# Properties

 Label 315.3.h.b Level $315$ Weight $3$ Character orbit 315.h Analytic conductor $8.583$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 315.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{4} + \beta q^{5} + 7 q^{7} - 7 q^{8}+O(q^{10})$$ q + q^2 - 3 * q^4 + b * q^5 + 7 * q^7 - 7 * q^8 $$q + q^{2} - 3 q^{4} + \beta q^{5} + 7 q^{7} - 7 q^{8} + \beta q^{10} - 2 q^{11} + 6 \beta q^{13} + 7 q^{14} + 5 q^{16} + 12 \beta q^{17} + 6 \beta q^{19} - 3 \beta q^{20} - 2 q^{22} - 26 q^{23} - 5 q^{25} + 6 \beta q^{26} - 21 q^{28} + 22 q^{29} + 24 \beta q^{31} + 33 q^{32} + 12 \beta q^{34} + 7 \beta q^{35} + 14 q^{37} + 6 \beta q^{38} - 7 \beta q^{40} - 12 \beta q^{41} - 34 q^{43} + 6 q^{44} - 26 q^{46} - 12 \beta q^{47} + 49 q^{49} - 5 q^{50} - 18 \beta q^{52} + 34 q^{53} - 2 \beta q^{55} - 49 q^{56} + 22 q^{58} - 18 \beta q^{59} - 42 \beta q^{61} + 24 \beta q^{62} + 13 q^{64} - 30 q^{65} + 14 q^{67} - 36 \beta q^{68} + 7 \beta q^{70} - 62 q^{71} + 24 \beta q^{73} + 14 q^{74} - 18 \beta q^{76} - 14 q^{77} + 38 q^{79} + 5 \beta q^{80} - 12 \beta q^{82} + 18 \beta q^{83} - 60 q^{85} - 34 q^{86} + 14 q^{88} - 12 \beta q^{89} + 42 \beta q^{91} + 78 q^{92} - 12 \beta q^{94} - 30 q^{95} + 12 \beta q^{97} + 49 q^{98} +O(q^{100})$$ q + q^2 - 3 * q^4 + b * q^5 + 7 * q^7 - 7 * q^8 + b * q^10 - 2 * q^11 + 6*b * q^13 + 7 * q^14 + 5 * q^16 + 12*b * q^17 + 6*b * q^19 - 3*b * q^20 - 2 * q^22 - 26 * q^23 - 5 * q^25 + 6*b * q^26 - 21 * q^28 + 22 * q^29 + 24*b * q^31 + 33 * q^32 + 12*b * q^34 + 7*b * q^35 + 14 * q^37 + 6*b * q^38 - 7*b * q^40 - 12*b * q^41 - 34 * q^43 + 6 * q^44 - 26 * q^46 - 12*b * q^47 + 49 * q^49 - 5 * q^50 - 18*b * q^52 + 34 * q^53 - 2*b * q^55 - 49 * q^56 + 22 * q^58 - 18*b * q^59 - 42*b * q^61 + 24*b * q^62 + 13 * q^64 - 30 * q^65 + 14 * q^67 - 36*b * q^68 + 7*b * q^70 - 62 * q^71 + 24*b * q^73 + 14 * q^74 - 18*b * q^76 - 14 * q^77 + 38 * q^79 + 5*b * q^80 - 12*b * q^82 + 18*b * q^83 - 60 * q^85 - 34 * q^86 + 14 * q^88 - 12*b * q^89 + 42*b * q^91 + 78 * q^92 - 12*b * q^94 - 30 * q^95 + 12*b * q^97 + 49 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8 $$2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8} - 4 q^{11} + 14 q^{14} + 10 q^{16} - 4 q^{22} - 52 q^{23} - 10 q^{25} - 42 q^{28} + 44 q^{29} + 66 q^{32} + 28 q^{37} - 68 q^{43} + 12 q^{44} - 52 q^{46} + 98 q^{49} - 10 q^{50} + 68 q^{53} - 98 q^{56} + 44 q^{58} + 26 q^{64} - 60 q^{65} + 28 q^{67} - 124 q^{71} + 28 q^{74} - 28 q^{77} + 76 q^{79} - 120 q^{85} - 68 q^{86} + 28 q^{88} + 156 q^{92} - 60 q^{95} + 98 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8 - 4 * q^11 + 14 * q^14 + 10 * q^16 - 4 * q^22 - 52 * q^23 - 10 * q^25 - 42 * q^28 + 44 * q^29 + 66 * q^32 + 28 * q^37 - 68 * q^43 + 12 * q^44 - 52 * q^46 + 98 * q^49 - 10 * q^50 + 68 * q^53 - 98 * q^56 + 44 * q^58 + 26 * q^64 - 60 * q^65 + 28 * q^67 - 124 * q^71 + 28 * q^74 - 28 * q^77 + 76 * q^79 - 120 * q^85 - 68 * q^86 + 28 * q^88 + 156 * q^92 - 60 * q^95 + 98 * q^98

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 − 2.23607i 2.23607i
1.00000 0 −3.00000 2.23607i 0 7.00000 −7.00000 0 2.23607i
181.2 1.00000 0 −3.00000 2.23607i 0 7.00000 −7.00000 0 2.23607i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.h.b 2
3.b odd 2 1 35.3.d.a 2
7.b odd 2 1 inner 315.3.h.b 2
12.b even 2 1 560.3.f.a 2
15.d odd 2 1 175.3.d.f 2
15.e even 4 2 175.3.c.d 4
21.c even 2 1 35.3.d.a 2
21.g even 6 2 245.3.h.b 4
21.h odd 6 2 245.3.h.b 4
84.h odd 2 1 560.3.f.a 2
105.g even 2 1 175.3.d.f 2
105.k odd 4 2 175.3.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 3.b odd 2 1
35.3.d.a 2 21.c even 2 1
175.3.c.d 4 15.e even 4 2
175.3.c.d 4 105.k odd 4 2
175.3.d.f 2 15.d odd 2 1
175.3.d.f 2 105.g even 2 1
245.3.h.b 4 21.g even 6 2
245.3.h.b 4 21.h odd 6 2
315.3.h.b 2 1.a even 1 1 trivial
315.3.h.b 2 7.b odd 2 1 inner
560.3.f.a 2 12.b even 2 1
560.3.f.a 2 84.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5$$
$7$ $$(T - 7)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 180$$
$17$ $$T^{2} + 720$$
$19$ $$T^{2} + 180$$
$23$ $$(T + 26)^{2}$$
$29$ $$(T - 22)^{2}$$
$31$ $$T^{2} + 2880$$
$37$ $$(T - 14)^{2}$$
$41$ $$T^{2} + 720$$
$43$ $$(T + 34)^{2}$$
$47$ $$T^{2} + 720$$
$53$ $$(T - 34)^{2}$$
$59$ $$T^{2} + 1620$$
$61$ $$T^{2} + 8820$$
$67$ $$(T - 14)^{2}$$
$71$ $$(T + 62)^{2}$$
$73$ $$T^{2} + 2880$$
$79$ $$(T - 38)^{2}$$
$83$ $$T^{2} + 1620$$
$89$ $$T^{2} + 720$$
$97$ $$T^{2} + 720$$