# Properties

 Label 315.4.a.o 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: 3.3.22952.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 18x + 14$$ x^3 - x^2 - 18*x + 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + 5 q^{5} + 7 q^{7} + (2 \beta_{2} - 5 \beta_1 + 23) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 5) * q^4 + 5 * q^5 + 7 * q^7 + (2*b2 - 5*b1 + 23) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + 5 q^{5} + 7 q^{7} + (2 \beta_{2} - 5 \beta_1 + 23) q^{8} + ( - 5 \beta_1 + 5) q^{10} + (4 \beta_{2} + 6 \beta_1 - 10) q^{11} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{13} + ( - 7 \beta_1 + 7) q^{14} + ( - 3 \beta_{2} - 24 \beta_1 + 47) q^{16} + ( - 12 \beta_1 + 82) q^{17} + (8 \beta_{2} - 6 \beta_1 - 28) q^{19} + (5 \beta_{2} - 10 \beta_1 + 25) q^{20} + ( - 6 \beta_{2} - 8 \beta_1 - 74) q^{22} + ( - 12 \beta_{2} + 12 \beta_1 + 10) q^{23} + 25 q^{25} + (2 \beta_{2} + 20 \beta_1 + 18) q^{26} + (7 \beta_{2} - 14 \beta_1 + 35) q^{28} + ( - 16 \beta_{2} + 32 \beta_1 - 16) q^{29} + (10 \beta_1 - 124) q^{31} + (8 \beta_{2} - 13 \beta_1 + 145) q^{32} + (12 \beta_{2} - 94 \beta_1 + 226) q^{34} + 35 q^{35} + ( - 40 \beta_1 + 114) q^{37} + (6 \beta_{2} - 26 \beta_1 + 60) q^{38} + (10 \beta_{2} - 25 \beta_1 + 115) q^{40} + ( - 20 \beta_{2} + 52 \beta_1 + 198) q^{41} + ( - 36 \beta_{2} + 52 \beta_1 - 28) q^{43} + ( - 24 \beta_{2} + 54 \beta_1 + 90) q^{44} + ( - 12 \beta_{2} + 74 \beta_1 - 158) q^{46} + (4 \beta_{2} + 8 \beta_1 + 308) q^{47} + 49 q^{49} + ( - 25 \beta_1 + 25) q^{50} + (12 \beta_{2} + 6 \beta_1 - 234) q^{52} + ( - 20 \beta_{2} - 54 \beta_1 + 124) q^{53} + (20 \beta_{2} + 30 \beta_1 - 50) q^{55} + (14 \beta_{2} - 35 \beta_1 + 161) q^{56} + ( - 32 \beta_{2} + 144 \beta_1 - 432) q^{58} + (72 \beta_{2} - 16 \beta_1 + 124) q^{59} + (120 \beta_1 + 310) q^{61} + ( - 10 \beta_{2} + 134 \beta_1 - 244) q^{62} + (37 \beta_{2} - 14 \beta_1 - 59) q^{64} + ( - 20 \beta_{2} - 10 \beta_1 + 10) q^{65} + (16 \beta_{2} + 208 \beta_1 - 92) q^{67} + (94 \beta_{2} - 296 \beta_1 + 722) q^{68} + ( - 35 \beta_1 + 35) q^{70} + ( - 4 \beta_{2} + 66 \beta_1 + 118) q^{71} + ( - 44 \beta_{2} - 22 \beta_1 - 82) q^{73} + (40 \beta_{2} - 154 \beta_1 + 594) q^{74} + ( - 38 \beta_{2} - 74 \beta_1 + 608) q^{76} + (28 \beta_{2} + 42 \beta_1 - 70) q^{77} + (108 \beta_{2} + 124 \beta_1 - 240) q^{79} + ( - 15 \beta_{2} - 120 \beta_1 + 235) q^{80} + ( - 52 \beta_{2} - 26 \beta_1 - 466) q^{82} + ( - 76 \beta_{2} - 128 \beta_1 + 384) q^{83} + ( - 60 \beta_1 + 410) q^{85} + ( - 52 \beta_{2} + 296 \beta_1 - 724) q^{86} + ( - 6 \beta_{2} + 172 \beta_1 - 14) q^{88} + ( - 32 \beta_{2} + 88 \beta_1 - 66) q^{89} + ( - 28 \beta_{2} - 14 \beta_1 + 14) q^{91} + (22 \beta_{2} + 208 \beta_1 - 1150) q^{92} + ( - 8 \beta_{2} - 324 \beta_1 + 220) q^{94} + (40 \beta_{2} - 30 \beta_1 - 140) q^{95} + (12 \beta_{2} + 206 \beta_1 - 1090) q^{97} + ( - 49 \beta_1 + 49) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 5) * q^4 + 5 * q^5 + 7 * q^7 + (2*b2 - 5*b1 + 23) * q^8 + (-5*b1 + 5) * q^10 + (4*b2 + 6*b1 - 10) * q^11 + (-4*b2 - 2*b1 + 2) * q^13 + (-7*b1 + 7) * q^14 + (-3*b2 - 24*b1 + 47) * q^16 + (-12*b1 + 82) * q^17 + (8*b2 - 6*b1 - 28) * q^19 + (5*b2 - 10*b1 + 25) * q^20 + (-6*b2 - 8*b1 - 74) * q^22 + (-12*b2 + 12*b1 + 10) * q^23 + 25 * q^25 + (2*b2 + 20*b1 + 18) * q^26 + (7*b2 - 14*b1 + 35) * q^28 + (-16*b2 + 32*b1 - 16) * q^29 + (10*b1 - 124) * q^31 + (8*b2 - 13*b1 + 145) * q^32 + (12*b2 - 94*b1 + 226) * q^34 + 35 * q^35 + (-40*b1 + 114) * q^37 + (6*b2 - 26*b1 + 60) * q^38 + (10*b2 - 25*b1 + 115) * q^40 + (-20*b2 + 52*b1 + 198) * q^41 + (-36*b2 + 52*b1 - 28) * q^43 + (-24*b2 + 54*b1 + 90) * q^44 + (-12*b2 + 74*b1 - 158) * q^46 + (4*b2 + 8*b1 + 308) * q^47 + 49 * q^49 + (-25*b1 + 25) * q^50 + (12*b2 + 6*b1 - 234) * q^52 + (-20*b2 - 54*b1 + 124) * q^53 + (20*b2 + 30*b1 - 50) * q^55 + (14*b2 - 35*b1 + 161) * q^56 + (-32*b2 + 144*b1 - 432) * q^58 + (72*b2 - 16*b1 + 124) * q^59 + (120*b1 + 310) * q^61 + (-10*b2 + 134*b1 - 244) * q^62 + (37*b2 - 14*b1 - 59) * q^64 + (-20*b2 - 10*b1 + 10) * q^65 + (16*b2 + 208*b1 - 92) * q^67 + (94*b2 - 296*b1 + 722) * q^68 + (-35*b1 + 35) * q^70 + (-4*b2 + 66*b1 + 118) * q^71 + (-44*b2 - 22*b1 - 82) * q^73 + (40*b2 - 154*b1 + 594) * q^74 + (-38*b2 - 74*b1 + 608) * q^76 + (28*b2 + 42*b1 - 70) * q^77 + (108*b2 + 124*b1 - 240) * q^79 + (-15*b2 - 120*b1 + 235) * q^80 + (-52*b2 - 26*b1 - 466) * q^82 + (-76*b2 - 128*b1 + 384) * q^83 + (-60*b1 + 410) * q^85 + (-52*b2 + 296*b1 - 724) * q^86 + (-6*b2 + 172*b1 - 14) * q^88 + (-32*b2 + 88*b1 - 66) * q^89 + (-28*b2 - 14*b1 + 14) * q^91 + (22*b2 + 208*b1 - 1150) * q^92 + (-8*b2 - 324*b1 + 220) * q^94 + (40*b2 - 30*b1 - 140) * q^95 + (12*b2 + 206*b1 - 1090) * q^97 + (-49*b1 + 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 $$3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28} - 32 q^{29} - 362 q^{31} + 430 q^{32} + 596 q^{34} + 105 q^{35} + 302 q^{37} + 160 q^{38} + 330 q^{40} + 626 q^{41} - 68 q^{43} + 300 q^{44} - 412 q^{46} + 936 q^{47} + 147 q^{49} + 50 q^{50} - 684 q^{52} + 298 q^{53} - 100 q^{55} + 462 q^{56} - 1184 q^{58} + 428 q^{59} + 1050 q^{61} - 608 q^{62} - 154 q^{64} - 52 q^{67} + 1964 q^{68} + 70 q^{70} + 416 q^{71} - 312 q^{73} + 1668 q^{74} + 1712 q^{76} - 140 q^{77} - 488 q^{79} + 570 q^{80} - 1476 q^{82} + 948 q^{83} + 1170 q^{85} - 1928 q^{86} + 124 q^{88} - 142 q^{89} - 3220 q^{92} + 328 q^{94} - 410 q^{95} - 3052 q^{97} + 98 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 + 10 * q^10 - 20 * q^11 + 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 70 * q^20 - 236 * q^22 + 30 * q^23 + 75 * q^25 + 76 * q^26 + 98 * q^28 - 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 + 105 * q^35 + 302 * q^37 + 160 * q^38 + 330 * q^40 + 626 * q^41 - 68 * q^43 + 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 50 * q^50 - 684 * q^52 + 298 * q^53 - 100 * q^55 + 462 * q^56 - 1184 * q^58 + 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 + 70 * q^70 + 416 * q^71 - 312 * q^73 + 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 570 * q^80 - 1476 * q^82 + 948 * q^83 + 1170 * q^85 - 1928 * q^86 + 124 * q^88 - 142 * q^89 - 3220 * q^92 + 328 * q^94 - 410 * q^95 - 3052 * q^97 + 98 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 18x + 14$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ b2 + 12

## 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
 4.37989 0.770205 −4.15010
−3.37989 0 3.42368 5.00000 0 7.00000 15.4675 0 −16.8995
1.2 0.229795 0 −7.94719 5.00000 0 7.00000 −3.66459 0 1.14898
1.3 5.15010 0 18.5235 5.00000 0 7.00000 54.1971 0 25.7505
 $$n$$: 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.o yes 3
3.b odd 2 1 315.4.a.n 3
5.b even 2 1 1575.4.a.bb 3
7.b odd 2 1 2205.4.a.bl 3
15.d odd 2 1 1575.4.a.be 3
21.c even 2 1 2205.4.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.n 3 3.b odd 2 1
315.4.a.o yes 3 1.a even 1 1 trivial
1575.4.a.bb 3 5.b even 2 1
1575.4.a.be 3 15.d odd 2 1
2205.4.a.bk 3 21.c even 2 1
2205.4.a.bl 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 17T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$3$ $$T^{3}$$
$5$ $$(T - 5)^{3}$$
$7$ $$(T - 7)^{3}$$
$11$ $$T^{3} + 20 T^{2} + \cdots - 32160$$
$13$ $$T^{3} - 1748T - 17328$$
$17$ $$T^{3} - 234 T^{2} + \cdots - 282328$$
$19$ $$T^{3} + 82 T^{2} + \cdots + 15280$$
$23$ $$T^{3} - 30 T^{2} + \cdots - 378280$$
$29$ $$T^{3} + 32 T^{2} + \cdots + 409600$$
$31$ $$T^{3} + 362 T^{2} + \cdots + 1543664$$
$37$ $$T^{3} - 302 T^{2} + \cdots + 1425496$$
$41$ $$T^{3} - 626 T^{2} + \cdots + 16079208$$
$43$ $$T^{3} + 68 T^{2} + \cdots - 10742464$$
$47$ $$T^{3} - 936 T^{2} + \cdots - 29520128$$
$53$ $$T^{3} - 298 T^{2} + \cdots + 19383056$$
$59$ $$T^{3} - 428 T^{2} + \cdots + 229584064$$
$61$ $$T^{3} - 1050 T^{2} + \cdots + 63221000$$
$67$ $$T^{3} + 52 T^{2} + \cdots - 93054016$$
$71$ $$T^{3} - 416 T^{2} + \cdots + 14345328$$
$73$ $$T^{3} + 312 T^{2} + \cdots - 43935536$$
$79$ $$T^{3} + 488 T^{2} + \cdots - 282894592$$
$83$ $$T^{3} - 948 T^{2} + \cdots + 431507200$$
$89$ $$T^{3} + 142 T^{2} + \cdots + 19655976$$
$97$ $$T^{3} + 3052 T^{2} + \cdots + 204161472$$