# Properties

 Label 240.2.bb.a Level $240$ Weight $2$ Character orbit 240.bb Analytic conductor $1.916$ Analytic rank $0$ Dimension $88$ 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.bb (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88 q - 4 q^{4} - 4 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88 q - 4 q^{4} - 4 q^{6} - 12 q^{10} + 4 q^{12} - 4 q^{15} - 12 q^{16} + 16 q^{18} - 8 q^{19} - 4 q^{21} - 20 q^{22} - 12 q^{24} - 36 q^{28} - 16 q^{30} - 16 q^{31} - 4 q^{33} + 28 q^{34} - 20 q^{36} + 24 q^{39} - 4 q^{40} - 20 q^{42} - 40 q^{43} + 8 q^{45} - 36 q^{46} + 16 q^{48} - 4 q^{51} + 24 q^{52} - 24 q^{54} + 12 q^{57} + 44 q^{58} - 56 q^{60} - 24 q^{61} - 32 q^{63} - 28 q^{64} + 12 q^{66} - 8 q^{67} - 12 q^{69} - 28 q^{70} + 64 q^{72} - 24 q^{75} - 36 q^{76} + 20 q^{78} - 8 q^{81} - 48 q^{82} + 48 q^{84} - 24 q^{85} - 12 q^{87} + 60 q^{88} + 76 q^{90} - 8 q^{91} - 20 q^{94} + 48 q^{96} - 8 q^{97} - 32 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
173.1 −1.41280 0.0631697i −1.53551 + 0.801371i 1.99202 + 0.178492i 1.97816 + 1.04254i 2.22000 1.03518i −0.592869 0.592869i −2.80305 0.378010i 1.71561 2.46103i −2.72889 1.59786i
173.2 −1.38946 0.263452i 0.0477618 + 1.73139i 1.86119 + 0.732110i −1.29291 1.82438i 0.389775 2.41828i −2.39784 2.39784i −2.39316 1.50757i −2.99544 + 0.165389i 1.31581 + 2.87553i
173.3 −1.37825 + 0.316900i 1.69498 + 0.356415i 1.79915 0.873536i 1.08277 1.95643i −2.44906 + 0.0459120i 2.05875 + 2.05875i −2.20285 + 1.77410i 2.74594 + 1.20823i −0.872339 + 3.03958i
173.4 −1.32357 0.498160i 1.72946 + 0.0947166i 1.50367 + 1.31870i −1.85549 + 1.24786i −2.24188 0.986912i 0.907692 + 0.907692i −1.33329 2.49446i 2.98206 + 0.327617i 3.07751 0.727289i
173.5 −1.30258 + 0.550725i −1.13657 1.30699i 1.39340 1.43472i 0.583249 2.15866i 2.20026 + 1.07651i −2.32976 2.32976i −1.02487 + 2.63622i −0.416423 + 2.97096i 0.429103 + 3.13303i
173.6 −1.29035 + 0.578793i 0.856888 1.50524i 1.33000 1.49369i 1.41205 + 1.73381i −0.234463 + 2.43824i 1.42263 + 1.42263i −0.851625 + 2.69717i −1.53149 2.57964i −2.82556 1.41993i
173.7 −1.28747 0.585178i −0.349092 1.69651i 1.31513 + 1.50679i −0.111699 + 2.23328i −0.543315 + 2.38847i −1.82036 1.82036i −0.811448 2.70953i −2.75627 + 1.18447i 1.45067 2.80990i
173.8 −1.23701 + 0.685419i 0.117548 + 1.72806i 1.06040 1.69575i −1.15738 + 1.91324i −1.32985 2.05706i 0.912923 + 0.912923i −0.149432 + 2.82448i −2.97237 + 0.406258i 0.120323 3.15999i
173.9 −1.18598 0.770357i −1.09486 1.34211i 0.813100 + 1.82726i 1.71343 1.43671i 0.264580 + 2.43516i 3.11495 + 3.11495i 0.443321 2.79347i −0.602544 + 2.93887i −3.13888 + 0.383960i
173.10 −1.06890 0.925990i 0.847085 + 1.51078i 0.285084 + 1.97958i 2.22651 + 0.206476i 0.493518 2.39926i 0.209149 + 0.209149i 1.52834 2.37995i −1.56489 + 2.55951i −2.18872 2.28243i
173.11 −0.984242 + 1.01551i 1.54229 0.788258i −0.0625351 1.99902i −2.21092 0.334403i −0.717498 + 2.34205i −2.87827 2.87827i 2.09158 + 1.90402i 1.75730 2.43144i 2.51567 1.91609i
173.12 −0.917499 + 1.07619i −1.42603 + 0.983076i −0.316391 1.97482i −0.322899 2.21263i 0.250400 2.43666i 1.41445 + 1.41445i 2.41558 + 1.47139i 1.06712 2.80379i 2.67748 + 1.68258i
173.13 −0.898057 1.09247i −1.70651 0.296326i −0.386989 + 1.96220i −2.17873 0.503115i 1.20882 + 2.13044i −2.12944 2.12944i 2.49119 1.33939i 2.82438 + 1.01137i 1.40699 + 2.83203i
173.14 −0.772133 1.18483i 1.05053 1.37710i −0.807621 + 1.82969i −0.966676 2.01632i −2.44276 0.181389i 0.166503 + 0.166503i 2.79145 0.455871i −0.792787 2.89335i −1.64258 + 2.70221i
173.15 −0.755591 + 1.19544i 1.16822 + 1.27877i −0.858166 1.80653i 2.22526 0.219616i −2.41140 + 0.430317i −1.05780 1.05780i 2.80803 + 0.339109i −0.270509 + 2.98778i −1.41885 + 2.82611i
173.16 −0.585912 + 1.28713i −0.720802 1.57494i −1.31341 1.50829i −2.13443 + 0.666497i 2.44948 0.00498857i 2.25736 + 2.25736i 2.71091 0.806809i −1.96089 + 2.27044i 0.392719 3.13780i
173.17 −0.490564 1.32640i 1.66945 0.461441i −1.51869 + 1.30137i 1.37523 + 1.76316i −1.43103 1.98800i −0.367568 0.367568i 2.47116 + 1.37599i 2.57414 1.54071i 1.66403 2.68905i
173.18 −0.418005 + 1.35103i −1.65521 + 0.510164i −1.65054 1.12947i −0.369196 + 2.20538i 0.00264245 2.44949i −2.84513 2.84513i 2.21588 1.75780i 2.47947 1.68886i −2.82520 1.42065i
173.19 −0.372166 1.36437i −1.33164 + 1.10758i −1.72298 + 1.01554i 1.55623 1.60566i 2.00674 + 1.40464i −0.854868 0.854868i 2.02681 + 1.97283i 0.546521 2.94980i −2.76989 1.52569i
173.20 −0.269844 + 1.38823i 0.147625 1.72575i −1.85437 0.749211i 2.12868 0.684619i 2.35590 + 0.670621i −1.09901 1.09901i 1.54047 2.37212i −2.95641 0.509529i 0.375997 + 3.13984i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.44 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
80.i odd 4 1 inner
240.bb 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.bb.a 88
3.b odd 2 1 inner 240.2.bb.a 88
4.b odd 2 1 960.2.bb.a 88
5.c odd 4 1 240.2.bf.a yes 88
12.b even 2 1 960.2.bb.a 88
15.e even 4 1 240.2.bf.a yes 88
16.e even 4 1 240.2.bf.a yes 88
16.f odd 4 1 960.2.bf.a 88
20.e even 4 1 960.2.bf.a 88
48.i odd 4 1 240.2.bf.a yes 88
48.k even 4 1 960.2.bf.a 88
60.l odd 4 1 960.2.bf.a 88
80.i odd 4 1 inner 240.2.bb.a 88
80.s even 4 1 960.2.bb.a 88
240.z odd 4 1 960.2.bb.a 88
240.bb even 4 1 inner 240.2.bb.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bb.a 88 1.a even 1 1 trivial
240.2.bb.a 88 3.b odd 2 1 inner
240.2.bb.a 88 80.i odd 4 1 inner
240.2.bb.a 88 240.bb even 4 1 inner
240.2.bf.a yes 88 5.c odd 4 1
240.2.bf.a yes 88 15.e even 4 1
240.2.bf.a yes 88 16.e even 4 1
240.2.bf.a yes 88 48.i odd 4 1
960.2.bb.a 88 4.b odd 2 1
960.2.bb.a 88 12.b even 2 1
960.2.bb.a 88 80.s even 4 1
960.2.bb.a 88 240.z odd 4 1
960.2.bf.a 88 16.f odd 4 1
960.2.bf.a 88 20.e even 4 1
960.2.bf.a 88 48.k even 4 1
960.2.bf.a 88 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(240, [\chi])$$.