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Properties

Label 882.3.s.e
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,3,Mod(557,882)]
 
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([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.557");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.s (of order \(6\), degree \(2\), not 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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{2}) q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{5} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{2}) q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{5} + 2 \beta_{6} q^{8} + (\beta_{5} - 2 \beta_1) q^{10} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{3} - 8) q^{13} + (4 \beta_1 - 4) q^{16} + (\beta_{4} - 13 \beta_{2}) q^{17} + (20 \beta_1 - 20) q^{19} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{20} + (2 \beta_{3} + 4) q^{22} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{2}) q^{23} + ( - 2 \beta_{5} + 33 \beta_1) q^{25} + ( - 2 \beta_{7} - 8 \beta_{6} - 8 \beta_{2}) q^{26} + (2 \beta_{7} + 19 \beta_{6} - 2 \beta_{4}) q^{29} + ( - 2 \beta_{5} - 4 \beta_1) q^{31} - 4 \beta_{2} q^{32} + ( - \beta_{3} - 26) q^{34} + (38 \beta_1 - 38) q^{37} - 20 \beta_{2} q^{38} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_1 + 4) q^{40} + ( - 3 \beta_{7} - 27 \beta_{6} + 3 \beta_{4}) q^{41} + ( - 6 \beta_{3} + 20) q^{43} + ( - 4 \beta_{7} + 4 \beta_{6} + 4 \beta_{2}) q^{44} + (2 \beta_{5} - 4 \beta_1) q^{46} + (12 \beta_{6} + 12 \beta_{2}) q^{47} + ( - 4 \beta_{7} + 33 \beta_{6} + 4 \beta_{4}) q^{50} + ( - 2 \beta_{5} - 16 \beta_1) q^{52} + (12 \beta_{4} + 3 \beta_{2}) q^{53} + ( - 4 \beta_{3} - 116) q^{55} + (2 \beta_{5} + 2 \beta_{3} + 38 \beta_1 - 38) q^{58} + ( - 4 \beta_{4} - 20 \beta_{2}) q^{59} + ( - 4 \beta_{5} - 4 \beta_{3} + \cdots - 58) q^{61}+ \cdots + (11 \beta_{3} - 72) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 8 q^{10} - 64 q^{13} - 16 q^{16} - 80 q^{19} + 32 q^{22} + 132 q^{25} - 16 q^{31} - 208 q^{34} - 152 q^{37} + 16 q^{40} + 160 q^{43} - 16 q^{46} - 64 q^{52} - 928 q^{55} - 152 q^{58} - 232 q^{61} - 64 q^{64} + 192 q^{67} + 96 q^{73} - 320 q^{76} - 304 q^{79} + 216 q^{82} + 656 q^{85} + 32 q^{88} + 96 q^{94} - 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 203\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} - 592 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 1066\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -92\nu^{6} + 880\nu^{4} - 5060\nu^{2} + 6624 ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 341\nu^{3} + 81\nu ) / 297 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -158\nu^{7} + 1210\nu^{5} - 8690\nu^{3} + 11376\nu ) / 1485 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} - 16\beta _1 + 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 22\beta_{6} + 5\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -31\beta_{7} + 158\beta_{6} + 158\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -55\beta_{3} - 592 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -203\beta_{4} + 1066\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−1.00781 − 0.581861i
2.23256 + 1.28897i
−2.23256 − 1.28897i
1.00781 + 0.581861i
−1.00781 + 0.581861i
2.23256 − 1.28897i
−2.23256 + 1.28897i
1.00781 − 0.581861i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −5.25600 + 3.03455i 0 0 2.82843i 0 4.29150 − 7.43310i
557.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 7.70549 − 4.44876i 0 0 2.82843i 0 −6.29150 + 10.8972i
557.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −7.70549 + 4.44876i 0 0 − 2.82843i 0 −6.29150 + 10.8972i
557.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 5.25600 − 3.03455i 0 0 − 2.82843i 0 4.29150 − 7.43310i
863.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −5.25600 − 3.03455i 0 0 − 2.82843i 0 4.29150 + 7.43310i
863.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 7.70549 + 4.44876i 0 0 − 2.82843i 0 −6.29150 − 10.8972i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.70549 − 4.44876i 0 0 2.82843i 0 −6.29150 − 10.8972i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 5.25600 + 3.03455i 0 0 2.82843i 0 4.29150 + 7.43310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.e 8
3.b odd 2 1 inner 882.3.s.e 8
7.b odd 2 1 882.3.s.i 8
7.c even 3 1 126.3.b.a ✓ 4
7.c even 3 1 inner 882.3.s.e 8
7.d odd 6 1 882.3.b.f 4
7.d odd 6 1 882.3.s.i 8
21.c even 2 1 882.3.s.i 8
21.g even 6 1 882.3.b.f 4
21.g even 6 1 882.3.s.i 8
21.h odd 6 1 126.3.b.a ✓ 4
21.h odd 6 1 inner 882.3.s.e 8
28.g odd 6 1 1008.3.d.a 4
35.j even 6 1 3150.3.e.e 4
35.l odd 12 2 3150.3.c.b 8
56.k odd 6 1 4032.3.d.j 4
56.p even 6 1 4032.3.d.i 4
63.g even 3 1 1134.3.q.c 8
63.h even 3 1 1134.3.q.c 8
63.j odd 6 1 1134.3.q.c 8
63.n odd 6 1 1134.3.q.c 8
84.n even 6 1 1008.3.d.a 4
105.o odd 6 1 3150.3.e.e 4
105.x even 12 2 3150.3.c.b 8
168.s odd 6 1 4032.3.d.i 4
168.v even 6 1 4032.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a ✓ 4 7.c even 3 1
126.3.b.a ✓ 4 21.h odd 6 1
882.3.b.f 4 7.d odd 6 1
882.3.b.f 4 21.g even 6 1
882.3.s.e 8 1.a even 1 1 trivial
882.3.s.e 8 3.b odd 2 1 inner
882.3.s.e 8 7.c even 3 1 inner
882.3.s.e 8 21.h odd 6 1 inner
882.3.s.i 8 7.b odd 2 1
882.3.s.i 8 7.d odd 6 1
882.3.s.i 8 21.c even 2 1
882.3.s.i 8 21.g even 6 1
1008.3.d.a 4 28.g odd 6 1
1008.3.d.a 4 84.n even 6 1
1134.3.q.c 8 63.g even 3 1
1134.3.q.c 8 63.h even 3 1
1134.3.q.c 8 63.j odd 6 1
1134.3.q.c 8 63.n odd 6 1
3150.3.c.b 8 35.l odd 12 2
3150.3.c.b 8 105.x even 12 2
3150.3.e.e 4 35.j even 6 1
3150.3.e.e 4 105.o odd 6 1
4032.3.d.i 4 56.p even 6 1
4032.3.d.i 4 168.s odd 6 1
4032.3.d.j 4 56.k odd 6 1
4032.3.d.j 4 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} - 116T_{5}^{6} + 10540T_{5}^{4} - 338256T_{5}^{2} + 8503056 \) Copy content Toggle raw display
\( T_{11}^{8} - 464T_{11}^{6} + 168640T_{11}^{4} - 21648384T_{11}^{2} + 2176782336 \) Copy content Toggle raw display
\( T_{13}^{2} + 16T_{13} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 116 T^{6} + \cdots + 8503056 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 2176782336 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16 T - 48)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 6324066576 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T + 400)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2176782336 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1892 T^{2} + 248004)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 38 T + 1444)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3924 T^{2} + 910116)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T - 3632)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} - 3392 T^{6} + \cdots + 84934656 \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{3} + \cdots + 2471184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 96 T^{3} + \cdots + 23658496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 464 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 48 T^{3} + \cdots + 4946176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 152 T^{3} + \cdots + 15872256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 18176 T^{2} + 53231616)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 196741925136 \) Copy content Toggle raw display
$97$ \( (T^{2} + 144 T - 8368)^{4} \) Copy content Toggle raw display
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