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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(49,464)] 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("464.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.u (of order \(7\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
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: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 65.2
Root \(2.06920 - 0.996473i\) of defining polynomial
Character \(\chi\) \(=\) 464.65
Dual form 464.2.u.h.257.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.06920 + 0.996473i) q^{3} +(-0.788529 - 3.45477i) q^{5} +(-3.72857 - 1.79558i) q^{7} +(1.41815 + 1.77830i) q^{9} +(1.14832 - 1.43995i) q^{11} +(2.09403 - 2.62583i) q^{13} +(1.81096 - 7.93435i) q^{15} +3.52078 q^{17} +(2.45556 - 1.18253i) q^{19} +(-5.92589 - 7.43083i) q^{21} +(-0.679736 + 2.97812i) q^{23} +(-6.80881 + 3.27895i) q^{25} +(-0.370748 - 1.62435i) q^{27} +(0.127372 + 5.38366i) q^{29} +(0.196643 + 0.861548i) q^{31} +(3.81096 - 1.83526i) q^{33} +(-3.26324 + 14.2972i) q^{35} +(3.04846 + 3.82264i) q^{37} +(6.94952 - 3.34671i) q^{39} -3.01488 q^{41} +(0.409830 - 1.79558i) q^{43} +(5.02538 - 6.30163i) q^{45} +(-1.25592 + 1.57487i) q^{47} +(6.31365 + 7.91707i) q^{49} +(7.28518 + 3.50836i) q^{51} +(1.47479 + 6.46147i) q^{53} +(-5.88016 - 2.83174i) q^{55} +6.25940 q^{57} -6.12406 q^{59} +(-1.64476 - 0.792074i) q^{61} +(-2.09457 - 9.17693i) q^{63} +(-10.7228 - 5.16384i) q^{65} +(0.0862879 + 0.108202i) q^{67} +(-4.37412 + 5.48497i) q^{69} +(8.17273 - 10.2483i) q^{71} +(3.42387 - 15.0009i) q^{73} -17.3562 q^{75} +(-6.86712 + 3.30703i) q^{77} +(9.90051 + 12.4148i) q^{79} +(2.36987 - 10.3831i) q^{81} +(-0.0422914 + 0.0203665i) q^{83} +(-2.77623 - 12.1635i) q^{85} +(-5.10111 + 11.2668i) q^{87} +(0.800961 + 3.50924i) q^{89} +(-12.5226 + 6.03056i) q^{91} +(-0.451617 + 1.97866i) q^{93} +(-6.02166 - 7.55092i) q^{95} +(-4.71887 + 2.27249i) q^{97} +4.18915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\)

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 0
\(3\) 2.06920 + 0.996473i 1.19465 + 0.575314i 0.922147 − 0.386840i \(-0.126434\pi\)
0.272505 + 0.962154i \(0.412148\pi\)
\(4\) 0 0
\(5\) −0.788529 − 3.45477i −0.352641 − 1.54502i −0.771058 − 0.636765i \(-0.780272\pi\)
0.418417 − 0.908255i \(-0.362585\pi\)
\(6\) 0 0
\(7\) −3.72857 − 1.79558i −1.40927 − 0.678666i −0.434247 − 0.900794i \(-0.642986\pi\)
−0.975018 + 0.222127i \(0.928700\pi\)
\(8\) 0 0
\(9\) 1.41815 + 1.77830i 0.472717 + 0.592768i
\(10\) 0 0
\(11\) 1.14832 − 1.43995i 0.346231 − 0.434160i −0.577975 − 0.816055i \(-0.696157\pi\)
0.924206 + 0.381895i \(0.124728\pi\)
\(12\) 0 0
\(13\) 2.09403 − 2.62583i 0.580778 − 0.728273i −0.401467 − 0.915873i \(-0.631500\pi\)
0.982245 + 0.187601i \(0.0600710\pi\)
\(14\) 0 0
\(15\) 1.81096 − 7.93435i 0.467589 − 2.04864i
\(16\) 0 0
\(17\) 3.52078 0.853914 0.426957 − 0.904272i \(-0.359586\pi\)
0.426957 + 0.904272i \(0.359586\pi\)
\(18\) 0 0
\(19\) 2.45556 − 1.18253i 0.563344 − 0.271292i −0.130463 − 0.991453i \(-0.541646\pi\)
0.693807 + 0.720161i \(0.255932\pi\)
\(20\) 0 0
\(21\) −5.92589 − 7.43083i −1.29313 − 1.62154i
\(22\) 0 0
\(23\) −0.679736 + 2.97812i −0.141735 + 0.620980i 0.853297 + 0.521425i \(0.174599\pi\)
−0.995032 + 0.0995556i \(0.968258\pi\)
\(24\) 0 0
\(25\) −6.80881 + 3.27895i −1.36176 + 0.655790i
\(26\) 0 0
\(27\) −0.370748 − 1.62435i −0.0713504 − 0.312607i
\(28\) 0 0
\(29\) 0.127372 + 5.38366i 0.0236524 + 0.999720i
\(30\) 0 0
\(31\) 0.196643 + 0.861548i 0.0353181 + 0.154739i 0.989512 − 0.144450i \(-0.0461412\pi\)
−0.954194 + 0.299188i \(0.903284\pi\)
\(32\) 0 0
\(33\) 3.81096 − 1.83526i 0.663404 − 0.319478i
\(34\) 0 0
\(35\) −3.26324 + 14.2972i −0.551589 + 2.41667i
\(36\) 0 0
\(37\) 3.04846 + 3.82264i 0.501163 + 0.628439i 0.966491 − 0.256700i \(-0.0826351\pi\)
−0.465328 + 0.885138i \(0.654064\pi\)
\(38\) 0 0
\(39\) 6.94952 − 3.34671i 1.11281 − 0.535903i
\(40\) 0 0
\(41\) −3.01488 −0.470846 −0.235423 − 0.971893i \(-0.575647\pi\)
−0.235423 + 0.971893i \(0.575647\pi\)
\(42\) 0 0
\(43\) 0.409830 − 1.79558i 0.0624985 − 0.273824i −0.934017 − 0.357228i \(-0.883722\pi\)
0.996516 + 0.0834039i \(0.0265792\pi\)
\(44\) 0 0
\(45\) 5.02538 − 6.30163i 0.749139 − 0.939391i
\(46\) 0 0
\(47\) −1.25592 + 1.57487i −0.183194 + 0.229718i −0.864946 − 0.501866i \(-0.832647\pi\)
0.681751 + 0.731584i \(0.261219\pi\)
\(48\) 0 0
\(49\) 6.31365 + 7.91707i 0.901950 + 1.13101i
\(50\) 0 0
\(51\) 7.28518 + 3.50836i 1.02013 + 0.491269i
\(52\) 0 0
\(53\) 1.47479 + 6.46147i 0.202578 + 0.887551i 0.969360 + 0.245644i \(0.0789993\pi\)
−0.766783 + 0.641907i \(0.778144\pi\)
\(54\) 0 0
\(55\) −5.88016 − 2.83174i −0.792881 − 0.381831i
\(56\) 0 0
\(57\) 6.25940 0.829077
\(58\) 0 0
\(59\) −6.12406 −0.797285 −0.398642 − 0.917106i \(-0.630519\pi\)
−0.398642 + 0.917106i \(0.630519\pi\)
\(60\) 0 0
\(61\) −1.64476 − 0.792074i −0.210590 − 0.101415i 0.325616 − 0.945502i \(-0.394428\pi\)
−0.536205 + 0.844088i \(0.680143\pi\)
\(62\) 0 0
\(63\) −2.09457 − 9.17693i −0.263892 − 1.15618i
\(64\) 0 0
\(65\) −10.7228 − 5.16384i −1.33000 − 0.640495i
\(66\) 0 0
\(67\) 0.0862879 + 0.108202i 0.0105417 + 0.0132189i 0.787074 − 0.616858i \(-0.211595\pi\)
−0.776533 + 0.630077i \(0.783023\pi\)
\(68\) 0 0
\(69\) −4.37412 + 5.48497i −0.526582 + 0.660313i
\(70\) 0 0
\(71\) 8.17273 − 10.2483i 0.969925 − 1.21625i −0.00640935 − 0.999979i \(-0.502040\pi\)
0.976334 − 0.216268i \(-0.0693884\pi\)
\(72\) 0 0
\(73\) 3.42387 − 15.0009i 0.400733 − 1.75573i −0.223709 − 0.974656i \(-0.571817\pi\)
0.624442 − 0.781071i \(-0.285326\pi\)
\(74\) 0 0
\(75\) −17.3562 −2.00412
\(76\) 0 0
\(77\) −6.86712 + 3.30703i −0.782581 + 0.376871i
\(78\) 0 0
\(79\) 9.90051 + 12.4148i 1.11389 + 1.39678i 0.908390 + 0.418123i \(0.137312\pi\)
0.205504 + 0.978656i \(0.434117\pi\)
\(80\) 0 0
\(81\) 2.36987 − 10.3831i 0.263319 − 1.15367i
\(82\) 0 0
\(83\) −0.0422914 + 0.0203665i −0.00464209 + 0.00223551i −0.436203 − 0.899848i \(-0.643677\pi\)
0.431561 + 0.902084i \(0.357963\pi\)
\(84\) 0 0
\(85\) −2.77623 − 12.1635i −0.301125 − 1.31931i
\(86\) 0 0
\(87\) −5.10111 + 11.2668i −0.546897 + 1.20793i
\(88\) 0 0
\(89\) 0.800961 + 3.50924i 0.0849017 + 0.371979i 0.999474 − 0.0324449i \(-0.0103293\pi\)
−0.914572 + 0.404423i \(0.867472\pi\)
\(90\) 0 0
\(91\) −12.5226 + 6.03056i −1.31272 + 0.632175i
\(92\) 0 0
\(93\) −0.451617 + 1.97866i −0.0468305 + 0.205178i
\(94\) 0 0
\(95\) −6.02166 − 7.55092i −0.617809 − 0.774708i
\(96\) 0 0
\(97\) −4.71887 + 2.27249i −0.479129 + 0.230736i −0.657829 − 0.753167i \(-0.728525\pi\)
0.178700 + 0.983904i \(0.442811\pi\)
\(98\) 0 0
\(99\) 4.18915 0.421025
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 464.2.u.h.65.2 12
4.3 odd 2 58.2.d.b.7.1 ✓ 12
12.11 even 2 522.2.k.h.181.2 12
29.25 even 7 inner 464.2.u.h.257.2 12
116.27 even 28 1682.2.b.i.1681.11 12
116.31 even 28 1682.2.b.i.1681.2 12
116.63 odd 14 1682.2.a.q.1.2 6
116.83 odd 14 58.2.d.b.25.1 yes 12
116.111 odd 14 1682.2.a.t.1.5 6
348.83 even 14 522.2.k.h.199.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.1 ✓ 12 4.3 odd 2
58.2.d.b.25.1 yes 12 116.83 odd 14
464.2.u.h.65.2 12 1.1 even 1 trivial
464.2.u.h.257.2 12 29.25 even 7 inner
522.2.k.h.181.2 12 12.11 even 2
522.2.k.h.199.2 12 348.83 even 14
1682.2.a.q.1.2 6 116.63 odd 14
1682.2.a.t.1.5 6 116.111 odd 14
1682.2.b.i.1681.2 12 116.31 even 28
1682.2.b.i.1681.11 12 116.27 even 28