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Properties

Label 817.1.bz.a
Level $817$
Weight $1$
Character orbit 817.bz
Analytic conductor $0.408$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [817,1,Mod(56,817)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(817, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 32]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("817.56");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 817 = 19 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 817.bz (of order \(42\), degree \(12\), 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: \(0.407736115321\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{42}^{12} q^{4} + ( - \zeta_{42}^{19} + \zeta_{42}^{10}) q^{5} + (\zeta_{42}^{6} - \zeta_{42}) q^{7} + \zeta_{42}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{42}^{12} q^{4} + ( - \zeta_{42}^{19} + \zeta_{42}^{10}) q^{5} + (\zeta_{42}^{6} - \zeta_{42}) q^{7} + \zeta_{42}^{2} q^{9} + ( - \zeta_{42}^{13} - \zeta_{42}^{5}) q^{11} - \zeta_{42}^{3} q^{16} + (\zeta_{42}^{18} + \zeta_{42}^{16}) q^{17} - \zeta_{42}^{19} q^{19} + (\zeta_{42}^{10} - \zeta_{42}) q^{20} + (\zeta_{42}^{6} - \zeta_{42}^{5}) q^{23} + (\zeta_{42}^{20} + \cdots + \zeta_{42}^{8}) q^{25} + \cdots + ( - \zeta_{42}^{15} - \zeta_{42}^{7}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} + 2 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} + 2 q^{5} - q^{7} + q^{9} + 2 q^{11} - 2 q^{16} - q^{17} + q^{19} + 2 q^{20} - q^{23} + 3 q^{25} - q^{28} + 4 q^{35} - 6 q^{36} + 12 q^{43} + 2 q^{44} + 10 q^{45} - 4 q^{47} - 7 q^{49} - 2 q^{55} - 8 q^{61} - q^{63} - 2 q^{64} - 8 q^{68} - 5 q^{73} + q^{76} - 6 q^{77} + 2 q^{80} + q^{81} - 8 q^{83} - 10 q^{85} - q^{92} + 2 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/817\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(476\)
\(\chi(n)\) \(-1\) \(\zeta_{42}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1
−0.733052 + 0.680173i
0.955573 − 0.294755i
0.826239 − 0.563320i
0.0747301 − 0.997204i
0.826239 + 0.563320i
0.365341 − 0.930874i
−0.733052 − 0.680173i
−0.988831 + 0.149042i
0.955573 + 0.294755i
0.0747301 + 0.997204i
−0.988831 − 0.149042i
0.365341 + 0.930874i
0 0 −0.900969 − 0.433884i 0.440071 + 0.0663300i 0 −0.955573 + 1.65510i 0 0.0747301 − 0.997204i 0
189.1 0 0 −0.900969 + 0.433884i −0.162592 + 0.414278i 0 0.733052 − 1.26968i 0 0.826239 − 0.563320i 0
246.1 0 0 0.623490 − 0.781831i 1.32091 + 1.22563i 0 −0.0747301 − 0.129436i 0 0.365341 − 0.930874i 0
341.1 0 0 0.623490 + 0.781831i −1.72188 − 0.531130i 0 −0.826239 − 1.43109i 0 −0.988831 − 0.149042i 0
455.1 0 0 0.623490 + 0.781831i 1.32091 − 1.22563i 0 −0.0747301 + 0.129436i 0 0.365341 + 0.930874i 0
531.1 0 0 −0.222521 − 0.974928i 0.0931869 + 1.24349i 0 0.988831 − 1.71271i 0 −0.733052 − 0.680173i 0
569.1 0 0 −0.900969 + 0.433884i 0.440071 − 0.0663300i 0 −0.955573 − 1.65510i 0 0.0747301 + 0.997204i 0
626.1 0 0 −0.222521 − 0.974928i 1.03030 − 0.702449i 0 −0.365341 − 0.632789i 0 0.955573 − 0.294755i 0
683.1 0 0 −0.900969 − 0.433884i −0.162592 − 0.414278i 0 0.733052 + 1.26968i 0 0.826239 + 0.563320i 0
702.1 0 0 0.623490 − 0.781831i −1.72188 + 0.531130i 0 −0.826239 + 1.43109i 0 −0.988831 + 0.149042i 0
740.1 0 0 −0.222521 + 0.974928i 1.03030 + 0.702449i 0 −0.365341 + 0.632789i 0 0.955573 + 0.294755i 0
797.1 0 0 −0.222521 + 0.974928i 0.0931869 − 1.24349i 0 0.988831 + 1.71271i 0 −0.733052 + 0.680173i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
43.g even 21 1 inner
817.bz odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 817.1.bz.a ✓ 12
19.b odd 2 1 CM 817.1.bz.a ✓ 12
43.g even 21 1 inner 817.1.bz.a ✓ 12
817.bz odd 42 1 inner 817.1.bz.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
817.1.bz.a ✓ 12 1.a even 1 1 trivial
817.1.bz.a ✓ 12 19.b odd 2 1 CM
817.1.bz.a ✓ 12 43.g even 21 1 inner
817.1.bz.a ✓ 12 817.bz odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(817, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( (T - 1)^{12} \) Copy content Toggle raw display
$47$ \( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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