Properties

 Label 550.6.b.a 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 29 i q^{3} - 16 q^{4} - 116 q^{6} - 230 i q^{7} - 64 i q^{8} - 598 q^{9} +O(q^{10})$$ q + 4*i * q^2 + 29*i * q^3 - 16 * q^4 - 116 * q^6 - 230*i * q^7 - 64*i * q^8 - 598 * q^9 $$q + 4 i q^{2} + 29 i q^{3} - 16 q^{4} - 116 q^{6} - 230 i q^{7} - 64 i q^{8} - 598 q^{9} + 121 q^{11} - 464 i q^{12} - 112 i q^{13} + 920 q^{14} + 256 q^{16} - 1142 i q^{17} - 2392 i q^{18} + 612 q^{19} + 6670 q^{21} + 484 i q^{22} + 1941 i q^{23} + 1856 q^{24} + 448 q^{26} - 10295 i q^{27} + 3680 i q^{28} - 1192 q^{29} - 1037 q^{31} + 1024 i q^{32} + 3509 i q^{33} + 4568 q^{34} + 9568 q^{36} + 8083 i q^{37} + 2448 i q^{38} + 3248 q^{39} - 10444 q^{41} + 26680 i q^{42} - 58 i q^{43} - 1936 q^{44} - 7764 q^{46} + 8656 i q^{47} + 7424 i q^{48} - 36093 q^{49} + 33118 q^{51} + 1792 i q^{52} + 20318 i q^{53} + 41180 q^{54} - 14720 q^{56} + 17748 i q^{57} - 4768 i q^{58} + 21351 q^{59} + 47044 q^{61} - 4148 i q^{62} + 137540 i q^{63} - 4096 q^{64} - 14036 q^{66} + 48093 i q^{67} + 18272 i q^{68} - 56289 q^{69} - 24967 q^{71} + 38272 i q^{72} + 42288 i q^{73} - 32332 q^{74} - 9792 q^{76} - 27830 i q^{77} + 12992 i q^{78} + 72410 q^{79} + 153241 q^{81} - 41776 i q^{82} + 15806 i q^{83} - 106720 q^{84} + 232 q^{86} - 34568 i q^{87} - 7744 i q^{88} + 114761 q^{89} - 25760 q^{91} - 31056 i q^{92} - 30073 i q^{93} - 34624 q^{94} - 29696 q^{96} - 5159 i q^{97} - 144372 i q^{98} - 72358 q^{99} +O(q^{100})$$ q + 4*i * q^2 + 29*i * q^3 - 16 * q^4 - 116 * q^6 - 230*i * q^7 - 64*i * q^8 - 598 * q^9 + 121 * q^11 - 464*i * q^12 - 112*i * q^13 + 920 * q^14 + 256 * q^16 - 1142*i * q^17 - 2392*i * q^18 + 612 * q^19 + 6670 * q^21 + 484*i * q^22 + 1941*i * q^23 + 1856 * q^24 + 448 * q^26 - 10295*i * q^27 + 3680*i * q^28 - 1192 * q^29 - 1037 * q^31 + 1024*i * q^32 + 3509*i * q^33 + 4568 * q^34 + 9568 * q^36 + 8083*i * q^37 + 2448*i * q^38 + 3248 * q^39 - 10444 * q^41 + 26680*i * q^42 - 58*i * q^43 - 1936 * q^44 - 7764 * q^46 + 8656*i * q^47 + 7424*i * q^48 - 36093 * q^49 + 33118 * q^51 + 1792*i * q^52 + 20318*i * q^53 + 41180 * q^54 - 14720 * q^56 + 17748*i * q^57 - 4768*i * q^58 + 21351 * q^59 + 47044 * q^61 - 4148*i * q^62 + 137540*i * q^63 - 4096 * q^64 - 14036 * q^66 + 48093*i * q^67 + 18272*i * q^68 - 56289 * q^69 - 24967 * q^71 + 38272*i * q^72 + 42288*i * q^73 - 32332 * q^74 - 9792 * q^76 - 27830*i * q^77 + 12992*i * q^78 + 72410 * q^79 + 153241 * q^81 - 41776*i * q^82 + 15806*i * q^83 - 106720 * q^84 + 232 * q^86 - 34568*i * q^87 - 7744*i * q^88 + 114761 * q^89 - 25760 * q^91 - 31056*i * q^92 - 30073*i * q^93 - 34624 * q^94 - 29696 * q^96 - 5159*i * q^97 - 144372*i * q^98 - 72358 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 232 q^{6} - 1196 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 232 * q^6 - 1196 * q^9 $$2 q - 32 q^{4} - 232 q^{6} - 1196 q^{9} + 242 q^{11} + 1840 q^{14} + 512 q^{16} + 1224 q^{19} + 13340 q^{21} + 3712 q^{24} + 896 q^{26} - 2384 q^{29} - 2074 q^{31} + 9136 q^{34} + 19136 q^{36} + 6496 q^{39} - 20888 q^{41} - 3872 q^{44} - 15528 q^{46} - 72186 q^{49} + 66236 q^{51} + 82360 q^{54} - 29440 q^{56} + 42702 q^{59} + 94088 q^{61} - 8192 q^{64} - 28072 q^{66} - 112578 q^{69} - 49934 q^{71} - 64664 q^{74} - 19584 q^{76} + 144820 q^{79} + 306482 q^{81} - 213440 q^{84} + 464 q^{86} + 229522 q^{89} - 51520 q^{91} - 69248 q^{94} - 59392 q^{96} - 144716 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 232 * q^6 - 1196 * q^9 + 242 * q^11 + 1840 * q^14 + 512 * q^16 + 1224 * q^19 + 13340 * q^21 + 3712 * q^24 + 896 * q^26 - 2384 * q^29 - 2074 * q^31 + 9136 * q^34 + 19136 * q^36 + 6496 * q^39 - 20888 * q^41 - 3872 * q^44 - 15528 * q^46 - 72186 * q^49 + 66236 * q^51 + 82360 * q^54 - 29440 * q^56 + 42702 * q^59 + 94088 * q^61 - 8192 * q^64 - 28072 * q^66 - 112578 * q^69 - 49934 * q^71 - 64664 * q^74 - 19584 * q^76 + 144820 * q^79 + 306482 * q^81 - 213440 * q^84 + 464 * q^86 + 229522 * q^89 - 51520 * q^91 - 69248 * q^94 - 59392 * q^96 - 144716 * 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 29.0000i −16.0000 0 −116.000 230.000i 64.0000i −598.000 0
199.2 4.00000i 29.0000i −16.0000 0 −116.000 230.000i 64.0000i −598.000 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.a 2
5.b even 2 1 inner 550.6.b.a 2
5.c odd 4 1 22.6.a.c 1
5.c odd 4 1 550.6.a.c 1
15.e even 4 1 198.6.a.b 1
20.e even 4 1 176.6.a.e 1
35.f even 4 1 1078.6.a.f 1
40.i odd 4 1 704.6.a.j 1
40.k even 4 1 704.6.a.a 1
55.e even 4 1 242.6.a.a 1

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

 $$T_{3}^{2} + 841$$ T3^2 + 841 $$T_{7}^{2} + 52900$$ T7^2 + 52900

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 841$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 52900$$
$11$ $$(T - 121)^{2}$$
$13$ $$T^{2} + 12544$$
$17$ $$T^{2} + 1304164$$
$19$ $$(T - 612)^{2}$$
$23$ $$T^{2} + 3767481$$
$29$ $$(T + 1192)^{2}$$
$31$ $$(T + 1037)^{2}$$
$37$ $$T^{2} + 65334889$$
$41$ $$(T + 10444)^{2}$$
$43$ $$T^{2} + 3364$$
$47$ $$T^{2} + 74926336$$
$53$ $$T^{2} + 412821124$$
$59$ $$(T - 21351)^{2}$$
$61$ $$(T - 47044)^{2}$$
$67$ $$T^{2} + 2312936649$$
$71$ $$(T + 24967)^{2}$$
$73$ $$T^{2} + 1788274944$$
$79$ $$(T - 72410)^{2}$$
$83$ $$T^{2} + 249829636$$
$89$ $$(T - 114761)^{2}$$
$97$ $$T^{2} + 26615281$$