Properties

Label 630.4.k.l
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 70)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 3 \beta_{3} + 5 \beta_{2} - \beta_1 + 4) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 3 \beta_{3} + 5 \beta_{2} - \beta_1 + 4) q^{7} + 8 q^{8} - 10 \beta_{2} q^{10} + (2 \beta_{3} + 34 \beta_{2} + 2 \beta_1) q^{11} + ( - 4 \beta_{3} - 26) q^{13} + (2 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 2) q^{14} + ( - 16 \beta_{2} - 16) q^{16} + ( - 8 \beta_{3} - 82 \beta_{2} - 8 \beta_1) q^{17} + (116 \beta_{2} - 4 \beta_1 + 116) q^{19} - 20 q^{20} + ( - 4 \beta_{3} + 68) q^{22} + ( - 99 \beta_{2} + 15 \beta_1 - 99) q^{23} + 25 \beta_{2} q^{25} + (52 \beta_{2} - 8 \beta_1 + 52) q^{26} + (8 \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 20) q^{28} + ( - 18 \beta_{3} + 9) q^{29} + (30 \beta_{3} + 98 \beta_{2} + 30 \beta_1) q^{31} + 32 \beta_{2} q^{32} + (16 \beta_{3} - 164) q^{34} + ( - 5 \beta_{3} + 20 \beta_{2} + 10 \beta_1 - 5) q^{35} + ( - 80 \beta_{2} - 12 \beta_1 - 80) q^{37} + (8 \beta_{3} - 232 \beta_{2} + 8 \beta_1) q^{38} + (40 \beta_{2} + 40) q^{40} + (38 \beta_{3} - 31) q^{41} + (25 \beta_{3} + 99) q^{43} + ( - 136 \beta_{2} - 8 \beta_1 - 136) q^{44} + ( - 30 \beta_{3} + 198 \beta_{2} - 30 \beta_1) q^{46} + (82 \beta_{2} + 8 \beta_1 + 82) q^{47} + ( - 4 \beta_{3} - 215 \beta_{2} + 22 \beta_1 + 129) q^{49} + 50 q^{50} + (16 \beta_{3} - 104 \beta_{2} + 16 \beta_1) q^{52} + (8 \beta_{3} - 20 \beta_{2} + 8 \beta_1) q^{53} + (10 \beta_{3} - 170) q^{55} + ( - 24 \beta_{3} + 40 \beta_{2} - 8 \beta_1 + 32) q^{56} + ( - 18 \beta_{2} - 36 \beta_1 - 18) q^{58} + (104 \beta_{3} + 40 \beta_{2} + 104 \beta_1) q^{59} + (87 \beta_{2} + 88 \beta_1 + 87) q^{61} + ( - 60 \beta_{3} + 196) q^{62} + 64 q^{64} + ( - 130 \beta_{2} + 20 \beta_1 - 130) q^{65} + (67 \beta_{3} - 527 \beta_{2} + 67 \beta_1) q^{67} + (328 \beta_{2} + 32 \beta_1 + 328) q^{68} + ( - 20 \beta_{3} + 10 \beta_{2} - 30 \beta_1 + 50) q^{70} + ( - 40 \beta_{3} + 416) q^{71} + ( - 48 \beta_{3} + 410 \beta_{2} - 48 \beta_1) q^{73} + (24 \beta_{3} + 160 \beta_{2} + 24 \beta_1) q^{74} + ( - 16 \beta_{3} - 464) q^{76} + (76 \beta_{3} + 242 \beta_{2} + 100 \beta_1 - 78) q^{77} + (288 \beta_{2} + 64 \beta_1 + 288) q^{79} - 80 \beta_{2} q^{80} + (62 \beta_{2} + 76 \beta_1 + 62) q^{82} + ( - 127 \beta_{3} + 149) q^{83} + ( - 40 \beta_{3} + 410) q^{85} + ( - 198 \beta_{2} + 50 \beta_1 - 198) q^{86} + (16 \beta_{3} + 272 \beta_{2} + 16 \beta_1) q^{88} + ( - 91 \beta_{2} + 76 \beta_1 - 91) q^{89} + (82 \beta_{3} - 314 \beta_{2} + 46 \beta_1 + 264) q^{91} + (60 \beta_{3} + 396) q^{92} + ( - 16 \beta_{3} - 164 \beta_{2} - 16 \beta_1) q^{94} + ( - 20 \beta_{3} + 580 \beta_{2} - 20 \beta_1) q^{95} + (48 \beta_{3} + 446) q^{97} + ( - 44 \beta_{3} - 258 \beta_{2} - 52 \beta_1 - 688) 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 \( 4 q - 4 q^{2} - 8 q^{4} + 10 q^{5} + 6 q^{7} + 32 q^{8} + 20 q^{10} - 68 q^{11} - 104 q^{13} + 24 q^{14} - 32 q^{16} + 164 q^{17} + 232 q^{19} - 80 q^{20} + 272 q^{22} - 198 q^{23} - 50 q^{25} + 104 q^{26} - 72 q^{28} + 36 q^{29} - 196 q^{31} - 64 q^{32} - 656 q^{34} - 60 q^{35} - 160 q^{37} + 464 q^{38} + 80 q^{40} - 124 q^{41} + 396 q^{43} - 272 q^{44} - 396 q^{46} + 164 q^{47} + 946 q^{49} + 200 q^{50} + 208 q^{52} + 40 q^{53} - 680 q^{55} + 48 q^{56} - 36 q^{58} - 80 q^{59} + 174 q^{61} + 784 q^{62} + 256 q^{64} - 260 q^{65} + 1054 q^{67} + 656 q^{68} + 180 q^{70} + 1664 q^{71} - 820 q^{73} - 320 q^{74} - 1856 q^{76} - 796 q^{77} + 576 q^{79} + 160 q^{80} + 124 q^{82} + 596 q^{83} + 1640 q^{85} - 396 q^{86} - 544 q^{88} - 182 q^{89} + 1684 q^{91} + 1584 q^{92} + 328 q^{94} - 1160 q^{95} + 1784 q^{97} - 2236 q^{98}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

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

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 −15.4558 + 10.2038i 8.00000 0 5.00000 8.66025i
361.2 −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
541.1 −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
541.2 −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
\(n\): 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.l 4
3.b odd 2 1 70.4.e.d 4
7.c even 3 1 inner 630.4.k.l 4
12.b even 2 1 560.4.q.j 4
15.d odd 2 1 350.4.e.h 4
15.e even 4 2 350.4.j.g 8
21.c even 2 1 490.4.e.u 4
21.g even 6 1 490.4.a.t 2
21.g even 6 1 490.4.e.u 4
21.h odd 6 1 70.4.e.d 4
21.h odd 6 1 490.4.a.r 2
84.n even 6 1 560.4.q.j 4
105.o odd 6 1 350.4.e.h 4
105.o odd 6 1 2450.4.a.bz 2
105.p even 6 1 2450.4.a.bv 2
105.x even 12 2 350.4.j.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.d 4 3.b odd 2 1
70.4.e.d 4 21.h odd 6 1
350.4.e.h 4 15.d odd 2 1
350.4.e.h 4 105.o odd 6 1
350.4.j.g 8 15.e even 4 2
350.4.j.g 8 105.x even 12 2
490.4.a.r 2 21.h odd 6 1
490.4.a.t 2 21.g even 6 1
490.4.e.u 4 21.c even 2 1
490.4.e.u 4 21.g even 6 1
560.4.q.j 4 12.b even 2 1
560.4.q.j 4 84.n even 6 1
630.4.k.l 4 1.a even 1 1 trivial
630.4.k.l 4 7.c even 3 1 inner
2450.4.a.bv 2 105.p even 6 1
2450.4.a.bz 2 105.o odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 68T_{11}^{3} + 3652T_{11}^{2} + 66096T_{11} + 944784 \) Copy content Toggle raw display
\( T_{13}^{2} + 52T_{13} - 60 \) 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
$11$ \( T^{4} + 68 T^{3} + 3652 T^{2} + \cdots + 944784 \) Copy content Toggle raw display
$13$ \( (T^{2} + 52 T - 60)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 164 T^{3} + \cdots + 14288400 \) Copy content Toggle raw display
$19$ \( T^{4} - 232 T^{3} + \cdots + 161798400 \) Copy content Toggle raw display
$23$ \( T^{4} + 198 T^{3} + 39753 T^{2} + \cdots + 301401 \) Copy content Toggle raw display
$29$ \( (T^{2} - 18 T - 14823)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 196 T^{3} + \cdots + 1010985616 \) Copy content Toggle raw display
$37$ \( T^{4} + 160 T^{3} + 25824 T^{2} + \cdots + 50176 \) Copy content Toggle raw display
$41$ \( (T^{2} + 62 T - 65463)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 198 T - 18949)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 164 T^{3} + \cdots + 14288400 \) Copy content Toggle raw display
$53$ \( T^{4} - 40 T^{3} + 4144 T^{2} + \cdots + 6471936 \) Copy content Toggle raw display
$59$ \( T^{4} + 80 T^{3} + \cdots + 245952516096 \) Copy content Toggle raw display
$61$ \( T^{4} - 174 T^{3} + \cdots + 121560309025 \) Copy content Toggle raw display
$67$ \( T^{4} - 1054 T^{3} + \cdots + 5074425225 \) Copy content Toggle raw display
$71$ \( (T^{2} - 832 T + 99456)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 820 T^{3} + \cdots + 3858397456 \) Copy content Toggle raw display
$79$ \( T^{4} - 576 T^{3} + \cdots + 11124342784 \) Copy content Toggle raw display
$83$ \( (T^{2} - 298 T - 719733)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 182 T^{3} + \cdots + 66262482225 \) Copy content Toggle raw display
$97$ \( (T^{2} - 892 T + 92932)^{2} \) Copy content Toggle raw display
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