# Properties

 Label 550.6.b.g Level $550$ Weight $6$ Character orbit 550.b Analytic conductor $88.211$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 550.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$88.2111008971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) 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.

 $$f(q)$$ $$=$$ $$q - 4 i q^{2} + 21 i q^{3} - 16 q^{4} + 84 q^{6} + 98 i q^{7} + 64 i q^{8} - 198 q^{9} +O(q^{10})$$ q - 4*i * q^2 + 21*i * q^3 - 16 * q^4 + 84 * q^6 + 98*i * q^7 + 64*i * q^8 - 198 * q^9 $$q - 4 i q^{2} + 21 i q^{3} - 16 q^{4} + 84 q^{6} + 98 i q^{7} + 64 i q^{8} - 198 q^{9} + 121 q^{11} - 336 i q^{12} - 824 i q^{13} + 392 q^{14} + 256 q^{16} + 978 i q^{17} + 792 i q^{18} + 2140 q^{19} - 2058 q^{21} - 484 i q^{22} - 3699 i q^{23} - 1344 q^{24} - 3296 q^{26} + 945 i q^{27} - 1568 i q^{28} - 3480 q^{29} - 7813 q^{31} - 1024 i q^{32} + 2541 i q^{33} + 3912 q^{34} + 3168 q^{36} - 13597 i q^{37} - 8560 i q^{38} + 17304 q^{39} + 6492 q^{41} + 8232 i q^{42} - 14234 i q^{43} - 1936 q^{44} - 14796 q^{46} - 20352 i q^{47} + 5376 i q^{48} + 7203 q^{49} - 20538 q^{51} + 13184 i q^{52} + 366 i q^{53} + 3780 q^{54} - 6272 q^{56} + 44940 i q^{57} + 13920 i q^{58} - 9825 q^{59} + 26132 q^{61} + 31252 i q^{62} - 19404 i q^{63} - 4096 q^{64} + 10164 q^{66} + 17093 i q^{67} - 15648 i q^{68} + 77679 q^{69} - 23583 q^{71} - 12672 i q^{72} + 35176 i q^{73} - 54388 q^{74} - 34240 q^{76} + 11858 i q^{77} - 69216 i q^{78} + 42490 q^{79} - 67959 q^{81} - 25968 i q^{82} - 22674 i q^{83} + 32928 q^{84} - 56936 q^{86} - 73080 i q^{87} + 7744 i q^{88} + 17145 q^{89} + 80752 q^{91} + 59184 i q^{92} - 164073 i q^{93} - 81408 q^{94} + 21504 q^{96} - 30727 i q^{97} - 28812 i q^{98} - 23958 q^{99} +O(q^{100})$$ q - 4*i * q^2 + 21*i * q^3 - 16 * q^4 + 84 * q^6 + 98*i * q^7 + 64*i * q^8 - 198 * q^9 + 121 * q^11 - 336*i * q^12 - 824*i * q^13 + 392 * q^14 + 256 * q^16 + 978*i * q^17 + 792*i * q^18 + 2140 * q^19 - 2058 * q^21 - 484*i * q^22 - 3699*i * q^23 - 1344 * q^24 - 3296 * q^26 + 945*i * q^27 - 1568*i * q^28 - 3480 * q^29 - 7813 * q^31 - 1024*i * q^32 + 2541*i * q^33 + 3912 * q^34 + 3168 * q^36 - 13597*i * q^37 - 8560*i * q^38 + 17304 * q^39 + 6492 * q^41 + 8232*i * q^42 - 14234*i * q^43 - 1936 * q^44 - 14796 * q^46 - 20352*i * q^47 + 5376*i * q^48 + 7203 * q^49 - 20538 * q^51 + 13184*i * q^52 + 366*i * q^53 + 3780 * q^54 - 6272 * q^56 + 44940*i * q^57 + 13920*i * q^58 - 9825 * q^59 + 26132 * q^61 + 31252*i * q^62 - 19404*i * q^63 - 4096 * q^64 + 10164 * q^66 + 17093*i * q^67 - 15648*i * q^68 + 77679 * q^69 - 23583 * q^71 - 12672*i * q^72 + 35176*i * q^73 - 54388 * q^74 - 34240 * q^76 + 11858*i * q^77 - 69216*i * q^78 + 42490 * q^79 - 67959 * q^81 - 25968*i * q^82 - 22674*i * q^83 + 32928 * q^84 - 56936 * q^86 - 73080*i * q^87 + 7744*i * q^88 + 17145 * q^89 + 80752 * q^91 + 59184*i * q^92 - 164073*i * q^93 - 81408 * q^94 + 21504 * q^96 - 30727*i * q^97 - 28812*i * q^98 - 23958 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} + 168 q^{6} - 396 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9 $$2 q - 32 q^{4} + 168 q^{6} - 396 q^{9} + 242 q^{11} + 784 q^{14} + 512 q^{16} + 4280 q^{19} - 4116 q^{21} - 2688 q^{24} - 6592 q^{26} - 6960 q^{29} - 15626 q^{31} + 7824 q^{34} + 6336 q^{36} + 34608 q^{39} + 12984 q^{41} - 3872 q^{44} - 29592 q^{46} + 14406 q^{49} - 41076 q^{51} + 7560 q^{54} - 12544 q^{56} - 19650 q^{59} + 52264 q^{61} - 8192 q^{64} + 20328 q^{66} + 155358 q^{69} - 47166 q^{71} - 108776 q^{74} - 68480 q^{76} + 84980 q^{79} - 135918 q^{81} + 65856 q^{84} - 113872 q^{86} + 34290 q^{89} + 161504 q^{91} - 162816 q^{94} + 43008 q^{96} - 47916 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9 + 242 * q^11 + 784 * q^14 + 512 * q^16 + 4280 * q^19 - 4116 * q^21 - 2688 * q^24 - 6592 * q^26 - 6960 * q^29 - 15626 * q^31 + 7824 * q^34 + 6336 * q^36 + 34608 * q^39 + 12984 * q^41 - 3872 * q^44 - 29592 * q^46 + 14406 * q^49 - 41076 * q^51 + 7560 * q^54 - 12544 * q^56 - 19650 * q^59 + 52264 * q^61 - 8192 * q^64 + 20328 * q^66 + 155358 * q^69 - 47166 * q^71 - 108776 * q^74 - 68480 * q^76 + 84980 * q^79 - 135918 * q^81 + 65856 * q^84 - 113872 * q^86 + 34290 * q^89 + 161504 * q^91 - 162816 * q^94 + 43008 * q^96 - 47916 * 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
4.00000i 21.0000i −16.0000 0 84.0000 98.0000i 64.0000i −198.000 0
199.2 4.00000i 21.0000i −16.0000 0 84.0000 98.0000i 64.0000i −198.000 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 550.6.b.g 2
5.b even 2 1 inner 550.6.b.g 2
5.c odd 4 1 22.6.a.a 1
5.c odd 4 1 550.6.a.g 1
15.e even 4 1 198.6.a.d 1
20.e even 4 1 176.6.a.d 1
35.f even 4 1 1078.6.a.b 1
40.i odd 4 1 704.6.a.i 1
40.k even 4 1 704.6.a.b 1
55.e even 4 1 242.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.a 1 5.c odd 4 1
176.6.a.d 1 20.e even 4 1
198.6.a.d 1 15.e even 4 1
242.6.a.c 1 55.e even 4 1
550.6.a.g 1 5.c odd 4 1
550.6.b.g 2 1.a even 1 1 trivial
550.6.b.g 2 5.b even 2 1 inner
704.6.a.b 1 40.k even 4 1
704.6.a.i 1 40.i odd 4 1
1078.6.a.b 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(550, [\chi])$$:

 $$T_{3}^{2} + 441$$ T3^2 + 441 $$T_{7}^{2} + 9604$$ T7^2 + 9604

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 441$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9604$$
$11$ $$(T - 121)^{2}$$
$13$ $$T^{2} + 678976$$
$17$ $$T^{2} + 956484$$
$19$ $$(T - 2140)^{2}$$
$23$ $$T^{2} + 13682601$$
$29$ $$(T + 3480)^{2}$$
$31$ $$(T + 7813)^{2}$$
$37$ $$T^{2} + 184878409$$
$41$ $$(T - 6492)^{2}$$
$43$ $$T^{2} + 202606756$$
$47$ $$T^{2} + 414203904$$
$53$ $$T^{2} + 133956$$
$59$ $$(T + 9825)^{2}$$
$61$ $$(T - 26132)^{2}$$
$67$ $$T^{2} + 292170649$$
$71$ $$(T + 23583)^{2}$$
$73$ $$T^{2} + 1237350976$$
$79$ $$(T - 42490)^{2}$$
$83$ $$T^{2} + 514110276$$
$89$ $$(T - 17145)^{2}$$
$97$ $$T^{2} + 944148529$$