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Properties

Label 465.2.t.d.119.12
Level $465$
Weight $2$
Character 465.119
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
Inner twists $8$

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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] = [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 119.12
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.12

$q$-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} +(-1.76870 + 1.36811i) q^{5} +(-2.76214 - 1.19910i) q^{6} +(-3.83754 - 2.21560i) q^{7} +1.69954 q^{8} +(2.04855 + 2.19167i) q^{9} +(3.07490 - 2.37846i) 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} +(-3.75372 + 0.953725i) q^{15} -4.99950 q^{16} +(-4.36900 - 2.52244i) q^{17} +(-3.56143 - 3.81024i) q^{18} +(2.25846 - 3.91177i) q^{19} +(-1.80834 + 1.39877i) q^{20} +(-4.56891 - 6.16700i) q^{21} +(-1.95149 - 3.38009i) q^{22} -3.41566i q^{23} +(2.70023 + 1.17222i) q^{24} +(1.25657 - 4.83953i) 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} +(2.09447 + 2.24080i) q^{36} +(3.79481 - 6.57281i) q^{37} +(-3.92636 + 6.80065i) q^{38} +(-0.548670 - 4.79076i) q^{39} +(-3.00597 + 2.32515i) 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} +(-6.62171 - 1.07377i) q^{45} +5.93816i q^{46} -6.70458 q^{47} +(-7.94319 - 3.44829i) q^{48} +(6.31780 + 10.9428i) q^{49} +(-2.18457 + 8.41356i) 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} +(-4.64531 - 1.90307i) q^{55} +(-6.52205 - 3.76551i) q^{56} +(6.28629 - 4.65729i) q^{57} +5.72053 q^{58} +(-9.97486 - 5.75899i) q^{59} +(-3.83786 + 0.975103i) q^{60} -10.8870i q^{61} +(9.61899 - 1.08157i) q^{62} +(-3.00552 - 12.9494i) q^{63} +0.797773 q^{64} +(5.76061 + 2.35997i) 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} +(5.33439 - 6.82233i) q^{75} +(2.30909 - 3.99945i) q^{76} -9.94815i q^{77} +(0.953867 + 8.32878i) q^{78} +(4.53284 + 2.61704i) q^{79} +(8.84259 - 6.83984i) 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} +(11.5119 + 1.86675i) 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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 \(0.710705\pi\)
\(3\) 1.58880 + 0.689727i 0.917293 + 0.398214i
\(4\) 1.02242 0.511208
\(5\) −1.76870 + 1.36811i −0.790985 + 0.611835i
\(6\) −2.76214 − 1.19910i −1.12764 − 0.489529i
\(7\) −3.83754 − 2.21560i −1.45045 − 0.837420i −0.451946 − 0.892045i \(-0.649270\pi\)
−0.998507 + 0.0546257i \(0.982603\pi\)
\(8\) 1.69954 0.600878
\(9\) 2.04855 + 2.19167i 0.682851 + 0.730558i
\(10\) 3.07490 − 2.37846i 0.972367 − 0.752137i
\(11\) 1.12251 + 1.94424i 0.338449 + 0.586212i 0.984141 − 0.177387i \(-0.0567643\pi\)
−0.645692 + 0.763598i \(0.723431\pi\)
\(12\) 1.62441 + 0.705187i 0.468927 + 0.203570i
\(13\) −1.39201 − 2.41104i −0.386075 − 0.668702i 0.605842 − 0.795585i \(-0.292836\pi\)
−0.991918 + 0.126883i \(0.959503\pi\)
\(14\) 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 −0.134305 − 0.990940i \(-0.542880\pi\)
−0.925332 + 0.379158i \(0.876214\pi\)
\(18\) −3.56143 − 3.81024i −0.839437 − 0.898083i
\(19\) 2.25846 − 3.91177i 0.518127 − 0.897422i −0.481652 − 0.876363i \(-0.659963\pi\)
0.999778 − 0.0210589i \(-0.00670377\pi\)
\(20\) −1.80834 + 1.39877i −0.404358 + 0.312775i
\(21\) −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
\(25\) 1.25657 − 4.83953i 0.251315 − 0.967905i
\(26\) 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 \(0.598828\pi\)
\(30\) 6.52588 − 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
\(36\) 2.09447 + 2.24080i 0.349079 + 0.373467i
\(37\) 3.79481 − 6.57281i 0.623863 − 1.08056i −0.364896 − 0.931048i \(-0.618895\pi\)
0.988759 − 0.149515i \(-0.0477712\pi\)
\(38\) −3.92636 + 6.80065i −0.636939 + 1.10321i
\(39\) −0.548670 − 4.79076i −0.0878574 − 0.767136i
\(40\) −3.00597 + 2.32515i −0.475286 + 0.367639i
\(41\) 0.237310 − 0.137011i 0.0370616 − 0.0213975i −0.481355 − 0.876526i \(-0.659855\pi\)
0.518416 + 0.855128i \(0.326522\pi\)
\(42\) 7.94309 + 10.7214i 1.22565 + 1.65435i
\(43\) 2.74312 − 4.75122i 0.418321 − 0.724554i −0.577449 − 0.816427i \(-0.695952\pi\)
0.995771 + 0.0918725i \(0.0292852\pi\)
\(44\) 1.14767 + 1.98782i 0.173018 + 0.299676i
\(45\) −6.62171 − 1.07377i −0.987106 − 0.160068i
\(46\) 5.93816i 0.875535i
\(47\) −6.70458 −0.977964 −0.488982 − 0.872294i \(-0.662632\pi\)
−0.488982 + 0.872294i \(0.662632\pi\)
\(48\) −7.94319 − 3.44829i −1.14650 − 0.497718i
\(49\) 6.31780 + 10.9428i 0.902543 + 1.56325i
\(50\) −2.18457 + 8.41356i −0.308944 + 1.18986i
\(51\) −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 \(-0.680708\pi\)
0.461325 + 0.887231i \(0.347374\pi\)
\(54\) −3.03036 − 8.51012i −0.412380 − 1.15808i
\(55\) −4.64531 − 1.90307i −0.626373 − 0.256609i
\(56\) −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 −0.318451 − 0.947939i \(-0.603163\pi\)
−0.980165 + 0.198183i \(0.936496\pi\)
\(60\) −3.83786 + 0.975103i −0.495466 + 0.125885i
\(61\) − 10.8870i − 1.39394i −0.717101 − 0.696970i \(-0.754531\pi\)
0.717101 − 0.696970i \(-0.245469\pi\)
\(62\) 9.61899 − 1.08157i 1.22161 − 0.137360i
\(63\) −3.00552 − 12.9494i −0.378661 − 1.63147i
\(64\) 0.797773 0.0997216
\(65\) 5.76061 + 2.35997i 0.714515 + 0.292719i
\(66\) −0.769191 − 6.71627i −0.0946809 − 0.826716i
\(67\) −11.4247 + 6.59604i −1.39575 + 0.805834i −0.993943 − 0.109893i \(-0.964949\pi\)
−0.401802 + 0.915727i \(0.631616\pi\)
\(68\) −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 −0.0282271 − 0.999602i \(-0.508986\pi\)
0.851567 + 0.524246i \(0.175653\pi\)
\(72\) 3.48160 + 3.72484i 0.410310 + 0.438976i
\(73\) 2.65062 + 4.59100i 0.310231 + 0.537336i 0.978412 − 0.206663i \(-0.0662602\pi\)
−0.668181 + 0.743999i \(0.732927\pi\)
\(74\) −6.59732 + 11.4269i −0.766923 + 1.32835i
\(75\) 5.33439 − 6.82233i 0.615963 − 0.787775i
\(76\) 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 0.732827 − 0.680415i \(-0.238200\pi\)
−0.222843 + 0.974854i \(0.571534\pi\)
\(80\) 8.84259 − 6.83984i 0.988632 − 0.764717i
\(81\) −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 \(-0.946384\pi\)
−0.347739 + 0.937591i \(0.613051\pi\)
\(84\) −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 \(0.523913\pi\)
\(90\) 11.5119 + 1.86675i 1.21346 + 0.196773i
\(91\) 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.119.12 yes 104
3.2 odd 2 inner 465.2.t.d.119.42 yes 104
5.4 even 2 inner 465.2.t.d.119.41 yes 104
15.14 odd 2 inner 465.2.t.d.119.11 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.11 yes 104
93.68 even 6 inner 465.2.t.d.254.41 yes 104
155.99 odd 6 inner 465.2.t.d.254.42 yes 104
465.254 even 6 inner 465.2.t.d.254.12 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.11 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.12 yes 104 1.1 even 1 trivial
465.2.t.d.119.41 yes 104 5.4 even 2 inner
465.2.t.d.119.42 yes 104 3.2 odd 2 inner
465.2.t.d.254.11 yes 104 31.6 odd 6 inner
465.2.t.d.254.12 yes 104 465.254 even 6 inner
465.2.t.d.254.41 yes 104 93.68 even 6 inner
465.2.t.d.254.42 yes 104 155.99 odd 6 inner