Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2240,4,Mod(1,2240)] 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("2240.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,10,0,14,0,12,0,-14] 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: \(132.164278413\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2240.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.65685 q^{3} +5.00000 q^{5} +7.00000 q^{7} -5.31371 q^{9} -52.2548 q^{11} -30.6569 q^{13} -23.2843 q^{15} +37.2254 q^{17} +80.2254 q^{19} -32.5980 q^{21} -25.8335 q^{23} +25.0000 q^{25} +150.480 q^{27} -20.9411 q^{29} +314.558 q^{31} +243.343 q^{33} +35.0000 q^{35} -197.147 q^{37} +142.765 q^{39} +11.3625 q^{41} -33.8335 q^{43} -26.5685 q^{45} +361.676 q^{47} +49.0000 q^{49} -173.353 q^{51} -153.019 q^{53} -261.274 q^{55} -373.598 q^{57} -616.000 q^{59} -15.2649 q^{61} -37.1960 q^{63} -153.284 q^{65} -166.510 q^{67} +120.303 q^{69} +952.000 q^{71} -148.489 q^{73} -116.421 q^{75} -365.784 q^{77} -857.725 q^{79} -557.294 q^{81} +660.528 q^{83} +186.127 q^{85} +97.5198 q^{87} -45.7746 q^{89} -214.598 q^{91} -1464.85 q^{93} +401.127 q^{95} +1682.13 q^{97} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} - 14 q^{11} - 50 q^{13} + 10 q^{15} - 50 q^{17} + 36 q^{19} + 14 q^{21} - 244 q^{23} + 50 q^{25} + 86 q^{27} + 26 q^{29} + 120 q^{31} + 498 q^{33} + 70 q^{35}+ \cdots + 940 q^{99}+O(q^{100}) \) 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
\(3\) −4.65685 −0.896212 −0.448106 − 0.893980i \(-0.647901\pi\)
−0.448106 + 0.893980i \(0.647901\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −5.31371 −0.196804
\(10\) 0 0
\(11\) −52.2548 −1.43231 −0.716156 − 0.697941i \(-0.754100\pi\)
−0.716156 + 0.697941i \(0.754100\pi\)
\(12\) 0 0
\(13\) −30.6569 −0.654052 −0.327026 − 0.945015i \(-0.606047\pi\)
−0.327026 + 0.945015i \(0.606047\pi\)
\(14\) 0 0
\(15\) −23.2843 −0.400798
\(16\) 0 0
\(17\) 37.2254 0.531087 0.265544 − 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) 0 0
\(19\) 80.2254 0.968683 0.484341 − 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(20\) 0 0
\(21\) −32.5980 −0.338736
\(22\) 0 0
\(23\) −25.8335 −0.234202 −0.117101 − 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 150.480 1.07259
\(28\) 0 0
\(29\) −20.9411 −0.134092 −0.0670460 − 0.997750i \(-0.521357\pi\)
−0.0670460 + 0.997750i \(0.521357\pi\)
\(30\) 0 0
\(31\) 314.558 1.82246 0.911232 − 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(32\) 0 0
\(33\) 243.343 1.28365
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −197.147 −0.875968 −0.437984 − 0.898983i \(-0.644307\pi\)
−0.437984 + 0.898983i \(0.644307\pi\)
\(38\) 0 0
\(39\) 142.765 0.586170
\(40\) 0 0
\(41\) 11.3625 0.0432810 0.0216405 − 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 − 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) 0 0
\(45\) −26.5685 −0.0880134
\(46\) 0 0
\(47\) 361.676 1.12247 0.561233 − 0.827658i \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −173.353 −0.475967
\(52\) 0 0
\(53\) −153.019 −0.396582 −0.198291 − 0.980143i \(-0.563539\pi\)
−0.198291 + 0.980143i \(0.563539\pi\)
\(54\) 0 0
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) −373.598 −0.868145
\(58\) 0 0
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −15.2649 −0.0320406 −0.0160203 − 0.999872i \(-0.505100\pi\)
−0.0160203 + 0.999872i \(0.505100\pi\)
\(62\) 0 0
\(63\) −37.1960 −0.0743849
\(64\) 0 0
\(65\) −153.284 −0.292501
\(66\) 0 0
\(67\) −166.510 −0.303618 −0.151809 − 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) 0 0
\(69\) 120.303 0.209895
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 − 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −148.489 −0.238074 −0.119037 − 0.992890i \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) 0 0
\(75\) −116.421 −0.179242
\(76\) 0 0
\(77\) −365.784 −0.541363
\(78\) 0 0
\(79\) −857.725 −1.22154 −0.610770 − 0.791808i \(-0.709140\pi\)
−0.610770 + 0.791808i \(0.709140\pi\)
\(80\) 0 0
\(81\) −557.294 −0.764464
\(82\) 0 0
\(83\) 660.528 0.873523 0.436761 − 0.899577i \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 0 0
\(85\) 186.127 0.237509
\(86\) 0 0
\(87\) 97.5198 0.120175
\(88\) 0 0
\(89\) −45.7746 −0.0545180 −0.0272590 − 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 0 0
\(91\) −214.598 −0.247209
\(92\) 0 0
\(93\) −1464.85 −1.63331
\(94\) 0 0
\(95\) 401.127 0.433208
\(96\) 0 0
\(97\) 1682.13 1.76076 0.880382 − 0.474265i \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) 0 0
\(99\) 277.667 0.281885
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 2240.4.a.bo.1.1 2
4.3 odd 2 2240.4.a.bn.1.2 2
8.3 odd 2 35.4.a.b.1.2 ✓ 2
8.5 even 2 560.4.a.r.1.2 2
24.11 even 2 315.4.a.f.1.1 2
40.3 even 4 175.4.b.c.99.1 4
40.19 odd 2 175.4.a.c.1.1 2
40.27 even 4 175.4.b.c.99.4 4
56.3 even 6 245.4.e.i.226.1 4
56.11 odd 6 245.4.e.h.226.1 4
56.19 even 6 245.4.e.i.116.1 4
56.27 even 2 245.4.a.k.1.2 2
56.51 odd 6 245.4.e.h.116.1 4
120.59 even 2 1575.4.a.z.1.2 2
168.83 odd 2 2205.4.a.u.1.1 2
280.139 even 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 ✓ 2 8.3 odd 2
175.4.a.c.1.1 2 40.19 odd 2
175.4.b.c.99.1 4 40.3 even 4
175.4.b.c.99.4 4 40.27 even 4
245.4.a.k.1.2 2 56.27 even 2
245.4.e.h.116.1 4 56.51 odd 6
245.4.e.h.226.1 4 56.11 odd 6
245.4.e.i.116.1 4 56.19 even 6
245.4.e.i.226.1 4 56.3 even 6
315.4.a.f.1.1 2 24.11 even 2
560.4.a.r.1.2 2 8.5 even 2
1225.4.a.m.1.1 2 280.139 even 2
1575.4.a.z.1.2 2 120.59 even 2
2205.4.a.u.1.1 2 168.83 odd 2
2240.4.a.bn.1.2 2 4.3 odd 2
2240.4.a.bo.1.1 2 1.1 even 1 trivial