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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(33,592)] 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("592.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3] 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 33.2
Root \(2.14169 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 592.33
Dual form 592.2.bc.d.305.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.238878 + 1.35474i) q^{3} +(0.266044 - 0.223238i) q^{5} +(0.365982 - 0.307095i) q^{7} +(1.04081 - 0.378824i) q^{9} +(1.29268 - 2.23899i) q^{11} +(4.21288 + 1.53336i) q^{13} +(0.365982 + 0.307095i) q^{15} +(-1.88864 + 0.687407i) q^{17} +(-0.611631 - 3.46873i) q^{19} +(0.503461 + 0.422454i) q^{21} +(2.60486 + 4.51175i) q^{23} +(-0.847296 + 4.80526i) q^{25} +(2.82530 + 4.89356i) q^{27} +(-0.114111 + 0.197646i) q^{29} +6.74658 q^{31} +(3.34205 + 1.21641i) q^{33} +(0.0288122 - 0.163402i) q^{35} +(2.42322 - 5.57925i) q^{37} +(-1.07095 + 6.07365i) q^{39} +(-10.0701 - 3.66521i) q^{41} -8.85889 q^{43} +(0.192334 - 0.333132i) q^{45} +(3.72890 + 6.45864i) q^{47} +(-1.17590 + 6.66887i) q^{49} +(-1.38241 - 2.39441i) q^{51} +(3.35194 + 2.81261i) q^{53} +(-0.155916 - 0.884246i) q^{55} +(4.55314 - 1.65721i) q^{57} +(-3.43016 - 2.87825i) q^{59} +(1.35175 + 0.491998i) q^{61} +(0.264583 - 0.458271i) q^{63} +(1.46312 - 0.532531i) q^{65} +(8.06111 - 6.76408i) q^{67} +(-5.49002 + 4.60667i) q^{69} +(-1.54193 - 8.74474i) q^{71} +7.18667 q^{73} -6.71229 q^{75} +(-0.214485 - 1.21641i) q^{77} +(-2.70857 + 2.27276i) q^{79} +(-3.40919 + 2.86065i) q^{81} +(1.32932 - 0.483834i) q^{83} +(-0.349006 + 0.604496i) q^{85} +(-0.295018 - 0.107378i) q^{87} +(2.92954 + 2.45818i) q^{89} +(2.01273 - 0.732572i) q^{91} +(1.61161 + 9.13989i) q^{93} +(-0.937074 - 0.786298i) q^{95} +(-9.24175 - 16.0072i) q^{97} +(0.497253 - 2.82006i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} - 33 q^{21} + 21 q^{23} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 42 q^{31} + 57 q^{33} + 9 q^{35} - 3 q^{37}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(1\) \(1\)

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
\(3\) 0.238878 + 1.35474i 0.137916 + 0.782162i 0.972784 + 0.231712i \(0.0744326\pi\)
−0.834868 + 0.550450i \(0.814456\pi\)
\(4\) 0 0
\(5\) 0.266044 − 0.223238i 0.118979 − 0.0998350i −0.581357 − 0.813649i \(-0.697478\pi\)
0.700336 + 0.713814i \(0.253034\pi\)
\(6\) 0 0
\(7\) 0.365982 − 0.307095i 0.138328 − 0.116071i −0.570998 − 0.820951i \(-0.693444\pi\)
0.709326 + 0.704880i \(0.248999\pi\)
\(8\) 0 0
\(9\) 1.04081 − 0.378824i 0.346937 − 0.126275i
\(10\) 0 0
\(11\) 1.29268 − 2.23899i 0.389758 − 0.675081i −0.602659 − 0.797999i \(-0.705892\pi\)
0.992417 + 0.122918i \(0.0392253\pi\)
\(12\) 0 0
\(13\) 4.21288 + 1.53336i 1.16844 + 0.425278i 0.852106 − 0.523369i \(-0.175325\pi\)
0.316336 + 0.948647i \(0.397547\pi\)
\(14\) 0 0
\(15\) 0.365982 + 0.307095i 0.0944962 + 0.0792917i
\(16\) 0 0
\(17\) −1.88864 + 0.687407i −0.458062 + 0.166721i −0.560737 − 0.827994i \(-0.689482\pi\)
0.102675 + 0.994715i \(0.467260\pi\)
\(18\) 0 0
\(19\) −0.611631 − 3.46873i −0.140318 − 0.795782i −0.971008 − 0.239047i \(-0.923165\pi\)
0.830690 − 0.556735i \(-0.187946\pi\)
\(20\) 0 0
\(21\) 0.503461 + 0.422454i 0.109864 + 0.0921869i
\(22\) 0 0
\(23\) 2.60486 + 4.51175i 0.543151 + 0.940765i 0.998721 + 0.0505649i \(0.0161022\pi\)
−0.455570 + 0.890200i \(0.650565\pi\)
\(24\) 0 0
\(25\) −0.847296 + 4.80526i −0.169459 + 0.961051i
\(26\) 0 0
\(27\) 2.82530 + 4.89356i 0.543729 + 0.941767i
\(28\) 0 0
\(29\) −0.114111 + 0.197646i −0.0211899 + 0.0367019i −0.876426 − 0.481537i \(-0.840079\pi\)
0.855236 + 0.518239i \(0.173412\pi\)
\(30\) 0 0
\(31\) 6.74658 1.21172 0.605861 − 0.795571i \(-0.292829\pi\)
0.605861 + 0.795571i \(0.292829\pi\)
\(32\) 0 0
\(33\) 3.34205 + 1.21641i 0.581776 + 0.211749i
\(34\) 0 0
\(35\) 0.0288122 − 0.163402i 0.00487015 − 0.0276200i
\(36\) 0 0
\(37\) 2.42322 − 5.57925i 0.398376 − 0.917222i
\(38\) 0 0
\(39\) −1.07095 + 6.07365i −0.171489 + 0.972563i
\(40\) 0 0
\(41\) −10.0701 − 3.66521i −1.57268 − 0.572410i −0.599086 − 0.800685i \(-0.704469\pi\)
−0.973597 + 0.228275i \(0.926691\pi\)
\(42\) 0 0
\(43\) −8.85889 −1.35097 −0.675484 − 0.737374i \(-0.736065\pi\)
−0.675484 + 0.737374i \(0.736065\pi\)
\(44\) 0 0
\(45\) 0.192334 − 0.333132i 0.0286715 − 0.0496604i
\(46\) 0 0
\(47\) 3.72890 + 6.45864i 0.543916 + 0.942090i 0.998674 + 0.0514750i \(0.0163923\pi\)
−0.454758 + 0.890615i \(0.650274\pi\)
\(48\) 0 0
\(49\) −1.17590 + 6.66887i −0.167986 + 0.952696i
\(50\) 0 0
\(51\) −1.38241 − 2.39441i −0.193577 − 0.335285i
\(52\) 0 0
\(53\) 3.35194 + 2.81261i 0.460424 + 0.386342i 0.843287 − 0.537464i \(-0.180617\pi\)
−0.382863 + 0.923805i \(0.625062\pi\)
\(54\) 0 0
\(55\) −0.155916 − 0.884246i −0.0210238 − 0.119232i
\(56\) 0 0
\(57\) 4.55314 − 1.65721i 0.603078 − 0.219502i
\(58\) 0 0
\(59\) −3.43016 − 2.87825i −0.446569 − 0.374716i 0.391592 − 0.920139i \(-0.371924\pi\)
−0.838161 + 0.545423i \(0.816369\pi\)
\(60\) 0 0
\(61\) 1.35175 + 0.491998i 0.173074 + 0.0629939i 0.427104 − 0.904203i \(-0.359534\pi\)
−0.254030 + 0.967196i \(0.581756\pi\)
\(62\) 0 0
\(63\) 0.264583 − 0.458271i 0.0333343 − 0.0577367i
\(64\) 0 0
\(65\) 1.46312 − 0.532531i 0.181477 − 0.0660523i
\(66\) 0 0
\(67\) 8.06111 − 6.76408i 0.984822 − 0.826363i 1.18267e−5 − 1.00000i \(-0.499996\pi\)
0.984810 + 0.173637i \(0.0555518\pi\)
\(68\) 0 0
\(69\) −5.49002 + 4.60667i −0.660921 + 0.554578i
\(70\) 0 0
\(71\) −1.54193 − 8.74474i −0.182994 − 1.03781i −0.928506 − 0.371318i \(-0.878906\pi\)
0.745512 − 0.666492i \(-0.232205\pi\)
\(72\) 0 0
\(73\) 7.18667 0.841136 0.420568 − 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(74\) 0 0
\(75\) −6.71229 −0.775069
\(76\) 0 0
\(77\) −0.214485 − 1.21641i −0.0244429 − 0.138622i
\(78\) 0 0
\(79\) −2.70857 + 2.27276i −0.304738 + 0.255705i −0.782313 − 0.622885i \(-0.785960\pi\)
0.477575 + 0.878591i \(0.341516\pi\)
\(80\) 0 0
\(81\) −3.40919 + 2.86065i −0.378799 + 0.317850i
\(82\) 0 0
\(83\) 1.32932 − 0.483834i 0.145912 − 0.0531077i −0.268032 − 0.963410i \(-0.586373\pi\)
0.413944 + 0.910302i \(0.364151\pi\)
\(84\) 0 0
\(85\) −0.349006 + 0.604496i −0.0378550 + 0.0655668i
\(86\) 0 0
\(87\) −0.295018 − 0.107378i −0.0316292 − 0.0115121i
\(88\) 0 0
\(89\) 2.92954 + 2.45818i 0.310531 + 0.260567i 0.784711 − 0.619861i \(-0.212811\pi\)
−0.474180 + 0.880428i \(0.657256\pi\)
\(90\) 0 0
\(91\) 2.01273 − 0.732572i 0.210991 − 0.0767944i
\(92\) 0 0
\(93\) 1.61161 + 9.13989i 0.167116 + 0.947762i
\(94\) 0 0
\(95\) −0.937074 − 0.786298i −0.0961417 − 0.0806725i
\(96\) 0 0
\(97\) −9.24175 − 16.0072i −0.938358 − 1.62528i −0.768534 − 0.639809i \(-0.779013\pi\)
−0.169824 − 0.985474i \(-0.554320\pi\)
\(98\) 0 0
\(99\) 0.497253 − 2.82006i 0.0499758 − 0.283427i
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.bc.d.33.2 12
4.3 odd 2 74.2.f.b.33.1 yes 12
12.11 even 2 666.2.x.g.181.1 12
37.9 even 9 inner 592.2.bc.d.305.2 12
148.3 odd 18 2738.2.a.q.1.3 6
148.71 odd 18 2738.2.a.t.1.3 6
148.83 odd 18 74.2.f.b.9.1 ✓ 12
444.83 even 18 666.2.x.g.379.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.9.1 ✓ 12 148.83 odd 18
74.2.f.b.33.1 yes 12 4.3 odd 2
592.2.bc.d.33.2 12 1.1 even 1 trivial
592.2.bc.d.305.2 12 37.9 even 9 inner
666.2.x.g.181.1 12 12.11 even 2
666.2.x.g.379.1 12 444.83 even 18
2738.2.a.q.1.3 6 148.3 odd 18
2738.2.a.t.1.3 6 148.71 odd 18