# Properties

 Label 240.2.bf.a Level $240$ Weight $2$ Character orbit 240.bf 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.bf (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.

 $$\operatorname{Tr}(f)(q) =$$ $$88 q - 4 q^{3} + 4 q^{4} - 4 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88 q - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{10} - 8 q^{12} - 8 q^{13} - 4 q^{15} - 12 q^{16} - 12 q^{18} + 8 q^{19} - 4 q^{21} - 4 q^{22} + 12 q^{24} - 4 q^{27} - 4 q^{28} - 28 q^{30} - 16 q^{31} - 4 q^{33} - 28 q^{34} - 20 q^{36} - 8 q^{37} - 24 q^{39} - 44 q^{40} + 40 q^{42} + 8 q^{45} - 36 q^{46} + 32 q^{48} - 4 q^{51} - 48 q^{52} + 24 q^{54} - 12 q^{57} - 20 q^{58} + 28 q^{60} - 24 q^{61} - 32 q^{63} + 28 q^{64} + 12 q^{66} + 12 q^{69} - 76 q^{70} + 24 q^{72} - 36 q^{75} - 36 q^{76} + 60 q^{78} - 8 q^{81} + 64 q^{82} - 48 q^{84} + 16 q^{85} + 12 q^{87} + 44 q^{88} - 56 q^{90} - 8 q^{91} - 16 q^{93} + 20 q^{94} + 48 q^{96} - 8 q^{97} + 32 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 −1.40756 0.137025i −0.623684 + 1.61586i 1.96245 + 0.385740i −2.19324 + 0.435524i 1.09929 2.18897i −2.31966 2.31966i −2.70941 0.811856i −2.22204 2.01558i 3.14680 0.312498i
53.2 −1.39624 0.224732i 1.59602 0.672846i 1.89899 + 0.627562i 1.06348 1.96698i −2.37964 + 0.580779i −3.58885 3.58885i −2.51042 1.30299i 2.09456 2.14775i −1.92693 + 2.50738i
53.3 −1.38823 + 0.269844i −1.72575 + 0.147625i 1.85437 0.749211i −0.684619 2.12868i 2.35590 0.670621i 1.09901 + 1.09901i −2.37212 + 1.54047i 2.95641 0.509529i 1.52482 + 2.77036i
53.4 −1.36437 0.372166i 1.10758 + 1.33164i 1.72298 + 1.01554i 1.60566 + 1.55623i −1.01556 2.22905i 0.854868 + 0.854868i −1.97283 2.02681i −0.546521 + 2.94980i −1.61153 2.72084i
53.5 −1.35103 + 0.418005i 0.510164 1.65521i 1.65054 1.12947i 2.20538 + 0.369196i 0.00264245 + 2.44949i 2.84513 + 2.84513i −1.75780 + 2.21588i −2.47947 1.68886i −3.13385 + 0.423066i
53.6 −1.32640 0.490564i −0.461441 1.66945i 1.51869 + 1.30137i −1.76316 + 1.37523i −0.206917 + 2.44073i 0.367568 + 0.367568i −1.37599 2.47116i −2.57414 + 1.54071i 3.01330 0.959166i
53.7 −1.28713 + 0.585912i −1.57494 0.720802i 1.31341 1.50829i 0.666497 + 2.13443i 2.44948 + 0.00498857i −2.25736 2.25736i −0.806809 + 2.71091i 1.96089 + 2.27044i −2.10846 2.35678i
53.8 −1.19544 + 0.755591i 1.27877 + 1.16822i 0.858166 1.80653i −0.219616 2.22526i −2.41140 0.430317i 1.05780 + 1.05780i 0.339109 + 2.80803i 0.270509 + 2.98778i 1.94392 + 2.49423i
53.9 −1.18483 0.772133i −1.37710 1.05053i 0.807621 + 1.82969i 2.01632 0.966676i 0.820472 + 2.30799i −0.166503 0.166503i 0.455871 2.79145i 0.792787 + 2.89335i −3.13539 0.411524i
53.10 −1.09247 0.898057i −0.296326 + 1.70651i 0.386989 + 1.96220i 0.503115 2.17873i 1.85627 1.59820i 2.12944 + 2.12944i 1.33939 2.49119i −2.82438 1.01137i −2.50626 + 1.92838i
53.11 −1.07619 + 0.917499i 0.983076 1.42603i 0.316391 1.97482i −2.21263 + 0.322899i 0.250400 + 2.43666i −1.41445 1.41445i 1.47139 + 2.41558i −1.06712 2.80379i 2.08496 2.37759i
53.12 −1.01551 + 0.984242i −0.788258 + 1.54229i 0.0625351 1.99902i −0.334403 + 2.21092i −0.717498 2.34205i 2.87827 + 2.87827i 1.90402 + 2.09158i −1.75730 2.43144i −1.83649 2.57435i
53.13 −0.925990 1.06890i 1.51078 0.847085i −0.285084 + 1.97958i −0.206476 + 2.22651i −2.30441 0.830473i −0.209149 0.209149i 2.37995 1.52834i 1.56489 2.55951i 2.57111 1.84103i
53.14 −0.770357 1.18598i −1.34211 + 1.09486i −0.813100 + 1.82726i 1.43671 + 1.71343i 2.33240 + 0.748285i −3.11495 3.11495i 2.79347 0.443321i 0.602544 2.93887i 0.925314 3.02387i
53.15 −0.685419 + 1.23701i 1.72806 + 0.117548i −1.06040 1.69575i 1.91324 + 1.15738i −1.32985 + 2.05706i −0.912923 0.912923i 2.82448 0.149432i 2.97237 + 0.406258i −2.74306 + 1.57341i
53.16 −0.585178 1.28747i −1.69651 + 0.349092i −1.31513 + 1.50679i −2.23328 0.111699i 1.44220 + 1.97991i 1.82036 + 1.82036i 2.70953 + 0.811448i 2.75627 1.18447i 1.16305 + 2.94063i
53.17 −0.578793 + 1.29035i −1.50524 + 0.856888i −1.33000 1.49369i 1.73381 1.41205i −0.234463 2.43824i −1.42263 1.42263i 2.69717 0.851625i 1.53149 2.57964i 0.818525 + 3.05451i
53.18 −0.550725 + 1.30258i −1.30699 1.13657i −1.39340 1.43472i −2.15866 0.583249i 2.20026 1.07651i 2.32976 + 2.32976i 2.63622 1.02487i 0.416423 + 2.97096i 1.94856 2.49061i
53.19 −0.498160 1.32357i 0.0947166 1.72946i −1.50367 + 1.31870i −1.24786 1.85549i −2.33624 + 0.736184i −0.907692 0.907692i 2.49446 + 1.33329i −2.98206 0.327617i −1.83424 + 2.57596i
53.20 −0.316900 + 1.37825i 0.356415 + 1.69498i −1.79915 0.873536i −1.95643 1.08277i −2.44906 0.0459120i −2.05875 2.05875i 1.77410 2.20285i −2.74594 + 1.20823i 2.11232 2.35331i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.44 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.t odd 4 1 inner
240.bf 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.bf.a yes 88
3.b odd 2 1 inner 240.2.bf.a yes 88
4.b odd 2 1 960.2.bf.a 88
5.c odd 4 1 240.2.bb.a 88
12.b even 2 1 960.2.bf.a 88
15.e even 4 1 240.2.bb.a 88
16.e even 4 1 240.2.bb.a 88
16.f odd 4 1 960.2.bb.a 88
20.e even 4 1 960.2.bb.a 88
48.i odd 4 1 240.2.bb.a 88
48.k even 4 1 960.2.bb.a 88
60.l odd 4 1 960.2.bb.a 88
80.j even 4 1 960.2.bf.a 88
80.t odd 4 1 inner 240.2.bf.a yes 88
240.bd odd 4 1 960.2.bf.a 88
240.bf even 4 1 inner 240.2.bf.a yes 88

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

## Hecke kernels

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