# Properties

 Label 315.4.a.i Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 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 $$\beta = \frac{1}{2}(1 + \sqrt{65})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 8) q^{4} - 5 q^{5} - 7 q^{7} + ( - \beta - 16) q^{8} +O(q^{10})$$ q - b * q^2 + (b + 8) * q^4 - 5 * q^5 - 7 * q^7 + (-b - 16) * q^8 $$q - \beta q^{2} + (\beta + 8) q^{4} - 5 q^{5} - 7 q^{7} + ( - \beta - 16) q^{8} + 5 \beta q^{10} + (2 \beta + 10) q^{11} + (2 \beta - 12) q^{13} + 7 \beta q^{14} + (9 \beta - 48) q^{16} + (16 \beta - 66) q^{17} + ( - 14 \beta + 58) q^{19} + ( - 5 \beta - 40) q^{20} + ( - 12 \beta - 32) q^{22} + (20 \beta - 140) q^{23} + 25 q^{25} + (10 \beta - 32) q^{26} + ( - 7 \beta - 56) q^{28} + (48 \beta + 74) q^{29} + (42 \beta + 54) q^{31} + (47 \beta - 16) q^{32} + (50 \beta - 256) q^{34} + 35 q^{35} + ( - 36 \beta - 30) q^{37} + ( - 44 \beta + 224) q^{38} + (5 \beta + 80) q^{40} + ( - 100 \beta + 138) q^{41} + ( - 32 \beta - 156) q^{43} + (28 \beta + 112) q^{44} + (120 \beta - 320) q^{46} + (48 \beta - 304) q^{47} + 49 q^{49} - 25 \beta q^{50} + (6 \beta - 64) q^{52} + ( - 86 \beta - 120) q^{53} + ( - 10 \beta - 50) q^{55} + (7 \beta + 112) q^{56} + ( - 122 \beta - 768) q^{58} + ( - 84 \beta + 464) q^{59} + (24 \beta - 114) q^{61} + ( - 96 \beta - 672) q^{62} + ( - 103 \beta - 368) q^{64} + ( - 10 \beta + 60) q^{65} + (64 \beta - 84) q^{67} + (78 \beta - 272) q^{68} - 35 \beta q^{70} + ( - 42 \beta - 814) q^{71} + ( - 202 \beta - 92) q^{73} + (66 \beta + 576) q^{74} + ( - 68 \beta + 240) q^{76} + ( - 14 \beta - 70) q^{77} + ( - 104 \beta - 392) q^{79} + ( - 45 \beta + 240) q^{80} + ( - 38 \beta + 1600) q^{82} + ( - 216 \beta - 356) q^{83} + ( - 80 \beta + 330) q^{85} + (188 \beta + 512) q^{86} + ( - 44 \beta - 192) q^{88} + ( - 8 \beta - 290) q^{89} + ( - 14 \beta + 84) q^{91} + (40 \beta - 800) q^{92} + (256 \beta - 768) q^{94} + (70 \beta - 290) q^{95} + (314 \beta + 104) q^{97} - 49 \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b + 8) * q^4 - 5 * q^5 - 7 * q^7 + (-b - 16) * q^8 + 5*b * q^10 + (2*b + 10) * q^11 + (2*b - 12) * q^13 + 7*b * q^14 + (9*b - 48) * q^16 + (16*b - 66) * q^17 + (-14*b + 58) * q^19 + (-5*b - 40) * q^20 + (-12*b - 32) * q^22 + (20*b - 140) * q^23 + 25 * q^25 + (10*b - 32) * q^26 + (-7*b - 56) * q^28 + (48*b + 74) * q^29 + (42*b + 54) * q^31 + (47*b - 16) * q^32 + (50*b - 256) * q^34 + 35 * q^35 + (-36*b - 30) * q^37 + (-44*b + 224) * q^38 + (5*b + 80) * q^40 + (-100*b + 138) * q^41 + (-32*b - 156) * q^43 + (28*b + 112) * q^44 + (120*b - 320) * q^46 + (48*b - 304) * q^47 + 49 * q^49 - 25*b * q^50 + (6*b - 64) * q^52 + (-86*b - 120) * q^53 + (-10*b - 50) * q^55 + (7*b + 112) * q^56 + (-122*b - 768) * q^58 + (-84*b + 464) * q^59 + (24*b - 114) * q^61 + (-96*b - 672) * q^62 + (-103*b - 368) * q^64 + (-10*b + 60) * q^65 + (64*b - 84) * q^67 + (78*b - 272) * q^68 - 35*b * q^70 + (-42*b - 814) * q^71 + (-202*b - 92) * q^73 + (66*b + 576) * q^74 + (-68*b + 240) * q^76 + (-14*b - 70) * q^77 + (-104*b - 392) * q^79 + (-45*b + 240) * q^80 + (-38*b + 1600) * q^82 + (-216*b - 356) * q^83 + (-80*b + 330) * q^85 + (188*b + 512) * q^86 + (-44*b - 192) * q^88 + (-8*b - 290) * q^89 + (-14*b + 84) * q^91 + (40*b - 800) * q^92 + (256*b - 768) * q^94 + (70*b - 290) * q^95 + (314*b + 104) * q^97 - 49*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8}+O(q^{10})$$ 2 * q - q^2 + 17 * q^4 - 10 * q^5 - 14 * q^7 - 33 * q^8 $$2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8} + 5 q^{10} + 22 q^{11} - 22 q^{13} + 7 q^{14} - 87 q^{16} - 116 q^{17} + 102 q^{19} - 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} - 54 q^{26} - 119 q^{28} + 196 q^{29} + 150 q^{31} + 15 q^{32} - 462 q^{34} + 70 q^{35} - 96 q^{37} + 404 q^{38} + 165 q^{40} + 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} - 560 q^{47} + 98 q^{49} - 25 q^{50} - 122 q^{52} - 326 q^{53} - 110 q^{55} + 231 q^{56} - 1658 q^{58} + 844 q^{59} - 204 q^{61} - 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} - 466 q^{68} - 35 q^{70} - 1670 q^{71} - 386 q^{73} + 1218 q^{74} + 412 q^{76} - 154 q^{77} - 888 q^{79} + 435 q^{80} + 3162 q^{82} - 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} - 588 q^{89} + 154 q^{91} - 1560 q^{92} - 1280 q^{94} - 510 q^{95} + 522 q^{97} - 49 q^{98}+O(q^{100})$$ 2 * q - q^2 + 17 * q^4 - 10 * q^5 - 14 * q^7 - 33 * q^8 + 5 * q^10 + 22 * q^11 - 22 * q^13 + 7 * q^14 - 87 * q^16 - 116 * q^17 + 102 * q^19 - 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 - 54 * q^26 - 119 * q^28 + 196 * q^29 + 150 * q^31 + 15 * q^32 - 462 * q^34 + 70 * q^35 - 96 * q^37 + 404 * q^38 + 165 * q^40 + 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 - 560 * q^47 + 98 * q^49 - 25 * q^50 - 122 * q^52 - 326 * q^53 - 110 * q^55 + 231 * q^56 - 1658 * q^58 + 844 * q^59 - 204 * q^61 - 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 - 466 * q^68 - 35 * q^70 - 1670 * q^71 - 386 * q^73 + 1218 * q^74 + 412 * q^76 - 154 * q^77 - 888 * q^79 + 435 * q^80 + 3162 * q^82 - 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 - 588 * q^89 + 154 * q^91 - 1560 * q^92 - 1280 * q^94 - 510 * q^95 + 522 * q^97 - 49 * 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
 4.53113 −3.53113
−4.53113 0 12.5311 −5.00000 0 −7.00000 −20.5311 0 22.6556
1.2 3.53113 0 4.46887 −5.00000 0 −7.00000 −12.4689 0 −17.6556
 $$n$$: 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.i 2
3.b odd 2 1 105.4.a.f 2
5.b even 2 1 1575.4.a.w 2
7.b odd 2 1 2205.4.a.z 2
12.b even 2 1 1680.4.a.bg 2
15.d odd 2 1 525.4.a.k 2
15.e even 4 2 525.4.d.h 4
21.c even 2 1 735.4.a.p 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 16$$
$3$ $$T^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} - 22T + 56$$
$13$ $$T^{2} + 22T + 56$$
$17$ $$T^{2} + 116T - 796$$
$19$ $$T^{2} - 102T - 584$$
$23$ $$T^{2} + 260T + 10400$$
$29$ $$T^{2} - 196T - 27836$$
$31$ $$T^{2} - 150T - 23040$$
$37$ $$T^{2} + 96T - 18756$$
$41$ $$T^{2} - 176T - 154756$$
$43$ $$T^{2} + 344T + 12944$$
$47$ $$T^{2} + 560T + 40960$$
$53$ $$T^{2} + 326T - 93616$$
$59$ $$T^{2} - 844T + 63424$$
$61$ $$T^{2} + 204T + 1044$$
$67$ $$T^{2} + 104T - 63856$$
$71$ $$T^{2} + 1670 T + 668560$$
$73$ $$T^{2} + 386T - 625816$$
$79$ $$T^{2} + 888T + 21376$$
$83$ $$T^{2} + 928T - 542864$$
$89$ $$T^{2} + 588T + 85396$$
$97$ $$T^{2} - 522 T - 1534064$$
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