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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(1141,1860)] 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("1860.1141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-6,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - 3 x^{11} + 29 x^{10} + 405 x^{8} - 117 x^{7} + 2515 x^{6} + 1476 x^{5} + 7995 x^{4} - 2175 x^{3} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1141.4
Root \(0.175815 - 0.304520i\) of defining polynomial
Character \(\chi\) \(=\) 1860.1141
Dual form 1860.2.q.i.1741.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(0.374535 + 0.648713i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.10496 - 1.91385i) q^{11} +(-1.83154 + 3.17232i) q^{13} -1.00000 q^{15} +(-0.147956 - 0.256267i) q^{17} +(-3.81747 - 6.61205i) q^{19} +(0.374535 - 0.648713i) q^{21} +7.43373 q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -4.95899 q^{29} +(4.23884 - 3.61001i) q^{31} -2.20992 q^{33} +0.749069 q^{35} +(-3.96542 - 6.86832i) q^{37} +3.66308 q^{39} +(1.23737 - 2.14318i) q^{41} +(1.58941 + 2.75294i) q^{43} +(0.500000 + 0.866025i) q^{45} -6.87300 q^{47} +(3.21945 - 5.57625i) q^{49} +(-0.147956 + 0.256267i) q^{51} +(5.96158 - 10.3258i) q^{53} +(-1.10496 - 1.91385i) q^{55} +(-3.81747 + 6.61205i) q^{57} +(4.18506 + 7.24874i) q^{59} -6.93085 q^{61} -0.749069 q^{63} +(1.83154 + 3.17232i) q^{65} +(1.12982 - 1.95691i) q^{67} +(-3.71686 - 6.43780i) q^{69} +(-4.95423 + 8.58098i) q^{71} +(6.47236 - 11.2105i) q^{73} +(-0.500000 + 0.866025i) q^{75} +1.65539 q^{77} +(-3.75986 - 6.51227i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-3.97565 + 6.88603i) q^{83} -0.295911 q^{85} +(2.47950 + 4.29461i) q^{87} -13.9114 q^{89} -2.74390 q^{91} +(-5.24579 - 1.86594i) q^{93} -7.63494 q^{95} +18.1889 q^{97} +(1.10496 + 1.91385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 6 q^{5} - 2 q^{7} - 6 q^{9} - 2 q^{13} - 12 q^{15} - 2 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} - 6 q^{25} + 12 q^{27} - 20 q^{29} + 7 q^{31} - 4 q^{35} + 3 q^{37} + 4 q^{39} - 9 q^{41}+ \cdots - 76 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) 0.374535 + 0.648713i 0.141561 + 0.245191i 0.928085 − 0.372370i \(-0.121455\pi\)
−0.786524 + 0.617560i \(0.788121\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.10496 − 1.91385i 0.333158 − 0.577047i −0.649971 − 0.759959i \(-0.725219\pi\)
0.983129 + 0.182912i \(0.0585523\pi\)
\(12\) 0 0
\(13\) −1.83154 + 3.17232i −0.507978 + 0.879844i 0.491979 + 0.870607i \(0.336274\pi\)
−0.999957 + 0.00923682i \(0.997060\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.147956 − 0.256267i −0.0358845 − 0.0621538i 0.847525 − 0.530755i \(-0.178092\pi\)
−0.883410 + 0.468601i \(0.844758\pi\)
\(18\) 0 0
\(19\) −3.81747 − 6.61205i −0.875787 − 1.51691i −0.855922 − 0.517105i \(-0.827010\pi\)
−0.0198655 − 0.999803i \(-0.506324\pi\)
\(20\) 0 0
\(21\) 0.374535 − 0.648713i 0.0817302 − 0.141561i
\(22\) 0 0
\(23\) 7.43373 1.55004 0.775020 − 0.631937i \(-0.217740\pi\)
0.775020 + 0.631937i \(0.217740\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.95899 −0.920862 −0.460431 − 0.887696i \(-0.652305\pi\)
−0.460431 + 0.887696i \(0.652305\pi\)
\(30\) 0 0
\(31\) 4.23884 − 3.61001i 0.761319 − 0.648378i
\(32\) 0 0
\(33\) −2.20992 −0.384698
\(34\) 0 0
\(35\) 0.749069 0.126616
\(36\) 0 0
\(37\) −3.96542 − 6.86832i −0.651912 − 1.12914i −0.982658 − 0.185425i \(-0.940634\pi\)
0.330747 − 0.943720i \(-0.392699\pi\)
\(38\) 0 0
\(39\) 3.66308 0.586562
\(40\) 0 0
\(41\) 1.23737 − 2.14318i 0.193244 − 0.334709i −0.753079 − 0.657930i \(-0.771432\pi\)
0.946324 + 0.323221i \(0.104766\pi\)
\(42\) 0 0
\(43\) 1.58941 + 2.75294i 0.242383 + 0.419820i 0.961393 − 0.275180i \(-0.0887376\pi\)
−0.719010 + 0.695000i \(0.755404\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −6.87300 −1.00253 −0.501265 − 0.865294i \(-0.667132\pi\)
−0.501265 + 0.865294i \(0.667132\pi\)
\(48\) 0 0
\(49\) 3.21945 − 5.57625i 0.459921 − 0.796607i
\(50\) 0 0
\(51\) −0.147956 + 0.256267i −0.0207179 + 0.0358845i
\(52\) 0 0
\(53\) 5.96158 − 10.3258i 0.818885 − 1.41835i −0.0876186 − 0.996154i \(-0.527926\pi\)
0.906504 − 0.422197i \(-0.138741\pi\)
\(54\) 0 0
\(55\) −1.10496 − 1.91385i −0.148993 − 0.258063i
\(56\) 0 0
\(57\) −3.81747 + 6.61205i −0.505636 + 0.875787i
\(58\) 0 0
\(59\) 4.18506 + 7.24874i 0.544849 + 0.943706i 0.998616 + 0.0525856i \(0.0167462\pi\)
−0.453768 + 0.891120i \(0.649920\pi\)
\(60\) 0 0
\(61\) −6.93085 −0.887404 −0.443702 − 0.896174i \(-0.646335\pi\)
−0.443702 + 0.896174i \(0.646335\pi\)
\(62\) 0 0
\(63\) −0.749069 −0.0943739
\(64\) 0 0
\(65\) 1.83154 + 3.17232i 0.227175 + 0.393478i
\(66\) 0 0
\(67\) 1.12982 − 1.95691i 0.138030 − 0.239075i −0.788721 − 0.614751i \(-0.789256\pi\)
0.926751 + 0.375677i \(0.122590\pi\)
\(68\) 0 0
\(69\) −3.71686 − 6.43780i −0.447458 − 0.775020i
\(70\) 0 0
\(71\) −4.95423 + 8.58098i −0.587959 + 1.01838i 0.406540 + 0.913633i \(0.366735\pi\)
−0.994499 + 0.104742i \(0.966598\pi\)
\(72\) 0 0
\(73\) 6.47236 − 11.2105i 0.757533 − 1.31209i −0.186572 − 0.982441i \(-0.559738\pi\)
0.944105 − 0.329644i \(-0.106929\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 1.65539 0.188649
\(78\) 0 0
\(79\) −3.75986 − 6.51227i −0.423017 − 0.732687i 0.573216 − 0.819404i \(-0.305696\pi\)
−0.996233 + 0.0867173i \(0.972362\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −3.97565 + 6.88603i −0.436384 + 0.755839i −0.997407 − 0.0719605i \(-0.977074\pi\)
0.561023 + 0.827800i \(0.310408\pi\)
\(84\) 0 0
\(85\) −0.295911 −0.0320961
\(86\) 0 0
\(87\) 2.47950 + 4.29461i 0.265830 + 0.460431i
\(88\) 0 0
\(89\) −13.9114 −1.47461 −0.737304 − 0.675561i \(-0.763901\pi\)
−0.737304 + 0.675561i \(0.763901\pi\)
\(90\) 0 0
\(91\) −2.74390 −0.287639
\(92\) 0 0
\(93\) −5.24579 − 1.86594i −0.543963 − 0.193489i
\(94\) 0 0
\(95\) −7.63494 −0.783328
\(96\) 0 0
\(97\) 18.1889 1.84680 0.923400 − 0.383838i \(-0.125398\pi\)
0.923400 + 0.383838i \(0.125398\pi\)
\(98\) 0 0
\(99\) 1.10496 + 1.91385i 0.111053 + 0.192349i
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 1860.2.q.i.1141.4 ✓ 12
31.5 even 3 inner 1860.2.q.i.1741.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.q.i.1141.4 ✓ 12 1.1 even 1 trivial
1860.2.q.i.1741.4 yes 12 31.5 even 3 inner