Properties

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

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 155)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_1 q^{3} - 2 q^{4} + ( - 4 \beta_{2} + 4) q^{6} + 2 \beta_1 q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_1 q^{3} - 2 q^{4} + ( - 4 \beta_{2} + 4) q^{6} + 2 \beta_1 q^{7} + \beta_{2} q^{9} - 5 \beta_{2} q^{11} + 2 \beta_1 q^{12} + ( - 3 \beta_{3} + 3 \beta_1) q^{13} + (8 \beta_{2} - 8) q^{14} - 4 q^{16} - 2 \beta_1 q^{17} + (\beta_{3} - \beta_1) q^{18} + ( - 4 \beta_{2} + 4) q^{19} - 8 \beta_{2} q^{21} + ( - 5 \beta_{3} + 5 \beta_1) q^{22} - \beta_{3} q^{23} + 12 \beta_{2} q^{26} + 2 \beta_{3} q^{27} - 4 \beta_1 q^{28} + 3 q^{29} + ( - \beta_{2} + 6) q^{31} - 4 \beta_{3} q^{32} + 5 \beta_{3} q^{33} + ( - 8 \beta_{2} + 8) q^{34} - 2 \beta_{2} q^{36} + 4 \beta_1 q^{37} + 4 \beta_1 q^{38} - 12 q^{39} + 3 \beta_{2} q^{41} + ( - 8 \beta_{3} + 8 \beta_1) q^{42} + 10 \beta_{2} q^{44} + 4 q^{46} + 4 \beta_{3} q^{47} + 4 \beta_1 q^{48} + 9 \beta_{2} q^{49} + 8 \beta_{2} q^{51} + (6 \beta_{3} - 6 \beta_1) q^{52} + ( - 3 \beta_{3} + 3 \beta_1) q^{53} - 8 q^{54} + (4 \beta_{3} - 4 \beta_1) q^{57} + 3 \beta_{3} q^{58} + ( - 13 \beta_{2} + 13) q^{59} + q^{61} + (5 \beta_{3} + \beta_1) q^{62} + 2 \beta_{3} q^{63} + 8 q^{64} - 20 q^{66} + (\beta_{3} - \beta_1) q^{67} + 4 \beta_1 q^{68} + (4 \beta_{2} - 4) q^{69} - 3 \beta_{2} q^{71} + (2 \beta_{3} - 2 \beta_1) q^{73} + (16 \beta_{2} - 16) q^{74} + (8 \beta_{2} - 8) q^{76} - 10 \beta_{3} q^{77} - 12 \beta_{3} q^{78} + ( - 3 \beta_{2} + 3) q^{79} + ( - 11 \beta_{2} + 11) q^{81} + (3 \beta_{3} - 3 \beta_1) q^{82} + ( - 2 \beta_{3} + 2 \beta_1) q^{83} + 16 \beta_{2} q^{84} - 3 \beta_1 q^{87} + 11 q^{89} + 24 q^{91} + 2 \beta_{3} q^{92} + (\beta_{3} - 6 \beta_1) q^{93} - 16 q^{94} + (16 \beta_{2} - 16) q^{96} + (9 \beta_{3} - 9 \beta_1) q^{98} + ( - 5 \beta_{2} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 8 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 8 q^{6} + 2 q^{9} - 10 q^{11} - 16 q^{14} - 16 q^{16} + 8 q^{19} - 16 q^{21} + 24 q^{26} + 12 q^{29} + 22 q^{31} + 16 q^{34} - 4 q^{36} - 48 q^{39} + 6 q^{41} + 20 q^{44} + 16 q^{46} + 18 q^{49} + 16 q^{51} - 32 q^{54} + 26 q^{59} + 4 q^{61} + 32 q^{64} - 80 q^{66} - 8 q^{69} - 6 q^{71} - 32 q^{74} - 16 q^{76} + 6 q^{79} + 22 q^{81} + 32 q^{84} + 44 q^{89} + 96 q^{91} - 64 q^{94} - 32 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

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
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
2.00000i −1.73205 + 1.00000i −2.00000 0 2.00000 + 3.46410i 3.46410 2.00000i 0 0.500000 0.866025i 0
149.2 2.00000i 1.73205 1.00000i −2.00000 0 2.00000 + 3.46410i −3.46410 + 2.00000i 0 0.500000 0.866025i 0
749.1 2.00000i 1.73205 + 1.00000i −2.00000 0 2.00000 3.46410i −3.46410 2.00000i 0 0.500000 + 0.866025i 0
749.2 2.00000i −1.73205 1.00000i −2.00000 0 2.00000 3.46410i 3.46410 + 2.00000i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
5.b even 2 1 inner 775.2.o.b 4
5.c odd 4 1 155.2.e.a 2
5.c odd 4 1 775.2.e.b 2
31.c even 3 1 inner 775.2.o.b 4
155.j even 6 1 inner 775.2.o.b 4
155.o odd 12 1 155.2.e.a 2
155.o odd 12 1 775.2.e.b 2
155.o odd 12 1 4805.2.a.c 1
155.p even 12 1 4805.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.a 2 5.c odd 4 1
155.2.e.a 2 155.o odd 12 1
775.2.e.b 2 5.c odd 4 1
775.2.e.b 2 155.o odd 12 1
775.2.o.b 4 1.a even 1 1 trivial
775.2.o.b 4 5.b even 2 1 inner
775.2.o.b 4 31.c even 3 1 inner
775.2.o.b 4 155.j even 6 1 inner
4805.2.a.a 1 155.p even 12 1
4805.2.a.c 1 155.o odd 12 1

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$61$ \( (T - 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T - 11)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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