# Properties

 Label 315.2.bl.b Level 315 Weight 2 Character orbit 315.bl Analytic conductor 2.515 Analytic rank 1 Dimension 2 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.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 1.73205i 0.500000 0.866025i −0.500000 + 0.866025i −1.50000 2.59808i 0.500000 2.59808i 1.73205i −3.00000 1.73205i
146.1 −1.50000 0.866025i 1.73205i 0.500000 + 0.866025i −0.500000 0.866025i −1.50000 + 2.59808i 0.500000 + 2.59808i 1.73205i −3.00000 1.73205i
 $$n$$: 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
63.o 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.bl.b 2
3.b odd 2 1 945.2.bl.h 2
7.b odd 2 1 315.2.bl.c yes 2
9.c even 3 1 945.2.bl.e 2
9.d odd 6 1 315.2.bl.c yes 2
21.c even 2 1 945.2.bl.e 2
63.l odd 6 1 945.2.bl.h 2
63.o even 6 1 inner 315.2.bl.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bl.b 2 1.a even 1 1 trivial
315.2.bl.b 2 63.o even 6 1 inner
315.2.bl.c yes 2 7.b odd 2 1
315.2.bl.c yes 2 9.d odd 6 1
945.2.bl.e 2 9.c even 3 1
945.2.bl.e 2 21.c even 2 1
945.2.bl.h 2 3.b odd 2 1
945.2.bl.h 2 63.l odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 3 T_{2} + 3$$ $$T_{11}^{2} + 9 T_{11} + 27$$ $$T_{13}^{2} + 9 T_{13} + 27$$ $$T_{17} - 3$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$( 1 - 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}$$
$31$ $$1 - 12 T + 79 T^{2} - 372 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 58 T^{2} + 2809 T^{4}$$
$59$ $$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4}$$
$71$ $$1 - 115 T^{2} + 5041 T^{4}$$
$73$ $$1 - 71 T^{2} + 5329 T^{4}$$
$79$ $$1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4}$$
$83$ $$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$1 + 21 T + 244 T^{2} + 2037 T^{3} + 9409 T^{4}$$