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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,5,Mod(17,80)] 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("80.17"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.26959704671\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.1
Root \(7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.5.p.e.33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-5.26209 + 5.26209i) q^{3} +(21.7863 - 12.2621i) q^{5} +(-4.21374 - 4.21374i) q^{7} +25.6209i q^{9} +152.621 q^{11} +(-136.573 + 136.573i) q^{13} +(-50.1170 + 179.165i) q^{15} +(348.669 + 348.669i) q^{17} +527.725i q^{19} +44.3461 q^{21} +(70.5445 - 70.5445i) q^{23} +(324.282 - 534.290i) q^{25} +(-561.048 - 561.048i) q^{27} +68.9670i q^{29} +1373.31 q^{31} +(-803.104 + 803.104i) q^{33} +(-143.471 - 40.1324i) q^{35} +(-1003.31 - 1003.31i) q^{37} -1437.31i q^{39} +663.379 q^{41} +(-1632.52 + 1632.52i) q^{43} +(314.165 + 558.183i) q^{45} +(-438.809 - 438.809i) q^{47} -2365.49i q^{49} -3669.46 q^{51} +(712.338 - 712.338i) q^{53} +(3325.04 - 1871.45i) q^{55} +(-2776.94 - 2776.94i) q^{57} -2918.34i q^{59} +1395.51 q^{61} +(107.960 - 107.960i) q^{63} +(-1300.74 + 4650.07i) q^{65} +(1691.51 + 1691.51i) q^{67} +742.423i q^{69} +3282.28 q^{71} +(-1465.81 + 1465.81i) q^{73} +(1105.08 + 4517.88i) q^{75} +(-643.104 - 643.104i) q^{77} -3389.80i q^{79} +3829.28 q^{81} +(-8690.02 + 8690.02i) q^{83} +(11871.6 + 3320.79i) q^{85} +(-362.910 - 362.910i) q^{87} -4274.77i q^{89} +1150.96 q^{91} +(-7226.49 + 7226.49i) q^{93} +(6471.01 + 11497.2i) q^{95} +(7215.47 + 7215.47i) q^{97} +3910.28i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 6 q^{5} - 110 q^{7} + 300 q^{11} - 360 q^{13} - 542 q^{15} + 960 q^{17} - 1996 q^{21} + 810 q^{23} + 1856 q^{25} - 2120 q^{27} + 836 q^{31} - 1660 q^{33} + 2562 q^{35} - 660 q^{37} + 2964 q^{41}+ \cdots - 3180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) −5.26209 + 5.26209i −0.584676 + 0.584676i −0.936185 − 0.351508i \(-0.885669\pi\)
0.351508 + 0.936185i \(0.385669\pi\)
\(4\) 0 0
\(5\) 21.7863 − 12.2621i 0.871450 − 0.490483i
\(6\) 0 0
\(7\) −4.21374 − 4.21374i −0.0859947 − 0.0859947i 0.662801 − 0.748796i \(-0.269368\pi\)
−0.748796 + 0.662801i \(0.769368\pi\)
\(8\) 0 0
\(9\) 25.6209i 0.316307i
\(10\) 0 0
\(11\) 152.621 1.26133 0.630665 − 0.776055i \(-0.282782\pi\)
0.630665 + 0.776055i \(0.282782\pi\)
\(12\) 0 0
\(13\) −136.573 + 136.573i −0.808121 + 0.808121i −0.984349 − 0.176228i \(-0.943610\pi\)
0.176228 + 0.984349i \(0.443610\pi\)
\(14\) 0 0
\(15\) −50.1170 + 179.165i −0.222742 + 0.796291i
\(16\) 0 0
\(17\) 348.669 + 348.669i 1.20647 + 1.20647i 0.972163 + 0.234305i \(0.0752813\pi\)
0.234305 + 0.972163i \(0.424719\pi\)
\(18\) 0 0
\(19\) 527.725i 1.46184i 0.682462 + 0.730921i \(0.260909\pi\)
−0.682462 + 0.730921i \(0.739091\pi\)
\(20\) 0 0
\(21\) 44.3461 0.100558
\(22\) 0 0
\(23\) 70.5445 − 70.5445i 0.133354 − 0.133354i −0.637279 − 0.770633i \(-0.719940\pi\)
0.770633 + 0.637279i \(0.219940\pi\)
\(24\) 0 0
\(25\) 324.282 − 534.290i 0.518852 − 0.854864i
\(26\) 0 0
\(27\) −561.048 − 561.048i −0.769614 − 0.769614i
\(28\) 0 0
\(29\) 68.9670i 0.0820059i 0.999159 + 0.0410030i \(0.0130553\pi\)
−0.999159 + 0.0410030i \(0.986945\pi\)
\(30\) 0 0
\(31\) 1373.31 1.42905 0.714523 − 0.699612i \(-0.246644\pi\)
0.714523 + 0.699612i \(0.246644\pi\)
\(32\) 0 0
\(33\) −803.104 + 803.104i −0.737470 + 0.737470i
\(34\) 0 0
\(35\) −143.471 − 40.1324i −0.117119 − 0.0327611i
\(36\) 0 0
\(37\) −1003.31 − 1003.31i −0.732875 − 0.732875i 0.238314 − 0.971188i \(-0.423405\pi\)
−0.971188 + 0.238314i \(0.923405\pi\)
\(38\) 0 0
\(39\) − 1437.31i − 0.944979i
\(40\) 0 0
\(41\) 663.379 0.394634 0.197317 − 0.980340i \(-0.436777\pi\)
0.197317 + 0.980340i \(0.436777\pi\)
\(42\) 0 0
\(43\) −1632.52 + 1632.52i −0.882920 + 0.882920i −0.993830 − 0.110910i \(-0.964623\pi\)
0.110910 + 0.993830i \(0.464623\pi\)
\(44\) 0 0
\(45\) 314.165 + 558.183i 0.155143 + 0.275646i
\(46\) 0 0
\(47\) −438.809 − 438.809i −0.198646 − 0.198646i 0.600773 − 0.799419i \(-0.294859\pi\)
−0.799419 + 0.600773i \(0.794859\pi\)
\(48\) 0 0
\(49\) − 2365.49i − 0.985210i
\(50\) 0 0
\(51\) −3669.46 −1.41079
\(52\) 0 0
\(53\) 712.338 − 712.338i 0.253591 − 0.253591i −0.568850 − 0.822441i \(-0.692611\pi\)
0.822441 + 0.568850i \(0.192611\pi\)
\(54\) 0 0
\(55\) 3325.04 − 1871.45i 1.09919 − 0.618661i
\(56\) 0 0
\(57\) −2776.94 − 2776.94i −0.854705 − 0.854705i
\(58\) 0 0
\(59\) − 2918.34i − 0.838363i −0.907903 − 0.419181i \(-0.862317\pi\)
0.907903 − 0.419181i \(-0.137683\pi\)
\(60\) 0 0
\(61\) 1395.51 0.375037 0.187518 − 0.982261i \(-0.439956\pi\)
0.187518 + 0.982261i \(0.439956\pi\)
\(62\) 0 0
\(63\) 107.960 − 107.960i 0.0272007 − 0.0272007i
\(64\) 0 0
\(65\) −1300.74 + 4650.07i −0.307868 + 1.10061i
\(66\) 0 0
\(67\) 1691.51 + 1691.51i 0.376813 + 0.376813i 0.869951 − 0.493138i \(-0.164150\pi\)
−0.493138 + 0.869951i \(0.664150\pi\)
\(68\) 0 0
\(69\) 742.423i 0.155938i
\(70\) 0 0
\(71\) 3282.28 0.651117 0.325558 − 0.945522i \(-0.394448\pi\)
0.325558 + 0.945522i \(0.394448\pi\)
\(72\) 0 0
\(73\) −1465.81 + 1465.81i −0.275063 + 0.275063i −0.831134 − 0.556072i \(-0.812308\pi\)
0.556072 + 0.831134i \(0.312308\pi\)
\(74\) 0 0
\(75\) 1105.08 + 4517.88i 0.196458 + 0.803179i
\(76\) 0 0
\(77\) −643.104 − 643.104i −0.108468 − 0.108468i
\(78\) 0 0
\(79\) − 3389.80i − 0.543150i −0.962417 − 0.271575i \(-0.912455\pi\)
0.962417 − 0.271575i \(-0.0875446\pi\)
\(80\) 0 0
\(81\) 3829.28 0.583643
\(82\) 0 0
\(83\) −8690.02 + 8690.02i −1.26143 + 1.26143i −0.311035 + 0.950398i \(0.600676\pi\)
−0.950398 + 0.311035i \(0.899324\pi\)
\(84\) 0 0
\(85\) 11871.6 + 3320.79i 1.64313 + 0.459624i
\(86\) 0 0
\(87\) −362.910 − 362.910i −0.0479469 − 0.0479469i
\(88\) 0 0
\(89\) − 4274.77i − 0.539675i −0.962906 − 0.269838i \(-0.913030\pi\)
0.962906 − 0.269838i \(-0.0869701\pi\)
\(90\) 0 0
\(91\) 1150.96 0.138988
\(92\) 0 0
\(93\) −7226.49 + 7226.49i −0.835529 + 0.835529i
\(94\) 0 0
\(95\) 6471.01 + 11497.2i 0.717010 + 1.27392i
\(96\) 0 0
\(97\) 7215.47 + 7215.47i 0.766869 + 0.766869i 0.977554 − 0.210685i \(-0.0675693\pi\)
−0.210685 + 0.977554i \(0.567569\pi\)
\(98\) 0 0
\(99\) 3910.28i 0.398967i
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 80.5.p.e.17.1 4
4.3 odd 2 20.5.f.a.17.2 yes 4
5.2 odd 4 400.5.p.h.193.2 4
5.3 odd 4 inner 80.5.p.e.33.1 4
5.4 even 2 400.5.p.h.257.2 4
8.3 odd 2 320.5.p.m.257.1 4
8.5 even 2 320.5.p.l.257.2 4
12.11 even 2 180.5.l.a.37.1 4
20.3 even 4 20.5.f.a.13.2 ✓ 4
20.7 even 4 100.5.f.c.93.1 4
20.19 odd 2 100.5.f.c.57.1 4
40.3 even 4 320.5.p.m.193.1 4
40.13 odd 4 320.5.p.l.193.2 4
60.23 odd 4 180.5.l.a.73.1 4
60.47 odd 4 900.5.l.a.793.2 4
60.59 even 2 900.5.l.a.757.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.2 ✓ 4 20.3 even 4
20.5.f.a.17.2 yes 4 4.3 odd 2
80.5.p.e.17.1 4 1.1 even 1 trivial
80.5.p.e.33.1 4 5.3 odd 4 inner
100.5.f.c.57.1 4 20.19 odd 2
100.5.f.c.93.1 4 20.7 even 4
180.5.l.a.37.1 4 12.11 even 2
180.5.l.a.73.1 4 60.23 odd 4
320.5.p.l.193.2 4 40.13 odd 4
320.5.p.l.257.2 4 8.5 even 2
320.5.p.m.193.1 4 40.3 even 4
320.5.p.m.257.1 4 8.3 odd 2
400.5.p.h.193.2 4 5.2 odd 4
400.5.p.h.257.2 4 5.4 even 2
900.5.l.a.757.2 4 60.59 even 2
900.5.l.a.793.2 4 60.47 odd 4