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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2368,2,Mod(961,2368)] 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("2368.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,0,0,8] 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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.3
Root \(-1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 2368.961
Dual form 2368.2.g.h.961.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.79129 q^{3} -0.791288i q^{5} +2.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +3.79129i q^{13} -1.41742i q^{15} +7.58258i q^{17} +1.58258i q^{19} +3.58258 q^{21} +3.79129i q^{23} +4.37386 q^{25} -5.00000 q^{27} -3.79129i q^{29} +8.37386i q^{31} -1.41742 q^{33} -1.58258i q^{35} +(-4.00000 + 4.58258i) q^{37} +6.79129i q^{39} +9.79129 q^{41} -6.00000i q^{43} -0.165151i q^{45} +7.58258 q^{47} -3.00000 q^{49} +13.5826i q^{51} +1.58258 q^{53} +0.626136i q^{55} +2.83485i q^{57} +1.58258i q^{59} -12.7913i q^{61} +0.417424 q^{63} +3.00000 q^{65} +6.37386 q^{67} +6.79129i q^{69} +9.16515 q^{71} -4.37386 q^{73} +7.83485 q^{75} -1.58258 q^{77} -8.20871i q^{79} -9.58258 q^{81} +15.1652 q^{83} +6.00000 q^{85} -6.79129i q^{87} +6.00000i q^{89} +7.58258i q^{91} +15.0000i q^{93} +1.25227 q^{95} -13.5826i q^{97} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 4 q^{21} - 10 q^{25} - 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 20 q^{63} + 12 q^{65} - 2 q^{67} + 10 q^{73} + 68 q^{75}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(1407\) \(1925\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.79129 1.03420 0.517100 − 0.855925i \(-0.327011\pi\)
0.517100 + 0.855925i \(0.327011\pi\)
\(4\) 0 0
\(5\) − 0.791288i − 0.353875i −0.984222 − 0.176937i \(-0.943381\pi\)
0.984222 − 0.176937i \(-0.0566190\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 0 0
\(13\) 3.79129i 1.05151i 0.850635 + 0.525757i \(0.176218\pi\)
−0.850635 + 0.525757i \(0.823782\pi\)
\(14\) 0 0
\(15\) − 1.41742i − 0.365977i
\(16\) 0 0
\(17\) 7.58258i 1.83904i 0.393038 + 0.919522i \(0.371424\pi\)
−0.393038 + 0.919522i \(0.628576\pi\)
\(18\) 0 0
\(19\) 1.58258i 0.363068i 0.983385 + 0.181534i \(0.0581062\pi\)
−0.983385 + 0.181534i \(0.941894\pi\)
\(20\) 0 0
\(21\) 3.58258 0.781782
\(22\) 0 0
\(23\) 3.79129i 0.790538i 0.918565 + 0.395269i \(0.129349\pi\)
−0.918565 + 0.395269i \(0.870651\pi\)
\(24\) 0 0
\(25\) 4.37386 0.874773
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) − 3.79129i − 0.704024i −0.935995 − 0.352012i \(-0.885498\pi\)
0.935995 − 0.352012i \(-0.114502\pi\)
\(30\) 0 0
\(31\) 8.37386i 1.50399i 0.659169 + 0.751995i \(0.270908\pi\)
−0.659169 + 0.751995i \(0.729092\pi\)
\(32\) 0 0
\(33\) −1.41742 −0.246742
\(34\) 0 0
\(35\) − 1.58258i − 0.267504i
\(36\) 0 0
\(37\) −4.00000 + 4.58258i −0.657596 + 0.753371i
\(38\) 0 0
\(39\) 6.79129i 1.08748i
\(40\) 0 0
\(41\) 9.79129 1.52914 0.764571 − 0.644539i \(-0.222951\pi\)
0.764571 + 0.644539i \(0.222951\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 − 0.457496i \(-0.848747\pi\)
0.889212 − 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) − 0.165151i − 0.0246193i
\(46\) 0 0
\(47\) 7.58258 1.10603 0.553016 − 0.833171i \(-0.313477\pi\)
0.553016 + 0.833171i \(0.313477\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 13.5826i 1.90194i
\(52\) 0 0
\(53\) 1.58258 0.217383 0.108692 − 0.994076i \(-0.465334\pi\)
0.108692 + 0.994076i \(0.465334\pi\)
\(54\) 0 0
\(55\) 0.626136i 0.0844282i
\(56\) 0 0
\(57\) 2.83485i 0.375485i
\(58\) 0 0
\(59\) 1.58258i 0.206034i 0.994680 + 0.103017i \(0.0328496\pi\)
−0.994680 + 0.103017i \(0.967150\pi\)
\(60\) 0 0
\(61\) − 12.7913i − 1.63776i −0.573967 − 0.818878i \(-0.694596\pi\)
0.573967 − 0.818878i \(-0.305404\pi\)
\(62\) 0 0
\(63\) 0.417424 0.0525905
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 6.37386 0.778691 0.389346 − 0.921092i \(-0.372701\pi\)
0.389346 + 0.921092i \(0.372701\pi\)
\(68\) 0 0
\(69\) 6.79129i 0.817575i
\(70\) 0 0
\(71\) 9.16515 1.08770 0.543852 − 0.839181i \(-0.316965\pi\)
0.543852 + 0.839181i \(0.316965\pi\)
\(72\) 0 0
\(73\) −4.37386 −0.511922 −0.255961 − 0.966687i \(-0.582392\pi\)
−0.255961 + 0.966687i \(0.582392\pi\)
\(74\) 0 0
\(75\) 7.83485 0.904690
\(76\) 0 0
\(77\) −1.58258 −0.180351
\(78\) 0 0
\(79\) − 8.20871i − 0.923552i −0.886997 − 0.461776i \(-0.847212\pi\)
0.886997 − 0.461776i \(-0.152788\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) 15.1652 1.66459 0.832296 − 0.554332i \(-0.187026\pi\)
0.832296 + 0.554332i \(0.187026\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) − 6.79129i − 0.728102i
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 7.58258i 0.794870i
\(92\) 0 0
\(93\) 15.0000i 1.55543i
\(94\) 0 0
\(95\) 1.25227 0.128480
\(96\) 0 0
\(97\) − 13.5826i − 1.37910i −0.724237 − 0.689551i \(-0.757808\pi\)
0.724237 − 0.689551i \(-0.242192\pi\)
\(98\) 0 0
\(99\) −0.165151 −0.0165983
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 2368.2.g.h.961.3 4
4.3 odd 2 2368.2.g.j.961.1 4
8.3 odd 2 74.2.b.a.73.2 ✓ 4
8.5 even 2 592.2.g.c.369.2 4
24.5 odd 2 5328.2.h.m.2737.2 4
24.11 even 2 666.2.c.b.73.3 4
37.36 even 2 inner 2368.2.g.h.961.4 4
40.3 even 4 1850.2.c.g.1849.3 4
40.19 odd 2 1850.2.d.e.1701.3 4
40.27 even 4 1850.2.c.h.1849.2 4
148.147 odd 2 2368.2.g.j.961.2 4
296.43 even 4 2738.2.a.h.1.1 2
296.147 odd 2 74.2.b.a.73.4 yes 4
296.179 even 4 2738.2.a.k.1.1 2
296.221 even 2 592.2.g.c.369.1 4
888.221 odd 2 5328.2.h.m.2737.3 4
888.443 even 2 666.2.c.b.73.2 4
1480.147 even 4 1850.2.c.g.1849.2 4
1480.443 even 4 1850.2.c.h.1849.3 4
1480.739 odd 2 1850.2.d.e.1701.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.2 ✓ 4 8.3 odd 2
74.2.b.a.73.4 yes 4 296.147 odd 2
592.2.g.c.369.1 4 296.221 even 2
592.2.g.c.369.2 4 8.5 even 2
666.2.c.b.73.2 4 888.443 even 2
666.2.c.b.73.3 4 24.11 even 2
1850.2.c.g.1849.2 4 1480.147 even 4
1850.2.c.g.1849.3 4 40.3 even 4
1850.2.c.h.1849.2 4 40.27 even 4
1850.2.c.h.1849.3 4 1480.443 even 4
1850.2.d.e.1701.1 4 1480.739 odd 2
1850.2.d.e.1701.3 4 40.19 odd 2
2368.2.g.h.961.3 4 1.1 even 1 trivial
2368.2.g.h.961.4 4 37.36 even 2 inner
2368.2.g.j.961.1 4 4.3 odd 2
2368.2.g.j.961.2 4 148.147 odd 2
2738.2.a.h.1.1 2 296.43 even 4
2738.2.a.k.1.1 2 296.179 even 4
5328.2.h.m.2737.2 4 24.5 odd 2
5328.2.h.m.2737.3 4 888.221 odd 2