# Properties

 Label 315.4.a.c 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: no (minimal twist has level 35) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 7 q^{4} + 5 q^{5} + 7 q^{7} + 15 q^{8}+O(q^{10})$$ q - q^2 - 7 * q^4 + 5 * q^5 + 7 * q^7 + 15 * q^8 $$q - q^{2} - 7 q^{4} + 5 q^{5} + 7 q^{7} + 15 q^{8} - 5 q^{10} - 12 q^{11} - 78 q^{13} - 7 q^{14} + 41 q^{16} + 94 q^{17} + 40 q^{19} - 35 q^{20} + 12 q^{22} - 32 q^{23} + 25 q^{25} + 78 q^{26} - 49 q^{28} + 50 q^{29} - 248 q^{31} - 161 q^{32} - 94 q^{34} + 35 q^{35} - 434 q^{37} - 40 q^{38} + 75 q^{40} - 402 q^{41} - 68 q^{43} + 84 q^{44} + 32 q^{46} - 536 q^{47} + 49 q^{49} - 25 q^{50} + 546 q^{52} - 22 q^{53} - 60 q^{55} + 105 q^{56} - 50 q^{58} + 560 q^{59} - 278 q^{61} + 248 q^{62} - 167 q^{64} - 390 q^{65} - 164 q^{67} - 658 q^{68} - 35 q^{70} - 672 q^{71} + 82 q^{73} + 434 q^{74} - 280 q^{76} - 84 q^{77} - 1000 q^{79} + 205 q^{80} + 402 q^{82} + 448 q^{83} + 470 q^{85} + 68 q^{86} - 180 q^{88} + 870 q^{89} - 546 q^{91} + 224 q^{92} + 536 q^{94} + 200 q^{95} + 1026 q^{97} - 49 q^{98}+O(q^{100})$$ q - q^2 - 7 * q^4 + 5 * q^5 + 7 * q^7 + 15 * q^8 - 5 * q^10 - 12 * q^11 - 78 * q^13 - 7 * q^14 + 41 * q^16 + 94 * q^17 + 40 * q^19 - 35 * q^20 + 12 * q^22 - 32 * q^23 + 25 * q^25 + 78 * q^26 - 49 * q^28 + 50 * q^29 - 248 * q^31 - 161 * q^32 - 94 * q^34 + 35 * q^35 - 434 * q^37 - 40 * q^38 + 75 * q^40 - 402 * q^41 - 68 * q^43 + 84 * q^44 + 32 * q^46 - 536 * q^47 + 49 * q^49 - 25 * q^50 + 546 * q^52 - 22 * q^53 - 60 * q^55 + 105 * q^56 - 50 * q^58 + 560 * q^59 - 278 * q^61 + 248 * q^62 - 167 * q^64 - 390 * q^65 - 164 * q^67 - 658 * q^68 - 35 * q^70 - 672 * q^71 + 82 * q^73 + 434 * q^74 - 280 * q^76 - 84 * q^77 - 1000 * q^79 + 205 * q^80 + 402 * q^82 + 448 * q^83 + 470 * q^85 + 68 * q^86 - 180 * q^88 + 870 * q^89 - 546 * q^91 + 224 * q^92 + 536 * q^94 + 200 * q^95 + 1026 * q^97 - 49 * 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
−1.00000 0 −7.00000 5.00000 0 7.00000 15.0000 0 −5.00000
 $$n$$: 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.c 1
3.b odd 2 1 35.4.a.a 1
5.b even 2 1 1575.4.a.g 1
7.b odd 2 1 2205.4.a.i 1
12.b even 2 1 560.4.a.p 1
15.d odd 2 1 175.4.a.a 1
15.e even 4 2 175.4.b.a 2
21.c even 2 1 245.4.a.d 1
21.g even 6 2 245.4.e.b 2
21.h odd 6 2 245.4.e.e 2
24.f even 2 1 2240.4.a.b 1
24.h odd 2 1 2240.4.a.bk 1
105.g even 2 1 1225.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.a 1 3.b odd 2 1
175.4.a.a 1 15.d odd 2 1
175.4.b.a 2 15.e even 4 2
245.4.a.d 1 21.c even 2 1
245.4.e.b 2 21.g even 6 2
245.4.e.e 2 21.h odd 6 2
315.4.a.c 1 1.a even 1 1 trivial
560.4.a.p 1 12.b even 2 1
1225.4.a.e 1 105.g even 2 1
1575.4.a.g 1 5.b even 2 1
2205.4.a.i 1 7.b odd 2 1
2240.4.a.b 1 24.f even 2 1
2240.4.a.bk 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 7$$
$11$ $$T + 12$$
$13$ $$T + 78$$
$17$ $$T - 94$$
$19$ $$T - 40$$
$23$ $$T + 32$$
$29$ $$T - 50$$
$31$ $$T + 248$$
$37$ $$T + 434$$
$41$ $$T + 402$$
$43$ $$T + 68$$
$47$ $$T + 536$$
$53$ $$T + 22$$
$59$ $$T - 560$$
$61$ $$T + 278$$
$67$ $$T + 164$$
$71$ $$T + 672$$
$73$ $$T - 82$$
$79$ $$T + 1000$$
$83$ $$T - 448$$
$89$ $$T - 870$$
$97$ $$T - 1026$$