Properties

 Label 552.2.b.c Level $552$ Weight $2$ Character orbit 552.b Analytic conductor $4.408$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$80$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 12q^{6} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 12q^{6} - 4q^{9} + 16q^{12} + 8q^{16} + 20q^{18} - 8q^{24} - 144q^{25} - 24q^{31} + 40q^{36} + 68q^{39} - 24q^{46} + 92q^{48} - 160q^{49} - 48q^{52} - 32q^{54} + 32q^{55} - 40q^{58} + 48q^{64} + 72q^{70} + 68q^{72} - 8q^{73} + 64q^{78} + 12q^{81} - 48q^{82} + 92q^{87} - 144q^{94} + 68q^{96} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
413.1 −1.41196 0.0797643i −0.807372 1.53237i 1.98728 + 0.225248i 2.83704i 1.01775 + 2.22805i 1.51036i −2.78799 0.476556i −1.69630 + 2.47438i −0.226295 + 4.00580i
413.2 −1.41196 0.0797643i −0.807372 1.53237i 1.98728 + 0.225248i 2.83704i 1.01775 + 2.22805i 1.51036i −2.78799 0.476556i −1.69630 + 2.47438i 0.226295 4.00580i
413.3 −1.41196 + 0.0797643i −0.807372 + 1.53237i 1.98728 0.225248i 2.83704i 1.01775 2.22805i 1.51036i −2.78799 + 0.476556i −1.69630 2.47438i 0.226295 + 4.00580i
413.4 −1.41196 + 0.0797643i −0.807372 + 1.53237i 1.98728 0.225248i 2.83704i 1.01775 2.22805i 1.51036i −2.78799 + 0.476556i −1.69630 2.47438i −0.226295 4.00580i
413.5 −1.36401 0.373449i 1.13395 1.30925i 1.72107 + 1.01878i 0.395927i −2.03567 + 1.36237i 4.71612i −1.96710 2.03236i −0.428296 2.96927i −0.147859 + 0.540050i
413.6 −1.36401 0.373449i 1.13395 1.30925i 1.72107 + 1.01878i 0.395927i −2.03567 + 1.36237i 4.71612i −1.96710 2.03236i −0.428296 2.96927i 0.147859 0.540050i
413.7 −1.36401 + 0.373449i 1.13395 + 1.30925i 1.72107 1.01878i 0.395927i −2.03567 1.36237i 4.71612i −1.96710 + 2.03236i −0.428296 + 2.96927i 0.147859 + 0.540050i
413.8 −1.36401 + 0.373449i 1.13395 + 1.30925i 1.72107 1.01878i 0.395927i −2.03567 1.36237i 4.71612i −1.96710 + 2.03236i −0.428296 + 2.96927i −0.147859 0.540050i
413.9 −1.32616 0.491215i 1.50954 + 0.849295i 1.51742 + 1.30286i 3.96539i −1.58470 1.86781i 0.287329i −1.37235 2.47319i 1.55740 + 2.56408i −1.94786 + 5.25875i
413.10 −1.32616 0.491215i 1.50954 + 0.849295i 1.51742 + 1.30286i 3.96539i −1.58470 1.86781i 0.287329i −1.37235 2.47319i 1.55740 + 2.56408i 1.94786 5.25875i
413.11 −1.32616 + 0.491215i 1.50954 0.849295i 1.51742 1.30286i 3.96539i −1.58470 + 1.86781i 0.287329i −1.37235 + 2.47319i 1.55740 2.56408i 1.94786 + 5.25875i
413.12 −1.32616 + 0.491215i 1.50954 0.849295i 1.51742 1.30286i 3.96539i −1.58470 + 1.86781i 0.287329i −1.37235 + 2.47319i 1.55740 2.56408i −1.94786 5.25875i
413.13 −1.25126 0.659042i −1.59997 0.663406i 1.13133 + 1.64927i 2.68496i 1.56477 + 1.88454i 3.86988i −0.328648 2.80927i 2.11979 + 2.12285i −1.76950 + 3.35959i
413.14 −1.25126 0.659042i −1.59997 0.663406i 1.13133 + 1.64927i 2.68496i 1.56477 + 1.88454i 3.86988i −0.328648 2.80927i 2.11979 + 2.12285i 1.76950 3.35959i
413.15 −1.25126 + 0.659042i −1.59997 + 0.663406i 1.13133 1.64927i 2.68496i 1.56477 1.88454i 3.86988i −0.328648 + 2.80927i 2.11979 2.12285i 1.76950 + 3.35959i
413.16 −1.25126 + 0.659042i −1.59997 + 0.663406i 1.13133 1.64927i 2.68496i 1.56477 1.88454i 3.86988i −0.328648 + 2.80927i 2.11979 2.12285i −1.76950 3.35959i
413.17 −1.02836 0.970807i 1.71832 0.217636i 0.115069 + 1.99669i 1.34112i −1.97835 1.44435i 2.80797i 1.82006 2.16503i 2.90527 0.747937i −1.30196 + 1.37916i
413.18 −1.02836 0.970807i 1.71832 0.217636i 0.115069 + 1.99669i 1.34112i −1.97835 1.44435i 2.80797i 1.82006 2.16503i 2.90527 0.747937i 1.30196 1.37916i
413.19 −1.02836 + 0.970807i 1.71832 + 0.217636i 0.115069 1.99669i 1.34112i −1.97835 + 1.44435i 2.80797i 1.82006 + 2.16503i 2.90527 + 0.747937i 1.30196 + 1.37916i
413.20 −1.02836 + 0.970807i 1.71832 + 0.217636i 0.115069 1.99669i 1.34112i −1.97835 + 1.44435i 2.80797i 1.82006 + 2.16503i 2.90527 + 0.747937i −1.30196 1.37916i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.80 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
23.b odd 2 1 inner
24.h odd 2 1 inner
69.c even 2 1 inner
184.e odd 2 1 inner
552.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.b.c 80
3.b odd 2 1 inner 552.2.b.c 80
4.b odd 2 1 2208.2.b.c 80
8.b even 2 1 inner 552.2.b.c 80
8.d odd 2 1 2208.2.b.c 80
12.b even 2 1 2208.2.b.c 80
23.b odd 2 1 inner 552.2.b.c 80
24.f even 2 1 2208.2.b.c 80
24.h odd 2 1 inner 552.2.b.c 80
69.c even 2 1 inner 552.2.b.c 80
92.b even 2 1 2208.2.b.c 80
184.e odd 2 1 inner 552.2.b.c 80
184.h even 2 1 2208.2.b.c 80
276.h odd 2 1 2208.2.b.c 80
552.b even 2 1 inner 552.2.b.c 80
552.h odd 2 1 2208.2.b.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.b.c 80 1.a even 1 1 trivial
552.2.b.c 80 3.b odd 2 1 inner
552.2.b.c 80 8.b even 2 1 inner
552.2.b.c 80 23.b odd 2 1 inner
552.2.b.c 80 24.h odd 2 1 inner
552.2.b.c 80 69.c even 2 1 inner
552.2.b.c 80 184.e odd 2 1 inner
552.2.b.c 80 552.b even 2 1 inner
2208.2.b.c 80 4.b odd 2 1
2208.2.b.c 80 8.d odd 2 1
2208.2.b.c 80 12.b even 2 1
2208.2.b.c 80 24.f even 2 1
2208.2.b.c 80 92.b even 2 1
2208.2.b.c 80 184.h even 2 1
2208.2.b.c 80 276.h odd 2 1
2208.2.b.c 80 552.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$:

 $$T_{5}^{20} + \cdots$$ $$T_{29}^{20} - \cdots$$