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Properties

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

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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: 8.0.621801639936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + 2 \beta_1 q^{4} - \beta_{2} q^{5} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + 2 \beta_1 q^{4} - \beta_{2} q^{5} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{8} + ( - \beta_{7} - \beta_{6}) q^{10} + 2 \beta_{5} q^{11} - \beta_{6} q^{13} + (4 \beta_1 - 4) q^{16} + 3 \beta_{3} q^{17} + 2 \beta_{7} q^{19} + (2 \beta_{3} - 2 \beta_{2}) q^{20} + 4 q^{22} - 30 \beta_{4} q^{23} + 49 \beta_1 q^{25} - 2 \beta_{2} q^{26} + ( - 11 \beta_{5} - 11 \beta_{4}) q^{29} + (2 \beta_{7} + 2 \beta_{6}) q^{31} - 4 \beta_{5} q^{32} + 3 \beta_{6} q^{34} + ( - 6 \beta_1 + 6) q^{37} - 4 \beta_{3} q^{38} - 2 \beta_{7} q^{40} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{41} - 68 q^{43} - 4 \beta_{4} q^{44} + 60 \beta_1 q^{46} + 8 \beta_{2} q^{47} + ( - 49 \beta_{5} - 49 \beta_{4}) q^{50} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{52} + 29 \beta_{5} q^{53} - 2 \beta_{6} q^{55} + (22 \beta_1 - 22) q^{58} - 8 \beta_{3} q^{59} - 8 \beta_{7} q^{61} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{62} - 8 q^{64} - 74 \beta_{4} q^{65} + 104 \beta_1 q^{67} + 6 \beta_{2} q^{68} + ( - 50 \beta_{5} - 50 \beta_{4}) q^{71} + (5 \beta_{7} + 5 \beta_{6}) q^{73} + 6 \beta_{5} q^{74} - 4 \beta_{6} q^{76} + ( - 20 \beta_1 + 20) q^{79} + 4 \beta_{3} q^{80} + 3 \beta_{7} q^{82} - 222 q^{85} + 68 \beta_{4} q^{86} + 8 \beta_1 q^{88} - 11 \beta_{2} q^{89} + ( - 60 \beta_{5} - 60 \beta_{4}) q^{92} + (8 \beta_{7} + 8 \beta_{6}) q^{94} + 148 \beta_{5} q^{95} + 13 \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} + 196 q^{25} + 24 q^{37} - 544 q^{43} + 240 q^{46} - 88 q^{58} - 64 q^{64} + 416 q^{67} + 80 q^{79} - 1776 q^{85} + 32 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 91\nu^{4} - 192\nu^{3} - 1187\nu^{2} + 1284\nu + 4425 ) / 2418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 91\nu^{4} + 192\nu^{3} + 2396\nu^{2} - 2493\nu - 15306 ) / 1209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -35\nu^{6} + 105\nu^{5} + 988\nu^{4} - 2151\nu^{3} - 8078\nu^{2} + 9171\nu + 4293 ) / 2418 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 724 \nu^{7} + 2534 \nu^{6} + 27020 \nu^{5} - 73885 \nu^{4} - 387688 \nu^{3} + 656684 \nu^{2} + \cdots - 872256 ) / 1374633 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1838 \nu^{7} + 6433 \nu^{6} + 45811 \nu^{5} - 130610 \nu^{4} - 262721 \nu^{3} + 527908 \nu^{2} + \cdots + 542487 ) / 2749266 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2896 \nu^{7} + 10136 \nu^{6} + 108080 \nu^{5} - 295540 \nu^{4} - 1550752 \nu^{3} + \cdots - 6238290 ) / 1374633 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8224 \nu^{7} + 28784 \nu^{6} + 291734 \nu^{5} - 801295 \nu^{4} - 3006376 \nu^{3} + \cdots - 1196985 ) / 1374633 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 4\beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 4\beta_{4} + 4\beta_{2} + 8\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 8\beta_{6} - 4\beta_{5} - 61\beta_{4} + 3\beta_{2} + 6\beta _1 + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} + 31\beta_{6} - 16\beta_{5} - 240\beta_{4} - 16\beta_{3} + 96\beta_{2} + 472\beta _1 + 382 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 220\beta_{7} + 309\beta_{6} - 776\beta_{5} - 2750\beta_{4} - 40\beta_{3} + 230\beta_{2} + 1160\beta _1 + 862 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 315\beta_{7} + 425\beta_{6} - 1144\beta_{5} - 3826\beta_{4} - 424\beta_{3} + 1054\beta_{2} + 7110\beta _1 + 1086 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5978 \beta_{7} + 5783 \beta_{6} - 31052 \beta_{5} - 59216 \beta_{4} - 2828 \beta_{3} + 6580 \beta_{2} + \cdots + 4650 ) / 4 \) 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
−3.76613 + 0.707107i
2.31664 + 0.707107i
4.76613 − 0.707107i
−1.31664 − 0.707107i
−3.76613 − 0.707107i
2.31664 − 0.707107i
4.76613 + 0.707107i
−1.31664 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 − 10.5357i
557.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 7.44983 − 4.30116i 0 0 2.82843i 0 −6.08276 + 10.5357i
557.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −7.44983 + 4.30116i 0 0 − 2.82843i 0 −6.08276 + 10.5357i
557.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 7.44983 − 4.30116i 0 0 − 2.82843i 0 6.08276 − 10.5357i
863.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −7.44983 − 4.30116i 0 0 − 2.82843i 0 6.08276 + 10.5357i
863.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 − 2.82843i 0 −6.08276 − 10.5357i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.44983 − 4.30116i 0 0 2.82843i 0 −6.08276 − 10.5357i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 + 10.5357i
\(n\): 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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.h 8
3.b odd 2 1 inner 882.3.s.h 8
7.b odd 2 1 inner 882.3.s.h 8
7.c even 3 1 882.3.b.g ✓ 4
7.c even 3 1 inner 882.3.s.h 8
7.d odd 6 1 882.3.b.g ✓ 4
7.d odd 6 1 inner 882.3.s.h 8
21.c even 2 1 inner 882.3.s.h 8
21.g even 6 1 882.3.b.g ✓ 4
21.g even 6 1 inner 882.3.s.h 8
21.h odd 6 1 882.3.b.g ✓ 4
21.h odd 6 1 inner 882.3.s.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.g ✓ 4 7.c even 3 1
882.3.b.g ✓ 4 7.d odd 6 1
882.3.b.g ✓ 4 21.g even 6 1
882.3.b.g ✓ 4 21.h odd 6 1
882.3.s.h 8 1.a even 1 1 trivial
882.3.s.h 8 3.b odd 2 1 inner
882.3.s.h 8 7.b odd 2 1 inner
882.3.s.h 8 7.c even 3 1 inner
882.3.s.h 8 7.d odd 6 1 inner
882.3.s.h 8 21.c even 2 1 inner
882.3.s.h 8 21.g even 6 1 inner
882.3.s.h 8 21.h odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - 74T_{5}^{2} + 5476 \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{2} - 148 \) 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
$5$ \( (T^{4} - 74 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 148)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 666 T^{2} + 443556)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 592 T^{2} + 350464)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1800 T^{2} + 3240000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 242)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 592 T^{2} + 350464)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 666)^{4} \) Copy content Toggle raw display
$43$ \( (T + 68)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4736 T^{2} + 22429696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1682 T^{2} + 2829124)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4736 T^{2} + 22429696)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 9472 T^{2} + 89718784)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 104 T + 10816)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 5000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3700 T^{2} + 13690000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 400)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 8954 T^{2} + 80174116)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 25012)^{4} \) Copy content Toggle raw display
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