# Properties

 Label 315.2.bz.a Level 315 Weight 2 Character orbit 315.bz Analytic conductor 2.515 Analytic rank 1 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.bz (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( -1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( -3 + \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( -1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( -3 + \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{8} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{10} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -2 - 2 \zeta_{12}^{3} ) q^{13} + ( 2 - \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( -2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{16} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( -2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{20} + ( 1 + \zeta_{12}^{3} ) q^{22} + ( -3 - 4 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( 4 + \zeta_{12}^{2} ) q^{28} + 3 \zeta_{12}^{3} q^{29} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{31} + ( 5 + 5 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{32} + 2 q^{34} + ( 2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{35} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + ( 3 + 3 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{38} + ( 3 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{41} + ( -4 - 5 \zeta_{12} + 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{43} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{44} + ( 7 + 4 \zeta_{12} - 7 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{46} + ( -4 + \zeta_{12} + 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( 8 - 5 \zeta_{12}^{2} ) q^{49} + ( -4 - 7 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{50} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{52} + ( -5 + 5 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{53} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{55} + ( -1 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{56} + ( -3 - 3 \zeta_{12} ) q^{58} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( -4 - 5 \zeta_{12} + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{61} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{62} + ( -1 + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{64} + ( -2 \zeta_{12} + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{65} + ( -1 - \zeta_{12} + 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{68} + ( -5 - 3 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{70} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{71} + ( 4 + 4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( -2 - 6 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{74} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{76} + ( 2 - \zeta_{12} - 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{77} + ( -2 - 5 \zeta_{12} + \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{79} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{80} + ( -1 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{82} + ( 1 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{83} + ( -2 - 2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( -4 \zeta_{12} - 9 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{86} + ( -3 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( -5 \zeta_{12} - 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{89} + ( 6 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{91} + ( 2 + 5 \zeta_{12} + 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{92} + ( 3 - \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} + ( -1 - \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{95} + ( 5 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{97} + ( -3 + 5 \zeta_{12} + 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 6q^{4} - 4q^{5} - 10q^{7} + 2q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 6q^{4} - 4q^{5} - 10q^{7} + 2q^{8} + 6q^{10} - 2q^{11} - 8q^{13} + 2q^{14} - 2q^{16} - 8q^{17} - 2q^{19} + 4q^{22} - 14q^{23} - 6q^{25} + 12q^{26} + 18q^{28} - 12q^{31} + 12q^{32} + 8q^{34} + 16q^{35} - 12q^{37} + 10q^{38} + 8q^{40} - 6q^{43} + 6q^{44} + 14q^{46} - 6q^{47} + 22q^{49} - 14q^{50} + 12q^{52} - 10q^{53} + 8q^{55} - 8q^{56} - 12q^{58} + 6q^{59} - 12q^{61} + 4q^{62} + 12q^{65} + 8q^{67} + 12q^{68} - 12q^{70} - 12q^{71} - 12q^{74} + 2q^{77} - 6q^{79} + 8q^{80} - 2q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 16q^{89} + 20q^{91} + 18q^{92} + 6q^{94} + 2q^{95} + 4q^{97} + 4q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
−0.500000 0.133975i 0 −1.50000 0.866025i −0.133975 2.23205i 0 −2.50000 + 0.866025i 1.36603 + 1.36603i 0 −0.232051 + 1.13397i
82.1 −0.500000 + 0.133975i 0 −1.50000 + 0.866025i −0.133975 + 2.23205i 0 −2.50000 0.866025i 1.36603 1.36603i 0 −0.232051 1.13397i
208.1 −0.500000 1.86603i 0 −1.50000 + 0.866025i −1.86603 + 1.23205i 0 −2.50000 0.866025i −0.366025 0.366025i 0 3.23205 + 2.86603i
262.1 −0.500000 + 1.86603i 0 −1.50000 0.866025i −1.86603 1.23205i 0 −2.50000 + 0.866025i −0.366025 + 0.366025i 0 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
35.k 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.bz.a 4
3.b odd 2 1 35.2.k.b yes 4
5.c odd 4 1 315.2.bz.b 4
7.d odd 6 1 315.2.bz.b 4
12.b even 2 1 560.2.ci.b 4
15.d odd 2 1 175.2.o.a 4
15.e even 4 1 35.2.k.a 4
15.e even 4 1 175.2.o.b 4
21.c even 2 1 245.2.l.b 4
21.g even 6 1 35.2.k.a 4
21.g even 6 1 245.2.f.a 4
21.h odd 6 1 245.2.f.b 4
21.h odd 6 1 245.2.l.a 4
35.k even 12 1 inner 315.2.bz.a 4
60.l odd 4 1 560.2.ci.a 4
84.j odd 6 1 560.2.ci.a 4
105.k odd 4 1 245.2.l.a 4
105.p even 6 1 175.2.o.b 4
105.w odd 12 1 35.2.k.b yes 4
105.w odd 12 1 175.2.o.a 4
105.w odd 12 1 245.2.f.b 4
105.x even 12 1 245.2.f.a 4
105.x even 12 1 245.2.l.b 4
420.br even 12 1 560.2.ci.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.k.a 4 15.e even 4 1
35.2.k.a 4 21.g even 6 1
35.2.k.b yes 4 3.b odd 2 1
35.2.k.b yes 4 105.w odd 12 1
175.2.o.a 4 15.d odd 2 1
175.2.o.a 4 105.w odd 12 1
175.2.o.b 4 15.e even 4 1
175.2.o.b 4 105.p even 6 1
245.2.f.a 4 21.g even 6 1
245.2.f.a 4 105.x even 12 1
245.2.f.b 4 21.h odd 6 1
245.2.f.b 4 105.w odd 12 1
245.2.l.a 4 21.h odd 6 1
245.2.l.a 4 105.k odd 4 1
245.2.l.b 4 21.c even 2 1
245.2.l.b 4 105.x even 12 1
315.2.bz.a 4 1.a even 1 1 trivial
315.2.bz.a 4 35.k even 12 1 inner
315.2.bz.b 4 5.c odd 4 1
315.2.bz.b 4 7.d odd 6 1
560.2.ci.a 4 60.l odd 4 1
560.2.ci.a 4 84.j odd 6 1
560.2.ci.b 4 12.b even 2 1
560.2.ci.b 4 420.br even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} + 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 884 T^{5} + 5780 T^{6} + 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 5198 T^{5} + 28037 T^{6} + 170338 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$
$31$ $$1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8}$$
$37$ $$1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8}$$
$43$ $$1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 31584 T^{5} + 198810 T^{6} + 622938 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} )$$
$59$ $$1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 6 T + 148 T^{2} + 426 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8}$$
$79$ $$1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} )$$
$97$ $$1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8}$$