# Properties

 Label 315.4.a.h Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $0$ 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: $$0$$ 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
 2.56155 −1.56155
−2.56155 0 −1.43845 −5.00000 0 −7.00000 24.1771 0 12.8078
1.2 1.56155 0 −5.56155 −5.00000 0 −7.00000 −21.1771 0 −7.80776
 $$n$$: 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.h 2
3.b odd 2 1 315.4.a.j yes 2
5.b even 2 1 1575.4.a.u 2
7.b odd 2 1 2205.4.a.y 2
15.d odd 2 1 1575.4.a.r 2
21.c even 2 1 2205.4.a.ba 2

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

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