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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.26
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.26

$q$-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+(0.926925 + 1.06809i) q^{2} -1.00000i q^{3} +(-0.281621 + 1.98007i) q^{4} +(2.48900 - 2.48900i) q^{5} +(1.06809 - 0.926925i) q^{6} +1.23837i q^{7} +(-2.37593 + 1.53458i) q^{8} -1.00000 q^{9} +(4.96558 + 0.351353i) q^{10} -5.59811i q^{11} +(1.98007 + 0.281621i) q^{12} +(3.22233 - 3.22233i) q^{13} +(-1.32269 + 1.14788i) q^{14} +(-2.48900 - 2.48900i) q^{15} +(-3.84138 - 1.11526i) q^{16} +(-2.22933 - 2.22933i) q^{17} +(-0.926925 - 1.06809i) q^{18} +(5.67997 + 5.67997i) q^{19} +(4.22744 + 5.62935i) q^{20} +1.23837 q^{21} +(5.97927 - 5.18903i) q^{22} +(-2.40735 + 2.40735i) q^{23} +(1.53458 + 2.37593i) q^{24} -7.39021i q^{25} +(6.42859 + 0.454873i) q^{26} +1.00000i q^{27} +(-2.45207 - 0.348751i) q^{28} +(-4.45747 - 4.45747i) q^{29} +(0.351353 - 4.96558i) q^{30} +(2.51566 + 2.51566i) q^{31} +(-2.36947 - 5.13669i) q^{32} -5.59811 q^{33} +(0.314699 - 4.44755i) q^{34} +(3.08230 + 3.08230i) q^{35} +(0.281621 - 1.98007i) q^{36} +(-0.507515 - 6.06155i) q^{37} +(-0.801799 + 11.3316i) q^{38} +(-3.22233 - 3.22233i) q^{39} +(-2.09412 + 9.73326i) q^{40} +10.0864i q^{41} +(1.14788 + 1.32269i) q^{42} +(7.24350 + 7.24350i) q^{43} +(11.0847 + 1.57655i) q^{44} +(-2.48900 + 2.48900i) q^{45} +(-4.80270 - 0.339828i) q^{46} -7.35728i q^{47} +(-1.11526 + 3.84138i) q^{48} +5.46644 q^{49} +(7.89339 - 6.85017i) q^{50} +(-2.22933 + 2.22933i) q^{51} +(5.47298 + 7.28793i) q^{52} +0.173792i q^{53} +(-1.06809 + 0.926925i) q^{54} +(-13.9337 - 13.9337i) q^{55} +(-1.90038 - 2.94229i) q^{56} +(5.67997 - 5.67997i) q^{57} +(0.629227 - 8.89270i) q^{58} +(3.79028 + 3.79028i) q^{59} +(5.62935 - 4.22744i) q^{60} +(-1.24228 - 1.24228i) q^{61} +(-0.355117 + 5.01877i) q^{62} -1.23837i q^{63} +(3.29011 - 7.29213i) q^{64} -16.0408i q^{65} +(-5.18903 - 5.97927i) q^{66} -2.06162i q^{67} +(5.04207 - 3.78642i) q^{68} +(2.40735 + 2.40735i) q^{69} +(-0.435106 + 6.14923i) q^{70} +12.8475i q^{71} +(2.37593 - 1.53458i) q^{72} +10.0534i q^{73} +(6.00384 - 6.16067i) q^{74} -7.39021 q^{75} +(-12.8463 + 9.64715i) q^{76} +6.93254 q^{77} +(0.454873 - 6.42859i) q^{78} +(0.543893 - 0.543893i) q^{79} +(-12.3371 + 6.78530i) q^{80} +1.00000 q^{81} +(-10.7731 + 9.34929i) q^{82} -16.5002 q^{83} +(-0.348751 + 2.45207i) q^{84} -11.0976 q^{85} +(-1.02251 + 14.4509i) q^{86} +(-4.45747 + 4.45747i) q^{87} +(8.59077 + 13.3007i) q^{88} +(9.27258 - 9.27258i) q^{89} +(-4.96558 - 0.351353i) q^{90} +(3.99045 + 3.99045i) q^{91} +(-4.08877 - 5.44469i) q^{92} +(2.51566 - 2.51566i) q^{93} +(7.85821 - 6.81964i) q^{94} +28.2748 q^{95} +(-5.13669 + 2.36947i) q^{96} +(5.10181 + 5.10181i) q^{97} +(5.06697 + 5.83863i) q^{98} +5.59811i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.926925 + 1.06809i 0.655435 + 0.755252i
\(3\) − 1.00000i − 0.577350i
\(4\) −0.281621 + 1.98007i −0.140811 + 0.990037i
\(5\) 2.48900 − 2.48900i 1.11311 − 1.11311i 0.120386 − 0.992727i \(-0.461587\pi\)
0.992727 − 0.120386i \(-0.0384132\pi\)
\(6\) 1.06809 − 0.926925i 0.436045 − 0.378415i
\(7\) 1.23837i 0.468060i 0.972229 + 0.234030i \(0.0751915\pi\)
−0.972229 + 0.234030i \(0.924809\pi\)
\(8\) −2.37593 + 1.53458i −0.840019 + 0.542557i
\(9\) −1.00000 −0.333333
\(10\) 4.96558 + 0.351353i 1.57025 + 0.111108i
\(11\) − 5.59811i − 1.68789i −0.536426 − 0.843947i \(-0.680226\pi\)
0.536426 − 0.843947i \(-0.319774\pi\)
\(12\) 1.98007 + 0.281621i 0.571598 + 0.0812970i
\(13\) 3.22233 − 3.22233i 0.893714 − 0.893714i −0.101156 − 0.994871i \(-0.532254\pi\)
0.994871 + 0.101156i \(0.0322542\pi\)
\(14\) −1.32269 + 1.14788i −0.353503 + 0.306783i
\(15\) −2.48900 − 2.48900i −0.642656 − 0.642656i
\(16\) −3.84138 − 1.11526i −0.960345 − 0.278815i
\(17\) −2.22933 − 2.22933i −0.540693 − 0.540693i 0.383039 − 0.923732i \(-0.374877\pi\)
−0.923732 + 0.383039i \(0.874877\pi\)
\(18\) −0.926925 − 1.06809i −0.218478 − 0.251751i
\(19\) 5.67997 + 5.67997i 1.30307 + 1.30307i 0.926309 + 0.376765i \(0.122963\pi\)
0.376765 + 0.926309i \(0.377037\pi\)
\(20\) 4.22744 + 5.62935i 0.945285 + 1.25876i
\(21\) 1.23837 0.270235
\(22\) 5.97927 − 5.18903i 1.27479 − 1.10630i
\(23\) −2.40735 + 2.40735i −0.501968 + 0.501968i −0.912049 − 0.410081i \(-0.865500\pi\)
0.410081 + 0.912049i \(0.365500\pi\)
\(24\) 1.53458 + 2.37593i 0.313245 + 0.484985i
\(25\) − 7.39021i − 1.47804i
\(26\) 6.42859 + 0.454873i 1.26075 + 0.0892079i
\(27\) 1.00000i 0.192450i
\(28\) −2.45207 − 0.348751i −0.463397 − 0.0659078i
\(29\) −4.45747 − 4.45747i −0.827731 − 0.827731i 0.159472 − 0.987202i \(-0.449021\pi\)
−0.987202 + 0.159472i \(0.949021\pi\)
\(30\) 0.351353 − 4.96558i 0.0641480 − 0.906586i
\(31\) 2.51566 + 2.51566i 0.451826 + 0.451826i 0.895960 − 0.444134i \(-0.146489\pi\)
−0.444134 + 0.895960i \(0.646489\pi\)
\(32\) −2.36947 − 5.13669i −0.418868 − 0.908047i
\(33\) −5.59811 −0.974506
\(34\) 0.314699 − 4.44755i 0.0539704 − 0.762749i
\(35\) 3.08230 + 3.08230i 0.521004 + 0.521004i
\(36\) 0.281621 − 1.98007i 0.0469368 − 0.330012i
\(37\) −0.507515 − 6.06155i −0.0834350 − 0.996513i
\(38\) −0.801799 + 11.3316i −0.130069 + 1.83823i
\(39\) −3.22233 − 3.22233i −0.515986 − 0.515986i
\(40\) −2.09412 + 9.73326i −0.331109 + 1.53896i
\(41\) 10.0864i 1.57522i 0.616172 + 0.787612i \(0.288683\pi\)
−0.616172 + 0.787612i \(0.711317\pi\)
\(42\) 1.14788 + 1.32269i 0.177121 + 0.204095i
\(43\) 7.24350 + 7.24350i 1.10462 + 1.10462i 0.993845 + 0.110779i \(0.0353345\pi\)
0.110779 + 0.993845i \(0.464666\pi\)
\(44\) 11.0847 + 1.57655i 1.67108 + 0.237673i
\(45\) −2.48900 + 2.48900i −0.371038 + 0.371038i
\(46\) −4.80270 − 0.339828i −0.708119 − 0.0501049i
\(47\) − 7.35728i − 1.07317i −0.843847 − 0.536585i \(-0.819714\pi\)
0.843847 − 0.536585i \(-0.180286\pi\)
\(48\) −1.11526 + 3.84138i −0.160974 + 0.554455i
\(49\) 5.46644 0.780919
\(50\) 7.89339 − 6.85017i 1.11629 − 0.968760i
\(51\) −2.22933 + 2.22933i −0.312169 + 0.312169i
\(52\) 5.47298 + 7.28793i 0.758965 + 1.01065i
\(53\) 0.173792i 0.0238721i 0.999929 + 0.0119361i \(0.00379946\pi\)
−0.999929 + 0.0119361i \(0.996201\pi\)
\(54\) −1.06809 + 0.926925i −0.145348 + 0.126138i
\(55\) −13.9337 − 13.9337i −1.87882 − 1.87882i
\(56\) −1.90038 − 2.94229i −0.253949 − 0.393180i
\(57\) 5.67997 − 5.67997i 0.752330 − 0.752330i
\(58\) 0.629227 − 8.89270i 0.0826216 − 1.16767i
\(59\) 3.79028 + 3.79028i 0.493452 + 0.493452i 0.909392 − 0.415940i \(-0.136547\pi\)
−0.415940 + 0.909392i \(0.636547\pi\)
\(60\) 5.62935 − 4.22744i 0.726746 − 0.545760i
\(61\) −1.24228 − 1.24228i −0.159058 − 0.159058i 0.623091 − 0.782149i \(-0.285877\pi\)
−0.782149 + 0.623091i \(0.785877\pi\)
\(62\) −0.355117 + 5.01877i −0.0450999 + 0.637385i
\(63\) − 1.23837i − 0.156020i
\(64\) 3.29011 − 7.29213i 0.411264 − 0.911516i
\(65\) − 16.0408i − 1.98961i
\(66\) −5.18903 − 5.97927i −0.638725 − 0.735998i
\(67\) − 2.06162i − 0.251867i −0.992039 − 0.125934i \(-0.959807\pi\)
0.992039 − 0.125934i \(-0.0401926\pi\)
\(68\) 5.04207 − 3.78642i 0.611441 − 0.459171i
\(69\) 2.40735 + 2.40735i 0.289811 + 0.289811i
\(70\) −0.435106 + 6.14923i −0.0520051 + 0.734974i
\(71\) 12.8475i 1.52471i 0.647157 + 0.762357i \(0.275958\pi\)
−0.647157 + 0.762357i \(0.724042\pi\)
\(72\) 2.37593 − 1.53458i 0.280006 − 0.180852i
\(73\) 10.0534i 1.17666i 0.808621 + 0.588330i \(0.200214\pi\)
−0.808621 + 0.588330i \(0.799786\pi\)
\(74\) 6.00384 − 6.16067i 0.697932 − 0.716164i
\(75\) −7.39021 −0.853348
\(76\) −12.8463 + 9.64715i −1.47358 + 1.10660i
\(77\) 6.93254 0.790037
\(78\) 0.454873 − 6.42859i 0.0515042 − 0.727895i
\(79\) 0.543893 − 0.543893i 0.0611928 − 0.0611928i −0.675848 − 0.737041i \(-0.736222\pi\)
0.737041 + 0.675848i \(0.236222\pi\)
\(80\) −12.3371 + 6.78530i −1.37933 + 0.758620i
\(81\) 1.00000 0.111111
\(82\) −10.7731 + 9.34929i −1.18969 + 1.03246i
\(83\) −16.5002 −1.81113 −0.905565 − 0.424208i \(-0.860553\pi\)
−0.905565 + 0.424208i \(0.860553\pi\)
\(84\) −0.348751 + 2.45207i −0.0380519 + 0.267542i
\(85\) −11.0976 −1.20371
\(86\) −1.02251 + 14.4509i −0.110260 + 1.55828i
\(87\) −4.45747 + 4.45747i −0.477891 + 0.477891i
\(88\) 8.59077 + 13.3007i 0.915779 + 1.41786i
\(89\) 9.27258 − 9.27258i 0.982891 − 0.982891i −0.0169650 − 0.999856i \(-0.505400\pi\)
0.999856 + 0.0169650i \(0.00540039\pi\)
\(90\) −4.96558 − 0.351353i −0.523418 − 0.0370359i
\(91\) 3.99045 + 3.99045i 0.418312 + 0.418312i
\(92\) −4.08877 − 5.44469i −0.426284 − 0.567649i
\(93\) 2.51566 − 2.51566i 0.260862 − 0.260862i
\(94\) 7.85821 − 6.81964i 0.810513 − 0.703392i
\(95\) 28.2748 2.90094
\(96\) −5.13669 + 2.36947i −0.524261 + 0.241833i
\(97\) 5.10181 + 5.10181i 0.518011 + 0.518011i 0.916969 − 0.398958i \(-0.130628\pi\)
−0.398958 + 0.916969i \(0.630628\pi\)
\(98\) 5.06697 + 5.83863i 0.511842 + 0.589791i
\(99\) 5.59811i 0.562632i
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.r.e.43.26 yes 72
8.3 odd 2 inner 888.2.r.e.43.9 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.9 yes 72
296.179 even 4 inner 888.2.r.e.475.26 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.9 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.26 yes 72 1.1 even 1 trivial
888.2.r.e.475.9 yes 72 37.31 odd 4 inner
888.2.r.e.475.26 yes 72 296.179 even 4 inner