# Properties

 Label 15.4.b.a Level 15 Weight 4 Character orbit 15.b Analytic conductor 0.885 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 15.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ \beta_{2} q^{3}$$ $$+ ( -5 + \beta_{3} ) q^{4}$$ $$+ ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( 5 - \beta_{3} ) q^{6}$$ $$+ ( 6 \beta_{1} - 4 \beta_{2} ) q^{7}$$ $$+ ( \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$-9 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ \beta_{2} q^{3}$$ $$+ ( -5 + \beta_{3} ) q^{4}$$ $$+ ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( 5 - \beta_{3} ) q^{6}$$ $$+ ( 6 \beta_{1} - 4 \beta_{2} ) q^{7}$$ $$+ ( \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$-9 q^{9}$$ $$+ ( 3 - 6 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{10}$$ $$+ ( -22 + 2 \beta_{3} ) q^{11}$$ $$-9 \beta_{1} q^{12}$$ $$+ ( -6 \beta_{1} + 16 \beta_{2} ) q^{13}$$ $$+ ( 58 - 2 \beta_{3} ) q^{14}$$ $$+ ( 13 + 9 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{15}$$ $$+ ( 13 - \beta_{3} ) q^{16}$$ $$+ ( -10 \beta_{1} - 12 \beta_{2} ) q^{17}$$ $$+ 9 \beta_{1} q^{18}$$ $$+ ( -24 - 8 \beta_{3} ) q^{19}$$ $$+ ( -102 + 9 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} ) q^{20}$$ $$+ ( 6 + 6 \beta_{3} ) q^{21}$$ $$+ ( 30 \beta_{1} + 16 \beta_{2} ) q^{22}$$ $$+ ( 8 \beta_{1} - 12 \beta_{2} ) q^{23}$$ $$+ ( -77 + \beta_{3} ) q^{24}$$ $$+ ( 67 - 24 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} ) q^{25}$$ $$+ ( 2 - 10 \beta_{3} ) q^{26}$$ $$-9 \beta_{2} q^{27}$$ $$+ ( -18 \beta_{1} - 48 \beta_{2} ) q^{28}$$ $$+ ( 152 + 14 \beta_{3} ) q^{29}$$ $$+ ( 102 - 9 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{30}$$ $$+ ( 24 + 4 \beta_{3} ) q^{31}$$ $$+ ( -9 \beta_{1} + 56 \beta_{2} ) q^{32}$$ $$+ ( -18 \beta_{1} - 12 \beta_{2} ) q^{33}$$ $$+ ( -190 + 22 \beta_{3} ) q^{34}$$ $$+ ( -70 + 60 \beta_{2} - 10 \beta_{3} ) q^{35}$$ $$+ ( 45 - 9 \beta_{3} ) q^{36}$$ $$+ ( -54 \beta_{1} - 24 \beta_{2} ) q^{37}$$ $$+ ( -8 \beta_{1} - 64 \beta_{2} ) q^{38}$$ $$+ ( -114 - 6 \beta_{3} ) q^{39}$$ $$+ ( 101 + 78 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} ) q^{40}$$ $$+ ( -206 + 4 \beta_{3} ) q^{41}$$ $$+ ( 18 \beta_{1} + 48 \beta_{2} ) q^{42}$$ $$+ ( 96 \beta_{1} - 28 \beta_{2} ) q^{43}$$ $$+ ( 294 - 30 \beta_{3} ) q^{44}$$ $$+ ( -18 - 9 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} ) q^{45}$$ $$+ ( 44 + 4 \beta_{3} ) q^{46}$$ $$+ ( 92 \beta_{1} - 76 \beta_{2} ) q^{47}$$ $$+ ( 9 \beta_{1} + 8 \beta_{2} ) q^{48}$$ $$+ ( -29 - 12 \beta_{3} ) q^{49}$$ $$+ ( -172 - 91 \beta_{1} - 48 \beta_{2} - 4 \beta_{3} ) q^{50}$$ $$+ ( 158 - 10 \beta_{3} ) q^{51}$$ $$+ ( -90 \beta_{1} + 48 \beta_{2} ) q^{52}$$ $$+ ( -82 \beta_{1} + 108 \beta_{2} ) q^{53}$$ $$+ ( -45 + 9 \beta_{3} ) q^{54}$$ $$+ ( -228 + 6 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{55}$$ $$+ ( -10 + 50 \beta_{3} ) q^{56}$$ $$+ ( 72 \beta_{1} - 64 \beta_{2} ) q^{57}$$ $$+ ( -96 \beta_{1} + 112 \beta_{2} ) q^{58}$$ $$+ ( 94 - 2 \beta_{3} ) q^{59}$$ $$+ ( 27 - 54 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{60}$$ $$+ ( 186 - 32 \beta_{3} ) q^{61}$$ $$+ ( -8 \beta_{1} + 32 \beta_{2} ) q^{62}$$ $$+ ( -54 \beta_{1} + 36 \beta_{2} ) q^{63}$$ $$+ ( 267 - 55 \beta_{3} ) q^{64}$$ $$+ ( 226 + 108 \beta_{1} - 96 \beta_{2} + 22 \beta_{3} ) q^{65}$$ $$+ ( -294 + 30 \beta_{3} ) q^{66}$$ $$+ ( -60 \beta_{1} - 92 \beta_{2} ) q^{67}$$ $$+ ( 198 \beta_{1} + 80 \beta_{2} ) q^{68}$$ $$+ ( 68 + 8 \beta_{3} ) q^{69}$$ $$+ ( 300 + 30 \beta_{1} - 80 \beta_{2} - 60 \beta_{3} ) q^{70}$$ $$+ ( 12 - 60 \beta_{3} ) q^{71}$$ $$+ ( -9 \beta_{1} - 72 \beta_{2} ) q^{72}$$ $$+ ( 108 \beta_{1} + 168 \beta_{2} ) q^{73}$$ $$+ ( -822 + 78 \beta_{3} ) q^{74}$$ $$+ ( -132 + 54 \beta_{1} + 37 \beta_{2} - 24 \beta_{3} ) q^{75}$$ $$+ ( -616 + 8 \beta_{3} ) q^{76}$$ $$+ ( -108 \beta_{1} - 48 \beta_{2} ) q^{77}$$ $$+ ( 90 \beta_{1} - 48 \beta_{2} ) q^{78}$$ $$+ ( -240 + 100 \beta_{3} ) q^{79}$$ $$+ ( 118 - \beta_{1} - 8 \beta_{2} - 14 \beta_{3} ) q^{80}$$ $$+ 81 q^{81}$$ $$+ ( 222 \beta_{1} + 32 \beta_{2} ) q^{82}$$ $$+ ( -208 \beta_{1} - 60 \beta_{2} ) q^{83}$$ $$+ ( 522 - 18 \beta_{3} ) q^{84}$$ $$+ ( -126 - 168 \beta_{1} - 44 \beta_{2} - 2 \beta_{3} ) q^{85}$$ $$+ ( 1108 - 68 \beta_{3} ) q^{86}$$ $$+ ( -126 \beta_{1} + 222 \beta_{2} ) q^{87}$$ $$+ ( -174 \beta_{1} - 112 \beta_{2} ) q^{88}$$ $$+ ( -534 - 48 \beta_{3} ) q^{89}$$ $$+ ( -27 + 54 \beta_{1} + 72 \beta_{2} - 9 \beta_{3} ) q^{90}$$ $$+ ( 444 + 84 \beta_{3} ) q^{91}$$ $$+ ( 36 \beta_{1} - 64 \beta_{2} ) q^{92}$$ $$+ ( -36 \beta_{1} + 44 \beta_{2} ) q^{93}$$ $$+ ( 816 - 16 \beta_{3} ) q^{94}$$ $$+ ( 688 - 136 \beta_{1} + 192 \beta_{2} + 16 \beta_{3} ) q^{95}$$ $$+ ( -459 - 9 \beta_{3} ) q^{96}$$ $$+ ( 240 \beta_{1} + 8 \beta_{2} ) q^{97}$$ $$+ ( -19 \beta_{1} - 96 \beta_{2} ) q^{98}$$ $$+ ( 198 - 18 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 18q^{4}$$ $$\mathstrut +\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 18q^{6}$$ $$\mathstrut -\mathstrut 36q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 18q^{4}$$ $$\mathstrut +\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 18q^{6}$$ $$\mathstrut -\mathstrut 36q^{9}$$ $$\mathstrut +\mathstrut 14q^{10}$$ $$\mathstrut -\mathstrut 84q^{11}$$ $$\mathstrut +\mathstrut 228q^{14}$$ $$\mathstrut +\mathstrut 54q^{15}$$ $$\mathstrut +\mathstrut 50q^{16}$$ $$\mathstrut -\mathstrut 112q^{19}$$ $$\mathstrut -\mathstrut 396q^{20}$$ $$\mathstrut +\mathstrut 36q^{21}$$ $$\mathstrut -\mathstrut 306q^{24}$$ $$\mathstrut +\mathstrut 256q^{25}$$ $$\mathstrut -\mathstrut 12q^{26}$$ $$\mathstrut +\mathstrut 636q^{29}$$ $$\mathstrut +\mathstrut 396q^{30}$$ $$\mathstrut +\mathstrut 104q^{31}$$ $$\mathstrut -\mathstrut 716q^{34}$$ $$\mathstrut -\mathstrut 300q^{35}$$ $$\mathstrut +\mathstrut 162q^{36}$$ $$\mathstrut -\mathstrut 468q^{39}$$ $$\mathstrut +\mathstrut 418q^{40}$$ $$\mathstrut -\mathstrut 816q^{41}$$ $$\mathstrut +\mathstrut 1116q^{44}$$ $$\mathstrut -\mathstrut 54q^{45}$$ $$\mathstrut +\mathstrut 184q^{46}$$ $$\mathstrut -\mathstrut 140q^{49}$$ $$\mathstrut -\mathstrut 696q^{50}$$ $$\mathstrut +\mathstrut 612q^{51}$$ $$\mathstrut -\mathstrut 162q^{54}$$ $$\mathstrut -\mathstrut 864q^{55}$$ $$\mathstrut +\mathstrut 60q^{56}$$ $$\mathstrut +\mathstrut 372q^{59}$$ $$\mathstrut +\mathstrut 126q^{60}$$ $$\mathstrut +\mathstrut 680q^{61}$$ $$\mathstrut +\mathstrut 958q^{64}$$ $$\mathstrut +\mathstrut 948q^{65}$$ $$\mathstrut -\mathstrut 1116q^{66}$$ $$\mathstrut +\mathstrut 288q^{69}$$ $$\mathstrut +\mathstrut 1080q^{70}$$ $$\mathstrut -\mathstrut 72q^{71}$$ $$\mathstrut -\mathstrut 3132q^{74}$$ $$\mathstrut -\mathstrut 576q^{75}$$ $$\mathstrut -\mathstrut 2448q^{76}$$ $$\mathstrut -\mathstrut 760q^{79}$$ $$\mathstrut +\mathstrut 444q^{80}$$ $$\mathstrut +\mathstrut 324q^{81}$$ $$\mathstrut +\mathstrut 2052q^{84}$$ $$\mathstrut -\mathstrut 508q^{85}$$ $$\mathstrut +\mathstrut 4296q^{86}$$ $$\mathstrut -\mathstrut 2232q^{89}$$ $$\mathstrut -\mathstrut 126q^{90}$$ $$\mathstrut +\mathstrut 1944q^{91}$$ $$\mathstrut +\mathstrut 3232q^{94}$$ $$\mathstrut +\mathstrut 2784q^{95}$$ $$\mathstrut -\mathstrut 1854q^{96}$$ $$\mathstrut +\mathstrut 756q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$21$$ $$x^{2}\mathstrut +\mathstrut$$ $$100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{3} + 33 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$32$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{2}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{1}$$$$)/3$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 3.00000i −14.1047 11.1047 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 52.2094i
4.2 1.70156i 3.00000i 5.10469 −8.10469 + 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 + 13.7906i
4.3 1.70156i 3.00000i 5.10469 −8.10469 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 13.7906i
4.4 4.70156i 3.00000i −14.1047 11.1047 + 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 + 52.2094i
 $$n$$: 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
5.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(15, [\chi])$$.