Properties

Label 726.2.e.b
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: \(\Q(\zeta_{10})\)
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.

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

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

\( T_{5} \)
\( T_{7}^{4} - 2 T_{7}^{3} + 4 T_{7}^{2} - 8 T_{7} + 16 \)
\( T_{13}^{4} + 4 T_{13}^{3} + 16 T_{13}^{2} + 64 T_{13} + 256 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} \)
$7$ \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 140 T^{5} - 147 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ 1
$13$ \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 520 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 6 T + 19 T^{2} + 12 T^{3} - 251 T^{4} + 204 T^{5} + 5491 T^{6} + 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 1672 T^{5} - 1083 T^{6} + 27436 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - 6 T + 23 T^{2} )^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 8 T + 33 T^{2} + 16 T^{3} - 895 T^{4} + 496 T^{5} + 31713 T^{6} + 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 10 T + 63 T^{2} - 260 T^{3} + 269 T^{4} - 9620 T^{5} + 86247 T^{6} - 506530 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 6 T - 5 T^{2} + 276 T^{3} - 1451 T^{4} + 11316 T^{5} - 8405 T^{6} - 413526 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 6 T - 11 T^{2} + 348 T^{3} - 1571 T^{4} + 16356 T^{5} - 24299 T^{6} - 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} - 205379 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 8 T + 3 T^{2} + 464 T^{3} - 3895 T^{4} + 28304 T^{5} + 11163 T^{6} - 1815848 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( 1 + 6 T - 35 T^{2} - 636 T^{3} - 1331 T^{4} - 45156 T^{5} - 176435 T^{6} + 2147466 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 2 T - 69 T^{2} + 284 T^{3} + 4469 T^{4} + 20732 T^{5} - 367701 T^{6} - 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 14 T + 117 T^{2} - 532 T^{3} - 1795 T^{4} - 42028 T^{5} + 730197 T^{6} - 6902546 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 12 T + 61 T^{2} - 264 T^{3} - 8231 T^{4} - 21912 T^{5} + 420229 T^{6} + 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{4} \)
$97$ \( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} + 2716 T^{5} + 931491 T^{6} + 12777422 T^{7} + 88529281 T^{8} \)
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