Properties

Label 630.2.j.i
Level $630$
Weight $2$
Character orbit 630.j
Analytic conductor $5.031$
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] = [630,2,Mod(211,630)]
 
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([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.211");
 
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.j (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{5} - \beta_{2}) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 1) q^{5} + \beta_{2} q^{6} - \beta_{3} q^{7} + q^{8} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{5} - \beta_{2}) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 1) q^{5} + \beta_{2} q^{6} - \beta_{3} q^{7} + q^{8} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9} + q^{10} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{11} - \beta_{5} q^{12} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{3} + 1) q^{14} - \beta_{5} q^{15} + \beta_{3} q^{16} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + \beta_1 - 3) q^{17} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1 + 1) q^{18} + ( - \beta_{5} + \beta_{4} + \beta_1 - 3) q^{19} + \beta_{3} q^{20} - \beta_{2} q^{21} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{22} + (3 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{23} + (\beta_{5} - \beta_{2}) q^{24} + \beta_{3} q^{25} + (2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{26} + (\beta_{4} - 4 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{27} - q^{28} + (\beta_{5} - 7 \beta_{3} + \beta_1) q^{29} + (\beta_{5} - \beta_{2}) q^{30} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{31} + ( - \beta_{3} - 1) q^{32} + (\beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{33} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{34} - q^{35} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{36} + (\beta_{5} - \beta_{2} - \beta_1 - 1) q^{37} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2}) q^{38} + (\beta_{5} - 6 \beta_{3} - \beta_{2} + 3 \beta_1 - 9) q^{39} + ( - \beta_{3} - 1) q^{40} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 4) q^{41} + \beta_{5} q^{42} + (4 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{4} + \beta_{2} - 2) q^{44} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{45} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 2 \beta_1 - 1) q^{46} + ( - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{47} + \beta_{2} q^{48} + ( - \beta_{3} - 1) q^{49} + ( - \beta_{3} - 1) q^{50} + ( - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{51} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{52} + (\beta_{5} + 4 \beta_{4} - 5 \beta_{2} - \beta_1 - 1) q^{53} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 4) q^{54} + ( - \beta_{4} + \beta_{2} - 2) q^{55} - \beta_{3} q^{56} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{57} + ( - \beta_{5} - \beta_{4} + 7 \beta_{3} + \beta_{2} - \beta_1 + 7) q^{58} + (6 \beta_{5} + 6 \beta_{4} + \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 1) q^{59} + \beta_{2} q^{60} + ( - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{4} + 2 \beta_{2} - 2) q^{62} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 - 1) q^{63} + q^{64} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{65} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{66} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{67} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{68} + ( - \beta_{5} - 4 \beta_{4} + 10 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 11) q^{69} - \beta_{3} q^{70} + (5 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} - 5 \beta_1 - 3) q^{71} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{72} + (2 \beta_{4} - 2 \beta_{2} - 3) q^{73} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{74} + \beta_{2} q^{75} + (3 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{76} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{77} + ( - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{78} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{79} + q^{80} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{81} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{82} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{5} + \beta_{2}) q^{84} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{85} + ( - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 7) q^{86} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 1) q^{87} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{88} - 2 q^{89} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{90} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{91} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{92} + (2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{94} + (3 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{95} - \beta_{5} q^{96} + ( - 8 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{97} + q^{98} + (3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - q^{3} - 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - q^{3} - 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{7} + 6 q^{8} + 5 q^{9} + 6 q^{10} + 7 q^{11} + 2 q^{12} + 3 q^{14} + 2 q^{15} - 3 q^{16} - 10 q^{17} + 2 q^{18} - 14 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} + 2 q^{23} - q^{24} - 3 q^{25} + 2 q^{27} - 6 q^{28} + 20 q^{29} - q^{30} + 8 q^{31} - 3 q^{32} + 13 q^{33} + 5 q^{34} - 6 q^{35} - 7 q^{36} - 8 q^{37} + 7 q^{38} - 34 q^{39} - 3 q^{40} + 7 q^{41} - 2 q^{42} + 15 q^{43} - 14 q^{44} - 7 q^{45} - 4 q^{46} - 2 q^{47} - q^{48} - 3 q^{49} - 3 q^{50} + q^{51} + 17 q^{54} - 14 q^{55} + 3 q^{56} - 13 q^{57} + 20 q^{58} + 7 q^{59} - q^{60} + 8 q^{61} - 16 q^{62} - 2 q^{63} + 6 q^{64} - 8 q^{66} - 3 q^{67} + 5 q^{68} + 26 q^{69} + 3 q^{70} - 32 q^{71} + 5 q^{72} - 14 q^{73} + 4 q^{74} - q^{75} + 7 q^{76} - 7 q^{77} + 38 q^{78} + 6 q^{79} + 6 q^{80} + 29 q^{81} - 14 q^{82} + 8 q^{83} + q^{84} + 5 q^{85} + 15 q^{86} - 4 q^{87} + 7 q^{88} - 12 q^{89} + 5 q^{90} + 2 q^{92} - 12 q^{93} - 2 q^{94} + 7 q^{95} + 2 q^{96} - 3 q^{97} + 6 q^{98} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
1.71903 + 0.211943i
0.403374 1.68443i
−1.62241 + 0.606458i
1.71903 0.211943i
0.403374 + 1.68443i
−1.62241 0.606458i
−0.500000 0.866025i −1.71903 + 0.211943i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.04307 + 1.38276i 0.500000 + 0.866025i 1.00000 2.91016 0.728674i 1.00000
211.2 −0.500000 0.866025i −0.403374 1.68443i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.25707 + 1.19154i 0.500000 + 0.866025i 1.00000 −2.67458 + 1.35891i 1.00000
211.3 −0.500000 0.866025i 1.62241 + 0.606458i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.285997 1.70828i 0.500000 + 0.866025i 1.00000 2.26442 + 1.96784i 1.00000
421.1 −0.500000 + 0.866025i −1.71903 0.211943i −0.500000 0.866025i −0.500000 0.866025i 1.04307 1.38276i 0.500000 0.866025i 1.00000 2.91016 + 0.728674i 1.00000
421.2 −0.500000 + 0.866025i −0.403374 + 1.68443i −0.500000 0.866025i −0.500000 0.866025i −1.25707 1.19154i 0.500000 0.866025i 1.00000 −2.67458 1.35891i 1.00000
421.3 −0.500000 + 0.866025i 1.62241 0.606458i −0.500000 0.866025i −0.500000 0.866025i −0.285997 + 1.70828i 0.500000 0.866025i 1.00000 2.26442 1.96784i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.j.i 6
3.b odd 2 1 1890.2.j.k 6
9.c even 3 1 inner 630.2.j.i 6
9.c even 3 1 5670.2.a.bs 3
9.d odd 6 1 1890.2.j.k 6
9.d odd 6 1 5670.2.a.bo 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.j.i 6 1.a even 1 1 trivial
630.2.j.i 6 9.c even 3 1 inner
1890.2.j.k 6 3.b odd 2 1
1890.2.j.k 6 9.d odd 6 1
5670.2.a.bo 3 9.d odd 6 1
5670.2.a.bs 3 9.c even 3 1

Hecke kernels

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

\( T_{11}^{6} - 7T_{11}^{5} + 38T_{11}^{4} - 79T_{11}^{3} + 128T_{11}^{2} + 11T_{11} + 1 \) Copy content Toggle raw display
\( T_{13}^{6} + 36T_{13}^{4} - 164T_{13}^{3} + 1296T_{13}^{2} - 2952T_{13} + 6724 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 2 T^{4} - 3 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 7 T^{5} + 38 T^{4} - 79 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} - 164 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$17$ \( (T^{3} + 5 T^{2} - 29 T - 141)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 7 T^{2} + 7 T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + 68 T^{4} + \cdots + 14884 \) Copy content Toggle raw display
$29$ \( T^{6} - 20 T^{5} + 272 T^{4} + \cdots + 68644 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + 64 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$37$ \( (T^{3} + 4 T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} + 82 T^{4} + \cdots + 23409 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + 222 T^{4} + \cdots + 196249 \) Copy content Toggle raw display
$47$ \( T^{6} + 2 T^{5} + 88 T^{4} - 216 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} - 126 T + 478)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 7 T^{5} + 214 T^{4} + \cdots + 1361889 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} + 54 T^{4} - 92 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} + 90 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$71$ \( (T^{3} + 16 T^{2} - 146 T - 2334)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 7 T^{2} - 5 T - 27)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 6 T^{5} + 120 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$83$ \( T^{6} - 8 T^{5} + 88 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T + 2)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + 294 T^{4} + \cdots + 3806401 \) Copy content Toggle raw display
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