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Properties

Label 25.3.f.a.2.4
Level $25$
Weight $3$
Character 25.2
Analytic conductor $0.681$
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] = [25,3,Mod(2,25)] 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("25.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.681200660901\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.4
Character \(\chi\) \(=\) 25.2
Dual form 25.3.f.a.13.4

$q$-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} +(-3.20727 + 3.83580i) q^{5} +(-0.828102 - 2.54863i) q^{6} +(3.62927 + 3.62927i) q^{7} +(-7.58652 - 3.86553i) q^{8} +(4.02758 - 5.54349i) q^{9} +(-6.88953 + 6.01004i) q^{10} +(5.24977 - 3.81418i) q^{11} +(0.150524 + 0.950373i) q^{12} +(-4.11948 + 0.652461i) q^{13} +(5.51633 + 7.59258i) q^{14} +(7.14285 + 1.63598i) q^{15} +(-10.4709 - 7.60754i) q^{16} +(-6.15907 + 12.0879i) q^{17} +(8.85947 - 8.85947i) q^{18} +(25.5969 - 8.31693i) q^{19} +(2.78092 - 1.74444i) q^{20} +(2.32445 - 7.15393i) q^{21} +(10.5721 - 5.38675i) q^{22} +(-5.03647 + 31.7990i) q^{23} +12.4786i q^{24} +(-4.42679 - 24.6050i) q^{25} -7.62639 q^{26} +(-22.9462 - 3.63433i) q^{27} +(-1.52986 - 3.00252i) q^{28} +(-52.2931 - 16.9911i) q^{29} +(12.4320 + 4.99773i) q^{30} +(8.09752 + 24.9216i) q^{31} +(7.34848 + 7.34848i) q^{32} +(-8.47360 - 4.31751i) q^{33} +(-14.5809 + 20.0689i) q^{34} +(-25.5612 + 2.28111i) q^{35} +(-3.63960 + 2.64432i) q^{36} +(-6.89764 - 43.5500i) q^{37} +(48.6069 - 7.69857i) q^{38} +(3.59290 + 4.94520i) q^{39} +(39.1595 - 16.7026i) q^{40} +(29.6072 + 21.5109i) q^{41} +(6.24428 - 12.2551i) q^{42} +(28.0309 - 28.0309i) q^{43} +(-4.05190 + 1.31654i) q^{44} +(8.34618 + 33.2285i) q^{45} +(-18.1917 + 55.9883i) q^{46} +(9.78846 - 4.98747i) q^{47} +(-2.96731 + 18.7348i) q^{48} -22.6568i q^{49} +(-0.956716 - 45.7027i) q^{50} +19.8826 q^{51} +(2.70466 + 0.428376i) q^{52} +(17.0182 + 33.4001i) q^{53} +(-40.4013 - 13.1272i) q^{54} +(-2.20700 + 32.3702i) q^{55} +(-13.5045 - 41.5626i) q^{56} +(-27.8914 - 27.8914i) q^{57} +(-89.5811 - 45.6438i) q^{58} +(-14.1810 + 19.5185i) q^{59} +(-4.12822 - 2.47073i) q^{60} +(-34.1559 + 24.8157i) q^{61} +(7.49548 + 47.3246i) q^{62} +(34.7360 - 5.50164i) q^{63} +(41.5995 + 57.2568i) q^{64} +(10.7096 - 17.8941i) q^{65} +(-14.0683 - 10.2212i) q^{66} +(-2.47613 + 4.85968i) q^{67} +(6.29831 - 6.29831i) q^{68} +(44.8749 - 14.5808i) q^{69} +(-46.8160 - 3.19191i) q^{70} +(33.2596 - 102.362i) q^{71} +(-51.9838 + 26.4871i) q^{72} +(-14.9176 + 94.1859i) q^{73} -80.6242i q^{74} +(-29.1844 + 22.1515i) q^{75} -17.6706 q^{76} +(32.8955 + 5.21014i) q^{77} +(5.07423 + 9.95874i) q^{78} +(106.211 + 34.5101i) q^{79} +(62.7640 - 15.7648i) q^{80} +(-8.53531 - 26.2690i) q^{81} +(47.3174 + 47.3174i) q^{82} +(-54.8539 - 27.9495i) q^{83} +(-2.90287 + 3.99546i) q^{84} +(-26.6128 - 62.3941i) q^{85} +(58.6417 - 42.6057i) q^{86} +(12.6059 + 79.5907i) q^{87} +(-54.5713 + 8.64325i) q^{88} +(-6.88788 - 9.48035i) q^{89} +(5.56844 + 62.3979i) q^{90} +(-17.3187 - 12.5827i) q^{91} +(9.59645 - 18.8341i) q^{92} +(27.1556 - 27.1556i) q^{93} +(19.1046 - 6.20745i) q^{94} +(-50.1941 + 124.859i) q^{95} +(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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 10 q^{2} - 10 q^{3} - 10 q^{4} - 10 q^{5} - 6 q^{6} - 10 q^{7} - 10 q^{8} - 10 q^{9} - 10 q^{10} - 6 q^{11} - 10 q^{12} - 10 q^{13} - 10 q^{14} - 10 q^{15} + 2 q^{16} + 60 q^{17} + 140 q^{18} + 90 q^{19}+ \cdots + 170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{20}\right)\)

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 \(-0.298878\pi\)
0.312366 + 0.949962i \(0.398878\pi\)
\(3\) −0.665351 − 1.30583i −0.221784 − 0.435275i 0.753126 − 0.657877i \(-0.228545\pi\)
−0.974910 + 0.222601i \(0.928545\pi\)
\(4\) −0.624420 − 0.202886i −0.156105 − 0.0507216i
\(5\) −3.20727 + 3.83580i −0.641455 + 0.767161i
\(6\) −0.828102 − 2.54863i −0.138017 − 0.424772i
\(7\) 3.62927 + 3.62927i 0.518467 + 0.518467i 0.917107 − 0.398640i \(-0.130518\pi\)
−0.398640 + 0.917107i \(0.630518\pi\)
\(8\) −7.58652 − 3.86553i −0.948315 − 0.483191i
\(9\) 4.02758 − 5.54349i 0.447509 − 0.615943i
\(10\) −6.88953 + 6.01004i −0.688953 + 0.601004i
\(11\) 5.24977 − 3.81418i 0.477252 − 0.346744i −0.323009 − 0.946396i \(-0.604694\pi\)
0.800261 + 0.599652i \(0.204694\pi\)
\(12\) 0.150524 + 0.950373i 0.0125437 + 0.0791978i
\(13\) −4.11948 + 0.652461i −0.316883 + 0.0501893i −0.312850 − 0.949803i \(-0.601284\pi\)
−0.00403283 + 0.999992i \(0.501284\pi\)
\(14\) 5.51633 + 7.59258i 0.394024 + 0.542327i
\(15\) 7.14285 + 1.63598i 0.476190 + 0.109065i
\(16\) −10.4709 − 7.60754i −0.654430 − 0.475471i
\(17\) −6.15907 + 12.0879i −0.362298 + 0.711051i −0.998152 − 0.0607586i \(-0.980648\pi\)
0.635854 + 0.771809i \(0.280648\pi\)
\(18\) 8.85947 − 8.85947i 0.492193 − 0.492193i
\(19\) 25.5969 − 8.31693i 1.34720 − 0.437733i 0.455453 − 0.890260i \(-0.349477\pi\)
0.891751 + 0.452527i \(0.149477\pi\)
\(20\) 2.78092 − 1.74444i 0.139046 − 0.0872220i
\(21\) 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 0.595960 + 0.803014i \(0.296772\pi\)
−0.814936 + 0.579551i \(0.803228\pi\)
\(24\) 12.4786i 0.519942i
\(25\) −4.42679 − 24.6050i −0.177071 − 0.984198i
\(26\) −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 −0.803258 − 0.595631i \(-0.796902\pi\)
−0.999953 + 0.00973242i \(0.996902\pi\)
\(30\) 12.4320 + 4.99773i 0.414400 + 0.166591i
\(31\) 8.09752 + 24.9216i 0.261210 + 0.803923i 0.992542 + 0.121901i \(0.0388991\pi\)
−0.731332 + 0.682022i \(0.761101\pi\)
\(32\) 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
\(35\) −25.5612 + 2.28111i −0.730321 + 0.0651744i
\(36\) −3.63960 + 2.64432i −0.101100 + 0.0734534i
\(37\) −6.89764 − 43.5500i −0.186423 − 1.17703i −0.886420 − 0.462882i \(-0.846816\pi\)
0.699997 − 0.714145i \(-0.253184\pi\)
\(38\) 48.6069 − 7.69857i 1.27913 − 0.202594i
\(39\) 3.59290 + 4.94520i 0.0921256 + 0.126800i
\(40\) 39.1595 − 16.7026i 0.978986 − 0.417565i
\(41\) 29.6072 + 21.5109i 0.722126 + 0.524655i 0.887063 − 0.461649i \(-0.152742\pi\)
−0.164937 + 0.986304i \(0.552742\pi\)
\(42\) 6.24428 − 12.2551i 0.148673 − 0.291788i
\(43\) 28.0309 − 28.0309i 0.651881 − 0.651881i −0.301565 − 0.953446i \(-0.597509\pi\)
0.953446 + 0.301565i \(0.0975090\pi\)
\(44\) −4.05190 + 1.31654i −0.0920887 + 0.0299214i
\(45\) 8.34618 + 33.2285i 0.185471 + 0.738411i
\(46\) −18.1917 + 55.9883i −0.395471 + 1.21714i
\(47\) 9.78846 − 4.98747i 0.208265 − 0.106116i −0.346747 − 0.937959i \(-0.612714\pi\)
0.555012 + 0.831842i \(0.312714\pi\)
\(48\) −2.96731 + 18.7348i −0.0618189 + 0.390309i
\(49\) − 22.6568i − 0.462383i
\(50\) −0.956716 − 45.7027i −0.0191343 − 0.914054i
\(51\) 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 \(-0.0349441\pi\)
−0.672883 + 0.739749i \(0.734944\pi\)
\(54\) −40.4013 − 13.1272i −0.748172 − 0.243096i
\(55\) −2.20700 + 32.3702i −0.0401273 + 0.588549i
\(56\) −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 −0.912105 − 0.409958i \(-0.865543\pi\)
0.671749 + 0.740779i \(0.265543\pi\)
\(60\) −4.12822 − 2.47073i −0.0688036 − 0.0411788i
\(61\) −34.1559 + 24.8157i −0.559933 + 0.406815i −0.831434 − 0.555623i \(-0.812480\pi\)
0.271502 + 0.962438i \(0.412480\pi\)
\(62\) 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
\(65\) 10.7096 − 17.8941i 0.164763 − 0.275294i
\(66\) −14.0683 − 10.2212i −0.213156 − 0.154867i
\(67\) −2.47613 + 4.85968i −0.0369572 + 0.0725326i −0.908747 − 0.417348i \(-0.862960\pi\)
0.871790 + 0.489881i \(0.162960\pi\)
\(68\) 6.29831 − 6.29831i 0.0926222 − 0.0926222i
\(69\) 44.8749 − 14.5808i 0.650361 − 0.211315i
\(70\) −46.8160 − 3.19191i −0.668800 − 0.0455988i
\(71\) 33.2596 − 102.362i 0.468445 − 1.44172i −0.386153 − 0.922435i \(-0.626196\pi\)
0.854598 − 0.519290i \(-0.173804\pi\)
\(72\) −51.9838 + 26.4871i −0.721998 + 0.367876i
\(73\) −14.9176 + 94.1859i −0.204350 + 1.29022i 0.645730 + 0.763565i \(0.276553\pi\)
−0.850081 + 0.526652i \(0.823447\pi\)
\(74\) − 80.6242i − 1.08952i
\(75\) −29.1844 + 22.1515i −0.389125 + 0.295354i
\(76\) −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 0.890821 − 0.454355i \(-0.150130\pi\)
0.453626 + 0.891192i \(0.350130\pi\)
\(80\) 62.7640 − 15.7648i 0.784550 − 0.197060i
\(81\) −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 \(-0.470940\pi\)
−0.752060 + 0.659095i \(0.770940\pi\)
\(84\) −2.90287 + 3.99546i −0.0345580 + 0.0475649i
\(85\) −26.6128 − 62.3941i −0.313092 − 0.734048i
\(86\) 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 0.768566 − 0.639771i \(-0.220971\pi\)
−0.845958 + 0.533250i \(0.820971\pi\)
\(90\) 5.56844 + 62.3979i 0.0618716 + 0.693310i
\(91\) −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
\(95\) −50.1941 + 124.859i −0.528359 + 1.31431i
\(96\) 4.70651 − 14.4851i 0.0490261 − 0.150887i
\(97\) 46.3527 − 23.6179i 0.477863 − 0.243483i −0.198431 − 0.980115i \(-0.563585\pi\)
0.676295 + 0.736631i \(0.263585\pi\)
\(98\) 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 25.3.f.a.2.4 ✓ 32
3.2 odd 2 225.3.r.a.127.1 32
4.3 odd 2 400.3.bg.c.177.3 32
5.2 odd 4 125.3.f.b.93.1 32
5.3 odd 4 125.3.f.a.93.4 32
5.4 even 2 125.3.f.c.32.1 32
25.9 even 10 125.3.f.b.82.1 32
25.12 odd 20 125.3.f.c.43.1 32
25.13 odd 20 inner 25.3.f.a.13.4 yes 32
25.16 even 5 125.3.f.a.82.4 32
75.38 even 20 225.3.r.a.163.1 32
100.63 even 20 400.3.bg.c.113.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.3.f.a.2.4 ✓ 32 1.1 even 1 trivial
25.3.f.a.13.4 yes 32 25.13 odd 20 inner
125.3.f.a.82.4 32 25.16 even 5
125.3.f.a.93.4 32 5.3 odd 4
125.3.f.b.82.1 32 25.9 even 10
125.3.f.b.93.1 32 5.2 odd 4
125.3.f.c.32.1 32 5.4 even 2
125.3.f.c.43.1 32 25.12 odd 20
225.3.r.a.127.1 32 3.2 odd 2
225.3.r.a.163.1 32 75.38 even 20
400.3.bg.c.113.3 32 100.63 even 20
400.3.bg.c.177.3 32 4.3 odd 2