# Properties

 Label 50.6.a.d Level $50$ Weight $6$ Character orbit 50.a Self dual yes Analytic conductor $8.019$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 50.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.01919099065$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + 64q^{8} + 333q^{9} + O(q^{10})$$ $$q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + 64q^{8} + 333q^{9} + 132q^{11} - 384q^{12} + 946q^{13} + 688q^{14} + 256q^{16} + 222q^{17} + 1332q^{18} + 500q^{19} - 4128q^{21} + 528q^{22} - 3564q^{23} - 1536q^{24} + 3784q^{26} - 2160q^{27} + 2752q^{28} + 2190q^{29} + 2312q^{31} + 1024q^{32} - 3168q^{33} + 888q^{34} + 5328q^{36} + 11242q^{37} + 2000q^{38} - 22704q^{39} + 1242q^{41} - 16512q^{42} - 20624q^{43} + 2112q^{44} - 14256q^{46} - 6588q^{47} - 6144q^{48} + 12777q^{49} - 5328q^{51} + 15136q^{52} + 21066q^{53} - 8640q^{54} + 11008q^{56} - 12000q^{57} + 8760q^{58} + 7980q^{59} + 16622q^{61} + 9248q^{62} + 57276q^{63} + 4096q^{64} - 12672q^{66} - 1808q^{67} + 3552q^{68} + 85536q^{69} - 24528q^{71} + 21312q^{72} - 20474q^{73} + 44968q^{74} + 8000q^{76} + 22704q^{77} - 90816q^{78} - 46240q^{79} - 29079q^{81} + 4968q^{82} + 51576q^{83} - 66048q^{84} - 82496q^{86} - 52560q^{87} + 8448q^{88} - 110310q^{89} + 162712q^{91} - 57024q^{92} - 55488q^{93} - 26352q^{94} - 24576q^{96} + 78382q^{97} + 51108q^{98} + 43956q^{99} + O(q^{100})$$

## 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.

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
4.00000 −24.0000 16.0000 0 −96.0000 172.000 64.0000 333.000 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 50.6.a.d 1
3.b odd 2 1 450.6.a.l 1
4.b odd 2 1 400.6.a.n 1
5.b even 2 1 10.6.a.b 1
5.c odd 4 2 50.6.b.a 2
15.d odd 2 1 90.6.a.d 1
15.e even 4 2 450.6.c.h 2
20.d odd 2 1 80.6.a.a 1
20.e even 4 2 400.6.c.b 2
35.c odd 2 1 490.6.a.a 1
40.e odd 2 1 320.6.a.o 1
40.f even 2 1 320.6.a.b 1
60.h even 2 1 720.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 5.b even 2 1
50.6.a.d 1 1.a even 1 1 trivial
50.6.b.a 2 5.c odd 4 2
80.6.a.a 1 20.d odd 2 1
90.6.a.d 1 15.d odd 2 1
320.6.a.b 1 40.f even 2 1
320.6.a.o 1 40.e odd 2 1
400.6.a.n 1 4.b odd 2 1
400.6.c.b 2 20.e even 4 2
450.6.a.l 1 3.b odd 2 1
450.6.c.h 2 15.e even 4 2
490.6.a.a 1 35.c odd 2 1
720.6.a.j 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 24$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(50))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$24 + T$$
$5$ $$T$$
$7$ $$-172 + T$$
$11$ $$-132 + T$$
$13$ $$-946 + T$$
$17$ $$-222 + T$$
$19$ $$-500 + T$$
$23$ $$3564 + T$$
$29$ $$-2190 + T$$
$31$ $$-2312 + T$$
$37$ $$-11242 + T$$
$41$ $$-1242 + T$$
$43$ $$20624 + T$$
$47$ $$6588 + T$$
$53$ $$-21066 + T$$
$59$ $$-7980 + T$$
$61$ $$-16622 + T$$
$67$ $$1808 + T$$
$71$ $$24528 + T$$
$73$ $$20474 + T$$
$79$ $$46240 + T$$
$83$ $$-51576 + T$$
$89$ $$110310 + T$$
$97$ $$-78382 + T$$