# Properties

 Label 35.3.l.a Level 35 Weight 3 Character orbit 35.l Analytic conductor 0.954 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{2} - 2q^{3} - 4q^{5} - 6q^{7} - 36q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{2} - 2q^{3} - 4q^{5} - 6q^{7} - 36q^{8} + 14q^{10} - 24q^{11} - 46q^{12} - 8q^{13} + 52q^{15} + 20q^{16} - 48q^{17} - 4q^{18} - 72q^{20} + 56q^{21} + 104q^{22} - 86q^{23} - 16q^{25} + 140q^{26} + 76q^{27} + 186q^{28} + 64q^{30} + 120q^{31} + 130q^{32} + 116q^{33} - 240q^{35} - 496q^{36} + 44q^{37} + 16q^{38} - 158q^{40} + 16q^{41} - 370q^{42} - 196q^{43} - 104q^{45} - 148q^{46} - 208q^{47} - 52q^{48} + 580q^{50} - 160q^{51} - 288q^{52} - 72q^{53} + 208q^{55} + 420q^{56} + 656q^{57} - 2q^{58} + 262q^{60} + 308q^{61} + 176q^{62} + 212q^{63} + 132q^{65} + 316q^{66} + 198q^{67} + 332q^{68} - 200q^{70} - 792q^{71} + 308q^{72} + 380q^{73} - 450q^{75} - 400q^{76} - 472q^{77} - 720q^{78} - 324q^{80} - 352q^{81} - 818q^{82} - 460q^{83} + 144q^{85} - 336q^{86} - 214q^{87} - 288q^{88} + 120q^{90} + 984q^{91} + 1372q^{92} - 68q^{93} - 88q^{95} + 816q^{96} - 72q^{97} + 482q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −3.54535 + 0.949975i −1.41502 0.379154i 8.20298 4.73599i 0.551807 4.96946i 5.37694 3.87123 5.83212i −14.2019 + 14.2019i −5.93570 3.42698i 2.76451 + 18.1427i
2.2 −1.89206 + 0.506976i 2.62470 + 0.703286i −0.141229 + 0.0815388i 4.09808 + 2.86456i −5.32264 2.16120 + 6.65802i 5.76622 5.76622i −1.39979 0.808169i −9.20609 3.34229i
2.3 −1.23172 + 0.330037i −4.01589 1.07605i −2.05590 + 1.18698i −1.75247 + 4.68282i 5.30157 −3.92739 5.79445i 5.74725 5.74725i 7.17525 + 4.14263i 0.613038 6.34629i
2.4 0.585559 0.156900i 4.00038 + 1.07190i −3.14584 + 1.81625i −4.51288 2.15265i 2.51064 3.65026 5.97291i −3.27174 + 3.27174i 7.05981 + 4.07598i −2.98031 0.552433i
2.5 1.75904 0.471334i −0.195743 0.0524492i −0.592022 + 0.341804i 4.78371 1.45469i −0.369042 −6.97731 0.563209i −6.03113 + 6.03113i −7.75866 4.47947i 7.72911 4.81359i
2.6 2.95850 0.792728i −2.36445 0.633552i 4.66022 2.69058i −4.16825 + 2.76146i −7.49746 5.78419 + 3.94249i 2.99127 2.99127i −2.60501 1.50400i −10.1427 + 11.4741i
18.1 −3.54535 0.949975i −1.41502 + 0.379154i 8.20298 + 4.73599i 0.551807 + 4.96946i 5.37694 3.87123 + 5.83212i −14.2019 14.2019i −5.93570 + 3.42698i 2.76451 18.1427i
18.2 −1.89206 0.506976i 2.62470 0.703286i −0.141229 0.0815388i 4.09808 2.86456i −5.32264 2.16120 6.65802i 5.76622 + 5.76622i −1.39979 + 0.808169i −9.20609 + 3.34229i
18.3 −1.23172 0.330037i −4.01589 + 1.07605i −2.05590 1.18698i −1.75247 4.68282i 5.30157 −3.92739 + 5.79445i 5.74725 + 5.74725i 7.17525 4.14263i 0.613038 + 6.34629i
18.4 0.585559 + 0.156900i 4.00038 1.07190i −3.14584 1.81625i −4.51288 + 2.15265i 2.51064 3.65026 + 5.97291i −3.27174 3.27174i 7.05981 4.07598i −2.98031 + 0.552433i
18.5 1.75904 + 0.471334i −0.195743 + 0.0524492i −0.592022 0.341804i 4.78371 + 1.45469i −0.369042 −6.97731 + 0.563209i −6.03113 6.03113i −7.75866 + 4.47947i 7.72911 + 4.81359i
18.6 2.95850 + 0.792728i −2.36445 + 0.633552i 4.66022 + 2.69058i −4.16825 2.76146i −7.49746 5.78419 3.94249i 2.99127 + 2.99127i −2.60501 + 1.50400i −10.1427 11.4741i
23.1 −0.792728 2.95850i 0.633552 2.36445i −4.66022 + 2.69058i −0.307372 + 4.99054i −7.49746 3.94249 5.78419i 2.99127 + 2.99127i 2.60501 + 1.50400i 15.0082 3.04678i
23.2 −0.471334 1.75904i 0.0524492 0.195743i 0.592022 0.341804i −1.13206 4.87016i −0.369042 −0.563209 + 6.97731i −6.03113 6.03113i 7.75866 + 4.47947i −8.03325 + 4.28681i
23.3 −0.156900 0.585559i −1.07190 + 4.00038i 3.14584 1.81625i 4.12069 + 2.83194i 2.51064 −5.97291 3.65026i −3.27174 3.27174i −7.05981 4.07598i 1.01173 2.85724i
23.4 0.330037 + 1.23172i 1.07605 4.01589i 2.05590 1.18698i −3.17921 + 3.85910i 5.30157 −5.79445 + 3.92739i 5.74725 + 5.74725i −7.17525 4.14263i −5.80257 2.64224i
23.5 0.506976 + 1.89206i −0.703286 + 2.62470i 0.141229 0.0815388i −4.52982 2.11677i −5.32264 6.65802 2.16120i 5.76622 + 5.76622i 1.39979 + 0.808169i 1.70854 9.64386i
23.6 0.949975 + 3.54535i 0.379154 1.41502i −8.20298 + 4.73599i 4.02777 2.96261i 5.37694 −5.83212 3.87123i −14.2019 14.2019i 5.93570 + 3.42698i 14.3298 + 11.4655i
32.1 −0.792728 + 2.95850i 0.633552 + 2.36445i −4.66022 2.69058i −0.307372 4.99054i −7.49746 3.94249 + 5.78419i 2.99127 2.99127i 2.60501 1.50400i 15.0082 + 3.04678i
32.2 −0.471334 + 1.75904i 0.0524492 + 0.195743i 0.592022 + 0.341804i −1.13206 + 4.87016i −0.369042 −0.563209 6.97731i −6.03113 + 6.03113i 7.75866 4.47947i −8.03325 4.28681i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.l.a 24
3.b odd 2 1 315.3.ca.a 24
5.b even 2 1 175.3.p.c 24
5.c odd 4 1 inner 35.3.l.a 24
5.c odd 4 1 175.3.p.c 24
7.b odd 2 1 245.3.m.b 24
7.c even 3 1 inner 35.3.l.a 24
7.c even 3 1 245.3.g.c 12
7.d odd 6 1 245.3.g.b 12
7.d odd 6 1 245.3.m.b 24
15.e even 4 1 315.3.ca.a 24
21.h odd 6 1 315.3.ca.a 24
35.f even 4 1 245.3.m.b 24
35.j even 6 1 175.3.p.c 24
35.k even 12 1 245.3.g.b 12
35.k even 12 1 245.3.m.b 24
35.l odd 12 1 inner 35.3.l.a 24
35.l odd 12 1 175.3.p.c 24
35.l odd 12 1 245.3.g.c 12
105.x even 12 1 315.3.ca.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.l.a 24 1.a even 1 1 trivial
35.3.l.a 24 5.c odd 4 1 inner
35.3.l.a 24 7.c even 3 1 inner
35.3.l.a 24 35.l odd 12 1 inner
175.3.p.c 24 5.b even 2 1
175.3.p.c 24 5.c odd 4 1
175.3.p.c 24 35.j even 6 1
175.3.p.c 24 35.l odd 12 1
245.3.g.b 12 7.d odd 6 1
245.3.g.b 12 35.k even 12 1
245.3.g.c 12 7.c even 3 1
245.3.g.c 12 35.l odd 12 1
245.3.m.b 24 7.b odd 2 1
245.3.m.b 24 7.d odd 6 1
245.3.m.b 24 35.f even 4 1
245.3.m.b 24 35.k even 12 1
315.3.ca.a 24 3.b odd 2 1
315.3.ca.a 24 15.e even 4 1
315.3.ca.a 24 21.h odd 6 1
315.3.ca.a 24 105.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database