Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-4,0,-25,0,3] 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: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.41740\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.84147 q^{3} -5.00000 q^{5} +8.97260 q^{7} +34.4886 q^{9} +28.9011 q^{11} +16.0879 q^{13} +39.2073 q^{15} +25.1771 q^{17} +35.6298 q^{19} -70.3583 q^{21} +23.0000 q^{23} +25.0000 q^{25} -58.7218 q^{27} +138.272 q^{29} -40.1277 q^{31} -226.627 q^{33} -44.8630 q^{35} +379.745 q^{37} -126.153 q^{39} +412.514 q^{41} +402.095 q^{43} -172.443 q^{45} -110.070 q^{47} -262.492 q^{49} -197.426 q^{51} -421.300 q^{53} -144.506 q^{55} -279.390 q^{57} -755.913 q^{59} -307.032 q^{61} +309.453 q^{63} -80.4396 q^{65} -319.974 q^{67} -180.354 q^{69} +554.138 q^{71} -705.131 q^{73} -196.037 q^{75} +259.318 q^{77} -1170.51 q^{79} -470.728 q^{81} +455.978 q^{83} -125.886 q^{85} -1084.26 q^{87} -1495.57 q^{89} +144.351 q^{91} +314.660 q^{93} -178.149 q^{95} +1041.24 q^{97} +996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35}+ \cdots - 2021 q^{99}+O(q^{100}) \) 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
\(3\) −7.84147 −1.50909 −0.754546 − 0.656248i \(-0.772143\pi\)
−0.754546 + 0.656248i \(0.772143\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 8.97260 0.484475 0.242237 − 0.970217i \(-0.422119\pi\)
0.242237 + 0.970217i \(0.422119\pi\)
\(8\) 0 0
\(9\) 34.4886 1.27736
\(10\) 0 0
\(11\) 28.9011 0.792183 0.396092 − 0.918211i \(-0.370366\pi\)
0.396092 + 0.918211i \(0.370366\pi\)
\(12\) 0 0
\(13\) 16.0879 0.343230 0.171615 − 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(14\) 0 0
\(15\) 39.2073 0.674886
\(16\) 0 0
\(17\) 25.1771 0.359197 0.179599 − 0.983740i \(-0.442520\pi\)
0.179599 + 0.983740i \(0.442520\pi\)
\(18\) 0 0
\(19\) 35.6298 0.430212 0.215106 − 0.976591i \(-0.430990\pi\)
0.215106 + 0.976591i \(0.430990\pi\)
\(20\) 0 0
\(21\) −70.3583 −0.731117
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −58.7218 −0.418556
\(28\) 0 0
\(29\) 138.272 0.885395 0.442698 − 0.896671i \(-0.354022\pi\)
0.442698 + 0.896671i \(0.354022\pi\)
\(30\) 0 0
\(31\) −40.1277 −0.232489 −0.116244 − 0.993221i \(-0.537086\pi\)
−0.116244 + 0.993221i \(0.537086\pi\)
\(32\) 0 0
\(33\) −226.627 −1.19548
\(34\) 0 0
\(35\) −44.8630 −0.216664
\(36\) 0 0
\(37\) 379.745 1.68729 0.843645 − 0.536902i \(-0.180405\pi\)
0.843645 + 0.536902i \(0.180405\pi\)
\(38\) 0 0
\(39\) −126.153 −0.517965
\(40\) 0 0
\(41\) 412.514 1.57132 0.785658 − 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) 0 0
\(43\) 402.095 1.42602 0.713011 − 0.701153i \(-0.247331\pi\)
0.713011 + 0.701153i \(0.247331\pi\)
\(44\) 0 0
\(45\) −172.443 −0.571251
\(46\) 0 0
\(47\) −110.070 −0.341603 −0.170801 − 0.985305i \(-0.554636\pi\)
−0.170801 + 0.985305i \(0.554636\pi\)
\(48\) 0 0
\(49\) −262.492 −0.765284
\(50\) 0 0
\(51\) −197.426 −0.542061
\(52\) 0 0
\(53\) −421.300 −1.09189 −0.545943 − 0.837822i \(-0.683829\pi\)
−0.545943 + 0.837822i \(0.683829\pi\)
\(54\) 0 0
\(55\) −144.506 −0.354275
\(56\) 0 0
\(57\) −279.390 −0.649229
\(58\) 0 0
\(59\) −755.913 −1.66799 −0.833996 − 0.551770i \(-0.813952\pi\)
−0.833996 + 0.551770i \(0.813952\pi\)
\(60\) 0 0
\(61\) −307.032 −0.644450 −0.322225 − 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) 0 0
\(63\) 309.453 0.618847
\(64\) 0 0
\(65\) −80.4396 −0.153497
\(66\) 0 0
\(67\) −319.974 −0.583448 −0.291724 − 0.956502i \(-0.594229\pi\)
−0.291724 + 0.956502i \(0.594229\pi\)
\(68\) 0 0
\(69\) −180.354 −0.314667
\(70\) 0 0
\(71\) 554.138 0.926254 0.463127 − 0.886292i \(-0.346727\pi\)
0.463127 + 0.886292i \(0.346727\pi\)
\(72\) 0 0
\(73\) −705.131 −1.13054 −0.565269 − 0.824907i \(-0.691228\pi\)
−0.565269 + 0.824907i \(0.691228\pi\)
\(74\) 0 0
\(75\) −196.037 −0.301818
\(76\) 0 0
\(77\) 259.318 0.383793
\(78\) 0 0
\(79\) −1170.51 −1.66699 −0.833495 − 0.552526i \(-0.813664\pi\)
−0.833495 + 0.552526i \(0.813664\pi\)
\(80\) 0 0
\(81\) −470.728 −0.645717
\(82\) 0 0
\(83\) 455.978 0.603014 0.301507 − 0.953464i \(-0.402510\pi\)
0.301507 + 0.953464i \(0.402510\pi\)
\(84\) 0 0
\(85\) −125.886 −0.160638
\(86\) 0 0
\(87\) −1084.26 −1.33614
\(88\) 0 0
\(89\) −1495.57 −1.78124 −0.890621 − 0.454746i \(-0.849730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(90\) 0 0
\(91\) 144.351 0.166286
\(92\) 0 0
\(93\) 314.660 0.350847
\(94\) 0 0
\(95\) −178.149 −0.192397
\(96\) 0 0
\(97\) 1041.24 1.08992 0.544958 − 0.838463i \(-0.316545\pi\)
0.544958 + 0.838463i \(0.316545\pi\)
\(98\) 0 0
\(99\) 996.760 1.01190
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 1840.4.a.n.1.2 5
4.3 odd 2 115.4.a.e.1.4 ✓ 5
12.11 even 2 1035.4.a.k.1.2 5
20.3 even 4 575.4.b.i.24.2 10
20.7 even 4 575.4.b.i.24.9 10
20.19 odd 2 575.4.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 ✓ 5 4.3 odd 2
575.4.a.j.1.2 5 20.19 odd 2
575.4.b.i.24.2 10 20.3 even 4
575.4.b.i.24.9 10 20.7 even 4
1035.4.a.k.1.2 5 12.11 even 2
1840.4.a.n.1.2 5 1.1 even 1 trivial