# Properties

 Label 350.2.q.c Level 350 Weight 2 Character orbit 350.q Analytic conductor 2.795 Analytic rank 0 Dimension 80 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.q (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{15})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 10q^{2} + 2q^{3} + 10q^{4} - 2q^{5} - 4q^{6} + 4q^{7} - 20q^{8} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 10q^{2} + 2q^{3} + 10q^{4} - 2q^{5} - 4q^{6} + 4q^{7} - 20q^{8} + 4q^{9} + 3q^{10} + 10q^{11} + 2q^{12} - 12q^{13} - 7q^{14} + 22q^{15} + 10q^{16} - 8q^{17} - 36q^{18} + 8q^{19} + 4q^{20} + 17q^{21} + 20q^{22} + 5q^{23} - 8q^{24} + 32q^{25} - 24q^{26} + 32q^{27} - 7q^{28} + 12q^{29} - 11q^{30} - 9q^{31} - 40q^{32} + 33q^{33} - 4q^{34} - 22q^{35} - 8q^{36} - 14q^{37} - 2q^{38} - 2q^{40} - 18q^{41} - 19q^{42} - 8q^{43} - 10q^{44} + 33q^{45} - 10q^{46} + 10q^{47} - 4q^{48} + 12q^{49} + 36q^{50} + 46q^{51} + 6q^{52} - 30q^{53} + 14q^{54} + 2q^{55} - q^{56} - 52q^{57} - 6q^{58} + 40q^{59} + 4q^{60} + 16q^{61} - 32q^{62} - 100q^{63} - 20q^{64} - 25q^{65} - 7q^{66} + q^{67} + 12q^{68} - 142q^{69} + 25q^{70} - 84q^{71} + 14q^{72} + 22q^{73} + 6q^{74} - 29q^{75} + 24q^{76} + 38q^{77} - 30q^{78} + 10q^{79} + 3q^{80} + 52q^{81} - 6q^{82} - 98q^{83} + 2q^{84} - 36q^{85} - 26q^{86} + 34q^{87} + 10q^{88} + 45q^{89} + 24q^{90} - 63q^{91} + 20q^{92} - 20q^{93} + 10q^{94} - 33q^{95} + 2q^{96} - 106q^{97} - 55q^{98} + 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 0.913545 + 0.406737i −1.89761 2.10751i 0.669131 + 0.743145i 1.19206 1.89182i −0.876351 2.69713i 2.35204 + 1.21158i 0.309017 + 0.951057i −0.527086 + 5.01489i 1.85848 1.24341i
11.2 0.913545 + 0.406737i −1.79467 1.99318i 0.669131 + 0.743145i −2.08698 + 0.802818i −0.828811 2.55082i −1.15917 + 2.37830i 0.309017 + 0.951057i −0.438351 + 4.17063i −2.23309 0.115440i
11.3 0.913545 + 0.406737i −0.699318 0.776671i 0.669131 + 0.743145i −2.20440 0.374997i −0.322958 0.993962i −0.649150 2.56488i 0.309017 + 0.951057i 0.199413 1.89729i −1.86129 1.23919i
11.4 0.913545 + 0.406737i −0.308075 0.342152i 0.669131 + 0.743145i 2.09051 + 0.793572i −0.142275 0.437877i −0.662723 + 2.56141i 0.309017 + 0.951057i 0.291428 2.77275i 1.58700 + 1.57525i
11.5 0.913545 + 0.406737i −0.138897 0.154261i 0.669131 + 0.743145i 1.77975 1.35369i −0.0641452 0.197419i 0.411507 2.61355i 0.309017 + 0.951057i 0.309081 2.94071i 2.17648 0.512766i
11.6 0.913545 + 0.406737i 0.352973 + 0.392016i 0.669131 + 0.743145i −0.665386 2.13477i 0.163009 + 0.501692i 2.31380 + 1.28309i 0.309017 + 0.951057i 0.284499 2.70682i 0.260430 2.22085i
11.7 0.913545 + 0.406737i 1.09566 + 1.21686i 0.669131 + 0.743145i −1.13462 + 1.92682i 0.505998 + 1.55730i −2.24539 1.39937i 0.309017 + 0.951057i 0.0333212 0.317030i −1.82023 + 1.29875i
11.8 0.913545 + 0.406737i 1.16133 + 1.28979i 0.669131 + 0.743145i 0.438531 + 2.19264i 0.536323 + 1.65063i 2.10056 1.60861i 0.309017 + 0.951057i −0.00127880 + 0.0121670i −0.491211 + 2.18145i
11.9 0.913545 + 0.406737i 1.76739 + 1.96288i 0.669131 + 0.743145i −2.20946 0.343957i 0.816212 + 2.51204i 0.680981 + 2.55661i 0.309017 + 0.951057i −0.415664 + 3.95478i −1.87854 1.21289i
11.10 0.913545 + 0.406737i 1.79948 + 1.99852i 0.669131 + 0.743145i 1.99096 1.01788i 0.831031 + 2.55765i −2.64246 + 0.131919i 0.309017 + 0.951057i −0.442385 + 4.20901i 2.23284 0.120079i
81.1 0.669131 0.743145i −0.326983 3.11103i −0.104528 0.994522i 1.99531 + 1.00932i −2.53074 1.83869i 0.935911 2.47469i −0.809017 0.587785i −6.63717 + 1.41077i 2.08519 0.807444i
81.2 0.669131 0.743145i −0.307151 2.92235i −0.104528 0.994522i −1.60236 + 1.55963i −2.37725 1.72718i −1.98259 + 1.75196i −0.809017 0.587785i −5.51135 + 1.17147i 0.0868405 + 2.23438i
81.3 0.669131 0.743145i −0.185703 1.76685i −0.104528 0.994522i −2.07165 0.841574i −1.43729 1.04425i 1.93450 1.80491i −0.809017 0.587785i −0.152831 + 0.0324853i −2.01162 + 0.976417i
81.4 0.669131 0.743145i −0.142789 1.35855i −0.104528 0.994522i 1.40758 1.73744i −1.10514 0.802933i 1.63333 + 2.08140i −0.809017 0.587785i 1.10918 0.235764i −0.349317 2.20861i
81.5 0.669131 0.743145i −0.0249244 0.237139i −0.104528 0.994522i 1.61270 + 1.54893i −0.192907 0.140155i −2.26242 + 1.37165i −0.809017 0.587785i 2.87883 0.611914i 2.23019 0.162036i
81.6 0.669131 0.743145i −0.00897919 0.0854313i −0.104528 0.994522i −0.515136 + 2.17592i −0.0694961 0.0504919i 2.53613 + 0.753683i −0.809017 0.587785i 2.92722 0.622201i 1.27233 + 1.83880i
81.7 0.669131 0.743145i 0.0735702 + 0.699974i −0.104528 0.994522i −1.29651 1.82183i 0.569410 + 0.413700i −2.28402 1.33539i −0.809017 0.587785i 2.44989 0.520741i −2.22142 0.255547i
81.8 0.669131 0.743145i 0.128261 + 1.22032i −0.104528 0.994522i 2.17491 0.519385i 0.992696 + 0.721236i −0.0870818 2.64432i −0.809017 0.587785i 1.46172 0.310698i 1.06932 1.96381i
81.9 0.669131 0.743145i 0.279306 + 2.65742i −0.104528 0.994522i −1.77018 + 1.36618i 2.16174 + 1.57059i −2.54508 + 0.722901i −0.809017 0.587785i −4.04940 + 0.860727i −0.169215 + 2.22966i
81.10 0.669131 0.743145i 0.306337 + 2.91460i −0.104528 0.994522i 0.374350 2.20451i 2.37095 + 1.72260i 2.62131 0.358777i −0.809017 0.587785i −5.46662 + 1.16197i −1.38778 1.75330i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.q.c 80
7.c even 3 1 inner 350.2.q.c 80
25.d even 5 1 inner 350.2.q.c 80
175.q even 15 1 inner 350.2.q.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.q.c 80 1.a even 1 1 trivial
350.2.q.c 80 7.c even 3 1 inner
350.2.q.c 80 25.d even 5 1 inner
350.2.q.c 80 175.q even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{80} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database