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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 325.7
Character \(\chi\) \(=\) 864.325
Dual form 864.2.v.b.109.7

$q$-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 0.603997 − 0.796987i \(-0.293574\pi\)
−0.136465 + 0.990645i \(0.543574\pi\)
\(6\) 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 −0.966702 − 0.255906i \(-0.917626\pi\)
0.864514 + 0.502608i \(0.167626\pi\)
\(12\) 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 \(0.239619\pi\)
−0.729787 + 0.683675i \(0.760381\pi\)
\(18\) 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 −0.962273 − 0.272085i \(-0.912287\pi\)
0.272085 + 0.962273i \(0.412287\pi\)
\(24\) 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 0.831590 − 0.555390i \(-0.187431\pi\)
−0.980743 + 0.195303i \(0.937431\pi\)
\(30\) 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 \(-0.461981\pi\)
0.992875 − 0.119157i \(-0.0380192\pi\)
\(42\) 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 \(0.102670\pi\)
−0.948431 + 0.316983i \(0.897330\pi\)
\(48\) 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 −0.961471 − 0.274907i \(-0.911353\pi\)
0.874251 + 0.485474i \(0.161353\pi\)
\(54\) 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 0.591432 − 0.806355i \(-0.298563\pi\)
−0.151973 + 0.988385i \(0.548563\pi\)
\(60\) 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 0.954514 − 0.298167i \(-0.0963753\pi\)
−0.298167 + 0.954514i \(0.596375\pi\)
\(72\) 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 −0.993053 − 0.117669i \(-0.962458\pi\)
−0.618990 + 0.785399i \(0.712458\pi\)
\(84\) 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 0.0899572 − 0.995946i \(-0.471327\pi\)
−0.995946 + 0.0899572i \(0.971327\pi\)
\(90\) 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.325.7 yes 128
3.2 odd 2 inner 864.2.v.b.325.26 yes 128
32.13 even 8 inner 864.2.v.b.109.7 ✓ 128
96.77 odd 8 inner 864.2.v.b.109.26 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.7 ✓ 128 32.13 even 8 inner
864.2.v.b.109.26 yes 128 96.77 odd 8 inner
864.2.v.b.325.7 yes 128 1.1 even 1 trivial
864.2.v.b.325.26 yes 128 3.2 odd 2 inner