# Properties

 Label 315.4.a.j Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} - 7 q^{7} + ( - 11 \beta + 4) q^{8}+O(q^{10})$$ q + b * q^2 + (b - 4) * q^4 + 5 * q^5 - 7 * q^7 + (-11*b + 4) * q^8 $$q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} - 7 q^{7} + ( - 11 \beta + 4) q^{8} + 5 \beta q^{10} + (4 \beta - 4) q^{11} - 22 \beta q^{13} - 7 \beta q^{14} + ( - 15 \beta - 12) q^{16} + (26 \beta - 42) q^{17} + ( - 44 \beta + 22) q^{19} + (5 \beta - 20) q^{20} + 16 q^{22} + ( - 14 \beta - 34) q^{23} + 25 q^{25} + ( - 22 \beta - 88) q^{26} + ( - 7 \beta + 28) q^{28} + ( - 30 \beta - 152) q^{29} + (18 \beta - 114) q^{31} + (61 \beta - 92) q^{32} + ( - 16 \beta + 104) q^{34} - 35 q^{35} + ( - 30 \beta + 18) q^{37} + ( - 22 \beta - 176) q^{38} + ( - 55 \beta + 20) q^{40} + (52 \beta - 114) q^{41} + (166 \beta - 60) q^{43} + ( - 16 \beta + 32) q^{44} + ( - 48 \beta - 56) q^{46} + (30 \beta - 272) q^{47} + 49 q^{49} + 25 \beta q^{50} + (66 \beta - 88) q^{52} + ( - 52 \beta - 378) q^{53} + (20 \beta - 20) q^{55} + (77 \beta - 28) q^{56} + ( - 182 \beta - 120) q^{58} - 284 q^{59} + (90 \beta - 354) q^{61} + ( - 96 \beta + 72) q^{62} + (89 \beta + 340) q^{64} - 110 \beta q^{65} + ( - 98 \beta + 396) q^{67} + ( - 120 \beta + 272) q^{68} - 35 \beta q^{70} + (162 \beta - 488) q^{71} + (290 \beta - 104) q^{73} + ( - 12 \beta - 120) q^{74} + (154 \beta - 264) q^{76} + ( - 28 \beta + 28) q^{77} + (328 \beta + 136) q^{79} + ( - 75 \beta - 60) q^{80} + ( - 62 \beta + 208) q^{82} + ( - 300 \beta + 284) q^{83} + (130 \beta - 210) q^{85} + (106 \beta + 664) q^{86} + (16 \beta - 192) q^{88} + (476 \beta - 202) q^{89} + 154 \beta q^{91} + (8 \beta + 80) q^{92} + ( - 242 \beta + 120) q^{94} + ( - 220 \beta + 110) q^{95} + (482 \beta + 572) q^{97} + 49 \beta q^{98} +O(q^{100})$$ q + b * q^2 + (b - 4) * q^4 + 5 * q^5 - 7 * q^7 + (-11*b + 4) * q^8 + 5*b * q^10 + (4*b - 4) * q^11 - 22*b * q^13 - 7*b * q^14 + (-15*b - 12) * q^16 + (26*b - 42) * q^17 + (-44*b + 22) * q^19 + (5*b - 20) * q^20 + 16 * q^22 + (-14*b - 34) * q^23 + 25 * q^25 + (-22*b - 88) * q^26 + (-7*b + 28) * q^28 + (-30*b - 152) * q^29 + (18*b - 114) * q^31 + (61*b - 92) * q^32 + (-16*b + 104) * q^34 - 35 * q^35 + (-30*b + 18) * q^37 + (-22*b - 176) * q^38 + (-55*b + 20) * q^40 + (52*b - 114) * q^41 + (166*b - 60) * q^43 + (-16*b + 32) * q^44 + (-48*b - 56) * q^46 + (30*b - 272) * q^47 + 49 * q^49 + 25*b * q^50 + (66*b - 88) * q^52 + (-52*b - 378) * q^53 + (20*b - 20) * q^55 + (77*b - 28) * q^56 + (-182*b - 120) * q^58 - 284 * q^59 + (90*b - 354) * q^61 + (-96*b + 72) * q^62 + (89*b + 340) * q^64 - 110*b * q^65 + (-98*b + 396) * q^67 + (-120*b + 272) * q^68 - 35*b * q^70 + (162*b - 488) * q^71 + (290*b - 104) * q^73 + (-12*b - 120) * q^74 + (154*b - 264) * q^76 + (-28*b + 28) * q^77 + (328*b + 136) * q^79 + (-75*b - 60) * q^80 + (-62*b + 208) * q^82 + (-300*b + 284) * q^83 + (130*b - 210) * q^85 + (106*b + 664) * q^86 + (16*b - 192) * q^88 + (476*b - 202) * q^89 + 154*b * q^91 + (8*b + 80) * q^92 + (-242*b + 120) * q^94 + (-220*b + 110) * q^95 + (482*b + 572) * q^97 + 49*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 14 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 14 * q^7 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 14 q^{7} - 3 q^{8} + 5 q^{10} - 4 q^{11} - 22 q^{13} - 7 q^{14} - 39 q^{16} - 58 q^{17} - 35 q^{20} + 32 q^{22} - 82 q^{23} + 50 q^{25} - 198 q^{26} + 49 q^{28} - 334 q^{29} - 210 q^{31} - 123 q^{32} + 192 q^{34} - 70 q^{35} + 6 q^{37} - 374 q^{38} - 15 q^{40} - 176 q^{41} + 46 q^{43} + 48 q^{44} - 160 q^{46} - 514 q^{47} + 98 q^{49} + 25 q^{50} - 110 q^{52} - 808 q^{53} - 20 q^{55} + 21 q^{56} - 422 q^{58} - 568 q^{59} - 618 q^{61} + 48 q^{62} + 769 q^{64} - 110 q^{65} + 694 q^{67} + 424 q^{68} - 35 q^{70} - 814 q^{71} + 82 q^{73} - 252 q^{74} - 374 q^{76} + 28 q^{77} + 600 q^{79} - 195 q^{80} + 354 q^{82} + 268 q^{83} - 290 q^{85} + 1434 q^{86} - 368 q^{88} + 72 q^{89} + 154 q^{91} + 168 q^{92} - 2 q^{94} + 1626 q^{97} + 49 q^{98}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 14 * q^7 - 3 * q^8 + 5 * q^10 - 4 * q^11 - 22 * q^13 - 7 * q^14 - 39 * q^16 - 58 * q^17 - 35 * q^20 + 32 * q^22 - 82 * q^23 + 50 * q^25 - 198 * q^26 + 49 * q^28 - 334 * q^29 - 210 * q^31 - 123 * q^32 + 192 * q^34 - 70 * q^35 + 6 * q^37 - 374 * q^38 - 15 * q^40 - 176 * q^41 + 46 * q^43 + 48 * q^44 - 160 * q^46 - 514 * q^47 + 98 * q^49 + 25 * q^50 - 110 * q^52 - 808 * q^53 - 20 * q^55 + 21 * q^56 - 422 * q^58 - 568 * q^59 - 618 * q^61 + 48 * q^62 + 769 * q^64 - 110 * q^65 + 694 * q^67 + 424 * q^68 - 35 * q^70 - 814 * q^71 + 82 * q^73 - 252 * q^74 - 374 * q^76 + 28 * q^77 + 600 * q^79 - 195 * q^80 + 354 * q^82 + 268 * q^83 - 290 * q^85 + 1434 * q^86 - 368 * q^88 + 72 * q^89 + 154 * q^91 + 168 * q^92 - 2 * q^94 + 1626 * 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
 −1.56155 2.56155
−1.56155 0 −5.56155 5.00000 0 −7.00000 21.1771 0 −7.80776
1.2 2.56155 0 −1.43845 5.00000 0 −7.00000 −24.1771 0 12.8078
 $$n$$: 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.j yes 2
3.b odd 2 1 315.4.a.h 2
5.b even 2 1 1575.4.a.r 2
7.b odd 2 1 2205.4.a.ba 2
15.d odd 2 1 1575.4.a.u 2
21.c even 2 1 2205.4.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 3.b odd 2 1
315.4.a.j yes 2 1.a even 1 1 trivial
1575.4.a.r 2 5.b even 2 1
1575.4.a.u 2 15.d odd 2 1
2205.4.a.y 2 21.c even 2 1
2205.4.a.ba 2 7.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 4T - 64$$
$13$ $$T^{2} + 22T - 1936$$
$17$ $$T^{2} + 58T - 2032$$
$19$ $$T^{2} - 8228$$
$23$ $$T^{2} + 82T + 848$$
$29$ $$T^{2} + 334T + 24064$$
$31$ $$T^{2} + 210T + 9648$$
$37$ $$T^{2} - 6T - 3816$$
$41$ $$T^{2} + 176T - 3748$$
$43$ $$T^{2} - 46T - 116584$$
$47$ $$T^{2} + 514T + 62224$$
$53$ $$T^{2} + 808T + 151724$$
$59$ $$(T + 284)^{2}$$
$61$ $$T^{2} + 618T + 61056$$
$67$ $$T^{2} - 694T + 79592$$
$71$ $$T^{2} + 814T + 54112$$
$73$ $$T^{2} - 82T - 355744$$
$79$ $$T^{2} - 600T - 367232$$
$83$ $$T^{2} - 268T - 364544$$
$89$ $$T^{2} - 72T - 961652$$
$97$ $$T^{2} - 1626 T - 326408$$