# Properties

 Label 13.4.e.a 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{4} + ( -8 + 16 \zeta_{6} ) q^{5} + ( -28 + 14 \zeta_{6} ) q^{6} + ( 26 - 13 \zeta_{6} ) q^{7} + ( -8 + 16 \zeta_{6} ) q^{8} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 - 2 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{4} + ( -8 + 16 \zeta_{6} ) q^{5} + ( -28 + 14 \zeta_{6} ) q^{6} + ( 26 - 13 \zeta_{6} ) q^{7} + ( -8 + 16 \zeta_{6} ) q^{8} -22 \zeta_{6} q^{9} + ( 48 - 48 \zeta_{6} ) q^{10} + ( -13 - 13 \zeta_{6} ) q^{11} + 28 q^{12} + ( -39 + 52 \zeta_{6} ) q^{13} -78 q^{14} + ( 56 + 56 \zeta_{6} ) q^{15} + ( 80 - 80 \zeta_{6} ) q^{16} -27 \zeta_{6} q^{17} + ( -44 + 88 \zeta_{6} ) q^{18} + ( -102 + 51 \zeta_{6} ) q^{19} + ( -64 + 32 \zeta_{6} ) q^{20} + ( 91 - 182 \zeta_{6} ) q^{21} + 78 \zeta_{6} q^{22} + ( -57 + 57 \zeta_{6} ) q^{23} + ( 56 + 56 \zeta_{6} ) q^{24} -67 q^{25} + ( 182 - 130 \zeta_{6} ) q^{26} + 35 q^{27} + ( 52 + 52 \zeta_{6} ) q^{28} + ( 69 - 69 \zeta_{6} ) q^{29} -336 \zeta_{6} q^{30} + ( 42 - 84 \zeta_{6} ) q^{31} + ( -192 + 96 \zeta_{6} ) q^{32} + ( -182 + 91 \zeta_{6} ) q^{33} + ( -54 + 108 \zeta_{6} ) q^{34} + 312 \zeta_{6} q^{35} + ( 88 - 88 \zeta_{6} ) q^{36} + ( -23 - 23 \zeta_{6} ) q^{37} + 306 q^{38} + ( 91 + 273 \zeta_{6} ) q^{39} -192 q^{40} + ( -227 - 227 \zeta_{6} ) q^{41} + ( -546 + 546 \zeta_{6} ) q^{42} + 85 \zeta_{6} q^{43} + ( 52 - 104 \zeta_{6} ) q^{44} + ( 352 - 176 \zeta_{6} ) q^{45} + ( 228 - 114 \zeta_{6} ) q^{46} + ( 198 - 396 \zeta_{6} ) q^{47} -560 \zeta_{6} q^{48} + ( 164 - 164 \zeta_{6} ) q^{49} + ( 134 + 134 \zeta_{6} ) q^{50} -189 q^{51} + ( -208 + 52 \zeta_{6} ) q^{52} + 426 q^{53} + ( -70 - 70 \zeta_{6} ) q^{54} + ( 312 - 312 \zeta_{6} ) q^{55} + 312 \zeta_{6} q^{56} + ( -357 + 714 \zeta_{6} ) q^{57} + ( -276 + 138 \zeta_{6} ) q^{58} + ( -22 + 11 \zeta_{6} ) q^{59} + ( -224 + 448 \zeta_{6} ) q^{60} + 17 \zeta_{6} q^{61} + ( -252 + 252 \zeta_{6} ) q^{62} + ( -286 - 286 \zeta_{6} ) q^{63} -64 q^{64} + ( -520 - 208 \zeta_{6} ) q^{65} + 546 q^{66} + ( 95 + 95 \zeta_{6} ) q^{67} + ( 108 - 108 \zeta_{6} ) q^{68} + 399 \zeta_{6} q^{69} + ( 624 - 1248 \zeta_{6} ) q^{70} + ( 674 - 337 \zeta_{6} ) q^{71} + ( 352 - 176 \zeta_{6} ) q^{72} + ( -580 + 1160 \zeta_{6} ) q^{73} + 138 \zeta_{6} q^{74} + ( -469 + 469 \zeta_{6} ) q^{75} + ( -204 - 204 \zeta_{6} ) q^{76} -507 q^{77} + ( 364 - 1274 \zeta_{6} ) q^{78} -1244 q^{79} + ( 640 + 640 \zeta_{6} ) q^{80} + ( 839 - 839 \zeta_{6} ) q^{81} + 1362 \zeta_{6} q^{82} + ( 246 - 492 \zeta_{6} ) q^{83} + ( 728 - 364 \zeta_{6} ) q^{84} + ( 432 - 216 \zeta_{6} ) q^{85} + ( 170 - 340 \zeta_{6} ) q^{86} -483 \zeta_{6} q^{87} + ( 312 - 312 \zeta_{6} ) q^{88} + ( 177 + 177 \zeta_{6} ) q^{89} -1056 q^{90} + ( -338 + 1183 \zeta_{6} ) q^{91} -228 q^{92} + ( -294 - 294 \zeta_{6} ) q^{93} + ( -1188 + 1188 \zeta_{6} ) q^{94} -1224 \zeta_{6} q^{95} + ( -672 + 1344 \zeta_{6} ) q^{96} + ( 1426 - 713 \zeta_{6} ) q^{97} + ( -656 + 328 \zeta_{6} ) q^{98} + ( -286 + 572 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9} + O(q^{10})$$ $$2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9} + 48 q^{10} - 39 q^{11} + 56 q^{12} - 26 q^{13} - 156 q^{14} + 168 q^{15} + 80 q^{16} - 27 q^{17} - 153 q^{19} - 96 q^{20} + 78 q^{22} - 57 q^{23} + 168 q^{24} - 134 q^{25} + 234 q^{26} + 70 q^{27} + 156 q^{28} + 69 q^{29} - 336 q^{30} - 288 q^{32} - 273 q^{33} + 312 q^{35} + 88 q^{36} - 69 q^{37} + 612 q^{38} + 455 q^{39} - 384 q^{40} - 681 q^{41} - 546 q^{42} + 85 q^{43} + 528 q^{45} + 342 q^{46} - 560 q^{48} + 164 q^{49} + 402 q^{50} - 378 q^{51} - 364 q^{52} + 852 q^{53} - 210 q^{54} + 312 q^{55} + 312 q^{56} - 414 q^{58} - 33 q^{59} + 17 q^{61} - 252 q^{62} - 858 q^{63} - 128 q^{64} - 1248 q^{65} + 1092 q^{66} + 285 q^{67} + 108 q^{68} + 399 q^{69} + 1011 q^{71} + 528 q^{72} + 138 q^{74} - 469 q^{75} - 612 q^{76} - 1014 q^{77} - 546 q^{78} - 2488 q^{79} + 1920 q^{80} + 839 q^{81} + 1362 q^{82} + 1092 q^{84} + 648 q^{85} - 483 q^{87} + 312 q^{88} + 531 q^{89} - 2112 q^{90} + 507 q^{91} - 456 q^{92} - 882 q^{93} - 1188 q^{94} - 1224 q^{95} + 2139 q^{97} - 984 q^{98} + O(q^{100})$$

## 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
−3.00000 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i −21.0000 + 12.1244i 19.5000 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
10.1 −3.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i −21.0000 12.1244i 19.5000 + 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
 $$n$$: 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.a 2
3.b odd 2 1 117.4.q.c 2
4.b odd 2 1 208.4.w.a 2
13.b even 2 1 169.4.e.b 2
13.c even 3 1 169.4.b.b 2
13.c even 3 1 169.4.e.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 inner 13.4.e.a 2
13.e even 6 1 169.4.b.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.h odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.i odd 6 1 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 6 T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$12 + 6 T + T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$192 + T^{2}$$
$7$ $$507 - 39 T + T^{2}$$
$11$ $$507 + 39 T + T^{2}$$
$13$ $$2197 + 26 T + T^{2}$$
$17$ $$729 + 27 T + T^{2}$$
$19$ $$7803 + 153 T + T^{2}$$
$23$ $$3249 + 57 T + T^{2}$$
$29$ $$4761 - 69 T + T^{2}$$
$31$ $$5292 + T^{2}$$
$37$ $$1587 + 69 T + T^{2}$$
$41$ $$154587 + 681 T + T^{2}$$
$43$ $$7225 - 85 T + T^{2}$$
$47$ $$117612 + T^{2}$$
$53$ $$( -426 + T )^{2}$$
$59$ $$363 + 33 T + T^{2}$$
$61$ $$289 - 17 T + T^{2}$$
$67$ $$27075 - 285 T + T^{2}$$
$71$ $$340707 - 1011 T + T^{2}$$
$73$ $$1009200 + T^{2}$$
$79$ $$( 1244 + T )^{2}$$
$83$ $$181548 + T^{2}$$
$89$ $$93987 - 531 T + T^{2}$$
$97$ $$1525107 - 2139 T + T^{2}$$