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Properties

Label 630.4.g.i
Level $630$
Weight $4$
Character orbit 630.g
Analytic conductor $37.171$
Analytic rank $0$
Dimension $6$
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] = [630,4,Mod(379,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.379");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.g (of order \(2\), degree \(1\), 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: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 30x^{3} + 2304x^{2} - 6048x + 7938 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{3} + 2 \beta_1 + 1) q^{5} + 7 \beta_1 q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{3} + 2 \beta_1 + 1) q^{5} + 7 \beta_1 q^{7} + 8 \beta_1 q^{8} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{10} + ( - 3 \beta_{5} - \beta_{4} + \cdots + 12) q^{11}+ \cdots + 98 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{4} + 8 q^{5} + 20 q^{10} + 56 q^{11} + 84 q^{14} + 96 q^{16} - 136 q^{19} - 32 q^{20} - 78 q^{25} - 128 q^{26} - 800 q^{29} - 96 q^{31} + 648 q^{34} - 70 q^{35} - 80 q^{40} + 1040 q^{41} - 224 q^{44} - 368 q^{46} - 294 q^{49} + 720 q^{50} - 1544 q^{55} - 336 q^{56} - 720 q^{59} + 316 q^{61} - 384 q^{64} - 688 q^{65} + 112 q^{70} - 168 q^{71} + 64 q^{74} + 544 q^{76} - 1712 q^{79} + 128 q^{80} + 1844 q^{85} - 192 q^{86} - 3040 q^{89} + 448 q^{91} - 624 q^{94} + 152 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} - 30x^{3} + 2304x^{2} - 6048x + 7938 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -23\nu^{5} + 16\nu^{4} + 5\nu^{3} - 810\nu^{2} - 52542\nu + 70119 ) / 72765 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 226\nu^{5} + 355\nu^{4} - 1405\nu^{3} - 14940\nu^{2} + 484344\nu + 132300 ) / 72765 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -80\nu^{5} + 213\nu^{4} + 235\nu^{3} + 5050\nu^{2} - 209370\nu + 477162 ) / 24255 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -80\nu^{5} + 213\nu^{4} + 235\nu^{3} + 5050\nu^{2} - 160860\nu + 477162 ) / 24255 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -26\nu^{5} - 23\nu^{4} - 145\nu^{3} + 1440\nu^{2} - 56574\nu + 21168 ) / 6615 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 66\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -47\beta_{5} + 3\beta_{4} + 3\beta_{3} - 47\beta_{2} + 60\beta _1 + 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -35\beta_{5} + 115\beta_{4} + 35\beta_{3} + 115\beta_{2} - 3048 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -105\beta_{5} - 2239\beta_{4} + 2239\beta_{3} + 105\beta_{2} - 3990\beta _1 + 3990 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.33768 − 1.33768i
−5.02443 + 5.02443i
4.68676 − 4.68676i
1.33768 + 1.33768i
−5.02443 − 5.02443i
4.68676 + 4.68676i
− 2.00000i 0 −4.00000 −7.02351 − 8.69887i 0 7.00000i 8.00000i 0 −17.3977 + 14.0470i
379.2 − 2.00000i 0 −4.00000 0.130709 + 11.1796i 0 7.00000i 8.00000i 0 22.3592 − 0.261418i
379.3 − 2.00000i 0 −4.00000 10.8928 + 2.51929i 0 7.00000i 8.00000i 0 5.03858 − 21.7856i
379.4 2.00000i 0 −4.00000 −7.02351 + 8.69887i 0 − 7.00000i − 8.00000i 0 −17.3977 − 14.0470i
379.5 2.00000i 0 −4.00000 0.130709 − 11.1796i 0 − 7.00000i − 8.00000i 0 22.3592 + 0.261418i
379.6 2.00000i 0 −4.00000 10.8928 − 2.51929i 0 − 7.00000i − 8.00000i 0 5.03858 + 21.7856i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.g.i yes 6
3.b odd 2 1 630.4.g.g ✓ 6
5.b even 2 1 inner 630.4.g.i yes 6
15.d odd 2 1 630.4.g.g ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.4.g.g ✓ 6 3.b odd 2 1
630.4.g.g ✓ 6 15.d odd 2 1
630.4.g.i yes 6 1.a even 1 1 trivial
630.4.g.i yes 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{3} - 28T_{11}^{2} - 2064T_{11} + 63840 \) acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 8 T^{5} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} - 28 T^{2} + \cdots + 63840)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 1136 T^{4} + \cdots + 5531904 \) Copy content Toggle raw display
$17$ \( T^{6} + 15900 T^{4} + \cdots + 305550400 \) Copy content Toggle raw display
$19$ \( (T^{3} + 68 T^{2} + \cdots + 3312)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 38572960000 \) Copy content Toggle raw display
$29$ \( (T^{3} + 400 T^{2} + \cdots - 12288)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 48 T^{2} + \cdots + 2987216)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 403072614400 \) Copy content Toggle raw display
$41$ \( (T^{3} - 520 T^{2} + \cdots + 51700080)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 314023350255616 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 578106241945600 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 578309766917184 \) Copy content Toggle raw display
$59$ \( (T^{3} + 360 T^{2} + \cdots + 27811328)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 158 T^{2} + \cdots + 49200792)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{3} + 84 T^{2} + \cdots - 10525536)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + 856 T^{2} + \cdots + 70541184)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1520 T^{2} + \cdots + 112732432)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
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