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Properties

Label 25.3.f.a.8.1
Level $25$
Weight $3$
Character 25.8
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 8.1
Character \(\chi\) \(=\) 25.8
Dual form 25.3.f.a.22.1

$q$-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+(-2.38234 - 1.21387i) q^{2} +(-3.57679 - 0.566508i) q^{3} +(1.85096 + 2.54762i) q^{4} +(-4.45026 - 2.27929i) q^{5} +(7.83348 + 5.69136i) q^{6} +(6.54971 - 6.54971i) q^{7} +(0.355933 + 2.24727i) q^{8} +(3.91299 + 1.27141i) q^{9} +(7.83532 + 10.8321i) q^{10} +(-3.18653 - 9.80714i) q^{11} +(-5.17723 - 10.1609i) q^{12} +(-11.3286 + 5.77220i) q^{13} +(-23.5541 + 7.65321i) q^{14} +(14.6264 + 10.6736i) q^{15} +(5.77235 - 17.7655i) q^{16} +(0.578287 - 0.0915916i) q^{17} +(-7.77878 - 7.77878i) q^{18} +(1.57500 - 2.16781i) q^{19} +(-2.43048 - 15.5565i) q^{20} +(-27.1374 + 19.7165i) q^{21} +(-4.31313 + 27.2320i) q^{22} +(16.5548 - 32.4907i) q^{23} -8.23965i q^{24} +(14.6097 + 20.2869i) q^{25} +33.9953 q^{26} +(15.7643 + 8.03233i) q^{27} +(28.8094 + 4.56297i) q^{28} +(9.15785 + 12.6047i) q^{29} +(-21.8888 - 43.1828i) q^{30} +(-15.1042 - 10.9739i) q^{31} +(-28.8811 + 28.8811i) q^{32} +(5.84174 + 36.8833i) q^{33} +(-1.48886 - 0.483759i) q^{34} +(-44.0766 + 14.2193i) q^{35} +(4.00371 + 12.3221i) q^{36} +(-26.1444 - 51.3113i) q^{37} +(-6.38363 + 3.25262i) q^{38} +(43.7900 - 14.2282i) q^{39} +(3.53818 - 10.8122i) q^{40} +(0.808618 - 2.48867i) q^{41} +(88.5838 - 14.0303i) q^{42} +(-7.76543 - 7.76543i) q^{43} +(19.0868 - 26.2707i) q^{44} +(-14.5159 - 14.5769i) q^{45} +(-78.8786 + 57.3086i) q^{46} +(-6.13916 + 38.7611i) q^{47} +(-30.7108 + 60.2733i) q^{48} -36.7975i q^{49} +(-10.1799 - 66.0645i) q^{50} -2.12030 q^{51} +(-35.6741 - 18.1769i) q^{52} +(39.0747 + 6.18882i) q^{53} +(-27.8059 - 38.2716i) q^{54} +(-8.17236 + 50.9074i) q^{55} +(17.0502 + 12.3877i) q^{56} +(-6.86154 + 6.86154i) q^{57} +(-6.51675 - 41.1452i) q^{58} +(0.494461 + 0.160660i) q^{59} +(-0.119521 + 57.0191i) q^{60} +(7.44472 + 22.9125i) q^{61} +(22.6627 + 44.4780i) q^{62} +(33.9563 - 17.3016i) q^{63} +(32.8008 - 10.6576i) q^{64} +(63.5717 + 0.133256i) q^{65} +(30.8543 - 94.9598i) q^{66} +(6.71462 - 1.06349i) q^{67} +(1.30372 + 1.30372i) q^{68} +(-77.6194 + 106.834i) q^{69} +(122.266 + 19.6278i) q^{70} +(80.2586 - 58.3113i) q^{71} +(-1.46444 + 9.24608i) q^{72} +(27.3185 - 53.6155i) q^{73} +153.977i q^{74} +(-40.7632 - 80.8383i) q^{75} +8.43802 q^{76} +(-85.1048 - 43.3631i) q^{77} +(-121.594 - 19.2586i) q^{78} +(2.28841 + 3.14973i) q^{79} +(-66.1811 + 65.9042i) q^{80} +(-81.7927 - 59.4258i) q^{81} +(-4.94732 + 4.94732i) q^{82} +(-10.3587 - 65.4021i) q^{83} +(-100.460 - 32.6415i) q^{84} +(-2.78229 - 0.910474i) q^{85} +(9.07375 + 27.9261i) q^{86} +(-25.6151 - 50.2724i) q^{87} +(20.9051 - 10.6517i) q^{88} +(-49.5405 + 16.0967i) q^{89} +(16.8876 + 52.3477i) q^{90} +(-36.3927 + 112.005i) q^{91} +(113.416 - 17.9634i) q^{92} +(47.8078 + 47.8078i) q^{93} +(61.6764 - 84.8903i) q^{94} +(-11.9502 + 6.05743i) q^{95} +(119.663 - 86.9404i) q^{96} +(0.610311 - 3.85335i) q^{97} +(-44.6672 + 87.6643i) q^{98} -42.4266i 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 \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{3}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38234 − 1.21387i −1.19117 − 0.606933i −0.257924 − 0.966165i \(-0.583039\pi\)
−0.933248 + 0.359233i \(0.883039\pi\)
\(3\) −3.57679 − 0.566508i −1.19226 − 0.188836i −0.471404 − 0.881917i \(-0.656253\pi\)
−0.720859 + 0.693081i \(0.756253\pi\)
\(4\) 1.85096 + 2.54762i 0.462739 + 0.636906i
\(5\) −4.45026 − 2.27929i −0.890053 − 0.455857i
\(6\) 7.83348 + 5.69136i 1.30558 + 0.948560i
\(7\) 6.54971 − 6.54971i 0.935673 − 0.935673i −0.0623792 − 0.998053i \(-0.519869\pi\)
0.998053 + 0.0623792i \(0.0198688\pi\)
\(8\) 0.355933 + 2.24727i 0.0444916 + 0.280909i
\(9\) 3.91299 + 1.27141i 0.434777 + 0.141268i
\(10\) 7.83532 + 10.8321i 0.783532 + 1.08321i
\(11\) −3.18653 − 9.80714i −0.289685 − 0.891558i −0.984955 − 0.172810i \(-0.944715\pi\)
0.695270 − 0.718748i \(-0.255285\pi\)
\(12\) −5.17723 − 10.1609i −0.431436 − 0.846741i
\(13\) −11.3286 + 5.77220i −0.871430 + 0.444016i −0.831720 − 0.555195i \(-0.812643\pi\)
−0.0397101 + 0.999211i \(0.512643\pi\)
\(14\) −23.5541 + 7.65321i −1.68244 + 0.546658i
\(15\) 14.6264 + 10.6736i 0.975095 + 0.711576i
\(16\) 5.77235 − 17.7655i 0.360772 − 1.11034i
\(17\) 0.578287 − 0.0915916i 0.0340169 − 0.00538774i −0.139403 − 0.990236i \(-0.544518\pi\)
0.173420 + 0.984848i \(0.444518\pi\)
\(18\) −7.77878 − 7.77878i −0.432154 − 0.432154i
\(19\) 1.57500 − 2.16781i 0.0828949 − 0.114095i −0.765556 − 0.643370i \(-0.777536\pi\)
0.848451 + 0.529275i \(0.177536\pi\)
\(20\) −2.43048 − 15.5565i −0.121524 − 0.777823i
\(21\) −27.1374 + 19.7165i −1.29226 + 0.938880i
\(22\) −4.31313 + 27.2320i −0.196051 + 1.23782i
\(23\) 16.5548 − 32.4907i 0.719775 − 1.41264i −0.183257 − 0.983065i \(-0.558664\pi\)
0.903032 − 0.429573i \(-0.141336\pi\)
\(24\) − 8.23965i − 0.343319i
\(25\) 14.6097 + 20.2869i 0.584388 + 0.811474i
\(26\) 33.9953 1.30751
\(27\) 15.7643 + 8.03233i 0.583864 + 0.297494i
\(28\) 28.8094 + 4.56297i 1.02891 + 0.162963i
\(29\) 9.15785 + 12.6047i 0.315788 + 0.434645i 0.937175 − 0.348859i \(-0.113431\pi\)
−0.621387 + 0.783504i \(0.713431\pi\)
\(30\) −21.8888 − 43.1828i −0.729628 − 1.43943i
\(31\) −15.1042 − 10.9739i −0.487233 − 0.353995i 0.316886 − 0.948464i \(-0.397363\pi\)
−0.804119 + 0.594468i \(0.797363\pi\)
\(32\) −28.8811 + 28.8811i −0.902536 + 0.902536i
\(33\) 5.84174 + 36.8833i 0.177022 + 1.11767i
\(34\) −1.48886 − 0.483759i −0.0437900 − 0.0142282i
\(35\) −44.0766 + 14.2193i −1.25933 + 0.406265i
\(36\) 4.00371 + 12.3221i 0.111214 + 0.342282i
\(37\) −26.1444 − 51.3113i −0.706606 − 1.38679i −0.912852 − 0.408292i \(-0.866125\pi\)
0.206246 − 0.978500i \(-0.433875\pi\)
\(38\) −6.38363 + 3.25262i −0.167990 + 0.0855953i
\(39\) 43.7900 − 14.2282i 1.12282 − 0.364826i
\(40\) 3.53818 − 10.8122i 0.0884544 − 0.270305i
\(41\) 0.808618 − 2.48867i 0.0197224 − 0.0606993i −0.940711 − 0.339209i \(-0.889841\pi\)
0.960433 + 0.278510i \(0.0898405\pi\)
\(42\) 88.5838 − 14.0303i 2.10914 − 0.334055i
\(43\) −7.76543 − 7.76543i −0.180591 − 0.180591i 0.611022 − 0.791614i \(-0.290759\pi\)
−0.791614 + 0.611022i \(0.790759\pi\)
\(44\) 19.0868 − 26.2707i 0.433790 − 0.597061i
\(45\) −14.5159 − 14.5769i −0.322577 − 0.323932i
\(46\) −78.8786 + 57.3086i −1.71475 + 1.24584i
\(47\) −6.13916 + 38.7611i −0.130620 + 0.824705i 0.832183 + 0.554501i \(0.187091\pi\)
−0.962803 + 0.270204i \(0.912909\pi\)
\(48\) −30.7108 + 60.2733i −0.639808 + 1.25569i
\(49\) − 36.7975i − 0.750969i
\(50\) −10.1799 − 66.0645i −0.203597 − 1.32129i
\(51\) −2.12030 −0.0415745
\(52\) −35.6741 − 18.1769i −0.686041 − 0.349555i
\(53\) 39.0747 + 6.18882i 0.737258 + 0.116770i 0.513760 − 0.857934i \(-0.328252\pi\)
0.223498 + 0.974704i \(0.428252\pi\)
\(54\) −27.8059 − 38.2716i −0.514924 − 0.708733i
\(55\) −8.17236 + 50.9074i −0.148588 + 0.925589i
\(56\) 17.0502 + 12.3877i 0.304468 + 0.221209i
\(57\) −6.86154 + 6.86154i −0.120378 + 0.120378i
\(58\) −6.51675 − 41.1452i −0.112358 − 0.709399i
\(59\) 0.494461 + 0.160660i 0.00838070 + 0.00272305i 0.313204 − 0.949686i \(-0.398598\pi\)
−0.304824 + 0.952409i \(0.598598\pi\)
\(60\) −0.119521 + 57.0191i −0.00199202 + 0.950318i
\(61\) 7.44472 + 22.9125i 0.122045 + 0.375615i 0.993351 − 0.115125i \(-0.0367267\pi\)
−0.871306 + 0.490739i \(0.836727\pi\)
\(62\) 22.6627 + 44.4780i 0.365527 + 0.717387i
\(63\) 33.9563 − 17.3016i 0.538989 − 0.274629i
\(64\) 32.8008 − 10.6576i 0.512512 − 0.166525i
\(65\) 63.5717 + 0.133256i 0.978027 + 0.00205010i
\(66\) 30.8543 − 94.9598i 0.467489 − 1.43878i
\(67\) 6.71462 − 1.06349i 0.100218 − 0.0158730i −0.106124 − 0.994353i \(-0.533844\pi\)
0.206342 + 0.978480i \(0.433844\pi\)
\(68\) 1.30372 + 1.30372i 0.0191724 + 0.0191724i
\(69\) −77.6194 + 106.834i −1.12492 + 1.54832i
\(70\) 122.266 + 19.6278i 1.74666 + 0.280398i
\(71\) 80.2586 − 58.3113i 1.13040 − 0.821286i 0.144650 − 0.989483i \(-0.453795\pi\)
0.985753 + 0.168197i \(0.0537945\pi\)
\(72\) −1.46444 + 9.24608i −0.0203394 + 0.128418i
\(73\) 27.3185 − 53.6155i 0.374226 − 0.734459i −0.624697 − 0.780867i \(-0.714777\pi\)
0.998923 + 0.0464080i \(0.0147774\pi\)
\(74\) 153.977i 2.08077i
\(75\) −40.7632 − 80.8383i −0.543510 − 1.07784i
\(76\) 8.43802 0.111027
\(77\) −85.1048 − 43.3631i −1.10526 − 0.563157i
\(78\) −121.594 − 19.2586i −1.55890 − 0.246905i
\(79\) 2.28841 + 3.14973i 0.0289672 + 0.0398699i 0.823255 − 0.567672i \(-0.192156\pi\)
−0.794288 + 0.607542i \(0.792156\pi\)
\(80\) −66.1811 + 65.9042i −0.827263 + 0.823802i
\(81\) −81.7927 − 59.4258i −1.00979 − 0.733652i
\(82\) −4.94732 + 4.94732i −0.0603332 + 0.0603332i
\(83\) −10.3587 − 65.4021i −0.124803 − 0.787977i −0.968107 − 0.250539i \(-0.919392\pi\)
0.843303 − 0.537438i \(-0.180608\pi\)
\(84\) −100.460 − 32.6415i −1.19596 − 0.388590i
\(85\) −2.78229 − 0.910474i −0.0327329 − 0.0107115i
\(86\) 9.07375 + 27.9261i 0.105509 + 0.324722i
\(87\) −25.6151 − 50.2724i −0.294426 − 0.577844i
\(88\) 20.9051 − 10.6517i 0.237558 − 0.121042i
\(89\) −49.5405 + 16.0967i −0.556635 + 0.180862i −0.573807 − 0.818991i \(-0.694534\pi\)
0.0171715 + 0.999853i \(0.494534\pi\)
\(90\) 16.8876 + 52.3477i 0.187640 + 0.581641i
\(91\) −36.3927 + 112.005i −0.399920 + 1.23083i
\(92\) 113.416 − 17.9634i 1.23278 − 0.195254i
\(93\) 47.8078 + 47.8078i 0.514063 + 0.514063i
\(94\) 61.6764 − 84.8903i 0.656132 − 0.903088i
\(95\) −11.9502 + 6.05743i −0.125792 + 0.0637624i
\(96\) 119.663 − 86.9404i 1.24649 − 0.905629i
\(97\) 0.610311 − 3.85335i 0.00629187 − 0.0397253i −0.984343 − 0.176265i \(-0.943598\pi\)
0.990635 + 0.136540i \(0.0435983\pi\)
\(98\) −44.6672 + 87.6643i −0.455788 + 0.894534i
\(99\) − 42.4266i − 0.428552i
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.8.1 ✓ 32
3.2 odd 2 225.3.r.a.208.4 32
4.3 odd 2 400.3.bg.c.33.3 32
5.2 odd 4 125.3.f.a.107.4 32
5.3 odd 4 125.3.f.b.107.1 32
5.4 even 2 125.3.f.c.18.4 32
25.3 odd 20 125.3.f.c.7.4 32
25.4 even 10 125.3.f.b.118.1 32
25.21 even 5 125.3.f.a.118.4 32
25.22 odd 20 inner 25.3.f.a.22.1 yes 32
75.47 even 20 225.3.r.a.172.4 32
100.47 even 20 400.3.bg.c.97.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.3.f.a.8.1 ✓ 32 1.1 even 1 trivial
25.3.f.a.22.1 yes 32 25.22 odd 20 inner
125.3.f.a.107.4 32 5.2 odd 4
125.3.f.a.118.4 32 25.21 even 5
125.3.f.b.107.1 32 5.3 odd 4
125.3.f.b.118.1 32 25.4 even 10
125.3.f.c.7.4 32 25.3 odd 20
125.3.f.c.18.4 32 5.4 even 2
225.3.r.a.172.4 32 75.47 even 20
225.3.r.a.208.4 32 3.2 odd 2
400.3.bg.c.33.3 32 4.3 odd 2
400.3.bg.c.97.3 32 100.47 even 20