Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2070,2,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,3,3,0,3,-3,0,-3,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.68740\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.59692 q^{7} -1.00000 q^{8} -1.00000 q^{10} -5.13163 q^{11} -1.22212 q^{13} +4.59692 q^{14} +1.00000 q^{16} +4.68740 q^{17} -4.59692 q^{19} +1.00000 q^{20} +5.13163 q^{22} +1.00000 q^{23} +1.00000 q^{25} +1.22212 q^{26} -4.59692 q^{28} -3.37480 q^{29} -0.777884 q^{31} -1.00000 q^{32} -4.68740 q^{34} -4.59692 q^{35} +5.81903 q^{37} +4.59692 q^{38} -1.00000 q^{40} +8.50643 q^{41} +8.00000 q^{43} -5.13163 q^{44} -1.00000 q^{46} +6.44423 q^{47} +14.1316 q^{49} -1.00000 q^{50} -1.22212 q^{52} +6.00000 q^{53} -5.13163 q^{55} +4.59692 q^{56} +3.37480 q^{58} -9.37480 q^{59} +10.9507 q^{61} +0.777884 q^{62} +1.00000 q^{64} -1.22212 q^{65} +15.6381 q^{67} +4.68740 q^{68} +4.59692 q^{70} -1.31260 q^{71} -4.44423 q^{73} -5.81903 q^{74} -4.59692 q^{76} +23.5897 q^{77} -4.88847 q^{79} +1.00000 q^{80} -8.50643 q^{82} +3.81903 q^{83} +4.68740 q^{85} -8.00000 q^{86} +5.13163 q^{88} -8.93057 q^{89} +5.61797 q^{91} +1.00000 q^{92} -6.44423 q^{94} -4.59692 q^{95} -18.0622 q^{97} -14.1316 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.59692 −1.73747 −0.868735 − 0.495277i \(-0.835067\pi\)
−0.868735 + 0.495277i \(0.835067\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −5.13163 −1.54725 −0.773623 − 0.633647i \(-0.781557\pi\)
−0.773623 + 0.633647i \(0.781557\pi\)
\(12\) 0 0
\(13\) −1.22212 −0.338954 −0.169477 − 0.985534i \(-0.554208\pi\)
−0.169477 + 0.985534i \(0.554208\pi\)
\(14\) 4.59692 1.22858
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.68740 1.13686 0.568431 − 0.822731i \(-0.307551\pi\)
0.568431 + 0.822731i \(0.307551\pi\)
\(18\) 0 0
\(19\) −4.59692 −1.05460 −0.527302 − 0.849678i \(-0.676796\pi\)
−0.527302 + 0.849678i \(0.676796\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 5.13163 1.09407
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.22212 0.239677
\(27\) 0 0
\(28\) −4.59692 −0.868735
\(29\) −3.37480 −0.626684 −0.313342 − 0.949640i \(-0.601449\pi\)
−0.313342 + 0.949640i \(0.601449\pi\)
\(30\) 0 0
\(31\) −0.777884 −0.139712 −0.0698560 − 0.997557i \(-0.522254\pi\)
−0.0698560 + 0.997557i \(0.522254\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.68740 −0.803882
\(35\) −4.59692 −0.777021
\(36\) 0 0
\(37\) 5.81903 0.956643 0.478321 − 0.878185i \(-0.341245\pi\)
0.478321 + 0.878185i \(0.341245\pi\)
\(38\) 4.59692 0.745718
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.50643 1.32848 0.664241 − 0.747519i \(-0.268755\pi\)
0.664241 + 0.747519i \(0.268755\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −5.13163 −0.773623
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.44423 0.939988 0.469994 − 0.882670i \(-0.344256\pi\)
0.469994 + 0.882670i \(0.344256\pi\)
\(48\) 0 0
\(49\) 14.1316 2.01880
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.22212 −0.169477
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −5.13163 −0.691949
\(56\) 4.59692 0.614289
\(57\) 0 0
\(58\) 3.37480 0.443133
\(59\) −9.37480 −1.22049 −0.610247 − 0.792211i \(-0.708930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(60\) 0 0
\(61\) 10.9507 1.40209 0.701044 − 0.713118i \(-0.252717\pi\)
0.701044 + 0.713118i \(0.252717\pi\)
\(62\) 0.777884 0.0987913
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.22212 −0.151585
\(66\) 0 0
\(67\) 15.6381 1.91049 0.955247 − 0.295810i \(-0.0955895\pi\)
0.955247 + 0.295810i \(0.0955895\pi\)
\(68\) 4.68740 0.568431
\(69\) 0 0
\(70\) 4.59692 0.549436
\(71\) −1.31260 −0.155777 −0.0778885 − 0.996962i \(-0.524818\pi\)
−0.0778885 + 0.996962i \(0.524818\pi\)
\(72\) 0 0
\(73\) −4.44423 −0.520158 −0.260079 − 0.965587i \(-0.583749\pi\)
−0.260079 + 0.965587i \(0.583749\pi\)
\(74\) −5.81903 −0.676449
\(75\) 0 0
\(76\) −4.59692 −0.527302
\(77\) 23.5897 2.68829
\(78\) 0 0
\(79\) −4.88847 −0.549995 −0.274998 − 0.961445i \(-0.588677\pi\)
−0.274998 + 0.961445i \(0.588677\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −8.50643 −0.939378
\(83\) 3.81903 0.419193 0.209597 − 0.977788i \(-0.432785\pi\)
0.209597 + 0.977788i \(0.432785\pi\)
\(84\) 0 0
\(85\) 4.68740 0.508420
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 5.13163 0.547034
\(89\) −8.93057 −0.946638 −0.473319 − 0.880891i \(-0.656944\pi\)
−0.473319 + 0.880891i \(0.656944\pi\)
\(90\) 0 0
\(91\) 5.61797 0.588923
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −6.44423 −0.664672
\(95\) −4.59692 −0.471634
\(96\) 0 0
\(97\) −18.0622 −1.83394 −0.916969 − 0.398958i \(-0.869372\pi\)
−0.916969 + 0.398958i \(0.869372\pi\)
\(98\) −14.1316 −1.42751
\(99\) 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 2070.2.a.z.1.1 3
3.2 odd 2 230.2.a.d.1.3 ✓ 3
12.11 even 2 1840.2.a.r.1.1 3
15.2 even 4 1150.2.b.j.599.4 6
15.8 even 4 1150.2.b.j.599.3 6
15.14 odd 2 1150.2.a.q.1.1 3
24.5 odd 2 7360.2.a.bz.1.1 3
24.11 even 2 7360.2.a.ce.1.3 3
60.59 even 2 9200.2.a.cf.1.3 3
69.68 even 2 5290.2.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.3 ✓ 3 3.2 odd 2
1150.2.a.q.1.1 3 15.14 odd 2
1150.2.b.j.599.3 6 15.8 even 4
1150.2.b.j.599.4 6 15.2 even 4
1840.2.a.r.1.1 3 12.11 even 2
2070.2.a.z.1.1 3 1.1 even 1 trivial
5290.2.a.r.1.3 3 69.68 even 2
7360.2.a.bz.1.1 3 24.5 odd 2
7360.2.a.ce.1.3 3 24.11 even 2
9200.2.a.cf.1.3 3 60.59 even 2