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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 443.3
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.2

$q$-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+(-1.41368 + 0.0387455i) q^{2} +(-1.13665 - 1.30691i) q^{3} +(1.99700 - 0.109548i) q^{4} -3.37843i q^{5} +(1.65751 + 1.80352i) q^{6} +1.96308i q^{7} +(-2.81888 + 0.232240i) q^{8} +(-0.416034 + 2.97101i) q^{9} +(0.130899 + 4.77603i) q^{10} -1.92138i q^{11} +(-2.41306 - 2.48538i) q^{12} +3.16618 q^{13} +(-0.0760606 - 2.77518i) q^{14} +(-4.41531 + 3.84011i) q^{15} +(3.97600 - 0.437533i) q^{16} +4.18199 q^{17} +(0.473027 - 4.21619i) q^{18} -7.44290i q^{19} +(-0.370099 - 6.74672i) q^{20} +(2.56558 - 2.23135i) q^{21} +(0.0744447 + 2.71622i) q^{22} -7.62446i q^{23} +(3.50761 + 3.42004i) q^{24} -6.41380 q^{25} +(-4.47598 + 0.122675i) q^{26} +(4.35574 - 2.83329i) q^{27} +(0.215051 + 3.92027i) q^{28} +1.08330i q^{29} +(6.09306 - 5.59977i) q^{30} -7.41964 q^{31} +(-5.60385 + 0.772585i) q^{32} +(-2.51107 + 2.18394i) q^{33} +(-5.91201 + 0.162033i) q^{34} +6.63214 q^{35} +(-0.505352 + 5.97868i) q^{36} +(2.35822 - 5.60703i) q^{37} +(0.288379 + 10.5219i) q^{38} +(-3.59886 - 4.13792i) q^{39} +(0.784608 + 9.52338i) q^{40} +3.85687i q^{41} +(-3.54045 + 3.25382i) q^{42} +9.68348i q^{43} +(-0.210482 - 3.83698i) q^{44} +(10.0374 + 1.40554i) q^{45} +(0.295413 + 10.7786i) q^{46} +7.60351 q^{47} +(-5.09115 - 4.69895i) q^{48} +3.14631 q^{49} +(9.06708 - 0.248506i) q^{50} +(-4.75348 - 5.46550i) q^{51} +(6.32286 - 0.346848i) q^{52} -11.2460 q^{53} +(-6.04785 + 4.17414i) q^{54} -6.49124 q^{55} +(-0.455907 - 5.53369i) q^{56} +(-9.72721 + 8.46000i) q^{57} +(-0.0419728 - 1.53144i) q^{58} -7.16102 q^{59} +(-8.39669 + 8.15238i) q^{60} -10.2370 q^{61} +(10.4890 - 0.287478i) q^{62} +(-5.83234 - 0.816710i) q^{63} +(7.89213 - 1.30931i) q^{64} -10.6967i q^{65} +(3.46524 - 3.18469i) q^{66} +0.861241 q^{67} +(8.35143 - 0.458128i) q^{68} +(-9.96449 + 8.66637i) q^{69} +(-9.37575 + 0.256966i) q^{70} -12.9620 q^{71} +(0.482760 - 8.47154i) q^{72} +12.4583 q^{73} +(-3.11653 + 8.01793i) q^{74} +(7.29028 + 8.38227i) q^{75} +(-0.815352 - 14.8635i) q^{76} +3.77182 q^{77} +(5.24797 + 5.71027i) q^{78} -12.0935 q^{79} +(-1.47818 - 13.4326i) q^{80} +(-8.65383 - 2.47209i) q^{81} +(-0.149436 - 5.45239i) q^{82} +6.03055i q^{83} +(4.87901 - 4.73705i) q^{84} -14.1286i q^{85} +(-0.375191 - 13.6894i) q^{86} +(1.41577 - 1.23133i) q^{87} +(0.446221 + 5.41612i) q^{88} +6.58152 q^{89} +(-14.2441 - 1.59809i) q^{90} +6.21548i q^{91} +(-0.835242 - 15.2260i) q^{92} +(8.43357 + 9.69681i) q^{93} +(-10.7490 + 0.294602i) q^{94} -25.1453 q^{95} +(7.37934 + 6.44557i) q^{96} -1.92844i q^{97} +(-4.44788 + 0.121905i) q^{98} +(5.70843 + 0.799358i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 4 q^{3} + 24 q^{4} - 4 q^{9} - 20 q^{10} - 18 q^{12} - 56 q^{16} + 72 q^{25} - 4 q^{27} - 64 q^{28} - 16 q^{33} - 8 q^{34} + 42 q^{36} + 44 q^{40} - 68 q^{46} - 94 q^{48} - 104 q^{49} - 28 q^{58}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41368 + 0.0387455i −0.999625 + 0.0273972i
\(3\) −1.13665 − 1.30691i −0.656248 − 0.754546i
\(4\) 1.99700 − 0.109548i 0.998499 − 0.0547738i
\(5\) − 3.37843i − 1.51088i −0.655217 − 0.755440i \(-0.727423\pi\)
0.655217 − 0.755440i \(-0.272577\pi\)
\(6\) 1.65751 + 1.80352i 0.676674 + 0.736283i
\(7\) 1.96308i 0.741976i 0.928638 + 0.370988i \(0.120981\pi\)
−0.928638 + 0.370988i \(0.879019\pi\)
\(8\) −2.81888 + 0.232240i −0.996623 + 0.0821094i
\(9\) −0.416034 + 2.97101i −0.138678 + 0.990338i
\(10\) 0.130899 + 4.77603i 0.0413939 + 1.51031i
\(11\) − 1.92138i − 0.579317i −0.957130 − 0.289658i \(-0.906458\pi\)
0.957130 − 0.289658i \(-0.0935417\pi\)
\(12\) −2.41306 − 2.48538i −0.696592 − 0.717468i
\(13\) 3.16618 0.878141 0.439071 − 0.898453i \(-0.355308\pi\)
0.439071 + 0.898453i \(0.355308\pi\)
\(14\) −0.0760606 − 2.77518i −0.0203281 − 0.741697i
\(15\) −4.41531 + 3.84011i −1.14003 + 0.991512i
\(16\) 3.97600 − 0.437533i 0.994000 − 0.109383i
\(17\) 4.18199 1.01428 0.507141 − 0.861863i \(-0.330702\pi\)
0.507141 + 0.861863i \(0.330702\pi\)
\(18\) 0.473027 − 4.21619i 0.111494 − 0.993765i
\(19\) − 7.44290i − 1.70752i −0.520668 − 0.853759i \(-0.674317\pi\)
0.520668 − 0.853759i \(-0.325683\pi\)
\(20\) −0.370099 − 6.74672i −0.0827567 − 1.50861i
\(21\) 2.56558 − 2.23135i 0.559854 − 0.486920i
\(22\) 0.0744447 + 2.71622i 0.0158717 + 0.579099i
\(23\) − 7.62446i − 1.58981i −0.606734 − 0.794905i \(-0.707521\pi\)
0.606734 − 0.794905i \(-0.292479\pi\)
\(24\) 3.50761 + 3.42004i 0.715987 + 0.698114i
\(25\) −6.41380 −1.28276
\(26\) −4.47598 + 0.122675i −0.877812 + 0.0240586i
\(27\) 4.35574 − 2.83329i 0.838262 − 0.545268i
\(28\) 0.215051 + 3.92027i 0.0406409 + 0.740862i
\(29\) 1.08330i 0.201163i 0.994929 + 0.100582i \(0.0320703\pi\)
−0.994929 + 0.100582i \(0.967930\pi\)
\(30\) 6.09306 − 5.59977i 1.11244 − 1.02237i
\(31\) −7.41964 −1.33261 −0.666303 − 0.745681i \(-0.732124\pi\)
−0.666303 + 0.745681i \(0.732124\pi\)
\(32\) −5.60385 + 0.772585i −0.990630 + 0.136575i
\(33\) −2.51107 + 2.18394i −0.437121 + 0.380175i
\(34\) −5.91201 + 0.162033i −1.01390 + 0.0277885i
\(35\) 6.63214 1.12104
\(36\) −0.505352 + 5.97868i −0.0842253 + 0.996447i
\(37\) 2.35822 − 5.60703i 0.387689 − 0.921790i
\(38\) 0.288379 + 10.5219i 0.0467812 + 1.70688i
\(39\) −3.59886 − 4.13792i −0.576278 − 0.662598i
\(40\) 0.784608 + 9.52338i 0.124057 + 1.50578i
\(41\) 3.85687i 0.602341i 0.953570 + 0.301171i \(0.0973774\pi\)
−0.953570 + 0.301171i \(0.902623\pi\)
\(42\) −3.54045 + 3.25382i −0.546304 + 0.502075i
\(43\) 9.68348i 1.47672i 0.674409 + 0.738358i \(0.264399\pi\)
−0.674409 + 0.738358i \(0.735601\pi\)
\(44\) −0.210482 − 3.83698i −0.0317314 − 0.578447i
\(45\) 10.0374 + 1.40554i 1.49628 + 0.209526i
\(46\) 0.295413 + 10.7786i 0.0435563 + 1.58921i
\(47\) 7.60351 1.10909 0.554543 − 0.832155i \(-0.312893\pi\)
0.554543 + 0.832155i \(0.312893\pi\)
\(48\) −5.09115 − 4.69895i −0.734845 − 0.678236i
\(49\) 3.14631 0.449472
\(50\) 9.06708 − 0.248506i 1.28228 − 0.0351441i
\(51\) −4.75348 − 5.46550i −0.665621 − 0.765322i
\(52\) 6.32286 − 0.346848i 0.876823 − 0.0480992i
\(53\) −11.2460 −1.54476 −0.772380 − 0.635160i \(-0.780934\pi\)
−0.772380 + 0.635160i \(0.780934\pi\)
\(54\) −6.04785 + 4.17414i −0.823008 + 0.568029i
\(55\) −6.49124 −0.875279
\(56\) −0.455907 − 5.53369i −0.0609231 − 0.739470i
\(57\) −9.72721 + 8.46000i −1.28840 + 1.12055i
\(58\) −0.0419728 − 1.53144i −0.00551130 − 0.201088i
\(59\) −7.16102 −0.932286 −0.466143 − 0.884710i \(-0.654357\pi\)
−0.466143 + 0.884710i \(0.654357\pi\)
\(60\) −8.39669 + 8.15238i −1.08401 + 1.05247i
\(61\) −10.2370 −1.31071 −0.655354 − 0.755322i \(-0.727480\pi\)
−0.655354 + 0.755322i \(0.727480\pi\)
\(62\) 10.4890 − 0.287478i 1.33211 − 0.0365097i
\(63\) −5.83234 − 0.816710i −0.734806 − 0.102896i
\(64\) 7.89213 − 1.30931i 0.986516 − 0.163664i
\(65\) − 10.6967i − 1.32677i
\(66\) 3.46524 − 3.18469i 0.426541 − 0.392008i
\(67\) 0.861241 0.105217 0.0526087 − 0.998615i \(-0.483246\pi\)
0.0526087 + 0.998615i \(0.483246\pi\)
\(68\) 8.35143 − 0.458128i 1.01276 − 0.0555562i
\(69\) −9.96449 + 8.66637i −1.19958 + 1.04331i
\(70\) −9.37575 + 0.256966i −1.12062 + 0.0307133i
\(71\) −12.9620 −1.53831 −0.769155 − 0.639062i \(-0.779323\pi\)
−0.769155 + 0.639062i \(0.779323\pi\)
\(72\) 0.482760 − 8.47154i 0.0568938 − 0.998380i
\(73\) 12.4583 1.45814 0.729070 − 0.684439i \(-0.239953\pi\)
0.729070 + 0.684439i \(0.239953\pi\)
\(74\) −3.11653 + 8.01793i −0.362289 + 0.932066i
\(75\) 7.29028 + 8.38227i 0.841809 + 0.967901i
\(76\) −0.815352 − 14.8635i −0.0935273 − 1.70495i
\(77\) 3.77182 0.429839
\(78\) 5.24797 + 5.71027i 0.594215 + 0.646560i
\(79\) −12.0935 −1.36062 −0.680312 − 0.732922i \(-0.738156\pi\)
−0.680312 + 0.732922i \(0.738156\pi\)
\(80\) −1.47818 − 13.4326i −0.165265 − 1.50181i
\(81\) −8.65383 − 2.47209i −0.961537 − 0.274676i
\(82\) −0.149436 − 5.45239i −0.0165025 − 0.602115i
\(83\) 6.03055i 0.661939i 0.943641 + 0.330970i \(0.107376\pi\)
−0.943641 + 0.330970i \(0.892624\pi\)
\(84\) 4.87901 − 4.73705i 0.532344 − 0.516854i
\(85\) − 14.1286i − 1.53246i
\(86\) −0.375191 − 13.6894i −0.0404579 − 1.47616i
\(87\) 1.41577 − 1.23133i 0.151787 − 0.132013i
\(88\) 0.446221 + 5.41612i 0.0475673 + 0.577361i
\(89\) 6.58152 0.697640 0.348820 − 0.937190i \(-0.386583\pi\)
0.348820 + 0.937190i \(0.386583\pi\)
\(90\) −14.2441 − 1.59809i −1.50146 − 0.168453i
\(91\) 6.21548i 0.651559i
\(92\) −0.835242 − 15.2260i −0.0870800 − 1.58742i
\(93\) 8.43357 + 9.69681i 0.874520 + 1.00551i
\(94\) −10.7490 + 0.294602i −1.10867 + 0.0303859i
\(95\) −25.1453 −2.57986
\(96\) 7.37934 + 6.44557i 0.753150 + 0.657848i
\(97\) − 1.92844i − 0.195804i −0.995196 − 0.0979019i \(-0.968787\pi\)
0.995196 − 0.0979019i \(-0.0312131\pi\)
\(98\) −4.44788 + 0.121905i −0.449303 + 0.0123143i
\(99\) 5.70843 + 0.799358i 0.573719 + 0.0803385i
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.c.d.443.3 yes 96
3.2 odd 2 inner 888.2.c.d.443.94 yes 96
8.3 odd 2 inner 888.2.c.d.443.1 ✓ 96
24.11 even 2 inner 888.2.c.d.443.96 yes 96
37.36 even 2 inner 888.2.c.d.443.93 yes 96
111.110 odd 2 inner 888.2.c.d.443.4 yes 96
296.147 odd 2 inner 888.2.c.d.443.95 yes 96
888.443 even 2 inner 888.2.c.d.443.2 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.1 ✓ 96 8.3 odd 2 inner
888.2.c.d.443.2 yes 96 888.443 even 2 inner
888.2.c.d.443.3 yes 96 1.1 even 1 trivial
888.2.c.d.443.4 yes 96 111.110 odd 2 inner
888.2.c.d.443.93 yes 96 37.36 even 2 inner
888.2.c.d.443.94 yes 96 3.2 odd 2 inner
888.2.c.d.443.95 yes 96 296.147 odd 2 inner
888.2.c.d.443.96 yes 96 24.11 even 2 inner