# Properties

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

# Related objects

## 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + 4 \zeta_{10} q^{5} + \zeta_{10} q^{6} -2 \zeta_{10}^{3} q^{7} -\zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + 4 \zeta_{10} q^{5} + \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} + 4 q^{10} + q^{12} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} -2 \zeta_{10}^{2} q^{14} + 4 \zeta_{10}^{3} q^{15} -\zeta_{10} q^{16} -2 \zeta_{10} q^{17} + \zeta_{10}^{3} q^{18} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{20} + 2 q^{21} -6 q^{23} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} + 11 \zeta_{10}^{2} q^{25} -4 \zeta_{10}^{3} q^{26} -\zeta_{10} q^{27} -2 \zeta_{10} q^{28} + 10 \zeta_{10}^{3} q^{29} + 4 \zeta_{10}^{2} q^{30} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{31} - q^{32} -2 q^{34} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{35} + \zeta_{10}^{2} q^{36} + 2 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} -4 \zeta_{10}^{3} q^{40} -2 \zeta_{10}^{2} q^{41} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{42} -4 q^{43} -4 q^{45} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{46} -2 \zeta_{10}^{2} q^{47} -\zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} + 11 \zeta_{10} q^{50} -2 \zeta_{10}^{3} q^{51} -4 \zeta_{10}^{2} q^{52} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{53} - q^{54} -2 q^{56} + 10 \zeta_{10}^{2} q^{58} + 4 \zeta_{10} q^{60} -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} + 16 q^{65} -12 q^{67} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{68} -6 \zeta_{10}^{2} q^{69} -8 \zeta_{10}^{3} q^{70} -2 \zeta_{10} q^{71} + \zeta_{10} q^{72} -6 \zeta_{10}^{3} q^{73} + 2 \zeta_{10}^{2} q^{74} + ( -11 + 11 \zeta_{10} - 11 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{75} + 4 q^{78} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{79} -4 \zeta_{10}^{2} q^{80} -\zeta_{10}^{3} q^{81} -2 \zeta_{10} q^{82} + 4 \zeta_{10} q^{83} -2 \zeta_{10}^{3} q^{84} -8 \zeta_{10}^{2} q^{85} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{86} -10 q^{87} + 10 q^{89} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{90} -8 \zeta_{10}^{2} q^{91} + 6 \zeta_{10}^{3} q^{92} + 8 \zeta_{10} q^{93} -2 \zeta_{10} q^{94} -\zeta_{10}^{2} q^{96} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{97} + 3 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{3} - q^{4} + 4q^{5} + q^{6} - 2q^{7} + q^{8} - q^{9} + O(q^{10})$$ $$4q + q^{2} - q^{3} - q^{4} + 4q^{5} + q^{6} - 2q^{7} + q^{8} - q^{9} + 16q^{10} + 4q^{12} + 4q^{13} + 2q^{14} + 4q^{15} - q^{16} - 2q^{17} + q^{18} + 4q^{20} + 8q^{21} - 24q^{23} + q^{24} - 11q^{25} - 4q^{26} - q^{27} - 2q^{28} + 10q^{29} - 4q^{30} + 8q^{31} - 4q^{32} - 8q^{34} + 8q^{35} - q^{36} + 2q^{37} + 4q^{39} - 4q^{40} + 2q^{41} + 2q^{42} - 16q^{43} - 16q^{45} - 6q^{46} + 2q^{47} - q^{48} + 3q^{49} + 11q^{50} - 2q^{51} + 4q^{52} - 4q^{53} - 4q^{54} - 8q^{56} - 10q^{58} + 4q^{60} - 8q^{61} - 8q^{62} - 2q^{63} - q^{64} + 64q^{65} - 48q^{67} - 2q^{68} + 6q^{69} - 8q^{70} - 2q^{71} + q^{72} - 6q^{73} - 2q^{74} - 11q^{75} + 16q^{78} + 10q^{79} + 4q^{80} - q^{81} - 2q^{82} + 4q^{83} - 2q^{84} + 8q^{85} - 4q^{86} - 40q^{87} + 40q^{89} - 4q^{90} + 8q^{91} + 6q^{92} + 8q^{93} - 2q^{94} + q^{96} + 2q^{97} + 12q^{98} + O(q^{100})$$

## 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 3.23607 + 2.35114i 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i 4.00000
493.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −1.23607 3.80423i −0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i 4.00000
511.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −1.23607 + 3.80423i −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i 4.00000
565.1 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 3.23607 2.35114i 0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.m 4
11.b odd 2 1 726.2.e.e 4
11.c even 5 1 726.2.a.d 1
11.c even 5 3 inner 726.2.e.m 4
11.d odd 10 1 66.2.a.c 1
11.d odd 10 3 726.2.e.e 4
33.f even 10 1 198.2.a.c 1
33.h odd 10 1 2178.2.a.m 1
44.g even 10 1 528.2.a.a 1
44.h odd 10 1 5808.2.a.b 1
55.h odd 10 1 1650.2.a.c 1
55.l even 20 2 1650.2.c.m 2
77.l even 10 1 3234.2.a.s 1
88.k even 10 1 2112.2.a.bd 1
88.p odd 10 1 2112.2.a.n 1
99.o odd 30 2 1782.2.e.l 2
99.p even 30 2 1782.2.e.n 2
132.n odd 10 1 1584.2.a.s 1
165.r even 10 1 4950.2.a.bo 1
165.u odd 20 2 4950.2.c.d 2
231.r odd 10 1 9702.2.a.a 1
264.r odd 10 1 6336.2.a.d 1
264.u even 10 1 6336.2.a.c 1

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$1 - 4 T + 11 T^{2} - 24 T^{3} + 41 T^{4} - 120 T^{5} + 275 T^{6} - 500 T^{7} + 625 T^{8}$$
$7$ $$1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 140 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8}$$
$11$ 1
$13$ $$1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 520 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 1020 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 6 T + 23 T^{2} )^{4}$$
$29$ $$1 - 10 T + 71 T^{2} - 420 T^{3} + 2141 T^{4} - 12180 T^{5} + 59711 T^{6} - 243890 T^{7} + 707281 T^{8}$$
$31$ $$1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 496 T^{5} + 31713 T^{6} - 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 - 2 T - 33 T^{2} + 140 T^{3} + 941 T^{4} + 5180 T^{5} - 45177 T^{6} - 101306 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 6396 T^{5} - 62197 T^{6} - 137842 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{4}$$
$47$ $$1 - 2 T - 43 T^{2} + 180 T^{3} + 1661 T^{4} + 8460 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 4 T - 37 T^{2} - 360 T^{3} + 521 T^{4} - 19080 T^{5} - 103933 T^{6} + 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4} - 205379 T^{6} + 12117361 T^{8}$$
$61$ $$1 + 8 T + 3 T^{2} - 464 T^{3} - 3895 T^{4} - 28304 T^{5} + 11163 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + 12 T + 67 T^{2} )^{4}$$
$71$ $$1 + 2 T - 67 T^{2} - 276 T^{3} + 4205 T^{4} - 19596 T^{5} - 337747 T^{6} + 715822 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 6 T - 37 T^{2} - 660 T^{3} - 1259 T^{4} - 48180 T^{5} - 197173 T^{6} + 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 4 T - 67 T^{2} + 600 T^{3} + 3161 T^{4} + 49800 T^{5} - 461563 T^{6} - 2287148 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{4}$$
$97$ $$1 - 2 T - 93 T^{2} + 380 T^{3} + 8261 T^{4} + 36860 T^{5} - 875037 T^{6} - 1825346 T^{7} + 88529281 T^{8}$$