# Properties

 Label 240.2.bk.b Level $240$ Weight $2$ Character orbit 240.bk Analytic conductor $1.916$ Analytic rank $0$ Dimension $60$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.bk (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$30$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 4 q^{3} + 8 q^{4} + 8 q^{6} + 16 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60 q + 4 q^{3} + 8 q^{4} + 8 q^{6} + 16 q^{7} + 4 q^{12} + 16 q^{13} - 52 q^{16} - 20 q^{18} + 20 q^{19} - 16 q^{21} - 24 q^{22} - 28 q^{24} - 20 q^{27} - 56 q^{28} - 12 q^{34} + 12 q^{36} + 24 q^{37} - 80 q^{39} - 8 q^{40} + 36 q^{42} - 8 q^{45} + 36 q^{46} + 56 q^{48} + 28 q^{49} - 36 q^{51} - 24 q^{52} + 92 q^{54} - 80 q^{58} + 28 q^{60} + 20 q^{61} - 64 q^{64} + 24 q^{66} - 40 q^{67} + 12 q^{69} - 16 q^{70} - 56 q^{72} + 4 q^{75} - 20 q^{76} - 8 q^{78} - 28 q^{81} - 32 q^{82} + 16 q^{84} - 20 q^{85} - 24 q^{87} + 8 q^{88} + 32 q^{90} - 16 q^{91} - 48 q^{93} + 28 q^{94} + 72 q^{96} - 8 q^{97} + 32 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.39972 + 0.201959i 1.67583 0.437717i 1.91842 0.565372i 0.707107 0.707107i −2.25729 + 0.951130i 2.03158 −2.57107 + 1.17881i 2.61681 1.46708i −0.846944 + 1.13256i
11.2 −1.38304 0.295312i −1.72570 + 0.148181i 1.82558 + 0.816854i −0.707107 + 0.707107i 2.43047 + 0.304680i −1.12315 −2.28362 1.66885i 2.95608 0.511431i 1.18677 0.769138i
11.3 −1.31930 + 0.509349i 0.956016 + 1.44431i 1.48113 1.34397i −0.707107 + 0.707107i −1.99693 1.41854i −2.89523 −1.26951 + 2.52752i −1.17207 + 2.76157i 0.572725 1.29305i
11.4 −1.24180 0.676700i −0.861442 + 1.50264i 1.08415 + 1.68066i 0.707107 0.707107i 2.08658 1.28304i 0.563322 −0.209007 2.82069i −1.51583 2.58887i −1.35659 + 0.399589i
11.5 −1.23685 0.685720i 1.73074 + 0.0673020i 1.05958 + 1.69626i −0.707107 + 0.707107i −2.09451 1.27005i 0.271924 −0.147371 2.82459i 2.99094 + 0.232965i 1.35946 0.389705i
11.6 −1.20816 + 0.735082i −1.18902 + 1.25946i 0.919310 1.77620i 0.707107 0.707107i 0.510724 2.39565i −2.92811 0.194974 + 2.82170i −0.172464 2.99504i −0.334518 + 1.37408i
11.7 −1.11727 + 0.867014i 0.773899 1.54954i 0.496574 1.93737i −0.707107 + 0.707107i 0.478822 + 2.40223i 1.63342 1.12492 + 2.59510i −1.80216 2.39838i 0.176956 1.40310i
11.8 −1.10098 0.887605i 0.807747 1.53217i 0.424316 + 1.95447i 0.707107 0.707107i −2.24928 + 0.969928i −5.10839 1.26763 2.52846i −1.69509 2.47521i −1.40614 + 0.150879i
11.9 −0.983055 1.01666i −1.40626 1.01116i −0.0672039 + 1.99887i 0.707107 0.707107i 0.354421 + 2.42371i 4.53819 2.09824 1.89668i 0.955115 + 2.84390i −1.41401 0.0237635i
11.10 −0.932052 + 1.06362i −1.67834 0.427993i −0.262558 1.98269i −0.707107 + 0.707107i 2.01952 1.38620i 2.37757 2.35354 + 1.56871i 2.63364 + 1.43663i −0.0930299 1.41115i
11.11 −0.774214 + 1.18347i 0.764773 + 1.55407i −0.801186 1.83251i 0.707107 0.707107i −2.43128 0.298096i 3.70081 2.78901 + 0.470579i −1.83024 + 2.37702i 0.289385 + 1.38429i
11.12 −0.772787 1.18440i −0.112911 + 1.72837i −0.805601 + 1.83058i −0.707107 + 0.707107i 2.13433 1.20193i −2.33502 2.79069 0.460493i −2.97450 0.390304i 1.38394 + 0.291053i
11.13 −0.468379 1.33440i 1.36262 + 1.06924i −1.56124 + 1.25001i 0.707107 0.707107i 0.788576 2.31908i 1.05330 2.39927 + 1.49784i 0.713439 + 2.91393i −1.27476 0.612368i
11.14 −0.371549 + 1.36453i −1.65322 0.516584i −1.72390 1.01398i 0.707107 0.707107i 1.31915 2.06394i −1.29120 2.02413 1.97558i 2.46628 + 1.70806i 0.702146 + 1.22760i
11.15 −0.0794625 1.41198i −1.53859 + 0.795457i −1.98737 + 0.224399i −0.707107 + 0.707107i 1.24543 + 2.10924i 3.51098 0.474768 + 2.78830i 1.73450 2.44776i 1.05461 + 0.942232i
11.16 0.0794625 + 1.41198i 0.795457 1.53859i −1.98737 + 0.224399i 0.707107 0.707107i 2.23566 + 1.00091i 3.51098 −0.474768 2.78830i −1.73450 2.44776i 1.05461 + 0.942232i
11.17 0.371549 1.36453i −0.516584 1.65322i −1.72390 1.01398i −0.707107 + 0.707107i −2.44781 + 0.0906432i −1.29120 −2.02413 + 1.97558i −2.46628 + 1.70806i 0.702146 + 1.22760i
11.18 0.468379 + 1.33440i 1.06924 + 1.36262i −1.56124 + 1.25001i −0.707107 + 0.707107i −1.31746 + 2.06502i 1.05330 −2.39927 1.49784i −0.713439 + 2.91393i −1.27476 0.612368i
11.19 0.772787 + 1.18440i 1.72837 0.112911i −0.805601 + 1.83058i 0.707107 0.707107i 1.46939 + 1.95982i −2.33502 −2.79069 + 0.460493i 2.97450 0.390304i 1.38394 + 0.291053i
11.20 0.774214 1.18347i 1.55407 + 0.764773i −0.801186 1.83251i −0.707107 + 0.707107i 2.10826 1.24709i 3.70081 −2.78901 0.470579i 1.83024 + 2.37702i 0.289385 + 1.38429i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.30 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.bk.b 60
3.b odd 2 1 inner 240.2.bk.b 60
4.b odd 2 1 960.2.bk.b 60
12.b even 2 1 960.2.bk.b 60
16.e even 4 1 960.2.bk.b 60
16.f odd 4 1 inner 240.2.bk.b 60
48.i odd 4 1 960.2.bk.b 60
48.k even 4 1 inner 240.2.bk.b 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bk.b 60 1.a even 1 1 trivial
240.2.bk.b 60 3.b odd 2 1 inner
240.2.bk.b 60 16.f odd 4 1 inner
240.2.bk.b 60 48.k even 4 1 inner
960.2.bk.b 60 4.b odd 2 1
960.2.bk.b 60 12.b even 2 1
960.2.bk.b 60 16.e even 4 1
960.2.bk.b 60 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{15} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.