Properties

 Label 770.2.o.a Level $770$ Weight $2$ Character orbit 770.o Analytic conductor $6.148$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.o (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 24 q^{2} - 24 q^{4} - 6 q^{5} - 4 q^{7} + 48 q^{8} - 28 q^{9}+O(q^{10})$$ 48 * q - 24 * q^2 - 24 * q^4 - 6 * q^5 - 4 * q^7 + 48 * q^8 - 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 24 q^{2} - 24 q^{4} - 6 q^{5} - 4 q^{7} + 48 q^{8} - 28 q^{9} + 6 q^{10} + q^{11} + 2 q^{14} + 4 q^{15} - 24 q^{16} - 28 q^{18} - 2 q^{22} - 6 q^{26} + 2 q^{28} - 2 q^{30} + 12 q^{31} - 24 q^{32} + 24 q^{33} - 8 q^{35} + 56 q^{36} - 6 q^{40} + 24 q^{43} + q^{44} + 6 q^{45} - 36 q^{49} + 6 q^{52} - 4 q^{56} - 4 q^{57} - 2 q^{60} - 52 q^{63} + 48 q^{64} - 24 q^{66} - 2 q^{70} - 40 q^{71} - 28 q^{72} + 42 q^{73} + 90 q^{75} + 19 q^{77} + 6 q^{80} - 72 q^{81} - 40 q^{85} - 12 q^{86} + 18 q^{87} + q^{88} + 60 q^{89} - 56 q^{91} + 24 q^{95} + 42 q^{98} - 58 q^{99}+O(q^{100})$$ 48 * q - 24 * q^2 - 24 * q^4 - 6 * q^5 - 4 * q^7 + 48 * q^8 - 28 * q^9 + 6 * q^10 + q^11 + 2 * q^14 + 4 * q^15 - 24 * q^16 - 28 * q^18 - 2 * q^22 - 6 * q^26 + 2 * q^28 - 2 * q^30 + 12 * q^31 - 24 * q^32 + 24 * q^33 - 8 * q^35 + 56 * q^36 - 6 * q^40 + 24 * q^43 + q^44 + 6 * q^45 - 36 * q^49 + 6 * q^52 - 4 * q^56 - 4 * q^57 - 2 * q^60 - 52 * q^63 + 48 * q^64 - 24 * q^66 - 2 * q^70 - 40 * q^71 - 28 * q^72 + 42 * q^73 + 90 * q^75 + 19 * q^77 + 6 * q^80 - 72 * q^81 - 40 * q^85 - 12 * q^86 + 18 * q^87 + q^88 + 60 * q^89 - 56 * q^91 + 24 * q^95 + 42 * q^98 - 58 * q^99

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
439.1 −0.500000 + 0.866025i −1.65185 2.86109i −0.500000 0.866025i −0.232176 2.22398i 3.30370 2.27237 + 1.35511i 1.00000 −3.95721 + 6.85408i 2.04211 + 0.910921i
439.2 −0.500000 + 0.866025i −1.64811 2.85461i −0.500000 0.866025i −0.0163520 + 2.23601i 3.29622 −1.01398 + 2.44374i 1.00000 −3.93253 + 6.81133i −1.92826 1.13217i
439.3 −0.500000 + 0.866025i −1.44024 2.49457i −0.500000 0.866025i −2.16254 0.568686i 2.88048 −0.391243 2.61666i 1.00000 −2.64857 + 4.58746i 1.57377 1.58848i
439.4 −0.500000 + 0.866025i −1.23964 2.14712i −0.500000 0.866025i 2.22778 + 0.192296i 2.47929 −2.62866 0.300206i 1.00000 −1.57343 + 2.72526i −1.28043 + 1.83317i
439.5 −0.500000 + 0.866025i −0.987813 1.71094i −0.500000 0.866025i 1.00552 + 1.99723i 1.97563 1.58012 2.12208i 1.00000 −0.451550 + 0.782108i −2.23241 0.127806i
439.6 −0.500000 + 0.866025i −0.781614 1.35379i −0.500000 0.866025i −0.438635 2.19262i 1.56323 −0.725302 + 2.54439i 1.00000 0.278160 0.481787i 2.11819 + 0.716443i
439.7 −0.500000 + 0.866025i −0.732537 1.26879i −0.500000 0.866025i −0.423475 + 2.19560i 1.46507 −1.43527 2.22261i 1.00000 0.426780 0.739205i −1.68971 1.46454i
439.8 −0.500000 + 0.866025i −0.730155 1.26467i −0.500000 0.866025i −2.18303 + 0.484147i 1.46031 −1.48697 + 2.18836i 1.00000 0.433747 0.751273i 0.672229 2.13263i
439.9 −0.500000 + 0.866025i −0.617381 1.06934i −0.500000 0.866025i −1.46901 1.68583i 1.23476 1.62005 2.09176i 1.00000 0.737681 1.27770i 2.19447 0.429285i
439.10 −0.500000 + 0.866025i −0.533793 0.924557i −0.500000 0.866025i 2.23597 + 0.0212957i 1.06759 1.22107 + 2.34712i 1.00000 0.930129 1.61103i −1.13643 + 1.92576i
439.11 −0.500000 + 0.866025i −0.340512 0.589784i −0.500000 0.866025i 1.37482 1.76348i 0.681024 2.52726 0.782908i 1.00000 1.26810 2.19642i 0.839811 + 2.07237i
439.12 −0.500000 + 0.866025i −0.0752187 0.130283i −0.500000 0.866025i 0.125277 2.23256i 0.150437 −2.53943 0.742498i 1.00000 1.48868 2.57848i 1.87081 + 1.22477i
439.13 −0.500000 + 0.866025i 0.0752187 + 0.130283i −0.500000 0.866025i −1.87081 + 1.22477i −0.150437 −2.53943 0.742498i 1.00000 1.48868 2.57848i −0.125277 2.23256i
439.14 −0.500000 + 0.866025i 0.340512 + 0.589784i −0.500000 0.866025i −0.839811 + 2.07237i −0.681024 2.52726 0.782908i 1.00000 1.26810 2.19642i −1.37482 1.76348i
439.15 −0.500000 + 0.866025i 0.533793 + 0.924557i −0.500000 0.866025i 1.13643 + 1.92576i −1.06759 1.22107 + 2.34712i 1.00000 0.930129 1.61103i −2.23597 + 0.0212957i
439.16 −0.500000 + 0.866025i 0.617381 + 1.06934i −0.500000 0.866025i −2.19447 0.429285i −1.23476 1.62005 2.09176i 1.00000 0.737681 1.27770i 1.46901 1.68583i
439.17 −0.500000 + 0.866025i 0.730155 + 1.26467i −0.500000 0.866025i −0.672229 2.13263i −1.46031 −1.48697 + 2.18836i 1.00000 0.433747 0.751273i 2.18303 + 0.484147i
439.18 −0.500000 + 0.866025i 0.732537 + 1.26879i −0.500000 0.866025i 1.68971 1.46454i −1.46507 −1.43527 2.22261i 1.00000 0.426780 0.739205i 0.423475 + 2.19560i
439.19 −0.500000 + 0.866025i 0.781614 + 1.35379i −0.500000 0.866025i −2.11819 + 0.716443i −1.56323 −0.725302 + 2.54439i 1.00000 0.278160 0.481787i 0.438635 2.19262i
439.20 −0.500000 + 0.866025i 0.987813 + 1.71094i −0.500000 0.866025i 2.23241 0.127806i −1.97563 1.58012 2.12208i 1.00000 −0.451550 + 0.782108i −1.00552 + 1.99723i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 549.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
55.d odd 2 1 inner
385.o even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.o.a 48
5.b even 2 1 770.2.o.b yes 48
7.d odd 6 1 inner 770.2.o.a 48
11.b odd 2 1 770.2.o.b yes 48
35.i odd 6 1 770.2.o.b yes 48
55.d odd 2 1 inner 770.2.o.a 48
77.i even 6 1 770.2.o.b yes 48
385.o even 6 1 inner 770.2.o.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.o.a 48 1.a even 1 1 trivial
770.2.o.a 48 7.d odd 6 1 inner
770.2.o.a 48 55.d odd 2 1 inner
770.2.o.a 48 385.o even 6 1 inner
770.2.o.b yes 48 5.b even 2 1
770.2.o.b yes 48 11.b odd 2 1
770.2.o.b yes 48 35.i odd 6 1
770.2.o.b yes 48 77.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{24} - 87 T_{17}^{22} + 5595 T_{17}^{20} + 2505 T_{17}^{19} - 137015 T_{17}^{18} - 66126 T_{17}^{17} + 2338454 T_{17}^{16} + 808092 T_{17}^{15} - 23570132 T_{17}^{14} - 6574584 T_{17}^{13} + 170955808 T_{17}^{12} + \cdots + 58982400$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.