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 28.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 57.28
Dual form 57.2.i.a.55.1

$q$-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 \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{9}\right)\)

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 0.851350 + 0.524597i \(0.175784\pi\)
−0.620585 + 0.784139i \(0.713105\pi\)
\(3\) −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 0.150565 − 0.988600i \(-0.451891\pi\)
−0.520121 + 0.854093i \(0.674113\pi\)
\(6\) −1.43969 − 1.20805i −0.587752 − 0.493183i
\(7\) 1.43969 − 2.49362i 0.544153 − 0.942500i −0.454507 − 0.890743i \(-0.650185\pi\)
0.998660 − 0.0517569i \(-0.0164821\pi\)
\(8\) 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 0.970641 − 0.240533i \(-0.0773222\pi\)
−0.693628 + 0.720333i \(0.743989\pi\)
\(12\) 0.766044 − 1.32683i 0.221138 − 0.383022i
\(13\) −2.11334 − 1.77330i −0.586135 − 0.491826i 0.300820 − 0.953681i \(-0.402740\pi\)
−0.886955 + 0.461855i \(0.847184\pi\)
\(14\) 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 −0.649286 − 0.760544i \(-0.724932\pi\)
0.350008 − 0.936747i \(-0.386179\pi\)
\(18\) 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 −0.894537 − 0.446995i \(-0.852494\pi\)
−0.397932 + 0.917415i \(0.630272\pi\)
\(24\) −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 0.891912 + 0.452209i \(0.149364\pi\)
−0.992788 + 0.119886i \(0.961747\pi\)
\(30\) 0.826352 + 1.43128i 0.150871 + 0.261315i
\(31\) −3.82635 + 6.62744i −0.687233 + 1.19032i 0.285496 + 0.958380i \(0.407842\pi\)
−0.972729 + 0.231943i \(0.925492\pi\)
\(32\) −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 −0.372465 − 0.928046i \(-0.621487\pi\)
0.849269 + 0.527960i \(0.177043\pi\)
\(42\) −5.08512 + 1.85083i −0.784651 + 0.285590i
\(43\) 10.7626 + 3.91728i 1.64129 + 0.597380i 0.987263 − 0.159094i \(-0.0508573\pi\)
0.654024 + 0.756474i \(0.273080\pi\)
\(44\) −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 −0.944014 − 0.329906i \(-0.892983\pi\)
0.999917 + 0.0128613i \(0.00409398\pi\)
\(48\) 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 −0.510482 − 0.859888i \(-0.670533\pi\)
0.161674 + 0.986844i \(0.448311\pi\)
\(54\) −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 −0.919146 − 0.393917i \(-0.871120\pi\)
0.728987 − 0.684527i \(-0.239991\pi\)
\(60\) −1.03209 + 0.866025i −0.133242 + 0.111803i
\(61\) −0.578726 + 0.210639i −0.0740982 + 0.0269696i −0.378803 − 0.925477i \(-0.623664\pi\)
0.304705 + 0.952447i \(0.401442\pi\)
\(62\) −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 0.920691 + 0.390292i \(0.127626\pi\)
−0.998654 + 0.0518592i \(0.983485\pi\)
\(68\) 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 0.722354 − 0.691523i \(-0.243060\pi\)
0.108853 + 0.994058i \(0.465282\pi\)
\(72\) 0.826352 − 0.300767i 0.0973865 − 0.0354458i
\(73\) −7.66637 + 6.43285i −0.897281 + 0.752908i −0.969657 − 0.244469i \(-0.921386\pi\)
0.0723759 + 0.997377i \(0.476942\pi\)
\(74\) 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 −0.570728 − 0.821139i \(-0.693339\pi\)
0.709559 + 0.704646i \(0.248894\pi\)
\(80\) 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 −0.891800 − 0.452430i \(-0.850557\pi\)
0.837716 + 0.546106i \(0.183890\pi\)
\(84\) −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 0.967294 + 0.253659i \(0.0816341\pi\)
0.417774 + 0.908551i \(0.362810\pi\)
\(90\) −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 −0.991594 − 0.129389i \(-0.958698\pi\)
0.887540 − 0.460731i \(-0.152413\pi\)
\(98\) 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.28.1 ✓ 6
3.2 odd 2 171.2.u.a.28.1 6
4.3 odd 2 912.2.bo.b.769.1 6
19.6 even 9 1083.2.a.m.1.3 3
19.13 odd 18 1083.2.a.n.1.1 3
19.17 even 9 inner 57.2.i.a.55.1 yes 6
57.17 odd 18 171.2.u.a.55.1 6
57.32 even 18 3249.2.a.x.1.3 3
57.44 odd 18 3249.2.a.w.1.1 3
76.55 odd 18 912.2.bo.b.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.i.a.28.1 ✓ 6 1.1 even 1 trivial
57.2.i.a.55.1 yes 6 19.17 even 9 inner
171.2.u.a.28.1 6 3.2 odd 2
171.2.u.a.55.1 6 57.17 odd 18
912.2.bo.b.625.1 6 76.55 odd 18
912.2.bo.b.769.1 6 4.3 odd 2
1083.2.a.m.1.3 3 19.6 even 9
1083.2.a.n.1.1 3 19.13 odd 18
3249.2.a.w.1.1 3 57.44 odd 18
3249.2.a.x.1.3 3 57.32 even 18