# Properties

 Label 315.2.cg.d 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.

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

## 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 1.50000 + 0.866025i 1.50000 0.866025i −1.00000 + 2.00000i 3.23205 + 0.866025i 0.500000 2.59808i −0.366025 + 0.366025i 1.50000 + 2.59808i −0.866025 + 4.23205i
187.1 0.133975 0.500000i 1.50000 0.866025i 1.50000 + 0.866025i −1.00000 + 2.00000i −0.232051 0.866025i 0.500000 + 2.59808i 1.36603 1.36603i 1.50000 2.59808i 0.866025 + 0.767949i
283.1 0.133975 + 0.500000i 1.50000 + 0.866025i 1.50000 0.866025i −1.00000 2.00000i −0.232051 + 0.866025i 0.500000 2.59808i 1.36603 + 1.36603i 1.50000 + 2.59808i 0.866025 0.767949i
313.1 1.86603 + 0.500000i 1.50000 0.866025i 1.50000 + 0.866025i −1.00000 2.00000i 3.23205 0.866025i 0.500000 + 2.59808i −0.366025 0.366025i 1.50000 2.59808i −0.866025 4.23205i
 $$n$$: 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.d yes 4
3.b odd 2 1 945.2.cj.b 4
5.c odd 4 1 315.2.cg.b yes 4
7.d odd 6 1 315.2.bs.d yes 4
9.c even 3 1 315.2.bs.a 4
9.d odd 6 1 945.2.bv.b 4
15.e even 4 1 945.2.cj.c 4
21.g even 6 1 945.2.bv.c 4
35.k even 12 1 315.2.bs.a 4
45.k odd 12 1 315.2.bs.d yes 4
45.l even 12 1 945.2.bv.c 4
63.k odd 6 1 315.2.cg.b yes 4
63.s even 6 1 945.2.cj.c 4
105.w odd 12 1 945.2.bv.b 4
315.bw odd 12 1 945.2.cj.b 4
315.cg even 12 1 inner 315.2.cg.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.a 4 9.c even 3 1
315.2.bs.a 4 35.k even 12 1
315.2.bs.d yes 4 7.d odd 6 1
315.2.bs.d yes 4 45.k odd 12 1
315.2.cg.b yes 4 5.c odd 4 1
315.2.cg.b yes 4 63.k odd 6 1
315.2.cg.d yes 4 1.a even 1 1 trivial
315.2.cg.d yes 4 315.cg even 12 1 inner
945.2.bv.b 4 9.d odd 6 1
945.2.bv.b 4 105.w odd 12 1
945.2.bv.c 4 21.g even 6 1
945.2.bv.c 4 45.l even 12 1
945.2.cj.b 4 3.b odd 2 1
945.2.cj.b 4 315.bw odd 12 1
945.2.cj.c 4 15.e even 4 1
945.2.cj.c 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 $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8}$$
$3$ $$( 1 - 3 T + 3 T^{2} )^{2}$$
$5$ $$( 1 + 2 T + 5 T^{2} )^{2}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{4}$$
$13$ $$1 - 10 T + 26 T^{2} + 108 T^{3} - 841 T^{4} + 1404 T^{5} + 4394 T^{6} - 21970 T^{7} + 28561 T^{8}$$
$17$ $$1 + 8 T + 80 T^{2} + 392 T^{3} + 2143 T^{4} + 6664 T^{5} + 23120 T^{6} + 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 - 4 T + 8 T^{2} - 4 T^{3} - 482 T^{4} - 92 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8}$$
$29$ $$1 + 24 T + 289 T^{2} + 2328 T^{3} + 14136 T^{4} + 67512 T^{5} + 243049 T^{6} + 585336 T^{7} + 707281 T^{8}$$
$31$ $$1 + 12 T + 118 T^{2} + 840 T^{3} + 5427 T^{4} + 26040 T^{5} + 113398 T^{6} + 357492 T^{7} + 923521 T^{8}$$
$37$ $$1 - 18 T + 90 T^{2} + 408 T^{3} - 6217 T^{4} + 15096 T^{5} + 123210 T^{6} - 911754 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 6 T + 9 T^{2} + 258 T^{3} - 2872 T^{4} + 11094 T^{5} + 16641 T^{6} - 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 6 T + 45 T^{2} + 378 T^{3} - 1816 T^{4} + 17766 T^{5} + 99405 T^{6} - 622938 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 4 T + 20 T^{2} + 244 T^{3} - 1097 T^{4} + 12932 T^{5} + 56180 T^{6} + 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 6 T + 56 T^{2} + 828 T^{3} - 5205 T^{4} + 48852 T^{5} + 194936 T^{6} - 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 24 T + 346 T^{2} - 3696 T^{3} + 31707 T^{4} - 225456 T^{5} + 1287466 T^{6} - 5447544 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 10 T + 74 T^{2} + 576 T^{3} + 2159 T^{4} + 38592 T^{5} + 332186 T^{6} + 3007630 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8}$$
$79$ $$1 + 12 T + 154 T^{2} + 1272 T^{3} + 8787 T^{4} + 100488 T^{5} + 961114 T^{6} + 5916468 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 26 T + 365 T^{2} + 3554 T^{3} + 32320 T^{4} + 294982 T^{5} + 2514485 T^{6} + 14866462 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 16 T + 62 T^{2} + 256 T^{3} + 6931 T^{4} + 22784 T^{5} + 491102 T^{6} + 11279504 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 34 T + 458 T^{2} - 2916 T^{3} + 15023 T^{4} - 282852 T^{5} + 4309322 T^{6} - 31030882 T^{7} + 88529281 T^{8}$$