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Properties

Label 882.2.f.j
Level $882$
Weight $2$
Character orbit 882.f
Analytic conductor $7.043$
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] = [882,2,Mod(295,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.f (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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

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

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
1.22474 + 0.707107i
−1.22474 − 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−0.500000 + 0.866025i −1.00000 − 1.41421i −0.500000 − 0.866025i −0.724745 − 1.25529i 1.72474 − 0.158919i 0 1.00000 −1.00000 + 2.82843i 1.44949
295.2 −0.500000 + 0.866025i −1.00000 + 1.41421i −0.500000 − 0.866025i 1.72474 + 2.98735i −0.724745 − 1.57313i 0 1.00000 −1.00000 − 2.82843i −3.44949
589.1 −0.500000 − 0.866025i −1.00000 − 1.41421i −0.500000 + 0.866025i 1.72474 − 2.98735i −0.724745 + 1.57313i 0 1.00000 −1.00000 + 2.82843i −3.44949
589.2 −0.500000 − 0.866025i −1.00000 + 1.41421i −0.500000 + 0.866025i −0.724745 + 1.25529i 1.72474 + 0.158919i 0 1.00000 −1.00000 − 2.82843i 1.44949
\(n\): e.g. 2-40 or 990-1000
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 882.2.f.j 4
3.b odd 2 1 2646.2.f.k 4
7.b odd 2 1 126.2.f.c ✓ 4
7.c even 3 1 882.2.e.n 4
7.c even 3 1 882.2.h.l 4
7.d odd 6 1 882.2.e.m 4
7.d odd 6 1 882.2.h.k 4
9.c even 3 1 inner 882.2.f.j 4
9.c even 3 1 7938.2.a.bn 2
9.d odd 6 1 2646.2.f.k 4
9.d odd 6 1 7938.2.a.bm 2
21.c even 2 1 378.2.f.d 4
21.g even 6 1 2646.2.e.l 4
21.g even 6 1 2646.2.h.m 4
21.h odd 6 1 2646.2.e.k 4
21.h odd 6 1 2646.2.h.n 4
28.d even 2 1 1008.2.r.e 4
63.g even 3 1 882.2.e.n 4
63.h even 3 1 882.2.h.l 4
63.i even 6 1 2646.2.h.m 4
63.j odd 6 1 2646.2.h.n 4
63.k odd 6 1 882.2.e.m 4
63.l odd 6 1 126.2.f.c ✓ 4
63.l odd 6 1 1134.2.a.p 2
63.n odd 6 1 2646.2.e.k 4
63.o even 6 1 378.2.f.d 4
63.o even 6 1 1134.2.a.i 2
63.s even 6 1 2646.2.e.l 4
63.t odd 6 1 882.2.h.k 4
84.h odd 2 1 3024.2.r.e 4
252.s odd 6 1 3024.2.r.e 4
252.s odd 6 1 9072.2.a.bd 2
252.bi even 6 1 1008.2.r.e 4
252.bi even 6 1 9072.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c ✓ 4 7.b odd 2 1
126.2.f.c ✓ 4 63.l odd 6 1
378.2.f.d 4 21.c even 2 1
378.2.f.d 4 63.o even 6 1
882.2.e.m 4 7.d odd 6 1
882.2.e.m 4 63.k odd 6 1
882.2.e.n 4 7.c even 3 1
882.2.e.n 4 63.g even 3 1
882.2.f.j 4 1.a even 1 1 trivial
882.2.f.j 4 9.c even 3 1 inner
882.2.h.k 4 7.d odd 6 1
882.2.h.k 4 63.t odd 6 1
882.2.h.l 4 7.c even 3 1
882.2.h.l 4 63.h even 3 1
1008.2.r.e 4 28.d even 2 1
1008.2.r.e 4 252.bi even 6 1
1134.2.a.i 2 63.o even 6 1
1134.2.a.p 2 63.l odd 6 1
2646.2.e.k 4 21.h odd 6 1
2646.2.e.k 4 63.n odd 6 1
2646.2.e.l 4 21.g even 6 1
2646.2.e.l 4 63.s even 6 1
2646.2.f.k 4 3.b odd 2 1
2646.2.f.k 4 9.d odd 6 1
2646.2.h.m 4 21.g even 6 1
2646.2.h.m 4 63.i even 6 1
2646.2.h.n 4 21.h odd 6 1
2646.2.h.n 4 63.j odd 6 1
3024.2.r.e 4 84.h odd 2 1
3024.2.r.e 4 252.s odd 6 1
7938.2.a.bm 2 9.d odd 6 1
7938.2.a.bn 2 9.c even 3 1
9072.2.a.bd 2 252.s odd 6 1
9072.2.a.bk 2 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 2T_{5}^{3} + 9T_{5}^{2} + 10T_{5} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 24T_{13}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 18 T^{3} + 249 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 216 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$71$ \( (T^{2} - 10 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 120)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
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