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(94,465)] 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("465.94"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.1016580161536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 94.3
Root \(-1.33253i\) of defining polynomial
Character \(\chi\) \(=\) 465.94
Dual form 465.2.c.a.94.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.39276i q^{2} +1.00000i q^{3} +0.0602300 q^{4} +(-1.33253 - 1.79565i) q^{5} +1.39276 q^{6} -0.392756i q^{7} -2.86940i q^{8} -1.00000 q^{9} +(-2.50091 + 1.85588i) q^{10} -2.96404 q^{11} +0.0602300i q^{12} -2.92963i q^{13} -0.547014 q^{14} +(1.79565 - 1.33253i) q^{15} -3.87591 q^{16} -0.739663i q^{17} +1.39276i q^{18} +3.50091 q^{19} +(-0.0802581 - 0.108152i) q^{20} +0.392756 q^{21} +4.12818i q^{22} -7.61557i q^{23} +2.86940 q^{24} +(-1.44875 + 4.78551i) q^{25} -4.08026 q^{26} -1.00000i q^{27} -0.0236557i q^{28} -0.883778 q^{29} +(-1.85588 - 2.50091i) q^{30} -1.00000 q^{31} -0.340595i q^{32} -2.96404i q^{33} -1.03017 q^{34} +(-0.705254 + 0.523358i) q^{35} -0.0602300 q^{36} -1.01775i q^{37} -4.87591i q^{38} +2.92963 q^{39} +(-5.15245 + 3.82355i) q^{40} -0.566932 q^{41} -0.547014i q^{42} -4.98562i q^{43} -0.178524 q^{44} +(1.33253 + 1.79565i) q^{45} -10.6066 q^{46} +2.22210i q^{47} -3.87591i q^{48} +6.84574 q^{49} +(6.66505 + 2.01775i) q^{50} +0.739663 q^{51} -0.176452i q^{52} +7.23391i q^{53} -1.39276 q^{54} +(3.94966 + 5.32238i) q^{55} -1.12697 q^{56} +3.50091i q^{57} +1.23089i q^{58} +4.56679 q^{59} +(0.108152 - 0.0802581i) q^{60} -2.23789 q^{61} +1.39276i q^{62} +0.392756i q^{63} -8.22619 q^{64} +(-5.26060 + 3.90381i) q^{65} -4.12818 q^{66} +3.28668i q^{67} -0.0445499i q^{68} +7.61557 q^{69} +(0.728910 + 0.982247i) q^{70} +9.31452 q^{71} +2.86940i q^{72} +15.6910i q^{73} -1.41748 q^{74} +(-4.78551 - 1.44875i) q^{75} +0.210860 q^{76} +1.16414i q^{77} -4.08026i q^{78} +10.7258 q^{79} +(5.16475 + 6.95980i) q^{80} +1.00000 q^{81} +0.789599i q^{82} -5.97573i q^{83} +0.0236557 q^{84} +(-1.32818 + 0.985620i) q^{85} -6.94375 q^{86} -0.883778i q^{87} +8.50500i q^{88} +2.02861 q^{89} +(2.50091 - 1.85588i) q^{90} -1.15063 q^{91} -0.458686i q^{92} -1.00000i q^{93} +3.09484 q^{94} +(-4.66505 - 6.28642i) q^{95} +0.340595 q^{96} +2.14846i q^{97} -9.53445i q^{98} +2.96404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 6 q^{6} - 10 q^{9} + 2 q^{10} - 32 q^{14} + 2 q^{15} + 6 q^{16} + 8 q^{19} + 28 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 12 q^{26} - 8 q^{29} + 4 q^{30} - 10 q^{31} - 12 q^{34} + 4 q^{35}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.39276i − 0.984827i −0.870361 − 0.492414i \(-0.836115\pi\)
0.870361 − 0.492414i \(-0.163885\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 0.0602300 0.0301150
\(5\) −1.33253 − 1.79565i −0.595924 − 0.803041i
\(6\) 1.39276 0.568590
\(7\) − 0.392756i − 0.148448i −0.997242 − 0.0742240i \(-0.976352\pi\)
0.997242 − 0.0742240i \(-0.0236480\pi\)
\(8\) − 2.86940i − 1.01449i
\(9\) −1.00000 −0.333333
\(10\) −2.50091 + 1.85588i −0.790857 + 0.586882i
\(11\) −2.96404 −0.893691 −0.446845 − 0.894611i \(-0.647453\pi\)
−0.446845 + 0.894611i \(0.647453\pi\)
\(12\) 0.0602300i 0.0173869i
\(13\) − 2.92963i − 0.812533i −0.913755 − 0.406266i \(-0.866831\pi\)
0.913755 − 0.406266i \(-0.133169\pi\)
\(14\) −0.547014 −0.146196
\(15\) 1.79565 − 1.33253i 0.463636 − 0.344057i
\(16\) −3.87591 −0.968978
\(17\) − 0.739663i − 0.179395i −0.995969 − 0.0896973i \(-0.971410\pi\)
0.995969 − 0.0896973i \(-0.0285900\pi\)
\(18\) 1.39276i 0.328276i
\(19\) 3.50091 0.803164 0.401582 − 0.915823i \(-0.368461\pi\)
0.401582 + 0.915823i \(0.368461\pi\)
\(20\) −0.0802581 − 0.108152i −0.0179463 − 0.0241836i
\(21\) 0.392756 0.0857064
\(22\) 4.12818i 0.880131i
\(23\) − 7.61557i − 1.58796i −0.607946 − 0.793979i \(-0.708006\pi\)
0.607946 − 0.793979i \(-0.291994\pi\)
\(24\) 2.86940 0.585713
\(25\) −1.44875 + 4.78551i −0.289750 + 0.957102i
\(26\) −4.08026 −0.800204
\(27\) − 1.00000i − 0.192450i
\(28\) − 0.0236557i − 0.00447051i
\(29\) −0.883778 −0.164114 −0.0820568 − 0.996628i \(-0.526149\pi\)
−0.0820568 + 0.996628i \(0.526149\pi\)
\(30\) −1.85588 − 2.50091i −0.338837 − 0.456601i
\(31\) −1.00000 −0.179605
\(32\) − 0.340595i − 0.0602093i
\(33\) − 2.96404i − 0.515973i
\(34\) −1.03017 −0.176673
\(35\) −0.705254 + 0.523358i −0.119210 + 0.0884636i
\(36\) −0.0602300 −0.0100383
\(37\) − 1.01775i − 0.167317i −0.996494 − 0.0836587i \(-0.973339\pi\)
0.996494 − 0.0836587i \(-0.0266606\pi\)
\(38\) − 4.87591i − 0.790977i
\(39\) 2.92963 0.469116
\(40\) −5.15245 + 3.82355i −0.814673 + 0.604556i
\(41\) −0.566932 −0.0885400 −0.0442700 − 0.999020i \(-0.514096\pi\)
−0.0442700 + 0.999020i \(0.514096\pi\)
\(42\) − 0.547014i − 0.0844060i
\(43\) − 4.98562i − 0.760300i −0.924925 − 0.380150i \(-0.875872\pi\)
0.924925 − 0.380150i \(-0.124128\pi\)
\(44\) −0.178524 −0.0269135
\(45\) 1.33253 + 1.79565i 0.198641 + 0.267680i
\(46\) −10.6066 −1.56386
\(47\) 2.22210i 0.324126i 0.986780 + 0.162063i \(0.0518148\pi\)
−0.986780 + 0.162063i \(0.948185\pi\)
\(48\) − 3.87591i − 0.559440i
\(49\) 6.84574 0.977963
\(50\) 6.66505 + 2.01775i 0.942581 + 0.285353i
\(51\) 0.739663 0.103573
\(52\) − 0.176452i − 0.0244694i
\(53\) 7.23391i 0.993654i 0.867850 + 0.496827i \(0.165502\pi\)
−0.867850 + 0.496827i \(0.834498\pi\)
\(54\) −1.39276 −0.189530
\(55\) 3.94966 + 5.32238i 0.532572 + 0.717670i
\(56\) −1.12697 −0.150598
\(57\) 3.50091i 0.463707i
\(58\) 1.23089i 0.161624i
\(59\) 4.56679 0.594545 0.297272 − 0.954793i \(-0.403923\pi\)
0.297272 + 0.954793i \(0.403923\pi\)
\(60\) 0.108152 − 0.0802581i 0.0139624 − 0.0103613i
\(61\) −2.23789 −0.286532 −0.143266 − 0.989684i \(-0.545760\pi\)
−0.143266 + 0.989684i \(0.545760\pi\)
\(62\) 1.39276i 0.176880i
\(63\) 0.392756i 0.0494826i
\(64\) −8.22619 −1.02827
\(65\) −5.26060 + 3.90381i −0.652497 + 0.484208i
\(66\) −4.12818 −0.508144
\(67\) 3.28668i 0.401531i 0.979639 + 0.200766i \(0.0643430\pi\)
−0.979639 + 0.200766i \(0.935657\pi\)
\(68\) − 0.0445499i − 0.00540247i
\(69\) 7.61557 0.916807
\(70\) 0.728910 + 0.982247i 0.0871214 + 0.117401i
\(71\) 9.31452 1.10543 0.552715 − 0.833370i \(-0.313592\pi\)
0.552715 + 0.833370i \(0.313592\pi\)
\(72\) 2.86940i 0.338162i
\(73\) 15.6910i 1.83649i 0.396011 + 0.918246i \(0.370394\pi\)
−0.396011 + 0.918246i \(0.629606\pi\)
\(74\) −1.41748 −0.164779
\(75\) −4.78551 − 1.44875i −0.552583 − 0.167287i
\(76\) 0.210860 0.0241873
\(77\) 1.16414i 0.132667i
\(78\) − 4.08026i − 0.461998i
\(79\) 10.7258 1.20675 0.603373 − 0.797459i \(-0.293823\pi\)
0.603373 + 0.797459i \(0.293823\pi\)
\(80\) 5.16475 + 6.95980i 0.577437 + 0.778129i
\(81\) 1.00000 0.111111
\(82\) 0.789599i 0.0871966i
\(83\) − 5.97573i − 0.655922i −0.944691 − 0.327961i \(-0.893639\pi\)
0.944691 − 0.327961i \(-0.106361\pi\)
\(84\) 0.0236557 0.00258105
\(85\) −1.32818 + 0.985620i −0.144061 + 0.106905i
\(86\) −6.94375 −0.748764
\(87\) − 0.883778i − 0.0947510i
\(88\) 8.50500i 0.906636i
\(89\) 2.02861 0.215033 0.107516 − 0.994203i \(-0.465710\pi\)
0.107516 + 0.994203i \(0.465710\pi\)
\(90\) 2.50091 − 1.85588i 0.263619 − 0.195627i
\(91\) −1.15063 −0.120619
\(92\) − 0.458686i − 0.0478214i
\(93\) − 1.00000i − 0.103695i
\(94\) 3.09484 0.319209
\(95\) −4.66505 − 6.28642i −0.478624 − 0.644973i
\(96\) 0.340595 0.0347619
\(97\) 2.14846i 0.218143i 0.994034 + 0.109072i \(0.0347878\pi\)
−0.994034 + 0.109072i \(0.965212\pi\)
\(98\) − 9.53445i − 0.963125i
\(99\) 2.96404 0.297897
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.c.a.94.3 ✓ 10
3.2 odd 2 1395.2.c.f.559.8 10
5.2 odd 4 2325.2.a.w.1.5 5
5.3 odd 4 2325.2.a.x.1.1 5
5.4 even 2 inner 465.2.c.a.94.8 yes 10
15.2 even 4 6975.2.a.bv.1.1 5
15.8 even 4 6975.2.a.bs.1.5 5
15.14 odd 2 1395.2.c.f.559.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.c.a.94.3 ✓ 10 1.1 even 1 trivial
465.2.c.a.94.8 yes 10 5.4 even 2 inner
1395.2.c.f.559.3 10 15.14 odd 2
1395.2.c.f.559.8 10 3.2 odd 2
2325.2.a.w.1.5 5 5.2 odd 4
2325.2.a.x.1.1 5 5.3 odd 4
6975.2.a.bs.1.5 5 15.8 even 4
6975.2.a.bv.1.1 5 15.2 even 4