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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 445.1
Character \(\chi\) \(=\) 1332.445
Dual form 1332.2.i.c.889.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.71232 - 0.260669i) q^{3} +(-1.51829 + 2.62976i) q^{5} +(2.42785 + 4.20516i) q^{7} +(2.86410 + 0.892698i) q^{9} +(-0.152741 - 0.264555i) q^{11} +(-2.86028 + 4.95415i) q^{13} +(3.28530 - 4.10723i) q^{15} +6.26916 q^{17} +7.36568 q^{19} +(-3.06111 - 7.83347i) q^{21} +(-2.49792 + 4.32653i) q^{23} +(-2.11042 - 3.65535i) q^{25} +(-4.67157 - 2.27517i) q^{27} +(-2.87610 - 4.98156i) q^{29} +(-0.0128401 + 0.0222397i) q^{31} +(0.192581 + 0.492818i) q^{33} -14.7447 q^{35} -1.00000 q^{37} +(6.18912 - 7.73753i) q^{39} +(0.533804 - 0.924576i) q^{41} +(2.63131 + 4.55757i) q^{43} +(-6.69612 + 6.17652i) q^{45} +(2.34054 + 4.05393i) q^{47} +(-8.28894 + 14.3569i) q^{49} +(-10.7348 - 1.63417i) q^{51} +1.34189 q^{53} +0.927621 q^{55} +(-12.6124 - 1.92000i) q^{57} +(6.84132 - 11.8495i) q^{59} +(-3.40664 - 5.90048i) q^{61} +(3.19968 + 14.2114i) q^{63} +(-8.68548 - 15.0437i) q^{65} +(-4.61142 + 7.98721i) q^{67} +(5.40504 - 6.75728i) q^{69} -0.254878 q^{71} +9.80884 q^{73} +(2.66088 + 6.80926i) q^{75} +(0.741665 - 1.28460i) q^{77} +(6.08367 + 10.5372i) q^{79} +(7.40618 + 5.11356i) q^{81} +(0.0754133 + 0.130620i) q^{83} +(-9.51842 + 16.4864i) q^{85} +(3.62629 + 9.27975i) q^{87} -10.4908 q^{89} -27.7774 q^{91} +(0.0277836 - 0.0347345i) q^{93} +(-11.1832 + 19.3700i) q^{95} +(-4.34637 - 7.52813i) q^{97} +(-0.201298 - 0.894064i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 34 q + q^{3} - 7 q^{5} - q^{7} + 13 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{15} + 12 q^{17} - 2 q^{19} + 7 q^{21} - 13 q^{23} - 22 q^{25} - 17 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} - 8 q^{35} - 34 q^{37}+ \cdots - 55 q^{99}+O(q^{100}) \) 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
\(3\) −1.71232 − 0.260669i −0.988610 − 0.150497i
\(4\) 0 0
\(5\) −1.51829 + 2.62976i −0.679001 + 1.17606i 0.296282 + 0.955101i \(0.404253\pi\)
−0.975282 + 0.220963i \(0.929080\pi\)
\(6\) 0 0
\(7\) 2.42785 + 4.20516i 0.917642 + 1.58940i 0.802987 + 0.595997i \(0.203243\pi\)
0.114655 + 0.993405i \(0.463424\pi\)
\(8\) 0 0
\(9\) 2.86410 + 0.892698i 0.954701 + 0.297566i
\(10\) 0 0
\(11\) −0.152741 − 0.264555i −0.0460531 − 0.0797663i 0.842080 − 0.539353i \(-0.181331\pi\)
−0.888133 + 0.459586i \(0.847998\pi\)
\(12\) 0 0
\(13\) −2.86028 + 4.95415i −0.793299 + 1.37403i 0.130614 + 0.991433i \(0.458305\pi\)
−0.923913 + 0.382602i \(0.875028\pi\)
\(14\) 0 0
\(15\) 3.28530 − 4.10723i 0.848261 − 1.06048i
\(16\) 0 0
\(17\) 6.26916 1.52050 0.760248 − 0.649633i \(-0.225078\pi\)
0.760248 + 0.649633i \(0.225078\pi\)
\(18\) 0 0
\(19\) 7.36568 1.68980 0.844902 − 0.534922i \(-0.179659\pi\)
0.844902 + 0.534922i \(0.179659\pi\)
\(20\) 0 0
\(21\) −3.06111 − 7.83347i −0.667990 − 1.70940i
\(22\) 0 0
\(23\) −2.49792 + 4.32653i −0.520852 + 0.902143i 0.478854 + 0.877895i \(0.341052\pi\)
−0.999706 + 0.0242481i \(0.992281\pi\)
\(24\) 0 0
\(25\) −2.11042 − 3.65535i −0.422083 − 0.731070i
\(26\) 0 0
\(27\) −4.67157 − 2.27517i −0.899045 − 0.437857i
\(28\) 0 0
\(29\) −2.87610 − 4.98156i −0.534079 − 0.925052i −0.999207 − 0.0398089i \(-0.987325\pi\)
0.465128 − 0.885243i \(-0.346008\pi\)
\(30\) 0 0
\(31\) −0.0128401 + 0.0222397i −0.00230615 + 0.00399436i −0.867176 − 0.498001i \(-0.834067\pi\)
0.864870 + 0.501996i \(0.167401\pi\)
\(32\) 0 0
\(33\) 0.192581 + 0.492818i 0.0335240 + 0.0857887i
\(34\) 0 0
\(35\) −14.7447 −2.49232
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 6.18912 − 7.73753i 0.991052 − 1.23900i
\(40\) 0 0
\(41\) 0.533804 − 0.924576i 0.0833662 − 0.144394i −0.821328 − 0.570457i \(-0.806766\pi\)
0.904694 + 0.426062i \(0.140100\pi\)
\(42\) 0 0
\(43\) 2.63131 + 4.55757i 0.401271 + 0.695022i 0.993880 − 0.110468i \(-0.0352351\pi\)
−0.592608 + 0.805491i \(0.701902\pi\)
\(44\) 0 0
\(45\) −6.69612 + 6.17652i −0.998199 + 0.920742i
\(46\) 0 0
\(47\) 2.34054 + 4.05393i 0.341403 + 0.591327i 0.984693 − 0.174295i \(-0.0557647\pi\)
−0.643291 + 0.765622i \(0.722431\pi\)
\(48\) 0 0
\(49\) −8.28894 + 14.3569i −1.18413 + 2.05098i
\(50\) 0 0
\(51\) −10.7348 − 1.63417i −1.50318 − 0.228830i
\(52\) 0 0
\(53\) 1.34189 0.184322 0.0921612 − 0.995744i \(-0.470623\pi\)
0.0921612 + 0.995744i \(0.470623\pi\)
\(54\) 0 0
\(55\) 0.927621 0.125080
\(56\) 0 0
\(57\) −12.6124 − 1.92000i −1.67056 − 0.254310i
\(58\) 0 0
\(59\) 6.84132 − 11.8495i 0.890665 − 1.54268i 0.0515845 − 0.998669i \(-0.483573\pi\)
0.839080 − 0.544008i \(-0.183094\pi\)
\(60\) 0 0
\(61\) −3.40664 − 5.90048i −0.436176 − 0.755479i 0.561215 − 0.827670i \(-0.310334\pi\)
−0.997391 + 0.0721915i \(0.977001\pi\)
\(62\) 0 0
\(63\) 3.19968 + 14.2114i 0.403122 + 1.79046i
\(64\) 0 0
\(65\) −8.68548 − 15.0437i −1.07730 − 1.86594i
\(66\) 0 0
\(67\) −4.61142 + 7.98721i −0.563374 + 0.975793i 0.433825 + 0.900997i \(0.357164\pi\)
−0.997199 + 0.0747956i \(0.976170\pi\)
\(68\) 0 0
\(69\) 5.40504 − 6.75728i 0.650690 − 0.813481i
\(70\) 0 0
\(71\) −0.254878 −0.0302485 −0.0151242 − 0.999886i \(-0.504814\pi\)
−0.0151242 + 0.999886i \(0.504814\pi\)
\(72\) 0 0
\(73\) 9.80884 1.14804 0.574019 − 0.818842i \(-0.305384\pi\)
0.574019 + 0.818842i \(0.305384\pi\)
\(74\) 0 0
\(75\) 2.66088 + 6.80926i 0.307252 + 0.786266i
\(76\) 0 0
\(77\) 0.741665 − 1.28460i 0.0845205 − 0.146394i
\(78\) 0 0
\(79\) 6.08367 + 10.5372i 0.684467 + 1.18553i 0.973604 + 0.228244i \(0.0732983\pi\)
−0.289137 + 0.957288i \(0.593368\pi\)
\(80\) 0 0
\(81\) 7.40618 + 5.11356i 0.822909 + 0.568173i
\(82\) 0 0
\(83\) 0.0754133 + 0.130620i 0.00827768 + 0.0143374i 0.870135 − 0.492814i \(-0.164032\pi\)
−0.861857 + 0.507152i \(0.830698\pi\)
\(84\) 0 0
\(85\) −9.51842 + 16.4864i −1.03242 + 1.78820i
\(86\) 0 0
\(87\) 3.62629 + 9.27975i 0.388779 + 0.994894i
\(88\) 0 0
\(89\) −10.4908 −1.11203 −0.556013 − 0.831174i \(-0.687669\pi\)
−0.556013 + 0.831174i \(0.687669\pi\)
\(90\) 0 0
\(91\) −27.7774 −2.91186
\(92\) 0 0
\(93\) 0.0277836 − 0.0347345i 0.00288102 − 0.00360180i
\(94\) 0 0
\(95\) −11.1832 + 19.3700i −1.14738 + 1.98732i
\(96\) 0 0
\(97\) −4.34637 − 7.52813i −0.441307 − 0.764365i 0.556480 − 0.830861i \(-0.312152\pi\)
−0.997787 + 0.0664955i \(0.978818\pi\)
\(98\) 0 0
\(99\) −0.201298 − 0.894064i −0.0202312 − 0.0898568i
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.445.1 ✓ 34
3.2 odd 2 3996.2.i.d.1333.14 34
9.2 odd 6 3996.2.i.d.2665.14 34
9.7 even 3 inner 1332.2.i.c.889.1 yes 34
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1332.2.i.c.445.1 ✓ 34 1.1 even 1 trivial
1332.2.i.c.889.1 yes 34 9.7 even 3 inner
3996.2.i.d.1333.14 34 3.2 odd 2
3996.2.i.d.2665.14 34 9.2 odd 6