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Properties

Label 775.1.w.b
Level $775$
Weight $1$
Character orbit 775.w
Analytic conductor $0.387$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,1,Mod(61,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.61");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 775.w (of order \(10\), degree \(4\), 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: \(0.386775384791\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{30}^{13} - \zeta_{30}^{5}) q^{2} + ( - \zeta_{30}^{11} + \cdots - \zeta_{30}^{3}) q^{4}+ \cdots + \zeta_{30}^{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{30}^{13} - \zeta_{30}^{5}) q^{2} + ( - \zeta_{30}^{11} + \cdots - \zeta_{30}^{3}) q^{4}+ \cdots + (\zeta_{30}^{14} + \cdots + \zeta_{30}^{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 5 q^{4} + q^{5} + 2 q^{7} + 8 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 5 q^{4} + q^{5} + 2 q^{7} + 8 q^{8} - 2 q^{9} - q^{10} - 7 q^{14} - 3 q^{16} + 2 q^{18} - 3 q^{19} + q^{25} - 5 q^{28} - 2 q^{31} - 10 q^{32} - q^{35} + 8 q^{38} - 4 q^{40} + 2 q^{41} + q^{45} - 4 q^{47} + 10 q^{49} - 6 q^{50} + 12 q^{56} + 2 q^{59} - 3 q^{62} + 2 q^{63} + 3 q^{64} + 6 q^{67} + q^{70} + 2 q^{71} + 8 q^{72} - 10 q^{76} - 6 q^{80} - 2 q^{81} - 2 q^{82} + 9 q^{90} - q^{94} - 6 q^{95} + 2 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1\) \(-\zeta_{30}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.104528 + 0.994522i
0.913545 − 0.406737i
−0.104528 − 0.994522i
0.913545 + 0.406737i
0.669131 − 0.743145i
−0.978148 − 0.207912i
0.669131 + 0.743145i
−0.978148 + 0.207912i
−1.47815 + 1.07394i 0 0.722562 − 2.22382i 0.913545 − 0.406737i 0 1.33826 0.755585 + 2.32545i −0.809017 − 0.587785i −0.913545 + 1.58231i
61.2 0.169131 − 0.122881i 0 −0.295511 + 0.909491i −0.104528 + 0.994522i 0 −1.95630 0.126381 + 0.388960i −0.809017 − 0.587785i 0.104528 + 0.181049i
216.1 −1.47815 − 1.07394i 0 0.722562 + 2.22382i 0.913545 + 0.406737i 0 1.33826 0.755585 − 2.32545i −0.809017 + 0.587785i −0.913545 − 1.58231i
216.2 0.169131 + 0.122881i 0 −0.295511 − 0.909491i −0.104528 − 0.994522i 0 −1.95630 0.126381 − 0.388960i −0.809017 + 0.587785i 0.104528 − 0.181049i
371.1 −0.604528 + 1.86055i 0 −2.28716 − 1.66172i −0.978148 − 0.207912i 0 1.82709 2.89169 − 2.10094i 0.309017 + 0.951057i 0.978148 − 1.69420i
371.2 0.413545 − 1.27276i 0 −0.639886 − 0.464905i 0.669131 − 0.743145i 0 −0.209057 0.226341 − 0.164446i 0.309017 + 0.951057i −0.669131 − 1.15897i
681.1 −0.604528 − 1.86055i 0 −2.28716 + 1.66172i −0.978148 + 0.207912i 0 1.82709 2.89169 + 2.10094i 0.309017 − 0.951057i 0.978148 + 1.69420i
681.2 0.413545 + 1.27276i 0 −0.639886 + 0.464905i 0.669131 + 0.743145i 0 −0.209057 0.226341 + 0.164446i 0.309017 − 0.951057i −0.669131 + 1.15897i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
25.d even 5 1 inner
775.w odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.w.b ✓ 8
5.b even 2 1 3875.1.w.c 8
5.c odd 4 2 3875.1.z.d 16
25.d even 5 1 inner 775.1.w.b ✓ 8
25.e even 10 1 3875.1.w.c 8
25.f odd 20 2 3875.1.z.d 16
31.b odd 2 1 CM 775.1.w.b ✓ 8
155.c odd 2 1 3875.1.w.c 8
155.f even 4 2 3875.1.z.d 16
775.w odd 10 1 inner 775.1.w.b ✓ 8
775.z odd 10 1 3875.1.w.c 8
775.ca even 20 2 3875.1.z.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.1.w.b ✓ 8 1.a even 1 1 trivial
775.1.w.b ✓ 8 25.d even 5 1 inner
775.1.w.b ✓ 8 31.b odd 2 1 CM
775.1.w.b ✓ 8 775.w odd 10 1 inner
3875.1.w.c 8 5.b even 2 1
3875.1.w.c 8 25.e even 10 1
3875.1.w.c 8 155.c odd 2 1
3875.1.w.c 8 775.z odd 10 1
3875.1.z.d 16 5.c odd 4 2
3875.1.z.d 16 25.f odd 20 2
3875.1.z.d 16 155.f even 4 2
3875.1.z.d 16 775.ca even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{7} + 8T_{2}^{6} + 11T_{2}^{5} + 15T_{2}^{4} + 11T_{2}^{3} + 18T_{2}^{2} - 7T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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