# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(199,550)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("550.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$88.2111008971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + i q^{3} - 16 q^{4} - 4 q^{6} + 166 i q^{7} - 64 i q^{8} + 242 q^{9} +O(q^{10})$$ q + 4*i * q^2 + i * q^3 - 16 * q^4 - 4 * q^6 + 166*i * q^7 - 64*i * q^8 + 242 * q^9 $$q + 4 i q^{2} + i q^{3} - 16 q^{4} - 4 q^{6} + 166 i q^{7} - 64 i q^{8} + 242 q^{9} - 121 q^{11} - 16 i q^{12} + 692 i q^{13} - 664 q^{14} + 256 q^{16} + 738 i q^{17} + 968 i q^{18} - 1424 q^{19} - 166 q^{21} - 484 i q^{22} - 1779 i q^{23} + 64 q^{24} - 2768 q^{26} + 485 i q^{27} - 2656 i q^{28} + 2064 q^{29} + 6245 q^{31} + 1024 i q^{32} - 121 i q^{33} - 2952 q^{34} - 3872 q^{36} + 14785 i q^{37} - 5696 i q^{38} - 692 q^{39} + 5304 q^{41} - 664 i q^{42} + 17798 i q^{43} + 1936 q^{44} + 7116 q^{46} + 17184 i q^{47} + 256 i q^{48} - 10749 q^{49} - 738 q^{51} - 11072 i q^{52} - 30726 i q^{53} - 1940 q^{54} + 10624 q^{56} - 1424 i q^{57} + 8256 i q^{58} + 34989 q^{59} - 45940 q^{61} + 24980 i q^{62} + 40172 i q^{63} - 4096 q^{64} + 484 q^{66} - 25343 i q^{67} - 11808 i q^{68} + 1779 q^{69} + 13311 q^{71} - 15488 i q^{72} - 53260 i q^{73} - 59140 q^{74} + 22784 q^{76} - 20086 i q^{77} - 2768 i q^{78} - 77234 q^{79} + 58321 q^{81} + 21216 i q^{82} + 55014 i q^{83} + 2656 q^{84} - 71192 q^{86} + 2064 i q^{87} + 7744 i q^{88} - 125415 q^{89} - 114872 q^{91} + 28464 i q^{92} + 6245 i q^{93} - 68736 q^{94} - 1024 q^{96} + 88807 i q^{97} - 42996 i q^{98} - 29282 q^{99} +O(q^{100})$$ q + 4*i * q^2 + i * q^3 - 16 * q^4 - 4 * q^6 + 166*i * q^7 - 64*i * q^8 + 242 * q^9 - 121 * q^11 - 16*i * q^12 + 692*i * q^13 - 664 * q^14 + 256 * q^16 + 738*i * q^17 + 968*i * q^18 - 1424 * q^19 - 166 * q^21 - 484*i * q^22 - 1779*i * q^23 + 64 * q^24 - 2768 * q^26 + 485*i * q^27 - 2656*i * q^28 + 2064 * q^29 + 6245 * q^31 + 1024*i * q^32 - 121*i * q^33 - 2952 * q^34 - 3872 * q^36 + 14785*i * q^37 - 5696*i * q^38 - 692 * q^39 + 5304 * q^41 - 664*i * q^42 + 17798*i * q^43 + 1936 * q^44 + 7116 * q^46 + 17184*i * q^47 + 256*i * q^48 - 10749 * q^49 - 738 * q^51 - 11072*i * q^52 - 30726*i * q^53 - 1940 * q^54 + 10624 * q^56 - 1424*i * q^57 + 8256*i * q^58 + 34989 * q^59 - 45940 * q^61 + 24980*i * q^62 + 40172*i * q^63 - 4096 * q^64 + 484 * q^66 - 25343*i * q^67 - 11808*i * q^68 + 1779 * q^69 + 13311 * q^71 - 15488*i * q^72 - 53260*i * q^73 - 59140 * q^74 + 22784 * q^76 - 20086*i * q^77 - 2768*i * q^78 - 77234 * q^79 + 58321 * q^81 + 21216*i * q^82 + 55014*i * q^83 + 2656 * q^84 - 71192 * q^86 + 2064*i * q^87 + 7744*i * q^88 - 125415 * q^89 - 114872 * q^91 + 28464*i * q^92 + 6245*i * q^93 - 68736 * q^94 - 1024 * q^96 + 88807*i * q^97 - 42996*i * q^98 - 29282 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 8 q^{6} + 484 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 8 * q^6 + 484 * q^9 $$2 q - 32 q^{4} - 8 q^{6} + 484 q^{9} - 242 q^{11} - 1328 q^{14} + 512 q^{16} - 2848 q^{19} - 332 q^{21} + 128 q^{24} - 5536 q^{26} + 4128 q^{29} + 12490 q^{31} - 5904 q^{34} - 7744 q^{36} - 1384 q^{39} + 10608 q^{41} + 3872 q^{44} + 14232 q^{46} - 21498 q^{49} - 1476 q^{51} - 3880 q^{54} + 21248 q^{56} + 69978 q^{59} - 91880 q^{61} - 8192 q^{64} + 968 q^{66} + 3558 q^{69} + 26622 q^{71} - 118280 q^{74} + 45568 q^{76} - 154468 q^{79} + 116642 q^{81} + 5312 q^{84} - 142384 q^{86} - 250830 q^{89} - 229744 q^{91} - 137472 q^{94} - 2048 q^{96} - 58564 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 8 * q^6 + 484 * q^9 - 242 * q^11 - 1328 * q^14 + 512 * q^16 - 2848 * q^19 - 332 * q^21 + 128 * q^24 - 5536 * q^26 + 4128 * q^29 + 12490 * q^31 - 5904 * q^34 - 7744 * q^36 - 1384 * q^39 + 10608 * q^41 + 3872 * q^44 + 14232 * q^46 - 21498 * q^49 - 1476 * q^51 - 3880 * q^54 + 21248 * q^56 + 69978 * q^59 - 91880 * q^61 - 8192 * q^64 + 968 * q^66 + 3558 * q^69 + 26622 * q^71 - 118280 * q^74 + 45568 * q^76 - 154468 * q^79 + 116642 * q^81 + 5312 * q^84 - 142384 * q^86 - 250830 * q^89 - 229744 * q^91 - 137472 * q^94 - 2048 * q^96 - 58564 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 1.00000i −16.0000 0 −4.00000 166.000i 64.0000i 242.000 0
199.2 4.00000i 1.00000i −16.0000 0 −4.00000 166.000i 64.0000i 242.000 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.f 2
5.b even 2 1 inner 550.6.b.f 2
5.c odd 4 1 22.6.a.b 1
5.c odd 4 1 550.6.a.f 1
15.e even 4 1 198.6.a.i 1
20.e even 4 1 176.6.a.b 1
35.f even 4 1 1078.6.a.a 1
40.i odd 4 1 704.6.a.e 1
40.k even 4 1 704.6.a.f 1
55.e even 4 1 242.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 5.c odd 4 1
176.6.a.b 1 20.e even 4 1
198.6.a.i 1 15.e even 4 1
242.6.a.d 1 55.e even 4 1
550.6.a.f 1 5.c odd 4 1
550.6.b.f 2 1.a even 1 1 trivial
550.6.b.f 2 5.b even 2 1 inner
704.6.a.e 1 40.i odd 4 1
704.6.a.f 1 40.k even 4 1
1078.6.a.a 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 $$S_{6}^{\mathrm{new}}(550, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 27556$$ T7^2 + 27556

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 27556$$
$11$ $$(T + 121)^{2}$$
$13$ $$T^{2} + 478864$$
$17$ $$T^{2} + 544644$$
$19$ $$(T + 1424)^{2}$$
$23$ $$T^{2} + 3164841$$
$29$ $$(T - 2064)^{2}$$
$31$ $$(T - 6245)^{2}$$
$37$ $$T^{2} + 218596225$$
$41$ $$(T - 5304)^{2}$$
$43$ $$T^{2} + 316768804$$
$47$ $$T^{2} + 295289856$$
$53$ $$T^{2} + 944087076$$
$59$ $$(T - 34989)^{2}$$
$61$ $$(T + 45940)^{2}$$
$67$ $$T^{2} + 642267649$$
$71$ $$(T - 13311)^{2}$$
$73$ $$T^{2} + 2836627600$$
$79$ $$(T + 77234)^{2}$$
$83$ $$T^{2} + 3026540196$$
$89$ $$(T + 125415)^{2}$$
$97$ $$T^{2} + 7886683249$$