# Properties

 Label 315.4.j.b Level $315$ Weight $4$ Character orbit 315.j Analytic conductor $18.586$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(46,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.46");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.j (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$18.5856016518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 5 \zeta_{6} q^{5} + ( - 14 \zeta_{6} - 7) q^{7} + 21 q^{8}+O(q^{10})$$ q + 3*z * q^2 + (z - 1) * q^4 + 5*z * q^5 + (-14*z - 7) * q^7 + 21 * q^8 $$q + 3 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 5 \zeta_{6} q^{5} + ( - 14 \zeta_{6} - 7) q^{7} + 21 q^{8} + (15 \zeta_{6} - 15) q^{10} + (45 \zeta_{6} - 45) q^{11} + 59 q^{13} + ( - 63 \zeta_{6} + 42) q^{14} + 71 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} + 121 \zeta_{6} q^{19} - 5 q^{20} - 135 q^{22} + 69 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 177 \zeta_{6} q^{26} + ( - 7 \zeta_{6} + 21) q^{28} + 162 q^{29} + ( - 88 \zeta_{6} + 88) q^{31} + (45 \zeta_{6} - 45) q^{32} - 162 q^{34} + ( - 105 \zeta_{6} + 70) q^{35} + 259 \zeta_{6} q^{37} + (363 \zeta_{6} - 363) q^{38} + 105 \zeta_{6} q^{40} - 195 q^{41} - 286 q^{43} - 45 \zeta_{6} q^{44} + (207 \zeta_{6} - 207) q^{46} + 45 \zeta_{6} q^{47} + (392 \zeta_{6} - 147) q^{49} - 75 q^{50} + (59 \zeta_{6} - 59) q^{52} + ( - 597 \zeta_{6} + 597) q^{53} - 225 q^{55} + ( - 294 \zeta_{6} - 147) q^{56} + 486 \zeta_{6} q^{58} + (360 \zeta_{6} - 360) q^{59} - 392 \zeta_{6} q^{61} + 264 q^{62} + 433 q^{64} + 295 \zeta_{6} q^{65} + ( - 280 \zeta_{6} + 280) q^{67} - 54 \zeta_{6} q^{68} + ( - 105 \zeta_{6} + 315) q^{70} - 48 q^{71} + (668 \zeta_{6} - 668) q^{73} + (777 \zeta_{6} - 777) q^{74} - 121 q^{76} + ( - 315 \zeta_{6} + 945) q^{77} - 782 \zeta_{6} q^{79} + (355 \zeta_{6} - 355) q^{80} - 585 \zeta_{6} q^{82} - 768 q^{83} - 270 q^{85} - 858 \zeta_{6} q^{86} + (945 \zeta_{6} - 945) q^{88} - 1194 \zeta_{6} q^{89} + ( - 826 \zeta_{6} - 413) q^{91} - 69 q^{92} + (135 \zeta_{6} - 135) q^{94} + (605 \zeta_{6} - 605) q^{95} + 902 q^{97} + (735 \zeta_{6} - 1176) q^{98} +O(q^{100})$$ q + 3*z * q^2 + (z - 1) * q^4 + 5*z * q^5 + (-14*z - 7) * q^7 + 21 * q^8 + (15*z - 15) * q^10 + (45*z - 45) * q^11 + 59 * q^13 + (-63*z + 42) * q^14 + 71*z * q^16 + (54*z - 54) * q^17 + 121*z * q^19 - 5 * q^20 - 135 * q^22 + 69*z * q^23 + (25*z - 25) * q^25 + 177*z * q^26 + (-7*z + 21) * q^28 + 162 * q^29 + (-88*z + 88) * q^31 + (45*z - 45) * q^32 - 162 * q^34 + (-105*z + 70) * q^35 + 259*z * q^37 + (363*z - 363) * q^38 + 105*z * q^40 - 195 * q^41 - 286 * q^43 - 45*z * q^44 + (207*z - 207) * q^46 + 45*z * q^47 + (392*z - 147) * q^49 - 75 * q^50 + (59*z - 59) * q^52 + (-597*z + 597) * q^53 - 225 * q^55 + (-294*z - 147) * q^56 + 486*z * q^58 + (360*z - 360) * q^59 - 392*z * q^61 + 264 * q^62 + 433 * q^64 + 295*z * q^65 + (-280*z + 280) * q^67 - 54*z * q^68 + (-105*z + 315) * q^70 - 48 * q^71 + (668*z - 668) * q^73 + (777*z - 777) * q^74 - 121 * q^76 + (-315*z + 945) * q^77 - 782*z * q^79 + (355*z - 355) * q^80 - 585*z * q^82 - 768 * q^83 - 270 * q^85 - 858*z * q^86 + (945*z - 945) * q^88 - 1194*z * q^89 + (-826*z - 413) * q^91 - 69 * q^92 + (135*z - 135) * q^94 + (605*z - 605) * q^95 + 902 * q^97 + (735*z - 1176) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - q^{4} + 5 q^{5} - 28 q^{7} + 42 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 - q^4 + 5 * q^5 - 28 * q^7 + 42 * q^8 $$2 q + 3 q^{2} - q^{4} + 5 q^{5} - 28 q^{7} + 42 q^{8} - 15 q^{10} - 45 q^{11} + 118 q^{13} + 21 q^{14} + 71 q^{16} - 54 q^{17} + 121 q^{19} - 10 q^{20} - 270 q^{22} + 69 q^{23} - 25 q^{25} + 177 q^{26} + 35 q^{28} + 324 q^{29} + 88 q^{31} - 45 q^{32} - 324 q^{34} + 35 q^{35} + 259 q^{37} - 363 q^{38} + 105 q^{40} - 390 q^{41} - 572 q^{43} - 45 q^{44} - 207 q^{46} + 45 q^{47} + 98 q^{49} - 150 q^{50} - 59 q^{52} + 597 q^{53} - 450 q^{55} - 588 q^{56} + 486 q^{58} - 360 q^{59} - 392 q^{61} + 528 q^{62} + 866 q^{64} + 295 q^{65} + 280 q^{67} - 54 q^{68} + 525 q^{70} - 96 q^{71} - 668 q^{73} - 777 q^{74} - 242 q^{76} + 1575 q^{77} - 782 q^{79} - 355 q^{80} - 585 q^{82} - 1536 q^{83} - 540 q^{85} - 858 q^{86} - 945 q^{88} - 1194 q^{89} - 1652 q^{91} - 138 q^{92} - 135 q^{94} - 605 q^{95} + 1804 q^{97} - 1617 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 - q^4 + 5 * q^5 - 28 * q^7 + 42 * q^8 - 15 * q^10 - 45 * q^11 + 118 * q^13 + 21 * q^14 + 71 * q^16 - 54 * q^17 + 121 * q^19 - 10 * q^20 - 270 * q^22 + 69 * q^23 - 25 * q^25 + 177 * q^26 + 35 * q^28 + 324 * q^29 + 88 * q^31 - 45 * q^32 - 324 * q^34 + 35 * q^35 + 259 * q^37 - 363 * q^38 + 105 * q^40 - 390 * q^41 - 572 * q^43 - 45 * q^44 - 207 * q^46 + 45 * q^47 + 98 * q^49 - 150 * q^50 - 59 * q^52 + 597 * q^53 - 450 * q^55 - 588 * q^56 + 486 * q^58 - 360 * q^59 - 392 * q^61 + 528 * q^62 + 866 * q^64 + 295 * q^65 + 280 * q^67 - 54 * q^68 + 525 * q^70 - 96 * q^71 - 668 * q^73 - 777 * q^74 - 242 * q^76 + 1575 * q^77 - 782 * q^79 - 355 * q^80 - 585 * q^82 - 1536 * q^83 - 540 * q^85 - 858 * q^86 - 945 * q^88 - 1194 * q^89 - 1652 * q^91 - 138 * q^92 - 135 * q^94 - 605 * q^95 + 1804 * q^97 - 1617 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 + 2.59808i 0 −0.500000 + 0.866025i 2.50000 + 4.33013i 0 −14.0000 12.1244i 21.0000 0 −7.50000 + 12.9904i
226.1 1.50000 2.59808i 0 −0.500000 0.866025i 2.50000 4.33013i 0 −14.0000 + 12.1244i 21.0000 0 −7.50000 12.9904i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.j.b 2
3.b odd 2 1 35.4.e.a 2
7.c even 3 1 inner 315.4.j.b 2
7.c even 3 1 2205.4.a.e 1
7.d odd 6 1 2205.4.a.g 1
12.b even 2 1 560.4.q.b 2
15.d odd 2 1 175.4.e.b 2
15.e even 4 2 175.4.k.b 4
21.c even 2 1 245.4.e.a 2
21.g even 6 1 245.4.a.f 1
21.g even 6 1 245.4.e.a 2
21.h odd 6 1 35.4.e.a 2
21.h odd 6 1 245.4.a.e 1
84.n even 6 1 560.4.q.b 2
105.o odd 6 1 175.4.e.b 2
105.o odd 6 1 1225.4.a.b 1
105.p even 6 1 1225.4.a.a 1
105.x even 12 2 175.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 3.b odd 2 1
35.4.e.a 2 21.h odd 6 1
175.4.e.b 2 15.d odd 2 1
175.4.e.b 2 105.o odd 6 1
175.4.k.b 4 15.e even 4 2
175.4.k.b 4 105.x even 12 2
245.4.a.e 1 21.h odd 6 1
245.4.a.f 1 21.g even 6 1
245.4.e.a 2 21.c even 2 1
245.4.e.a 2 21.g even 6 1
315.4.j.b 2 1.a even 1 1 trivial
315.4.j.b 2 7.c even 3 1 inner
560.4.q.b 2 12.b even 2 1
560.4.q.b 2 84.n even 6 1
1225.4.a.a 1 105.p even 6 1
1225.4.a.b 1 105.o odd 6 1
2205.4.a.e 1 7.c even 3 1
2205.4.a.g 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{2} - 3T_{2} + 9$$ T2^2 - 3*T2 + 9 $$T_{13} - 59$$ T13 - 59

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} + 28T + 343$$
$11$ $$T^{2} + 45T + 2025$$
$13$ $$(T - 59)^{2}$$
$17$ $$T^{2} + 54T + 2916$$
$19$ $$T^{2} - 121T + 14641$$
$23$ $$T^{2} - 69T + 4761$$
$29$ $$(T - 162)^{2}$$
$31$ $$T^{2} - 88T + 7744$$
$37$ $$T^{2} - 259T + 67081$$
$41$ $$(T + 195)^{2}$$
$43$ $$(T + 286)^{2}$$
$47$ $$T^{2} - 45T + 2025$$
$53$ $$T^{2} - 597T + 356409$$
$59$ $$T^{2} + 360T + 129600$$
$61$ $$T^{2} + 392T + 153664$$
$67$ $$T^{2} - 280T + 78400$$
$71$ $$(T + 48)^{2}$$
$73$ $$T^{2} + 668T + 446224$$
$79$ $$T^{2} + 782T + 611524$$
$83$ $$(T + 768)^{2}$$
$89$ $$T^{2} + 1194 T + 1425636$$
$97$ $$(T - 902)^{2}$$