Properties

Label 630.4.k.n
Level $630$
Weight $4$
Character orbit 630.k
Analytic conductor $37.171$
Analytic rank $0$
Dimension $4$
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(361,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.361");
 
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.k (of order \(3\), 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: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{46})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 46x^{2} + 2116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + ( - 4 \beta_{2} - 4) q^{4} - 5 \beta_{2} q^{5} + (3 \beta_{3} - 5 \beta_{2} + \beta_1 - 4) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + ( - 4 \beta_{2} - 4) q^{4} - 5 \beta_{2} q^{5} + (3 \beta_{3} - 5 \beta_{2} + \beta_1 - 4) q^{7} - 8 q^{8} + ( - 10 \beta_{2} - 10) q^{10} + ( - 10 \beta_{2} + 5 \beta_1 - 10) q^{11} + ( - \beta_{3} + 21) q^{13} + (4 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 10) q^{14} + 16 \beta_{2} q^{16} + (38 \beta_{2} + 11 \beta_1 + 38) q^{17} + ( - 2 \beta_{3} + 45 \beta_{2} - 2 \beta_1) q^{19} - 20 q^{20} + ( - 10 \beta_{3} - 20) q^{22} + (11 \beta_{3} - 22 \beta_{2} + 11 \beta_1) q^{23} + ( - 25 \beta_{2} - 25) q^{25} + ( - 2 \beta_{3} - 42 \beta_{2} - 2 \beta_1) q^{26} + ( - 4 \beta_{3} + 16 \beta_{2} + 8 \beta_1 - 4) q^{28} + ( - 5 \beta_{3} + 80) q^{29} + ( - 31 \beta_{2} - 8 \beta_1 - 31) q^{31} + (32 \beta_{2} + 32) q^{32} + ( - 22 \beta_{3} + 76) q^{34} + (10 \beta_{3} - 5 \beta_{2} + 15 \beta_1 - 25) q^{35} + ( - 31 \beta_{3} - 179 \beta_{2} - 31 \beta_1) q^{37} + (90 \beta_{2} - 4 \beta_1 + 90) q^{38} + 40 \beta_{2} q^{40} + ( - 27 \beta_{3} - 18) q^{41} + ( - 27 \beta_{3} - 67) q^{43} + ( - 20 \beta_{3} + 40 \beta_{2} - 20 \beta_1) q^{44} + ( - 44 \beta_{2} + 22 \beta_1 - 44) q^{46} + (36 \beta_{3} - 342 \beta_{2} + 36 \beta_1) q^{47} + ( - 4 \beta_{3} - 215 \beta_{2} + 22 \beta_1 + 129) q^{49} - 50 q^{50} + ( - 84 \beta_{2} - 4 \beta_1 - 84) q^{52} + (8 \beta_{2} + 50 \beta_1 + 8) q^{53} + ( - 25 \beta_{3} - 50) q^{55} + ( - 24 \beta_{3} + 40 \beta_{2} - 8 \beta_1 + 32) q^{56} + ( - 10 \beta_{3} - 160 \beta_{2} - 10 \beta_1) q^{58} + (276 \beta_{2} - 15 \beta_1 + 276) q^{59} + ( - 18 \beta_{3} + 656 \beta_{2} - 18 \beta_1) q^{61} + (16 \beta_{3} - 62) q^{62} + 64 q^{64} + ( - 5 \beta_{3} - 105 \beta_{2} - 5 \beta_1) q^{65} + ( - 97 \beta_{2} - 41 \beta_1 - 97) q^{67} + ( - 44 \beta_{3} - 152 \beta_{2} - 44 \beta_1) q^{68} + ( - 10 \beta_{3} + 40 \beta_{2} + 20 \beta_1 - 10) q^{70} + ( - 137 \beta_{3} - 190) q^{71} + ( - 99 \beta_{2} + 11 \beta_1 - 99) q^{73} + ( - 358 \beta_{2} - 62 \beta_1 - 358) q^{74} + (8 \beta_{3} + 180) q^{76} + ( - 35 \beta_{3} - 420 \beta_{2} - 700) q^{77} + (166 \beta_{3} + 63 \beta_{2} + 166 \beta_1) q^{79} + (80 \beta_{2} + 80) q^{80} + ( - 54 \beta_{3} + 36 \beta_{2} - 54 \beta_1) q^{82} + ( - 147 \beta_{3} - 432) q^{83} + ( - 55 \beta_{3} + 190) q^{85} + ( - 54 \beta_{3} + 134 \beta_{2} - 54 \beta_1) q^{86} + (80 \beta_{2} - 40 \beta_1 + 80) q^{88} + ( - 157 \beta_{3} + 92 \beta_{2} - 157 \beta_1) q^{89} + (62 \beta_{3} - 59 \beta_{2} + 16 \beta_1 - 176) q^{91} + ( - 44 \beta_{3} - 88) q^{92} + ( - 684 \beta_{2} + 72 \beta_1 - 684) q^{94} + (225 \beta_{2} - 10 \beta_1 + 225) q^{95} + (30 \beta_{3} + 284) q^{97} + ( - 52 \beta_{3} - 688 \beta_{2} - 8 \beta_1 - 430) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} - 6 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} - 6 q^{7} - 32 q^{8} - 20 q^{10} - 20 q^{11} + 84 q^{13} - 36 q^{14} - 32 q^{16} + 76 q^{17} - 90 q^{19} - 80 q^{20} - 80 q^{22} + 44 q^{23} - 50 q^{25} + 84 q^{26} - 48 q^{28} + 320 q^{29} - 62 q^{31} + 64 q^{32} + 304 q^{34} - 90 q^{35} + 358 q^{37} + 180 q^{38} - 80 q^{40} - 72 q^{41} - 268 q^{43} - 80 q^{44} - 88 q^{46} + 684 q^{47} + 946 q^{49} - 200 q^{50} - 168 q^{52} + 16 q^{53} - 200 q^{55} + 48 q^{56} + 320 q^{58} + 552 q^{59} - 1312 q^{61} - 248 q^{62} + 256 q^{64} + 210 q^{65} - 194 q^{67} + 304 q^{68} - 120 q^{70} - 760 q^{71} - 198 q^{73} - 716 q^{74} + 720 q^{76} - 1960 q^{77} - 126 q^{79} + 160 q^{80} - 72 q^{82} - 1728 q^{83} + 760 q^{85} - 268 q^{86} + 160 q^{88} - 184 q^{89} - 586 q^{91} - 352 q^{92} - 1368 q^{94} + 450 q^{95} + 1136 q^{97} - 344 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 46x^{2} + 2116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 46 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 46\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 46\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
3.39116 5.87367i
−3.39116 + 5.87367i
3.39116 + 5.87367i
−3.39116 5.87367i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 −18.4558 1.54354i −8.00000 0 −5.00000 + 8.66025i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 15.4558 + 10.2038i −8.00000 0 −5.00000 + 8.66025i
541.1 1.00000 1.73205i 0 −2.00000 3.46410i 2.50000 4.33013i 0 −18.4558 + 1.54354i −8.00000 0 −5.00000 8.66025i
541.2 1.00000 1.73205i 0 −2.00000 3.46410i 2.50000 4.33013i 0 15.4558 10.2038i −8.00000 0 −5.00000 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.k.n 4
3.b odd 2 1 210.4.i.h 4
7.c even 3 1 inner 630.4.k.n 4
21.g even 6 1 1470.4.a.bp 2
21.h odd 6 1 210.4.i.h 4
21.h odd 6 1 1470.4.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.i.h 4 3.b odd 2 1
210.4.i.h 4 21.h odd 6 1
630.4.k.n 4 1.a even 1 1 trivial
630.4.k.n 4 7.c even 3 1 inner
1470.4.a.bo 2 21.h odd 6 1
1470.4.a.bp 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{4} + 20T_{11}^{3} + 1450T_{11}^{2} - 21000T_{11} + 1102500 \) Copy content Toggle raw display
\( T_{13}^{2} - 42T_{13} + 395 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} - 455 T^{2} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} + 20 T^{3} + 1450 T^{2} + \cdots + 1102500 \) Copy content Toggle raw display
$13$ \( (T^{2} - 42 T + 395)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 76 T^{3} + 9898 T^{2} + \cdots + 16990884 \) Copy content Toggle raw display
$19$ \( T^{4} + 90 T^{3} + 6259 T^{2} + \cdots + 3389281 \) Copy content Toggle raw display
$23$ \( T^{4} - 44 T^{3} + 7018 T^{2} + \cdots + 25826724 \) Copy content Toggle raw display
$29$ \( (T^{2} - 160 T + 5250)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 62 T^{3} + 5827 T^{2} + \cdots + 3932289 \) Copy content Toggle raw display
$37$ \( T^{4} - 358 T^{3} + \cdots + 147987225 \) Copy content Toggle raw display
$41$ \( (T^{2} + 36 T - 33210)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 134 T - 29045)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 684 T^{3} + \cdots + 3288793104 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + \cdots + 13210284096 \) Copy content Toggle raw display
$59$ \( T^{4} - 552 T^{3} + \cdots + 4333062276 \) Copy content Toggle raw display
$61$ \( T^{4} + 1312 T^{3} + \cdots + 172583746624 \) Copy content Toggle raw display
$67$ \( T^{4} + 194 T^{3} + \cdots + 4612718889 \) Copy content Toggle raw display
$71$ \( (T^{2} + 380 T - 827274)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 198 T^{3} + \cdots + 17935225 \) Copy content Toggle raw display
$79$ \( T^{4} + 126 T^{3} + \cdots + 1596702650449 \) Copy content Toggle raw display
$83$ \( (T^{2} + 864 T - 807390)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 184 T^{3} + \cdots + 1266502652100 \) Copy content Toggle raw display
$97$ \( (T^{2} - 568 T + 39256)^{2} \) Copy content Toggle raw display
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