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Properties

Label 888.2.r.e.43.13
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 43.13
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.13

$q$-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+(-0.615171 + 1.27341i) q^{2} -1.00000i q^{3} +(-1.24313 - 1.56673i) q^{4} +(-0.125657 + 0.125657i) q^{5} +(1.27341 + 0.615171i) q^{6} -2.31726i q^{7} +(2.75982 - 0.619204i) q^{8} -1.00000 q^{9} +(-0.0827118 - 0.237313i) q^{10} -2.93113i q^{11} +(-1.56673 + 1.24313i) q^{12} +(-1.37090 + 1.37090i) q^{13} +(2.95082 + 1.42551i) q^{14} +(0.125657 + 0.125657i) q^{15} +(-0.909260 + 3.89529i) q^{16} +(-5.24248 - 5.24248i) q^{17} +(0.615171 - 1.27341i) q^{18} +(2.39931 + 2.39931i) q^{19} +(0.353078 + 0.0406622i) q^{20} -2.31726 q^{21} +(3.73253 + 1.80315i) q^{22} +(-4.68535 + 4.68535i) q^{23} +(-0.619204 - 2.75982i) q^{24} +4.96842i q^{25} +(-0.902377 - 2.58906i) q^{26} +1.00000i q^{27} +(-3.63051 + 2.88066i) q^{28} +(-3.81328 - 3.81328i) q^{29} +(-0.237313 + 0.0827118i) q^{30} +(4.99158 + 4.99158i) q^{31} +(-4.40093 - 3.55412i) q^{32} -2.93113 q^{33} +(9.90083 - 3.45079i) q^{34} +(0.291180 + 0.291180i) q^{35} +(1.24313 + 1.56673i) q^{36} +(-6.00921 - 0.943046i) q^{37} +(-4.53128 + 1.57931i) q^{38} +(1.37090 + 1.37090i) q^{39} +(-0.268983 + 0.424597i) q^{40} -5.64781i q^{41} +(1.42551 - 2.95082i) q^{42} +(-5.86247 - 5.86247i) q^{43} +(-4.59228 + 3.64378i) q^{44} +(0.125657 - 0.125657i) q^{45} +(-3.08406 - 8.84864i) q^{46} +8.48216i q^{47} +(3.89529 + 0.909260i) q^{48} +1.63030 q^{49} +(-6.32682 - 3.05643i) q^{50} +(-5.24248 + 5.24248i) q^{51} +(3.85204 + 0.443620i) q^{52} +6.86334i q^{53} +(-1.27341 - 0.615171i) q^{54} +(0.368317 + 0.368317i) q^{55} +(-1.43486 - 6.39522i) q^{56} +(2.39931 - 2.39931i) q^{57} +(7.20167 - 2.51003i) q^{58} +(-6.32620 - 6.32620i) q^{59} +(0.0406622 - 0.353078i) q^{60} +(-8.73477 - 8.73477i) q^{61} +(-9.42699 + 3.28564i) q^{62} +2.31726i q^{63} +(7.23317 - 3.41778i) q^{64} -0.344527i q^{65} +(1.80315 - 3.73253i) q^{66} -4.42553i q^{67} +(-1.69645 + 14.7306i) q^{68} +(4.68535 + 4.68535i) q^{69} +(-0.549916 + 0.191665i) q^{70} -9.41207i q^{71} +(-2.75982 + 0.619204i) q^{72} -15.4599i q^{73} +(4.89758 - 7.07204i) q^{74} +4.96842 q^{75} +(0.776408 - 6.74171i) q^{76} -6.79220 q^{77} +(-2.58906 + 0.902377i) q^{78} +(-8.88567 + 8.88567i) q^{79} +(-0.375215 - 0.603724i) q^{80} +1.00000 q^{81} +(7.19196 + 3.47437i) q^{82} -13.9706 q^{83} +(2.88066 + 3.63051i) q^{84} +1.31751 q^{85} +(11.0717 - 3.85889i) q^{86} +(-3.81328 + 3.81328i) q^{87} +(-1.81497 - 8.08939i) q^{88} +(-4.91828 + 4.91828i) q^{89} +(0.0827118 + 0.237313i) q^{90} +(3.17674 + 3.17674i) q^{91} +(13.1651 + 1.51616i) q^{92} +(4.99158 - 4.99158i) q^{93} +(-10.8012 - 5.21798i) q^{94} -0.602979 q^{95} +(-3.55412 + 4.40093i) q^{96} +(6.59196 + 6.59196i) q^{97} +(-1.00291 + 2.07603i) q^{98} +2.93113i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.615171 + 1.27341i −0.434992 + 0.900434i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.24313 − 1.56673i −0.621565 − 0.783363i
\(5\) −0.125657 + 0.125657i −0.0561955 + 0.0561955i −0.734646 − 0.678451i \(-0.762652\pi\)
0.678451 + 0.734646i \(0.262652\pi\)
\(6\) 1.27341 + 0.615171i 0.519866 + 0.251143i
\(7\) − 2.31726i − 0.875842i −0.899013 − 0.437921i \(-0.855715\pi\)
0.899013 − 0.437921i \(-0.144285\pi\)
\(8\) 2.75982 − 0.619204i 0.975742 − 0.218922i
\(9\) −1.00000 −0.333333
\(10\) −0.0827118 − 0.237313i −0.0261558 − 0.0750449i
\(11\) − 2.93113i − 0.883770i −0.897072 − 0.441885i \(-0.854310\pi\)
0.897072 − 0.441885i \(-0.145690\pi\)
\(12\) −1.56673 + 1.24313i −0.452275 + 0.358860i
\(13\) −1.37090 + 1.37090i −0.380220 + 0.380220i −0.871181 − 0.490961i \(-0.836646\pi\)
0.490961 + 0.871181i \(0.336646\pi\)
\(14\) 2.95082 + 1.42551i 0.788639 + 0.380984i
\(15\) 0.125657 + 0.125657i 0.0324445 + 0.0324445i
\(16\) −0.909260 + 3.89529i −0.227315 + 0.973821i
\(17\) −5.24248 − 5.24248i −1.27149 − 1.27149i −0.945308 − 0.326180i \(-0.894238\pi\)
−0.326180 − 0.945308i \(-0.605762\pi\)
\(18\) 0.615171 − 1.27341i 0.144997 − 0.300145i
\(19\) 2.39931 + 2.39931i 0.550439 + 0.550439i 0.926567 − 0.376129i \(-0.122745\pi\)
−0.376129 + 0.926567i \(0.622745\pi\)
\(20\) 0.353078 + 0.0406622i 0.0789506 + 0.00909234i
\(21\) −2.31726 −0.505668
\(22\) 3.73253 + 1.80315i 0.795777 + 0.384433i
\(23\) −4.68535 + 4.68535i −0.976963 + 0.976963i −0.999741 − 0.0227779i \(-0.992749\pi\)
0.0227779 + 0.999741i \(0.492749\pi\)
\(24\) −0.619204 − 2.75982i −0.126395 − 0.563345i
\(25\) 4.96842i 0.993684i
\(26\) −0.902377 − 2.58906i −0.176971 − 0.507756i
\(27\) 1.00000i 0.192450i
\(28\) −3.63051 + 2.88066i −0.686103 + 0.544393i
\(29\) −3.81328 − 3.81328i −0.708108 − 0.708108i 0.258029 − 0.966137i \(-0.416927\pi\)
−0.966137 + 0.258029i \(0.916927\pi\)
\(30\) −0.237313 + 0.0827118i −0.0433272 + 0.0151010i
\(31\) 4.99158 + 4.99158i 0.896515 + 0.896515i 0.995126 − 0.0986112i \(-0.0314400\pi\)
−0.0986112 + 0.995126i \(0.531440\pi\)
\(32\) −4.40093 − 3.55412i −0.777982 − 0.628286i
\(33\) −2.93113 −0.510245
\(34\) 9.90083 − 3.45079i 1.69798 − 0.591805i
\(35\) 0.291180 + 0.291180i 0.0492184 + 0.0492184i
\(36\) 1.24313 + 1.56673i 0.207188 + 0.261121i
\(37\) −6.00921 − 0.943046i −0.987909 − 0.155036i
\(38\) −4.53128 + 1.57931i −0.735070 + 0.256198i
\(39\) 1.37090 + 1.37090i 0.219520 + 0.219520i
\(40\) −0.268983 + 0.424597i −0.0425299 + 0.0671347i
\(41\) − 5.64781i − 0.882040i −0.897497 − 0.441020i \(-0.854617\pi\)
0.897497 − 0.441020i \(-0.145383\pi\)
\(42\) 1.42551 − 2.95082i 0.219961 − 0.455321i
\(43\) −5.86247 − 5.86247i −0.894018 − 0.894018i 0.100880 − 0.994899i \(-0.467834\pi\)
−0.994899 + 0.100880i \(0.967834\pi\)
\(44\) −4.59228 + 3.64378i −0.692313 + 0.549320i
\(45\) 0.125657 − 0.125657i 0.0187318 − 0.0187318i
\(46\) −3.08406 − 8.84864i −0.454720 − 1.30466i
\(47\) 8.48216i 1.23725i 0.785686 + 0.618625i \(0.212310\pi\)
−0.785686 + 0.618625i \(0.787690\pi\)
\(48\) 3.89529 + 0.909260i 0.562236 + 0.131240i
\(49\) 1.63030 0.232900
\(50\) −6.32682 − 3.05643i −0.894747 − 0.432244i
\(51\) −5.24248 + 5.24248i −0.734094 + 0.734094i
\(52\) 3.85204 + 0.443620i 0.534181 + 0.0615190i
\(53\) 6.86334i 0.942753i 0.881932 + 0.471376i \(0.156243\pi\)
−0.881932 + 0.471376i \(0.843757\pi\)
\(54\) −1.27341 − 0.615171i −0.173289 − 0.0837142i
\(55\) 0.368317 + 0.368317i 0.0496639 + 0.0496639i
\(56\) −1.43486 − 6.39522i −0.191741 − 0.854597i
\(57\) 2.39931 − 2.39931i 0.317796 − 0.317796i
\(58\) 7.20167 − 2.51003i 0.945626 − 0.329584i
\(59\) −6.32620 − 6.32620i −0.823601 − 0.823601i 0.163021 − 0.986623i \(-0.447876\pi\)
−0.986623 + 0.163021i \(0.947876\pi\)
\(60\) 0.0406622 − 0.353078i 0.00524946 − 0.0455821i
\(61\) −8.73477 − 8.73477i −1.11837 − 1.11837i −0.991980 − 0.126393i \(-0.959660\pi\)
−0.126393 − 0.991980i \(-0.540340\pi\)
\(62\) −9.42699 + 3.28564i −1.19723 + 0.417276i
\(63\) 2.31726i 0.291947i
\(64\) 7.23317 − 3.41778i 0.904146 − 0.427223i
\(65\) − 0.344527i − 0.0427333i
\(66\) 1.80315 − 3.73253i 0.221952 − 0.459442i
\(67\) − 4.42553i − 0.540665i −0.962767 − 0.270333i \(-0.912866\pi\)
0.962767 − 0.270333i \(-0.0871336\pi\)
\(68\) −1.69645 + 14.7306i −0.205725 + 1.78635i
\(69\) 4.68535 + 4.68535i 0.564050 + 0.564050i
\(70\) −0.549916 + 0.191665i −0.0657275 + 0.0229083i
\(71\) − 9.41207i − 1.11701i −0.829502 − 0.558503i \(-0.811376\pi\)
0.829502 − 0.558503i \(-0.188624\pi\)
\(72\) −2.75982 + 0.619204i −0.325247 + 0.0729739i
\(73\) − 15.4599i − 1.80945i −0.425999 − 0.904724i \(-0.640077\pi\)
0.425999 − 0.904724i \(-0.359923\pi\)
\(74\) 4.89758 − 7.07204i 0.569332 − 0.822108i
\(75\) 4.96842 0.573704
\(76\) 0.776408 − 6.74171i 0.0890601 − 0.773327i
\(77\) −6.79220 −0.774044
\(78\) −2.58906 + 0.902377i −0.293153 + 0.102174i
\(79\) −8.88567 + 8.88567i −0.999716 + 0.999716i −1.00000 0.000283549i \(-0.999910\pi\)
0.000283549 1.00000i \(0.499910\pi\)
\(80\) −0.375215 − 0.603724i −0.0419503 − 0.0674984i
\(81\) 1.00000 0.111111
\(82\) 7.19196 + 3.47437i 0.794219 + 0.383680i
\(83\) −13.9706 −1.53347 −0.766734 − 0.641965i \(-0.778119\pi\)
−0.766734 + 0.641965i \(0.778119\pi\)
\(84\) 2.88066 + 3.63051i 0.314305 + 0.396121i
\(85\) 1.31751 0.142904
\(86\) 11.0717 − 3.85889i 1.19390 − 0.416114i
\(87\) −3.81328 + 3.81328i −0.408826 + 0.408826i
\(88\) −1.81497 − 8.08939i −0.193477 − 0.862332i
\(89\) −4.91828 + 4.91828i −0.521336 + 0.521336i −0.917975 − 0.396639i \(-0.870177\pi\)
0.396639 + 0.917975i \(0.370177\pi\)
\(90\) 0.0827118 + 0.237313i 0.00871859 + 0.0250150i
\(91\) 3.17674 + 3.17674i 0.333013 + 0.333013i
\(92\) 13.1651 + 1.51616i 1.37256 + 0.158071i
\(93\) 4.99158 − 4.99158i 0.517603 − 0.517603i
\(94\) −10.8012 − 5.21798i −1.11406 − 0.538194i
\(95\) −0.602979 −0.0618643
\(96\) −3.55412 + 4.40093i −0.362741 + 0.449168i
\(97\) 6.59196 + 6.59196i 0.669313 + 0.669313i 0.957557 − 0.288244i \(-0.0930715\pi\)
−0.288244 + 0.957557i \(0.593072\pi\)
\(98\) −1.00291 + 2.07603i −0.101310 + 0.209711i
\(99\) 2.93113i 0.294590i
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.43.13 yes 72
8.3 odd 2 inner 888.2.r.e.43.4 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.4 yes 72
296.179 even 4 inner 888.2.r.e.475.13 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.4 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.13 yes 72 1.1 even 1 trivial
888.2.r.e.475.4 yes 72 37.31 odd 4 inner
888.2.r.e.475.13 yes 72 296.179 even 4 inner