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Properties

Label 57.2.e.b.49.3
Level $57$
Weight $2$
Character 57.49
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [57,2,Mod(7,57)] 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("57.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.3
Root \(0.403374 - 1.68443i\) of defining polynomial
Character \(\chi\) \(=\) 57.49
Dual form 57.2.e.b.7.3

$q$-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+(1.25707 - 2.17731i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-2.16044 - 3.74200i) q^{4} +(-1.66044 + 2.87597i) q^{5} +(1.25707 + 2.17731i) q^{6} +2.32088 q^{7} -5.83502 q^{8} +(-0.500000 - 0.866025i) q^{9} +(4.17458 + 7.23058i) q^{10} -1.70739 q^{11} +4.32088 q^{12} +(-2.01414 - 3.48859i) q^{13} +(2.91751 - 5.05328i) q^{14} +(-1.66044 - 2.87597i) q^{15} +(-3.01414 + 5.22064i) q^{16} -2.51414 q^{18} +(0.193252 + 4.35461i) q^{19} +14.3492 q^{20} +(-1.16044 + 2.00994i) q^{21} +(-2.14631 + 3.71751i) q^{22} +(1.17458 + 2.03443i) q^{23} +(2.91751 - 5.05328i) q^{24} +(-3.01414 - 5.22064i) q^{25} -10.1276 q^{26} +1.00000 q^{27} +(-5.01414 - 8.68474i) q^{28} +(-3.32088 - 5.75194i) q^{29} -8.34916 q^{30} +6.70739 q^{31} +(1.74293 + 3.01885i) q^{32} +(0.853695 - 1.47864i) q^{33} +(-3.85369 + 6.67479i) q^{35} +(-2.16044 + 3.74200i) q^{36} -1.00000 q^{37} +(9.72426 + 5.05328i) q^{38} +4.02827 q^{39} +(9.68872 - 16.7813i) q^{40} +(3.32088 - 5.75194i) q^{41} +(2.91751 + 5.05328i) q^{42} +(-0.353695 + 0.612617i) q^{43} +(3.68872 + 6.38904i) q^{44} +3.32088 q^{45} +5.90611 q^{46} +(3.00000 + 5.19615i) q^{47} +(-3.01414 - 5.22064i) q^{48} -1.61350 q^{49} -15.1559 q^{50} +(-8.70285 + 15.0738i) q^{52} +(4.98133 + 8.62791i) q^{53} +(1.25707 - 2.17731i) q^{54} +(2.83502 - 4.91040i) q^{55} -13.5424 q^{56} +(-3.86783 - 2.00994i) q^{57} -16.6983 q^{58} +(-0.853695 + 1.47864i) q^{59} +(-7.17458 + 12.4267i) q^{60} +(-1.69325 - 2.93280i) q^{61} +(8.43165 - 14.6040i) q^{62} +(-1.16044 - 2.00994i) q^{63} -3.29261 q^{64} +13.3774 q^{65} +(-2.14631 - 3.71751i) q^{66} +(4.18872 + 7.25507i) q^{67} -2.34916 q^{69} +(9.68872 + 16.7813i) q^{70} +(4.70739 - 8.15344i) q^{71} +(2.91751 + 5.05328i) q^{72} +(-5.82088 + 10.0821i) q^{73} +(-1.25707 + 2.17731i) q^{74} +6.02827 q^{75} +(15.8774 - 10.1310i) q^{76} -3.96265 q^{77} +(5.06382 - 8.77079i) q^{78} +(1.67458 - 2.90046i) q^{79} +(-10.0096 - 17.3371i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-8.34916 - 14.4612i) q^{82} -10.0565 q^{83} +10.0283 q^{84} +(0.889237 + 1.54020i) q^{86} +6.64177 q^{87} +9.96265 q^{88} +(1.33956 + 2.32018i) q^{89} +(4.17458 - 7.23058i) q^{90} +(-4.67458 - 8.09661i) q^{91} +(5.07522 - 8.79054i) q^{92} +(-3.35369 + 5.80877i) q^{93} +15.0848 q^{94} +(-12.8446 - 6.67479i) q^{95} -3.48586 q^{96} +(-8.86330 + 15.3517i) q^{97} +(-2.02827 + 3.51307i) q^{98} +(0.853695 + 1.47864i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 3 q^{3} - 5 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - 3 q^{9} + 4 q^{10} + 10 q^{12} + q^{13} + 3 q^{14} - 2 q^{15} - 5 q^{16} - 2 q^{18} + 4 q^{19} + 44 q^{20} + q^{21} - 18 q^{22}+ \cdots + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.25707 − 2.17731i 0.888882 − 1.53959i 0.0476826 − 0.998863i \(-0.484816\pi\)
0.841199 − 0.540726i \(-0.181850\pi\)
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −2.16044 − 3.74200i −1.08022 − 1.87100i
\(5\) −1.66044 + 2.87597i −0.742572 + 1.28617i 0.208748 + 0.977969i \(0.433061\pi\)
−0.951320 + 0.308204i \(0.900272\pi\)
\(6\) 1.25707 + 2.17731i 0.513196 + 0.888882i
\(7\) 2.32088 0.877212 0.438606 − 0.898679i \(-0.355472\pi\)
0.438606 + 0.898679i \(0.355472\pi\)
\(8\) −5.83502 −2.06299
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 4.17458 + 7.23058i 1.32012 + 2.28651i
\(11\) −1.70739 −0.514797 −0.257399 − 0.966305i \(-0.582865\pi\)
−0.257399 + 0.966305i \(0.582865\pi\)
\(12\) 4.32088 1.24733
\(13\) −2.01414 − 3.48859i −0.558621 − 0.967560i −0.997612 − 0.0690685i \(-0.977997\pi\)
0.438991 − 0.898492i \(-0.355336\pi\)
\(14\) 2.91751 − 5.05328i 0.779738 − 1.35055i
\(15\) −1.66044 − 2.87597i −0.428724 − 0.742572i
\(16\) −3.01414 + 5.22064i −0.753534 + 1.30516i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) −2.51414 −0.592588
\(19\) 0.193252 + 4.35461i 0.0443351 + 0.999017i
\(20\) 14.3492 3.20857
\(21\) −1.16044 + 2.00994i −0.253229 + 0.438606i
\(22\) −2.14631 + 3.71751i −0.457594 + 0.792576i
\(23\) 1.17458 + 2.03443i 0.244917 + 0.424208i 0.962108 − 0.272668i \(-0.0879061\pi\)
−0.717191 + 0.696876i \(0.754573\pi\)
\(24\) 2.91751 − 5.05328i 0.595534 − 1.03150i
\(25\) −3.01414 − 5.22064i −0.602827 − 1.04413i
\(26\) −10.1276 −1.98619
\(27\) 1.00000 0.192450
\(28\) −5.01414 − 8.68474i −0.947583 − 1.64126i
\(29\) −3.32088 − 5.75194i −0.616673 − 1.06811i −0.990089 − 0.140444i \(-0.955147\pi\)
0.373416 − 0.927664i \(-0.378186\pi\)
\(30\) −8.34916 −1.52434
\(31\) 6.70739 1.20468 0.602341 − 0.798239i \(-0.294235\pi\)
0.602341 + 0.798239i \(0.294235\pi\)
\(32\) 1.74293 + 3.01885i 0.308110 + 0.533662i
\(33\) 0.853695 − 1.47864i 0.148609 − 0.257399i
\(34\) 0 0
\(35\) −3.85369 + 6.67479i −0.651393 + 1.12825i
\(36\) −2.16044 + 3.74200i −0.360074 + 0.623666i
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 9.72426 + 5.05328i 1.57748 + 0.819750i
\(39\) 4.02827 0.645040
\(40\) 9.68872 − 16.7813i 1.53192 − 2.65336i
\(41\) 3.32088 − 5.75194i 0.518635 − 0.898302i −0.481131 − 0.876649i \(-0.659774\pi\)
0.999766 − 0.0216532i \(-0.00689298\pi\)
\(42\) 2.91751 + 5.05328i 0.450182 + 0.779738i
\(43\) −0.353695 + 0.612617i −0.0539379 + 0.0934232i −0.891734 − 0.452561i \(-0.850511\pi\)
0.837796 + 0.545984i \(0.183844\pi\)
\(44\) 3.68872 + 6.38904i 0.556095 + 0.963185i
\(45\) 3.32088 0.495048
\(46\) 5.90611 0.870808
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) −3.01414 − 5.22064i −0.435053 − 0.753534i
\(49\) −1.61350 −0.230499
\(50\) −15.1559 −2.14337
\(51\) 0 0
\(52\) −8.70285 + 15.0738i −1.20687 + 2.09036i
\(53\) 4.98133 + 8.62791i 0.684238 + 1.18513i 0.973676 + 0.227938i \(0.0731983\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(54\) 1.25707 − 2.17731i 0.171065 − 0.296294i
\(55\) 2.83502 − 4.91040i 0.382274 − 0.662118i
\(56\) −13.5424 −1.80968
\(57\) −3.86783 − 2.00994i −0.512307 − 0.266224i
\(58\) −16.6983 −2.19260
\(59\) −0.853695 + 1.47864i −0.111142 + 0.192503i −0.916231 − 0.400651i \(-0.868784\pi\)
0.805089 + 0.593154i \(0.202117\pi\)
\(60\) −7.17458 + 12.4267i −0.926234 + 1.60428i
\(61\) −1.69325 − 2.93280i −0.216799 − 0.375506i 0.737029 − 0.675861i \(-0.236228\pi\)
−0.953828 + 0.300355i \(0.902895\pi\)
\(62\) 8.43165 − 14.6040i 1.07082 − 1.85472i
\(63\) −1.16044 − 2.00994i −0.146202 − 0.253229i
\(64\) −3.29261 −0.411576
\(65\) 13.3774 1.65927
\(66\) −2.14631 − 3.71751i −0.264192 − 0.457594i
\(67\) 4.18872 + 7.25507i 0.511733 + 0.886348i 0.999907 + 0.0136016i \(0.00432967\pi\)
−0.488174 + 0.872746i \(0.662337\pi\)
\(68\) 0 0
\(69\) −2.34916 −0.282805
\(70\) 9.68872 + 16.7813i 1.15802 + 2.00575i
\(71\) 4.70739 − 8.15344i 0.558664 − 0.967635i −0.438944 − 0.898514i \(-0.644647\pi\)
0.997608 − 0.0691206i \(-0.0220193\pi\)
\(72\) 2.91751 + 5.05328i 0.343832 + 0.595534i
\(73\) −5.82088 + 10.0821i −0.681283 + 1.18002i 0.293307 + 0.956018i \(0.405244\pi\)
−0.974590 + 0.223998i \(0.928089\pi\)
\(74\) −1.25707 + 2.17731i −0.146131 + 0.253107i
\(75\) 6.02827 0.696085
\(76\) 15.8774 − 10.1310i 1.82127 − 1.16211i
\(77\) −3.96265 −0.451586
\(78\) 5.06382 − 8.77079i 0.573364 − 0.993096i
\(79\) 1.67458 − 2.90046i 0.188405 − 0.326327i −0.756314 − 0.654209i \(-0.773002\pi\)
0.944719 + 0.327882i \(0.106335\pi\)
\(80\) −10.0096 − 17.3371i −1.11911 − 1.93835i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −8.34916 − 14.4612i −0.922010 − 1.59697i
\(83\) −10.0565 −1.10385 −0.551925 − 0.833894i \(-0.686106\pi\)
−0.551925 + 0.833894i \(0.686106\pi\)
\(84\) 10.0283 1.09417
\(85\) 0 0
\(86\) 0.889237 + 1.54020i 0.0958889 + 0.166084i
\(87\) 6.64177 0.712072
\(88\) 9.96265 1.06202
\(89\) 1.33956 + 2.32018i 0.141993 + 0.245939i 0.928247 − 0.371964i \(-0.121316\pi\)
−0.786254 + 0.617903i \(0.787982\pi\)
\(90\) 4.17458 − 7.23058i 0.440039 − 0.762170i
\(91\) −4.67458 − 8.09661i −0.490029 − 0.848755i
\(92\) 5.07522 − 8.79054i 0.529128 − 0.916477i
\(93\) −3.35369 + 5.80877i −0.347762 + 0.602341i
\(94\) 15.0848 1.55588
\(95\) −12.8446 − 6.67479i −1.31783 − 0.684820i
\(96\) −3.48586 −0.355774
\(97\) −8.86330 + 15.3517i −0.899931 + 1.55873i −0.0723511 + 0.997379i \(0.523050\pi\)
−0.827580 + 0.561347i \(0.810283\pi\)
\(98\) −2.02827 + 3.51307i −0.204887 + 0.354874i
\(99\) 0.853695 + 1.47864i 0.0857995 + 0.148609i
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 57.2.e.b.49.3 yes 6
3.2 odd 2 171.2.f.b.163.1 6
4.3 odd 2 912.2.q.l.49.1 6
12.11 even 2 2736.2.s.z.1873.3 6
19.7 even 3 inner 57.2.e.b.7.3 ✓ 6
19.8 odd 6 1083.2.a.o.1.3 3
19.11 even 3 1083.2.a.l.1.1 3
57.8 even 6 3249.2.a.t.1.1 3
57.11 odd 6 3249.2.a.y.1.3 3
57.26 odd 6 171.2.f.b.64.1 6
76.7 odd 6 912.2.q.l.577.1 6
228.83 even 6 2736.2.s.z.577.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 19.7 even 3 inner
57.2.e.b.49.3 yes 6 1.1 even 1 trivial
171.2.f.b.64.1 6 57.26 odd 6
171.2.f.b.163.1 6 3.2 odd 2
912.2.q.l.49.1 6 4.3 odd 2
912.2.q.l.577.1 6 76.7 odd 6
1083.2.a.l.1.1 3 19.11 even 3
1083.2.a.o.1.3 3 19.8 odd 6
2736.2.s.z.577.3 6 228.83 even 6
2736.2.s.z.1873.3 6 12.11 even 2
3249.2.a.t.1.1 3 57.8 even 6
3249.2.a.y.1.3 3 57.11 odd 6