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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 877.1
Character \(\chi\) \(=\) 888.877
Dual form 888.2.bh.a.565.1

$q$-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+(-1.41421 + 0.00150368i) q^{2} +(0.866025 + 0.500000i) q^{3} +(2.00000 - 0.00425304i) q^{4} +(-1.17926 - 0.680844i) q^{5} +(-1.22550 - 0.705804i) q^{6} +(-0.930629 + 1.61190i) q^{7} +(-2.82841 + 0.00902206i) q^{8} +(0.500000 + 0.866025i) q^{9} +(1.66874 + 0.961084i) q^{10} +5.10656i q^{11} +(1.73417 + 0.996314i) q^{12} +(-0.598536 - 0.345565i) q^{13} +(1.31368 - 2.28096i) q^{14} +(-0.680844 - 1.17926i) q^{15} +(3.99996 - 0.0170121i) q^{16} +(-2.81775 - 4.88049i) q^{17} +(-0.708409 - 1.22399i) q^{18} +(3.01549 + 1.74099i) q^{19} +(-2.36140 - 1.35667i) q^{20} +(-1.61190 + 0.930629i) q^{21} +(-0.00767862 - 7.22176i) q^{22} -6.73165 q^{23} +(-2.45399 - 1.40639i) q^{24} +(-1.57290 - 2.72435i) q^{25} +(0.846976 + 0.487802i) q^{26} +1.00000i q^{27} +(-1.85440 + 3.22774i) q^{28} -0.624204i q^{29} +(0.964631 + 1.66669i) q^{30} -2.96728 q^{31} +(-5.65677 + 0.0300734i) q^{32} +(-2.55328 + 4.42241i) q^{33} +(3.99224 + 6.89782i) q^{34} +(2.19490 - 1.26723i) q^{35} +(1.00368 + 1.72992i) q^{36} +(-5.59382 + 2.38939i) q^{37} +(-4.26716 - 2.45760i) q^{38} +(-0.345565 - 0.598536i) q^{39} +(3.34156 + 1.91507i) q^{40} +(-1.48334 + 2.56922i) q^{41} +(2.27817 - 1.31853i) q^{42} -0.0870114i q^{43} +(0.0217184 + 10.2131i) q^{44} -1.36169i q^{45} +(9.51999 - 0.0101222i) q^{46} -4.40028 q^{47} +(3.47258 + 1.98525i) q^{48} +(1.76786 + 3.06202i) q^{49} +(2.22852 + 3.85045i) q^{50} -5.63551i q^{51} +(-1.19854 - 0.688582i) q^{52} +(0.220539 - 0.127328i) q^{53} +(-0.00150368 - 1.41421i) q^{54} +(3.47677 - 6.02194i) q^{55} +(2.61766 - 4.56750i) q^{56} +(1.74099 + 3.01549i) q^{57} +(0.000938603 + 0.882758i) q^{58} +(-11.3517 + 6.55393i) q^{59} +(-1.36670 - 2.35561i) q^{60} +(-3.79455 - 2.19079i) q^{61} +(4.19637 - 0.00446184i) q^{62} -1.86126 q^{63} +(7.99984 - 0.0510362i) q^{64} +(0.470551 + 0.815018i) q^{65} +(3.60423 - 6.25807i) q^{66} +(9.62036 + 5.55432i) q^{67} +(-5.65625 - 9.74898i) q^{68} +(-5.82978 - 3.36583i) q^{69} +(-3.10215 + 1.79543i) q^{70} +(5.16295 - 8.94249i) q^{71} +(-1.42202 - 2.44497i) q^{72} -0.235595 q^{73} +(7.90726 - 3.38752i) q^{74} -3.14581i q^{75} +(6.03837 + 3.46915i) q^{76} +(-8.23125 - 4.75231i) q^{77} +(0.489602 + 0.845937i) q^{78} +(-5.43775 + 9.41846i) q^{79} +(-4.72856 - 2.70329i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(2.09390 - 3.63566i) q^{82} +(-7.23425 + 4.17669i) q^{83} +(-3.21983 + 1.86811i) q^{84} +7.67379i q^{85} +(0.000130837 + 0.123053i) q^{86} +(0.312102 - 0.540577i) q^{87} +(-0.0460717 - 14.4435i) q^{88} +(2.86850 + 4.96840i) q^{89} +(0.00204754 + 1.92572i) q^{90} +(1.11403 - 0.643185i) q^{91} +(-13.4633 + 0.0286300i) q^{92} +(-2.56974 - 1.48364i) q^{93} +(6.22293 - 0.00661660i) q^{94} +(-2.37069 - 4.10615i) q^{95} +(-4.91395 - 2.80234i) q^{96} -11.9129 q^{97} +(-2.50473 - 4.32769i) q^{98} +(-4.42241 + 2.55328i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 + 0.00150368i −0.999999 + 0.00106326i
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 2.00000 − 0.00425304i 0.999998 − 0.00212652i
\(5\) −1.17926 − 0.680844i −0.527379 − 0.304482i 0.212569 − 0.977146i \(-0.431817\pi\)
−0.739949 + 0.672663i \(0.765150\pi\)
\(6\) −1.22550 − 0.705804i −0.500307 − 0.288143i
\(7\) −0.930629 + 1.61190i −0.351745 + 0.609240i −0.986555 − 0.163428i \(-0.947745\pi\)
0.634811 + 0.772668i \(0.281078\pi\)
\(8\) −2.82841 + 0.00902206i −0.999995 + 0.00318978i
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 1.66874 + 0.961084i 0.527703 + 0.303922i
\(11\) 5.10656i 1.53969i 0.638233 + 0.769843i \(0.279666\pi\)
−0.638233 + 0.769843i \(0.720334\pi\)
\(12\) 1.73417 + 0.996314i 0.500613 + 0.287611i
\(13\) −0.598536 − 0.345565i −0.166004 − 0.0958424i 0.414696 − 0.909960i \(-0.363888\pi\)
−0.580700 + 0.814117i \(0.697221\pi\)
\(14\) 1.31368 − 2.28096i 0.351097 − 0.609613i
\(15\) −0.680844 − 1.17926i −0.175793 − 0.304482i
\(16\) 3.99996 − 0.0170121i 0.999991 − 0.00425303i
\(17\) −2.81775 − 4.88049i −0.683405 − 1.18369i −0.973935 − 0.226827i \(-0.927165\pi\)
0.290530 − 0.956866i \(-0.406168\pi\)
\(18\) −0.708409 − 1.22399i −0.166974 − 0.288498i
\(19\) 3.01549 + 1.74099i 0.691801 + 0.399411i 0.804286 − 0.594242i \(-0.202548\pi\)
−0.112486 + 0.993653i \(0.535881\pi\)
\(20\) −2.36140 − 1.35667i −0.528025 − 0.303360i
\(21\) −1.61190 + 0.930629i −0.351745 + 0.203080i
\(22\) −0.00767862 − 7.22176i −0.00163709 − 1.53969i
\(23\) −6.73165 −1.40365 −0.701823 − 0.712351i \(-0.747630\pi\)
−0.701823 + 0.712351i \(0.747630\pi\)
\(24\) −2.45399 − 1.40639i −0.500918 − 0.287079i
\(25\) −1.57290 − 2.72435i −0.314581 − 0.544870i
\(26\) 0.846976 + 0.487802i 0.166106 + 0.0956658i
\(27\) 1.00000i 0.192450i
\(28\) −1.85440 + 3.22774i −0.350448 + 0.609986i
\(29\) − 0.624204i − 0.115912i −0.998319 − 0.0579559i \(-0.981542\pi\)
0.998319 − 0.0579559i \(-0.0184583\pi\)
\(30\) 0.964631 + 1.66669i 0.176117 + 0.304295i
\(31\) −2.96728 −0.532940 −0.266470 − 0.963843i \(-0.585857\pi\)
−0.266470 + 0.963843i \(0.585857\pi\)
\(32\) −5.65677 + 0.0300734i −0.999986 + 0.00531628i
\(33\) −2.55328 + 4.42241i −0.444469 + 0.769843i
\(34\) 3.99224 + 6.89782i 0.684664 + 1.18297i
\(35\) 2.19490 − 1.26723i 0.371006 − 0.214200i
\(36\) 1.00368 + 1.72992i 0.167280 + 0.288320i
\(37\) −5.59382 + 2.38939i −0.919618 + 0.392813i
\(38\) −4.26716 − 2.45760i −0.692225 − 0.398676i
\(39\) −0.345565 − 0.598536i −0.0553346 − 0.0958424i
\(40\) 3.34156 + 1.91507i 0.528348 + 0.302799i
\(41\) −1.48334 + 2.56922i −0.231659 + 0.401245i −0.958297 − 0.285776i \(-0.907749\pi\)
0.726637 + 0.687021i \(0.241082\pi\)
\(42\) 2.27817 − 1.31853i 0.351529 − 0.203454i
\(43\) − 0.0870114i − 0.0132691i −0.999978 − 0.00663455i \(-0.997888\pi\)
0.999978 − 0.00663455i \(-0.00211186\pi\)
\(44\) 0.0217184 + 10.2131i 0.00327417 + 1.53968i
\(45\) − 1.36169i − 0.202988i
\(46\) 9.51999 − 0.0101222i 1.40365 − 0.00149244i
\(47\) −4.40028 −0.641847 −0.320923 − 0.947105i \(-0.603993\pi\)
−0.320923 + 0.947105i \(0.603993\pi\)
\(48\) 3.47258 + 1.98525i 0.501223 + 0.286546i
\(49\) 1.76786 + 3.06202i 0.252551 + 0.437432i
\(50\) 2.22852 + 3.85045i 0.315160 + 0.544535i
\(51\) − 5.63551i − 0.789129i
\(52\) −1.19854 − 0.688582i −0.166207 − 0.0954892i
\(53\) 0.220539 − 0.127328i 0.0302934 − 0.0174899i −0.484777 − 0.874638i \(-0.661099\pi\)
0.515070 + 0.857148i \(0.327766\pi\)
\(54\) −0.00150368 − 1.41421i −0.000204625 − 0.192450i
\(55\) 3.47677 − 6.02194i 0.468807 − 0.811998i
\(56\) 2.61766 − 4.56750i 0.349800 − 0.610359i
\(57\) 1.74099 + 3.01549i 0.230600 + 0.399411i
\(58\) 0.000938603 0.882758i 0.000123245 0.115912i
\(59\) −11.3517 + 6.55393i −1.47787 + 0.853249i −0.999687 − 0.0250103i \(-0.992038\pi\)
−0.478184 + 0.878260i \(0.658705\pi\)
\(60\) −1.36670 − 2.35561i −0.176440 − 0.304108i
\(61\) −3.79455 − 2.19079i −0.485843 − 0.280502i 0.237005 − 0.971508i \(-0.423834\pi\)
−0.722848 + 0.691007i \(0.757167\pi\)
\(62\) 4.19637 − 0.00446184i 0.532940 − 0.000566654i
\(63\) −1.86126 −0.234496
\(64\) 7.99984 − 0.0510362i 0.999980 − 0.00637952i
\(65\) 0.470551 + 0.815018i 0.0583647 + 0.101091i
\(66\) 3.60423 − 6.25807i 0.443650 − 0.770315i
\(67\) 9.62036 + 5.55432i 1.17531 + 0.678568i 0.954926 − 0.296844i \(-0.0959341\pi\)
0.220389 + 0.975412i \(0.429267\pi\)
\(68\) −5.65625 − 9.74898i −0.685921 − 1.18224i
\(69\) −5.82978 − 3.36583i −0.701823 − 0.405198i
\(70\) −3.10215 + 1.79543i −0.370778 + 0.214595i
\(71\) 5.16295 − 8.94249i 0.612729 − 1.06128i −0.378049 − 0.925785i \(-0.623405\pi\)
0.990778 − 0.135492i \(-0.0432616\pi\)
\(72\) −1.42202 − 2.44497i −0.167587 − 0.288142i
\(73\) −0.235595 −0.0275743 −0.0137871 − 0.999905i \(-0.504389\pi\)
−0.0137871 + 0.999905i \(0.504389\pi\)
\(74\) 7.90726 − 3.38752i 0.919200 − 0.393791i
\(75\) − 3.14581i − 0.363247i
\(76\) 6.03837 + 3.46915i 0.692648 + 0.397939i
\(77\) −8.23125 − 4.75231i −0.938038 − 0.541576i
\(78\) 0.489602 + 0.845937i 0.0554365 + 0.0957835i
\(79\) −5.43775 + 9.41846i −0.611795 + 1.05966i 0.379143 + 0.925338i \(0.376219\pi\)
−0.990938 + 0.134322i \(0.957114\pi\)
\(80\) −4.72856 − 2.70329i −0.528669 − 0.302237i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 2.09390 − 3.63566i 0.231232 − 0.401492i
\(83\) −7.23425 + 4.17669i −0.794062 + 0.458452i −0.841391 − 0.540428i \(-0.818262\pi\)
0.0473288 + 0.998879i \(0.484929\pi\)
\(84\) −3.21983 + 1.86811i −0.351312 + 0.203827i
\(85\) 7.67379i 0.832340i
\(86\) 0.000130837 0.123053i 1.41085e−5 0.0132691i
\(87\) 0.312102 − 0.540577i 0.0334609 − 0.0579559i
\(88\) −0.0460717 − 14.4435i −0.00491126 − 1.53968i
\(89\) 2.86850 + 4.96840i 0.304061 + 0.526649i 0.977052 − 0.213002i \(-0.0683240\pi\)
−0.672991 + 0.739651i \(0.734991\pi\)
\(90\) 0.00204754 + 1.92572i 0.000215830 + 0.202988i
\(91\) 1.11403 − 0.643185i 0.116782 − 0.0674241i
\(92\) −13.4633 + 0.0286300i −1.40364 + 0.00298488i
\(93\) −2.56974 − 1.48364i −0.266470 − 0.153847i
\(94\) 6.22293 − 0.00661660i 0.641846 − 0.000682451i
\(95\) −2.37069 − 4.10615i −0.243227 − 0.421282i
\(96\) −4.91395 − 2.80234i −0.501528 − 0.286013i
\(97\) −11.9129 −1.20957 −0.604785 − 0.796389i \(-0.706741\pi\)
−0.604785 + 0.796389i \(0.706741\pi\)
\(98\) −2.50473 − 4.32769i −0.253016 − 0.437163i
\(99\) −4.42241 + 2.55328i −0.444469 + 0.256614i
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.877.1 yes 152
8.5 even 2 inner 888.2.bh.a.877.51 yes 152
37.10 even 3 inner 888.2.bh.a.565.51 yes 152
296.269 even 6 inner 888.2.bh.a.565.1 ✓ 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.1 ✓ 152 296.269 even 6 inner
888.2.bh.a.565.51 yes 152 37.10 even 3 inner
888.2.bh.a.877.1 yes 152 1.1 even 1 trivial
888.2.bh.a.877.51 yes 152 8.5 even 2 inner