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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 254.11
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.11

$q$-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} +(1.39172 - 1.03108i) q^{3} +1.02242 q^{4} +(2.06916 + 0.847683i) q^{5} +(-2.41952 + 1.79253i) q^{6} +(3.83754 - 2.21560i) q^{7} +1.69954 q^{8} +(0.873767 - 2.86994i) q^{9} +(-3.59726 - 1.47371i) q^{10} +(-1.12251 + 1.94424i) q^{11} +(1.42292 - 1.05419i) q^{12} +(1.39201 - 2.41104i) q^{13} +(-6.67160 + 3.85185i) q^{14} +(3.75372 - 0.953725i) q^{15} -4.99950 q^{16} +(-4.36900 + 2.52244i) q^{17} +(-1.51905 + 4.98941i) q^{18} +(2.25846 + 3.91177i) q^{19} +(2.11554 + 0.866684i) q^{20} +(3.05632 - 7.04029i) q^{21} +(1.95149 - 3.38009i) q^{22} +3.41566i q^{23} +(2.36528 - 1.75235i) 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} +(-6.52588 + 1.65806i) 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} +(0.893353 - 2.93427i) 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} +(-5.31345 + 12.2396i) q^{42} +(-2.74312 - 4.75122i) q^{43} +(-1.14767 + 1.98782i) q^{44} +(4.24076 - 5.19769i) q^{45} -5.93816i q^{46} -6.70458 q^{47} +(-6.95790 + 5.15486i) q^{48} +(6.31780 - 10.9428i) q^{49} +(-6.19408 - 6.09867i) q^{50} +(-3.47959 + 8.01529i) 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} +(7.17648 + 3.11544i) q^{57} -5.72053 q^{58} +(9.97486 - 5.75899i) q^{59} +(3.83786 - 0.975103i) 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} +(3.52181 + 4.75365i) q^{69} +(-17.0698 + 2.31470i) q^{70} +(-6.93758 - 4.00542i) q^{71} +(1.48500 - 4.87757i) q^{72} +(-2.65062 + 4.59100i) q^{73} +(6.59732 + 11.4269i) q^{74} +(8.57551 + 1.20855i) 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} +(-7.47306 - 5.01531i) q^{81} +(0.412565 + 0.238195i) q^{82} +(-12.1496 - 7.01455i) q^{83} +(3.12483 - 7.19810i) q^{84} +(-11.1784 + 1.51582i) q^{85} +(4.76894 + 8.26004i) q^{86} +(4.57942 - 3.39273i) q^{87} +(-1.90775 + 3.30432i) q^{88} +1.41614 q^{89} +(-7.37261 + 9.03622i) q^{90} -12.3366i q^{91} +3.49223i q^{92} +(-8.34170 + 4.83901i) q^{93} +11.6560 q^{94} +(1.35718 + 10.0085i) q^{95} +(7.36581 - 5.45706i) q^{96} +17.5285i q^{97} +(-10.9836 + 19.0241i) q^{98} +(4.59904 + 4.92035i) 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 \(-0.710705\pi\)
−0.614656 + 0.788795i \(0.710705\pi\)
\(3\) 1.39172 − 1.03108i 0.803510 − 0.595292i
\(4\) 1.02242 0.511208
\(5\) 2.06916 + 0.847683i 0.925358 + 0.379095i
\(6\) −2.41952 + 1.79253i −0.987764 + 0.731799i
\(7\) 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
\(9\) 0.873767 − 2.86994i 0.291256 − 0.956645i
\(10\) −3.59726 − 1.47371i −1.13755 − 0.466027i
\(11\) −1.12251 + 1.94424i −0.338449 + 0.586212i −0.984141 − 0.177387i \(-0.943236\pi\)
0.645692 + 0.763598i \(0.276569\pi\)
\(12\) 1.42292 − 1.05419i 0.410760 − 0.304318i
\(13\) 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
\(15\) 3.75372 − 0.953725i 0.969206 − 0.246251i
\(16\) −4.99950 −1.24987
\(17\) −4.36900 + 2.52244i −1.05964 + 0.611782i −0.925332 − 0.379158i \(-0.876214\pi\)
−0.134305 + 0.990940i \(0.542880\pi\)
\(18\) −1.51905 + 4.98941i −0.358044 + 1.17602i
\(19\) 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
\(21\) 3.05632 − 7.04029i 0.666944 − 1.53632i
\(22\) 1.95149 − 3.38009i 0.416060 − 0.720637i
\(23\) 3.41566i 0.712215i 0.934445 + 0.356108i \(0.115896\pi\)
−0.934445 + 0.356108i \(0.884104\pi\)
\(24\) 2.36528 − 1.75235i 0.482812 − 0.357698i
\(25\) 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 \(-0.401172\pi\)
0.305513 + 0.952188i \(0.401172\pi\)
\(30\) −6.52588 + 1.65806i −1.19146 + 0.302719i
\(31\) −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
\(36\) 0.893353 − 2.93427i 0.148892 − 0.489044i
\(37\) −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 0.481355 − 0.876526i \(-0.340145\pi\)
−0.518416 + 0.855128i \(0.673478\pi\)
\(42\) −5.31345 + 12.2396i −0.819883 + 1.88861i
\(43\) −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
\(45\) 4.24076 − 5.19769i 0.632176 − 0.774825i
\(46\) − 5.93816i − 0.875535i
\(47\) −6.70458 −0.977964 −0.488982 − 0.872294i \(-0.662632\pi\)
−0.488982 + 0.872294i \(0.662632\pi\)
\(48\) −6.95790 + 5.15486i −1.00429 + 0.744040i
\(49\) 6.31780 − 10.9428i 0.902543 − 1.56325i
\(50\) −6.19408 − 6.09867i −0.875975 − 0.862482i
\(51\) −3.47959 + 8.01529i −0.487240 + 1.12237i
\(52\) 1.42322 − 2.46508i 0.197365 − 0.341845i
\(53\) −0.556042 − 0.321031i −0.0763782 − 0.0440970i 0.461325 − 0.887231i \(-0.347374\pi\)
−0.537703 + 0.843134i \(0.680708\pi\)
\(54\) 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
\(57\) 7.17648 + 3.11544i 0.950548 + 0.412651i
\(58\) −5.72053 −0.751142
\(59\) 9.97486 − 5.75899i 1.29862 − 0.749756i 0.318451 − 0.947939i \(-0.396837\pi\)
0.980165 + 0.198183i \(0.0635041\pi\)
\(60\) 3.83786 − 0.975103i 0.495466 − 0.125885i
\(61\) 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
\(69\) 3.52181 + 4.75365i 0.423976 + 0.572272i
\(70\) −17.0698 + 2.31470i −2.04023 + 0.276659i
\(71\) −6.93758 − 4.00542i −0.823340 − 0.475355i 0.0282271 − 0.999602i \(-0.491014\pi\)
−0.851567 + 0.524246i \(0.824347\pi\)
\(72\) 1.48500 − 4.87757i 0.175009 − 0.574827i
\(73\) −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
\(75\) 8.57551 + 1.20855i 0.990215 + 0.139552i
\(76\) 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
\(81\) −7.47306 − 5.01531i −0.830340 − 0.557257i
\(82\) 0.412565 + 0.238195i 0.0455602 + 0.0263042i
\(83\) −12.1496 − 7.01455i −1.33359 − 0.769947i −0.347739 − 0.937591i \(-0.613051\pi\)
−0.985847 + 0.167645i \(0.946384\pi\)
\(84\) 3.12483 − 7.19810i 0.340947 − 0.785377i
\(85\) −11.1784 + 1.51582i −1.21247 + 0.164413i
\(86\) 4.76894 + 8.26004i 0.514247 + 0.890703i
\(87\) 4.57942 − 3.39273i 0.490966 − 0.363739i
\(88\) −1.90775 + 3.30432i −0.203367 + 0.352242i
\(89\) 1.41614 0.150110 0.0750552 − 0.997179i \(-0.476087\pi\)
0.0750552 + 0.997179i \(0.476087\pi\)
\(90\) −7.37261 + 9.03622i −0.777141 + 0.952502i
\(91\) − 12.3366i − 1.29323i
\(92\) 3.49223i 0.364090i
\(93\) −8.34170 + 4.83901i −0.864994 + 0.501782i
\(94\) 11.6560 1.20222
\(95\) 1.35718 + 10.0085i 0.139244 + 1.02686i
\(96\) 7.36581 − 5.45706i 0.751769 − 0.556959i
\(97\) 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
\(99\) 4.59904 + 4.92035i 0.462221 + 0.494514i
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.254.11 yes 104
3.2 odd 2 inner 465.2.t.d.254.41 yes 104
5.4 even 2 inner 465.2.t.d.254.42 yes 104
15.14 odd 2 inner 465.2.t.d.254.12 yes 104
31.26 odd 6 inner 465.2.t.d.119.12 yes 104
93.26 even 6 inner 465.2.t.d.119.42 yes 104
155.119 odd 6 inner 465.2.t.d.119.41 yes 104
465.119 even 6 inner 465.2.t.d.119.11 ✓ 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.11 ✓ 104 465.119 even 6 inner
465.2.t.d.119.12 yes 104 31.26 odd 6 inner
465.2.t.d.119.41 yes 104 155.119 odd 6 inner
465.2.t.d.119.42 yes 104 93.26 even 6 inner
465.2.t.d.254.11 yes 104 1.1 even 1 trivial
465.2.t.d.254.12 yes 104 15.14 odd 2 inner
465.2.t.d.254.41 yes 104 3.2 odd 2 inner
465.2.t.d.254.42 yes 104 5.4 even 2 inner