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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 109.10
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.934820 + 1.06118i) q^{2} +(-0.252222 - 1.98403i) q^{4} +(0.930451 - 0.385405i) q^{5} +(-0.938500 + 0.938500i) q^{7} +(2.34121 + 1.58706i) q^{8} +(-0.460818 + 1.34766i) q^{10} +(-0.846222 - 2.04296i) q^{11} +(-5.64483 - 2.33816i) q^{13} +(-0.118592 - 1.87325i) q^{14} +(-3.87277 + 1.00083i) q^{16} +7.71645i q^{17} +(-3.36643 - 1.39442i) q^{19} +(-0.999337 - 1.74884i) q^{20} +(2.95902 + 1.01180i) q^{22} +(5.65026 + 5.65026i) q^{23} +(-2.81833 + 2.81833i) q^{25} +(7.75812 - 3.80444i) q^{26} +(2.09872 + 1.62530i) q^{28} +(-0.949070 + 2.29126i) q^{29} +5.39395 q^{31} +(2.55827 - 5.04532i) q^{32} +(-8.18857 - 7.21350i) q^{34} +(-0.511525 + 1.23493i) q^{35} +(-4.01544 + 1.66325i) q^{37} +(4.62674 - 2.26887i) q^{38} +(2.79004 + 0.574368i) q^{40} +(-6.02596 - 6.02596i) q^{41} +(3.38507 + 8.17229i) q^{43} +(-3.83986 + 2.19421i) q^{44} +(-11.2779 + 0.713987i) q^{46} +6.68653i q^{47} +5.23844i q^{49} +(-0.356135 - 5.62540i) q^{50} +(-3.21524 + 11.7893i) q^{52} +(0.589451 + 1.42306i) q^{53} +(-1.57474 - 1.57474i) q^{55} +(-3.68668 + 0.707766i) q^{56} +(-1.54424 - 3.14905i) q^{58} +(-7.21416 + 2.98820i) q^{59} +(-3.05674 + 7.37963i) q^{61} +(-5.04238 + 5.72397i) q^{62} +(2.96249 + 7.43126i) q^{64} -6.15337 q^{65} +(3.77223 - 9.10696i) q^{67} +(15.3097 - 1.94626i) q^{68} +(-0.832305 - 1.69726i) q^{70} +(-1.95353 + 1.95353i) q^{71} +(-8.58563 - 8.58563i) q^{73} +(1.98870 - 5.81596i) q^{74} +(-1.91749 + 7.03081i) q^{76} +(2.71150 + 1.12314i) q^{77} -11.7674i q^{79} +(-3.21769 + 2.42381i) q^{80} +(12.0278 - 0.761461i) q^{82} +(-6.05690 - 2.50885i) q^{83} +(2.97396 + 7.17978i) q^{85} +(-11.8367 - 4.04744i) q^{86} +(1.26112 - 6.12599i) q^{88} +(-5.36845 + 5.36845i) q^{89} +(7.49204 - 3.10330i) q^{91} +(9.78518 - 12.6354i) q^{92} +(-7.09564 - 6.25071i) q^{94} -3.66972 q^{95} -8.31108 q^{97} +(-5.55894 - 4.89700i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.934820 + 1.06118i −0.661018 + 0.750370i
\(3\) 0 0
\(4\) −0.252222 − 1.98403i −0.126111 − 0.992016i
\(5\) 0.930451 − 0.385405i 0.416110 − 0.172359i −0.164798 − 0.986327i \(-0.552697\pi\)
0.580909 + 0.813969i \(0.302697\pi\)
\(6\) 0 0
\(7\) −0.938500 + 0.938500i −0.354720 + 0.354720i −0.861862 − 0.507143i \(-0.830702\pi\)
0.507143 + 0.861862i \(0.330702\pi\)
\(8\) 2.34121 + 1.58706i 0.827741 + 0.561110i
\(9\) 0 0
\(10\) −0.460818 + 1.34766i −0.145724 + 0.426169i
\(11\) −0.846222 − 2.04296i −0.255145 − 0.615976i 0.743459 − 0.668781i \(-0.233184\pi\)
−0.998605 + 0.0528055i \(0.983184\pi\)
\(12\) 0 0
\(13\) −5.64483 − 2.33816i −1.56559 − 0.648490i −0.579543 − 0.814941i \(-0.696769\pi\)
−0.986050 + 0.166451i \(0.946769\pi\)
\(14\) −0.118592 − 1.87325i −0.0316951 − 0.500647i
\(15\) 0 0
\(16\) −3.87277 + 1.00083i −0.968192 + 0.250209i
\(17\) 7.71645i 1.87151i 0.352645 + 0.935757i \(0.385282\pi\)
−0.352645 + 0.935757i \(0.614718\pi\)
\(18\) 0 0
\(19\) −3.36643 − 1.39442i −0.772312 − 0.319902i −0.0385035 − 0.999258i \(-0.512259\pi\)
−0.733809 + 0.679356i \(0.762259\pi\)
\(20\) −0.999337 − 1.74884i −0.223459 − 0.391052i
\(21\) 0 0
\(22\) 2.95902 + 1.01180i 0.630865 + 0.215717i
\(23\) 5.65026 + 5.65026i 1.17816 + 1.17816i 0.980212 + 0.197948i \(0.0634278\pi\)
0.197948 + 0.980212i \(0.436572\pi\)
\(24\) 0 0
\(25\) −2.81833 + 2.81833i −0.563666 + 0.563666i
\(26\) 7.75812 − 3.80444i 1.52149 − 0.746111i
\(27\) 0 0
\(28\) 2.09872 + 1.62530i 0.396622 + 0.307154i
\(29\) −0.949070 + 2.29126i −0.176238 + 0.425476i −0.987172 − 0.159662i \(-0.948960\pi\)
0.810934 + 0.585138i \(0.198960\pi\)
\(30\) 0 0
\(31\) 5.39395 0.968782 0.484391 − 0.874852i \(-0.339041\pi\)
0.484391 + 0.874852i \(0.339041\pi\)
\(32\) 2.55827 − 5.04532i 0.452243 − 0.891895i
\(33\) 0 0
\(34\) −8.18857 − 7.21350i −1.40433 − 1.23710i
\(35\) −0.511525 + 1.23493i −0.0864636 + 0.208741i
\(36\) 0 0
\(37\) −4.01544 + 1.66325i −0.660134 + 0.273436i −0.687495 − 0.726189i \(-0.741290\pi\)
0.0273609 + 0.999626i \(0.491290\pi\)
\(38\) 4.62674 − 2.26887i 0.750557 − 0.368059i
\(39\) 0 0
\(40\) 2.79004 + 0.574368i 0.441144 + 0.0908155i
\(41\) −6.02596 − 6.02596i −0.941096 − 0.941096i 0.0572629 − 0.998359i \(-0.481763\pi\)
−0.998359 + 0.0572629i \(0.981763\pi\)
\(42\) 0 0
\(43\) 3.38507 + 8.17229i 0.516219 + 1.24626i 0.940210 + 0.340596i \(0.110629\pi\)
−0.423991 + 0.905666i \(0.639371\pi\)
\(44\) −3.83986 + 2.19421i −0.578881 + 0.330790i
\(45\) 0 0
\(46\) −11.2779 + 0.713987i −1.66284 + 0.105272i
\(47\) 6.68653i 0.975331i 0.873031 + 0.487666i \(0.162151\pi\)
−0.873031 + 0.487666i \(0.837849\pi\)
\(48\) 0 0
\(49\) 5.23844i 0.748348i
\(50\) −0.356135 − 5.62540i −0.0503650 − 0.795552i
\(51\) 0 0
\(52\) −3.21524 + 11.7893i −0.445874 + 1.63488i
\(53\) 0.589451 + 1.42306i 0.0809673 + 0.195472i 0.959179 − 0.282800i \(-0.0912632\pi\)
−0.878212 + 0.478272i \(0.841263\pi\)
\(54\) 0 0
\(55\) −1.57474 − 1.57474i −0.212337 − 0.212337i
\(56\) −3.68668 + 0.707766i −0.492653 + 0.0945792i
\(57\) 0 0
\(58\) −1.54424 − 3.14905i −0.202768 − 0.413491i
\(59\) −7.21416 + 2.98820i −0.939204 + 0.389031i −0.799163 − 0.601115i \(-0.794723\pi\)
−0.140041 + 0.990146i \(0.544723\pi\)
\(60\) 0 0
\(61\) −3.05674 + 7.37963i −0.391376 + 0.944865i 0.598265 + 0.801298i \(0.295857\pi\)
−0.989641 + 0.143566i \(0.954143\pi\)
\(62\) −5.04238 + 5.72397i −0.640382 + 0.726946i
\(63\) 0 0
\(64\) 2.96249 + 7.43126i 0.370311 + 0.928908i
\(65\) −6.15337 −0.763232
\(66\) 0 0
\(67\) 3.77223 − 9.10696i 0.460851 − 1.11259i −0.507198 − 0.861830i \(-0.669319\pi\)
0.968049 − 0.250762i \(-0.0806813\pi\)
\(68\) 15.3097 − 1.94626i 1.85657 − 0.236019i
\(69\) 0 0
\(70\) −0.832305 − 1.69726i −0.0994795 − 0.202861i
\(71\) −1.95353 + 1.95353i −0.231842 + 0.231842i −0.813461 − 0.581619i \(-0.802419\pi\)
0.581619 + 0.813461i \(0.302419\pi\)
\(72\) 0 0
\(73\) −8.58563 − 8.58563i −1.00487 − 1.00487i −0.999988 − 0.00488401i \(-0.998445\pi\)
−0.00488401 − 0.999988i \(-0.501555\pi\)
\(74\) 1.98870 − 5.81596i 0.231182 − 0.676091i
\(75\) 0 0
\(76\) −1.91749 + 7.03081i −0.219951 + 0.806489i
\(77\) 2.71150 + 1.12314i 0.309004 + 0.127994i
\(78\) 0 0
\(79\) − 11.7674i − 1.32394i −0.749531 − 0.661969i \(-0.769721\pi\)
0.749531 − 0.661969i \(-0.230279\pi\)
\(80\) −3.21769 + 2.42381i −0.359749 + 0.270991i
\(81\) 0 0
\(82\) 12.0278 − 0.761461i 1.32825 − 0.0840894i
\(83\) −6.05690 − 2.50885i −0.664831 − 0.275382i 0.0246390 − 0.999696i \(-0.492156\pi\)
−0.689470 + 0.724314i \(0.742156\pi\)
\(84\) 0 0
\(85\) 2.97396 + 7.17978i 0.322571 + 0.778756i
\(86\) −11.8367 − 4.04744i −1.27639 − 0.436446i
\(87\) 0 0
\(88\) 1.26112 − 6.12599i 0.134436 − 0.653033i
\(89\) −5.36845 + 5.36845i −0.569055 + 0.569055i −0.931864 − 0.362809i \(-0.881818\pi\)
0.362809 + 0.931864i \(0.381818\pi\)
\(90\) 0 0
\(91\) 7.49204 − 3.10330i 0.785379 − 0.325315i
\(92\) 9.78518 − 12.6354i 1.02018 − 1.31733i
\(93\) 0 0
\(94\) −7.09564 − 6.25071i −0.731860 − 0.644711i
\(95\) −3.66972 −0.376505
\(96\) 0 0
\(97\) −8.31108 −0.843862 −0.421931 − 0.906628i \(-0.638648\pi\)
−0.421931 + 0.906628i \(0.638648\pi\)
\(98\) −5.55894 − 4.89700i −0.561538 − 0.494671i
\(99\) 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.109.10 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.23 yes 128
32.5 even 8 inner 864.2.v.a.325.10 yes 128
96.5 odd 8 inner 864.2.v.a.325.23 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.10 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.23 yes 128 3.2 odd 2 inner
864.2.v.a.325.10 yes 128 32.5 even 8 inner
864.2.v.a.325.23 yes 128 96.5 odd 8 inner