# Properties

 Label 950.2.n.b Level $950$ Weight $2$ Character orbit 950.n Analytic conductor $7.586$ Analytic rank $0$ Dimension $96$ 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: $$96$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96q + 24q^{4} + 8q^{5} - 6q^{6} + 34q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q + 24q^{4} + 8q^{5} - 6q^{6} + 34q^{9} - 24q^{11} + 10q^{12} + 10q^{14} - 8q^{15} - 24q^{16} + 30q^{17} + 24q^{19} + 2q^{20} - 24q^{24} - 60q^{25} + 84q^{26} - 30q^{27} - 10q^{28} - 4q^{29} + 16q^{30} - 14q^{31} + 100q^{33} + 8q^{34} + 42q^{35} - 34q^{36} - 30q^{37} + 32q^{39} + 12q^{41} + 10q^{42} + 4q^{44} - 18q^{45} - 10q^{46} + 10q^{48} - 132q^{49} - 36q^{50} + 36q^{51} + 30q^{53} + 24q^{54} - 4q^{55} - 10q^{56} + 60q^{58} + 16q^{59} + 8q^{60} + 42q^{61} - 110q^{63} + 24q^{64} + 12q^{65} - 20q^{66} + 130q^{67} - 8q^{69} + 20q^{70} - 8q^{71} - 120q^{73} - 124q^{74} - 24q^{75} + 96q^{76} - 50q^{78} + 4q^{79} - 2q^{80} - 10q^{81} - 70q^{83} + 52q^{85} - 44q^{86} - 70q^{87} + 10q^{88} - 26q^{89} + 32q^{90} - 4q^{91} - 10q^{92} + 10q^{94} + 2q^{95} - 6q^{96} - 10q^{97} - 60q^{98} - 184q^{99} + O(q^{100})$$

## 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 −3.22850 1.04900i −0.309017 + 0.951057i −1.55488 1.60697i 1.04900 + 3.22850i 3.46343i 0.951057 0.309017i 6.89577 + 5.01007i −0.386131 + 2.20248i
39.2 −0.587785 0.809017i −3.16378 1.02797i −0.309017 + 0.951057i 1.22578 1.87015i 1.02797 + 3.16378i 4.52249i 0.951057 0.309017i 6.52572 + 4.74121i −2.23348 + 0.107573i
39.3 −0.587785 0.809017i −2.09906 0.682024i −0.309017 + 0.951057i −0.131723 + 2.23218i 0.682024 + 2.09906i 4.35569i 0.951057 0.309017i 1.51382 + 1.09986i 1.88330 1.20548i
39.4 −0.587785 0.809017i −2.06198 0.669976i −0.309017 + 0.951057i 2.16827 + 0.546445i 0.669976 + 2.06198i 0.754312i 0.951057 0.309017i 1.37582 + 0.999595i −0.832394 2.07536i
39.5 −0.587785 0.809017i −1.16860 0.379700i −0.309017 + 0.951057i −2.18761 0.463006i 0.379700 + 1.16860i 0.0227859i 0.951057 0.309017i −1.20561 0.875924i 0.911264 + 2.04196i
39.6 −0.587785 0.809017i −0.408708 0.132797i −0.309017 + 0.951057i 1.37111 1.76637i 0.132797 + 0.408708i 1.90044i 0.951057 0.309017i −2.27764 1.65481i −2.23494 0.0710083i
39.7 −0.587785 0.809017i 0.0436702 + 0.0141893i −0.309017 + 0.951057i 0.459049 + 2.18844i −0.0141893 0.0436702i 2.50954i 0.951057 0.309017i −2.42535 1.76212i 1.50066 1.65771i
39.8 −0.587785 0.809017i 0.702911 + 0.228390i −0.309017 + 0.951057i −0.520909 2.17455i −0.228390 0.702911i 2.52633i 0.951057 0.309017i −1.98513 1.44228i −1.45306 + 1.69959i
39.9 −0.587785 0.809017i 1.21854 + 0.395928i −0.309017 + 0.951057i 1.81457 + 1.30665i −0.395928 1.21854i 1.92349i 0.951057 0.309017i −1.09897 0.798446i −0.00947820 2.23605i
39.10 −0.587785 0.809017i 1.40124 + 0.455290i −0.309017 + 0.951057i −1.43867 + 1.71179i −0.455290 1.40124i 2.83916i 0.951057 0.309017i −0.670872 0.487417i 2.23050 + 0.157745i
39.11 −0.587785 0.809017i 1.83314 + 0.595623i −0.309017 + 0.951057i 2.21079 0.335262i −0.595623 1.83314i 3.44879i 0.951057 0.309017i 0.578586 + 0.420367i −1.57070 1.59151i
39.12 −0.587785 0.809017i 2.95991 + 0.961734i −0.309017 + 0.951057i −2.05251 0.887239i −0.961734 2.95991i 0.626185i 0.951057 0.309017i 5.40910 + 3.92994i 0.488645 + 2.18202i
39.13 0.587785 + 0.809017i −3.02842 0.983992i −0.309017 + 0.951057i 1.06990 + 1.96349i −0.983992 3.02842i 2.61307i −0.951057 + 0.309017i 5.77601 + 4.19652i −0.959627 + 2.01968i
39.14 0.587785 + 0.809017i −2.52973 0.821960i −0.309017 + 0.951057i 1.13666 1.92562i −0.821960 2.52973i 1.29013i −0.951057 + 0.309017i 3.29688 + 2.39532i 2.22597 0.212273i
39.15 0.587785 + 0.809017i −2.08312 0.676848i −0.309017 + 0.951057i −1.46374 1.69040i −0.676848 2.08312i 1.17098i −0.951057 + 0.309017i 1.45423 + 1.05656i 0.507200 2.17779i
39.16 0.587785 + 0.809017i −1.43673 0.466821i −0.309017 + 0.951057i −1.88711 + 1.19951i −0.466821 1.43673i 3.41486i −0.951057 + 0.309017i −0.580791 0.421969i −2.07964 0.821650i
39.17 0.587785 + 0.809017i −0.0948098 0.0308056i −0.309017 + 0.951057i 1.43256 1.71691i −0.0308056 0.0948098i 2.23805i −0.951057 + 0.309017i −2.41901 1.75751i 2.23105 + 0.149792i
39.18 0.587785 + 0.809017i −0.0145515 0.00472808i −0.309017 + 0.951057i 1.99515 + 1.00964i −0.00472808 0.0145515i 0.890552i −0.951057 + 0.309017i −2.42686 1.76322i 0.355906 + 2.20756i
39.19 0.587785 + 0.809017i 0.219108 + 0.0711926i −0.309017 + 0.951057i −0.684773 + 2.12863i 0.0711926 + 0.219108i 4.22635i −0.951057 + 0.309017i −2.38411 1.73216i −2.12460 + 0.697187i
39.20 0.587785 + 0.809017i 1.16397 + 0.378196i −0.309017 + 0.951057i −2.21303 + 0.320187i 0.378196 + 1.16397i 3.41155i −0.951057 + 0.309017i −1.21526 0.882938i −1.55982 1.60217i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 609.24 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.b 96
25.e even 10 1 inner 950.2.n.b 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.n.b 96 1.a even 1 1 trivial
950.2.n.b 96 25.e even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!97$$$$T_{3}^{76} -$$$$43\!\cdots\!20$$$$T_{3}^{75} -$$$$85\!\cdots\!51$$$$T_{3}^{74} +$$$$37\!\cdots\!80$$$$T_{3}^{73} +$$$$56\!\cdots\!23$$$$T_{3}^{72} -$$$$29\!\cdots\!30$$$$T_{3}^{71} -$$$$32\!\cdots\!09$$$$T_{3}^{70} +$$$$21\!\cdots\!70$$$$T_{3}^{69} +$$$$15\!\cdots\!53$$$$T_{3}^{68} -$$$$14\!\cdots\!70$$$$T_{3}^{67} -$$$$58\!\cdots\!97$$$$T_{3}^{66} +$$$$85\!\cdots\!20$$$$T_{3}^{65} +$$$$11\!\cdots\!98$$$$T_{3}^{64} -$$$$47\!\cdots\!50$$$$T_{3}^{63} +$$$$61\!\cdots\!28$$$$T_{3}^{62} +$$$$24\!\cdots\!20$$$$T_{3}^{61} -$$$$91\!\cdots\!27$$$$T_{3}^{60} -$$$$11\!\cdots\!70$$$$T_{3}^{59} +$$$$70\!\cdots\!28$$$$T_{3}^{58} +$$$$48\!\cdots\!60$$$$T_{3}^{57} -$$$$42\!\cdots\!70$$$$T_{3}^{56} -$$$$18\!\cdots\!10$$$$T_{3}^{55} +$$$$21\!\cdots\!32$$$$T_{3}^{54} +$$$$67\!\cdots\!00$$$$T_{3}^{53} -$$$$95\!\cdots\!19$$$$T_{3}^{52} -$$$$21\!\cdots\!90$$$$T_{3}^{51} +$$$$37\!\cdots\!69$$$$T_{3}^{50} +$$$$59\!\cdots\!70$$$$T_{3}^{49} -$$$$13\!\cdots\!54$$$$T_{3}^{48} -$$$$14\!\cdots\!10$$$$T_{3}^{47} +$$$$41\!\cdots\!49$$$$T_{3}^{46} +$$$$29\!\cdots\!70$$$$T_{3}^{45} -$$$$11\!\cdots\!53$$$$T_{3}^{44} -$$$$41\!\cdots\!90$$$$T_{3}^{43} +$$$$28\!\cdots\!48$$$$T_{3}^{42} +$$$$16\!\cdots\!60$$$$T_{3}^{41} -$$$$60\!\cdots\!70$$$$T_{3}^{40} +$$$$11\!\cdots\!80$$$$T_{3}^{39} +$$$$11\!\cdots\!83$$$$T_{3}^{38} -$$$$50\!\cdots\!30$$$$T_{3}^{37} -$$$$18\!\cdots\!20$$$$T_{3}^{36} +$$$$13\!\cdots\!50$$$$T_{3}^{35} +$$$$23\!\cdots\!22$$$$T_{3}^{34} -$$$$25\!\cdots\!50$$$$T_{3}^{33} -$$$$22\!\cdots\!49$$$$T_{3}^{32} +$$$$35\!\cdots\!90$$$$T_{3}^{31} +$$$$18\!\cdots\!99$$$$T_{3}^{30} -$$$$45\!\cdots\!90$$$$T_{3}^{29} -$$$$33\!\cdots\!74$$$$T_{3}^{28} +$$$$43\!\cdots\!50$$$$T_{3}^{27} -$$$$16\!\cdots\!80$$$$T_{3}^{26} -$$$$23\!\cdots\!40$$$$T_{3}^{25} +$$$$22\!\cdots\!49$$$$T_{3}^{24} -$$$$32\!\cdots\!50$$$$T_{3}^{23} -$$$$12\!\cdots\!78$$$$T_{3}^{22} -$$$$22\!\cdots\!40$$$$T_{3}^{21} +$$$$21\!\cdots\!48$$$$T_{3}^{20} -$$$$20\!\cdots\!40$$$$T_{3}^{19} -$$$$42\!\cdots\!68$$$$T_{3}^{18} +$$$$14\!\cdots\!60$$$$T_{3}^{17} +$$$$36\!\cdots\!20$$$$T_{3}^{16} -$$$$25\!\cdots\!40$$$$T_{3}^{15} +$$$$44\!\cdots\!48$$$$T_{3}^{14} -$$$$48\!\cdots\!40$$$$T_{3}^{13} +$$$$49\!\cdots\!64$$$$T_{3}^{12} -$$$$21\!\cdots\!40$$$$T_{3}^{11} +$$$$30\!\cdots\!56$$$$T_{3}^{10} +$$$$45\!\cdots\!20$$$$T_{3}^{9} -$$$$53\!\cdots\!36$$$$T_{3}^{8} +$$$$40\!\cdots\!40$$$$T_{3}^{7} +$$$$49\!\cdots\!88$$$$T_{3}^{6} -$$$$32\!\cdots\!20$$$$T_{3}^{5} - 215404622848 T_{3}^{4} + 22720706560 T_{3}^{3} + 380170240 T_{3}^{2} + 1597440 T_{3} + 4096$$">$$T_{3}^{96} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.