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Properties

Label 726.2.e.g
Level $726$
Weight $2$
Character orbit 726.e
Analytic conductor $5.797$
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] = [726,2,Mod(487,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.487");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.e (of order \(5\), degree \(4\), 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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} + 4 q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} + 4 q^{7} - q^{8} - q^{9} + 8 q^{10} - 4 q^{12} + 6 q^{13} + 4 q^{14} + 2 q^{15} - q^{16} - 2 q^{17} - q^{18} - 4 q^{19} - 2 q^{20} + 16 q^{21} + 16 q^{23} + q^{24} + q^{25} + 6 q^{26} + q^{27} + 4 q^{28} - 6 q^{29} + 2 q^{30} + 4 q^{32} + 8 q^{34} + 8 q^{35} - q^{36} - 6 q^{37} - 4 q^{38} - 6 q^{39} - 2 q^{40} + 6 q^{41} - 4 q^{42} + 16 q^{43} + 8 q^{45} - 4 q^{46} + 12 q^{47} + q^{48} - 9 q^{49} + q^{50} + 2 q^{51} + 6 q^{52} - 2 q^{53} - 4 q^{54} - 16 q^{56} + 4 q^{57} - 6 q^{58} - 12 q^{59} + 2 q^{60} + 14 q^{61} + 4 q^{63} - q^{64} - 48 q^{65} + 16 q^{67} - 2 q^{68} + 4 q^{69} + 8 q^{70} + 12 q^{71} - q^{72} + 6 q^{73} - 6 q^{74} - q^{75} + 16 q^{76} + 24 q^{78} + 4 q^{79} - 2 q^{80} - q^{81} + 6 q^{82} - 4 q^{83} - 4 q^{84} - 4 q^{85} - 4 q^{86} - 24 q^{87} + 40 q^{89} - 2 q^{90} - 24 q^{91} - 4 q^{92} + 12 q^{94} - 8 q^{95} + q^{96} + 14 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(607\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
0.809017 + 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.809017 − 0.587785i
−0.809017 + 0.587785i −0.309017 − 0.951057i 0.309017 − 0.951057i −1.61803 − 1.17557i 0.809017 + 0.587785i −1.23607 + 3.80423i 0.309017 + 0.951057i −0.809017 + 0.587785i 2.00000
493.1 0.309017 − 0.951057i 0.809017 − 0.587785i −0.809017 − 0.587785i 0.618034 + 1.90211i −0.309017 − 0.951057i 3.23607 + 2.35114i −0.809017 + 0.587785i 0.309017 − 0.951057i 2.00000
511.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.618034 − 1.90211i −0.309017 + 0.951057i 3.23607 − 2.35114i −0.809017 − 0.587785i 0.309017 + 0.951057i 2.00000
565.1 −0.809017 − 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 − 0.587785i −1.23607 − 3.80423i 0.309017 − 0.951057i −0.809017 − 0.587785i 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.e.g 4
11.b odd 2 1 726.2.e.o 4
11.c even 5 1 66.2.a.b ✓ 1
11.c even 5 3 inner 726.2.e.g 4
11.d odd 10 1 726.2.a.c 1
11.d odd 10 3 726.2.e.o 4
33.f even 10 1 2178.2.a.g 1
33.h odd 10 1 198.2.a.a 1
44.g even 10 1 5808.2.a.bc 1
44.h odd 10 1 528.2.a.j 1
55.j even 10 1 1650.2.a.k 1
55.k odd 20 2 1650.2.c.e 2
77.j odd 10 1 3234.2.a.t 1
88.l odd 10 1 2112.2.a.e 1
88.o even 10 1 2112.2.a.r 1
99.m even 15 2 1782.2.e.e 2
99.n odd 30 2 1782.2.e.v 2
132.o even 10 1 1584.2.a.f 1
165.o odd 10 1 4950.2.a.bu 1
165.v even 20 2 4950.2.c.p 2
231.u even 10 1 9702.2.a.x 1
264.t odd 10 1 6336.2.a.bw 1
264.w even 10 1 6336.2.a.cj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b ✓ 1 11.c even 5 1
198.2.a.a 1 33.h odd 10 1
528.2.a.j 1 44.h odd 10 1
726.2.a.c 1 11.d odd 10 1
726.2.e.g 4 1.a even 1 1 trivial
726.2.e.g 4 11.c even 5 3 inner
726.2.e.o 4 11.b odd 2 1
726.2.e.o 4 11.d odd 10 3
1584.2.a.f 1 132.o even 10 1
1650.2.a.k 1 55.j even 10 1
1650.2.c.e 2 55.k odd 20 2
1782.2.e.e 2 99.m even 15 2
1782.2.e.v 2 99.n odd 30 2
2112.2.a.e 1 88.l odd 10 1
2112.2.a.r 1 88.o even 10 1
2178.2.a.g 1 33.f even 10 1
3234.2.a.t 1 77.j odd 10 1
4950.2.a.bu 1 165.o odd 10 1
4950.2.c.p 2 165.v even 20 2
5808.2.a.bc 1 44.g even 10 1
6336.2.a.bw 1 264.t odd 10 1
6336.2.a.cj 1 264.w even 10 1
9702.2.a.x 1 231.u even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\):

\( T_{5}^{4} + 2T_{5}^{3} + 4T_{5}^{2} + 8T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} + 16T_{7}^{2} - 64T_{7} + 256 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} + 36T_{13}^{2} - 216T_{13} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$61$ \( T^{4} - 14 T^{3} + \cdots + 38416 \) Copy content Toggle raw display
$67$ \( (T - 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 38416 \) Copy content Toggle raw display
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