↑

Properties

Label 775.2.o.d
Level $775$
Weight $2$
Character orbit 775.o
Analytic conductor $6.188$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(149,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.o (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 31)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{6} - \beta_1) q^{3} + ( - 2 \beta_{7} - 1) q^{4} + ( - 2 \beta_{7} + 2 \beta_{4} + \cdots - 3) q^{6}+ \cdots + 2 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{6} - \beta_1) q^{3} + ( - 2 \beta_{7} - 1) q^{4} + ( - 2 \beta_{7} + 2 \beta_{4} + \cdots - 3) q^{6}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{4} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 12 q^{6} + 4 q^{11} - 12 q^{14} + 24 q^{16} + 12 q^{19} + 12 q^{21} + 20 q^{24} + 12 q^{26} + 32 q^{29} - 40 q^{31} - 4 q^{34} - 32 q^{36} + 24 q^{39} - 4 q^{41} - 52 q^{44} + 32 q^{46} - 16 q^{49} + 4 q^{51} + 8 q^{54} + 20 q^{56} - 12 q^{59} + 56 q^{64} - 120 q^{66} + 16 q^{69} + 28 q^{71} - 4 q^{74} + 4 q^{76} - 44 q^{79} + 4 q^{81} - 44 q^{84} + 52 q^{86} + 32 q^{89} + 24 q^{91} - 32 q^{94} + 4 q^{96} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
− 2.41421i 2.09077 − 1.20711i −3.82843 0 −2.91421 − 5.04757i 2.09077 − 1.20711i 4.41421i 1.41421 − 2.44949i 0
149.2 − 0.414214i 0.358719 − 0.207107i 1.82843 0 −0.0857864 − 0.148586i 0.358719 − 0.207107i − 1.58579i −1.41421 + 2.44949i 0
149.3 0.414214i −0.358719 + 0.207107i 1.82843 0 −0.0857864 − 0.148586i −0.358719 + 0.207107i 1.58579i −1.41421 + 2.44949i 0
149.4 2.41421i −2.09077 + 1.20711i −3.82843 0 −2.91421 − 5.04757i −2.09077 + 1.20711i − 4.41421i 1.41421 − 2.44949i 0
749.1 − 2.41421i −2.09077 − 1.20711i −3.82843 0 −2.91421 + 5.04757i −2.09077 − 1.20711i 4.41421i 1.41421 + 2.44949i 0
749.2 − 0.414214i −0.358719 − 0.207107i 1.82843 0 −0.0857864 + 0.148586i −0.358719 − 0.207107i − 1.58579i −1.41421 − 2.44949i 0
749.3 0.414214i 0.358719 + 0.207107i 1.82843 0 −0.0857864 + 0.148586i 0.358719 + 0.207107i 1.58579i −1.41421 − 2.44949i 0
749.4 2.41421i 2.09077 + 1.20711i −3.82843 0 −2.91421 + 5.04757i 2.09077 + 1.20711i − 4.41421i 1.41421 + 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.o.d 8
5.b even 2 1 inner 775.2.o.d 8
5.c odd 4 1 31.2.c.a ✓ 4
5.c odd 4 1 775.2.e.e 4
15.e even 4 1 279.2.h.c 4
20.e even 4 1 496.2.i.h 4
31.c even 3 1 inner 775.2.o.d 8
155.f even 4 1 961.2.c.a 4
155.j even 6 1 inner 775.2.o.d 8
155.o odd 12 1 31.2.c.a ✓ 4
155.o odd 12 1 775.2.e.e 4
155.o odd 12 1 961.2.a.a 2
155.p even 12 1 961.2.a.c 2
155.p even 12 1 961.2.c.a 4
155.r even 20 4 961.2.g.r 16
155.s odd 20 4 961.2.g.o 16
155.w odd 60 4 961.2.d.l 8
155.w odd 60 4 961.2.g.o 16
155.x even 60 4 961.2.d.i 8
155.x even 60 4 961.2.g.r 16
465.bc odd 12 1 8649.2.a.k 2
465.be even 12 1 279.2.h.c 4
465.be even 12 1 8649.2.a.l 2
620.be even 12 1 496.2.i.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a ✓ 4 5.c odd 4 1
31.2.c.a ✓ 4 155.o odd 12 1
279.2.h.c 4 15.e even 4 1
279.2.h.c 4 465.be even 12 1
496.2.i.h 4 20.e even 4 1
496.2.i.h 4 620.be even 12 1
775.2.e.e 4 5.c odd 4 1
775.2.e.e 4 155.o odd 12 1
775.2.o.d 8 1.a even 1 1 trivial
775.2.o.d 8 5.b even 2 1 inner
775.2.o.d 8 31.c even 3 1 inner
775.2.o.d 8 155.j even 6 1 inner
961.2.a.a 2 155.o odd 12 1
961.2.a.c 2 155.p even 12 1
961.2.c.a 4 155.f even 4 1
961.2.c.a 4 155.p even 12 1
961.2.d.i 8 155.x even 60 4
961.2.d.l 8 155.w odd 60 4
961.2.g.o 16 155.s odd 20 4
961.2.g.o 16 155.w odd 60 4
961.2.g.r 16 155.r even 20 4
961.2.g.r 16 155.x even 60 4
8649.2.a.k 2 465.bc odd 12 1
8649.2.a.l 2 465.be even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{6} + 35T_{7}^{4} - 6T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$17$ \( T^{8} - 34 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 29 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 31)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 198 T^{6} + \cdots + 88529281 \) Copy content Toggle raw display
$47$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 34 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 38 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$71$ \( (T^{4} - 14 T^{3} + 197 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( (T^{4} + 22 T^{3} + \cdots + 10609)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 118 T^{6} + \cdots + 2825761 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 56)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 144 T^{2} + 3136)^{2} \) Copy content Toggle raw display
show more
show less