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Properties

Label 578.2.d.e
Level $578$
Weight $2$
Character orbit 578.d
Analytic conductor $4.615$
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] = [578,2,Mod(155,578)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(578, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("578.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.d (of order \(8\), degree \(4\), 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: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 34)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{5} q^{3} - \beta_{4} q^{4} - \beta_{3} q^{6} + 2 \beta_{7} q^{7} + \beta_{2} q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{5} q^{3} - \beta_{4} q^{4} - \beta_{3} q^{6} + 2 \beta_{7} q^{7} + \beta_{2} q^{8} - \beta_{2} q^{9} - 3 \beta_{3} q^{11} + \beta_1 q^{12} + 2 \beta_{4} q^{13} - 2 \beta_{5} q^{14} - q^{16} + q^{18} - 4 \beta_{6} q^{19} - 8 \beta_{4} q^{21} + 3 \beta_1 q^{22} + \beta_{7} q^{24} - 5 \beta_{2} q^{25} - 2 \beta_{2} q^{26} + 2 \beta_{7} q^{27} + 2 \beta_{3} q^{28} - 2 \beta_{5} q^{31} - \beta_{6} q^{32} + 12 q^{33} + \beta_{6} q^{36} + 2 \beta_{5} q^{37} + 4 \beta_{4} q^{38} - 2 \beta_1 q^{39} - 3 \beta_{7} q^{41} + 8 \beta_{2} q^{42} - 8 \beta_{2} q^{43} + 3 \beta_{7} q^{44} - \beta_{5} q^{48} - 9 \beta_{6} q^{49} + 5 q^{50} + 2 q^{52} - 6 \beta_{6} q^{53} - 2 \beta_{5} q^{54} - 2 \beta_1 q^{56} + 4 \beta_{3} q^{57} - 2 \beta_{7} q^{61} + 2 \beta_{3} q^{62} + 2 \beta_1 q^{63} + \beta_{4} q^{64} + 12 \beta_{6} q^{66} - 8 q^{67} - \beta_{4} q^{72} + \beta_1 q^{73} - 2 \beta_{3} q^{74} - 5 \beta_{7} q^{75} - 4 \beta_{2} q^{76} + 24 \beta_{2} q^{77} - 2 \beta_{7} q^{78} - 4 \beta_{3} q^{79} - 11 \beta_{4} q^{81} + 3 \beta_{5} q^{82} - 8 q^{84} + 8 q^{86} - 3 \beta_{5} q^{88} + 6 \beta_{4} q^{89} - 4 \beta_{3} q^{91} + 8 \beta_{2} q^{93} + \beta_{3} q^{96} - 7 \beta_1 q^{97} + 9 \beta_{4} q^{98} + 3 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{16} + 8 q^{18} + 96 q^{33} + 40 q^{50} + 16 q^{52} - 64 q^{67} - 64 q^{84} + 64 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{16}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{16}^{7} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( \beta_{7} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.382683 − 0.923880i
0.382683 + 0.923880i
−0.382683 + 0.923880i
0.382683 − 0.923880i
0.923880 + 0.382683i
−0.923880 − 0.382683i
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.707107 + 0.707107i −1.84776 + 0.765367i 1.00000i 0 −1.84776 − 0.765367i 1.53073 − 3.69552i −0.707107 + 0.707107i 0.707107 − 0.707107i 0
155.2 0.707107 + 0.707107i 1.84776 − 0.765367i 1.00000i 0 1.84776 + 0.765367i −1.53073 + 3.69552i −0.707107 + 0.707107i 0.707107 − 0.707107i 0
179.1 0.707107 − 0.707107i −1.84776 − 0.765367i − 1.00000i 0 −1.84776 + 0.765367i 1.53073 + 3.69552i −0.707107 − 0.707107i 0.707107 + 0.707107i 0
179.2 0.707107 − 0.707107i 1.84776 + 0.765367i − 1.00000i 0 1.84776 − 0.765367i −1.53073 − 3.69552i −0.707107 − 0.707107i 0.707107 + 0.707107i 0
399.1 −0.707107 + 0.707107i −0.765367 + 1.84776i − 1.00000i 0 −0.765367 − 1.84776i −3.69552 + 1.53073i 0.707107 + 0.707107i −0.707107 − 0.707107i 0
399.2 −0.707107 + 0.707107i 0.765367 − 1.84776i − 1.00000i 0 0.765367 + 1.84776i 3.69552 − 1.53073i 0.707107 + 0.707107i −0.707107 − 0.707107i 0
423.1 −0.707107 − 0.707107i −0.765367 − 1.84776i 1.00000i 0 −0.765367 + 1.84776i −3.69552 − 1.53073i 0.707107 − 0.707107i −0.707107 + 0.707107i 0
423.2 −0.707107 − 0.707107i 0.765367 + 1.84776i 1.00000i 0 0.765367 − 1.84776i 3.69552 + 1.53073i 0.707107 − 0.707107i −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 578.2.d.e 8
17.b even 2 1 inner 578.2.d.e 8
17.c even 4 2 inner 578.2.d.e 8
17.d even 8 4 inner 578.2.d.e 8
17.e odd 16 1 34.2.a.a ✓ 1
17.e odd 16 1 578.2.a.a 1
17.e odd 16 2 578.2.b.a 2
17.e odd 16 4 578.2.c.e 4
51.i even 16 1 306.2.a.a 1
51.i even 16 1 5202.2.a.d 1
68.i even 16 1 272.2.a.d 1
68.i even 16 1 4624.2.a.a 1
85.o even 16 1 850.2.c.b 2
85.p odd 16 1 850.2.a.e 1
85.r even 16 1 850.2.c.b 2
119.p even 16 1 1666.2.a.m 1
136.q odd 16 1 1088.2.a.l 1
136.s even 16 1 1088.2.a.d 1
187.m even 16 1 4114.2.a.a 1
204.t odd 16 1 2448.2.a.k 1
221.y odd 16 1 5746.2.a.b 1
255.be even 16 1 7650.2.a.ci 1
340.bg even 16 1 6800.2.a.b 1
408.bg odd 16 1 9792.2.a.bj 1
408.bm even 16 1 9792.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a ✓ 1 17.e odd 16 1
272.2.a.d 1 68.i even 16 1
306.2.a.a 1 51.i even 16 1
578.2.a.a 1 17.e odd 16 1
578.2.b.a 2 17.e odd 16 2
578.2.c.e 4 17.e odd 16 4
578.2.d.e 8 1.a even 1 1 trivial
578.2.d.e 8 17.b even 2 1 inner
578.2.d.e 8 17.c even 4 2 inner
578.2.d.e 8 17.d even 8 4 inner
850.2.a.e 1 85.p odd 16 1
850.2.c.b 2 85.o even 16 1
850.2.c.b 2 85.r even 16 1
1088.2.a.d 1 136.s even 16 1
1088.2.a.l 1 136.q odd 16 1
1666.2.a.m 1 119.p even 16 1
2448.2.a.k 1 204.t odd 16 1
4114.2.a.a 1 187.m even 16 1
4624.2.a.a 1 68.i even 16 1
5202.2.a.d 1 51.i even 16 1
5746.2.a.b 1 221.y odd 16 1
6800.2.a.b 1 340.bg even 16 1
7650.2.a.ci 1 255.be even 16 1
9792.2.a.y 1 408.bm even 16 1
9792.2.a.bj 1 408.bg odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 256 \) acting on \(S_{2}^{\mathrm{new}}(578, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 256 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 65536 \) Copy content Toggle raw display
$11$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 65536 \) Copy content Toggle raw display
$37$ \( T^{8} + 65536 \) Copy content Toggle raw display
$41$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 65536 \) Copy content Toggle raw display
$67$ \( (T + 8)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 256 \) Copy content Toggle raw display
$79$ \( T^{8} + 16777216 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 1475789056 \) Copy content Toggle raw display
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