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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 109.3
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.33723 - 0.460244i) q^{2} +(1.57635 + 1.23090i) q^{4} +(-2.37116 + 0.982165i) q^{5} +(2.29537 - 2.29537i) q^{7} +(-1.54142 - 2.37150i) q^{8} +(3.62281 - 0.222067i) q^{10} +(0.00833427 + 0.0201207i) q^{11} +(-4.05547 - 1.67983i) q^{13} +(-4.12585 + 2.01300i) q^{14} +(0.969765 + 3.88066i) q^{16} +1.80284i q^{17} +(0.650151 + 0.269301i) q^{19} +(-4.94672 - 1.37042i) q^{20} +(-0.00188438 - 0.0307418i) q^{22} +(1.25771 + 1.25771i) q^{23} +(1.12220 - 1.12220i) q^{25} +(4.64996 + 4.11282i) q^{26} +(6.44367 - 0.792934i) q^{28} +(0.655744 - 1.58311i) q^{29} -1.65118 q^{31} +(0.489257 - 5.63566i) q^{32} +(0.829744 - 2.41080i) q^{34} +(-3.18824 + 7.69710i) q^{35} +(-11.0433 + 4.57430i) q^{37} +(-0.745455 - 0.659345i) q^{38} +(5.98416 + 4.10926i) q^{40} +(-3.27522 - 3.27522i) q^{41} +(-2.93287 - 7.08058i) q^{43} +(-0.0116289 + 0.0419760i) q^{44} +(-1.10299 - 2.26070i) q^{46} -9.31235i q^{47} -3.53740i q^{49} +(-2.01712 + 0.984150i) q^{50} +(-4.32514 - 7.63989i) q^{52} +(-5.35156 - 12.9198i) q^{53} +(-0.0395237 - 0.0395237i) q^{55} +(-8.98159 - 1.90533i) q^{56} +(-1.60549 + 1.81517i) q^{58} +(-8.36913 + 3.46661i) q^{59} +(-0.0511774 + 0.123553i) q^{61} +(2.20800 + 0.759946i) q^{62} +(-3.24802 + 7.31097i) q^{64} +11.2660 q^{65} +(-2.29687 + 5.54514i) q^{67} +(-2.21911 + 2.84190i) q^{68} +(7.80595 - 8.82540i) q^{70} +(-10.9401 + 10.9401i) q^{71} +(5.54998 + 5.54998i) q^{73} +(16.8727 - 1.03425i) q^{74} +(0.693383 + 1.22478i) q^{76} +(0.0653146 + 0.0270542i) q^{77} -1.71204i q^{79} +(-6.11092 - 8.24919i) q^{80} +(2.87231 + 5.88711i) q^{82} +(-8.31348 - 3.44356i) q^{83} +(-1.77068 - 4.27481i) q^{85} +(0.663121 + 10.8182i) q^{86} +(0.0348696 - 0.0507793i) q^{88} +(8.03010 - 8.03010i) q^{89} +(-13.1646 + 5.45296i) q^{91} +(0.434477 + 3.53072i) q^{92} +(-4.28595 + 12.4527i) q^{94} -1.80611 q^{95} -12.8284 q^{97} +(-1.62807 + 4.73031i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33723 − 0.460244i −0.945562 − 0.325442i
\(3\) 0 0
\(4\) 1.57635 + 1.23090i 0.788175 + 0.615451i
\(5\) −2.37116 + 0.982165i −1.06041 + 0.439238i −0.843598 − 0.536975i \(-0.819567\pi\)
−0.216815 + 0.976213i \(0.569567\pi\)
\(6\) 0 0
\(7\) 2.29537 − 2.29537i 0.867566 − 0.867566i −0.124636 − 0.992203i \(-0.539776\pi\)
0.992203 + 0.124636i \(0.0397763\pi\)
\(8\) −1.54142 − 2.37150i −0.544976 − 0.838452i
\(9\) 0 0
\(10\) 3.62281 − 0.222067i 1.14563 − 0.0702238i
\(11\) 0.00833427 + 0.0201207i 0.00251288 + 0.00606663i 0.925131 − 0.379648i \(-0.123955\pi\)
−0.922618 + 0.385715i \(0.873955\pi\)
\(12\) 0 0
\(13\) −4.05547 − 1.67983i −1.12479 − 0.465902i −0.258781 − 0.965936i \(-0.583321\pi\)
−0.866005 + 0.500035i \(0.833321\pi\)
\(14\) −4.12585 + 2.01300i −1.10268 + 0.537996i
\(15\) 0 0
\(16\) 0.969765 + 3.88066i 0.242441 + 0.970166i
\(17\) 1.80284i 0.437252i 0.975809 + 0.218626i \(0.0701575\pi\)
−0.975809 + 0.218626i \(0.929843\pi\)
\(18\) 0 0
\(19\) 0.650151 + 0.269301i 0.149155 + 0.0617820i 0.456012 − 0.889974i \(-0.349277\pi\)
−0.306857 + 0.951756i \(0.599277\pi\)
\(20\) −4.94672 − 1.37042i −1.10612 − 0.306436i
\(21\) 0 0
\(22\) −0.00188438 − 0.0307418i −0.000401750 − 0.00655417i
\(23\) 1.25771 + 1.25771i 0.262251 + 0.262251i 0.825968 − 0.563717i \(-0.190629\pi\)
−0.563717 + 0.825968i \(0.690629\pi\)
\(24\) 0 0
\(25\) 1.12220 − 1.12220i 0.224440 − 0.224440i
\(26\) 4.64996 + 4.11282i 0.911931 + 0.806591i
\(27\) 0 0
\(28\) 6.44367 − 0.792934i 1.21774 − 0.149850i
\(29\) 0.655744 − 1.58311i 0.121769 − 0.293976i −0.851227 − 0.524797i \(-0.824141\pi\)
0.972996 + 0.230821i \(0.0741413\pi\)
\(30\) 0 0
\(31\) −1.65118 −0.296561 −0.148280 − 0.988945i \(-0.547374\pi\)
−0.148280 + 0.988945i \(0.547374\pi\)
\(32\) 0.489257 − 5.63566i 0.0864892 − 0.996253i
\(33\) 0 0
\(34\) 0.829744 − 2.41080i 0.142300 − 0.413449i
\(35\) −3.18824 + 7.69710i −0.538911 + 1.30105i
\(36\) 0 0
\(37\) −11.0433 + 4.57430i −1.81551 + 0.752011i −0.836577 + 0.547850i \(0.815446\pi\)
−0.978937 + 0.204161i \(0.934554\pi\)
\(38\) −0.745455 − 0.659345i −0.120929 − 0.106960i
\(39\) 0 0
\(40\) 5.98416 + 4.10926i 0.946179 + 0.649732i
\(41\) −3.27522 − 3.27522i −0.511503 − 0.511503i 0.403484 − 0.914987i \(-0.367799\pi\)
−0.914987 + 0.403484i \(0.867799\pi\)
\(42\) 0 0
\(43\) −2.93287 − 7.08058i −0.447259 − 1.07978i −0.973345 − 0.229347i \(-0.926341\pi\)
0.526086 − 0.850432i \(-0.323659\pi\)
\(44\) −0.0116289 + 0.0419760i −0.00175312 + 0.00632812i
\(45\) 0 0
\(46\) −1.10299 − 2.26070i −0.162627 − 0.333322i
\(47\) − 9.31235i − 1.35835i −0.733978 − 0.679173i \(-0.762339\pi\)
0.733978 − 0.679173i \(-0.237661\pi\)
\(48\) 0 0
\(49\) − 3.53740i − 0.505343i
\(50\) −2.01712 + 0.984150i −0.285264 + 0.139180i
\(51\) 0 0
\(52\) −4.32514 − 7.63989i −0.599789 − 1.05946i
\(53\) −5.35156 − 12.9198i −0.735094 − 1.77467i −0.624815 − 0.780773i \(-0.714826\pi\)
−0.110279 − 0.993901i \(-0.535174\pi\)
\(54\) 0 0
\(55\) −0.0395237 − 0.0395237i −0.00532938 − 0.00532938i
\(56\) −8.98159 − 1.90533i −1.20022 − 0.254610i
\(57\) 0 0
\(58\) −1.60549 + 1.81517i −0.210812 + 0.238344i
\(59\) −8.36913 + 3.46661i −1.08957 + 0.451314i −0.853856 − 0.520509i \(-0.825742\pi\)
−0.235712 + 0.971823i \(0.575742\pi\)
\(60\) 0 0
\(61\) −0.0511774 + 0.123553i −0.00655260 + 0.0158194i −0.927122 − 0.374760i \(-0.877725\pi\)
0.920569 + 0.390579i \(0.127725\pi\)
\(62\) 2.20800 + 0.759946i 0.280417 + 0.0965133i
\(63\) 0 0
\(64\) −3.24802 + 7.31097i −0.406003 + 0.913872i
\(65\) 11.2660 1.39738
\(66\) 0 0
\(67\) −2.29687 + 5.54514i −0.280608 + 0.677447i −0.999850 − 0.0173122i \(-0.994489\pi\)
0.719242 + 0.694759i \(0.244489\pi\)
\(68\) −2.21911 + 2.84190i −0.269107 + 0.344631i
\(69\) 0 0
\(70\) 7.80595 − 8.82540i 0.932989 − 1.05484i
\(71\) −10.9401 + 10.9401i −1.29835 + 1.29835i −0.368875 + 0.929479i \(0.620257\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(72\) 0 0
\(73\) 5.54998 + 5.54998i 0.649577 + 0.649577i 0.952891 − 0.303314i \(-0.0980931\pi\)
−0.303314 + 0.952891i \(0.598093\pi\)
\(74\) 16.8727 − 1.03425i 1.96142 − 0.120229i
\(75\) 0 0
\(76\) 0.693383 + 1.22478i 0.0795364 + 0.140492i
\(77\) 0.0653146 + 0.0270542i 0.00744329 + 0.00308311i
\(78\) 0 0
\(79\) − 1.71204i − 0.192620i −0.995351 − 0.0963100i \(-0.969296\pi\)
0.995351 − 0.0963100i \(-0.0307040\pi\)
\(80\) −6.11092 − 8.24919i −0.683221 − 0.922288i
\(81\) 0 0
\(82\) 2.87231 + 5.88711i 0.317194 + 0.650123i
\(83\) −8.31348 − 3.44356i −0.912523 − 0.377979i −0.123501 − 0.992344i \(-0.539412\pi\)
−0.789022 + 0.614365i \(0.789412\pi\)
\(84\) 0 0
\(85\) −1.77068 − 4.27481i −0.192057 − 0.463668i
\(86\) 0.663121 + 10.8182i 0.0715062 + 1.16655i
\(87\) 0 0
\(88\) 0.0348696 − 0.0507793i 0.00371712 − 0.00541309i
\(89\) 8.03010 − 8.03010i 0.851189 − 0.851189i −0.139090 − 0.990280i \(-0.544418\pi\)
0.990280 + 0.139090i \(0.0444178\pi\)
\(90\) 0 0
\(91\) −13.1646 + 5.45296i −1.38003 + 0.571626i
\(92\) 0.434477 + 3.53072i 0.0452973 + 0.368103i
\(93\) 0 0
\(94\) −4.28595 + 12.4527i −0.442062 + 1.28440i
\(95\) −1.80611 −0.185303
\(96\) 0 0
\(97\) −12.8284 −1.30252 −0.651261 − 0.758854i \(-0.725760\pi\)
−0.651261 + 0.758854i \(0.725760\pi\)
\(98\) −1.62807 + 4.73031i −0.164460 + 0.477833i
\(99\) 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.109.3 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.30 yes 128
32.5 even 8 inner 864.2.v.b.325.3 yes 128
96.5 odd 8 inner 864.2.v.b.325.30 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.3 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.30 yes 128 3.2 odd 2 inner
864.2.v.b.325.3 yes 128 32.5 even 8 inner
864.2.v.b.325.30 yes 128 96.5 odd 8 inner