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 1.2
Root \(-2.14510\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.60147 q^{5} +2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{10} +1.00000 q^{11} +4.89167 q^{13} +2.89167 q^{14} +1.00000 q^{16} +5.74657 q^{17} -1.00000 q^{19} +1.60147 q^{20} +1.00000 q^{22} +7.74657 q^{23} -2.43531 q^{25} +4.89167 q^{26} +2.89167 q^{28} -5.34803 q^{29} -7.43531 q^{31} +1.00000 q^{32} +5.74657 q^{34} +4.63091 q^{35} -7.49314 q^{37} -1.00000 q^{38} +1.60147 q^{40} +2.05783 q^{41} -10.6887 q^{43} +1.00000 q^{44} +7.74657 q^{46} +7.08727 q^{47} +1.36176 q^{49} -2.43531 q^{50} +4.89167 q^{52} -8.32698 q^{53} +1.60147 q^{55} +2.89167 q^{56} -5.34803 q^{58} +2.54364 q^{59} +13.4931 q^{61} -7.43531 q^{62} +1.00000 q^{64} +7.83384 q^{65} -10.3848 q^{67} +5.74657 q^{68} +4.63091 q^{70} -14.9789 q^{71} -3.74657 q^{73} -7.49314 q^{74} -1.00000 q^{76} +2.89167 q^{77} -8.69607 q^{79} +1.60147 q^{80} +2.05783 q^{82} +6.10833 q^{83} +9.20293 q^{85} -10.6887 q^{86} +1.00000 q^{88} -3.20293 q^{89} +14.1451 q^{91} +7.74657 q^{92} +7.08727 q^{94} -1.60147 q^{95} -6.98627 q^{97} +1.36176 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) \) 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
\(5\) 1.60147 0.716197 0.358099 − 0.933684i \(-0.383425\pi\)
0.358099 + 0.933684i \(0.383425\pi\)
\(6\) 0 0
\(7\) 2.89167 1.09295 0.546474 − 0.837476i \(-0.315970\pi\)
0.546474 + 0.837476i \(0.315970\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.60147 0.506428
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.89167 1.35671 0.678353 − 0.734736i \(-0.262694\pi\)
0.678353 + 0.734736i \(0.262694\pi\)
\(14\) 2.89167 0.772832
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.74657 1.39375 0.696874 − 0.717194i \(-0.254574\pi\)
0.696874 + 0.717194i \(0.254574\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.60147 0.358099
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.74657 1.61527 0.807636 − 0.589682i \(-0.200747\pi\)
0.807636 + 0.589682i \(0.200747\pi\)
\(24\) 0 0
\(25\) −2.43531 −0.487062
\(26\) 4.89167 0.959336
\(27\) 0 0
\(28\) 2.89167 0.546474
\(29\) −5.34803 −0.993105 −0.496552 − 0.868007i \(-0.665401\pi\)
−0.496552 + 0.868007i \(0.665401\pi\)
\(30\) 0 0
\(31\) −7.43531 −1.33542 −0.667710 − 0.744421i \(-0.732726\pi\)
−0.667710 + 0.744421i \(0.732726\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.74657 0.985528
\(35\) 4.63091 0.782767
\(36\) 0 0
\(37\) −7.49314 −1.23186 −0.615932 − 0.787799i \(-0.711220\pi\)
−0.615932 + 0.787799i \(0.711220\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.60147 0.253214
\(41\) 2.05783 0.321379 0.160689 − 0.987005i \(-0.448628\pi\)
0.160689 + 0.987005i \(0.448628\pi\)
\(42\) 0 0
\(43\) −10.6887 −1.63002 −0.815009 − 0.579449i \(-0.803268\pi\)
−0.815009 + 0.579449i \(0.803268\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 7.74657 1.14217
\(47\) 7.08727 1.03379 0.516893 − 0.856050i \(-0.327089\pi\)
0.516893 + 0.856050i \(0.327089\pi\)
\(48\) 0 0
\(49\) 1.36176 0.194537
\(50\) −2.43531 −0.344405
\(51\) 0 0
\(52\) 4.89167 0.678353
\(53\) −8.32698 −1.14380 −0.571899 − 0.820324i \(-0.693793\pi\)
−0.571899 + 0.820324i \(0.693793\pi\)
\(54\) 0 0
\(55\) 1.60147 0.215942
\(56\) 2.89167 0.386416
\(57\) 0 0
\(58\) −5.34803 −0.702231
\(59\) 2.54364 0.331153 0.165577 − 0.986197i \(-0.447051\pi\)
0.165577 + 0.986197i \(0.447051\pi\)
\(60\) 0 0
\(61\) 13.4931 1.72762 0.863810 − 0.503818i \(-0.168072\pi\)
0.863810 + 0.503818i \(0.168072\pi\)
\(62\) −7.43531 −0.944285
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.83384 0.971669
\(66\) 0 0
\(67\) −10.3848 −1.26871 −0.634353 − 0.773043i \(-0.718733\pi\)
−0.634353 + 0.773043i \(0.718733\pi\)
\(68\) 5.74657 0.696874
\(69\) 0 0
\(70\) 4.63091 0.553500
\(71\) −14.9789 −1.77767 −0.888837 − 0.458224i \(-0.848486\pi\)
−0.888837 + 0.458224i \(0.848486\pi\)
\(72\) 0 0
\(73\) −3.74657 −0.438503 −0.219251 − 0.975668i \(-0.570361\pi\)
−0.219251 + 0.975668i \(0.570361\pi\)
\(74\) −7.49314 −0.871059
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 2.89167 0.329536
\(78\) 0 0
\(79\) −8.69607 −0.978384 −0.489192 − 0.872176i \(-0.662708\pi\)
−0.489192 + 0.872176i \(0.662708\pi\)
\(80\) 1.60147 0.179049
\(81\) 0 0
\(82\) 2.05783 0.227249
\(83\) 6.10833 0.670476 0.335238 − 0.942133i \(-0.391183\pi\)
0.335238 + 0.942133i \(0.391183\pi\)
\(84\) 0 0
\(85\) 9.20293 0.998198
\(86\) −10.6887 −1.15260
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −3.20293 −0.339510 −0.169755 − 0.985486i \(-0.554298\pi\)
−0.169755 + 0.985486i \(0.554298\pi\)
\(90\) 0 0
\(91\) 14.1451 1.48281
\(92\) 7.74657 0.807636
\(93\) 0 0
\(94\) 7.08727 0.730997
\(95\) −1.60147 −0.164307
\(96\) 0 0
\(97\) −6.98627 −0.709349 −0.354674 − 0.934990i \(-0.615408\pi\)
−0.354674 + 0.934990i \(0.615408\pi\)
\(98\) 1.36176 0.137559
\(99\) 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.1.2 3
3.2 odd 2 418.2.a.g.1.1 ✓ 3
12.11 even 2 3344.2.a.q.1.3 3
33.32 even 2 4598.2.a.bo.1.1 3
57.56 even 2 7942.2.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.1 ✓ 3 3.2 odd 2
3344.2.a.q.1.3 3 12.11 even 2
3762.2.a.bg.1.2 3 1.1 even 1 trivial
4598.2.a.bo.1.1 3 33.32 even 2
7942.2.a.bi.1.3 3 57.56 even 2