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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 445.5
Character \(\chi\) \(=\) 1332.445
Dual form 1332.2.i.c.889.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.16045 + 1.28583i) q^{3} +(-1.56351 + 2.70808i) q^{5} +(-1.04639 - 1.81240i) q^{7} +(-0.306720 - 2.98428i) q^{9} +(2.17264 + 3.76313i) q^{11} +(-2.20670 + 3.82211i) q^{13} +(-1.66776 - 5.15300i) q^{15} +1.61956 q^{17} -5.97187 q^{19} +(3.54472 + 0.757716i) q^{21} +(-0.990360 + 1.71535i) q^{23} +(-2.38914 - 4.13811i) q^{25} +(4.19321 + 3.06871i) q^{27} +(-2.89726 - 5.01820i) q^{29} +(-2.94935 + 5.10842i) q^{31} +(-7.35998 - 1.57326i) q^{33} +6.54417 q^{35} -1.00000 q^{37} +(-2.35383 - 7.27280i) q^{39} +(2.16451 - 3.74904i) q^{41} +(-2.14805 - 3.72053i) q^{43} +(8.56124 + 3.83533i) q^{45} +(-5.18914 - 8.98786i) q^{47} +(1.31014 - 2.26923i) q^{49} +(-1.87942 + 2.08248i) q^{51} +6.87138 q^{53} -13.5878 q^{55} +(6.93005 - 7.67882i) q^{57} +(2.82035 - 4.88499i) q^{59} +(3.79862 + 6.57940i) q^{61} +(-5.08776 + 3.67862i) q^{63} +(-6.90040 - 11.9518i) q^{65} +(4.31888 - 7.48053i) q^{67} +(-1.05639 - 3.26402i) q^{69} +4.08791 q^{71} -0.0702391 q^{73} +(8.09339 + 1.73004i) q^{75} +(4.54686 - 7.87539i) q^{77} +(-3.38025 - 5.85477i) q^{79} +(-8.81185 + 1.83068i) q^{81} +(-4.13970 - 7.17017i) q^{83} +(-2.53221 + 4.38591i) q^{85} +(9.81467 + 2.09798i) q^{87} +4.39297 q^{89} +9.23625 q^{91} +(-3.14600 - 9.72042i) q^{93} +(9.33710 - 16.1723i) q^{95} +(-0.333957 - 0.578430i) q^{97} +(10.5638 - 7.63800i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 34 q + q^{3} - 7 q^{5} - q^{7} + 13 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{15} + 12 q^{17} - 2 q^{19} + 7 q^{21} - 13 q^{23} - 22 q^{25} - 17 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} - 8 q^{35} - 34 q^{37}+ \cdots - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.16045 + 1.28583i −0.669985 + 0.742375i
\(4\) 0 0
\(5\) −1.56351 + 2.70808i −0.699224 + 1.21109i 0.269512 + 0.962997i \(0.413138\pi\)
−0.968736 + 0.248094i \(0.920196\pi\)
\(6\) 0 0
\(7\) −1.04639 − 1.81240i −0.395498 − 0.685023i 0.597667 − 0.801745i \(-0.296095\pi\)
−0.993165 + 0.116722i \(0.962761\pi\)
\(8\) 0 0
\(9\) −0.306720 − 2.98428i −0.102240 − 0.994760i
\(10\) 0 0
\(11\) 2.17264 + 3.76313i 0.655076 + 1.13463i 0.981875 + 0.189531i \(0.0606969\pi\)
−0.326798 + 0.945094i \(0.605970\pi\)
\(12\) 0 0
\(13\) −2.20670 + 3.82211i −0.612028 + 1.06006i 0.378871 + 0.925450i \(0.376313\pi\)
−0.990898 + 0.134613i \(0.957021\pi\)
\(14\) 0 0
\(15\) −1.66776 − 5.15300i −0.430614 − 1.33050i
\(16\) 0 0
\(17\) 1.61956 0.392802 0.196401 − 0.980524i \(-0.437075\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(18\) 0 0
\(19\) −5.97187 −1.37004 −0.685021 − 0.728524i \(-0.740207\pi\)
−0.685021 + 0.728524i \(0.740207\pi\)
\(20\) 0 0
\(21\) 3.54472 + 0.757716i 0.773521 + 0.165347i
\(22\) 0 0
\(23\) −0.990360 + 1.71535i −0.206504 + 0.357676i −0.950611 − 0.310385i \(-0.899542\pi\)
0.744107 + 0.668061i \(0.232875\pi\)
\(24\) 0 0
\(25\) −2.38914 − 4.13811i −0.477828 − 0.827623i
\(26\) 0 0
\(27\) 4.19321 + 3.06871i 0.806984 + 0.590574i
\(28\) 0 0
\(29\) −2.89726 − 5.01820i −0.538007 − 0.931856i −0.999011 − 0.0444580i \(-0.985844\pi\)
0.461004 − 0.887398i \(-0.347489\pi\)
\(30\) 0 0
\(31\) −2.94935 + 5.10842i −0.529718 + 0.917499i 0.469681 + 0.882836i \(0.344369\pi\)
−0.999399 + 0.0346629i \(0.988964\pi\)
\(32\) 0 0
\(33\) −7.35998 − 1.57326i −1.28121 − 0.273870i
\(34\) 0 0
\(35\) 6.54417 1.10617
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −2.35383 − 7.27280i −0.376914 − 1.16458i
\(40\) 0 0
\(41\) 2.16451 − 3.74904i 0.338039 − 0.585501i −0.646025 − 0.763317i \(-0.723570\pi\)
0.984064 + 0.177815i \(0.0569030\pi\)
\(42\) 0 0
\(43\) −2.14805 − 3.72053i −0.327574 − 0.567375i 0.654456 − 0.756100i \(-0.272898\pi\)
−0.982030 + 0.188725i \(0.939564\pi\)
\(44\) 0 0
\(45\) 8.56124 + 3.83533i 1.27623 + 0.571738i
\(46\) 0 0
\(47\) −5.18914 − 8.98786i −0.756914 − 1.31101i −0.944417 − 0.328750i \(-0.893373\pi\)
0.187503 − 0.982264i \(-0.439961\pi\)
\(48\) 0 0
\(49\) 1.31014 − 2.26923i 0.187163 − 0.324175i
\(50\) 0 0
\(51\) −1.87942 + 2.08248i −0.263171 + 0.291606i
\(52\) 0 0
\(53\) 6.87138 0.943857 0.471928 − 0.881637i \(-0.343558\pi\)
0.471928 + 0.881637i \(0.343558\pi\)
\(54\) 0 0
\(55\) −13.5878 −1.83218
\(56\) 0 0
\(57\) 6.93005 − 7.67882i 0.917907 − 1.01708i
\(58\) 0 0
\(59\) 2.82035 − 4.88499i 0.367178 − 0.635971i −0.621945 − 0.783061i \(-0.713657\pi\)
0.989123 + 0.147090i \(0.0469907\pi\)
\(60\) 0 0
\(61\) 3.79862 + 6.57940i 0.486363 + 0.842406i 0.999877 − 0.0156754i \(-0.00498985\pi\)
−0.513514 + 0.858081i \(0.671657\pi\)
\(62\) 0 0
\(63\) −5.08776 + 3.67862i −0.640997 + 0.463462i
\(64\) 0 0
\(65\) −6.90040 − 11.9518i −0.855889 − 1.48244i
\(66\) 0 0
\(67\) 4.31888 − 7.48053i 0.527636 − 0.913892i −0.471845 − 0.881681i \(-0.656412\pi\)
0.999481 − 0.0322106i \(-0.0102547\pi\)
\(68\) 0 0
\(69\) −1.05639 − 3.26402i −0.127175 − 0.392941i
\(70\) 0 0
\(71\) 4.08791 0.485146 0.242573 − 0.970133i \(-0.422009\pi\)
0.242573 + 0.970133i \(0.422009\pi\)
\(72\) 0 0
\(73\) −0.0702391 −0.00822086 −0.00411043 − 0.999992i \(-0.501308\pi\)
−0.00411043 + 0.999992i \(0.501308\pi\)
\(74\) 0 0
\(75\) 8.09339 + 1.73004i 0.934544 + 0.199767i
\(76\) 0 0
\(77\) 4.54686 − 7.87539i 0.518163 − 0.897484i
\(78\) 0 0
\(79\) −3.38025 − 5.85477i −0.380308 − 0.658713i 0.610798 − 0.791786i \(-0.290849\pi\)
−0.991106 + 0.133073i \(0.957515\pi\)
\(80\) 0 0
\(81\) −8.81185 + 1.83068i −0.979094 + 0.203408i
\(82\) 0 0
\(83\) −4.13970 − 7.17017i −0.454391 − 0.787028i 0.544262 − 0.838915i \(-0.316810\pi\)
−0.998653 + 0.0518871i \(0.983476\pi\)
\(84\) 0 0
\(85\) −2.53221 + 4.38591i −0.274656 + 0.475719i
\(86\) 0 0
\(87\) 9.81467 + 2.09798i 1.05224 + 0.224927i
\(88\) 0 0
\(89\) 4.39297 0.465654 0.232827 − 0.972518i \(-0.425202\pi\)
0.232827 + 0.972518i \(0.425202\pi\)
\(90\) 0 0
\(91\) 9.23625 0.968223
\(92\) 0 0
\(93\) −3.14600 − 9.72042i −0.326225 − 1.00796i
\(94\) 0 0
\(95\) 9.33710 − 16.1723i 0.957966 − 1.65925i
\(96\) 0 0
\(97\) −0.333957 − 0.578430i −0.0339082 − 0.0587307i 0.848573 − 0.529078i \(-0.177462\pi\)
−0.882481 + 0.470347i \(0.844129\pi\)
\(98\) 0 0
\(99\) 10.5638 − 7.63800i 1.06170 − 0.767648i
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.445.5 ✓ 34
3.2 odd 2 3996.2.i.d.1333.15 34
9.2 odd 6 3996.2.i.d.2665.15 34
9.7 even 3 inner 1332.2.i.c.889.5 yes 34
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1332.2.i.c.445.5 ✓ 34 1.1 even 1 trivial
1332.2.i.c.889.5 yes 34 9.7 even 3 inner
3996.2.i.d.1333.15 34 3.2 odd 2
3996.2.i.d.2665.15 34 9.2 odd 6