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Properties

Label 888.2.r.e.43.17
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
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(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 43.17
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.17

$q$-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+(-0.0909954 - 1.41128i) q^{2} -1.00000i q^{3} +(-1.98344 + 0.256841i) q^{4} +(1.25714 - 1.25714i) q^{5} +(-1.41128 + 0.0909954i) q^{6} -5.01253i q^{7} +(0.542959 + 2.77582i) q^{8} -1.00000 q^{9} +(-1.88857 - 1.65978i) q^{10} -5.64096i q^{11} +(0.256841 + 1.98344i) q^{12} +(1.56784 - 1.56784i) q^{13} +(-7.07409 + 0.456117i) q^{14} +(-1.25714 - 1.25714i) q^{15} +(3.86807 - 1.01886i) q^{16} +(3.02871 + 3.02871i) q^{17} +(0.0909954 + 1.41128i) q^{18} +(4.79021 + 4.79021i) q^{19} +(-2.17057 + 2.81634i) q^{20} -5.01253 q^{21} +(-7.96098 + 0.513301i) q^{22} +(1.14779 - 1.14779i) q^{23} +(2.77582 - 0.542959i) q^{24} +1.83922i q^{25} +(-2.35533 - 2.06999i) q^{26} +1.00000i q^{27} +(1.28742 + 9.94204i) q^{28} +(0.724508 + 0.724508i) q^{29} +(-1.65978 + 1.88857i) q^{30} +(-1.02483 - 1.02483i) q^{31} +(-1.78987 - 5.36622i) q^{32} -5.64096 q^{33} +(3.99877 - 4.54997i) q^{34} +(-6.30143 - 6.30143i) q^{35} +(1.98344 - 0.256841i) q^{36} +(-5.74482 + 1.99926i) q^{37} +(6.32445 - 7.19623i) q^{38} +(-1.56784 - 1.56784i) q^{39} +(4.17216 + 2.80701i) q^{40} -0.114263i q^{41} +(0.456117 + 7.07409i) q^{42} +(-6.45665 - 6.45665i) q^{43} +(1.44883 + 11.1885i) q^{44} +(-1.25714 + 1.25714i) q^{45} +(-1.72430 - 1.51542i) q^{46} +2.26650i q^{47} +(-1.01886 - 3.86807i) q^{48} -18.1254 q^{49} +(2.59566 - 0.167360i) q^{50} +(3.02871 - 3.02871i) q^{51} +(-2.70702 + 3.51239i) q^{52} +3.83199i q^{53} +(1.41128 - 0.0909954i) q^{54} +(-7.09145 - 7.09145i) q^{55} +(13.9139 - 2.72159i) q^{56} +(4.79021 - 4.79021i) q^{57} +(0.956559 - 1.08841i) q^{58} +(5.30169 + 5.30169i) q^{59} +(2.81634 + 2.17057i) q^{60} +(9.25481 + 9.25481i) q^{61} +(-1.35308 + 1.53959i) q^{62} +5.01253i q^{63} +(-7.41039 + 3.01431i) q^{64} -3.94197i q^{65} +(0.513301 + 7.96098i) q^{66} +8.52951i q^{67} +(-6.78516 - 5.22937i) q^{68} +(-1.14779 - 1.14779i) q^{69} +(-8.31969 + 9.46650i) q^{70} +9.67018i q^{71} +(-0.542959 - 2.77582i) q^{72} -13.9333i q^{73} +(3.34427 + 7.92564i) q^{74} +1.83922 q^{75} +(-10.7314 - 8.27077i) q^{76} -28.2754 q^{77} +(-2.06999 + 2.35533i) q^{78} +(-2.40809 + 2.40809i) q^{79} +(3.58184 - 6.14353i) q^{80} +1.00000 q^{81} +(-0.161258 + 0.0103975i) q^{82} +17.2438 q^{83} +(9.94204 - 1.28742i) q^{84} +7.61501 q^{85} +(-8.52463 + 9.69968i) q^{86} +(0.724508 - 0.724508i) q^{87} +(15.6583 - 3.06281i) q^{88} +(1.97427 - 1.97427i) q^{89} +(1.88857 + 1.65978i) q^{90} +(-7.85882 - 7.85882i) q^{91} +(-1.98178 + 2.57138i) q^{92} +(-1.02483 + 1.02483i) q^{93} +(3.19867 - 0.206241i) q^{94} +12.0439 q^{95} +(-5.36622 + 1.78987i) q^{96} +(-3.62186 - 3.62186i) q^{97} +(1.64933 + 25.5801i) q^{98} +5.64096i 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0909954 − 1.41128i −0.0643435 − 0.997928i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.98344 + 0.256841i −0.991720 + 0.128420i
\(5\) 1.25714 − 1.25714i 0.562208 − 0.562208i −0.367726 − 0.929934i \(-0.619864\pi\)
0.929934 + 0.367726i \(0.119864\pi\)
\(6\) −1.41128 + 0.0909954i −0.576154 + 0.0371487i
\(7\) − 5.01253i − 1.89456i −0.320412 − 0.947278i \(-0.603821\pi\)
0.320412 − 0.947278i \(-0.396179\pi\)
\(8\) 0.542959 + 2.77582i 0.191965 + 0.981402i
\(9\) −1.00000 −0.333333
\(10\) −1.88857 − 1.65978i −0.597218 − 0.524869i
\(11\) − 5.64096i − 1.70081i −0.526127 − 0.850406i \(-0.676356\pi\)
0.526127 − 0.850406i \(-0.323644\pi\)
\(12\) 0.256841 + 1.98344i 0.0741435 + 0.572570i
\(13\) 1.56784 − 1.56784i 0.434840 − 0.434840i −0.455431 − 0.890271i \(-0.650515\pi\)
0.890271 + 0.455431i \(0.150515\pi\)
\(14\) −7.07409 + 0.456117i −1.89063 + 0.121902i
\(15\) −1.25714 − 1.25714i −0.324591 − 0.324591i
\(16\) 3.86807 − 1.01886i 0.967016 − 0.254714i
\(17\) 3.02871 + 3.02871i 0.734571 + 0.734571i 0.971522 − 0.236951i \(-0.0761481\pi\)
−0.236951 + 0.971522i \(0.576148\pi\)
\(18\) 0.0909954 + 1.41128i 0.0214478 + 0.332643i
\(19\) 4.79021 + 4.79021i 1.09895 + 1.09895i 0.994534 + 0.104415i \(0.0332972\pi\)
0.104415 + 0.994534i \(0.466703\pi\)
\(20\) −2.17057 + 2.81634i −0.485354 + 0.629752i
\(21\) −5.01253 −1.09382
\(22\) −7.96098 + 0.513301i −1.69729 + 0.109436i
\(23\) 1.14779 − 1.14779i 0.239331 − 0.239331i −0.577242 − 0.816573i \(-0.695871\pi\)
0.816573 + 0.577242i \(0.195871\pi\)
\(24\) 2.77582 − 0.542959i 0.566613 − 0.110831i
\(25\) 1.83922i 0.367844i
\(26\) −2.35533 − 2.06999i −0.461918 − 0.405959i
\(27\) 1.00000i 0.192450i
\(28\) 1.28742 + 9.94204i 0.243299 + 1.87887i
\(29\) 0.724508 + 0.724508i 0.134538 + 0.134538i 0.771169 − 0.636631i \(-0.219673\pi\)
−0.636631 + 0.771169i \(0.719673\pi\)
\(30\) −1.65978 + 1.88857i −0.303033 + 0.344804i
\(31\) −1.02483 − 1.02483i −0.184066 − 0.184066i 0.609059 − 0.793125i \(-0.291547\pi\)
−0.793125 + 0.609059i \(0.791547\pi\)
\(32\) −1.78987 − 5.36622i −0.316407 − 0.948623i
\(33\) −5.64096 −0.981964
\(34\) 3.99877 − 4.54997i 0.685784 − 0.780313i
\(35\) −6.30143 − 6.30143i −1.06514 − 1.06514i
\(36\) 1.98344 − 0.256841i 0.330573 − 0.0428068i
\(37\) −5.74482 + 1.99926i −0.944443 + 0.328676i
\(38\) 6.32445 − 7.19623i 1.02596 − 1.16738i
\(39\) −1.56784 − 1.56784i −0.251055 − 0.251055i
\(40\) 4.17216 + 2.80701i 0.659677 + 0.443828i
\(41\) − 0.114263i − 0.0178450i −0.999960 − 0.00892248i \(-0.997160\pi\)
0.999960 − 0.00892248i \(-0.00284015\pi\)
\(42\) 0.456117 + 7.07409i 0.0703804 + 1.09156i
\(43\) −6.45665 − 6.45665i −0.984629 − 0.984629i 0.0152544 − 0.999884i \(-0.495144\pi\)
−0.999884 + 0.0152544i \(0.995144\pi\)
\(44\) 1.44883 + 11.1885i 0.218419 + 1.68673i
\(45\) −1.25714 + 1.25714i −0.187403 + 0.187403i
\(46\) −1.72430 − 1.51542i −0.254235 − 0.223436i
\(47\) 2.26650i 0.330603i 0.986243 + 0.165302i \(0.0528597\pi\)
−0.986243 + 0.165302i \(0.947140\pi\)
\(48\) −1.01886 − 3.86807i −0.147059 − 0.558307i
\(49\) −18.1254 −2.58934
\(50\) 2.59566 − 0.167360i 0.367081 − 0.0236683i
\(51\) 3.02871 − 3.02871i 0.424104 − 0.424104i
\(52\) −2.70702 + 3.51239i −0.375397 + 0.487081i
\(53\) 3.83199i 0.526365i 0.964746 + 0.263182i \(0.0847721\pi\)
−0.964746 + 0.263182i \(0.915228\pi\)
\(54\) 1.41128 − 0.0909954i 0.192051 − 0.0123829i
\(55\) −7.09145 − 7.09145i −0.956211 − 0.956211i
\(56\) 13.9139 − 2.72159i 1.85932 − 0.363688i
\(57\) 4.79021 − 4.79021i 0.634479 − 0.634479i
\(58\) 0.956559 − 1.08841i 0.125602 − 0.142916i
\(59\) 5.30169 + 5.30169i 0.690221 + 0.690221i 0.962280 − 0.272059i \(-0.0877047\pi\)
−0.272059 + 0.962280i \(0.587705\pi\)
\(60\) 2.81634 + 2.17057i 0.363588 + 0.280219i
\(61\) 9.25481 + 9.25481i 1.18496 + 1.18496i 0.978444 + 0.206514i \(0.0662118\pi\)
0.206514 + 0.978444i \(0.433788\pi\)
\(62\) −1.35308 + 1.53959i −0.171841 + 0.195528i
\(63\) 5.01253i 0.631519i
\(64\) −7.41039 + 3.01431i −0.926299 + 0.376789i
\(65\) − 3.94197i − 0.488941i
\(66\) 0.513301 + 7.96098i 0.0631830 + 0.979929i
\(67\) 8.52951i 1.04205i 0.853543 + 0.521023i \(0.174449\pi\)
−0.853543 + 0.521023i \(0.825551\pi\)
\(68\) −6.78516 − 5.22937i −0.822822 − 0.634154i
\(69\) −1.14779 − 1.14779i −0.138178 − 0.138178i
\(70\) −8.31969 + 9.46650i −0.994394 + 1.13146i
\(71\) 9.67018i 1.14764i 0.818982 + 0.573820i \(0.194539\pi\)
−0.818982 + 0.573820i \(0.805461\pi\)
\(72\) −0.542959 − 2.77582i −0.0639883 − 0.327134i
\(73\) − 13.9333i − 1.63077i −0.578920 − 0.815384i \(-0.696526\pi\)
0.578920 − 0.815384i \(-0.303474\pi\)
\(74\) 3.34427 + 7.92564i 0.388764 + 0.921337i
\(75\) 1.83922 0.212375
\(76\) −10.7314 − 8.27077i −1.23098 − 0.948722i
\(77\) −28.2754 −3.22228
\(78\) −2.06999 + 2.35533i −0.234381 + 0.266688i
\(79\) −2.40809 + 2.40809i −0.270932 + 0.270932i −0.829475 − 0.558544i \(-0.811360\pi\)
0.558544 + 0.829475i \(0.311360\pi\)
\(80\) 3.58184 − 6.14353i 0.400462 − 0.686867i
\(81\) 1.00000 0.111111
\(82\) −0.161258 + 0.0103975i −0.0178080 + 0.00114821i
\(83\) 17.2438 1.89275 0.946375 − 0.323069i \(-0.104715\pi\)
0.946375 + 0.323069i \(0.104715\pi\)
\(84\) 9.94204 − 1.28742i 1.08477 − 0.140469i
\(85\) 7.61501 0.825963
\(86\) −8.52463 + 9.69968i −0.919234 + 1.04594i
\(87\) 0.724508 − 0.724508i 0.0776754 − 0.0776754i
\(88\) 15.6583 − 3.06281i 1.66918 − 0.326496i
\(89\) 1.97427 − 1.97427i 0.209272 − 0.209272i −0.594686 − 0.803958i \(-0.702724\pi\)
0.803958 + 0.594686i \(0.202724\pi\)
\(90\) 1.88857 + 1.65978i 0.199073 + 0.174956i
\(91\) −7.85882 − 7.85882i −0.823828 − 0.823828i
\(92\) −1.98178 + 2.57138i −0.206615 + 0.268085i
\(93\) −1.02483 + 1.02483i −0.106270 + 0.106270i
\(94\) 3.19867 − 0.206241i 0.329918 − 0.0212722i
\(95\) 12.0439 1.23568
\(96\) −5.36622 + 1.78987i −0.547688 + 0.182678i
\(97\) −3.62186 − 3.62186i −0.367745 − 0.367745i 0.498909 − 0.866654i \(-0.333734\pi\)
−0.866654 + 0.498909i \(0.833734\pi\)
\(98\) 1.64933 + 25.5801i 0.166607 + 2.58398i
\(99\) 5.64096i 0.566937i
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.43.17 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.35 yes 72
37.31 odd 4 inner 888.2.r.e.475.35 yes 72
296.179 even 4 inner 888.2.r.e.475.17 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.17 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.35 yes 72 8.3 odd 2 inner
888.2.r.e.475.17 yes 72 296.179 even 4 inner
888.2.r.e.475.35 yes 72 37.31 odd 4 inner