Properties

Label 475.2.n.a
Level $475$
Weight $2$
Character orbit 475.n
Analytic conductor $3.793$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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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: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 16 q^{4} - 3 q^{5} + 6 q^{6} + 8 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 16 q^{4} - 3 q^{5} + 6 q^{6} + 8 q^{9} - 36 q^{10} + 20 q^{11} + 45 q^{12} - 10 q^{14} - 20 q^{16} - 15 q^{17} + 20 q^{19} + 12 q^{20} + 16 q^{21} + 15 q^{23} + 72 q^{24} + 41 q^{25} - 84 q^{26} + 15 q^{27} + 30 q^{28} - 24 q^{29} - 40 q^{30} + 8 q^{31} - 75 q^{33} - 24 q^{34} - 33 q^{35} - 32 q^{36} - 15 q^{37} - 30 q^{39} - 28 q^{40} + 13 q^{41} - 130 q^{42} - 24 q^{44} + 6 q^{45} + 30 q^{46} + 145 q^{48} - 28 q^{49} + 77 q^{50} - 36 q^{51} - 5 q^{52} - 10 q^{53} + 15 q^{54} - 8 q^{55} + 48 q^{56} - 60 q^{58} - 19 q^{59} - 110 q^{60} + 8 q^{61} + 110 q^{62} + 55 q^{63} + 16 q^{64} - 43 q^{65} - 17 q^{66} - 65 q^{67} - 42 q^{69} + 4 q^{70} + 18 q^{71} + 100 q^{73} + 22 q^{74} + 115 q^{75} + 64 q^{76} - 145 q^{78} - 16 q^{79} - 97 q^{80} + q^{81} - 70 q^{83} - 46 q^{84} - 16 q^{85} + 64 q^{86} + 10 q^{87} + 30 q^{88} + 4 q^{89} - 8 q^{90} + 16 q^{91} - 135 q^{92} + 38 q^{94} - 2 q^{95} + 50 q^{96} + 150 q^{97} + 130 q^{98} + 178 q^{99} + O(q^{100}) \)

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 −1.60597 2.21043i −1.56040 0.507006i −1.68883 + 5.19768i 0.411675 2.19785i 1.38526 + 4.26341i 3.04333i 9.00431 2.92568i −0.249248 0.181089i −5.51933 + 2.61970i
39.2 −1.39149 1.91522i 2.70917 + 0.880261i −1.11379 + 3.42788i 2.14567 + 0.629357i −2.08388 6.41351i 1.72629i 3.61202 1.17362i 4.13767 + 3.00619i −1.78032 4.98517i
39.3 −1.21720 1.67534i −0.191192 0.0621221i −0.707134 + 2.17633i −2.22482 0.223992i 0.128644 + 0.395926i 3.63478i 0.567865 0.184511i −2.39436 1.73960i 2.33280 + 3.99997i
39.4 −1.12280 1.54541i 1.41655 + 0.460266i −0.509559 + 1.56826i −0.295058 2.21652i −0.879213 2.70594i 3.52405i −0.637733 + 0.207212i −0.632270 0.459371i −3.09413 + 2.94469i
39.5 −1.02546 1.41142i −2.62954 0.854391i −0.322513 + 0.992591i 1.35649 + 1.77762i 1.49058 + 4.58753i 0.182993i −1.58676 + 0.515570i 3.75747 + 2.72996i 1.11795 3.73745i
39.6 −0.819032 1.12730i 0.641372 + 0.208394i 0.0180399 0.0555211i −0.501037 + 2.17921i −0.290381 0.893701i 0.0684183i −2.72781 + 0.886319i −2.05912 1.49604i 2.86699 1.22002i
39.7 −0.708585 0.975283i −1.48229 0.481625i 0.168949 0.519971i −1.59939 + 1.56267i 0.580606 + 1.78692i 2.31635i −2.91986 + 0.948721i −0.461836 0.335543i 2.65735 + 0.452578i
39.8 −0.396268 0.545416i 2.29043 + 0.744207i 0.477583 1.46985i −1.89679 1.18414i −0.501723 1.54414i 2.02861i −2.27328 + 0.738634i 2.26518 + 1.64575i 0.105787 + 1.50378i
39.9 −0.325781 0.448399i −0.381722 0.124029i 0.523105 1.60995i −0.781184 2.09517i 0.0687433 + 0.211570i 3.07504i −1.94657 + 0.632479i −2.29672 1.66867i −0.684979 + 1.03285i
39.10 0.0162182 + 0.0223225i 2.53599 + 0.823993i 0.617799 1.90139i 2.18453 0.477335i 0.0227357 + 0.0699732i 0.768841i 0.104947 0.0340992i 3.32522 + 2.41592i 0.0460844 + 0.0410225i
39.11 0.203802 + 0.280510i −1.02562 0.333243i 0.580884 1.78778i 1.88943 + 1.19586i −0.115545 0.355611i 4.66677i 1.27939 0.415700i −1.48621 1.07980i 0.0496198 + 0.773721i
39.12 0.302513 + 0.416374i 2.54210 + 0.825977i 0.536181 1.65020i −1.61099 + 1.55071i 0.425102 + 1.30833i 4.22504i 1.82825 0.594036i 3.35296 + 2.43607i −1.13302 0.201663i
39.13 0.470002 + 0.646902i −1.99652 0.648707i 0.420454 1.29402i −0.877494 2.05670i −0.518716 1.59644i 0.826548i 2.55568 0.830390i 1.13820 + 0.826953i 0.918057 1.53430i
39.14 0.494729 + 0.680936i −1.00441 0.326353i 0.399117 1.22836i 2.12122 0.707408i −0.274686 0.845396i 5.02465i 2.63486 0.856119i −1.52472 1.10777i 1.53113 + 1.09444i
39.15 0.636441 + 0.875986i 0.575766 + 0.187078i 0.255740 0.787086i −2.22529 0.219241i 0.202564 + 0.623427i 2.44664i 2.91181 0.946103i −2.13054 1.54793i −1.22422 2.08886i
39.16 1.06731 + 1.46902i 1.88156 + 0.611355i −0.400851 + 1.23369i −0.796184 + 2.08952i 1.11011 + 3.41656i 1.51173i 1.21373 0.394364i 0.739455 + 0.537245i −3.91933 + 1.06055i
39.17 1.10380 + 1.51924i −1.16023 0.376980i −0.471705 + 1.45176i 2.05956 0.870765i −0.707927 2.17878i 1.60835i 0.845716 0.274790i −1.22304 0.888592i 3.59623 + 2.16782i
39.18 1.30771 + 1.79991i 1.66715 + 0.541690i −0.911532 + 2.80541i 0.941634 2.02813i 1.20516 + 3.70909i 2.10218i −2.00966 + 0.652977i 0.0589103 + 0.0428008i 4.88184 0.957357i
39.19 1.49224 + 2.05389i −0.805296 0.261657i −1.37366 + 4.22769i 0.616846 + 2.14930i −0.664281 2.04445i 1.89983i −5.90407 + 1.91835i −1.84701 1.34193i −3.49395 + 4.47421i
39.20 1.51783 + 2.08911i −2.90483 0.943838i −1.44255 + 4.43972i −2.22781 + 0.191942i −2.43726 7.50111i 3.43848i −6.55283 + 2.12914i 5.12018 + 3.72003i −3.78243 4.36282i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.20
Significant digits:
Format:

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.a 80
25.e even 10 1 inner 475.2.n.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.n.a 80 1.a even 1 1 trivial
475.2.n.a 80 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(23\!\cdots\!41\)\( T_{2}^{52} + 541404866505 T_{2}^{51} - \)\(79\!\cdots\!08\)\( T_{2}^{50} - \)\(22\!\cdots\!10\)\( T_{2}^{49} + \)\(24\!\cdots\!20\)\( T_{2}^{48} + \)\(82\!\cdots\!10\)\( T_{2}^{47} - \)\(68\!\cdots\!33\)\( T_{2}^{46} - \)\(26\!\cdots\!60\)\( T_{2}^{45} + \)\(17\!\cdots\!13\)\( T_{2}^{44} + \)\(77\!\cdots\!45\)\( T_{2}^{43} - \)\(39\!\cdots\!67\)\( T_{2}^{42} - \)\(19\!\cdots\!20\)\( T_{2}^{41} + \)\(78\!\cdots\!31\)\( T_{2}^{40} + \)\(43\!\cdots\!65\)\( T_{2}^{39} - \)\(13\!\cdots\!69\)\( T_{2}^{38} - \)\(83\!\cdots\!55\)\( T_{2}^{37} + \)\(21\!\cdots\!08\)\( T_{2}^{36} + \)\(13\!\cdots\!00\)\( T_{2}^{35} - \)\(29\!\cdots\!85\)\( T_{2}^{34} - \)\(17\!\cdots\!95\)\( T_{2}^{33} + \)\(36\!\cdots\!98\)\( T_{2}^{32} + \)\(19\!\cdots\!45\)\( T_{2}^{31} - \)\(39\!\cdots\!28\)\( T_{2}^{30} - \)\(18\!\cdots\!70\)\( T_{2}^{29} + \)\(38\!\cdots\!20\)\( T_{2}^{28} + \)\(13\!\cdots\!00\)\( T_{2}^{27} - \)\(31\!\cdots\!47\)\( T_{2}^{26} - \)\(73\!\cdots\!50\)\( T_{2}^{25} + \)\(21\!\cdots\!61\)\( T_{2}^{24} + \)\(29\!\cdots\!65\)\( T_{2}^{23} - \)\(12\!\cdots\!90\)\( T_{2}^{22} - \)\(13\!\cdots\!50\)\( T_{2}^{21} + \)\(60\!\cdots\!26\)\( T_{2}^{20} + \)\(49\!\cdots\!95\)\( T_{2}^{19} - \)\(24\!\cdots\!19\)\( T_{2}^{18} - \)\(30\!\cdots\!65\)\( T_{2}^{17} + \)\(99\!\cdots\!41\)\( T_{2}^{16} + \)\(12\!\cdots\!95\)\( T_{2}^{15} - \)\(17\!\cdots\!60\)\( T_{2}^{14} + \)\(19\!\cdots\!25\)\( T_{2}^{13} + \)\(64\!\cdots\!60\)\( T_{2}^{12} + 412280400375 T_{2}^{11} - 932541734675 T_{2}^{10} + 61740112175 T_{2}^{9} + 123520868325 T_{2}^{8} - 25960937625 T_{2}^{7} - 5471960250 T_{2}^{6} + 1193167750 T_{2}^{5} + 296597750 T_{2}^{4} + 18340625 T_{2}^{3} + 153750 T_{2}^{2} - 5000 T_{2} + 625 \)">\(T_{2}^{80} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).