# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$0.767024830075$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2}$$ $$+ ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{3}$$ $$-5 \beta_{1} q^{4}$$ $$+ ( -10 - 5 \beta_{3} ) q^{5}$$ $$+ ( 10 \beta_{1} + 14 \beta_{2} ) q^{6}$$ $$+ ( -\beta_{1} - 7 \beta_{2} ) q^{7}$$ $$+ ( -4 + 7 \beta_{3} ) q^{8}$$ $$+ ( -15 \beta_{1} - 10 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2}$$ $$+ ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{3}$$ $$-5 \beta_{1} q^{4}$$ $$+ ( -10 - 5 \beta_{3} ) q^{5}$$ $$+ ( 10 \beta_{1} + 14 \beta_{2} ) q^{6}$$ $$+ ( -\beta_{1} - 7 \beta_{2} ) q^{7}$$ $$+ ( -4 + 7 \beta_{3} ) q^{8}$$ $$+ ( -15 \beta_{1} - 10 \beta_{2} ) q^{9}$$ $$+ ( -5 \beta_{1} - 5 \beta_{3} ) q^{10}$$ $$+ ( -1 + 15 \beta_{1} + \beta_{2} + 15 \beta_{3} ) q^{11}$$ $$+ ( 60 - 20 \beta_{3} ) q^{12}$$ $$+ ( 49 + 5 \beta_{1} - 40 \beta_{2} - 2 \beta_{3} ) q^{13}$$ $$+ ( -18 + 10 \beta_{3} ) q^{14}$$ $$+ ( -50 - 10 \beta_{1} + 50 \beta_{2} - 10 \beta_{3} ) q^{15}$$ $$+ ( -36 - 15 \beta_{1} + 36 \beta_{2} - 15 \beta_{3} ) q^{16}$$ $$+ ( 16 \beta_{1} - 43 \beta_{2} ) q^{17}$$ $$+ ( -80 + 55 \beta_{3} ) q^{18}$$ $$+ ( -5 \beta_{1} + 73 \beta_{2} ) q^{19}$$ $$+ ( 25 \beta_{1} - 100 \beta_{2} ) q^{20}$$ $$+ ( 19 - 25 \beta_{3} ) q^{21}$$ $$+ ( 46 \beta_{1} + 62 \beta_{2} ) q^{22}$$ $$+ ( -89 - 33 \beta_{1} + 89 \beta_{2} - 33 \beta_{3} ) q^{23}$$ $$+ ( 88 - 40 \beta_{1} - 88 \beta_{2} - 40 \beta_{3} ) q^{24}$$ $$+ ( 75 + 75 \beta_{3} ) q^{25}$$ $$+ ( 46 - 55 \beta_{1} - 106 \beta_{2} - 30 \beta_{3} ) q^{26}$$ $$+ ( 163 - 9 \beta_{3} ) q^{27}$$ $$+ ( -20 + 40 \beta_{1} + 20 \beta_{2} + 40 \beta_{3} ) q^{28}$$ $$+ ( 13 + 60 \beta_{1} - 13 \beta_{2} + 60 \beta_{3} ) q^{29}$$ $$+ ( 20 \beta_{1} + 60 \beta_{2} ) q^{30}$$ $$+ ( -120 - 100 \beta_{3} ) q^{31}$$ $$+ ( -65 \beta_{1} - 20 \beta_{2} ) q^{32}$$ $$+ ( -63 \beta_{1} - 181 \beta_{2} ) q^{33}$$ $$+ ( -22 - 5 \beta_{3} ) q^{34}$$ $$+ ( -30 \beta_{1} + 50 \beta_{2} ) q^{35}$$ $$+ ( -300 + 125 \beta_{1} + 300 \beta_{2} + 125 \beta_{3} ) q^{36}$$ $$+ ( 73 - 44 \beta_{1} - 73 \beta_{2} - 44 \beta_{3} ) q^{37}$$ $$+ ( 126 - 58 \beta_{3} ) q^{38}$$ $$+ ( -93 + 155 \beta_{1} + 73 \beta_{2} + 55 \beta_{3} ) q^{39}$$ $$+ ( -100 - 15 \beta_{3} ) q^{40}$$ $$+ ( -259 + 20 \beta_{1} + 259 \beta_{2} + 20 \beta_{3} ) q^{41}$$ $$+ ( 138 - 94 \beta_{1} - 138 \beta_{2} - 94 \beta_{3} ) q^{42}$$ $$+ ( -97 \beta_{1} - 179 \beta_{2} ) q^{43}$$ $$+ ( 300 - 80 \beta_{3} ) q^{44}$$ $$+ ( 25 \beta_{1} - 200 \beta_{2} ) q^{45}$$ $$+ ( -10 \beta_{1} + 46 \beta_{2} ) q^{46}$$ $$+ ( 100 + 140 \beta_{3} ) q^{47}$$ $$+ ( -48 \beta_{1} + 144 \beta_{2} ) q^{48}$$ $$+ ( 290 + 15 \beta_{1} - 290 \beta_{2} + 15 \beta_{3} ) q^{49}$$ $$+ ( -150 + 150 \beta_{1} + 150 \beta_{2} + 150 \beta_{3} ) q^{50}$$ $$+ ( -149 - 65 \beta_{3} ) q^{51}$$ $$+ ( 100 - 80 \beta_{1} - 140 \beta_{2} + 175 \beta_{3} ) q^{52}$$ $$+ ( 190 - 165 \beta_{3} ) q^{53}$$ $$+ ( 362 - 190 \beta_{1} - 362 \beta_{2} - 190 \beta_{3} ) q^{54}$$ $$+ ( -290 - 70 \beta_{1} + 290 \beta_{2} - 70 \beta_{3} ) q^{55}$$ $$+ ( 60 \beta_{1} + 56 \beta_{2} ) q^{56}$$ $$+ ( -13 + 199 \beta_{3} ) q^{57}$$ $$+ ( 167 \beta_{1} + 214 \beta_{2} ) q^{58}$$ $$+ ( 55 \beta_{1} + 377 \beta_{2} ) q^{59}$$ $$+ ( -200 - 200 \beta_{3} ) q^{60}$$ $$+ ( 200 \beta_{1} - 351 \beta_{2} ) q^{61}$$ $$+ ( 160 - 180 \beta_{1} - 160 \beta_{2} - 180 \beta_{3} ) q^{62}$$ $$+ ( -130 + 130 \beta_{1} + 130 \beta_{2} + 130 \beta_{3} ) q^{63}$$ $$+ ( -588 + 95 \beta_{3} ) q^{64}$$ $$+ ( -450 - 225 \beta_{1} + 500 \beta_{2} - 235 \beta_{3} ) q^{65}$$ $$+ ( -614 + 370 \beta_{3} ) q^{66}$$ $$+ ( 283 + 91 \beta_{1} - 283 \beta_{2} + 91 \beta_{3} ) q^{67}$$ $$+ ( 320 + 135 \beta_{1} - 320 \beta_{2} + 135 \beta_{3} ) q^{68}$$ $$+ ( -135 \beta_{1} + 307 \beta_{2} ) q^{69}$$ $$+ ( -20 + 40 \beta_{3} ) q^{70}$$ $$+ ( -105 \beta_{1} - 11 \beta_{2} ) q^{71}$$ $$+ ( 235 \beta_{1} + 460 \beta_{2} ) q^{72}$$ $$+ ( 250 - 85 \beta_{3} ) q^{73}$$ $$+ ( -205 \beta_{1} - 322 \beta_{2} ) q^{74}$$ $$+ ( 825 - 75 \beta_{1} - 825 \beta_{2} - 75 \beta_{3} ) q^{75}$$ $$+ ( -100 - 340 \beta_{1} + 100 \beta_{2} - 340 \beta_{3} ) q^{76}$$ $$+ ( 67 - 121 \beta_{3} ) q^{77}$$ $$+ ( 360 + 258 \beta_{1} + 406 \beta_{2} - 280 \beta_{3} ) q^{78}$$ $$+ ( 140 + 40 \beta_{3} ) q^{79}$$ $$+ ( 660 + 255 \beta_{1} - 660 \beta_{2} + 255 \beta_{3} ) q^{80}$$ $$+ ( -1 + 120 \beta_{1} + \beta_{2} + 120 \beta_{3} ) q^{81}$$ $$+ ( 319 \beta_{1} + 598 \beta_{2} ) q^{82}$$ $$+ ( 180 + 100 \beta_{3} ) q^{83}$$ $$+ ( -220 \beta_{1} - 500 \beta_{2} ) q^{84}$$ $$+ ( -295 \beta_{1} + 750 \beta_{2} ) q^{85}$$ $$+ ( -746 + 470 \beta_{3} ) q^{86}$$ $$+ ( -201 \beta_{1} - 707 \beta_{2} ) q^{87}$$ $$+ ( 424 - 172 \beta_{1} - 424 \beta_{2} - 172 \beta_{3} ) q^{88}$$ $$+ ( -523 - 125 \beta_{1} + 523 \beta_{2} - 125 \beta_{3} ) q^{89}$$ $$+ ( -300 + 125 \beta_{3} ) q^{90}$$ $$+ ( -260 - 65 \beta_{1} - 91 \beta_{2} ) q^{91}$$ $$+ ( -660 - 280 \beta_{3} ) q^{92}$$ $$+ ( -1080 + 40 \beta_{1} + 1080 \beta_{2} + 40 \beta_{3} ) q^{93}$$ $$+ ( -360 + 320 \beta_{1} + 360 \beta_{2} + 320 \beta_{3} ) q^{94}$$ $$+ ( 390 \beta_{1} - 830 \beta_{2} ) q^{95}$$ $$+ ( 800 - 320 \beta_{3} ) q^{96}$$ $$+ ( 469 \beta_{1} - 27 \beta_{2} ) q^{97}$$ $$+ ( -245 \beta_{1} - 520 \beta_{2} ) q^{98}$$ $$+ ( 910 - 390 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 5q^{2}$$ $$\mathstrut -\mathstrut 5q^{3}$$ $$\mathstrut -\mathstrut 5q^{4}$$ $$\mathstrut -\mathstrut 30q^{5}$$ $$\mathstrut +\mathstrut 38q^{6}$$ $$\mathstrut -\mathstrut 15q^{7}$$ $$\mathstrut -\mathstrut 30q^{8}$$ $$\mathstrut -\mathstrut 35q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 5q^{2}$$ $$\mathstrut -\mathstrut 5q^{3}$$ $$\mathstrut -\mathstrut 5q^{4}$$ $$\mathstrut -\mathstrut 30q^{5}$$ $$\mathstrut +\mathstrut 38q^{6}$$ $$\mathstrut -\mathstrut 15q^{7}$$ $$\mathstrut -\mathstrut 30q^{8}$$ $$\mathstrut -\mathstrut 35q^{9}$$ $$\mathstrut +\mathstrut 5q^{10}$$ $$\mathstrut -\mathstrut 17q^{11}$$ $$\mathstrut +\mathstrut 280q^{12}$$ $$\mathstrut +\mathstrut 125q^{13}$$ $$\mathstrut -\mathstrut 92q^{14}$$ $$\mathstrut -\mathstrut 90q^{15}$$ $$\mathstrut -\mathstrut 57q^{16}$$ $$\mathstrut -\mathstrut 70q^{17}$$ $$\mathstrut -\mathstrut 430q^{18}$$ $$\mathstrut +\mathstrut 141q^{19}$$ $$\mathstrut -\mathstrut 175q^{20}$$ $$\mathstrut +\mathstrut 126q^{21}$$ $$\mathstrut +\mathstrut 170q^{22}$$ $$\mathstrut -\mathstrut 145q^{23}$$ $$\mathstrut +\mathstrut 216q^{24}$$ $$\mathstrut +\mathstrut 150q^{25}$$ $$\mathstrut -\mathstrut 23q^{26}$$ $$\mathstrut +\mathstrut 670q^{27}$$ $$\mathstrut -\mathstrut 80q^{28}$$ $$\mathstrut -\mathstrut 34q^{29}$$ $$\mathstrut +\mathstrut 140q^{30}$$ $$\mathstrut -\mathstrut 280q^{31}$$ $$\mathstrut -\mathstrut 105q^{32}$$ $$\mathstrut -\mathstrut 425q^{33}$$ $$\mathstrut -\mathstrut 78q^{34}$$ $$\mathstrut +\mathstrut 70q^{35}$$ $$\mathstrut -\mathstrut 725q^{36}$$ $$\mathstrut +\mathstrut 190q^{37}$$ $$\mathstrut +\mathstrut 620q^{38}$$ $$\mathstrut -\mathstrut 181q^{39}$$ $$\mathstrut -\mathstrut 370q^{40}$$ $$\mathstrut -\mathstrut 538q^{41}$$ $$\mathstrut +\mathstrut 370q^{42}$$ $$\mathstrut -\mathstrut 455q^{43}$$ $$\mathstrut +\mathstrut 1360q^{44}$$ $$\mathstrut -\mathstrut 375q^{45}$$ $$\mathstrut +\mathstrut 82q^{46}$$ $$\mathstrut +\mathstrut 120q^{47}$$ $$\mathstrut +\mathstrut 240q^{48}$$ $$\mathstrut +\mathstrut 565q^{49}$$ $$\mathstrut -\mathstrut 450q^{50}$$ $$\mathstrut -\mathstrut 466q^{51}$$ $$\mathstrut -\mathstrut 310q^{52}$$ $$\mathstrut +\mathstrut 1090q^{53}$$ $$\mathstrut +\mathstrut 914q^{54}$$ $$\mathstrut -\mathstrut 510q^{55}$$ $$\mathstrut +\mathstrut 172q^{56}$$ $$\mathstrut -\mathstrut 450q^{57}$$ $$\mathstrut +\mathstrut 595q^{58}$$ $$\mathstrut +\mathstrut 809q^{59}$$ $$\mathstrut -\mathstrut 400q^{60}$$ $$\mathstrut -\mathstrut 502q^{61}$$ $$\mathstrut +\mathstrut 500q^{62}$$ $$\mathstrut -\mathstrut 390q^{63}$$ $$\mathstrut -\mathstrut 2542q^{64}$$ $$\mathstrut -\mathstrut 555q^{65}$$ $$\mathstrut -\mathstrut 3196q^{66}$$ $$\mathstrut +\mathstrut 475q^{67}$$ $$\mathstrut +\mathstrut 505q^{68}$$ $$\mathstrut +\mathstrut 479q^{69}$$ $$\mathstrut -\mathstrut 160q^{70}$$ $$\mathstrut -\mathstrut 127q^{71}$$ $$\mathstrut +\mathstrut 1155q^{72}$$ $$\mathstrut +\mathstrut 1170q^{73}$$ $$\mathstrut -\mathstrut 849q^{74}$$ $$\mathstrut +\mathstrut 1725q^{75}$$ $$\mathstrut +\mathstrut 140q^{76}$$ $$\mathstrut +\mathstrut 510q^{77}$$ $$\mathstrut +\mathstrut 3070q^{78}$$ $$\mathstrut +\mathstrut 480q^{79}$$ $$\mathstrut +\mathstrut 1065q^{80}$$ $$\mathstrut -\mathstrut 122q^{81}$$ $$\mathstrut +\mathstrut 1515q^{82}$$ $$\mathstrut +\mathstrut 520q^{83}$$ $$\mathstrut -\mathstrut 1220q^{84}$$ $$\mathstrut +\mathstrut 1205q^{85}$$ $$\mathstrut -\mathstrut 3924q^{86}$$ $$\mathstrut -\mathstrut 1615q^{87}$$ $$\mathstrut +\mathstrut 1020q^{88}$$ $$\mathstrut -\mathstrut 921q^{89}$$ $$\mathstrut -\mathstrut 1450q^{90}$$ $$\mathstrut -\mathstrut 1287q^{91}$$ $$\mathstrut -\mathstrut 2080q^{92}$$ $$\mathstrut -\mathstrut 2200q^{93}$$ $$\mathstrut -\mathstrut 1040q^{94}$$ $$\mathstrut -\mathstrut 1270q^{95}$$ $$\mathstrut +\mathstrut 3840q^{96}$$ $$\mathstrut +\mathstrut 415q^{97}$$ $$\mathstrut -\mathstrut 1285q^{98}$$ $$\mathstrut +\mathstrut 4420q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut +\mathstrut$$ $$5$$ $$x^{2}\mathstrut +\mathstrut$$ $$4$$ $$x\mathstrut +\mathstrut$$ $$16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4$$ $$\nu^{3}$$ $$=$$ $$5$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.780776 + 1.35234i 1.28078 − 2.21837i −0.780776 − 1.35234i 1.28078 + 2.21837i
0.219224 + 0.379706i 1.84233 + 3.19101i 3.90388 6.76172i −17.8078 −0.807764 + 1.39909i −2.71922 + 4.70983i 6.93087 6.71165 11.6249i −3.90388 6.76172i
3.2 2.28078 + 3.95042i −4.34233 7.52113i −6.40388 + 11.0918i 2.80776 19.8078 34.3081i −4.78078 + 8.28055i −21.9309 −24.2116 + 41.9358i 6.40388 + 11.0918i
9.1 0.219224 0.379706i 1.84233 3.19101i 3.90388 + 6.76172i −17.8078 −0.807764 1.39909i −2.71922 4.70983i 6.93087 6.71165 + 11.6249i −3.90388 + 6.76172i
9.2 2.28078 3.95042i −4.34233 + 7.52113i −6.40388 11.0918i 2.80776 19.8078 + 34.3081i −4.78078 8.28055i −21.9309 −24.2116 41.9358i 6.40388 11.0918i
 $$n$$: 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.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4}$$ $$\mathstrut -\mathstrut 5 T_{2}^{3}$$ $$\mathstrut +\mathstrut 23 T_{2}^{2}$$ $$\mathstrut -\mathstrut 10 T_{2}$$ $$\mathstrut +\mathstrut 4$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.