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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 47.8
Character \(\chi\) \(=\) 160.47
Dual form 160.4.o.a.143.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.102537 - 0.102537i) q^{3} +(2.04880 + 10.9910i) q^{5} +(15.5472 - 15.5472i) q^{7} +26.9790i q^{9} -20.0976 q^{11} +(-9.16919 - 9.16919i) q^{13} +(1.33707 + 0.916911i) q^{15} +(74.0631 + 74.0631i) q^{17} +111.980i q^{19} -3.18834i q^{21} +(75.1676 + 75.1676i) q^{23} +(-116.605 + 45.0368i) q^{25} +(5.53486 + 5.53486i) q^{27} +203.367 q^{29} -120.164i q^{31} +(-2.06075 + 2.06075i) q^{33} +(202.733 + 139.027i) q^{35} +(64.2751 - 64.2751i) q^{37} -1.88037 q^{39} -41.9562 q^{41} +(-133.823 + 133.823i) q^{43} +(-296.526 + 55.2745i) q^{45} +(-280.893 + 280.893i) q^{47} -140.433i q^{49} +15.1885 q^{51} +(-14.0325 - 14.0325i) q^{53} +(-41.1759 - 220.893i) q^{55} +(11.4821 + 11.4821i) q^{57} -774.159i q^{59} -159.120i q^{61} +(419.448 + 419.448i) q^{63} +(81.9928 - 119.565i) q^{65} +(-425.342 - 425.342i) q^{67} +15.4150 q^{69} -843.169i q^{71} +(721.403 - 721.403i) q^{73} +(-7.33839 + 16.5743i) q^{75} +(-312.462 + 312.462i) q^{77} -249.583 q^{79} -727.297 q^{81} +(294.739 - 294.739i) q^{83} +(-662.288 + 965.770i) q^{85} +(20.8527 - 20.8527i) q^{87} +316.918i q^{89} -285.111 q^{91} +(-12.3213 - 12.3213i) q^{93} +(-1230.77 + 229.424i) q^{95} +(-142.522 - 142.522i) q^{97} -542.212i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{3} + 8 q^{11} + 48 q^{17} + 40 q^{25} - 104 q^{27} - 112 q^{33} + 460 q^{35} - 8 q^{41} + 868 q^{43} - 1480 q^{51} + 104 q^{57} + 520 q^{65} + 1852 q^{67} - 744 q^{73} - 3300 q^{75} - 1240 q^{81}+ \cdots - 584 q^{97}+O(q^{100}) \) 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
\(3\) 0.102537 − 0.102537i 0.0197333 − 0.0197333i −0.697171 − 0.716905i \(-0.745558\pi\)
0.716905 + 0.697171i \(0.245558\pi\)
\(4\) 0 0
\(5\) 2.04880 + 10.9910i 0.183250 + 0.983066i
\(6\) 0 0
\(7\) 15.5472 − 15.5472i 0.839471 − 0.839471i −0.149318 − 0.988789i \(-0.547708\pi\)
0.988789 + 0.149318i \(0.0477078\pi\)
\(8\) 0 0
\(9\) 26.9790i 0.999221i
\(10\) 0 0
\(11\) −20.0976 −0.550877 −0.275439 − 0.961319i \(-0.588823\pi\)
−0.275439 + 0.961319i \(0.588823\pi\)
\(12\) 0 0
\(13\) −9.16919 − 9.16919i −0.195621 − 0.195621i 0.602499 − 0.798120i \(-0.294172\pi\)
−0.798120 + 0.602499i \(0.794172\pi\)
\(14\) 0 0
\(15\) 1.33707 + 0.916911i 0.0230153 + 0.0157830i
\(16\) 0 0
\(17\) 74.0631 + 74.0631i 1.05664 + 1.05664i 0.998296 + 0.0583477i \(0.0185832\pi\)
0.0583477 + 0.998296i \(0.481417\pi\)
\(18\) 0 0
\(19\) 111.980i 1.35210i 0.736855 + 0.676050i \(0.236310\pi\)
−0.736855 + 0.676050i \(0.763690\pi\)
\(20\) 0 0
\(21\) − 3.18834i − 0.0331311i
\(22\) 0 0
\(23\) 75.1676 + 75.1676i 0.681458 + 0.681458i 0.960329 − 0.278871i \(-0.0899602\pi\)
−0.278871 + 0.960329i \(0.589960\pi\)
\(24\) 0 0
\(25\) −116.605 + 45.0368i −0.932839 + 0.360294i
\(26\) 0 0
\(27\) 5.53486 + 5.53486i 0.0394513 + 0.0394513i
\(28\) 0 0
\(29\) 203.367 1.30222 0.651108 − 0.758985i \(-0.274305\pi\)
0.651108 + 0.758985i \(0.274305\pi\)
\(30\) 0 0
\(31\) − 120.164i − 0.696196i −0.937458 − 0.348098i \(-0.886828\pi\)
0.937458 − 0.348098i \(-0.113172\pi\)
\(32\) 0 0
\(33\) −2.06075 + 2.06075i −0.0108706 + 0.0108706i
\(34\) 0 0
\(35\) 202.733 + 139.027i 0.979089 + 0.671423i
\(36\) 0 0
\(37\) 64.2751 − 64.2751i 0.285588 − 0.285588i −0.549745 − 0.835333i \(-0.685275\pi\)
0.835333 + 0.549745i \(0.185275\pi\)
\(38\) 0 0
\(39\) −1.88037 −0.00772051
\(40\) 0 0
\(41\) −41.9562 −0.159816 −0.0799080 − 0.996802i \(-0.525463\pi\)
−0.0799080 + 0.996802i \(0.525463\pi\)
\(42\) 0 0
\(43\) −133.823 + 133.823i −0.474602 + 0.474602i −0.903400 − 0.428798i \(-0.858937\pi\)
0.428798 + 0.903400i \(0.358937\pi\)
\(44\) 0 0
\(45\) −296.526 + 55.2745i −0.982301 + 0.183108i
\(46\) 0 0
\(47\) −280.893 + 280.893i −0.871755 + 0.871755i −0.992664 − 0.120909i \(-0.961419\pi\)
0.120909 + 0.992664i \(0.461419\pi\)
\(48\) 0 0
\(49\) − 140.433i − 0.409424i
\(50\) 0 0
\(51\) 15.1885 0.0417022
\(52\) 0 0
\(53\) −14.0325 − 14.0325i −0.0363683 − 0.0363683i 0.688689 − 0.725057i \(-0.258187\pi\)
−0.725057 + 0.688689i \(0.758187\pi\)
\(54\) 0 0
\(55\) −41.1759 − 220.893i −0.100948 − 0.541549i
\(56\) 0 0
\(57\) 11.4821 + 11.4821i 0.0266814 + 0.0266814i
\(58\) 0 0
\(59\) − 774.159i − 1.70825i −0.520066 − 0.854126i \(-0.674093\pi\)
0.520066 − 0.854126i \(-0.325907\pi\)
\(60\) 0 0
\(61\) − 159.120i − 0.333988i −0.985958 − 0.166994i \(-0.946594\pi\)
0.985958 − 0.166994i \(-0.0534061\pi\)
\(62\) 0 0
\(63\) 419.448 + 419.448i 0.838818 + 0.838818i
\(64\) 0 0
\(65\) 81.9928 − 119.565i 0.156461 − 0.228156i
\(66\) 0 0
\(67\) −425.342 − 425.342i −0.775579 − 0.775579i 0.203496 − 0.979076i \(-0.434770\pi\)
−0.979076 + 0.203496i \(0.934770\pi\)
\(68\) 0 0
\(69\) 15.4150 0.0268948
\(70\) 0 0
\(71\) − 843.169i − 1.40938i −0.709517 − 0.704689i \(-0.751087\pi\)
0.709517 − 0.704689i \(-0.248913\pi\)
\(72\) 0 0
\(73\) 721.403 − 721.403i 1.15663 − 1.15663i 0.171432 − 0.985196i \(-0.445161\pi\)
0.985196 − 0.171432i \(-0.0548393\pi\)
\(74\) 0 0
\(75\) −7.33839 + 16.5743i −0.0112982 + 0.0255178i
\(76\) 0 0
\(77\) −312.462 + 312.462i −0.462446 + 0.462446i
\(78\) 0 0
\(79\) −249.583 −0.355447 −0.177724 − 0.984080i \(-0.556873\pi\)
−0.177724 + 0.984080i \(0.556873\pi\)
\(80\) 0 0
\(81\) −727.297 −0.997664
\(82\) 0 0
\(83\) 294.739 − 294.739i 0.389781 − 0.389781i −0.484828 − 0.874609i \(-0.661118\pi\)
0.874609 + 0.484828i \(0.161118\pi\)
\(84\) 0 0
\(85\) −662.288 + 965.770i −0.845121 + 1.23238i
\(86\) 0 0
\(87\) 20.8527 − 20.8527i 0.0256970 − 0.0256970i
\(88\) 0 0
\(89\) 316.918i 0.377452i 0.982030 + 0.188726i \(0.0604358\pi\)
−0.982030 + 0.188726i \(0.939564\pi\)
\(90\) 0 0
\(91\) −285.111 −0.328437
\(92\) 0 0
\(93\) −12.3213 − 12.3213i −0.0137383 − 0.0137383i
\(94\) 0 0
\(95\) −1230.77 + 229.424i −1.32920 + 0.247773i
\(96\) 0 0
\(97\) −142.522 − 142.522i −0.149184 − 0.149184i 0.628569 − 0.777754i \(-0.283641\pi\)
−0.777754 + 0.628569i \(0.783641\pi\)
\(98\) 0 0
\(99\) − 542.212i − 0.550448i
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.47.8 32
4.3 odd 2 40.4.k.a.27.3 yes 32
5.3 odd 4 inner 160.4.o.a.143.7 32
8.3 odd 2 inner 160.4.o.a.47.7 32
8.5 even 2 40.4.k.a.27.10 yes 32
20.3 even 4 40.4.k.a.3.10 yes 32
20.7 even 4 200.4.k.j.43.7 32
20.19 odd 2 200.4.k.j.107.14 32
40.3 even 4 inner 160.4.o.a.143.8 32
40.13 odd 4 40.4.k.a.3.3 ✓ 32
40.29 even 2 200.4.k.j.107.7 32
40.37 odd 4 200.4.k.j.43.14 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.k.a.3.3 ✓ 32 40.13 odd 4
40.4.k.a.3.10 yes 32 20.3 even 4
40.4.k.a.27.3 yes 32 4.3 odd 2
40.4.k.a.27.10 yes 32 8.5 even 2
160.4.o.a.47.7 32 8.3 odd 2 inner
160.4.o.a.47.8 32 1.1 even 1 trivial
160.4.o.a.143.7 32 5.3 odd 4 inner
160.4.o.a.143.8 32 40.3 even 4 inner
200.4.k.j.43.7 32 20.7 even 4
200.4.k.j.43.14 32 40.37 odd 4
200.4.k.j.107.7 32 40.29 even 2
200.4.k.j.107.14 32 20.19 odd 2