# Properties

 Label 50.6.b.a Level $50$ Weight $6$ Character orbit 50.b Analytic conductor $8.019$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 50.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01919099065$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 24 i q^{3} -16 q^{4} -96 q^{6} + 172 i q^{7} -64 i q^{8} -333 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 24 i q^{3} -16 q^{4} -96 q^{6} + 172 i q^{7} -64 i q^{8} -333 q^{9} + 132 q^{11} -384 i q^{12} -946 i q^{13} -688 q^{14} + 256 q^{16} + 222 i q^{17} -1332 i q^{18} -500 q^{19} -4128 q^{21} + 528 i q^{22} + 3564 i q^{23} + 1536 q^{24} + 3784 q^{26} -2160 i q^{27} -2752 i q^{28} -2190 q^{29} + 2312 q^{31} + 1024 i q^{32} + 3168 i q^{33} -888 q^{34} + 5328 q^{36} + 11242 i q^{37} -2000 i q^{38} + 22704 q^{39} + 1242 q^{41} -16512 i q^{42} + 20624 i q^{43} -2112 q^{44} -14256 q^{46} -6588 i q^{47} + 6144 i q^{48} -12777 q^{49} -5328 q^{51} + 15136 i q^{52} -21066 i q^{53} + 8640 q^{54} + 11008 q^{56} -12000 i q^{57} -8760 i q^{58} -7980 q^{59} + 16622 q^{61} + 9248 i q^{62} -57276 i q^{63} -4096 q^{64} -12672 q^{66} -1808 i q^{67} -3552 i q^{68} -85536 q^{69} -24528 q^{71} + 21312 i q^{72} + 20474 i q^{73} -44968 q^{74} + 8000 q^{76} + 22704 i q^{77} + 90816 i q^{78} + 46240 q^{79} -29079 q^{81} + 4968 i q^{82} -51576 i q^{83} + 66048 q^{84} -82496 q^{86} -52560 i q^{87} -8448 i q^{88} + 110310 q^{89} + 162712 q^{91} -57024 i q^{92} + 55488 i q^{93} + 26352 q^{94} -24576 q^{96} + 78382 i q^{97} -51108 i q^{98} -43956 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} - 192q^{6} - 666q^{9} + O(q^{10})$$ $$2q - 32q^{4} - 192q^{6} - 666q^{9} + 264q^{11} - 1376q^{14} + 512q^{16} - 1000q^{19} - 8256q^{21} + 3072q^{24} + 7568q^{26} - 4380q^{29} + 4624q^{31} - 1776q^{34} + 10656q^{36} + 45408q^{39} + 2484q^{41} - 4224q^{44} - 28512q^{46} - 25554q^{49} - 10656q^{51} + 17280q^{54} + 22016q^{56} - 15960q^{59} + 33244q^{61} - 8192q^{64} - 25344q^{66} - 171072q^{69} - 49056q^{71} - 89936q^{74} + 16000q^{76} + 92480q^{79} - 58158q^{81} + 132096q^{84} - 164992q^{86} + 220620q^{89} + 325424q^{91} + 52704q^{94} - 49152q^{96} - 87912q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
4.00000i 24.0000i −16.0000 0 −96.0000 172.000i 64.0000i −333.000 0
49.2 4.00000i 24.0000i −16.0000 0 −96.0000 172.000i 64.0000i −333.000 0
 $$n$$: 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 50.6.b.a 2
3.b odd 2 1 450.6.c.h 2
4.b odd 2 1 400.6.c.b 2
5.b even 2 1 inner 50.6.b.a 2
5.c odd 4 1 10.6.a.b 1
5.c odd 4 1 50.6.a.d 1
15.d odd 2 1 450.6.c.h 2
15.e even 4 1 90.6.a.d 1
15.e even 4 1 450.6.a.l 1
20.d odd 2 1 400.6.c.b 2
20.e even 4 1 80.6.a.a 1
20.e even 4 1 400.6.a.n 1
35.f even 4 1 490.6.a.a 1
40.i odd 4 1 320.6.a.b 1
40.k even 4 1 320.6.a.o 1
60.l odd 4 1 720.6.a.j 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 576$$ acting on $$S_{6}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$576 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$29584 + T^{2}$$
$11$ $$( -132 + T )^{2}$$
$13$ $$894916 + T^{2}$$
$17$ $$49284 + T^{2}$$
$19$ $$( 500 + T )^{2}$$
$23$ $$12702096 + T^{2}$$
$29$ $$( 2190 + T )^{2}$$
$31$ $$( -2312 + T )^{2}$$
$37$ $$126382564 + T^{2}$$
$41$ $$( -1242 + T )^{2}$$
$43$ $$425349376 + T^{2}$$
$47$ $$43401744 + T^{2}$$
$53$ $$443776356 + T^{2}$$
$59$ $$( 7980 + T )^{2}$$
$61$ $$( -16622 + T )^{2}$$
$67$ $$3268864 + T^{2}$$
$71$ $$( 24528 + T )^{2}$$
$73$ $$419184676 + T^{2}$$
$79$ $$( -46240 + T )^{2}$$
$83$ $$2660083776 + T^{2}$$
$89$ $$( -110310 + T )^{2}$$
$97$ $$6143737924 + T^{2}$$
show more
show less