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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 109.2
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.39716 + 0.218946i) q^{2} +(1.90413 - 0.611807i) q^{4} +(-2.13352 + 0.883735i) q^{5} +(0.426205 - 0.426205i) q^{7} +(-2.52642 + 1.27169i) q^{8} +(2.78739 - 1.70185i) q^{10} +(-0.144287 - 0.348341i) q^{11} +(0.0788424 + 0.0326576i) q^{13} +(-0.502162 + 0.688794i) q^{14} +(3.25138 - 2.32991i) q^{16} -2.60658i q^{17} +(-2.91702 - 1.20827i) q^{19} +(-3.52182 + 2.98805i) q^{20} +(0.277861 + 0.455097i) q^{22} +(3.52617 + 3.52617i) q^{23} +(0.235404 - 0.235404i) q^{25} +(-0.117306 - 0.0283657i) q^{26} +(0.550793 - 1.07230i) q^{28} +(3.48138 - 8.40479i) q^{29} +3.54897 q^{31} +(-4.03259 + 3.96715i) q^{32} +(0.570702 + 3.64182i) q^{34} +(-0.532667 + 1.28597i) q^{35} +(6.59514 - 2.73180i) q^{37} +(4.34010 + 1.04948i) q^{38} +(4.26634 - 4.94587i) q^{40} +(3.51963 + 3.51963i) q^{41} +(-2.74958 - 6.63808i) q^{43} +(-0.487858 - 0.575008i) q^{44} +(-5.69867 - 4.15459i) q^{46} -3.40368i q^{47} +6.63670i q^{49} +(-0.277357 + 0.380439i) q^{50} +(0.170106 + 0.0139478i) q^{52} +(-1.68239 - 4.06164i) q^{53} +(0.615681 + 0.615681i) q^{55} +(-0.534770 + 1.61878i) q^{56} +(-3.02385 + 12.5051i) q^{58} +(2.70405 - 1.12005i) q^{59} +(2.88389 - 6.96232i) q^{61} +(-4.95849 + 0.777033i) q^{62} +(4.76559 - 6.42567i) q^{64} -0.197073 q^{65} +(-1.00349 + 2.42263i) q^{67} +(-1.59473 - 4.96326i) q^{68} +(0.462664 - 1.91334i) q^{70} +(9.57606 - 9.57606i) q^{71} +(7.59378 + 7.59378i) q^{73} +(-8.61637 + 5.26075i) q^{74} +(-6.29361 - 0.516044i) q^{76} +(-0.209961 - 0.0869686i) q^{77} -13.3063i q^{79} +(-4.87788 + 7.84429i) q^{80} +(-5.68810 - 4.14688i) q^{82} +(2.41425 + 1.00002i) q^{83} +(2.30353 + 5.56121i) q^{85} +(5.29500 + 8.67247i) q^{86} +(0.807513 + 0.696565i) q^{88} +(5.44699 - 5.44699i) q^{89} +(0.0475219 - 0.0196842i) q^{91} +(8.87160 + 4.55693i) q^{92} +(0.745224 + 4.75550i) q^{94} +7.29133 q^{95} +5.79690 q^{97} +(-1.45308 - 9.27254i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39716 + 0.218946i −0.987943 + 0.154818i
\(3\) 0 0
\(4\) 1.90413 − 0.611807i 0.952063 − 0.305903i
\(5\) −2.13352 + 0.883735i −0.954141 + 0.395218i −0.804786 − 0.593565i \(-0.797720\pi\)
−0.149355 + 0.988784i \(0.547720\pi\)
\(6\) 0 0
\(7\) 0.426205 − 0.426205i 0.161091 − 0.161091i −0.621959 − 0.783050i \(-0.713663\pi\)
0.783050 + 0.621959i \(0.213663\pi\)
\(8\) −2.52642 + 1.27169i −0.893224 + 0.449612i
\(9\) 0 0
\(10\) 2.78739 − 1.70185i 0.881450 − 0.538172i
\(11\) −0.144287 − 0.348341i −0.0435043 − 0.105029i 0.900634 − 0.434579i \(-0.143103\pi\)
−0.944138 + 0.329550i \(0.893103\pi\)
\(12\) 0 0
\(13\) 0.0788424 + 0.0326576i 0.0218669 + 0.00905758i 0.393590 − 0.919286i \(-0.371233\pi\)
−0.371723 + 0.928344i \(0.621233\pi\)
\(14\) −0.502162 + 0.688794i −0.134208 + 0.184088i
\(15\) 0 0
\(16\) 3.25138 − 2.32991i 0.812846 − 0.582478i
\(17\) − 2.60658i − 0.632189i −0.948728 − 0.316095i \(-0.897628\pi\)
0.948728 − 0.316095i \(-0.102372\pi\)
\(18\) 0 0
\(19\) −2.91702 − 1.20827i −0.669211 − 0.277196i 0.0220975 − 0.999756i \(-0.492966\pi\)
−0.691309 + 0.722559i \(0.742966\pi\)
\(20\) −3.52182 + 2.98805i −0.787503 + 0.668147i
\(21\) 0 0
\(22\) 0.277861 + 0.455097i 0.0592401 + 0.0970270i
\(23\) 3.52617 + 3.52617i 0.735257 + 0.735257i 0.971656 − 0.236399i \(-0.0759672\pi\)
−0.236399 + 0.971656i \(0.575967\pi\)
\(24\) 0 0
\(25\) 0.235404 − 0.235404i 0.0470809 − 0.0470809i
\(26\) −0.117306 − 0.0283657i −0.0230056 − 0.00556297i
\(27\) 0 0
\(28\) 0.550793 − 1.07230i 0.104090 − 0.202646i
\(29\) 3.48138 − 8.40479i 0.646475 − 1.56073i −0.171317 − 0.985216i \(-0.554802\pi\)
0.817792 − 0.575513i \(-0.195198\pi\)
\(30\) 0 0
\(31\) 3.54897 0.637414 0.318707 − 0.947853i \(-0.396751\pi\)
0.318707 + 0.947853i \(0.396751\pi\)
\(32\) −4.03259 + 3.96715i −0.712867 + 0.701299i
\(33\) 0 0
\(34\) 0.570702 + 3.64182i 0.0978745 + 0.624567i
\(35\) −0.532667 + 1.28597i −0.0900372 + 0.217369i
\(36\) 0 0
\(37\) 6.59514 − 2.73180i 1.08423 − 0.449105i 0.232242 − 0.972658i \(-0.425394\pi\)
0.851993 + 0.523553i \(0.175394\pi\)
\(38\) 4.34010 + 1.04948i 0.704058 + 0.170248i
\(39\) 0 0
\(40\) 4.26634 − 4.94587i 0.674567 − 0.782011i
\(41\) 3.51963 + 3.51963i 0.549674 + 0.549674i 0.926346 − 0.376673i \(-0.122932\pi\)
−0.376673 + 0.926346i \(0.622932\pi\)
\(42\) 0 0
\(43\) −2.74958 − 6.63808i −0.419307 − 1.01230i −0.982549 − 0.186005i \(-0.940446\pi\)
0.563241 − 0.826292i \(-0.309554\pi\)
\(44\) −0.487858 − 0.575008i −0.0735474 − 0.0866857i
\(45\) 0 0
\(46\) −5.69867 − 4.15459i −0.840223 − 0.612561i
\(47\) − 3.40368i − 0.496478i −0.968699 − 0.248239i \(-0.920148\pi\)
0.968699 − 0.248239i \(-0.0798519\pi\)
\(48\) 0 0
\(49\) 6.63670i 0.948100i
\(50\) −0.277357 + 0.380439i −0.0392243 + 0.0538022i
\(51\) 0 0
\(52\) 0.170106 + 0.0139478i 0.0235894 + 0.00193421i
\(53\) −1.68239 − 4.06164i −0.231094 − 0.557910i 0.765213 − 0.643777i \(-0.222634\pi\)
−0.996307 + 0.0858677i \(0.972634\pi\)
\(54\) 0 0
\(55\) 0.615681 + 0.615681i 0.0830184 + 0.0830184i
\(56\) −0.534770 + 1.61878i −0.0714617 + 0.216318i
\(57\) 0 0
\(58\) −3.02385 + 12.5051i −0.397051 + 1.64200i
\(59\) 2.70405 − 1.12005i 0.352037 − 0.145819i −0.199655 − 0.979866i \(-0.563982\pi\)
0.551693 + 0.834048i \(0.313982\pi\)
\(60\) 0 0
\(61\) 2.88389 − 6.96232i 0.369244 − 0.891434i −0.624631 − 0.780920i \(-0.714750\pi\)
0.993875 − 0.110513i \(-0.0352495\pi\)
\(62\) −4.95849 + 0.777033i −0.629728 + 0.0986833i
\(63\) 0 0
\(64\) 4.76559 − 6.42567i 0.595698 − 0.803208i
\(65\) −0.197073 −0.0244439
\(66\) 0 0
\(67\) −1.00349 + 2.42263i −0.122595 + 0.295971i −0.973248 − 0.229757i \(-0.926207\pi\)
0.850653 + 0.525728i \(0.176207\pi\)
\(68\) −1.59473 − 4.96326i −0.193389 − 0.601884i
\(69\) 0 0
\(70\) 0.462664 − 1.91334i 0.0552989 − 0.228688i
\(71\) 9.57606 − 9.57606i 1.13647 − 1.13647i 0.147391 − 0.989078i \(-0.452912\pi\)
0.989078 − 0.147391i \(-0.0470877\pi\)
\(72\) 0 0
\(73\) 7.59378 + 7.59378i 0.888784 + 0.888784i 0.994406 − 0.105622i \(-0.0336833\pi\)
−0.105622 + 0.994406i \(0.533683\pi\)
\(74\) −8.61637 + 5.26075i −1.00163 + 0.611549i
\(75\) 0 0
\(76\) −6.29361 − 0.516044i −0.721926 − 0.0591943i
\(77\) −0.209961 − 0.0869686i −0.0239272 − 0.00991099i
\(78\) 0 0
\(79\) − 13.3063i − 1.49708i −0.663090 − 0.748540i \(-0.730755\pi\)
0.663090 − 0.748540i \(-0.269245\pi\)
\(80\) −4.87788 + 7.84429i −0.545364 + 0.877018i
\(81\) 0 0
\(82\) −5.68810 − 4.14688i −0.628146 − 0.457947i
\(83\) 2.41425 + 1.00002i 0.264998 + 0.109766i 0.511227 − 0.859446i \(-0.329191\pi\)
−0.246228 + 0.969212i \(0.579191\pi\)
\(84\) 0 0
\(85\) 2.30353 + 5.56121i 0.249853 + 0.603198i
\(86\) 5.29500 + 8.67247i 0.570974 + 0.935176i
\(87\) 0 0
\(88\) 0.807513 + 0.696565i 0.0860812 + 0.0742541i
\(89\) 5.44699 − 5.44699i 0.577380 − 0.577380i −0.356801 − 0.934181i \(-0.616132\pi\)
0.934181 + 0.356801i \(0.116132\pi\)
\(90\) 0 0
\(91\) 0.0475219 − 0.0196842i 0.00498165 − 0.00206347i
\(92\) 8.87160 + 4.55693i 0.924929 + 0.475093i
\(93\) 0 0
\(94\) 0.745224 + 4.75550i 0.0768639 + 0.490492i
\(95\) 7.29133 0.748075
\(96\) 0 0
\(97\) 5.79690 0.588586 0.294293 − 0.955715i \(-0.404916\pi\)
0.294293 + 0.955715i \(0.404916\pi\)
\(98\) −1.45308 − 9.27254i −0.146783 − 0.936668i
\(99\) 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.109.2 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.31 yes 128
32.5 even 8 inner 864.2.v.a.325.2 yes 128
96.5 odd 8 inner 864.2.v.a.325.31 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.2 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.31 yes 128 3.2 odd 2 inner
864.2.v.a.325.2 yes 128 32.5 even 8 inner
864.2.v.a.325.31 yes 128 96.5 odd 8 inner