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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 109.6
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.14607 - 0.828563i) q^{2} +(0.626968 + 1.89919i) q^{4} +(-2.89399 + 1.19873i) q^{5} +(-3.03933 + 3.03933i) q^{7} +(0.855044 - 2.69609i) q^{8} +(4.30995 + 1.02402i) q^{10} +(-2.07833 - 5.01753i) q^{11} +(0.154565 + 0.0640231i) q^{13} +(6.00158 - 0.965021i) q^{14} +(-3.21382 + 2.38146i) q^{16} +4.77723i q^{17} +(-3.30151 - 1.36753i) q^{19} +(-4.09106 - 4.74467i) q^{20} +(-1.77542 + 7.47248i) q^{22} +(1.87996 + 1.87996i) q^{23} +(3.40271 - 3.40271i) q^{25} +(-0.124096 - 0.201442i) q^{26} +(-7.67783 - 3.86670i) q^{28} +(1.81073 - 4.37148i) q^{29} +9.71647 q^{31} +(5.65646 - 0.0664742i) q^{32} +(3.95824 - 5.47506i) q^{34} +(5.15247 - 12.4392i) q^{35} +(8.56912 - 3.54945i) q^{37} +(2.65069 + 4.30280i) q^{38} +(0.757396 + 8.82744i) q^{40} +(-3.98159 - 3.98159i) q^{41} +(-0.172185 - 0.415692i) q^{43} +(8.22618 - 7.09296i) q^{44} +(-0.596908 - 3.71224i) q^{46} -6.11328i q^{47} -11.4751i q^{49} +(-6.71911 + 1.08040i) q^{50} +(-0.0246842 + 0.333689i) q^{52} +(2.21266 + 5.34183i) q^{53} +(12.0293 + 12.0293i) q^{55} +(5.59555 + 10.7931i) q^{56} +(-5.69727 + 3.50974i) q^{58} +(2.09529 - 0.867896i) q^{59} +(-3.46853 + 8.37378i) q^{61} +(-11.1358 - 8.05070i) q^{62} +(-6.53780 - 4.61055i) q^{64} -0.524058 q^{65} +(4.88800 - 11.8007i) q^{67} +(-9.07286 + 2.99517i) q^{68} +(-16.2117 + 9.98705i) q^{70} +(-8.20687 + 8.20687i) q^{71} +(-1.88142 - 1.88142i) q^{73} +(-12.7618 - 3.03213i) q^{74} +(0.527254 - 7.12759i) q^{76} +(21.5667 + 8.93321i) q^{77} -15.0641i q^{79} +(6.44605 - 10.7444i) q^{80} +(1.26420 + 7.86220i) q^{82} +(1.42473 + 0.590143i) q^{83} +(-5.72662 - 13.8253i) q^{85} +(-0.147090 + 0.619080i) q^{86} +(-15.3048 + 1.31315i) q^{88} +(-1.32189 + 1.32189i) q^{89} +(-0.664364 + 0.275188i) q^{91} +(-2.39172 + 4.74908i) q^{92} +(-5.06523 + 7.00626i) q^{94} +11.1939 q^{95} -1.06320 q^{97} +(-9.50784 + 13.1513i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14607 − 0.828563i −0.810396 − 0.585882i
\(3\) 0 0
\(4\) 0.626968 + 1.89919i 0.313484 + 0.949593i
\(5\) −2.89399 + 1.19873i −1.29423 + 0.536089i −0.920244 − 0.391344i \(-0.872010\pi\)
−0.373989 + 0.927433i \(0.622010\pi\)
\(6\) 0 0
\(7\) −3.03933 + 3.03933i −1.14876 + 1.14876i −0.161964 + 0.986797i \(0.551783\pi\)
−0.986797 + 0.161964i \(0.948217\pi\)
\(8\) 0.855044 − 2.69609i 0.302304 − 0.953212i
\(9\) 0 0
\(10\) 4.30995 + 1.02402i 1.36293 + 0.323824i
\(11\) −2.07833 − 5.01753i −0.626639 − 1.51284i −0.843774 − 0.536699i \(-0.819671\pi\)
0.217134 − 0.976142i \(-0.430329\pi\)
\(12\) 0 0
\(13\) 0.154565 + 0.0640231i 0.0428687 + 0.0177568i 0.404015 − 0.914752i \(-0.367614\pi\)
−0.361146 + 0.932509i \(0.617614\pi\)
\(14\) 6.00158 − 0.965021i 1.60399 − 0.257913i
\(15\) 0 0
\(16\) −3.21382 + 2.38146i −0.803456 + 0.595365i
\(17\) 4.77723i 1.15865i 0.815097 + 0.579325i \(0.196684\pi\)
−0.815097 + 0.579325i \(0.803316\pi\)
\(18\) 0 0
\(19\) −3.30151 − 1.36753i −0.757419 − 0.313733i −0.0296543 − 0.999560i \(-0.509441\pi\)
−0.727765 + 0.685827i \(0.759441\pi\)
\(20\) −4.09106 − 4.74467i −0.914788 − 1.06094i
\(21\) 0 0
\(22\) −1.77542 + 7.47248i −0.378521 + 1.59314i
\(23\) 1.87996 + 1.87996i 0.391999 + 0.391999i 0.875399 − 0.483400i \(-0.160598\pi\)
−0.483400 + 0.875399i \(0.660598\pi\)
\(24\) 0 0
\(25\) 3.40271 − 3.40271i 0.680542 − 0.680542i
\(26\) −0.124096 − 0.201442i −0.0243373 − 0.0395061i
\(27\) 0 0
\(28\) −7.67783 − 3.86670i −1.45097 − 0.730737i
\(29\) 1.81073 − 4.37148i 0.336243 − 0.811763i −0.661826 − 0.749657i \(-0.730218\pi\)
0.998070 − 0.0621060i \(-0.0197817\pi\)
\(30\) 0 0
\(31\) 9.71647 1.74513 0.872565 − 0.488499i \(-0.162455\pi\)
0.872565 + 0.488499i \(0.162455\pi\)
\(32\) 5.65646 − 0.0664742i 0.999931 − 0.0117511i
\(33\) 0 0
\(34\) 3.95824 − 5.47506i 0.678832 − 0.938965i
\(35\) 5.15247 − 12.4392i 0.870926 − 2.10260i
\(36\) 0 0
\(37\) 8.56912 − 3.54945i 1.40876 − 0.583526i 0.456747 − 0.889597i \(-0.349015\pi\)
0.952009 + 0.306071i \(0.0990146\pi\)
\(38\) 2.65069 + 4.30280i 0.429999 + 0.698006i
\(39\) 0 0
\(40\) 0.757396 + 8.82744i 0.119755 + 1.39574i
\(41\) −3.98159 − 3.98159i −0.621821 − 0.621821i 0.324176 − 0.945997i \(-0.394913\pi\)
−0.945997 + 0.324176i \(0.894913\pi\)
\(42\) 0 0
\(43\) −0.172185 − 0.415692i −0.0262580 − 0.0633924i 0.910207 − 0.414154i \(-0.135923\pi\)
−0.936465 + 0.350762i \(0.885923\pi\)
\(44\) 8.22618 − 7.09296i 1.24014 − 1.06930i
\(45\) 0 0
\(46\) −0.596908 − 3.71224i −0.0880093 − 0.547340i
\(47\) − 6.11328i − 0.891713i −0.895104 − 0.445856i \(-0.852899\pi\)
0.895104 − 0.445856i \(-0.147101\pi\)
\(48\) 0 0
\(49\) − 11.4751i − 1.63930i
\(50\) −6.71911 + 1.08040i −0.950226 + 0.152791i
\(51\) 0 0
\(52\) −0.0246842 + 0.333689i −0.00342309 + 0.0462743i
\(53\) 2.21266 + 5.34183i 0.303932 + 0.733757i 0.999877 + 0.0156620i \(0.00498558\pi\)
−0.695945 + 0.718095i \(0.745014\pi\)
\(54\) 0 0
\(55\) 12.0293 + 12.0293i 1.62204 + 1.62204i
\(56\) 5.59555 + 10.7931i 0.747737 + 1.44229i
\(57\) 0 0
\(58\) −5.69727 + 3.50974i −0.748088 + 0.460851i
\(59\) 2.09529 − 0.867896i 0.272783 − 0.112990i −0.242099 − 0.970252i \(-0.577836\pi\)
0.514882 + 0.857261i \(0.327836\pi\)
\(60\) 0 0
\(61\) −3.46853 + 8.37378i −0.444100 + 1.07215i 0.530396 + 0.847750i \(0.322043\pi\)
−0.974496 + 0.224403i \(0.927957\pi\)
\(62\) −11.1358 − 8.05070i −1.41425 − 1.02244i
\(63\) 0 0
\(64\) −6.53780 − 4.61055i −0.817225 − 0.576319i
\(65\) −0.524058 −0.0650014
\(66\) 0 0
\(67\) 4.88800 − 11.8007i 0.597164 − 1.44168i −0.279296 − 0.960205i \(-0.590101\pi\)
0.876459 − 0.481476i \(-0.159899\pi\)
\(68\) −9.07286 + 2.99517i −1.10025 + 0.363218i
\(69\) 0 0
\(70\) −16.2117 + 9.98705i −1.93767 + 1.19368i
\(71\) −8.20687 + 8.20687i −0.973976 + 0.973976i −0.999670 − 0.0256934i \(-0.991821\pi\)
0.0256934 + 0.999670i \(0.491821\pi\)
\(72\) 0 0
\(73\) −1.88142 − 1.88142i −0.220204 − 0.220204i 0.588380 − 0.808584i \(-0.299766\pi\)
−0.808584 + 0.588380i \(0.799766\pi\)
\(74\) −12.7618 − 3.03213i −1.48353 − 0.352478i
\(75\) 0 0
\(76\) 0.527254 − 7.12759i 0.0604802 − 0.817590i
\(77\) 21.5667 + 8.93321i 2.45775 + 1.01803i
\(78\) 0 0
\(79\) − 15.0641i − 1.69485i −0.530917 − 0.847424i \(-0.678152\pi\)
0.530917 − 0.847424i \(-0.321848\pi\)
\(80\) 6.44605 − 10.7444i 0.720690 − 1.20126i
\(81\) 0 0
\(82\) 1.26420 + 7.86220i 0.139607 + 0.868235i
\(83\) 1.42473 + 0.590143i 0.156385 + 0.0647766i 0.459503 − 0.888176i \(-0.348028\pi\)
−0.303118 + 0.952953i \(0.598028\pi\)
\(84\) 0 0
\(85\) −5.72662 − 13.8253i −0.621139 − 1.49956i
\(86\) −0.147090 + 0.619080i −0.0158611 + 0.0667571i
\(87\) 0 0
\(88\) −15.3048 + 1.31315i −1.63149 + 0.139983i
\(89\) −1.32189 + 1.32189i −0.140120 + 0.140120i −0.773687 − 0.633568i \(-0.781590\pi\)
0.633568 + 0.773687i \(0.281590\pi\)
\(90\) 0 0
\(91\) −0.664364 + 0.275188i −0.0696442 + 0.0288476i
\(92\) −2.39172 + 4.74908i −0.249354 + 0.495125i
\(93\) 0 0
\(94\) −5.06523 + 7.00626i −0.522439 + 0.722641i
\(95\) 11.1939 1.14847
\(96\) 0 0
\(97\) −1.06320 −0.107952 −0.0539759 − 0.998542i \(-0.517189\pi\)
−0.0539759 + 0.998542i \(0.517189\pi\)
\(98\) −9.50784 + 13.1513i −0.960437 + 1.32848i
\(99\) 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.109.6 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.27 yes 128
32.5 even 8 inner 864.2.v.b.325.6 yes 128
96.5 odd 8 inner 864.2.v.b.325.27 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.6 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.27 yes 128 3.2 odd 2 inner
864.2.v.b.325.6 yes 128 32.5 even 8 inner
864.2.v.b.325.27 yes 128 96.5 odd 8 inner