Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,-5,0,2,0,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-3.36002\) 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)
 
\(f(q)\) \(=\) \(q+3.36002 q^{3} -1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} +5.48021 q^{11} -1.04937 q^{13} -3.36002 q^{15} -6.74222 q^{17} +1.55049 q^{19} +6.40939 q^{21} +1.00000 q^{23} +1.00000 q^{25} +17.7736 q^{27} -3.38219 q^{29} -10.9327 q^{31} +18.4136 q^{33} -1.90754 q^{35} +5.26201 q^{37} -3.52589 q^{39} +6.09801 q^{41} -8.28974 q^{45} -0.403830 q^{47} -3.36128 q^{49} -22.6540 q^{51} +5.88332 q^{53} -5.48021 q^{55} +5.20968 q^{57} -9.60111 q^{59} -7.09927 q^{61} +15.8130 q^{63} +1.04937 q^{65} -13.7971 q^{67} +3.36002 q^{69} -0.478950 q^{71} +2.40383 q^{73} +3.36002 q^{75} +10.4537 q^{77} +4.24037 q^{79} +34.8505 q^{81} +11.2620 q^{83} +6.74222 q^{85} -11.3642 q^{87} +4.90495 q^{89} -2.00171 q^{91} -36.7340 q^{93} -1.55049 q^{95} -12.3433 q^{97} +45.4295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41}+ \cdots + 65 q^{99}+O(q^{100}) \) 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
\(3\) 3.36002 1.93991 0.969954 − 0.243287i \(-0.0782256\pi\)
0.969954 + 0.243287i \(0.0782256\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.90754 0.720984 0.360492 − 0.932762i \(-0.382609\pi\)
0.360492 + 0.932762i \(0.382609\pi\)
\(8\) 0 0
\(9\) 8.28974 2.76325
\(10\) 0 0
\(11\) 5.48021 1.65234 0.826172 − 0.563418i \(-0.190514\pi\)
0.826172 + 0.563418i \(0.190514\pi\)
\(12\) 0 0
\(13\) −1.04937 −0.291041 −0.145521 − 0.989355i \(-0.546486\pi\)
−0.145521 + 0.989355i \(0.546486\pi\)
\(14\) 0 0
\(15\) −3.36002 −0.867554
\(16\) 0 0
\(17\) −6.74222 −1.63523 −0.817614 − 0.575767i \(-0.804703\pi\)
−0.817614 + 0.575767i \(0.804703\pi\)
\(18\) 0 0
\(19\) 1.55049 0.355707 0.177853 − 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(20\) 0 0
\(21\) 6.40939 1.39864
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 17.7736 3.42054
\(28\) 0 0
\(29\) −3.38219 −0.628058 −0.314029 − 0.949413i \(-0.601679\pi\)
−0.314029 + 0.949413i \(0.601679\pi\)
\(30\) 0 0
\(31\) −10.9327 −1.96357 −0.981784 − 0.190000i \(-0.939151\pi\)
−0.981784 + 0.190000i \(0.939151\pi\)
\(32\) 0 0
\(33\) 18.4136 3.20540
\(34\) 0 0
\(35\) −1.90754 −0.322434
\(36\) 0 0
\(37\) 5.26201 0.865069 0.432534 − 0.901617i \(-0.357619\pi\)
0.432534 + 0.901617i \(0.357619\pi\)
\(38\) 0 0
\(39\) −3.52589 −0.564594
\(40\) 0 0
\(41\) 6.09801 0.952350 0.476175 − 0.879351i \(-0.342023\pi\)
0.476175 + 0.879351i \(0.342023\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −8.28974 −1.23576
\(46\) 0 0
\(47\) −0.403830 −0.0589046 −0.0294523 − 0.999566i \(-0.509376\pi\)
−0.0294523 + 0.999566i \(0.509376\pi\)
\(48\) 0 0
\(49\) −3.36128 −0.480183
\(50\) 0 0
\(51\) −22.6540 −3.17219
\(52\) 0 0
\(53\) 5.88332 0.808136 0.404068 − 0.914729i \(-0.367596\pi\)
0.404068 + 0.914729i \(0.367596\pi\)
\(54\) 0 0
\(55\) −5.48021 −0.738951
\(56\) 0 0
\(57\) 5.20968 0.690039
\(58\) 0 0
\(59\) −9.60111 −1.24996 −0.624979 − 0.780641i \(-0.714893\pi\)
−0.624979 + 0.780641i \(0.714893\pi\)
\(60\) 0 0
\(61\) −7.09927 −0.908968 −0.454484 − 0.890755i \(-0.650176\pi\)
−0.454484 + 0.890755i \(0.650176\pi\)
\(62\) 0 0
\(63\) 15.8130 1.99226
\(64\) 0 0
\(65\) 1.04937 0.130158
\(66\) 0 0
\(67\) −13.7971 −1.68559 −0.842794 − 0.538236i \(-0.819091\pi\)
−0.842794 + 0.538236i \(0.819091\pi\)
\(68\) 0 0
\(69\) 3.36002 0.404499
\(70\) 0 0
\(71\) −0.478950 −0.0568409 −0.0284205 − 0.999596i \(-0.509048\pi\)
−0.0284205 + 0.999596i \(0.509048\pi\)
\(72\) 0 0
\(73\) 2.40383 0.281347 0.140673 − 0.990056i \(-0.455073\pi\)
0.140673 + 0.990056i \(0.455073\pi\)
\(74\) 0 0
\(75\) 3.36002 0.387982
\(76\) 0 0
\(77\) 10.4537 1.19131
\(78\) 0 0
\(79\) 4.24037 0.477079 0.238540 − 0.971133i \(-0.423331\pi\)
0.238540 + 0.971133i \(0.423331\pi\)
\(80\) 0 0
\(81\) 34.8505 3.87228
\(82\) 0 0
\(83\) 11.2620 1.23617 0.618083 − 0.786113i \(-0.287910\pi\)
0.618083 + 0.786113i \(0.287910\pi\)
\(84\) 0 0
\(85\) 6.74222 0.731296
\(86\) 0 0
\(87\) −11.3642 −1.21837
\(88\) 0 0
\(89\) 4.90495 0.519924 0.259962 − 0.965619i \(-0.416290\pi\)
0.259962 + 0.965619i \(0.416290\pi\)
\(90\) 0 0
\(91\) −2.00171 −0.209836
\(92\) 0 0
\(93\) −36.7340 −3.80914
\(94\) 0 0
\(95\) −1.55049 −0.159077
\(96\) 0 0
\(97\) −12.3433 −1.25327 −0.626637 − 0.779311i \(-0.715569\pi\)
−0.626637 + 0.779311i \(0.715569\pi\)
\(98\) 0 0
\(99\) 45.4295 4.56583
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 1840.2.a.v.1.5 5
4.3 odd 2 920.2.a.j.1.1 ✓ 5
5.4 even 2 9200.2.a.cu.1.1 5
8.3 odd 2 7360.2.a.co.1.5 5
8.5 even 2 7360.2.a.cp.1.1 5
12.11 even 2 8280.2.a.bs.1.3 5
20.3 even 4 4600.2.e.u.4049.1 10
20.7 even 4 4600.2.e.u.4049.10 10
20.19 odd 2 4600.2.a.be.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.1 ✓ 5 4.3 odd 2
1840.2.a.v.1.5 5 1.1 even 1 trivial
4600.2.a.be.1.5 5 20.19 odd 2
4600.2.e.u.4049.1 10 20.3 even 4
4600.2.e.u.4049.10 10 20.7 even 4
7360.2.a.co.1.5 5 8.3 odd 2
7360.2.a.cp.1.1 5 8.5 even 2
8280.2.a.bs.1.3 5 12.11 even 2
9200.2.a.cu.1.1 5 5.4 even 2