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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(445,1332)] 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("1332.445"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [34,0,1,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 889.3
Character \(\chi\) \(=\) 1332.889
Dual form 1332.2.i.c.445.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.63601 + 0.568754i) q^{3} +(-0.132836 - 0.230079i) q^{5} +(1.19699 - 2.07324i) q^{7} +(2.35304 - 1.86097i) q^{9} +(3.00400 - 5.20308i) q^{11} +(0.513191 + 0.888874i) q^{13} +(0.348179 + 0.300859i) q^{15} -4.90997 q^{17} -7.64925 q^{19} +(-0.779115 + 4.07263i) q^{21} +(3.43727 + 5.95353i) q^{23} +(2.46471 - 4.26900i) q^{25} +(-2.79115 + 4.38286i) q^{27} +(-2.58430 + 4.47614i) q^{29} +(-1.42091 - 2.46110i) q^{31} +(-1.95529 + 10.2208i) q^{33} -0.636013 q^{35} -1.00000 q^{37} +(-1.34514 - 1.16232i) q^{39} +(-1.99912 - 3.46258i) q^{41} +(4.77526 - 8.27100i) q^{43} +(-0.740738 - 0.294180i) q^{45} +(-0.799817 + 1.38532i) q^{47} +(0.634439 + 1.09888i) q^{49} +(8.03274 - 2.79256i) q^{51} -11.1446 q^{53} -1.59616 q^{55} +(12.5142 - 4.35054i) q^{57} +(-5.44831 - 9.43676i) q^{59} +(1.50686 - 2.60997i) q^{61} +(-1.04169 - 7.10598i) q^{63} +(0.136341 - 0.236149i) q^{65} +(-4.90402 - 8.49400i) q^{67} +(-9.00949 - 7.78505i) q^{69} -6.65856 q^{71} +0.732804 q^{73} +(-1.60427 + 8.38593i) q^{75} +(-7.19150 - 12.4561i) q^{77} +(6.64553 - 11.5104i) q^{79} +(2.07358 - 8.75787i) q^{81} +(5.04894 - 8.74502i) q^{83} +(0.652221 + 1.12968i) q^{85} +(1.68211 - 8.79283i) q^{87} -9.35589 q^{89} +2.45714 q^{91} +(3.72439 + 3.21822i) q^{93} +(1.01610 + 1.75993i) q^{95} +(-3.92783 + 6.80320i) q^{97} +(-2.61426 - 17.8334i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 34 q + q^{3} - 7 q^{5} - q^{7} + 13 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{15} + 12 q^{17} - 2 q^{19} + 7 q^{21} - 13 q^{23} - 22 q^{25} - 17 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} - 8 q^{35} - 34 q^{37}+ \cdots - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times\).

\(n\) \(667\) \(1037\) \(1297\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −1.63601 + 0.568754i −0.944549 + 0.328370i
\(4\) 0 0
\(5\) −0.132836 − 0.230079i −0.0594061 − 0.102894i 0.834793 − 0.550564i \(-0.185587\pi\)
−0.894199 + 0.447670i \(0.852254\pi\)
\(6\) 0 0
\(7\) 1.19699 − 2.07324i 0.452419 − 0.783613i −0.546117 − 0.837709i \(-0.683895\pi\)
0.998536 + 0.0540965i \(0.0172278\pi\)
\(8\) 0 0
\(9\) 2.35304 − 1.86097i 0.784346 − 0.620324i
\(10\) 0 0
\(11\) 3.00400 − 5.20308i 0.905740 − 1.56879i 0.0858195 − 0.996311i \(-0.472649\pi\)
0.819921 − 0.572477i \(-0.194018\pi\)
\(12\) 0 0
\(13\) 0.513191 + 0.888874i 0.142334 + 0.246529i 0.928375 − 0.371645i \(-0.121206\pi\)
−0.786041 + 0.618174i \(0.787873\pi\)
\(14\) 0 0
\(15\) 0.348179 + 0.300859i 0.0898994 + 0.0776816i
\(16\) 0 0
\(17\) −4.90997 −1.19084 −0.595421 − 0.803414i \(-0.703015\pi\)
−0.595421 + 0.803414i \(0.703015\pi\)
\(18\) 0 0
\(19\) −7.64925 −1.75486 −0.877429 − 0.479706i \(-0.840743\pi\)
−0.877429 + 0.479706i \(0.840743\pi\)
\(20\) 0 0
\(21\) −0.779115 + 4.07263i −0.170017 + 0.888721i
\(22\) 0 0
\(23\) 3.43727 + 5.95353i 0.716721 + 1.24140i 0.962292 + 0.272018i \(0.0876910\pi\)
−0.245572 + 0.969378i \(0.578976\pi\)
\(24\) 0 0
\(25\) 2.46471 − 4.26900i 0.492942 − 0.853800i
\(26\) 0 0
\(27\) −2.79115 + 4.38286i −0.537157 + 0.843482i
\(28\) 0 0
\(29\) −2.58430 + 4.47614i −0.479893 + 0.831198i −0.999734 − 0.0230644i \(-0.992658\pi\)
0.519841 + 0.854263i \(0.325991\pi\)
\(30\) 0 0
\(31\) −1.42091 − 2.46110i −0.255204 − 0.442026i 0.709747 − 0.704457i \(-0.248809\pi\)
−0.964951 + 0.262431i \(0.915476\pi\)
\(32\) 0 0
\(33\) −1.95529 + 10.2208i −0.340373 + 1.77922i
\(34\) 0 0
\(35\) −0.636013 −0.107506
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −1.34514 − 1.16232i −0.215394 − 0.186121i
\(40\) 0 0
\(41\) −1.99912 − 3.46258i −0.312211 − 0.540765i 0.666630 − 0.745389i \(-0.267736\pi\)
−0.978841 + 0.204624i \(0.934403\pi\)
\(42\) 0 0
\(43\) 4.77526 − 8.27100i 0.728221 − 1.26131i −0.229414 − 0.973329i \(-0.573681\pi\)
0.957635 − 0.287986i \(-0.0929857\pi\)
\(44\) 0 0
\(45\) −0.740738 − 0.294180i −0.110423 − 0.0438538i
\(46\) 0 0
\(47\) −0.799817 + 1.38532i −0.116665 + 0.202070i −0.918444 − 0.395550i \(-0.870554\pi\)
0.801779 + 0.597621i \(0.203887\pi\)
\(48\) 0 0
\(49\) 0.634439 + 1.09888i 0.0906342 + 0.156983i
\(50\) 0 0
\(51\) 8.03274 − 2.79256i 1.12481 − 0.391037i
\(52\) 0 0
\(53\) −11.1446 −1.53083 −0.765415 − 0.643537i \(-0.777466\pi\)
−0.765415 + 0.643537i \(0.777466\pi\)
\(54\) 0 0
\(55\) −1.59616 −0.215226
\(56\) 0 0
\(57\) 12.5142 − 4.35054i 1.65755 − 0.576243i
\(58\) 0 0
\(59\) −5.44831 − 9.43676i −0.709310 − 1.22856i −0.965113 − 0.261832i \(-0.915673\pi\)
0.255803 − 0.966729i \(-0.417660\pi\)
\(60\) 0 0
\(61\) 1.50686 − 2.60997i 0.192934 − 0.334172i −0.753287 − 0.657692i \(-0.771533\pi\)
0.946221 + 0.323520i \(0.104866\pi\)
\(62\) 0 0
\(63\) −1.04169 − 7.10598i −0.131240 − 0.895270i
\(64\) 0 0
\(65\) 0.136341 − 0.236149i 0.0169110 − 0.0292907i
\(66\) 0 0
\(67\) −4.90402 − 8.49400i −0.599121 − 1.03771i −0.992951 − 0.118525i \(-0.962184\pi\)
0.393830 − 0.919183i \(-0.371150\pi\)
\(68\) 0 0
\(69\) −9.00949 − 7.78505i −1.08462 − 0.937210i
\(70\) 0 0
\(71\) −6.65856 −0.790226 −0.395113 − 0.918633i \(-0.629295\pi\)
−0.395113 + 0.918633i \(0.629295\pi\)
\(72\) 0 0
\(73\) 0.732804 0.0857683 0.0428841 − 0.999080i \(-0.486345\pi\)
0.0428841 + 0.999080i \(0.486345\pi\)
\(74\) 0 0
\(75\) −1.60427 + 8.38593i −0.185245 + 0.968324i
\(76\) 0 0
\(77\) −7.19150 − 12.4561i −0.819548 − 1.41950i
\(78\) 0 0
\(79\) 6.64553 − 11.5104i 0.747680 − 1.29502i −0.201252 − 0.979540i \(-0.564501\pi\)
0.948932 − 0.315481i \(-0.102166\pi\)
\(80\) 0 0
\(81\) 2.07358 − 8.75787i 0.230397 − 0.973097i
\(82\) 0 0
\(83\) 5.04894 − 8.74502i 0.554193 − 0.959891i −0.443773 − 0.896139i \(-0.646360\pi\)
0.997966 − 0.0637513i \(-0.0203064\pi\)
\(84\) 0 0
\(85\) 0.652221 + 1.12968i 0.0707433 + 0.122531i
\(86\) 0 0
\(87\) 1.68211 − 8.79283i 0.180341 − 0.942690i
\(88\) 0 0
\(89\) −9.35589 −0.991723 −0.495861 − 0.868402i \(-0.665148\pi\)
−0.495861 + 0.868402i \(0.665148\pi\)
\(90\) 0 0
\(91\) 2.45714 0.257578
\(92\) 0 0
\(93\) 3.72439 + 3.21822i 0.386201 + 0.333714i
\(94\) 0 0
\(95\) 1.01610 + 1.75993i 0.104249 + 0.180565i
\(96\) 0 0
\(97\) −3.92783 + 6.80320i −0.398811 + 0.690760i −0.993579 − 0.113137i \(-0.963910\pi\)
0.594769 + 0.803897i \(0.297244\pi\)
\(98\) 0 0
\(99\) −2.61426 − 17.8334i −0.262743 − 1.79232i
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 1332.2.i.c.889.3 yes 34
3.2 odd 2 3996.2.i.d.2665.9 34
9.4 even 3 inner 1332.2.i.c.445.3 ✓ 34
9.5 odd 6 3996.2.i.d.1333.9 34
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1332.2.i.c.445.3 ✓ 34 9.4 even 3 inner
1332.2.i.c.889.3 yes 34 1.1 even 1 trivial
3996.2.i.d.1333.9 34 9.5 odd 6
3996.2.i.d.2665.9 34 3.2 odd 2