Properties

Label 726.2.e.o
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: \(x^{4} - x^{3} + x^{2} - x + 1\)
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.

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

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

Hecke kernels

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

\( T_{5}^{4} + 2 T_{5}^{3} + 4 T_{5}^{2} + 8 T_{5} + 16 \)
\( T_{7}^{4} + 4 T_{7}^{3} + 16 T_{7}^{2} + 64 T_{7} + 256 \)
\( T_{13}^{4} + 6 T_{13}^{3} + 36 T_{13}^{2} + 216 T_{13} + 1296 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( -4 + T )^{4} \)
$29$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 38416 + 2744 T + 196 T^{2} + 14 T^{3} + T^{4} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( 20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( ( -10 + T )^{4} \)
$97$ \( 38416 - 2744 T + 196 T^{2} - 14 T^{3} + T^{4} \)
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