Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [133,2,Mod(39,133)] 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("133.39"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(133, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: \(1.06201034688\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 58.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 133.58
Dual form 133.2.f.a.39.1

$q$-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 + 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-1.50000 + 2.59808i) q^{5} +4.00000 q^{6} +(-2.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{10} +(-2.00000 - 3.46410i) q^{11} +(-2.00000 + 3.46410i) q^{12} -6.00000 q^{13} +(5.00000 - 1.73205i) q^{14} +6.00000 q^{15} +(2.00000 - 3.46410i) q^{16} +(3.50000 + 6.06218i) q^{17} +(-1.00000 - 1.73205i) q^{18} +(-0.500000 + 0.866025i) q^{19} +6.00000 q^{20} +(-1.00000 + 5.19615i) q^{21} +8.00000 q^{22} +(-1.50000 + 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(6.00000 - 10.3923i) q^{26} -4.00000 q^{27} +(-1.00000 + 5.19615i) q^{28} +(-6.00000 + 10.3923i) q^{30} +(4.00000 + 6.92820i) q^{32} +(-4.00000 + 6.92820i) q^{33} -14.0000 q^{34} +(7.50000 - 2.59808i) q^{35} +2.00000 q^{36} +(1.00000 - 1.73205i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(6.00000 + 10.3923i) q^{39} -4.00000 q^{41} +(-8.00000 - 6.92820i) q^{42} +5.00000 q^{43} +(-4.00000 + 6.92820i) q^{44} +(-1.50000 - 2.59808i) q^{45} +(-3.00000 - 5.19615i) q^{46} +(-2.00000 + 3.46410i) q^{47} -8.00000 q^{48} +(1.00000 + 6.92820i) q^{49} +8.00000 q^{50} +(7.00000 - 12.1244i) q^{51} +(6.00000 + 10.3923i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(4.00000 - 6.92820i) q^{54} +12.0000 q^{55} +2.00000 q^{57} +(-4.00000 - 6.92820i) q^{59} +(-6.00000 - 10.3923i) q^{60} +(1.00000 - 1.73205i) q^{61} +(2.50000 - 0.866025i) q^{63} -8.00000 q^{64} +(9.00000 - 15.5885i) q^{65} +(-8.00000 - 13.8564i) q^{66} +(-4.00000 - 6.92820i) q^{67} +(7.00000 - 12.1244i) q^{68} +6.00000 q^{69} +(-3.00000 + 15.5885i) q^{70} -2.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(2.00000 + 3.46410i) q^{74} +(-4.00000 + 6.92820i) q^{75} +2.00000 q^{76} +(-2.00000 + 10.3923i) q^{77} -24.0000 q^{78} +(-6.00000 + 10.3923i) q^{79} +(6.00000 + 10.3923i) q^{80} +(5.50000 + 9.52628i) q^{81} +(4.00000 - 6.92820i) q^{82} -3.00000 q^{83} +(10.0000 - 3.46410i) q^{84} -21.0000 q^{85} +(-5.00000 + 8.66025i) q^{86} +(4.00000 - 6.92820i) q^{89} +6.00000 q^{90} +(12.0000 + 10.3923i) q^{91} +6.00000 q^{92} +(-4.00000 - 6.92820i) q^{94} +(-1.50000 - 2.59808i) q^{95} +(8.00000 - 13.8564i) q^{96} +4.00000 q^{97} +(-13.0000 - 5.19615i) q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{5} + 8 q^{6} - 4 q^{7} - q^{9} - 6 q^{10} - 4 q^{11} - 4 q^{12} - 12 q^{13} + 10 q^{14} + 12 q^{15} + 4 q^{16} + 7 q^{17} - 2 q^{18} - q^{19} + 12 q^{20} - 2 q^{21}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.73205i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) −1.00000 − 1.73205i −0.577350 − 1.00000i −0.995782 − 0.0917517i \(-0.970753\pi\)
0.418432 − 0.908248i \(-0.362580\pi\)
\(4\) −1.00000 − 1.73205i −0.500000 − 0.866025i
\(5\) −1.50000 + 2.59808i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 4.00000 1.63299
\(7\) −2.00000 − 1.73205i −0.755929 − 0.654654i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −3.00000 − 5.19615i −0.948683 − 1.64317i
\(11\) −2.00000 − 3.46410i −0.603023 − 1.04447i −0.992361 − 0.123371i \(-0.960630\pi\)
0.389338 − 0.921095i \(-0.372704\pi\)
\(12\) −2.00000 + 3.46410i −0.577350 + 1.00000i
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 5.00000 − 1.73205i 1.33631 − 0.462910i
\(15\) 6.00000 1.54919
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) 3.50000 + 6.06218i 0.848875 + 1.47029i 0.882213 + 0.470850i \(0.156053\pi\)
−0.0333386 + 0.999444i \(0.510614\pi\)
\(18\) −1.00000 − 1.73205i −0.235702 − 0.408248i
\(19\) −0.500000 + 0.866025i −0.114708 + 0.198680i
\(20\) 6.00000 1.34164
\(21\) −1.00000 + 5.19615i −0.218218 + 1.13389i
\(22\) 8.00000 1.70561
\(23\) −1.50000 + 2.59808i −0.312772 + 0.541736i −0.978961 − 0.204046i \(-0.934591\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(24\) 0 0
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) 6.00000 − 10.3923i 1.17670 − 2.03810i
\(27\) −4.00000 −0.769800
\(28\) −1.00000 + 5.19615i −0.188982 + 0.981981i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −6.00000 + 10.3923i −1.09545 + 1.89737i
\(31\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 4.00000 + 6.92820i 0.707107 + 1.22474i
\(33\) −4.00000 + 6.92820i −0.696311 + 1.20605i
\(34\) −14.0000 −2.40098
\(35\) 7.50000 − 2.59808i 1.26773 − 0.439155i
\(36\) 2.00000 0.333333
\(37\) 1.00000 − 1.73205i 0.164399 − 0.284747i −0.772043 − 0.635571i \(-0.780765\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(38\) −1.00000 − 1.73205i −0.162221 − 0.280976i
\(39\) 6.00000 + 10.3923i 0.960769 + 1.66410i
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −8.00000 − 6.92820i −1.23443 − 1.06904i
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −4.00000 + 6.92820i −0.603023 + 1.04447i
\(45\) −1.50000 − 2.59808i −0.223607 − 0.387298i
\(46\) −3.00000 − 5.19615i −0.442326 − 0.766131i
\(47\) −2.00000 + 3.46410i −0.291730 + 0.505291i −0.974219 − 0.225605i \(-0.927564\pi\)
0.682489 + 0.730896i \(0.260898\pi\)
\(48\) −8.00000 −1.15470
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 8.00000 1.13137
\(51\) 7.00000 − 12.1244i 0.980196 − 1.69775i
\(52\) 6.00000 + 10.3923i 0.832050 + 1.44115i
\(53\) −3.00000 − 5.19615i −0.412082 − 0.713746i 0.583036 − 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 4.00000 − 6.92820i 0.544331 − 0.942809i
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −4.00000 − 6.92820i −0.520756 − 0.901975i −0.999709 − 0.0241347i \(-0.992317\pi\)
0.478953 − 0.877841i \(-0.341016\pi\)
\(60\) −6.00000 − 10.3923i −0.774597 − 1.34164i
\(61\) 1.00000 − 1.73205i 0.128037 − 0.221766i −0.794879 − 0.606768i \(-0.792466\pi\)
0.922916 + 0.385002i \(0.125799\pi\)
\(62\) 0 0
\(63\) 2.50000 − 0.866025i 0.314970 − 0.109109i
\(64\) −8.00000 −1.00000
\(65\) 9.00000 − 15.5885i 1.11631 − 1.93351i
\(66\) −8.00000 − 13.8564i −0.984732 − 1.70561i
\(67\) −4.00000 − 6.92820i −0.488678 − 0.846415i 0.511237 − 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) 7.00000 − 12.1244i 0.848875 − 1.47029i
\(69\) 6.00000 0.722315
\(70\) −3.00000 + 15.5885i −0.358569 + 1.86318i
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −5.00000 − 8.66025i −0.585206 − 1.01361i −0.994850 − 0.101361i \(-0.967680\pi\)
0.409644 − 0.912245i \(-0.365653\pi\)
\(74\) 2.00000 + 3.46410i 0.232495 + 0.402694i
\(75\) −4.00000 + 6.92820i −0.461880 + 0.800000i
\(76\) 2.00000 0.229416
\(77\) −2.00000 + 10.3923i −0.227921 + 1.18431i
\(78\) −24.0000 −2.71746
\(79\) −6.00000 + 10.3923i −0.675053 + 1.16923i 0.301401 + 0.953498i \(0.402546\pi\)
−0.976453 + 0.215728i \(0.930788\pi\)
\(80\) 6.00000 + 10.3923i 0.670820 + 1.16190i
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 4.00000 − 6.92820i 0.441726 − 0.765092i
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 10.0000 − 3.46410i 1.09109 − 0.377964i
\(85\) −21.0000 −2.27777
\(86\) −5.00000 + 8.66025i −0.539164 + 0.933859i
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000 − 6.92820i 0.423999 − 0.734388i −0.572327 − 0.820025i \(-0.693959\pi\)
0.996326 + 0.0856373i \(0.0272926\pi\)
\(90\) 6.00000 0.632456
\(91\) 12.0000 + 10.3923i 1.25794 + 1.08941i
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −4.00000 − 6.92820i −0.412568 − 0.714590i
\(95\) −1.50000 − 2.59808i −0.153897 − 0.266557i
\(96\) 8.00000 − 13.8564i 0.816497 − 1.41421i
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) −13.0000 − 5.19615i −1.31320 − 0.524891i
\(99\) 4.00000 0.402015
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 133.2.f.a.58.1 yes 2
3.2 odd 2 1197.2.j.c.856.1 2
7.2 even 3 931.2.a.c.1.1 1
7.3 odd 6 931.2.f.a.704.1 2
7.4 even 3 inner 133.2.f.a.39.1 ✓ 2
7.5 odd 6 931.2.a.b.1.1 1
7.6 odd 2 931.2.f.a.324.1 2
21.2 odd 6 8379.2.a.a.1.1 1
21.5 even 6 8379.2.a.d.1.1 1
21.11 odd 6 1197.2.j.c.172.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
133.2.f.a.39.1 ✓ 2 7.4 even 3 inner
133.2.f.a.58.1 yes 2 1.1 even 1 trivial
931.2.a.b.1.1 1 7.5 odd 6
931.2.a.c.1.1 1 7.2 even 3
931.2.f.a.324.1 2 7.6 odd 2
931.2.f.a.704.1 2 7.3 odd 6
1197.2.j.c.172.1 2 21.11 odd 6
1197.2.j.c.856.1 2 3.2 odd 2
8379.2.a.a.1.1 1 21.2 odd 6
8379.2.a.d.1.1 1 21.5 even 6