# Properties

 Label 13.4.c.a Level 13 Weight 4 Character orbit 13.c 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.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.767024830075$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -4 + 4 \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -8 \zeta_{6} q^{4} + 17 q^{5} -8 \zeta_{6} q^{6} -20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -4 + 4 \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -8 \zeta_{6} q^{4} + 17 q^{5} -8 \zeta_{6} q^{6} -20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} + ( -68 + 68 \zeta_{6} ) q^{10} + ( 32 - 32 \zeta_{6} ) q^{11} + 16 q^{12} + ( -39 - 13 \zeta_{6} ) q^{13} + 80 q^{14} + ( -34 + 34 \zeta_{6} ) q^{15} + ( 64 - 64 \zeta_{6} ) q^{16} + 13 \zeta_{6} q^{17} -92 q^{18} -30 \zeta_{6} q^{19} -136 \zeta_{6} q^{20} + 40 q^{21} + 128 \zeta_{6} q^{22} + ( -78 + 78 \zeta_{6} ) q^{23} + 164 q^{25} + ( 208 - 156 \zeta_{6} ) q^{26} -100 q^{27} + ( -160 + 160 \zeta_{6} ) q^{28} + ( -197 + 197 \zeta_{6} ) q^{29} -136 \zeta_{6} q^{30} -74 q^{31} + 256 \zeta_{6} q^{32} + 64 \zeta_{6} q^{33} -52 q^{34} -340 \zeta_{6} q^{35} + ( 184 - 184 \zeta_{6} ) q^{36} + ( 227 - 227 \zeta_{6} ) q^{37} + 120 q^{38} + ( 104 - 78 \zeta_{6} ) q^{39} + ( 165 - 165 \zeta_{6} ) q^{41} + ( -160 + 160 \zeta_{6} ) q^{42} + 156 \zeta_{6} q^{43} -256 q^{44} + 391 \zeta_{6} q^{45} -312 \zeta_{6} q^{46} -162 q^{47} + 128 \zeta_{6} q^{48} + ( -57 + 57 \zeta_{6} ) q^{49} + ( -656 + 656 \zeta_{6} ) q^{50} -26 q^{51} + ( -104 + 416 \zeta_{6} ) q^{52} + 93 q^{53} + ( 400 - 400 \zeta_{6} ) q^{54} + ( 544 - 544 \zeta_{6} ) q^{55} + 60 q^{57} -788 \zeta_{6} q^{58} + 864 \zeta_{6} q^{59} + 272 q^{60} -145 \zeta_{6} q^{61} + ( 296 - 296 \zeta_{6} ) q^{62} + ( 460 - 460 \zeta_{6} ) q^{63} -512 q^{64} + ( -663 - 221 \zeta_{6} ) q^{65} -256 q^{66} + ( -862 + 862 \zeta_{6} ) q^{67} + ( 104 - 104 \zeta_{6} ) q^{68} -156 \zeta_{6} q^{69} + 1360 q^{70} -654 \zeta_{6} q^{71} + 215 q^{73} + 908 \zeta_{6} q^{74} + ( -328 + 328 \zeta_{6} ) q^{75} + ( -240 + 240 \zeta_{6} ) q^{76} -640 q^{77} + ( -104 + 416 \zeta_{6} ) q^{78} -76 q^{79} + ( 1088 - 1088 \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 660 \zeta_{6} q^{82} + 628 q^{83} -320 \zeta_{6} q^{84} + 221 \zeta_{6} q^{85} -624 q^{86} -394 \zeta_{6} q^{87} + ( 266 - 266 \zeta_{6} ) q^{89} -1564 q^{90} + ( -260 + 1040 \zeta_{6} ) q^{91} + 624 q^{92} + ( 148 - 148 \zeta_{6} ) q^{93} + ( 648 - 648 \zeta_{6} ) q^{94} -510 \zeta_{6} q^{95} -512 q^{96} -238 \zeta_{6} q^{97} -228 \zeta_{6} q^{98} + 736 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 2q^{3} - 8q^{4} + 34q^{5} - 8q^{6} - 20q^{7} + 23q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 2q^{3} - 8q^{4} + 34q^{5} - 8q^{6} - 20q^{7} + 23q^{9} - 68q^{10} + 32q^{11} + 32q^{12} - 91q^{13} + 160q^{14} - 34q^{15} + 64q^{16} + 13q^{17} - 184q^{18} - 30q^{19} - 136q^{20} + 80q^{21} + 128q^{22} - 78q^{23} + 328q^{25} + 260q^{26} - 200q^{27} - 160q^{28} - 197q^{29} - 136q^{30} - 148q^{31} + 256q^{32} + 64q^{33} - 104q^{34} - 340q^{35} + 184q^{36} + 227q^{37} + 240q^{38} + 130q^{39} + 165q^{41} - 160q^{42} + 156q^{43} - 512q^{44} + 391q^{45} - 312q^{46} - 324q^{47} + 128q^{48} - 57q^{49} - 656q^{50} - 52q^{51} + 208q^{52} + 186q^{53} + 400q^{54} + 544q^{55} + 120q^{57} - 788q^{58} + 864q^{59} + 544q^{60} - 145q^{61} + 296q^{62} + 460q^{63} - 1024q^{64} - 1547q^{65} - 512q^{66} - 862q^{67} + 104q^{68} - 156q^{69} + 2720q^{70} - 654q^{71} + 430q^{73} + 908q^{74} - 328q^{75} - 240q^{76} - 1280q^{77} + 208q^{78} - 152q^{79} + 1088q^{80} - 421q^{81} + 660q^{82} + 1256q^{83} - 320q^{84} + 221q^{85} - 1248q^{86} - 394q^{87} + 266q^{89} - 3128q^{90} + 520q^{91} + 1248q^{92} + 148q^{93} + 648q^{94} - 510q^{95} - 1024q^{96} - 238q^{97} - 228q^{98} + 1472q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 − 0.866025i 0.5 + 0.866025i
−2.00000 3.46410i −1.00000 1.73205i −4.00000 + 6.92820i 17.0000 −4.00000 + 6.92820i −10.0000 + 17.3205i 0 11.5000 19.9186i −34.0000 58.8897i
9.1 −2.00000 + 3.46410i −1.00000 + 1.73205i −4.00000 6.92820i 17.0000 −4.00000 6.92820i −10.0000 17.3205i 0 11.5000 + 19.9186i −34.0000 + 58.8897i
 $$n$$: 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4 T_{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 8 T^{2} + 32 T^{3} + 64 T^{4}$$
$3$ $$1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4}$$
$5$ $$( 1 - 17 T + 125 T^{2} )^{2}$$
$7$ $$( 1 - 17 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} )$$
$11$ $$1 - 32 T - 307 T^{2} - 42592 T^{3} + 1771561 T^{4}$$
$13$ $$1 + 91 T + 2197 T^{2}$$
$17$ $$1 - 13 T - 4744 T^{2} - 63869 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 30 T - 5959 T^{2} + 205770 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 197 T + 14420 T^{2} + 4804633 T^{3} + 594823321 T^{4}$$
$31$ $$( 1 + 74 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 227 T + 876 T^{2} - 11498231 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 165 T - 41696 T^{2} - 11371965 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 156 T - 55171 T^{2} - 12403092 T^{3} + 6321363049 T^{4}$$
$47$ $$( 1 + 162 T + 103823 T^{2} )^{2}$$
$53$ $$( 1 - 93 T + 148877 T^{2} )^{2}$$
$59$ $$1 - 864 T + 541117 T^{2} - 177447456 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 862 T + 442281 T^{2} + 259257706 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 654 T + 69805 T^{2} + 234073794 T^{3} + 128100283921 T^{4}$$
$73$ $$( 1 - 215 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 + 76 T + 493039 T^{2} )^{2}$$
$83$ $$( 1 - 628 T + 571787 T^{2} )^{2}$$
$89$ $$1 - 266 T - 634213 T^{2} - 187521754 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 238 T - 856029 T^{2} + 217216174 T^{3} + 832972004929 T^{4}$$