# Properties

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

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

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{2}$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.86603 0.500000i 1.73205 1.50000 0.866025i 2.00000 1.00000i 3.23205 0.866025i −2.50000 + 0.866025i −0.366025 + 0.366025i 3.00000 3.23205 2.86603i
187.1 0.133975 0.500000i −1.73205 1.50000 + 0.866025i 2.00000 1.00000i −0.232051 + 0.866025i −2.50000 0.866025i 1.36603 1.36603i 3.00000 −0.232051 1.13397i
283.1 0.133975 + 0.500000i −1.73205 1.50000 0.866025i 2.00000 + 1.00000i −0.232051 0.866025i −2.50000 + 0.866025i 1.36603 + 1.36603i 3.00000 −0.232051 + 1.13397i
313.1 1.86603 + 0.500000i 1.73205 1.50000 + 0.866025i 2.00000 + 1.00000i 3.23205 + 0.866025i −2.50000 0.866025i −0.366025 0.366025i 3.00000 3.23205 + 2.86603i
 $$n$$: 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.cg 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.cg.c yes 4
3.b odd 2 1 945.2.cj.a 4
5.c odd 4 1 315.2.cg.a yes 4
7.d odd 6 1 315.2.bs.b 4
9.c even 3 1 315.2.bs.c yes 4
9.d odd 6 1 945.2.bv.d 4
15.e even 4 1 945.2.cj.d 4
21.g even 6 1 945.2.bv.a 4
35.k even 12 1 315.2.bs.c yes 4
45.k odd 12 1 315.2.bs.b 4
45.l even 12 1 945.2.bv.a 4
63.k odd 6 1 315.2.cg.a yes 4
63.s even 6 1 945.2.cj.d 4
105.w odd 12 1 945.2.bv.d 4
315.bw odd 12 1 945.2.cj.a 4
315.cg even 12 1 inner 315.2.cg.c yes 4

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

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{11}^{2} + 4 T_{11} - 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 - 4 T + 5 T^{2} )^{2}$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 + 8 T + 20 T^{2} - 36 T^{3} - 361 T^{4} - 468 T^{5} + 3380 T^{6} + 17576 T^{7} + 28561 T^{8}$$
$17$ $$1 - 4 T + 8 T^{2} + 104 T^{3} - 497 T^{4} + 1768 T^{5} + 2312 T^{6} - 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 10 T + 40 T^{2} - 220 T^{3} + 1339 T^{4} - 4180 T^{5} + 14440 T^{6} - 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 - 10 T + 50 T^{2} - 340 T^{3} + 2191 T^{4} - 7820 T^{5} + 26450 T^{6} - 121670 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 12 T + 77 T^{2} - 348 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 35712 T^{5} + 165292 T^{6} + 536238 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 28536 T^{5} + 178186 T^{6} + 827052 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 6 T + 45 T^{2} - 366 T^{3} + 1328 T^{4} - 15738 T^{5} + 83205 T^{6} - 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 24 T + 369 T^{2} - 3792 T^{3} + 30092 T^{4} - 178224 T^{5} + 815121 T^{6} - 2491752 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 10 T + 74 T^{2} + 520 T^{3} + 2551 T^{4} + 27560 T^{5} + 207866 T^{6} + 1488770 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8}$$
$61$ $$1 + 36 T + 661 T^{2} + 8244 T^{3} + 75072 T^{4} + 502884 T^{5} + 2459581 T^{6} + 8171316 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 16 T + 65 T^{2} - 960 T^{3} - 14284 T^{4} - 64320 T^{5} + 291785 T^{6} + 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{4}$$
$73$ $$1 + 18 T + 90 T^{2} - 840 T^{3} - 14929 T^{4} - 61320 T^{5} + 479610 T^{6} + 7002306 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 12 T + 154 T^{2} + 1272 T^{3} + 8787 T^{4} + 100488 T^{5} + 961114 T^{6} + 5916468 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 8 T - 103 T^{2} + 88 T^{3} + 14272 T^{4} + 7832 T^{5} - 815863 T^{6} - 5639752 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 14 T + 50 T^{2} - 1500 T^{3} - 23089 T^{4} - 145500 T^{5} + 470450 T^{6} + 12777422 T^{7} + 88529281 T^{8}$$