Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,16] 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 325.26
Character \(\chi\) \(=\) 864.325
Dual form 864.2.v.b.109.26

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.12159 - 0.861417i) q^{2} +(0.515922 - 1.93231i) q^{4} +(-1.04543 - 0.433032i) q^{5} +(2.00753 + 2.00753i) q^{7} +(-1.08587 - 2.61168i) q^{8} +(-1.54557 + 0.414869i) q^{10} +(0.338917 - 0.818217i) q^{11} +(1.78349 - 0.738744i) q^{13} +(3.98095 + 0.522303i) q^{14} +(-3.46765 - 1.99384i) q^{16} -5.63773i q^{17} +(4.50203 - 1.86480i) q^{19} +(-1.37612 + 1.79669i) q^{20} +(-0.324701 - 1.20965i) q^{22} +(3.31003 - 3.31003i) q^{23} +(-2.63012 - 2.63012i) q^{25} +(1.36397 - 2.36489i) q^{26} +(4.91490 - 2.84344i) q^{28} +(0.803214 + 1.93913i) q^{29} -2.71880 q^{31} +(-5.60681 + 0.750818i) q^{32} +(-4.85643 - 6.32321i) q^{34} +(-1.22941 - 2.96807i) q^{35} +(5.51487 + 2.28433i) q^{37} +(3.44305 - 5.96967i) q^{38} +(0.00426366 + 3.20056i) q^{40} +(-7.12048 + 7.12048i) q^{41} +(-1.98667 + 4.79625i) q^{43} +(-1.40619 - 1.07703i) q^{44} +(0.861176 - 6.56380i) q^{46} -4.34625i q^{47} +1.06036i q^{49} +(-5.21554 - 0.684284i) q^{50} +(-0.507343 - 3.82738i) q^{52} +(0.634970 - 1.53295i) q^{53} +(-0.708629 + 0.708629i) q^{55} +(3.06311 - 7.42295i) q^{56} +(2.57128 + 1.48300i) q^{58} +(-3.37555 - 1.39820i) q^{59} +(4.15501 + 10.0311i) q^{61} +(-3.04937 + 2.34202i) q^{62} +(-5.64176 + 5.67191i) q^{64} -2.18442 q^{65} +(4.91364 + 11.8626i) q^{67} +(-10.8938 - 2.90863i) q^{68} +(-3.93564 - 2.26991i) q^{70} +(-5.53047 - 5.53047i) q^{71} +(1.43402 - 1.43402i) q^{73} +(8.15317 - 2.18852i) q^{74} +(-1.28068 - 9.66142i) q^{76} +(2.32298 - 0.962211i) q^{77} -3.15375i q^{79} +(2.76179 + 3.58603i) q^{80} +(-1.85255 + 14.1200i) q^{82} +(14.6864 - 6.08331i) q^{83} +(-2.44132 + 5.89387i) q^{85} +(1.90334 + 7.09077i) q^{86} +(-2.50494 + 0.00333699i) q^{88} +(8.54708 + 8.54708i) q^{89} +(5.06346 + 2.09735i) q^{91} +(-4.68828 - 8.10372i) q^{92} +(-3.74393 - 4.87471i) q^{94} -5.51409 q^{95} -13.8914 q^{97} +(0.913414 + 1.18929i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.12159 − 0.861417i 0.793083 − 0.609114i
\(3\) 0 0
\(4\) 0.515922 − 1.93231i 0.257961 − 0.966155i
\(5\) −1.04543 − 0.433032i −0.467532 − 0.193658i 0.136465 − 0.990645i \(-0.456426\pi\)
−0.603997 + 0.796987i \(0.706426\pi\)
\(6\) 0 0
\(7\) 2.00753 + 2.00753i 0.758775 + 0.758775i 0.976100 − 0.217324i \(-0.0697328\pi\)
−0.217324 + 0.976100i \(0.569733\pi\)
\(8\) −1.08587 − 2.61168i −0.383914 − 0.923369i
\(9\) 0 0
\(10\) −1.54557 + 0.414869i −0.488751 + 0.131193i
\(11\) 0.338917 − 0.818217i 0.102187 − 0.246702i −0.864514 − 0.502608i \(-0.832374\pi\)
0.966702 + 0.255906i \(0.0823739\pi\)
\(12\) 0 0
\(13\) 1.78349 − 0.738744i 0.494650 − 0.204891i −0.121391 − 0.992605i \(-0.538735\pi\)
0.616041 + 0.787714i \(0.288735\pi\)
\(14\) 3.98095 + 0.522303i 1.06395 + 0.139591i
\(15\) 0 0
\(16\) −3.46765 − 1.99384i −0.866912 − 0.498461i
\(17\) − 5.63773i − 1.36735i −0.729787 − 0.683675i \(-0.760381\pi\)
0.729787 − 0.683675i \(-0.239619\pi\)
\(18\) 0 0
\(19\) 4.50203 − 1.86480i 1.03284 − 0.427815i 0.199101 − 0.979979i \(-0.436198\pi\)
0.833736 + 0.552164i \(0.186198\pi\)
\(20\) −1.37612 + 1.79669i −0.307709 + 0.401752i
\(21\) 0 0
\(22\) −0.324701 − 1.20965i −0.0692265 − 0.257899i
\(23\) 3.31003 − 3.31003i 0.690188 − 0.690188i −0.272085 − 0.962273i \(-0.587713\pi\)
0.962273 + 0.272085i \(0.0877132\pi\)
\(24\) 0 0
\(25\) −2.63012 − 2.63012i −0.526024 − 0.526024i
\(26\) 1.36397 − 2.36489i 0.267497 − 0.463794i
\(27\) 0 0
\(28\) 4.91490 − 2.84344i 0.928829 − 0.537360i
\(29\) 0.803214 + 1.93913i 0.149153 + 0.360088i 0.980743 − 0.195303i \(-0.0625689\pi\)
−0.831590 + 0.555390i \(0.812569\pi\)
\(30\) 0 0
\(31\) −2.71880 −0.488310 −0.244155 − 0.969736i \(-0.578511\pi\)
−0.244155 + 0.969736i \(0.578511\pi\)
\(32\) −5.60681 + 0.750818i −0.991153 + 0.132727i
\(33\) 0 0
\(34\) −4.85643 − 6.32321i −0.832872 − 1.08442i
\(35\) −1.22941 − 2.96807i −0.207809 − 0.501695i
\(36\) 0 0
\(37\) 5.51487 + 2.28433i 0.906638 + 0.375542i 0.786769 − 0.617248i \(-0.211752\pi\)
0.119870 + 0.992790i \(0.461752\pi\)
\(38\) 3.44305 − 5.96967i 0.558537 − 0.968408i
\(39\) 0 0
\(40\) 0.00426366 + 3.20056i 0.000674144 + 0.506052i
\(41\) −7.12048 + 7.12048i −1.11203 + 1.11203i −0.119157 + 0.992875i \(0.538019\pi\)
−0.992875 + 0.119157i \(0.961981\pi\)
\(42\) 0 0
\(43\) −1.98667 + 4.79625i −0.302965 + 0.731421i 0.696933 + 0.717136i \(0.254547\pi\)
−0.999898 + 0.0142852i \(0.995453\pi\)
\(44\) −1.40619 − 1.07703i −0.211992 − 0.162368i
\(45\) 0 0
\(46\) 0.861176 − 6.56380i 0.126973 − 0.967780i
\(47\) − 4.34625i − 0.633966i −0.948431 − 0.316983i \(-0.897330\pi\)
0.948431 − 0.316983i \(-0.102670\pi\)
\(48\) 0 0
\(49\) 1.06036i 0.151480i
\(50\) −5.21554 − 0.684284i −0.737589 − 0.0967723i
\(51\) 0 0
\(52\) −0.507343 − 3.82738i −0.0703558 − 0.530763i
\(53\) 0.634970 − 1.53295i 0.0872199 − 0.210567i −0.874251 − 0.485474i \(-0.838647\pi\)
0.961471 + 0.274907i \(0.0886470\pi\)
\(54\) 0 0
\(55\) −0.708629 + 0.708629i −0.0955515 + 0.0955515i
\(56\) 3.06311 − 7.42295i 0.409325 − 0.991934i
\(57\) 0 0
\(58\) 2.57128 + 1.48300i 0.337625 + 0.194728i
\(59\) −3.37555 − 1.39820i −0.439459 − 0.182030i 0.151973 − 0.988385i \(-0.451437\pi\)
−0.591432 + 0.806355i \(0.701437\pi\)
\(60\) 0 0
\(61\) 4.15501 + 10.0311i 0.531994 + 1.28435i 0.930201 + 0.367052i \(0.119633\pi\)
−0.398206 + 0.917296i \(0.630367\pi\)
\(62\) −3.04937 + 2.34202i −0.387271 + 0.297437i
\(63\) 0 0
\(64\) −5.64176 + 5.67191i −0.705220 + 0.708988i
\(65\) −2.18442 −0.270943
\(66\) 0 0
\(67\) 4.91364 + 11.8626i 0.600296 + 1.44924i 0.873277 + 0.487224i \(0.161991\pi\)
−0.272981 + 0.962020i \(0.588009\pi\)
\(68\) −10.8938 − 2.90863i −1.32107 − 0.352723i
\(69\) 0 0
\(70\) −3.93564 − 2.26991i −0.470399 − 0.271306i
\(71\) −5.53047 − 5.53047i −0.656347 − 0.656347i 0.298167 − 0.954514i \(-0.403625\pi\)
−0.954514 + 0.298167i \(0.903625\pi\)
\(72\) 0 0
\(73\) 1.43402 − 1.43402i 0.167839 − 0.167839i −0.618190 − 0.786029i \(-0.712134\pi\)
0.786029 + 0.618190i \(0.212134\pi\)
\(74\) 8.15317 − 2.18852i 0.947787 − 0.254410i
\(75\) 0 0
\(76\) −1.28068 − 9.66142i −0.146904 − 1.10824i
\(77\) 2.32298 − 0.962211i 0.264728 − 0.109654i
\(78\) 0 0
\(79\) − 3.15375i − 0.354824i −0.984137 − 0.177412i \(-0.943227\pi\)
0.984137 − 0.177412i \(-0.0567725\pi\)
\(80\) 2.76179 + 3.58603i 0.308778 + 0.400931i
\(81\) 0 0
\(82\) −1.85255 + 14.1200i −0.204580 + 1.55929i
\(83\) 14.6864 − 6.08331i 1.61204 − 0.667730i 0.618990 − 0.785399i \(-0.287542\pi\)
0.993053 + 0.117669i \(0.0375422\pi\)
\(84\) 0 0
\(85\) −2.44132 + 5.89387i −0.264798 + 0.639280i
\(86\) 1.90334 + 7.09077i 0.205243 + 0.764617i
\(87\) 0 0
\(88\) −2.50494 + 0.00333699i −0.267028 + 0.000355724i
\(89\) 8.54708 + 8.54708i 0.905988 + 0.905988i 0.995946 − 0.0899572i \(-0.0286730\pi\)
−0.0899572 + 0.995946i \(0.528673\pi\)
\(90\) 0 0
\(91\) 5.06346 + 2.09735i 0.530794 + 0.219862i
\(92\) −4.68828 − 8.10372i −0.488787 − 0.844871i
\(93\) 0 0
\(94\) −3.74393 − 4.87471i −0.386157 − 0.502788i
\(95\) −5.51409 −0.565734
\(96\) 0 0
\(97\) −13.8914 −1.41046 −0.705231 − 0.708977i \(-0.749157\pi\)
−0.705231 + 0.708977i \(0.749157\pi\)
\(98\) 0.913414 + 1.18929i 0.0922688 + 0.120137i
\(99\) 0 0
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 864.2.v.b.325.26 yes 128
3.2 odd 2 inner 864.2.v.b.325.7 yes 128
32.13 even 8 inner 864.2.v.b.109.26 yes 128
96.77 odd 8 inner 864.2.v.b.109.7 ✓ 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.7 ✓ 128 96.77 odd 8 inner
864.2.v.b.109.26 yes 128 32.13 even 8 inner
864.2.v.b.325.7 yes 128 3.2 odd 2 inner
864.2.v.b.325.26 yes 128 1.1 even 1 trivial