# Properties

 Label 315.2.k.a Level 315 Weight 2 Character orbit 315.k Analytic conductor 2.515 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + 2 \beta_{2} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{6} + ( -3 - 2 \beta_{2} ) q^{7} -3 q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + 2 \beta_{2} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{6} + ( -3 - 2 \beta_{2} ) q^{7} -3 q^{8} -3 q^{9} -\beta_{1} q^{10} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{13} + ( 2 - 3 \beta_{1} - 2 \beta_{3} ) q^{14} + ( -1 - 2 \beta_{2} ) q^{15} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{16} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{17} -3 \beta_{1} q^{18} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{20} + ( 1 - 4 \beta_{2} ) q^{21} + ( 4 - 4 \beta_{3} ) q^{23} + ( -3 - 6 \beta_{2} ) q^{24} + q^{25} + ( -3 + 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{26} + ( -3 - 6 \beta_{2} ) q^{27} + ( 5 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{28} + ( 4 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{29} + ( 2 - \beta_{1} - 2 \beta_{3} ) q^{30} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{31} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{32} + ( 1 - \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{34} + ( 3 + 2 \beta_{2} ) q^{35} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{36} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{37} + ( -7 + \beta_{3} ) q^{38} + ( -5 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{39} + 3 q^{40} + ( -3 - 3 \beta_{2} ) q^{41} + ( 4 + \beta_{1} - 4 \beta_{3} ) q^{42} + ( 4 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{43} + 3 q^{45} + ( 12 + 4 \beta_{1} + 12 \beta_{2} ) q^{46} + ( 3 - 4 \beta_{1} + 3 \beta_{2} ) q^{47} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{48} + ( 5 + 8 \beta_{2} ) q^{49} + \beta_{1} q^{50} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -12 + 5 \beta_{3} ) q^{52} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 6 - 3 \beta_{1} - 6 \beta_{3} ) q^{54} + ( 9 + 6 \beta_{2} ) q^{56} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{57} + ( 13 - \beta_{3} ) q^{58} + ( -2 + 2 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{60} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{61} + ( -7 + \beta_{3} ) q^{62} + ( 9 + 6 \beta_{2} ) q^{63} + ( -5 + 6 \beta_{3} ) q^{64} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{65} + ( 4 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{67} + ( -4 + \beta_{3} ) q^{68} + ( -4 + 8 \beta_{1} + 4 \beta_{3} ) q^{69} + ( -2 + 3 \beta_{1} + 2 \beta_{3} ) q^{70} + 9 q^{72} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( -7 + \beta_{3} ) q^{74} + ( 1 + 2 \beta_{2} ) q^{75} + ( -5 - 3 \beta_{1} - 5 \beta_{2} ) q^{76} + ( -15 - 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{78} + ( -5 + 6 \beta_{1} - 5 \beta_{2} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{80} + 9 q^{81} + ( 3 - 3 \beta_{1} - 3 \beta_{3} ) q^{82} + ( -4 + 4 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 3 + 5 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{84} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 17 - 5 \beta_{3} ) q^{86} + ( -2 + 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{87} + 3 \beta_{2} q^{89} + 3 \beta_{1} q^{90} + ( 3 - 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{91} + ( -8 + 8 \beta_{1} + 12 \beta_{2} + 8 \beta_{3} ) q^{92} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{93} + ( 1 - \beta_{1} - 12 \beta_{2} - \beta_{3} ) q^{94} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{95} + ( -7 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{96} + ( 2 - 2 \beta_{1} + 11 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -8 + 5 \beta_{1} + 8 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - 3q^{4} - 4q^{5} - 3q^{6} - 8q^{7} - 12q^{8} - 12q^{9} + O(q^{10})$$ $$4q + q^{2} - 3q^{4} - 4q^{5} - 3q^{6} - 8q^{7} - 12q^{8} - 12q^{9} - q^{10} - 9q^{12} + 4q^{13} + q^{14} + 3q^{16} - 4q^{17} - 3q^{18} + 3q^{20} + 12q^{21} + 8q^{23} + 4q^{25} - 15q^{26} + 15q^{28} - 2q^{29} + 3q^{30} - 7q^{32} - 11q^{34} + 8q^{35} + 9q^{36} - 26q^{38} - 12q^{39} + 12q^{40} - 6q^{41} + 9q^{42} + 6q^{43} + 12q^{45} + 28q^{46} + 2q^{47} - 9q^{48} + 4q^{49} + q^{50} + 12q^{51} - 38q^{52} - 8q^{53} + 9q^{54} + 24q^{56} + 50q^{58} + 16q^{59} + 9q^{60} + 6q^{61} - 26q^{62} + 24q^{63} - 8q^{64} - 4q^{65} + 6q^{67} - 14q^{68} - q^{70} + 36q^{72} - 26q^{74} - 13q^{76} - 45q^{78} - 4q^{79} - 3q^{80} + 36q^{81} + 3q^{82} + 14q^{83} + 9q^{84} + 4q^{85} + 58q^{86} - 6q^{87} - 6q^{89} + 3q^{90} + 4q^{91} - 32q^{92} + 25q^{94} - 21q^{96} - 20q^{97} - 11q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu - 3$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 7$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −0.651388 − 1.12824i 1.15139 + 1.99426i −0.651388 + 1.12824i 1.15139 − 1.99426i
−0.651388 1.12824i 1.73205i 0.151388 0.262211i −1.00000 1.95416 1.12824i −2.00000 1.73205i −3.00000 −3.00000 0.651388 + 1.12824i
16.2 1.15139 + 1.99426i 1.73205i −1.65139 + 2.86029i −1.00000 −3.45416 + 1.99426i −2.00000 1.73205i −3.00000 −3.00000 −1.15139 1.99426i
256.1 −0.651388 + 1.12824i 1.73205i 0.151388 + 0.262211i −1.00000 1.95416 + 1.12824i −2.00000 + 1.73205i −3.00000 −3.00000 0.651388 1.12824i
256.2 1.15139 1.99426i 1.73205i −1.65139 2.86029i −1.00000 −3.45416 1.99426i −2.00000 + 1.73205i −3.00000 −3.00000 −1.15139 + 1.99426i
 $$n$$: 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
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.k.a 4
3.b odd 2 1 945.2.k.a 4
7.c even 3 1 315.2.l.a yes 4
9.c even 3 1 315.2.l.a yes 4
9.d odd 6 1 945.2.l.a 4
21.h odd 6 1 945.2.l.a 4
63.g even 3 1 inner 315.2.k.a 4
63.n odd 6 1 945.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.a 4 1.a even 1 1 trivial
315.2.k.a 4 63.g even 3 1 inner
315.2.l.a yes 4 7.c even 3 1
315.2.l.a yes 4 9.c even 3 1
945.2.k.a 4 3.b odd 2 1
945.2.k.a 4 63.n odd 6 1
945.2.l.a 4 9.d odd 6 1
945.2.l.a 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{3} - 5 T^{4} + 6 T^{5} - 8 T^{7} + 16 T^{8}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$( 1 + 4 T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} )^{4}$$
$13$ $$1 - 4 T - T^{2} + 36 T^{3} - 88 T^{4} + 468 T^{5} - 169 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 612 T^{5} - 2601 T^{6} + 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 25 T^{2} + 264 T^{4} - 9025 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 4 T - 2 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 2958 T^{5} - 2523 T^{6} + 48778 T^{7} + 707281 T^{8}$$
$31$ $$1 - 49 T^{2} + 1440 T^{4} - 47089 T^{6} + 923521 T^{8}$$
$37$ $$1 - 61 T^{2} + 2352 T^{4} - 83509 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 6 T - 7 T^{2} + 258 T^{3} - 1548 T^{4} + 11094 T^{5} - 12943 T^{6} - 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 2 T - 39 T^{2} + 102 T^{3} - 548 T^{4} + 4794 T^{5} - 86151 T^{6} - 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 8 T - 45 T^{2} + 24 T^{3} + 5680 T^{4} + 1272 T^{5} - 126405 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 16 T + 87 T^{2} - 816 T^{3} + 9976 T^{4} - 48144 T^{5} + 302847 T^{6} - 3286064 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 6 T - 43 T^{2} + 258 T^{3} + 324 T^{4} + 15738 T^{5} - 160003 T^{6} - 1361886 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 6 T - 55 T^{2} + 258 T^{3} + 1380 T^{4} + 17286 T^{5} - 246895 T^{6} - 1804578 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$1 - 133 T^{2} + 12360 T^{4} - 708757 T^{6} + 28398241 T^{8}$$
$79$ $$1 + 4 T - 29 T^{2} - 452 T^{3} - 5480 T^{4} - 35708 T^{5} - 180989 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 14 T + 33 T^{2} + 42 T^{3} + 3412 T^{4} + 3486 T^{5} + 227337 T^{6} - 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 20 T + 119 T^{2} + 1740 T^{3} + 30752 T^{4} + 168780 T^{5} + 1119671 T^{6} + 18253460 T^{7} + 88529281 T^{8}$$