Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,4,Mod(47,160)] 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("160.47"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.1
Character \(\chi\) \(=\) 160.47
Dual form 160.4.o.a.143.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-6.15076 + 6.15076i) q^{3} +(-11.1441 + 0.899439i) q^{5} +(-16.5614 + 16.5614i) q^{7} -48.6638i q^{9} +11.0706 q^{11} +(-4.95962 - 4.95962i) q^{13} +(63.0125 - 74.0770i) q^{15} +(68.6010 + 68.6010i) q^{17} -29.4303i q^{19} -203.730i q^{21} +(-70.5372 - 70.5372i) q^{23} +(123.382 - 20.0469i) q^{25} +(133.249 + 133.249i) q^{27} +39.6115 q^{29} -167.596i q^{31} +(-68.0928 + 68.0928i) q^{33} +(169.666 - 199.458i) q^{35} +(38.3756 - 38.3756i) q^{37} +61.0109 q^{39} -305.291 q^{41} +(114.783 - 114.783i) q^{43} +(43.7701 + 542.314i) q^{45} +(-335.637 + 335.637i) q^{47} -205.559i q^{49} -843.897 q^{51} +(-455.381 - 455.381i) q^{53} +(-123.372 + 9.95735i) q^{55} +(181.019 + 181.019i) q^{57} -170.086i q^{59} +512.994i q^{61} +(805.939 + 805.939i) q^{63} +(59.7314 + 50.8096i) q^{65} +(476.312 + 476.312i) q^{67} +867.716 q^{69} -627.703i q^{71} +(-2.93764 + 2.93764i) q^{73} +(-635.590 + 882.197i) q^{75} +(-183.345 + 183.345i) q^{77} +132.081 q^{79} -325.240 q^{81} +(111.150 - 111.150i) q^{83} +(-826.199 - 702.794i) q^{85} +(-243.641 + 243.641i) q^{87} -836.950i q^{89} +164.276 q^{91} +(1030.84 + 1030.84i) q^{93} +(26.4708 + 327.975i) q^{95} +(-485.063 - 485.063i) q^{97} -538.738i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{3} + 8 q^{11} + 48 q^{17} + 40 q^{25} - 104 q^{27} - 112 q^{33} + 460 q^{35} - 8 q^{41} + 868 q^{43} - 1480 q^{51} + 104 q^{57} + 520 q^{65} + 1852 q^{67} - 744 q^{73} - 3300 q^{75} - 1240 q^{81}+ \cdots - 584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) −6.15076 + 6.15076i −1.18371 + 1.18371i −0.204940 + 0.978774i \(0.565700\pi\)
−0.978774 + 0.204940i \(0.934300\pi\)
\(4\) 0 0
\(5\) −11.1441 + 0.899439i −0.996759 + 0.0804483i
\(6\) 0 0
\(7\) −16.5614 + 16.5614i −0.894231 + 0.894231i −0.994918 − 0.100687i \(-0.967896\pi\)
0.100687 + 0.994918i \(0.467896\pi\)
\(8\) 0 0
\(9\) − 48.6638i − 1.80236i
\(10\) 0 0
\(11\) 11.0706 0.303447 0.151724 − 0.988423i \(-0.451518\pi\)
0.151724 + 0.988423i \(0.451518\pi\)
\(12\) 0 0
\(13\) −4.95962 − 4.95962i −0.105812 − 0.105812i 0.652219 − 0.758031i \(-0.273838\pi\)
−0.758031 + 0.652219i \(0.773838\pi\)
\(14\) 0 0
\(15\) 63.0125 − 74.0770i 1.08465 − 1.27511i
\(16\) 0 0
\(17\) 68.6010 + 68.6010i 0.978717 + 0.978717i 0.999778 − 0.0210615i \(-0.00670459\pi\)
−0.0210615 + 0.999778i \(0.506705\pi\)
\(18\) 0 0
\(19\) − 29.4303i − 0.355357i −0.984089 − 0.177678i \(-0.943141\pi\)
0.984089 − 0.177678i \(-0.0568587\pi\)
\(20\) 0 0
\(21\) − 203.730i − 2.11703i
\(22\) 0 0
\(23\) −70.5372 − 70.5372i −0.639480 − 0.639480i 0.310947 − 0.950427i \(-0.399354\pi\)
−0.950427 + 0.310947i \(0.899354\pi\)
\(24\) 0 0
\(25\) 123.382 − 20.0469i 0.987056 − 0.160375i
\(26\) 0 0
\(27\) 133.249 + 133.249i 0.949767 + 0.949767i
\(28\) 0 0
\(29\) 39.6115 0.253644 0.126822 − 0.991925i \(-0.459522\pi\)
0.126822 + 0.991925i \(0.459522\pi\)
\(30\) 0 0
\(31\) − 167.596i − 0.971002i −0.874236 − 0.485501i \(-0.838637\pi\)
0.874236 − 0.485501i \(-0.161363\pi\)
\(32\) 0 0
\(33\) −68.0928 + 68.0928i −0.359195 + 0.359195i
\(34\) 0 0
\(35\) 169.666 − 199.458i 0.819393 − 0.963272i
\(36\) 0 0
\(37\) 38.3756 − 38.3756i 0.170511 − 0.170511i −0.616693 − 0.787204i \(-0.711528\pi\)
0.787204 + 0.616693i \(0.211528\pi\)
\(38\) 0 0
\(39\) 61.0109 0.250501
\(40\) 0 0
\(41\) −305.291 −1.16289 −0.581445 − 0.813586i \(-0.697512\pi\)
−0.581445 + 0.813586i \(0.697512\pi\)
\(42\) 0 0
\(43\) 114.783 − 114.783i 0.407077 − 0.407077i −0.473641 − 0.880718i \(-0.657061\pi\)
0.880718 + 0.473641i \(0.157061\pi\)
\(44\) 0 0
\(45\) 43.7701 + 542.314i 0.144997 + 1.79652i
\(46\) 0 0
\(47\) −335.637 + 335.637i −1.04165 + 1.04165i −0.0425604 + 0.999094i \(0.513551\pi\)
−0.999094 + 0.0425604i \(0.986449\pi\)
\(48\) 0 0
\(49\) − 205.559i − 0.599297i
\(50\) 0 0
\(51\) −843.897 −2.31704
\(52\) 0 0
\(53\) −455.381 − 455.381i −1.18022 − 1.18022i −0.979688 − 0.200527i \(-0.935734\pi\)
−0.200527 − 0.979688i \(-0.564266\pi\)
\(54\) 0 0
\(55\) −123.372 + 9.95735i −0.302464 + 0.0244118i
\(56\) 0 0
\(57\) 181.019 + 181.019i 0.420641 + 0.420641i
\(58\) 0 0
\(59\) − 170.086i − 0.375311i −0.982235 − 0.187655i \(-0.939911\pi\)
0.982235 − 0.187655i \(-0.0600888\pi\)
\(60\) 0 0
\(61\) 512.994i 1.07676i 0.842703 + 0.538378i \(0.180963\pi\)
−0.842703 + 0.538378i \(0.819037\pi\)
\(62\) 0 0
\(63\) 805.939 + 805.939i 1.61173 + 1.61173i
\(64\) 0 0
\(65\) 59.7314 + 50.8096i 0.113981 + 0.0969563i
\(66\) 0 0
\(67\) 476.312 + 476.312i 0.868519 + 0.868519i 0.992308 − 0.123790i \(-0.0395048\pi\)
−0.123790 + 0.992308i \(0.539505\pi\)
\(68\) 0 0
\(69\) 867.716 1.51392
\(70\) 0 0
\(71\) − 627.703i − 1.04922i −0.851343 − 0.524610i \(-0.824211\pi\)
0.851343 − 0.524610i \(-0.175789\pi\)
\(72\) 0 0
\(73\) −2.93764 + 2.93764i −0.00470992 + 0.00470992i −0.709458 − 0.704748i \(-0.751060\pi\)
0.704748 + 0.709458i \(0.251060\pi\)
\(74\) 0 0
\(75\) −635.590 + 882.197i −0.978555 + 1.35823i
\(76\) 0 0
\(77\) −183.345 + 183.345i −0.271352 + 0.271352i
\(78\) 0 0
\(79\) 132.081 0.188105 0.0940523 − 0.995567i \(-0.470018\pi\)
0.0940523 + 0.995567i \(0.470018\pi\)
\(80\) 0 0
\(81\) −325.240 −0.446145
\(82\) 0 0
\(83\) 111.150 − 111.150i 0.146991 − 0.146991i −0.629781 − 0.776772i \(-0.716855\pi\)
0.776772 + 0.629781i \(0.216855\pi\)
\(84\) 0 0
\(85\) −826.199 − 702.794i −1.05428 − 0.896808i
\(86\) 0 0
\(87\) −243.641 + 243.641i −0.300242 + 0.300242i
\(88\) 0 0
\(89\) − 836.950i − 0.996815i −0.866943 − 0.498407i \(-0.833918\pi\)
0.866943 − 0.498407i \(-0.166082\pi\)
\(90\) 0 0
\(91\) 164.276 0.189240
\(92\) 0 0
\(93\) 1030.84 + 1030.84i 1.14939 + 1.14939i
\(94\) 0 0
\(95\) 26.4708 + 327.975i 0.0285878 + 0.354205i
\(96\) 0 0
\(97\) −485.063 − 485.063i −0.507739 − 0.507739i 0.406093 − 0.913832i \(-0.366891\pi\)
−0.913832 + 0.406093i \(0.866891\pi\)
\(98\) 0 0
\(99\) − 538.738i − 0.546922i
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 160.4.o.a.47.1 32
4.3 odd 2 40.4.k.a.27.1 yes 32
5.3 odd 4 inner 160.4.o.a.143.2 32
8.3 odd 2 inner 160.4.o.a.47.2 32
8.5 even 2 40.4.k.a.27.8 yes 32
20.3 even 4 40.4.k.a.3.8 yes 32
20.7 even 4 200.4.k.j.43.9 32
20.19 odd 2 200.4.k.j.107.16 32
40.3 even 4 inner 160.4.o.a.143.1 32
40.13 odd 4 40.4.k.a.3.1 ✓ 32
40.29 even 2 200.4.k.j.107.9 32
40.37 odd 4 200.4.k.j.43.16 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.k.a.3.1 ✓ 32 40.13 odd 4
40.4.k.a.3.8 yes 32 20.3 even 4
40.4.k.a.27.1 yes 32 4.3 odd 2
40.4.k.a.27.8 yes 32 8.5 even 2
160.4.o.a.47.1 32 1.1 even 1 trivial
160.4.o.a.47.2 32 8.3 odd 2 inner
160.4.o.a.143.1 32 40.3 even 4 inner
160.4.o.a.143.2 32 5.3 odd 4 inner
200.4.k.j.43.9 32 20.7 even 4
200.4.k.j.43.16 32 40.37 odd 4
200.4.k.j.107.9 32 40.29 even 2
200.4.k.j.107.16 32 20.19 odd 2