Properties

Label 2600.1.ck.d
Level $2600$
Weight $1$
Character orbit 2600.ck
Analytic conductor $1.298$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -104
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2600,1,Mod(571,2600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2600.571");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2600.ck (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: \(1.29756903285\)
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.

\(f(q)\) \(=\) \( q + \zeta_{30}^{9} q^{2} + \zeta_{30}^{3} q^{3} - \zeta_{30}^{3} q^{4} - \zeta_{30}^{10} q^{5} + \zeta_{30}^{12} q^{6} + (\zeta_{30}^{11} - \zeta_{30}^{4}) q^{7} - \zeta_{30}^{12} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30}^{9} q^{2} + \zeta_{30}^{3} q^{3} - \zeta_{30}^{3} q^{4} - \zeta_{30}^{10} q^{5} + \zeta_{30}^{12} q^{6} + (\zeta_{30}^{11} - \zeta_{30}^{4}) q^{7} - \zeta_{30}^{12} q^{8} + \zeta_{30}^{4} q^{10} - \zeta_{30}^{6} q^{12} - \zeta_{30}^{6} q^{13} + ( - \zeta_{30}^{13} - \zeta_{30}^{5}) q^{14} - \zeta_{30}^{13} q^{15} + \zeta_{30}^{6} q^{16} + ( - \zeta_{30}^{5} + \zeta_{30}^{4}) q^{17} + \zeta_{30}^{13} q^{20} + (\zeta_{30}^{14} - \zeta_{30}^{7}) q^{21} + q^{24} - \zeta_{30}^{5} q^{25} + q^{26} - \zeta_{30}^{9} q^{27} + ( - \zeta_{30}^{14} + \zeta_{30}^{7}) q^{28} + \zeta_{30}^{7} q^{30} + ( - \zeta_{30}^{6} + \zeta_{30}^{3}) q^{31} - q^{32} + ( - \zeta_{30}^{14} + \zeta_{30}^{13}) q^{34} + (\zeta_{30}^{14} + \zeta_{30}^{6}) q^{35} + ( - \zeta_{30}^{8} - \zeta_{30}^{4}) q^{37} - \zeta_{30}^{9} q^{39} - \zeta_{30}^{7} q^{40} + ( - \zeta_{30}^{8} + \zeta_{30}) q^{42} + ( - \zeta_{30}^{11} + \zeta_{30}^{4}) q^{43} + (\zeta_{30}^{11} - \zeta_{30}^{10}) q^{47} + \zeta_{30}^{9} q^{48} + (\zeta_{30}^{8} - \zeta_{30}^{7} + 1) q^{49} - \zeta_{30}^{14} q^{50} + ( - \zeta_{30}^{8} + \zeta_{30}^{7}) q^{51} + \zeta_{30}^{9} q^{52} + \zeta_{30}^{3} q^{54} + (\zeta_{30}^{8} - \zeta_{30}) q^{56} - \zeta_{30} q^{60} + (\zeta_{30}^{12} + 1) q^{62} - \zeta_{30}^{9} q^{64} - \zeta_{30} q^{65} + (\zeta_{30}^{8} - \zeta_{30}^{7}) q^{68} + ( - \zeta_{30}^{8} - 1) q^{70} + (\zeta_{30}^{5} + \zeta_{30}) q^{71} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{74} - \zeta_{30}^{8} q^{75} + \zeta_{30}^{3} q^{78} + \zeta_{30} q^{80} - \zeta_{30}^{12} q^{81} + (\zeta_{30}^{10} + \zeta_{30}^{2}) q^{84} + ( - \zeta_{30}^{14} - 1) q^{85} + (\zeta_{30}^{13} + \zeta_{30}^{5}) q^{86} + (\zeta_{30}^{10} + \zeta_{30}^{2}) q^{91} + ( - \zeta_{30}^{9} + \zeta_{30}^{6}) q^{93} + ( - \zeta_{30}^{5} + \zeta_{30}^{4}) q^{94} - \zeta_{30}^{3} q^{96} + (\zeta_{30}^{9} - \zeta_{30}^{2} + \zeta_{30}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + q^{10} + 2 q^{12} + 2 q^{13} - 3 q^{14} + q^{15} - 2 q^{16} - 3 q^{17} - q^{20} + 2 q^{21} + 8 q^{24} - 4 q^{25} + 8 q^{26} - 2 q^{27} - 2 q^{28} - q^{30} + 4 q^{31} - 8 q^{32} - 2 q^{34} - q^{35} - 2 q^{37} - 2 q^{39} + q^{40} - 2 q^{42} + 2 q^{43} + 3 q^{47} + 2 q^{48} + 10 q^{49} - q^{50} - 2 q^{51} + 2 q^{52} + 2 q^{54} + 2 q^{56} + q^{60} + 6 q^{62} - 2 q^{64} + q^{65} + 2 q^{68} - 9 q^{70} + 3 q^{71} + 2 q^{74} - q^{75} + 2 q^{78} - q^{80} + 2 q^{81} - 3 q^{84} - 9 q^{85} + 3 q^{86} - 3 q^{91} - 4 q^{93} - 3 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).

\(n\) \(1301\) \(1601\) \(1951\) \(1977\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{30}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
−0.978148 0.207912i
0.669131 0.743145i
−0.104528 0.994522i
0.913545 + 0.406737i
0.913545 0.406737i
−0.104528 + 0.994522i
0.669131 + 0.743145i
−0.978148 + 0.207912i
−0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 0.500000 0.866025i −0.809017 + 0.587785i −1.33826 0.809017 0.587785i 0 0.669131 + 0.743145i
571.2 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 0.500000 + 0.866025i −0.809017 + 0.587785i 1.95630 0.809017 0.587785i 0 −0.978148 + 0.207912i
1091.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.500000 0.866025i 0.309017 0.951057i −1.82709 −0.309017 + 0.951057i 0 0.913545 0.406737i
1091.2 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.500000 + 0.866025i 0.309017 0.951057i 0.209057 −0.309017 + 0.951057i 0 −0.104528 + 0.994522i
1611.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.500000 0.866025i 0.309017 + 0.951057i 0.209057 −0.309017 0.951057i 0 −0.104528 0.994522i
1611.2 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.500000 + 0.866025i 0.309017 + 0.951057i −1.82709 −0.309017 0.951057i 0 0.913545 + 0.406737i
2131.1 −0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 0.500000 0.866025i −0.809017 0.587785i 1.95630 0.809017 + 0.587785i 0 −0.978148 0.207912i
2131.2 −0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 0.500000 + 0.866025i −0.809017 0.587785i −1.33826 0.809017 + 0.587785i 0 0.669131 0.743145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
25.d even 5 1 inner
2600.ck odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.ck.d yes 8
8.d odd 2 1 2600.1.ck.c 8
13.b even 2 1 2600.1.ck.c 8
25.d even 5 1 inner 2600.1.ck.d yes 8
104.h odd 2 1 CM 2600.1.ck.d yes 8
200.n odd 10 1 2600.1.ck.c 8
325.q even 10 1 2600.1.ck.c 8
2600.ck odd 10 1 inner 2600.1.ck.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2600.1.ck.c 8 8.d odd 2 1
2600.1.ck.c 8 13.b even 2 1
2600.1.ck.c 8 200.n odd 10 1
2600.1.ck.c 8 325.q even 10 1
2600.1.ck.d yes 8 1.a even 1 1 trivial
2600.1.ck.d yes 8 25.d even 5 1 inner
2600.1.ck.d yes 8 104.h odd 2 1 CM
2600.1.ck.d yes 8 2600.ck odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\):

\( T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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