# Properties

 Label 315.4.a.a Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 7 q^{7} - 45 q^{8}+O(q^{10})$$ q - 5 * q^2 + 17 * q^4 - 5 * q^5 + 7 * q^7 - 45 * q^8 $$q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 7 q^{7} - 45 q^{8} + 25 q^{10} - 12 q^{11} + 30 q^{13} - 35 q^{14} + 89 q^{16} + 134 q^{17} - 92 q^{19} - 85 q^{20} + 60 q^{22} - 112 q^{23} + 25 q^{25} - 150 q^{26} + 119 q^{28} + 58 q^{29} - 224 q^{31} - 85 q^{32} - 670 q^{34} - 35 q^{35} - 146 q^{37} + 460 q^{38} + 225 q^{40} - 18 q^{41} + 340 q^{43} - 204 q^{44} + 560 q^{46} - 208 q^{47} + 49 q^{49} - 125 q^{50} + 510 q^{52} + 754 q^{53} + 60 q^{55} - 315 q^{56} - 290 q^{58} - 380 q^{59} + 718 q^{61} + 1120 q^{62} - 287 q^{64} - 150 q^{65} + 412 q^{67} + 2278 q^{68} + 175 q^{70} + 960 q^{71} + 1066 q^{73} + 730 q^{74} - 1564 q^{76} - 84 q^{77} + 896 q^{79} - 445 q^{80} + 90 q^{82} - 436 q^{83} - 670 q^{85} - 1700 q^{86} + 540 q^{88} + 1038 q^{89} + 210 q^{91} - 1904 q^{92} + 1040 q^{94} + 460 q^{95} - 702 q^{97} - 245 q^{98}+O(q^{100})$$ q - 5 * q^2 + 17 * q^4 - 5 * q^5 + 7 * q^7 - 45 * q^8 + 25 * q^10 - 12 * q^11 + 30 * q^13 - 35 * q^14 + 89 * q^16 + 134 * q^17 - 92 * q^19 - 85 * q^20 + 60 * q^22 - 112 * q^23 + 25 * q^25 - 150 * q^26 + 119 * q^28 + 58 * q^29 - 224 * q^31 - 85 * q^32 - 670 * q^34 - 35 * q^35 - 146 * q^37 + 460 * q^38 + 225 * q^40 - 18 * q^41 + 340 * q^43 - 204 * q^44 + 560 * q^46 - 208 * q^47 + 49 * q^49 - 125 * q^50 + 510 * q^52 + 754 * q^53 + 60 * q^55 - 315 * q^56 - 290 * q^58 - 380 * q^59 + 718 * q^61 + 1120 * q^62 - 287 * q^64 - 150 * q^65 + 412 * q^67 + 2278 * q^68 + 175 * q^70 + 960 * q^71 + 1066 * q^73 + 730 * q^74 - 1564 * q^76 - 84 * q^77 + 896 * q^79 - 445 * q^80 + 90 * q^82 - 436 * q^83 - 670 * q^85 - 1700 * q^86 + 540 * q^88 + 1038 * q^89 + 210 * q^91 - 1904 * q^92 + 1040 * q^94 + 460 * q^95 - 702 * q^97 - 245 * 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
−5.00000 0 17.0000 −5.00000 0 7.00000 −45.0000 0 25.0000
 $$n$$: 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.a 1
3.b odd 2 1 105.4.a.b 1
5.b even 2 1 1575.4.a.l 1
7.b odd 2 1 2205.4.a.b 1
12.b even 2 1 1680.4.a.u 1
15.d odd 2 1 525.4.a.a 1
15.e even 4 2 525.4.d.a 2
21.c even 2 1 735.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 3.b odd 2 1
315.4.a.a 1 1.a even 1 1 trivial
525.4.a.a 1 15.d odd 2 1
525.4.d.a 2 15.e even 4 2
735.4.a.j 1 21.c even 2 1
1575.4.a.l 1 5.b even 2 1
1680.4.a.u 1 12.b even 2 1
2205.4.a.b 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 7$$
$11$ $$T + 12$$
$13$ $$T - 30$$
$17$ $$T - 134$$
$19$ $$T + 92$$
$23$ $$T + 112$$
$29$ $$T - 58$$
$31$ $$T + 224$$
$37$ $$T + 146$$
$41$ $$T + 18$$
$43$ $$T - 340$$
$47$ $$T + 208$$
$53$ $$T - 754$$
$59$ $$T + 380$$
$61$ $$T - 718$$
$67$ $$T - 412$$
$71$ $$T - 960$$
$73$ $$T - 1066$$
$79$ $$T - 896$$
$83$ $$T + 436$$
$89$ $$T - 1038$$
$97$ $$T + 702$$