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,-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 109.8
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.01339 - 0.986432i) q^{2} +(0.0539048 + 1.99927i) q^{4} +(-2.55155 + 1.05689i) q^{5} +(1.06497 - 1.06497i) q^{7} +(1.91752 - 2.07921i) q^{8} +(3.62825 + 1.44589i) q^{10} +(-0.641664 - 1.54911i) q^{11} +(0.200618 + 0.0830988i) q^{13} +(-2.12975 + 0.0287062i) q^{14} +(-3.99419 + 0.215541i) q^{16} +2.47031i q^{17} +(4.02517 + 1.66728i) q^{19} +(-2.25054 - 5.04427i) q^{20} +(-0.877841 + 2.20281i) q^{22} +(-2.90736 - 2.90736i) q^{23} +(1.85785 - 1.85785i) q^{25} +(-0.121332 - 0.282107i) q^{26} +(2.18658 + 2.07177i) q^{28} +(-1.64618 + 3.97423i) q^{29} -0.400720 q^{31} +(4.26027 + 3.72157i) q^{32} +(2.43680 - 2.50338i) q^{34} +(-1.59177 + 3.84288i) q^{35} +(-4.73288 + 1.96042i) q^{37} +(-2.43439 - 5.66015i) q^{38} +(-2.69516 + 7.33180i) q^{40} +(2.07457 + 2.07457i) q^{41} +(3.45626 + 8.34415i) q^{43} +(3.06251 - 1.36637i) q^{44} +(0.0783673 + 5.81418i) q^{46} +6.94628i q^{47} +4.73166i q^{49} +(-3.71536 + 0.0500780i) q^{50} +(-0.155323 + 0.405570i) q^{52} +(4.57054 + 11.0343i) q^{53} +(3.27447 + 3.27447i) q^{55} +(-0.172195 - 4.25641i) q^{56} +(5.58852 - 2.40358i) q^{58} +(-5.70502 + 2.36310i) q^{59} +(-0.321300 + 0.775687i) q^{61} +(0.406085 + 0.395283i) q^{62} +(-0.646231 - 7.97386i) q^{64} -0.599712 q^{65} +(-5.91697 + 14.2848i) q^{67} +(-4.93883 + 0.133162i) q^{68} +(5.40383 - 2.32415i) q^{70} +(7.28346 - 7.28346i) q^{71} +(-7.84498 - 7.84498i) q^{73} +(6.73006 + 2.68199i) q^{74} +(-3.11637 + 8.13729i) q^{76} +(-2.33312 - 0.966410i) q^{77} -1.11506i q^{79} +(9.96356 - 4.77136i) q^{80} +(-0.0559198 - 4.14877i) q^{82} +(12.4078 + 5.13949i) q^{83} +(-2.61084 - 6.30312i) q^{85} +(4.72841 - 11.8652i) q^{86} +(-4.45134 - 1.63630i) q^{88} +(7.58120 - 7.58120i) q^{89} +(0.302151 - 0.125155i) q^{91} +(5.65588 - 5.96932i) q^{92} +(6.85203 - 7.03927i) q^{94} -12.0325 q^{95} +11.3315 q^{97} +(4.66746 - 4.79500i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.01339 − 0.986432i −0.716573 − 0.697513i
\(3\) 0 0
\(4\) 0.0539048 + 1.99927i 0.0269524 + 0.999637i
\(5\) −2.55155 + 1.05689i −1.14109 + 0.472653i −0.871535 − 0.490333i \(-0.836875\pi\)
−0.269551 + 0.962986i \(0.586875\pi\)
\(6\) 0 0
\(7\) 1.06497 − 1.06497i 0.402522 − 0.402522i −0.476599 − 0.879121i \(-0.658131\pi\)
0.879121 + 0.476599i \(0.158131\pi\)
\(8\) 1.91752 − 2.07921i 0.677946 − 0.735112i
\(9\) 0 0
\(10\) 3.62825 + 1.44589i 1.14735 + 0.457232i
\(11\) −0.641664 − 1.54911i −0.193469 − 0.467075i 0.797141 − 0.603793i \(-0.206345\pi\)
−0.990610 + 0.136718i \(0.956345\pi\)
\(12\) 0 0
\(13\) 0.200618 + 0.0830988i 0.0556415 + 0.0230474i 0.410331 − 0.911937i \(-0.365413\pi\)
−0.354689 + 0.934984i \(0.615413\pi\)
\(14\) −2.12975 + 0.0287062i −0.569201 + 0.00767205i
\(15\) 0 0
\(16\) −3.99419 + 0.215541i −0.998547 + 0.0538852i
\(17\) 2.47031i 0.599139i 0.954074 + 0.299569i \(0.0968430\pi\)
−0.954074 + 0.299569i \(0.903157\pi\)
\(18\) 0 0
\(19\) 4.02517 + 1.66728i 0.923437 + 0.382500i 0.793185 − 0.608981i \(-0.208421\pi\)
0.130252 + 0.991481i \(0.458421\pi\)
\(20\) −2.25054 − 5.04427i −0.503237 − 1.12793i
\(21\) 0 0
\(22\) −0.877841 + 2.20281i −0.187156 + 0.469640i
\(23\) −2.90736 − 2.90736i −0.606225 − 0.606225i 0.335732 − 0.941958i \(-0.391016\pi\)
−0.941958 + 0.335732i \(0.891016\pi\)
\(24\) 0 0
\(25\) 1.85785 − 1.85785i 0.371570 − 0.371570i
\(26\) −0.121332 − 0.282107i −0.0237953 − 0.0553258i
\(27\) 0 0
\(28\) 2.18658 + 2.07177i 0.413225 + 0.391527i
\(29\) −1.64618 + 3.97423i −0.305688 + 0.737995i 0.694147 + 0.719833i \(0.255782\pi\)
−0.999835 + 0.0181624i \(0.994218\pi\)
\(30\) 0 0
\(31\) −0.400720 −0.0719715 −0.0359857 − 0.999352i \(-0.511457\pi\)
−0.0359857 + 0.999352i \(0.511457\pi\)
\(32\) 4.26027 + 3.72157i 0.753117 + 0.657887i
\(33\) 0 0
\(34\) 2.43680 − 2.50338i 0.417907 − 0.429326i
\(35\) −1.59177 + 3.84288i −0.269059 + 0.649566i
\(36\) 0 0
\(37\) −4.73288 + 1.96042i −0.778080 + 0.322291i −0.736140 − 0.676829i \(-0.763354\pi\)
−0.0419397 + 0.999120i \(0.513354\pi\)
\(38\) −2.43439 − 5.66015i −0.394911 − 0.918198i
\(39\) 0 0
\(40\) −2.69516 + 7.33180i −0.426142 + 1.15926i
\(41\) 2.07457 + 2.07457i 0.323994 + 0.323994i 0.850297 − 0.526303i \(-0.176422\pi\)
−0.526303 + 0.850297i \(0.676422\pi\)
\(42\) 0 0
\(43\) 3.45626 + 8.34415i 0.527075 + 1.27247i 0.933431 + 0.358757i \(0.116799\pi\)
−0.406356 + 0.913715i \(0.633201\pi\)
\(44\) 3.06251 − 1.36637i 0.461691 − 0.205987i
\(45\) 0 0
\(46\) 0.0783673 + 5.81418i 0.0115546 + 0.857254i
\(47\) 6.94628i 1.01322i 0.862176 + 0.506610i \(0.169101\pi\)
−0.862176 + 0.506610i \(0.830899\pi\)
\(48\) 0 0
\(49\) 4.73166i 0.675952i
\(50\) −3.71536 + 0.0500780i −0.525431 + 0.00708210i
\(51\) 0 0
\(52\) −0.155323 + 0.405570i −0.0215394 + 0.0562424i
\(53\) 4.57054 + 11.0343i 0.627813 + 1.51567i 0.842335 + 0.538955i \(0.181181\pi\)
−0.214522 + 0.976719i \(0.568819\pi\)
\(54\) 0 0
\(55\) 3.27447 + 3.27447i 0.441529 + 0.441529i
\(56\) −0.172195 − 4.25641i −0.0230106 − 0.568787i
\(57\) 0 0
\(58\) 5.58852 − 2.40358i 0.733808 − 0.315606i
\(59\) −5.70502 + 2.36310i −0.742730 + 0.307649i −0.721771 − 0.692131i \(-0.756672\pi\)
−0.0209586 + 0.999780i \(0.506672\pi\)
\(60\) 0 0
\(61\) −0.321300 + 0.775687i −0.0411383 + 0.0993166i −0.943113 − 0.332472i \(-0.892117\pi\)
0.901975 + 0.431789i \(0.142117\pi\)
\(62\) 0.406085 + 0.395283i 0.0515728 + 0.0502010i
\(63\) 0 0
\(64\) −0.646231 − 7.97386i −0.0807788 − 0.996732i
\(65\) −0.599712 −0.0743852
\(66\) 0 0
\(67\) −5.91697 + 14.2848i −0.722873 + 1.74517i −0.0578745 + 0.998324i \(0.518432\pi\)
−0.664998 + 0.746845i \(0.731568\pi\)
\(68\) −4.93883 + 0.133162i −0.598921 + 0.0161482i
\(69\) 0 0
\(70\) 5.40383 − 2.32415i 0.645881 − 0.277789i
\(71\) 7.28346 − 7.28346i 0.864387 − 0.864387i −0.127457 − 0.991844i \(-0.540681\pi\)
0.991844 + 0.127457i \(0.0406814\pi\)
\(72\) 0 0
\(73\) −7.84498 − 7.84498i −0.918185 − 0.918185i 0.0787121 − 0.996897i \(-0.474919\pi\)
−0.996897 + 0.0787121i \(0.974919\pi\)
\(74\) 6.73006 + 2.68199i 0.782353 + 0.311776i
\(75\) 0 0
\(76\) −3.11637 + 8.13729i −0.357472 + 0.933411i
\(77\) −2.33312 − 0.966410i −0.265884 − 0.110133i
\(78\) 0 0
\(79\) − 1.11506i − 0.125454i −0.998031 − 0.0627269i \(-0.980020\pi\)
0.998031 − 0.0627269i \(-0.0199797\pi\)
\(80\) 9.96356 − 4.77136i 1.11396 − 0.533454i
\(81\) 0 0
\(82\) −0.0559198 − 4.14877i −0.00617531 − 0.458155i
\(83\) 12.4078 + 5.13949i 1.36193 + 0.564132i 0.939590 − 0.342303i \(-0.111207\pi\)
0.422345 + 0.906435i \(0.361207\pi\)
\(84\) 0 0
\(85\) −2.61084 − 6.30312i −0.283185 − 0.683669i
\(86\) 4.72841 − 11.8652i 0.509877 − 1.27946i
\(87\) 0 0
\(88\) −4.45134 − 1.63630i −0.474514 − 0.174430i
\(89\) 7.58120 − 7.58120i 0.803606 − 0.803606i −0.180052 − 0.983657i \(-0.557626\pi\)
0.983657 + 0.180052i \(0.0576265\pi\)
\(90\) 0 0
\(91\) 0.302151 − 0.125155i 0.0316740 − 0.0131198i
\(92\) 5.65588 − 5.96932i 0.589666 − 0.622344i
\(93\) 0 0
\(94\) 6.85203 − 7.03927i 0.706733 − 0.726045i
\(95\) −12.0325 −1.23451
\(96\) 0 0
\(97\) 11.3315 1.15054 0.575268 − 0.817965i \(-0.304898\pi\)
0.575268 + 0.817965i \(0.304898\pi\)
\(98\) 4.66746 − 4.79500i 0.471485 − 0.484368i
\(99\) 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.109.8 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.25 yes 128
32.5 even 8 inner 864.2.v.a.325.8 yes 128
96.5 odd 8 inner 864.2.v.a.325.25 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.8 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.25 yes 128 3.2 odd 2 inner
864.2.v.a.325.8 yes 128 32.5 even 8 inner
864.2.v.a.325.25 yes 128 96.5 odd 8 inner