# Properties

 Label 50.4.a.b Level $50$ Weight $4$ Character orbit 50.a Self dual yes Analytic conductor $2.950$ 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] = [50,4,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, 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("50.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 50.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: $$2.95009550029$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 26 q^{7} - 8 q^{8} - 23 q^{9}+O(q^{10})$$ q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^6 - 26 * q^7 - 8 * q^8 - 23 * q^9 $$q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 26 q^{7} - 8 q^{8} - 23 q^{9} - 28 q^{11} - 8 q^{12} - 12 q^{13} + 52 q^{14} + 16 q^{16} + 64 q^{17} + 46 q^{18} - 60 q^{19} + 52 q^{21} + 56 q^{22} + 58 q^{23} + 16 q^{24} + 24 q^{26} + 100 q^{27} - 104 q^{28} + 90 q^{29} - 128 q^{31} - 32 q^{32} + 56 q^{33} - 128 q^{34} - 92 q^{36} - 236 q^{37} + 120 q^{38} + 24 q^{39} + 242 q^{41} - 104 q^{42} - 362 q^{43} - 112 q^{44} - 116 q^{46} - 226 q^{47} - 32 q^{48} + 333 q^{49} - 128 q^{51} - 48 q^{52} + 108 q^{53} - 200 q^{54} + 208 q^{56} + 120 q^{57} - 180 q^{58} - 20 q^{59} + 542 q^{61} + 256 q^{62} + 598 q^{63} + 64 q^{64} - 112 q^{66} + 434 q^{67} + 256 q^{68} - 116 q^{69} - 1128 q^{71} + 184 q^{72} - 632 q^{73} + 472 q^{74} - 240 q^{76} + 728 q^{77} - 48 q^{78} - 720 q^{79} + 421 q^{81} - 484 q^{82} + 478 q^{83} + 208 q^{84} + 724 q^{86} - 180 q^{87} + 224 q^{88} - 490 q^{89} + 312 q^{91} + 232 q^{92} + 256 q^{93} + 452 q^{94} + 64 q^{96} - 1456 q^{97} - 666 q^{98} + 644 q^{99}+O(q^{100})$$ q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^6 - 26 * q^7 - 8 * q^8 - 23 * q^9 - 28 * q^11 - 8 * q^12 - 12 * q^13 + 52 * q^14 + 16 * q^16 + 64 * q^17 + 46 * q^18 - 60 * q^19 + 52 * q^21 + 56 * q^22 + 58 * q^23 + 16 * q^24 + 24 * q^26 + 100 * q^27 - 104 * q^28 + 90 * q^29 - 128 * q^31 - 32 * q^32 + 56 * q^33 - 128 * q^34 - 92 * q^36 - 236 * q^37 + 120 * q^38 + 24 * q^39 + 242 * q^41 - 104 * q^42 - 362 * q^43 - 112 * q^44 - 116 * q^46 - 226 * q^47 - 32 * q^48 + 333 * q^49 - 128 * q^51 - 48 * q^52 + 108 * q^53 - 200 * q^54 + 208 * q^56 + 120 * q^57 - 180 * q^58 - 20 * q^59 + 542 * q^61 + 256 * q^62 + 598 * q^63 + 64 * q^64 - 112 * q^66 + 434 * q^67 + 256 * q^68 - 116 * q^69 - 1128 * q^71 + 184 * q^72 - 632 * q^73 + 472 * q^74 - 240 * q^76 + 728 * q^77 - 48 * q^78 - 720 * q^79 + 421 * q^81 - 484 * q^82 + 478 * q^83 + 208 * q^84 + 724 * q^86 - 180 * q^87 + 224 * q^88 - 490 * q^89 + 312 * q^91 + 232 * q^92 + 256 * q^93 + 452 * q^94 + 64 * q^96 - 1456 * q^97 - 666 * q^98 + 644 * 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 −2.00000 4.00000 0 4.00000 −26.0000 −8.00000 −23.0000 0
 $$n$$: 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 50.4.a.b 1
3.b odd 2 1 450.4.a.k 1
4.b odd 2 1 400.4.a.n 1
5.b even 2 1 50.4.a.d 1
5.c odd 4 2 10.4.b.a 2
7.b odd 2 1 2450.4.a.o 1
8.b even 2 1 1600.4.a.bh 1
8.d odd 2 1 1600.4.a.t 1
15.d odd 2 1 450.4.a.j 1
15.e even 4 2 90.4.c.b 2
20.d odd 2 1 400.4.a.h 1
20.e even 4 2 80.4.c.a 2
35.c odd 2 1 2450.4.a.bb 1
35.f even 4 2 490.4.c.b 2
40.e odd 2 1 1600.4.a.bg 1
40.f even 2 1 1600.4.a.u 1
40.i odd 4 2 320.4.c.d 2
40.k even 4 2 320.4.c.c 2
60.l odd 4 2 720.4.f.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(50))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T + 26$$
$11$ $$T + 28$$
$13$ $$T + 12$$
$17$ $$T - 64$$
$19$ $$T + 60$$
$23$ $$T - 58$$
$29$ $$T - 90$$
$31$ $$T + 128$$
$37$ $$T + 236$$
$41$ $$T - 242$$
$43$ $$T + 362$$
$47$ $$T + 226$$
$53$ $$T - 108$$
$59$ $$T + 20$$
$61$ $$T - 542$$
$67$ $$T - 434$$
$71$ $$T + 1128$$
$73$ $$T + 632$$
$79$ $$T + 720$$
$83$ $$T - 478$$
$89$ $$T + 490$$
$97$ $$T + 1456$$