# Properties

 Label 726.2.e.g 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: $$\Q(\zeta_{10})$$ 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: 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.g 4
11.b odd 2 1 726.2.e.o 4
11.c even 5 1 66.2.a.b 1
11.c even 5 3 inner 726.2.e.g 4
11.d odd 10 1 726.2.a.c 1
11.d odd 10 3 726.2.e.o 4
33.f even 10 1 2178.2.a.g 1
33.h odd 10 1 198.2.a.a 1
44.g even 10 1 5808.2.a.bc 1
44.h odd 10 1 528.2.a.j 1
55.j even 10 1 1650.2.a.k 1
55.k odd 20 2 1650.2.c.e 2
77.j odd 10 1 3234.2.a.t 1
88.l odd 10 1 2112.2.a.e 1
88.o even 10 1 2112.2.a.r 1
99.m even 15 2 1782.2.e.e 2
99.n odd 30 2 1782.2.e.v 2
132.o even 10 1 1584.2.a.f 1
165.o odd 10 1 4950.2.a.bu 1
165.v even 20 2 4950.2.c.p 2
231.u even 10 1 9702.2.a.x 1
264.t odd 10 1 6336.2.a.bw 1
264.w even 10 1 6336.2.a.cj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 11.c even 5 1
198.2.a.a 1 33.h odd 10 1
528.2.a.j 1 44.h odd 10 1
726.2.a.c 1 11.d odd 10 1
726.2.e.g 4 1.a even 1 1 trivial
726.2.e.g 4 11.c even 5 3 inner
726.2.e.o 4 11.b odd 2 1
726.2.e.o 4 11.d odd 10 3
1584.2.a.f 1 132.o even 10 1
1650.2.a.k 1 55.j even 10 1
1650.2.c.e 2 55.k odd 20 2
1782.2.e.e 2 99.m even 15 2
1782.2.e.v 2 99.n odd 30 2
2112.2.a.e 1 88.l odd 10 1
2112.2.a.r 1 88.o even 10 1
2178.2.a.g 1 33.f even 10 1
3234.2.a.t 1 77.j odd 10 1
4950.2.a.bu 1 165.o odd 10 1
4950.2.c.p 2 165.v even 20 2
5808.2.a.bc 1 44.g even 10 1
6336.2.a.bw 1 264.t odd 10 1
6336.2.a.cj 1 264.w even 10 1
9702.2.a.x 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 $$S_{2}^{\mathrm{new}}(726, [\chi])$$:

 $$T_{5}^{4} + 2 T_{5}^{3} + 4 T_{5}^{2} + 8 T_{5} + 16$$ $$T_{7}^{4} - 4 T_{7}^{3} + 16 T_{7}^{2} - 64 T_{7} + 256$$ $$T_{13}^{4} - 6 T_{13}^{3} + 36 T_{13}^{2} - 216 T_{13} + 1296$$

## Hecke characteristic polynomials

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