# Properties

 Label 60.3.l.a Level $60$ Weight $3$ Character orbit 60.l Analytic conductor $1.635$ Analytic rank $0$ Dimension $40$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 60.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63488158616$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{6} - 12q^{10} - 20q^{12} - 8q^{13} - 36q^{16} - 24q^{18} - 24q^{21} - 76q^{22} - 8q^{25} - 84q^{28} + 68q^{30} - 40q^{33} + 172q^{36} - 40q^{37} + 104q^{40} + 236q^{42} - 104q^{45} + 240q^{46} + 196q^{48} + 304q^{52} - 72q^{57} + 180q^{58} - 284q^{60} + 48q^{61} - 552q^{66} - 372q^{70} - 600q^{72} + 104q^{73} - 736q^{76} - 408q^{78} + 72q^{81} - 720q^{82} + 216q^{85} - 580q^{88} + 528q^{90} + 368q^{93} + 884q^{96} + 72q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −1.99497 0.141758i −2.06649 + 2.17477i 3.95981 + 0.565605i −3.07600 3.94185i 4.43087 4.04566i −5.18766 5.18766i −7.81952 1.68970i −0.459255 8.98827i 5.57774 + 8.29993i
23.2 −1.84549 + 0.770813i −2.78107 1.12501i 2.81170 2.84506i 3.86232 + 3.17529i 5.99962 0.0674770i 4.75159 + 4.75159i −2.99596 + 7.41783i 6.46869 + 6.25748i −9.57545 2.88285i
23.3 −1.81610 0.837725i 2.69303 + 1.32197i 2.59643 + 3.04278i −3.21472 + 3.82956i −3.78336 4.65685i 3.54241 + 3.54241i −2.16636 7.70110i 5.50478 + 7.12021i 9.04638 4.26182i
23.4 −1.75394 0.961083i 0.903948 2.86057i 2.15264 + 3.37137i 4.95584 0.663068i −4.33472 + 4.14852i −7.30016 7.30016i −0.535443 7.98206i −7.36576 5.17162i −9.32953 3.59999i
23.5 −1.68375 + 1.07935i 2.99716 + 0.130491i 1.67002 3.63470i 1.65103 4.71955i −5.18731 + 3.01526i 1.91561 + 1.91561i 1.11122 + 7.92245i 8.96594 + 0.782204i 2.31412 + 9.72856i
23.6 −1.07935 + 1.68375i 0.130491 + 2.99716i −1.67002 3.63470i −1.65103 + 4.71955i −5.18731 3.01526i −1.91561 1.91561i 7.92245 + 1.11122i −8.96594 + 0.782204i −6.16449 7.87394i
23.7 −0.961083 1.75394i −0.903948 + 2.86057i −2.15264 + 3.37137i 4.95584 0.663068i 5.88605 1.16377i 7.30016 + 7.30016i 7.98206 + 0.535443i −7.36576 5.17162i −5.92596 8.05500i
23.8 −0.837725 1.81610i −2.69303 1.32197i −2.59643 + 3.04278i −3.21472 + 3.82956i −0.144815 + 5.99825i −3.54241 3.54241i 7.70110 + 2.16636i 5.50478 + 7.12021i 9.64792 + 2.63013i
23.9 −0.770813 + 1.84549i −1.12501 2.78107i −2.81170 2.84506i −3.86232 3.17529i 5.99962 + 0.0674770i −4.75159 4.75159i 7.41783 2.99596i −6.46869 + 6.25748i 8.83710 4.68034i
23.10 −0.141758 1.99497i 2.06649 2.17477i −3.95981 + 0.565605i −3.07600 3.94185i −4.63154 3.81429i 5.18766 + 5.18766i 1.68970 + 7.81952i −0.459255 8.98827i −7.42783 + 6.69532i
23.11 0.141758 + 1.99497i 2.17477 2.06649i −3.95981 + 0.565605i 3.07600 + 3.94185i 4.43087 + 4.04566i 5.18766 + 5.18766i −1.68970 7.81952i 0.459255 8.98827i −7.42783 + 6.69532i
23.12 0.770813 1.84549i 2.78107 + 1.12501i −2.81170 2.84506i 3.86232 + 3.17529i 4.21989 4.26527i −4.75159 4.75159i −7.41783 + 2.99596i 6.46869 + 6.25748i 8.83710 4.68034i
23.13 0.837725 + 1.81610i 1.32197 + 2.69303i −2.59643 + 3.04278i 3.21472 3.82956i −3.78336 + 4.65685i −3.54241 3.54241i −7.70110 2.16636i −5.50478 + 7.12021i 9.64792 + 2.63013i
23.14 0.961083 + 1.75394i −2.86057 + 0.903948i −2.15264 + 3.37137i −4.95584 + 0.663068i −4.33472 4.14852i 7.30016 + 7.30016i −7.98206 0.535443i 7.36576 5.17162i −5.92596 8.05500i
23.15 1.07935 1.68375i −2.99716 0.130491i −1.67002 3.63470i 1.65103 4.71955i −3.45469 + 4.90562i −1.91561 1.91561i −7.92245 1.11122i 8.96594 + 0.782204i −6.16449 7.87394i
23.16 1.68375 1.07935i −0.130491 2.99716i 1.67002 3.63470i −1.65103 + 4.71955i −3.45469 4.90562i 1.91561 + 1.91561i −1.11122 7.92245i −8.96594 + 0.782204i 2.31412 + 9.72856i
23.17 1.75394 + 0.961083i 2.86057 0.903948i 2.15264 + 3.37137i −4.95584 + 0.663068i 5.88605 + 1.16377i −7.30016 7.30016i 0.535443 + 7.98206i 7.36576 5.17162i −9.32953 3.59999i
23.18 1.81610 + 0.837725i −1.32197 2.69303i 2.59643 + 3.04278i 3.21472 3.82956i −0.144815 5.99825i 3.54241 + 3.54241i 2.16636 + 7.70110i −5.50478 + 7.12021i 9.04638 4.26182i
23.19 1.84549 0.770813i 1.12501 + 2.78107i 2.81170 2.84506i −3.86232 3.17529i 4.21989 + 4.26527i 4.75159 + 4.75159i 2.99596 7.41783i −6.46869 + 6.25748i −9.57545 2.88285i
23.20 1.99497 + 0.141758i −2.17477 + 2.06649i 3.95981 + 0.565605i 3.07600 + 3.94185i −4.63154 + 3.81429i −5.18766 5.18766i 7.81952 + 1.68970i 0.459255 8.98827i 5.57774 + 8.29993i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.20 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
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.l.a 40
3.b odd 2 1 inner 60.3.l.a 40
4.b odd 2 1 inner 60.3.l.a 40
5.b even 2 1 300.3.l.g 40
5.c odd 4 1 inner 60.3.l.a 40
5.c odd 4 1 300.3.l.g 40
12.b even 2 1 inner 60.3.l.a 40
15.d odd 2 1 300.3.l.g 40
15.e even 4 1 inner 60.3.l.a 40
15.e even 4 1 300.3.l.g 40
20.d odd 2 1 300.3.l.g 40
20.e even 4 1 inner 60.3.l.a 40
20.e even 4 1 300.3.l.g 40
60.h even 2 1 300.3.l.g 40
60.l odd 4 1 inner 60.3.l.a 40
60.l odd 4 1 300.3.l.g 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.l.a 40 1.a even 1 1 trivial
60.3.l.a 40 3.b odd 2 1 inner
60.3.l.a 40 4.b odd 2 1 inner
60.3.l.a 40 5.c odd 4 1 inner
60.3.l.a 40 12.b even 2 1 inner
60.3.l.a 40 15.e even 4 1 inner
60.3.l.a 40 20.e even 4 1 inner
60.3.l.a 40 60.l odd 4 1 inner
300.3.l.g 40 5.b even 2 1
300.3.l.g 40 5.c odd 4 1
300.3.l.g 40 15.d odd 2 1
300.3.l.g 40 15.e even 4 1
300.3.l.g 40 20.d odd 2 1
300.3.l.g 40 20.e even 4 1
300.3.l.g 40 60.h even 2 1
300.3.l.g 40 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(60, [\chi])$$.