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Properties

Label 592.2.i.g.433.2
Level $592$
Weight $2$
Character 592.433
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 433.2
Root \(-0.740597 + 1.28275i\) of defining polynomial
Character \(\chi\) \(=\) 592.433
Dual form 592.2.i.g.417.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.403032 - 0.698071i) q^{3} +(0.0969683 - 0.167954i) q^{5} +(-2.07816 + 3.59948i) q^{7} +(1.17513 + 2.03539i) q^{9} +(1.00000 - 1.73205i) q^{13} +(-0.0781626 - 0.135382i) q^{15} +(2.65633 + 4.60089i) q^{17} +(-2.15633 + 3.73486i) q^{19} +(1.67513 + 2.90141i) q^{21} -5.76845 q^{23} +(2.48119 + 4.29755i) q^{25} +4.31265 q^{27} -2.76845 q^{29} +5.61213 q^{31} +(0.403032 + 0.698071i) q^{35} +(2.29090 - 5.63487i) q^{37} +(-0.806063 - 1.39614i) q^{39} +(0.981194 - 1.69948i) q^{41} +9.89446 q^{43} +0.455802 q^{45} +1.61213 q^{47} +(-5.13752 - 8.89844i) q^{49} +4.28233 q^{51} +(-0.675131 - 1.16936i) q^{53} +(1.73813 + 3.01054i) q^{57} +(2.15633 + 3.73486i) q^{59} +(-6.38423 + 11.0578i) q^{61} -9.76845 q^{63} +(-0.193937 - 0.335908i) q^{65} +(0.387873 - 0.671816i) q^{67} +(-2.32487 + 4.02679i) q^{69} +(5.04055 - 8.73049i) q^{71} -12.0508 q^{73} +4.00000 q^{75} +(-5.50659 + 9.53769i) q^{79} +(-1.78726 + 3.09562i) q^{81} +(-4.14117 - 7.17271i) q^{83} +1.03032 q^{85} +(-1.11577 + 1.93258i) q^{87} +(-7.74965 - 13.4228i) q^{89} +(4.15633 + 7.19897i) q^{91} +(2.26187 - 3.91767i) q^{93} +(0.418190 + 0.724327i) q^{95} +7.18664 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + q^{5} - 2 q^{7} - 3 q^{9} + 6 q^{13} + 10 q^{15} - 5 q^{17} + 8 q^{19} - 12 q^{23} + 4 q^{25} - 16 q^{27} + 6 q^{29} + 32 q^{31} + 2 q^{35} + 15 q^{37} - 4 q^{39} - 5 q^{41} + 20 q^{43}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.403032 − 0.698071i 0.232690 − 0.403032i −0.725909 − 0.687791i \(-0.758580\pi\)
0.958599 + 0.284760i \(0.0919138\pi\)
\(4\) 0 0
\(5\) 0.0969683 − 0.167954i 0.0433655 − 0.0751113i −0.843528 − 0.537085i \(-0.819525\pi\)
0.886893 + 0.461974i \(0.152859\pi\)
\(6\) 0 0
\(7\) −2.07816 + 3.59948i −0.785472 + 1.36048i 0.143245 + 0.989687i \(0.454246\pi\)
−0.928717 + 0.370790i \(0.879087\pi\)
\(8\) 0 0
\(9\) 1.17513 + 2.03539i 0.391710 + 0.678462i
\(10\) 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 −0.693375 − 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) −0.0781626 − 0.135382i −0.0201815 − 0.0349554i
\(16\) 0 0
\(17\) 2.65633 + 4.60089i 0.644253 + 1.11588i 0.984473 + 0.175534i \(0.0561652\pi\)
−0.340220 + 0.940346i \(0.610502\pi\)
\(18\) 0 0
\(19\) −2.15633 + 3.73486i −0.494695 + 0.856837i −0.999981 − 0.00611498i \(-0.998054\pi\)
0.505286 + 0.862952i \(0.331387\pi\)
\(20\) 0 0
\(21\) 1.67513 + 2.90141i 0.365544 + 0.633140i
\(22\) 0 0
\(23\) −5.76845 −1.20281 −0.601403 − 0.798946i \(-0.705391\pi\)
−0.601403 + 0.798946i \(0.705391\pi\)
\(24\) 0 0
\(25\) 2.48119 + 4.29755i 0.496239 + 0.859511i
\(26\) 0 0
\(27\) 4.31265 0.829970
\(28\) 0 0
\(29\) −2.76845 −0.514089 −0.257044 − 0.966400i \(-0.582749\pi\)
−0.257044 + 0.966400i \(0.582749\pi\)
\(30\) 0 0
\(31\) 5.61213 1.00797 0.503984 − 0.863713i \(-0.331867\pi\)
0.503984 + 0.863713i \(0.331867\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.403032 + 0.698071i 0.0681248 + 0.117996i
\(36\) 0 0
\(37\) 2.29090 − 5.63487i 0.376622 − 0.926367i
\(38\) 0 0
\(39\) −0.806063 − 1.39614i −0.129073 − 0.223562i
\(40\) 0 0
\(41\) 0.981194 − 1.69948i 0.153237 − 0.265414i −0.779179 − 0.626802i \(-0.784364\pi\)
0.932416 + 0.361388i \(0.117697\pi\)
\(42\) 0 0
\(43\) 9.89446 1.50889 0.754446 − 0.656363i \(-0.227906\pi\)
0.754446 + 0.656363i \(0.227906\pi\)
\(44\) 0 0
\(45\) 0.455802 0.0679469
\(46\) 0 0
\(47\) 1.61213 0.235153 0.117576 − 0.993064i \(-0.462487\pi\)
0.117576 + 0.993064i \(0.462487\pi\)
\(48\) 0 0
\(49\) −5.13752 − 8.89844i −0.733931 − 1.27121i
\(50\) 0 0
\(51\) 4.28233 0.599647
\(52\) 0 0
\(53\) −0.675131 − 1.16936i −0.0927364 − 0.160624i 0.815925 − 0.578157i \(-0.196228\pi\)
−0.908662 + 0.417533i \(0.862895\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.73813 + 3.01054i 0.230222 + 0.398755i
\(58\) 0 0
\(59\) 2.15633 + 3.73486i 0.280730 + 0.486238i 0.971565 − 0.236774i \(-0.0760902\pi\)
−0.690835 + 0.723012i \(0.742757\pi\)
\(60\) 0 0
\(61\) −6.38423 + 11.0578i −0.817416 + 1.41581i 0.0901634 + 0.995927i \(0.471261\pi\)
−0.907580 + 0.419880i \(0.862072\pi\)
\(62\) 0 0
\(63\) −9.76845 −1.23071
\(64\) 0 0
\(65\) −0.193937 − 0.335908i −0.0240549 − 0.0416643i
\(66\) 0 0
\(67\) 0.387873 − 0.671816i 0.0473862 − 0.0820754i −0.841359 − 0.540476i \(-0.818244\pi\)
0.888746 + 0.458401i \(0.151578\pi\)
\(68\) 0 0
\(69\) −2.32487 + 4.02679i −0.279881 + 0.484769i
\(70\) 0 0
\(71\) 5.04055 − 8.73049i 0.598203 − 1.03612i −0.394883 − 0.918731i \(-0.629215\pi\)
0.993086 − 0.117387i \(-0.0374518\pi\)
\(72\) 0 0
\(73\) −12.0508 −1.41044 −0.705219 − 0.708990i \(-0.749151\pi\)
−0.705219 + 0.708990i \(0.749151\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50659 + 9.53769i −0.619539 + 1.07307i 0.370030 + 0.929020i \(0.379347\pi\)
−0.989570 + 0.144054i \(0.953986\pi\)
\(80\) 0 0
\(81\) −1.78726 + 3.09562i −0.198584 + 0.343958i
\(82\) 0 0
\(83\) −4.14117 − 7.17271i −0.454552 − 0.787307i 0.544110 − 0.839014i \(-0.316867\pi\)
−0.998662 + 0.0517064i \(0.983534\pi\)
\(84\) 0 0
\(85\) 1.03032 0.111754
\(86\) 0 0
\(87\) −1.11577 + 1.93258i −0.119624 + 0.207194i
\(88\) 0 0
\(89\) −7.74965 − 13.4228i −0.821461 − 1.42281i −0.904594 − 0.426274i \(-0.859826\pi\)
0.0831334 − 0.996538i \(-0.473507\pi\)
\(90\) 0 0
\(91\) 4.15633 + 7.19897i 0.435701 + 0.754657i
\(92\) 0 0
\(93\) 2.26187 − 3.91767i 0.234544 − 0.406243i
\(94\) 0 0
\(95\) 0.418190 + 0.724327i 0.0429054 + 0.0743144i
\(96\) 0 0
\(97\) 7.18664 0.729693 0.364846 − 0.931068i \(-0.381121\pi\)
0.364846 + 0.931068i \(0.381121\pi\)
\(98\) 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.433.2 6
4.3 odd 2 296.2.i.b.137.2 yes 6
12.11 even 2 2664.2.r.i.433.2 6
37.10 even 3 inner 592.2.i.g.417.2 6
148.47 odd 6 296.2.i.b.121.2 ✓ 6
444.47 even 6 2664.2.r.i.1009.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.2 ✓ 6 148.47 odd 6
296.2.i.b.137.2 yes 6 4.3 odd 2
592.2.i.g.417.2 6 37.10 even 3 inner
592.2.i.g.433.2 6 1.1 even 1 trivial
2664.2.r.i.433.2 6 12.11 even 2
2664.2.r.i.1009.2 6 444.47 even 6