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Properties

Label 2064.2.d.a.431.8
Level $2064$
Weight $2$
Character 2064.431
Analytic conductor $16.481$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2064,2,Mod(431,2064)] 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("2064.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2064, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2064 = 2^{4} \cdot 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2064.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.4811229772\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.8
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2064.431
Dual form 2064.2.d.a.431.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.41421 + 1.00000i) q^{3} +1.93185i q^{5} -4.46410i q^{7} +(1.00000 + 2.82843i) q^{9} +2.96713 q^{11} +5.73205 q^{13} +(-1.93185 + 2.73205i) q^{15} -6.31319i q^{17} -5.00000i q^{19} +(4.46410 - 6.31319i) q^{21} -6.31319 q^{23} +1.26795 q^{25} +(-1.41421 + 5.00000i) q^{27} +1.27551i q^{29} -10.1962i q^{31} +(4.19615 + 2.96713i) q^{33} +8.62398 q^{35} -3.26795 q^{37} +(8.10634 + 5.73205i) q^{39} -3.10583i q^{41} +1.00000i q^{43} +(-5.46410 + 1.93185i) q^{45} -5.13922 q^{47} -12.9282 q^{49} +(6.31319 - 8.92820i) q^{51} +10.5558i q^{53} +5.73205i q^{55} +(5.00000 - 7.07107i) q^{57} +14.0406 q^{59} -1.07180 q^{61} +(12.6264 - 4.46410i) q^{63} +11.0735i q^{65} +8.00000i q^{67} +(-8.92820 - 6.31319i) q^{69} -0.378937 q^{71} +1.80385 q^{73} +(1.79315 + 1.26795i) q^{75} -13.2456i q^{77} -1.80385i q^{79} +(-7.00000 + 5.65685i) q^{81} +12.1087 q^{83} +12.1962 q^{85} +(-1.27551 + 1.80385i) q^{87} +13.0053i q^{89} -25.5885i q^{91} +(10.1962 - 14.4195i) q^{93} +9.65926 q^{95} +10.2679 q^{97} +(2.96713 + 8.39230i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + 32 q^{13} + 8 q^{21} + 24 q^{25} - 8 q^{33} - 40 q^{37} - 16 q^{45} - 48 q^{49} + 40 q^{57} - 64 q^{61} - 16 q^{69} + 56 q^{73} - 56 q^{81} + 56 q^{85} + 40 q^{93} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2064\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(517\) \(689\) \(1807\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) 1.41421 + 1.00000i 0.816497 + 0.577350i
\(4\) 0 0
\(5\) 1.93185i 0.863950i 0.901886 + 0.431975i \(0.142183\pi\)
−0.901886 + 0.431975i \(0.857817\pi\)
\(6\) 0 0
\(7\) − 4.46410i − 1.68727i −0.536916 − 0.843636i \(-0.680411\pi\)
0.536916 − 0.843636i \(-0.319589\pi\)
\(8\) 0 0
\(9\) 1.00000 + 2.82843i 0.333333 + 0.942809i
\(10\) 0 0
\(11\) 2.96713 0.894623 0.447311 − 0.894378i \(-0.352382\pi\)
0.447311 + 0.894378i \(0.352382\pi\)
\(12\) 0 0
\(13\) 5.73205 1.58978 0.794892 − 0.606750i \(-0.207527\pi\)
0.794892 + 0.606750i \(0.207527\pi\)
\(14\) 0 0
\(15\) −1.93185 + 2.73205i −0.498802 + 0.705412i
\(16\) 0 0
\(17\) − 6.31319i − 1.53117i −0.643332 − 0.765587i \(-0.722449\pi\)
0.643332 − 0.765587i \(-0.277551\pi\)
\(18\) 0 0
\(19\) − 5.00000i − 1.14708i −0.819178 − 0.573539i \(-0.805570\pi\)
0.819178 − 0.573539i \(-0.194430\pi\)
\(20\) 0 0
\(21\) 4.46410 − 6.31319i 0.974147 − 1.37765i
\(22\) 0 0
\(23\) −6.31319 −1.31639 −0.658196 − 0.752847i \(-0.728680\pi\)
−0.658196 + 0.752847i \(0.728680\pi\)
\(24\) 0 0
\(25\) 1.26795 0.253590
\(26\) 0 0
\(27\) −1.41421 + 5.00000i −0.272166 + 0.962250i
\(28\) 0 0
\(29\) 1.27551i 0.236857i 0.992963 + 0.118428i \(0.0377856\pi\)
−0.992963 + 0.118428i \(0.962214\pi\)
\(30\) 0 0
\(31\) − 10.1962i − 1.83128i −0.401996 − 0.915642i \(-0.631683\pi\)
0.401996 − 0.915642i \(-0.368317\pi\)
\(32\) 0 0
\(33\) 4.19615 + 2.96713i 0.730456 + 0.516511i
\(34\) 0 0
\(35\) 8.62398 1.45772
\(36\) 0 0
\(37\) −3.26795 −0.537248 −0.268624 − 0.963245i \(-0.586569\pi\)
−0.268624 + 0.963245i \(0.586569\pi\)
\(38\) 0 0
\(39\) 8.10634 + 5.73205i 1.29805 + 0.917863i
\(40\) 0 0
\(41\) − 3.10583i − 0.485049i −0.970145 − 0.242524i \(-0.922025\pi\)
0.970145 − 0.242524i \(-0.0779755\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i
\(44\) 0 0
\(45\) −5.46410 + 1.93185i −0.814540 + 0.287983i
\(46\) 0 0
\(47\) −5.13922 −0.749632 −0.374816 − 0.927099i \(-0.622294\pi\)
−0.374816 + 0.927099i \(0.622294\pi\)
\(48\) 0 0
\(49\) −12.9282 −1.84689
\(50\) 0 0
\(51\) 6.31319 − 8.92820i 0.884024 − 1.25020i
\(52\) 0 0
\(53\) 10.5558i 1.44996i 0.688772 + 0.724978i \(0.258150\pi\)
−0.688772 + 0.724978i \(0.741850\pi\)
\(54\) 0 0
\(55\) 5.73205i 0.772910i
\(56\) 0 0
\(57\) 5.00000 − 7.07107i 0.662266 − 0.936586i
\(58\) 0 0
\(59\) 14.0406 1.82793 0.913965 − 0.405793i \(-0.133004\pi\)
0.913965 + 0.405793i \(0.133004\pi\)
\(60\) 0 0
\(61\) −1.07180 −0.137230 −0.0686148 − 0.997643i \(-0.521858\pi\)
−0.0686148 + 0.997643i \(0.521858\pi\)
\(62\) 0 0
\(63\) 12.6264 − 4.46410i 1.59078 − 0.562424i
\(64\) 0 0
\(65\) 11.0735i 1.37350i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −8.92820 − 6.31319i −1.07483 − 0.760019i
\(70\) 0 0
\(71\) −0.378937 −0.0449716 −0.0224858 − 0.999747i \(-0.507158\pi\)
−0.0224858 + 0.999747i \(0.507158\pi\)
\(72\) 0 0
\(73\) 1.80385 0.211124 0.105562 − 0.994413i \(-0.466336\pi\)
0.105562 + 0.994413i \(0.466336\pi\)
\(74\) 0 0
\(75\) 1.79315 + 1.26795i 0.207055 + 0.146410i
\(76\) 0 0
\(77\) − 13.2456i − 1.50947i
\(78\) 0 0
\(79\) − 1.80385i − 0.202949i −0.994838 − 0.101474i \(-0.967644\pi\)
0.994838 − 0.101474i \(-0.0323560\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 0 0
\(83\) 12.1087 1.32911 0.664554 − 0.747240i \(-0.268622\pi\)
0.664554 + 0.747240i \(0.268622\pi\)
\(84\) 0 0
\(85\) 12.1962 1.32286
\(86\) 0 0
\(87\) −1.27551 + 1.80385i −0.136749 + 0.193393i
\(88\) 0 0
\(89\) 13.0053i 1.37856i 0.724494 + 0.689281i \(0.242073\pi\)
−0.724494 + 0.689281i \(0.757927\pi\)
\(90\) 0 0
\(91\) − 25.5885i − 2.68240i
\(92\) 0 0
\(93\) 10.1962 − 14.4195i 1.05729 − 1.49524i
\(94\) 0 0
\(95\) 9.65926 0.991019
\(96\) 0 0
\(97\) 10.2679 1.04255 0.521276 − 0.853388i \(-0.325456\pi\)
0.521276 + 0.853388i \(0.325456\pi\)
\(98\) 0 0
\(99\) 2.96713 + 8.39230i 0.298208 + 0.843458i
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 2064.2.d.a.431.8 yes 8
3.2 odd 2 inner 2064.2.d.a.431.3 yes 8
4.3 odd 2 inner 2064.2.d.a.431.2 ✓ 8
12.11 even 2 inner 2064.2.d.a.431.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2064.2.d.a.431.2 ✓ 8 4.3 odd 2 inner
2064.2.d.a.431.3 yes 8 3.2 odd 2 inner
2064.2.d.a.431.5 yes 8 12.11 even 2 inner
2064.2.d.a.431.8 yes 8 1.1 even 1 trivial