Properties

Label 726.2.e.k
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
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

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.

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

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 0 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i 0
493.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0 −0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i 0
511.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0 −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i 0
565.1 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 0 0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i 0
\(n\): 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.k 4
11.b odd 2 1 726.2.e.b 4
11.c even 5 1 66.2.a.a 1
11.c even 5 3 inner 726.2.e.k 4
11.d odd 10 1 726.2.a.i 1
11.d odd 10 3 726.2.e.b 4
33.f even 10 1 2178.2.a.b 1
33.h odd 10 1 198.2.a.e 1
44.g even 10 1 5808.2.a.l 1
44.h odd 10 1 528.2.a.d 1
55.j even 10 1 1650.2.a.m 1
55.k odd 20 2 1650.2.c.d 2
77.j odd 10 1 3234.2.a.d 1
88.l odd 10 1 2112.2.a.v 1
88.o even 10 1 2112.2.a.i 1
99.m even 15 2 1782.2.e.s 2
99.n odd 30 2 1782.2.e.f 2
132.o even 10 1 1584.2.a.h 1
165.o odd 10 1 4950.2.a.g 1
165.v even 20 2 4950.2.c.r 2
231.u even 10 1 9702.2.a.bu 1
264.t odd 10 1 6336.2.a.bj 1
264.w even 10 1 6336.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 11.c even 5 1
198.2.a.e 1 33.h odd 10 1
528.2.a.d 1 44.h odd 10 1
726.2.a.i 1 11.d odd 10 1
726.2.e.b 4 11.b odd 2 1
726.2.e.b 4 11.d odd 10 3
726.2.e.k 4 1.a even 1 1 trivial
726.2.e.k 4 11.c even 5 3 inner
1584.2.a.h 1 132.o even 10 1
1650.2.a.m 1 55.j even 10 1
1650.2.c.d 2 55.k odd 20 2
1782.2.e.f 2 99.n odd 30 2
1782.2.e.s 2 99.m even 15 2
2112.2.a.i 1 88.o even 10 1
2112.2.a.v 1 88.l odd 10 1
2178.2.a.b 1 33.f even 10 1
3234.2.a.d 1 77.j odd 10 1
4950.2.a.g 1 165.o odd 10 1
4950.2.c.r 2 165.v even 20 2
5808.2.a.l 1 44.g even 10 1
6336.2.a.bf 1 264.w even 10 1
6336.2.a.bj 1 264.t odd 10 1
9702.2.a.bu 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 \(S_{2}^{\mathrm{new}}(726, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 64T_{13} + 256 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$23$ \( (T - 6)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
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