# 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.

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

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

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

## 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 $$S_{3}^{\mathrm{new}}(13, [\chi])$$.