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Properties

Label 490.2.l.a
Level $490$
Weight $2$
Character orbit 490.l
Analytic conductor $3.913$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(117,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.l (of order \(12\), degree \(4\), 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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{48}^{10} q^{2} + (\zeta_{48}^{15} + \cdots - \zeta_{48}^{5}) q^{3} + \cdots + ( - \zeta_{48}^{14} + \cdots - \zeta_{48}^{4}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{10} q^{2} + (\zeta_{48}^{15} + \cdots - \zeta_{48}^{5}) q^{3} + \cdots + ( - 4 \zeta_{48}^{12} + \cdots - 2 \zeta_{48}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{15} + 8 q^{16} - 8 q^{18} - 32 q^{22} + 8 q^{23} - 16 q^{30} + 16 q^{36} - 32 q^{37} + 64 q^{43} + 16 q^{46} + 64 q^{50} + 32 q^{51} + 32 q^{53} - 16 q^{57} + 16 q^{58} + 8 q^{60} - 8 q^{65} - 16 q^{67} - 32 q^{71} - 8 q^{72} - 80 q^{78} - 24 q^{81} - 96 q^{85} - 16 q^{88} - 16 q^{92} + 16 q^{93} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1 - \zeta_{48}^{8}\) \(-\zeta_{48}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−0.793353 + 0.608761i
0.793353 − 0.608761i
0.608761 + 0.793353i
−0.608761 − 0.793353i
0.991445 − 0.130526i
−0.991445 + 0.130526i
0.130526 + 0.991445i
−0.130526 − 0.991445i
0.991445 + 0.130526i
−0.991445 − 0.130526i
0.130526 − 0.991445i
−0.130526 + 0.991445i
−0.793353 − 0.608761i
0.793353 + 0.608761i
0.608761 − 0.793353i
−0.608761 + 0.793353i
−0.965926 + 0.258819i −0.198092 + 0.739288i 0.866025 − 0.500000i −0.977945 + 2.01088i − 0.765367i 0 −0.707107 + 0.707107i 2.09077 + 1.20711i 0.424170 − 2.19547i
117.2 −0.965926 + 0.258819i 0.198092 − 0.739288i 0.866025 − 0.500000i 0.977945 − 2.01088i 0.765367i 0 −0.707107 + 0.707107i 2.09077 + 1.20711i −0.424170 + 2.19547i
117.3 0.965926 − 0.258819i −0.478235 + 1.78480i 0.866025 − 0.500000i 2.01088 + 0.977945i 1.84776i 0 0.707107 − 0.707107i −0.358719 − 0.207107i 2.19547 + 0.424170i
117.4 0.965926 − 0.258819i 0.478235 − 1.78480i 0.866025 − 0.500000i −2.01088 − 0.977945i − 1.84776i 0 0.707107 − 0.707107i −0.358719 − 0.207107i −2.19547 − 0.424170i
227.1 −0.258819 + 0.965926i −1.78480 + 0.478235i −0.866025 − 0.500000i 1.85236 − 1.25250i − 1.84776i 0 0.707107 − 0.707107i 0.358719 − 0.207107i 0.730392 + 2.11342i
227.2 −0.258819 + 0.965926i 1.78480 − 0.478235i −0.866025 − 0.500000i −1.85236 + 1.25250i 1.84776i 0 0.707107 − 0.707107i 0.358719 − 0.207107i −0.730392 − 2.11342i
227.3 0.258819 − 0.965926i −0.739288 + 0.198092i −0.866025 − 0.500000i 1.25250 + 1.85236i 0.765367i 0 −0.707107 + 0.707107i −2.09077 + 1.20711i 2.11342 − 0.730392i
227.4 0.258819 − 0.965926i 0.739288 − 0.198092i −0.866025 − 0.500000i −1.25250 − 1.85236i − 0.765367i 0 −0.707107 + 0.707107i −2.09077 + 1.20711i −2.11342 + 0.730392i
313.1 −0.258819 − 0.965926i −1.78480 − 0.478235i −0.866025 + 0.500000i 1.85236 + 1.25250i 1.84776i 0 0.707107 + 0.707107i 0.358719 + 0.207107i 0.730392 − 2.11342i
313.2 −0.258819 − 0.965926i 1.78480 + 0.478235i −0.866025 + 0.500000i −1.85236 − 1.25250i − 1.84776i 0 0.707107 + 0.707107i 0.358719 + 0.207107i −0.730392 + 2.11342i
313.3 0.258819 + 0.965926i −0.739288 − 0.198092i −0.866025 + 0.500000i 1.25250 − 1.85236i − 0.765367i 0 −0.707107 − 0.707107i −2.09077 − 1.20711i 2.11342 + 0.730392i
313.4 0.258819 + 0.965926i 0.739288 + 0.198092i −0.866025 + 0.500000i −1.25250 + 1.85236i 0.765367i 0 −0.707107 − 0.707107i −2.09077 − 1.20711i −2.11342 − 0.730392i
423.1 −0.965926 − 0.258819i −0.198092 − 0.739288i 0.866025 + 0.500000i −0.977945 − 2.01088i 0.765367i 0 −0.707107 − 0.707107i 2.09077 − 1.20711i 0.424170 + 2.19547i
423.2 −0.965926 − 0.258819i 0.198092 + 0.739288i 0.866025 + 0.500000i 0.977945 + 2.01088i − 0.765367i 0 −0.707107 − 0.707107i 2.09077 − 1.20711i −0.424170 − 2.19547i
423.3 0.965926 + 0.258819i −0.478235 − 1.78480i 0.866025 + 0.500000i 2.01088 − 0.977945i − 1.84776i 0 0.707107 + 0.707107i −0.358719 + 0.207107i 2.19547 − 0.424170i
423.4 0.965926 + 0.258819i 0.478235 + 1.78480i 0.866025 + 0.500000i −2.01088 + 0.977945i 1.84776i 0 0.707107 + 0.707107i −0.358719 + 0.207107i −2.19547 + 0.424170i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.l.a 16
5.c odd 4 1 inner 490.2.l.a 16
7.b odd 2 1 inner 490.2.l.a 16
7.c even 3 1 70.2.g.a ✓ 8
7.c even 3 1 inner 490.2.l.a 16
7.d odd 6 1 70.2.g.a ✓ 8
7.d odd 6 1 inner 490.2.l.a 16
21.g even 6 1 630.2.p.a 8
21.h odd 6 1 630.2.p.a 8
28.f even 6 1 560.2.bj.c 8
28.g odd 6 1 560.2.bj.c 8
35.f even 4 1 inner 490.2.l.a 16
35.i odd 6 1 350.2.g.a 8
35.j even 6 1 350.2.g.a 8
35.k even 12 1 70.2.g.a ✓ 8
35.k even 12 1 350.2.g.a 8
35.k even 12 1 inner 490.2.l.a 16
35.l odd 12 1 70.2.g.a ✓ 8
35.l odd 12 1 350.2.g.a 8
35.l odd 12 1 inner 490.2.l.a 16
105.w odd 12 1 630.2.p.a 8
105.x even 12 1 630.2.p.a 8
140.w even 12 1 560.2.bj.c 8
140.x odd 12 1 560.2.bj.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.g.a ✓ 8 7.c even 3 1
70.2.g.a ✓ 8 7.d odd 6 1
70.2.g.a ✓ 8 35.k even 12 1
70.2.g.a ✓ 8 35.l odd 12 1
350.2.g.a 8 35.i odd 6 1
350.2.g.a 8 35.j even 6 1
350.2.g.a 8 35.k even 12 1
350.2.g.a 8 35.l odd 12 1
490.2.l.a 16 1.a even 1 1 trivial
490.2.l.a 16 5.c odd 4 1 inner
490.2.l.a 16 7.b odd 2 1 inner
490.2.l.a 16 7.c even 3 1 inner
490.2.l.a 16 7.d odd 6 1 inner
490.2.l.a 16 35.f even 4 1 inner
490.2.l.a 16 35.k even 12 1 inner
490.2.l.a 16 35.l odd 12 1 inner
560.2.bj.c 8 28.f even 6 1
560.2.bj.c 8 28.g odd 6 1
560.2.bj.c 8 140.w even 12 1
560.2.bj.c 8 140.x odd 12 1
630.2.p.a 8 21.g even 6 1
630.2.p.a 8 21.h odd 6 1
630.2.p.a 8 105.w odd 12 1
630.2.p.a 8 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 12T_{3}^{12} + 140T_{3}^{8} - 48T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 12 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} + 48 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{2} + 64)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1548 T^{4} + 334084)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 268435456 \) Copy content Toggle raw display
$19$ \( (T^{8} + 52 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 16 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 16 T^{7} + \cdots + 614656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{2} + 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 32)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} - 192 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( (T^{8} - 16 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 100 T^{6} + \cdots + 1562500)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 148 T^{6} + \cdots + 19518724)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 8 T^{7} + \cdots + 342102016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 2)^{8} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 6044831973376 \) Copy content Toggle raw display
$79$ \( (T^{8} - 108 T^{6} + \cdots + 4477456)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 6732 T^{4} + 11303044)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 416 T^{6} + \cdots + 1368408064)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 62208 T^{4} + 107495424)^{2} \) Copy content Toggle raw display
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