# Properties

 Label 10.4.a.a Level $10$ Weight $4$ Character orbit 10.a Self dual yes Analytic conductor $0.590$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,4,Mod(1,10)]

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([0]))

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

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

chi := DirichletCharacter("10.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$10 = 2 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 10.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: $$0.590019100057$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 8 q^{3} + 4 q^{4} + 5 q^{5} - 16 q^{6} - 4 q^{7} + 8 q^{8} + 37 q^{9}+O(q^{10})$$ q + 2 * q^2 - 8 * q^3 + 4 * q^4 + 5 * q^5 - 16 * q^6 - 4 * q^7 + 8 * q^8 + 37 * q^9 $$q + 2 q^{2} - 8 q^{3} + 4 q^{4} + 5 q^{5} - 16 q^{6} - 4 q^{7} + 8 q^{8} + 37 q^{9} + 10 q^{10} + 12 q^{11} - 32 q^{12} - 58 q^{13} - 8 q^{14} - 40 q^{15} + 16 q^{16} + 66 q^{17} + 74 q^{18} - 100 q^{19} + 20 q^{20} + 32 q^{21} + 24 q^{22} + 132 q^{23} - 64 q^{24} + 25 q^{25} - 116 q^{26} - 80 q^{27} - 16 q^{28} - 90 q^{29} - 80 q^{30} + 152 q^{31} + 32 q^{32} - 96 q^{33} + 132 q^{34} - 20 q^{35} + 148 q^{36} - 34 q^{37} - 200 q^{38} + 464 q^{39} + 40 q^{40} - 438 q^{41} + 64 q^{42} + 32 q^{43} + 48 q^{44} + 185 q^{45} + 264 q^{46} - 204 q^{47} - 128 q^{48} - 327 q^{49} + 50 q^{50} - 528 q^{51} - 232 q^{52} + 222 q^{53} - 160 q^{54} + 60 q^{55} - 32 q^{56} + 800 q^{57} - 180 q^{58} + 420 q^{59} - 160 q^{60} + 902 q^{61} + 304 q^{62} - 148 q^{63} + 64 q^{64} - 290 q^{65} - 192 q^{66} - 1024 q^{67} + 264 q^{68} - 1056 q^{69} - 40 q^{70} + 432 q^{71} + 296 q^{72} + 362 q^{73} - 68 q^{74} - 200 q^{75} - 400 q^{76} - 48 q^{77} + 928 q^{78} - 160 q^{79} + 80 q^{80} - 359 q^{81} - 876 q^{82} + 72 q^{83} + 128 q^{84} + 330 q^{85} + 64 q^{86} + 720 q^{87} + 96 q^{88} + 810 q^{89} + 370 q^{90} + 232 q^{91} + 528 q^{92} - 1216 q^{93} - 408 q^{94} - 500 q^{95} - 256 q^{96} + 1106 q^{97} - 654 q^{98} + 444 q^{99}+O(q^{100})$$ q + 2 * q^2 - 8 * q^3 + 4 * q^4 + 5 * q^5 - 16 * q^6 - 4 * q^7 + 8 * q^8 + 37 * q^9 + 10 * q^10 + 12 * q^11 - 32 * q^12 - 58 * q^13 - 8 * q^14 - 40 * q^15 + 16 * q^16 + 66 * q^17 + 74 * q^18 - 100 * q^19 + 20 * q^20 + 32 * q^21 + 24 * q^22 + 132 * q^23 - 64 * q^24 + 25 * q^25 - 116 * q^26 - 80 * q^27 - 16 * q^28 - 90 * q^29 - 80 * q^30 + 152 * q^31 + 32 * q^32 - 96 * q^33 + 132 * q^34 - 20 * q^35 + 148 * q^36 - 34 * q^37 - 200 * q^38 + 464 * q^39 + 40 * q^40 - 438 * q^41 + 64 * q^42 + 32 * q^43 + 48 * q^44 + 185 * q^45 + 264 * q^46 - 204 * q^47 - 128 * q^48 - 327 * q^49 + 50 * q^50 - 528 * q^51 - 232 * q^52 + 222 * q^53 - 160 * q^54 + 60 * q^55 - 32 * q^56 + 800 * q^57 - 180 * q^58 + 420 * q^59 - 160 * q^60 + 902 * q^61 + 304 * q^62 - 148 * q^63 + 64 * q^64 - 290 * q^65 - 192 * q^66 - 1024 * q^67 + 264 * q^68 - 1056 * q^69 - 40 * q^70 + 432 * q^71 + 296 * q^72 + 362 * q^73 - 68 * q^74 - 200 * q^75 - 400 * q^76 - 48 * q^77 + 928 * q^78 - 160 * q^79 + 80 * q^80 - 359 * q^81 - 876 * q^82 + 72 * q^83 + 128 * q^84 + 330 * q^85 + 64 * q^86 + 720 * q^87 + 96 * q^88 + 810 * q^89 + 370 * q^90 + 232 * q^91 + 528 * q^92 - 1216 * q^93 - 408 * q^94 - 500 * q^95 - 256 * q^96 + 1106 * q^97 - 654 * q^98 + 444 * q^99

## 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
 0
2.00000 −8.00000 4.00000 5.00000 −16.0000 −4.00000 8.00000 37.0000 10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## 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 10.4.a.a 1
3.b odd 2 1 90.4.a.a 1
4.b odd 2 1 80.4.a.f 1
5.b even 2 1 50.4.a.c 1
5.c odd 4 2 50.4.b.a 2
7.b odd 2 1 490.4.a.o 1
7.c even 3 2 490.4.e.i 2
7.d odd 6 2 490.4.e.a 2
8.b even 2 1 320.4.a.m 1
8.d odd 2 1 320.4.a.b 1
9.c even 3 2 810.4.e.c 2
9.d odd 6 2 810.4.e.w 2
11.b odd 2 1 1210.4.a.b 1
12.b even 2 1 720.4.a.j 1
13.b even 2 1 1690.4.a.a 1
15.d odd 2 1 450.4.a.q 1
15.e even 4 2 450.4.c.d 2
16.e even 4 2 1280.4.d.j 2
16.f odd 4 2 1280.4.d.g 2
20.d odd 2 1 400.4.a.b 1
20.e even 4 2 400.4.c.c 2
35.c odd 2 1 2450.4.a.b 1
40.e odd 2 1 1600.4.a.bx 1
40.f even 2 1 1600.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 1.a even 1 1 trivial
50.4.a.c 1 5.b even 2 1
50.4.b.a 2 5.c odd 4 2
80.4.a.f 1 4.b odd 2 1
90.4.a.a 1 3.b odd 2 1
320.4.a.b 1 8.d odd 2 1
320.4.a.m 1 8.b even 2 1
400.4.a.b 1 20.d odd 2 1
400.4.c.c 2 20.e even 4 2
450.4.a.q 1 15.d odd 2 1
450.4.c.d 2 15.e even 4 2
490.4.a.o 1 7.b odd 2 1
490.4.e.a 2 7.d odd 6 2
490.4.e.i 2 7.c even 3 2
720.4.a.j 1 12.b even 2 1
810.4.e.c 2 9.c even 3 2
810.4.e.w 2 9.d odd 6 2
1210.4.a.b 1 11.b odd 2 1
1280.4.d.g 2 16.f odd 4 2
1280.4.d.j 2 16.e even 4 2
1600.4.a.d 1 40.f even 2 1
1600.4.a.bx 1 40.e odd 2 1
1690.4.a.a 1 13.b even 2 1
2450.4.a.b 1 35.c odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(10))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 8$$
$5$ $$T - 5$$
$7$ $$T + 4$$
$11$ $$T - 12$$
$13$ $$T + 58$$
$17$ $$T - 66$$
$19$ $$T + 100$$
$23$ $$T - 132$$
$29$ $$T + 90$$
$31$ $$T - 152$$
$37$ $$T + 34$$
$41$ $$T + 438$$
$43$ $$T - 32$$
$47$ $$T + 204$$
$53$ $$T - 222$$
$59$ $$T - 420$$
$61$ $$T - 902$$
$67$ $$T + 1024$$
$71$ $$T - 432$$
$73$ $$T - 362$$
$79$ $$T + 160$$
$83$ $$T - 72$$
$89$ $$T - 810$$
$97$ $$T - 1106$$