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Properties

Label 888.2.bh.a.565.10
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 565.10
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.10

$q$-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+(-1.32259 + 0.500752i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(1.49849 - 1.32458i) q^{4} +(0.648798 - 0.374584i) q^{5} +(0.895021 - 1.09496i) q^{6} +(-1.94971 - 3.37700i) q^{7} +(-1.31861 + 2.50225i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.670520 + 0.820308i) q^{10} -5.14163i q^{11} +(-0.635443 + 1.89637i) q^{12} +(-4.03597 + 2.33017i) q^{13} +(4.26971 + 3.49007i) q^{14} +(-0.374584 + 0.648798i) q^{15} +(0.490968 - 3.96975i) q^{16} +(-0.316631 + 0.548421i) q^{17} +(-0.227631 + 1.39577i) q^{18} +(-6.93762 + 4.00544i) q^{19} +(0.476053 - 1.42070i) q^{20} +(3.37700 + 1.94971i) q^{21} +(2.57468 + 6.80027i) q^{22} +4.05437 q^{23} +(-0.109180 - 2.82632i) q^{24} +(-2.21937 + 3.84407i) q^{25} +(4.17110 - 5.10288i) q^{26} +1.00000i q^{27} +(-7.39474 - 2.47786i) q^{28} +4.92152i q^{29} +(0.170534 - 1.04567i) q^{30} +8.73743 q^{31} +(1.33851 + 5.49622i) q^{32} +(2.57081 + 4.45278i) q^{33} +(0.144150 - 0.883890i) q^{34} +(-2.52994 - 1.46066i) q^{35} +(-0.397874 - 1.96002i) q^{36} +(-4.25602 + 4.34584i) q^{37} +(7.16990 - 8.77159i) q^{38} +(2.33017 - 4.03597i) q^{39} +(0.0817938 + 2.11739i) q^{40} +(0.764557 + 1.32425i) q^{41} +(-5.44271 - 0.887630i) q^{42} +8.90966i q^{43} +(-6.81051 - 7.70470i) q^{44} -0.749167i q^{45} +(-5.36227 + 2.03023i) q^{46} -1.04591 q^{47} +(1.55969 + 3.68339i) q^{48} +(-4.10275 + 7.10617i) q^{49} +(1.01040 - 6.19549i) q^{50} -0.633262i q^{51} +(-2.96138 + 8.83772i) q^{52} +(0.950878 + 0.548990i) q^{53} +(-0.500752 - 1.32259i) q^{54} +(-1.92597 - 3.33588i) q^{55} +(11.0210 - 0.425737i) q^{56} +(4.00544 - 6.93762i) q^{57} +(-2.46446 - 6.50915i) q^{58} +(8.30004 + 4.79203i) q^{59} +(0.298074 + 1.46839i) q^{60} +(4.66958 - 2.69599i) q^{61} +(-11.5560 + 4.37529i) q^{62} -3.89942 q^{63} +(-4.52255 - 6.59898i) q^{64} +(-1.74569 + 3.02362i) q^{65} +(-5.62988 - 4.60187i) q^{66} +(-1.16826 + 0.674494i) q^{67} +(0.251958 + 1.24121i) q^{68} +(-3.51118 + 2.02718i) q^{69} +(4.07750 + 0.664983i) q^{70} +(-2.32191 - 4.02166i) q^{71} +(1.50771 + 2.39307i) q^{72} -6.48731 q^{73} +(3.45278 - 7.87898i) q^{74} -4.43875i q^{75} +(-5.09046 + 15.1916i) q^{76} +(-17.3633 + 10.0247i) q^{77} +(-1.06084 + 6.50478i) q^{78} +(-0.327537 - 0.567311i) q^{79} +(-1.16847 - 2.75948i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-1.67432 - 1.36859i) q^{82} +(-8.56975 - 4.94775i) q^{83} +(7.64296 - 1.55148i) q^{84} +0.474419i q^{85} +(-4.46153 - 11.7838i) q^{86} +(-2.46076 - 4.26216i) q^{87} +(12.8657 + 6.77979i) q^{88} +(-6.26061 + 10.8437i) q^{89} +(0.375147 + 0.990842i) q^{90} +(15.7380 + 9.08632i) q^{91} +(6.07544 - 5.37034i) q^{92} +(-7.56684 + 4.36872i) q^{93} +(1.38331 - 0.523742i) q^{94} +(-3.00074 + 5.19744i) q^{95} +(-3.90729 - 4.09060i) q^{96} -19.3925 q^{97} +(1.86783 - 11.4530i) q^{98} +(-4.45278 - 2.57081i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32259 + 0.500752i −0.935213 + 0.354085i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 1.49849 − 1.32458i 0.749247 − 0.662291i
\(5\) 0.648798 − 0.374584i 0.290151 − 0.167519i −0.347859 − 0.937547i \(-0.613091\pi\)
0.638010 + 0.770028i \(0.279758\pi\)
\(6\) 0.895021 − 1.09496i 0.365391 − 0.447015i
\(7\) −1.94971 − 3.37700i −0.736922 − 1.27639i −0.953875 − 0.300204i \(-0.902945\pi\)
0.216953 − 0.976182i \(-0.430388\pi\)
\(8\) −1.31861 + 2.50225i −0.466198 + 0.884680i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −0.670520 + 0.820308i −0.212037 + 0.259404i
\(11\) − 5.14163i − 1.55026i −0.631802 − 0.775130i \(-0.717684\pi\)
0.631802 − 0.775130i \(-0.282316\pi\)
\(12\) −0.635443 + 1.89637i −0.183437 + 0.547434i
\(13\) −4.03597 + 2.33017i −1.11938 + 0.646273i −0.941242 − 0.337733i \(-0.890340\pi\)
−0.178135 + 0.984006i \(0.557007\pi\)
\(14\) 4.26971 + 3.49007i 1.14113 + 0.932759i
\(15\) −0.374584 + 0.648798i −0.0967171 + 0.167519i
\(16\) 0.490968 − 3.96975i 0.122742 − 0.992439i
\(17\) −0.316631 + 0.548421i −0.0767942 + 0.133012i −0.901865 − 0.432018i \(-0.857802\pi\)
0.825071 + 0.565029i \(0.191135\pi\)
\(18\) −0.227631 + 1.39577i −0.0536532 + 0.328987i
\(19\) −6.93762 + 4.00544i −1.59160 + 0.918911i −0.598567 + 0.801073i \(0.704263\pi\)
−0.993033 + 0.117838i \(0.962404\pi\)
\(20\) 0.476053 − 1.42070i 0.106449 − 0.317677i
\(21\) 3.37700 + 1.94971i 0.736922 + 0.425462i
\(22\) 2.57468 + 6.80027i 0.548924 + 1.44982i
\(23\) 4.05437 0.845394 0.422697 − 0.906271i \(-0.361083\pi\)
0.422697 + 0.906271i \(0.361083\pi\)
\(24\) −0.109180 − 2.82632i −0.0222862 − 0.576920i
\(25\) −2.21937 + 3.84407i −0.443875 + 0.768814i
\(26\) 4.17110 − 5.10288i 0.818021 − 1.00076i
\(27\) 1.00000i 0.192450i
\(28\) −7.39474 − 2.47786i −1.39747 − 0.468272i
\(29\) 4.92152i 0.913903i 0.889492 + 0.456951i \(0.151059\pi\)
−0.889492 + 0.456951i \(0.848941\pi\)
\(30\) 0.170534 − 1.04567i 0.0311351 − 0.190912i
\(31\) 8.73743 1.56929 0.784644 − 0.619946i \(-0.212846\pi\)
0.784644 + 0.619946i \(0.212846\pi\)
\(32\) 1.33851 + 5.49622i 0.236618 + 0.971603i
\(33\) 2.57081 + 4.45278i 0.447521 + 0.775130i
\(34\) 0.144150 − 0.883890i 0.0247215 − 0.151586i
\(35\) −2.52994 − 1.46066i −0.427637 − 0.246897i
\(36\) −0.397874 − 1.96002i −0.0663124 − 0.326671i
\(37\) −4.25602 + 4.34584i −0.699685 + 0.714452i
\(38\) 7.16990 − 8.77159i 1.16311 − 1.42294i
\(39\) 2.33017 − 4.03597i 0.373126 − 0.646273i
\(40\) 0.0817938 + 2.11739i 0.0129327 + 0.334788i
\(41\) 0.764557 + 1.32425i 0.119404 + 0.206813i 0.919532 − 0.393016i \(-0.128568\pi\)
−0.800128 + 0.599830i \(0.795235\pi\)
\(42\) −5.44271 − 0.887630i −0.839829 − 0.136964i
\(43\) 8.90966i 1.35871i 0.733810 + 0.679355i \(0.237740\pi\)
−0.733810 + 0.679355i \(0.762260\pi\)
\(44\) −6.81051 − 7.70470i −1.02672 − 1.16153i
\(45\) − 0.749167i − 0.111679i
\(46\) −5.36227 + 2.03023i −0.790623 + 0.299342i
\(47\) −1.04591 −0.152562 −0.0762808 − 0.997086i \(-0.524305\pi\)
−0.0762808 + 0.997086i \(0.524305\pi\)
\(48\) 1.55969 + 3.68339i 0.225121 + 0.531652i
\(49\) −4.10275 + 7.10617i −0.586107 + 1.01517i
\(50\) 1.01040 − 6.19549i 0.142892 − 0.876174i
\(51\) − 0.633262i − 0.0886744i
\(52\) −2.96138 + 8.83772i −0.410670 + 1.22557i
\(53\) 0.950878 + 0.548990i 0.130613 + 0.0754095i 0.563883 − 0.825855i \(-0.309307\pi\)
−0.433270 + 0.901264i \(0.642640\pi\)
\(54\) −0.500752 − 1.32259i −0.0681438 − 0.179982i
\(55\) −1.92597 − 3.33588i −0.259698 − 0.449810i
\(56\) 11.0210 − 0.425737i 1.47274 − 0.0568916i
\(57\) 4.00544 − 6.93762i 0.530533 − 0.918911i
\(58\) −2.46446 − 6.50915i −0.323600 − 0.854694i
\(59\) 8.30004 + 4.79203i 1.08057 + 0.623870i 0.931051 − 0.364889i \(-0.118893\pi\)
0.149523 + 0.988758i \(0.452226\pi\)
\(60\) 0.298074 + 1.46839i 0.0384812 + 0.189568i
\(61\) 4.66958 − 2.69599i 0.597879 − 0.345186i −0.170328 − 0.985387i \(-0.554483\pi\)
0.768207 + 0.640202i \(0.221149\pi\)
\(62\) −11.5560 + 4.37529i −1.46762 + 0.555662i
\(63\) −3.89942 −0.491281
\(64\) −4.52255 − 6.59898i −0.565319 − 0.824873i
\(65\) −1.74569 + 3.02362i −0.216526 + 0.375034i
\(66\) −5.62988 − 4.60187i −0.692990 − 0.566451i
\(67\) −1.16826 + 0.674494i −0.142725 + 0.0824025i −0.569662 − 0.821879i \(-0.692926\pi\)
0.426937 + 0.904281i \(0.359593\pi\)
\(68\) 0.251958 + 1.24121i 0.0305544 + 0.150519i
\(69\) −3.51118 + 2.02718i −0.422697 + 0.244044i
\(70\) 4.07750 + 0.664983i 0.487355 + 0.0794807i
\(71\) −2.32191 − 4.02166i −0.275559 − 0.477283i 0.694717 − 0.719283i \(-0.255530\pi\)
−0.970276 + 0.242001i \(0.922196\pi\)
\(72\) 1.50771 + 2.39307i 0.177686 + 0.282027i
\(73\) −6.48731 −0.759282 −0.379641 − 0.925134i \(-0.623953\pi\)
−0.379641 + 0.925134i \(0.623953\pi\)
\(74\) 3.45278 − 7.87898i 0.401378 − 0.915913i
\(75\) − 4.43875i − 0.512543i
\(76\) −5.09046 + 15.1916i −0.583915 + 1.74259i
\(77\) −17.3633 + 10.0247i −1.97873 + 1.14242i
\(78\) −1.06084 + 6.50478i −0.120116 + 0.736521i
\(79\) −0.327537 − 0.567311i −0.0368508 − 0.0638274i 0.847012 − 0.531574i \(-0.178399\pi\)
−0.883863 + 0.467747i \(0.845066\pi\)
\(80\) −1.16847 − 2.75948i −0.130638 − 0.308519i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −1.67432 − 1.36859i −0.184897 − 0.151135i
\(83\) −8.56975 − 4.94775i −0.940652 − 0.543086i −0.0504874 − 0.998725i \(-0.516077\pi\)
−0.890165 + 0.455639i \(0.849411\pi\)
\(84\) 7.64296 − 1.55148i 0.833916 − 0.169280i
\(85\) 0.474419i 0.0514579i
\(86\) −4.46153 − 11.7838i −0.481099 − 1.27068i
\(87\) −2.46076 − 4.26216i −0.263821 − 0.456951i
\(88\) 12.8657 + 6.77979i 1.37148 + 0.722728i
\(89\) −6.26061 + 10.8437i −0.663623 + 1.14943i 0.316033 + 0.948748i \(0.397649\pi\)
−0.979657 + 0.200681i \(0.935684\pi\)
\(90\) 0.375147 + 0.990842i 0.0395440 + 0.104444i
\(91\) 15.7380 + 9.08632i 1.64979 + 0.952505i
\(92\) 6.07544 − 5.37034i 0.633409 − 0.559897i
\(93\) −7.56684 + 4.36872i −0.784644 + 0.453015i
\(94\) 1.38331 − 0.523742i 0.142678 − 0.0540199i
\(95\) −3.00074 + 5.19744i −0.307870 + 0.533246i
\(96\) −3.90729 − 4.09060i −0.398787 − 0.417496i
\(97\) −19.3925 −1.96901 −0.984506 − 0.175349i \(-0.943895\pi\)
−0.984506 + 0.175349i \(0.943895\pi\)
\(98\) 1.86783 − 11.4530i 0.188679 − 1.15693i
\(99\) −4.45278 − 2.57081i −0.447521 − 0.258377i
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.565.10 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.41 yes 152
37.26 even 3 inner 888.2.bh.a.877.41 yes 152
296.285 even 6 inner 888.2.bh.a.877.10 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.10 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.41 yes 152 8.5 even 2 inner
888.2.bh.a.877.10 yes 152 296.285 even 6 inner
888.2.bh.a.877.41 yes 152 37.26 even 3 inner