Properties

Label 651.1.co.a
Level $651$
Weight $1$
Character orbit 651.co
Analytic conductor $0.325$
Analytic rank $0$
Dimension $8$
Projective image $D_{30}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [651,1,Mod(17,651)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(651, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 5, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("651.17");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 651 = 3 \cdot 7 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 651.co (of order \(30\), degree \(8\), 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: \(0.324891323224\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{30}^{9} q^{3} - \zeta_{30}^{7} q^{4} + \zeta_{30} q^{7} - \zeta_{30}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30}^{9} q^{3} - \zeta_{30}^{7} q^{4} + \zeta_{30} q^{7} - \zeta_{30}^{3} q^{9} - \zeta_{30} q^{12} + ( - \zeta_{30}^{12} + \zeta_{30}^{11}) q^{13} + \zeta_{30}^{14} q^{16} + (\zeta_{30}^{13} + \zeta_{30}^{4}) q^{19} - \zeta_{30}^{10} q^{21} - q^{25} + \zeta_{30}^{12} q^{27} - \zeta_{30}^{8} q^{28} - \zeta_{30}^{14} q^{31} + \zeta_{30}^{10} q^{36} + ( - \zeta_{30}^{11} + \zeta_{30}^{9}) q^{37} + ( - \zeta_{30}^{6} + \zeta_{30}^{5}) q^{39} + ( - \zeta_{30}^{5} - \zeta_{30}^{2}) q^{43} + \zeta_{30}^{8} q^{48} + \zeta_{30}^{2} q^{49} + ( - \zeta_{30}^{4} + \zeta_{30}^{3}) q^{52} + ( - \zeta_{30}^{13} + \zeta_{30}^{7}) q^{57} + (\zeta_{30}^{8} + \zeta_{30}^{2}) q^{61} - \zeta_{30}^{4} q^{63} + \zeta_{30}^{6} q^{64} + (\zeta_{30}^{7} + \zeta_{30}^{3}) q^{67} + ( - \zeta_{30}^{13} + \zeta_{30}^{6}) q^{73} + \zeta_{30}^{9} q^{75} + ( - \zeta_{30}^{11} + \zeta_{30}^{5}) q^{76} + ( - \zeta_{30}^{6} - \zeta_{30}^{5}) q^{79} + \zeta_{30}^{6} q^{81} - \zeta_{30}^{2} q^{84} + ( - \zeta_{30}^{13} + \zeta_{30}^{12}) q^{91} - \zeta_{30}^{8} q^{93} + ( - \zeta_{30}^{14} + \zeta_{30}^{10}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + q^{4} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + q^{4} - q^{7} - 2 q^{9} + q^{12} + q^{13} + q^{16} + 4 q^{21} - 8 q^{25} - 2 q^{27} - q^{28} - q^{31} - 4 q^{36} + 3 q^{37} + 6 q^{39} - 5 q^{43} + q^{48} + q^{49} + q^{52} + 2 q^{61} - q^{63} - 2 q^{64} + q^{67} - q^{73} + 2 q^{75} + 5 q^{76} - 2 q^{79} - 2 q^{81} - q^{84} - q^{91} - q^{93} - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/651\mathbb{Z}\right)^\times\).

\(n\) \(94\) \(127\) \(218\)
\(\chi(n)\) \(\zeta_{30}^{5}\) \(\zeta_{30}^{13}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.669131 + 0.743145i
0.913545 0.406737i
−0.104528 + 0.994522i
0.669131 0.743145i
−0.978148 + 0.207912i
−0.104528 0.994522i
−0.978148 0.207912i
0.913545 + 0.406737i
0 0.309017 + 0.951057i 0.913545 0.406737i 0 0 −0.669131 0.743145i 0 −0.809017 + 0.587785i 0
353.1 0 −0.809017 + 0.587785i −0.978148 0.207912i 0 0 −0.913545 + 0.406737i 0 0.309017 0.951057i 0
362.1 0 −0.809017 + 0.587785i 0.669131 0.743145i 0 0 0.104528 0.994522i 0 0.309017 0.951057i 0
383.1 0 0.309017 0.951057i 0.913545 + 0.406737i 0 0 −0.669131 + 0.743145i 0 −0.809017 0.587785i 0
425.1 0 0.309017 + 0.951057i −0.104528 + 0.994522i 0 0 0.978148 0.207912i 0 −0.809017 + 0.587785i 0
437.1 0 −0.809017 0.587785i 0.669131 + 0.743145i 0 0 0.104528 + 0.994522i 0 0.309017 + 0.951057i 0
458.1 0 0.309017 0.951057i −0.104528 0.994522i 0 0 0.978148 + 0.207912i 0 −0.809017 0.587785i 0
509.1 0 −0.809017 0.587785i −0.978148 + 0.207912i 0 0 −0.913545 0.406737i 0 0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
217.bf even 30 1 inner
651.co odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 651.1.co.a yes 8
3.b odd 2 1 CM 651.1.co.a yes 8
7.d odd 6 1 651.1.ch.a 8
21.g even 6 1 651.1.ch.a 8
31.h odd 30 1 651.1.ch.a 8
93.p even 30 1 651.1.ch.a 8
217.bf even 30 1 inner 651.1.co.a yes 8
651.co odd 30 1 inner 651.1.co.a yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
651.1.ch.a 8 7.d odd 6 1
651.1.ch.a 8 21.g even 6 1
651.1.ch.a 8 31.h odd 30 1
651.1.ch.a 8 93.p even 30 1
651.1.co.a yes 8 1.a even 1 1 trivial
651.1.co.a yes 8 3.b odd 2 1 CM
651.1.co.a yes 8 217.bf even 30 1 inner
651.1.co.a yes 8 651.co odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(651, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - T^{7} - 4 T^{5} - 6 T^{4} + 11 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{5} - 5 T^{4} + 25 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + T^{6} + 6 T^{5} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} + 15 T^{6} + 30 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{7} + 5 T^{6} - 4 T^{5} + 19 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + 4 T^{5} - 6 T^{4} - 11 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + 6 T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 5 T^{7} + 12 T^{6} + 15 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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