# Properties

 Label 550.2.b.a 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: no (minimal twist has level 110) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} - i q^{8} + 2 q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 - q^6 + 3*i * q^7 - i * q^8 + 2 * q^9 $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} - i q^{8} + 2 q^{9} + q^{11} - i q^{12} + 6 i q^{13} - 3 q^{14} + q^{16} - 7 i q^{17} + 2 i q^{18} - 5 q^{19} - 3 q^{21} + i q^{22} + 6 i q^{23} + q^{24} - 6 q^{26} + 5 i q^{27} - 3 i q^{28} - 5 q^{29} - 3 q^{31} + i q^{32} + i q^{33} + 7 q^{34} - 2 q^{36} + 3 i q^{37} - 5 i q^{38} - 6 q^{39} + 2 q^{41} - 3 i q^{42} - 4 i q^{43} - q^{44} - 6 q^{46} - 2 i q^{47} + i q^{48} - 2 q^{49} + 7 q^{51} - 6 i q^{52} + i q^{53} - 5 q^{54} + 3 q^{56} - 5 i q^{57} - 5 i q^{58} + 10 q^{59} + 7 q^{61} - 3 i q^{62} + 6 i q^{63} - q^{64} - q^{66} + 8 i q^{67} + 7 i q^{68} - 6 q^{69} + 7 q^{71} - 2 i q^{72} - 14 i q^{73} - 3 q^{74} + 5 q^{76} + 3 i q^{77} - 6 i q^{78} - 10 q^{79} + q^{81} + 2 i q^{82} + 6 i q^{83} + 3 q^{84} + 4 q^{86} - 5 i q^{87} - i q^{88} + 15 q^{89} - 18 q^{91} - 6 i q^{92} - 3 i q^{93} + 2 q^{94} - q^{96} - 12 i q^{97} - 2 i q^{98} + 2 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 - q^6 + 3*i * q^7 - i * q^8 + 2 * q^9 + q^11 - i * q^12 + 6*i * q^13 - 3 * q^14 + q^16 - 7*i * q^17 + 2*i * q^18 - 5 * q^19 - 3 * q^21 + i * q^22 + 6*i * q^23 + q^24 - 6 * q^26 + 5*i * q^27 - 3*i * q^28 - 5 * q^29 - 3 * q^31 + i * q^32 + i * q^33 + 7 * q^34 - 2 * q^36 + 3*i * q^37 - 5*i * q^38 - 6 * q^39 + 2 * q^41 - 3*i * q^42 - 4*i * q^43 - q^44 - 6 * q^46 - 2*i * q^47 + i * q^48 - 2 * q^49 + 7 * q^51 - 6*i * q^52 + i * q^53 - 5 * q^54 + 3 * q^56 - 5*i * q^57 - 5*i * q^58 + 10 * q^59 + 7 * q^61 - 3*i * q^62 + 6*i * q^63 - q^64 - q^66 + 8*i * q^67 + 7*i * q^68 - 6 * q^69 + 7 * q^71 - 2*i * q^72 - 14*i * q^73 - 3 * q^74 + 5 * q^76 + 3*i * q^77 - 6*i * q^78 - 10 * q^79 + q^81 + 2*i * q^82 + 6*i * q^83 + 3 * q^84 + 4 * q^86 - 5*i * q^87 - i * q^88 + 15 * q^89 - 18 * q^91 - 6*i * q^92 - 3*i * q^93 + 2 * q^94 - q^96 - 12*i * q^97 - 2*i * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 2 q^{11} - 6 q^{14} + 2 q^{16} - 10 q^{19} - 6 q^{21} + 2 q^{24} - 12 q^{26} - 10 q^{29} - 6 q^{31} + 14 q^{34} - 4 q^{36} - 12 q^{39} + 4 q^{41} - 2 q^{44} - 12 q^{46} - 4 q^{49} + 14 q^{51} - 10 q^{54} + 6 q^{56} + 20 q^{59} + 14 q^{61} - 2 q^{64} - 2 q^{66} - 12 q^{69} + 14 q^{71} - 6 q^{74} + 10 q^{76} - 20 q^{79} + 2 q^{81} + 6 q^{84} + 8 q^{86} + 30 q^{89} - 36 q^{91} + 4 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 + 2 * q^11 - 6 * q^14 + 2 * q^16 - 10 * q^19 - 6 * q^21 + 2 * q^24 - 12 * q^26 - 10 * q^29 - 6 * q^31 + 14 * q^34 - 4 * q^36 - 12 * q^39 + 4 * q^41 - 2 * q^44 - 12 * q^46 - 4 * q^49 + 14 * q^51 - 10 * q^54 + 6 * q^56 + 20 * q^59 + 14 * q^61 - 2 * q^64 - 2 * q^66 - 12 * q^69 + 14 * q^71 - 6 * q^74 + 10 * q^76 - 20 * q^79 + 2 * q^81 + 6 * q^84 + 8 * q^86 + 30 * q^89 - 36 * q^91 + 4 * q^94 - 2 * q^96 + 4 * 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 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i 2.00000 0
199.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i 2.00000 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.a 2
3.b odd 2 1 4950.2.c.m 2
4.b odd 2 1 4400.2.b.i 2
5.b even 2 1 inner 550.2.b.a 2
5.c odd 4 1 110.2.a.b 1
5.c odd 4 1 550.2.a.f 1
15.d odd 2 1 4950.2.c.m 2
15.e even 4 1 990.2.a.d 1
15.e even 4 1 4950.2.a.bc 1
20.d odd 2 1 4400.2.b.i 2
20.e even 4 1 880.2.a.i 1
20.e even 4 1 4400.2.a.l 1
35.f even 4 1 5390.2.a.bf 1
40.i odd 4 1 3520.2.a.y 1
40.k even 4 1 3520.2.a.h 1
55.e even 4 1 1210.2.a.b 1
55.e even 4 1 6050.2.a.bj 1
60.l odd 4 1 7920.2.a.d 1
220.i odd 4 1 9680.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 5.c odd 4 1
550.2.a.f 1 5.c odd 4 1
550.2.b.a 2 1.a even 1 1 trivial
550.2.b.a 2 5.b even 2 1 inner
880.2.a.i 1 20.e even 4 1
990.2.a.d 1 15.e even 4 1
1210.2.a.b 1 55.e even 4 1
3520.2.a.h 1 40.k even 4 1
3520.2.a.y 1 40.i odd 4 1
4400.2.a.l 1 20.e even 4 1
4400.2.b.i 2 4.b odd 2 1
4400.2.b.i 2 20.d odd 2 1
4950.2.a.bc 1 15.e even 4 1
4950.2.c.m 2 3.b odd 2 1
4950.2.c.m 2 15.d odd 2 1
5390.2.a.bf 1 35.f even 4 1
6050.2.a.bj 1 55.e even 4 1
7920.2.a.d 1 60.l odd 4 1
9680.2.a.x 1 220.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 9$$ T7^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 49$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 1$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 7)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 144$$