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Properties

Label 888.2.bh.a.565.7
Level $888$
Weight $2$
Character 888.565
Analytic conductor $7.091$
Analytic rank $0$
Dimension $152$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), 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: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 565.7
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.7

$q$-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+(-1.39353 + 0.241013i) q^{2} +(0.866025 - 0.500000i) q^{3} +(1.88383 - 0.671716i) q^{4} +(3.27063 - 1.88830i) q^{5} +(-1.08632 + 0.905486i) q^{6} +(1.78004 + 3.08311i) q^{7} +(-2.46327 + 1.39008i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-4.10260 + 3.41966i) q^{10} -5.07427i q^{11} +(1.29558 - 1.52364i) q^{12} +(3.96777 - 2.29079i) q^{13} +(-3.22360 - 3.86738i) q^{14} +(1.88830 - 3.27063i) q^{15} +(3.09759 - 2.53079i) q^{16} +(-0.609944 + 1.05645i) q^{17} +(-0.488039 + 1.32733i) q^{18} +(-5.74922 + 3.31931i) q^{19} +(4.89289 - 5.75416i) q^{20} +(3.08311 + 1.78004i) q^{21} +(1.22297 + 7.07112i) q^{22} +5.41167 q^{23} +(-1.43821 + 2.43548i) q^{24} +(4.63135 - 8.02173i) q^{25} +(-4.97708 + 4.14856i) q^{26} -1.00000i q^{27} +(5.42426 + 4.61237i) q^{28} -1.51562i q^{29} +(-1.84313 + 5.01281i) q^{30} -8.76260 q^{31} +(-3.70662 + 4.27328i) q^{32} +(-2.53714 - 4.39445i) q^{33} +(0.595353 - 1.61920i) q^{34} +(11.6437 + 6.72248i) q^{35} +(0.360189 - 1.96730i) q^{36} +(-1.37721 + 5.92480i) q^{37} +(7.21168 - 6.01119i) q^{38} +(2.29079 - 3.96777i) q^{39} +(-5.43154 + 9.19782i) q^{40} +(3.57339 + 6.18930i) q^{41} +(-4.72541 - 1.73745i) q^{42} +1.28968i q^{43} +(-3.40847 - 9.55904i) q^{44} -3.77660i q^{45} +(-7.54130 + 1.30429i) q^{46} -7.91073 q^{47} +(1.41720 - 3.74053i) q^{48} +(-2.83706 + 4.91393i) q^{49} +(-4.52055 + 12.2947i) q^{50} +1.21989i q^{51} +(5.93582 - 6.98067i) q^{52} +(-5.52210 - 3.18819i) q^{53} +(0.241013 + 1.39353i) q^{54} +(-9.58174 - 16.5961i) q^{55} +(-8.67048 - 5.12013i) q^{56} +(-3.31931 + 5.74922i) q^{57} +(0.365284 + 2.11205i) q^{58} +(-0.528051 - 0.304871i) q^{59} +(1.36029 - 7.42970i) q^{60} +(-0.636936 + 0.367735i) q^{61} +(12.2109 - 2.11190i) q^{62} +3.56007 q^{63} +(4.13535 - 6.84827i) q^{64} +(8.65140 - 14.9847i) q^{65} +(4.59468 + 5.51229i) q^{66} +(-9.32688 + 5.38488i) q^{67} +(-0.439390 + 2.39988i) q^{68} +(4.68665 - 2.70584i) q^{69} +(-17.8460 - 6.56167i) q^{70} +(-1.33077 - 2.30496i) q^{71} +(-0.0277875 + 2.82829i) q^{72} +8.70866 q^{73} +(0.491220 - 8.58829i) q^{74} -9.26269i q^{75} +(-8.60089 + 10.1149i) q^{76} +(15.6446 - 9.03239i) q^{77} +(-2.23599 + 6.08130i) q^{78} +(4.92286 + 8.52665i) q^{79} +(5.35219 - 14.1265i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-6.47132 - 7.76371i) q^{82} +(9.04080 + 5.21971i) q^{83} +(7.00373 + 1.28230i) q^{84} +4.60703i q^{85} +(-0.310831 - 1.79721i) q^{86} +(-0.757808 - 1.31256i) q^{87} +(7.05365 + 12.4993i) q^{88} +(-3.93082 + 6.80838i) q^{89} +(0.910210 + 5.26278i) q^{90} +(14.1255 + 8.15539i) q^{91} +(10.1946 - 3.63511i) q^{92} +(-7.58864 + 4.38130i) q^{93} +(11.0238 - 1.90659i) q^{94} +(-12.5357 + 21.7125i) q^{95} +(-1.07339 + 5.55408i) q^{96} -1.32010 q^{97} +(2.76919 - 7.53146i) q^{98} +(-4.39445 - 2.53714i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39353 + 0.241013i −0.985371 + 0.170422i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 1.88383 − 0.671716i 0.941913 − 0.335858i
\(5\) 3.27063 − 1.88830i 1.46267 − 0.844473i 0.463536 − 0.886078i \(-0.346580\pi\)
0.999134 + 0.0416051i \(0.0132471\pi\)
\(6\) −1.08632 + 0.905486i −0.443489 + 0.369663i
\(7\) 1.78004 + 3.08311i 0.672791 + 1.16531i 0.977109 + 0.212737i \(0.0682378\pi\)
−0.304319 + 0.952570i \(0.598429\pi\)
\(8\) −2.46327 + 1.39008i −0.870896 + 0.491468i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −4.10260 + 3.41966i −1.29736 + 1.08139i
\(11\) − 5.07427i − 1.52995i −0.644060 − 0.764975i \(-0.722751\pi\)
0.644060 − 0.764975i \(-0.277249\pi\)
\(12\) 1.29558 − 1.52364i 0.374002 − 0.439836i
\(13\) 3.96777 − 2.29079i 1.10046 − 0.635352i 0.164119 − 0.986441i \(-0.447522\pi\)
0.936342 + 0.351089i \(0.114188\pi\)
\(14\) −3.22360 − 3.86738i −0.861543 − 1.03360i
\(15\) 1.88830 − 3.27063i 0.487557 − 0.844473i
\(16\) 3.09759 − 2.53079i 0.774399 − 0.632698i
\(17\) −0.609944 + 1.05645i −0.147933 + 0.256228i −0.930463 − 0.366385i \(-0.880595\pi\)
0.782530 + 0.622613i \(0.213929\pi\)
\(18\) −0.488039 + 1.32733i −0.115032 + 0.312856i
\(19\) −5.74922 + 3.31931i −1.31896 + 0.761503i −0.983562 − 0.180573i \(-0.942205\pi\)
−0.335400 + 0.942076i \(0.608871\pi\)
\(20\) 4.89289 − 5.75416i 1.09408 − 1.28667i
\(21\) 3.08311 + 1.78004i 0.672791 + 0.388436i
\(22\) 1.22297 + 7.07112i 0.260737 + 1.50757i
\(23\) 5.41167 1.12841 0.564206 − 0.825634i \(-0.309182\pi\)
0.564206 + 0.825634i \(0.309182\pi\)
\(24\) −1.43821 + 2.43548i −0.293573 + 0.497140i
\(25\) 4.63135 − 8.02173i 0.926269 − 1.60435i
\(26\) −4.97708 + 4.14856i −0.976085 + 0.813600i
\(27\) − 1.00000i − 0.192450i
\(28\) 5.42426 + 4.61237i 1.02509 + 0.871655i
\(29\) − 1.51562i − 0.281443i −0.990049 − 0.140721i \(-0.955058\pi\)
0.990049 − 0.140721i \(-0.0449422\pi\)
\(30\) −1.84313 + 5.01281i −0.336507 + 0.915210i
\(31\) −8.76260 −1.57381 −0.786905 − 0.617074i \(-0.788318\pi\)
−0.786905 + 0.617074i \(0.788318\pi\)
\(32\) −3.70662 + 4.27328i −0.655244 + 0.755417i
\(33\) −2.53714 − 4.39445i −0.441659 − 0.764975i
\(34\) 0.595353 − 1.61920i 0.102102 − 0.277690i
\(35\) 11.6437 + 6.72248i 1.96814 + 1.13631i
\(36\) 0.360189 − 1.96730i 0.0600315 − 0.327883i
\(37\) −1.37721 + 5.92480i −0.226412 + 0.974032i
\(38\) 7.21168 − 6.01119i 1.16989 − 0.975143i
\(39\) 2.29079 − 3.96777i 0.366820 − 0.635352i
\(40\) −5.43154 + 9.19782i −0.858802 + 1.45430i
\(41\) 3.57339 + 6.18930i 0.558070 + 0.966606i 0.997658 + 0.0684062i \(0.0217914\pi\)
−0.439587 + 0.898200i \(0.644875\pi\)
\(42\) −4.72541 − 1.73745i −0.729146 − 0.268095i
\(43\) 1.28968i 0.196675i 0.995153 + 0.0983375i \(0.0313525\pi\)
−0.995153 + 0.0983375i \(0.968648\pi\)
\(44\) −3.40847 − 9.55904i −0.513846 − 1.44108i
\(45\) − 3.77660i − 0.562982i
\(46\) −7.54130 + 1.30429i −1.11190 + 0.192306i
\(47\) −7.91073 −1.15390 −0.576949 − 0.816780i \(-0.695757\pi\)
−0.576949 + 0.816780i \(0.695757\pi\)
\(48\) 1.41720 − 3.74053i 0.204555 − 0.539899i
\(49\) −2.83706 + 4.91393i −0.405294 + 0.701990i
\(50\) −4.52055 + 12.2947i −0.639303 + 1.73873i
\(51\) 1.21989i 0.170818i
\(52\) 5.93582 − 6.98067i 0.823150 − 0.968045i
\(53\) −5.52210 − 3.18819i −0.758519 − 0.437931i 0.0702447 − 0.997530i \(-0.477622\pi\)
−0.828764 + 0.559599i \(0.810955\pi\)
\(54\) 0.241013 + 1.39353i 0.0327978 + 0.189635i
\(55\) −9.58174 − 16.5961i −1.29200 − 2.23781i
\(56\) −8.67048 − 5.12013i −1.15864 − 0.684206i
\(57\) −3.31931 + 5.74922i −0.439654 + 0.761503i
\(58\) 0.365284 + 2.11205i 0.0479641 + 0.277326i
\(59\) −0.528051 − 0.304871i −0.0687464 − 0.0396908i 0.465233 − 0.885188i \(-0.345971\pi\)
−0.533979 + 0.845498i \(0.679304\pi\)
\(60\) 1.36029 − 7.42970i 0.175613 − 0.959170i
\(61\) −0.636936 + 0.367735i −0.0815513 + 0.0470837i −0.540221 − 0.841523i \(-0.681659\pi\)
0.458670 + 0.888607i \(0.348326\pi\)
\(62\) 12.2109 − 2.11190i 1.55079 − 0.268212i
\(63\) 3.56007 0.448527
\(64\) 4.13535 − 6.84827i 0.516919 − 0.856034i
\(65\) 8.65140 − 14.9847i 1.07307 − 1.85862i
\(66\) 4.59468 + 5.51229i 0.565566 + 0.678516i
\(67\) −9.32688 + 5.38488i −1.13946 + 0.657867i −0.946297 − 0.323299i \(-0.895208\pi\)
−0.193163 + 0.981167i \(0.561875\pi\)
\(68\) −0.439390 + 2.39988i −0.0532839 + 0.291029i
\(69\) 4.68665 − 2.70584i 0.564206 − 0.325744i
\(70\) −17.8460 − 6.56167i −2.13300 − 0.784269i
\(71\) −1.33077 − 2.30496i −0.157933 − 0.273548i 0.776190 − 0.630499i \(-0.217150\pi\)
−0.934123 + 0.356951i \(0.883816\pi\)
\(72\) −0.0277875 + 2.82829i −0.00327479 + 0.333317i
\(73\) 8.70866 1.01927 0.509636 − 0.860390i \(-0.329780\pi\)
0.509636 + 0.860390i \(0.329780\pi\)
\(74\) 0.491220 − 8.58829i 0.0571031 − 0.998368i
\(75\) − 9.26269i − 1.06956i
\(76\) −8.60089 + 10.1149i −0.986589 + 1.16025i
\(77\) 15.6446 − 9.03239i 1.78286 − 1.02934i
\(78\) −2.23599 + 6.08130i −0.253176 + 0.688572i
\(79\) 4.92286 + 8.52665i 0.553866 + 0.959323i 0.997991 + 0.0633585i \(0.0201812\pi\)
−0.444125 + 0.895965i \(0.646486\pi\)
\(80\) 5.35219 − 14.1265i 0.598393 − 1.57939i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −6.47132 − 7.76371i −0.714637 − 0.857358i
\(83\) 9.04080 + 5.21971i 0.992356 + 0.572937i 0.905978 − 0.423325i \(-0.139137\pi\)
0.0863785 + 0.996262i \(0.472471\pi\)
\(84\) 7.00373 + 1.28230i 0.764169 + 0.139910i
\(85\) 4.60703i 0.499702i
\(86\) −0.310831 − 1.79721i −0.0335178 − 0.193798i
\(87\) −0.757808 − 1.31256i −0.0812455 − 0.140721i
\(88\) 7.05365 + 12.4993i 0.751921 + 1.33243i
\(89\) −3.93082 + 6.80838i −0.416666 + 0.721687i −0.995602 − 0.0936866i \(-0.970135\pi\)
0.578936 + 0.815373i \(0.303468\pi\)
\(90\) 0.910210 + 5.26278i 0.0959446 + 0.554746i
\(91\) 14.1255 + 8.15539i 1.48076 + 0.854917i
\(92\) 10.1946 − 3.63511i 1.06287 − 0.378986i
\(93\) −7.58864 + 4.38130i −0.786905 + 0.454320i
\(94\) 11.0238 − 1.90659i 1.13702 − 0.196650i
\(95\) −12.5357 + 21.7125i −1.28614 + 2.22765i
\(96\) −1.07339 + 5.55408i −0.109552 + 0.566861i
\(97\) −1.32010 −0.134036 −0.0670178 − 0.997752i \(-0.521348\pi\)
−0.0670178 + 0.997752i \(0.521348\pi\)
\(98\) 2.76919 − 7.53146i 0.279731 − 0.760792i
\(99\) −4.39445 − 2.53714i −0.441659 − 0.254992i
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.bh.a.565.7 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.48 yes 152
37.26 even 3 inner 888.2.bh.a.877.48 yes 152
296.285 even 6 inner 888.2.bh.a.877.7 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.7 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.48 yes 152 8.5 even 2 inner
888.2.bh.a.877.7 yes 152 296.285 even 6 inner
888.2.bh.a.877.48 yes 152 37.26 even 3 inner