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Properties

Label 630.4.g.h
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: \( x^{6} - 2x^{5} + 2x^{4} - 10x^{3} + 7744x^{2} - 16368x + 17298 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 4 q^{4} + \beta_{3} q^{5} + 7 \beta_1 q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 4 q^{4} + \beta_{3} q^{5} + 7 \beta_1 q^{7} + 8 \beta_1 q^{8} + 2 \beta_{2} q^{10} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 12) q^{11}+ \cdots + 98 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{4} + 2 q^{5} - 76 q^{11} + 84 q^{14} + 96 q^{16} - 108 q^{19} - 8 q^{20} - 18 q^{25} - 136 q^{26} + 292 q^{29} + 340 q^{31} - 224 q^{34} - 492 q^{41} + 304 q^{44} + 224 q^{46} - 294 q^{49} - 80 q^{50} - 716 q^{55} - 336 q^{56} + 1136 q^{59} + 836 q^{61} - 384 q^{64} + 40 q^{65} + 28 q^{70} - 1660 q^{71} + 784 q^{74} + 432 q^{76} + 1112 q^{79} + 32 q^{80} - 928 q^{85} - 48 q^{86} + 1940 q^{89} + 476 q^{91} - 256 q^{94} + 1996 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} - 10x^{3} + 7744x^{2} - 16368x + 17298 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1585\nu^{5} + 1496\nu^{4} + 85\nu^{3} - 134810\nu^{2} - 12265870\nu + 12980289 ) / 13274355 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -85408\nu^{5} + 226337\nu^{4} + 163705\nu^{3} + 5850970\nu^{2} - 754597642\nu + 1480551723 ) / 92920485 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -559\nu^{5} - 499\nu^{4} + 6784\nu^{3} + 31342\nu^{2} - 4235170\nu - 569523 ) / 599487 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3113\nu^{5} + 7639\nu^{4} + 5300\nu^{3} + 158300\nu^{2} - 21116702\nu + 47693331 ) / 2997435 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -98986\nu^{5} - 42247\nu^{4} - 1058315\nu^{3} + 5918860\nu^{2} - 752072134\nu + 111784047 ) / 92920485 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{5} + 3\beta_{4} + 3\beta_{3} + 4\beta_{2} - 116\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -87\beta_{5} + 7\beta_{4} + 87\beta_{3} + 7\beta_{2} - 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -265\beta_{5} + 365\beta_{4} - 365\beta_{3} + 265\beta_{2} - 10058 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -595\beta_{5} - 7649\beta_{4} - 595\beta_{3} + 7649\beta_{2} + 854\beta _1 - 854 ) / 2 \) 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
1.05830 − 1.05830i
6.59951 − 6.59951i
−6.65781 + 6.65781i
1.05830 + 1.05830i
6.59951 + 6.59951i
−6.65781 − 6.65781i
− 2.00000i 0 −4.00000 −8.44426 + 7.32765i 0 7.00000i 8.00000i 0 14.6553 + 16.8885i
379.2 − 2.00000i 0 −4.00000 −1.07000 − 11.1290i 0 7.00000i 8.00000i 0 −22.2580 + 2.14000i
379.3 − 2.00000i 0 −4.00000 10.5143 + 3.80137i 0 7.00000i 8.00000i 0 7.60273 − 21.0285i
379.4 2.00000i 0 −4.00000 −8.44426 − 7.32765i 0 − 7.00000i − 8.00000i 0 14.6553 − 16.8885i
379.5 2.00000i 0 −4.00000 −1.07000 + 11.1290i 0 − 7.00000i − 8.00000i 0 −22.2580 − 2.14000i
379.6 2.00000i 0 −4.00000 10.5143 − 3.80137i 0 − 7.00000i − 8.00000i 0 7.60273 + 21.0285i
\(n\): 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.h 6
3.b odd 2 1 210.4.g.c ✓ 6
5.b even 2 1 inner 630.4.g.h 6
15.d odd 2 1 210.4.g.c ✓ 6
15.e even 4 1 1050.4.a.bj 3
15.e even 4 1 1050.4.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.g.c ✓ 6 3.b odd 2 1
210.4.g.c ✓ 6 15.d odd 2 1
630.4.g.h 6 1.a even 1 1 trivial
630.4.g.h 6 5.b even 2 1 inner
1050.4.a.bj 3 15.e even 4 1
1050.4.a.bm 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{3} + 38T_{11}^{2} - 228T_{11} - 4920 \) 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
$5$ \( T^{6} - 2 T^{5} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 38 T^{2} + \cdots - 4920)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 1116 T^{4} + \cdots + 13601344 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 33089065216 \) Copy content Toggle raw display
$19$ \( (T^{3} + 54 T^{2} + \cdots - 102200)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 240413702400 \) Copy content Toggle raw display
$29$ \( (T^{3} - 146 T^{2} + \cdots - 29400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 170 T^{2} + \cdots + 17227656)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 309555029118976 \) Copy content Toggle raw display
$41$ \( (T^{3} + 246 T^{2} + \cdots - 19590264)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 140991238144 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 90774247112704 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 119854440000 \) Copy content Toggle raw display
$59$ \( (T^{3} - 568 T^{2} + \cdots + 17555200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 418 T^{2} + \cdots + 144292584)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 204274984550400 \) Copy content Toggle raw display
$71$ \( (T^{3} + 830 T^{2} + \cdots - 33568440)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{3} - 556 T^{2} + \cdots - 36948800)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 707624692678656 \) Copy content Toggle raw display
$89$ \( (T^{3} - 970 T^{2} + \cdots + 2284547560)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
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