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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [192,13,Mod(65,192)] 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("192.65"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1350,0,0,0,-80500,0,759618] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(175.486812917\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.2
Root \(5.09902i\) of defining polynomial
Character \(\chi\) \(=\) 192.65
Dual form 192.13.e.c.65.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-675.000 + 275.347i) q^{3} +21109.9i q^{5} -40250.0 q^{7} +(379809. - 371719. i) q^{9} +1.16105e6i q^{11} -1.28405e6 q^{13} +(-5.81256e6 - 1.42492e7i) q^{15} -1.48445e7i q^{17} +5.33436e7 q^{19} +(2.71688e7 - 1.10827e7i) q^{21} +1.07466e8i q^{23} -2.01489e8 q^{25} +(-1.54019e8 + 3.55489e8i) q^{27} +1.20239e8i q^{29} -6.65262e7 q^{31} +(-3.19691e8 - 7.83707e8i) q^{33} -8.49675e8i q^{35} -2.22873e9 q^{37} +(8.66734e8 - 3.53559e8i) q^{39} +8.21168e9i q^{41} +8.97722e9 q^{43} +(7.84696e9 + 8.01775e9i) q^{45} -1.07692e9i q^{47} -1.22212e10 q^{49} +(4.08739e9 + 1.00200e10i) q^{51} +4.11442e10i q^{53} -2.45096e10 q^{55} +(-3.60069e10 + 1.46880e10i) q^{57} +4.61074e10i q^{59} +4.06799e10 q^{61} +(-1.52873e10 + 1.49617e10i) q^{63} -2.71062e10i q^{65} +1.21177e11 q^{67} +(-2.95906e10 - 7.25399e10i) q^{69} +4.48565e10i q^{71} -6.09562e10 q^{73} +(1.36005e11 - 5.54794e10i) q^{75} -4.67321e10i q^{77} +2.52325e11 q^{79} +(6.08022e9 - 2.82364e11i) q^{81} +4.10810e11i q^{83} +3.13367e11 q^{85} +(-3.31076e10 - 8.11616e10i) q^{87} -1.12519e11i q^{89} +5.16830e10 q^{91} +(4.49052e10 - 1.83178e10i) q^{93} +1.12608e12i q^{95} +6.53818e11 q^{97} +(4.31583e11 + 4.40976e11i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1350 q^{3} - 80500 q^{7} + 759618 q^{9} - 2568100 q^{13} - 11625120 q^{15} + 106687156 q^{19} + 54337500 q^{21} - 402977950 q^{25} - 308038950 q^{27} - 133052404 q^{31} - 639381600 q^{33} - 4457452900 q^{37}+ \cdots + 863165160000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −675.000 + 275.347i −0.925926 + 0.377705i
\(4\) 0 0
\(5\) 21109.9i 1.35104i 0.737343 + 0.675518i \(0.236080\pi\)
−0.737343 + 0.675518i \(0.763920\pi\)
\(6\) 0 0
\(7\) −40250.0 −0.342119 −0.171060 − 0.985261i \(-0.554719\pi\)
−0.171060 + 0.985261i \(0.554719\pi\)
\(8\) 0 0
\(9\) 379809. − 371719.i 0.714678 − 0.699454i
\(10\) 0 0
\(11\) 1.16105e6i 0.655381i 0.944785 + 0.327690i \(0.106270\pi\)
−0.944785 + 0.327690i \(0.893730\pi\)
\(12\) 0 0
\(13\) −1.28405e6 −0.266025 −0.133012 − 0.991114i \(-0.542465\pi\)
−0.133012 + 0.991114i \(0.542465\pi\)
\(14\) 0 0
\(15\) −5.81256e6 − 1.42492e7i −0.510293 − 1.25096i
\(16\) 0 0
\(17\) − 1.48445e7i − 0.614996i −0.951549 − 0.307498i \(-0.900508\pi\)
0.951549 − 0.307498i \(-0.0994918\pi\)
\(18\) 0 0
\(19\) 5.33436e7 1.13386 0.566931 − 0.823765i \(-0.308130\pi\)
0.566931 + 0.823765i \(0.308130\pi\)
\(20\) 0 0
\(21\) 2.71688e7 − 1.10827e7i 0.316777 − 0.129220i
\(22\) 0 0
\(23\) 1.07466e8i 0.725949i 0.931799 + 0.362974i \(0.118239\pi\)
−0.931799 + 0.362974i \(0.881761\pi\)
\(24\) 0 0
\(25\) −2.01489e8 −0.825299
\(26\) 0 0
\(27\) −1.54019e8 + 3.55489e8i −0.397551 + 0.917580i
\(28\) 0 0
\(29\) 1.20239e8i 0.202143i 0.994879 + 0.101072i \(0.0322271\pi\)
−0.994879 + 0.101072i \(0.967773\pi\)
\(30\) 0 0
\(31\) −6.65262e7 −0.0749588 −0.0374794 − 0.999297i \(-0.511933\pi\)
−0.0374794 + 0.999297i \(0.511933\pi\)
\(32\) 0 0
\(33\) −3.19691e8 − 7.83707e8i −0.247541 − 0.606834i
\(34\) 0 0
\(35\) − 8.49675e8i − 0.462216i
\(36\) 0 0
\(37\) −2.22873e9 −0.868653 −0.434327 − 0.900755i \(-0.643014\pi\)
−0.434327 + 0.900755i \(0.643014\pi\)
\(38\) 0 0
\(39\) 8.66734e8 − 3.53559e8i 0.246319 − 0.100479i
\(40\) 0 0
\(41\) 8.21168e9i 1.72874i 0.502858 + 0.864369i \(0.332282\pi\)
−0.502858 + 0.864369i \(0.667718\pi\)
\(42\) 0 0
\(43\) 8.97722e9 1.42014 0.710070 − 0.704131i \(-0.248663\pi\)
0.710070 + 0.704131i \(0.248663\pi\)
\(44\) 0 0
\(45\) 7.84696e9 + 8.01775e9i 0.944988 + 0.965555i
\(46\) 0 0
\(47\) − 1.07692e9i − 0.0999075i −0.998752 − 0.0499538i \(-0.984093\pi\)
0.998752 − 0.0499538i \(-0.0159074\pi\)
\(48\) 0 0
\(49\) −1.22212e10 −0.882954
\(50\) 0 0
\(51\) 4.08739e9 + 1.00200e10i 0.232287 + 0.569441i
\(52\) 0 0
\(53\) 4.11442e10i 1.85632i 0.372181 + 0.928160i \(0.378610\pi\)
−0.372181 + 0.928160i \(0.621390\pi\)
\(54\) 0 0
\(55\) −2.45096e10 −0.885443
\(56\) 0 0
\(57\) −3.60069e10 + 1.46880e10i −1.04987 + 0.428266i
\(58\) 0 0
\(59\) 4.61074e10i 1.09310i 0.837428 + 0.546548i \(0.184058\pi\)
−0.837428 + 0.546548i \(0.815942\pi\)
\(60\) 0 0
\(61\) 4.06799e10 0.789589 0.394795 − 0.918769i \(-0.370816\pi\)
0.394795 + 0.918769i \(0.370816\pi\)
\(62\) 0 0
\(63\) −1.52873e10 + 1.49617e10i −0.244505 + 0.239297i
\(64\) 0 0
\(65\) − 2.71062e10i − 0.359409i
\(66\) 0 0
\(67\) 1.21177e11 1.33959 0.669793 − 0.742548i \(-0.266383\pi\)
0.669793 + 0.742548i \(0.266383\pi\)
\(68\) 0 0
\(69\) −2.95906e10 − 7.25399e10i −0.274195 − 0.672175i
\(70\) 0 0
\(71\) 4.48565e10i 0.350167i 0.984554 + 0.175083i \(0.0560195\pi\)
−0.984554 + 0.175083i \(0.943980\pi\)
\(72\) 0 0
\(73\) −6.09562e10 −0.402792 −0.201396 − 0.979510i \(-0.564548\pi\)
−0.201396 + 0.979510i \(0.564548\pi\)
\(74\) 0 0
\(75\) 1.36005e11 − 5.54794e10i 0.764166 − 0.311720i
\(76\) 0 0
\(77\) − 4.67321e10i − 0.224218i
\(78\) 0 0
\(79\) 2.52325e11 1.03800 0.519000 − 0.854774i \(-0.326304\pi\)
0.519000 + 0.854774i \(0.326304\pi\)
\(80\) 0 0
\(81\) 6.08022e9 − 2.82364e11i 0.0215283 − 0.999768i
\(82\) 0 0
\(83\) 4.10810e11i 1.25653i 0.778000 + 0.628264i \(0.216234\pi\)
−0.778000 + 0.628264i \(0.783766\pi\)
\(84\) 0 0
\(85\) 3.13367e11 0.830882
\(86\) 0 0
\(87\) −3.31076e10 − 8.11616e10i −0.0763505 − 0.187170i
\(88\) 0 0
\(89\) − 1.12519e11i − 0.226404i −0.993572 − 0.113202i \(-0.963889\pi\)
0.993572 − 0.113202i \(-0.0361108\pi\)
\(90\) 0 0
\(91\) 5.16830e10 0.0910122
\(92\) 0 0
\(93\) 4.49052e10 − 1.83178e10i 0.0694063 − 0.0283123i
\(94\) 0 0
\(95\) 1.12608e12i 1.53189i
\(96\) 0 0
\(97\) 6.53818e11 0.784922 0.392461 − 0.919769i \(-0.371624\pi\)
0.392461 + 0.919769i \(0.371624\pi\)
\(98\) 0 0
\(99\) 4.31583e11 + 4.40976e11i 0.458409 + 0.468386i
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 192.13.e.c.65.2 2
3.2 odd 2 inner 192.13.e.c.65.1 2
4.3 odd 2 192.13.e.d.65.1 2
8.3 odd 2 3.13.b.b.2.1 ✓ 2
8.5 even 2 48.13.e.b.17.1 2
12.11 even 2 192.13.e.d.65.2 2
24.5 odd 2 48.13.e.b.17.2 2
24.11 even 2 3.13.b.b.2.2 yes 2
40.3 even 4 75.13.d.b.74.1 4
40.19 odd 2 75.13.c.c.26.2 2
40.27 even 4 75.13.d.b.74.4 4
72.11 even 6 81.13.d.c.53.2 4
72.43 odd 6 81.13.d.c.53.1 4
72.59 even 6 81.13.d.c.26.1 4
72.67 odd 6 81.13.d.c.26.2 4
120.59 even 2 75.13.c.c.26.1 2
120.83 odd 4 75.13.d.b.74.3 4
120.107 odd 4 75.13.d.b.74.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.13.b.b.2.1 ✓ 2 8.3 odd 2
3.13.b.b.2.2 yes 2 24.11 even 2
48.13.e.b.17.1 2 8.5 even 2
48.13.e.b.17.2 2 24.5 odd 2
75.13.c.c.26.1 2 120.59 even 2
75.13.c.c.26.2 2 40.19 odd 2
75.13.d.b.74.1 4 40.3 even 4
75.13.d.b.74.2 4 120.107 odd 4
75.13.d.b.74.3 4 120.83 odd 4
75.13.d.b.74.4 4 40.27 even 4
81.13.d.c.26.1 4 72.59 even 6
81.13.d.c.26.2 4 72.67 odd 6
81.13.d.c.53.1 4 72.43 odd 6
81.13.d.c.53.2 4 72.11 even 6
192.13.e.c.65.1 2 3.2 odd 2 inner
192.13.e.c.65.2 2 1.1 even 1 trivial
192.13.e.d.65.1 2 4.3 odd 2
192.13.e.d.65.2 2 12.11 even 2