# Properties

 Label 13.3.d.a Level 13 Weight 3 Character orbit 13.d Analytic conductor 0.354 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 13.d (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.354224343668$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -1 - \beta_{1} + \beta_{3} ) q^{3} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{6} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{7} + ( 9 - 9 \beta_{2} + 3 \beta_{3} ) q^{8} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -1 - \beta_{1} + \beta_{3} ) q^{3} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{6} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{7} + ( 9 - 9 \beta_{2} + 3 \beta_{3} ) q^{8} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{10} + ( -1 + \beta_{2} - 4 \beta_{3} ) q^{11} + ( -\beta_{1} + 17 \beta_{2} - \beta_{3} ) q^{12} + ( 2 - 3 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{14} + ( -7 - 5 \beta_{1} - 7 \beta_{2} ) q^{15} + ( -21 + 4 \beta_{1} - 4 \beta_{3} ) q^{16} + ( 6 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{17} + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{18} + 2 \beta_{1} q^{19} + ( 4 - 4 \beta_{2} - 5 \beta_{3} ) q^{20} + ( 8 - 8 \beta_{2} - 7 \beta_{3} ) q^{21} + ( 22 - 5 \beta_{1} + 5 \beta_{3} ) q^{22} + ( -3 \beta_{1} + 18 \beta_{2} - 3 \beta_{3} ) q^{23} + ( 6 - 6 \beta_{2} + 15 \beta_{3} ) q^{24} + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{25} + ( -22 + 7 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} ) q^{26} + ( -13 + 5 \beta_{1} - 5 \beta_{3} ) q^{27} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{28} + ( 10 - 5 \beta_{1} + 5 \beta_{3} ) q^{29} + ( -2 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} ) q^{30} + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{31} + ( 5 - 17 \beta_{1} + 5 \beta_{2} ) q^{32} + ( -19 + 19 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -27 + 27 \beta_{2} - 9 \beta_{3} ) q^{34} + ( -17 - 5 \beta_{1} + 5 \beta_{3} ) q^{35} + ( 2 \beta_{1} - 34 \beta_{2} + 2 \beta_{3} ) q^{36} + ( 10 - 10 \beta_{2} - 9 \beta_{3} ) q^{37} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 23 + 11 \beta_{1} + 15 \beta_{2} + 10 \beta_{3} ) q^{39} + ( 21 + 3 \beta_{1} - 3 \beta_{3} ) q^{40} + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{41} + ( 19 + \beta_{1} - \beta_{3} ) q^{42} + ( -3 \beta_{1} - 21 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -43 + 16 \beta_{1} - 43 \beta_{2} ) q^{44} + ( 14 + 10 \beta_{1} + 14 \beta_{2} ) q^{45} + ( 33 - 33 \beta_{2} + 24 \beta_{3} ) q^{46} + ( -1 + \beta_{2} + 23 \beta_{3} ) q^{47} + ( -19 + 17 \beta_{1} - 17 \beta_{3} ) q^{48} + ( -6 \beta_{1} + 26 \beta_{2} - 6 \beta_{3} ) q^{49} + ( -32 + 32 \beta_{2} - 20 \beta_{3} ) q^{50} + ( -9 \beta_{1} - 63 \beta_{2} - 9 \beta_{3} ) q^{51} + ( 20 - 30 \beta_{1} + 56 \beta_{2} + 7 \beta_{3} ) q^{52} + ( -20 - 5 \beta_{1} + 5 \beta_{3} ) q^{53} + ( 38 - 23 \beta_{1} + 38 \beta_{2} ) q^{54} + ( 16 + 7 \beta_{1} - 7 \beta_{3} ) q^{55} + 39 \beta_{2} q^{56} + ( -10 - 2 \beta_{1} - 10 \beta_{2} ) q^{57} + ( -35 + 20 \beta_{1} - 35 \beta_{2} ) q^{58} + ( 14 - 14 \beta_{2} - 28 \beta_{3} ) q^{59} + ( -29 + 29 \beta_{2} + 13 \beta_{3} ) q^{60} + ( -74 - 2 \beta_{1} + 2 \beta_{3} ) q^{61} + ( 16 \beta_{1} - 50 \beta_{2} + 16 \beta_{3} ) q^{62} + ( -16 + 16 \beta_{2} + 14 \beta_{3} ) q^{63} + ( 6 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} ) q^{64} + ( 14 - 8 \beta_{1} - 31 \beta_{2} - 12 \beta_{3} ) q^{65} + ( 28 - 17 \beta_{1} + 17 \beta_{3} ) q^{66} + ( -21 + 38 \beta_{1} - 21 \beta_{2} ) q^{67} + ( 111 - 12 \beta_{1} + 12 \beta_{3} ) q^{68} + ( -15 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} ) q^{69} + ( -8 - 7 \beta_{1} - 8 \beta_{2} ) q^{70} + ( 71 + 13 \beta_{1} + 71 \beta_{2} ) q^{71} + ( -12 + 12 \beta_{2} - 30 \beta_{3} ) q^{72} -20 \beta_{3} q^{73} + ( 25 + \beta_{1} - \beta_{3} ) q^{74} + ( 8 \beta_{1} - 28 \beta_{2} + 8 \beta_{3} ) q^{75} + ( 20 - 20 \beta_{2} + 6 \beta_{3} ) q^{76} + ( 11 \beta_{1} + 14 \beta_{2} + 11 \beta_{3} ) q^{77} + ( -58 + 22 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{78} + ( 16 - 11 \beta_{1} + 11 \beta_{3} ) q^{79} + ( -22 - 5 \beta_{1} - 22 \beta_{2} ) q^{80} + ( -55 - 10 \beta_{1} + 10 \beta_{3} ) q^{81} + ( 10 \beta_{1} - 26 \beta_{2} + 10 \beta_{3} ) q^{82} + ( -13 - 20 \beta_{1} - 13 \beta_{2} ) q^{83} + ( -46 - 11 \beta_{1} - 46 \beta_{2} ) q^{84} + ( -36 + 36 \beta_{2} + 27 \beta_{3} ) q^{85} + ( -6 + 6 \beta_{2} - 15 \beta_{3} ) q^{86} + ( 40 - 5 \beta_{1} + 5 \beta_{3} ) q^{87} + ( -39 \beta_{1} + 78 \beta_{2} - 39 \beta_{3} ) q^{88} + ( 50 - 50 \beta_{2} + 26 \beta_{3} ) q^{89} + ( 4 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} ) q^{90} + ( 39 + 13 \beta_{1} + 26 \beta_{2} - 13 \beta_{3} ) q^{91} + ( -114 + 45 \beta_{1} - 45 \beta_{3} ) q^{92} + ( 20 - 14 \beta_{1} + 20 \beta_{2} ) q^{93} + ( -113 + 22 \beta_{1} - 22 \beta_{3} ) q^{94} + ( 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 80 + 7 \beta_{1} + 80 \beta_{2} ) q^{96} + ( -17 - 66 \beta_{1} - 17 \beta_{2} ) q^{97} + ( 56 - 56 \beta_{2} + 38 \beta_{3} ) q^{98} + ( 38 - 38 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} + 8q^{5} - 16q^{6} - 12q^{7} + 36q^{8} + 8q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} + 8q^{5} - 16q^{6} - 12q^{7} + 36q^{8} + 8q^{9} - 4q^{11} + 8q^{13} + 4q^{14} - 28q^{15} - 84q^{16} + 32q^{18} + 16q^{20} + 32q^{21} + 88q^{22} + 24q^{24} - 88q^{26} - 52q^{27} + 4q^{28} + 40q^{29} + 40q^{31} + 20q^{32} - 76q^{33} - 108q^{34} - 68q^{35} + 40q^{37} + 92q^{39} + 84q^{40} + 32q^{41} + 76q^{42} - 172q^{44} + 56q^{45} + 132q^{46} - 4q^{47} - 76q^{48} - 128q^{50} + 80q^{52} - 80q^{53} + 152q^{54} + 64q^{55} - 40q^{57} - 140q^{58} + 56q^{59} - 116q^{60} - 296q^{61} - 64q^{63} + 56q^{65} + 112q^{66} - 84q^{67} + 444q^{68} - 32q^{70} + 284q^{71} - 48q^{72} + 100q^{74} + 80q^{76} - 232q^{78} + 64q^{79} - 88q^{80} - 220q^{81} - 52q^{83} - 184q^{84} - 144q^{85} - 24q^{86} + 160q^{87} + 200q^{89} + 156q^{91} - 456q^{92} + 80q^{93} - 452q^{94} + 320q^{96} - 68q^{97} + 224q^{98} + 152q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character Values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
−2.58114 + 2.58114i 2.16228 9.32456i 0.418861 0.418861i −5.58114 + 5.58114i −1.41886 1.41886i 13.7434 + 13.7434i −4.32456 2.16228i
5.2 0.581139 0.581139i −4.16228 3.32456i 3.58114 3.58114i −2.41886 + 2.41886i −4.58114 4.58114i 4.25658 + 4.25658i 8.32456 4.16228i
8.1 −2.58114 2.58114i 2.16228 9.32456i 0.418861 + 0.418861i −5.58114 5.58114i −1.41886 + 1.41886i 13.7434 13.7434i −4.32456 2.16228i
8.2 0.581139 + 0.581139i −4.16228 3.32456i 3.58114 + 3.58114i −2.41886 2.41886i −4.58114 + 4.58114i 4.25658 4.25658i 8.32456 4.16228i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(13, [\chi])$$.