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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 475.4
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.4

$q$-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+(-1.27341 - 0.615171i) q^{2} +1.00000i q^{3} +(1.24313 + 1.56673i) q^{4} +(0.125657 + 0.125657i) q^{5} +(0.615171 - 1.27341i) q^{6} -2.31726i q^{7} +(-0.619204 - 2.75982i) 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} +(-1.42551 + 2.95082i) q^{14} +(-0.125657 + 0.125657i) q^{15} +(-0.909260 + 3.89529i) q^{16} +(-5.24248 + 5.24248i) q^{17} +(1.27341 + 0.615171i) q^{18} +(2.39931 - 2.39931i) q^{19} +(-0.0406622 + 0.353078i) q^{20} +2.31726 q^{21} +(1.80315 - 3.73253i) q^{22} +(4.68535 + 4.68535i) q^{23} +(2.75982 - 0.619204i) 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} +(3.55412 - 4.40093i) 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} +(-2.95082 - 1.42551i) 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} +(-3.05643 + 6.32682i) q^{50} +(-5.24248 - 5.24248i) q^{51} +(-0.443620 + 3.85204i) q^{52} +6.86334i q^{53} +(-0.615171 + 1.27341i) q^{54} +(-0.368317 + 0.368317i) q^{55} +(-6.39522 + 1.43486i) q^{56} +(2.39931 + 2.39931i) q^{57} +(-7.20167 + 2.51003i) q^{58} +(-6.32620 + 6.32620i) q^{59} +(-0.353078 - 0.0406622i) 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} +(3.73253 + 1.80315i) q^{66} +4.42553i q^{67} +(-14.7306 - 1.69645i) q^{68} +(-4.68535 + 4.68535i) q^{69} +(-0.549916 + 0.191665i) q^{70} -9.41207i q^{71} +(0.619204 + 2.75982i) q^{72} +15.4599i q^{73} +(-8.23231 - 2.49581i) q^{74} +4.96842 q^{75} +(6.74171 + 0.776408i) q^{76} +6.79220 q^{77} +(2.58906 - 0.902377i) q^{78} +(8.88567 + 8.88567i) q^{79} +(-0.603724 + 0.375215i) q^{80} +1.00000 q^{81} +(3.47437 - 7.19196i) 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} +(8.08939 - 1.81497i) q^{88} +(-4.91828 - 4.91828i) q^{89} +(0.0827118 + 0.237313i) q^{90} +(3.17674 - 3.17674i) q^{91} +(-1.51616 + 13.1651i) q^{92} +(-4.99158 - 4.99158i) q^{93} +(5.21798 - 10.8012i) q^{94} +0.602979 q^{95} +(4.40093 + 3.55412i) q^{96} +(6.59196 - 6.59196i) q^{97} +(-2.07603 - 1.00291i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27341 − 0.615171i −0.900434 − 0.434992i
\(3\) 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 \(-0.237348\pi\)
−0.678451 + 0.734646i \(0.737348\pi\)
\(6\) 0.615171 − 1.27341i 0.251143 − 0.519866i
\(7\) − 2.31726i − 0.875842i −0.899013 − 0.437921i \(-0.855715\pi\)
0.899013 − 0.437921i \(-0.144285\pi\)
\(8\) −0.619204 − 2.75982i −0.218922 − 0.975742i
\(9\) −1.00000 −0.333333
\(10\) −0.0827118 − 0.237313i −0.0261558 − 0.0750449i
\(11\) 2.93113i 0.883770i 0.897072 + 0.441885i \(0.145690\pi\)
−0.897072 + 0.441885i \(0.854310\pi\)
\(12\) −1.56673 + 1.24313i −0.452275 + 0.358860i
\(13\) 1.37090 + 1.37090i 0.380220 + 0.380220i 0.871181 − 0.490961i \(-0.163354\pi\)
−0.490961 + 0.871181i \(0.663354\pi\)
\(14\) −1.42551 + 2.95082i −0.380984 + 0.788639i
\(15\) −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 −0.326180 + 0.945308i \(0.605762\pi\)
−0.945308 + 0.326180i \(0.894238\pi\)
\(18\) 1.27341 + 0.615171i 0.300145 + 0.144997i
\(19\) 2.39931 − 2.39931i 0.550439 − 0.550439i −0.376129 − 0.926567i \(-0.622745\pi\)
0.926567 + 0.376129i \(0.122745\pi\)
\(20\) −0.0406622 + 0.353078i −0.00909234 + 0.0789506i
\(21\) 2.31726 0.505668
\(22\) 1.80315 − 3.73253i 0.384433 − 0.795777i
\(23\) 4.68535 + 4.68535i 0.976963 + 0.976963i 0.999741 − 0.0227779i \(-0.00725106\pi\)
−0.0227779 + 0.999741i \(0.507251\pi\)
\(24\) 2.75982 − 0.619204i 0.563345 − 0.126395i
\(25\) − 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 \(-0.583073\pi\)
0.966137 + 0.258029i \(0.0830731\pi\)
\(30\) 0.237313 − 0.0827118i 0.0433272 − 0.0151010i
\(31\) −4.99158 + 4.99158i −0.896515 + 0.896515i −0.995126 − 0.0986112i \(-0.968560\pi\)
0.0986112 + 0.995126i \(0.468560\pi\)
\(32\) 3.55412 − 4.40093i 0.628286 − 0.777982i
\(33\) −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 \(0.145383\pi\)
−0.897497 + 0.441020i \(0.854617\pi\)
\(42\) −2.95082 − 1.42551i −0.455321 − 0.219961i
\(43\) −5.86247 + 5.86247i −0.894018 + 0.894018i −0.994899 − 0.100880i \(-0.967834\pi\)
0.100880 + 0.994899i \(0.467834\pi\)
\(44\) −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
\(50\) −3.05643 + 6.32682i −0.432244 + 0.894747i
\(51\) −5.24248 − 5.24248i −0.734094 − 0.734094i
\(52\) −0.443620 + 3.85204i −0.0615190 + 0.534181i
\(53\) 6.86334i 0.942753i 0.881932 + 0.471376i \(0.156243\pi\)
−0.881932 + 0.471376i \(0.843757\pi\)
\(54\) −0.615171 + 1.27341i −0.0837142 + 0.173289i
\(55\) −0.368317 + 0.368317i −0.0496639 + 0.0496639i
\(56\) −6.39522 + 1.43486i −0.854597 + 0.191741i
\(57\) 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 −0.986623 − 0.163021i \(-0.947876\pi\)
0.163021 + 0.986623i \(0.447876\pi\)
\(60\) −0.353078 − 0.0406622i −0.0455821 − 0.00524946i
\(61\) 8.73477 − 8.73477i 1.11837 − 1.11837i 0.126393 − 0.991980i \(-0.459660\pi\)
0.991980 − 0.126393i \(-0.0403399\pi\)
\(62\) 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
\(66\) 3.73253 + 1.80315i 0.459442 + 0.221952i
\(67\) 4.42553i 0.540665i 0.962767 + 0.270333i \(0.0871336\pi\)
−0.962767 + 0.270333i \(0.912866\pi\)
\(68\) −14.7306 − 1.69645i −1.78635 − 0.205725i
\(69\) −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 \(-0.188624\pi\)
\(72\) 0.619204 + 2.75982i 0.0729739 + 0.325247i
\(73\) 15.4599i 1.80945i 0.425999 + 0.904724i \(0.359923\pi\)
−0.425999 + 0.904724i \(0.640077\pi\)
\(74\) −8.23231 − 2.49581i −0.956986 − 0.290133i
\(75\) 4.96842 0.573704
\(76\) 6.74171 + 0.776408i 0.773327 + 0.0890601i
\(77\) 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 \(-9.02565e-5\pi\)
−0.000283549 1.00000i \(0.500090\pi\)
\(80\) −0.603724 + 0.375215i −0.0674984 + 0.0419503i
\(81\) 1.00000 0.111111
\(82\) 3.47437 − 7.19196i 0.383680 − 0.794219i
\(83\) −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
\(88\) 8.08939 − 1.81497i 0.862332 − 0.193477i
\(89\) −4.91828 − 4.91828i −0.521336 − 0.521336i 0.396639 − 0.917975i \(-0.370177\pi\)
−0.917975 + 0.396639i \(0.870177\pi\)
\(90\) 0.0827118 + 0.237313i 0.00871859 + 0.0250150i
\(91\) 3.17674 − 3.17674i 0.333013 − 0.333013i
\(92\) −1.51616 + 13.1651i −0.158071 + 1.37256i
\(93\) −4.99158 − 4.99158i −0.517603 − 0.517603i
\(94\) 5.21798 − 10.8012i 0.538194 − 1.11406i
\(95\) 0.602979 0.0618643
\(96\) 4.40093 + 3.55412i 0.449168 + 0.362741i
\(97\) 6.59196 − 6.59196i 0.669313 − 0.669313i −0.288244 − 0.957557i \(-0.593072\pi\)
0.957557 + 0.288244i \(0.0930715\pi\)
\(98\) −2.07603 − 1.00291i −0.209711 − 0.101310i
\(99\) − 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.475.4 yes 72
8.3 odd 2 inner 888.2.r.e.475.13 yes 72
37.6 odd 4 inner 888.2.r.e.43.13 yes 72
296.43 even 4 inner 888.2.r.e.43.4 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.4 ✓ 72 296.43 even 4 inner
888.2.r.e.43.13 yes 72 37.6 odd 4 inner
888.2.r.e.475.4 yes 72 1.1 even 1 trivial
888.2.r.e.475.13 yes 72 8.3 odd 2 inner