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 565.16
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.16

$q$-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+(-1.13796 - 0.839665i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.589926 + 1.91102i) q^{4} +(2.72899 - 1.57558i) q^{5} +(1.40534 + 0.158189i) q^{6} +(0.116457 + 0.201710i) q^{7} +(0.933300 - 2.67001i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-4.42846 - 0.498480i) q^{10} -5.08806i q^{11} +(-1.46640 - 1.36003i) q^{12} +(1.08082 - 0.624014i) q^{13} +(0.0368445 - 0.327324i) q^{14} +(-1.57558 + 2.72899i) q^{15} +(-3.30398 + 2.25472i) q^{16} +(1.12317 - 1.94539i) q^{17} +(-1.29615 + 0.565674i) q^{18} +(-1.62145 + 0.936144i) q^{19} +(4.62087 + 4.28567i) q^{20} +(-0.201710 - 0.116457i) q^{21} +(-4.27226 + 5.79003i) q^{22} -6.63320 q^{23} +(0.526744 + 2.77895i) q^{24} +(2.46493 - 4.26939i) q^{25} +(-1.75390 - 0.197424i) q^{26} +1.00000i q^{27} +(-0.316770 + 0.341546i) q^{28} -2.37390i q^{29} +(4.08440 - 1.78253i) q^{30} +1.78662 q^{31} +(5.65301 + 0.208444i) q^{32} +(2.54403 + 4.40639i) q^{33} +(-2.91160 + 1.27069i) q^{34} +(0.635622 + 0.366976i) q^{35} +(1.94995 + 0.444618i) q^{36} +(-4.68370 - 3.88111i) q^{37} +(2.63120 + 0.296175i) q^{38} +(-0.624014 + 1.08082i) q^{39} +(-1.65986 - 8.75693i) q^{40} +(3.60687 + 6.24728i) q^{41} +(0.131754 + 0.301893i) q^{42} -0.945339i q^{43} +(9.72337 - 3.00158i) q^{44} -3.15117i q^{45} +(7.54835 + 5.56967i) q^{46} +7.36826 q^{47} +(1.73397 - 3.60463i) q^{48} +(3.47288 - 6.01520i) q^{49} +(-6.38986 + 2.78870i) q^{50} +2.24634i q^{51} +(1.83011 + 1.69735i) q^{52} +(1.10258 + 0.636576i) q^{53} +(0.839665 - 1.13796i) q^{54} +(-8.01667 - 13.8853i) q^{55} +(0.647257 - 0.122686i) q^{56} +(0.936144 - 1.62145i) q^{57} +(-1.99328 + 2.70141i) q^{58} +(-1.74163 - 1.00553i) q^{59} +(-6.14463 - 1.40107i) q^{60} +(0.425169 - 0.245471i) q^{61} +(-2.03311 - 1.50016i) q^{62} +0.232914 q^{63} +(-6.25790 - 4.98384i) q^{64} +(1.96637 - 3.40586i) q^{65} +(0.804875 - 7.15045i) q^{66} +(-12.4948 + 7.21388i) q^{67} +(4.38025 + 0.998762i) q^{68} +(5.74452 - 3.31660i) q^{69} +(-0.415178 - 0.951315i) q^{70} +(-1.83051 - 3.17053i) q^{71} +(-1.84565 - 2.14327i) q^{72} -4.45771 q^{73} +(2.07105 + 8.34930i) q^{74} +4.92987i q^{75} +(-2.74552 - 2.54636i) q^{76} +(1.02631 - 0.592541i) q^{77} +(1.61763 - 0.705976i) q^{78} +(-8.78978 - 15.2244i) q^{79} +(-5.46403 + 11.3588i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(1.14113 - 10.1377i) q^{82} +(-7.69737 - 4.44408i) q^{83} +(0.103558 - 0.454172i) q^{84} -7.07859i q^{85} +(-0.793768 + 1.07576i) q^{86} +(1.18695 + 2.05586i) q^{87} +(-13.5852 - 4.74868i) q^{88} +(8.59719 - 14.8908i) q^{89} +(-2.64593 + 3.58592i) q^{90} +(0.251739 + 0.145342i) q^{91} +(-3.91310 - 12.6762i) q^{92} +(-1.54726 + 0.893309i) q^{93} +(-8.38481 - 6.18687i) q^{94} +(-2.94995 + 5.10946i) q^{95} +(-4.99987 + 2.64599i) q^{96} +4.14227 q^{97} +(-9.00276 + 3.92903i) q^{98} +(-4.40639 - 2.54403i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13796 − 0.839665i −0.804662 − 0.593733i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 0.589926 + 1.91102i 0.294963 + 0.955509i
\(5\) 2.72899 − 1.57558i 1.22044 − 0.704623i 0.255430 − 0.966827i \(-0.417783\pi\)
0.965012 + 0.262205i \(0.0844496\pi\)
\(6\) 1.40534 + 0.158189i 0.573727 + 0.0645804i
\(7\) 0.116457 + 0.201710i 0.0440167 + 0.0762392i 0.887194 − 0.461396i \(-0.152651\pi\)
−0.843178 + 0.537635i \(0.819318\pi\)
\(8\) 0.933300 − 2.67001i 0.329971 − 0.943991i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −4.42846 − 0.498480i −1.40040 − 0.157633i
\(11\) − 5.08806i − 1.53411i −0.641583 − 0.767054i \(-0.721722\pi\)
0.641583 − 0.767054i \(-0.278278\pi\)
\(12\) −1.46640 − 1.36003i −0.423313 − 0.392606i
\(13\) 1.08082 − 0.624014i 0.299766 − 0.173070i −0.342572 − 0.939492i \(-0.611298\pi\)
0.642338 + 0.766422i \(0.277965\pi\)
\(14\) 0.0368445 − 0.327324i 0.00984710 − 0.0874809i
\(15\) −1.57558 + 2.72899i −0.406814 + 0.704623i
\(16\) −3.30398 + 2.25472i −0.825994 + 0.563679i
\(17\) 1.12317 − 1.94539i 0.272408 − 0.471825i −0.697070 − 0.717003i \(-0.745513\pi\)
0.969478 + 0.245178i \(0.0788465\pi\)
\(18\) −1.29615 + 0.565674i −0.305506 + 0.133331i
\(19\) −1.62145 + 0.936144i −0.371986 + 0.214766i −0.674326 − 0.738434i \(-0.735566\pi\)
0.302340 + 0.953200i \(0.402232\pi\)
\(20\) 4.62087 + 4.28567i 1.03326 + 0.958306i
\(21\) −0.201710 − 0.116457i −0.0440167 − 0.0254131i
\(22\) −4.27226 + 5.79003i −0.910850 + 1.23444i
\(23\) −6.63320 −1.38312 −0.691559 − 0.722320i \(-0.743076\pi\)
−0.691559 + 0.722320i \(0.743076\pi\)
\(24\) 0.526744 + 2.77895i 0.107521 + 0.567250i
\(25\) 2.46493 − 4.26939i 0.492987 − 0.853878i
\(26\) −1.75390 − 0.197424i −0.343968 − 0.0387181i
\(27\) 1.00000i 0.192450i
\(28\) −0.316770 + 0.341546i −0.0598639 + 0.0645461i
\(29\) − 2.37390i − 0.440822i −0.975407 − 0.220411i \(-0.929260\pi\)
0.975407 − 0.220411i \(-0.0707399\pi\)
\(30\) 4.08440 − 1.78253i 0.745706 − 0.325445i
\(31\) 1.78662 0.320886 0.160443 − 0.987045i \(-0.448708\pi\)
0.160443 + 0.987045i \(0.448708\pi\)
\(32\) 5.65301 + 0.208444i 0.999321 + 0.0368481i
\(33\) 2.54403 + 4.40639i 0.442859 + 0.767054i
\(34\) −2.91160 + 1.27069i −0.499335 + 0.217922i
\(35\) 0.635622 + 0.366976i 0.107440 + 0.0620303i
\(36\) 1.94995 + 0.444618i 0.324992 + 0.0741030i
\(37\) −4.68370 − 3.88111i −0.769995 − 0.638050i
\(38\) 2.63120 + 0.296175i 0.426837 + 0.0480459i
\(39\) −0.624014 + 1.08082i −0.0999222 + 0.173070i
\(40\) −1.65986 − 8.75693i −0.262447 − 1.38459i
\(41\) 3.60687 + 6.24728i 0.563298 + 0.975661i 0.997206 + 0.0747037i \(0.0238011\pi\)
−0.433908 + 0.900957i \(0.642866\pi\)
\(42\) 0.131754 + 0.301893i 0.0203300 + 0.0465831i
\(43\) − 0.945339i − 0.144163i −0.997399 − 0.0720815i \(-0.977036\pi\)
0.997399 − 0.0720815i \(-0.0229642\pi\)
\(44\) 9.72337 − 3.00158i 1.46585 − 0.452505i
\(45\) − 3.15117i − 0.469749i
\(46\) 7.54835 + 5.56967i 1.11294 + 0.821203i
\(47\) 7.36826 1.07477 0.537385 − 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(48\) 1.73397 − 3.60463i 0.250277 − 0.520283i
\(49\) 3.47288 − 6.01520i 0.496125 − 0.859314i
\(50\) −6.38986 + 2.78870i −0.903663 + 0.394381i
\(51\) 2.24634i 0.314550i
\(52\) 1.83011 + 1.69735i 0.253790 + 0.235380i
\(53\) 1.10258 + 0.636576i 0.151451 + 0.0874405i 0.573811 − 0.818988i \(-0.305465\pi\)
−0.422359 + 0.906429i \(0.638798\pi\)
\(54\) 0.839665 − 1.13796i 0.114264 − 0.154857i
\(55\) −8.01667 − 13.8853i −1.08097 − 1.87229i
\(56\) 0.647257 − 0.122686i 0.0864933 − 0.0163946i
\(57\) 0.936144 − 1.62145i 0.123995 − 0.214766i
\(58\) −1.99328 + 2.70141i −0.261731 + 0.354713i
\(59\) −1.74163 − 1.00553i −0.226741 − 0.130909i 0.382327 − 0.924027i \(-0.375123\pi\)
−0.609068 + 0.793118i \(0.708456\pi\)
\(60\) −6.14463 − 1.40107i −0.793268 − 0.180877i
\(61\) 0.425169 − 0.245471i 0.0544373 − 0.0314294i −0.472534 − 0.881312i \(-0.656661\pi\)
0.526972 + 0.849883i \(0.323327\pi\)
\(62\) −2.03311 − 1.50016i −0.258205 − 0.190521i
\(63\) 0.232914 0.0293445
\(64\) −6.25790 − 4.98384i −0.782238 − 0.622980i
\(65\) 1.96637 − 3.40586i 0.243898 − 0.422445i
\(66\) 0.804875 − 7.15045i 0.0990732 − 0.880159i
\(67\) −12.4948 + 7.21388i −1.52648 + 0.881315i −0.526977 + 0.849880i \(0.676675\pi\)
−0.999506 + 0.0314355i \(0.989992\pi\)
\(68\) 4.38025 + 0.998762i 0.531183 + 0.121118i
\(69\) 5.74452 − 3.31660i 0.691559 − 0.399272i
\(70\) −0.415178 − 0.951315i −0.0496232 − 0.113704i
\(71\) −1.83051 − 3.17053i −0.217241 − 0.376273i 0.736722 − 0.676195i \(-0.236372\pi\)
−0.953964 + 0.299923i \(0.903039\pi\)
\(72\) −1.84565 − 2.14327i −0.217512 − 0.252586i
\(73\) −4.45771 −0.521735 −0.260868 − 0.965375i \(-0.584009\pi\)
−0.260868 + 0.965375i \(0.584009\pi\)
\(74\) 2.07105 + 8.34930i 0.240755 + 0.970586i
\(75\) 4.92987i 0.569252i
\(76\) −2.74552 − 2.54636i −0.314933 − 0.292088i
\(77\) 1.02631 − 0.592541i 0.116959 − 0.0675263i
\(78\) 1.61763 − 0.705976i 0.183161 − 0.0799361i
\(79\) −8.78978 − 15.2244i −0.988928 − 1.71287i −0.622979 − 0.782238i \(-0.714078\pi\)
−0.365949 − 0.930635i \(-0.619255\pi\)
\(80\) −5.46403 + 11.3588i −0.610897 + 1.26995i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.14113 − 10.1377i 0.126017 − 1.11953i
\(83\) −7.69737 − 4.44408i −0.844897 − 0.487801i 0.0140291 − 0.999902i \(-0.495534\pi\)
−0.858926 + 0.512100i \(0.828868\pi\)
\(84\) 0.103558 − 0.454172i 0.0112991 − 0.0495542i
\(85\) − 7.07859i − 0.767781i
\(86\) −0.793768 + 1.07576i −0.0855942 + 0.116002i
\(87\) 1.18695 + 2.05586i 0.127254 + 0.220411i
\(88\) −13.5852 − 4.74868i −1.44818 − 0.506211i
\(89\) 8.59719 − 14.8908i 0.911300 − 1.57842i 0.0990699 − 0.995080i \(-0.468413\pi\)
0.812230 − 0.583337i \(-0.198253\pi\)
\(90\) −2.64593 + 3.58592i −0.278905 + 0.377989i
\(91\) 0.251739 + 0.145342i 0.0263895 + 0.0152360i
\(92\) −3.91310 − 12.6762i −0.407969 − 1.32158i
\(93\) −1.54726 + 0.893309i −0.160443 + 0.0926319i
\(94\) −8.38481 − 6.18687i −0.864827 − 0.638126i
\(95\) −2.94995 + 5.10946i −0.302658 + 0.524219i
\(96\) −4.99987 + 2.64599i −0.510298 + 0.270055i
\(97\) 4.14227 0.420584 0.210292 − 0.977639i \(-0.432559\pi\)
0.210292 + 0.977639i \(0.432559\pi\)
\(98\) −9.00276 + 3.92903i −0.909416 + 0.396892i
\(99\) −4.40639 − 2.54403i −0.442859 − 0.255685i
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.565.16 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.66 yes 152
37.26 even 3 inner 888.2.bh.a.877.66 yes 152
296.285 even 6 inner 888.2.bh.a.877.16 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.16 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.66 yes 152 8.5 even 2 inner
888.2.bh.a.877.16 yes 152 296.285 even 6 inner
888.2.bh.a.877.66 yes 152 37.26 even 3 inner