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Properties

Label 125.3.f.c.43.1
Level $125$
Weight $3$
Character 125.43
Analytic conductor $3.406$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [125,3,Mod(7,125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("125.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(125, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([17])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 125.f (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,10,10,-10,0,-6,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.40600330450\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 43.1
Character \(\chi\) \(=\) 125.43
Dual form 125.3.f.c.32.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.80600 + 0.286042i) q^{2} +(0.665351 - 1.30583i) q^{3} +(-0.624420 + 0.202886i) q^{4} +(-0.828102 + 2.54863i) q^{6} +(-3.62927 + 3.62927i) q^{7} +(7.58652 - 3.86553i) q^{8} +(4.02758 + 5.54349i) q^{9} +(5.24977 + 3.81418i) q^{11} +(-0.150524 + 0.950373i) q^{12} +(4.11948 + 0.652461i) q^{13} +(5.51633 - 7.59258i) q^{14} +(-10.4709 + 7.60754i) q^{16} +(6.15907 + 12.0879i) q^{17} +(-8.85947 - 8.85947i) q^{18} +(25.5969 + 8.31693i) q^{19} +(2.32445 + 7.15393i) q^{21} +(-10.5721 - 5.38675i) q^{22} +(5.03647 + 31.7990i) q^{23} -12.4786i q^{24} -7.62639 q^{26} +(22.9462 - 3.63433i) q^{27} +(1.52986 - 3.00252i) q^{28} +(-52.2931 + 16.9911i) q^{29} +(8.09752 - 24.9216i) q^{31} +(-7.34848 + 7.34848i) q^{32} +(8.47360 - 4.31751i) q^{33} +(-14.5809 - 20.0689i) q^{34} +(-3.63960 - 2.64432i) q^{36} +(6.89764 - 43.5500i) q^{37} +(-48.6069 - 7.69857i) q^{38} +(3.59290 - 4.94520i) q^{39} +(29.6072 - 21.5109i) q^{41} +(-6.24428 - 12.2551i) q^{42} +(-28.0309 - 28.0309i) q^{43} +(-4.05190 - 1.31654i) q^{44} +(-18.1917 - 55.9883i) q^{46} +(-9.78846 - 4.98747i) q^{47} +(2.96731 + 18.7348i) q^{48} +22.6568i q^{49} +19.8826 q^{51} +(-2.70466 + 0.428376i) q^{52} +(-17.0182 + 33.4001i) q^{53} +(-40.4013 + 13.1272i) q^{54} +(-13.5045 + 41.5626i) q^{56} +(27.8914 - 27.8914i) q^{57} +(89.5811 - 45.6438i) q^{58} +(-14.1810 - 19.5185i) q^{59} +(-34.1559 - 24.8157i) q^{61} +(-7.49548 + 47.3246i) q^{62} +(-34.7360 - 5.50164i) q^{63} +(41.5995 - 57.2568i) q^{64} +(-14.0683 + 10.2212i) q^{66} +(2.47613 + 4.85968i) q^{67} +(-6.29831 - 6.29831i) q^{68} +(44.8749 + 14.5808i) q^{69} +(33.2596 + 102.362i) q^{71} +(51.9838 + 26.4871i) q^{72} +(14.9176 + 94.1859i) q^{73} +80.6242i q^{74} -17.6706 q^{76} +(-32.8955 + 5.21014i) q^{77} +(-5.07423 + 9.95874i) q^{78} +(106.211 - 34.5101i) q^{79} +(-8.53531 + 26.2690i) q^{81} +(-47.3174 + 47.3174i) q^{82} +(54.8539 - 27.9495i) q^{83} +(-2.90287 - 3.99546i) q^{84} +(58.6417 + 42.6057i) q^{86} +(-12.6059 + 79.5907i) q^{87} +(54.5713 + 8.64325i) q^{88} +(-6.88788 + 9.48035i) q^{89} +(-17.3187 + 12.5827i) q^{91} +(-9.59645 - 18.8341i) q^{92} +(-27.1556 - 27.1556i) q^{93} +(19.1046 + 6.20745i) q^{94} +(4.70651 + 14.4851i) q^{96} +(-46.3527 - 23.6179i) q^{97} +(-6.48079 - 40.9181i) q^{98} +44.4640i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 10 q^{2} + 10 q^{3} - 10 q^{4} - 6 q^{6} + 10 q^{7} + 10 q^{8} - 10 q^{9} - 6 q^{11} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 2 q^{16} - 60 q^{17} - 140 q^{18} + 90 q^{19} - 6 q^{21} - 70 q^{22} - 10 q^{23}+ \cdots - 170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{19}{20}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80600 + 0.286042i −0.902999 + 0.143021i −0.590632 − 0.806941i \(-0.701122\pi\)
−0.312366 + 0.949962i \(0.601122\pi\)
\(3\) 0.665351 − 1.30583i 0.221784 − 0.435275i −0.753126 − 0.657877i \(-0.771455\pi\)
0.974910 + 0.222601i \(0.0714549\pi\)
\(4\) −0.624420 + 0.202886i −0.156105 + 0.0507216i
\(5\) 0 0
\(6\) −0.828102 + 2.54863i −0.138017 + 0.424772i
\(7\) −3.62927 + 3.62927i −0.518467 + 0.518467i −0.917107 − 0.398640i \(-0.869482\pi\)
0.398640 + 0.917107i \(0.369482\pi\)
\(8\) 7.58652 − 3.86553i 0.948315 − 0.483191i
\(9\) 4.02758 + 5.54349i 0.447509 + 0.615943i
\(10\) 0 0
\(11\) 5.24977 + 3.81418i 0.477252 + 0.346744i 0.800261 − 0.599652i \(-0.204694\pi\)
−0.323009 + 0.946396i \(0.604694\pi\)
\(12\) −0.150524 + 0.950373i −0.0125437 + 0.0791978i
\(13\) 4.11948 + 0.652461i 0.316883 + 0.0501893i 0.312850 − 0.949803i \(-0.398716\pi\)
0.00403283 + 0.999992i \(0.498716\pi\)
\(14\) 5.51633 − 7.59258i 0.394024 − 0.542327i
\(15\) 0 0
\(16\) −10.4709 + 7.60754i −0.654430 + 0.475471i
\(17\) 6.15907 + 12.0879i 0.362298 + 0.711051i 0.998152 − 0.0607586i \(-0.0193520\pi\)
−0.635854 + 0.771809i \(0.719352\pi\)
\(18\) −8.85947 − 8.85947i −0.492193 − 0.492193i
\(19\) 25.5969 + 8.31693i 1.34720 + 0.437733i 0.891751 − 0.452527i \(-0.149477\pi\)
0.455453 + 0.890260i \(0.349477\pi\)
\(20\) 0 0
\(21\) 2.32445 + 7.15393i 0.110688 + 0.340663i
\(22\) −10.5721 − 5.38675i −0.480550 − 0.244852i
\(23\) 5.03647 + 31.7990i 0.218977 + 1.38256i 0.814936 + 0.579551i \(0.196772\pi\)
−0.595960 + 0.803014i \(0.703228\pi\)
\(24\) − 12.4786i − 0.519942i
\(25\) 0 0
\(26\) −7.62639 −0.293323
\(27\) 22.9462 − 3.63433i 0.849861 − 0.134605i
\(28\) 1.52986 − 3.00252i 0.0546378 − 0.107233i
\(29\) −52.2931 + 16.9911i −1.80321 + 0.585899i −0.999953 − 0.00973242i \(-0.996902\pi\)
−0.803258 + 0.595631i \(0.796902\pi\)
\(30\) 0 0
\(31\) 8.09752 − 24.9216i 0.261210 − 0.803923i −0.731332 − 0.682022i \(-0.761101\pi\)
0.992542 − 0.121901i \(-0.0388991\pi\)
\(32\) −7.34848 + 7.34848i −0.229640 + 0.229640i
\(33\) 8.47360 − 4.31751i 0.256776 − 0.130834i
\(34\) −14.5809 − 20.0689i −0.428850 − 0.590262i
\(35\) 0 0
\(36\) −3.63960 − 2.64432i −0.101100 − 0.0734534i
\(37\) 6.89764 − 43.5500i 0.186423 − 1.17703i −0.699997 − 0.714145i \(-0.746816\pi\)
0.886420 − 0.462882i \(-0.153184\pi\)
\(38\) −48.6069 − 7.69857i −1.27913 − 0.202594i
\(39\) 3.59290 − 4.94520i 0.0921256 − 0.126800i
\(40\) 0 0
\(41\) 29.6072 − 21.5109i 0.722126 − 0.524655i −0.164937 − 0.986304i \(-0.552742\pi\)
0.887063 + 0.461649i \(0.152742\pi\)
\(42\) −6.24428 − 12.2551i −0.148673 − 0.291788i
\(43\) −28.0309 − 28.0309i −0.651881 − 0.651881i 0.301565 − 0.953446i \(-0.402491\pi\)
−0.953446 + 0.301565i \(0.902491\pi\)
\(44\) −4.05190 − 1.31654i −0.0920887 − 0.0299214i
\(45\) 0 0
\(46\) −18.1917 − 55.9883i −0.395471 − 1.21714i
\(47\) −9.78846 − 4.98747i −0.208265 − 0.106116i 0.346747 − 0.937959i \(-0.387286\pi\)
−0.555012 + 0.831842i \(0.687286\pi\)
\(48\) 2.96731 + 18.7348i 0.0618189 + 0.390309i
\(49\) 22.6568i 0.462383i
\(50\) 0 0
\(51\) 19.8826 0.389855
\(52\) −2.70466 + 0.428376i −0.0520126 + 0.00823799i
\(53\) −17.0182 + 33.4001i −0.321098 + 0.630190i −0.993980 − 0.109560i \(-0.965056\pi\)
0.672883 + 0.739749i \(0.265056\pi\)
\(54\) −40.4013 + 13.1272i −0.748172 + 0.243096i
\(55\) 0 0
\(56\) −13.5045 + 41.5626i −0.241152 + 0.742189i
\(57\) 27.8914 − 27.8914i 0.489322 − 0.489322i
\(58\) 89.5811 − 45.6438i 1.54450 − 0.786963i
\(59\) −14.1810 − 19.5185i −0.240356 − 0.330821i 0.671749 − 0.740779i \(-0.265543\pi\)
−0.912105 + 0.409958i \(0.865543\pi\)
\(60\) 0 0
\(61\) −34.1559 − 24.8157i −0.559933 − 0.406815i 0.271502 − 0.962438i \(-0.412480\pi\)
−0.831434 + 0.555623i \(0.812480\pi\)
\(62\) −7.49548 + 47.3246i −0.120895 + 0.763300i
\(63\) −34.7360 − 5.50164i −0.551365 − 0.0873276i
\(64\) 41.5995 − 57.2568i 0.649993 − 0.894638i
\(65\) 0 0
\(66\) −14.0683 + 10.2212i −0.213156 + 0.154867i
\(67\) 2.47613 + 4.85968i 0.0369572 + 0.0725326i 0.908747 − 0.417348i \(-0.137040\pi\)
−0.871790 + 0.489881i \(0.837040\pi\)
\(68\) −6.29831 − 6.29831i −0.0926222 − 0.0926222i
\(69\) 44.8749 + 14.5808i 0.650361 + 0.211315i
\(70\) 0 0
\(71\) 33.2596 + 102.362i 0.468445 + 1.44172i 0.854598 + 0.519290i \(0.173804\pi\)
−0.386153 + 0.922435i \(0.626196\pi\)
\(72\) 51.9838 + 26.4871i 0.721998 + 0.367876i
\(73\) 14.9176 + 94.1859i 0.204350 + 1.29022i 0.850081 + 0.526652i \(0.176553\pi\)
−0.645730 + 0.763565i \(0.723447\pi\)
\(74\) 80.6242i 1.08952i
\(75\) 0 0
\(76\) −17.6706 −0.232508
\(77\) −32.8955 + 5.21014i −0.427215 + 0.0676642i
\(78\) −5.07423 + 9.95874i −0.0650542 + 0.127676i
\(79\) 106.211 − 34.5101i 1.34445 − 0.436837i 0.453626 − 0.891192i \(-0.350130\pi\)
0.890821 + 0.454355i \(0.150130\pi\)
\(80\) 0 0
\(81\) −8.53531 + 26.2690i −0.105374 + 0.324308i
\(82\) −47.3174 + 47.3174i −0.577042 + 0.577042i
\(83\) 54.8539 − 27.9495i 0.660891 − 0.336741i −0.0911693 − 0.995835i \(-0.529060\pi\)
0.752060 + 0.659095i \(0.229060\pi\)
\(84\) −2.90287 − 3.99546i −0.0345580 − 0.0475649i
\(85\) 0 0
\(86\) 58.6417 + 42.6057i 0.681880 + 0.495415i
\(87\) −12.6059 + 79.5907i −0.144896 + 0.914835i
\(88\) 54.5713 + 8.64325i 0.620129 + 0.0982187i
\(89\) −6.88788 + 9.48035i −0.0773919 + 0.106521i −0.845958 − 0.533250i \(-0.820971\pi\)
0.768566 + 0.639771i \(0.220971\pi\)
\(90\) 0 0
\(91\) −17.3187 + 12.5827i −0.190315 + 0.138272i
\(92\) −9.59645 − 18.8341i −0.104309 − 0.204718i
\(93\) −27.1556 − 27.1556i −0.291995 − 0.291995i
\(94\) 19.1046 + 6.20745i 0.203240 + 0.0660367i
\(95\) 0 0
\(96\) 4.70651 + 14.4851i 0.0490261 + 0.150887i
\(97\) −46.3527 − 23.6179i −0.477863 − 0.243483i 0.198431 − 0.980115i \(-0.436415\pi\)
−0.676295 + 0.736631i \(0.736415\pi\)
\(98\) −6.48079 − 40.9181i −0.0661305 − 0.417532i
\(99\) 44.4640i 0.449131i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 125.3.f.c.43.1 32
5.2 odd 4 125.3.f.b.82.1 32
5.3 odd 4 125.3.f.a.82.4 32
5.4 even 2 25.3.f.a.13.4 yes 32
15.14 odd 2 225.3.r.a.163.1 32
20.19 odd 2 400.3.bg.c.113.3 32
25.2 odd 20 inner 125.3.f.c.32.1 32
25.11 even 5 125.3.f.b.93.1 32
25.14 even 10 125.3.f.a.93.4 32
25.23 odd 20 25.3.f.a.2.4 ✓ 32
75.23 even 20 225.3.r.a.127.1 32
100.23 even 20 400.3.bg.c.177.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.3.f.a.2.4 ✓ 32 25.23 odd 20
25.3.f.a.13.4 yes 32 5.4 even 2
125.3.f.a.82.4 32 5.3 odd 4
125.3.f.a.93.4 32 25.14 even 10
125.3.f.b.82.1 32 5.2 odd 4
125.3.f.b.93.1 32 25.11 even 5
125.3.f.c.32.1 32 25.2 odd 20 inner
125.3.f.c.43.1 32 1.1 even 1 trivial
225.3.r.a.127.1 32 75.23 even 20
225.3.r.a.163.1 32 15.14 odd 2
400.3.bg.c.113.3 32 20.19 odd 2
400.3.bg.c.177.3 32 100.23 even 20