# Properties

 Label 8.4.b.a Level $8$ Weight $4$ Character orbit 8.b Analytic conductor $0.472$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,4,Mod(5,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("8.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$, degree $$1$$, 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: $$0.472015280046$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 2$$ x^2 - x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + 2 \beta q^{3} + (2 \beta - 6) q^{4} - 4 \beta q^{5} + ( - 2 \beta + 14) q^{6} - 8 q^{7} + (4 \beta + 20) q^{8} - q^{9} +O(q^{10})$$ q + (-b - 1) * q^2 + 2*b * q^3 + (2*b - 6) * q^4 - 4*b * q^5 + (-2*b + 14) * q^6 - 8 * q^7 + (4*b + 20) * q^8 - q^9 $$q + ( - \beta - 1) q^{2} + 2 \beta q^{3} + (2 \beta - 6) q^{4} - 4 \beta q^{5} + ( - 2 \beta + 14) q^{6} - 8 q^{7} + (4 \beta + 20) q^{8} - q^{9} + (4 \beta - 28) q^{10} - 6 \beta q^{11} + ( - 12 \beta - 28) q^{12} + 20 \beta q^{13} + (8 \beta + 8) q^{14} + 56 q^{15} + ( - 24 \beta + 8) q^{16} - 14 q^{17} + (\beta + 1) q^{18} - 14 \beta q^{19} + (24 \beta + 56) q^{20} - 16 \beta q^{21} + (6 \beta - 42) q^{22} - 152 q^{23} + (40 \beta - 56) q^{24} + 13 q^{25} + ( - 20 \beta + 140) q^{26} + 52 \beta q^{27} + ( - 16 \beta + 48) q^{28} - 60 \beta q^{29} + ( - 56 \beta - 56) q^{30} + 224 q^{31} + (16 \beta - 176) q^{32} + 84 q^{33} + (14 \beta + 14) q^{34} + 32 \beta q^{35} + ( - 2 \beta + 6) q^{36} + 92 \beta q^{37} + (14 \beta - 98) q^{38} - 280 q^{39} + ( - 80 \beta + 112) q^{40} - 70 q^{41} + (16 \beta - 112) q^{42} - 166 \beta q^{43} + (36 \beta + 84) q^{44} + 4 \beta q^{45} + (152 \beta + 152) q^{46} + 336 q^{47} + (16 \beta + 336) q^{48} - 279 q^{49} + ( - 13 \beta - 13) q^{50} - 28 \beta q^{51} + ( - 120 \beta - 280) q^{52} + 12 \beta q^{53} + ( - 52 \beta + 364) q^{54} - 168 q^{55} + ( - 32 \beta - 160) q^{56} + 196 q^{57} + (60 \beta - 420) q^{58} + 202 \beta q^{59} + (112 \beta - 336) q^{60} + 36 \beta q^{61} + ( - 224 \beta - 224) q^{62} + 8 q^{63} + (160 \beta + 288) q^{64} + 560 q^{65} + ( - 84 \beta - 84) q^{66} + 66 \beta q^{67} + ( - 28 \beta + 84) q^{68} - 304 \beta q^{69} + ( - 32 \beta + 224) q^{70} - 72 q^{71} + ( - 4 \beta - 20) q^{72} - 294 q^{73} + ( - 92 \beta + 644) q^{74} + 26 \beta q^{75} + (84 \beta + 196) q^{76} + 48 \beta q^{77} + (280 \beta + 280) q^{78} - 464 q^{79} + ( - 32 \beta - 672) q^{80} - 755 q^{81} + (70 \beta + 70) q^{82} - 206 \beta q^{83} + (96 \beta + 224) q^{84} + 56 \beta q^{85} + (166 \beta - 1162) q^{86} + 840 q^{87} + ( - 120 \beta + 168) q^{88} + 266 q^{89} + ( - 4 \beta + 28) q^{90} - 160 \beta q^{91} + ( - 304 \beta + 912) q^{92} + 448 \beta q^{93} + ( - 336 \beta - 336) q^{94} - 392 q^{95} + ( - 352 \beta - 224) q^{96} + 994 q^{97} + (279 \beta + 279) q^{98} + 6 \beta q^{99} +O(q^{100})$$ q + (-b - 1) * q^2 + 2*b * q^3 + (2*b - 6) * q^4 - 4*b * q^5 + (-2*b + 14) * q^6 - 8 * q^7 + (4*b + 20) * q^8 - q^9 + (4*b - 28) * q^10 - 6*b * q^11 + (-12*b - 28) * q^12 + 20*b * q^13 + (8*b + 8) * q^14 + 56 * q^15 + (-24*b + 8) * q^16 - 14 * q^17 + (b + 1) * q^18 - 14*b * q^19 + (24*b + 56) * q^20 - 16*b * q^21 + (6*b - 42) * q^22 - 152 * q^23 + (40*b - 56) * q^24 + 13 * q^25 + (-20*b + 140) * q^26 + 52*b * q^27 + (-16*b + 48) * q^28 - 60*b * q^29 + (-56*b - 56) * q^30 + 224 * q^31 + (16*b - 176) * q^32 + 84 * q^33 + (14*b + 14) * q^34 + 32*b * q^35 + (-2*b + 6) * q^36 + 92*b * q^37 + (14*b - 98) * q^38 - 280 * q^39 + (-80*b + 112) * q^40 - 70 * q^41 + (16*b - 112) * q^42 - 166*b * q^43 + (36*b + 84) * q^44 + 4*b * q^45 + (152*b + 152) * q^46 + 336 * q^47 + (16*b + 336) * q^48 - 279 * q^49 + (-13*b - 13) * q^50 - 28*b * q^51 + (-120*b - 280) * q^52 + 12*b * q^53 + (-52*b + 364) * q^54 - 168 * q^55 + (-32*b - 160) * q^56 + 196 * q^57 + (60*b - 420) * q^58 + 202*b * q^59 + (112*b - 336) * q^60 + 36*b * q^61 + (-224*b - 224) * q^62 + 8 * q^63 + (160*b + 288) * q^64 + 560 * q^65 + (-84*b - 84) * q^66 + 66*b * q^67 + (-28*b + 84) * q^68 - 304*b * q^69 + (-32*b + 224) * q^70 - 72 * q^71 + (-4*b - 20) * q^72 - 294 * q^73 + (-92*b + 644) * q^74 + 26*b * q^75 + (84*b + 196) * q^76 + 48*b * q^77 + (280*b + 280) * q^78 - 464 * q^79 + (-32*b - 672) * q^80 - 755 * q^81 + (70*b + 70) * q^82 - 206*b * q^83 + (96*b + 224) * q^84 + 56*b * q^85 + (166*b - 1162) * q^86 + 840 * q^87 + (-120*b + 168) * q^88 + 266 * q^89 + (-4*b + 28) * q^90 - 160*b * q^91 + (-304*b + 912) * q^92 + 448*b * q^93 + (-336*b - 336) * q^94 - 392 * q^95 + (-352*b - 224) * q^96 + 994 * q^97 + (279*b + 279) * q^98 + 6*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 12 q^{4} + 28 q^{6} - 16 q^{7} + 40 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 12 * q^4 + 28 * q^6 - 16 * q^7 + 40 * q^8 - 2 * q^9 $$2 q - 2 q^{2} - 12 q^{4} + 28 q^{6} - 16 q^{7} + 40 q^{8} - 2 q^{9} - 56 q^{10} - 56 q^{12} + 16 q^{14} + 112 q^{15} + 16 q^{16} - 28 q^{17} + 2 q^{18} + 112 q^{20} - 84 q^{22} - 304 q^{23} - 112 q^{24} + 26 q^{25} + 280 q^{26} + 96 q^{28} - 112 q^{30} + 448 q^{31} - 352 q^{32} + 168 q^{33} + 28 q^{34} + 12 q^{36} - 196 q^{38} - 560 q^{39} + 224 q^{40} - 140 q^{41} - 224 q^{42} + 168 q^{44} + 304 q^{46} + 672 q^{47} + 672 q^{48} - 558 q^{49} - 26 q^{50} - 560 q^{52} + 728 q^{54} - 336 q^{55} - 320 q^{56} + 392 q^{57} - 840 q^{58} - 672 q^{60} - 448 q^{62} + 16 q^{63} + 576 q^{64} + 1120 q^{65} - 168 q^{66} + 168 q^{68} + 448 q^{70} - 144 q^{71} - 40 q^{72} - 588 q^{73} + 1288 q^{74} + 392 q^{76} + 560 q^{78} - 928 q^{79} - 1344 q^{80} - 1510 q^{81} + 140 q^{82} + 448 q^{84} - 2324 q^{86} + 1680 q^{87} + 336 q^{88} + 532 q^{89} + 56 q^{90} + 1824 q^{92} - 672 q^{94} - 784 q^{95} - 448 q^{96} + 1988 q^{97} + 558 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 12 * q^4 + 28 * q^6 - 16 * q^7 + 40 * q^8 - 2 * q^9 - 56 * q^10 - 56 * q^12 + 16 * q^14 + 112 * q^15 + 16 * q^16 - 28 * q^17 + 2 * q^18 + 112 * q^20 - 84 * q^22 - 304 * q^23 - 112 * q^24 + 26 * q^25 + 280 * q^26 + 96 * q^28 - 112 * q^30 + 448 * q^31 - 352 * q^32 + 168 * q^33 + 28 * q^34 + 12 * q^36 - 196 * q^38 - 560 * q^39 + 224 * q^40 - 140 * q^41 - 224 * q^42 + 168 * q^44 + 304 * q^46 + 672 * q^47 + 672 * q^48 - 558 * q^49 - 26 * q^50 - 560 * q^52 + 728 * q^54 - 336 * q^55 - 320 * q^56 + 392 * q^57 - 840 * q^58 - 672 * q^60 - 448 * q^62 + 16 * q^63 + 576 * q^64 + 1120 * q^65 - 168 * q^66 + 168 * q^68 + 448 * q^70 - 144 * q^71 - 40 * q^72 - 588 * q^73 + 1288 * q^74 + 392 * q^76 + 560 * q^78 - 928 * q^79 - 1344 * q^80 - 1510 * q^81 + 140 * q^82 + 448 * q^84 - 2324 * q^86 + 1680 * q^87 + 336 * q^88 + 532 * q^89 + 56 * q^90 + 1824 * q^92 - 672 * q^94 - 784 * q^95 - 448 * q^96 + 1988 * q^97 + 558 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 + 1.32288i 0.5 − 1.32288i
−1.00000 2.64575i 5.29150i −6.00000 + 5.29150i 10.5830i 14.0000 5.29150i −8.00000 20.0000 + 10.5830i −1.00000 −28.0000 + 10.5830i
5.2 −1.00000 + 2.64575i 5.29150i −6.00000 5.29150i 10.5830i 14.0000 + 5.29150i −8.00000 20.0000 10.5830i −1.00000 −28.0000 10.5830i
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.4.b.a 2
3.b odd 2 1 72.4.d.b 2
4.b odd 2 1 32.4.b.a 2
5.b even 2 1 200.4.d.a 2
5.c odd 4 2 200.4.f.a 4
8.b even 2 1 inner 8.4.b.a 2
8.d odd 2 1 32.4.b.a 2
12.b even 2 1 288.4.d.a 2
16.e even 4 2 256.4.a.l 2
16.f odd 4 2 256.4.a.j 2
20.d odd 2 1 800.4.d.a 2
20.e even 4 2 800.4.f.a 4
24.f even 2 1 288.4.d.a 2
24.h odd 2 1 72.4.d.b 2
40.e odd 2 1 800.4.d.a 2
40.f even 2 1 200.4.d.a 2
40.i odd 4 2 200.4.f.a 4
40.k even 4 2 800.4.f.a 4
48.i odd 4 2 2304.4.a.bn 2
48.k even 4 2 2304.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 1.a even 1 1 trivial
8.4.b.a 2 8.b even 2 1 inner
32.4.b.a 2 4.b odd 2 1
32.4.b.a 2 8.d odd 2 1
72.4.d.b 2 3.b odd 2 1
72.4.d.b 2 24.h odd 2 1
200.4.d.a 2 5.b even 2 1
200.4.d.a 2 40.f even 2 1
200.4.f.a 4 5.c odd 4 2
200.4.f.a 4 40.i odd 4 2
256.4.a.j 2 16.f odd 4 2
256.4.a.l 2 16.e even 4 2
288.4.d.a 2 12.b even 2 1
288.4.d.a 2 24.f even 2 1
800.4.d.a 2 20.d odd 2 1
800.4.d.a 2 40.e odd 2 1
800.4.f.a 4 20.e even 4 2
800.4.f.a 4 40.k even 4 2
2304.4.a.v 2 48.k even 4 2
2304.4.a.bn 2 48.i odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 8$$
$3$ $$T^{2} + 28$$
$5$ $$T^{2} + 112$$
$7$ $$(T + 8)^{2}$$
$11$ $$T^{2} + 252$$
$13$ $$T^{2} + 2800$$
$17$ $$(T + 14)^{2}$$
$19$ $$T^{2} + 1372$$
$23$ $$(T + 152)^{2}$$
$29$ $$T^{2} + 25200$$
$31$ $$(T - 224)^{2}$$
$37$ $$T^{2} + 59248$$
$41$ $$(T + 70)^{2}$$
$43$ $$T^{2} + 192892$$
$47$ $$(T - 336)^{2}$$
$53$ $$T^{2} + 1008$$
$59$ $$T^{2} + 285628$$
$61$ $$T^{2} + 9072$$
$67$ $$T^{2} + 30492$$
$71$ $$(T + 72)^{2}$$
$73$ $$(T + 294)^{2}$$
$79$ $$(T + 464)^{2}$$
$83$ $$T^{2} + 297052$$
$89$ $$(T - 266)^{2}$$
$97$ $$(T - 994)^{2}$$