# Properties

 Label 13.4.e.b Level 13 Weight 4 Character orbit 13.e Analytic conductor 0.767 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 13.e (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.767024830075$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -5 \zeta_{6} q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{6} + ( -16 + 8 \zeta_{6} ) q^{7} + ( 13 - 26 \zeta_{6} ) q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -5 \zeta_{6} q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{6} + ( -16 + 8 \zeta_{6} ) q^{7} + ( 13 - 26 \zeta_{6} ) q^{8} + 23 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{10} + ( 8 + 8 \zeta_{6} ) q^{11} + 10 q^{12} + ( 39 + 13 \zeta_{6} ) q^{13} -24 q^{14} + ( 2 + 2 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -117 \zeta_{6} q^{17} + ( -23 + 46 \zeta_{6} ) q^{18} + ( -132 + 66 \zeta_{6} ) q^{19} + ( -10 + 5 \zeta_{6} ) q^{20} + ( 16 - 32 \zeta_{6} ) q^{21} + 24 \zeta_{6} q^{22} + ( 78 - 78 \zeta_{6} ) q^{23} + ( 26 + 26 \zeta_{6} ) q^{24} + 122 q^{25} + ( 26 + 65 \zeta_{6} ) q^{26} -100 q^{27} + ( 40 + 40 \zeta_{6} ) q^{28} + ( 141 - 141 \zeta_{6} ) q^{29} + 6 \zeta_{6} q^{30} + ( -90 + 180 \zeta_{6} ) q^{31} + ( -210 + 105 \zeta_{6} ) q^{32} + ( -32 + 16 \zeta_{6} ) q^{33} + ( 117 - 234 \zeta_{6} ) q^{34} + 24 \zeta_{6} q^{35} + ( 115 - 115 \zeta_{6} ) q^{36} + ( -83 - 83 \zeta_{6} ) q^{37} -198 q^{38} + ( -104 + 78 \zeta_{6} ) q^{39} -39 q^{40} + ( 157 + 157 \zeta_{6} ) q^{41} + ( 48 - 48 \zeta_{6} ) q^{42} -104 \zeta_{6} q^{43} + ( 40 - 80 \zeta_{6} ) q^{44} + ( 46 - 23 \zeta_{6} ) q^{45} + ( 156 - 78 \zeta_{6} ) q^{46} + ( -174 + 348 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{48} + ( -151 + 151 \zeta_{6} ) q^{49} + ( 122 + 122 \zeta_{6} ) q^{50} + 234 q^{51} + ( 65 - 260 \zeta_{6} ) q^{52} + 93 q^{53} + ( -100 - 100 \zeta_{6} ) q^{54} + ( 24 - 24 \zeta_{6} ) q^{55} + 312 \zeta_{6} q^{56} + ( 132 - 264 \zeta_{6} ) q^{57} + ( 282 - 141 \zeta_{6} ) q^{58} + ( -328 + 164 \zeta_{6} ) q^{59} + ( 10 - 20 \zeta_{6} ) q^{60} -145 \zeta_{6} q^{61} + ( -270 + 270 \zeta_{6} ) q^{62} + ( -184 - 184 \zeta_{6} ) q^{63} -307 q^{64} + ( 65 - 91 \zeta_{6} ) q^{65} -48 q^{66} + ( -454 - 454 \zeta_{6} ) q^{67} + ( -585 + 585 \zeta_{6} ) q^{68} + 156 \zeta_{6} q^{69} + ( -24 + 48 \zeta_{6} ) q^{70} + ( 1220 - 610 \zeta_{6} ) q^{71} + ( 598 - 299 \zeta_{6} ) q^{72} + ( -265 + 530 \zeta_{6} ) q^{73} -249 \zeta_{6} q^{74} + ( -244 + 244 \zeta_{6} ) q^{75} + ( 330 + 330 \zeta_{6} ) q^{76} -192 q^{77} + ( -182 + 52 \zeta_{6} ) q^{78} + 1276 q^{79} + ( 1 + \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 471 \zeta_{6} q^{82} + ( 456 - 912 \zeta_{6} ) q^{83} + ( -160 + 80 \zeta_{6} ) q^{84} + ( -234 + 117 \zeta_{6} ) q^{85} + ( 104 - 208 \zeta_{6} ) q^{86} + 282 \zeta_{6} q^{87} + ( 312 - 312 \zeta_{6} ) q^{88} + ( -564 - 564 \zeta_{6} ) q^{89} + 69 q^{90} + ( -728 + 208 \zeta_{6} ) q^{91} -390 q^{92} + ( -180 - 180 \zeta_{6} ) q^{93} + ( -522 + 522 \zeta_{6} ) q^{94} + 198 \zeta_{6} q^{95} + ( 210 - 420 \zeta_{6} ) q^{96} + ( 232 - 116 \zeta_{6} ) q^{97} + ( -302 + 151 \zeta_{6} ) q^{98} + ( -184 + 368 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} - 2q^{3} - 5q^{4} - 6q^{6} - 24q^{7} + 23q^{9} + O(q^{10})$$ $$2q + 3q^{2} - 2q^{3} - 5q^{4} - 6q^{6} - 24q^{7} + 23q^{9} + 3q^{10} + 24q^{11} + 20q^{12} + 91q^{13} - 48q^{14} + 6q^{15} - q^{16} - 117q^{17} - 198q^{19} - 15q^{20} + 24q^{22} + 78q^{23} + 78q^{24} + 244q^{25} + 117q^{26} - 200q^{27} + 120q^{28} + 141q^{29} + 6q^{30} - 315q^{32} - 48q^{33} + 24q^{35} + 115q^{36} - 249q^{37} - 396q^{38} - 130q^{39} - 78q^{40} + 471q^{41} + 48q^{42} - 104q^{43} + 69q^{45} + 234q^{46} - 2q^{48} - 151q^{49} + 366q^{50} + 468q^{51} - 130q^{52} + 186q^{53} - 300q^{54} + 24q^{55} + 312q^{56} + 423q^{58} - 492q^{59} - 145q^{61} - 270q^{62} - 552q^{63} - 614q^{64} + 39q^{65} - 96q^{66} - 1362q^{67} - 585q^{68} + 156q^{69} + 1830q^{71} + 897q^{72} - 249q^{74} - 244q^{75} + 990q^{76} - 384q^{77} - 312q^{78} + 2552q^{79} + 3q^{80} - 421q^{81} + 471q^{82} - 240q^{84} - 351q^{85} + 282q^{87} + 312q^{88} - 1692q^{89} + 138q^{90} - 1248q^{91} - 780q^{92} - 540q^{93} - 522q^{94} + 198q^{95} + 348q^{97} - 453q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 + 0.866025i −1.00000 + 1.73205i −2.50000 4.33013i 1.73205i −3.00000 + 1.73205i −12.0000 + 6.92820i 22.5167i 11.5000 + 19.9186i 1.50000 2.59808i
10.1 1.50000 0.866025i −1.00000 1.73205i −2.50000 + 4.33013i 1.73205i −3.00000 1.73205i −12.0000 6.92820i 22.5167i 11.5000 19.9186i 1.50000 + 2.59808i
 $$n$$: 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.4.e.b 2
3.b odd 2 1 117.4.q.a 2
4.b odd 2 1 208.4.w.b 2
13.b even 2 1 169.4.e.a 2
13.c even 3 1 169.4.b.d 2
13.c even 3 1 169.4.e.a 2
13.d odd 4 2 169.4.c.h 4
13.e even 6 1 inner 13.4.e.b 2
13.e even 6 1 169.4.b.d 2
13.f odd 12 2 169.4.a.i 2
13.f odd 12 2 169.4.c.h 4
39.h odd 6 1 117.4.q.a 2
39.k even 12 2 1521.4.a.o 2
52.i odd 6 1 208.4.w.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 1.a even 1 1 trivial
13.4.e.b 2 13.e even 6 1 inner
117.4.q.a 2 3.b odd 2 1
117.4.q.a 2 39.h odd 6 1
169.4.a.i 2 13.f odd 12 2
169.4.b.d 2 13.c even 3 1
169.4.b.d 2 13.e even 6 1
169.4.c.h 4 13.d odd 4 2
169.4.c.h 4 13.f odd 12 2
169.4.e.a 2 13.b even 2 1
169.4.e.a 2 13.c even 3 1
208.4.w.b 2 4.b odd 2 1
208.4.w.b 2 52.i odd 6 1
1521.4.a.o 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3 T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 11 T^{2} - 24 T^{3} + 64 T^{4}$$
$3$ $$1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4}$$
$5$ $$1 - 247 T^{2} + 15625 T^{4}$$
$7$ $$1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4}$$
$11$ $$1 - 24 T + 1523 T^{2} - 31944 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 91 T + 2197 T^{2}$$
$17$ $$1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4}$$
$31$ $$( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} )$$
$37$ $$1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 116818 T^{2} + 10779215329 T^{4}$$
$53$ $$( 1 - 93 T + 148877 T^{2} )^{2}$$
$59$ $$1 + 492 T + 286067 T^{2} + 101046468 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 1830 T + 1474211 T^{2} - 654977130 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 567359 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 - 1276 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 519766 T^{2} + 326940373369 T^{4}$$
$89$ $$1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4}$$