# Properties

 Label 15.4.e.a Level 15 Weight 4 Character orbit 15.e Analytic conductor 0.885 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.28356903014400.8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{3} q^{2}$$ $$+ ( -1 - \beta_{2} + \beta_{5} ) q^{3}$$ $$+ ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4}$$ $$+ ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5}$$ $$+ ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6}$$ $$+ ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7}$$ $$+ ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8}$$ $$+ ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{3} q^{2}$$ $$+ ( -1 - \beta_{2} + \beta_{5} ) q^{3}$$ $$+ ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4}$$ $$+ ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5}$$ $$+ ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6}$$ $$+ ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7}$$ $$+ ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8}$$ $$+ ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9}$$ $$+ ( -15 - 10 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{10}$$ $$+ ( -7 \beta_{1} - 7 \beta_{3} + 2 \beta_{6} ) q^{11}$$ $$+ ( 17 - 17 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{12}$$ $$+ ( 8 + 8 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{13}$$ $$+ ( 7 \beta_{1} - 7 \beta_{3} - 8 \beta_{7} ) q^{14}$$ $$+ ( 10 + 9 \beta_{1} + 25 \beta_{2} - 12 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{15}$$ $$+ ( 39 + 7 \beta_{4} - 7 \beta_{5} ) q^{16}$$ $$+ ( 20 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{17}$$ $$+ ( -30 - 21 \beta_{1} - 30 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} ) q^{18}$$ $$+ ( -12 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{19}$$ $$+ ( 2 \beta_{1} + 19 \beta_{3} + 2 \beta_{6} - 6 \beta_{7} ) q^{20}$$ $$+ ( -62 + 6 \beta_{1} + 6 \beta_{3} - \beta_{4} + \beta_{5} - 11 \beta_{6} ) q^{21}$$ $$+ ( -65 + 65 \beta_{2} - 10 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} ) q^{22}$$ $$+ ( 14 \beta_{1} + 9 \beta_{6} - 9 \beta_{7} ) q^{23}$$ $$+ ( -3 \beta_{1} + 39 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 21 \beta_{7} ) q^{24}$$ $$+ ( -20 - 85 \beta_{2} + 20 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{25}$$ $$+ ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{6} ) q^{26}$$ $$+ ( 87 - 87 \beta_{2} - 18 \beta_{3} - 3 \beta_{4} - 6 \beta_{6} - 9 \beta_{7} ) q^{27}$$ $$+ ( 63 + 63 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{28}$$ $$+ ( -3 \beta_{1} + 3 \beta_{3} + 22 \beta_{7} ) q^{29}$$ $$+ ( 85 - 15 \beta_{1} + 105 \beta_{2} - 10 \beta_{4} - 20 \beta_{5} + 25 \beta_{6} - 15 \beta_{7} ) q^{30}$$ $$+ ( 62 - 30 \beta_{4} + 30 \beta_{5} ) q^{31}$$ $$+ ( -43 \beta_{3} - 30 \beta_{6} - 30 \beta_{7} ) q^{32}$$ $$+ ( -25 + 36 \beta_{1} - 25 \beta_{2} - 20 \beta_{5} + 27 \beta_{6} - 27 \beta_{7} ) q^{33}$$ $$+ ( -166 \beta_{2} + 34 \beta_{4} + 34 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} ) q^{34}$$ $$+ ( -33 \beta_{1} - 31 \beta_{3} + 17 \beta_{6} + 19 \beta_{7} ) q^{35}$$ $$+ ( -105 + 6 \beta_{1} + 6 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 24 \beta_{6} ) q^{36}$$ $$+ ( -146 + 146 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{37}$$ $$+ ( -12 \beta_{1} - 12 \beta_{6} + 12 \beta_{7} ) q^{38}$$ $$+ ( -3 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} ) q^{39}$$ $$+ ( -70 - 55 \beta_{2} - 15 \beta_{4} - 35 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{40}$$ $$+ ( 32 \beta_{1} + 32 \beta_{3} - 52 \beta_{6} ) q^{41}$$ $$+ ( 65 - 65 \beta_{2} + 66 \beta_{3} - 10 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} ) q^{42}$$ $$+ ( 78 + 78 \beta_{2} - 38 \beta_{5} + 19 \beta_{6} - 19 \beta_{7} ) q^{43}$$ $$+ ( -29 \beta_{1} + 29 \beta_{3} + 36 \beta_{7} ) q^{44}$$ $$+ ( 30 + 12 \beta_{1} + 90 \beta_{2} + 69 \beta_{3} + 45 \beta_{4} + 15 \beta_{5} - 18 \beta_{6} + 24 \beta_{7} ) q^{45}$$ $$+ ( 108 + 32 \beta_{4} - 32 \beta_{5} ) q^{46}$$ $$+ ( 2 \beta_{3} + 21 \beta_{6} + 21 \beta_{7} ) q^{47}$$ $$+ ( -151 + 21 \beta_{1} - 151 \beta_{2} + 46 \beta_{5} - 21 \beta_{6} + 21 \beta_{7} ) q^{48}$$ $$+ ( 85 \beta_{2} - 16 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} ) q^{49}$$ $$+ ( 105 \beta_{1} - 40 \beta_{3} - 20 \beta_{6} - 40 \beta_{7} ) q^{50}$$ $$+ ( -106 - 81 \beta_{1} - 81 \beta_{3} + \beta_{4} - \beta_{5} - 19 \beta_{6} ) q^{51}$$ $$+ ( 24 - 24 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} ) q^{52}$$ $$+ ( -130 \beta_{1} + \beta_{6} - \beta_{7} ) q^{53}$$ $$+ ( 81 \beta_{1} + 147 \beta_{2} - 81 \beta_{3} - 33 \beta_{4} - 33 \beta_{5} + 33 \beta_{6} - 27 \beta_{7} ) q^{54}$$ $$+ ( 35 - 145 \beta_{2} - 10 \beta_{4} + 80 \beta_{5} - 35 \beta_{6} + 35 \beta_{7} ) q^{55}$$ $$+ ( -3 \beta_{1} - 3 \beta_{3} + 68 \beta_{6} ) q^{56}$$ $$+ ( 84 - 84 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + 24 \beta_{6} + 18 \beta_{7} ) q^{57}$$ $$+ ( -5 - 5 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{58}$$ $$+ ( 39 \beta_{1} - 39 \beta_{3} - 136 \beta_{7} ) q^{59}$$ $$+ ( 55 - 69 \beta_{1} - 70 \beta_{2} - 33 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 34 \beta_{6} - 18 \beta_{7} ) q^{60}$$ $$+ 2 q^{61}$$ $$+ ( 58 \beta_{3} + 60 \beta_{6} + 60 \beta_{7} ) q^{62}$$ $$+ ( 183 - 6 \beta_{1} + 183 \beta_{2} - 18 \beta_{5} - 48 \beta_{6} + 48 \beta_{7} ) q^{63}$$ $$+ ( 15 \beta_{2} - 47 \beta_{4} - 47 \beta_{5} + 47 \beta_{6} - 47 \beta_{7} ) q^{64}$$ $$+ ( 8 \beta_{1} + 6 \beta_{3} - 17 \beta_{6} + 31 \beta_{7} ) q^{65}$$ $$+ ( 290 - 15 \beta_{1} - 15 \beta_{3} + 70 \beta_{4} - 70 \beta_{5} - 40 \beta_{6} ) q^{66}$$ $$+ ( 84 - 84 \beta_{2} + 134 \beta_{4} - 67 \beta_{6} + 67 \beta_{7} ) q^{67}$$ $$+ ( 142 \beta_{1} + 12 \beta_{6} - 12 \beta_{7} ) q^{68}$$ $$+ ( -69 \beta_{1} - 148 \beta_{2} + 69 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} + 13 \beta_{6} - 12 \beta_{7} ) q^{69}$$ $$+ ( -295 + 315 \beta_{2} - 30 \beta_{4} + 40 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{70}$$ $$+ ( 116 \beta_{1} + 116 \beta_{3} + 74 \beta_{6} ) q^{71}$$ $$+ ( -210 + 210 \beta_{2} - 27 \beta_{3} + 60 \beta_{4} - 60 \beta_{6} ) q^{72}$$ $$+ ( -317 - 317 \beta_{2} + 12 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{73}$$ $$+ ( -158 \beta_{1} + 158 \beta_{3} + 12 \beta_{7} ) q^{74}$$ $$+ ( -385 + 45 \beta_{1} - 55 \beta_{2} + 15 \beta_{3} - 90 \beta_{4} - 5 \beta_{5} + 30 \beta_{6} + 30 \beta_{7} ) q^{75}$$ $$+ ( -180 + 12 \beta_{4} - 12 \beta_{5} ) q^{76}$$ $$+ ( -74 \beta_{3} - 78 \beta_{6} - 78 \beta_{7} ) q^{77}$$ $$+ ( -40 + 12 \beta_{1} - 40 \beta_{2} - 20 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} ) q^{78}$$ $$+ ( 298 \beta_{2} - 56 \beta_{4} - 56 \beta_{5} + 56 \beta_{6} - 56 \beta_{7} ) q^{79}$$ $$+ ( -197 \beta_{1} + 46 \beta_{3} - 22 \beta_{6} + 46 \beta_{7} ) q^{80}$$ $$+ ( -9 + 72 \beta_{1} + 72 \beta_{3} - 90 \beta_{4} + 90 \beta_{5} + 108 \beta_{6} ) q^{81}$$ $$+ ( 340 - 340 \beta_{2} - 40 \beta_{4} + 20 \beta_{6} - 20 \beta_{7} ) q^{82}$$ $$+ ( 86 \beta_{1} - 7 \beta_{6} + 7 \beta_{7} ) q^{83}$$ $$+ ( -3 \beta_{1} - 94 \beta_{2} + 3 \beta_{3} + 62 \beta_{4} + 62 \beta_{5} - 62 \beta_{6} - 6 \beta_{7} ) q^{84}$$ $$+ ( 650 + 60 \beta_{2} + 30 \beta_{4} - 100 \beta_{5} + 35 \beta_{6} - 35 \beta_{7} ) q^{85}$$ $$+ ( -154 \beta_{1} - 154 \beta_{3} - 76 \beta_{6} ) q^{86}$$ $$+ ( -195 + 195 \beta_{2} - 84 \beta_{3} + 60 \beta_{4} - 3 \beta_{6} + 57 \beta_{7} ) q^{87}$$ $$+ ( 295 + 295 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{88}$$ $$+ ( 276 \beta_{1} - 276 \beta_{3} + 126 \beta_{7} ) q^{89}$$ $$+ ( 90 + 30 \beta_{1} - 615 \beta_{2} - 90 \beta_{3} + 105 \beta_{4} + 45 \beta_{5} - 45 \beta_{6} - 15 \beta_{7} ) q^{90}$$ $$+ ( -144 + 38 \beta_{4} - 38 \beta_{5} ) q^{91}$$ $$+ ( -124 \beta_{3} + 8 \beta_{6} + 8 \beta_{7} ) q^{92}$$ $$+ ( 418 - 90 \beta_{1} + 418 \beta_{2} + 32 \beta_{5} + 90 \beta_{6} - 90 \beta_{7} ) q^{93}$$ $$+ ( 24 \beta_{2} + 44 \beta_{4} + 44 \beta_{5} - 44 \beta_{6} + 44 \beta_{7} ) q^{94}$$ $$+ ( -6 \beta_{1} + 78 \beta_{3} - 6 \beta_{6} - 72 \beta_{7} ) q^{95}$$ $$+ ( 497 + 219 \beta_{1} + 219 \beta_{3} - 47 \beta_{4} + 47 \beta_{5} - 4 \beta_{6} ) q^{96}$$ $$+ ( 179 - 179 \beta_{2} - 236 \beta_{4} + 118 \beta_{6} - 118 \beta_{7} ) q^{97}$$ $$+ ( -149 \beta_{1} - 32 \beta_{6} + 32 \beta_{7} ) q^{98}$$ $$+ ( -129 \beta_{1} - 540 \beta_{2} + 129 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} + 66 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut -\mathstrut 16q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut -\mathstrut 16q^{7}$$ $$\mathstrut -\mathstrut 100q^{10}$$ $$\mathstrut +\mathstrut 132q^{12}$$ $$\mathstrut +\mathstrut 68q^{13}$$ $$\mathstrut +\mathstrut 90q^{15}$$ $$\mathstrut +\mathstrut 284q^{16}$$ $$\mathstrut -\mathstrut 240q^{18}$$ $$\mathstrut -\mathstrut 492q^{21}$$ $$\mathstrut -\mathstrut 500q^{22}$$ $$\mathstrut -\mathstrut 220q^{25}$$ $$\mathstrut +\mathstrut 702q^{27}$$ $$\mathstrut +\mathstrut 508q^{28}$$ $$\mathstrut +\mathstrut 660q^{30}$$ $$\mathstrut +\mathstrut 616q^{31}$$ $$\mathstrut -\mathstrut 240q^{33}$$ $$\mathstrut -\mathstrut 804q^{36}$$ $$\mathstrut -\mathstrut 1156q^{37}$$ $$\mathstrut -\mathstrut 600q^{40}$$ $$\mathstrut +\mathstrut 540q^{42}$$ $$\mathstrut +\mathstrut 548q^{43}$$ $$\mathstrut +\mathstrut 180q^{45}$$ $$\mathstrut +\mathstrut 736q^{46}$$ $$\mathstrut -\mathstrut 1116q^{48}$$ $$\mathstrut -\mathstrut 852q^{51}$$ $$\mathstrut +\mathstrut 224q^{52}$$ $$\mathstrut +\mathstrut 460q^{55}$$ $$\mathstrut +\mathstrut 684q^{57}$$ $$\mathstrut +\mathstrut 60q^{58}$$ $$\mathstrut +\mathstrut 540q^{60}$$ $$\mathstrut +\mathstrut 16q^{61}$$ $$\mathstrut +\mathstrut 1428q^{63}$$ $$\mathstrut +\mathstrut 2040q^{66}$$ $$\mathstrut +\mathstrut 404q^{67}$$ $$\mathstrut -\mathstrut 2220q^{70}$$ $$\mathstrut -\mathstrut 1800q^{72}$$ $$\mathstrut -\mathstrut 2512q^{73}$$ $$\mathstrut -\mathstrut 2910q^{75}$$ $$\mathstrut -\mathstrut 1488q^{76}$$ $$\mathstrut -\mathstrut 360q^{78}$$ $$\mathstrut +\mathstrut 288q^{81}$$ $$\mathstrut +\mathstrut 2800q^{82}$$ $$\mathstrut +\mathstrut 4940q^{85}$$ $$\mathstrut -\mathstrut 1680q^{87}$$ $$\mathstrut +\mathstrut 2460q^{88}$$ $$\mathstrut +\mathstrut 600q^{90}$$ $$\mathstrut -\mathstrut 1304q^{91}$$ $$\mathstrut +\mathstrut 3408q^{93}$$ $$\mathstrut +\mathstrut 4164q^{96}$$ $$\mathstrut +\mathstrut 1904q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut +\mathstrut$$ $$209$$ $$x^{4}\mathstrut +\mathstrut$$ $$1600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 249 \nu^{2}$$$$)/680$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 249 \nu^{3}$$$$)/680$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} - 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu - 4520$$$$)/1360$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} + 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu + 4520$$$$)/1360$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} - 40 \nu^{5} - 2557 \nu^{3} - 7240 \nu$$$$)/2720$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{7} + 40 \nu^{5} - 2557 \nu^{3} + 7240 \nu$$$$)/2720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$17$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$113$$ $$\nu^{5}$$ $$=$$ $$34$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$34$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$181$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-$$$$249$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$249$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$249$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$249$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1561$$ $$\beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-$$$$498$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$498$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$2557$$ $$\beta_{3}$$

## Character Values

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

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −2.66260 − 2.66260i −1.18766 − 1.18766i 1.18766 + 1.18766i 2.66260 + 2.66260i −2.66260 + 2.66260i −1.18766 + 1.18766i 1.18766 − 1.18766i 2.66260 − 2.66260i
−2.66260 + 2.66260i −4.37420 + 2.80471i 6.17891i 9.55729 + 5.80157i 4.17891 19.1146i 9.35782 + 9.35782i −4.84884 4.84884i 11.2672 24.5367i −40.8945 + 10.0000i
2.2 −1.18766 + 1.18766i 5.11173 + 0.932827i 5.17891i −2.48157 10.9015i −7.17891 + 4.96314i −13.3578 13.3578i −15.6521 15.6521i 25.2597 + 9.53673i 15.8945 + 10.0000i
2.3 1.18766 1.18766i −0.932827 5.11173i 5.17891i 2.48157 + 10.9015i −7.17891 4.96314i −13.3578 13.3578i 15.6521 + 15.6521i −25.2597 + 9.53673i 15.8945 + 10.0000i
2.4 2.66260 2.66260i −2.80471 + 4.37420i 6.17891i −9.55729 5.80157i 4.17891 + 19.1146i 9.35782 + 9.35782i 4.84884 + 4.84884i −11.2672 24.5367i −40.8945 + 10.0000i
8.1 −2.66260 2.66260i −4.37420 2.80471i 6.17891i 9.55729 5.80157i 4.17891 + 19.1146i 9.35782 9.35782i −4.84884 + 4.84884i 11.2672 + 24.5367i −40.8945 10.0000i
8.2 −1.18766 1.18766i 5.11173 0.932827i 5.17891i −2.48157 + 10.9015i −7.17891 4.96314i −13.3578 + 13.3578i −15.6521 + 15.6521i 25.2597 9.53673i 15.8945 10.0000i
8.3 1.18766 + 1.18766i −0.932827 + 5.11173i 5.17891i 2.48157 10.9015i −7.17891 + 4.96314i −13.3578 + 13.3578i 15.6521 15.6521i −25.2597 9.53673i 15.8945 10.0000i
8.4 2.66260 + 2.66260i −2.80471 4.37420i 6.17891i −9.55729 + 5.80157i 4.17891 19.1146i 9.35782 9.35782i 4.84884 4.84884i −11.2672 + 24.5367i −40.8945 10.0000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.4 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
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes

## Hecke kernels

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