.2.l.e yes 8 55.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{8} + 496T_{7}^{4} + 20736 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 496 T^{4} + 20736 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 496 T^{4} + 20736 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 48)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 12784 T^{4} + 1679616 \) Copy content Toggle raw display
$41$ \( (T^{4} + 76 T^{2} + 144)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 496 T^{4} + 20736 \) Copy content Toggle raw display
$47$ \( T^{8} + 28784 T^{4} + 181063936 \) Copy content Toggle raw display
$53$ \( T^{8} + 7936 T^{4} + 5308416 \) Copy content Toggle raw display
$59$ \( (T^{4} + 28 T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 188 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 30464 T^{4} + 65536 \) Copy content Toggle raw display
$71$ \( (T^{2} + 14 T + 36)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 7936 T^{4} + 5308416 \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 43264)^{2} \) Copy content Toggle raw display
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