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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 109.5
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.29834 - 0.560644i) q^{2} +(1.37136 + 1.45581i) q^{4} +(1.49059 - 0.617423i) q^{5} +(-3.02902 + 3.02902i) q^{7} +(-0.964291 - 2.65897i) q^{8} +(-2.28144 - 0.0340682i) q^{10} +(0.783887 + 1.89247i) q^{11} +(-2.34011 - 0.969307i) q^{13} +(5.63089 - 2.23448i) q^{14} +(-0.238764 + 3.99287i) q^{16} -0.0122868i q^{17} +(-4.36871 - 1.80958i) q^{19} +(2.94298 + 1.32331i) q^{20} +(0.0432534 - 2.89655i) q^{22} +(-4.89425 - 4.89425i) q^{23} +(-1.69489 + 1.69489i) q^{25} +(2.49482 + 2.57046i) q^{26} +(-8.56354 - 0.255811i) q^{28} +(-0.488562 + 1.17949i) q^{29} -7.33135 q^{31} +(2.54857 - 5.05022i) q^{32} +(-0.00688851 + 0.0159524i) q^{34} +(-2.64484 + 6.38521i) q^{35} +(8.98691 - 3.72250i) q^{37} +(4.65752 + 4.79873i) q^{38} +(-3.07907 - 3.36807i) q^{40} +(-0.293164 - 0.293164i) q^{41} +(0.807094 + 1.94850i) q^{43} +(-1.68009 + 3.73644i) q^{44} +(3.61045 + 9.09831i) q^{46} -10.6583i q^{47} -11.3499i q^{49} +(3.15076 - 1.25030i) q^{50} +(-1.79800 - 4.73603i) q^{52} +(-4.68985 - 11.3223i) q^{53} +(2.33691 + 2.33691i) q^{55} +(10.9749 + 5.13323i) q^{56} +(1.29559 - 1.25747i) q^{58} +(-5.64978 + 2.34021i) q^{59} +(-2.02067 + 4.87832i) q^{61} +(9.51857 + 4.11028i) q^{62} +(-6.14029 + 5.12805i) q^{64} -4.08662 q^{65} +(-3.76193 + 9.08210i) q^{67} +(0.0178872 - 0.0168496i) q^{68} +(7.01372 - 6.80734i) q^{70} +(-6.19966 + 6.19966i) q^{71} +(-0.256907 - 0.256907i) q^{73} +(-13.7550 - 0.205400i) q^{74} +(-3.35665 - 8.84158i) q^{76} +(-8.10674 - 3.35792i) q^{77} +16.9687i q^{79} +(2.10939 + 6.09915i) q^{80} +(0.216265 + 0.544985i) q^{82} +(-0.950366 - 0.393655i) q^{83} +(-0.00758614 - 0.0183146i) q^{85} +(0.0445339 - 2.98230i) q^{86} +(4.27614 - 3.90923i) q^{88} +(-7.66699 + 7.66699i) q^{89} +(10.0243 - 4.15220i) q^{91} +(0.413336 - 13.8368i) q^{92} +(-5.97552 + 13.8381i) q^{94} -7.62923 q^{95} -6.43958 q^{97} +(-6.36326 + 14.7360i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29834 − 0.560644i −0.918063 − 0.396435i
\(3\) 0 0
\(4\) 1.37136 + 1.45581i 0.685678 + 0.727905i
\(5\) 1.49059 − 0.617423i 0.666612 − 0.276120i −0.0236057 − 0.999721i \(-0.507515\pi\)
0.690218 + 0.723602i \(0.257515\pi\)
\(6\) 0 0
\(7\) −3.02902 + 3.02902i −1.14486 + 1.14486i −0.157312 + 0.987549i \(0.550283\pi\)
−0.987549 + 0.157312i \(0.949717\pi\)
\(8\) −0.964291 − 2.65897i −0.340928 − 0.940089i
\(9\) 0 0
\(10\) −2.28144 − 0.0340682i −0.721455 − 0.0107733i
\(11\) 0.783887 + 1.89247i 0.236351 + 0.570601i 0.996900 − 0.0786789i \(-0.0250702\pi\)
−0.760549 + 0.649280i \(0.775070\pi\)
\(12\) 0 0
\(13\) −2.34011 − 0.969307i −0.649031 − 0.268837i 0.0337836 − 0.999429i \(-0.489244\pi\)
−0.682815 + 0.730592i \(0.739244\pi\)
\(14\) 5.63089 − 2.23448i 1.50492 − 0.597191i
\(15\) 0 0
\(16\) −0.238764 + 3.99287i −0.0596910 + 0.998217i
\(17\) − 0.0122868i − 0.00297998i −0.999999 − 0.00148999i \(-0.999526\pi\)
0.999999 − 0.00148999i \(-0.000474279\pi\)
\(18\) 0 0
\(19\) −4.36871 − 1.80958i −1.00225 − 0.415146i −0.179628 − 0.983735i \(-0.557490\pi\)
−0.822622 + 0.568589i \(0.807490\pi\)
\(20\) 2.94298 + 1.32331i 0.658070 + 0.295901i
\(21\) 0 0
\(22\) 0.0432534 − 2.89655i 0.00922165 − 0.617546i
\(23\) −4.89425 − 4.89425i −1.02052 − 1.02052i −0.999785 − 0.0207358i \(-0.993399\pi\)
−0.0207358 − 0.999785i \(-0.506601\pi\)
\(24\) 0 0
\(25\) −1.69489 + 1.69489i −0.338977 + 0.338977i
\(26\) 2.49482 + 2.57046i 0.489274 + 0.504108i
\(27\) 0 0
\(28\) −8.56354 − 0.255811i −1.61836 − 0.0483438i
\(29\) −0.488562 + 1.17949i −0.0907236 + 0.219026i −0.962728 − 0.270472i \(-0.912820\pi\)
0.872004 + 0.489499i \(0.162820\pi\)
\(30\) 0 0
\(31\) −7.33135 −1.31675 −0.658375 − 0.752690i \(-0.728756\pi\)
−0.658375 + 0.752690i \(0.728756\pi\)
\(32\) 2.54857 − 5.05022i 0.450529 − 0.892762i
\(33\) 0 0
\(34\) −0.00688851 + 0.0159524i −0.00118137 + 0.00273581i
\(35\) −2.64484 + 6.38521i −0.447060 + 1.07930i
\(36\) 0 0
\(37\) 8.98691 − 3.72250i 1.47744 − 0.611975i 0.508898 − 0.860827i \(-0.330053\pi\)
0.968542 + 0.248852i \(0.0800532\pi\)
\(38\) 4.65752 + 4.79873i 0.755550 + 0.778457i
\(39\) 0 0
\(40\) −3.07907 − 3.36807i −0.486844 − 0.532538i
\(41\) −0.293164 − 0.293164i −0.0457844 − 0.0457844i 0.683844 − 0.729628i \(-0.260307\pi\)
−0.729628 + 0.683844i \(0.760307\pi\)
\(42\) 0 0
\(43\) 0.807094 + 1.94850i 0.123081 + 0.297143i 0.973396 − 0.229131i \(-0.0735885\pi\)
−0.850315 + 0.526274i \(0.823588\pi\)
\(44\) −1.68009 + 3.73644i −0.253283 + 0.563290i
\(45\) 0 0
\(46\) 3.61045 + 9.09831i 0.532332 + 1.34147i
\(47\) − 10.6583i − 1.55468i −0.629084 − 0.777338i \(-0.716570\pi\)
0.629084 − 0.777338i \(-0.283430\pi\)
\(48\) 0 0
\(49\) − 11.3499i − 1.62141i
\(50\) 3.15076 − 1.25030i 0.445585 − 0.176820i
\(51\) 0 0
\(52\) −1.79800 − 4.73603i −0.249338 − 0.656769i
\(53\) −4.68985 − 11.3223i −0.644200 − 1.55524i −0.820962 − 0.570983i \(-0.806562\pi\)
0.176761 − 0.984254i \(-0.443438\pi\)
\(54\) 0 0
\(55\) 2.33691 + 2.33691i 0.315109 + 0.315109i
\(56\) 10.9749 + 5.13323i 1.46659 + 0.685956i
\(57\) 0 0
\(58\) 1.29559 − 1.25747i 0.170120 − 0.165114i
\(59\) −5.64978 + 2.34021i −0.735539 + 0.304670i −0.718826 − 0.695190i \(-0.755320\pi\)
−0.0167128 + 0.999860i \(0.505320\pi\)
\(60\) 0 0
\(61\) −2.02067 + 4.87832i −0.258720 + 0.624605i −0.998854 − 0.0478538i \(-0.984762\pi\)
0.740134 + 0.672459i \(0.234762\pi\)
\(62\) 9.51857 + 4.11028i 1.20886 + 0.522006i
\(63\) 0 0
\(64\) −6.14029 + 5.12805i −0.767536 + 0.641006i
\(65\) −4.08662 −0.506883
\(66\) 0 0
\(67\) −3.76193 + 9.08210i −0.459593 + 1.10956i 0.508970 + 0.860784i \(0.330027\pi\)
−0.968562 + 0.248771i \(0.919973\pi\)
\(68\) 0.0178872 − 0.0168496i 0.00216914 − 0.00204331i
\(69\) 0 0
\(70\) 7.01372 − 6.80734i 0.838300 − 0.813633i
\(71\) −6.19966 + 6.19966i −0.735764 + 0.735764i −0.971755 − 0.235991i \(-0.924166\pi\)
0.235991 + 0.971755i \(0.424166\pi\)
\(72\) 0 0
\(73\) −0.256907 − 0.256907i −0.0300687 − 0.0300687i 0.691913 − 0.721981i \(-0.256768\pi\)
−0.721981 + 0.691913i \(0.756768\pi\)
\(74\) −13.7550 − 0.205400i −1.59899 − 0.0238773i
\(75\) 0 0
\(76\) −3.35665 − 8.84158i −0.385034 − 1.01420i
\(77\) −8.10674 − 3.35792i −0.923849 − 0.382671i
\(78\) 0 0
\(79\) 16.9687i 1.90913i 0.298010 + 0.954563i \(0.403677\pi\)
−0.298010 + 0.954563i \(0.596323\pi\)
\(80\) 2.10939 + 6.09915i 0.235837 + 0.681906i
\(81\) 0 0
\(82\) 0.216265 + 0.544985i 0.0238824 + 0.0601836i
\(83\) −0.950366 − 0.393655i −0.104316 − 0.0432092i 0.329915 − 0.944011i \(-0.392980\pi\)
−0.434231 + 0.900801i \(0.642980\pi\)
\(84\) 0 0
\(85\) −0.00758614 − 0.0183146i −0.000822833 − 0.00198649i
\(86\) 0.0445339 − 2.98230i 0.00480222 − 0.321590i
\(87\) 0 0
\(88\) 4.27614 − 3.90923i 0.455838 − 0.416725i
\(89\) −7.66699 + 7.66699i −0.812699 + 0.812699i −0.985038 − 0.172338i \(-0.944868\pi\)
0.172338 + 0.985038i \(0.444868\pi\)
\(90\) 0 0
\(91\) 10.0243 − 4.15220i 1.05083 − 0.435269i
\(92\) 0.413336 − 13.8368i 0.0430933 − 1.44259i
\(93\) 0 0
\(94\) −5.97552 + 13.8381i −0.616328 + 1.42729i
\(95\) −7.62923 −0.782742
\(96\) 0 0
\(97\) −6.43958 −0.653840 −0.326920 − 0.945052i \(-0.606011\pi\)
−0.326920 + 0.945052i \(0.606011\pi\)
\(98\) −6.36326 + 14.7360i −0.642786 + 1.48856i
\(99\) 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.109.5 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.28 yes 128
32.5 even 8 inner 864.2.v.a.325.5 yes 128
96.5 odd 8 inner 864.2.v.a.325.28 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.5 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.28 yes 128 3.2 odd 2 inner
864.2.v.a.325.5 yes 128 32.5 even 8 inner
864.2.v.a.325.28 yes 128 96.5 odd 8 inner