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Properties

Label 630.2.m.c
Level $630$
Weight $2$
Character orbit 630.m
Analytic conductor $5.031$
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] = [630,2,Mod(197,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.m (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{4} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{5} + \beta_{3} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{4} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{5} + \beta_{3} q^{7} + \beta_{3} q^{8} + \beta_{2} q^{10} + (2 \beta_{5} - \beta_{4} - \beta_{2}) q^{11} + (\beta_{6} + \beta_{3} + \beta_{2}) q^{13} - q^{14} - q^{16} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots - \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{13} - 8 q^{14} - 8 q^{16} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 4 q^{25} - 24 q^{29} - 8 q^{31} + 4 q^{35} - 4 q^{37} + 4 q^{40} - 12 q^{43} - 24 q^{44} - 12 q^{47} + 32 q^{50} + 4 q^{52} + 32 q^{53} - 24 q^{55} + 12 q^{58} - 16 q^{59} + 4 q^{62} + 32 q^{65} + 20 q^{67} + 36 q^{73} - 40 q^{74} - 4 q^{77} - 12 q^{82} + 56 q^{83} + 32 q^{85} - 4 q^{88} - 72 q^{89} + 8 q^{92} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-\beta_{5}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
− 1.69230i
0.692297i
2.16053i
− 3.16053i
1.69230i
− 0.692297i
− 2.16053i
3.16053i
−0.707107 + 0.707107i 0 − 1.00000i −0.489528 + 2.18183i 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −1.19663 − 1.88893i
197.2 −0.707107 + 0.707107i 0 − 1.00000i 1.19663 − 1.88893i 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0.489528 + 2.18183i
197.3 0.707107 − 0.707107i 0 − 1.00000i −2.23483 + 0.0743018i 0 −0.707107 − 0.707107i −0.707107 − 0.707107i 0 −1.52773 + 1.63280i
197.4 0.707107 − 0.707107i 0 − 1.00000i 1.52773 + 1.63280i 0 −0.707107 − 0.707107i −0.707107 − 0.707107i 0 2.23483 + 0.0743018i
323.1 −0.707107 − 0.707107i 0 1.00000i −0.489528 − 2.18183i 0 0.707107 − 0.707107i 0.707107 − 0.707107i 0 −1.19663 + 1.88893i
323.2 −0.707107 − 0.707107i 0 1.00000i 1.19663 + 1.88893i 0 0.707107 − 0.707107i 0.707107 − 0.707107i 0 0.489528 − 2.18183i
323.3 0.707107 + 0.707107i 0 1.00000i −2.23483 − 0.0743018i 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 −1.52773 − 1.63280i
323.4 0.707107 + 0.707107i 0 1.00000i 1.52773 − 1.63280i 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 2.23483 − 0.0743018i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.m.c ✓ 8
3.b odd 2 1 630.2.m.d yes 8
5.b even 2 1 3150.2.m.i 8
5.c odd 4 1 630.2.m.d yes 8
5.c odd 4 1 3150.2.m.j 8
15.d odd 2 1 3150.2.m.j 8
15.e even 4 1 inner 630.2.m.c ✓ 8
15.e even 4 1 3150.2.m.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.c ✓ 8 1.a even 1 1 trivial
630.2.m.c ✓ 8 15.e even 4 1 inner
630.2.m.d yes 8 3.b odd 2 1
630.2.m.d yes 8 5.c odd 4 1
3150.2.m.i 8 5.b even 2 1
3150.2.m.i 8 15.e even 4 1
3150.2.m.j 8 5.c odd 4 1
3150.2.m.j 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{8} + 76T_{11}^{6} + 1668T_{11}^{4} + 9728T_{11}^{2} + 16384 \) Copy content Toggle raw display
\( T_{17}^{8} + 64T_{17}^{5} + 1040T_{17}^{4} + 2304T_{17}^{3} + 2048T_{17}^{2} - 8192T_{17} + 16384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 76 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{5} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 2104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 14 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 3136 \) Copy content Toggle raw display
$41$ \( T^{8} + 252 T^{6} + \cdots + 341056 \) Copy content Toggle raw display
$43$ \( T^{8} + 12 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 2458624 \) Copy content Toggle raw display
$53$ \( T^{8} - 32 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 20 T^{7} + \cdots + 200704 \) Copy content Toggle raw display
$71$ \( T^{8} + 352 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$73$ \( T^{8} - 36 T^{7} + \cdots + 8340544 \) Copy content Toggle raw display
$79$ \( T^{8} + 352 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$83$ \( T^{8} - 56 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$89$ \( (T^{4} + 36 T^{3} + \cdots - 8248)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots + 1784896 \) Copy content Toggle raw display
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