# 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 $$2$$, degree $$1$$, minimal)

## 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: $$3^{2}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 24q^{4} + 60q^{6} - 144q^{9} + O(q^{10})$$ $$4q - 24q^{4} + 60q^{6} - 144q^{9} + 80q^{10} + 120q^{15} - 496q^{16} + 1232q^{19} + 468q^{21} - 1320q^{24} - 2180q^{25} + 2340q^{30} + 128q^{31} + 3520q^{34} + 864q^{36} - 7488q^{39} - 1760q^{40} + 7020q^{45} - 5240q^{46} + 8668q^{49} + 5280q^{51} - 9180q^{54} - 9360q^{55} - 720q^{60} - 3712q^{61} + 17056q^{64} + 9360q^{66} - 7860q^{69} - 4680q^{70} + 9360q^{75} - 7392q^{76} + 32q^{79} - 15876q^{81} - 2808q^{84} + 7040q^{85} - 2880q^{90} + 14976q^{91} - 30920q^{94} + 13680q^{96} + 28080q^{99} + O(q^{100})$$

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

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

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.d.c 4
3.b odd 2 1 inner 15.5.d.c 4
4.b odd 2 1 240.5.c.c 4
5.b even 2 1 inner 15.5.d.c 4
5.c odd 4 2 75.5.c.h 4
12.b even 2 1 240.5.c.c 4
15.d odd 2 1 inner 15.5.d.c 4
15.e even 4 2 75.5.c.h 4
20.d odd 2 1 240.5.c.c 4
60.h even 2 1 240.5.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.d.c 4 1.a even 1 1 trivial
15.5.d.c 4 3.b odd 2 1 inner
15.5.d.c 4 5.b even 2 1 inner
15.5.d.c 4 15.d odd 2 1 inner
75.5.c.h 4 5.c odd 4 2
75.5.c.h 4 15.e even 4 2
240.5.c.c 4 4.b odd 2 1
240.5.c.c 4 12.b even 2 1
240.5.c.c 4 20.d odd 2 1
240.5.c.c 4 60.h even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 22 T^{2} + 256 T^{4} )^{2}$$
$3$ $$1 + 72 T^{2} + 6561 T^{4}$$
$5$ $$1 + 1090 T^{2} + 390625 T^{4}$$
$7$ $$( 1 - 4568 T^{2} + 5764801 T^{4} )^{2}$$
$11$ $$( 1 - 19922 T^{2} + 214358881 T^{4} )^{2}$$
$13$ $$( 1 + 2782 T^{2} + 815730721 T^{4} )^{2}$$
$17$ $$( 1 + 89602 T^{2} + 6975757441 T^{4} )^{2}$$
$19$ $$( 1 - 308 T + 130321 T^{2} )^{4}$$
$23$ $$( 1 + 388072 T^{2} + 78310985281 T^{4} )^{2}$$
$29$ $$( 1 - 1377122 T^{2} + 500246412961 T^{4} )^{2}$$
$31$ $$( 1 - 32 T + 923521 T^{2} )^{4}$$
$37$ $$( 1 - 2097218 T^{2} + 3512479453921 T^{4} )^{2}$$
$41$ $$( 1 - 1324862 T^{2} + 7984925229121 T^{4} )^{2}$$
$43$ $$( 1 - 154328 T^{2} + 11688200277601 T^{4} )^{2}$$
$47$ $$( 1 + 3784072 T^{2} + 23811286661761 T^{4} )^{2}$$
$53$ $$( 1 + 13773922 T^{2} + 62259690411361 T^{4} )^{2}$$
$59$ $$( 1 - 8500562 T^{2} + 146830437604321 T^{4} )^{2}$$
$61$ $$( 1 + 928 T + 13845841 T^{2} )^{4}$$
$67$ $$( 1 - 33618968 T^{2} + 406067677556641 T^{4} )^{2}$$
$71$ $$( 1 - 29257922 T^{2} + 645753531245761 T^{4} )^{2}$$
$73$ $$( 1 - 38971298 T^{2} + 806460091894081 T^{4} )^{2}$$
$79$ $$( 1 - 8 T + 38950081 T^{2} )^{4}$$
$83$ $$( 1 + 75232552 T^{2} + 2252292232139041 T^{4} )^{2}$$
$89$ $$( 1 - 39222722 T^{2} + 3936588805702081 T^{4} )^{2}$$
$97$ $$( 1 - 170764898 T^{2} + 7837433594376961 T^{4} )^{2}$$