Properties

Label 816.2.c.f
Level $816$
Weight $2$
Character orbit 816.c
Analytic conductor $6.516$
Analytic rank $0$
Dimension $6$
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] = [816,2,Mod(577,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.c (of order \(2\), degree \(1\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 408)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{7} - q^{9} + \beta_{5} q^{11} + ( - \beta_{4} + 2 \beta_1) q^{13} + ( - \beta_1 - 1) q^{15} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{4} - 2 \beta_1 - 2) q^{19} + (\beta_{4} - \beta_1 - 1) q^{21} + (3 \beta_{3} + \beta_{2}) q^{23} + (\beta_{4} - 2 \beta_1 - 1) q^{25} - \beta_{2} q^{27} + ( - \beta_{5} + \beta_{3} + 3 \beta_{2}) q^{29} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{31} + \beta_{4} q^{33} + (2 \beta_{4} - 4 \beta_1 - 4) q^{35} + (\beta_{5} - \beta_{3} - 3 \beta_{2}) q^{37} + (\beta_{5} - 2 \beta_{3}) q^{39} + (\beta_{5} - 4 \beta_{2}) q^{41} + ( - 3 \beta_{4} + 2 \beta_1 + 2) q^{43} + (\beta_{3} - \beta_{2}) q^{45} + 2 \beta_{4} q^{47} + ( - 4 \beta_1 - 5) q^{49} + (\beta_{5} + \beta_{2} + \beta_1 + 1) q^{51} + 6 q^{53} + (\beta_{4} - 2 \beta_1 + 2) q^{55} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{57} + 4 \beta_1 q^{59} + ( - \beta_{5} + \beta_{3} + 3 \beta_{2}) q^{61} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{63} + (3 \beta_{5} - 4 \beta_{3} + 8 \beta_{2}) q^{65} + 4 q^{67} + (3 \beta_1 - 1) q^{69} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{71} + (2 \beta_{5} + 8 \beta_{2}) q^{73} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2}) q^{75} + ( - 2 \beta_{4} - 8) q^{77} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2}) q^{79} + q^{81} + (2 \beta_{4} - 4) q^{83} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 6) q^{85} + ( - \beta_{4} + \beta_1 - 3) q^{87} + ( - 4 \beta_1 - 6) q^{89} + ( - 6 \beta_{3} + 14 \beta_{2}) q^{91} + (\beta_{4} - \beta_1 - 1) q^{93} + ( - \beta_{5} + 2 \beta_{3} - 14 \beta_{2}) q^{95} + (2 \beta_{5} - 2 \beta_{3} + 10 \beta_{2}) q^{97} - \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} - 2 q^{13} - 6 q^{15} + 4 q^{17} - 14 q^{19} - 4 q^{21} - 4 q^{25} + 2 q^{33} - 20 q^{35} + 6 q^{43} + 4 q^{47} - 30 q^{49} + 6 q^{51} + 36 q^{53} + 14 q^{55} + 24 q^{67} - 6 q^{69} - 52 q^{77} + 6 q^{81} - 20 q^{83} + 34 q^{85} - 20 q^{87} - 36 q^{89} - 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} - 3\nu^{4} + 8\nu^{3} - 11\nu^{2} + 8\nu - 20 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - 11\beta_{2} - 4\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + 3\beta_{4} - 2\beta_{3} + 8\beta_{2} - 7\beta _1 + 7 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.671462 + 1.24464i
1.40680 + 0.144584i
0.264658 1.38923i
0.264658 + 1.38923i
1.40680 0.144584i
−0.671462 1.24464i
0 1.00000i 0 3.48929i 0 4.68585i 0 −1.00000 0
577.2 0 1.00000i 0 1.28917i 0 3.62721i 0 −1.00000 0
577.3 0 1.00000i 0 1.77846i 0 0.941367i 0 −1.00000 0
577.4 0 1.00000i 0 1.77846i 0 0.941367i 0 −1.00000 0
577.5 0 1.00000i 0 1.28917i 0 3.62721i 0 −1.00000 0
577.6 0 1.00000i 0 3.48929i 0 4.68585i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.c.f 6
3.b odd 2 1 2448.2.c.r 6
4.b odd 2 1 408.2.c.b 6
8.b even 2 1 3264.2.c.n 6
8.d odd 2 1 3264.2.c.o 6
12.b even 2 1 1224.2.c.h 6
17.b even 2 1 inner 816.2.c.f 6
51.c odd 2 1 2448.2.c.r 6
68.d odd 2 1 408.2.c.b 6
68.f odd 4 1 6936.2.a.bb 3
68.f odd 4 1 6936.2.a.be 3
136.e odd 2 1 3264.2.c.o 6
136.h even 2 1 3264.2.c.n 6
204.h even 2 1 1224.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.c.b 6 4.b odd 2 1
408.2.c.b 6 68.d odd 2 1
816.2.c.f 6 1.a even 1 1 trivial
816.2.c.f 6 17.b even 2 1 inner
1224.2.c.h 6 12.b even 2 1
1224.2.c.h 6 204.h even 2 1
2448.2.c.r 6 3.b odd 2 1
2448.2.c.r 6 51.c odd 2 1
3264.2.c.n 6 8.b even 2 1
3264.2.c.n 6 136.h even 2 1
3264.2.c.o 6 8.d odd 2 1
3264.2.c.o 6 136.e odd 2 1
6936.2.a.bb 3 68.f odd 4 1
6936.2.a.be 3 68.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 17T_{5}^{4} + 64T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 17 T^{4} + 64 T^{2} + 64 \) Copy content Toggle raw display
$7$ \( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{6} + 33 T^{4} + 224 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{3} + T^{2} - 32 T - 76)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} - 9 T^{4} + 120 T^{3} + \cdots + 4913 \) Copy content Toggle raw display
$19$ \( (T^{3} + 7 T^{2} - 40 T - 272)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 129 T^{4} + 3648 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{6} + 89 T^{4} + 768 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( (T^{3} - 3 T^{2} - 136 T + 592)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 2 T^{2} - 64 T - 128)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{6} \) Copy content Toggle raw display
$59$ \( (T^{3} - 112 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T - 4)^{6} \) Copy content Toggle raw display
$71$ \( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{6} + 292 T^{4} + 20480 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( T^{6} + 272 T^{4} + 18496 T^{2} + \cdots + 369664 \) Copy content Toggle raw display
$83$ \( (T^{3} + 10 T^{2} - 32 T - 352)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} - 4 T - 584)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 400 T^{4} + 29696 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
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