# Properties

 Label 15.5.c.a Level 15 Weight 5 Character orbit 15.c Analytic conductor 1.551 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.55054944626$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 1 - \beta_{2} ) q^{3}$$ $$+ ( -8 + \beta_{2} + \beta_{4} ) q^{4}$$ $$+ \beta_{3} q^{5}$$ $$+ ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6}$$ $$+ ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7}$$ $$+ ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8}$$ $$+ ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 1 - \beta_{2} ) q^{3}$$ $$+ ( -8 + \beta_{2} + \beta_{4} ) q^{4}$$ $$+ \beta_{3} q^{5}$$ $$+ ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6}$$ $$+ ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7}$$ $$+ ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8}$$ $$+ ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9}$$ $$+ ( 7 - 5 \beta_{2} - \beta_{4} + \beta_{5} ) q^{10}$$ $$+ ( 4 - 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{5} ) q^{11}$$ $$+ ( -73 + 3 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{12}$$ $$+ ( -70 + 2 \beta_{2} + 2 \beta_{4} ) q^{13}$$ $$+ ( -2 + 36 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{14}$$ $$+ ( 9 - 15 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} ) q^{15}$$ $$+ ( 126 - 31 \beta_{2} + \beta_{4} + 8 \beta_{5} ) q^{16}$$ $$+ ( -8 - 20 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} - 8 \beta_{5} ) q^{17}$$ $$+ ( 194 + 63 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{18}$$ $$+ ( -34 + 26 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} ) q^{19}$$ $$+ ( 5 + 35 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + 5 \beta_{5} ) q^{20}$$ $$+ ( -150 + 21 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{21}$$ $$+ ( 64 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{22}$$ $$+ ( 7 - 72 \beta_{1} + 14 \beta_{2} - 48 \beta_{3} + 7 \beta_{5} ) q^{23}$$ $$+ ( -132 - 144 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} ) q^{24}$$ $$-125 q^{25}$$ $$+ ( -2 - 104 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} ) q^{26}$$ $$+ ( -70 + 18 \beta_{1} - 29 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} - 5 \beta_{5} ) q^{27}$$ $$+ ( -622 + 22 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} ) q^{28}$$ $$+ ( -2 + 146 \beta_{1} - 4 \beta_{2} - 48 \beta_{3} - 2 \beta_{5} ) q^{29}$$ $$+ ( 365 - 30 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 15 \beta_{4} ) q^{30}$$ $$+ ( 636 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{31}$$ $$+ ( 15 + 53 \beta_{1} + 30 \beta_{2} - 42 \beta_{3} + 15 \beta_{5} ) q^{32}$$ $$+ ( 744 + 102 \beta_{1} + 12 \beta_{2} + 42 \beta_{3} + 36 \beta_{4} + 10 \beta_{5} ) q^{33}$$ $$+ ( 490 - 114 \beta_{2} - 26 \beta_{4} + 22 \beta_{5} ) q^{34}$$ $$+ ( -15 + 20 \beta_{1} - 30 \beta_{2} + 20 \beta_{3} - 15 \beta_{5} ) q^{35}$$ $$+ ( -1252 + 126 \beta_{1} + 73 \beta_{2} - 42 \beta_{3} + 21 \beta_{4} - 26 \beta_{5} ) q^{36}$$ $$+ ( 306 - 58 \beta_{2} + 54 \beta_{4} + 28 \beta_{5} ) q^{37}$$ $$+ ( -26 - 134 \beta_{1} - 52 \beta_{2} + 120 \beta_{3} - 26 \beta_{5} ) q^{38}$$ $$+ ( -200 + 6 \beta_{1} + 74 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} ) q^{39}$$ $$+ ( -764 + 35 \beta_{2} + 27 \beta_{4} - 2 \beta_{5} ) q^{40}$$ $$+ ( -14 + 182 \beta_{1} - 28 \beta_{2} - 90 \beta_{3} - 14 \beta_{5} ) q^{41}$$ $$+ ( -318 - 36 \beta_{1} - 6 \beta_{2} - 126 \beta_{3} + 42 \beta_{5} ) q^{42}$$ $$+ ( -1277 - 168 \beta_{2} - 52 \beta_{4} + 29 \beta_{5} ) q^{43}$$ $$+ ( 34 - 22 \beta_{1} + 68 \beta_{2} + 132 \beta_{3} + 34 \beta_{5} ) q^{44}$$ $$+ ( 310 - 45 \beta_{1} + 5 \beta_{2} + 24 \beta_{3} + 15 \beta_{4} - 25 \beta_{5} ) q^{45}$$ $$+ ( 1420 + 224 \beta_{2} - 24 \beta_{4} - 62 \beta_{5} ) q^{46}$$ $$+ ( 31 - 212 \beta_{1} + 62 \beta_{2} + 72 \beta_{3} + 31 \beta_{5} ) q^{47}$$ $$+ ( 2405 - 213 \beta_{1} - 92 \beta_{2} - 78 \beta_{3} - 66 \beta_{4} - 27 \beta_{5} ) q^{48}$$ $$+ ( -187 + 126 \beta_{2} + 14 \beta_{4} - 28 \beta_{5} ) q^{49}$$ $$-125 \beta_{1} q^{50}$$ $$+ ( -1434 - 366 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} - 54 \beta_{4} - 32 \beta_{5} ) q^{51}$$ $$+ ( 1452 - 148 \beta_{2} - 84 \beta_{4} + 16 \beta_{5} ) q^{52}$$ $$+ ( 48 + 340 \beta_{1} + 96 \beta_{2} + 6 \beta_{3} + 48 \beta_{5} ) q^{53}$$ $$+ ( -92 + 333 \beta_{1} - 214 \beta_{2} - 129 \beta_{3} - 30 \beta_{4} + 77 \beta_{5} ) q^{54}$$ $$+ ( -210 + 50 \beta_{2} - 70 \beta_{4} - 30 \beta_{5} ) q^{55}$$ $$+ ( -54 - 78 \beta_{1} - 108 \beta_{2} + 192 \beta_{3} - 54 \beta_{5} ) q^{56}$$ $$+ ( -1922 + 168 \beta_{1} + 14 \beta_{2} + 42 \beta_{3} + 66 \beta_{4} + 12 \beta_{5} ) q^{57}$$ $$+ ( -3848 + 370 \beta_{2} + 194 \beta_{4} - 44 \beta_{5} ) q^{58}$$ $$+ ( -54 - 758 \beta_{1} - 108 \beta_{2} - 96 \beta_{3} - 54 \beta_{5} ) q^{59}$$ $$+ ( 858 + 360 \beta_{1} + 15 \beta_{2} - 60 \beta_{3} + 21 \beta_{4} + 34 \beta_{5} ) q^{60}$$ $$+ ( 1048 - 150 \beta_{2} + 122 \beta_{4} + 68 \beta_{5} ) q^{61}$$ $$+ ( -30 + 582 \beta_{1} - 60 \beta_{2} + 132 \beta_{3} - 30 \beta_{5} ) q^{62}$$ $$+ ( 2547 - 342 \beta_{1} + 180 \beta_{2} - 72 \beta_{3} + 18 \beta_{4} + 63 \beta_{5} ) q^{63}$$ $$+ ( 510 - 113 \beta_{2} + 111 \beta_{4} + 56 \beta_{5} ) q^{64}$$ $$+ ( 10 + 70 \beta_{1} + 20 \beta_{2} - 70 \beta_{3} + 10 \beta_{5} ) q^{65}$$ $$+ ( -2150 + 210 \beta_{1} - 100 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{66}$$ $$+ ( 2299 + 32 \beta_{2} - 140 \beta_{4} - 43 \beta_{5} ) q^{67}$$ $$+ ( -14 + 854 \beta_{1} - 28 \beta_{2} - 456 \beta_{3} - 14 \beta_{5} ) q^{68}$$ $$+ ( 870 + 693 \beta_{1} + 21 \beta_{2} + 279 \beta_{3} - 81 \beta_{4} - 147 \beta_{5} ) q^{69}$$ $$+ ( -400 - 200 \beta_{2} + 50 \beta_{5} ) q^{70}$$ $$+ ( 78 + 236 \beta_{1} + 156 \beta_{2} + 120 \beta_{3} + 78 \beta_{5} ) q^{71}$$ $$+ ( -339 - 783 \beta_{1} + 150 \beta_{2} + 342 \beta_{3} + 72 \beta_{4} - 147 \beta_{5} ) q^{72}$$ $$+ ( 6 - 308 \beta_{2} - 196 \beta_{4} + 28 \beta_{5} ) q^{73}$$ $$+ ( 58 - 304 \beta_{1} + 116 \beta_{2} - 180 \beta_{3} + 58 \beta_{5} ) q^{74}$$ $$+ ( -125 + 125 \beta_{2} ) q^{75}$$ $$+ ( 3408 - 526 \beta_{2} - 222 \beta_{4} + 76 \beta_{5} ) q^{76}$$ $$+ ( -68 - 576 \beta_{1} - 136 \beta_{2} - 516 \beta_{3} - 68 \beta_{5} ) q^{77}$$ $$+ ( -264 - 546 \beta_{1} - 6 \beta_{2} + 294 \beta_{3} + 18 \beta_{4} - 86 \beta_{5} ) q^{78}$$ $$+ ( -2594 + 386 \beta_{2} - 38 \beta_{4} - 106 \beta_{5} ) q^{79}$$ $$+ ( 45 - 685 \beta_{1} + 90 \beta_{2} + 70 \beta_{3} + 45 \beta_{5} ) q^{80}$$ $$+ ( 907 - 828 \beta_{1} + 2 \beta_{2} + 60 \beta_{3} - 30 \beta_{4} + 176 \beta_{5} ) q^{81}$$ $$+ ( -5054 + 520 \beta_{2} + 272 \beta_{4} - 62 \beta_{5} ) q^{82}$$ $$+ ( -21 - 288 \beta_{1} - 42 \beta_{2} + 408 \beta_{3} - 21 \beta_{5} ) q^{83}$$ $$+ ( -2250 + 156 \beta_{1} + 594 \beta_{2} + 66 \beta_{3} + 42 \beta_{4} + 24 \beta_{5} ) q^{84}$$ $$+ ( -498 + 20 \beta_{2} + 164 \beta_{4} + 36 \beta_{5} ) q^{85}$$ $$+ ( 168 - 74 \beta_{1} + 336 \beta_{2} - 834 \beta_{3} + 168 \beta_{5} ) q^{86}$$ $$+ ( -804 + 1104 \beta_{1} - 6 \beta_{2} - 456 \beta_{3} - 162 \beta_{4} + 44 \beta_{5} ) q^{87}$$ $$+ ( 2612 + 70 \beta_{2} - 186 \beta_{4} - 64 \beta_{5} ) q^{88}$$ $$+ ( -156 - 12 \beta_{1} - 312 \beta_{2} + 396 \beta_{3} - 156 \beta_{5} ) q^{89}$$ $$+ ( 1133 - 395 \beta_{2} + 210 \beta_{3} - 69 \beta_{4} + 19 \beta_{5} ) q^{90}$$ $$+ ( -1946 - 64 \beta_{2} + 104 \beta_{4} + 42 \beta_{5} ) q^{91}$$ $$+ ( -112 - 6 \beta_{1} - 224 \beta_{2} + 204 \beta_{3} - 112 \beta_{5} ) q^{92}$$ $$+ ( -1578 + 210 \beta_{1} - 672 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{93}$$ $$+ ( 5716 - 324 \beta_{2} - 284 \beta_{4} + 10 \beta_{5} ) q^{94}$$ $$+ ( -20 + 610 \beta_{1} - 40 \beta_{2} + 8 \beta_{3} - 20 \beta_{5} ) q^{95}$$ $$+ ( 2412 + 1194 \beta_{1} + 45 \beta_{2} - 24 \beta_{3} + 9 \beta_{4} + 14 \beta_{5} ) q^{96}$$ $$+ ( 1822 + 712 \beta_{2} + 216 \beta_{4} - 124 \beta_{5} ) q^{97}$$ $$+ ( -126 - 733 \beta_{1} - 252 \beta_{2} + 588 \beta_{3} - 126 \beta_{5} ) q^{98}$$ $$+ ( 1424 - 342 \beta_{1} - 632 \beta_{2} - 564 \beta_{3} + 300 \beta_{4} + 64 \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut +\mathstrut 8q^{3}$$ $$\mathstrut -\mathstrut 50q^{4}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 76q^{7}$$ $$\mathstrut +\mathstrut 118q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut +\mathstrut 8q^{3}$$ $$\mathstrut -\mathstrut 50q^{4}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 76q^{7}$$ $$\mathstrut +\mathstrut 118q^{9}$$ $$\mathstrut +\mathstrut 50q^{10}$$ $$\mathstrut -\mathstrut 452q^{12}$$ $$\mathstrut -\mathstrut 424q^{13}$$ $$\mathstrut +\mathstrut 50q^{15}$$ $$\mathstrut +\mathstrut 802q^{16}$$ $$\mathstrut +\mathstrut 1160q^{18}$$ $$\mathstrut -\mathstrut 244q^{19}$$ $$\mathstrut -\mathstrut 876q^{21}$$ $$\mathstrut +\mathstrut 340q^{22}$$ $$\mathstrut -\mathstrut 786q^{24}$$ $$\mathstrut -\mathstrut 750q^{25}$$ $$\mathstrut -\mathstrut 352q^{27}$$ $$\mathstrut -\mathstrut 3764q^{28}$$ $$\mathstrut +\mathstrut 2200q^{30}$$ $$\mathstrut +\mathstrut 3772q^{31}$$ $$\mathstrut +\mathstrut 4420q^{33}$$ $$\mathstrut +\mathstrut 3124q^{34}$$ $$\mathstrut -\mathstrut 7606q^{36}$$ $$\mathstrut +\mathstrut 1896q^{37}$$ $$\mathstrut -\mathstrut 1336q^{39}$$ $$\mathstrut -\mathstrut 4650q^{40}$$ $$\mathstrut -\mathstrut 1980q^{42}$$ $$\mathstrut -\mathstrut 7384q^{43}$$ $$\mathstrut +\mathstrut 1900q^{45}$$ $$\mathstrut +\mathstrut 8196q^{46}$$ $$\mathstrut +\mathstrut 14668q^{48}$$ $$\mathstrut -\mathstrut 1318q^{49}$$ $$\mathstrut -\mathstrut 8492q^{51}$$ $$\mathstrut +\mathstrut 8976q^{52}$$ $$\mathstrut -\mathstrut 278q^{54}$$ $$\mathstrut -\mathstrut 1300q^{55}$$ $$\mathstrut -\mathstrut 11584q^{57}$$ $$\mathstrut -\mathstrut 23740q^{58}$$ $$\mathstrut +\mathstrut 5050q^{60}$$ $$\mathstrut +\mathstrut 6452q^{61}$$ $$\mathstrut +\mathstrut 14796q^{63}$$ $$\mathstrut +\mathstrut 3174q^{64}$$ $$\mathstrut -\mathstrut 12760q^{66}$$ $$\mathstrut +\mathstrut 13816q^{67}$$ $$\mathstrut +\mathstrut 5472q^{69}$$ $$\mathstrut -\mathstrut 2100q^{70}$$ $$\mathstrut -\mathstrut 2040q^{72}$$ $$\mathstrut +\mathstrut 596q^{73}$$ $$\mathstrut -\mathstrut 1000q^{75}$$ $$\mathstrut +\mathstrut 21348q^{76}$$ $$\mathstrut -\mathstrut 1400q^{78}$$ $$\mathstrut -\mathstrut 16124q^{79}$$ $$\mathstrut +\mathstrut 5086q^{81}$$ $$\mathstrut -\mathstrut 31240q^{82}$$ $$\mathstrut -\mathstrut 14736q^{84}$$ $$\mathstrut -\mathstrut 3100q^{85}$$ $$\mathstrut -\mathstrut 4900q^{87}$$ $$\mathstrut +\mathstrut 15660q^{88}$$ $$\mathstrut +\mathstrut 7550q^{90}$$ $$\mathstrut -\mathstrut 11632q^{91}$$ $$\mathstrut -\mathstrut 8184q^{93}$$ $$\mathstrut +\mathstrut 34924q^{94}$$ $$\mathstrut +\mathstrut 14354q^{96}$$ $$\mathstrut +\mathstrut 9756q^{97}$$ $$\mathstrut +\mathstrut 9680q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut +\mathstrut$$ $$73$$ $$x^{4}\mathstrut +\mathstrut$$ $$1096$$ $$x^{2}\mathstrut +\mathstrut$$ $$180$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 71 \nu^{3} - 47 \nu^{2} + 1002 \nu - 114$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 79 \nu^{3} + 1330 \nu$$$$)/48$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 71 \nu^{3} + 95 \nu^{2} - 1002 \nu + 1266$$$$)/48$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} + \nu^{4} + 142 \nu^{3} + 47 \nu^{2} + 2004 \nu + 90$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$24$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{5}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$41$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{4}$$ $$=$$ $$8$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$47$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$79$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1022$$ $$\nu^{5}$$ $$=$$ $$79$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$426$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$158$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1909$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$79$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 − 7.20990i − 4.56632i − 0.407512i 0.407512i 4.56632i 7.20990i
7.20990i 8.77108 + 2.01697i −35.9827 11.1803i 14.5422 63.2386i 23.3388 144.073i 72.8637 + 35.3820i 80.6091
11.2 4.56632i −7.98405 4.15391i −4.85128 11.1803i −18.9681 + 36.4577i 61.6068 50.9086i 46.4900 + 66.3301i −51.0530
11.3 0.407512i 3.21297 + 8.40695i 15.8339 11.1803i 3.42594 1.30932i −46.9457 12.9727i −60.3537 + 54.0225i −4.55613
11.4 0.407512i 3.21297 8.40695i 15.8339 11.1803i 3.42594 + 1.30932i −46.9457 12.9727i −60.3537 54.0225i −4.55613
11.5 4.56632i −7.98405 + 4.15391i −4.85128 11.1803i −18.9681 36.4577i 61.6068 50.9086i 46.4900 66.3301i −51.0530
11.6 7.20990i 8.77108 2.01697i −35.9827 11.1803i 14.5422 + 63.2386i 23.3388 144.073i 72.8637 35.3820i 80.6091
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.6 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

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(15, [\chi])$$.