Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,5,Mod(193,320)] 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("320.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,10,0,6,0,110] 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: \(33.0783881868\)
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 257.1
Root \(7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 320.257
Dual form 320.5.p.m.193.1

$q$-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 \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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 0.748796 − 0.662801i \(-0.230632\pi\)
−0.662801 + 0.748796i \(0.730632\pi\)
\(8\) 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 −0.176228 − 0.984349i \(-0.556390\pi\)
0.984349 + 0.176228i \(0.0563896\pi\)
\(14\) 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 −0.770633 − 0.637279i \(-0.780060\pi\)
0.637279 + 0.770633i \(0.280060\pi\)
\(24\) 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 \(-0.986945\pi\)
0.999159 − 0.0410030i \(-0.0130553\pi\)
\(30\) 0 0
\(31\) −1373.31 −1.42905 −0.714523 − 0.699612i \(-0.753356\pi\)
−0.714523 + 0.699612i \(0.753356\pi\)
\(32\) 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 0.971188 − 0.238314i \(-0.0765946\pi\)
−0.238314 + 0.971188i \(0.576595\pi\)
\(38\) 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 0.799419 − 0.600773i \(-0.205141\pi\)
−0.600773 + 0.799419i \(0.705141\pi\)
\(48\) 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 −0.822441 − 0.568850i \(-0.807389\pi\)
0.568850 + 0.822441i \(0.307389\pi\)
\(54\) 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 \(-0.560044\pi\)
−0.187518 + 0.982261i \(0.560044\pi\)
\(62\) 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 \(-0.605552\pi\)
−0.325558 + 0.945522i \(0.605552\pi\)
\(72\) 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 \(0.0875446\pi\)
−0.962417 + 0.271575i \(0.912455\pi\)
\(80\) 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 320.5.p.m.257.1 4
4.3 odd 2 320.5.p.l.257.2 4
5.3 odd 4 inner 320.5.p.m.193.1 4
8.3 odd 2 80.5.p.e.17.1 4
8.5 even 2 20.5.f.a.17.2 yes 4
20.3 even 4 320.5.p.l.193.2 4
24.5 odd 2 180.5.l.a.37.1 4
40.3 even 4 80.5.p.e.33.1 4
40.13 odd 4 20.5.f.a.13.2 ✓ 4
40.19 odd 2 400.5.p.h.257.2 4
40.27 even 4 400.5.p.h.193.2 4
40.29 even 2 100.5.f.c.57.1 4
40.37 odd 4 100.5.f.c.93.1 4
120.29 odd 2 900.5.l.a.757.2 4
120.53 even 4 180.5.l.a.73.1 4
120.77 even 4 900.5.l.a.793.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.2 ✓ 4 40.13 odd 4
20.5.f.a.17.2 yes 4 8.5 even 2
80.5.p.e.17.1 4 8.3 odd 2
80.5.p.e.33.1 4 40.3 even 4
100.5.f.c.57.1 4 40.29 even 2
100.5.f.c.93.1 4 40.37 odd 4
180.5.l.a.37.1 4 24.5 odd 2
180.5.l.a.73.1 4 120.53 even 4
320.5.p.l.193.2 4 20.3 even 4
320.5.p.l.257.2 4 4.3 odd 2
320.5.p.m.193.1 4 5.3 odd 4 inner
320.5.p.m.257.1 4 1.1 even 1 trivial
400.5.p.h.193.2 4 40.27 even 4
400.5.p.h.257.2 4 40.19 odd 2
900.5.l.a.757.2 4 120.29 odd 2
900.5.l.a.793.2 4 120.77 even 4