# Properties

 Label 10.4.b.a Level $10$ Weight $4$ Character orbit 10.b Analytic conductor $0.590$ 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] = [10,4,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$10 = 2 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 10.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: $$0.590019100057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - \beta q^{3} - 4 q^{4} + ( - 5 \beta - 5) q^{5} + 4 q^{6} + 13 \beta q^{7} - 4 \beta q^{8} + 23 q^{9} +O(q^{10})$$ q + b * q^2 - b * q^3 - 4 * q^4 + (-5*b - 5) * q^5 + 4 * q^6 + 13*b * q^7 - 4*b * q^8 + 23 * q^9 $$q + \beta q^{2} - \beta q^{3} - 4 q^{4} + ( - 5 \beta - 5) q^{5} + 4 q^{6} + 13 \beta q^{7} - 4 \beta q^{8} + 23 q^{9} + ( - 5 \beta + 20) q^{10} - 28 q^{11} + 4 \beta q^{12} - 6 \beta q^{13} - 52 q^{14} + (5 \beta - 20) q^{15} + 16 q^{16} - 32 \beta q^{17} + 23 \beta q^{18} + 60 q^{19} + (20 \beta + 20) q^{20} + 52 q^{21} - 28 \beta q^{22} + 29 \beta q^{23} - 16 q^{24} + (50 \beta - 75) q^{25} + 24 q^{26} - 50 \beta q^{27} - 52 \beta q^{28} - 90 q^{29} + ( - 20 \beta - 20) q^{30} - 128 q^{31} + 16 \beta q^{32} + 28 \beta q^{33} + 128 q^{34} + ( - 65 \beta + 260) q^{35} - 92 q^{36} + 118 \beta q^{37} + 60 \beta q^{38} - 24 q^{39} + (20 \beta - 80) q^{40} + 242 q^{41} + 52 \beta q^{42} - 181 \beta q^{43} + 112 q^{44} + ( - 115 \beta - 115) q^{45} - 116 q^{46} + 113 \beta q^{47} - 16 \beta q^{48} - 333 q^{49} + ( - 75 \beta - 200) q^{50} - 128 q^{51} + 24 \beta q^{52} + 54 \beta q^{53} + 200 q^{54} + (140 \beta + 140) q^{55} + 208 q^{56} - 60 \beta q^{57} - 90 \beta q^{58} + 20 q^{59} + ( - 20 \beta + 80) q^{60} + 542 q^{61} - 128 \beta q^{62} + 299 \beta q^{63} - 64 q^{64} + (30 \beta - 120) q^{65} - 112 q^{66} - 217 \beta q^{67} + 128 \beta q^{68} + 116 q^{69} + (260 \beta + 260) q^{70} - 1128 q^{71} - 92 \beta q^{72} - 316 \beta q^{73} - 472 q^{74} + (75 \beta + 200) q^{75} - 240 q^{76} - 364 \beta q^{77} - 24 \beta q^{78} + 720 q^{79} + ( - 80 \beta - 80) q^{80} + 421 q^{81} + 242 \beta q^{82} + 239 \beta q^{83} - 208 q^{84} + (160 \beta - 640) q^{85} + 724 q^{86} + 90 \beta q^{87} + 112 \beta q^{88} + 490 q^{89} + ( - 115 \beta + 460) q^{90} + 312 q^{91} - 116 \beta q^{92} + 128 \beta q^{93} - 452 q^{94} + ( - 300 \beta - 300) q^{95} + 64 q^{96} + 728 \beta q^{97} - 333 \beta q^{98} - 644 q^{99} +O(q^{100})$$ q + b * q^2 - b * q^3 - 4 * q^4 + (-5*b - 5) * q^5 + 4 * q^6 + 13*b * q^7 - 4*b * q^8 + 23 * q^9 + (-5*b + 20) * q^10 - 28 * q^11 + 4*b * q^12 - 6*b * q^13 - 52 * q^14 + (5*b - 20) * q^15 + 16 * q^16 - 32*b * q^17 + 23*b * q^18 + 60 * q^19 + (20*b + 20) * q^20 + 52 * q^21 - 28*b * q^22 + 29*b * q^23 - 16 * q^24 + (50*b - 75) * q^25 + 24 * q^26 - 50*b * q^27 - 52*b * q^28 - 90 * q^29 + (-20*b - 20) * q^30 - 128 * q^31 + 16*b * q^32 + 28*b * q^33 + 128 * q^34 + (-65*b + 260) * q^35 - 92 * q^36 + 118*b * q^37 + 60*b * q^38 - 24 * q^39 + (20*b - 80) * q^40 + 242 * q^41 + 52*b * q^42 - 181*b * q^43 + 112 * q^44 + (-115*b - 115) * q^45 - 116 * q^46 + 113*b * q^47 - 16*b * q^48 - 333 * q^49 + (-75*b - 200) * q^50 - 128 * q^51 + 24*b * q^52 + 54*b * q^53 + 200 * q^54 + (140*b + 140) * q^55 + 208 * q^56 - 60*b * q^57 - 90*b * q^58 + 20 * q^59 + (-20*b + 80) * q^60 + 542 * q^61 - 128*b * q^62 + 299*b * q^63 - 64 * q^64 + (30*b - 120) * q^65 - 112 * q^66 - 217*b * q^67 + 128*b * q^68 + 116 * q^69 + (260*b + 260) * q^70 - 1128 * q^71 - 92*b * q^72 - 316*b * q^73 - 472 * q^74 + (75*b + 200) * q^75 - 240 * q^76 - 364*b * q^77 - 24*b * q^78 + 720 * q^79 + (-80*b - 80) * q^80 + 421 * q^81 + 242*b * q^82 + 239*b * q^83 - 208 * q^84 + (160*b - 640) * q^85 + 724 * q^86 + 90*b * q^87 + 112*b * q^88 + 490 * q^89 + (-115*b + 460) * q^90 + 312 * q^91 - 116*b * q^92 + 128*b * q^93 - 452 * q^94 + (-300*b - 300) * q^95 + 64 * q^96 + 728*b * q^97 - 333*b * q^98 - 644 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9 $$2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9} + 40 q^{10} - 56 q^{11} - 104 q^{14} - 40 q^{15} + 32 q^{16} + 120 q^{19} + 40 q^{20} + 104 q^{21} - 32 q^{24} - 150 q^{25} + 48 q^{26} - 180 q^{29} - 40 q^{30} - 256 q^{31} + 256 q^{34} + 520 q^{35} - 184 q^{36} - 48 q^{39} - 160 q^{40} + 484 q^{41} + 224 q^{44} - 230 q^{45} - 232 q^{46} - 666 q^{49} - 400 q^{50} - 256 q^{51} + 400 q^{54} + 280 q^{55} + 416 q^{56} + 40 q^{59} + 160 q^{60} + 1084 q^{61} - 128 q^{64} - 240 q^{65} - 224 q^{66} + 232 q^{69} + 520 q^{70} - 2256 q^{71} - 944 q^{74} + 400 q^{75} - 480 q^{76} + 1440 q^{79} - 160 q^{80} + 842 q^{81} - 416 q^{84} - 1280 q^{85} + 1448 q^{86} + 980 q^{89} + 920 q^{90} + 624 q^{91} - 904 q^{94} - 600 q^{95} + 128 q^{96} - 1288 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9 + 40 * q^10 - 56 * q^11 - 104 * q^14 - 40 * q^15 + 32 * q^16 + 120 * q^19 + 40 * q^20 + 104 * q^21 - 32 * q^24 - 150 * q^25 + 48 * q^26 - 180 * q^29 - 40 * q^30 - 256 * q^31 + 256 * q^34 + 520 * q^35 - 184 * q^36 - 48 * q^39 - 160 * q^40 + 484 * q^41 + 224 * q^44 - 230 * q^45 - 232 * q^46 - 666 * q^49 - 400 * q^50 - 256 * q^51 + 400 * q^54 + 280 * q^55 + 416 * q^56 + 40 * q^59 + 160 * q^60 + 1084 * q^61 - 128 * q^64 - 240 * q^65 - 224 * q^66 + 232 * q^69 + 520 * q^70 - 2256 * q^71 - 944 * q^74 + 400 * q^75 - 480 * q^76 + 1440 * q^79 - 160 * q^80 + 842 * q^81 - 416 * q^84 - 1280 * q^85 + 1448 * q^86 + 980 * q^89 + 920 * q^90 + 624 * q^91 - 904 * q^94 - 600 * q^95 + 128 * q^96 - 1288 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 − 1.00000i 1.00000i
2.00000i 2.00000i −4.00000 −5.00000 + 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 + 10.0000i
9.2 2.00000i 2.00000i −4.00000 −5.00000 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 10.0000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.4.b.a 2
3.b odd 2 1 90.4.c.b 2
4.b odd 2 1 80.4.c.a 2
5.b even 2 1 inner 10.4.b.a 2
5.c odd 4 1 50.4.a.b 1
5.c odd 4 1 50.4.a.d 1
7.b odd 2 1 490.4.c.b 2
8.b even 2 1 320.4.c.d 2
8.d odd 2 1 320.4.c.c 2
12.b even 2 1 720.4.f.f 2
15.d odd 2 1 90.4.c.b 2
15.e even 4 1 450.4.a.j 1
15.e even 4 1 450.4.a.k 1
20.d odd 2 1 80.4.c.a 2
20.e even 4 1 400.4.a.h 1
20.e even 4 1 400.4.a.n 1
35.c odd 2 1 490.4.c.b 2
35.f even 4 1 2450.4.a.o 1
35.f even 4 1 2450.4.a.bb 1
40.e odd 2 1 320.4.c.c 2
40.f even 2 1 320.4.c.d 2
40.i odd 4 1 1600.4.a.u 1
40.i odd 4 1 1600.4.a.bh 1
40.k even 4 1 1600.4.a.t 1
40.k even 4 1 1600.4.a.bg 1
60.h even 2 1 720.4.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 1.a even 1 1 trivial
10.4.b.a 2 5.b even 2 1 inner
50.4.a.b 1 5.c odd 4 1
50.4.a.d 1 5.c odd 4 1
80.4.c.a 2 4.b odd 2 1
80.4.c.a 2 20.d odd 2 1
90.4.c.b 2 3.b odd 2 1
90.4.c.b 2 15.d odd 2 1
320.4.c.c 2 8.d odd 2 1
320.4.c.c 2 40.e odd 2 1
320.4.c.d 2 8.b even 2 1
320.4.c.d 2 40.f even 2 1
400.4.a.h 1 20.e even 4 1
400.4.a.n 1 20.e even 4 1
450.4.a.j 1 15.e even 4 1
450.4.a.k 1 15.e even 4 1
490.4.c.b 2 7.b odd 2 1
490.4.c.b 2 35.c odd 2 1
720.4.f.f 2 12.b even 2 1
720.4.f.f 2 60.h even 2 1
1600.4.a.t 1 40.k even 4 1
1600.4.a.u 1 40.i odd 4 1
1600.4.a.bg 1 40.k even 4 1
1600.4.a.bh 1 40.i odd 4 1
2450.4.a.o 1 35.f even 4 1
2450.4.a.bb 1 35.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(10, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 10T + 125$$
$7$ $$T^{2} + 676$$
$11$ $$(T + 28)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 4096$$
$19$ $$(T - 60)^{2}$$
$23$ $$T^{2} + 3364$$
$29$ $$(T + 90)^{2}$$
$31$ $$(T + 128)^{2}$$
$37$ $$T^{2} + 55696$$
$41$ $$(T - 242)^{2}$$
$43$ $$T^{2} + 131044$$
$47$ $$T^{2} + 51076$$
$53$ $$T^{2} + 11664$$
$59$ $$(T - 20)^{2}$$
$61$ $$(T - 542)^{2}$$
$67$ $$T^{2} + 188356$$
$71$ $$(T + 1128)^{2}$$
$73$ $$T^{2} + 399424$$
$79$ $$(T - 720)^{2}$$
$83$ $$T^{2} + 228484$$
$89$ $$(T - 490)^{2}$$
$97$ $$T^{2} + 2119936$$