# Properties

 Label 315.4.a.k 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 5 q^{5} - 7 q^{7} + ( - 5 \beta + 9) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - 5 * q^5 - 7 * q^7 + (-5*b + 9) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 5 q^{5} - 7 q^{7} + ( - 5 \beta + 9) q^{8} + ( - 5 \beta - 5) q^{10} + ( - 20 \beta + 8) q^{11} + ( - 2 \beta - 38) q^{13} + ( - 7 \beta - 7) q^{14} + ( - 12 \beta - 39) q^{16} + (2 \beta + 62) q^{17} + ( - 16 \beta - 48) q^{19} + ( - 10 \beta - 5) q^{20} + ( - 12 \beta - 152) q^{22} + (34 \beta + 8) q^{23} + 25 q^{25} + ( - 40 \beta - 54) q^{26} + ( - 14 \beta - 7) q^{28} + (54 \beta - 94) q^{29} + (18 \beta - 60) q^{31} + ( - 11 \beta - 207) q^{32} + (64 \beta + 78) q^{34} + 35 q^{35} + ( - 66 \beta - 66) q^{37} + ( - 64 \beta - 176) q^{38} + (25 \beta - 45) q^{40} + ( - 80 \beta - 50) q^{41} + (62 \beta - 268) q^{43} + ( - 4 \beta - 312) q^{44} + (42 \beta + 280) q^{46} + (42 \beta + 464) q^{47} + 49 q^{49} + (25 \beta + 25) q^{50} + ( - 78 \beta - 70) q^{52} + ( - 64 \beta - 442) q^{53} + (100 \beta - 40) q^{55} + (35 \beta - 63) q^{56} + ( - 40 \beta + 338) q^{58} + (204 \beta - 52) q^{59} + ( - 42 \beta - 234) q^{61} + ( - 42 \beta + 84) q^{62} + ( - 122 \beta + 17) q^{64} + (10 \beta + 190) q^{65} + (38 \beta - 844) q^{67} + (126 \beta + 94) q^{68} + (35 \beta + 35) q^{70} + (150 \beta + 68) q^{71} + (322 \beta + 254) q^{73} + ( - 132 \beta - 594) q^{74} + ( - 112 \beta - 304) q^{76} + (140 \beta - 56) q^{77} + ( - 232 \beta - 216) q^{79} + (60 \beta + 195) q^{80} + ( - 130 \beta - 690) q^{82} + (84 \beta + 292) q^{83} + ( - 10 \beta - 310) q^{85} + ( - 206 \beta + 228) q^{86} + ( - 220 \beta + 872) q^{88} + ( - 112 \beta + 702) q^{89} + (14 \beta + 266) q^{91} + (50 \beta + 552) q^{92} + (506 \beta + 800) q^{94} + (80 \beta + 240) q^{95} + (46 \beta - 594) q^{97} + (49 \beta + 49) q^{98}+O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - 5 * q^5 - 7 * q^7 + (-5*b + 9) * q^8 + (-5*b - 5) * q^10 + (-20*b + 8) * q^11 + (-2*b - 38) * q^13 + (-7*b - 7) * q^14 + (-12*b - 39) * q^16 + (2*b + 62) * q^17 + (-16*b - 48) * q^19 + (-10*b - 5) * q^20 + (-12*b - 152) * q^22 + (34*b + 8) * q^23 + 25 * q^25 + (-40*b - 54) * q^26 + (-14*b - 7) * q^28 + (54*b - 94) * q^29 + (18*b - 60) * q^31 + (-11*b - 207) * q^32 + (64*b + 78) * q^34 + 35 * q^35 + (-66*b - 66) * q^37 + (-64*b - 176) * q^38 + (25*b - 45) * q^40 + (-80*b - 50) * q^41 + (62*b - 268) * q^43 + (-4*b - 312) * q^44 + (42*b + 280) * q^46 + (42*b + 464) * q^47 + 49 * q^49 + (25*b + 25) * q^50 + (-78*b - 70) * q^52 + (-64*b - 442) * q^53 + (100*b - 40) * q^55 + (35*b - 63) * q^56 + (-40*b + 338) * q^58 + (204*b - 52) * q^59 + (-42*b - 234) * q^61 + (-42*b + 84) * q^62 + (-122*b + 17) * q^64 + (10*b + 190) * q^65 + (38*b - 844) * q^67 + (126*b + 94) * q^68 + (35*b + 35) * q^70 + (150*b + 68) * q^71 + (322*b + 254) * q^73 + (-132*b - 594) * q^74 + (-112*b - 304) * q^76 + (140*b - 56) * q^77 + (-232*b - 216) * q^79 + (60*b + 195) * q^80 + (-130*b - 690) * q^82 + (84*b + 292) * q^83 + (-10*b - 310) * q^85 + (-206*b + 228) * q^86 + (-220*b + 872) * q^88 + (-112*b + 702) * q^89 + (14*b + 266) * q^91 + (50*b + 552) * q^92 + (506*b + 800) * q^94 + (80*b + 240) * q^95 + (46*b - 594) * q^97 + (49*b + 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8} - 10 q^{10} + 16 q^{11} - 76 q^{13} - 14 q^{14} - 78 q^{16} + 124 q^{17} - 96 q^{19} - 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} - 108 q^{26} - 14 q^{28} - 188 q^{29} - 120 q^{31} - 414 q^{32} + 156 q^{34} + 70 q^{35} - 132 q^{37} - 352 q^{38} - 90 q^{40} - 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} + 928 q^{47} + 98 q^{49} + 50 q^{50} - 140 q^{52} - 884 q^{53} - 80 q^{55} - 126 q^{56} + 676 q^{58} - 104 q^{59} - 468 q^{61} + 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} + 188 q^{68} + 70 q^{70} + 136 q^{71} + 508 q^{73} - 1188 q^{74} - 608 q^{76} - 112 q^{77} - 432 q^{79} + 390 q^{80} - 1380 q^{82} + 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} + 1404 q^{89} + 532 q^{91} + 1104 q^{92} + 1600 q^{94} + 480 q^{95} - 1188 q^{97} + 98 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8 - 10 * q^10 + 16 * q^11 - 76 * q^13 - 14 * q^14 - 78 * q^16 + 124 * q^17 - 96 * q^19 - 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 - 108 * q^26 - 14 * q^28 - 188 * q^29 - 120 * q^31 - 414 * q^32 + 156 * q^34 + 70 * q^35 - 132 * q^37 - 352 * q^38 - 90 * q^40 - 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 + 928 * q^47 + 98 * q^49 + 50 * q^50 - 140 * q^52 - 884 * q^53 - 80 * q^55 - 126 * q^56 + 676 * q^58 - 104 * q^59 - 468 * q^61 + 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 + 188 * q^68 + 70 * q^70 + 136 * q^71 + 508 * q^73 - 1188 * q^74 - 608 * q^76 - 112 * q^77 - 432 * q^79 + 390 * q^80 - 1380 * q^82 + 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 + 1404 * q^89 + 532 * q^91 + 1104 * q^92 + 1600 * q^94 + 480 * q^95 - 1188 * q^97 + 98 * 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
 −1.41421 1.41421
−1.82843 0 −4.65685 −5.00000 0 −7.00000 23.1421 0 9.14214
1.2 3.82843 0 6.65685 −5.00000 0 −7.00000 −5.14214 0 −19.1421
 $$n$$: 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.k 2
3.b odd 2 1 105.4.a.e 2
5.b even 2 1 1575.4.a.q 2
7.b odd 2 1 2205.4.a.bb 2
12.b even 2 1 1680.4.a.bo 2
15.d odd 2 1 525.4.a.l 2
15.e even 4 2 525.4.d.l 4
21.c even 2 1 735.4.a.o 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 7$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 7$$
$3$ $$T^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} - 16T - 3136$$
$13$ $$T^{2} + 76T + 1412$$
$17$ $$T^{2} - 124T + 3812$$
$19$ $$T^{2} + 96T + 256$$
$23$ $$T^{2} - 16T - 9184$$
$29$ $$T^{2} + 188T - 14492$$
$31$ $$T^{2} + 120T + 1008$$
$37$ $$T^{2} + 132T - 30492$$
$41$ $$T^{2} + 100T - 48700$$
$43$ $$T^{2} + 536T + 41072$$
$47$ $$T^{2} - 928T + 201184$$
$53$ $$T^{2} + 884T + 162596$$
$59$ $$T^{2} + 104T - 330224$$
$61$ $$T^{2} + 468T + 40644$$
$67$ $$T^{2} + 1688 T + 700784$$
$71$ $$T^{2} - 136T - 175376$$
$73$ $$T^{2} - 508T - 764956$$
$79$ $$T^{2} + 432T - 383936$$
$83$ $$T^{2} - 584T + 28816$$
$89$ $$T^{2} - 1404 T + 392452$$
$97$ $$T^{2} + 1188 T + 335908$$