# Properties

 Label 350.2.x.a Level 350 Weight 2 Character orbit 350.x Analytic conductor 2.795 Analytic rank 0 Dimension 320 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.x (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$20$$ over $$\Q(\zeta_{60})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$320q + 12q^{5} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$320q + 12q^{5} - 8q^{7} + 12q^{10} - 16q^{15} - 40q^{16} + 36q^{17} + 8q^{18} - 72q^{22} + 44q^{23} - 12q^{25} - 24q^{28} - 80q^{29} + 20q^{30} - 48q^{33} - 28q^{35} + 80q^{36} - 4q^{37} - 24q^{38} - 40q^{39} - 36q^{42} + 88q^{43} - 228q^{45} - 12q^{47} + 32q^{50} - 52q^{53} + 152q^{57} + 32q^{58} - 120q^{59} - 8q^{60} + 136q^{63} + 8q^{65} - 32q^{67} - 144q^{68} + 92q^{70} + 8q^{72} + 12q^{73} - 432q^{75} + 144q^{77} - 16q^{78} + 12q^{80} - 40q^{81} - 192q^{82} + 60q^{84} - 24q^{85} + 24q^{87} + 4q^{88} - 300q^{89} - 8q^{92} - 68q^{93} + 20q^{95} - 40q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −0.544639 + 0.838671i −1.15652 + 3.01283i −0.406737 0.913545i −1.77963 1.35386i −1.89689 2.61084i −0.00275235 + 2.64575i 0.987688 + 0.156434i −5.51019 4.96140i 2.10469 0.755156i
3.2 −0.544639 + 0.838671i −0.817783 + 2.13040i −0.406737 0.913545i 2.21313 0.319493i −1.34131 1.84615i 2.63815 0.200418i 0.987688 + 0.156434i −1.64039 1.47701i −0.937405 + 2.03009i
3.3 −0.544639 + 0.838671i −0.476017 + 1.24007i −0.406737 0.913545i 1.14772 + 1.91905i −0.780750 1.07461i −2.35038 + 1.21478i 0.987688 + 0.156434i 0.918261 + 0.826806i −2.23454 0.0826285i
3.4 −0.544639 + 0.838671i −0.316438 + 0.824348i −0.406737 0.913545i 0.514287 2.17612i −0.519012 0.714359i −2.61096 0.427657i 0.987688 + 0.156434i 1.65002 + 1.48568i 1.54495 + 1.61652i
3.5 −0.544639 + 0.838671i −0.255867 + 0.666556i −0.406737 0.913545i −2.11039 0.739089i −0.419666 0.577620i 2.27088 1.35761i 0.987688 + 0.156434i 1.85061 + 1.66629i 1.76925 1.36739i
3.6 −0.544639 + 0.838671i 0.172615 0.449679i −0.406737 0.913545i −2.02394 + 0.950624i 0.283119 + 0.389680i −1.46618 2.20234i 0.987688 + 0.156434i 2.05702 + 1.85215i 0.305054 2.21516i
3.7 −0.544639 + 0.838671i 0.375167 0.977343i −0.406737 0.913545i 1.91741 + 1.15045i 0.615338 + 0.846940i 2.64280 + 0.124862i 0.987688 + 0.156434i 1.41499 + 1.27406i −2.00915 + 0.981495i
3.8 −0.544639 + 0.838671i 0.637721 1.66132i −0.406737 0.913545i −1.80729 + 1.31671i 1.04597 + 1.43966i −0.155100 + 2.64120i 0.987688 + 0.156434i −0.123863 0.111527i −0.119964 2.23285i
3.9 −0.544639 + 0.838671i 0.828696 2.15883i −0.406737 0.913545i 1.91840 1.14880i 1.35920 + 1.87078i −1.79641 1.94240i 0.987688 + 0.156434i −1.74436 1.57063i −0.0813718 + 2.23459i
3.10 −0.544639 + 0.838671i 1.00842 2.62703i −0.406737 0.913545i −0.333527 2.21105i 1.65399 + 2.27652i 2.63222 + 0.267259i 0.987688 + 0.156434i −3.65494 3.29092i 2.03600 + 0.924507i
3.11 0.544639 0.838671i −1.20499 + 3.13910i −0.406737 0.913545i 1.05534 + 1.97136i 1.97639 + 2.72026i 2.18246 + 1.49561i −0.987688 0.156434i −6.17252 5.55776i 2.22810 + 0.188596i
3.12 0.544639 0.838671i −1.02776 + 2.67740i −0.406737 0.913545i −1.46986 1.68509i 1.68570 + 2.32016i 0.0482142 2.64531i −0.987688 0.156434i −3.88274 3.49603i −2.21377 + 0.314962i
3.13 0.544639 0.838671i −0.439528 + 1.14501i −0.406737 0.913545i −2.19147 + 0.444347i 0.720902 + 0.992236i −2.02541 + 1.70227i −0.987688 0.156434i 1.11157 + 1.00086i −0.820901 + 2.07993i
3.14 0.544639 0.838671i −0.327039 + 0.851967i −0.406737 0.913545i −1.80326 + 1.32222i 0.536401 + 0.738293i 2.28229 1.33834i −0.987688 0.156434i 1.61054 + 1.45014i 0.126777 + 2.23247i
3.15 0.544639 0.838671i −0.110635 + 0.288213i −0.406737 0.913545i 1.72772 1.41950i 0.181460 + 0.249758i 2.11003 + 1.59618i −0.987688 0.156434i 2.15861 + 1.94362i −0.249509 2.22210i
3.16 0.544639 0.838671i −0.0130481 + 0.0339916i −0.406737 0.913545i 0.859199 + 2.06441i 0.0214012 + 0.0294562i −1.30429 + 2.30192i −0.987688 0.156434i 2.22845 + 2.00650i 2.19931 + 0.403772i
3.17 0.544639 0.838671i 0.309740 0.806901i −0.406737 0.913545i −0.415279 2.19717i −0.508027 0.699240i −1.18335 2.36636i −0.987688 0.156434i 1.67428 + 1.50753i −2.06888 0.848381i
3.18 0.544639 0.838671i 0.822850 2.14360i −0.406737 0.913545i −2.04873 0.895939i −1.34962 1.85759i 2.63801 + 0.202309i −0.987688 0.156434i −1.68849 1.52033i −1.86722 + 1.23025i
3.19 0.544639 0.838671i 0.894901 2.33130i −0.406737 0.913545i 0.406439 + 2.19882i −1.46779 2.02024i −1.33531 2.28406i −0.987688 0.156434i −2.40466 2.16517i 2.06545 + 0.856694i
3.20 0.544639 0.838671i 1.09550 2.85388i −0.406737 0.913545i 1.64939 1.50980i −1.79681 2.47310i −1.61036 + 2.09922i −0.987688 0.156434i −4.71508 4.24548i −0.367903 2.20559i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 327.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.x.a 320
7.d odd 6 1 inner 350.2.x.a 320
25.f odd 20 1 inner 350.2.x.a 320
175.x even 60 1 inner 350.2.x.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.x.a 320 1.a even 1 1 trivial
350.2.x.a 320 7.d odd 6 1 inner
350.2.x.a 320 25.f odd 20 1 inner
350.2.x.a 320 175.x even 60 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database