# Properties

 Label 315.2.bh.b 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: $$\Z[a_1, a_2, a_3]$$ 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.

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

## 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
−1.73205 1.00000i −0.866025 + 1.50000i 1.00000 + 1.73205i 0.133975 + 2.23205i 3.00000 1.73205i 0.866025 + 0.500000i 0 −1.50000 2.59808i 2.00000 4.00000i
169.2 1.73205 + 1.00000i 0.866025 1.50000i 1.00000 + 1.73205i 1.86603 + 1.23205i 3.00000 1.73205i −0.866025 0.500000i 0 −1.50000 2.59808i 2.00000 + 4.00000i
274.1 −1.73205 + 1.00000i −0.866025 1.50000i 1.00000 1.73205i 0.133975 2.23205i 3.00000 + 1.73205i 0.866025 0.500000i 0 −1.50000 + 2.59808i 2.00000 + 4.00000i
274.2 1.73205 1.00000i 0.866025 + 1.50000i 1.00000 1.73205i 1.86603 1.23205i 3.00000 + 1.73205i −0.866025 + 0.500000i 0 −1.50000 + 2.59808i 2.00000 4.00000i
 $$n$$: 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.b 4
3.b odd 2 1 945.2.bh.b 4
5.b even 2 1 inner 315.2.bh.b 4
9.c even 3 1 inner 315.2.bh.b 4
9.d odd 6 1 945.2.bh.b 4
15.d odd 2 1 945.2.bh.b 4
45.h odd 6 1 945.2.bh.b 4
45.j even 6 1 inner 315.2.bh.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 6 T + 19 T^{2} )^{4}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 8 T + 23 T^{2} + 328 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 82 T^{2} + 4875 T^{4} + 151618 T^{6} + 3418801 T^{8}$$
$47$ $$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 53 T^{2} )^{4}$$
$59$ $$( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 7 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 65 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 123 T^{2} + 8240 T^{4} - 847347 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 12 T + 89 T^{2} )^{4}$$
$97$ $$1 + 145 T^{2} + 11616 T^{4} + 1364305 T^{6} + 88529281 T^{8}$$