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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 109.15
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.114947 + 1.40953i) q^{2} +(-1.97357 - 0.324044i) q^{4} +(-2.86147 + 1.18526i) q^{5} +(-2.81522 + 2.81522i) q^{7} +(0.683608 - 2.74457i) q^{8} +(-1.34175 - 4.16959i) q^{10} +(-1.61200 - 3.89171i) q^{11} +(0.251988 + 0.104377i) q^{13} +(-3.64455 - 4.29176i) q^{14} +(3.78999 + 1.27905i) q^{16} -4.45400i q^{17} +(6.24427 + 2.58646i) q^{19} +(6.03140 - 1.41196i) q^{20} +(5.67079 - 1.82483i) q^{22} +(1.13166 + 1.13166i) q^{23} +(3.24764 - 3.24764i) q^{25} +(-0.176088 + 0.343189i) q^{26} +(6.46831 - 4.64380i) q^{28} +(-1.24837 + 3.01383i) q^{29} -6.19011 q^{31} +(-2.23851 + 5.19510i) q^{32} +(6.27807 + 0.511975i) q^{34} +(4.71891 - 11.3925i) q^{35} +(3.63120 - 1.50409i) q^{37} +(-4.36347 + 8.50420i) q^{38} +(1.29691 + 8.66377i) q^{40} +(2.17515 + 2.17515i) q^{41} +(2.48476 + 5.99874i) q^{43} +(1.92031 + 8.20293i) q^{44} +(-1.72519 + 1.46503i) q^{46} -5.97132i q^{47} -8.85098i q^{49} +(4.20436 + 4.95097i) q^{50} +(-0.463495 - 0.287651i) q^{52} +(-4.91142 - 11.8572i) q^{53} +(9.22538 + 9.22538i) q^{55} +(5.80208 + 9.65110i) q^{56} +(-4.10459 - 2.10605i) q^{58} +(8.72352 - 3.61340i) q^{59} +(4.24159 - 10.2401i) q^{61} +(0.711536 - 8.72517i) q^{62} +(-7.06536 - 3.75243i) q^{64} -0.844772 q^{65} +(5.46508 - 13.1939i) q^{67} +(-1.44329 + 8.79031i) q^{68} +(15.5156 + 7.96100i) q^{70} +(-4.47445 + 4.47445i) q^{71} +(-7.25497 - 7.25497i) q^{73} +(1.70267 + 5.29119i) q^{74} +(-11.4854 - 7.12799i) q^{76} +(15.4942 + 6.41790i) q^{77} -6.73384i q^{79} +(-12.3610 + 0.832159i) q^{80} +(-3.31598 + 2.81593i) q^{82} +(-11.1232 - 4.60738i) q^{83} +(5.27915 + 12.7450i) q^{85} +(-8.74105 + 2.81282i) q^{86} +(-11.7831 + 1.76384i) q^{88} +(0.420332 - 0.420332i) q^{89} +(-1.00325 + 0.415559i) q^{91} +(-1.86670 - 2.60011i) q^{92} +(8.41678 + 0.686387i) q^{94} -20.9334 q^{95} +0.576440 q^{97} +(12.4758 + 1.01740i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.114947 + 1.40953i −0.0812800 + 0.996691i
\(3\) 0 0
\(4\) −1.97357 − 0.324044i −0.986787 − 0.162022i
\(5\) −2.86147 + 1.18526i −1.27969 + 0.530065i −0.915896 − 0.401415i \(-0.868518\pi\)
−0.363793 + 0.931480i \(0.618518\pi\)
\(6\) 0 0
\(7\) −2.81522 + 2.81522i −1.06405 + 1.06405i −0.0662521 + 0.997803i \(0.521104\pi\)
−0.997803 + 0.0662521i \(0.978896\pi\)
\(8\) 0.683608 − 2.74457i 0.241692 − 0.970353i
\(9\) 0 0
\(10\) −1.34175 − 4.16959i −0.424298 − 1.31854i
\(11\) −1.61200 − 3.89171i −0.486036 − 1.17339i −0.956698 − 0.291081i \(-0.905985\pi\)
0.470663 − 0.882313i \(-0.344015\pi\)
\(12\) 0 0
\(13\) 0.251988 + 0.104377i 0.0698890 + 0.0289490i 0.417354 − 0.908744i \(-0.362957\pi\)
−0.347465 + 0.937693i \(0.612957\pi\)
\(14\) −3.64455 − 4.29176i −0.974048 − 1.14702i
\(15\) 0 0
\(16\) 3.78999 + 1.27905i 0.947498 + 0.319763i
\(17\) − 4.45400i − 1.08025i −0.841583 − 0.540127i \(-0.818376\pi\)
0.841583 − 0.540127i \(-0.181624\pi\)
\(18\) 0 0
\(19\) 6.24427 + 2.58646i 1.43253 + 0.593375i 0.957976 − 0.286850i \(-0.0926080\pi\)
0.474558 + 0.880224i \(0.342608\pi\)
\(20\) 6.03140 − 1.41196i 1.34866 − 0.315723i
\(21\) 0 0
\(22\) 5.67079 − 1.82483i 1.20902 − 0.389054i
\(23\) 1.13166 + 1.13166i 0.235967 + 0.235967i 0.815178 − 0.579211i \(-0.196639\pi\)
−0.579211 + 0.815178i \(0.696639\pi\)
\(24\) 0 0
\(25\) 3.24764 − 3.24764i 0.649529 − 0.649529i
\(26\) −0.176088 + 0.343189i −0.0345338 + 0.0673048i
\(27\) 0 0
\(28\) 6.46831 − 4.64380i 1.22240 − 0.877595i
\(29\) −1.24837 + 3.01383i −0.231816 + 0.559653i −0.996391 − 0.0848813i \(-0.972949\pi\)
0.764575 + 0.644535i \(0.222949\pi\)
\(30\) 0 0
\(31\) −6.19011 −1.11178 −0.555888 − 0.831257i \(-0.687622\pi\)
−0.555888 + 0.831257i \(0.687622\pi\)
\(32\) −2.23851 + 5.19510i −0.395717 + 0.918372i
\(33\) 0 0
\(34\) 6.27807 + 0.511975i 1.07668 + 0.0878031i
\(35\) 4.71891 − 11.3925i 0.797642 − 1.92568i
\(36\) 0 0
\(37\) 3.63120 − 1.50409i 0.596966 − 0.247271i −0.0636785 − 0.997970i \(-0.520283\pi\)
0.660644 + 0.750699i \(0.270283\pi\)
\(38\) −4.36347 + 8.50420i −0.707848 + 1.37956i
\(39\) 0 0
\(40\) 1.29691 + 8.66377i 0.205059 + 1.36986i
\(41\) 2.17515 + 2.17515i 0.339702 + 0.339702i 0.856255 − 0.516553i \(-0.172785\pi\)
−0.516553 + 0.856255i \(0.672785\pi\)
\(42\) 0 0
\(43\) 2.48476 + 5.99874i 0.378922 + 0.914799i 0.992168 + 0.124907i \(0.0398634\pi\)
−0.613246 + 0.789892i \(0.710137\pi\)
\(44\) 1.92031 + 8.20293i 0.289498 + 1.23664i
\(45\) 0 0
\(46\) −1.72519 + 1.46503i −0.254365 + 0.216006i
\(47\) − 5.97132i − 0.871007i −0.900187 − 0.435503i \(-0.856570\pi\)
0.900187 − 0.435503i \(-0.143430\pi\)
\(48\) 0 0
\(49\) − 8.85098i − 1.26443i
\(50\) 4.20436 + 4.95097i 0.594586 + 0.700173i
\(51\) 0 0
\(52\) −0.463495 − 0.287651i −0.0642752 − 0.0398901i
\(53\) −4.91142 − 11.8572i −0.674635 − 1.62871i −0.773640 − 0.633626i \(-0.781566\pi\)
0.0990049 − 0.995087i \(-0.468434\pi\)
\(54\) 0 0
\(55\) 9.22538 + 9.22538i 1.24395 + 1.24395i
\(56\) 5.80208 + 9.65110i 0.775335 + 1.28968i
\(57\) 0 0
\(58\) −4.10459 − 2.10605i −0.538960 − 0.276538i
\(59\) 8.72352 − 3.61340i 1.13571 − 0.470425i 0.265989 − 0.963976i \(-0.414301\pi\)
0.869717 + 0.493551i \(0.164301\pi\)
\(60\) 0 0
\(61\) 4.24159 − 10.2401i 0.543080 − 1.31111i −0.379460 − 0.925208i \(-0.623890\pi\)
0.922540 − 0.385902i \(-0.126110\pi\)
\(62\) 0.711536 − 8.72517i 0.0903651 − 1.10810i
\(63\) 0 0
\(64\) −7.06536 − 3.75243i −0.883170 − 0.469053i
\(65\) −0.844772 −0.104781
\(66\) 0 0
\(67\) 5.46508 − 13.1939i 0.667666 − 1.61189i −0.117840 − 0.993033i \(-0.537597\pi\)
0.785505 − 0.618855i \(-0.212403\pi\)
\(68\) −1.44329 + 8.79031i −0.175025 + 1.06598i
\(69\) 0 0
\(70\) 15.5156 + 7.96100i 1.85447 + 0.951522i
\(71\) −4.47445 + 4.47445i −0.531019 + 0.531019i −0.920876 − 0.389856i \(-0.872525\pi\)
0.389856 + 0.920876i \(0.372525\pi\)
\(72\) 0 0
\(73\) −7.25497 − 7.25497i −0.849130 − 0.849130i 0.140895 − 0.990025i \(-0.455002\pi\)
−0.990025 + 0.140895i \(0.955002\pi\)
\(74\) 1.70267 + 5.29119i 0.197932 + 0.615089i
\(75\) 0 0
\(76\) −11.4854 − 7.12799i −1.31747 − 0.817637i
\(77\) 15.4942 + 6.41790i 1.76572 + 0.731387i
\(78\) 0 0
\(79\) − 6.73384i − 0.757616i −0.925475 − 0.378808i \(-0.876334\pi\)
0.925475 − 0.378808i \(-0.123666\pi\)
\(80\) −12.3610 + 0.832159i −1.38200 + 0.0930382i
\(81\) 0 0
\(82\) −3.31598 + 2.81593i −0.366189 + 0.310967i
\(83\) −11.1232 − 4.60738i −1.22093 − 0.505726i −0.323225 − 0.946322i \(-0.604767\pi\)
−0.897706 + 0.440596i \(0.854767\pi\)
\(84\) 0 0
\(85\) 5.27915 + 12.7450i 0.572605 + 1.38239i
\(86\) −8.74105 + 2.81282i −0.942571 + 0.303314i
\(87\) 0 0
\(88\) −11.7831 + 1.76384i −1.25608 + 0.188026i
\(89\) 0.420332 − 0.420332i 0.0445552 − 0.0445552i −0.684478 − 0.729033i \(-0.739970\pi\)
0.729033 + 0.684478i \(0.239970\pi\)
\(90\) 0 0
\(91\) −1.00325 + 0.415559i −0.105169 + 0.0435625i
\(92\) −1.86670 − 2.60011i −0.194617 − 0.271081i
\(93\) 0 0
\(94\) 8.41678 + 0.686387i 0.868125 + 0.0707954i
\(95\) −20.9334 −2.14772
\(96\) 0 0
\(97\) 0.576440 0.0585286 0.0292643 − 0.999572i \(-0.490684\pi\)
0.0292643 + 0.999572i \(0.490684\pi\)
\(98\) 12.4758 + 1.01740i 1.26024 + 0.102773i
\(99\) 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.109.15 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.18 yes 128
32.5 even 8 inner 864.2.v.a.325.15 yes 128
96.5 odd 8 inner 864.2.v.a.325.18 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.15 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.18 yes 128 3.2 odd 2 inner
864.2.v.a.325.15 yes 128 32.5 even 8 inner
864.2.v.a.325.18 yes 128 96.5 odd 8 inner