Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [648,3,Mod(377,648)] 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("648.377"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-24] 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: no
Analytic conductor: \(17.6567211305\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 648.593
Dual form 648.3.m.a.377.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(6.12372 - 3.53553i) q^{5} +(-6.00000 + 10.3923i) q^{7} +(4.89898 + 2.82843i) q^{11} +(4.00000 + 6.92820i) q^{13} +9.89949i q^{17} -16.0000 q^{19} +(-34.2929 + 19.7990i) q^{23} +(12.5000 - 21.6506i) q^{25} +(25.7196 + 14.8492i) q^{29} +(2.00000 + 3.46410i) q^{31} +84.8528i q^{35} +30.0000 q^{37} +(-18.3712 + 10.6066i) q^{41} +(4.00000 - 6.92820i) q^{43} +(14.6969 + 8.48528i) q^{47} +(-47.5000 - 82.2724i) q^{49} +49.4975i q^{53} +40.0000 q^{55} +(68.5857 - 39.5980i) q^{59} +(7.00000 - 12.1244i) q^{61} +(48.9898 + 28.2843i) q^{65} +(44.0000 + 76.2102i) q^{67} -28.2843i q^{71} -80.0000 q^{73} +(-58.7878 + 33.9411i) q^{77} +(-50.0000 + 86.6025i) q^{79} +(112.677 + 65.0538i) q^{83} +(35.0000 + 60.6218i) q^{85} +148.492i q^{89} -96.0000 q^{91} +(-97.9796 + 56.5685i) q^{95} +(56.0000 - 96.9948i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{7} + 16 q^{13} - 64 q^{19} + 50 q^{25} + 8 q^{31} + 120 q^{37} + 16 q^{43} - 190 q^{49} + 160 q^{55} + 28 q^{61} + 176 q^{67} - 320 q^{73} - 200 q^{79} + 140 q^{85} - 384 q^{91} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 6.12372 − 3.53553i 1.22474 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(6\) 0 0
\(7\) −6.00000 + 10.3923i −0.857143 + 1.48461i 0.0174999 + 0.999847i \(0.494429\pi\)
−0.874643 + 0.484768i \(0.838904\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898 + 2.82843i 0.445362 + 0.257130i 0.705869 − 0.708342i \(-0.250557\pi\)
−0.260508 + 0.965472i \(0.583890\pi\)
\(12\) 0 0
\(13\) 4.00000 + 6.92820i 0.307692 + 0.532939i 0.977857 − 0.209274i \(-0.0671099\pi\)
−0.670165 + 0.742212i \(0.733777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.89949i 0.582323i 0.956674 + 0.291162i \(0.0940417\pi\)
−0.956674 + 0.291162i \(0.905958\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −34.2929 + 19.7990i −1.49099 + 0.860826i −0.999947 − 0.0103075i \(-0.996719\pi\)
−0.491047 + 0.871133i \(0.663386\pi\)
\(24\) 0 0
\(25\) 12.5000 − 21.6506i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 25.7196 + 14.8492i 0.886884 + 0.512043i 0.872922 − 0.487860i \(-0.162222\pi\)
0.0139622 + 0.999903i \(0.495556\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.0645161 + 0.111745i 0.896479 − 0.443086i \(-0.146116\pi\)
−0.831963 + 0.554831i \(0.812783\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 84.8528i 2.42437i
\(36\) 0 0
\(37\) 30.0000 0.810811 0.405405 − 0.914137i \(-0.367130\pi\)
0.405405 + 0.914137i \(0.367130\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −18.3712 + 10.6066i −0.448077 + 0.258698i −0.707018 − 0.707196i \(-0.749960\pi\)
0.258940 + 0.965893i \(0.416627\pi\)
\(42\) 0 0
\(43\) 4.00000 − 6.92820i 0.0930233 − 0.161121i −0.815759 − 0.578392i \(-0.803680\pi\)
0.908782 + 0.417271i \(0.137014\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 14.6969 + 8.48528i 0.312701 + 0.180538i 0.648134 − 0.761526i \(-0.275549\pi\)
−0.335434 + 0.942064i \(0.608883\pi\)
\(48\) 0 0
\(49\) −47.5000 − 82.2724i −0.969388 − 1.67903i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 49.4975i 0.933915i 0.884280 + 0.466957i \(0.154650\pi\)
−0.884280 + 0.466957i \(0.845350\pi\)
\(54\) 0 0
\(55\) 40.0000 0.727273
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 68.5857 − 39.5980i 1.16247 − 0.671152i 0.210575 − 0.977578i \(-0.432466\pi\)
0.951895 + 0.306425i \(0.0991330\pi\)
\(60\) 0 0
\(61\) 7.00000 − 12.1244i 0.114754 − 0.198760i −0.802927 − 0.596077i \(-0.796725\pi\)
0.917681 + 0.397317i \(0.130059\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 48.9898 + 28.2843i 0.753689 + 0.435143i
\(66\) 0 0
\(67\) 44.0000 + 76.2102i 0.656716 + 1.13747i 0.981461 + 0.191664i \(0.0613885\pi\)
−0.324744 + 0.945802i \(0.605278\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 28.2843i − 0.398370i −0.979962 − 0.199185i \(-0.936171\pi\)
0.979962 − 0.199185i \(-0.0638295\pi\)
\(72\) 0 0
\(73\) −80.0000 −1.09589 −0.547945 − 0.836514i \(-0.684590\pi\)
−0.547945 + 0.836514i \(0.684590\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −58.7878 + 33.9411i −0.763477 + 0.440794i
\(78\) 0 0
\(79\) −50.0000 + 86.6025i −0.632911 + 1.09623i 0.354042 + 0.935229i \(0.384807\pi\)
−0.986954 + 0.161005i \(0.948526\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 112.677 + 65.0538i 1.35755 + 0.783781i 0.989293 − 0.145944i \(-0.0466218\pi\)
0.368256 + 0.929725i \(0.379955\pi\)
\(84\) 0 0
\(85\) 35.0000 + 60.6218i 0.411765 + 0.713197i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 148.492i 1.66845i 0.551421 + 0.834227i \(0.314086\pi\)
−0.551421 + 0.834227i \(0.685914\pi\)
\(90\) 0 0
\(91\) −96.0000 −1.05495
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −97.9796 + 56.5685i −1.03136 + 0.595458i
\(96\) 0 0
\(97\) 56.0000 − 96.9948i 0.577320 − 0.999947i −0.418466 − 0.908233i \(-0.637432\pi\)
0.995785 − 0.0917143i \(-0.0292346\pi\)
\(98\) 0 0
\(99\) 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 648.3.m.a.593.2 4
3.2 odd 2 inner 648.3.m.a.593.1 4
4.3 odd 2 1296.3.q.k.593.2 4
9.2 odd 6 72.3.e.a.17.1 ✓ 2
9.4 even 3 inner 648.3.m.a.377.1 4
9.5 odd 6 inner 648.3.m.a.377.2 4
9.7 even 3 72.3.e.a.17.2 yes 2
12.11 even 2 1296.3.q.k.593.1 4
36.7 odd 6 144.3.e.a.17.2 2
36.11 even 6 144.3.e.a.17.1 2
36.23 even 6 1296.3.q.k.1025.2 4
36.31 odd 6 1296.3.q.k.1025.1 4
45.2 even 12 1800.3.c.a.449.4 4
45.7 odd 12 1800.3.c.a.449.3 4
45.29 odd 6 1800.3.l.a.1601.2 2
45.34 even 6 1800.3.l.a.1601.1 2
45.38 even 12 1800.3.c.a.449.2 4
45.43 odd 12 1800.3.c.a.449.1 4
63.20 even 6 3528.3.d.a.1961.2 2
63.34 odd 6 3528.3.d.a.1961.1 2
72.11 even 6 576.3.e.a.449.2 2
72.29 odd 6 576.3.e.h.449.2 2
72.43 odd 6 576.3.e.a.449.1 2
72.61 even 6 576.3.e.h.449.1 2
144.11 even 12 2304.3.h.h.2177.4 4
144.29 odd 12 2304.3.h.a.2177.1 4
144.43 odd 12 2304.3.h.h.2177.2 4
144.61 even 12 2304.3.h.a.2177.3 4
144.83 even 12 2304.3.h.h.2177.1 4
144.101 odd 12 2304.3.h.a.2177.4 4
144.115 odd 12 2304.3.h.h.2177.3 4
144.133 even 12 2304.3.h.a.2177.2 4
180.7 even 12 3600.3.c.c.449.2 4
180.43 even 12 3600.3.c.c.449.4 4
180.47 odd 12 3600.3.c.c.449.1 4
180.79 odd 6 3600.3.l.l.1601.2 2
180.83 odd 12 3600.3.c.c.449.3 4
180.119 even 6 3600.3.l.l.1601.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.e.a.17.1 ✓ 2 9.2 odd 6
72.3.e.a.17.2 yes 2 9.7 even 3
144.3.e.a.17.1 2 36.11 even 6
144.3.e.a.17.2 2 36.7 odd 6
576.3.e.a.449.1 2 72.43 odd 6
576.3.e.a.449.2 2 72.11 even 6
576.3.e.h.449.1 2 72.61 even 6
576.3.e.h.449.2 2 72.29 odd 6
648.3.m.a.377.1 4 9.4 even 3 inner
648.3.m.a.377.2 4 9.5 odd 6 inner
648.3.m.a.593.1 4 3.2 odd 2 inner
648.3.m.a.593.2 4 1.1 even 1 trivial
1296.3.q.k.593.1 4 12.11 even 2
1296.3.q.k.593.2 4 4.3 odd 2
1296.3.q.k.1025.1 4 36.31 odd 6
1296.3.q.k.1025.2 4 36.23 even 6
1800.3.c.a.449.1 4 45.43 odd 12
1800.3.c.a.449.2 4 45.38 even 12
1800.3.c.a.449.3 4 45.7 odd 12
1800.3.c.a.449.4 4 45.2 even 12
1800.3.l.a.1601.1 2 45.34 even 6
1800.3.l.a.1601.2 2 45.29 odd 6
2304.3.h.a.2177.1 4 144.29 odd 12
2304.3.h.a.2177.2 4 144.133 even 12
2304.3.h.a.2177.3 4 144.61 even 12
2304.3.h.a.2177.4 4 144.101 odd 12
2304.3.h.h.2177.1 4 144.83 even 12
2304.3.h.h.2177.2 4 144.43 odd 12
2304.3.h.h.2177.3 4 144.115 odd 12
2304.3.h.h.2177.4 4 144.11 even 12
3528.3.d.a.1961.1 2 63.34 odd 6
3528.3.d.a.1961.2 2 63.20 even 6
3600.3.c.c.449.1 4 180.47 odd 12
3600.3.c.c.449.2 4 180.7 even 12
3600.3.c.c.449.3 4 180.83 odd 12
3600.3.c.c.449.4 4 180.43 even 12
3600.3.l.l.1601.1 2 180.119 even 6
3600.3.l.l.1601.2 2 180.79 odd 6