Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3762,2,Mod(1,3762)] 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("3762.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3762, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,3,0,-6,3,0,3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.72545 q^{5} -4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{10} +1.00000 q^{11} -2.67750 q^{13} -4.67750 q^{14} +1.00000 q^{16} -0.201472 q^{17} -1.00000 q^{19} -2.72545 q^{20} +1.00000 q^{22} +1.79853 q^{23} +2.42807 q^{25} -2.67750 q^{26} -4.67750 q^{28} +4.92692 q^{29} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} +4.40294 q^{37} -1.00000 q^{38} -2.72545 q^{40} -4.97487 q^{41} -11.7734 q^{43} +1.00000 q^{44} +1.79853 q^{46} +12.4989 q^{47} +14.8790 q^{49} +2.42807 q^{50} -2.67750 q^{52} +4.10557 q^{53} -2.72545 q^{55} -4.67750 q^{56} +4.92692 q^{58} +5.24943 q^{59} +1.59706 q^{61} -2.57193 q^{62} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +12.7483 q^{70} -12.8214 q^{71} +2.20147 q^{73} +4.40294 q^{74} -1.00000 q^{76} -4.67750 q^{77} +11.8538 q^{79} -2.72545 q^{80} -4.97487 q^{82} +13.6775 q^{83} +0.549103 q^{85} -11.7734 q^{86} +1.00000 q^{88} +5.45090 q^{89} +12.5240 q^{91} +1.79853 q^{92} +12.4989 q^{94} +2.72545 q^{95} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.72545 −1.21886 −0.609429 − 0.792841i \(-0.708601\pi\)
−0.609429 + 0.792841i \(0.708601\pi\)
\(6\) 0 0
\(7\) −4.67750 −1.76793 −0.883964 − 0.467556i \(-0.845135\pi\)
−0.883964 + 0.467556i \(0.845135\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.72545 −0.861863
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.67750 −0.742604 −0.371302 − 0.928512i \(-0.621089\pi\)
−0.371302 + 0.928512i \(0.621089\pi\)
\(14\) −4.67750 −1.25011
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.201472 −0.0488642 −0.0244321 − 0.999701i \(-0.507778\pi\)
−0.0244321 + 0.999701i \(0.507778\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.72545 −0.609429
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.79853 0.375019 0.187509 − 0.982263i \(-0.439958\pi\)
0.187509 + 0.982263i \(0.439958\pi\)
\(24\) 0 0
\(25\) 2.42807 0.485614
\(26\) −2.67750 −0.525100
\(27\) 0 0
\(28\) −4.67750 −0.883964
\(29\) 4.92692 0.914906 0.457453 − 0.889234i \(-0.348762\pi\)
0.457453 + 0.889234i \(0.348762\pi\)
\(30\) 0 0
\(31\) −2.57193 −0.461932 −0.230966 − 0.972962i \(-0.574189\pi\)
−0.230966 + 0.972962i \(0.574189\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.201472 −0.0345522
\(35\) 12.7483 2.15485
\(36\) 0 0
\(37\) 4.40294 0.723840 0.361920 − 0.932209i \(-0.382121\pi\)
0.361920 + 0.932209i \(0.382121\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.72545 −0.430931
\(41\) −4.97487 −0.776945 −0.388472 − 0.921460i \(-0.626997\pi\)
−0.388472 + 0.921460i \(0.626997\pi\)
\(42\) 0 0
\(43\) −11.7734 −1.79543 −0.897713 − 0.440580i \(-0.854773\pi\)
−0.897713 + 0.440580i \(0.854773\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 1.79853 0.265178
\(47\) 12.4989 1.82314 0.911572 − 0.411140i \(-0.134869\pi\)
0.911572 + 0.411140i \(0.134869\pi\)
\(48\) 0 0
\(49\) 14.8790 2.12557
\(50\) 2.42807 0.343381
\(51\) 0 0
\(52\) −2.67750 −0.371302
\(53\) 4.10557 0.563943 0.281971 − 0.959423i \(-0.409012\pi\)
0.281971 + 0.959423i \(0.409012\pi\)
\(54\) 0 0
\(55\) −2.72545 −0.367499
\(56\) −4.67750 −0.625057
\(57\) 0 0
\(58\) 4.92692 0.646936
\(59\) 5.24943 0.683417 0.341708 − 0.939806i \(-0.388994\pi\)
0.341708 + 0.939806i \(0.388994\pi\)
\(60\) 0 0
\(61\) 1.59706 0.204482 0.102241 − 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) −2.57193 −0.326635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.29738 0.905128
\(66\) 0 0
\(67\) 9.08044 1.10935 0.554676 − 0.832066i \(-0.312842\pi\)
0.554676 + 0.832066i \(0.312842\pi\)
\(68\) −0.201472 −0.0244321
\(69\) 0 0
\(70\) 12.7483 1.52371
\(71\) −12.8214 −1.52161 −0.760807 − 0.648978i \(-0.775197\pi\)
−0.760807 + 0.648978i \(0.775197\pi\)
\(72\) 0 0
\(73\) 2.20147 0.257663 0.128831 − 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) 4.40294 0.511832
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −4.67750 −0.533050
\(78\) 0 0
\(79\) 11.8538 1.33366 0.666831 − 0.745209i \(-0.267650\pi\)
0.666831 + 0.745209i \(0.267650\pi\)
\(80\) −2.72545 −0.304714
\(81\) 0 0
\(82\) −4.97487 −0.549383
\(83\) 13.6775 1.50130 0.750650 − 0.660700i \(-0.229740\pi\)
0.750650 + 0.660700i \(0.229740\pi\)
\(84\) 0 0
\(85\) 0.549103 0.0595585
\(86\) −11.7734 −1.26956
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 5.45090 0.577794 0.288897 − 0.957360i \(-0.406711\pi\)
0.288897 + 0.957360i \(0.406711\pi\)
\(90\) 0 0
\(91\) 12.5240 1.31287
\(92\) 1.79853 0.187509
\(93\) 0 0
\(94\) 12.4989 1.28916
\(95\) 2.72545 0.279625
\(96\) 0 0
\(97\) 16.8059 1.70638 0.853190 − 0.521601i \(-0.174665\pi\)
0.853190 + 0.521601i \(0.174665\pi\)
\(98\) 14.8790 1.50300
\(99\) 0 0
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 3762.2.a.bg.1.1 3
3.2 odd 2 418.2.a.g.1.2 ✓ 3
12.11 even 2 3344.2.a.q.1.2 3
33.32 even 2 4598.2.a.bo.1.2 3
57.56 even 2 7942.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 ✓ 3 3.2 odd 2
3344.2.a.q.1.2 3 12.11 even 2
3762.2.a.bg.1.1 3 1.1 even 1 trivial
4598.2.a.bo.1.2 3 33.32 even 2
7942.2.a.bi.1.2 3 57.56 even 2