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Properties

Label 726.2.e.j
Level $726$
Weight $2$
Character orbit 726.e
Analytic conductor $5.797$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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 \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - q^{4} - 5 q^{5} + q^{6} + 3 q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} - q^{4} - 5 q^{5} + q^{6} + 3 q^{7} + q^{8} - q^{9} - 10 q^{10} + 4 q^{12} - 6 q^{13} - 3 q^{14} - q^{16} - 4 q^{17} + q^{18} - 16 q^{19} - 5 q^{20} - 2 q^{21} + 4 q^{23} + q^{24} + 10 q^{25} - 4 q^{26} - q^{27} - 2 q^{28} + 4 q^{29} + 2 q^{31} - 4 q^{32} - 16 q^{34} - 5 q^{35} - q^{36} - 20 q^{37} - 14 q^{38} - 6 q^{39} - 16 q^{41} + 2 q^{42} - 28 q^{43} + 10 q^{45} - 14 q^{46} - 4 q^{47} - q^{48} + 8 q^{49} + 15 q^{50} - 4 q^{51} + 4 q^{52} - 4 q^{54} + 2 q^{56} + 14 q^{57} - 4 q^{58} + 15 q^{59} - 5 q^{60} + 12 q^{61} - 7 q^{62} + 3 q^{63} - q^{64} + 20 q^{65} - 16 q^{67} - 4 q^{68} - 16 q^{69} - 5 q^{70} - 6 q^{71} + q^{72} + 18 q^{73} + 20 q^{74} - 15 q^{75} + 4 q^{76} - 4 q^{78} + 11 q^{79} - q^{81} - 14 q^{82} - 3 q^{83} + 3 q^{84} - 2 q^{86} + 14 q^{87} + 36 q^{89} + 5 q^{90} - 2 q^{91} - 16 q^{92} + 2 q^{93} + 4 q^{94} - 30 q^{95} + q^{96} + 11 q^{97} + 22 q^{98}+O(q^{100}) \) 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 −2.92705 − 2.12663i 0.809017 + 0.587785i 0.190983 − 0.587785i −0.309017 − 0.951057i −0.809017 + 0.587785i −3.61803
493.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 − 0.587785i 0.427051 + 1.31433i −0.309017 − 0.951057i 1.30902 + 0.951057i 0.809017 − 0.587785i 0.309017 − 0.951057i −1.38197
511.1 −0.309017 − 0.951057i −0.809017 − 0.587785i −0.809017 + 0.587785i 0.427051 − 1.31433i −0.309017 + 0.951057i 1.30902 − 0.951057i 0.809017 + 0.587785i 0.309017 + 0.951057i −1.38197
565.1 0.809017 + 0.587785i 0.309017 − 0.951057i 0.309017 + 0.951057i −2.92705 + 2.12663i 0.809017 − 0.587785i 0.190983 + 0.587785i −0.309017 + 0.951057i −0.809017 − 0.587785i −3.61803
\(n\): 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 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.e.j 4
11.b odd 2 1 726.2.e.a 4
11.c even 5 2 66.2.e.b ✓ 4
11.c even 5 1 726.2.a.k 2
11.c even 5 1 inner 726.2.e.j 4
11.d odd 10 1 726.2.a.m 2
11.d odd 10 1 726.2.e.a 4
11.d odd 10 2 726.2.e.c 4
33.f even 10 1 2178.2.a.o 2
33.h odd 10 2 198.2.f.a 4
33.h odd 10 1 2178.2.a.v 2
44.g even 10 1 5808.2.a.by 2
44.h odd 10 2 528.2.y.g 4
44.h odd 10 1 5808.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.b ✓ 4 11.c even 5 2
198.2.f.a 4 33.h odd 10 2
528.2.y.g 4 44.h odd 10 2
726.2.a.k 2 11.c even 5 1
726.2.a.m 2 11.d odd 10 1
726.2.e.a 4 11.b odd 2 1
726.2.e.a 4 11.d odd 10 1
726.2.e.c 4 11.d odd 10 2
726.2.e.j 4 1.a even 1 1 trivial
726.2.e.j 4 11.c even 5 1 inner
2178.2.a.o 2 33.f even 10 1
2178.2.a.v 2 33.h odd 10 1
5808.2.a.by 2 44.g even 10 1
5808.2.a.bz 2 44.h odd 10 1

Hecke kernels

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

\( T_{5}^{4} + 5T_{5}^{3} + 10T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} + 4T_{7}^{2} - 2T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} + 16T_{13}^{2} + 16T_{13} + 16 \) 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
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{4} + 20 T^{3} + \cdots + 6400 \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T + 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{4} - 15 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} - 18 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 11 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
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