# Properties

 Label 315.4.a.m 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 4) q^{2} + ( - 7 \beta + 12) q^{4} + 5 q^{5} - 7 q^{7} + ( - 25 \beta + 44) q^{8}+O(q^{10})$$ q + (-b + 4) * q^2 + (-7*b + 12) * q^4 + 5 * q^5 - 7 * q^7 + (-25*b + 44) * q^8 $$q + ( - \beta + 4) q^{2} + ( - 7 \beta + 12) q^{4} + 5 q^{5} - 7 q^{7} + ( - 25 \beta + 44) q^{8} + ( - 5 \beta + 20) q^{10} + ( - 10 \beta + 18) q^{11} + (22 \beta - 4) q^{13} + (7 \beta - 28) q^{14} + ( - 63 \beta + 180) q^{16} + (28 \beta - 22) q^{17} + (26 \beta + 74) q^{19} + ( - 35 \beta + 60) q^{20} + ( - 48 \beta + 112) q^{22} + (56 \beta - 120) q^{23} + 25 q^{25} + (70 \beta - 104) q^{26} + (49 \beta - 84) q^{28} + ( - 84 \beta + 58) q^{29} + ( - 18 \beta + 174) q^{31} + ( - 169 \beta + 620) q^{32} + (106 \beta - 200) q^{34} - 35 q^{35} + ( - 24 \beta - 54) q^{37} + (4 \beta + 192) q^{38} + ( - 125 \beta + 220) q^{40} + (140 \beta - 170) q^{41} + (68 \beta + 148) q^{43} + ( - 176 \beta + 496) q^{44} + (288 \beta - 704) q^{46} + (108 \beta - 200) q^{47} + 49 q^{49} + ( - 25 \beta + 100) q^{50} + (138 \beta - 664) q^{52} + (214 \beta - 124) q^{53} + ( - 50 \beta + 90) q^{55} + (175 \beta - 308) q^{56} + ( - 310 \beta + 568) q^{58} + (36 \beta - 200) q^{59} + (252 \beta + 270) q^{61} + ( - 228 \beta + 768) q^{62} + ( - 623 \beta + 1716) q^{64} + (110 \beta - 20) q^{65} + ( - 28 \beta - 380) q^{67} + (294 \beta - 1048) q^{68} + (35 \beta - 140) q^{70} + ( - 330 \beta - 62) q^{71} + (178 \beta + 300) q^{73} + ( - 18 \beta - 120) q^{74} + ( - 388 \beta + 160) q^{76} + (70 \beta - 126) q^{77} + ( - 88 \beta + 248) q^{79} + ( - 315 \beta + 900) q^{80} + (590 \beta - 1240) q^{82} + ( - 264 \beta - 436) q^{83} + (140 \beta - 110) q^{85} + (56 \beta + 320) q^{86} + ( - 640 \beta + 1792) q^{88} + ( - 728 \beta + 346) q^{89} + ( - 154 \beta + 28) q^{91} + (1120 \beta - 3008) q^{92} + (524 \beta - 1232) q^{94} + (130 \beta + 370) q^{95} + ( - 146 \beta - 176) q^{97} + ( - 49 \beta + 196) q^{98}+O(q^{100})$$ q + (-b + 4) * q^2 + (-7*b + 12) * q^4 + 5 * q^5 - 7 * q^7 + (-25*b + 44) * q^8 + (-5*b + 20) * q^10 + (-10*b + 18) * q^11 + (22*b - 4) * q^13 + (7*b - 28) * q^14 + (-63*b + 180) * q^16 + (28*b - 22) * q^17 + (26*b + 74) * q^19 + (-35*b + 60) * q^20 + (-48*b + 112) * q^22 + (56*b - 120) * q^23 + 25 * q^25 + (70*b - 104) * q^26 + (49*b - 84) * q^28 + (-84*b + 58) * q^29 + (-18*b + 174) * q^31 + (-169*b + 620) * q^32 + (106*b - 200) * q^34 - 35 * q^35 + (-24*b - 54) * q^37 + (4*b + 192) * q^38 + (-125*b + 220) * q^40 + (140*b - 170) * q^41 + (68*b + 148) * q^43 + (-176*b + 496) * q^44 + (288*b - 704) * q^46 + (108*b - 200) * q^47 + 49 * q^49 + (-25*b + 100) * q^50 + (138*b - 664) * q^52 + (214*b - 124) * q^53 + (-50*b + 90) * q^55 + (175*b - 308) * q^56 + (-310*b + 568) * q^58 + (36*b - 200) * q^59 + (252*b + 270) * q^61 + (-228*b + 768) * q^62 + (-623*b + 1716) * q^64 + (110*b - 20) * q^65 + (-28*b - 380) * q^67 + (294*b - 1048) * q^68 + (35*b - 140) * q^70 + (-330*b - 62) * q^71 + (178*b + 300) * q^73 + (-18*b - 120) * q^74 + (-388*b + 160) * q^76 + (70*b - 126) * q^77 + (-88*b + 248) * q^79 + (-315*b + 900) * q^80 + (590*b - 1240) * q^82 + (-264*b - 436) * q^83 + (140*b - 110) * q^85 + (56*b + 320) * q^86 + (-640*b + 1792) * q^88 + (-728*b + 346) * q^89 + (-154*b + 28) * q^91 + (1120*b - 3008) * q^92 + (524*b - 1232) * q^94 + (130*b + 370) * q^95 + (-146*b - 176) * q^97 + (-49*b + 196) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8}+O(q^{10})$$ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 $$2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8} + 35 q^{10} + 26 q^{11} + 14 q^{13} - 49 q^{14} + 297 q^{16} - 16 q^{17} + 174 q^{19} + 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} - 138 q^{26} - 119 q^{28} + 32 q^{29} + 330 q^{31} + 1071 q^{32} - 294 q^{34} - 70 q^{35} - 132 q^{37} + 388 q^{38} + 315 q^{40} - 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} - 292 q^{47} + 98 q^{49} + 175 q^{50} - 1190 q^{52} - 34 q^{53} + 130 q^{55} - 441 q^{56} + 826 q^{58} - 364 q^{59} + 792 q^{61} + 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} - 1802 q^{68} - 245 q^{70} - 454 q^{71} + 778 q^{73} - 258 q^{74} - 68 q^{76} - 182 q^{77} + 408 q^{79} + 1485 q^{80} - 1890 q^{82} - 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} - 36 q^{89} - 98 q^{91} - 4896 q^{92} - 1940 q^{94} + 870 q^{95} - 498 q^{97} + 343 q^{98}+O(q^{100})$$ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 + 35 * q^10 + 26 * q^11 + 14 * q^13 - 49 * q^14 + 297 * q^16 - 16 * q^17 + 174 * q^19 + 85 * q^20 + 176 * q^22 - 184 * q^23 + 50 * q^25 - 138 * q^26 - 119 * q^28 + 32 * q^29 + 330 * q^31 + 1071 * q^32 - 294 * q^34 - 70 * q^35 - 132 * q^37 + 388 * q^38 + 315 * q^40 - 200 * q^41 + 364 * q^43 + 816 * q^44 - 1120 * q^46 - 292 * q^47 + 98 * q^49 + 175 * q^50 - 1190 * q^52 - 34 * q^53 + 130 * q^55 - 441 * q^56 + 826 * q^58 - 364 * q^59 + 792 * q^61 + 1308 * q^62 + 2809 * q^64 + 70 * q^65 - 788 * q^67 - 1802 * q^68 - 245 * q^70 - 454 * q^71 + 778 * q^73 - 258 * q^74 - 68 * q^76 - 182 * q^77 + 408 * q^79 + 1485 * q^80 - 1890 * q^82 - 1136 * q^83 - 80 * q^85 + 696 * q^86 + 2944 * q^88 - 36 * q^89 - 98 * q^91 - 4896 * q^92 - 1940 * q^94 + 870 * q^95 - 498 * q^97 + 343 * 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
 2.56155 −1.56155
1.43845 0 −5.93087 5.00000 0 −7.00000 −20.0388 0 7.19224
1.2 5.56155 0 22.9309 5.00000 0 −7.00000 83.0388 0 27.8078
 $$n$$: 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.m 2
3.b odd 2 1 105.4.a.c 2
5.b even 2 1 1575.4.a.m 2
7.b odd 2 1 2205.4.a.bh 2
12.b even 2 1 1680.4.a.bk 2
15.d odd 2 1 525.4.a.p 2
15.e even 4 2 525.4.d.i 4
21.c even 2 1 735.4.a.k 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 7T_{2} + 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 7T + 8$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} - 26T - 256$$
$13$ $$T^{2} - 14T - 2008$$
$17$ $$T^{2} + 16T - 3268$$
$19$ $$T^{2} - 174T + 4696$$
$23$ $$T^{2} + 184T - 4864$$
$29$ $$T^{2} - 32T - 29732$$
$31$ $$T^{2} - 330T + 25848$$
$37$ $$T^{2} + 132T + 1908$$
$41$ $$T^{2} + 200T - 73300$$
$43$ $$T^{2} - 364T + 13472$$
$47$ $$T^{2} + 292T - 28256$$
$53$ $$T^{2} + 34T - 194344$$
$59$ $$T^{2} + 364T + 27616$$
$61$ $$T^{2} - 792T - 113076$$
$67$ $$T^{2} + 788T + 151904$$
$71$ $$T^{2} + 454T - 411296$$
$73$ $$T^{2} - 778T + 16664$$
$79$ $$T^{2} - 408T + 8704$$
$83$ $$T^{2} + 1136T + 26416$$
$89$ $$T^{2} + 36T - 2252108$$
$97$ $$T^{2} + 498T - 28592$$