# Properties

 Label 475.2.n.b Level $475$ Weight $2$ Character orbit 475.n Analytic conductor $3.793$ Analytic rank $0$ Dimension $96$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.n (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 26 q^{4} - 4 q^{5} - 2 q^{6} + 28 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96 q + 26 q^{4} - 4 q^{5} - 2 q^{6} + 28 q^{9} + 28 q^{10} - 15 q^{11} - 85 q^{12} + 10 q^{14} - 10 q^{15} - 42 q^{16} + 20 q^{17} - 24 q^{19} - 16 q^{21} - 35 q^{23} - 24 q^{24} - 8 q^{25} + 28 q^{26} + 15 q^{27} + 30 q^{28} + 28 q^{29} - 64 q^{30} - 8 q^{31} + 25 q^{33} - 8 q^{34} + 33 q^{35} - 42 q^{36} - 55 q^{37} - 6 q^{39} - 48 q^{40} - 27 q^{41} + 210 q^{42} - 4 q^{44} + 15 q^{45} + 10 q^{46} - 115 q^{48} - 150 q^{49} + 9 q^{50} + 60 q^{51} - 5 q^{52} + 40 q^{53} + 47 q^{54} + 33 q^{55} - 12 q^{56} + 60 q^{58} + 25 q^{59} + 170 q^{60} + 26 q^{61} - 110 q^{62} - 30 q^{63} + 62 q^{64} - 15 q^{65} - 41 q^{66} + 35 q^{67} + 14 q^{69} - 20 q^{70} - 38 q^{71} - 60 q^{73} + 6 q^{74} - 151 q^{75} - 104 q^{76} + 115 q^{78} + 8 q^{79} - 63 q^{80} - 67 q^{81} + 160 q^{83} + 18 q^{84} - 8 q^{85} - 10 q^{87} - 120 q^{88} + 76 q^{89} + 108 q^{90} - 8 q^{91} + 85 q^{92} + 58 q^{94} + q^{95} - 6 q^{96} - 10 q^{97} + 10 q^{98} - 112 q^{99} + O(q^{100})$$

## 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 −1.64403 2.26281i 1.92696 + 0.626107i −1.79945 + 5.53815i −1.43116 1.71807i −1.75121 5.38968i 2.71859i 10.1699 3.30442i 0.894106 + 0.649606i −1.53480 + 6.06301i
39.2 −1.49821 2.06211i −0.334580 0.108712i −1.38963 + 4.27683i 0.909940 + 2.04255i 0.277096 + 0.852813i 3.13524i 6.05293 1.96672i −2.32693 1.69061i 2.84868 4.93656i
39.3 −1.47440 2.02934i −2.71260 0.881376i −1.32633 + 4.08203i −1.86244 + 1.23746i 2.21085 + 6.80430i 0.624081i 5.46813 1.77670i 4.15431 + 3.01828i 5.25723 + 1.95502i
39.4 −1.33322 1.83502i 1.54929 + 0.503395i −0.971794 + 2.99087i −1.83686 + 1.27512i −1.14181 3.51412i 1.90146i 2.46954 0.802403i −0.280156 0.203545i 4.78883 + 1.67066i
39.5 −1.09699 1.50988i −2.32300 0.754787i −0.458306 + 1.41052i −0.339804 2.21010i 1.40867 + 4.33543i 1.57904i −0.917463 + 0.298102i 2.39956 + 1.74338i −2.96421 + 2.93752i
39.6 −0.905108 1.24577i 2.85833 + 0.928728i −0.114698 + 0.353005i 0.555661 2.16593i −1.43011 4.40143i 3.69151i −2.38541 + 0.775067i 4.88047 + 3.54587i −3.20119 + 1.26817i
39.7 −0.883547 1.21610i 1.73129 + 0.562532i −0.0802061 + 0.246849i 0.949947 + 2.02425i −0.845587 2.60245i 4.12637i −2.48816 + 0.808453i 0.253887 + 0.184460i 1.62237 2.94375i
39.8 −0.695076 0.956690i −0.0684675 0.0222464i 0.185909 0.572168i 2.00266 0.994659i 0.0263072 + 0.0809651i 2.31826i −2.92592 + 0.950690i −2.42286 1.76031i −2.34358 1.22456i
39.9 −0.568591 0.782598i −1.99600 0.648541i 0.328870 1.01216i −2.03411 0.928656i 0.627363 + 1.93082i 4.53281i −2.81910 + 0.915982i 1.13637 + 0.825624i 0.429811 + 2.11992i
39.10 −0.262787 0.361695i 2.64793 + 0.860365i 0.556267 1.71202i 0.0123725 + 2.23603i −0.384652 1.18384i 4.32436i −1.61580 + 0.525007i 3.84426 + 2.79302i 0.805512 0.592076i
39.11 −0.222497 0.306241i −2.07855 0.675363i 0.573755 1.76584i 1.71339 + 1.43676i 0.255648 + 0.786804i 3.25699i −1.38845 + 0.451133i 1.43722 + 1.04420i 0.0587711 0.844386i
39.12 −0.183620 0.252731i 0.168087 + 0.0546149i 0.587877 1.80930i −2.21699 0.291495i −0.0170613 0.0525093i 2.35604i −1.15942 + 0.376718i −2.40178 1.74500i 0.333414 + 0.613827i
39.13 0.0240095 + 0.0330462i −0.866799 0.281640i 0.617518 1.90053i −0.855159 + 2.06608i −0.0115043 0.0354065i 1.23310i 0.155328 0.0504691i −1.75503 1.27510i −0.0888082 + 0.0213458i
39.14 0.0974656 + 0.134150i −3.23712 1.05180i 0.609537 1.87596i 1.77115 1.36492i −0.174408 0.536773i 2.20218i 0.626475 0.203554i 6.94558 + 5.04626i 0.355731 + 0.104567i
39.15 0.216810 + 0.298413i 1.05547 + 0.342943i 0.575990 1.77272i 0.566211 2.16319i 0.126498 + 0.389319i 0.311701i 1.35549 0.440426i −1.43064 1.03942i 0.768285 0.300037i
39.16 0.702847 + 0.967387i 1.21108 + 0.393505i 0.176192 0.542263i 1.31036 + 1.81189i 0.470535 + 1.44816i 1.16501i 2.92288 0.949700i −1.11518 0.810223i −0.831819 + 2.54111i
39.17 0.861798 + 1.18616i 0.988821 + 0.321288i −0.0462526 + 0.142351i 1.82580 + 1.29092i 0.471065 + 1.44979i 2.09401i 2.58012 0.838333i −1.55251 1.12796i 0.0422287 + 3.27820i
39.18 0.923507 + 1.27110i 3.21001 + 1.04299i −0.144793 + 0.445626i −1.48513 1.67165i 1.63872 + 5.04345i 3.13596i 2.28838 0.743540i 6.78926 + 4.93269i 0.753303 3.43152i
39.19 1.07816 + 1.48396i −2.46206 0.799971i −0.421671 + 1.29777i −1.24655 1.85637i −1.46737 4.51609i 3.88486i 1.10853 0.360183i 2.99472 + 2.17579i 1.41081 3.85129i
39.20 1.11654 + 1.53678i −0.629163 0.204428i −0.497012 + 1.52965i −2.15934 + 0.580726i −0.388324 1.19514i 3.69899i 0.707528 0.229890i −2.07300 1.50612i −3.30344 2.67004i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.24 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 475.2.n.b 96
25.e even 10 1 inner 475.2.n.b 96

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$27\!\cdots\!50$$$$T_{2}^{72} - 542493177045 T_{2}^{71} -$$$$15\!\cdots\!46$$$$T_{2}^{70} +$$$$34\!\cdots\!95$$$$T_{2}^{69} +$$$$75\!\cdots\!41$$$$T_{2}^{68} -$$$$19\!\cdots\!80$$$$T_{2}^{67} -$$$$35\!\cdots\!90$$$$T_{2}^{66} +$$$$10\!\cdots\!75$$$$T_{2}^{65} +$$$$15\!\cdots\!00$$$$T_{2}^{64} -$$$$49\!\cdots\!15$$$$T_{2}^{63} -$$$$60\!\cdots\!52$$$$T_{2}^{62} +$$$$21\!\cdots\!90$$$$T_{2}^{61} +$$$$22\!\cdots\!41$$$$T_{2}^{60} -$$$$83\!\cdots\!80$$$$T_{2}^{59} -$$$$77\!\cdots\!41$$$$T_{2}^{58} +$$$$30\!\cdots\!95$$$$T_{2}^{57} +$$$$24\!\cdots\!54$$$$T_{2}^{56} -$$$$97\!\cdots\!50$$$$T_{2}^{55} -$$$$73\!\cdots\!97$$$$T_{2}^{54} +$$$$28\!\cdots\!25$$$$T_{2}^{53} +$$$$20\!\cdots\!94$$$$T_{2}^{52} -$$$$76\!\cdots\!15$$$$T_{2}^{51} -$$$$51\!\cdots\!16$$$$T_{2}^{50} +$$$$18\!\cdots\!95$$$$T_{2}^{49} +$$$$12\!\cdots\!33$$$$T_{2}^{48} -$$$$38\!\cdots\!20$$$$T_{2}^{47} -$$$$26\!\cdots\!08$$$$T_{2}^{46} +$$$$72\!\cdots\!45$$$$T_{2}^{45} +$$$$53\!\cdots\!51$$$$T_{2}^{44} -$$$$11\!\cdots\!90$$$$T_{2}^{43} -$$$$99\!\cdots\!40$$$$T_{2}^{42} +$$$$13\!\cdots\!30$$$$T_{2}^{41} +$$$$16\!\cdots\!38$$$$T_{2}^{40} -$$$$95\!\cdots\!40$$$$T_{2}^{39} -$$$$25\!\cdots\!10$$$$T_{2}^{38} -$$$$98\!\cdots\!25$$$$T_{2}^{37} +$$$$35\!\cdots\!54$$$$T_{2}^{36} +$$$$53\!\cdots\!10$$$$T_{2}^{35} -$$$$44\!\cdots\!53$$$$T_{2}^{34} -$$$$12\!\cdots\!95$$$$T_{2}^{33} +$$$$47\!\cdots\!17$$$$T_{2}^{32} +$$$$20\!\cdots\!60$$$$T_{2}^{31} -$$$$43\!\cdots\!60$$$$T_{2}^{30} -$$$$26\!\cdots\!30$$$$T_{2}^{29} +$$$$33\!\cdots\!66$$$$T_{2}^{28} +$$$$28\!\cdots\!45$$$$T_{2}^{27} -$$$$18\!\cdots\!99$$$$T_{2}^{26} -$$$$24\!\cdots\!35$$$$T_{2}^{25} +$$$$43\!\cdots\!07$$$$T_{2}^{24} +$$$$14\!\cdots\!75$$$$T_{2}^{23} +$$$$23\!\cdots\!44$$$$T_{2}^{22} -$$$$51\!\cdots\!60$$$$T_{2}^{21} -$$$$23\!\cdots\!86$$$$T_{2}^{20} +$$$$81\!\cdots\!00$$$$T_{2}^{19} +$$$$85\!\cdots\!06$$$$T_{2}^{18} +$$$$73\!\cdots\!55$$$$T_{2}^{17} -$$$$13\!\cdots\!70$$$$T_{2}^{16} -$$$$41\!\cdots\!60$$$$T_{2}^{15} +$$$$64\!\cdots\!55$$$$T_{2}^{14} +$$$$39\!\cdots\!55$$$$T_{2}^{13} -$$$$50\!\cdots\!61$$$$T_{2}^{12} -$$$$24\!\cdots\!15$$$$T_{2}^{11} +$$$$28\!\cdots\!57$$$$T_{2}^{10} +$$$$18\!\cdots\!90$$$$T_{2}^{9} +$$$$51\!\cdots\!17$$$$T_{2}^{8} +$$$$76\!\cdots\!10$$$$T_{2}^{7} +$$$$96\!\cdots\!36$$$$T_{2}^{6} +$$$$21\!\cdots\!60$$$$T_{2}^{5} + 388680416144 T_{2}^{4} + 28887981920 T_{2}^{3} + 333658368 T_{2}^{2} - 7185280 T_{2} + 3041536$$">$$T_{2}^{96} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.