Properties

Label 726.2.e.f
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + ( - 3 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{5} + \zeta_{10} q^{6} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + ( - 3 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{5} + \zeta_{10} q^{6} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 1) q^{10} - q^{12} + ( - 2 \zeta_{10}^{3} + 2) q^{13} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} + \zeta_{10}) q^{14} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{15} - \zeta_{10} q^{16} + (4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{17} - \zeta_{10}^{3} q^{18} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{19} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{20} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 4) q^{21} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{23} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + ( - 3 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 3 \zeta_{10}) q^{25} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{26} + \zeta_{10} q^{27} + ( - \zeta_{10}^{2} - 3 \zeta_{10} - 1) q^{28} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10}) q^{30} + (8 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 8) q^{31} + q^{32} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2}) q^{34} + (7 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} - 7) q^{35} + \zeta_{10}^{2} q^{36} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{37} + (2 \zeta_{10}^{2} + 2) q^{38} + ( - 2 \zeta_{10}^{2} - 2) q^{39} + ( - \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{40} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{41} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 4) q^{42} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{43} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 1) q^{45} + ( - 2 \zeta_{10}^{3} + 2) q^{46} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{47} + \zeta_{10}^{3} q^{48} + ( - 7 \zeta_{10}^{2} - 3 \zeta_{10} - 7) q^{49} + (3 \zeta_{10}^{2} - 8 \zeta_{10} + 3) q^{50} + ( - 4 \zeta_{10} + 4) q^{51} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{52} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{53} - q^{54} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 4) q^{56} + (2 \zeta_{10}^{3} - 2) q^{57} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{58} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{59} + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{60} + ( - 2 \zeta_{10}^{2} + 10 \zeta_{10} - 2) q^{61} + ( - 8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{62} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} + \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 8) q^{65} + 4 q^{67} + ( - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{68} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{69} + ( - 7 \zeta_{10}^{3} - 10 \zeta_{10} + 10) q^{70} + ( - 4 \zeta_{10}^{2} - 6 \zeta_{10} - 4) q^{71} - \zeta_{10} q^{72} + (7 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{73} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 4 \zeta_{10}) q^{74} + ( - 5 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 5) q^{75} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{76} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{78} + ( - 5 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 5) q^{79} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{80} - \zeta_{10}^{3} q^{81} + (2 \zeta_{10}^{2} + 2) q^{82} + ( - 7 \zeta_{10}^{2} + 12 \zeta_{10} - 7) q^{83} + (4 \zeta_{10}^{3} + \zeta_{10} - 1) q^{84} + (8 \zeta_{10}^{3} - 20 \zeta_{10}^{2} + 8 \zeta_{10}) q^{85} + ( - 2 \zeta_{10}^{3} + 2) q^{86} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{87} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 8) q^{89} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{90} + ( - 8 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 8 \zeta_{10}) q^{91} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{92} + (\zeta_{10}^{2} + 7 \zeta_{10} + 1) q^{93} + ( - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{94} + (8 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{95} - \zeta_{10}^{2} q^{96} + (7 \zeta_{10}^{3} - 7) q^{97} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 10) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} - q^{4} - 7 q^{5} + q^{6} - q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{3} - q^{4} - 7 q^{5} + q^{6} - q^{7} - q^{8} - q^{9} - 2 q^{10} - 4 q^{12} + 6 q^{13} - q^{14} - 8 q^{15} - q^{16} + 8 q^{17} - q^{18} - 4 q^{19} - 7 q^{20} - 14 q^{21} - 4 q^{23} + q^{24} - 14 q^{25} - 4 q^{26} + q^{27} - 6 q^{28} + 4 q^{29} - 8 q^{30} - 10 q^{31} + 4 q^{32} + 8 q^{34} - 27 q^{35} - q^{36} - 16 q^{37} + 6 q^{38} - 6 q^{39} + 8 q^{40} - 4 q^{41} + 6 q^{42} - 4 q^{43} - 2 q^{45} + 6 q^{46} + 12 q^{47} + q^{48} - 24 q^{49} + q^{50} + 12 q^{51} - 4 q^{52} + 8 q^{53} - 4 q^{54} + 14 q^{56} - 6 q^{57} + 4 q^{58} - 7 q^{59} + 7 q^{60} + 4 q^{61} - 5 q^{62} - q^{63} - q^{64} - 28 q^{65} + 16 q^{67} + 8 q^{68} + 4 q^{69} + 23 q^{70} - 18 q^{71} - q^{72} - 14 q^{73} - 16 q^{74} - q^{75} - 4 q^{76} + 4 q^{78} + 19 q^{79} + 8 q^{80} - q^{81} + 6 q^{82} - 9 q^{83} + q^{84} + 36 q^{85} + 6 q^{86} + 6 q^{87} - 20 q^{89} - 7 q^{90} - 14 q^{91} - 4 q^{92} + 10 q^{93} - 8 q^{94} + 2 q^{95} + q^{96} - 21 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −2.30902 1.67760i 0.809017 + 0.587785i 1.42705 4.39201i 0.309017 + 0.951057i −0.809017 + 0.587785i 2.85410
493.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −1.19098 3.66547i −0.309017 0.951057i −1.92705 1.40008i −0.809017 + 0.587785i 0.309017 0.951057i −3.85410
511.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −1.19098 + 3.66547i −0.309017 + 0.951057i −1.92705 + 1.40008i −0.809017 0.587785i 0.309017 + 0.951057i −3.85410
565.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −2.30902 + 1.67760i 0.809017 0.587785i 1.42705 + 4.39201i 0.309017 0.951057i −0.809017 0.587785i 2.85410
\(n\): 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.f 4
11.b odd 2 1 726.2.e.n 4
11.c even 5 2 66.2.e.a 4
11.c even 5 1 726.2.a.l 2
11.c even 5 1 inner 726.2.e.f 4
11.d odd 10 1 726.2.a.j 2
11.d odd 10 1 726.2.e.n 4
11.d odd 10 2 726.2.e.r 4
33.f even 10 1 2178.2.a.bb 2
33.h odd 10 2 198.2.f.c 4
33.h odd 10 1 2178.2.a.t 2
44.g even 10 1 5808.2.a.cg 2
44.h odd 10 2 528.2.y.d 4
44.h odd 10 1 5808.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.a 4 11.c even 5 2
198.2.f.c 4 33.h odd 10 2
528.2.y.d 4 44.h odd 10 2
726.2.a.j 2 11.d odd 10 1
726.2.a.l 2 11.c even 5 1
726.2.e.f 4 1.a even 1 1 trivial
726.2.e.f 4 11.c even 5 1 inner
726.2.e.n 4 11.b odd 2 1
726.2.e.n 4 11.d odd 10 1
726.2.e.r 4 11.d odd 10 2
2178.2.a.t 2 33.h odd 10 1
2178.2.a.bb 2 33.f even 10 1
5808.2.a.cb 2 44.h odd 10 1
5808.2.a.cg 2 44.g even 10 1

Hecke kernels

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

\( T_{5}^{4} + 7T_{5}^{3} + 34T_{5}^{2} + 88T_{5} + 121 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} + 16T_{7}^{2} + 66T_{7} + 121 \) 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

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + 16 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 6 T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 96 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$67$ \( (T - 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 124 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{4} - 19 T^{3} + 186 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$83$ \( T^{4} + 9 T^{3} + 256 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 21 T^{3} + 196 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
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