# Properties

 Label 315.2.bb.b Level 315 Weight 2 Character orbit 315.bb Analytic conductor 2.515 Analytic rank 0 Dimension 24 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} - 12q^{10} - 36q^{19} + 12q^{25} - 60q^{31} + 96q^{40} - 24q^{46} + 36q^{49} + 48q^{61} + 48q^{64} - 48q^{70} - 60q^{79} - 72q^{85} + 60q^{91} + 48q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1 −1.28469 + 2.22514i 0 −2.30084 3.98517i 1.33613 + 1.79297i 0 2.64140 0.151755i 6.68468 0 −5.70613 + 0.669669i
89.2 −1.28469 + 2.22514i 0 −2.30084 3.98517i 2.22083 + 0.260635i 0 −2.64140 + 0.151755i 6.68468 0 −3.43302 + 4.60682i
89.3 −0.956572 + 1.65683i 0 −0.830062 1.43771i −2.17319 0.526555i 0 −1.11878 2.39757i −0.650234 0 2.95122 3.09692i
89.4 −0.956572 + 1.65683i 0 −0.830062 1.43771i −1.54260 1.61876i 0 1.11878 + 2.39757i −0.650234 0 4.15762 1.00738i
89.5 −0.659204 + 1.14177i 0 0.130901 + 0.226727i −0.729935 + 2.11357i 0 −2.12635 1.57437i −2.98198 0 −1.93205 2.22670i
89.6 −0.659204 + 1.14177i 0 0.130901 + 0.226727i 1.46544 1.68893i 0 2.12635 + 1.57437i −2.98198 0 0.962352 + 2.78655i
89.7 0.659204 1.14177i 0 0.130901 + 0.226727i −1.46544 + 1.68893i 0 2.12635 + 1.57437i 2.98198 0 0.962352 + 2.78655i
89.8 0.659204 1.14177i 0 0.130901 + 0.226727i 0.729935 2.11357i 0 −2.12635 1.57437i 2.98198 0 −1.93205 2.22670i
89.9 0.956572 1.65683i 0 −0.830062 1.43771i 1.54260 + 1.61876i 0 1.11878 + 2.39757i 0.650234 0 4.15762 1.00738i
89.10 0.956572 1.65683i 0 −0.830062 1.43771i 2.17319 + 0.526555i 0 −1.11878 2.39757i 0.650234 0 2.95122 3.09692i
89.11 1.28469 2.22514i 0 −2.30084 3.98517i −2.22083 0.260635i 0 −2.64140 + 0.151755i −6.68468 0 −3.43302 + 4.60682i
89.12 1.28469 2.22514i 0 −2.30084 3.98517i −1.33613 1.79297i 0 2.64140 0.151755i −6.68468 0 −5.70613 + 0.669669i
269.1 −1.28469 2.22514i 0 −2.30084 + 3.98517i 1.33613 1.79297i 0 2.64140 + 0.151755i 6.68468 0 −5.70613 0.669669i
269.2 −1.28469 2.22514i 0 −2.30084 + 3.98517i 2.22083 0.260635i 0 −2.64140 0.151755i 6.68468 0 −3.43302 4.60682i
269.3 −0.956572 1.65683i 0 −0.830062 + 1.43771i −2.17319 + 0.526555i 0 −1.11878 + 2.39757i −0.650234 0 2.95122 + 3.09692i
269.4 −0.956572 1.65683i 0 −0.830062 + 1.43771i −1.54260 + 1.61876i 0 1.11878 2.39757i −0.650234 0 4.15762 + 1.00738i
269.5 −0.659204 1.14177i 0 0.130901 0.226727i −0.729935 2.11357i 0 −2.12635 + 1.57437i −2.98198 0 −1.93205 + 2.22670i
269.6 −0.659204 1.14177i 0 0.130901 0.226727i 1.46544 + 1.68893i 0 2.12635 1.57437i −2.98198 0 0.962352 2.78655i
269.7 0.659204 + 1.14177i 0 0.130901 0.226727i −1.46544 1.68893i 0 2.12635 1.57437i 2.98198 0 0.962352 2.78655i
269.8 0.659204 + 1.14177i 0 0.130901 0.226727i 0.729935 + 2.11357i 0 −2.12635 + 1.57437i 2.98198 0 −1.93205 + 2.22670i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p 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.bb.b 24
3.b odd 2 1 inner 315.2.bb.b 24
5.b even 2 1 inner 315.2.bb.b 24
5.c odd 4 2 1575.2.bk.i 24
7.c even 3 1 2205.2.g.b 24
7.d odd 6 1 inner 315.2.bb.b 24
7.d odd 6 1 2205.2.g.b 24
15.d odd 2 1 inner 315.2.bb.b 24
15.e even 4 2 1575.2.bk.i 24
21.g even 6 1 inner 315.2.bb.b 24
21.g even 6 1 2205.2.g.b 24
21.h odd 6 1 2205.2.g.b 24
35.i odd 6 1 inner 315.2.bb.b 24
35.i odd 6 1 2205.2.g.b 24
35.j even 6 1 2205.2.g.b 24
35.k even 12 2 1575.2.bk.i 24
105.o odd 6 1 2205.2.g.b 24
105.p even 6 1 inner 315.2.bb.b 24
105.p even 6 1 2205.2.g.b 24
105.w odd 12 2 1575.2.bk.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.b 24 1.a even 1 1 trivial
315.2.bb.b 24 3.b odd 2 1 inner
315.2.bb.b 24 5.b even 2 1 inner
315.2.bb.b 24 7.d odd 6 1 inner
315.2.bb.b 24 15.d odd 2 1 inner
315.2.bb.b 24 21.g even 6 1 inner
315.2.bb.b 24 35.i odd 6 1 inner
315.2.bb.b 24 105.p even 6 1 inner
1575.2.bk.i 24 5.c odd 4 2
1575.2.bk.i 24 15.e even 4 2
1575.2.bk.i 24 35.k even 12 2
1575.2.bk.i 24 105.w odd 12 2
2205.2.g.b 24 7.c even 3 1
2205.2.g.b 24 7.d odd 6 1
2205.2.g.b 24 21.g even 6 1
2205.2.g.b 24 21.h odd 6 1
2205.2.g.b 24 35.i odd 6 1
2205.2.g.b 24 35.j even 6 1
2205.2.g.b 24 105.o odd 6 1
2205.2.g.b 24 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 12 T_{2}^{10} + 102 T_{2}^{8} + 420 T_{2}^{6} + 1260 T_{2}^{4} + 1764 T_{2}^{2} + 1764$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database