# Properties

 Label 315.2.bh.a Level 315 Weight 2 Character orbit 315.bh 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.bh (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 1 + \zeta_{12}^{2} ) q^{6} + \zeta_{12} q^{7} -3 \zeta_{12}^{3} q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 1 + \zeta_{12}^{2} ) q^{6} + \zeta_{12} q^{7} -3 \zeta_{12}^{3} q^{8} + 3 q^{9} + ( -1 + 2 \zeta_{12}^{3} ) q^{10} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{12} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{13} + \zeta_{12}^{2} q^{14} + ( -2 + 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{15} + ( 1 - \zeta_{12}^{2} ) q^{16} + 7 \zeta_{12}^{3} q^{17} + 3 \zeta_{12} q^{18} -6 q^{19} + ( 2 + \zeta_{12} - 2 \zeta_{12}^{2} ) q^{20} + ( 1 + \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{22} + ( 3 - 6 \zeta_{12}^{2} ) q^{24} + ( -3 - 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} - q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} -\zeta_{12}^{3} q^{28} + ( 10 - 10 \zeta_{12}^{2} ) q^{29} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{30} -8 \zeta_{12}^{2} q^{31} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{32} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{33} + ( -7 + 7 \zeta_{12}^{2} ) q^{34} + ( -1 + 2 \zeta_{12}^{3} ) q^{35} -3 \zeta_{12}^{2} q^{36} + 8 \zeta_{12}^{3} q^{37} -6 \zeta_{12} q^{38} + ( -2 + \zeta_{12}^{2} ) q^{39} + ( 6 \zeta_{12} + 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{40} -2 \zeta_{12}^{2} q^{41} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{42} -4 \zeta_{12} q^{43} -3 q^{44} + ( -3 \zeta_{12} + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{45} -9 \zeta_{12} q^{47} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{48} + \zeta_{12}^{2} q^{49} + ( -3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + ( -7 + 14 \zeta_{12}^{2} ) q^{51} + \zeta_{12} q^{52} -6 \zeta_{12}^{3} q^{53} + ( 3 + 3 \zeta_{12}^{2} ) q^{54} + ( 6 + 3 \zeta_{12}^{3} ) q^{55} + ( 3 - 3 \zeta_{12}^{2} ) q^{56} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{57} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{58} + 4 \zeta_{12}^{2} q^{59} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{60} + ( -10 + 10 \zeta_{12}^{2} ) q^{61} -8 \zeta_{12}^{3} q^{62} + 3 \zeta_{12} q^{63} -7 q^{64} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{65} + ( 6 - 3 \zeta_{12}^{2} ) q^{66} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{68} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{70} -5 q^{71} -9 \zeta_{12}^{3} q^{72} + 3 \zeta_{12}^{3} q^{73} + ( -8 + 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} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{77} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{78} + ( 7 - 7 \zeta_{12}^{2} ) q^{79} + ( 2 + \zeta_{12}^{3} ) q^{80} + 9 q^{81} -2 \zeta_{12}^{3} q^{82} + 5 \zeta_{12} q^{83} + ( 1 - 2 \zeta_{12}^{2} ) q^{84} + ( -14 \zeta_{12} - 7 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{85} -4 \zeta_{12}^{2} q^{86} + ( 10 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{87} -9 \zeta_{12} q^{88} + 18 q^{89} + ( -3 + 6 \zeta_{12}^{3} ) q^{90} - q^{91} + ( -8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{93} -9 \zeta_{12}^{2} q^{94} + ( 6 \zeta_{12} - 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{95} + ( -10 + 5 \zeta_{12}^{2} ) q^{96} + 5 \zeta_{12} q^{97} + \zeta_{12}^{3} q^{98} + ( 9 - 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + 4q^{5} + 6q^{6} + 12q^{9} + O(q^{10})$$ $$4q - 2q^{4} + 4q^{5} + 6q^{6} + 12q^{9} - 4q^{10} + 6q^{11} + 2q^{14} - 6q^{15} + 2q^{16} - 24q^{19} + 4q^{20} + 6q^{21} - 6q^{25} - 4q^{26} + 20q^{29} - 16q^{31} - 14q^{34} - 4q^{35} - 6q^{36} - 6q^{39} + 6q^{40} - 4q^{41} - 12q^{44} + 12q^{45} + 2q^{49} - 8q^{50} + 18q^{54} + 24q^{55} + 6q^{56} + 8q^{59} + 6q^{60} - 20q^{61} - 28q^{64} + 2q^{65} + 18q^{66} - 4q^{70} - 20q^{71} - 16q^{74} - 24q^{75} + 12q^{76} + 14q^{79} + 8q^{80} + 36q^{81} - 14q^{85} - 8q^{86} + 72q^{89} - 12q^{90} - 4q^{91} - 18q^{94} - 24q^{95} - 30q^{96} + 18q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −1.73205 −0.500000 0.866025i 1.86603 + 1.23205i 1.50000 + 0.866025i −0.866025 0.500000i 3.00000i 3.00000 −1.00000 2.00000i
169.2 0.866025 + 0.500000i 1.73205 −0.500000 0.866025i 0.133975 + 2.23205i 1.50000 + 0.866025i 0.866025 + 0.500000i 3.00000i 3.00000 −1.00000 + 2.00000i
274.1 −0.866025 + 0.500000i −1.73205 −0.500000 + 0.866025i 1.86603 1.23205i 1.50000 0.866025i −0.866025 + 0.500000i 3.00000i 3.00000 −1.00000 + 2.00000i
274.2 0.866025 0.500000i 1.73205 −0.500000 + 0.866025i 0.133975 2.23205i 1.50000 0.866025i 0.866025 0.500000i 3.00000i 3.00000 −1.00000 2.00000i
 $$n$$: 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
9.c even 3 1 inner
45.j 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.bh.a 4
3.b odd 2 1 945.2.bh.a 4
5.b even 2 1 inner 315.2.bh.a 4
9.c even 3 1 inner 315.2.bh.a 4
9.d odd 6 1 945.2.bh.a 4
15.d odd 2 1 945.2.bh.a 4
45.h odd 6 1 945.2.bh.a 4
45.j even 6 1 inner 315.2.bh.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bh.a 4 1.a even 1 1 trivial
315.2.bh.a 4 5.b even 2 1 inner
315.2.bh.a 4 9.c even 3 1 inner
315.2.bh.a 4 45.j even 6 1 inner
945.2.bh.a 4 3.b odd 2 1
945.2.bh.a 4 9.d odd 6 1
945.2.bh.a 4 15.d odd 2 1
945.2.bh.a 4 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 + 25 T^{2} + 456 T^{4} + 4225 T^{6} + 28561 T^{8}$$
$17$ $$( 1 + 15 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 6 T + 19 T^{2} )^{4}$$
$23$ $$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$
$47$ $$1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 5 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 137 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 141 T^{2} + 12992 T^{4} + 971349 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 18 T + 89 T^{2} )^{4}$$
$97$ $$( 1 + 2 T^{2} + 9409 T^{4} )( 1 + 167 T^{2} + 9409 T^{4} )$$