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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(121,1332)] 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("1332.121"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [74] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(74\)
Relative dimension: \(37\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.3
Character \(\chi\) \(=\) 1332.121
Dual form 1332.2.l.b.1321.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.65481 - 0.511467i) q^{3} -3.76461 q^{5} +(1.95890 - 3.39292i) q^{7} +(2.47680 + 1.69276i) q^{9} +(2.75612 + 4.77374i) q^{11} -6.83676 q^{13} +(6.22973 + 1.92548i) q^{15} +(-2.88006 + 4.98841i) q^{17} +(1.05568 + 1.82850i) q^{19} +(-4.97699 + 4.61273i) q^{21} +(3.63205 - 6.29090i) q^{23} +9.17231 q^{25} +(-3.23285 - 4.06801i) q^{27} +(0.0749313 + 0.129785i) q^{29} +(-0.655989 + 1.13621i) q^{31} +(-2.11925 - 9.30931i) q^{33} +(-7.37452 + 12.7730i) q^{35} +(6.04251 - 0.698648i) q^{37} +(11.3135 + 3.49678i) q^{39} -0.518626 q^{41} +(-3.47525 - 6.01931i) q^{43} +(-9.32420 - 6.37260i) q^{45} +(0.557495 + 0.965609i) q^{47} +(-4.17461 - 7.23064i) q^{49} +(7.31737 - 6.78183i) q^{51} +(1.90355 - 3.29705i) q^{53} +(-10.3757 - 17.9713i) q^{55} +(-0.811741 - 3.56577i) q^{57} +(1.95924 + 3.39350i) q^{59} +(3.73588 - 6.47073i) q^{61} +(10.5952 - 5.08764i) q^{63} +25.7378 q^{65} -2.49399 q^{67} +(-9.22796 + 8.55258i) q^{69} +(4.44336 + 7.69612i) q^{71} +10.3273 q^{73} +(-15.1784 - 4.69134i) q^{75} +21.5959 q^{77} +(3.14988 - 5.45576i) q^{79} +(3.26910 + 8.38528i) q^{81} +10.2050 q^{83} +(10.8423 - 18.7794i) q^{85} +(-0.0576165 - 0.253094i) q^{87} +(7.59908 - 13.1620i) q^{89} +(-13.3926 + 23.1966i) q^{91} +(1.66667 - 1.54469i) q^{93} +(-3.97424 - 6.88358i) q^{95} +(0.214638 + 0.371764i) q^{97} +(-1.25445 + 16.4891i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 74 q + 5 q^{3} + 6 q^{5} - q^{7} - q^{9} + q^{11} - 4 q^{13} - 2 q^{15} + 3 q^{17} + q^{19} - 3 q^{21} - 17 q^{23} + 96 q^{25} + 17 q^{27} + 6 q^{29} + 3 q^{31} - 16 q^{33} - 6 q^{35} + 5 q^{37} + 35 q^{39}+ \cdots - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.65481 − 0.511467i −0.955406 − 0.295296i
\(4\) 0 0
\(5\) −3.76461 −1.68359 −0.841793 − 0.539800i \(-0.818500\pi\)
−0.841793 + 0.539800i \(0.818500\pi\)
\(6\) 0 0
\(7\) 1.95890 − 3.39292i 0.740396 − 1.28240i −0.211919 − 0.977287i \(-0.567971\pi\)
0.952315 − 0.305117i \(-0.0986955\pi\)
\(8\) 0 0
\(9\) 2.47680 + 1.69276i 0.825601 + 0.564254i
\(10\) 0 0
\(11\) 2.75612 + 4.77374i 0.831002 + 1.43934i 0.897244 + 0.441534i \(0.145566\pi\)
−0.0662424 + 0.997804i \(0.521101\pi\)
\(12\) 0 0
\(13\) −6.83676 −1.89618 −0.948088 − 0.318009i \(-0.896986\pi\)
−0.948088 + 0.318009i \(0.896986\pi\)
\(14\) 0 0
\(15\) 6.22973 + 1.92548i 1.60851 + 0.497156i
\(16\) 0 0
\(17\) −2.88006 + 4.98841i −0.698517 + 1.20987i 0.270463 + 0.962730i \(0.412823\pi\)
−0.968980 + 0.247137i \(0.920510\pi\)
\(18\) 0 0
\(19\) 1.05568 + 1.82850i 0.242190 + 0.419486i 0.961338 − 0.275371i \(-0.0888008\pi\)
−0.719148 + 0.694857i \(0.755467\pi\)
\(20\) 0 0
\(21\) −4.97699 + 4.61273i −1.08607 + 1.00658i
\(22\) 0 0
\(23\) 3.63205 − 6.29090i 0.757336 − 1.31174i −0.186869 − 0.982385i \(-0.559834\pi\)
0.944205 − 0.329359i \(-0.106833\pi\)
\(24\) 0 0
\(25\) 9.17231 1.83446
\(26\) 0 0
\(27\) −3.23285 − 4.06801i −0.622162 − 0.782888i
\(28\) 0 0
\(29\) 0.0749313 + 0.129785i 0.0139144 + 0.0241004i 0.872899 − 0.487901i \(-0.162237\pi\)
−0.858984 + 0.512002i \(0.828904\pi\)
\(30\) 0 0
\(31\) −0.655989 + 1.13621i −0.117819 + 0.204069i −0.918903 − 0.394483i \(-0.870924\pi\)
0.801084 + 0.598552i \(0.204257\pi\)
\(32\) 0 0
\(33\) −2.11925 − 9.30931i −0.368914 − 1.62054i
\(34\) 0 0
\(35\) −7.37452 + 12.7730i −1.24652 + 2.15904i
\(36\) 0 0
\(37\) 6.04251 − 0.698648i 0.993382 − 0.114857i
\(38\) 0 0
\(39\) 11.3135 + 3.49678i 1.81162 + 0.559932i
\(40\) 0 0
\(41\) −0.518626 −0.0809957 −0.0404979 − 0.999180i \(-0.512894\pi\)
−0.0404979 + 0.999180i \(0.512894\pi\)
\(42\) 0 0
\(43\) −3.47525 − 6.01931i −0.529970 − 0.917936i −0.999389 − 0.0349597i \(-0.988870\pi\)
0.469418 − 0.882976i \(-0.344464\pi\)
\(44\) 0 0
\(45\) −9.32420 − 6.37260i −1.38997 − 0.949971i
\(46\) 0 0
\(47\) 0.557495 + 0.965609i 0.0813190 + 0.140849i 0.903817 − 0.427920i \(-0.140753\pi\)
−0.822498 + 0.568768i \(0.807420\pi\)
\(48\) 0 0
\(49\) −4.17461 − 7.23064i −0.596373 − 1.03295i
\(50\) 0 0
\(51\) 7.31737 − 6.78183i 1.02464 − 0.949646i
\(52\) 0 0
\(53\) 1.90355 − 3.29705i 0.261473 − 0.452885i −0.705160 − 0.709048i \(-0.749125\pi\)
0.966634 + 0.256163i \(0.0824584\pi\)
\(54\) 0 0
\(55\) −10.3757 − 17.9713i −1.39906 − 2.42325i
\(56\) 0 0
\(57\) −0.811741 − 3.56577i −0.107518 − 0.472297i
\(58\) 0 0
\(59\) 1.95924 + 3.39350i 0.255071 + 0.441796i 0.964915 − 0.262563i \(-0.0845679\pi\)
−0.709844 + 0.704359i \(0.751235\pi\)
\(60\) 0 0
\(61\) 3.73588 − 6.47073i 0.478330 − 0.828492i −0.521361 − 0.853336i \(-0.674576\pi\)
0.999691 + 0.0248442i \(0.00790898\pi\)
\(62\) 0 0
\(63\) 10.5952 − 5.08764i 1.33487 − 0.640982i
\(64\) 0 0
\(65\) 25.7378 3.19238
\(66\) 0 0
\(67\) −2.49399 −0.304689 −0.152345 − 0.988327i \(-0.548682\pi\)
−0.152345 + 0.988327i \(0.548682\pi\)
\(68\) 0 0
\(69\) −9.22796 + 8.55258i −1.11092 + 1.02961i
\(70\) 0 0
\(71\) 4.44336 + 7.69612i 0.527329 + 0.913361i 0.999493 + 0.0318501i \(0.0101399\pi\)
−0.472163 + 0.881511i \(0.656527\pi\)
\(72\) 0 0
\(73\) 10.3273 1.20872 0.604361 − 0.796711i \(-0.293428\pi\)
0.604361 + 0.796711i \(0.293428\pi\)
\(74\) 0 0
\(75\) −15.1784 − 4.69134i −1.75266 − 0.541709i
\(76\) 0 0
\(77\) 21.5959 2.46108
\(78\) 0 0
\(79\) 3.14988 − 5.45576i 0.354389 − 0.613821i −0.632624 − 0.774459i \(-0.718022\pi\)
0.987013 + 0.160639i \(0.0513554\pi\)
\(80\) 0 0
\(81\) 3.26910 + 8.38528i 0.363234 + 0.931698i
\(82\) 0 0
\(83\) 10.2050 1.12015 0.560073 − 0.828443i \(-0.310773\pi\)
0.560073 + 0.828443i \(0.310773\pi\)
\(84\) 0 0
\(85\) 10.8423 − 18.7794i 1.17601 − 2.03692i
\(86\) 0 0
\(87\) −0.0576165 − 0.253094i −0.00617714 − 0.0271346i
\(88\) 0 0
\(89\) 7.59908 − 13.1620i 0.805501 − 1.39517i −0.110452 − 0.993881i \(-0.535230\pi\)
0.915953 − 0.401286i \(-0.131437\pi\)
\(90\) 0 0
\(91\) −13.3926 + 23.1966i −1.40392 + 2.43166i
\(92\) 0 0
\(93\) 1.66667 − 1.54469i 0.172826 − 0.160177i
\(94\) 0 0
\(95\) −3.97424 − 6.88358i −0.407748 − 0.706241i
\(96\) 0 0
\(97\) 0.214638 + 0.371764i 0.0217932 + 0.0377469i 0.876716 − 0.481008i \(-0.159729\pi\)
−0.854923 + 0.518755i \(0.826396\pi\)
\(98\) 0 0
\(99\) −1.25445 + 16.4891i −0.126077 + 1.65722i
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.l.b.121.3 yes 74
3.2 odd 2 3996.2.l.b.1009.35 74
9.2 odd 6 3996.2.k.b.2341.3 74
9.7 even 3 1332.2.k.b.565.22 ✓ 74
37.26 even 3 1332.2.k.b.877.22 yes 74
111.26 odd 6 3996.2.k.b.1765.3 74
333.137 odd 6 3996.2.l.b.3097.35 74
333.322 even 3 inner 1332.2.l.b.1321.3 yes 74
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1332.2.k.b.565.22 ✓ 74 9.7 even 3
1332.2.k.b.877.22 yes 74 37.26 even 3
1332.2.l.b.121.3 yes 74 1.1 even 1 trivial
1332.2.l.b.1321.3 yes 74 333.322 even 3 inner
3996.2.k.b.1765.3 74 111.26 odd 6
3996.2.k.b.2341.3 74 9.2 odd 6
3996.2.l.b.1009.35 74 3.2 odd 2
3996.2.l.b.3097.35 74 333.137 odd 6