# Properties

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

# Related objects

## 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -26 i q^{3} -16 q^{4} + 104 q^{6} + 22 i q^{7} -64 i q^{8} -433 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} -26 i q^{3} -16 q^{4} + 104 q^{6} + 22 i q^{7} -64 i q^{8} -433 q^{9} -768 q^{11} + 416 i q^{12} -46 i q^{13} -88 q^{14} + 256 q^{16} -378 i q^{17} -1732 i q^{18} -1100 q^{19} + 572 q^{21} -3072 i q^{22} -1986 i q^{23} -1664 q^{24} + 184 q^{26} + 4940 i q^{27} -352 i q^{28} + 5610 q^{29} -3988 q^{31} + 1024 i q^{32} + 19968 i q^{33} + 1512 q^{34} + 6928 q^{36} + 142 i q^{37} -4400 i q^{38} -1196 q^{39} + 1542 q^{41} + 2288 i q^{42} -5026 i q^{43} + 12288 q^{44} + 7944 q^{46} -24738 i q^{47} -6656 i q^{48} + 16323 q^{49} -9828 q^{51} + 736 i q^{52} -14166 i q^{53} -19760 q^{54} + 1408 q^{56} + 28600 i q^{57} + 22440 i q^{58} -28380 q^{59} + 5522 q^{61} -15952 i q^{62} -9526 i q^{63} -4096 q^{64} -79872 q^{66} + 24742 i q^{67} + 6048 i q^{68} -51636 q^{69} + 42372 q^{71} + 27712 i q^{72} -52126 i q^{73} -568 q^{74} + 17600 q^{76} -16896 i q^{77} -4784 i q^{78} + 39640 q^{79} + 23221 q^{81} + 6168 i q^{82} -59826 i q^{83} -9152 q^{84} + 20104 q^{86} -145860 i q^{87} + 49152 i q^{88} -57690 q^{89} + 1012 q^{91} + 31776 i q^{92} + 103688 i q^{93} + 98952 q^{94} + 26624 q^{96} + 144382 i q^{97} + 65292 i q^{98} + 332544 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + 208q^{6} - 866q^{9} + O(q^{10})$$ $$2q - 32q^{4} + 208q^{6} - 866q^{9} - 1536q^{11} - 176q^{14} + 512q^{16} - 2200q^{19} + 1144q^{21} - 3328q^{24} + 368q^{26} + 11220q^{29} - 7976q^{31} + 3024q^{34} + 13856q^{36} - 2392q^{39} + 3084q^{41} + 24576q^{44} + 15888q^{46} + 32646q^{49} - 19656q^{51} - 39520q^{54} + 2816q^{56} - 56760q^{59} + 11044q^{61} - 8192q^{64} - 159744q^{66} - 103272q^{69} + 84744q^{71} - 1136q^{74} + 35200q^{76} + 79280q^{79} + 46442q^{81} - 18304q^{84} + 40208q^{86} - 115380q^{89} + 2024q^{91} + 197904q^{94} + 53248q^{96} + 665088q^{99} + O(q^{100})$$

## 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 26.0000i −16.0000 0 104.000 22.0000i 64.0000i −433.000 0
49.2 4.00000i 26.0000i −16.0000 0 104.000 22.0000i 64.0000i −433.000 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.d 2
3.b odd 2 1 450.6.c.o 2
4.b odd 2 1 400.6.c.a 2
5.b even 2 1 inner 50.6.b.d 2
5.c odd 4 1 10.6.a.a 1
5.c odd 4 1 50.6.a.g 1
15.d odd 2 1 450.6.c.o 2
15.e even 4 1 90.6.a.f 1
15.e even 4 1 450.6.a.h 1
20.d odd 2 1 400.6.c.a 2
20.e even 4 1 80.6.a.h 1
20.e even 4 1 400.6.a.a 1
35.f even 4 1 490.6.a.j 1
40.i odd 4 1 320.6.a.p 1
40.k even 4 1 320.6.a.a 1
60.l odd 4 1 720.6.a.r 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 676$$ acting on $$S_{6}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$676 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$484 + T^{2}$$
$11$ $$( 768 + T )^{2}$$
$13$ $$2116 + T^{2}$$
$17$ $$142884 + T^{2}$$
$19$ $$( 1100 + T )^{2}$$
$23$ $$3944196 + T^{2}$$
$29$ $$( -5610 + T )^{2}$$
$31$ $$( 3988 + T )^{2}$$
$37$ $$20164 + T^{2}$$
$41$ $$( -1542 + T )^{2}$$
$43$ $$25260676 + T^{2}$$
$47$ $$611968644 + T^{2}$$
$53$ $$200675556 + T^{2}$$
$59$ $$( 28380 + T )^{2}$$
$61$ $$( -5522 + T )^{2}$$
$67$ $$612166564 + T^{2}$$
$71$ $$( -42372 + T )^{2}$$
$73$ $$2717119876 + T^{2}$$
$79$ $$( -39640 + T )^{2}$$
$83$ $$3579150276 + T^{2}$$
$89$ $$( 57690 + T )^{2}$$
$97$ $$20846161924 + T^{2}$$