# Properties

 Label 60.5.l.a Level $60$ Weight $5$ Character orbit 60.l Analytic conductor $6.202$ Analytic rank $0$ Dimension $88$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.20219778503$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 4q^{6} + 148q^{10} - 164q^{12} - 8q^{13} - 484q^{16} + 456q^{18} + 600q^{21} - 348q^{22} - 8q^{25} - 148q^{28} + 68q^{30} - 328q^{33} - 1556q^{36} + 2232q^{37} + 2024q^{40} - 2116q^{42} - 2504q^{45} - 4432q^{46} + 1540q^{48} + 80q^{52} + 3192q^{57} - 1948q^{58} + 2836q^{60} - 7568q^{61} + 3384q^{66} + 8348q^{70} + 5256q^{72} + 5592q^{73} - 256q^{76} + 13176q^{78} - 12648q^{81} + 20400q^{82} + 5816q^{85} + 7932q^{88} - 13152q^{90} + 3680q^{93} - 18028q^{96} - 15208q^{97} + O(q^{100})$$

## 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 −3.99427 + 0.214108i 8.89373 1.37898i 15.9083 1.71041i 16.1632 + 19.0723i −35.2287 + 7.41222i −4.24582 4.24582i −63.1758 + 10.2379i 77.1968 24.5285i −68.6436 72.7190i
23.2 −3.98340 0.364047i −6.64241 6.07276i 15.7349 + 2.90029i −22.0852 + 11.7150i 24.2486 + 26.6084i −39.7479 39.7479i −61.6227 17.2813i 7.24316 + 80.6755i 92.2391 38.6257i
23.3 −3.94073 + 0.686026i −4.52298 + 7.78092i 15.0587 5.40689i −10.9064 + 22.4956i 12.4859 33.7654i 53.2499 + 53.2499i −55.6332 + 31.6378i −40.0853 70.3859i 27.5466 96.1311i
23.4 −3.91278 + 0.830775i 1.90378 8.79634i 14.6196 6.50127i 6.71772 24.0805i −0.141262 + 35.9997i 17.5991 + 17.5991i −51.8022 + 37.5837i −73.7513 33.4925i −6.27943 + 99.8026i
23.5 −3.80860 1.22253i 4.13002 + 7.99643i 13.0108 + 9.31227i 3.82277 24.7060i −5.95367 35.5043i 27.1938 + 27.1938i −38.1685 51.3728i −46.8859 + 66.0508i −44.7633 + 89.4217i
23.6 −3.70170 1.51572i −8.98881 0.448719i 11.4052 + 11.2215i 24.1558 6.44182i 32.5938 + 15.2855i 0.578123 + 0.578123i −25.2101 58.8256i 80.5973 + 8.06689i −99.1816 12.7677i
23.7 −3.48777 + 1.95843i 6.74616 + 5.95729i 8.32911 13.6611i −24.5931 4.49233i −35.1960 7.56581i −44.1332 44.1332i −2.29574 + 63.9588i 10.0213 + 80.3777i 94.5729 32.4955i
23.8 −3.32846 + 2.21841i −5.44127 + 7.16886i 6.15731 14.7678i 23.0805 9.60676i 2.20756 35.9323i −46.8712 46.8712i 12.2666 + 62.8134i −21.7852 78.0154i −55.5109 + 83.1778i
23.9 −3.26396 2.31226i 7.54313 4.90930i 5.30687 + 15.0943i −23.1720 9.38384i −35.9721 1.41794i −14.0617 14.0617i 17.5805 61.5380i 32.7976 74.0629i 53.9347 + 84.2083i
23.10 −2.92347 + 2.73008i −8.80598 1.85868i 1.09333 15.9626i −14.3650 20.4608i 30.8183 18.6073i 27.9806 + 27.9806i 40.3829 + 49.6510i 74.0906 + 32.7349i 97.8554 + 20.5990i
23.11 −2.89513 2.76011i 0.700985 + 8.97266i 0.763541 + 15.9818i 6.70624 + 24.0837i 22.7361 27.9118i −58.7764 58.7764i 41.9010 48.3767i −80.0172 + 12.5794i 47.0584 88.2355i
23.12 −2.76011 2.89513i −0.700985 8.97266i −0.763541 + 15.9818i 6.70624 + 24.0837i −24.0422 + 26.7950i 58.7764 + 58.7764i 48.3767 41.9010i −80.0172 + 12.5794i 51.2155 85.8893i
23.13 −2.73008 + 2.92347i −1.85868 8.80598i −1.09333 15.9626i 14.3650 + 20.4608i 30.8183 + 18.6073i −27.9806 27.9806i 49.6510 + 40.3829i −74.0906 + 32.7349i −99.0343 13.8641i
23.14 −2.31226 3.26396i −7.54313 + 4.90930i −5.30687 + 15.0943i −23.1720 9.38384i 33.4655 + 13.2689i 14.0617 + 14.0617i 61.5380 17.5805i 32.7976 74.0629i 22.9514 + 97.3305i
23.15 −2.21841 + 3.32846i 7.16886 5.44127i −6.15731 14.7678i −23.0805 + 9.60676i 2.20756 + 35.9323i 46.8712 + 46.8712i 62.8134 + 12.2666i 21.7852 78.0154i 19.2263 98.1343i
23.16 −1.95843 + 3.48777i 5.95729 + 6.74616i −8.32911 13.6611i 24.5931 + 4.49233i −35.1960 + 7.56581i 44.1332 + 44.1332i 63.9588 2.29574i −10.0213 + 80.3777i −63.8320 + 76.9771i
23.17 −1.51572 3.70170i 8.98881 + 0.448719i −11.4052 + 11.2215i 24.1558 6.44182i −11.9635 33.9540i −0.578123 0.578123i 58.8256 + 25.2101i 80.5973 + 8.06689i −60.4591 79.6536i
23.18 −1.22253 3.80860i −4.13002 7.99643i −13.0108 + 9.31227i 3.82277 24.7060i −25.4061 + 25.5055i −27.1938 27.1938i 51.3728 + 38.1685i −46.8859 + 66.0508i −98.7687 + 15.6445i
23.19 −0.830775 + 3.91278i −8.79634 + 1.90378i −14.6196 6.50127i −6.71772 + 24.0805i −0.141262 35.9997i −17.5991 17.5991i 37.5837 51.8022i 73.7513 33.4925i −88.6408 46.2905i
23.20 −0.686026 + 3.94073i 7.78092 4.52298i −15.0587 5.40689i 10.9064 22.4956i 12.4859 + 33.7654i −53.2499 53.2499i 31.6378 55.6332i 40.0853 70.3859i 81.1669 + 58.4117i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.44 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
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 60.5.l.a 88
3.b odd 2 1 inner 60.5.l.a 88
4.b odd 2 1 inner 60.5.l.a 88
5.c odd 4 1 inner 60.5.l.a 88
12.b even 2 1 inner 60.5.l.a 88
15.e even 4 1 inner 60.5.l.a 88
20.e even 4 1 inner 60.5.l.a 88
60.l odd 4 1 inner 60.5.l.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.l.a 88 1.a even 1 1 trivial
60.5.l.a 88 3.b odd 2 1 inner
60.5.l.a 88 4.b odd 2 1 inner
60.5.l.a 88 5.c odd 4 1 inner
60.5.l.a 88 12.b even 2 1 inner
60.5.l.a 88 15.e even 4 1 inner
60.5.l.a 88 20.e even 4 1 inner
60.5.l.a 88 60.l odd 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(60, [\chi])$$.