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Properties

Label 882.2.u.e
Level $882$
Weight $2$
Character orbit 882.u
Analytic conductor $7.043$
Analytic rank $0$
Dimension $12$
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(127,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.u (of order \(7\), degree \(6\), 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: 12.0.7877952219361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{3} q^{4} + (\beta_{11} + \beta_{7} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots - \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{3} q^{4} + (\beta_{11} + \beta_{7} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{9} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 2 q^{4} + q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 2 q^{4} + q^{5} - 2 q^{8} + q^{10} + 6 q^{11} - q^{13} - 2 q^{16} + 7 q^{17} + 20 q^{19} + 8 q^{20} - 8 q^{22} + 21 q^{23} + 13 q^{25} - 8 q^{26} + 7 q^{28} - 3 q^{29} - 2 q^{31} - 2 q^{32} - 7 q^{34} - 28 q^{35} + 33 q^{37} + 6 q^{38} - 6 q^{40} + 14 q^{41} + 13 q^{43} - q^{44} - 7 q^{46} + 40 q^{47} + 14 q^{49} + 6 q^{50} + 13 q^{52} + 8 q^{53} + 4 q^{55} - 3 q^{58} - 6 q^{59} + 8 q^{61} - 9 q^{62} - 2 q^{64} - 17 q^{65} - 78 q^{67} - 28 q^{68} + 14 q^{70} + 2 q^{71} + 11 q^{73} - 2 q^{74} - 8 q^{76} - 14 q^{77} + 44 q^{79} - 6 q^{80} + 14 q^{82} + 46 q^{83} + 49 q^{85} - 22 q^{86} - q^{88} + 30 q^{89} - 35 q^{91} - 7 q^{92} - 16 q^{94} + 11 q^{95} - 30 q^{97} - 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} + \cdots - 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} - \nu^{10} + 18 \nu^{9} - 11 \nu^{8} - 36 \nu^{7} + 50 \nu^{6} + 49 \nu^{5} - 94 \nu^{4} + \cdots - 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} - 22 \nu^{9} + 57 \nu^{8} - 4 \nu^{7} - 166 \nu^{6} + 165 \nu^{5} + 186 \nu^{4} + \cdots - 352 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{10} + \nu^{9} + 12 \nu^{8} - 19 \nu^{7} - 18 \nu^{6} + 62 \nu^{5} - 3 \nu^{4} - 108 \nu^{3} + \cdots - 112 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{11} - 21 \nu^{10} + 6 \nu^{9} + 77 \nu^{8} - 112 \nu^{7} - 70 \nu^{6} + 305 \nu^{5} + \cdots - 416 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{11} + 11 \nu^{10} - 10 \nu^{9} - 35 \nu^{8} + 72 \nu^{7} + 2 \nu^{6} - 167 \nu^{5} + \cdots + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 31 \nu^{8} + 19 \nu^{7} + 60 \nu^{6} - 93 \nu^{5} - 31 \nu^{4} + \cdots + 112 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15 \nu^{11} + 15 \nu^{10} + 62 \nu^{9} - 135 \nu^{8} - 32 \nu^{7} + 370 \nu^{6} - 275 \nu^{5} + \cdots + 608 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13 \nu^{11} + 25 \nu^{10} + 54 \nu^{9} - 197 \nu^{8} + 76 \nu^{7} + 478 \nu^{6} - 625 \nu^{5} + \cdots + 1248 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} - 5 \beta_{6} + \beta_{5} + 5 \beta_{3} + \cdots + 4 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} - 4 \beta_{10} + \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + \beta_{6} + 4 \beta_{5} - 7 \beta_{4} + \cdots + 9 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - \beta_{8} - 3 \beta_{7} + 12 \beta_{6} + 6 \beta_{5} + \cdots + 3 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} - 12 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} + \cdots - 16 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} - 3\beta_{10} + 3\beta_{9} - 2\beta_{8} + \beta_{7} + 3\beta_{6} + 2\beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13 \beta_{11} - 22 \beta_{10} + 9 \beta_{9} - 20 \beta_{8} - 11 \beta_{7} + 2 \beta_{6} - 13 \beta_{5} + \cdots - 17 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 31 \beta_{11} - 11 \beta_{10} + 8 \beta_{9} - 31 \beta_{8} + 47 \beta_{7} - 27 \beta_{6} + 39 \beta_{5} + \cdots + 16 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} + \beta_{8} - 5 \beta_{6} - 2 \beta_{5} - 9 \beta_{4} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 48 \beta_{11} + 50 \beta_{10} + 12 \beta_{9} - 50 \beta_{8} - 24 \beta_{7} - 37 \beta_{6} + 13 \beta_{5} + \cdots + 31 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 66 \beta_{11} - 10 \beta_{10} - 50 \beta_{9} + 17 \beta_{8} - 103 \beta_{7} + 20 \beta_{6} + \cdots - 93 ) / 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.639551 − 1.26134i
1.38491 + 0.286410i
−1.25719 − 0.647667i
0.911180 + 1.08155i
1.23295 + 0.692694i
−1.41140 + 0.0891373i
1.23295 − 0.692694i
−1.41140 − 0.0891373i
−1.25719 + 0.647667i
0.911180 − 1.08155i
0.639551 + 1.26134i
1.38491 − 0.286410i
−0.900969 − 0.433884i 0 0.623490 + 0.781831i −0.393821 − 1.72544i 0 −2.46938 + 0.949814i −0.222521 − 0.974928i 0 −0.393821 + 1.72544i
127.2 −0.900969 − 0.433884i 0 0.623490 + 0.781831i −0.0621061 − 0.272104i 0 2.63695 − 0.215673i −0.222521 − 0.974928i 0 −0.0621061 + 0.272104i
253.1 0.623490 + 0.781831i 0 −0.222521 + 0.974928i −0.740920 + 0.356808i 0 2.11950 + 1.58357i −0.900969 + 0.433884i 0 −0.740920 − 0.356808i
253.2 0.623490 + 0.781831i 0 −0.222521 + 0.974928i 3.16635 − 1.52483i 0 0.0834024 − 2.64444i −0.900969 + 0.433884i 0 3.16635 + 1.52483i
379.1 −0.222521 − 0.974928i 0 −0.900969 + 0.433884i −2.38348 − 2.98879i 0 0.253488 − 2.63358i 0.623490 + 0.781831i 0 −2.38348 + 2.98879i
379.2 −0.222521 − 0.974928i 0 −0.900969 + 0.433884i 0.913978 + 1.14609i 0 −2.62396 − 0.338895i 0.623490 + 0.781831i 0 0.913978 − 1.14609i
505.1 −0.222521 + 0.974928i 0 −0.900969 − 0.433884i −2.38348 + 2.98879i 0 0.253488 + 2.63358i 0.623490 − 0.781831i 0 −2.38348 − 2.98879i
505.2 −0.222521 + 0.974928i 0 −0.900969 − 0.433884i 0.913978 − 1.14609i 0 −2.62396 + 0.338895i 0.623490 − 0.781831i 0 0.913978 + 1.14609i
631.1 0.623490 − 0.781831i 0 −0.222521 − 0.974928i −0.740920 − 0.356808i 0 2.11950 − 1.58357i −0.900969 − 0.433884i 0 −0.740920 + 0.356808i
631.2 0.623490 − 0.781831i 0 −0.222521 − 0.974928i 3.16635 + 1.52483i 0 0.0834024 + 2.64444i −0.900969 − 0.433884i 0 3.16635 − 1.52483i
757.1 −0.900969 + 0.433884i 0 0.623490 − 0.781831i −0.393821 + 1.72544i 0 −2.46938 − 0.949814i −0.222521 + 0.974928i 0 −0.393821 − 1.72544i
757.2 −0.900969 + 0.433884i 0 0.623490 − 0.781831i −0.0621061 + 0.272104i 0 2.63695 + 0.215673i −0.222521 + 0.974928i 0 −0.0621061 − 0.272104i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.u.e 12
3.b odd 2 1 294.2.i.c ✓ 12
49.e even 7 1 inner 882.2.u.e 12
147.l odd 14 1 294.2.i.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.i.c ✓ 12 3.b odd 2 1
294.2.i.c ✓ 12 147.l odd 14 1
882.2.u.e 12 1.a even 1 1 trivial
882.2.u.e 12 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - T_{5}^{11} - T_{5}^{10} - 39 T_{5}^{9} + 153 T_{5}^{8} - 5 T_{5}^{7} + 497 T_{5}^{6} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{12} - 7 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{12} + T^{11} + \cdots + 63001 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{11} + \cdots + 7529536 \) Copy content Toggle raw display
$19$ \( (T^{3} - 5 T^{2} + 6 T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} - 21 T^{11} + \cdots + 153664 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 322624 \) Copy content Toggle raw display
$31$ \( (T^{6} + T^{5} + \cdots - 10933)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 33 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$41$ \( T^{12} - 14 T^{11} + \cdots + 153664 \) Copy content Toggle raw display
$43$ \( T^{12} - 13 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2164854784 \) Copy content Toggle raw display
$53$ \( T^{12} - 8 T^{11} + \cdots + 322624 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2396298304 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 4294574089 \) Copy content Toggle raw display
$67$ \( (T^{6} + 39 T^{5} + \cdots - 283576)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1036916560681 \) Copy content Toggle raw display
$79$ \( (T^{6} - 22 T^{5} + \cdots + 156773)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 101767744 \) Copy content Toggle raw display
$89$ \( T^{12} - 30 T^{11} + \cdots + 3182656 \) Copy content Toggle raw display
$97$ \( (T^{6} + 15 T^{5} + \cdots + 536243)^{2} \) Copy content Toggle raw display
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