Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(417,592)] 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("592.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.1
Root \(-0.827721 + 1.43366i\) of defining polynomial
Character \(\chi\) \(=\) 592.417
Dual form 592.2.i.e.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.52569 - 2.64257i) q^{3} +(-0.629755 - 1.09077i) q^{5} +(1.52569 + 2.64257i) q^{7} +(-3.15544 + 5.46539i) q^{9} -5.31088 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-1.92162 + 3.32834i) q^{15} +(-2.55137 + 4.41911i) q^{17} +(2.39593 + 4.14988i) q^{19} +(4.65544 - 8.06346i) q^{21} -1.74049 q^{23} +(1.70682 - 2.95629i) q^{25} +10.1027 q^{27} -4.05137 q^{29} +0.791864 q^{31} +(8.10275 + 14.0344i) q^{33} +(1.92162 - 3.32834i) q^{35} +(-1.37024 + 5.92642i) q^{37} +(-3.05137 + 5.28514i) q^{39} +(0.104068 + 0.180251i) q^{41} +4.36226 q^{43} +7.94863 q^{45} +3.20814 q^{47} +(-1.15544 + 2.00128i) q^{49} +15.5704 q^{51} +(-5.65544 + 9.79551i) q^{53} +(3.34456 + 5.79294i) q^{55} +(7.31088 - 12.6628i) q^{57} +(-6.36226 + 11.0198i) q^{59} +(-3.02569 - 5.24064i) q^{61} -19.2569 q^{63} +(-1.25951 + 2.18154i) q^{65} +(-4.65544 - 8.06346i) q^{67} +(2.65544 + 4.59936i) q^{69} +(-3.52569 - 6.10667i) q^{71} -3.31088 q^{73} -10.4163 q^{75} +(-8.10275 - 14.0344i) q^{77} +(-5.70682 - 9.88450i) q^{79} +(-5.94731 - 10.3010i) q^{81} +(-1.78520 + 3.09205i) q^{83} +6.42697 q^{85} +(6.18113 + 10.7060i) q^{87} +(-1.63642 + 2.83436i) q^{89} +(3.05137 - 5.28514i) q^{91} +(-1.20814 - 2.09255i) q^{93} +(3.01770 - 5.22681i) q^{95} +2.20814 q^{97} +(16.7582 - 29.0260i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{5} - 7 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 3 q^{17} + 8 q^{19} + 16 q^{21} - 16 q^{23} - 20 q^{25} + 24 q^{27} - 6 q^{29} - 8 q^{31} + 12 q^{33} - 4 q^{35} - 11 q^{37} + 7 q^{41} - 16 q^{43}+ \cdots + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.52569 − 2.64257i −0.880856 − 1.52569i −0.850391 − 0.526151i \(-0.823635\pi\)
−0.0304649 − 0.999536i \(-0.509699\pi\)
\(4\) 0 0
\(5\) −0.629755 − 1.09077i −0.281635 − 0.487806i 0.690153 − 0.723664i \(-0.257543\pi\)
−0.971788 + 0.235858i \(0.924210\pi\)
\(6\) 0 0
\(7\) 1.52569 + 2.64257i 0.576656 + 0.998797i 0.995860 + 0.0909046i \(0.0289758\pi\)
−0.419204 + 0.907892i \(0.637691\pi\)
\(8\) 0 0
\(9\) −3.15544 + 5.46539i −1.05181 + 1.82180i
\(10\) 0 0
\(11\) −5.31088 −1.60129 −0.800646 − 0.599138i \(-0.795510\pi\)
−0.800646 + 0.599138i \(0.795510\pi\)
\(12\) 0 0
\(13\) −1.00000 − 1.73205i −0.277350 − 0.480384i 0.693375 − 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) −1.92162 + 3.32834i −0.496160 + 0.859374i
\(16\) 0 0
\(17\) −2.55137 + 4.41911i −0.618799 + 1.07179i 0.370906 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143121i \(0.954286\pi\)
\(18\) 0 0
\(19\) 2.39593 + 4.14988i 0.549664 + 0.952047i 0.998297 + 0.0583306i \(0.0185778\pi\)
−0.448633 + 0.893716i \(0.648089\pi\)
\(20\) 0 0
\(21\) 4.65544 − 8.06346i 1.01590 − 1.75959i
\(22\) 0 0
\(23\) −1.74049 −0.362917 −0.181459 − 0.983399i \(-0.558082\pi\)
−0.181459 + 0.983399i \(0.558082\pi\)
\(24\) 0 0
\(25\) 1.70682 − 2.95629i 0.341363 − 0.591259i
\(26\) 0 0
\(27\) 10.1027 1.94427
\(28\) 0 0
\(29\) −4.05137 −0.752321 −0.376161 − 0.926554i \(-0.622756\pi\)
−0.376161 + 0.926554i \(0.622756\pi\)
\(30\) 0 0
\(31\) 0.791864 0.142223 0.0711115 − 0.997468i \(-0.477345\pi\)
0.0711115 + 0.997468i \(0.477345\pi\)
\(32\) 0 0
\(33\) 8.10275 + 14.0344i 1.41051 + 2.44307i
\(34\) 0 0
\(35\) 1.92162 − 3.32834i 0.324813 − 0.562592i
\(36\) 0 0
\(37\) −1.37024 + 5.92642i −0.225267 + 0.974297i
\(38\) 0 0
\(39\) −3.05137 + 5.28514i −0.488611 + 0.846299i
\(40\) 0 0
\(41\) 0.104068 + 0.180251i 0.0162527 + 0.0281505i 0.874037 − 0.485859i \(-0.161493\pi\)
−0.857785 + 0.514009i \(0.828160\pi\)
\(42\) 0 0
\(43\) 4.36226 0.665238 0.332619 − 0.943061i \(-0.392068\pi\)
0.332619 + 0.943061i \(0.392068\pi\)
\(44\) 0 0
\(45\) 7.94863 1.18491
\(46\) 0 0
\(47\) 3.20814 0.467955 0.233977 − 0.972242i \(-0.424826\pi\)
0.233977 + 0.972242i \(0.424826\pi\)
\(48\) 0 0
\(49\) −1.15544 + 2.00128i −0.165063 + 0.285898i
\(50\) 0 0
\(51\) 15.5704 2.18029
\(52\) 0 0
\(53\) −5.65544 + 9.79551i −0.776835 + 1.34552i 0.156923 + 0.987611i \(0.449843\pi\)
−0.933757 + 0.357906i \(0.883491\pi\)
\(54\) 0 0
\(55\) 3.34456 + 5.79294i 0.450980 + 0.781120i
\(56\) 0 0
\(57\) 7.31088 − 12.6628i 0.968350 − 1.67723i
\(58\) 0 0
\(59\) −6.36226 + 11.0198i −0.828296 + 1.43465i 0.0710788 + 0.997471i \(0.477356\pi\)
−0.899374 + 0.437179i \(0.855978\pi\)
\(60\) 0 0
\(61\) −3.02569 − 5.24064i −0.387400 − 0.670996i 0.604699 − 0.796454i \(-0.293293\pi\)
−0.992099 + 0.125458i \(0.959960\pi\)
\(62\) 0 0
\(63\) −19.2569 −2.42614
\(64\) 0 0
\(65\) −1.25951 + 2.18154i −0.156223 + 0.270586i
\(66\) 0 0
\(67\) −4.65544 − 8.06346i −0.568753 − 0.985109i −0.996690 − 0.0813001i \(-0.974093\pi\)
0.427937 − 0.903809i \(-0.359241\pi\)
\(68\) 0 0
\(69\) 2.65544 + 4.59936i 0.319678 + 0.553698i
\(70\) 0 0
\(71\) −3.52569 − 6.10667i −0.418422 − 0.724728i 0.577359 − 0.816491i \(-0.304083\pi\)
−0.995781 + 0.0917622i \(0.970750\pi\)
\(72\) 0 0
\(73\) −3.31088 −0.387510 −0.193755 − 0.981050i \(-0.562067\pi\)
−0.193755 + 0.981050i \(0.562067\pi\)
\(74\) 0 0
\(75\) −10.4163 −1.20277
\(76\) 0 0
\(77\) −8.10275 − 14.0344i −0.923394 − 1.59937i
\(78\) 0 0
\(79\) −5.70682 − 9.88450i −0.642067 − 1.11209i −0.984971 − 0.172722i \(-0.944744\pi\)
0.342904 − 0.939371i \(-0.388590\pi\)
\(80\) 0 0
\(81\) −5.94731 − 10.3010i −0.660812 − 1.14456i
\(82\) 0 0
\(83\) −1.78520 + 3.09205i −0.195951 + 0.339397i −0.947212 − 0.320608i \(-0.896113\pi\)
0.751261 + 0.660005i \(0.229446\pi\)
\(84\) 0 0
\(85\) 6.42697 0.697102
\(86\) 0 0
\(87\) 6.18113 + 10.7060i 0.662687 + 1.14781i
\(88\) 0 0
\(89\) −1.63642 + 2.83436i −0.173460 + 0.300442i −0.939627 − 0.342199i \(-0.888828\pi\)
0.766167 + 0.642641i \(0.222162\pi\)
\(90\) 0 0
\(91\) 3.05137 − 5.28514i 0.319871 − 0.554033i
\(92\) 0 0
\(93\) −1.20814 − 2.09255i −0.125278 − 0.216988i
\(94\) 0 0
\(95\) 3.01770 − 5.22681i 0.309610 − 0.536260i
\(96\) 0 0
\(97\) 2.20814 0.224202 0.112101 − 0.993697i \(-0.464242\pi\)
0.112101 + 0.993697i \(0.464242\pi\)
\(98\) 0 0
\(99\) 16.7582 − 29.0260i 1.68426 − 2.91723i
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 592.2.i.e.417.1 6
4.3 odd 2 74.2.c.c.47.3 ✓ 6
12.11 even 2 666.2.f.j.343.2 6
37.26 even 3 inner 592.2.i.e.433.1 6
148.27 odd 6 2738.2.a.n.1.1 3
148.47 odd 6 2738.2.a.o.1.1 3
148.63 odd 6 74.2.c.c.63.3 yes 6
444.359 even 6 666.2.f.j.433.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.c.47.3 ✓ 6 4.3 odd 2
74.2.c.c.63.3 yes 6 148.63 odd 6
592.2.i.e.417.1 6 1.1 even 1 trivial
592.2.i.e.433.1 6 37.26 even 3 inner
666.2.f.j.343.2 6 12.11 even 2
666.2.f.j.433.2 6 444.359 even 6
2738.2.a.n.1.1 3 148.27 odd 6
2738.2.a.o.1.1 3 148.47 odd 6