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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,-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: \(6.89907473464\)
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 325.19
Character \(\chi\) \(=\) 864.325
Dual form 864.2.v.a.109.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.139754 + 1.40729i) q^{2} +(-1.96094 + 0.393350i) q^{4} +(0.345772 + 0.143223i) q^{5} +(0.299071 + 0.299071i) q^{7} +(-0.827608 - 2.70464i) q^{8} +(-0.153234 + 0.506617i) q^{10} +(-1.61654 + 3.90267i) q^{11} +(-1.58168 + 0.655155i) q^{13} +(-0.379084 + 0.462677i) q^{14} +(3.69055 - 1.54267i) q^{16} +1.95201i q^{17} +(0.0281853 - 0.0116747i) q^{19} +(-0.734373 - 0.144843i) q^{20} +(-5.71812 - 1.72953i) q^{22} +(-2.80070 + 2.80070i) q^{23} +(-3.43649 - 3.43649i) q^{25} +(-1.14304 - 2.13433i) q^{26} +(-0.704100 - 0.468820i) q^{28} +(1.57266 + 3.79673i) q^{29} -7.50880 q^{31} +(2.68676 + 4.97809i) q^{32} +(-2.74704 + 0.272801i) q^{34} +(0.0605764 + 0.146244i) q^{35} +(-2.16498 - 0.896764i) q^{37} +(0.0203688 + 0.0380333i) q^{38} +(0.101204 - 1.05372i) q^{40} +(-6.80246 + 6.80246i) q^{41} +(-2.40817 + 5.81383i) q^{43} +(1.63482 - 8.28877i) q^{44} +(-4.33282 - 3.55000i) q^{46} +2.05564i q^{47} -6.82111i q^{49} +(4.35588 - 5.31640i) q^{50} +(2.84388 - 1.90687i) q^{52} +(1.75463 - 4.23606i) q^{53} +(-1.11791 + 1.11791i) q^{55} +(0.561366 - 1.05639i) q^{56} +(-5.12331 + 2.74379i) q^{58} +(11.3824 + 4.71475i) q^{59} +(-2.70584 - 6.53249i) q^{61} +(-1.04939 - 10.5671i) q^{62} +(-6.63013 + 4.47676i) q^{64} -0.640735 q^{65} +(0.0765846 + 0.184891i) q^{67} +(-0.767822 - 3.82776i) q^{68} +(-0.197343 + 0.105687i) q^{70} +(-4.37932 - 4.37932i) q^{71} +(2.81966 - 2.81966i) q^{73} +(0.959443 - 3.17208i) q^{74} +(-0.0506774 + 0.0339801i) q^{76} +(-1.65064 + 0.683717i) q^{77} +6.03128i q^{79} +(1.49703 - 0.00483868i) q^{80} +(-10.5237 - 8.62238i) q^{82} +(-2.55943 + 1.06015i) q^{83} +(-0.279573 + 0.674948i) q^{85} +(-8.51831 - 2.57649i) q^{86} +(11.8932 + 1.14227i) q^{88} +(-2.16284 - 2.16284i) q^{89} +(-0.668975 - 0.277098i) q^{91} +(4.39035 - 6.59366i) q^{92} +(-2.89288 + 0.287284i) q^{94} +0.0114178 q^{95} +1.88742 q^{97} +(9.59929 - 0.953280i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.139754 + 1.40729i 0.0988213 + 0.995105i
\(3\) 0 0
\(4\) −1.96094 + 0.393350i −0.980469 + 0.196675i
\(5\) 0.345772 + 0.143223i 0.154634 + 0.0640514i 0.458658 − 0.888613i \(-0.348330\pi\)
−0.304024 + 0.952664i \(0.598330\pi\)
\(6\) 0 0
\(7\) 0.299071 + 0.299071i 0.113038 + 0.113038i 0.761364 − 0.648325i \(-0.224530\pi\)
−0.648325 + 0.761364i \(0.724530\pi\)
\(8\) −0.827608 − 2.70464i −0.292604 − 0.956234i
\(9\) 0 0
\(10\) −0.153234 + 0.506617i −0.0484568 + 0.160207i
\(11\) −1.61654 + 3.90267i −0.487405 + 1.17670i 0.468616 + 0.883402i \(0.344753\pi\)
−0.956021 + 0.293298i \(0.905247\pi\)
\(12\) 0 0
\(13\) −1.58168 + 0.655155i −0.438680 + 0.181707i −0.591082 − 0.806611i \(-0.701299\pi\)
0.152402 + 0.988319i \(0.451299\pi\)
\(14\) −0.379084 + 0.462677i −0.101314 + 0.123656i
\(15\) 0 0
\(16\) 3.69055 − 1.54267i 0.922638 − 0.385668i
\(17\) 1.95201i 0.473431i 0.971579 + 0.236715i \(0.0760709\pi\)
−0.971579 + 0.236715i \(0.923929\pi\)
\(18\) 0 0
\(19\) 0.0281853 − 0.0116747i 0.00646615 − 0.00267837i −0.379448 − 0.925213i \(-0.623886\pi\)
0.385914 + 0.922535i \(0.373886\pi\)
\(20\) −0.734373 − 0.144843i −0.164211 − 0.0323878i
\(21\) 0 0
\(22\) −5.71812 − 1.72953i −1.21911 − 0.368737i
\(23\) −2.80070 + 2.80070i −0.583987 + 0.583987i −0.935996 − 0.352009i \(-0.885499\pi\)
0.352009 + 0.935996i \(0.385499\pi\)
\(24\) 0 0
\(25\) −3.43649 − 3.43649i −0.687298 − 0.687298i
\(26\) −1.14304 − 2.13433i −0.224169 − 0.418577i
\(27\) 0 0
\(28\) −0.704100 − 0.468820i −0.133062 − 0.0885987i
\(29\) 1.57266 + 3.79673i 0.292035 + 0.705034i 0.999999 − 0.00121375i \(-0.000386349\pi\)
−0.707965 + 0.706248i \(0.750386\pi\)
\(30\) 0 0
\(31\) −7.50880 −1.34862 −0.674310 − 0.738449i \(-0.735559\pi\)
−0.674310 + 0.738449i \(0.735559\pi\)
\(32\) 2.68676 + 4.97809i 0.474956 + 0.880010i
\(33\) 0 0
\(34\) −2.74704 + 0.272801i −0.471113 + 0.0467850i
\(35\) 0.0605764 + 0.146244i 0.0102393 + 0.0247198i
\(36\) 0 0
\(37\) −2.16498 − 0.896764i −0.355921 − 0.147427i 0.197557 − 0.980291i \(-0.436699\pi\)
−0.553477 + 0.832864i \(0.686699\pi\)
\(38\) 0.0203688 + 0.0380333i 0.00330425 + 0.00616982i
\(39\) 0 0
\(40\) 0.101204 − 1.05372i 0.0160017 − 0.166608i
\(41\) −6.80246 + 6.80246i −1.06237 + 1.06237i −0.0644453 + 0.997921i \(0.520528\pi\)
−0.997921 + 0.0644453i \(0.979472\pi\)
\(42\) 0 0
\(43\) −2.40817 + 5.81383i −0.367242 + 0.886601i 0.626958 + 0.779053i \(0.284300\pi\)
−0.994200 + 0.107548i \(0.965700\pi\)
\(44\) 1.63482 − 8.28877i 0.246458 − 1.24958i
\(45\) 0 0
\(46\) −4.33282 − 3.55000i −0.638839 − 0.523418i
\(47\) 2.05564i 0.299846i 0.988698 + 0.149923i \(0.0479025\pi\)
−0.988698 + 0.149923i \(0.952097\pi\)
\(48\) 0 0
\(49\) − 6.82111i − 0.974445i
\(50\) 4.35588 − 5.31640i 0.616014 − 0.751853i
\(51\) 0 0
\(52\) 2.84388 − 1.90687i 0.394375 − 0.264436i
\(53\) 1.75463 − 4.23606i 0.241017 − 0.581867i −0.756367 − 0.654147i \(-0.773028\pi\)
0.997384 + 0.0722803i \(0.0230276\pi\)
\(54\) 0 0
\(55\) −1.11791 + 1.11791i −0.150739 + 0.150739i
\(56\) 0.561366 − 1.05639i 0.0750157 − 0.141167i
\(57\) 0 0
\(58\) −5.12331 + 2.74379i −0.672724 + 0.360278i
\(59\) 11.3824 + 4.71475i 1.48187 + 0.613809i 0.969529 − 0.244977i \(-0.0787803\pi\)
0.512336 + 0.858785i \(0.328780\pi\)
\(60\) 0 0
\(61\) −2.70584 − 6.53249i −0.346448 − 0.836399i −0.997034 − 0.0769664i \(-0.975477\pi\)
0.650586 − 0.759433i \(-0.274523\pi\)
\(62\) −1.04939 − 10.5671i −0.133272 − 1.34202i
\(63\) 0 0
\(64\) −6.63013 + 4.47676i −0.828766 + 0.559595i
\(65\) −0.640735 −0.0794734
\(66\) 0 0
\(67\) 0.0765846 + 0.184891i 0.00935629 + 0.0225881i 0.928489 − 0.371361i \(-0.121109\pi\)
−0.919132 + 0.393949i \(0.871109\pi\)
\(68\) −0.767822 − 3.82776i −0.0931121 − 0.464184i
\(69\) 0 0
\(70\) −0.197343 + 0.105687i −0.0235870 + 0.0126320i
\(71\) −4.37932 − 4.37932i −0.519730 − 0.519730i 0.397760 − 0.917490i \(-0.369788\pi\)
−0.917490 + 0.397760i \(0.869788\pi\)
\(72\) 0 0
\(73\) 2.81966 − 2.81966i 0.330016 − 0.330016i −0.522576 − 0.852593i \(-0.675029\pi\)
0.852593 + 0.522576i \(0.175029\pi\)
\(74\) 0.959443 − 3.17208i 0.111533 − 0.368747i
\(75\) 0 0
\(76\) −0.0506774 + 0.0339801i −0.00581309 + 0.00389779i
\(77\) −1.65064 + 0.683717i −0.188108 + 0.0779168i
\(78\) 0 0
\(79\) 6.03128i 0.678572i 0.940683 + 0.339286i \(0.110185\pi\)
−0.940683 + 0.339286i \(0.889815\pi\)
\(80\) 1.49703 − 0.00483868i 0.167374 − 0.000540981i
\(81\) 0 0
\(82\) −10.5237 − 8.62238i −1.16215 − 0.952182i
\(83\) −2.55943 + 1.06015i −0.280934 + 0.116367i −0.518701 − 0.854955i \(-0.673584\pi\)
0.237768 + 0.971322i \(0.423584\pi\)
\(84\) 0 0
\(85\) −0.279573 + 0.674948i −0.0303239 + 0.0732084i
\(86\) −8.51831 − 2.57649i −0.918553 − 0.277830i
\(87\) 0 0
\(88\) 11.8932 + 1.14227i 1.26782 + 0.121767i
\(89\) −2.16284 − 2.16284i −0.229261 − 0.229261i 0.583123 − 0.812384i \(-0.301831\pi\)
−0.812384 + 0.583123i \(0.801831\pi\)
\(90\) 0 0
\(91\) −0.668975 − 0.277098i −0.0701276 − 0.0290478i
\(92\) 4.39035 − 6.59366i 0.457725 − 0.687437i
\(93\) 0 0
\(94\) −2.89288 + 0.287284i −0.298378 + 0.0296311i
\(95\) 0.0114178 0.00117144
\(96\) 0 0
\(97\) 1.88742 0.191639 0.0958194 − 0.995399i \(-0.469453\pi\)
0.0958194 + 0.995399i \(0.469453\pi\)
\(98\) 9.59929 − 0.953280i 0.969675 − 0.0962958i
\(99\) 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.a.325.19 yes 128
3.2 odd 2 inner 864.2.v.a.325.14 yes 128
32.13 even 8 inner 864.2.v.a.109.19 yes 128
96.77 odd 8 inner 864.2.v.a.109.14 ✓ 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.14 ✓ 128 96.77 odd 8 inner
864.2.v.a.109.19 yes 128 32.13 even 8 inner
864.2.v.a.325.14 yes 128 3.2 odd 2 inner
864.2.v.a.325.19 yes 128 1.1 even 1 trivial