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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,2,Mod(63,800)] 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("800.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 0, 19])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bq (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,0,2,0,-4] 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: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 223.6
Character \(\chi\) \(=\) 800.223
Dual form 800.2.bq.c.287.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.440304 + 0.864144i) q^{3} +(-2.21992 - 0.268223i) q^{5} +(-2.34980 - 2.34980i) q^{7} +(1.21048 - 1.66608i) q^{9} +(1.76565 + 2.43020i) q^{11} +(0.910159 + 5.74652i) q^{13} +(-0.745656 - 2.03643i) q^{15} +(3.39204 + 1.72833i) q^{17} +(0.0821413 + 0.252805i) q^{19} +(0.995939 - 3.06519i) q^{21} +(-1.20596 + 7.61411i) q^{23} +(4.85611 + 1.19087i) q^{25} +(4.84645 + 0.767602i) q^{27} +(3.49891 + 1.13687i) q^{29} +(-2.21186 + 0.718676i) q^{31} +(-1.32263 + 2.59580i) q^{33} +(4.58609 + 5.84663i) q^{35} +(10.1766 - 1.61182i) q^{37} +(-4.56507 + 3.31672i) q^{39} +(-1.02740 - 0.746453i) q^{41} +(-2.20292 + 2.20292i) q^{43} +(-3.13405 + 3.37389i) q^{45} +(-4.22246 + 2.15145i) q^{47} +4.04308i q^{49} +3.69220i q^{51} +(5.44391 - 2.77381i) q^{53} +(-3.26776 - 5.86845i) q^{55} +(-0.182293 + 0.182293i) q^{57} +(-9.55473 - 6.94192i) q^{59} +(-5.05755 + 3.67453i) q^{61} +(-6.75932 + 1.07057i) q^{63} +(-0.479132 - 13.0009i) q^{65} +(-6.95421 + 13.6484i) q^{67} +(-7.11068 + 2.31040i) q^{69} +(8.86316 + 2.87981i) q^{71} +(-5.17821 - 0.820148i) q^{73} +(1.10908 + 4.72073i) q^{75} +(1.56157 - 9.85939i) q^{77} +(3.72879 - 11.4760i) q^{79} +(-0.438568 - 1.34977i) q^{81} +(7.11391 + 3.62472i) q^{83} +(-7.06648 - 4.74658i) q^{85} +(0.558168 + 3.52413i) q^{87} +(4.48956 + 6.17934i) q^{89} +(11.3645 - 15.6418i) q^{91} +(-1.59493 - 1.59493i) q^{93} +(-0.114539 - 0.583240i) q^{95} +(-1.61088 - 3.16152i) q^{97} +6.18619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 2 q^{5} - 4 q^{7} + 4 q^{13} - 22 q^{15} + 8 q^{17} + 18 q^{19} - 16 q^{21} - 8 q^{23} + 40 q^{25} - 18 q^{27} + 20 q^{31} + 44 q^{33} - 38 q^{35} - 10 q^{37} + 36 q^{39} - 16 q^{41} - 32 q^{43}+ \cdots - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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
\(3\) 0.440304 + 0.864144i 0.254209 + 0.498914i 0.982478 − 0.186378i \(-0.0596748\pi\)
−0.728269 + 0.685292i \(0.759675\pi\)
\(4\) 0 0
\(5\) −2.21992 − 0.268223i −0.992780 − 0.119953i
\(6\) 0 0
\(7\) −2.34980 − 2.34980i −0.888139 − 0.888139i 0.106205 − 0.994344i \(-0.466130\pi\)
−0.994344 + 0.106205i \(0.966130\pi\)
\(8\) 0 0
\(9\) 1.21048 − 1.66608i 0.403492 − 0.555360i
\(10\) 0 0
\(11\) 1.76565 + 2.43020i 0.532362 + 0.732734i 0.987488 − 0.157693i \(-0.0504057\pi\)
−0.455126 + 0.890427i \(0.650406\pi\)
\(12\) 0 0
\(13\) 0.910159 + 5.74652i 0.252433 + 1.59380i 0.709724 + 0.704480i \(0.248820\pi\)
−0.457292 + 0.889317i \(0.651180\pi\)
\(14\) 0 0
\(15\) −0.745656 − 2.03643i −0.192528 − 0.525805i
\(16\) 0 0
\(17\) 3.39204 + 1.72833i 0.822690 + 0.419181i 0.814059 − 0.580781i \(-0.197253\pi\)
0.00863008 + 0.999963i \(0.497253\pi\)
\(18\) 0 0
\(19\) 0.0821413 + 0.252805i 0.0188445 + 0.0579974i 0.960037 − 0.279875i \(-0.0902929\pi\)
−0.941192 + 0.337872i \(0.890293\pi\)
\(20\) 0 0
\(21\) 0.995939 − 3.06519i 0.217332 − 0.668879i
\(22\) 0 0
\(23\) −1.20596 + 7.61411i −0.251459 + 1.58765i 0.461950 + 0.886906i \(0.347150\pi\)
−0.713410 + 0.700747i \(0.752850\pi\)
\(24\) 0 0
\(25\) 4.85611 + 1.19087i 0.971222 + 0.238174i
\(26\) 0 0
\(27\) 4.84645 + 0.767602i 0.932699 + 0.147725i
\(28\) 0 0
\(29\) 3.49891 + 1.13687i 0.649732 + 0.211111i 0.615296 − 0.788296i \(-0.289036\pi\)
0.0344361 + 0.999407i \(0.489036\pi\)
\(30\) 0 0
\(31\) −2.21186 + 0.718676i −0.397261 + 0.129078i −0.500833 − 0.865544i \(-0.666973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(32\) 0 0
\(33\) −1.32263 + 2.59580i −0.230240 + 0.451871i
\(34\) 0 0
\(35\) 4.58609 + 5.84663i 0.775191 + 0.988262i
\(36\) 0 0
\(37\) 10.1766 − 1.61182i 1.67303 − 0.264981i 0.753339 − 0.657632i \(-0.228442\pi\)
0.919688 + 0.392651i \(0.128442\pi\)
\(38\) 0 0
\(39\) −4.56507 + 3.31672i −0.730997 + 0.531100i
\(40\) 0 0
\(41\) −1.02740 − 0.746453i −0.160454 − 0.116576i 0.504661 − 0.863317i \(-0.331617\pi\)
−0.665115 + 0.746741i \(0.731617\pi\)
\(42\) 0 0
\(43\) −2.20292 + 2.20292i −0.335942 + 0.335942i −0.854838 − 0.518895i \(-0.826343\pi\)
0.518895 + 0.854838i \(0.326343\pi\)
\(44\) 0 0
\(45\) −3.13405 + 3.37389i −0.467196 + 0.502950i
\(46\) 0 0
\(47\) −4.22246 + 2.15145i −0.615909 + 0.313822i −0.733967 − 0.679185i \(-0.762333\pi\)
0.118057 + 0.993007i \(0.462333\pi\)
\(48\) 0 0
\(49\) 4.04308i 0.577583i
\(50\) 0 0
\(51\) 3.69220i 0.517011i
\(52\) 0 0
\(53\) 5.44391 − 2.77381i 0.747779 − 0.381012i −0.0381964 − 0.999270i \(-0.512161\pi\)
0.785975 + 0.618258i \(0.212161\pi\)
\(54\) 0 0
\(55\) −3.26776 − 5.86845i −0.440625 − 0.791302i
\(56\) 0 0
\(57\) −0.182293 + 0.182293i −0.0241453 + 0.0241453i
\(58\) 0 0
\(59\) −9.55473 − 6.94192i −1.24392 − 0.903761i −0.246067 − 0.969253i \(-0.579138\pi\)
−0.997853 + 0.0654918i \(0.979138\pi\)
\(60\) 0 0
\(61\) −5.05755 + 3.67453i −0.647553 + 0.470475i −0.862437 − 0.506165i \(-0.831063\pi\)
0.214884 + 0.976640i \(0.431063\pi\)
\(62\) 0 0
\(63\) −6.75932 + 1.07057i −0.851594 + 0.134879i
\(64\) 0 0
\(65\) −0.479132 − 13.0009i −0.0594290 − 1.61257i
\(66\) 0 0
\(67\) −6.95421 + 13.6484i −0.849592 + 1.66742i −0.110437 + 0.993883i \(0.535225\pi\)
−0.739155 + 0.673536i \(0.764775\pi\)
\(68\) 0 0
\(69\) −7.11068 + 2.31040i −0.856026 + 0.278140i
\(70\) 0 0
\(71\) 8.86316 + 2.87981i 1.05186 + 0.341771i 0.783399 − 0.621519i \(-0.213484\pi\)
0.268464 + 0.963290i \(0.413484\pi\)
\(72\) 0 0
\(73\) −5.17821 − 0.820148i −0.606064 − 0.0959911i −0.154142 − 0.988049i \(-0.549261\pi\)
−0.451922 + 0.892058i \(0.649261\pi\)
\(74\) 0 0
\(75\) 1.10908 + 4.72073i 0.128065 + 0.545103i
\(76\) 0 0
\(77\) 1.56157 − 9.85939i 0.177958 − 1.12358i
\(78\) 0 0
\(79\) 3.72879 − 11.4760i 0.419522 − 1.29116i −0.488621 − 0.872496i \(-0.662500\pi\)
0.908143 − 0.418660i \(-0.137500\pi\)
\(80\) 0 0
\(81\) −0.438568 − 1.34977i −0.0487298 − 0.149975i
\(82\) 0 0
\(83\) 7.11391 + 3.62472i 0.780853 + 0.397865i 0.798512 − 0.601980i \(-0.205621\pi\)
−0.0176584 + 0.999844i \(0.505621\pi\)
\(84\) 0 0
\(85\) −7.06648 − 4.74658i −0.766467 − 0.514839i
\(86\) 0 0
\(87\) 0.558168 + 3.52413i 0.0598419 + 0.377827i
\(88\) 0 0
\(89\) 4.48956 + 6.17934i 0.475892 + 0.655009i 0.977709 − 0.209965i \(-0.0673349\pi\)
−0.501817 + 0.864974i \(0.667335\pi\)
\(90\) 0 0
\(91\) 11.3645 − 15.6418i 1.19132 − 1.63971i
\(92\) 0 0
\(93\) −1.59493 − 1.59493i −0.165386 − 0.165386i
\(94\) 0 0
\(95\) −0.114539 − 0.583240i −0.0117515 − 0.0598391i
\(96\) 0 0
\(97\) −1.61088 − 3.16152i −0.163560 − 0.321004i 0.794651 − 0.607066i \(-0.207654\pi\)
−0.958211 + 0.286062i \(0.907654\pi\)
\(98\) 0 0
\(99\) 6.18619 0.621735
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 800.2.bq.c.223.6 ✓ 64
4.3 odd 2 800.2.bq.d.223.3 yes 64
25.12 odd 20 800.2.bq.d.287.3 yes 64
100.87 even 20 inner 800.2.bq.c.287.6 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.bq.c.223.6 ✓ 64 1.1 even 1 trivial
800.2.bq.c.287.6 yes 64 100.87 even 20 inner
800.2.bq.d.223.3 yes 64 4.3 odd 2
800.2.bq.d.287.3 yes 64 25.12 odd 20