# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + q^{4} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + q^{4} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{10} -6 \zeta_{12}^{2} q^{11} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{12} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + ( 1 + 2 \zeta_{12}^{2} ) q^{14} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{15} - q^{16} -2 \zeta_{12} q^{17} -3 \zeta_{12} q^{18} -6 \zeta_{12}^{2} q^{19} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{20} + ( 4 + \zeta_{12}^{2} ) q^{21} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{22} -3 \zeta_{12} q^{23} + ( -6 + 3 \zeta_{12}^{2} ) q^{24} + ( -4 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + 4 \zeta_{12}^{2} q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{28} + ( -2 + 2 \zeta_{12}^{2} ) q^{29} + ( 1 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{30} -4 q^{31} + 5 \zeta_{12}^{3} q^{32} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{33} + ( 2 - 2 \zeta_{12}^{2} ) q^{34} + ( 6 + \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} + ( -3 + 3 \zeta_{12}^{2} ) q^{36} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{37} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{38} + ( 4 + 4 \zeta_{12}^{2} ) q^{39} + ( -6 - 3 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{40} -2 \zeta_{12}^{2} q^{41} + ( -\zeta_{12} + 5 \zeta_{12}^{3} ) q^{42} + \zeta_{12} q^{43} -6 \zeta_{12}^{2} q^{44} + ( -6 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{45} + ( 3 - 3 \zeta_{12}^{2} ) q^{46} + 3 \zeta_{12}^{3} q^{47} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + ( 2 - 4 \zeta_{12}^{2} ) q^{51} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} + 12 \zeta_{12} q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{54} + ( 6 - 12 \zeta_{12}^{3} ) q^{55} + ( 3 + 6 \zeta_{12}^{2} ) q^{56} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{57} -2 \zeta_{12} q^{58} -4 q^{59} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{60} + 7 q^{61} -4 \zeta_{12}^{3} q^{62} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} -7 q^{64} + ( 8 + 4 \zeta_{12}^{3} ) q^{65} + ( 6 + 6 \zeta_{12}^{2} ) q^{66} + 7 \zeta_{12}^{3} q^{67} -2 \zeta_{12} q^{68} + ( 3 - 6 \zeta_{12}^{2} ) q^{69} + ( -3 + 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} + 4 q^{71} -9 \zeta_{12} q^{72} -8 \zeta_{12}^{2} q^{74} + ( -4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{75} -6 \zeta_{12}^{2} q^{76} + ( -18 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{78} + 14 q^{79} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{82} + 4 \zeta_{12} q^{83} + ( 4 + \zeta_{12}^{2} ) q^{84} + ( 2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( -1 + \zeta_{12}^{2} ) q^{86} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 18 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{88} -3 \zeta_{12}^{2} q^{89} + ( 3 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{90} + ( 8 - 12 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12} q^{92} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} -3 q^{94} + ( 6 - 12 \zeta_{12}^{3} ) q^{95} + ( -10 + 5 \zeta_{12}^{2} ) q^{96} -2 \zeta_{12} q^{97} + ( 8 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{98} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 2q^{5} - 6q^{6} - 6q^{9} + O(q^{10})$$ $$4q + 4q^{4} - 2q^{5} - 6q^{6} - 6q^{9} - 4q^{10} - 12q^{11} + 8q^{14} - 4q^{16} - 12q^{19} - 2q^{20} + 18q^{21} - 18q^{24} + 6q^{25} + 8q^{26} - 4q^{29} - 16q^{31} + 4q^{34} + 20q^{35} - 6q^{36} + 24q^{39} - 12q^{40} - 4q^{41} - 12q^{44} - 6q^{45} + 6q^{46} - 4q^{49} - 8q^{50} + 24q^{55} + 24q^{56} - 16q^{59} + 28q^{61} - 28q^{64} + 32q^{65} + 36q^{66} - 10q^{70} + 16q^{71} - 16q^{74} - 24q^{75} - 12q^{76} + 56q^{79} + 2q^{80} - 18q^{81} + 18q^{84} - 8q^{85} - 2q^{86} - 6q^{89} - 12q^{90} + 8q^{91} - 12q^{94} + 24q^{95} - 30q^{96} + 72q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
184.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
1.00000i 0.866025 1.50000i 1.00000 1.23205 1.86603i −1.50000 0.866025i 1.73205 + 2.00000i 3.00000i −1.50000 2.59808i −1.86603 1.23205i
184.2 1.00000i −0.866025 + 1.50000i 1.00000 −2.23205 + 0.133975i −1.50000 0.866025i −1.73205 2.00000i 3.00000i −1.50000 2.59808i −0.133975 2.23205i
214.1 1.00000i −0.866025 1.50000i 1.00000 −2.23205 0.133975i −1.50000 + 0.866025i −1.73205 + 2.00000i 3.00000i −1.50000 + 2.59808i −0.133975 + 2.23205i
214.2 1.00000i 0.866025 + 1.50000i 1.00000 1.23205 + 1.86603i −1.50000 + 0.866025i 1.73205 2.00000i 3.00000i −1.50000 + 2.59808i −1.86603 + 1.23205i
 $$n$$: 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
5.b even 2 1 inner
63.h even 3 1 inner
315.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.r.a 4
3.b odd 2 1 945.2.r.a 4
5.b even 2 1 inner 315.2.r.a 4
7.c even 3 1 315.2.bo.a yes 4
9.c even 3 1 315.2.bo.a yes 4
9.d odd 6 1 945.2.bo.a 4
15.d odd 2 1 945.2.r.a 4
21.h odd 6 1 945.2.bo.a 4
35.j even 6 1 315.2.bo.a yes 4
45.h odd 6 1 945.2.bo.a 4
45.j even 6 1 315.2.bo.a yes 4
63.h even 3 1 inner 315.2.r.a 4
63.j odd 6 1 945.2.r.a 4
105.o odd 6 1 945.2.bo.a 4
315.r even 6 1 inner 315.2.r.a 4
315.br odd 6 1 945.2.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.a 4 1.a even 1 1 trivial
315.2.r.a 4 5.b even 2 1 inner
315.2.r.a 4 63.h even 3 1 inner
315.2.r.a 4 315.r even 6 1 inner
315.2.bo.a yes 4 7.c even 3 1
315.2.bo.a yes 4 9.c even 3 1
315.2.bo.a yes 4 35.j even 6 1
315.2.bo.a yes 4 45.j even 6 1
945.2.r.a 4 3.b odd 2 1
945.2.r.a 4 15.d odd 2 1
945.2.r.a 4 63.j odd 6 1
945.2.r.a 4 315.br odd 6 1
945.2.bo.a 4 9.d odd 6 1
945.2.bo.a 4 21.h odd 6 1
945.2.bo.a 4 45.h odd 6 1
945.2.bo.a 4 105.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T^{2} + 4 T^{4} )^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{4}$$
$37$ $$1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 - 85 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{4}$$
$61$ $$( 1 - 7 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 85 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 4 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 73 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{4}$$
$83$ $$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$