# Properties

 Label 315.3.h.a Level $315$ Weight $3$ Character orbit 315.h Analytic conductor $8.583$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,3,Mod(181,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.181");

S:= CuspForms(chi, 3);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + \beta q^{5} + ( - 3 \beta - 2) q^{7} + 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + b * q^5 + (-3*b - 2) * q^7 + 8 * q^8 $$q - 2 q^{2} + \beta q^{5} + ( - 3 \beta - 2) q^{7} + 8 q^{8} - 2 \beta q^{10} + q^{11} + 9 \beta q^{13} + (6 \beta + 4) q^{14} - 16 q^{16} - 3 \beta q^{17} - 6 \beta q^{19} - 2 q^{22} - 8 q^{23} - 5 q^{25} - 18 \beta q^{26} - 41 q^{29} - 18 \beta q^{31} + 6 \beta q^{34} + ( - 2 \beta + 15) q^{35} - 28 q^{37} + 12 \beta q^{38} + 8 \beta q^{40} + 6 \beta q^{41} - 82 q^{43} + 16 q^{46} + 9 \beta q^{47} + (12 \beta - 41) q^{49} + 10 q^{50} - 74 q^{53} + \beta q^{55} + ( - 24 \beta - 16) q^{56} + 82 q^{58} - 42 \beta q^{59} - 36 \beta q^{61} + 36 \beta q^{62} + 64 q^{64} - 45 q^{65} + 2 q^{67} + (4 \beta - 30) q^{70} - 14 q^{71} + 30 \beta q^{73} + 56 q^{74} + ( - 3 \beta - 2) q^{77} - 19 q^{79} - 16 \beta q^{80} - 12 \beta q^{82} + 42 \beta q^{83} + 15 q^{85} + 164 q^{86} + 8 q^{88} + 48 \beta q^{89} + ( - 18 \beta + 135) q^{91} - 18 \beta q^{94} + 30 q^{95} + 27 \beta q^{97} + ( - 24 \beta + 82) q^{98} +O(q^{100})$$ q - 2 * q^2 + b * q^5 + (-3*b - 2) * q^7 + 8 * q^8 - 2*b * q^10 + q^11 + 9*b * q^13 + (6*b + 4) * q^14 - 16 * q^16 - 3*b * q^17 - 6*b * q^19 - 2 * q^22 - 8 * q^23 - 5 * q^25 - 18*b * q^26 - 41 * q^29 - 18*b * q^31 + 6*b * q^34 + (-2*b + 15) * q^35 - 28 * q^37 + 12*b * q^38 + 8*b * q^40 + 6*b * q^41 - 82 * q^43 + 16 * q^46 + 9*b * q^47 + (12*b - 41) * q^49 + 10 * q^50 - 74 * q^53 + b * q^55 + (-24*b - 16) * q^56 + 82 * q^58 - 42*b * q^59 - 36*b * q^61 + 36*b * q^62 + 64 * q^64 - 45 * q^65 + 2 * q^67 + (4*b - 30) * q^70 - 14 * q^71 + 30*b * q^73 + 56 * q^74 + (-3*b - 2) * q^77 - 19 * q^79 - 16*b * q^80 - 12*b * q^82 + 42*b * q^83 + 15 * q^85 + 164 * q^86 + 8 * q^88 + 48*b * q^89 + (-18*b + 135) * q^91 - 18*b * q^94 + 30 * q^95 + 27*b * q^97 + (-24*b + 82) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 4 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 4 * q^2 - 4 * q^7 + 16 * q^8 $$2 q - 4 q^{2} - 4 q^{7} + 16 q^{8} + 2 q^{11} + 8 q^{14} - 32 q^{16} - 4 q^{22} - 16 q^{23} - 10 q^{25} - 82 q^{29} + 30 q^{35} - 56 q^{37} - 164 q^{43} + 32 q^{46} - 82 q^{49} + 20 q^{50} - 148 q^{53} - 32 q^{56} + 164 q^{58} + 128 q^{64} - 90 q^{65} + 4 q^{67} - 60 q^{70} - 28 q^{71} + 112 q^{74} - 4 q^{77} - 38 q^{79} + 30 q^{85} + 328 q^{86} + 16 q^{88} + 270 q^{91} + 60 q^{95} + 164 q^{98}+O(q^{100})$$ 2 * q - 4 * q^2 - 4 * q^7 + 16 * q^8 + 2 * q^11 + 8 * q^14 - 32 * q^16 - 4 * q^22 - 16 * q^23 - 10 * q^25 - 82 * q^29 + 30 * q^35 - 56 * q^37 - 164 * q^43 + 32 * q^46 - 82 * q^49 + 20 * q^50 - 148 * q^53 - 32 * q^56 + 164 * q^58 + 128 * q^64 - 90 * q^65 + 4 * q^67 - 60 * q^70 - 28 * q^71 + 112 * q^74 - 4 * q^77 - 38 * q^79 + 30 * q^85 + 328 * q^86 + 16 * q^88 + 270 * q^91 + 60 * q^95 + 164 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
−2.00000 0 0 2.23607i 0 −2.00000 + 6.70820i 8.00000 0 4.47214i
181.2 −2.00000 0 0 2.23607i 0 −2.00000 6.70820i 8.00000 0 4.47214i
 $$n$$: 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.a 2
3.b odd 2 1 35.3.d.b 2
7.b odd 2 1 inner 315.3.h.a 2
12.b even 2 1 560.3.f.b 2
15.d odd 2 1 175.3.d.c 2
15.e even 4 2 175.3.c.c 4
21.c even 2 1 35.3.d.b 2
21.g even 6 2 245.3.h.a 4
21.h odd 6 2 245.3.h.a 4
84.h odd 2 1 560.3.f.b 2
105.g even 2 1 175.3.d.c 2
105.k odd 4 2 175.3.c.c 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5$$
$7$ $$T^{2} + 4T + 49$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 405$$
$17$ $$T^{2} + 45$$
$19$ $$T^{2} + 180$$
$23$ $$(T + 8)^{2}$$
$29$ $$(T + 41)^{2}$$
$31$ $$T^{2} + 1620$$
$37$ $$(T + 28)^{2}$$
$41$ $$T^{2} + 180$$
$43$ $$(T + 82)^{2}$$
$47$ $$T^{2} + 405$$
$53$ $$(T + 74)^{2}$$
$59$ $$T^{2} + 8820$$
$61$ $$T^{2} + 6480$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} + 4500$$
$79$ $$(T + 19)^{2}$$
$83$ $$T^{2} + 8820$$
$89$ $$T^{2} + 11520$$
$97$ $$T^{2} + 3645$$