# Properties

 Label 315.2.bs.c Level 315 Weight 2 Character orbit 315.bs 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.bs (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

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

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( 2 - \zeta_{12}^{2} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} ) q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{8} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( 2 - \zeta_{12}^{2} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} ) q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{8} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{10} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + 3 \zeta_{12}^{2} q^{12} + ( 2 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( 2 - \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( -2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{15} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{16} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{17} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} + ( -\zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( -2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{20} + ( 4 + \zeta_{12}^{2} ) q^{21} + ( 4 + 2 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{22} + ( -2 + 2 \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{24} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{26} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -5 + 4 \zeta_{12}^{2} ) q^{28} + ( -4 - 4 \zeta_{12}^{2} ) q^{29} + ( 3 + \zeta_{12} - 3 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{30} + ( 3 - 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{31} + ( -4 - \zeta_{12} - \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{32} + ( 2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{33} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{34} + ( -6 + \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} + ( 3 + 3 \zeta_{12}^{2} ) q^{36} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( 4 - \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{38} + ( 2 - 2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{39} + ( -2 + 2 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + ( 4 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{41} + ( 1 - 4 \zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{42} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{43} + ( -4 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{44} + ( 3 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{45} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{46} + ( 6 + 9 \zeta_{12} - 9 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{47} + ( 2 + 6 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{48} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( 1 + 4 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{51} + ( 2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( -1 + \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{53} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{54} + ( -6 - 8 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{55} + ( -2 - \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{56} + ( 5 - 3 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{57} + ( -4 + 4 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + ( -6 + 3 \zeta_{12}^{3} ) q^{60} + ( -6 + 12 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{61} + ( -1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{62} + ( 9 - 3 \zeta_{12}^{2} ) q^{63} + ( -1 + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{64} + ( 2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( 2 - 2 \zeta_{12} - 10 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{66} + ( 5 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{68} + ( -1 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{69} + ( 4 + 8 \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{70} -6 q^{71} + ( 3 - 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{72} + ( -5 - 5 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{74} + ( -3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{75} + ( -10 + 3 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{76} + ( -4 - 2 \zeta_{12} + 6 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{77} + ( -6 - 4 \zeta_{12} + 6 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{78} + ( -2 + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{79} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{83} + ( -6 + 9 \zeta_{12}^{2} ) q^{84} + ( 6 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{85} + ( -3 + 3 \zeta_{12}^{2} ) q^{86} -12 q^{87} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( 3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{89} + ( 3 - 3 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{90} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( -4 - 4 \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{92} + ( 5 \zeta_{12} - 9 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{93} + ( -15 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{94} + ( -9 + 4 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{95} + ( -9 - 6 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{96} + ( -3 + 3 \zeta_{12} + 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{97} + ( 8 + 3 \zeta_{12} - 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{98} + ( 6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 6q^{3} - 4q^{5} - 6q^{6} + 8q^{7} + 2q^{8} + 6q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 6q^{3} - 4q^{5} - 6q^{6} + 8q^{7} + 2q^{8} + 6q^{9} + 6q^{10} + 4q^{11} + 6q^{12} + 4q^{13} + 2q^{14} + 4q^{16} + 4q^{17} - 12q^{18} + 10q^{19} - 12q^{20} + 18q^{21} + 4q^{22} - 2q^{23} + 6q^{24} - 6q^{25} - 12q^{26} - 12q^{28} - 24q^{29} + 6q^{30} - 18q^{32} + 12q^{33} - 4q^{34} - 20q^{35} + 18q^{36} - 6q^{37} + 10q^{38} - 10q^{40} + 12q^{41} - 6q^{42} - 12q^{43} - 12q^{44} + 12q^{45} + 8q^{46} + 6q^{47} + 6q^{48} + 4q^{49} + 10q^{50} + 12q^{52} - 10q^{53} - 18q^{54} - 16q^{55} - 2q^{56} + 30q^{57} - 24q^{60} - 8q^{62} + 30q^{63} + 12q^{65} - 12q^{66} + 14q^{67} + 12q^{68} + 6q^{69} + 18q^{70} - 24q^{71} + 12q^{72} - 18q^{73} - 18q^{74} - 18q^{75} - 30q^{76} - 4q^{77} - 12q^{78} + 8q^{80} - 18q^{81} + 12q^{82} - 2q^{83} - 6q^{84} + 4q^{85} - 6q^{86} - 48q^{87} - 4q^{88} + 8q^{89} + 20q^{91} - 18q^{92} - 18q^{93} - 60q^{94} - 40q^{95} - 30q^{96} - 2q^{97} + 22q^{98} + 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{2}$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
52.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 1.36603i 1.50000 0.866025i 1.73205i −0.133975 + 2.23205i −3.23205 0.866025i 2.00000 + 1.73205i −0.366025 + 0.366025i 1.50000 2.59808i 3.23205 2.86603i
103.1 −1.36603 + 1.36603i 1.50000 + 0.866025i 1.73205i −0.133975 2.23205i −3.23205 + 0.866025i 2.00000 1.73205i −0.366025 0.366025i 1.50000 + 2.59808i 3.23205 + 2.86603i
178.1 0.366025 0.366025i 1.50000 0.866025i 1.73205i −1.86603 + 1.23205i 0.232051 0.866025i 2.00000 + 1.73205i 1.36603 + 1.36603i 1.50000 2.59808i −0.232051 + 1.13397i
292.1 0.366025 + 0.366025i 1.50000 + 0.866025i 1.73205i −1.86603 1.23205i 0.232051 + 0.866025i 2.00000 1.73205i 1.36603 1.36603i 1.50000 + 2.59808i −0.232051 1.13397i
 $$n$$: 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
315.bs even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bs.c yes 4
3.b odd 2 1 945.2.bv.d 4
5.c odd 4 1 315.2.bs.b 4
7.d odd 6 1 315.2.cg.a yes 4
9.c even 3 1 315.2.cg.c yes 4
9.d odd 6 1 945.2.cj.a 4
15.e even 4 1 945.2.bv.a 4
21.g even 6 1 945.2.cj.d 4
35.k even 12 1 315.2.cg.c yes 4
45.k odd 12 1 315.2.cg.a yes 4
45.l even 12 1 945.2.cj.d 4
63.i even 6 1 945.2.bv.a 4
63.t odd 6 1 315.2.bs.b 4
105.w odd 12 1 945.2.cj.a 4
315.bs even 12 1 inner 315.2.bs.c yes 4
315.bu odd 12 1 945.2.bv.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.b 4 5.c odd 4 1
315.2.bs.b 4 63.t odd 6 1
315.2.bs.c yes 4 1.a even 1 1 trivial
315.2.bs.c yes 4 315.bs even 12 1 inner
315.2.cg.a yes 4 7.d odd 6 1
315.2.cg.a yes 4 45.k odd 12 1
315.2.cg.c yes 4 9.c even 3 1
315.2.cg.c yes 4 35.k even 12 1
945.2.bv.a 4 15.e even 4 1
945.2.bv.a 4 63.i even 6 1
945.2.bv.d 4 3.b odd 2 1
945.2.bv.d 4 315.bu odd 12 1
945.2.cj.a 4 9.d odd 6 1
945.2.cj.a 4 105.w odd 12 1
945.2.cj.d 4 21.g even 6 1
945.2.cj.d 4 45.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{11}^{4} - 4 T_{11}^{3} + 24 T_{11}^{2} + 32 T_{11} + 64$$ $$T_{13}^{4} - 4 T_{13}^{3} + 20 T_{13}^{2} - 32 T_{13} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ $$( 1 - 3 T + 3 T^{2} )^{2}$$
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$1 - 4 T + 2 T^{2} + 32 T^{3} - 101 T^{4} + 352 T^{5} + 242 T^{6} - 5324 T^{7} + 14641 T^{8}$$
$13$ $$1 - 4 T + 20 T^{2} - 84 T^{3} + 263 T^{4} - 1092 T^{5} + 3380 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 - 4 T + 8 T^{2} + 104 T^{3} - 497 T^{4} + 1768 T^{5} + 2312 T^{6} - 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 10 T + 40 T^{2} - 220 T^{3} + 1339 T^{4} - 4180 T^{5} + 14440 T^{6} - 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T + 17 T^{2} - 142 T^{3} - 224 T^{4} - 3266 T^{5} + 8993 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 20 T^{2} - 678 T^{4} - 19220 T^{6} + 923521 T^{8}$$
$37$ $$1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 12 T + 106 T^{2} - 696 T^{3} + 3651 T^{4} - 28536 T^{5} + 178186 T^{6} - 827052 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 12 T + 45 T^{2} - 204 T^{3} - 3316 T^{4} - 8772 T^{5} + 83205 T^{6} + 954084 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 6 T + 18 T^{2} + 420 T^{3} - 3889 T^{4} + 19740 T^{5} + 39762 T^{6} - 622938 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 10 T + 74 T^{2} + 520 T^{3} + 2551 T^{4} + 27560 T^{5} + 207866 T^{6} + 1488770 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$1 - 26 T^{2} + 7179 T^{4} - 96746 T^{6} + 13845841 T^{8}$$
$67$ $$1 - 14 T + 98 T^{2} - 1092 T^{3} + 12047 T^{4} - 73164 T^{5} + 439922 T^{6} - 4210682 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{4}$$
$73$ $$1 + 18 T + 90 T^{2} - 840 T^{3} - 14929 T^{4} - 61320 T^{5} + 479610 T^{6} + 7002306 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 164 T^{2} + 16134 T^{4} - 1023524 T^{6} + 38950081 T^{8}$$
$83$ $$1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 8 T - 103 T^{2} + 88 T^{3} + 14272 T^{4} + 7832 T^{5} - 815863 T^{6} - 5639752 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 2 T + 50 T^{2} - 1188 T^{3} - 4465 T^{4} - 115236 T^{5} + 470450 T^{6} + 1825346 T^{7} + 88529281 T^{8}$$