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 1.5
Root \(-0.595043\) 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)
 
\(f(q)\) \(=\) \(q+7.11323 q^{3} -5.00000 q^{5} -13.7888 q^{7} +23.5981 q^{9} -24.2317 q^{11} +3.05016 q^{13} -35.5662 q^{15} +63.1126 q^{17} +2.07770 q^{19} -98.0832 q^{21} +23.0000 q^{23} +25.0000 q^{25} -24.1987 q^{27} -8.16397 q^{29} +156.989 q^{31} -172.366 q^{33} +68.9442 q^{35} +302.801 q^{37} +21.6965 q^{39} -42.7514 q^{41} -215.265 q^{43} -117.990 q^{45} -247.096 q^{47} -152.868 q^{49} +448.935 q^{51} +600.400 q^{53} +121.159 q^{55} +14.7792 q^{57} -92.2014 q^{59} +532.635 q^{61} -325.390 q^{63} -15.2508 q^{65} -30.3010 q^{67} +163.604 q^{69} +736.349 q^{71} +349.936 q^{73} +177.831 q^{75} +334.127 q^{77} -301.545 q^{79} -809.279 q^{81} -139.488 q^{83} -315.563 q^{85} -58.0722 q^{87} +859.551 q^{89} -42.0581 q^{91} +1116.70 q^{93} -10.3885 q^{95} -927.475 q^{97} -571.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35}+ \cdots - 2021 q^{99}+O(q^{100}) \) 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
\(3\) 7.11323 1.36894 0.684471 − 0.729040i \(-0.260033\pi\)
0.684471 + 0.729040i \(0.260033\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −13.7888 −0.744527 −0.372263 − 0.928127i \(-0.621418\pi\)
−0.372263 + 0.928127i \(0.621418\pi\)
\(8\) 0 0
\(9\) 23.5981 0.874003
\(10\) 0 0
\(11\) −24.2317 −0.664195 −0.332098 − 0.943245i \(-0.607756\pi\)
−0.332098 + 0.943245i \(0.607756\pi\)
\(12\) 0 0
\(13\) 3.05016 0.0650739 0.0325370 − 0.999471i \(-0.489641\pi\)
0.0325370 + 0.999471i \(0.489641\pi\)
\(14\) 0 0
\(15\) −35.5662 −0.612210
\(16\) 0 0
\(17\) 63.1126 0.900415 0.450208 − 0.892924i \(-0.351350\pi\)
0.450208 + 0.892924i \(0.351350\pi\)
\(18\) 0 0
\(19\) 2.07770 0.0250872 0.0125436 − 0.999921i \(-0.496007\pi\)
0.0125436 + 0.999921i \(0.496007\pi\)
\(20\) 0 0
\(21\) −98.0832 −1.01921
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −24.1987 −0.172483
\(28\) 0 0
\(29\) −8.16397 −0.0522762 −0.0261381 − 0.999658i \(-0.508321\pi\)
−0.0261381 + 0.999658i \(0.508321\pi\)
\(30\) 0 0
\(31\) 156.989 0.909553 0.454776 − 0.890606i \(-0.349719\pi\)
0.454776 + 0.890606i \(0.349719\pi\)
\(32\) 0 0
\(33\) −172.366 −0.909245
\(34\) 0 0
\(35\) 68.9442 0.332963
\(36\) 0 0
\(37\) 302.801 1.34541 0.672706 − 0.739910i \(-0.265132\pi\)
0.672706 + 0.739910i \(0.265132\pi\)
\(38\) 0 0
\(39\) 21.6965 0.0890824
\(40\) 0 0
\(41\) −42.7514 −0.162845 −0.0814225 − 0.996680i \(-0.525946\pi\)
−0.0814225 + 0.996680i \(0.525946\pi\)
\(42\) 0 0
\(43\) −215.265 −0.763434 −0.381717 − 0.924279i \(-0.624667\pi\)
−0.381717 + 0.924279i \(0.624667\pi\)
\(44\) 0 0
\(45\) −117.990 −0.390866
\(46\) 0 0
\(47\) −247.096 −0.766866 −0.383433 − 0.923569i \(-0.625258\pi\)
−0.383433 + 0.923569i \(0.625258\pi\)
\(48\) 0 0
\(49\) −152.868 −0.445680
\(50\) 0 0
\(51\) 448.935 1.23262
\(52\) 0 0
\(53\) 600.400 1.55606 0.778031 − 0.628225i \(-0.216218\pi\)
0.778031 + 0.628225i \(0.216218\pi\)
\(54\) 0 0
\(55\) 121.159 0.297037
\(56\) 0 0
\(57\) 14.7792 0.0343430
\(58\) 0 0
\(59\) −92.2014 −0.203451 −0.101725 − 0.994813i \(-0.532436\pi\)
−0.101725 + 0.994813i \(0.532436\pi\)
\(60\) 0 0
\(61\) 532.635 1.11798 0.558991 − 0.829174i \(-0.311189\pi\)
0.558991 + 0.829174i \(0.311189\pi\)
\(62\) 0 0
\(63\) −325.390 −0.650719
\(64\) 0 0
\(65\) −15.2508 −0.0291019
\(66\) 0 0
\(67\) −30.3010 −0.0552515 −0.0276258 − 0.999618i \(-0.508795\pi\)
−0.0276258 + 0.999618i \(0.508795\pi\)
\(68\) 0 0
\(69\) 163.604 0.285444
\(70\) 0 0
\(71\) 736.349 1.23083 0.615413 − 0.788205i \(-0.288989\pi\)
0.615413 + 0.788205i \(0.288989\pi\)
\(72\) 0 0
\(73\) 349.936 0.561053 0.280527 − 0.959846i \(-0.409491\pi\)
0.280527 + 0.959846i \(0.409491\pi\)
\(74\) 0 0
\(75\) 177.831 0.273788
\(76\) 0 0
\(77\) 334.127 0.494511
\(78\) 0 0
\(79\) −301.545 −0.429449 −0.214725 − 0.976675i \(-0.568885\pi\)
−0.214725 + 0.976675i \(0.568885\pi\)
\(80\) 0 0
\(81\) −809.279 −1.11012
\(82\) 0 0
\(83\) −139.488 −0.184468 −0.0922340 − 0.995737i \(-0.529401\pi\)
−0.0922340 + 0.995737i \(0.529401\pi\)
\(84\) 0 0
\(85\) −315.563 −0.402678
\(86\) 0 0
\(87\) −58.0722 −0.0715631
\(88\) 0 0
\(89\) 859.551 1.02373 0.511866 − 0.859065i \(-0.328954\pi\)
0.511866 + 0.859065i \(0.328954\pi\)
\(90\) 0 0
\(91\) −42.0581 −0.0484493
\(92\) 0 0
\(93\) 1116.70 1.24513
\(94\) 0 0
\(95\) −10.3885 −0.0112193
\(96\) 0 0
\(97\) −927.475 −0.970833 −0.485417 − 0.874283i \(-0.661332\pi\)
−0.485417 + 0.874283i \(0.661332\pi\)
\(98\) 0 0
\(99\) −571.823 −0.580508
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.1.5 5
4.3 odd 2 115.4.a.e.1.3 ✓ 5
12.11 even 2 1035.4.a.k.1.3 5
20.3 even 4 575.4.b.i.24.5 10
20.7 even 4 575.4.b.i.24.6 10
20.19 odd 2 575.4.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.3 ✓ 5 4.3 odd 2
575.4.a.j.1.3 5 20.19 odd 2
575.4.b.i.24.5 10 20.3 even 4
575.4.b.i.24.6 10 20.7 even 4
1035.4.a.k.1.3 5 12.11 even 2
1840.4.a.n.1.5 5 1.1 even 1 trivial