# Properties

 Label 315.4.a.p Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.14360.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 17x - 14$$ x^3 - 17*x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + 7 q^{7} + (3 \beta_{2} - \beta_1 + 4) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 4) * q^4 - 5 * q^5 + 7 * q^7 + (3*b2 - b1 + 4) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + 7 q^{7} + (3 \beta_{2} - \beta_1 + 4) q^{8} + (5 \beta_1 - 5) q^{10} + ( - \beta_{2} - 3 \beta_1 + 25) q^{11} + (5 \beta_{2} - 13 \beta_1 + 13) q^{13} + ( - 7 \beta_1 + 7) q^{14} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + ( - 11 \beta_{2} - 13 \beta_1 + 21) q^{17} + (6 \beta_{2} - 10 \beta_1 + 54) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 20) q^{20} + (\beta_{2} - 20 \beta_1 + 61) q^{22} + ( - 2 \beta_{2} + 14 \beta_1 + 42) q^{23} + 25 q^{25} + (23 \beta_{2} - 38 \beta_1 + 141) q^{26} + (7 \beta_{2} - 7 \beta_1 + 28) q^{28} + ( - 17 \beta_{2} - 19 \beta_1 - 105) q^{29} + ( - 4 \beta_{2} + 24 \beta_1 + 108) q^{31} + ( - 15 \beta_{2} + 39 \beta_1 + 66) q^{32} + ( - 9 \beta_{2} + 34 \beta_1 + 197) q^{34} - 35 q^{35} + ( - 12 \beta_{2} + 16 \beta_1 - 14) q^{37} + (22 \beta_{2} - 84 \beta_1 + 146) q^{38} + ( - 15 \beta_{2} + 5 \beta_1 - 20) q^{40} + ( - 2 \beta_{2} - 10 \beta_1 - 120) q^{41} + ( - 34 \beta_{2} + 30 \beta_1 + 6) q^{43} + (30 \beta_{2} - 42 \beta_1 + 78) q^{44} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + (13 \beta_{2} + 51 \beta_1 + 239) q^{47} + 49 q^{49} + ( - 25 \beta_1 + 25) q^{50} + (44 \beta_{2} - 152 \beta_1 + 386) q^{52} + (22 \beta_{2} + 130 \beta_1 - 44) q^{53} + (5 \beta_{2} + 15 \beta_1 - 125) q^{55} + (21 \beta_{2} - 7 \beta_1 + 28) q^{56} + ( - 15 \beta_{2} + 190 \beta_1 + 155) q^{58} + ( - 48 \beta_{2} + 176 \beta_1 + 76) q^{59} + ( - 26 \beta_{2} - 34 \beta_1 + 416) q^{61} + ( - 32 \beta_{2} - 88 \beta_1 - 144) q^{62} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + ( - 25 \beta_{2} + 65 \beta_1 - 65) q^{65} + (108 \beta_{2} - 12 \beta_1 + 32) q^{67} + (36 \beta_{2} - 48 \beta_1 - 318) q^{68} + (35 \beta_1 - 35) q^{70} + (40 \beta_{2} - 72 \beta_1 + 32) q^{71} + (76 \beta_{2} + 124 \beta_1 + 78) q^{73} + ( - 40 \beta_{2} + 74 \beta_1 - 154) q^{74} + (80 \beta_{2} - 176 \beta_1 + 572) q^{76} + ( - 7 \beta_{2} - 21 \beta_1 + 175) q^{77} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + (5 \beta_{2} + 55 \beta_1 + 130) q^{80} + (6 \beta_{2} + 130 \beta_1 - 4) q^{82} + ( - 8 \beta_{2} + 160 \beta_1 + 556) q^{83} + (55 \beta_{2} + 65 \beta_1 - 105) q^{85} + ( - 98 \beta_{2} + 164 \beta_1 - 222) q^{86} + (94 \beta_{2} - 68 \beta_1 - 38) q^{88} + (82 \beta_{2} + 98 \beta_1 - 108) q^{89} + (35 \beta_{2} - 91 \beta_1 + 91) q^{91} + (12 \beta_{2} + 84 \beta_1 - 36) q^{92} + ( - 25 \beta_{2} - 304 \beta_1 - 361) q^{94} + ( - 30 \beta_{2} + 50 \beta_1 - 270) q^{95} + ( - 65 \beta_{2} - 87 \beta_1 + 55) q^{97} + ( - 49 \beta_1 + 49) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 4) * q^4 - 5 * q^5 + 7 * q^7 + (3*b2 - b1 + 4) * q^8 + (5*b1 - 5) * q^10 + (-b2 - 3*b1 + 25) * q^11 + (5*b2 - 13*b1 + 13) * q^13 + (-7*b1 + 7) * q^14 + (-b2 - 11*b1 - 26) * q^16 + (-11*b2 - 13*b1 + 21) * q^17 + (6*b2 - 10*b1 + 54) * q^19 + (-5*b2 + 5*b1 - 20) * q^20 + (b2 - 20*b1 + 61) * q^22 + (-2*b2 + 14*b1 + 42) * q^23 + 25 * q^25 + (23*b2 - 38*b1 + 141) * q^26 + (7*b2 - 7*b1 + 28) * q^28 + (-17*b2 - 19*b1 - 105) * q^29 + (-4*b2 + 24*b1 + 108) * q^31 + (-15*b2 + 39*b1 + 66) * q^32 + (-9*b2 + 34*b1 + 197) * q^34 - 35 * q^35 + (-12*b2 + 16*b1 - 14) * q^37 + (22*b2 - 84*b1 + 146) * q^38 + (-15*b2 + 5*b1 - 20) * q^40 + (-2*b2 - 10*b1 - 120) * q^41 + (-34*b2 + 30*b1 + 6) * q^43 + (30*b2 - 42*b1 + 78) * q^44 + (-18*b2 - 32*b1 - 106) * q^46 + (13*b2 + 51*b1 + 239) * q^47 + 49 * q^49 + (-25*b1 + 25) * q^50 + (44*b2 - 152*b1 + 386) * q^52 + (22*b2 + 130*b1 - 44) * q^53 + (5*b2 + 15*b1 - 125) * q^55 + (21*b2 - 7*b1 + 28) * q^56 + (-15*b2 + 190*b1 + 155) * q^58 + (-48*b2 + 176*b1 + 76) * q^59 + (-26*b2 - 34*b1 + 416) * q^61 + (-32*b2 - 88*b1 - 144) * q^62 + (-61*b2 + 97*b1 - 110) * q^64 + (-25*b2 + 65*b1 - 65) * q^65 + (108*b2 - 12*b1 + 32) * q^67 + (36*b2 - 48*b1 - 318) * q^68 + (35*b1 - 35) * q^70 + (40*b2 - 72*b1 + 32) * q^71 + (76*b2 + 124*b1 + 78) * q^73 + (-40*b2 + 74*b1 - 154) * q^74 + (80*b2 - 176*b1 + 572) * q^76 + (-7*b2 - 21*b1 + 175) * q^77 + (-89*b2 - 83*b1 - 315) * q^79 + (5*b2 + 55*b1 + 130) * q^80 + (6*b2 + 130*b1 - 4) * q^82 + (-8*b2 + 160*b1 + 556) * q^83 + (55*b2 + 65*b1 - 105) * q^85 + (-98*b2 + 164*b1 - 222) * q^86 + (94*b2 - 68*b1 - 38) * q^88 + (82*b2 + 98*b1 - 108) * q^89 + (35*b2 - 91*b1 + 91) * q^91 + (12*b2 + 84*b1 - 36) * q^92 + (-25*b2 - 304*b1 - 361) * q^94 + (-30*b2 + 50*b1 - 270) * q^95 + (-65*b2 - 87*b1 + 55) * q^97 + (-49*b1 + 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 13 * q^4 - 15 * q^5 + 21 * q^7 + 15 * q^8 $$3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8} - 15 q^{10} + 74 q^{11} + 44 q^{13} + 21 q^{14} - 79 q^{16} + 52 q^{17} + 168 q^{19} - 65 q^{20} + 184 q^{22} + 124 q^{23} + 75 q^{25} + 446 q^{26} + 91 q^{28} - 332 q^{29} + 320 q^{31} + 183 q^{32} + 582 q^{34} - 105 q^{35} - 54 q^{37} + 460 q^{38} - 75 q^{40} - 362 q^{41} - 16 q^{43} + 264 q^{44} - 336 q^{46} + 730 q^{47} + 147 q^{49} + 75 q^{50} + 1202 q^{52} - 110 q^{53} - 370 q^{55} + 105 q^{56} + 450 q^{58} + 180 q^{59} + 1222 q^{61} - 464 q^{62} - 391 q^{64} - 220 q^{65} + 204 q^{67} - 918 q^{68} - 105 q^{70} + 136 q^{71} + 310 q^{73} - 502 q^{74} + 1796 q^{76} + 518 q^{77} - 1034 q^{79} + 395 q^{80} - 6 q^{82} + 1660 q^{83} - 260 q^{85} - 764 q^{86} - 20 q^{88} - 242 q^{89} + 308 q^{91} - 96 q^{92} - 1108 q^{94} - 840 q^{95} + 100 q^{97} + 147 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 13 * q^4 - 15 * q^5 + 21 * q^7 + 15 * q^8 - 15 * q^10 + 74 * q^11 + 44 * q^13 + 21 * q^14 - 79 * q^16 + 52 * q^17 + 168 * q^19 - 65 * q^20 + 184 * q^22 + 124 * q^23 + 75 * q^25 + 446 * q^26 + 91 * q^28 - 332 * q^29 + 320 * q^31 + 183 * q^32 + 582 * q^34 - 105 * q^35 - 54 * q^37 + 460 * q^38 - 75 * q^40 - 362 * q^41 - 16 * q^43 + 264 * q^44 - 336 * q^46 + 730 * q^47 + 147 * q^49 + 75 * q^50 + 1202 * q^52 - 110 * q^53 - 370 * q^55 + 105 * q^56 + 450 * q^58 + 180 * q^59 + 1222 * q^61 - 464 * q^62 - 391 * q^64 - 220 * q^65 + 204 * q^67 - 918 * q^68 - 105 * q^70 + 136 * q^71 + 310 * q^73 - 502 * q^74 + 1796 * q^76 + 518 * q^77 - 1034 * q^79 + 395 * q^80 - 6 * q^82 + 1660 * q^83 - 260 * q^85 - 764 * q^86 - 20 * q^88 - 242 * q^89 + 308 * q^91 - 96 * q^92 - 1108 * q^94 - 840 * q^95 + 100 * q^97 + 147 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 17x - 14$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 11$$ v^2 - v - 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11

## 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
 4.48565 −0.861086 −3.62456
−3.48565 0 4.14976 −5.00000 0 7.00000 13.4206 0 17.4283
1.2 1.86109 0 −4.53636 −5.00000 0 7.00000 −23.3312 0 −9.30543
1.3 4.62456 0 13.3866 −5.00000 0 7.00000 24.9107 0 −23.1228
 $$n$$: 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.p 3
3.b odd 2 1 35.4.a.c 3
5.b even 2 1 1575.4.a.ba 3
7.b odd 2 1 2205.4.a.bm 3
12.b even 2 1 560.4.a.u 3
15.d odd 2 1 175.4.a.f 3
15.e even 4 2 175.4.b.e 6
21.c even 2 1 245.4.a.l 3
21.g even 6 2 245.4.e.n 6
21.h odd 6 2 245.4.e.m 6
24.f even 2 1 2240.4.a.bv 3
24.h odd 2 1 2240.4.a.bt 3
105.g even 2 1 1225.4.a.y 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 3T_{2}^{2} - 14T_{2} + 30$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3 T^{2} + \cdots + 30$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$(T - 7)^{3}$$
$11$ $$T^{3} - 74 T^{2} + \cdots - 7692$$
$13$ $$T^{3} - 44 T^{2} + \cdots - 44870$$
$17$ $$T^{3} - 52 T^{2} + \cdots + 56706$$
$19$ $$T^{3} - 168 T^{2} + \cdots - 28720$$
$23$ $$T^{3} - 124 T^{2} + \cdots + 94368$$
$29$ $$T^{3} + 332 T^{2} + \cdots - 2565450$$
$31$ $$T^{3} - 320 T^{2} + \cdots + 50176$$
$37$ $$T^{3} + 54 T^{2} + \cdots + 25736$$
$41$ $$T^{3} + 362 T^{2} + \cdots + 1536192$$
$43$ $$T^{3} + 16 T^{2} + \cdots - 1524560$$
$47$ $$T^{3} - 730 T^{2} + \cdots - 4968912$$
$53$ $$T^{3} + 110 T^{2} + \cdots - 90318336$$
$59$ $$T^{3} - 180 T^{2} + \cdots + 202459200$$
$61$ $$T^{3} - 1222 T^{2} + \cdots - 38393792$$
$67$ $$T^{3} - 204 T^{2} + \cdots + 324944128$$
$71$ $$T^{3} - 136 T^{2} + \cdots - 15575040$$
$73$ $$T^{3} - 310 T^{2} + \cdots + 48718616$$
$79$ $$T^{3} + 1034 T^{2} + \cdots - 343615600$$
$83$ $$T^{3} - 1660 T^{2} + \cdots + 42727104$$
$89$ $$T^{3} + 242 T^{2} + \cdots + 6359520$$
$97$ $$T^{3} - 100 T^{2} + \cdots - 1978018$$