Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,2,0,-1,0,-3,0,-2,0,-1,0,-1,0,-5,0,10,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.1011730757\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4640)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9280.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.61803 q^{3} +1.00000 q^{5} -3.85410 q^{7} -0.381966 q^{9} +1.23607 q^{11} -6.09017 q^{13} -1.61803 q^{15} -1.38197 q^{17} +7.23607 q^{19} +6.23607 q^{21} +0.854102 q^{23} +1.00000 q^{25} +5.47214 q^{27} -1.00000 q^{29} -0.618034 q^{31} -2.00000 q^{33} -3.85410 q^{35} -4.76393 q^{37} +9.85410 q^{39} +9.70820 q^{41} -5.38197 q^{43} -0.381966 q^{45} +8.00000 q^{47} +7.85410 q^{49} +2.23607 q^{51} +6.32624 q^{53} +1.23607 q^{55} -11.7082 q^{57} -11.6180 q^{59} -8.85410 q^{61} +1.47214 q^{63} -6.09017 q^{65} +6.47214 q^{67} -1.38197 q^{69} +4.94427 q^{71} +13.0902 q^{73} -1.61803 q^{75} -4.76393 q^{77} +14.0902 q^{79} -7.70820 q^{81} +2.29180 q^{83} -1.38197 q^{85} +1.61803 q^{87} +11.7082 q^{89} +23.4721 q^{91} +1.00000 q^{93} +7.23607 q^{95} -15.7984 q^{97} -0.472136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - 2 q^{11} - q^{13} - q^{15} - 5 q^{17} + 10 q^{19} + 8 q^{21} - 5 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} + q^{31} - 4 q^{33} - q^{35} - 14 q^{37} + 13 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) −1.61803 −0.934172 −0.467086 − 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.85410 −1.45671 −0.728357 − 0.685198i \(-0.759716\pi\)
−0.728357 + 0.685198i \(0.759716\pi\)
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) 1.23607 0.372689 0.186344 − 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 0 0
\(13\) −6.09017 −1.68911 −0.844555 − 0.535469i \(-0.820135\pi\)
−0.844555 + 0.535469i \(0.820135\pi\)
\(14\) 0 0
\(15\) −1.61803 −0.417775
\(16\) 0 0
\(17\) −1.38197 −0.335176 −0.167588 − 0.985857i \(-0.553598\pi\)
−0.167588 + 0.985857i \(0.553598\pi\)
\(18\) 0 0
\(19\) 7.23607 1.66007 0.830034 − 0.557713i \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) 0 0
\(21\) 6.23607 1.36082
\(22\) 0 0
\(23\) 0.854102 0.178093 0.0890463 − 0.996027i \(-0.471618\pi\)
0.0890463 + 0.996027i \(0.471618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.47214 1.05311
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −0.618034 −0.111002 −0.0555011 − 0.998459i \(-0.517676\pi\)
−0.0555011 + 0.998459i \(0.517676\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −3.85410 −0.651462
\(36\) 0 0
\(37\) −4.76393 −0.783186 −0.391593 − 0.920139i \(-0.628076\pi\)
−0.391593 + 0.920139i \(0.628076\pi\)
\(38\) 0 0
\(39\) 9.85410 1.57792
\(40\) 0 0
\(41\) 9.70820 1.51617 0.758083 − 0.652158i \(-0.226136\pi\)
0.758083 + 0.652158i \(0.226136\pi\)
\(42\) 0 0
\(43\) −5.38197 −0.820742 −0.410371 − 0.911919i \(-0.634601\pi\)
−0.410371 + 0.911919i \(0.634601\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 7.85410 1.12201
\(50\) 0 0
\(51\) 2.23607 0.313112
\(52\) 0 0
\(53\) 6.32624 0.868976 0.434488 − 0.900678i \(-0.356929\pi\)
0.434488 + 0.900678i \(0.356929\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) 0 0
\(57\) −11.7082 −1.55079
\(58\) 0 0
\(59\) −11.6180 −1.51254 −0.756270 − 0.654260i \(-0.772980\pi\)
−0.756270 + 0.654260i \(0.772980\pi\)
\(60\) 0 0
\(61\) −8.85410 −1.13365 −0.566826 − 0.823838i \(-0.691829\pi\)
−0.566826 + 0.823838i \(0.691829\pi\)
\(62\) 0 0
\(63\) 1.47214 0.185472
\(64\) 0 0
\(65\) −6.09017 −0.755393
\(66\) 0 0
\(67\) 6.47214 0.790697 0.395349 − 0.918531i \(-0.370624\pi\)
0.395349 + 0.918531i \(0.370624\pi\)
\(68\) 0 0
\(69\) −1.38197 −0.166369
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 − 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 0 0
\(73\) 13.0902 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(74\) 0 0
\(75\) −1.61803 −0.186834
\(76\) 0 0
\(77\) −4.76393 −0.542900
\(78\) 0 0
\(79\) 14.0902 1.58527 0.792634 − 0.609698i \(-0.208709\pi\)
0.792634 + 0.609698i \(0.208709\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 2.29180 0.251557 0.125779 − 0.992058i \(-0.459857\pi\)
0.125779 + 0.992058i \(0.459857\pi\)
\(84\) 0 0
\(85\) −1.38197 −0.149895
\(86\) 0 0
\(87\) 1.61803 0.173471
\(88\) 0 0
\(89\) 11.7082 1.24107 0.620534 − 0.784180i \(-0.286916\pi\)
0.620534 + 0.784180i \(0.286916\pi\)
\(90\) 0 0
\(91\) 23.4721 2.46055
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 7.23607 0.742405
\(96\) 0 0
\(97\) −15.7984 −1.60408 −0.802041 − 0.597269i \(-0.796252\pi\)
−0.802041 + 0.597269i \(0.796252\pi\)
\(98\) 0 0
\(99\) −0.472136 −0.0474514
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 9280.2.a.y.1.1 2
4.3 odd 2 9280.2.a.bd.1.2 2
8.3 odd 2 4640.2.a.g.1.1 ✓ 2
8.5 even 2 4640.2.a.i.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.g.1.1 ✓ 2 8.3 odd 2
4640.2.a.i.1.2 yes 2 8.5 even 2
9280.2.a.y.1.1 2 1.1 even 1 trivial
9280.2.a.bd.1.2 2 4.3 odd 2