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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 109.3
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.38295 + 0.295740i) q^{2} +(1.82508 - 0.817983i) q^{4} +(3.70533 - 1.53480i) q^{5} +(1.70420 - 1.70420i) q^{7} +(-2.28207 + 1.67097i) q^{8} +(-4.67037 + 3.21836i) q^{10} +(-0.539908 - 1.30345i) q^{11} +(4.74278 + 1.96452i) q^{13} +(-1.85282 + 2.86082i) q^{14} +(2.66181 - 2.98576i) q^{16} +5.18418i q^{17} +(6.56852 + 2.72077i) q^{19} +(5.50707 - 5.83202i) q^{20} +(1.13215 + 1.64293i) q^{22} +(-3.07987 - 3.07987i) q^{23} +(7.83834 - 7.83834i) q^{25} +(-7.14000 - 1.31420i) q^{26} +(1.71629 - 4.50431i) q^{28} +(-2.51674 + 6.07595i) q^{29} -7.47332 q^{31} +(-2.79812 + 4.91635i) q^{32} +(-1.53317 - 7.16943i) q^{34} +(3.69903 - 8.93025i) q^{35} +(-4.40696 + 1.82542i) q^{37} +(-9.88854 - 1.82010i) q^{38} +(-5.89122 + 9.69403i) q^{40} +(-1.33547 - 1.33547i) q^{41} +(-2.02307 - 4.88412i) q^{43} +(-2.05157 - 1.93726i) q^{44} +(5.17013 + 3.34845i) q^{46} +0.857860i q^{47} +1.19138i q^{49} +(-8.52189 + 13.1581i) q^{50} +(10.2629 - 0.294109i) q^{52} +(-3.35470 - 8.09895i) q^{53} +(-4.00107 - 4.00107i) q^{55} +(-1.04143 + 6.73679i) q^{56} +(1.68362 - 9.14701i) q^{58} +(2.86121 - 1.18515i) q^{59} +(-2.84041 + 6.85735i) q^{61} +(10.3352 - 2.21016i) q^{62} +(2.41569 - 7.62656i) q^{64} +20.5887 q^{65} +(0.0611610 - 0.147656i) q^{67} +(4.24057 + 9.46152i) q^{68} +(-2.47453 + 13.4440i) q^{70} +(4.00830 - 4.00830i) q^{71} +(0.651872 + 0.651872i) q^{73} +(5.55474 - 3.82778i) q^{74} +(14.2136 - 0.407327i) q^{76} +(-3.14146 - 1.30124i) q^{77} -9.89973i q^{79} +(5.28033 - 15.1486i) q^{80} +(2.24183 + 1.45193i) q^{82} +(-9.41435 - 3.89955i) q^{83} +(7.95666 + 19.2091i) q^{85} +(4.24222 + 6.15617i) q^{86} +(3.41014 + 2.07240i) q^{88} +(4.18011 - 4.18011i) q^{89} +(11.4306 - 4.73472i) q^{91} +(-8.14028 - 3.10171i) q^{92} +(-0.253703 - 1.18637i) q^{94} +28.5144 q^{95} -9.70998 q^{97} +(-0.352338 - 1.64761i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38295 + 0.295740i −0.977890 + 0.209119i
\(3\) 0 0
\(4\) 1.82508 − 0.817983i 0.912538 − 0.408992i
\(5\) 3.70533 − 1.53480i 1.65707 − 0.686383i 0.659226 − 0.751945i \(-0.270884\pi\)
0.997848 + 0.0655623i \(0.0208841\pi\)
\(6\) 0 0
\(7\) 1.70420 − 1.70420i 0.644129 − 0.644129i −0.307439 − 0.951568i \(-0.599472\pi\)
0.951568 + 0.307439i \(0.0994721\pi\)
\(8\) −2.28207 + 1.67097i −0.806834 + 0.590778i
\(9\) 0 0
\(10\) −4.67037 + 3.21836i −1.47690 + 1.01773i
\(11\) −0.539908 − 1.30345i −0.162788 − 0.393006i 0.821346 − 0.570430i \(-0.193223\pi\)
−0.984134 + 0.177424i \(0.943223\pi\)
\(12\) 0 0
\(13\) 4.74278 + 1.96452i 1.31541 + 0.544861i 0.926459 − 0.376397i \(-0.122837\pi\)
0.388953 + 0.921258i \(0.372837\pi\)
\(14\) −1.85282 + 2.86082i −0.495187 + 0.764587i
\(15\) 0 0
\(16\) 2.66181 − 2.98576i 0.665452 − 0.746441i
\(17\) 5.18418i 1.25735i 0.777669 + 0.628674i \(0.216402\pi\)
−0.777669 + 0.628674i \(0.783598\pi\)
\(18\) 0 0
\(19\) 6.56852 + 2.72077i 1.50692 + 0.624187i 0.974920 − 0.222556i \(-0.0714401\pi\)
0.532001 + 0.846744i \(0.321440\pi\)
\(20\) 5.50707 − 5.83202i 1.23142 − 1.30408i
\(21\) 0 0
\(22\) 1.13215 + 1.64293i 0.241374 + 0.350274i
\(23\) −3.07987 − 3.07987i −0.642197 − 0.642197i 0.308898 − 0.951095i \(-0.400040\pi\)
−0.951095 + 0.308898i \(0.900040\pi\)
\(24\) 0 0
\(25\) 7.83834 − 7.83834i 1.56767 − 1.56767i
\(26\) −7.14000 − 1.31420i −1.40027 − 0.257736i
\(27\) 0 0
\(28\) 1.71629 − 4.50431i 0.324349 − 0.851235i
\(29\) −2.51674 + 6.07595i −0.467347 + 1.12828i 0.497969 + 0.867195i \(0.334079\pi\)
−0.965317 + 0.261082i \(0.915921\pi\)
\(30\) 0 0
\(31\) −7.47332 −1.34225 −0.671124 − 0.741345i \(-0.734188\pi\)
−0.671124 + 0.741345i \(0.734188\pi\)
\(32\) −2.79812 + 4.91635i −0.494643 + 0.869096i
\(33\) 0 0
\(34\) −1.53317 − 7.16943i −0.262936 − 1.22955i
\(35\) 3.69903 − 8.93025i 0.625250 − 1.50949i
\(36\) 0 0
\(37\) −4.40696 + 1.82542i −0.724501 + 0.300098i −0.714290 − 0.699850i \(-0.753250\pi\)
−0.0102106 + 0.999948i \(0.503250\pi\)
\(38\) −9.88854 − 1.82010i −1.60413 − 0.295260i
\(39\) 0 0
\(40\) −5.89122 + 9.69403i −0.931484 + 1.53276i
\(41\) −1.33547 − 1.33547i −0.208565 − 0.208565i 0.595092 − 0.803657i \(-0.297115\pi\)
−0.803657 + 0.595092i \(0.797115\pi\)
\(42\) 0 0
\(43\) −2.02307 − 4.88412i −0.308515 − 0.744821i −0.999754 − 0.0221943i \(-0.992935\pi\)
0.691239 − 0.722626i \(-0.257065\pi\)
\(44\) −2.05157 − 1.93726i −0.309286 − 0.292054i
\(45\) 0 0
\(46\) 5.17013 + 3.34845i 0.762294 + 0.493702i
\(47\) 0.857860i 0.125132i 0.998041 + 0.0625658i \(0.0199283\pi\)
−0.998041 + 0.0625658i \(0.980072\pi\)
\(48\) 0 0
\(49\) 1.19138i 0.170197i
\(50\) −8.52189 + 13.1581i −1.20518 + 1.86084i
\(51\) 0 0
\(52\) 10.2629 − 0.294109i 1.42321 − 0.0407856i
\(53\) −3.35470 − 8.09895i −0.460803 − 1.11248i −0.968068 − 0.250687i \(-0.919344\pi\)
0.507265 − 0.861790i \(-0.330656\pi\)
\(54\) 0 0
\(55\) −4.00107 − 4.00107i −0.539504 − 0.539504i
\(56\) −1.04143 + 6.73679i −0.139168 + 0.900242i
\(57\) 0 0
\(58\) 1.68362 − 9.14701i 0.221070 − 1.20106i
\(59\) 2.86121 − 1.18515i 0.372497 − 0.154293i −0.188578 − 0.982058i \(-0.560388\pi\)
0.561075 + 0.827765i \(0.310388\pi\)
\(60\) 0 0
\(61\) −2.84041 + 6.85735i −0.363677 + 0.877994i 0.631079 + 0.775718i \(0.282612\pi\)
−0.994756 + 0.102275i \(0.967388\pi\)
\(62\) 10.3352 − 2.21016i 1.31257 − 0.280690i
\(63\) 0 0
\(64\) 2.41569 − 7.62656i 0.301962 − 0.953320i
\(65\) 20.5887 2.55372
\(66\) 0 0
\(67\) 0.0611610 − 0.147656i 0.00747201 − 0.0180390i −0.920100 − 0.391685i \(-0.871892\pi\)
0.927572 + 0.373646i \(0.121892\pi\)
\(68\) 4.24057 + 9.46152i 0.514245 + 1.14738i
\(69\) 0 0
\(70\) −2.47453 + 13.4440i −0.295763 + 1.60686i
\(71\) 4.00830 − 4.00830i 0.475697 − 0.475697i −0.428055 − 0.903753i \(-0.640801\pi\)
0.903753 + 0.428055i \(0.140801\pi\)
\(72\) 0 0
\(73\) 0.651872 + 0.651872i 0.0762959 + 0.0762959i 0.744225 − 0.667929i \(-0.232819\pi\)
−0.667929 + 0.744225i \(0.732819\pi\)
\(74\) 5.55474 − 3.82778i 0.645726 − 0.444970i
\(75\) 0 0
\(76\) 14.2136 − 0.407327i 1.63041 − 0.0467236i
\(77\) −3.14146 − 1.30124i −0.358003 − 0.148290i
\(78\) 0 0
\(79\) − 9.89973i − 1.11381i −0.830577 − 0.556903i \(-0.811989\pi\)
0.830577 − 0.556903i \(-0.188011\pi\)
\(80\) 5.28033 − 15.1486i 0.590359 − 1.69366i
\(81\) 0 0
\(82\) 2.24183 + 1.45193i 0.247568 + 0.160339i
\(83\) −9.41435 − 3.89955i −1.03336 − 0.428031i −0.199436 − 0.979911i \(-0.563911\pi\)
−0.833924 + 0.551879i \(0.813911\pi\)
\(84\) 0 0
\(85\) 7.95666 + 19.2091i 0.863021 + 2.08352i
\(86\) 4.24222 + 6.15617i 0.457450 + 0.663836i
\(87\) 0 0
\(88\) 3.41014 + 2.07240i 0.363522 + 0.220918i
\(89\) 4.18011 − 4.18011i 0.443091 − 0.443091i −0.449959 − 0.893049i \(-0.648561\pi\)
0.893049 + 0.449959i \(0.148561\pi\)
\(90\) 0 0
\(91\) 11.4306 − 4.73472i 1.19825 − 0.496333i
\(92\) −8.14028 − 3.10171i −0.848683 − 0.323376i
\(93\) 0 0
\(94\) −0.253703 − 1.18637i −0.0261675 − 0.122365i
\(95\) 28.5144 2.92551
\(96\) 0 0
\(97\) −9.70998 −0.985899 −0.492949 − 0.870058i \(-0.664081\pi\)
−0.492949 + 0.870058i \(0.664081\pi\)
\(98\) −0.352338 − 1.64761i −0.0355915 − 0.166434i
\(99\) 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.109.3 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.30 yes 128
32.5 even 8 inner 864.2.v.a.325.3 yes 128
96.5 odd 8 inner 864.2.v.a.325.30 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.3 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.30 yes 128 3.2 odd 2 inner
864.2.v.a.325.3 yes 128 32.5 even 8 inner
864.2.v.a.325.30 yes 128 96.5 odd 8 inner