# Properties

 Label 315.2.z.a Level 315 Weight 2 Character orbit 315.z Analytic conductor 2.515 Analytic rank 0 Dimension 8 CM discriminant -35 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.31116960000.2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + ( 2 + 2 \beta_{4} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{2} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} + ( 2 + 2 \beta_{4} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{2} - \beta_{7} ) q^{9} + ( 3 + \beta_{2} + \beta_{4} ) q^{11} -2 \beta_{1} q^{12} + ( -\beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{13} + ( 3 + 3 \beta_{4} + \beta_{7} ) q^{15} + 4 \beta_{4} q^{16} + ( 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{20} + ( \beta_{2} + 3 \beta_{4} ) q^{21} -5 \beta_{4} q^{25} + ( 3 \beta_{3} + \beta_{5} ) q^{27} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{28} + ( -2 \beta_{2} + \beta_{4} ) q^{29} + ( -2 \beta_{1} - 3 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{33} + ( -1 - 2 \beta_{2} - 2 \beta_{7} ) q^{35} -2 \beta_{7} q^{36} + ( 3 + 3 \beta_{2} - 6 \beta_{4} + 2 \beta_{7} ) q^{39} + ( 4 + 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{7} ) q^{44} + ( -2 \beta_{1} - 3 \beta_{6} ) q^{45} + ( \beta_{1} + \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{47} + ( -4 \beta_{1} - 4 \beta_{5} ) q^{48} + ( -7 - 7 \beta_{4} ) q^{49} + ( -6 - 2 \beta_{2} - 9 \beta_{4} + \beta_{7} ) q^{51} + ( 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{52} + ( 3 \beta_{1} + 4 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{55} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{60} + ( -4 \beta_{1} - 3 \beta_{3} - 4 \beta_{5} + 3 \beta_{6} ) q^{63} -8 q^{64} + ( -15 + \beta_{2} - 8 \beta_{4} ) q^{65} + ( 2 \beta_{1} - 6 \beta_{3} + 6 \beta_{5} + 4 \beta_{6} ) q^{68} + ( -5 + 2 \beta_{2} - 12 \beta_{4} + 2 \beta_{7} ) q^{71} + ( -6 \beta_{1} - 5 \beta_{3} - \beta_{5} + 6 \beta_{6} ) q^{73} + ( 5 \beta_{1} + 5 \beta_{5} ) q^{75} + ( 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - 5 \beta_{6} ) q^{77} + ( -3 - 3 \beta_{2} - \beta_{4} - 6 \beta_{7} ) q^{79} + ( 4 \beta_{3} + 4 \beta_{5} ) q^{80} + ( -9 - \beta_{2} - \beta_{7} ) q^{81} + ( \beta_{1} + \beta_{3} - 5 \beta_{5} + 5 \beta_{6} ) q^{83} + ( -6 + 2 \beta_{2} + 2 \beta_{7} ) q^{84} + ( 4 + 6 \beta_{2} + \beta_{4} + 3 \beta_{7} ) q^{85} + ( \beta_{1} + 6 \beta_{3} + \beta_{5} - 6 \beta_{6} ) q^{87} + ( 5 - 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{91} + ( -5 \beta_{1} + 5 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -\beta_{2} - 9 \beta_{4} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + 4q^{9} + O(q^{10})$$ $$8q + 8q^{4} + 4q^{9} + 18q^{11} + 10q^{15} - 16q^{16} - 14q^{21} + 20q^{25} + 4q^{36} + 38q^{39} - 28q^{49} - 10q^{51} - 20q^{60} - 64q^{64} - 90q^{65} - 2q^{79} - 68q^{81} - 56q^{84} + 10q^{85} + 28q^{91} + 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} - 8 x^{4} + 9 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 8 \nu^{5} + 64 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} + 8 \nu^{2} - 81$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 8 \nu^{5} + 8 \nu^{3} - 81 \nu$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 17 \nu$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} - 8 \nu^{2} + 9$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 3 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 9 \beta_{4} + 9$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{6} + 8 \beta_{5} - 3 \beta_{3} + 8 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{7} + 8 \beta_{2} - 9$$ $$\nu^{7}$$ $$=$$ $$24 \beta_{6} - 17 \beta_{1}$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
104.1
 1.62968 − 0.586627i −0.306808 − 1.70466i 0.306808 + 1.70466i −1.62968 + 0.586627i −0.306808 + 1.70466i 1.62968 + 0.586627i −1.62968 − 0.586627i 0.306808 − 1.70466i
0 −1.32288 1.11803i 1.00000 1.73205i 1.93649 + 1.11803i 0 1.32288 + 2.29129i 0 0.500000 + 2.95804i 0
104.2 0 −1.32288 + 1.11803i 1.00000 1.73205i −1.93649 1.11803i 0 1.32288 + 2.29129i 0 0.500000 2.95804i 0
104.3 0 1.32288 1.11803i 1.00000 1.73205i 1.93649 + 1.11803i 0 −1.32288 2.29129i 0 0.500000 2.95804i 0
104.4 0 1.32288 + 1.11803i 1.00000 1.73205i −1.93649 1.11803i 0 −1.32288 2.29129i 0 0.500000 + 2.95804i 0
209.1 0 −1.32288 1.11803i 1.00000 + 1.73205i −1.93649 + 1.11803i 0 1.32288 2.29129i 0 0.500000 + 2.95804i 0
209.2 0 −1.32288 + 1.11803i 1.00000 + 1.73205i 1.93649 1.11803i 0 1.32288 2.29129i 0 0.500000 2.95804i 0
209.3 0 1.32288 1.11803i 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.32288 + 2.29129i 0 0.500000 2.95804i 0
209.4 0 1.32288 + 1.11803i 1.00000 + 1.73205i 1.93649 1.11803i 0 −1.32288 + 2.29129i 0 0.500000 + 2.95804i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.4 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.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
63.o even 6 1 inner
315.z 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.z.a 8
3.b odd 2 1 945.2.z.a 8
5.b even 2 1 inner 315.2.z.a 8
7.b odd 2 1 inner 315.2.z.a 8
9.c even 3 1 945.2.z.a 8
9.d odd 6 1 inner 315.2.z.a 8
15.d odd 2 1 945.2.z.a 8
21.c even 2 1 945.2.z.a 8
35.c odd 2 1 CM 315.2.z.a 8
45.h odd 6 1 inner 315.2.z.a 8
45.j even 6 1 945.2.z.a 8
63.l odd 6 1 945.2.z.a 8
63.o even 6 1 inner 315.2.z.a 8
105.g even 2 1 945.2.z.a 8
315.z even 6 1 inner 315.2.z.a 8
315.bg odd 6 1 945.2.z.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.z.a 8 1.a even 1 1 trivial
315.2.z.a 8 5.b even 2 1 inner
315.2.z.a 8 7.b odd 2 1 inner
315.2.z.a 8 9.d odd 6 1 inner
315.2.z.a 8 35.c odd 2 1 CM
315.2.z.a 8 45.h odd 6 1 inner
315.2.z.a 8 63.o even 6 1 inner
315.2.z.a 8 315.z even 6 1 inner
945.2.z.a 8 3.b odd 2 1
945.2.z.a 8 9.c even 3 1
945.2.z.a 8 15.d odd 2 1
945.2.z.a 8 21.c even 2 1
945.2.z.a 8 45.j even 6 1
945.2.z.a 8 63.l odd 6 1
945.2.z.a 8 105.g even 2 1
945.2.z.a 8 315.bg odd 6 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$
$3$ $$( 1 - T^{2} + 9 T^{4} )^{2}$$
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{4}( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 19 T^{2} + 169 T^{4} )^{2}( 1 - 19 T^{2} + 192 T^{4} - 3211 T^{6} + 28561 T^{8} )$$
$17$ $$( 1 + 29 T^{2} + 552 T^{4} + 8381 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$( 1 - 23 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 31 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{8}$$
$41$ $$( 1 - 41 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 43 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 31 T^{2} + 2209 T^{4} )^{2}( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} )$$
$53$ $$( 1 + 53 T^{2} )^{8}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 67 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 12 T + 73 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}( 1 + 12 T + 73 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + T + 79 T^{2} )^{4}( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 86 T^{2} + 6889 T^{4} )^{2}( 1 + 86 T^{2} + 507 T^{4} + 592454 T^{6} + 47458321 T^{8} )$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 - 149 T^{2} + 9409 T^{4} )^{2}( 1 + 149 T^{2} + 12792 T^{4} + 1401941 T^{6} + 88529281 T^{8} )$$