# Properties

 Label 550.2.b.d Level $550$ Weight $2$ Character orbit 550.b Analytic conductor $4.392$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.39177211117$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + 4 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 + 4*i * q^7 - i * q^8 - q^9 $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + 4 i q^{7} - i q^{8} - q^{9} - q^{11} + 2 i q^{12} + 5 i q^{13} - 4 q^{14} + q^{16} - i q^{18} + 7 q^{19} + 8 q^{21} - i q^{22} + 3 i q^{23} - 2 q^{24} - 5 q^{26} - 4 i q^{27} - 4 i q^{28} - 3 q^{29} + 5 q^{31} + i q^{32} + 2 i q^{33} + q^{36} + 4 i q^{37} + 7 i q^{38} + 10 q^{39} + 12 q^{41} + 8 i q^{42} + 5 i q^{43} + q^{44} - 3 q^{46} - 2 i q^{48} - 9 q^{49} - 5 i q^{52} + 6 i q^{53} + 4 q^{54} + 4 q^{56} - 14 i q^{57} - 3 i q^{58} - 12 q^{59} - 10 q^{61} + 5 i q^{62} - 4 i q^{63} - q^{64} - 2 q^{66} - 14 i q^{67} + 6 q^{69} + 3 q^{71} + i q^{72} + 8 i q^{73} - 4 q^{74} - 7 q^{76} - 4 i q^{77} + 10 i q^{78} + 4 q^{79} - 11 q^{81} + 12 i q^{82} - 15 i q^{83} - 8 q^{84} - 5 q^{86} + 6 i q^{87} + i q^{88} - 3 q^{89} - 20 q^{91} - 3 i q^{92} - 10 i q^{93} + 2 q^{96} + 13 i q^{97} - 9 i q^{98} + q^{99} +O(q^{100})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 + 4*i * q^7 - i * q^8 - q^9 - q^11 + 2*i * q^12 + 5*i * q^13 - 4 * q^14 + q^16 - i * q^18 + 7 * q^19 + 8 * q^21 - i * q^22 + 3*i * q^23 - 2 * q^24 - 5 * q^26 - 4*i * q^27 - 4*i * q^28 - 3 * q^29 + 5 * q^31 + i * q^32 + 2*i * q^33 + q^36 + 4*i * q^37 + 7*i * q^38 + 10 * q^39 + 12 * q^41 + 8*i * q^42 + 5*i * q^43 + q^44 - 3 * q^46 - 2*i * q^48 - 9 * q^49 - 5*i * q^52 + 6*i * q^53 + 4 * q^54 + 4 * q^56 - 14*i * q^57 - 3*i * q^58 - 12 * q^59 - 10 * q^61 + 5*i * q^62 - 4*i * q^63 - q^64 - 2 * q^66 - 14*i * q^67 + 6 * q^69 + 3 * q^71 + i * q^72 + 8*i * q^73 - 4 * q^74 - 7 * q^76 - 4*i * q^77 + 10*i * q^78 + 4 * q^79 - 11 * q^81 + 12*i * q^82 - 15*i * q^83 - 8 * q^84 - 5 * q^86 + 6*i * q^87 + i * q^88 - 3 * q^89 - 20 * q^91 - 3*i * q^92 - 10*i * q^93 + 2 * q^96 + 13*i * q^97 - 9*i * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} + 14 q^{19} + 16 q^{21} - 4 q^{24} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 2 q^{36} + 20 q^{39} + 24 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49} + 8 q^{54} + 8 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} - 4 q^{66} + 12 q^{69} + 6 q^{71} - 8 q^{74} - 14 q^{76} + 8 q^{79} - 22 q^{81} - 16 q^{84} - 10 q^{86} - 6 q^{89} - 40 q^{91} + 4 q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 2 * q^11 - 8 * q^14 + 2 * q^16 + 14 * q^19 + 16 * q^21 - 4 * q^24 - 10 * q^26 - 6 * q^29 + 10 * q^31 + 2 * q^36 + 20 * q^39 + 24 * q^41 + 2 * q^44 - 6 * q^46 - 18 * q^49 + 8 * q^54 + 8 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 4 * q^66 + 12 * q^69 + 6 * q^71 - 8 * q^74 - 14 * q^76 + 8 * q^79 - 22 * q^81 - 16 * q^84 - 10 * q^86 - 6 * q^89 - 40 * q^91 + 4 * q^96 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 2.00000 4.00000i 1.00000i −1.00000 0
199.2 1.00000i 2.00000i −1.00000 0 2.00000 4.00000i 1.00000i −1.00000 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 550.2.b.d 2
3.b odd 2 1 4950.2.c.ba 2
4.b odd 2 1 4400.2.b.e 2
5.b even 2 1 inner 550.2.b.d 2
5.c odd 4 1 550.2.a.a 1
5.c odd 4 1 550.2.a.m yes 1
15.d odd 2 1 4950.2.c.ba 2
15.e even 4 1 4950.2.a.u 1
15.e even 4 1 4950.2.a.y 1
20.d odd 2 1 4400.2.b.e 2
20.e even 4 1 4400.2.a.d 1
20.e even 4 1 4400.2.a.bc 1
55.e even 4 1 6050.2.a.n 1
55.e even 4 1 6050.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 5.c odd 4 1
550.2.a.m yes 1 5.c odd 4 1
550.2.b.d 2 1.a even 1 1 trivial
550.2.b.d 2 5.b even 2 1 inner
4400.2.a.d 1 20.e even 4 1
4400.2.a.bc 1 20.e even 4 1
4400.2.b.e 2 4.b odd 2 1
4400.2.b.e 2 20.d odd 2 1
4950.2.a.u 1 15.e even 4 1
4950.2.a.y 1 15.e even 4 1
4950.2.c.ba 2 3.b odd 2 1
4950.2.c.ba 2 15.d odd 2 1
6050.2.a.n 1 55.e even 4 1
6050.2.a.bb 1 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(550, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 16$$ T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2}$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 12)^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 225$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} + 169$$