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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1320,2,Mod(529,1320)] 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("1320.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-2,0,0,0,-6,0,-6] 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: no
Analytic conductor: \(10.5402530668\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 529.1
Root \(-0.854638 + 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 1320.529
Dual form 1320.2.d.a.529.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.70928i q^{7} -1.00000 q^{9} -1.00000 q^{11} +2.63090i q^{13} +(-0.539189 + 2.17009i) q^{15} -2.78765i q^{17} +5.41855 q^{19} +1.70928 q^{21} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +2.92162 q^{29} +2.34017 q^{31} +1.00000i q^{33} +(0.921622 - 3.70928i) q^{35} +1.26180i q^{37} +2.63090 q^{39} +9.75872 q^{41} -1.70928i q^{43} +(2.17009 + 0.539189i) q^{45} -3.41855i q^{47} +4.07838 q^{49} -2.78765 q^{51} +2.92162i q^{53} +(2.17009 + 0.539189i) q^{55} -5.41855i q^{57} +2.92162 q^{59} -2.00000 q^{61} -1.70928i q^{63} +(1.41855 - 5.70928i) q^{65} -6.83710i q^{67} -6.92162 q^{71} +10.0494i q^{73} +(2.34017 - 4.41855i) q^{75} -1.70928i q^{77} +14.0989 q^{79} +1.00000 q^{81} -15.3112i q^{83} +(-1.50307 + 6.04945i) q^{85} -2.92162i q^{87} +17.0205 q^{89} -4.49693 q^{91} -2.34017i q^{93} +(-11.7587 - 2.92162i) q^{95} +0.680346i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9} - 6 q^{11} + 4 q^{19} - 4 q^{21} - 2 q^{25} + 24 q^{29} - 8 q^{31} + 12 q^{35} + 8 q^{39} + 8 q^{41} + 2 q^{45} + 18 q^{49} + 4 q^{51} + 2 q^{55} + 24 q^{59} - 12 q^{61} - 20 q^{65}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) −2.17009 − 0.539189i −0.970492 − 0.241133i
\(6\) 0 0
\(7\) 1.70928i 0.646045i 0.946391 + 0.323023i \(0.104699\pi\)
−0.946391 + 0.323023i \(0.895301\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.63090i 0.729680i 0.931070 + 0.364840i \(0.118876\pi\)
−0.931070 + 0.364840i \(0.881124\pi\)
\(14\) 0 0
\(15\) −0.539189 + 2.17009i −0.139218 + 0.560314i
\(16\) 0 0
\(17\) − 2.78765i − 0.676105i −0.941127 − 0.338053i \(-0.890232\pi\)
0.941127 − 0.338053i \(-0.109768\pi\)
\(18\) 0 0
\(19\) 5.41855 1.24310 0.621550 − 0.783374i \(-0.286503\pi\)
0.621550 + 0.783374i \(0.286503\pi\)
\(20\) 0 0
\(21\) 1.70928 0.372994
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 4.41855 + 2.34017i 0.883710 + 0.468035i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.92162 0.542532 0.271266 − 0.962504i \(-0.412558\pi\)
0.271266 + 0.962504i \(0.412558\pi\)
\(30\) 0 0
\(31\) 2.34017 0.420307 0.210154 − 0.977668i \(-0.432604\pi\)
0.210154 + 0.977668i \(0.432604\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0.921622 − 3.70928i 0.155783 − 0.626982i
\(36\) 0 0
\(37\) 1.26180i 0.207438i 0.994607 + 0.103719i \(0.0330742\pi\)
−0.994607 + 0.103719i \(0.966926\pi\)
\(38\) 0 0
\(39\) 2.63090 0.421281
\(40\) 0 0
\(41\) 9.75872 1.52406 0.762028 − 0.647544i \(-0.224204\pi\)
0.762028 + 0.647544i \(0.224204\pi\)
\(42\) 0 0
\(43\) − 1.70928i − 0.260662i −0.991471 − 0.130331i \(-0.958396\pi\)
0.991471 − 0.130331i \(-0.0416040\pi\)
\(44\) 0 0
\(45\) 2.17009 + 0.539189i 0.323497 + 0.0803775i
\(46\) 0 0
\(47\) − 3.41855i − 0.498647i −0.968420 − 0.249323i \(-0.919792\pi\)
0.968420 − 0.249323i \(-0.0802082\pi\)
\(48\) 0 0
\(49\) 4.07838 0.582625
\(50\) 0 0
\(51\) −2.78765 −0.390350
\(52\) 0 0
\(53\) 2.92162i 0.401316i 0.979661 + 0.200658i \(0.0643079\pi\)
−0.979661 + 0.200658i \(0.935692\pi\)
\(54\) 0 0
\(55\) 2.17009 + 0.539189i 0.292614 + 0.0727042i
\(56\) 0 0
\(57\) − 5.41855i − 0.717705i
\(58\) 0 0
\(59\) 2.92162 0.380363 0.190181 − 0.981749i \(-0.439092\pi\)
0.190181 + 0.981749i \(0.439092\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) − 1.70928i − 0.215348i
\(64\) 0 0
\(65\) 1.41855 − 5.70928i 0.175950 − 0.708148i
\(66\) 0 0
\(67\) − 6.83710i − 0.835285i −0.908611 − 0.417642i \(-0.862856\pi\)
0.908611 − 0.417642i \(-0.137144\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.92162 −0.821445 −0.410723 − 0.911760i \(-0.634724\pi\)
−0.410723 + 0.911760i \(0.634724\pi\)
\(72\) 0 0
\(73\) 10.0494i 1.17620i 0.808789 + 0.588099i \(0.200124\pi\)
−0.808789 + 0.588099i \(0.799876\pi\)
\(74\) 0 0
\(75\) 2.34017 − 4.41855i 0.270220 − 0.510210i
\(76\) 0 0
\(77\) − 1.70928i − 0.194790i
\(78\) 0 0
\(79\) 14.0989 1.58625 0.793125 − 0.609059i \(-0.208453\pi\)
0.793125 + 0.609059i \(0.208453\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 15.3112i − 1.68063i −0.542101 − 0.840314i \(-0.682371\pi\)
0.542101 − 0.840314i \(-0.317629\pi\)
\(84\) 0 0
\(85\) −1.50307 + 6.04945i −0.163031 + 0.656155i
\(86\) 0 0
\(87\) − 2.92162i − 0.313231i
\(88\) 0 0
\(89\) 17.0205 1.80417 0.902086 − 0.431557i \(-0.142036\pi\)
0.902086 + 0.431557i \(0.142036\pi\)
\(90\) 0 0
\(91\) −4.49693 −0.471406
\(92\) 0 0
\(93\) − 2.34017i − 0.242665i
\(94\) 0 0
\(95\) −11.7587 − 2.92162i −1.20642 − 0.299752i
\(96\) 0 0
\(97\) 0.680346i 0.0690787i 0.999403 + 0.0345393i \(0.0109964\pi\)
−0.999403 + 0.0345393i \(0.989004\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
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 1320.2.d.a.529.1 ✓ 6
3.2 odd 2 3960.2.d.e.3169.6 6
4.3 odd 2 2640.2.d.g.529.4 6
5.2 odd 4 6600.2.a.bp.1.1 3
5.3 odd 4 6600.2.a.bt.1.3 3
5.4 even 2 inner 1320.2.d.a.529.4 yes 6
15.14 odd 2 3960.2.d.e.3169.5 6
20.19 odd 2 2640.2.d.g.529.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1320.2.d.a.529.1 ✓ 6 1.1 even 1 trivial
1320.2.d.a.529.4 yes 6 5.4 even 2 inner
2640.2.d.g.529.1 6 20.19 odd 2
2640.2.d.g.529.4 6 4.3 odd 2
3960.2.d.e.3169.5 6 15.14 odd 2
3960.2.d.e.3169.6 6 3.2 odd 2
6600.2.a.bp.1.1 3 5.2 odd 4
6600.2.a.bt.1.3 3 5.3 odd 4