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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 109.5
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.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)
 
\(f(q)\) \(=\) \(q+(-1.19327 + 0.759019i) q^{2} +(0.847781 - 1.81143i) q^{4} +(2.08348 - 0.863006i) q^{5} +(-2.85037 + 2.85037i) q^{7} +(0.363276 + 2.80500i) q^{8} +(-1.83111 + 2.61120i) q^{10} +(-1.60584 - 3.87685i) q^{11} +(-4.07396 - 1.68749i) q^{13} +(1.23777 - 5.56473i) q^{14} +(-2.56253 - 3.07139i) q^{16} -2.43357i q^{17} +(5.59078 + 2.31578i) q^{19} +(0.203063 - 4.50571i) q^{20} +(4.85880 + 3.40725i) q^{22} +(-3.00850 - 3.00850i) q^{23} +(0.0605754 - 0.0605754i) q^{25} +(6.14217 - 1.07858i) q^{26} +(2.74674 + 7.57972i) q^{28} +(3.80631 - 9.18924i) q^{29} -8.41072 q^{31} +(5.38903 + 1.71998i) q^{32} +(1.84713 + 2.90390i) q^{34} +(-3.47880 + 8.39856i) q^{35} +(-0.565027 + 0.234042i) q^{37} +(-8.42902 + 1.48016i) q^{38} +(3.17761 + 5.53065i) q^{40} +(-0.301316 - 0.301316i) q^{41} +(-2.97363 - 7.17897i) q^{43} +(-8.38402 - 0.377851i) q^{44} +(5.87346 + 1.30644i) q^{46} +8.16538i q^{47} -9.24917i q^{49} +(-0.0263049 + 0.118261i) q^{50} +(-6.51059 + 5.94906i) q^{52} +(-4.15977 - 10.0426i) q^{53} +(-6.69148 - 6.69148i) q^{55} +(-9.03075 - 6.95981i) q^{56} +(2.43286 + 13.8543i) q^{58} +(-1.89160 + 0.783525i) q^{59} +(2.77585 - 6.70150i) q^{61} +(10.0362 - 6.38389i) q^{62} +(-7.73606 + 2.03798i) q^{64} -9.94433 q^{65} +(-3.15764 + 7.62322i) q^{67} +(-4.40823 - 2.06313i) q^{68} +(-2.22352 - 12.6622i) q^{70} +(-3.64048 + 3.64048i) q^{71} +(2.01970 + 2.01970i) q^{73} +(0.496587 - 0.708141i) q^{74} +(8.93461 - 8.16401i) q^{76} +(15.6277 + 6.47319i) q^{77} -14.0780i q^{79} +(-7.98961 - 4.18769i) q^{80} +(0.588256 + 0.130847i) q^{82} +(9.22437 + 3.82086i) q^{83} +(-2.10018 - 5.07029i) q^{85} +(8.99730 + 6.30940i) q^{86} +(10.2912 - 5.91275i) q^{88} +(6.86945 - 6.86945i) q^{89} +(16.4222 - 6.80232i) q^{91} +(-8.00023 + 2.89913i) q^{92} +(-6.19768 - 9.74350i) q^{94} +13.6468 q^{95} -0.463135 q^{97} +(7.02029 + 11.0367i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.19327 + 0.759019i −0.843769 + 0.536707i
\(3\) 0 0
\(4\) 0.847781 − 1.81143i 0.423891 − 0.905713i
\(5\) 2.08348 − 0.863006i 0.931760 − 0.385948i 0.135414 − 0.990789i \(-0.456763\pi\)
0.796346 + 0.604841i \(0.206763\pi\)
\(6\) 0 0
\(7\) −2.85037 + 2.85037i −1.07734 + 1.07734i −0.0805896 + 0.996747i \(0.525680\pi\)
−0.996747 + 0.0805896i \(0.974320\pi\)
\(8\) 0.363276 + 2.80500i 0.128437 + 0.991718i
\(9\) 0 0
\(10\) −1.83111 + 2.61120i −0.579049 + 0.825733i
\(11\) −1.60584 − 3.87685i −0.484180 − 1.16891i −0.957606 − 0.288080i \(-0.906983\pi\)
0.473427 − 0.880833i \(-0.343017\pi\)
\(12\) 0 0
\(13\) −4.07396 − 1.68749i −1.12991 − 0.468026i −0.262162 − 0.965024i \(-0.584436\pi\)
−0.867752 + 0.496998i \(0.834436\pi\)
\(14\) 1.23777 − 5.56473i 0.330808 − 1.48724i
\(15\) 0 0
\(16\) −2.56253 − 3.07139i −0.640634 − 0.767847i
\(17\) − 2.43357i − 0.590227i −0.955462 − 0.295114i \(-0.904642\pi\)
0.955462 − 0.295114i \(-0.0953575\pi\)
\(18\) 0 0
\(19\) 5.59078 + 2.31578i 1.28261 + 0.531275i 0.916775 − 0.399403i \(-0.130783\pi\)
0.365837 + 0.930679i \(0.380783\pi\)
\(20\) 0.203063 − 4.50571i 0.0454064 − 1.00751i
\(21\) 0 0
\(22\) 4.85880 + 3.40725i 1.03590 + 0.726429i
\(23\) −3.00850 − 3.00850i −0.627316 − 0.627316i 0.320076 − 0.947392i \(-0.396292\pi\)
−0.947392 + 0.320076i \(0.896292\pi\)
\(24\) 0 0
\(25\) 0.0605754 − 0.0605754i 0.0121151 − 0.0121151i
\(26\) 6.14217 − 1.07858i 1.20458 − 0.211528i
\(27\) 0 0
\(28\) 2.74674 + 7.57972i 0.519086 + 1.43243i
\(29\) 3.80631 − 9.18924i 0.706813 − 1.70640i −0.00100092 − 0.999999i \(-0.500319\pi\)
0.707814 − 0.706399i \(-0.249681\pi\)
\(30\) 0 0
\(31\) −8.41072 −1.51061 −0.755305 − 0.655374i \(-0.772511\pi\)
−0.755305 + 0.655374i \(0.772511\pi\)
\(32\) 5.38903 + 1.71998i 0.952655 + 0.304052i
\(33\) 0 0
\(34\) 1.84713 + 2.90390i 0.316779 + 0.498015i
\(35\) −3.47880 + 8.39856i −0.588024 + 1.41962i
\(36\) 0 0
\(37\) −0.565027 + 0.234042i −0.0928899 + 0.0384762i −0.428644 − 0.903473i \(-0.641009\pi\)
0.335755 + 0.941949i \(0.391009\pi\)
\(38\) −8.42902 + 1.48016i −1.36737 + 0.240114i
\(39\) 0 0
\(40\) 3.17761 + 5.53065i 0.502424 + 0.874473i
\(41\) −0.301316 − 0.301316i −0.0470577 − 0.0470577i 0.683186 − 0.730244i \(-0.260594\pi\)
−0.730244 + 0.683186i \(0.760594\pi\)
\(42\) 0 0
\(43\) −2.97363 − 7.17897i −0.453474 − 1.09478i −0.970992 − 0.239110i \(-0.923144\pi\)
0.517519 − 0.855672i \(-0.326856\pi\)
\(44\) −8.38402 − 0.377851i −1.26394 − 0.0569632i
\(45\) 0 0
\(46\) 5.87346 + 1.30644i 0.865994 + 0.192624i
\(47\) 8.16538i 1.19104i 0.803339 + 0.595522i \(0.203055\pi\)
−0.803339 + 0.595522i \(0.796945\pi\)
\(48\) 0 0
\(49\) − 9.24917i − 1.32131i
\(50\) −0.0263049 + 0.118261i −0.00372007 + 0.0167246i
\(51\) 0 0
\(52\) −6.51059 + 5.94906i −0.902857 + 0.824986i
\(53\) −4.15977 − 10.0426i −0.571389 − 1.37945i −0.900373 − 0.435119i \(-0.856706\pi\)
0.328984 − 0.944336i \(-0.393294\pi\)
\(54\) 0 0
\(55\) −6.69148 − 6.69148i −0.902279 − 0.902279i
\(56\) −9.03075 − 6.95981i −1.20678 − 0.930044i
\(57\) 0 0
\(58\) 2.43286 + 13.8543i 0.319450 + 1.81916i
\(59\) −1.89160 + 0.783525i −0.246265 + 0.102006i −0.502402 − 0.864634i \(-0.667550\pi\)
0.256137 + 0.966640i \(0.417550\pi\)
\(60\) 0 0
\(61\) 2.77585 − 6.70150i 0.355412 − 0.858040i −0.640521 − 0.767941i \(-0.721282\pi\)
0.995933 − 0.0900992i \(-0.0287184\pi\)
\(62\) 10.0362 − 6.38389i 1.27460 − 0.810755i
\(63\) 0 0
\(64\) −7.73606 + 2.03798i −0.967008 + 0.254747i
\(65\) −9.94433 −1.23344
\(66\) 0 0
\(67\) −3.15764 + 7.62322i −0.385767 + 0.931325i 0.605059 + 0.796181i \(0.293150\pi\)
−0.990826 + 0.135144i \(0.956850\pi\)
\(68\) −4.40823 − 2.06313i −0.534577 − 0.250192i
\(69\) 0 0
\(70\) −2.22352 − 12.6622i −0.265762 − 1.51342i
\(71\) −3.64048 + 3.64048i −0.432045 + 0.432045i −0.889324 − 0.457278i \(-0.848824\pi\)
0.457278 + 0.889324i \(0.348824\pi\)
\(72\) 0 0
\(73\) 2.01970 + 2.01970i 0.236388 + 0.236388i 0.815353 − 0.578965i \(-0.196543\pi\)
−0.578965 + 0.815353i \(0.696543\pi\)
\(74\) 0.496587 − 0.708141i 0.0577271 − 0.0823197i
\(75\) 0 0
\(76\) 8.93461 − 8.16401i 1.02487 − 0.936477i
\(77\) 15.6277 + 6.47319i 1.78094 + 0.737688i
\(78\) 0 0
\(79\) − 14.0780i − 1.58390i −0.610585 − 0.791951i \(-0.709065\pi\)
0.610585 − 0.791951i \(-0.290935\pi\)
\(80\) −7.98961 − 4.18769i −0.893266 − 0.468198i
\(81\) 0 0
\(82\) 0.588256 + 0.130847i 0.0649620 + 0.0144496i
\(83\) 9.22437 + 3.82086i 1.01251 + 0.419394i 0.826369 − 0.563130i \(-0.190403\pi\)
0.186138 + 0.982524i \(0.440403\pi\)
\(84\) 0 0
\(85\) −2.10018 − 5.07029i −0.227797 − 0.549951i
\(86\) 8.99730 + 6.30940i 0.970204 + 0.680360i
\(87\) 0 0
\(88\) 10.2912 − 5.91275i 1.09704 − 0.630302i
\(89\) 6.86945 − 6.86945i 0.728160 − 0.728160i −0.242093 − 0.970253i \(-0.577834\pi\)
0.970253 + 0.242093i \(0.0778339\pi\)
\(90\) 0 0
\(91\) 16.4222 − 6.80232i 1.72152 − 0.713076i
\(92\) −8.00023 + 2.89913i −0.834082 + 0.302255i
\(93\) 0 0
\(94\) −6.19768 − 9.74350i −0.639242 − 1.00496i
\(95\) 13.6468 1.40013
\(96\) 0 0
\(97\) −0.463135 −0.0470242 −0.0235121 − 0.999724i \(-0.507485\pi\)
−0.0235121 + 0.999724i \(0.507485\pi\)
\(98\) 7.02029 + 11.0367i 0.709157 + 1.11488i
\(99\) 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.109.5 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.28 yes 128
32.5 even 8 inner 864.2.v.b.325.5 yes 128
96.5 odd 8 inner 864.2.v.b.325.28 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.5 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.28 yes 128 3.2 odd 2 inner
864.2.v.b.325.5 yes 128 32.5 even 8 inner
864.2.v.b.325.28 yes 128 96.5 odd 8 inner