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Properties

Label 65.2.o.a.32.4
Level $65$
Weight $2$
Character 65.32
Analytic conductor $0.519$
Analytic rank $0$
Dimension $20$
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] = [65,2,Mod(2,65)] 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("65.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(65, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.o (of order \(12\), degree \(4\), 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: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 32.4
Root \(-1.02262i\) of defining polynomial
Character \(\chi\) \(=\) 65.32
Dual form 65.2.o.a.63.4

$q$-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+(0.511309 - 0.885613i) q^{2} +(-0.721300 - 2.69193i) q^{3} +(0.477126 + 0.826407i) q^{4} +(-1.69584 + 1.45744i) q^{5} +(-2.75281 - 0.737614i) q^{6} +(0.834479 - 0.481787i) q^{7} +3.02107 q^{8} +(-4.12812 + 2.38337i) q^{9} +(0.423625 + 2.24706i) q^{10} +(1.60661 - 0.430490i) q^{11} +(1.88048 - 1.88048i) q^{12} +(1.82127 + 3.11175i) q^{13} -0.985368i q^{14} +(5.14652 + 3.51384i) q^{15} +(0.590448 - 1.02269i) q^{16} +(-7.00342 - 1.87656i) q^{17} +4.87456i q^{18} +(-0.707496 + 2.64041i) q^{19} +(-2.01357 - 0.706075i) q^{20} +(-1.89884 - 1.89884i) q^{21} +(0.440226 - 1.64295i) q^{22} +(-3.72214 + 0.997344i) q^{23} +(-2.17910 - 8.13250i) q^{24} +(0.751762 - 4.94316i) q^{25} +(3.68704 - 0.0218799i) q^{26} +(3.48159 + 3.48159i) q^{27} +(0.796304 + 0.459747i) q^{28} +(0.253107 + 0.146132i) q^{29} +(5.74336 - 2.76117i) q^{30} +(-0.125649 + 0.125649i) q^{31} +(2.41727 + 4.18683i) q^{32} +(-2.31769 - 4.01436i) q^{33} +(-5.24282 + 5.24282i) q^{34} +(-0.712972 + 2.03323i) q^{35} +(-3.93927 - 2.27434i) q^{36} +(3.53443 + 2.04061i) q^{37} +(1.97663 + 1.97663i) q^{38} +(7.06291 - 7.14724i) q^{39} +(-5.12326 + 4.40302i) q^{40} +(-1.79277 - 6.69071i) q^{41} +(-2.65254 + 0.710745i) q^{42} +(2.05706 - 7.67707i) q^{43} +(1.12232 + 1.12232i) q^{44} +(3.52703 - 10.0583i) q^{45} +(-1.01990 + 3.80633i) q^{46} -7.84582i q^{47} +(-3.17888 - 0.851780i) q^{48} +(-3.03576 + 5.25810i) q^{49} +(-3.99335 - 3.19325i) q^{50} +20.2063i q^{51} +(-1.70259 + 2.98981i) q^{52} +(-1.99855 + 1.99855i) q^{53} +(4.86351 - 1.30317i) q^{54} +(-2.09714 + 3.07157i) q^{55} +(2.52102 - 1.45551i) q^{56} +7.61811 q^{57} +(0.258832 - 0.149437i) q^{58} +(4.87924 + 1.30739i) q^{59} +(-0.448318 + 5.92967i) q^{60} +(-1.04169 - 1.80425i) q^{61} +(0.0470311 + 0.175522i) q^{62} +(-2.29655 + 3.97775i) q^{63} +7.30568 q^{64} +(-7.62376 - 2.62264i) q^{65} -4.74023 q^{66} +(-3.64915 + 6.32050i) q^{67} +(-1.79071 - 6.68304i) q^{68} +(5.36956 + 9.30034i) q^{69} +(1.43611 + 1.67103i) q^{70} +(12.6082 + 3.37837i) q^{71} +(-12.4713 + 7.20034i) q^{72} +3.22747 q^{73} +(3.61437 - 2.08676i) q^{74} +(-13.8489 + 1.54181i) q^{75} +(-2.51962 + 0.675130i) q^{76} +(1.13328 - 1.13328i) q^{77} +(-2.71836 - 9.90945i) q^{78} -13.5845i q^{79} +(0.489192 + 2.59485i) q^{80} +(-0.289196 + 0.500902i) q^{81} +(-6.84204 - 1.83332i) q^{82} +8.56854i q^{83} +(0.663230 - 2.47521i) q^{84} +(14.6117 - 7.02469i) q^{85} +(-5.74712 - 5.74712i) q^{86} +(0.210809 - 0.786751i) q^{87} +(4.85368 - 1.30054i) q^{88} +(0.134207 + 0.500868i) q^{89} +(-7.10435 - 8.26648i) q^{90} +(3.01901 + 1.71922i) q^{91} +(-2.60014 - 2.60014i) q^{92} +(0.428870 + 0.247608i) q^{93} +(-6.94836 - 4.01164i) q^{94} +(-2.64843 - 5.50885i) q^{95} +(9.52707 - 9.52707i) q^{96} +(-3.75660 - 6.50662i) q^{97} +(3.10442 + 5.37702i) q^{98} +(-5.60626 + 5.60626i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 8 q^{6} - 6 q^{7} + 12 q^{8} - 12 q^{9} - 10 q^{10} - 16 q^{11} + 24 q^{12} + 2 q^{13} + 12 q^{15} - 2 q^{16} - 10 q^{17} + 20 q^{19} + 14 q^{20} + 4 q^{21}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{5}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.511309 − 0.885613i 0.361550 − 0.626223i −0.626666 − 0.779288i \(-0.715581\pi\)
0.988216 + 0.153065i \(0.0489144\pi\)
\(3\) −0.721300 − 2.69193i −0.416443 − 1.55418i −0.781929 − 0.623368i \(-0.785764\pi\)
0.365486 − 0.930817i \(-0.380903\pi\)
\(4\) 0.477126 + 0.826407i 0.238563 + 0.413204i
\(5\) −1.69584 + 1.45744i −0.758404 + 0.651785i
\(6\) −2.75281 − 0.737614i −1.12383 − 0.301130i
\(7\) 0.834479 − 0.481787i 0.315404 − 0.182098i −0.333938 − 0.942595i \(-0.608378\pi\)
0.649342 + 0.760497i \(0.275044\pi\)
\(8\) 3.02107 1.06811
\(9\) −4.12812 + 2.38337i −1.37604 + 0.794457i
\(10\) 0.423625 + 2.24706i 0.133962 + 0.710583i
\(11\) 1.60661 − 0.430490i 0.484411 − 0.129797i −0.00834492 − 0.999965i \(-0.502656\pi\)
0.492756 + 0.870168i \(0.335990\pi\)
\(12\) 1.88048 − 1.88048i 0.542847 − 0.542847i
\(13\) 1.82127 + 3.11175i 0.505130 + 0.863043i
\(14\) − 0.985368i − 0.263351i
\(15\) 5.14652 + 3.51384i 1.32883 + 0.907268i
\(16\) 0.590448 − 1.02269i 0.147612 − 0.255671i
\(17\) −7.00342 − 1.87656i −1.69858 − 0.455133i −0.725998 − 0.687697i \(-0.758622\pi\)
−0.972581 + 0.232564i \(0.925289\pi\)
\(18\) 4.87456i 1.14894i
\(19\) −0.707496 + 2.64041i −0.162311 + 0.605752i 0.836057 + 0.548642i \(0.184855\pi\)
−0.998368 + 0.0571095i \(0.981812\pi\)
\(20\) −2.01357 − 0.706075i −0.450247 − 0.157883i
\(21\) −1.89884 − 1.89884i −0.414362 − 0.414362i
\(22\) 0.440226 − 1.64295i 0.0938565 − 0.350277i
\(23\) −3.72214 + 0.997344i −0.776120 + 0.207961i −0.625073 − 0.780566i \(-0.714931\pi\)
−0.151046 + 0.988527i \(0.548264\pi\)
\(24\) −2.17910 − 8.13250i −0.444806 − 1.66004i
\(25\) 0.751762 − 4.94316i 0.150352 − 0.988632i
\(26\) 3.68704 − 0.0218799i 0.723087 − 0.00429099i
\(27\) 3.48159 + 3.48159i 0.670033 + 0.670033i
\(28\) 0.796304 + 0.459747i 0.150487 + 0.0868839i
\(29\) 0.253107 + 0.146132i 0.0470008 + 0.0271360i 0.523316 − 0.852139i \(-0.324695\pi\)
−0.476315 + 0.879274i \(0.658028\pi\)
\(30\) 5.74336 − 2.76117i 1.04859 − 0.504118i
\(31\) −0.125649 + 0.125649i −0.0225673 + 0.0225673i −0.718300 − 0.695733i \(-0.755080\pi\)
0.695733 + 0.718300i \(0.255080\pi\)
\(32\) 2.41727 + 4.18683i 0.427317 + 0.740134i
\(33\) −2.31769 − 4.01436i −0.403458 − 0.698811i
\(34\) −5.24282 + 5.24282i −0.899136 + 0.899136i
\(35\) −0.712972 + 2.03323i −0.120514 + 0.343679i
\(36\) −3.93927 − 2.27434i −0.656545 − 0.379057i
\(37\) 3.53443 + 2.04061i 0.581057 + 0.335474i 0.761553 − 0.648102i \(-0.224437\pi\)
−0.180496 + 0.983576i \(0.557770\pi\)
\(38\) 1.97663 + 1.97663i 0.320652 + 0.320652i
\(39\) 7.06291 − 7.14724i 1.13097 − 1.14447i
\(40\) −5.12326 + 4.40302i −0.810059 + 0.696178i
\(41\) −1.79277 − 6.69071i −0.279984 − 1.04491i −0.952427 − 0.304765i \(-0.901422\pi\)
0.672444 − 0.740148i \(-0.265245\pi\)
\(42\) −2.65254 + 0.710745i −0.409295 + 0.109670i
\(43\) 2.05706 − 7.67707i 0.313699 − 1.17074i −0.611495 − 0.791248i \(-0.709432\pi\)
0.925194 − 0.379494i \(-0.123902\pi\)
\(44\) 1.12232 + 1.12232i 0.169195 + 0.169195i
\(45\) 3.52703 − 10.0583i 0.525779 − 1.49940i
\(46\) −1.01990 + 3.80633i −0.150376 + 0.561212i
\(47\) − 7.84582i − 1.14443i −0.820103 − 0.572215i \(-0.806084\pi\)
0.820103 − 0.572215i \(-0.193916\pi\)
\(48\) −3.17888 − 0.851780i −0.458832 − 0.122944i
\(49\) −3.03576 + 5.25810i −0.433680 + 0.751156i
\(50\) −3.99335 − 3.19325i −0.564744 − 0.451594i
\(51\) 20.2063i 2.82944i
\(52\) −1.70259 + 2.98981i −0.236107 + 0.414612i
\(53\) −1.99855 + 1.99855i −0.274522 + 0.274522i −0.830918 − 0.556395i \(-0.812184\pi\)
0.556395 + 0.830918i \(0.312184\pi\)
\(54\) 4.86351 − 1.30317i 0.661840 − 0.177339i
\(55\) −2.09714 + 3.07157i −0.282779 + 0.414171i
\(56\) 2.52102 − 1.45551i 0.336886 − 0.194501i
\(57\) 7.61811 1.00904
\(58\) 0.258832 − 0.149437i 0.0339863 − 0.0196220i
\(59\) 4.87924 + 1.30739i 0.635223 + 0.170207i 0.562039 − 0.827111i \(-0.310017\pi\)
0.0731843 + 0.997318i \(0.476684\pi\)
\(60\) −0.448318 + 5.92967i −0.0578776 + 0.765517i
\(61\) −1.04169 − 1.80425i −0.133374 − 0.231011i 0.791601 − 0.611038i \(-0.209248\pi\)
−0.924975 + 0.380027i \(0.875915\pi\)
\(62\) 0.0470311 + 0.175522i 0.00597296 + 0.0222914i
\(63\) −2.29655 + 3.97775i −0.289339 + 0.501149i
\(64\) 7.30568 0.913209
\(65\) −7.62376 − 2.62264i −0.945611 − 0.325299i
\(66\) −4.74023 −0.583482
\(67\) −3.64915 + 6.32050i −0.445814 + 0.772173i −0.998109 − 0.0614765i \(-0.980419\pi\)
0.552294 + 0.833649i \(0.313752\pi\)
\(68\) −1.79071 − 6.68304i −0.217156 − 0.810437i
\(69\) 5.36956 + 9.30034i 0.646419 + 1.11963i
\(70\) 1.43611 + 1.67103i 0.171648 + 0.199726i
\(71\) 12.6082 + 3.37837i 1.49632 + 0.400939i 0.911867 − 0.410486i \(-0.134641\pi\)
0.584457 + 0.811425i \(0.301308\pi\)
\(72\) −12.4713 + 7.20034i −1.46976 + 0.848568i
\(73\) 3.22747 0.377746 0.188873 − 0.982001i \(-0.439517\pi\)
0.188873 + 0.982001i \(0.439517\pi\)
\(74\) 3.61437 − 2.08676i 0.420163 − 0.242581i
\(75\) −13.8489 + 1.54181i −1.59913 + 0.178033i
\(76\) −2.51962 + 0.675130i −0.289020 + 0.0774427i
\(77\) 1.13328 − 1.13328i 0.129149 − 0.129149i
\(78\) −2.71836 − 9.90945i −0.307793 − 1.12202i
\(79\) − 13.5845i − 1.52838i −0.644992 − 0.764190i \(-0.723139\pi\)
0.644992 − 0.764190i \(-0.276861\pi\)
\(80\) 0.489192 + 2.59485i 0.0546934 + 0.290113i
\(81\) −0.289196 + 0.500902i −0.0321329 + 0.0556558i
\(82\) −6.84204 − 1.83332i −0.755577 − 0.202456i
\(83\) 8.56854i 0.940519i 0.882528 + 0.470260i \(0.155840\pi\)
−0.882528 + 0.470260i \(0.844160\pi\)
\(84\) 0.663230 − 2.47521i 0.0723643 − 0.270067i
\(85\) 14.6117 − 7.02469i 1.58486 − 0.761934i
\(86\) −5.74712 − 5.74712i −0.619728 − 0.619728i
\(87\) 0.210809 − 0.786751i 0.0226011 − 0.0843486i
\(88\) 4.85368 − 1.30054i 0.517404 − 0.138638i
\(89\) 0.134207 + 0.500868i 0.0142259 + 0.0530919i 0.972674 − 0.232175i \(-0.0745843\pi\)
−0.958448 + 0.285267i \(0.907918\pi\)
\(90\) −7.10435 − 8.26648i −0.748864 − 0.871363i
\(91\) 3.01901 + 1.71922i 0.316479 + 0.180223i
\(92\) −2.60014 − 2.60014i −0.271084 − 0.271084i
\(93\) 0.428870 + 0.247608i 0.0444718 + 0.0256758i
\(94\) −6.94836 − 4.01164i −0.716669 − 0.413769i
\(95\) −2.64843 − 5.50885i −0.271723 − 0.565196i
\(96\) 9.52707 − 9.52707i 0.972353 − 0.972353i
\(97\) −3.75660 − 6.50662i −0.381425 − 0.660648i 0.609841 − 0.792524i \(-0.291233\pi\)
−0.991266 + 0.131876i \(0.957900\pi\)
\(98\) 3.10442 + 5.37702i 0.313594 + 0.543161i
\(99\) −5.60626 + 5.60626i −0.563450 + 0.563450i
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 65.2.o.a.32.4 ✓ 20
3.2 odd 2 585.2.cf.a.487.2 20
5.2 odd 4 325.2.x.b.318.4 20
5.3 odd 4 65.2.t.a.58.2 yes 20
5.4 even 2 325.2.s.b.32.2 20
13.2 odd 12 845.2.t.g.427.4 20
13.3 even 3 845.2.o.e.587.4 20
13.4 even 6 845.2.k.d.577.8 20
13.5 odd 4 845.2.t.e.657.2 20
13.6 odd 12 845.2.f.d.437.8 20
13.7 odd 12 845.2.f.e.437.3 20
13.8 odd 4 845.2.t.f.657.4 20
13.9 even 3 845.2.k.e.577.3 20
13.10 even 6 845.2.o.f.587.2 20
13.11 odd 12 65.2.t.a.37.2 yes 20
13.12 even 2 845.2.o.g.357.2 20
15.8 even 4 585.2.dp.a.253.4 20
39.11 even 12 585.2.dp.a.37.4 20
65.3 odd 12 845.2.t.f.418.4 20
65.8 even 4 845.2.o.e.488.4 20
65.18 even 4 845.2.o.f.488.2 20
65.23 odd 12 845.2.t.e.418.2 20
65.24 odd 12 325.2.x.b.232.4 20
65.28 even 12 845.2.o.g.258.2 20
65.33 even 12 845.2.k.e.268.3 20
65.37 even 12 325.2.s.b.193.2 20
65.38 odd 4 845.2.t.g.188.4 20
65.43 odd 12 845.2.f.d.408.3 20
65.48 odd 12 845.2.f.e.408.8 20
65.58 even 12 845.2.k.d.268.8 20
65.63 even 12 inner 65.2.o.a.63.4 yes 20
195.128 odd 12 585.2.cf.a.388.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.o.a.32.4 ✓ 20 1.1 even 1 trivial
65.2.o.a.63.4 yes 20 65.63 even 12 inner
65.2.t.a.37.2 yes 20 13.11 odd 12
65.2.t.a.58.2 yes 20 5.3 odd 4
325.2.s.b.32.2 20 5.4 even 2
325.2.s.b.193.2 20 65.37 even 12
325.2.x.b.232.4 20 65.24 odd 12
325.2.x.b.318.4 20 5.2 odd 4
585.2.cf.a.388.2 20 195.128 odd 12
585.2.cf.a.487.2 20 3.2 odd 2
585.2.dp.a.37.4 20 39.11 even 12
585.2.dp.a.253.4 20 15.8 even 4
845.2.f.d.408.3 20 65.43 odd 12
845.2.f.d.437.8 20 13.6 odd 12
845.2.f.e.408.8 20 65.48 odd 12
845.2.f.e.437.3 20 13.7 odd 12
845.2.k.d.268.8 20 65.58 even 12
845.2.k.d.577.8 20 13.4 even 6
845.2.k.e.268.3 20 65.33 even 12
845.2.k.e.577.3 20 13.9 even 3
845.2.o.e.488.4 20 65.8 even 4
845.2.o.e.587.4 20 13.3 even 3
845.2.o.f.488.2 20 65.18 even 4
845.2.o.f.587.2 20 13.10 even 6
845.2.o.g.258.2 20 65.28 even 12
845.2.o.g.357.2 20 13.12 even 2
845.2.t.e.418.2 20 65.23 odd 12
845.2.t.e.657.2 20 13.5 odd 4
845.2.t.f.418.4 20 65.3 odd 12
845.2.t.f.657.4 20 13.8 odd 4
845.2.t.g.188.4 20 65.38 odd 4
845.2.t.g.427.4 20 13.2 odd 12