Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,-3,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.523976 q^{3} +1.00000 q^{4} +2.72545 q^{5} +0.523976 q^{6} -4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{9} +2.72545 q^{10} -1.00000 q^{11} +0.523976 q^{12} +2.67750 q^{13} -4.67750 q^{14} +1.42807 q^{15} +1.00000 q^{16} +0.201472 q^{17} -2.72545 q^{18} +2.72545 q^{20} -2.45090 q^{21} -1.00000 q^{22} -1.79853 q^{23} +0.523976 q^{24} +2.42807 q^{25} +2.67750 q^{26} -3.00000 q^{27} -4.67750 q^{28} +4.92692 q^{29} +1.42807 q^{30} +2.57193 q^{31} +1.00000 q^{32} -0.523976 q^{33} +0.201472 q^{34} -12.7483 q^{35} -2.72545 q^{36} -4.40294 q^{37} +1.40294 q^{39} +2.72545 q^{40} -4.97487 q^{41} -2.45090 q^{42} -11.7734 q^{43} -1.00000 q^{44} -7.42807 q^{45} -1.79853 q^{46} -12.4989 q^{47} +0.523976 q^{48} +14.8790 q^{49} +2.42807 q^{50} +0.105567 q^{51} +2.67750 q^{52} +4.10557 q^{53} -3.00000 q^{54} -2.72545 q^{55} -4.67750 q^{56} +4.92692 q^{58} +5.24943 q^{59} +1.42807 q^{60} +1.59706 q^{61} +2.57193 q^{62} +12.7483 q^{63} +1.00000 q^{64} +7.29738 q^{65} -0.523976 q^{66} -9.08044 q^{67} +0.201472 q^{68} -0.942386 q^{69} -12.7483 q^{70} -12.8214 q^{71} -2.72545 q^{72} +2.20147 q^{73} -4.40294 q^{74} +1.27225 q^{75} +4.67750 q^{77} +1.40294 q^{78} -11.8538 q^{79} +2.72545 q^{80} +6.60442 q^{81} -4.97487 q^{82} -13.6775 q^{83} -2.45090 q^{84} +0.549103 q^{85} -11.7734 q^{86} +2.58159 q^{87} -1.00000 q^{88} +5.45090 q^{89} -7.42807 q^{90} -12.5240 q^{91} -1.79853 q^{92} +1.34763 q^{93} -12.4989 q^{94} +0.523976 q^{96} -16.8059 q^{97} +14.8790 q^{98} +2.72545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) 0.523976 0.302518 0.151259 − 0.988494i \(-0.451667\pi\)
0.151259 + 0.988494i \(0.451667\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.72545 1.21886 0.609429 − 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(6\) 0.523976 0.213912
\(7\) −4.67750 −1.76793 −0.883964 − 0.467556i \(-0.845135\pi\)
−0.883964 + 0.467556i \(0.845135\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.72545 −0.908483
\(10\) 2.72545 0.861863
\(11\) −1.00000 −0.301511
\(12\) 0.523976 0.151259
\(13\) 2.67750 0.742604 0.371302 − 0.928512i \(-0.378911\pi\)
0.371302 + 0.928512i \(0.378911\pi\)
\(14\) −4.67750 −1.25011
\(15\) 1.42807 0.368726
\(16\) 1.00000 0.250000
\(17\) 0.201472 0.0488642 0.0244321 − 0.999701i \(-0.492222\pi\)
0.0244321 + 0.999701i \(0.492222\pi\)
\(18\) −2.72545 −0.642394
\(19\) 0 0
\(20\) 2.72545 0.609429
\(21\) −2.45090 −0.534830
\(22\) −1.00000 −0.213201
\(23\) −1.79853 −0.375019 −0.187509 − 0.982263i \(-0.560042\pi\)
−0.187509 + 0.982263i \(0.560042\pi\)
\(24\) 0.523976 0.106956
\(25\) 2.42807 0.485614
\(26\) 2.67750 0.525100
\(27\) −3.00000 −0.577350
\(28\) −4.67750 −0.883964
\(29\) 4.92692 0.914906 0.457453 − 0.889234i \(-0.348762\pi\)
0.457453 + 0.889234i \(0.348762\pi\)
\(30\) 1.42807 0.260729
\(31\) 2.57193 0.461932 0.230966 − 0.972962i \(-0.425811\pi\)
0.230966 + 0.972962i \(0.425811\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.523976 −0.0912126
\(34\) 0.201472 0.0345522
\(35\) −12.7483 −2.15485
\(36\) −2.72545 −0.454241
\(37\) −4.40294 −0.723840 −0.361920 − 0.932209i \(-0.617879\pi\)
−0.361920 + 0.932209i \(0.617879\pi\)
\(38\) 0 0
\(39\) 1.40294 0.224651
\(40\) 2.72545 0.430931
\(41\) −4.97487 −0.776945 −0.388472 − 0.921460i \(-0.626997\pi\)
−0.388472 + 0.921460i \(0.626997\pi\)
\(42\) −2.45090 −0.378182
\(43\) −11.7734 −1.79543 −0.897713 − 0.440580i \(-0.854773\pi\)
−0.897713 + 0.440580i \(0.854773\pi\)
\(44\) −1.00000 −0.150756
\(45\) −7.42807 −1.10731
\(46\) −1.79853 −0.265178
\(47\) −12.4989 −1.82314 −0.911572 − 0.411140i \(-0.865131\pi\)
−0.911572 + 0.411140i \(0.865131\pi\)
\(48\) 0.523976 0.0756295
\(49\) 14.8790 2.12557
\(50\) 2.42807 0.343381
\(51\) 0.105567 0.0147823
\(52\) 2.67750 0.371302
\(53\) 4.10557 0.563943 0.281971 − 0.959423i \(-0.409012\pi\)
0.281971 + 0.959423i \(0.409012\pi\)
\(54\) −3.00000 −0.408248
\(55\) −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 \(0.388994\pi\)
\(60\) 1.42807 0.184363
\(61\) 1.59706 0.204482 0.102241 − 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) 2.57193 0.326635
\(63\) 12.7483 1.60613
\(64\) 1.00000 0.125000
\(65\) 7.29738 0.905128
\(66\) −0.523976 −0.0644970
\(67\) −9.08044 −1.10935 −0.554676 − 0.832066i \(-0.687158\pi\)
−0.554676 + 0.832066i \(0.687158\pi\)
\(68\) 0.201472 0.0244321
\(69\) −0.942386 −0.113450
\(70\) −12.7483 −1.52371
\(71\) −12.8214 −1.52161 −0.760807 − 0.648978i \(-0.775197\pi\)
−0.760807 + 0.648978i \(0.775197\pi\)
\(72\) −2.72545 −0.321197
\(73\) 2.20147 0.257663 0.128831 − 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) −4.40294 −0.511832
\(75\) 1.27225 0.146907
\(76\) 0 0
\(77\) 4.67750 0.533050
\(78\) 1.40294 0.158852
\(79\) −11.8538 −1.33366 −0.666831 − 0.745209i \(-0.732350\pi\)
−0.666831 + 0.745209i \(0.732350\pi\)
\(80\) 2.72545 0.304714
\(81\) 6.60442 0.733824
\(82\) −4.97487 −0.549383
\(83\) −13.6775 −1.50130 −0.750650 − 0.660700i \(-0.770260\pi\)
−0.750650 + 0.660700i \(0.770260\pi\)
\(84\) −2.45090 −0.267415
\(85\) 0.549103 0.0595585
\(86\) −11.7734 −1.26956
\(87\) 2.58159 0.276776
\(88\) −1.00000 −0.106600
\(89\) 5.45090 0.577794 0.288897 − 0.957360i \(-0.406711\pi\)
0.288897 + 0.957360i \(0.406711\pi\)
\(90\) −7.42807 −0.782987
\(91\) −12.5240 −1.31287
\(92\) −1.79853 −0.187509
\(93\) 1.34763 0.139743
\(94\) −12.4989 −1.28916
\(95\) 0 0
\(96\) 0.523976 0.0534781
\(97\) −16.8059 −1.70638 −0.853190 − 0.521601i \(-0.825335\pi\)
−0.853190 + 0.521601i \(0.825335\pi\)
\(98\) 14.8790 1.50300
\(99\) 2.72545 0.273918
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 7942.2.a.bi.1.2 3
19.18 odd 2 418.2.a.g.1.2 ✓ 3
57.56 even 2 3762.2.a.bg.1.1 3
76.75 even 2 3344.2.a.q.1.2 3
209.208 even 2 4598.2.a.bo.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 ✓ 3 19.18 odd 2
3344.2.a.q.1.2 3 76.75 even 2
3762.2.a.bg.1.1 3 57.56 even 2
4598.2.a.bo.1.2 3 209.208 even 2
7942.2.a.bi.1.2 3 1.1 even 1 trivial