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(4,57)] 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("57.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(18)) 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.i (of order \(9\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 55.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 57.55
Dual form 57.2.i.a.28.1

$q$-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+(0.326352 - 1.85083i) q^{2} +(-0.766044 - 0.642788i) q^{3} +(-1.43969 - 0.524005i) q^{4} +(-0.826352 + 0.300767i) q^{5} +(-1.43969 + 1.20805i) q^{6} +(1.43969 + 2.49362i) q^{7} +(0.439693 - 0.761570i) q^{8} +(0.173648 + 0.984808i) q^{9} +(0.286989 + 1.62760i) q^{10} +(0.918748 - 1.59132i) q^{11} +(0.766044 + 1.32683i) q^{12} +(-2.11334 + 1.77330i) q^{13} +(5.08512 - 1.85083i) q^{14} +(0.826352 + 0.300767i) q^{15} +(-3.61334 - 3.03195i) q^{16} +(-1.23396 + 6.99811i) q^{17} +1.87939 q^{18} +(3.93969 - 1.86516i) q^{19} +1.34730 q^{20} +(0.500000 - 2.83564i) q^{21} +(-2.64543 - 2.21978i) q^{22} +(-6.19846 - 2.25606i) q^{23} +(-0.826352 + 0.300767i) q^{24} +(-3.23783 + 2.71686i) q^{25} +(2.59240 + 4.49016i) q^{26} +(0.500000 - 0.866025i) q^{27} +(-0.766044 - 4.34445i) q^{28} +(-0.543233 - 3.08083i) q^{29} +(0.826352 - 1.43128i) q^{30} +(-3.82635 - 6.62744i) q^{31} +(-5.44356 + 4.56769i) q^{32} +(-1.72668 + 0.628461i) q^{33} +(12.5496 + 4.56769i) q^{34} +(-1.93969 - 1.62760i) q^{35} +(0.266044 - 1.50881i) q^{36} +2.83750 q^{37} +(-2.16637 - 7.90041i) q^{38} +2.75877 q^{39} +(-0.134285 + 0.761570i) q^{40} +(3.05303 + 2.56180i) q^{41} +(-5.08512 - 1.85083i) q^{42} +(10.7626 - 3.91728i) q^{43} +(-2.15657 + 1.80958i) q^{44} +(-0.439693 - 0.761570i) q^{45} +(-6.19846 + 10.7361i) q^{46} +(0.383256 + 2.17355i) q^{47} +(0.819078 + 4.64522i) q^{48} +(-0.645430 + 1.11792i) q^{49} +(3.97178 + 6.87933i) q^{50} +(5.44356 - 4.56769i) q^{51} +(3.97178 - 1.44561i) q^{52} +(-2.53936 - 0.924252i) q^{53} +(-1.43969 - 1.20805i) q^{54} +(-0.280592 + 1.59132i) q^{55} +2.53209 q^{56} +(-4.21688 - 1.10359i) q^{57} -5.87939 q^{58} +(-1.46064 + 8.28368i) q^{59} +(-1.03209 - 0.866025i) q^{60} +(-0.578726 - 0.210639i) q^{61} +(-13.5150 + 4.91906i) q^{62} +(-2.20574 + 1.85083i) q^{63} +(1.96064 + 3.39592i) q^{64} +(1.21301 - 2.10100i) q^{65} +(0.599670 + 3.40090i) q^{66} +(-0.638156 - 3.61916i) q^{67} +(5.44356 - 9.42853i) q^{68} +(3.29813 + 5.71253i) q^{69} +(-3.64543 + 3.05888i) q^{70} +(7.00387 - 2.54920i) q^{71} +(0.826352 + 0.300767i) q^{72} +(-7.66637 - 6.43285i) q^{73} +(0.926022 - 5.25173i) q^{74} +4.22668 q^{75} +(-6.64930 + 0.620838i) q^{76} +5.29086 q^{77} +(0.900330 - 5.10602i) q^{78} +(1.23396 + 1.03541i) q^{79} +(3.89780 + 1.41868i) q^{80} +(-0.939693 + 0.342020i) q^{81} +(5.73783 - 4.81461i) q^{82} +(-0.492726 - 0.853427i) q^{83} +(-2.20574 + 3.82045i) q^{84} +(-1.08512 - 6.15403i) q^{85} +(-3.73783 - 21.1983i) q^{86} +(-1.56418 + 2.70924i) q^{87} +(-0.807934 - 1.39938i) q^{88} +(13.0667 - 10.9643i) q^{89} +(-1.55303 + 0.565258i) q^{90} +(-7.46451 - 2.71686i) q^{91} +(7.74170 + 6.49605i) q^{92} +(-1.32888 + 7.53644i) q^{93} +4.14796 q^{94} +(-2.69459 + 2.72621i) q^{95} +7.10607 q^{96} +(-1.02481 + 5.81201i) q^{97} +(1.85844 + 1.55942i) q^{98} +(1.72668 + 0.628461i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} - 6 q^{10} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 6 q^{15} - 15 q^{16} - 12 q^{17} + 18 q^{19} + 6 q^{20} + 3 q^{21} - 9 q^{23} - 6 q^{24} + 12 q^{26}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.326352 − 1.85083i 0.230766 − 1.30874i −0.620585 − 0.784139i \(-0.713105\pi\)
0.851350 − 0.524597i \(-0.175784\pi\)
\(3\) −0.766044 − 0.642788i −0.442276 − 0.371114i
\(4\) −1.43969 − 0.524005i −0.719846 − 0.262003i
\(5\) −0.826352 + 0.300767i −0.369556 + 0.134507i −0.520121 − 0.854093i \(-0.674113\pi\)
0.150565 + 0.988600i \(0.451891\pi\)
\(6\) −1.43969 + 1.20805i −0.587752 + 0.493183i
\(7\) 1.43969 + 2.49362i 0.544153 + 0.942500i 0.998660 + 0.0517569i \(0.0164821\pi\)
−0.454507 + 0.890743i \(0.650185\pi\)
\(8\) 0.439693 − 0.761570i 0.155455 − 0.269256i
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0.286989 + 1.62760i 0.0907539 + 0.514691i
\(11\) 0.918748 − 1.59132i 0.277013 − 0.479801i −0.693628 − 0.720333i \(-0.743989\pi\)
0.970641 + 0.240533i \(0.0773222\pi\)
\(12\) 0.766044 + 1.32683i 0.221138 + 0.383022i
\(13\) −2.11334 + 1.77330i −0.586135 + 0.491826i −0.886955 − 0.461855i \(-0.847184\pi\)
0.300820 + 0.953681i \(0.402740\pi\)
\(14\) 5.08512 − 1.85083i 1.35906 − 0.494656i
\(15\) 0.826352 + 0.300767i 0.213363 + 0.0776578i
\(16\) −3.61334 − 3.03195i −0.903335 − 0.757988i
\(17\) −1.23396 + 6.99811i −0.299278 + 1.69729i 0.350008 + 0.936747i \(0.386179\pi\)
−0.649286 + 0.760544i \(0.724932\pi\)
\(18\) 1.87939 0.442975
\(19\) 3.93969 − 1.86516i 0.903827 − 0.427897i
\(20\) 1.34730 0.301265
\(21\) 0.500000 − 2.83564i 0.109109 − 0.618788i
\(22\) −2.64543 − 2.21978i −0.564008 − 0.473258i
\(23\) −6.19846 − 2.25606i −1.29247 − 0.470420i −0.397932 − 0.917415i \(-0.630272\pi\)
−0.894537 + 0.446995i \(0.852494\pi\)
\(24\) −0.826352 + 0.300767i −0.168678 + 0.0613939i
\(25\) −3.23783 + 2.71686i −0.647565 + 0.543372i
\(26\) 2.59240 + 4.49016i 0.508411 + 0.880593i
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) −0.766044 − 4.34445i −0.144769 − 0.821025i
\(29\) −0.543233 − 3.08083i −0.100876 − 0.572096i −0.992788 − 0.119886i \(-0.961747\pi\)
0.891912 − 0.452209i \(-0.149364\pi\)
\(30\) 0.826352 − 1.43128i 0.150871 − 0.261315i
\(31\) −3.82635 − 6.62744i −0.687233 − 1.19032i −0.972729 − 0.231943i \(-0.925492\pi\)
0.285496 − 0.958380i \(-0.407842\pi\)
\(32\) −5.44356 + 4.56769i −0.962295 + 0.807461i
\(33\) −1.72668 + 0.628461i −0.300577 + 0.109401i
\(34\) 12.5496 + 4.56769i 2.15224 + 0.783353i
\(35\) −1.93969 − 1.62760i −0.327868 − 0.275114i
\(36\) 0.266044 − 1.50881i 0.0443407 − 0.251469i
\(37\) 2.83750 0.466481 0.233241 − 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(38\) −2.16637 − 7.90041i −0.351432 − 1.28162i
\(39\) 2.75877 0.441757
\(40\) −0.134285 + 0.761570i −0.0212324 + 0.120415i
\(41\) 3.05303 + 2.56180i 0.476804 + 0.400086i 0.849269 − 0.527960i \(-0.177043\pi\)
−0.372465 + 0.928046i \(0.621487\pi\)
\(42\) −5.08512 − 1.85083i −0.784651 − 0.285590i
\(43\) 10.7626 − 3.91728i 1.64129 − 0.597380i 0.654024 − 0.756474i \(-0.273080\pi\)
0.987263 + 0.159094i \(0.0508573\pi\)
\(44\) −2.15657 + 1.80958i −0.325116 + 0.272805i
\(45\) −0.439693 − 0.761570i −0.0655455 − 0.113528i
\(46\) −6.19846 + 10.7361i −0.913914 + 1.58294i
\(47\) 0.383256 + 2.17355i 0.0559036 + 0.317045i 0.999917 − 0.0128613i \(-0.00409398\pi\)
−0.944014 + 0.329906i \(0.892983\pi\)
\(48\) 0.819078 + 4.64522i 0.118224 + 0.670480i
\(49\) −0.645430 + 1.11792i −0.0922042 + 0.159702i
\(50\) 3.97178 + 6.87933i 0.561695 + 0.972884i
\(51\) 5.44356 − 4.56769i 0.762251 − 0.639605i
\(52\) 3.97178 − 1.44561i 0.550787 − 0.200470i
\(53\) −2.53936 − 0.924252i −0.348808 − 0.126956i 0.161674 − 0.986844i \(-0.448311\pi\)
−0.510482 + 0.859888i \(0.670533\pi\)
\(54\) −1.43969 − 1.20805i −0.195917 − 0.164394i
\(55\) −0.280592 + 1.59132i −0.0378351 + 0.214573i
\(56\) 2.53209 0.338365
\(57\) −4.21688 − 1.10359i −0.558540 − 0.146174i
\(58\) −5.87939 −0.772001
\(59\) −1.46064 + 8.28368i −0.190159 + 1.07844i 0.728987 + 0.684527i \(0.239991\pi\)
−0.919146 + 0.393917i \(0.871120\pi\)
\(60\) −1.03209 − 0.866025i −0.133242 − 0.111803i
\(61\) −0.578726 − 0.210639i −0.0740982 − 0.0269696i 0.304705 − 0.952447i \(-0.401442\pi\)
−0.378803 + 0.925477i \(0.623664\pi\)
\(62\) −13.5150 + 4.91906i −1.71641 + 0.624722i
\(63\) −2.20574 + 1.85083i −0.277897 + 0.233183i
\(64\) 1.96064 + 3.39592i 0.245080 + 0.424490i
\(65\) 1.21301 − 2.10100i 0.150456 − 0.260597i
\(66\) 0.599670 + 3.40090i 0.0738143 + 0.418622i
\(67\) −0.638156 − 3.61916i −0.0779631 − 0.442151i −0.998654 − 0.0518592i \(-0.983485\pi\)
0.920691 − 0.390292i \(-0.127626\pi\)
\(68\) 5.44356 − 9.42853i 0.660129 − 1.14338i
\(69\) 3.29813 + 5.71253i 0.397049 + 0.687708i
\(70\) −3.64543 + 3.05888i −0.435712 + 0.365606i
\(71\) 7.00387 − 2.54920i 0.831206 − 0.302534i 0.108853 − 0.994058i \(-0.465282\pi\)
0.722354 + 0.691523i \(0.243060\pi\)
\(72\) 0.826352 + 0.300767i 0.0973865 + 0.0354458i
\(73\) −7.66637 − 6.43285i −0.897281 − 0.752908i 0.0723759 − 0.997377i \(-0.476942\pi\)
−0.969657 + 0.244469i \(0.921386\pi\)
\(74\) 0.926022 − 5.25173i 0.107648 − 0.610501i
\(75\) 4.22668 0.488055
\(76\) −6.64930 + 0.620838i −0.762727 + 0.0712149i
\(77\) 5.29086 0.602949
\(78\) 0.900330 − 5.10602i 0.101942 − 0.578143i
\(79\) 1.23396 + 1.03541i 0.138831 + 0.116493i 0.709559 − 0.704646i \(-0.248894\pi\)
−0.570728 + 0.821139i \(0.693339\pi\)
\(80\) 3.89780 + 1.41868i 0.435788 + 0.158614i
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 5.73783 − 4.81461i 0.633637 − 0.531684i
\(83\) −0.492726 − 0.853427i −0.0540837 − 0.0936757i 0.837716 − 0.546106i \(-0.183890\pi\)
−0.891800 + 0.452430i \(0.850557\pi\)
\(84\) −2.20574 + 3.82045i −0.240666 + 0.416845i
\(85\) −1.08512 − 6.15403i −0.117698 − 0.667499i
\(86\) −3.73783 − 21.1983i −0.403060 − 2.28587i
\(87\) −1.56418 + 2.70924i −0.167697 + 0.290461i
\(88\) −0.807934 − 1.39938i −0.0861260 − 0.149175i
\(89\) 13.0667 − 10.9643i 1.38507 − 1.16221i 0.417774 − 0.908551i \(-0.362810\pi\)
0.967294 − 0.253659i \(-0.0816341\pi\)
\(90\) −1.55303 + 0.565258i −0.163704 + 0.0595834i
\(91\) −7.46451 − 2.71686i −0.782493 − 0.284804i
\(92\) 7.74170 + 6.49605i 0.807128 + 0.677261i
\(93\) −1.32888 + 7.53644i −0.137798 + 0.781493i
\(94\) 4.14796 0.427829
\(95\) −2.69459 + 2.72621i −0.276459 + 0.279703i
\(96\) 7.10607 0.725260
\(97\) −1.02481 + 5.81201i −0.104054 + 0.590121i 0.887540 + 0.460731i \(0.152413\pi\)
−0.991594 + 0.129389i \(0.958698\pi\)
\(98\) 1.85844 + 1.55942i 0.187731 + 0.157525i
\(99\) 1.72668 + 0.628461i 0.173538 + 0.0631627i
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.i.a.55.1 yes 6
3.2 odd 2 171.2.u.a.55.1 6
4.3 odd 2 912.2.bo.b.625.1 6
19.3 odd 18 1083.2.a.n.1.1 3
19.9 even 9 inner 57.2.i.a.28.1 ✓ 6
19.16 even 9 1083.2.a.m.1.3 3
57.35 odd 18 3249.2.a.w.1.1 3
57.41 even 18 3249.2.a.x.1.3 3
57.47 odd 18 171.2.u.a.28.1 6
76.47 odd 18 912.2.bo.b.769.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.i.a.28.1 ✓ 6 19.9 even 9 inner
57.2.i.a.55.1 yes 6 1.1 even 1 trivial
171.2.u.a.28.1 6 57.47 odd 18
171.2.u.a.55.1 6 3.2 odd 2
912.2.bo.b.625.1 6 4.3 odd 2
912.2.bo.b.769.1 6 76.47 odd 18
1083.2.a.m.1.3 3 19.16 even 9
1083.2.a.n.1.1 3 19.3 odd 18
3249.2.a.w.1.1 3 57.35 odd 18
3249.2.a.x.1.3 3 57.41 even 18