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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 475.9
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.9

$q$-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+(-1.06809 + 0.926925i) q^{2} +1.00000i q^{3} +(0.281621 - 1.98007i) q^{4} +(-2.48900 - 2.48900i) q^{5} +(-0.926925 - 1.06809i) q^{6} +1.23837i q^{7} +(1.53458 + 2.37593i) 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} +(-1.14788 - 1.32269i) q^{14} +(2.48900 - 2.48900i) q^{15} +(-3.84138 - 1.11526i) q^{16} +(-2.22933 + 2.22933i) q^{17} +(1.06809 - 0.926925i) q^{18} +(5.67997 - 5.67997i) q^{19} +(-5.62935 + 4.22744i) q^{20} -1.23837 q^{21} +(-5.18903 - 5.97927i) q^{22} +(2.40735 + 2.40735i) q^{23} +(-2.37593 + 1.53458i) 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} +(5.13669 - 2.36947i) 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} +(1.32269 - 1.14788i) 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} +(-6.85017 - 7.89339i) q^{50} +(-2.22933 - 2.22933i) q^{51} +(-7.28793 + 5.47298i) q^{52} +0.173792i q^{53} +(0.926925 + 1.06809i) q^{54} +(13.9337 - 13.9337i) q^{55} +(-2.94229 + 1.90038i) q^{56} +(5.67997 + 5.67997i) q^{57} +(-0.629227 + 8.89270i) q^{58} +(3.79028 - 3.79028i) q^{59} +(-4.22744 - 5.62935i) 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} +(5.97927 - 5.18903i) q^{66} +2.06162i q^{67} +(3.78642 + 5.04207i) q^{68} +(-2.40735 + 2.40735i) q^{69} +(-0.435106 + 6.14923i) q^{70} +12.8475i q^{71} +(-1.53458 - 2.37593i) q^{72} -10.0534i q^{73} +(5.07653 + 6.94470i) q^{74} -7.39021 q^{75} +(-9.64715 - 12.8463i) q^{76} -6.93254 q^{77} +(-0.454873 + 6.42859i) q^{78} +(-0.543893 - 0.543893i) q^{79} +(6.78530 + 12.3371i) q^{80} +1.00000 q^{81} +(9.34929 + 10.7731i) 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} +(-13.3007 + 8.59077i) q^{88} +(9.27258 + 9.27258i) q^{89} +(-4.96558 - 0.351353i) q^{90} +(3.99045 - 3.99045i) q^{91} +(5.44469 - 4.08877i) q^{92} +(-2.51566 - 2.51566i) q^{93} +(6.81964 + 7.85821i) q^{94} -28.2748 q^{95} +(2.36947 + 5.13669i) q^{96} +(5.10181 - 5.10181i) q^{97} +(-5.83863 + 5.06697i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06809 + 0.926925i −0.755252 + 0.655435i
\(3\) 1.00000i 0.577350i
\(4\) 0.281621 − 1.98007i 0.140811 − 0.990037i
\(5\) −2.48900 − 2.48900i −1.11311 − 1.11311i −0.992727 − 0.120386i \(-0.961587\pi\)
−0.120386 − 0.992727i \(-0.538413\pi\)
\(6\) −0.926925 − 1.06809i −0.378415 − 0.436045i
\(7\) 1.23837i 0.468060i 0.972229 + 0.234030i \(0.0751915\pi\)
−0.972229 + 0.234030i \(0.924809\pi\)
\(8\) 1.53458 + 2.37593i 0.542557 + 0.840019i
\(9\) −1.00000 −0.333333
\(10\) 4.96558 + 0.351353i 1.57025 + 0.111108i
\(11\) 5.59811i 1.68789i 0.536426 + 0.843947i \(0.319774\pi\)
−0.536426 + 0.843947i \(0.680226\pi\)
\(12\) 1.98007 + 0.281621i 0.571598 + 0.0812970i
\(13\) −3.22233 − 3.22233i −0.893714 − 0.893714i 0.101156 − 0.994871i \(-0.467746\pi\)
−0.994871 + 0.101156i \(0.967746\pi\)
\(14\) −1.14788 − 1.32269i −0.306783 − 0.353503i
\(15\) 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 −0.923732 − 0.383039i \(-0.874877\pi\)
0.383039 + 0.923732i \(0.374877\pi\)
\(18\) 1.06809 − 0.926925i 0.251751 − 0.218478i
\(19\) 5.67997 − 5.67997i 1.30307 − 1.30307i 0.376765 − 0.926309i \(-0.377037\pi\)
0.926309 − 0.376765i \(-0.122963\pi\)
\(20\) −5.62935 + 4.22744i −1.25876 + 0.945285i
\(21\) −1.23837 −0.270235
\(22\) −5.18903 − 5.97927i −1.10630 − 1.27479i
\(23\) 2.40735 + 2.40735i 0.501968 + 0.501968i 0.912049 − 0.410081i \(-0.134500\pi\)
−0.410081 + 0.912049i \(0.634500\pi\)
\(24\) −2.37593 + 1.53458i −0.484985 + 0.313245i
\(25\) 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 \(-0.550979\pi\)
0.987202 + 0.159472i \(0.0509791\pi\)
\(30\) −0.351353 + 4.96558i −0.0641480 + 0.906586i
\(31\) −2.51566 + 2.51566i −0.451826 + 0.451826i −0.895960 − 0.444134i \(-0.853511\pi\)
0.444134 + 0.895960i \(0.353511\pi\)
\(32\) 5.13669 − 2.36947i 0.908047 − 0.418868i
\(33\) −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 \(-0.711317\pi\)
0.616172 − 0.787612i \(-0.288683\pi\)
\(42\) 1.32269 − 1.14788i 0.204095 − 0.177121i
\(43\) 7.24350 − 7.24350i 1.10462 − 1.10462i 0.110779 − 0.993845i \(-0.464666\pi\)
0.993845 − 0.110779i \(-0.0353345\pi\)
\(44\) 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
\(50\) −6.85017 − 7.89339i −0.968760 − 1.11629i
\(51\) −2.22933 − 2.22933i −0.312169 − 0.312169i
\(52\) −7.28793 + 5.47298i −1.01065 + 0.758965i
\(53\) 0.173792i 0.0238721i 0.999929 + 0.0119361i \(0.00379946\pi\)
−0.999929 + 0.0119361i \(0.996201\pi\)
\(54\) 0.926925 + 1.06809i 0.126138 + 0.145348i
\(55\) 13.9337 − 13.9337i 1.87882 − 1.87882i
\(56\) −2.94229 + 1.90038i −0.393180 + 0.253949i
\(57\) 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 −0.415940 − 0.909392i \(-0.636547\pi\)
0.909392 + 0.415940i \(0.136547\pi\)
\(60\) −4.22744 − 5.62935i −0.545760 − 0.726746i
\(61\) 1.24228 − 1.24228i 0.159058 − 0.159058i −0.623091 − 0.782149i \(-0.714123\pi\)
0.782149 + 0.623091i \(0.214123\pi\)
\(62\) 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
\(66\) 5.97927 − 5.18903i 0.735998 − 0.638725i
\(67\) 2.06162i 0.251867i 0.992039 + 0.125934i \(0.0401926\pi\)
−0.992039 + 0.125934i \(0.959807\pi\)
\(68\) 3.78642 + 5.04207i 0.459171 + 0.611441i
\(69\) −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 \(0.724042\pi\)
\(72\) −1.53458 − 2.37593i −0.180852 − 0.280006i
\(73\) − 10.0534i − 1.17666i −0.808621 − 0.588330i \(-0.799786\pi\)
0.808621 − 0.588330i \(-0.200214\pi\)
\(74\) 5.07653 + 6.94470i 0.590135 + 0.807305i
\(75\) −7.39021 −0.853348
\(76\) −9.64715 − 12.8463i −1.10660 − 1.47358i
\(77\) −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 \(-0.263778\pi\)
−0.737041 + 0.675848i \(0.763778\pi\)
\(80\) 6.78530 + 12.3371i 0.758620 + 1.37933i
\(81\) 1.00000 0.111111
\(82\) 9.34929 + 10.7731i 1.03246 + 1.18969i
\(83\) −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
\(88\) −13.3007 + 8.59077i −1.41786 + 0.915779i
\(89\) 9.27258 + 9.27258i 0.982891 + 0.982891i 0.999856 − 0.0169650i \(-0.00540039\pi\)
−0.0169650 + 0.999856i \(0.505400\pi\)
\(90\) −4.96558 − 0.351353i −0.523418 − 0.0370359i
\(91\) 3.99045 − 3.99045i 0.418312 − 0.418312i
\(92\) 5.44469 − 4.08877i 0.567649 − 0.426284i
\(93\) −2.51566 − 2.51566i −0.260862 − 0.260862i
\(94\) 6.81964 + 7.85821i 0.703392 + 0.810513i
\(95\) −28.2748 −2.90094
\(96\) 2.36947 + 5.13669i 0.241833 + 0.524261i
\(97\) 5.10181 − 5.10181i 0.518011 − 0.518011i −0.398958 − 0.916969i \(-0.630628\pi\)
0.916969 + 0.398958i \(0.130628\pi\)
\(98\) −5.83863 + 5.06697i −0.589791 + 0.511842i
\(99\) − 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.475.9 yes 72
8.3 odd 2 inner 888.2.r.e.475.26 yes 72
37.6 odd 4 inner 888.2.r.e.43.26 yes 72
296.43 even 4 inner 888.2.r.e.43.9 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.9 ✓ 72 296.43 even 4 inner
888.2.r.e.43.26 yes 72 37.6 odd 4 inner
888.2.r.e.475.9 yes 72 1.1 even 1 trivial
888.2.r.e.475.26 yes 72 8.3 odd 2 inner