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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 254.42
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.42

$q$-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} +(1.76870 + 1.36811i) q^{5} +(-2.41952 + 1.79253i) q^{6} +(-3.83754 + 2.21560i) q^{7} -1.69954 q^{8} +(0.873767 - 2.86994i) q^{9} +(3.07490 + 2.37846i) 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} +(-3.87215 - 0.0803602i) q^{15} -4.99950 q^{16} +(4.36900 - 2.52244i) q^{17} +(1.51905 - 4.98941i) q^{18} +(2.25846 + 3.91177i) q^{19} +(1.80834 + 1.39877i) q^{20} +(3.05632 - 7.04029i) q^{21} +(-1.95149 + 3.38009i) q^{22} -3.41566i q^{23} +(2.36528 - 1.75235i) 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} +(-6.73177 - 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} +(0.893353 - 2.93427i) q^{36} +(3.79481 + 6.57281i) q^{37} +(3.92636 + 6.80065i) q^{38} +(-0.548670 - 4.79076i) q^{39} +(-3.00597 - 2.32515i) 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} +(5.47180 - 3.88064i) q^{45} -5.93816i q^{46} +6.70458 q^{47} +(6.95790 - 5.15486i) q^{48} +(6.31780 - 10.9428i) q^{49} +(2.18457 + 8.41356i) 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} +(-4.64531 + 1.90307i) q^{55} +(6.52205 - 3.76551i) q^{56} +(-7.17648 - 3.11544i) q^{57} +5.72053 q^{58} +(9.97486 - 5.75899i) q^{59} +(-3.95894 - 0.0821615i) q^{60} +10.8870i q^{61} +(-9.61899 - 1.08157i) q^{62} +(3.00552 + 12.9494i) q^{63} +0.797773 q^{64} +(-5.76061 + 2.35997i) 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} +(-6.73872 - 5.43964i) 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} +(-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} +(9.51278 - 6.74653i) 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 \(-0.289295\pi\)
0.614656 + 0.788795i \(0.289295\pi\)
\(3\) −1.39172 + 1.03108i −0.803510 + 0.595292i
\(4\) 1.02242 0.511208
\(5\) 1.76870 + 1.36811i 0.790985 + 0.611835i
\(6\) −2.41952 + 1.79253i −0.987764 + 0.731799i
\(7\) −3.83754 + 2.21560i −1.45045 + 0.837420i −0.998507 − 0.0546257i \(-0.982603\pi\)
−0.451946 + 0.892045i \(0.649270\pi\)
\(8\) −1.69954 −0.600878
\(9\) 0.873767 − 2.86994i 0.291256 − 0.956645i
\(10\) 3.07490 + 2.37846i 0.972367 + 0.752137i
\(11\) −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 −0.991918 − 0.126883i \(-0.959503\pi\)
0.605842 + 0.795585i \(0.292836\pi\)
\(14\) −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 \(0.123786\pi\)
\(18\) 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 \(0.659963\pi\)
\(20\) 1.80834 + 1.39877i 0.404358 + 0.312775i
\(21\) 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 \(-0.884104\pi\)
0.934445 − 0.356108i \(-0.115896\pi\)
\(24\) 2.36528 − 1.75235i 0.482812 − 0.357698i
\(25\) 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 \(-0.401172\pi\)
0.305513 + 0.952188i \(0.401172\pi\)
\(30\) −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
\(36\) 0.893353 − 2.93427i 0.148892 − 0.489044i
\(37\) 3.79481 + 6.57281i 0.623863 + 1.08056i 0.988759 + 0.149515i \(0.0477712\pi\)
−0.364896 + 0.931048i \(0.618895\pi\)
\(38\) 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 \(-0.340145\pi\)
−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 0.995771 − 0.0918725i \(-0.0292852\pi\)
−0.577449 + 0.816427i \(0.695952\pi\)
\(44\) −1.14767 + 1.98782i −0.173018 + 0.299676i
\(45\) 5.47180 − 3.88064i 0.815688 − 0.578492i
\(46\) − 5.93816i − 0.875535i
\(47\) 6.70458 0.977964 0.488982 − 0.872294i \(-0.337368\pi\)
0.488982 + 0.872294i \(0.337368\pi\)
\(48\) 6.95790 − 5.15486i 1.00429 − 0.744040i
\(49\) 6.31780 − 10.9428i 0.902543 − 1.56325i
\(50\) 2.18457 + 8.41356i 0.308944 + 1.18986i
\(51\) −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 0.537703 − 0.843134i \(-0.319292\pi\)
−0.461325 + 0.887231i \(0.652626\pi\)
\(54\) 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
\(57\) −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 \(0.0635041\pi\)
\(60\) −3.95894 − 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
\(65\) −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 −0.401802 − 0.915727i \(-0.631616\pi\)
−0.993943 + 0.109893i \(0.964949\pi\)
\(68\) 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 −0.668181 − 0.743999i \(-0.732927\pi\)
0.978412 + 0.206663i \(0.0662602\pi\)
\(74\) 6.59732 + 11.4269i 0.766923 + 1.32835i
\(75\) −6.73872 − 5.43964i −0.778120 − 0.628116i
\(76\) 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 \(0.238200\pi\)
\(80\) −8.84259 − 6.83984i −0.988632 − 0.764717i
\(81\) −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 0.985847 − 0.167645i \(-0.0536161\pi\)
0.347739 + 0.937591i \(0.386949\pi\)
\(84\) 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 \(0.476087\pi\)
\(90\) 9.51278 − 6.74653i 1.00274 − 0.711146i
\(91\) − 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 \(-0.650792\pi\)
0.456207 − 0.889873i \(-0.349208\pi\)
\(98\) 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.254.42 yes 104
3.2 odd 2 inner 465.2.t.d.254.12 yes 104
5.4 even 2 inner 465.2.t.d.254.11 yes 104
15.14 odd 2 inner 465.2.t.d.254.41 yes 104
31.26 odd 6 inner 465.2.t.d.119.41 yes 104
93.26 even 6 inner 465.2.t.d.119.11 ✓ 104
155.119 odd 6 inner 465.2.t.d.119.12 yes 104
465.119 even 6 inner 465.2.t.d.119.42 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.11 ✓ 104 93.26 even 6 inner
465.2.t.d.119.12 yes 104 155.119 odd 6 inner
465.2.t.d.119.41 yes 104 31.26 odd 6 inner
465.2.t.d.119.42 yes 104 465.119 even 6 inner
465.2.t.d.254.11 yes 104 5.4 even 2 inner
465.2.t.d.254.12 yes 104 3.2 odd 2 inner
465.2.t.d.254.41 yes 104 15.14 odd 2 inner
465.2.t.d.254.42 yes 104 1.1 even 1 trivial