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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 443.4
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.1

$q$-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} +(1.55623 - 1.89160i) 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} +(-2.12673 + 2.73442i) 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} +(0.703254 + 4.18395i) 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} +(2.90057 - 3.94800i) 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} +(-6.39064 - 5.25762i) 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} +(-1.15629 - 5.88753i) 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} +(3.71336 + 3.05501i) 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} +(-3.94752 + 5.69360i) 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} +(-6.26741 - 3.83661i) 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} +(9.23804 + 7.18500i) 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} +(3.63447 + 2.99011i) 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} +(1.86274 + 8.27830i) q^{72} +12.4583 q^{73} +(3.55102 + 7.83519i) q^{74} +(7.29028 - 8.38227i) q^{75} +(0.815352 + 14.8635i) q^{76} -3.77182 q^{77} +(-4.92731 + 5.98915i) 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} +(-5.36789 - 4.17494i) 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} +(14.1352 - 2.37590i) 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} +(5.35994 - 8.20189i) 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 \(-0.272577\pi\)
\(6\) 1.55623 − 1.89160i 0.635329 − 0.772242i
\(7\) 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 \(0.0935417\pi\)
−0.957130 + 0.289658i \(0.906458\pi\)
\(12\) −2.12673 + 2.73442i −0.613933 + 0.789358i
\(13\) −3.16618 −0.878141 −0.439071 − 0.898453i \(-0.644692\pi\)
−0.439071 + 0.898453i \(0.644692\pi\)
\(14\) −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 \(0.330702\pi\)
\(18\) 0.703254 + 4.18395i 0.165759 + 0.986166i
\(19\) 7.44290i 1.70752i 0.520668 + 0.853759i \(0.325683\pi\)
−0.520668 + 0.853759i \(0.674317\pi\)
\(20\) −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 \(-0.292479\pi\)
\(24\) 2.90057 − 3.94800i 0.592076 − 0.805882i
\(25\) −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 \(0.967930\pi\)
\(30\) −6.39064 − 5.25762i −1.16677 − 0.959906i
\(31\) 7.41964 1.33261 0.666303 − 0.745681i \(-0.267876\pi\)
0.666303 + 0.745681i \(0.267876\pi\)
\(32\) −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
\(36\) −1.15629 − 5.88753i −0.192714 − 0.981255i
\(37\) −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 \(-0.902623\pi\)
0.953570 − 0.301171i \(-0.0973774\pi\)
\(42\) 3.71336 + 3.05501i 0.572985 + 0.471399i
\(43\) − 9.68348i − 1.47672i −0.674409 − 0.738358i \(-0.735601\pi\)
0.674409 − 0.738358i \(-0.264399\pi\)
\(44\) 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 \(-0.687107\pi\)
−0.554543 + 0.832155i \(0.687107\pi\)
\(48\) −3.94752 + 5.69360i −0.569775 + 0.821801i
\(49\) 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 \(-0.219066\pi\)
0.772380 + 0.635160i \(0.219066\pi\)
\(54\) −6.26741 − 3.83661i −0.852886 − 0.522097i
\(55\) 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 \(0.654357\pi\)
\(60\) 9.23804 + 7.18500i 1.19263 + 0.927580i
\(61\) 10.2370 1.31071 0.655354 − 0.755322i \(-0.272520\pi\)
0.655354 + 0.755322i \(0.272520\pi\)
\(62\) −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
\(66\) 3.63447 + 2.99011i 0.447373 + 0.368057i
\(67\) 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 \(-0.220677\pi\)
0.769155 + 0.639062i \(0.220677\pi\)
\(72\) 1.86274 + 8.27830i 0.219526 + 0.975607i
\(73\) 12.4583 1.45814 0.729070 − 0.684439i \(-0.239953\pi\)
0.729070 + 0.684439i \(0.239953\pi\)
\(74\) 3.55102 + 7.83519i 0.412798 + 0.910823i
\(75\) 7.29028 − 8.38227i 0.841809 − 0.967901i
\(76\) 0.815352 + 14.8635i 0.0935273 + 1.70495i
\(77\) −3.77182 −0.429839
\(78\) −4.92731 + 5.98915i −0.557908 + 0.678137i
\(79\) 12.0935 1.36062 0.680312 − 0.732922i \(-0.261844\pi\)
0.680312 + 0.732922i \(0.261844\pi\)
\(80\) −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 \(-0.892624\pi\)
0.943641 − 0.330970i \(-0.107376\pi\)
\(84\) −5.36789 − 4.17494i −0.585684 − 0.455523i
\(85\) − 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 \(0.386583\pi\)
\(90\) 14.1352 − 2.37590i 1.48998 − 0.250441i
\(91\) − 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
\(96\) 5.35994 − 8.20189i 0.547046 − 0.837102i
\(97\) 1.92844i 0.195804i 0.995196 + 0.0979019i \(0.0312131\pi\)
−0.995196 + 0.0979019i \(0.968787\pi\)
\(98\) −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.443.4 yes 96
3.2 odd 2 inner 888.2.c.d.443.93 yes 96
8.3 odd 2 inner 888.2.c.d.443.2 yes 96
24.11 even 2 inner 888.2.c.d.443.95 yes 96
37.36 even 2 inner 888.2.c.d.443.94 yes 96
111.110 odd 2 inner 888.2.c.d.443.3 yes 96
296.147 odd 2 inner 888.2.c.d.443.96 yes 96
888.443 even 2 inner 888.2.c.d.443.1 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.1 ✓ 96 888.443 even 2 inner
888.2.c.d.443.2 yes 96 8.3 odd 2 inner
888.2.c.d.443.3 yes 96 111.110 odd 2 inner
888.2.c.d.443.4 yes 96 1.1 even 1 trivial
888.2.c.d.443.93 yes 96 3.2 odd 2 inner
888.2.c.d.443.94 yes 96 37.36 even 2 inner
888.2.c.d.443.95 yes 96 24.11 even 2 inner
888.2.c.d.443.96 yes 96 296.147 odd 2 inner