# Properties

 Label 15.5.d.c Level 15 Weight 5 Character orbit 15.d Analytic conductor 1.551 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.55054944626$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{10}, \sqrt{-26})$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( \beta_{1} + \beta_{2} ) q^{3}$$ $$-6 q^{4}$$ $$+ ( 2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( 15 + \beta_{3} ) q^{6}$$ $$+ ( -2 \beta_{1} + \beta_{2} ) q^{7}$$ $$-22 \beta_{2} q^{8}$$ $$+ ( -36 + 3 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( \beta_{1} + \beta_{2} ) q^{3}$$ $$-6 q^{4}$$ $$+ ( 2 \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( 15 + \beta_{3} ) q^{6}$$ $$+ ( -2 \beta_{1} + \beta_{2} ) q^{7}$$ $$-22 \beta_{2} q^{8}$$ $$+ ( -36 + 3 \beta_{3} ) q^{9}$$ $$+ ( 20 - 10 \beta_{1} + 5 \beta_{2} ) q^{10}$$ $$-4 \beta_{3} q^{11}$$ $$+ ( -6 \beta_{1} - 6 \beta_{2} ) q^{12}$$ $$+ ( 32 \beta_{1} - 16 \beta_{2} ) q^{13}$$ $$-2 \beta_{3} q^{14}$$ $$+ ( 30 - 15 \beta_{1} + 66 \beta_{2} + 2 \beta_{3} ) q^{15}$$ $$-124 q^{16}$$ $$+ 88 \beta_{2} q^{17}$$ $$+ ( 30 \beta_{1} - 51 \beta_{2} ) q^{18}$$ $$+ 308 q^{19}$$ $$+ ( -12 \beta_{2} + 6 \beta_{3} ) q^{20}$$ $$+ ( 117 - 3 \beta_{3} ) q^{21}$$ $$+ ( -40 \beta_{1} + 20 \beta_{2} ) q^{22}$$ $$-131 \beta_{2} q^{23}$$ $$+ ( -330 - 22 \beta_{3} ) q^{24}$$ $$+ ( -545 - 40 \beta_{1} + 20 \beta_{2} ) q^{25}$$ $$+ 32 \beta_{3} q^{26}$$ $$+ ( 9 \beta_{1} - 234 \beta_{2} ) q^{27}$$ $$+ ( 12 \beta_{1} - 6 \beta_{2} ) q^{28}$$ $$+ 8 \beta_{3} q^{29}$$ $$+ ( 585 + 20 \beta_{1} + 20 \beta_{2} - 15 \beta_{3} ) q^{30}$$ $$+ 32 q^{31}$$ $$+ 228 \beta_{2} q^{32}$$ $$+ ( -60 \beta_{1} + 264 \beta_{2} ) q^{33}$$ $$+ 880 q^{34}$$ $$+ ( -117 \beta_{2} - 4 \beta_{3} ) q^{35}$$ $$+ ( 216 - 18 \beta_{3} ) q^{36}$$ $$+ ( 168 \beta_{1} - 84 \beta_{2} ) q^{37}$$ $$+ 308 \beta_{2} q^{38}$$ $$+ ( -1872 + 48 \beta_{3} ) q^{39}$$ $$+ ( -440 + 220 \beta_{1} - 110 \beta_{2} ) q^{40}$$ $$-86 \beta_{3} q^{41}$$ $$+ ( -30 \beta_{1} + 132 \beta_{2} ) q^{42}$$ $$+ ( -338 \beta_{1} + 169 \beta_{2} ) q^{43}$$ $$+ 24 \beta_{3} q^{44}$$ $$+ ( 1755 + 60 \beta_{1} - 102 \beta_{2} + 36 \beta_{3} ) q^{45}$$ $$-1310 q^{46}$$ $$-773 \beta_{2} q^{47}$$ $$+ ( -124 \beta_{1} - 124 \beta_{2} ) q^{48}$$ $$+ 2167 q^{49}$$ $$+ ( -545 \beta_{2} - 40 \beta_{3} ) q^{50}$$ $$+ ( 1320 + 88 \beta_{3} ) q^{51}$$ $$+ ( -192 \beta_{1} + 96 \beta_{2} ) q^{52}$$ $$+ 448 \beta_{2} q^{53}$$ $$+ ( -2295 + 9 \beta_{3} ) q^{54}$$ $$+ ( -2340 - 80 \beta_{1} + 40 \beta_{2} ) q^{55}$$ $$+ 44 \beta_{3} q^{56}$$ $$+ ( 308 \beta_{1} + 308 \beta_{2} ) q^{57}$$ $$+ ( 80 \beta_{1} - 40 \beta_{2} ) q^{58}$$ $$-164 \beta_{3} q^{59}$$ $$+ ( -180 + 90 \beta_{1} - 396 \beta_{2} - 12 \beta_{3} ) q^{60}$$ $$-928 q^{61}$$ $$+ 32 \beta_{2} q^{62}$$ $$+ ( 72 \beta_{1} + 315 \beta_{2} ) q^{63}$$ $$+ 4264 q^{64}$$ $$+ ( 1872 \beta_{2} + 64 \beta_{3} ) q^{65}$$ $$+ ( 2340 - 60 \beta_{3} ) q^{66}$$ $$+ ( 338 \beta_{1} - 169 \beta_{2} ) q^{67}$$ $$-528 \beta_{2} q^{68}$$ $$+ ( -1965 - 131 \beta_{3} ) q^{69}$$ $$+ ( -1170 - 40 \beta_{1} + 20 \beta_{2} ) q^{70}$$ $$+ 192 \beta_{3} q^{71}$$ $$+ ( -660 \beta_{1} + 1122 \beta_{2} ) q^{72}$$ $$+ ( 552 \beta_{1} - 276 \beta_{2} ) q^{73}$$ $$+ 168 \beta_{3} q^{74}$$ $$+ ( 2340 - 545 \beta_{1} - 545 \beta_{2} - 60 \beta_{3} ) q^{75}$$ $$-1848 q^{76}$$ $$-468 \beta_{2} q^{77}$$ $$+ ( 480 \beta_{1} - 2112 \beta_{2} ) q^{78}$$ $$+ 8 q^{79}$$ $$+ ( -248 \beta_{2} + 124 \beta_{3} ) q^{80}$$ $$+ ( -3969 - 216 \beta_{3} ) q^{81}$$ $$+ ( -860 \beta_{1} + 430 \beta_{2} ) q^{82}$$ $$-1403 \beta_{2} q^{83}$$ $$+ ( -702 + 18 \beta_{3} ) q^{84}$$ $$+ ( 1760 - 880 \beta_{1} + 440 \beta_{2} ) q^{85}$$ $$-338 \beta_{3} q^{86}$$ $$+ ( 120 \beta_{1} - 528 \beta_{2} ) q^{87}$$ $$+ ( 880 \beta_{1} - 440 \beta_{2} ) q^{88}$$ $$+ 384 \beta_{3} q^{89}$$ $$+ ( -720 + 360 \beta_{1} + 1575 \beta_{2} + 60 \beta_{3} ) q^{90}$$ $$+ 3744 q^{91}$$ $$+ 786 \beta_{2} q^{92}$$ $$+ ( 32 \beta_{1} + 32 \beta_{2} ) q^{93}$$ $$-7730 q^{94}$$ $$+ ( 616 \beta_{2} - 308 \beta_{3} ) q^{95}$$ $$+ ( 3420 + 228 \beta_{3} ) q^{96}$$ $$+ ( 328 \beta_{1} - 164 \beta_{2} ) q^{97}$$ $$+ 2167 \beta_{2} q^{98}$$ $$+ ( 7020 + 144 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut +\mathstrut 60q^{6}$$ $$\mathstrut -\mathstrut 144q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut +\mathstrut 60q^{6}$$ $$\mathstrut -\mathstrut 144q^{9}$$ $$\mathstrut +\mathstrut 80q^{10}$$ $$\mathstrut +\mathstrut 120q^{15}$$ $$\mathstrut -\mathstrut 496q^{16}$$ $$\mathstrut +\mathstrut 1232q^{19}$$ $$\mathstrut +\mathstrut 468q^{21}$$ $$\mathstrut -\mathstrut 1320q^{24}$$ $$\mathstrut -\mathstrut 2180q^{25}$$ $$\mathstrut +\mathstrut 2340q^{30}$$ $$\mathstrut +\mathstrut 128q^{31}$$ $$\mathstrut +\mathstrut 3520q^{34}$$ $$\mathstrut +\mathstrut 864q^{36}$$ $$\mathstrut -\mathstrut 7488q^{39}$$ $$\mathstrut -\mathstrut 1760q^{40}$$ $$\mathstrut +\mathstrut 7020q^{45}$$ $$\mathstrut -\mathstrut 5240q^{46}$$ $$\mathstrut +\mathstrut 8668q^{49}$$ $$\mathstrut +\mathstrut 5280q^{51}$$ $$\mathstrut -\mathstrut 9180q^{54}$$ $$\mathstrut -\mathstrut 9360q^{55}$$ $$\mathstrut -\mathstrut 720q^{60}$$ $$\mathstrut -\mathstrut 3712q^{61}$$ $$\mathstrut +\mathstrut 17056q^{64}$$ $$\mathstrut +\mathstrut 9360q^{66}$$ $$\mathstrut -\mathstrut 7860q^{69}$$ $$\mathstrut -\mathstrut 4680q^{70}$$ $$\mathstrut +\mathstrut 9360q^{75}$$ $$\mathstrut -\mathstrut 7392q^{76}$$ $$\mathstrut +\mathstrut 32q^{79}$$ $$\mathstrut -\mathstrut 15876q^{81}$$ $$\mathstrut -\mathstrut 2808q^{84}$$ $$\mathstrut +\mathstrut 7040q^{85}$$ $$\mathstrut -\mathstrut 2880q^{90}$$ $$\mathstrut +\mathstrut 14976q^{91}$$ $$\mathstrut -\mathstrut 30920q^{94}$$ $$\mathstrut +\mathstrut 13680q^{96}$$ $$\mathstrut +\mathstrut 28080q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$8$$ $$x^{2}\mathstrut +\mathstrut$$ $$81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 26 \nu$$$$)/9$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$12$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$26$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}$$$$)/3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 −1.58114 − 2.54951i −1.58114 + 2.54951i 1.58114 − 2.54951i 1.58114 + 2.54951i
−3.16228 −4.74342 7.64853i −6.00000 −6.32456 24.1868i 15.0000 + 24.1868i 15.2971i 69.5701 −36.0000 + 72.5603i 20.0000 + 76.4853i
14.2 −3.16228 −4.74342 + 7.64853i −6.00000 −6.32456 + 24.1868i 15.0000 24.1868i 15.2971i 69.5701 −36.0000 72.5603i 20.0000 76.4853i
14.3 3.16228 4.74342 7.64853i −6.00000 6.32456 + 24.1868i 15.0000 24.1868i 15.2971i −69.5701 −36.0000 72.5603i 20.0000 + 76.4853i
14.4 3.16228 4.74342 + 7.64853i −6.00000 6.32456 24.1868i 15.0000 + 24.1868i 15.2971i −69.5701 −36.0000 + 72.5603i 20.0000 76.4853i
 $$n$$: 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
3.b Odd 1 yes
5.b Even 1 yes
15.d Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2}$$ $$\mathstrut -\mathstrut 10$$ acting on $$S_{5}^{\mathrm{new}}(15, [\chi])$$.