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Properties

Label 650.2.n.c
Level $650$
Weight $2$
Character orbit 650.n
Analytic conductor $5.190$
Analytic rank $0$
Dimension $8$
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] = [650,2,Mod(49,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.n (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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \cdots - 1) q^{3}+ \cdots + ( - 6 \beta_{7} - 2 \beta_{6} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 2 q^{9} + 12 q^{11} - 6 q^{13} - 4 q^{16} + 4 q^{18} - 6 q^{19} - 12 q^{22} + 24 q^{23} + 6 q^{26} + 12 q^{29} - 4 q^{32} - 2 q^{36} + 10 q^{39} - 24 q^{41} + 6 q^{42} + 18 q^{43} - 24 q^{46} - 24 q^{47} + 10 q^{49} + 18 q^{54} + 24 q^{57} + 12 q^{58} - 30 q^{59} - 10 q^{61} + 24 q^{62} + 8 q^{64} - 6 q^{67} + 18 q^{69} - 2 q^{72} - 24 q^{73} + 6 q^{76} - 20 q^{78} + 4 q^{79} + 8 q^{81} + 24 q^{82} + 12 q^{83} - 6 q^{84} - 48 q^{87} + 12 q^{88} + 6 q^{89} + 24 q^{91} - 12 q^{93} + 12 q^{94} + 36 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 3\beta_{6} - 5\beta_{5} + \beta_{4} - 4\beta_{3} + 3\beta_{2} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 4\beta_{4} + 2\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.20036 + 0.747754i
−1.27597 + 0.609843i
1.40994 − 0.109843i
0.665665 − 1.24775i
1.20036 − 0.747754i
−1.27597 − 0.609843i
1.40994 + 0.109843i
0.665665 + 1.24775i
−0.500000 − 0.866025i −2.16117 + 1.24775i −0.500000 + 0.866025i 0 2.16117 + 1.24775i 0.165665 − 0.286941i 1.00000 1.61378 − 2.79515i 0
49.2 −0.500000 − 0.866025i −0.190254 + 0.109843i −0.500000 + 0.866025i 0 0.190254 + 0.109843i 0.909941 − 1.57606i 1.00000 −1.47587 + 2.55628i 0
49.3 −0.500000 − 0.866025i 1.05628 − 0.609843i −0.500000 + 0.866025i 0 −1.05628 − 0.609843i −1.77597 + 3.07606i 1.00000 −0.756182 + 1.30975i 0
49.4 −0.500000 − 0.866025i 1.29515 − 0.747754i −0.500000 + 0.866025i 0 −1.29515 − 0.747754i 0.700360 − 1.21306i 1.00000 −0.381728 + 0.661173i 0
199.1 −0.500000 + 0.866025i −2.16117 − 1.24775i −0.500000 − 0.866025i 0 2.16117 − 1.24775i 0.165665 + 0.286941i 1.00000 1.61378 + 2.79515i 0
199.2 −0.500000 + 0.866025i −0.190254 − 0.109843i −0.500000 − 0.866025i 0 0.190254 − 0.109843i 0.909941 + 1.57606i 1.00000 −1.47587 − 2.55628i 0
199.3 −0.500000 + 0.866025i 1.05628 + 0.609843i −0.500000 − 0.866025i 0 −1.05628 + 0.609843i −1.77597 − 3.07606i 1.00000 −0.756182 − 1.30975i 0
199.4 −0.500000 + 0.866025i 1.29515 + 0.747754i −0.500000 − 0.866025i 0 −1.29515 + 0.747754i 0.700360 + 1.21306i 1.00000 −0.381728 − 0.661173i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.n.c 8
5.b even 2 1 650.2.n.f 8
5.c odd 4 1 650.2.m.b ✓ 8
5.c odd 4 1 650.2.m.d yes 8
13.e even 6 1 650.2.n.f 8
65.l even 6 1 inner 650.2.n.c 8
65.o even 12 1 8450.2.a.cl 4
65.o even 12 1 8450.2.a.co 4
65.r odd 12 1 650.2.m.b ✓ 8
65.r odd 12 1 650.2.m.d yes 8
65.t even 12 1 8450.2.a.ch 4
65.t even 12 1 8450.2.a.ck 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.m.b ✓ 8 5.c odd 4 1
650.2.m.b ✓ 8 65.r odd 12 1
650.2.m.d yes 8 5.c odd 4 1
650.2.m.d yes 8 65.r odd 12 1
650.2.n.c 8 1.a even 1 1 trivial
650.2.n.c 8 65.l even 6 1 inner
650.2.n.f 8 5.b even 2 1
650.2.n.f 8 13.e even 6 1
8450.2.a.ch 4 65.t even 12 1
8450.2.a.ck 4 65.t even 12 1
8450.2.a.cl 4 65.o even 12 1
8450.2.a.co 4 65.o even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 5T_{3}^{6} + 24T_{3}^{4} - 30T_{3}^{3} + 7T_{3}^{2} + 6T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 9 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots + 42849 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 27 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 84681 \) Copy content Toggle raw display
$23$ \( T^{8} - 24 T^{7} + \cdots + 1521 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{7} + \cdots + 576081 \) Copy content Toggle raw display
$31$ \( T^{8} + 120 T^{6} + \cdots + 389376 \) Copy content Toggle raw display
$37$ \( T^{8} + 15 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 558009 \) Copy content Toggle raw display
$43$ \( T^{8} - 18 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} + \cdots - 531)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 216 T^{6} + \cdots + 257049 \) Copy content Toggle raw display
$59$ \( T^{8} + 30 T^{7} + \cdots + 178929 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots + 5031049 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + \cdots + 8071281 \) Copy content Toggle raw display
$71$ \( T^{8} - 180 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$73$ \( (T^{4} + 12 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6 T^{3} - 42 T^{2} + \cdots - 99)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 6 T^{7} + \cdots + 11881809 \) Copy content Toggle raw display
$97$ \( T^{8} - 36 T^{7} + \cdots + 7469289 \) Copy content Toggle raw display
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