# Properties

 Label 90.4.a.a Level $90$ Weight $4$ Character orbit 90.a Self dual yes Analytic conductor $5.310$ Analytic rank $1$ 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] = [90,4,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("90.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 90.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: $$5.31017190052$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 8 q^{8} + 10 q^{10} - 12 q^{11} - 58 q^{13} + 8 q^{14} + 16 q^{16} - 66 q^{17} - 100 q^{19} - 20 q^{20} + 24 q^{22} - 132 q^{23} + 25 q^{25} + 116 q^{26} - 16 q^{28} + 90 q^{29} + 152 q^{31} - 32 q^{32} + 132 q^{34} + 20 q^{35} - 34 q^{37} + 200 q^{38} + 40 q^{40} + 438 q^{41} + 32 q^{43} - 48 q^{44} + 264 q^{46} + 204 q^{47} - 327 q^{49} - 50 q^{50} - 232 q^{52} - 222 q^{53} + 60 q^{55} + 32 q^{56} - 180 q^{58} - 420 q^{59} + 902 q^{61} - 304 q^{62} + 64 q^{64} + 290 q^{65} - 1024 q^{67} - 264 q^{68} - 40 q^{70} - 432 q^{71} + 362 q^{73} + 68 q^{74} - 400 q^{76} + 48 q^{77} - 160 q^{79} - 80 q^{80} - 876 q^{82} - 72 q^{83} + 330 q^{85} - 64 q^{86} + 96 q^{88} - 810 q^{89} + 232 q^{91} - 528 q^{92} - 408 q^{94} + 500 q^{95} + 1106 q^{97} + 654 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 8 * q^8 + 10 * q^10 - 12 * q^11 - 58 * q^13 + 8 * q^14 + 16 * q^16 - 66 * q^17 - 100 * q^19 - 20 * q^20 + 24 * q^22 - 132 * q^23 + 25 * q^25 + 116 * q^26 - 16 * q^28 + 90 * q^29 + 152 * q^31 - 32 * q^32 + 132 * q^34 + 20 * q^35 - 34 * q^37 + 200 * q^38 + 40 * q^40 + 438 * q^41 + 32 * q^43 - 48 * q^44 + 264 * q^46 + 204 * q^47 - 327 * q^49 - 50 * q^50 - 232 * q^52 - 222 * q^53 + 60 * q^55 + 32 * q^56 - 180 * q^58 - 420 * q^59 + 902 * q^61 - 304 * q^62 + 64 * q^64 + 290 * q^65 - 1024 * q^67 - 264 * q^68 - 40 * q^70 - 432 * q^71 + 362 * q^73 + 68 * q^74 - 400 * q^76 + 48 * q^77 - 160 * q^79 - 80 * q^80 - 876 * q^82 - 72 * q^83 + 330 * q^85 - 64 * q^86 + 96 * q^88 - 810 * q^89 + 232 * q^91 - 528 * q^92 - 408 * q^94 + 500 * q^95 + 1106 * q^97 + 654 * 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
 0
−2.00000 0 4.00000 −5.00000 0 −4.00000 −8.00000 0 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$

## 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 90.4.a.a 1
3.b odd 2 1 10.4.a.a 1
4.b odd 2 1 720.4.a.j 1
5.b even 2 1 450.4.a.q 1
5.c odd 4 2 450.4.c.d 2
9.c even 3 2 810.4.e.w 2
9.d odd 6 2 810.4.e.c 2
12.b even 2 1 80.4.a.f 1
15.d odd 2 1 50.4.a.c 1
15.e even 4 2 50.4.b.a 2
21.c even 2 1 490.4.a.o 1
21.g even 6 2 490.4.e.a 2
21.h odd 6 2 490.4.e.i 2
24.f even 2 1 320.4.a.b 1
24.h odd 2 1 320.4.a.m 1
33.d even 2 1 1210.4.a.b 1
39.d odd 2 1 1690.4.a.a 1
48.i odd 4 2 1280.4.d.j 2
48.k even 4 2 1280.4.d.g 2
60.h even 2 1 400.4.a.b 1
60.l odd 4 2 400.4.c.c 2
105.g even 2 1 2450.4.a.b 1
120.i odd 2 1 1600.4.a.d 1
120.m even 2 1 1600.4.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 3.b odd 2 1
50.4.a.c 1 15.d odd 2 1
50.4.b.a 2 15.e even 4 2
80.4.a.f 1 12.b even 2 1
90.4.a.a 1 1.a even 1 1 trivial
320.4.a.b 1 24.f even 2 1
320.4.a.m 1 24.h odd 2 1
400.4.a.b 1 60.h even 2 1
400.4.c.c 2 60.l odd 4 2
450.4.a.q 1 5.b even 2 1
450.4.c.d 2 5.c odd 4 2
490.4.a.o 1 21.c even 2 1
490.4.e.a 2 21.g even 6 2
490.4.e.i 2 21.h odd 6 2
720.4.a.j 1 4.b odd 2 1
810.4.e.c 2 9.d odd 6 2
810.4.e.w 2 9.c even 3 2
1210.4.a.b 1 33.d even 2 1
1280.4.d.g 2 48.k even 4 2
1280.4.d.j 2 48.i odd 4 2
1600.4.a.d 1 120.i odd 2 1
1600.4.a.bx 1 120.m even 2 1
1690.4.a.a 1 39.d odd 2 1
2450.4.a.b 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(90))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

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