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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3840,2,Mod(769,3840)] 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("3840.769"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,0,0,0,0,2,0,0,0,8,0,-8,0,0,0,2,0,0,0,0, 0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.6
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 3840.769
Dual form 3840.2.f.h.769.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(2.21432 - 0.311108i) q^{5} +1.37778i q^{7} -1.00000 q^{9} -4.42864 q^{11} +2.42864i q^{13} +(0.311108 + 2.21432i) q^{15} -1.37778i q^{17} +5.80642 q^{19} -1.37778 q^{21} +1.80642i q^{23} +(4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +4.42864 q^{29} +3.05086 q^{31} -4.42864i q^{33} +(0.428639 + 3.05086i) q^{35} -9.18421i q^{37} -2.42864 q^{39} +2.00000 q^{41} +7.61285i q^{43} +(-2.21432 + 0.311108i) q^{45} +7.05086i q^{47} +5.10171 q^{49} +1.37778 q^{51} +12.2351i q^{53} +(-9.80642 + 1.37778i) q^{55} +5.80642i q^{57} -9.67307 q^{59} -4.00000 q^{61} -1.37778i q^{63} +(0.755569 + 5.37778i) q^{65} +10.1017i q^{67} -1.80642 q^{69} -11.6128 q^{71} +5.24443i q^{73} +(1.37778 + 4.80642i) q^{75} -6.10171i q^{77} +6.66370 q^{79} +1.00000 q^{81} +10.1017i q^{83} +(-0.428639 - 3.05086i) q^{85} +4.42864i q^{87} -3.51114 q^{89} -3.34614 q^{91} +3.05086i q^{93} +(12.8573 - 1.80642i) q^{95} +5.24443i q^{97} +4.42864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 2 q^{15} + 8 q^{19} - 8 q^{21} + 2 q^{25} - 8 q^{31} - 24 q^{35} + 12 q^{39} + 12 q^{41} - 22 q^{49} + 8 q^{51} - 32 q^{55} - 32 q^{59} - 24 q^{61} + 4 q^{65} + 16 q^{69} - 16 q^{71} + 8 q^{75}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.21432 − 0.311108i 0.990274 − 0.139132i
\(6\) 0 0
\(7\) 1.37778i 0.520754i 0.965507 + 0.260377i \(0.0838468\pi\)
−0.965507 + 0.260377i \(0.916153\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.42864 −1.33529 −0.667643 − 0.744482i \(-0.732697\pi\)
−0.667643 + 0.744482i \(0.732697\pi\)
\(12\) 0 0
\(13\) 2.42864i 0.673583i 0.941579 + 0.336792i \(0.109342\pi\)
−0.941579 + 0.336792i \(0.890658\pi\)
\(14\) 0 0
\(15\) 0.311108 + 2.21432i 0.0803277 + 0.571735i
\(16\) 0 0
\(17\) − 1.37778i − 0.334162i −0.985943 − 0.167081i \(-0.946566\pi\)
0.985943 − 0.167081i \(-0.0534341\pi\)
\(18\) 0 0
\(19\) 5.80642 1.33208 0.666042 − 0.745914i \(-0.267987\pi\)
0.666042 + 0.745914i \(0.267987\pi\)
\(20\) 0 0
\(21\) −1.37778 −0.300657
\(22\) 0 0
\(23\) 1.80642i 0.376665i 0.982105 + 0.188333i \(0.0603083\pi\)
−0.982105 + 0.188333i \(0.939692\pi\)
\(24\) 0 0
\(25\) 4.80642 − 1.37778i 0.961285 − 0.275557i
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 4.42864 0.822378 0.411189 − 0.911550i \(-0.365114\pi\)
0.411189 + 0.911550i \(0.365114\pi\)
\(30\) 0 0
\(31\) 3.05086 0.547950 0.273975 − 0.961737i \(-0.411661\pi\)
0.273975 + 0.961737i \(0.411661\pi\)
\(32\) 0 0
\(33\) − 4.42864i − 0.770927i
\(34\) 0 0
\(35\) 0.428639 + 3.05086i 0.0724533 + 0.515689i
\(36\) 0 0
\(37\) − 9.18421i − 1.50987i −0.655797 − 0.754937i \(-0.727667\pi\)
0.655797 − 0.754937i \(-0.272333\pi\)
\(38\) 0 0
\(39\) −2.42864 −0.388894
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 7.61285i 1.16095i 0.814279 + 0.580474i \(0.197133\pi\)
−0.814279 + 0.580474i \(0.802867\pi\)
\(44\) 0 0
\(45\) −2.21432 + 0.311108i −0.330091 + 0.0463772i
\(46\) 0 0
\(47\) 7.05086i 1.02847i 0.857648 + 0.514236i \(0.171925\pi\)
−0.857648 + 0.514236i \(0.828075\pi\)
\(48\) 0 0
\(49\) 5.10171 0.728816
\(50\) 0 0
\(51\) 1.37778 0.192928
\(52\) 0 0
\(53\) 12.2351i 1.68062i 0.542110 + 0.840308i \(0.317626\pi\)
−0.542110 + 0.840308i \(0.682374\pi\)
\(54\) 0 0
\(55\) −9.80642 + 1.37778i −1.32230 + 0.185780i
\(56\) 0 0
\(57\) 5.80642i 0.769080i
\(58\) 0 0
\(59\) −9.67307 −1.25933 −0.629663 − 0.776868i \(-0.716807\pi\)
−0.629663 + 0.776868i \(0.716807\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) − 1.37778i − 0.173585i
\(64\) 0 0
\(65\) 0.755569 + 5.37778i 0.0937168 + 0.667032i
\(66\) 0 0
\(67\) 10.1017i 1.23412i 0.786916 + 0.617060i \(0.211676\pi\)
−0.786916 + 0.617060i \(0.788324\pi\)
\(68\) 0 0
\(69\) −1.80642 −0.217468
\(70\) 0 0
\(71\) −11.6128 −1.37819 −0.689096 − 0.724670i \(-0.741992\pi\)
−0.689096 + 0.724670i \(0.741992\pi\)
\(72\) 0 0
\(73\) 5.24443i 0.613814i 0.951739 + 0.306907i \(0.0992941\pi\)
−0.951739 + 0.306907i \(0.900706\pi\)
\(74\) 0 0
\(75\) 1.37778 + 4.80642i 0.159093 + 0.554998i
\(76\) 0 0
\(77\) − 6.10171i − 0.695354i
\(78\) 0 0
\(79\) 6.66370 0.749725 0.374863 − 0.927080i \(-0.377690\pi\)
0.374863 + 0.927080i \(0.377690\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.1017i 1.10881i 0.832248 + 0.554403i \(0.187054\pi\)
−0.832248 + 0.554403i \(0.812946\pi\)
\(84\) 0 0
\(85\) −0.428639 − 3.05086i −0.0464925 − 0.330912i
\(86\) 0 0
\(87\) 4.42864i 0.474800i
\(88\) 0 0
\(89\) −3.51114 −0.372180 −0.186090 − 0.982533i \(-0.559582\pi\)
−0.186090 + 0.982533i \(0.559582\pi\)
\(90\) 0 0
\(91\) −3.34614 −0.350771
\(92\) 0 0
\(93\) 3.05086i 0.316359i
\(94\) 0 0
\(95\) 12.8573 − 1.80642i 1.31913 − 0.185335i
\(96\) 0 0
\(97\) 5.24443i 0.532491i 0.963905 + 0.266246i \(0.0857832\pi\)
−0.963905 + 0.266246i \(0.914217\pi\)
\(98\) 0 0
\(99\) 4.42864 0.445095
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 3840.2.f.h.769.6 6
4.3 odd 2 3840.2.f.e.769.3 6
5.4 even 2 inner 3840.2.f.h.769.3 6
8.3 odd 2 3840.2.f.f.769.4 6
8.5 even 2 3840.2.f.g.769.1 6
16.3 odd 4 1920.2.d.e.1729.3 yes 6
16.5 even 4 1920.2.d.f.1729.4 yes 6
16.11 odd 4 1920.2.d.d.1729.4 yes 6
16.13 even 4 1920.2.d.c.1729.3 ✓ 6
20.19 odd 2 3840.2.f.e.769.6 6
40.19 odd 2 3840.2.f.f.769.1 6
40.29 even 2 3840.2.f.g.769.4 6
80.19 odd 4 1920.2.d.d.1729.3 yes 6
80.29 even 4 1920.2.d.f.1729.3 yes 6
80.59 odd 4 1920.2.d.e.1729.4 yes 6
80.69 even 4 1920.2.d.c.1729.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1920.2.d.c.1729.3 ✓ 6 16.13 even 4
1920.2.d.c.1729.4 yes 6 80.69 even 4
1920.2.d.d.1729.3 yes 6 80.19 odd 4
1920.2.d.d.1729.4 yes 6 16.11 odd 4
1920.2.d.e.1729.3 yes 6 16.3 odd 4
1920.2.d.e.1729.4 yes 6 80.59 odd 4
1920.2.d.f.1729.3 yes 6 80.29 even 4
1920.2.d.f.1729.4 yes 6 16.5 even 4
3840.2.f.e.769.3 6 4.3 odd 2
3840.2.f.e.769.6 6 20.19 odd 2
3840.2.f.f.769.1 6 40.19 odd 2
3840.2.f.f.769.4 6 8.3 odd 2
3840.2.f.g.769.1 6 8.5 even 2
3840.2.f.g.769.4 6 40.29 even 2
3840.2.f.h.769.3 6 5.4 even 2 inner
3840.2.f.h.769.6 6 1.1 even 1 trivial