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Properties

Label 592.2.i.g.417.3
Level $592$
Weight $2$
Character 592.417
Analytic conductor $4.727$
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] = [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 = [6,0,2] 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: 6.0.591408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) 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 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.3
Root \(0.155554 + 0.269427i\) of defining polynomial
Character \(\chi\) \(=\) 592.417
Dual form 592.2.i.g.433.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.45161 + 2.51426i) q^{3} +(-0.951606 - 1.64823i) q^{5} +(0.762714 + 1.32106i) q^{7} +(-2.71432 + 4.70134i) q^{9} +(1.00000 + 1.73205i) q^{13} +(2.76271 - 4.78516i) q^{15} +(-3.02543 + 5.24019i) q^{17} +(3.52543 + 6.10622i) q^{19} +(-2.21432 + 3.83531i) q^{21} -4.28100 q^{23} +(0.688892 - 1.19320i) q^{25} -7.05086 q^{27} -1.28100 q^{29} +9.80642 q^{31} +(1.45161 - 2.51426i) q^{35} +(-0.854818 - 6.02240i) q^{37} +(-2.90321 + 5.02851i) q^{39} +(-0.811108 - 1.40488i) q^{41} -7.76049 q^{43} +10.3319 q^{45} +5.80642 q^{47} +(2.33654 - 4.04700i) q^{49} -17.5669 q^{51} +(3.21432 - 5.56737i) q^{53} +(-10.2351 + 17.7276i) q^{57} +(-3.52543 + 6.10622i) q^{59} +(-5.64050 - 9.76963i) q^{61} -8.28100 q^{63} +(1.90321 - 3.29646i) q^{65} +(-3.80642 - 6.59292i) q^{67} +(-6.21432 - 10.7635i) q^{69} +(-1.38493 - 2.39877i) q^{71} +11.2859 q^{73} +4.00000 q^{75} +(7.95407 + 13.7768i) q^{79} +(-2.09210 - 3.62363i) q^{81} +(6.78346 - 11.7493i) q^{83} +11.5161 q^{85} +(-1.85950 - 3.22075i) q^{87} +(-4.46989 + 7.74207i) q^{89} +(-1.52543 + 2.64212i) q^{91} +(14.2351 + 24.6559i) q^{93} +(6.70964 - 11.6214i) q^{95} +11.9906 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + q^{5} - 2 q^{7} - 3 q^{9} + 6 q^{13} + 10 q^{15} - 5 q^{17} + 8 q^{19} - 12 q^{23} + 4 q^{25} - 16 q^{27} + 6 q^{29} + 32 q^{31} + 2 q^{35} + 15 q^{37} - 4 q^{39} - 5 q^{41} + 20 q^{43}+ \cdots + 18 q^{97}+O(q^{100}) \) 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
\(3\) 1.45161 + 2.51426i 0.838085 + 1.45161i 0.891494 + 0.453033i \(0.149658\pi\)
−0.0534085 + 0.998573i \(0.517009\pi\)
\(4\) 0 0
\(5\) −0.951606 − 1.64823i −0.425571 − 0.737111i 0.570902 − 0.821018i \(-0.306593\pi\)
−0.996474 + 0.0839071i \(0.973260\pi\)
\(6\) 0 0
\(7\) 0.762714 + 1.32106i 0.288279 + 0.499313i 0.973399 − 0.229117i \(-0.0735837\pi\)
−0.685120 + 0.728430i \(0.740250\pi\)
\(8\) 0 0
\(9\) −2.71432 + 4.70134i −0.904773 + 1.56711i
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 − 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 2.76271 − 4.78516i 0.713330 − 1.23552i
\(16\) 0 0
\(17\) −3.02543 + 5.24019i −0.733774 + 1.27093i 0.221485 + 0.975164i \(0.428910\pi\)
−0.955259 + 0.295770i \(0.904424\pi\)
\(18\) 0 0
\(19\) 3.52543 + 6.10622i 0.808789 + 1.40086i 0.913703 + 0.406382i \(0.133210\pi\)
−0.104915 + 0.994481i \(0.533457\pi\)
\(20\) 0 0
\(21\) −2.21432 + 3.83531i −0.483204 + 0.836934i
\(22\) 0 0
\(23\) −4.28100 −0.892649 −0.446325 − 0.894871i \(-0.647267\pi\)
−0.446325 + 0.894871i \(0.647267\pi\)
\(24\) 0 0
\(25\) 0.688892 − 1.19320i 0.137778 − 0.238639i
\(26\) 0 0
\(27\) −7.05086 −1.35694
\(28\) 0 0
\(29\) −1.28100 −0.237875 −0.118938 − 0.992902i \(-0.537949\pi\)
−0.118938 + 0.992902i \(0.537949\pi\)
\(30\) 0 0
\(31\) 9.80642 1.76129 0.880643 − 0.473781i \(-0.157111\pi\)
0.880643 + 0.473781i \(0.157111\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.45161 − 2.51426i 0.245366 − 0.424987i
\(36\) 0 0
\(37\) −0.854818 − 6.02240i −0.140531 − 0.990076i
\(38\) 0 0
\(39\) −2.90321 + 5.02851i −0.464886 + 0.805206i
\(40\) 0 0
\(41\) −0.811108 − 1.40488i −0.126674 − 0.219405i 0.795712 − 0.605675i \(-0.207097\pi\)
−0.922386 + 0.386270i \(0.873763\pi\)
\(42\) 0 0
\(43\) −7.76049 −1.18346 −0.591732 − 0.806135i \(-0.701556\pi\)
−0.591732 + 0.806135i \(0.701556\pi\)
\(44\) 0 0
\(45\) 10.3319 1.54018
\(46\) 0 0
\(47\) 5.80642 0.846954 0.423477 − 0.905907i \(-0.360809\pi\)
0.423477 + 0.905907i \(0.360809\pi\)
\(48\) 0 0
\(49\) 2.33654 − 4.04700i 0.333791 − 0.578143i
\(50\) 0 0
\(51\) −17.5669 −2.45986
\(52\) 0 0
\(53\) 3.21432 − 5.56737i 0.441521 − 0.764736i −0.556282 − 0.830994i \(-0.687772\pi\)
0.997803 + 0.0662573i \(0.0211058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.2351 + 17.7276i −1.35567 + 2.34808i
\(58\) 0 0
\(59\) −3.52543 + 6.10622i −0.458972 + 0.794962i −0.998907 − 0.0467444i \(-0.985115\pi\)
0.539935 + 0.841707i \(0.318449\pi\)
\(60\) 0 0
\(61\) −5.64050 − 9.76963i −0.722192 − 1.25087i −0.960119 − 0.279590i \(-0.909801\pi\)
0.237928 − 0.971283i \(-0.423532\pi\)
\(62\) 0 0
\(63\) −8.28100 −1.04331
\(64\) 0 0
\(65\) 1.90321 − 3.29646i 0.236064 − 0.408876i
\(66\) 0 0
\(67\) −3.80642 − 6.59292i −0.465029 − 0.805453i 0.534174 − 0.845374i \(-0.320623\pi\)
−0.999203 + 0.0399211i \(0.987289\pi\)
\(68\) 0 0
\(69\) −6.21432 − 10.7635i −0.748116 − 1.29578i
\(70\) 0 0
\(71\) −1.38493 − 2.39877i −0.164361 − 0.284681i 0.772067 − 0.635541i \(-0.219223\pi\)
−0.936428 + 0.350859i \(0.885889\pi\)
\(72\) 0 0
\(73\) 11.2859 1.32092 0.660458 − 0.750863i \(-0.270362\pi\)
0.660458 + 0.750863i \(0.270362\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.95407 + 13.7768i 0.894902 + 1.55002i 0.833925 + 0.551877i \(0.186088\pi\)
0.0609772 + 0.998139i \(0.480578\pi\)
\(80\) 0 0
\(81\) −2.09210 − 3.62363i −0.232456 − 0.402626i
\(82\) 0 0
\(83\) 6.78346 − 11.7493i 0.744581 − 1.28965i −0.205809 − 0.978592i \(-0.565983\pi\)
0.950390 − 0.311060i \(-0.100684\pi\)
\(84\) 0 0
\(85\) 11.5161 1.24909
\(86\) 0 0
\(87\) −1.85950 − 3.22075i −0.199360 − 0.345301i
\(88\) 0 0
\(89\) −4.46989 + 7.74207i −0.473807 + 0.820658i −0.999550 − 0.0299852i \(-0.990454\pi\)
0.525743 + 0.850643i \(0.323787\pi\)
\(90\) 0 0
\(91\) −1.52543 + 2.64212i −0.159908 + 0.276969i
\(92\) 0 0
\(93\) 14.2351 + 24.6559i 1.47611 + 2.55669i
\(94\) 0 0
\(95\) 6.70964 − 11.6214i 0.688394 − 1.19233i
\(96\) 0 0
\(97\) 11.9906 1.21746 0.608732 − 0.793376i \(-0.291678\pi\)
0.608732 + 0.793376i \(0.291678\pi\)
\(98\) 0 0
\(99\) 0 0
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.g.417.3 6
4.3 odd 2 296.2.i.b.121.1 ✓ 6
12.11 even 2 2664.2.r.i.1009.3 6
37.26 even 3 inner 592.2.i.g.433.3 6
148.63 odd 6 296.2.i.b.137.1 yes 6
444.359 even 6 2664.2.r.i.433.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.1 ✓ 6 4.3 odd 2
296.2.i.b.137.1 yes 6 148.63 odd 6
592.2.i.g.417.3 6 1.1 even 1 trivial
592.2.i.g.433.3 6 37.26 even 3 inner
2664.2.r.i.433.3 6 444.359 even 6
2664.2.r.i.1009.3 6 12.11 even 2