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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(473,888)] 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("888.473"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.br (of order \(12\), degree \(4\), 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: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 569.32
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.32

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36692 + 1.06373i) q^{3} +(0.356362 - 1.32996i) q^{5} +(0.659764 + 1.14274i) q^{7} +(0.736949 + 2.90808i) q^{9} -0.353852 q^{11} +(-0.106594 - 0.0285618i) q^{13} +(1.90184 - 1.43888i) q^{15} +(-0.268822 + 0.0720307i) q^{17} +(7.97106 + 2.13584i) q^{19} +(-0.313728 + 2.26385i) q^{21} +(-0.593239 - 0.593239i) q^{23} +(2.68832 + 1.55210i) q^{25} +(-2.08606 + 4.75903i) q^{27} +(-1.88653 + 1.88653i) q^{29} +(3.87869 + 3.87869i) q^{31} +(-0.483688 - 0.376404i) q^{33} +(1.75492 - 0.470230i) q^{35} +(-5.93717 - 1.32288i) q^{37} +(-0.115324 - 0.152430i) q^{39} +(0.379388 + 0.657119i) q^{41} +(4.16777 - 4.16777i) q^{43} +(4.13025 + 0.0562150i) q^{45} -6.18133i q^{47} +(2.62942 - 4.55430i) q^{49} +(-0.444080 - 0.187494i) q^{51} +(-3.52233 - 2.03362i) q^{53} +(-0.126100 + 0.470610i) q^{55} +(8.62385 + 11.3986i) q^{57} +(-5.80978 + 1.55673i) q^{59} +(-1.41803 + 5.29217i) q^{61} +(-2.83697 + 2.76079i) q^{63} +(-0.0759724 + 0.131588i) q^{65} +(4.53482 - 2.61818i) q^{67} +(-0.179864 - 1.44196i) q^{69} +(5.08729 - 2.93715i) q^{71} -1.03868i q^{73} +(2.02370 + 4.98125i) q^{75} +(-0.233459 - 0.404363i) q^{77} +(0.472722 + 0.126665i) q^{79} +(-7.91381 + 4.28620i) q^{81} +(-11.5472 - 6.66677i) q^{83} +0.383192i q^{85} +(-4.58550 + 0.571976i) q^{87} +(4.27017 + 15.9365i) q^{89} +(-0.0376881 - 0.140654i) q^{91} +(1.17598 + 9.42774i) q^{93} +(5.68117 - 9.84008i) q^{95} +(-7.97830 + 7.97830i) q^{97} +(-0.260771 - 1.02903i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\)

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
\(3\) 1.36692 + 1.06373i 0.789192 + 0.614146i
\(4\) 0 0
\(5\) 0.356362 − 1.32996i 0.159370 − 0.594777i −0.839321 − 0.543636i \(-0.817047\pi\)
0.998691 − 0.0511417i \(-0.0162860\pi\)
\(6\) 0 0
\(7\) 0.659764 + 1.14274i 0.249367 + 0.431917i 0.963350 − 0.268246i \(-0.0864441\pi\)
−0.713983 + 0.700163i \(0.753111\pi\)
\(8\) 0 0
\(9\) 0.736949 + 2.90808i 0.245650 + 0.969359i
\(10\) 0 0
\(11\) −0.353852 −0.106690 −0.0533452 − 0.998576i \(-0.516988\pi\)
−0.0533452 + 0.998576i \(0.516988\pi\)
\(12\) 0 0
\(13\) −0.106594 − 0.0285618i −0.0295639 − 0.00792163i 0.244007 − 0.969774i \(-0.421538\pi\)
−0.273571 + 0.961852i \(0.588205\pi\)
\(14\) 0 0
\(15\) 1.90184 − 1.43888i 0.491054 − 0.371517i
\(16\) 0 0
\(17\) −0.268822 + 0.0720307i −0.0651989 + 0.0174700i −0.291271 − 0.956641i \(-0.594078\pi\)
0.226072 + 0.974111i \(0.427411\pi\)
\(18\) 0 0
\(19\) 7.97106 + 2.13584i 1.82869 + 0.489995i 0.997791 − 0.0664333i \(-0.0211620\pi\)
0.830896 + 0.556428i \(0.187829\pi\)
\(20\) 0 0
\(21\) −0.313728 + 2.26385i −0.0684611 + 0.494013i
\(22\) 0 0
\(23\) −0.593239 − 0.593239i −0.123699 − 0.123699i 0.642547 − 0.766246i \(-0.277878\pi\)
−0.766246 + 0.642547i \(0.777878\pi\)
\(24\) 0 0
\(25\) 2.68832 + 1.55210i 0.537664 + 0.310420i
\(26\) 0 0
\(27\) −2.08606 + 4.75903i −0.401463 + 0.915875i
\(28\) 0 0
\(29\) −1.88653 + 1.88653i −0.350320 + 0.350320i −0.860229 − 0.509909i \(-0.829679\pi\)
0.509909 + 0.860229i \(0.329679\pi\)
\(30\) 0 0
\(31\) 3.87869 + 3.87869i 0.696633 + 0.696633i 0.963683 − 0.267050i \(-0.0860489\pi\)
−0.267050 + 0.963683i \(0.586049\pi\)
\(32\) 0 0
\(33\) −0.483688 − 0.376404i −0.0841993 − 0.0655235i
\(34\) 0 0
\(35\) 1.75492 − 0.470230i 0.296636 − 0.0794834i
\(36\) 0 0
\(37\) −5.93717 − 1.32288i −0.976065 − 0.217481i
\(38\) 0 0
\(39\) −0.115324 − 0.152430i −0.0184666 − 0.0244083i
\(40\) 0 0
\(41\) 0.379388 + 0.657119i 0.0592504 + 0.102625i 0.894129 − 0.447809i \(-0.147796\pi\)
−0.834879 + 0.550434i \(0.814462\pi\)
\(42\) 0 0
\(43\) 4.16777 − 4.16777i 0.635579 − 0.635579i −0.313883 − 0.949462i \(-0.601630\pi\)
0.949462 + 0.313883i \(0.101630\pi\)
\(44\) 0 0
\(45\) 4.13025 + 0.0562150i 0.615702 + 0.00838004i
\(46\) 0 0
\(47\) − 6.18133i − 0.901639i −0.892615 − 0.450820i \(-0.851132\pi\)
0.892615 − 0.450820i \(-0.148868\pi\)
\(48\) 0 0
\(49\) 2.62942 − 4.55430i 0.375632 − 0.650614i
\(50\) 0 0
\(51\) −0.444080 − 0.187494i −0.0621836 − 0.0262545i
\(52\) 0 0
\(53\) −3.52233 − 2.03362i −0.483829 − 0.279339i 0.238182 − 0.971221i \(-0.423449\pi\)
−0.722011 + 0.691882i \(0.756782\pi\)
\(54\) 0 0
\(55\) −0.126100 + 0.470610i −0.0170033 + 0.0634571i
\(56\) 0 0
\(57\) 8.62385 + 11.3986i 1.14226 + 1.50978i
\(58\) 0 0
\(59\) −5.80978 + 1.55673i −0.756369 + 0.202669i −0.616341 − 0.787479i \(-0.711386\pi\)
−0.140028 + 0.990148i \(0.544719\pi\)
\(60\) 0 0
\(61\) −1.41803 + 5.29217i −0.181561 + 0.677593i 0.813780 + 0.581173i \(0.197406\pi\)
−0.995341 + 0.0964204i \(0.969261\pi\)
\(62\) 0 0
\(63\) −2.83697 + 2.76079i −0.357425 + 0.347826i
\(64\) 0 0
\(65\) −0.0759724 + 0.131588i −0.00942321 + 0.0163215i
\(66\) 0 0
\(67\) 4.53482 − 2.61818i 0.554016 − 0.319861i −0.196724 − 0.980459i \(-0.563030\pi\)
0.750740 + 0.660598i \(0.229697\pi\)
\(68\) 0 0
\(69\) −0.179864 − 1.44196i −0.0216531 − 0.173592i
\(70\) 0 0
\(71\) 5.08729 − 2.93715i 0.603750 − 0.348575i −0.166765 − 0.985997i \(-0.553332\pi\)
0.770516 + 0.637421i \(0.219999\pi\)
\(72\) 0 0
\(73\) − 1.03868i − 0.121568i −0.998151 − 0.0607842i \(-0.980640\pi\)
0.998151 − 0.0607842i \(-0.0193601\pi\)
\(74\) 0 0
\(75\) 2.02370 + 4.98125i 0.233677 + 0.575186i
\(76\) 0 0
\(77\) −0.233459 − 0.404363i −0.0266051 − 0.0460814i
\(78\) 0 0
\(79\) 0.472722 + 0.126665i 0.0531853 + 0.0142510i 0.285314 − 0.958434i \(-0.407902\pi\)
−0.232128 + 0.972685i \(0.574569\pi\)
\(80\) 0 0
\(81\) −7.91381 + 4.28620i −0.879313 + 0.476245i
\(82\) 0 0
\(83\) −11.5472 − 6.66677i −1.26747 − 0.731773i −0.292960 − 0.956125i \(-0.594640\pi\)
−0.974508 + 0.224351i \(0.927974\pi\)
\(84\) 0 0
\(85\) 0.383192i 0.0415631i
\(86\) 0 0
\(87\) −4.58550 + 0.571976i −0.491617 + 0.0613223i
\(88\) 0 0
\(89\) 4.27017 + 15.9365i 0.452637 + 1.68927i 0.694942 + 0.719065i \(0.255430\pi\)
−0.242305 + 0.970200i \(0.577904\pi\)
\(90\) 0 0
\(91\) −0.0376881 − 0.140654i −0.00395079 − 0.0147445i
\(92\) 0 0
\(93\) 1.17598 + 9.42774i 0.121943 + 0.977611i
\(94\) 0 0
\(95\) 5.68117 − 9.84008i 0.582876 − 1.00957i
\(96\) 0 0
\(97\) −7.97830 + 7.97830i −0.810074 + 0.810074i −0.984645 − 0.174571i \(-0.944146\pi\)
0.174571 + 0.984645i \(0.444146\pi\)
\(98\) 0 0
\(99\) −0.260771 − 1.02903i −0.0262085 − 0.103421i
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 888.2.br.a.569.32 yes 152
3.2 odd 2 inner 888.2.br.a.569.19 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.19 yes 152
111.8 even 12 inner 888.2.br.a.785.32 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.19 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.32 yes 152 1.1 even 1 trivial
888.2.br.a.785.19 yes 152 37.8 odd 12 inner
888.2.br.a.785.32 yes 152 111.8 even 12 inner