# Properties

 Label 950.2.n.a Level $950$ Weight $2$ Character orbit 950.n Analytic conductor $7.586$ Analytic rank $0$ Dimension $88$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.n (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q + 22q^{4} + 10q^{5} + 2q^{6} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q + 22q^{4} + 10q^{5} + 2q^{6} + 24q^{9} + 26q^{11} + 10q^{12} - 10q^{14} + 12q^{15} - 22q^{16} - 40q^{17} - 22q^{19} + 10q^{23} + 8q^{24} + 6q^{25} - 28q^{26} - 30q^{27} - 10q^{28} - 4q^{29} - 4q^{30} + 2q^{31} - 8q^{34} - 48q^{35} - 24q^{36} + 50q^{37} + 8q^{39} + 32q^{41} + 10q^{42} + 4q^{44} - 8q^{45} + 10q^{46} + 10q^{48} - 56q^{49} + 28q^{50} - 60q^{51} - 70q^{53} - 8q^{54} + 4q^{55} + 10q^{56} - 60q^{58} - 28q^{59} - 12q^{60} - 58q^{61} + 60q^{63} + 22q^{64} - 24q^{65} + 4q^{66} - 70q^{67} - 8q^{69} - 4q^{70} + 48q^{71} + 40q^{73} + 52q^{74} + 108q^{75} - 88q^{76} - 50q^{78} - 20q^{79} + 24q^{81} - 80q^{83} + 30q^{85} + 20q^{86} + 70q^{87} + 10q^{88} - 62q^{89} - 104q^{90} + 20q^{91} - 10q^{92} - 10q^{94} + 2q^{96} - 10q^{97} + 60q^{98} + 156q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1 −0.587785 0.809017i −2.56558 0.833609i −0.309017 + 0.951057i 0.0605232 + 2.23525i 0.833609 + 2.56558i 3.84106i 0.951057 0.309017i 3.46027 + 2.51403i 1.77278 1.36281i
39.2 −0.587785 0.809017i −2.20724 0.717177i −0.309017 + 0.951057i 2.23210 0.133199i 0.717177 + 2.20724i 2.34643i 0.951057 0.309017i 1.93053 + 1.40261i −1.41975 1.72751i
39.3 −0.587785 0.809017i −2.20015 0.714873i −0.309017 + 0.951057i −2.17200 0.531443i 0.714873 + 2.20015i 2.86995i 0.951057 0.309017i 1.90257 + 1.38230i 0.846721 + 2.06956i
39.4 −0.587785 0.809017i −1.06179 0.344997i −0.309017 + 0.951057i −0.518181 2.17520i 0.344997 + 1.06179i 0.133310i 0.951057 0.309017i −1.41867 1.03073i −1.45519 + 1.69777i
39.5 −0.587785 0.809017i −0.812784 0.264090i −0.309017 + 0.951057i 0.775807 + 2.09717i 0.264090 + 0.812784i 3.74573i 0.951057 0.309017i −1.83618 1.33406i 1.24064 1.86033i
39.6 −0.587785 0.809017i −0.273661 0.0889179i −0.309017 + 0.951057i −1.60391 + 1.55803i 0.0889179 + 0.273661i 0.224849i 0.951057 0.309017i −2.36007 1.71469i 2.20323 + 0.381801i
39.7 −0.587785 0.809017i 0.236785 + 0.0769362i −0.309017 + 0.951057i −1.89595 1.18549i −0.0769362 0.236785i 4.79883i 0.951057 0.309017i −2.37690 1.72692i 0.155332 + 2.23067i
39.8 −0.587785 0.809017i 1.25142 + 0.406611i −0.309017 + 0.951057i 1.47560 1.68006i −0.406611 1.25142i 2.56926i 0.951057 0.309017i −1.02633 0.745671i −2.22653 0.206269i
39.9 −0.587785 0.809017i 1.63623 + 0.531642i −0.309017 + 0.951057i 1.99525 1.00944i −0.531642 1.63623i 3.69408i 0.951057 0.309017i −0.0324603 0.0235838i −1.98943 1.02086i
39.10 −0.587785 0.809017i 2.87975 + 0.935689i −0.309017 + 0.951057i 0.587147 + 2.15760i −0.935689 2.87975i 1.61619i 0.951057 0.309017i 4.99042 + 3.62575i 1.40042 1.74322i
39.11 −0.587785 0.809017i 2.95005 + 0.958531i −0.309017 + 0.951057i 1.23590 1.86348i −0.958531 2.95005i 2.65378i 0.951057 0.309017i 5.35699 + 3.89208i −2.23403 + 0.0954627i
39.12 0.587785 + 0.809017i −3.09795 1.00658i −0.309017 + 0.951057i −0.367092 + 2.20573i −1.00658 3.09795i 4.43412i −0.951057 + 0.309017i 6.15701 + 4.47333i −2.00024 + 0.999512i
39.13 0.587785 + 0.809017i −1.88808 0.613474i −0.309017 + 0.951057i 2.21981 + 0.269141i −0.613474 1.88808i 0.715850i −0.951057 + 0.309017i 0.761435 + 0.553215i 1.08703 + 1.95406i
39.14 0.587785 + 0.809017i −1.78167 0.578899i −0.309017 + 0.951057i 0.452104 2.18989i −0.578899 1.78167i 4.62971i −0.951057 + 0.309017i 0.412164 + 0.299454i 2.03740 0.921423i
39.15 0.587785 + 0.809017i −1.70705 0.554656i −0.309017 + 0.951057i −2.20234 + 0.386932i −0.554656 1.70705i 2.40506i −0.951057 + 0.309017i 0.179341 + 0.130299i −1.60753 1.55429i
39.16 0.587785 + 0.809017i −1.33501 0.433772i −0.309017 + 0.951057i 0.407477 + 2.19863i −0.433772 1.33501i 1.42943i −0.951057 + 0.309017i −0.832948 0.605172i −1.53922 + 1.62198i
39.17 0.587785 + 0.809017i −0.284260 0.0923618i −0.309017 + 0.951057i 0.810167 2.08414i −0.0923618 0.284260i 5.14607i −0.951057 + 0.309017i −2.35478 1.71085i 2.16231 0.569586i
39.18 0.587785 + 0.809017i 0.285827 + 0.0928709i −0.309017 + 0.951057i 2.23251 0.126065i 0.0928709 + 0.285827i 0.304227i −0.951057 + 0.309017i −2.35398 1.71027i 1.41423 + 1.73204i
39.19 0.587785 + 0.809017i 0.718550 + 0.233471i −0.309017 + 0.951057i −1.61612 1.54537i 0.233471 + 0.718550i 0.567373i −0.951057 + 0.309017i −1.96525 1.42783i 0.300299 2.21581i
39.20 0.587785 + 0.809017i 1.55730 + 0.505997i −0.309017 + 0.951057i −0.854432 + 2.06638i 0.505997 + 1.55730i 0.994490i −0.951057 + 0.309017i −0.257907 0.187380i −2.17396 + 0.523340i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 609.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.n.a 88
25.e even 10 1 inner 950.2.n.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.n.a 88 1.a even 1 1 trivial
950.2.n.a 88 25.e even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!11$$$$T_{3}^{66} +$$$$37\!\cdots\!40$$$$T_{3}^{65} +$$$$13\!\cdots\!67$$$$T_{3}^{64} -$$$$22\!\cdots\!10$$$$T_{3}^{63} -$$$$80\!\cdots\!37$$$$T_{3}^{62} +$$$$12\!\cdots\!10$$$$T_{3}^{61} +$$$$44\!\cdots\!81$$$$T_{3}^{60} -$$$$58\!\cdots\!70$$$$T_{3}^{59} -$$$$22\!\cdots\!01$$$$T_{3}^{58} +$$$$25\!\cdots\!20$$$$T_{3}^{57} +$$$$10\!\cdots\!42$$$$T_{3}^{56} -$$$$94\!\cdots\!50$$$$T_{3}^{55} -$$$$46\!\cdots\!84$$$$T_{3}^{54} +$$$$30\!\cdots\!40$$$$T_{3}^{53} +$$$$18\!\cdots\!89$$$$T_{3}^{52} -$$$$73\!\cdots\!70$$$$T_{3}^{51} -$$$$66\!\cdots\!64$$$$T_{3}^{50} +$$$$11\!\cdots\!20$$$$T_{3}^{49} +$$$$21\!\cdots\!42$$$$T_{3}^{48} +$$$$16\!\cdots\!50$$$$T_{3}^{47} -$$$$63\!\cdots\!64$$$$T_{3}^{46} -$$$$21\!\cdots\!00$$$$T_{3}^{45} +$$$$16\!\cdots\!97$$$$T_{3}^{44} +$$$$10\!\cdots\!30$$$$T_{3}^{43} -$$$$35\!\cdots\!59$$$$T_{3}^{42} -$$$$32\!\cdots\!50$$$$T_{3}^{41} +$$$$68\!\cdots\!82$$$$T_{3}^{40} +$$$$85\!\cdots\!30$$$$T_{3}^{39} -$$$$11\!\cdots\!99$$$$T_{3}^{38} -$$$$19\!\cdots\!30$$$$T_{3}^{37} +$$$$13\!\cdots\!59$$$$T_{3}^{36} +$$$$36\!\cdots\!50$$$$T_{3}^{35} -$$$$94\!\cdots\!84$$$$T_{3}^{34} -$$$$53\!\cdots\!00$$$$T_{3}^{33} +$$$$38\!\cdots\!42$$$$T_{3}^{32} +$$$$74\!\cdots\!80$$$$T_{3}^{31} +$$$$13\!\cdots\!39$$$$T_{3}^{30} -$$$$88\!\cdots\!50$$$$T_{3}^{29} -$$$$38\!\cdots\!84$$$$T_{3}^{28} +$$$$87\!\cdots\!10$$$$T_{3}^{27} +$$$$63\!\cdots\!58$$$$T_{3}^{26} -$$$$56\!\cdots\!30$$$$T_{3}^{25} -$$$$47\!\cdots\!33$$$$T_{3}^{24} +$$$$67\!\cdots\!30$$$$T_{3}^{23} +$$$$80\!\cdots\!99$$$$T_{3}^{22} -$$$$10\!\cdots\!90$$$$T_{3}^{21} -$$$$43\!\cdots\!18$$$$T_{3}^{20} +$$$$72\!\cdots\!90$$$$T_{3}^{19} +$$$$31\!\cdots\!16$$$$T_{3}^{18} +$$$$17\!\cdots\!20$$$$T_{3}^{17} -$$$$18\!\cdots\!19$$$$T_{3}^{16} -$$$$60\!\cdots\!70$$$$T_{3}^{15} +$$$$75\!\cdots\!54$$$$T_{3}^{14} +$$$$63\!\cdots\!60$$$$T_{3}^{13} +$$$$75\!\cdots\!68$$$$T_{3}^{12} -$$$$90\!\cdots\!40$$$$T_{3}^{11} -$$$$23\!\cdots\!72$$$$T_{3}^{10} +$$$$10\!\cdots\!60$$$$T_{3}^{9} +$$$$36\!\cdots\!68$$$$T_{3}^{8} -$$$$13\!\cdots\!20$$$$T_{3}^{7} -$$$$52\!\cdots\!68$$$$T_{3}^{6} +$$$$14\!\cdots\!60$$$$T_{3}^{5} +$$$$59\!\cdots\!12$$$$T_{3}^{4} -$$$$81\!\cdots\!20$$$$T_{3}^{3} -$$$$35\!\cdots\!00$$$$T_{3}^{2} +$$$$20\!\cdots\!80$$$$T_{3} +$$$$94\!\cdots\!16$$">$$T_{3}^{88} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.