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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 254.41
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.41

$q$-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+1.73851 q^{2} +(-1.58880 + 0.689727i) q^{3} +1.02242 q^{4} +(-2.06916 - 0.847683i) q^{5} +(-2.76214 + 1.19910i) q^{6} +(3.83754 - 2.21560i) q^{7} -1.69954 q^{8} +(2.04855 - 2.19167i) q^{9} +(-3.59726 - 1.47371i) q^{10} +(1.12251 - 1.94424i) q^{11} +(-1.62441 + 0.705187i) q^{12} +(1.39201 - 2.41104i) q^{13} +(6.67160 - 3.85185i) q^{14} +(3.87215 - 0.0803602i) q^{15} -4.99950 q^{16} +(4.36900 - 2.52244i) q^{17} +(3.56143 - 3.81024i) q^{18} +(2.25846 + 3.91177i) q^{19} +(-2.11554 - 0.866684i) q^{20} +(-4.56891 + 6.16700i) q^{21} +(1.95149 - 3.38009i) q^{22} -3.41566i q^{23} +(2.70023 - 1.17222i) q^{24} +(3.56287 + 3.50799i) q^{25} +(2.42003 - 4.19161i) q^{26} +(-1.74308 + 4.89507i) q^{27} +(3.92356 - 2.26527i) q^{28} -3.29048 q^{29} +(6.73177 - 0.139707i) q^{30} +(-5.53290 - 0.622128i) q^{31} -5.29259 q^{32} +(-0.442443 + 3.86323i) q^{33} +(7.59554 - 4.38529i) q^{34} +(-9.81862 + 1.33143i) q^{35} +(2.09447 - 2.24080i) q^{36} +(-3.79481 - 6.57281i) q^{37} +(3.92636 + 6.80065i) q^{38} +(-0.548670 + 4.79076i) q^{39} +(3.51663 + 1.44067i) q^{40} +(0.237310 + 0.137011i) q^{41} +(-7.94309 + 10.7214i) q^{42} +(-2.74312 - 4.75122i) q^{43} +(1.14767 - 1.98782i) q^{44} +(-6.09663 + 2.79840i) q^{45} -5.93816i q^{46} +6.70458 q^{47} +(7.94319 - 3.44829i) q^{48} +(6.31780 - 10.9428i) q^{49} +(6.19408 + 6.09867i) q^{50} +(-5.20165 + 7.02106i) q^{51} +(1.42322 - 2.46508i) q^{52} +(0.556042 + 0.321031i) q^{53} +(-3.03036 + 8.51012i) q^{54} +(-3.97076 + 3.07142i) q^{55} +(-6.52205 + 3.76551i) q^{56} +(-6.28629 - 4.65729i) q^{57} -5.72053 q^{58} +(-9.97486 + 5.75899i) q^{59} +(3.95894 - 0.0821615i) q^{60} +10.8870i q^{61} +(-9.61899 - 1.08157i) q^{62} +(3.00552 - 12.9494i) q^{63} +0.797773 q^{64} +(-4.92410 + 3.80884i) q^{65} +(-0.769191 + 6.71627i) q^{66} +(11.4247 + 6.59604i) q^{67} +(4.46693 - 2.57898i) q^{68} +(2.35588 + 5.42680i) q^{69} +(-17.0698 + 2.31470i) q^{70} +(6.93758 + 4.00542i) q^{71} +(-3.48160 + 3.72484i) q^{72} +(-2.65062 + 4.59100i) q^{73} +(-6.59732 - 11.4269i) q^{74} +(-8.08023 - 3.11608i) q^{75} +(2.30909 + 3.99945i) q^{76} -9.94815i q^{77} +(-0.953867 + 8.32878i) q^{78} +(4.53284 - 2.61704i) q^{79} +(10.3448 + 4.23799i) q^{80} +(-0.606857 - 8.97952i) q^{81} +(0.412565 + 0.238195i) q^{82} +(12.1496 + 7.01455i) q^{83} +(-4.67132 + 6.30523i) q^{84} +(-11.1784 + 1.51582i) q^{85} +(-4.76894 - 8.26004i) q^{86} +(5.22790 - 2.26953i) q^{87} +(-1.90775 + 3.30432i) q^{88} -1.41614 q^{89} +(-10.5991 + 4.86505i) q^{90} -12.3366i q^{91} -3.49223i q^{92} +(9.21975 - 2.82775i) q^{93} +11.6560 q^{94} +(-1.35718 - 10.0085i) q^{95} +(8.40886 - 3.65044i) q^{96} +17.5285i q^{97} +(10.9836 - 19.0241i) q^{98} +(-1.96162 - 6.44306i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73851 1.22931 0.614656 − 0.788795i \(-0.289295\pi\)
0.614656 + 0.788795i \(0.289295\pi\)
\(3\) −1.58880 + 0.689727i −0.917293 + 0.398214i
\(4\) 1.02242 0.511208
\(5\) −2.06916 − 0.847683i −0.925358 − 0.379095i
\(6\) −2.76214 + 1.19910i −1.12764 + 0.489529i
\(7\) 3.83754 − 2.21560i 1.45045 − 0.837420i 0.451946 − 0.892045i \(-0.350730\pi\)
0.998507 + 0.0546257i \(0.0173966\pi\)
\(8\) −1.69954 −0.600878
\(9\) 2.04855 − 2.19167i 0.682851 − 0.730558i
\(10\) −3.59726 − 1.47371i −1.13755 − 0.466027i
\(11\) 1.12251 − 1.94424i 0.338449 − 0.586212i −0.645692 − 0.763598i \(-0.723431\pi\)
0.984141 + 0.177387i \(0.0567643\pi\)
\(12\) −1.62441 + 0.705187i −0.468927 + 0.203570i
\(13\) 1.39201 − 2.41104i 0.386075 − 0.668702i −0.605842 − 0.795585i \(-0.707164\pi\)
0.991918 + 0.126883i \(0.0404972\pi\)
\(14\) 6.67160 − 3.85185i 1.78306 − 1.02945i
\(15\) 3.87215 − 0.0803602i 0.999785 − 0.0207489i
\(16\) −4.99950 −1.24987
\(17\) 4.36900 − 2.52244i 1.05964 − 0.611782i 0.134305 − 0.990940i \(-0.457120\pi\)
0.925332 + 0.379158i \(0.123786\pi\)
\(18\) 3.56143 − 3.81024i 0.839437 − 0.898083i
\(19\) 2.25846 + 3.91177i 0.518127 + 0.897422i 0.999778 + 0.0210589i \(0.00670377\pi\)
−0.481652 + 0.876363i \(0.659963\pi\)
\(20\) −2.11554 − 0.866684i −0.473050 − 0.193796i
\(21\) −4.56891 + 6.16700i −0.997018 + 1.34575i
\(22\) 1.95149 − 3.38009i 0.416060 − 0.720637i
\(23\) − 3.41566i − 0.712215i −0.934445 − 0.356108i \(-0.884104\pi\)
0.934445 − 0.356108i \(-0.115896\pi\)
\(24\) 2.70023 − 1.17222i 0.551181 − 0.239278i
\(25\) 3.56287 + 3.50799i 0.712573 + 0.701598i
\(26\) 2.42003 − 4.19161i 0.474607 − 0.822043i
\(27\) −1.74308 + 4.89507i −0.335456 + 0.942056i
\(28\) 3.92356 − 2.26527i 0.741483 − 0.428095i
\(29\) −3.29048 −0.611026 −0.305513 − 0.952188i \(-0.598828\pi\)
−0.305513 + 0.952188i \(0.598828\pi\)
\(30\) 6.73177 − 0.139707i 1.22905 − 0.0255069i
\(31\) −5.53290 − 0.622128i −0.993738 − 0.111737i
\(32\) −5.29259 −0.935607
\(33\) −0.442443 + 3.86323i −0.0770194 + 0.672503i
\(34\) 7.59554 − 4.38529i 1.30262 − 0.752071i
\(35\) −9.81862 + 1.33143i −1.65965 + 0.225052i
\(36\) 2.09447 − 2.24080i 0.349079 − 0.373467i
\(37\) −3.79481 − 6.57281i −0.623863 − 1.08056i −0.988759 − 0.149515i \(-0.952229\pi\)
0.364896 − 0.931048i \(-0.381105\pi\)
\(38\) 3.92636 + 6.80065i 0.636939 + 1.10321i
\(39\) −0.548670 + 4.79076i −0.0878574 + 0.767136i
\(40\) 3.51663 + 1.44067i 0.556027 + 0.227790i
\(41\) 0.237310 + 0.137011i 0.0370616 + 0.0213975i 0.518416 − 0.855128i \(-0.326522\pi\)
−0.481355 + 0.876526i \(0.659855\pi\)
\(42\) −7.94309 + 10.7214i −1.22565 + 1.65435i
\(43\) −2.74312 − 4.75122i −0.418321 − 0.724554i 0.577449 − 0.816427i \(-0.304048\pi\)
−0.995771 + 0.0918725i \(0.970715\pi\)
\(44\) 1.14767 − 1.98782i 0.173018 − 0.299676i
\(45\) −6.09663 + 2.79840i −0.908833 + 0.417161i
\(46\) − 5.93816i − 0.875535i
\(47\) 6.70458 0.977964 0.488982 − 0.872294i \(-0.337368\pi\)
0.488982 + 0.872294i \(0.337368\pi\)
\(48\) 7.94319 − 3.44829i 1.14650 − 0.497718i
\(49\) 6.31780 − 10.9428i 0.902543 − 1.56325i
\(50\) 6.19408 + 6.09867i 0.875975 + 0.862482i
\(51\) −5.20165 + 7.02106i −0.728377 + 0.983145i
\(52\) 1.42322 − 2.46508i 0.197365 − 0.341845i
\(53\) 0.556042 + 0.321031i 0.0763782 + 0.0440970i 0.537703 − 0.843134i \(-0.319292\pi\)
−0.461325 + 0.887231i \(0.652626\pi\)
\(54\) −3.03036 + 8.51012i −0.412380 + 1.15808i
\(55\) −3.97076 + 3.07142i −0.535417 + 0.414151i
\(56\) −6.52205 + 3.76551i −0.871546 + 0.503187i
\(57\) −6.28629 − 4.65729i −0.832640 − 0.616873i
\(58\) −5.72053 −0.751142
\(59\) −9.97486 + 5.75899i −1.29862 + 0.749756i −0.980165 − 0.198183i \(-0.936496\pi\)
−0.318451 + 0.947939i \(0.603163\pi\)
\(60\) 3.95894 − 0.0821615i 0.511097 − 0.0106070i
\(61\) 10.8870i 1.39394i 0.717101 + 0.696970i \(0.245469\pi\)
−0.717101 + 0.696970i \(0.754531\pi\)
\(62\) −9.61899 − 1.08157i −1.22161 − 0.137360i
\(63\) 3.00552 − 12.9494i 0.378661 − 1.63147i
\(64\) 0.797773 0.0997216
\(65\) −4.92410 + 3.80884i −0.610760 + 0.472429i
\(66\) −0.769191 + 6.71627i −0.0946809 + 0.826716i
\(67\) 11.4247 + 6.59604i 1.39575 + 0.805834i 0.993943 − 0.109893i \(-0.0350507\pi\)
0.401802 + 0.915727i \(0.368384\pi\)
\(68\) 4.46693 − 2.57898i 0.541694 − 0.312747i
\(69\) 2.35588 + 5.42680i 0.283614 + 0.653310i
\(70\) −17.0698 + 2.31470i −2.04023 + 0.276659i
\(71\) 6.93758 + 4.00542i 0.823340 + 0.475355i 0.851567 − 0.524246i \(-0.175653\pi\)
−0.0282271 + 0.999602i \(0.508986\pi\)
\(72\) −3.48160 + 3.72484i −0.410310 + 0.438976i
\(73\) −2.65062 + 4.59100i −0.310231 + 0.537336i −0.978412 − 0.206663i \(-0.933740\pi\)
0.668181 + 0.743999i \(0.267073\pi\)
\(74\) −6.59732 − 11.4269i −0.766923 − 1.32835i
\(75\) −8.08023 − 3.11608i −0.933024 − 0.359814i
\(76\) 2.30909 + 3.99945i 0.264870 + 0.458769i
\(77\) − 9.94815i − 1.13370i
\(78\) −0.953867 + 8.32878i −0.108004 + 0.943049i
\(79\) 4.53284 − 2.61704i 0.509985 − 0.294440i −0.222843 − 0.974854i \(-0.571534\pi\)
0.732827 + 0.680415i \(0.238200\pi\)
\(80\) 10.3448 + 4.23799i 1.15658 + 0.473822i
\(81\) −0.606857 − 8.97952i −0.0674285 − 0.997724i
\(82\) 0.412565 + 0.238195i 0.0455602 + 0.0263042i
\(83\) 12.1496 + 7.01455i 1.33359 + 0.769947i 0.985847 − 0.167645i \(-0.0536161\pi\)
0.347739 + 0.937591i \(0.386949\pi\)
\(84\) −4.67132 + 6.30523i −0.509683 + 0.687957i
\(85\) −11.1784 + 1.51582i −1.21247 + 0.164413i
\(86\) −4.76894 − 8.26004i −0.514247 − 0.890703i
\(87\) 5.22790 − 2.26953i 0.560490 − 0.243319i
\(88\) −1.90775 + 3.30432i −0.203367 + 0.352242i
\(89\) −1.41614 −0.150110 −0.0750552 − 0.997179i \(-0.523913\pi\)
−0.0750552 + 0.997179i \(0.523913\pi\)
\(90\) −10.5991 + 4.86505i −1.11724 + 0.512821i
\(91\) − 12.3366i − 1.29323i
\(92\) − 3.49223i − 0.364090i
\(93\) 9.21975 − 2.82775i 0.956044 − 0.293224i
\(94\) 11.6560 1.20222
\(95\) −1.35718 − 10.0085i −0.139244 − 1.02686i
\(96\) 8.40886 − 3.65044i 0.858225 − 0.372572i
\(97\) 17.5285i 1.77975i 0.456207 + 0.889873i \(0.349208\pi\)
−0.456207 + 0.889873i \(0.650792\pi\)
\(98\) 10.9836 − 19.0241i 1.10951 − 1.92172i
\(99\) −1.96162 − 6.44306i −0.197151 − 0.647552i
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 465.2.t.d.254.41 yes 104
3.2 odd 2 inner 465.2.t.d.254.11 yes 104
5.4 even 2 inner 465.2.t.d.254.12 yes 104
15.14 odd 2 inner 465.2.t.d.254.42 yes 104
31.26 odd 6 inner 465.2.t.d.119.42 yes 104
93.26 even 6 inner 465.2.t.d.119.12 yes 104
155.119 odd 6 inner 465.2.t.d.119.11 ✓ 104
465.119 even 6 inner 465.2.t.d.119.41 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.11 ✓ 104 155.119 odd 6 inner
465.2.t.d.119.12 yes 104 93.26 even 6 inner
465.2.t.d.119.41 yes 104 465.119 even 6 inner
465.2.t.d.119.42 yes 104 31.26 odd 6 inner
465.2.t.d.254.11 yes 104 3.2 odd 2 inner
465.2.t.d.254.12 yes 104 5.4 even 2 inner
465.2.t.d.254.41 yes 104 1.1 even 1 trivial
465.2.t.d.254.42 yes 104 15.14 odd 2 inner