# Properties

 Label 315.2.i.a Level 315 Weight 2 Character orbit 315.i Analytic conductor 2.515 Analytic rank 1 Dimension 2 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( -1 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( -1 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} -2 q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + ( -2 + 4 \zeta_{6} ) q^{12} -6 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( 1 - 2 \zeta_{6} ) q^{15} + ( 4 - 4 \zeta_{6} ) q^{16} -2 q^{17} -6 q^{18} -2 q^{19} + ( 2 - 2 \zeta_{6} ) q^{20} + ( 2 - \zeta_{6} ) q^{21} -6 \zeta_{6} q^{22} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 12 q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + 2 q^{28} + ( 1 - \zeta_{6} ) q^{29} + ( 2 + 2 \zeta_{6} ) q^{30} -10 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( 6 - 3 \zeta_{6} ) q^{33} + ( 4 - 4 \zeta_{6} ) q^{34} - q^{35} + ( 6 - 6 \zeta_{6} ) q^{36} -2 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( -6 + 12 \zeta_{6} ) q^{39} + 6 \zeta_{6} q^{41} + ( -2 + 4 \zeta_{6} ) q^{42} + ( -4 + 4 \zeta_{6} ) q^{43} + 6 q^{44} + ( -3 + 3 \zeta_{6} ) q^{45} + 8 q^{46} + ( -7 + 7 \zeta_{6} ) q^{47} + ( -8 + 4 \zeta_{6} ) q^{48} -\zeta_{6} q^{49} -2 \zeta_{6} q^{50} + ( 2 + 2 \zeta_{6} ) q^{51} + ( -12 + 12 \zeta_{6} ) q^{52} -4 q^{53} + ( 6 + 6 \zeta_{6} ) q^{54} -3 q^{55} + ( 2 + 2 \zeta_{6} ) q^{57} + 2 \zeta_{6} q^{58} -14 \zeta_{6} q^{59} + ( -4 + 2 \zeta_{6} ) q^{60} + ( -4 + 4 \zeta_{6} ) q^{61} + 20 q^{62} -3 q^{63} -8 q^{64} + ( 6 - 6 \zeta_{6} ) q^{65} + ( -6 + 12 \zeta_{6} ) q^{66} + 2 \zeta_{6} q^{67} + 4 \zeta_{6} q^{68} + ( -4 + 8 \zeta_{6} ) q^{69} + ( 2 - 2 \zeta_{6} ) q^{70} -9 q^{71} + 13 q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + ( 2 - \zeta_{6} ) q^{75} + 4 \zeta_{6} q^{76} -3 \zeta_{6} q^{77} + ( -12 - 12 \zeta_{6} ) q^{78} + ( -17 + 17 \zeta_{6} ) q^{79} + 4 q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -12 q^{82} + ( -13 + 13 \zeta_{6} ) q^{83} + ( -2 - 2 \zeta_{6} ) q^{84} -2 \zeta_{6} q^{85} -8 \zeta_{6} q^{86} + ( -2 + \zeta_{6} ) q^{87} + 6 q^{89} -6 \zeta_{6} q^{90} + 6 q^{91} + ( -8 + 8 \zeta_{6} ) q^{92} + ( -10 + 20 \zeta_{6} ) q^{93} -14 \zeta_{6} q^{94} -2 \zeta_{6} q^{95} + ( 8 - 16 \zeta_{6} ) q^{96} + ( 9 - 9 \zeta_{6} ) q^{97} + 2 q^{98} -9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 3q^{3} - 2q^{4} + q^{5} + 6q^{6} - q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 3q^{3} - 2q^{4} + q^{5} + 6q^{6} - q^{7} + 3q^{9} - 4q^{10} - 3q^{11} - 6q^{13} - 2q^{14} + 4q^{16} - 4q^{17} - 12q^{18} - 4q^{19} + 2q^{20} + 3q^{21} - 6q^{22} - 4q^{23} - q^{25} + 24q^{26} + 4q^{28} + q^{29} + 6q^{30} - 10q^{31} + 8q^{32} + 9q^{33} + 4q^{34} - 2q^{35} + 6q^{36} - 4q^{37} + 4q^{38} + 6q^{41} - 4q^{43} + 12q^{44} - 3q^{45} + 16q^{46} - 7q^{47} - 12q^{48} - q^{49} - 2q^{50} + 6q^{51} - 12q^{52} - 8q^{53} + 18q^{54} - 6q^{55} + 6q^{57} + 2q^{58} - 14q^{59} - 6q^{60} - 4q^{61} + 40q^{62} - 6q^{63} - 16q^{64} + 6q^{65} + 2q^{67} + 4q^{68} + 2q^{70} - 18q^{71} + 26q^{73} + 4q^{74} + 3q^{75} + 4q^{76} - 3q^{77} - 36q^{78} - 17q^{79} + 8q^{80} - 9q^{81} - 24q^{82} - 13q^{83} - 6q^{84} - 2q^{85} - 8q^{86} - 3q^{87} + 12q^{89} - 6q^{90} + 12q^{91} - 8q^{92} - 14q^{94} - 2q^{95} + 9q^{97} + 4q^{98} - 18q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i −1.50000 0.866025i −1.00000 1.73205i 0.500000 + 0.866025i 3.00000 1.73205i −0.500000 + 0.866025i 0 1.50000 + 2.59808i −2.00000
211.1 −1.00000 1.73205i −1.50000 + 0.866025i −1.00000 + 1.73205i 0.500000 0.866025i 3.00000 + 1.73205i −0.500000 0.866025i 0 1.50000 2.59808i −2.00000
 $$n$$: 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
9.c 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.i.a 2
3.b odd 2 1 945.2.i.b 2
9.c even 3 1 inner 315.2.i.a 2
9.c even 3 1 2835.2.a.h 1
9.d odd 6 1 945.2.i.b 2
9.d odd 6 1 2835.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.a 2 1.a even 1 1 trivial
315.2.i.a 2 9.c even 3 1 inner
945.2.i.b 2 3.b odd 2 1
945.2.i.b 2 9.d odd 6 1
2835.2.a.b 1 9.d odd 6 1
2835.2.a.h 1 9.c even 3 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 9 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 13 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4}$$