# Properties

 Label 315.4.a.g 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: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + 5 q^{5} + 7 q^{7} + ( - \beta - 25) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (3*b + 3) * q^4 + 5 * q^5 + 7 * q^7 + (-b - 25) * q^8 $$q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + 5 q^{5} + 7 q^{7} + ( - \beta - 25) q^{8} + ( - 5 \beta - 5) q^{10} + (2 \beta - 32) q^{11} + ( - 10 \beta + 2) q^{13} + ( - 7 \beta - 7) q^{14} + (3 \beta + 11) q^{16} + (12 \beta - 26) q^{17} + ( - 2 \beta - 60) q^{19} + (15 \beta + 15) q^{20} + (28 \beta + 12) q^{22} + (48 \beta - 32) q^{23} + 25 q^{25} + (18 \beta + 98) q^{26} + (21 \beta + 21) q^{28} + ( - 12 \beta - 170) q^{29} + ( - 38 \beta + 52) q^{31} + ( - 9 \beta + 159) q^{32} + (2 \beta - 94) q^{34} + 35 q^{35} + (80 \beta - 134) q^{37} + (64 \beta + 80) q^{38} + ( - 5 \beta - 125) q^{40} + (108 \beta - 62) q^{41} + (100 \beta - 248) q^{43} + ( - 84 \beta - 36) q^{44} + ( - 64 \beta - 448) q^{46} + ( - 140 \beta + 164) q^{47} + 49 q^{49} + ( - 25 \beta - 25) q^{50} + ( - 54 \beta - 294) q^{52} + ( - 58 \beta - 462) q^{53} + (10 \beta - 160) q^{55} + ( - 7 \beta - 175) q^{56} + (194 \beta + 290) q^{58} + ( - 76 \beta - 220) q^{59} + ( - 84 \beta - 398) q^{61} + (24 \beta + 328) q^{62} + ( - 165 \beta - 157) q^{64} + ( - 50 \beta + 10) q^{65} + ( - 228 \beta - 64) q^{67} + ( - 6 \beta + 282) q^{68} + ( - 35 \beta - 35) q^{70} + ( - 86 \beta - 112) q^{71} + ( - 38 \beta + 182) q^{73} + ( - 26 \beta - 666) q^{74} + ( - 192 \beta - 240) q^{76} + (14 \beta - 224) q^{77} + ( - 8 \beta + 920) q^{79} + (15 \beta + 55) q^{80} + ( - 154 \beta - 1018) q^{82} + (224 \beta + 228) q^{83} + (60 \beta - 130) q^{85} + (48 \beta - 752) q^{86} + ( - 20 \beta + 780) q^{88} + ( - 336 \beta - 230) q^{89} + ( - 70 \beta + 14) q^{91} + (192 \beta + 1344) q^{92} + (116 \beta + 1236) q^{94} + ( - 10 \beta - 300) q^{95} + (278 \beta - 474) q^{97} + ( - 49 \beta - 49) q^{98}+O(q^{100})$$ q + (-b - 1) * q^2 + (3*b + 3) * q^4 + 5 * q^5 + 7 * q^7 + (-b - 25) * q^8 + (-5*b - 5) * q^10 + (2*b - 32) * q^11 + (-10*b + 2) * q^13 + (-7*b - 7) * q^14 + (3*b + 11) * q^16 + (12*b - 26) * q^17 + (-2*b - 60) * q^19 + (15*b + 15) * q^20 + (28*b + 12) * q^22 + (48*b - 32) * q^23 + 25 * q^25 + (18*b + 98) * q^26 + (21*b + 21) * q^28 + (-12*b - 170) * q^29 + (-38*b + 52) * q^31 + (-9*b + 159) * q^32 + (2*b - 94) * q^34 + 35 * q^35 + (80*b - 134) * q^37 + (64*b + 80) * q^38 + (-5*b - 125) * q^40 + (108*b - 62) * q^41 + (100*b - 248) * q^43 + (-84*b - 36) * q^44 + (-64*b - 448) * q^46 + (-140*b + 164) * q^47 + 49 * q^49 + (-25*b - 25) * q^50 + (-54*b - 294) * q^52 + (-58*b - 462) * q^53 + (10*b - 160) * q^55 + (-7*b - 175) * q^56 + (194*b + 290) * q^58 + (-76*b - 220) * q^59 + (-84*b - 398) * q^61 + (24*b + 328) * q^62 + (-165*b - 157) * q^64 + (-50*b + 10) * q^65 + (-228*b - 64) * q^67 + (-6*b + 282) * q^68 + (-35*b - 35) * q^70 + (-86*b - 112) * q^71 + (-38*b + 182) * q^73 + (-26*b - 666) * q^74 + (-192*b - 240) * q^76 + (14*b - 224) * q^77 + (-8*b + 920) * q^79 + (15*b + 55) * q^80 + (-154*b - 1018) * q^82 + (224*b + 228) * q^83 + (60*b - 130) * q^85 + (48*b - 752) * q^86 + (-20*b + 780) * q^88 + (-336*b - 230) * q^89 + (-70*b + 14) * q^91 + (192*b + 1344) * q^92 + (116*b + 1236) * q^94 + (-10*b - 300) * q^95 + (278*b - 474) * q^97 + (-49*b - 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8 $$2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8} - 15 q^{10} - 62 q^{11} - 6 q^{13} - 21 q^{14} + 25 q^{16} - 40 q^{17} - 122 q^{19} + 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} + 214 q^{26} + 63 q^{28} - 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{34} + 70 q^{35} - 188 q^{37} + 224 q^{38} - 255 q^{40} - 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} + 188 q^{47} + 98 q^{49} - 75 q^{50} - 642 q^{52} - 982 q^{53} - 310 q^{55} - 357 q^{56} + 774 q^{58} - 516 q^{59} - 880 q^{61} + 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} + 558 q^{68} - 105 q^{70} - 310 q^{71} + 326 q^{73} - 1358 q^{74} - 672 q^{76} - 434 q^{77} + 1832 q^{79} + 125 q^{80} - 2190 q^{82} + 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} - 796 q^{89} - 42 q^{91} + 2880 q^{92} + 2588 q^{94} - 610 q^{95} - 670 q^{97} - 147 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8 - 15 * q^10 - 62 * q^11 - 6 * q^13 - 21 * q^14 + 25 * q^16 - 40 * q^17 - 122 * q^19 + 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 + 214 * q^26 + 63 * q^28 - 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^34 + 70 * q^35 - 188 * q^37 + 224 * q^38 - 255 * q^40 - 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 + 188 * q^47 + 98 * q^49 - 75 * q^50 - 642 * q^52 - 982 * q^53 - 310 * q^55 - 357 * q^56 + 774 * q^58 - 516 * q^59 - 880 * q^61 + 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 + 558 * q^68 - 105 * q^70 - 310 * q^71 + 326 * q^73 - 1358 * q^74 - 672 * q^76 - 434 * q^77 + 1832 * q^79 + 125 * q^80 - 2190 * q^82 + 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 - 796 * q^89 - 42 * q^91 + 2880 * q^92 + 2588 * q^94 - 610 * q^95 - 670 * q^97 - 147 * 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
 3.70156 −2.70156
−4.70156 0 14.1047 5.00000 0 7.00000 −28.7016 0 −23.5078
1.2 1.70156 0 −5.10469 5.00000 0 7.00000 −22.2984 0 8.50781
 $$n$$: 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.g 2
3.b odd 2 1 105.4.a.g 2
5.b even 2 1 1575.4.a.y 2
7.b odd 2 1 2205.4.a.v 2
12.b even 2 1 1680.4.a.y 2
15.d odd 2 1 525.4.a.i 2
15.e even 4 2 525.4.d.j 4
21.c even 2 1 735.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 3.b odd 2 1
315.4.a.g 2 1.a even 1 1 trivial
525.4.a.i 2 15.d odd 2 1
525.4.d.j 4 15.e even 4 2
735.4.a.q 2 21.c even 2 1
1575.4.a.y 2 5.b even 2 1
1680.4.a.y 2 12.b even 2 1
2205.4.a.v 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 8$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} + 62T + 920$$
$13$ $$T^{2} + 6T - 1016$$
$17$ $$T^{2} + 40T - 1076$$
$19$ $$T^{2} + 122T + 3680$$
$23$ $$T^{2} + 16T - 23552$$
$29$ $$T^{2} + 352T + 29500$$
$31$ $$T^{2} - 66T - 13712$$
$37$ $$T^{2} + 188T - 56764$$
$41$ $$T^{2} + 16T - 119492$$
$43$ $$T^{2} + 396T - 63296$$
$47$ $$T^{2} - 188T - 192064$$
$53$ $$T^{2} + 982T + 206600$$
$59$ $$T^{2} + 516T + 7360$$
$61$ $$T^{2} + 880T + 121276$$
$67$ $$T^{2} + 356T - 501152$$
$71$ $$T^{2} + 310T - 51784$$
$73$ $$T^{2} - 326T + 11768$$
$79$ $$T^{2} - 1832 T + 838400$$
$83$ $$T^{2} - 680T - 398704$$
$89$ $$T^{2} + 796T - 998780$$
$97$ $$T^{2} + 670T - 679936$$