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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 109.1
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41128 - 0.0910249i) q^{2} +(1.98343 + 0.256923i) q^{4} +(2.84116 - 1.17684i) q^{5} +(-0.237847 + 0.237847i) q^{7} +(-2.77579 - 0.543133i) q^{8} +(-4.11679 + 1.40224i) q^{10} +(-1.66757 - 4.02588i) q^{11} +(1.18188 + 0.489551i) q^{13} +(0.357318 - 0.314019i) q^{14} +(3.86798 + 1.01918i) q^{16} -5.92975i q^{17} +(-7.48264 - 3.09941i) q^{19} +(5.93759 - 1.60423i) q^{20} +(1.98696 + 5.83344i) q^{22} +(-2.53252 - 2.53252i) q^{23} +(3.15166 - 3.15166i) q^{25} +(-1.62340 - 0.798474i) q^{26} +(-0.532860 + 0.410644i) q^{28} +(-0.277228 + 0.669289i) q^{29} +3.89737 q^{31} +(-5.36604 - 1.79043i) q^{32} +(-0.539755 + 8.36855i) q^{34} +(-0.395851 + 0.955668i) q^{35} +(0.757860 - 0.313916i) q^{37} +(10.2780 + 5.05525i) q^{38} +(-8.52563 + 1.72355i) q^{40} +(-3.40516 - 3.40516i) q^{41} +(2.78682 + 6.72797i) q^{43} +(-2.27317 - 8.41349i) q^{44} +(3.34357 + 3.80462i) q^{46} -1.68151i q^{47} +6.88686i q^{49} +(-4.73476 + 4.16100i) q^{50} +(2.21840 + 1.27464i) q^{52} +(5.04548 + 12.1809i) q^{53} +(-9.47568 - 9.47568i) q^{55} +(0.789394 - 0.531030i) q^{56} +(0.452169 - 0.919320i) q^{58} +(1.95599 - 0.810198i) q^{59} +(2.31485 - 5.58854i) q^{61} +(-5.50029 - 0.354758i) q^{62} +(7.41001 + 3.01524i) q^{64} +3.93403 q^{65} +(4.21245 - 10.1698i) q^{67} +(1.52349 - 11.7612i) q^{68} +(0.645646 - 1.31268i) q^{70} +(-6.76159 + 6.76159i) q^{71} +(-5.49242 - 5.49242i) q^{73} +(-1.09813 + 0.374039i) q^{74} +(-14.0450 - 8.06993i) q^{76} +(1.35417 + 0.560915i) q^{77} -12.8822i q^{79} +(12.1889 - 1.65637i) q^{80} +(4.49568 + 5.11559i) q^{82} +(-2.57136 - 1.06509i) q^{83} +(-6.97840 - 16.8473i) q^{85} +(-3.32057 - 9.74873i) q^{86} +(2.44225 + 12.0807i) q^{88} +(12.2809 - 12.2809i) q^{89} +(-0.397544 + 0.164668i) q^{91} +(-4.37240 - 5.67373i) q^{92} +(-0.153059 + 2.37308i) q^{94} -24.9069 q^{95} +12.6111 q^{97} +(0.626876 - 9.71929i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41128 − 0.0910249i −0.997926 − 0.0643643i
\(3\) 0 0
\(4\) 1.98343 + 0.256923i 0.991714 + 0.128462i
\(5\) 2.84116 − 1.17684i 1.27060 − 0.526301i 0.357456 − 0.933930i \(-0.383644\pi\)
0.913148 + 0.407629i \(0.133644\pi\)
\(6\) 0 0
\(7\) −0.237847 + 0.237847i −0.0898976 + 0.0898976i −0.750625 − 0.660728i \(-0.770248\pi\)
0.660728 + 0.750625i \(0.270248\pi\)
\(8\) −2.77579 − 0.543133i −0.981390 − 0.192026i
\(9\) 0 0
\(10\) −4.11679 + 1.40224i −1.30184 + 0.443428i
\(11\) −1.66757 − 4.02588i −0.502793 − 1.21385i −0.947957 − 0.318400i \(-0.896855\pi\)
0.445164 − 0.895449i \(-0.353145\pi\)
\(12\) 0 0
\(13\) 1.18188 + 0.489551i 0.327794 + 0.135777i 0.540511 − 0.841337i \(-0.318231\pi\)
−0.212716 + 0.977114i \(0.568231\pi\)
\(14\) 0.357318 − 0.314019i 0.0954974 − 0.0839250i
\(15\) 0 0
\(16\) 3.86798 + 1.01918i 0.966995 + 0.254795i
\(17\) − 5.92975i − 1.43818i −0.694919 − 0.719088i \(-0.744560\pi\)
0.694919 − 0.719088i \(-0.255440\pi\)
\(18\) 0 0
\(19\) −7.48264 − 3.09941i −1.71664 − 0.711054i −0.999906 − 0.0137030i \(-0.995638\pi\)
−0.716730 − 0.697351i \(-0.754362\pi\)
\(20\) 5.93759 − 1.60423i 1.32769 − 0.358716i
\(21\) 0 0
\(22\) 1.98696 + 5.83344i 0.423622 + 1.24369i
\(23\) −2.53252 − 2.53252i −0.528066 − 0.528066i 0.391929 − 0.919995i \(-0.371808\pi\)
−0.919995 + 0.391929i \(0.871808\pi\)
\(24\) 0 0
\(25\) 3.15166 − 3.15166i 0.630333 − 0.630333i
\(26\) −1.62340 − 0.798474i −0.318376 − 0.156594i
\(27\) 0 0
\(28\) −0.532860 + 0.410644i −0.100701 + 0.0776043i
\(29\) −0.277228 + 0.669289i −0.0514800 + 0.124284i −0.947527 − 0.319675i \(-0.896426\pi\)
0.896047 + 0.443959i \(0.146426\pi\)
\(30\) 0 0
\(31\) 3.89737 0.699989 0.349994 − 0.936752i \(-0.386184\pi\)
0.349994 + 0.936752i \(0.386184\pi\)
\(32\) −5.36604 − 1.79043i −0.948590 − 0.316506i
\(33\) 0 0
\(34\) −0.539755 + 8.36855i −0.0925673 + 1.43519i
\(35\) −0.395851 + 0.955668i −0.0669110 + 0.161537i
\(36\) 0 0
\(37\) 0.757860 − 0.313916i 0.124591 − 0.0516075i −0.319517 − 0.947581i \(-0.603521\pi\)
0.444108 + 0.895973i \(0.353521\pi\)
\(38\) 10.2780 + 5.05525i 1.66731 + 0.820070i
\(39\) 0 0
\(40\) −8.52563 + 1.72355i −1.34802 + 0.272517i
\(41\) −3.40516 − 3.40516i −0.531796 − 0.531796i 0.389311 − 0.921107i \(-0.372713\pi\)
−0.921107 + 0.389311i \(0.872713\pi\)
\(42\) 0 0
\(43\) 2.78682 + 6.72797i 0.424986 + 1.02601i 0.980856 + 0.194737i \(0.0623853\pi\)
−0.555870 + 0.831269i \(0.687615\pi\)
\(44\) −2.27317 − 8.41349i −0.342694 − 1.26838i
\(45\) 0 0
\(46\) 3.34357 + 3.80462i 0.492983 + 0.560960i
\(47\) − 1.68151i − 0.245273i −0.992452 − 0.122637i \(-0.960865\pi\)
0.992452 − 0.122637i \(-0.0391350\pi\)
\(48\) 0 0
\(49\) 6.88686i 0.983837i
\(50\) −4.73476 + 4.16100i −0.669597 + 0.588455i
\(51\) 0 0
\(52\) 2.21840 + 1.27464i 0.307636 + 0.176761i
\(53\) 5.04548 + 12.1809i 0.693050 + 1.67317i 0.738544 + 0.674205i \(0.235514\pi\)
−0.0454947 + 0.998965i \(0.514486\pi\)
\(54\) 0 0
\(55\) −9.47568 − 9.47568i −1.27770 − 1.27770i
\(56\) 0.789394 − 0.531030i 0.105487 − 0.0709619i
\(57\) 0 0
\(58\) 0.452169 − 0.919320i 0.0593727 − 0.120713i
\(59\) 1.95599 − 0.810198i 0.254648 − 0.105479i −0.251708 − 0.967803i \(-0.580992\pi\)
0.506356 + 0.862325i \(0.330992\pi\)
\(60\) 0 0
\(61\) 2.31485 − 5.58854i 0.296386 − 0.715539i −0.703602 − 0.710594i \(-0.748426\pi\)
0.999988 − 0.00494445i \(-0.00157387\pi\)
\(62\) −5.50029 − 0.354758i −0.698537 − 0.0450543i
\(63\) 0 0
\(64\) 7.41001 + 3.01524i 0.926252 + 0.376906i
\(65\) 3.93403 0.487956
\(66\) 0 0
\(67\) 4.21245 − 10.1698i 0.514633 − 1.24243i −0.426528 − 0.904474i \(-0.640263\pi\)
0.941161 − 0.337959i \(-0.109737\pi\)
\(68\) 1.52349 − 11.7612i 0.184751 − 1.42626i
\(69\) 0 0
\(70\) 0.645646 − 1.31268i 0.0771695 − 0.156896i
\(71\) −6.76159 + 6.76159i −0.802453 + 0.802453i −0.983478 − 0.181025i \(-0.942058\pi\)
0.181025 + 0.983478i \(0.442058\pi\)
\(72\) 0 0
\(73\) −5.49242 − 5.49242i −0.642839 − 0.642839i 0.308413 − 0.951252i \(-0.400202\pi\)
−0.951252 + 0.308413i \(0.900202\pi\)
\(74\) −1.09813 + 0.374039i −0.127655 + 0.0434812i
\(75\) 0 0
\(76\) −14.0450 − 8.06993i −1.61107 − 0.925684i
\(77\) 1.35417 + 0.560915i 0.154322 + 0.0639222i
\(78\) 0 0
\(79\) − 12.8822i − 1.44936i −0.689085 − 0.724680i \(-0.741988\pi\)
0.689085 − 0.724680i \(-0.258012\pi\)
\(80\) 12.1889 − 1.65637i 1.36277 − 0.185188i
\(81\) 0 0
\(82\) 4.49568 + 5.11559i 0.496465 + 0.564922i
\(83\) −2.57136 − 1.06509i −0.282244 − 0.116909i 0.237071 − 0.971492i \(-0.423813\pi\)
−0.519314 + 0.854583i \(0.673813\pi\)
\(84\) 0 0
\(85\) −6.97840 − 16.8473i −0.756914 − 1.82735i
\(86\) −3.32057 − 9.74873i −0.358066 − 1.05123i
\(87\) 0 0
\(88\) 2.44225 + 12.0807i 0.260345 + 1.28781i
\(89\) 12.2809 − 12.2809i 1.30177 − 1.30177i 0.374571 − 0.927198i \(-0.377790\pi\)
0.927198 − 0.374571i \(-0.122210\pi\)
\(90\) 0 0
\(91\) −0.397544 + 0.164668i −0.0416739 + 0.0172619i
\(92\) −4.37240 − 5.67373i −0.455855 − 0.591527i
\(93\) 0 0
\(94\) −0.153059 + 2.37308i −0.0157869 + 0.244765i
\(95\) −24.9069 −2.55539
\(96\) 0 0
\(97\) 12.6111 1.28046 0.640230 − 0.768183i \(-0.278839\pi\)
0.640230 + 0.768183i \(0.278839\pi\)
\(98\) 0.626876 − 9.71929i 0.0633240 − 0.981797i
\(99\) 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.109.1 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.32 yes 128
32.5 even 8 inner 864.2.v.b.325.1 yes 128
96.5 odd 8 inner 864.2.v.b.325.32 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.1 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.32 yes 128 3.2 odd 2 inner
864.2.v.b.325.1 yes 128 32.5 even 8 inner
864.2.v.b.325.32 yes 128 96.5 odd 8 inner