# Properties

 Label 315.4.a.l 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: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{2} + (4 \beta + 1) q^{4} + 5 q^{5} - 7 q^{7} + (\beta + 6) q^{8}+O(q^{10})$$ q + (b + 2) * q^2 + (4*b + 1) * q^4 + 5 * q^5 - 7 * q^7 + (b + 6) * q^8 $$q + (\beta + 2) q^{2} + (4 \beta + 1) q^{4} + 5 q^{5} - 7 q^{7} + (\beta + 6) q^{8} + (5 \beta + 10) q^{10} + ( - 2 \beta + 46) q^{11} + (38 \beta + 4) q^{13} + ( - 7 \beta - 14) q^{14} + ( - 24 \beta + 9) q^{16} + (44 \beta + 22) q^{17} + ( - 26 \beta - 54) q^{19} + (20 \beta + 5) q^{20} + (42 \beta + 82) q^{22} + ( - 20 \beta + 160) q^{23} + 25 q^{25} + (80 \beta + 198) q^{26} + ( - 28 \beta - 7) q^{28} + (12 \beta + 118) q^{29} + ( - 102 \beta - 30) q^{31} + ( - 47 \beta - 150) q^{32} + (110 \beta + 264) q^{34} - 35 q^{35} + ( - 24 \beta + 102) q^{37} + ( - 106 \beta - 238) q^{38} + (5 \beta + 30) q^{40} + ( - 80 \beta - 22) q^{41} + ( - 128 \beta + 68) q^{43} + (182 \beta + 6) q^{44} + (120 \beta + 220) q^{46} + ( - 168 \beta - 200) q^{47} + 49 q^{49} + (25 \beta + 50) q^{50} + (54 \beta + 764) q^{52} + (86 \beta - 8) q^{53} + ( - 10 \beta + 230) q^{55} + ( - 7 \beta - 42) q^{56} + (142 \beta + 296) q^{58} + ( - 36 \beta + 232) q^{59} + ( - 84 \beta - 342) q^{61} + ( - 234 \beta - 570) q^{62} + ( - 52 \beta - 607) q^{64} + (190 \beta + 20) q^{65} + ( - 164 \beta + 368) q^{67} + (132 \beta + 902) q^{68} + ( - 35 \beta - 70) q^{70} + ( - 138 \beta + 370) q^{71} + (122 \beta + 212) q^{73} + (54 \beta + 84) q^{74} + ( - 242 \beta - 574) q^{76} + (14 \beta - 322) q^{77} + (484 \beta - 204) q^{79} + ( - 120 \beta + 45) q^{80} + ( - 182 \beta - 444) q^{82} + ( - 84 \beta - 304) q^{83} + (220 \beta + 110) q^{85} + ( - 188 \beta - 504) q^{86} + (34 \beta + 266) q^{88} + ( - 112 \beta + 666) q^{89} + ( - 266 \beta - 28) q^{91} + (620 \beta - 240) q^{92} + ( - 536 \beta - 1240) q^{94} + ( - 130 \beta - 270) q^{95} + (86 \beta - 1224) q^{97} + (49 \beta + 98) q^{98}+O(q^{100})$$ q + (b + 2) * q^2 + (4*b + 1) * q^4 + 5 * q^5 - 7 * q^7 + (b + 6) * q^8 + (5*b + 10) * q^10 + (-2*b + 46) * q^11 + (38*b + 4) * q^13 + (-7*b - 14) * q^14 + (-24*b + 9) * q^16 + (44*b + 22) * q^17 + (-26*b - 54) * q^19 + (20*b + 5) * q^20 + (42*b + 82) * q^22 + (-20*b + 160) * q^23 + 25 * q^25 + (80*b + 198) * q^26 + (-28*b - 7) * q^28 + (12*b + 118) * q^29 + (-102*b - 30) * q^31 + (-47*b - 150) * q^32 + (110*b + 264) * q^34 - 35 * q^35 + (-24*b + 102) * q^37 + (-106*b - 238) * q^38 + (5*b + 30) * q^40 + (-80*b - 22) * q^41 + (-128*b + 68) * q^43 + (182*b + 6) * q^44 + (120*b + 220) * q^46 + (-168*b - 200) * q^47 + 49 * q^49 + (25*b + 50) * q^50 + (54*b + 764) * q^52 + (86*b - 8) * q^53 + (-10*b + 230) * q^55 + (-7*b - 42) * q^56 + (142*b + 296) * q^58 + (-36*b + 232) * q^59 + (-84*b - 342) * q^61 + (-234*b - 570) * q^62 + (-52*b - 607) * q^64 + (190*b + 20) * q^65 + (-164*b + 368) * q^67 + (132*b + 902) * q^68 + (-35*b - 70) * q^70 + (-138*b + 370) * q^71 + (122*b + 212) * q^73 + (54*b + 84) * q^74 + (-242*b - 574) * q^76 + (14*b - 322) * q^77 + (484*b - 204) * q^79 + (-120*b + 45) * q^80 + (-182*b - 444) * q^82 + (-84*b - 304) * q^83 + (220*b + 110) * q^85 + (-188*b - 504) * q^86 + (34*b + 266) * q^88 + (-112*b + 666) * q^89 + (-266*b - 28) * q^91 + (620*b - 240) * q^92 + (-536*b - 1240) * q^94 + (-130*b - 270) * q^95 + (86*b - 1224) * q^97 + (49*b + 98) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8 $$2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8} + 20 q^{10} + 92 q^{11} + 8 q^{13} - 28 q^{14} + 18 q^{16} + 44 q^{17} - 108 q^{19} + 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} + 396 q^{26} - 14 q^{28} + 236 q^{29} - 60 q^{31} - 300 q^{32} + 528 q^{34} - 70 q^{35} + 204 q^{37} - 476 q^{38} + 60 q^{40} - 44 q^{41} + 136 q^{43} + 12 q^{44} + 440 q^{46} - 400 q^{47} + 98 q^{49} + 100 q^{50} + 1528 q^{52} - 16 q^{53} + 460 q^{55} - 84 q^{56} + 592 q^{58} + 464 q^{59} - 684 q^{61} - 1140 q^{62} - 1214 q^{64} + 40 q^{65} + 736 q^{67} + 1804 q^{68} - 140 q^{70} + 740 q^{71} + 424 q^{73} + 168 q^{74} - 1148 q^{76} - 644 q^{77} - 408 q^{79} + 90 q^{80} - 888 q^{82} - 608 q^{83} + 220 q^{85} - 1008 q^{86} + 532 q^{88} + 1332 q^{89} - 56 q^{91} - 480 q^{92} - 2480 q^{94} - 540 q^{95} - 2448 q^{97} + 196 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8 + 20 * q^10 + 92 * q^11 + 8 * q^13 - 28 * q^14 + 18 * q^16 + 44 * q^17 - 108 * q^19 + 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 + 396 * q^26 - 14 * q^28 + 236 * q^29 - 60 * q^31 - 300 * q^32 + 528 * q^34 - 70 * q^35 + 204 * q^37 - 476 * q^38 + 60 * q^40 - 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 - 400 * q^47 + 98 * q^49 + 100 * q^50 + 1528 * q^52 - 16 * q^53 + 460 * q^55 - 84 * q^56 + 592 * q^58 + 464 * q^59 - 684 * q^61 - 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 + 1804 * q^68 - 140 * q^70 + 740 * q^71 + 424 * q^73 + 168 * q^74 - 1148 * q^76 - 644 * q^77 - 408 * q^79 + 90 * q^80 - 888 * q^82 - 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 + 1332 * q^89 - 56 * q^91 - 480 * q^92 - 2480 * q^94 - 540 * q^95 - 2448 * q^97 + 196 * 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
 −0.618034 1.61803
−0.236068 0 −7.94427 5.00000 0 −7.00000 3.76393 0 −1.18034
1.2 4.23607 0 9.94427 5.00000 0 −7.00000 8.23607 0 21.1803
 $$n$$: 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.l 2
3.b odd 2 1 105.4.a.d 2
5.b even 2 1 1575.4.a.n 2
7.b odd 2 1 2205.4.a.be 2
12.b even 2 1 1680.4.a.bd 2
15.d odd 2 1 525.4.a.o 2
15.e even 4 2 525.4.d.k 4
21.c even 2 1 735.4.a.m 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T - 1$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} - 92T + 2096$$
$13$ $$T^{2} - 8T - 7204$$
$17$ $$T^{2} - 44T - 9196$$
$19$ $$T^{2} + 108T - 464$$
$23$ $$T^{2} - 320T + 23600$$
$29$ $$T^{2} - 236T + 13204$$
$31$ $$T^{2} + 60T - 51120$$
$37$ $$T^{2} - 204T + 7524$$
$41$ $$T^{2} + 44T - 31516$$
$43$ $$T^{2} - 136T - 77296$$
$47$ $$T^{2} + 400T - 101120$$
$53$ $$T^{2} + 16T - 36916$$
$59$ $$T^{2} - 464T + 47344$$
$61$ $$T^{2} + 684T + 81684$$
$67$ $$T^{2} - 736T + 944$$
$71$ $$T^{2} - 740T + 41680$$
$73$ $$T^{2} - 424T - 29476$$
$79$ $$T^{2} + 408 T - 1129664$$
$83$ $$T^{2} + 608T + 57136$$
$89$ $$T^{2} - 1332 T + 380836$$
$97$ $$T^{2} + 2448 T + 1461196$$