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Properties

Label 125.3.f.b.82.1
Level $125$
Weight $3$
Character 125.82
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 = [32,0,0,10,0,-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 82.1
Character \(\chi\) \(=\) 125.82
Dual form 125.3.f.b.93.1

$q$-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+(-0.286042 - 1.80600i) q^{2} +(-1.30583 - 0.665351i) q^{3} +(0.624420 - 0.202886i) q^{4} +(-0.828102 + 2.54863i) q^{6} +(-3.62927 - 3.62927i) q^{7} +(-3.86553 - 7.58652i) q^{8} +(-4.02758 - 5.54349i) q^{9} +(5.24977 + 3.81418i) q^{11} +(-0.950373 - 0.150524i) q^{12} +(0.652461 - 4.11948i) q^{13} +(-5.51633 + 7.59258i) q^{14} +(-10.4709 + 7.60754i) q^{16} +(-12.0879 + 6.15907i) q^{17} +(-8.85947 + 8.85947i) q^{18} +(-25.5969 - 8.31693i) q^{19} +(2.32445 + 7.15393i) q^{21} +(5.38675 - 10.5721i) q^{22} +(31.7990 - 5.03647i) q^{23} +12.4786i q^{24} -7.62639 q^{26} +(3.63433 + 22.9462i) q^{27} +(-3.00252 - 1.52986i) q^{28} +(52.2931 - 16.9911i) q^{29} +(8.09752 - 24.9216i) q^{31} +(-7.34848 - 7.34848i) q^{32} +(-4.31751 - 8.47360i) q^{33} +(14.5809 + 20.0689i) q^{34} +(-3.63960 - 2.64432i) q^{36} +(43.5500 + 6.89764i) q^{37} +(-7.69857 + 48.6069i) q^{38} +(-3.59290 + 4.94520i) q^{39} +(29.6072 - 21.5109i) q^{41} +(12.2551 - 6.24428i) q^{42} +(-28.0309 + 28.0309i) q^{43} +(4.05190 + 1.31654i) q^{44} +(-18.1917 - 55.9883i) q^{46} +(4.98747 - 9.78846i) q^{47} +(18.7348 - 2.96731i) q^{48} -22.6568i q^{49} +19.8826 q^{51} +(-0.428376 - 2.70466i) q^{52} +(33.4001 + 17.0182i) q^{53} +(40.4013 - 13.1272i) q^{54} +(-13.5045 + 41.5626i) q^{56} +(27.8914 + 27.8914i) q^{57} +(-45.6438 - 89.5811i) q^{58} +(14.1810 + 19.5185i) q^{59} +(-34.1559 - 24.8157i) q^{61} +(-47.3246 - 7.49548i) q^{62} +(-5.50164 + 34.7360i) q^{63} +(-41.5995 + 57.2568i) q^{64} +(-14.0683 + 10.2212i) q^{66} +(-4.85968 + 2.47613i) q^{67} +(-6.29831 + 6.29831i) q^{68} +(-44.8749 - 14.5808i) q^{69} +(33.2596 + 102.362i) q^{71} +(-26.4871 + 51.9838i) q^{72} +(94.1859 - 14.9176i) q^{73} -80.6242i q^{74} -17.6706 q^{76} +(-5.21014 - 32.8955i) q^{77} +(9.95874 + 5.07423i) q^{78} +(-106.211 + 34.5101i) q^{79} +(-8.53531 + 26.2690i) q^{81} +(-47.3174 - 47.3174i) q^{82} +(-27.9495 - 54.8539i) q^{83} +(2.90287 + 3.99546i) q^{84} +(58.6417 + 42.6057i) q^{86} +(-79.5907 - 12.6059i) q^{87} +(8.64325 - 54.5713i) q^{88} +(6.88788 - 9.48035i) q^{89} +(-17.3187 + 12.5827i) q^{91} +(18.8341 - 9.59645i) q^{92} +(-27.1556 + 27.1556i) q^{93} +(-19.1046 - 6.20745i) q^{94} +(4.70651 + 14.4851i) q^{96} +(23.6179 - 46.3527i) q^{97} +(-40.9181 + 6.48079i) q^{98} -44.4640i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 10 q^{4} - 6 q^{6} + 10 q^{7} - 30 q^{8} + 10 q^{9} - 6 q^{11} - 80 q^{12} + 10 q^{14} + 2 q^{16} - 140 q^{18} - 90 q^{19} - 6 q^{21} + 150 q^{22} + 160 q^{23} + 4 q^{26} - 210 q^{27} - 160 q^{28}+ \cdots - 170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{9}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.286042 − 1.80600i −0.143021 − 0.902999i −0.949962 − 0.312366i \(-0.898878\pi\)
0.806941 − 0.590632i \(-0.201122\pi\)
\(3\) −1.30583 − 0.665351i −0.435275 − 0.221784i 0.222601 − 0.974910i \(-0.428545\pi\)
−0.657877 + 0.753126i \(0.728545\pi\)
\(4\) 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 0.398640 − 0.917107i \(-0.369482\pi\)
−0.917107 + 0.398640i \(0.869482\pi\)
\(8\) −3.86553 − 7.58652i −0.483191 − 0.948315i
\(9\) −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 \(0.604694\pi\)
\(12\) −0.950373 − 0.150524i −0.0791978 − 0.0125437i
\(13\) 0.652461 − 4.11948i 0.0501893 − 0.316883i −0.949803 − 0.312850i \(-0.898716\pi\)
0.999992 − 0.00403283i \(-0.00128369\pi\)
\(14\) −5.51633 + 7.59258i −0.394024 + 0.542327i
\(15\) 0 0
\(16\) −10.4709 + 7.60754i −0.654430 + 0.475471i
\(17\) −12.0879 + 6.15907i −0.711051 + 0.362298i −0.771809 − 0.635854i \(-0.780648\pi\)
0.0607586 + 0.998152i \(0.480648\pi\)
\(18\) −8.85947 + 8.85947i −0.492193 + 0.492193i
\(19\) −25.5969 − 8.31693i −1.34720 − 0.437733i −0.455453 − 0.890260i \(-0.650523\pi\)
−0.891751 + 0.452527i \(0.850523\pi\)
\(20\) 0 0
\(21\) 2.32445 + 7.15393i 0.110688 + 0.340663i
\(22\) 5.38675 − 10.5721i 0.244852 − 0.480550i
\(23\) 31.7990 − 5.03647i 1.38256 − 0.218977i 0.579551 − 0.814936i \(-0.303228\pi\)
0.803014 + 0.595960i \(0.203228\pi\)
\(24\) 12.4786i 0.519942i
\(25\) 0 0
\(26\) −7.62639 −0.293323
\(27\) 3.63433 + 22.9462i 0.134605 + 0.849861i
\(28\) −3.00252 − 1.52986i −0.107233 − 0.0546378i
\(29\) 52.2931 − 16.9911i 1.80321 − 0.585899i 0.803258 − 0.595631i \(-0.203098\pi\)
0.999953 + 0.00973242i \(0.00309797\pi\)
\(30\) 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
\(33\) −4.31751 − 8.47360i −0.130834 − 0.256776i
\(34\) 14.5809 + 20.0689i 0.428850 + 0.590262i
\(35\) 0 0
\(36\) −3.63960 − 2.64432i −0.101100 − 0.0734534i
\(37\) 43.5500 + 6.89764i 1.17703 + 0.186423i 0.714145 − 0.699997i \(-0.246816\pi\)
0.462882 + 0.886420i \(0.346816\pi\)
\(38\) −7.69857 + 48.6069i −0.202594 + 1.27913i
\(39\) −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 \(0.152742\pi\)
\(42\) 12.2551 − 6.24428i 0.291788 − 0.148673i
\(43\) −28.0309 + 28.0309i −0.651881 + 0.651881i −0.953446 − 0.301565i \(-0.902491\pi\)
0.301565 + 0.953446i \(0.402491\pi\)
\(44\) 4.05190 + 1.31654i 0.0920887 + 0.0299214i
\(45\) 0 0
\(46\) −18.1917 − 55.9883i −0.395471 − 1.21714i
\(47\) 4.98747 − 9.78846i 0.106116 − 0.208265i −0.831842 − 0.555012i \(-0.812714\pi\)
0.937959 + 0.346747i \(0.112714\pi\)
\(48\) 18.7348 − 2.96731i 0.390309 − 0.0618189i
\(49\) − 22.6568i − 0.462383i
\(50\) 0 0
\(51\) 19.8826 0.389855
\(52\) −0.428376 − 2.70466i −0.00823799 − 0.0520126i
\(53\) 33.4001 + 17.0182i 0.630190 + 0.321098i 0.739749 − 0.672883i \(-0.234944\pi\)
−0.109560 + 0.993980i \(0.534944\pi\)
\(54\) 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
\(58\) −45.6438 − 89.5811i −0.786963 − 1.54450i
\(59\) 14.1810 + 19.5185i 0.240356 + 0.330821i 0.912105 − 0.409958i \(-0.134457\pi\)
−0.671749 + 0.740779i \(0.734457\pi\)
\(60\) 0 0
\(61\) −34.1559 − 24.8157i −0.559933 − 0.406815i 0.271502 − 0.962438i \(-0.412480\pi\)
−0.831434 + 0.555623i \(0.812480\pi\)
\(62\) −47.3246 − 7.49548i −0.763300 − 0.120895i
\(63\) −5.50164 + 34.7360i −0.0873276 + 0.551365i
\(64\) −41.5995 + 57.2568i −0.649993 + 0.894638i
\(65\) 0 0
\(66\) −14.0683 + 10.2212i −0.213156 + 0.154867i
\(67\) −4.85968 + 2.47613i −0.0725326 + 0.0369572i −0.489881 − 0.871790i \(-0.662960\pi\)
0.417348 + 0.908747i \(0.362960\pi\)
\(68\) −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 \(0.626196\pi\)
\(72\) −26.4871 + 51.9838i −0.367876 + 0.721998i
\(73\) 94.1859 − 14.9176i 1.29022 − 0.204350i 0.526652 − 0.850081i \(-0.323447\pi\)
0.763565 + 0.645730i \(0.223447\pi\)
\(74\) − 80.6242i − 1.08952i
\(75\) 0 0
\(76\) −17.6706 −0.232508
\(77\) −5.21014 − 32.8955i −0.0676642 − 0.427215i
\(78\) 9.95874 + 5.07423i 0.127676 + 0.0650542i
\(79\) −106.211 + 34.5101i −1.34445 + 0.436837i −0.890821 − 0.454355i \(-0.849870\pi\)
−0.453626 + 0.891192i \(0.649870\pi\)
\(80\) 0 0
\(81\) −8.53531 + 26.2690i −0.105374 + 0.324308i
\(82\) −47.3174 − 47.3174i −0.577042 − 0.577042i
\(83\) −27.9495 − 54.8539i −0.336741 − 0.660891i 0.659095 − 0.752060i \(-0.270940\pi\)
−0.995835 + 0.0911693i \(0.970940\pi\)
\(84\) 2.90287 + 3.99546i 0.0345580 + 0.0475649i
\(85\) 0 0
\(86\) 58.6417 + 42.6057i 0.681880 + 0.495415i
\(87\) −79.5907 − 12.6059i −0.914835 − 0.144896i
\(88\) 8.64325 − 54.5713i 0.0982187 − 0.620129i
\(89\) 6.88788 − 9.48035i 0.0773919 − 0.106521i −0.768566 − 0.639771i \(-0.779029\pi\)
0.845958 + 0.533250i \(0.179029\pi\)
\(90\) 0 0
\(91\) −17.3187 + 12.5827i −0.190315 + 0.138272i
\(92\) 18.8341 − 9.59645i 0.204718 − 0.104309i
\(93\) −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
\(97\) 23.6179 − 46.3527i 0.243483 − 0.477863i −0.736631 − 0.676295i \(-0.763585\pi\)
0.980115 + 0.198431i \(0.0635847\pi\)
\(98\) −40.9181 + 6.48079i −0.417532 + 0.0661305i
\(99\) − 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.b.82.1 32
5.2 odd 4 25.3.f.a.13.4 yes 32
5.3 odd 4 125.3.f.c.43.1 32
5.4 even 2 125.3.f.a.82.4 32
15.2 even 4 225.3.r.a.163.1 32
20.7 even 4 400.3.bg.c.113.3 32
25.2 odd 20 125.3.f.a.93.4 32
25.11 even 5 125.3.f.c.32.1 32
25.14 even 10 25.3.f.a.2.4 ✓ 32
25.23 odd 20 inner 125.3.f.b.93.1 32
75.14 odd 10 225.3.r.a.127.1 32
100.39 odd 10 400.3.bg.c.177.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.3.f.a.2.4 ✓ 32 25.14 even 10
25.3.f.a.13.4 yes 32 5.2 odd 4
125.3.f.a.82.4 32 5.4 even 2
125.3.f.a.93.4 32 25.2 odd 20
125.3.f.b.82.1 32 1.1 even 1 trivial
125.3.f.b.93.1 32 25.23 odd 20 inner
125.3.f.c.32.1 32 25.11 even 5
125.3.f.c.43.1 32 5.3 odd 4
225.3.r.a.127.1 32 75.14 odd 10
225.3.r.a.163.1 32 15.2 even 4
400.3.bg.c.113.3 32 20.7 even 4
400.3.bg.c.177.3 32 100.39 odd 10