# Properties

 Label 315.4.a.b Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## 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: yes Analytic conductor: $$18.5856016518$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} - 5 q^{5} + 7 q^{7} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 - 5 * q^5 + 7 * q^7 + 21 * q^8 $$q - 3 q^{2} + q^{4} - 5 q^{5} + 7 q^{7} + 21 q^{8} + 15 q^{10} - 60 q^{11} + 38 q^{13} - 21 q^{14} - 71 q^{16} + 84 q^{17} + 110 q^{19} - 5 q^{20} + 180 q^{22} - 120 q^{23} + 25 q^{25} - 114 q^{26} + 7 q^{28} - 162 q^{29} + 236 q^{31} + 45 q^{32} - 252 q^{34} - 35 q^{35} - 376 q^{37} - 330 q^{38} - 105 q^{40} + 126 q^{41} - 34 q^{43} - 60 q^{44} + 360 q^{46} + 6 q^{47} + 49 q^{49} - 75 q^{50} + 38 q^{52} - 582 q^{53} + 300 q^{55} + 147 q^{56} + 486 q^{58} - 492 q^{59} - 880 q^{61} - 708 q^{62} + 433 q^{64} - 190 q^{65} - 826 q^{67} + 84 q^{68} + 105 q^{70} + 666 q^{71} - 826 q^{73} + 1128 q^{74} + 110 q^{76} - 420 q^{77} - 592 q^{79} + 355 q^{80} - 378 q^{82} - 792 q^{83} - 420 q^{85} + 102 q^{86} - 1260 q^{88} - 1002 q^{89} + 266 q^{91} - 120 q^{92} - 18 q^{94} - 550 q^{95} + 1442 q^{97} - 147 q^{98}+O(q^{100})$$ q - 3 * q^2 + q^4 - 5 * q^5 + 7 * q^7 + 21 * q^8 + 15 * q^10 - 60 * q^11 + 38 * q^13 - 21 * q^14 - 71 * q^16 + 84 * q^17 + 110 * q^19 - 5 * q^20 + 180 * q^22 - 120 * q^23 + 25 * q^25 - 114 * q^26 + 7 * q^28 - 162 * q^29 + 236 * q^31 + 45 * q^32 - 252 * q^34 - 35 * q^35 - 376 * q^37 - 330 * q^38 - 105 * q^40 + 126 * q^41 - 34 * q^43 - 60 * q^44 + 360 * q^46 + 6 * q^47 + 49 * q^49 - 75 * q^50 + 38 * q^52 - 582 * q^53 + 300 * q^55 + 147 * q^56 + 486 * q^58 - 492 * q^59 - 880 * q^61 - 708 * q^62 + 433 * q^64 - 190 * q^65 - 826 * q^67 + 84 * q^68 + 105 * q^70 + 666 * q^71 - 826 * q^73 + 1128 * q^74 + 110 * q^76 - 420 * q^77 - 592 * q^79 + 355 * q^80 - 378 * q^82 - 792 * q^83 - 420 * q^85 + 102 * q^86 - 1260 * q^88 - 1002 * q^89 + 266 * q^91 - 120 * q^92 - 18 * q^94 - 550 * q^95 + 1442 * q^97 - 147 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−3.00000 0 1.00000 −5.00000 0 7.00000 21.0000 0 15.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$+1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.a.b 1
3.b odd 2 1 315.4.a.e yes 1
5.b even 2 1 1575.4.a.i 1
7.b odd 2 1 2205.4.a.f 1
15.d odd 2 1 1575.4.a.c 1
21.c even 2 1 2205.4.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.b 1 1.a even 1 1 trivial
315.4.a.e yes 1 3.b odd 2 1
1575.4.a.c 1 15.d odd 2 1
1575.4.a.i 1 5.b even 2 1
2205.4.a.f 1 7.b odd 2 1
2205.4.a.p 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 7$$
$11$ $$T + 60$$
$13$ $$T - 38$$
$17$ $$T - 84$$
$19$ $$T - 110$$
$23$ $$T + 120$$
$29$ $$T + 162$$
$31$ $$T - 236$$
$37$ $$T + 376$$
$41$ $$T - 126$$
$43$ $$T + 34$$
$47$ $$T - 6$$
$53$ $$T + 582$$
$59$ $$T + 492$$
$61$ $$T + 880$$
$67$ $$T + 826$$
$71$ $$T - 666$$
$73$ $$T + 826$$
$79$ $$T + 592$$
$83$ $$T + 792$$
$89$ $$T + 1002$$
$97$ $$T - 1442$$