Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(1,1776)] 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("1776.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,-2,0,-4,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.1814313990\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 111)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.37033\) of defining polynomial
Character \(\chi\) \(=\) 1776.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.78220 q^{5} +4.06064 q^{7} +1.00000 q^{9} +2.24441 q^{11} +5.56441 q^{13} -1.78220 q^{15} -0.462211 q^{17} -2.49624 q^{19} -4.06064 q^{21} +7.59843 q^{23} -1.82375 q^{25} -1.00000 q^{27} +5.01909 q^{29} -6.06064 q^{31} -2.24441 q^{33} +7.23689 q^{35} -1.00000 q^{37} -5.56441 q^{39} -4.80130 q^{41} -11.5419 q^{43} +1.78220 q^{45} +5.31999 q^{47} +9.48883 q^{49} +0.462211 q^{51} -1.23689 q^{53} +4.00000 q^{55} +2.49624 q^{57} -14.8886 q^{59} -7.48130 q^{61} +4.06064 q^{63} +9.91690 q^{65} -4.06064 q^{67} -7.59843 q^{69} +9.80882 q^{71} +0.183770 q^{73} +1.82375 q^{75} +9.11376 q^{77} +6.98507 q^{79} +1.00000 q^{81} +0.435595 q^{83} -0.823754 q^{85} -5.01909 q^{87} -5.90349 q^{89} +22.5951 q^{91} +6.06064 q^{93} -4.44880 q^{95} +9.19687 q^{97} +2.24441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{19} + 4 q^{21} + 10 q^{23} - 4 q^{27} - 2 q^{29} - 4 q^{31} + 16 q^{35} - 4 q^{37} - 4 q^{39} + 12 q^{41} - 4 q^{43} - 2 q^{45}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.78220 0.797025 0.398513 − 0.917163i \(-0.369527\pi\)
0.398513 + 0.917163i \(0.369527\pi\)
\(6\) 0 0
\(7\) 4.06064 1.53478 0.767390 − 0.641181i \(-0.221555\pi\)
0.767390 + 0.641181i \(0.221555\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.24441 0.676716 0.338358 − 0.941017i \(-0.390128\pi\)
0.338358 + 0.941017i \(0.390128\pi\)
\(12\) 0 0
\(13\) 5.56441 1.54329 0.771644 − 0.636054i \(-0.219435\pi\)
0.771644 + 0.636054i \(0.219435\pi\)
\(14\) 0 0
\(15\) −1.78220 −0.460163
\(16\) 0 0
\(17\) −0.462211 −0.112103 −0.0560513 − 0.998428i \(-0.517851\pi\)
−0.0560513 + 0.998428i \(0.517851\pi\)
\(18\) 0 0
\(19\) −2.49624 −0.572676 −0.286338 − 0.958129i \(-0.592438\pi\)
−0.286338 + 0.958129i \(0.592438\pi\)
\(20\) 0 0
\(21\) −4.06064 −0.886105
\(22\) 0 0
\(23\) 7.59843 1.58438 0.792191 − 0.610273i \(-0.208940\pi\)
0.792191 + 0.610273i \(0.208940\pi\)
\(24\) 0 0
\(25\) −1.82375 −0.364751
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.01909 0.932022 0.466011 − 0.884779i \(-0.345691\pi\)
0.466011 + 0.884779i \(0.345691\pi\)
\(30\) 0 0
\(31\) −6.06064 −1.08852 −0.544262 − 0.838915i \(-0.683190\pi\)
−0.544262 + 0.838915i \(0.683190\pi\)
\(32\) 0 0
\(33\) −2.24441 −0.390702
\(34\) 0 0
\(35\) 7.23689 1.22326
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −5.56441 −0.891018
\(40\) 0 0
\(41\) −4.80130 −0.749836 −0.374918 − 0.927058i \(-0.622329\pi\)
−0.374918 + 0.927058i \(0.622329\pi\)
\(42\) 0 0
\(43\) −11.5419 −1.76013 −0.880065 − 0.474853i \(-0.842501\pi\)
−0.880065 + 0.474853i \(0.842501\pi\)
\(44\) 0 0
\(45\) 1.78220 0.265675
\(46\) 0 0
\(47\) 5.31999 0.776001 0.388000 − 0.921659i \(-0.373166\pi\)
0.388000 + 0.921659i \(0.373166\pi\)
\(48\) 0 0
\(49\) 9.48883 1.35555
\(50\) 0 0
\(51\) 0.462211 0.0647225
\(52\) 0 0
\(53\) −1.23689 −0.169900 −0.0849500 − 0.996385i \(-0.527073\pi\)
−0.0849500 + 0.996385i \(0.527073\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 2.49624 0.330635
\(58\) 0 0
\(59\) −14.8886 −1.93832 −0.969162 − 0.246423i \(-0.920745\pi\)
−0.969162 + 0.246423i \(0.920745\pi\)
\(60\) 0 0
\(61\) −7.48130 −0.957883 −0.478941 − 0.877847i \(-0.658979\pi\)
−0.478941 + 0.877847i \(0.658979\pi\)
\(62\) 0 0
\(63\) 4.06064 0.511593
\(64\) 0 0
\(65\) 9.91690 1.23004
\(66\) 0 0
\(67\) −4.06064 −0.496087 −0.248043 − 0.968749i \(-0.579788\pi\)
−0.248043 + 0.968749i \(0.579788\pi\)
\(68\) 0 0
\(69\) −7.59843 −0.914744
\(70\) 0 0
\(71\) 9.80882 1.16409 0.582046 − 0.813156i \(-0.302252\pi\)
0.582046 + 0.813156i \(0.302252\pi\)
\(72\) 0 0
\(73\) 0.183770 0.0215086 0.0107543 − 0.999942i \(-0.496577\pi\)
0.0107543 + 0.999942i \(0.496577\pi\)
\(74\) 0 0
\(75\) 1.82375 0.210589
\(76\) 0 0
\(77\) 9.11376 1.03861
\(78\) 0 0
\(79\) 6.98507 0.785881 0.392941 − 0.919564i \(-0.371458\pi\)
0.392941 + 0.919564i \(0.371458\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.435595 0.0478127 0.0239064 − 0.999714i \(-0.492390\pi\)
0.0239064 + 0.999714i \(0.492390\pi\)
\(84\) 0 0
\(85\) −0.823754 −0.0893486
\(86\) 0 0
\(87\) −5.01909 −0.538103
\(88\) 0 0
\(89\) −5.90349 −0.625769 −0.312884 − 0.949791i \(-0.601295\pi\)
−0.312884 + 0.949791i \(0.601295\pi\)
\(90\) 0 0
\(91\) 22.5951 2.36861
\(92\) 0 0
\(93\) 6.06064 0.628459
\(94\) 0 0
\(95\) −4.44880 −0.456438
\(96\) 0 0
\(97\) 9.19687 0.933800 0.466900 − 0.884310i \(-0.345371\pi\)
0.466900 + 0.884310i \(0.345371\pi\)
\(98\) 0 0
\(99\) 2.24441 0.225572
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 1776.2.a.u.1.4 4
3.2 odd 2 5328.2.a.bs.1.1 4
4.3 odd 2 111.2.a.b.1.3 ✓ 4
8.3 odd 2 7104.2.a.cc.1.1 4
8.5 even 2 7104.2.a.cf.1.1 4
12.11 even 2 333.2.a.g.1.2 4
20.19 odd 2 2775.2.a.x.1.2 4
28.27 even 2 5439.2.a.u.1.3 4
60.59 even 2 8325.2.a.bv.1.3 4
148.147 odd 2 4107.2.a.i.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.2.a.b.1.3 ✓ 4 4.3 odd 2
333.2.a.g.1.2 4 12.11 even 2
1776.2.a.u.1.4 4 1.1 even 1 trivial
2775.2.a.x.1.2 4 20.19 odd 2
4107.2.a.i.1.2 4 148.147 odd 2
5328.2.a.bs.1.1 4 3.2 odd 2
5439.2.a.u.1.3 4 28.27 even 2
7104.2.a.cc.1.1 4 8.3 odd 2
7104.2.a.cf.1.1 4 8.5 even 2
8325.2.a.bv.1.3 4 60.59 even 2