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Properties

Label 1176.3.z.c
Level $1176$
Weight $3$
Character orbit 1176.z
Analytic conductor $32.044$
Analytic rank $0$
Dimension $8$
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] = [1176,3,Mod(313,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.313");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1176.z (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: \(32.0436790888\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{5} - \beta_1 + 1) q^{5} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{5} - \beta_1 + 1) q^{5} + 3 \beta_1 q^{9} + (\beta_{7} + 2 \beta_{5} - \beta_{4} + \cdots - 5) q^{11}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{5} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 6 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 6 q^{5} + 12 q^{9} - 22 q^{11} - 12 q^{15} - 36 q^{17} - 42 q^{19} + 48 q^{23} + 42 q^{25} + 68 q^{29} + 60 q^{31} + 66 q^{33} - 118 q^{37} - 18 q^{39} - 92 q^{43} + 18 q^{45} + 12 q^{47} + 36 q^{51} + 10 q^{53} + 84 q^{57} + 54 q^{59} - 24 q^{61} - 148 q^{65} + 22 q^{67} - 392 q^{71} + 138 q^{73} - 126 q^{75} + 164 q^{79} - 36 q^{81} + 200 q^{85} - 102 q^{87} + 60 q^{89} - 60 q^{93} - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} + \cdots - 1034839 ) / 658490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 149\nu^{7} + 24\nu^{6} - 928\nu^{5} - 1609\nu^{4} + 5872\nu^{3} + 12912\nu^{2} + 57887\nu - 46552 ) / 9407 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6546 \nu^{7} - 1812 \nu^{6} + 70064 \nu^{5} + 173218 \nu^{4} - 443336 \nu^{3} - 974856 \nu^{2} + \cdots + 3514676 ) / 329245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11533 \nu^{7} - 30909 \nu^{6} - 121832 \nu^{5} + 29696 \nu^{4} + 997968 \nu^{3} + 162453 \nu^{2} + \cdots - 2603393 ) / 329245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2924 \nu^{7} - 2812 \nu^{6} - 16696 \nu^{5} - 26272 \nu^{4} + 189984 \nu^{3} + 39299 \nu^{2} + \cdots - 691564 ) / 47035 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45244 \nu^{7} + 5583 \nu^{6} - 215876 \nu^{5} - 916372 \nu^{4} + 1365974 \nu^{3} + 3003654 \nu^{2} + \cdots - 10829159 ) / 658490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + 2\beta_{2} - 8\beta _1 + 8 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 4\beta_{4} - 4\beta_{3} + 32\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 9\beta_{6} - 7\beta_{5} - 9\beta_{4} - 6\beta_{3} + 9\beta_{2} - 9\beta _1 + 99 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 23\beta_{4} + 10\beta_{2} + 121\beta _1 - 121 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33\beta_{6} - 49\beta_{5} + 141\beta_{4} - 141\beta_{3} + 736\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -91\beta_{7} + 75\beta_{6} - 91\beta_{5} - 75\beta_{4} + 160\beta_{3} + 75\beta_{2} - 75\beta _1 - 344 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\beta_{7} + 207\beta_{4} - 8\beta_{2} + 734\beta _1 - 734 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
313.1
−1.33172 + 1.34622i
2.40015 − 0.808379i
−1.90015 + 1.67440i
1.83172 − 0.480194i
−1.33172 − 1.34622i
2.40015 + 0.808379i
−1.90015 − 1.67440i
1.83172 + 0.480194i
0 −1.50000 + 0.866025i 0 −5.30550 − 3.06313i 0 0 0 1.50000 − 2.59808i 0
313.2 0 −1.50000 + 0.866025i 0 −3.18140 − 1.83678i 0 0 0 1.50000 − 2.59808i 0
313.3 0 −1.50000 + 0.866025i 0 4.68140 + 2.70281i 0 0 0 1.50000 − 2.59808i 0
313.4 0 −1.50000 + 0.866025i 0 6.80550 + 3.92916i 0 0 0 1.50000 − 2.59808i 0
913.1 0 −1.50000 − 0.866025i 0 −5.30550 + 3.06313i 0 0 0 1.50000 + 2.59808i 0
913.2 0 −1.50000 − 0.866025i 0 −3.18140 + 1.83678i 0 0 0 1.50000 + 2.59808i 0
913.3 0 −1.50000 − 0.866025i 0 4.68140 − 2.70281i 0 0 0 1.50000 + 2.59808i 0
913.4 0 −1.50000 − 0.866025i 0 6.80550 − 3.92916i 0 0 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 313.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.3.z.c 8
7.b odd 2 1 168.3.z.b ✓ 8
7.c even 3 1 168.3.z.b ✓ 8
7.c even 3 1 1176.3.f.c 8
7.d odd 6 1 1176.3.f.c 8
7.d odd 6 1 inner 1176.3.z.c 8
21.c even 2 1 504.3.by.c 8
21.g even 6 1 3528.3.f.b 8
21.h odd 6 1 504.3.by.c 8
21.h odd 6 1 3528.3.f.b 8
28.d even 2 1 336.3.bh.g 8
28.f even 6 1 2352.3.f.g 8
28.g odd 6 1 336.3.bh.g 8
28.g odd 6 1 2352.3.f.g 8
84.h odd 2 1 1008.3.cg.p 8
84.n even 6 1 1008.3.cg.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b ✓ 8 7.b odd 2 1
168.3.z.b ✓ 8 7.c even 3 1
336.3.bh.g 8 28.d even 2 1
336.3.bh.g 8 28.g odd 6 1
504.3.by.c 8 21.c even 2 1
504.3.by.c 8 21.h odd 6 1
1008.3.cg.p 8 84.h odd 2 1
1008.3.cg.p 8 84.n even 6 1
1176.3.f.c 8 7.c even 3 1
1176.3.f.c 8 7.d odd 6 1
1176.3.z.c 8 1.a even 1 1 trivial
1176.3.z.c 8 7.d odd 6 1 inner
2352.3.f.g 8 28.f even 6 1
2352.3.f.g 8 28.g odd 6 1
3528.3.f.b 8 21.g even 6 1
3528.3.f.b 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 6T_{5}^{7} - 53T_{5}^{6} + 390T_{5}^{5} + 2861T_{5}^{4} - 13260T_{5}^{3} - 48268T_{5}^{2} + 195024T_{5} + 913936 \) acting on \(S_{3}^{\mathrm{new}}(1176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 22 T^{7} + \cdots + 17875984 \) Copy content Toggle raw display
$13$ \( T^{8} + 262 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3470623744 \) Copy content Toggle raw display
$19$ \( T^{8} + 42 T^{7} + \cdots + 17272336 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22620160000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 34 T^{3} + \cdots + 224128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 25912950625 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 17069945104 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 580010189056 \) Copy content Toggle raw display
$43$ \( (T^{4} + 46 T^{3} + \cdots + 1658308)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 52408029184 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1491466217536 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1048985640000 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 43785853599744 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 95387087104 \) Copy content Toggle raw display
$71$ \( (T^{4} + 196 T^{3} + \cdots - 13209344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 622497709609216 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 839297841424 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 723343446016 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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