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Properties

Label 2064.2.d.a.431.6
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 431.6
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2064.431
Dual form 2064.2.d.a.431.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.00000i) q^{3} +0.517638i q^{5} -2.46410i q^{7} +(1.00000 - 2.82843i) q^{9} -4.38134 q^{11} +2.26795 q^{13} +(0.517638 + 0.732051i) q^{15} -3.48477i q^{17} +5.00000i q^{19} +(-2.46410 - 3.48477i) q^{21} +3.48477 q^{23} +4.73205 q^{25} +(-1.41421 - 5.00000i) q^{27} -8.62398i q^{29} -0.196152i q^{31} +(-6.19615 + 4.38134i) q^{33} +1.27551 q^{35} -6.73205 q^{37} +(3.20736 - 2.26795i) q^{39} -11.5911i q^{41} -1.00000i q^{43} +(1.46410 + 0.517638i) q^{45} -7.58871 q^{47} +0.928203 q^{49} +(-3.48477 - 4.92820i) q^{51} -0.757875i q^{53} -2.26795i q^{55} +(5.00000 + 7.07107i) q^{57} -5.55532 q^{59} -14.9282 q^{61} +(-6.96953 - 2.46410i) q^{63} +1.17398i q^{65} -8.00000i q^{67} +(4.92820 - 3.48477i) q^{69} -5.27792 q^{71} +12.1962 q^{73} +(6.69213 - 4.73205i) q^{75} +10.7961i q^{77} +12.1962i q^{79} +(-7.00000 - 5.65685i) q^{81} -5.03768 q^{83} +1.80385 q^{85} +(-8.62398 - 12.1962i) q^{87} +1.69161i q^{89} -5.58846i q^{91} +(-0.196152 - 0.277401i) q^{93} -2.58819 q^{95} +13.7321 q^{97} +(-4.38134 + 12.3923i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + 32 q^{13} + 8 q^{21} + 24 q^{25} - 8 q^{33} - 40 q^{37} - 16 q^{45} - 48 q^{49} + 40 q^{57} - 64 q^{61} - 16 q^{69} + 56 q^{73} - 56 q^{81} + 56 q^{85} + 40 q^{93} + 96 q^{97}+O(q^{100}) \) 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
\(5\) 0.517638i 0.231495i 0.993279 + 0.115747i \(0.0369263\pi\)
−0.993279 + 0.115747i \(0.963074\pi\)
\(6\) 0 0
\(7\) − 2.46410i − 0.931343i −0.884958 − 0.465671i \(-0.845813\pi\)
0.884958 − 0.465671i \(-0.154187\pi\)
\(8\) 0 0
\(9\) 1.00000 − 2.82843i 0.333333 − 0.942809i
\(10\) 0 0
\(11\) −4.38134 −1.32102 −0.660512 − 0.750815i \(-0.729661\pi\)
−0.660512 + 0.750815i \(0.729661\pi\)
\(12\) 0 0
\(13\) 2.26795 0.629016 0.314508 − 0.949255i \(-0.398160\pi\)
0.314508 + 0.949255i \(0.398160\pi\)
\(14\) 0 0
\(15\) 0.517638 + 0.732051i 0.133654 + 0.189015i
\(16\) 0 0
\(17\) − 3.48477i − 0.845180i −0.906321 − 0.422590i \(-0.861121\pi\)
0.906321 − 0.422590i \(-0.138879\pi\)
\(18\) 0 0
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) −2.46410 − 3.48477i −0.537711 − 0.760438i
\(22\) 0 0
\(23\) 3.48477 0.726624 0.363312 − 0.931668i \(-0.381646\pi\)
0.363312 + 0.931668i \(0.381646\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 0 0
\(27\) −1.41421 − 5.00000i −0.272166 − 0.962250i
\(28\) 0 0
\(29\) − 8.62398i − 1.60143i −0.599043 − 0.800717i \(-0.704452\pi\)
0.599043 − 0.800717i \(-0.295548\pi\)
\(30\) 0 0
\(31\) − 0.196152i − 0.0352300i −0.999845 − 0.0176150i \(-0.994393\pi\)
0.999845 − 0.0176150i \(-0.00560732\pi\)
\(32\) 0 0
\(33\) −6.19615 + 4.38134i −1.07861 + 0.762694i
\(34\) 0 0
\(35\) 1.27551 0.215601
\(36\) 0 0
\(37\) −6.73205 −1.10674 −0.553371 − 0.832935i \(-0.686659\pi\)
−0.553371 + 0.832935i \(0.686659\pi\)
\(38\) 0 0
\(39\) 3.20736 − 2.26795i 0.513589 − 0.363163i
\(40\) 0 0
\(41\) − 11.5911i − 1.81023i −0.425170 − 0.905114i \(-0.639786\pi\)
0.425170 − 0.905114i \(-0.360214\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i
\(44\) 0 0
\(45\) 1.46410 + 0.517638i 0.218255 + 0.0771649i
\(46\) 0 0
\(47\) −7.58871 −1.10693 −0.553463 − 0.832874i \(-0.686694\pi\)
−0.553463 + 0.832874i \(0.686694\pi\)
\(48\) 0 0
\(49\) 0.928203 0.132600
\(50\) 0 0
\(51\) −3.48477 − 4.92820i −0.487965 − 0.690086i
\(52\) 0 0
\(53\) − 0.757875i − 0.104102i −0.998644 − 0.0520511i \(-0.983424\pi\)
0.998644 − 0.0520511i \(-0.0165759\pi\)
\(54\) 0 0
\(55\) − 2.26795i − 0.305810i
\(56\) 0 0
\(57\) 5.00000 + 7.07107i 0.662266 + 0.936586i
\(58\) 0 0
\(59\) −5.55532 −0.723241 −0.361620 − 0.932325i \(-0.617776\pi\)
−0.361620 + 0.932325i \(0.617776\pi\)
\(60\) 0 0
\(61\) −14.9282 −1.91136 −0.955680 − 0.294407i \(-0.904878\pi\)
−0.955680 + 0.294407i \(0.904878\pi\)
\(62\) 0 0
\(63\) −6.96953 − 2.46410i −0.878078 − 0.310448i
\(64\) 0 0
\(65\) 1.17398i 0.145614i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 4.92820 − 3.48477i 0.593286 − 0.419517i
\(70\) 0 0
\(71\) −5.27792 −0.626373 −0.313187 − 0.949692i \(-0.601397\pi\)
−0.313187 + 0.949692i \(0.601397\pi\)
\(72\) 0 0
\(73\) 12.1962 1.42745 0.713726 − 0.700425i \(-0.247006\pi\)
0.713726 + 0.700425i \(0.247006\pi\)
\(74\) 0 0
\(75\) 6.69213 − 4.73205i 0.772741 − 0.546410i
\(76\) 0 0
\(77\) 10.7961i 1.23033i
\(78\) 0 0
\(79\) 12.1962i 1.37217i 0.727519 + 0.686087i \(0.240673\pi\)
−0.727519 + 0.686087i \(0.759327\pi\)
\(80\) 0 0
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 0 0
\(83\) −5.03768 −0.552957 −0.276479 − 0.961020i \(-0.589167\pi\)
−0.276479 + 0.961020i \(0.589167\pi\)
\(84\) 0 0
\(85\) 1.80385 0.195655
\(86\) 0 0
\(87\) −8.62398 − 12.1962i −0.924588 − 1.30756i
\(88\) 0 0
\(89\) 1.69161i 0.179311i 0.995973 + 0.0896554i \(0.0285766\pi\)
−0.995973 + 0.0896554i \(0.971423\pi\)
\(90\) 0 0
\(91\) − 5.58846i − 0.585830i
\(92\) 0 0
\(93\) −0.196152 − 0.277401i −0.0203401 − 0.0287652i
\(94\) 0 0
\(95\) −2.58819 −0.265543
\(96\) 0 0
\(97\) 13.7321 1.39428 0.697139 − 0.716936i \(-0.254456\pi\)
0.697139 + 0.716936i \(0.254456\pi\)
\(98\) 0 0
\(99\) −4.38134 + 12.3923i −0.440341 + 1.24547i
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.431.6 yes 8
3.2 odd 2 inner 2064.2.d.a.431.1 ✓ 8
4.3 odd 2 inner 2064.2.d.a.431.4 yes 8
12.11 even 2 inner 2064.2.d.a.431.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2064.2.d.a.431.1 ✓ 8 3.2 odd 2 inner
2064.2.d.a.431.4 yes 8 4.3 odd 2 inner
2064.2.d.a.431.6 yes 8 1.1 even 1 trivial
2064.2.d.a.431.7 yes 8 12.11 even 2 inner