# Properties

 Label 50.6.b.c 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: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -11 i q^{3} -16 q^{4} + 44 q^{6} + 142 i q^{7} -64 i q^{8} + 122 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} -11 i q^{3} -16 q^{4} + 44 q^{6} + 142 i q^{7} -64 i q^{8} + 122 q^{9} + 777 q^{11} + 176 i q^{12} + 884 i q^{13} -568 q^{14} + 256 q^{16} + 27 i q^{17} + 488 i q^{18} -1145 q^{19} + 1562 q^{21} + 3108 i q^{22} + 1854 i q^{23} -704 q^{24} -3536 q^{26} -4015 i q^{27} -2272 i q^{28} + 4920 q^{29} + 1802 q^{31} + 1024 i q^{32} -8547 i q^{33} -108 q^{34} -1952 q^{36} -13178 i q^{37} -4580 i q^{38} + 9724 q^{39} -15123 q^{41} + 6248 i q^{42} + 7844 i q^{43} -12432 q^{44} -7416 q^{46} + 6732 i q^{47} -2816 i q^{48} -3357 q^{49} + 297 q^{51} -14144 i q^{52} + 3414 i q^{53} + 16060 q^{54} + 9088 q^{56} + 12595 i q^{57} + 19680 i q^{58} -33960 q^{59} + 47402 q^{61} + 7208 i q^{62} + 17324 i q^{63} -4096 q^{64} + 34188 q^{66} + 13177 i q^{67} -432 i q^{68} + 20394 q^{69} -7548 q^{71} -7808 i q^{72} -59821 i q^{73} + 52712 q^{74} + 18320 q^{76} + 110334 i q^{77} + 38896 i q^{78} -75830 q^{79} -14519 q^{81} -60492 i q^{82} + 46299 i q^{83} -24992 q^{84} -31376 q^{86} -54120 i q^{87} -49728 i q^{88} + 30585 q^{89} -125528 q^{91} -29664 i q^{92} -19822 i q^{93} -26928 q^{94} + 11264 q^{96} -104018 i q^{97} -13428 i q^{98} + 94794 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + 88q^{6} + 244q^{9} + O(q^{10})$$ $$2q - 32q^{4} + 88q^{6} + 244q^{9} + 1554q^{11} - 1136q^{14} + 512q^{16} - 2290q^{19} + 3124q^{21} - 1408q^{24} - 7072q^{26} + 9840q^{29} + 3604q^{31} - 216q^{34} - 3904q^{36} + 19448q^{39} - 30246q^{41} - 24864q^{44} - 14832q^{46} - 6714q^{49} + 594q^{51} + 32120q^{54} + 18176q^{56} - 67920q^{59} + 94804q^{61} - 8192q^{64} + 68376q^{66} + 40788q^{69} - 15096q^{71} + 105424q^{74} + 36640q^{76} - 151660q^{79} - 29038q^{81} - 49984q^{84} - 62752q^{86} + 61170q^{89} - 251056q^{91} - 53856q^{94} + 22528q^{96} + 189588q^{99} + O(q^{100})$$

## 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 11.0000i −16.0000 0 44.0000 142.000i 64.0000i 122.000 0
49.2 4.00000i 11.0000i −16.0000 0 44.0000 142.000i 64.0000i 122.000 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.c 2
3.b odd 2 1 450.6.c.a 2
4.b odd 2 1 400.6.c.g 2
5.b even 2 1 inner 50.6.b.c 2
5.c odd 4 1 50.6.a.a 1
5.c odd 4 1 50.6.a.f yes 1
15.d odd 2 1 450.6.c.a 2
15.e even 4 1 450.6.a.j 1
15.e even 4 1 450.6.a.n 1
20.d odd 2 1 400.6.c.g 2
20.e even 4 1 400.6.a.e 1
20.e even 4 1 400.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 5.c odd 4 1
50.6.a.f yes 1 5.c odd 4 1
50.6.b.c 2 1.a even 1 1 trivial
50.6.b.c 2 5.b even 2 1 inner
400.6.a.e 1 20.e even 4 1
400.6.a.j 1 20.e even 4 1
400.6.c.g 2 4.b odd 2 1
400.6.c.g 2 20.d odd 2 1
450.6.a.j 1 15.e even 4 1
450.6.a.n 1 15.e even 4 1
450.6.c.a 2 3.b odd 2 1
450.6.c.a 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 121$$ acting on $$S_{6}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$121 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$20164 + T^{2}$$
$11$ $$( -777 + T )^{2}$$
$13$ $$781456 + T^{2}$$
$17$ $$729 + T^{2}$$
$19$ $$( 1145 + T )^{2}$$
$23$ $$3437316 + T^{2}$$
$29$ $$( -4920 + T )^{2}$$
$31$ $$( -1802 + T )^{2}$$
$37$ $$173659684 + T^{2}$$
$41$ $$( 15123 + T )^{2}$$
$43$ $$61528336 + T^{2}$$
$47$ $$45319824 + T^{2}$$
$53$ $$11655396 + T^{2}$$
$59$ $$( 33960 + T )^{2}$$
$61$ $$( -47402 + T )^{2}$$
$67$ $$173633329 + T^{2}$$
$71$ $$( 7548 + T )^{2}$$
$73$ $$3578552041 + T^{2}$$
$79$ $$( 75830 + T )^{2}$$
$83$ $$2143597401 + T^{2}$$
$89$ $$( -30585 + T )^{2}$$
$97$ $$10819744324 + T^{2}$$