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Properties

Label 8464.2.a.cc.1.6
Level $8464$
Weight $2$
Character 8464.1
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,0,0,0,0,0,18,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.299900807424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 14x^{6} + 30x^{5} + 37x^{4} - 88x^{3} + 24x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 4232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.04100\) of defining polynomial
Character \(\chi\) \(=\) 8464.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.308608 q^{3} +1.98578 q^{5} -2.43259 q^{7} -2.90476 q^{9} -4.25235 q^{11} -4.90476 q^{13} +0.612826 q^{15} -1.89807 q^{17} -6.15042 q^{19} -0.750716 q^{21} -1.05668 q^{25} -1.82225 q^{27} +8.05668 q^{29} +6.96145 q^{31} -1.31231 q^{33} -4.83058 q^{35} +5.61589 q^{37} -1.51365 q^{39} -3.43947 q^{41} -8.82627 q^{43} -5.76821 q^{45} +2.13086 q^{47} -1.08251 q^{49} -0.585757 q^{51} +7.26964 q^{53} -8.44423 q^{55} -1.89807 q^{57} -2.65284 q^{59} +10.3151 q^{61} +7.06609 q^{63} -9.73977 q^{65} +3.71783 q^{67} -10.9789 q^{71} +9.61722 q^{73} -0.326101 q^{75} +10.3442 q^{77} +8.50471 q^{79} +8.15192 q^{81} +1.89807 q^{83} -3.76914 q^{85} +2.48635 q^{87} +12.1348 q^{89} +11.9313 q^{91} +2.14835 q^{93} -12.2134 q^{95} -13.0284 q^{97} +12.3521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 18 q^{9} + 2 q^{13} + 34 q^{25} + 10 q^{27} + 22 q^{29} - 28 q^{31} + 16 q^{35} + 8 q^{39} + 6 q^{41} - 12 q^{47} + 8 q^{49} - 32 q^{55} + 58 q^{59} - 60 q^{71} + 68 q^{73} - 54 q^{75} + 8 q^{77}+ \cdots - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.308608 0.178175 0.0890873 − 0.996024i \(-0.471605\pi\)
0.0890873 + 0.996024i \(0.471605\pi\)
\(4\) 0 0
\(5\) 1.98578 0.888067 0.444034 − 0.896010i \(-0.353547\pi\)
0.444034 + 0.896010i \(0.353547\pi\)
\(6\) 0 0
\(7\) −2.43259 −0.919432 −0.459716 − 0.888066i \(-0.652049\pi\)
−0.459716 + 0.888066i \(0.652049\pi\)
\(8\) 0 0
\(9\) −2.90476 −0.968254
\(10\) 0 0
\(11\) −4.25235 −1.28213 −0.641066 − 0.767485i \(-0.721508\pi\)
−0.641066 + 0.767485i \(0.721508\pi\)
\(12\) 0 0
\(13\) −4.90476 −1.36034 −0.680168 − 0.733056i \(-0.738093\pi\)
−0.680168 + 0.733056i \(0.738093\pi\)
\(14\) 0 0
\(15\) 0.612826 0.158231
\(16\) 0 0
\(17\) −1.89807 −0.460349 −0.230174 − 0.973149i \(-0.573930\pi\)
−0.230174 + 0.973149i \(0.573930\pi\)
\(18\) 0 0
\(19\) −6.15042 −1.41100 −0.705501 − 0.708708i \(-0.749278\pi\)
−0.705501 + 0.708708i \(0.749278\pi\)
\(20\) 0 0
\(21\) −0.750716 −0.163820
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −1.05668 −0.211337
\(26\) 0 0
\(27\) −1.82225 −0.350693
\(28\) 0 0
\(29\) 8.05668 1.49609 0.748044 − 0.663649i \(-0.230993\pi\)
0.748044 + 0.663649i \(0.230993\pi\)
\(30\) 0 0
\(31\) 6.96145 1.25031 0.625156 − 0.780500i \(-0.285035\pi\)
0.625156 + 0.780500i \(0.285035\pi\)
\(32\) 0 0
\(33\) −1.31231 −0.228444
\(34\) 0 0
\(35\) −4.83058 −0.816518
\(36\) 0 0
\(37\) 5.61589 0.923247 0.461624 − 0.887076i \(-0.347267\pi\)
0.461624 + 0.887076i \(0.347267\pi\)
\(38\) 0 0
\(39\) −1.51365 −0.242377
\(40\) 0 0
\(41\) −3.43947 −0.537155 −0.268577 − 0.963258i \(-0.586554\pi\)
−0.268577 + 0.963258i \(0.586554\pi\)
\(42\) 0 0
\(43\) −8.82627 −1.34599 −0.672997 − 0.739645i \(-0.734993\pi\)
−0.672997 + 0.739645i \(0.734993\pi\)
\(44\) 0 0
\(45\) −5.76821 −0.859874
\(46\) 0 0
\(47\) 2.13086 0.310818 0.155409 − 0.987850i \(-0.450330\pi\)
0.155409 + 0.987850i \(0.450330\pi\)
\(48\) 0 0
\(49\) −1.08251 −0.154644
\(50\) 0 0
\(51\) −0.585757 −0.0820224
\(52\) 0 0
\(53\) 7.26964 0.998562 0.499281 − 0.866440i \(-0.333597\pi\)
0.499281 + 0.866440i \(0.333597\pi\)
\(54\) 0 0
\(55\) −8.44423 −1.13862
\(56\) 0 0
\(57\) −1.89807 −0.251405
\(58\) 0 0
\(59\) −2.65284 −0.345370 −0.172685 − 0.984977i \(-0.555244\pi\)
−0.172685 + 0.984977i \(0.555244\pi\)
\(60\) 0 0
\(61\) 10.3151 1.32071 0.660354 − 0.750954i \(-0.270406\pi\)
0.660354 + 0.750954i \(0.270406\pi\)
\(62\) 0 0
\(63\) 7.06609 0.890244
\(64\) 0 0
\(65\) −9.73977 −1.20807
\(66\) 0 0
\(67\) 3.71783 0.454205 0.227103 − 0.973871i \(-0.427075\pi\)
0.227103 + 0.973871i \(0.427075\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.9789 −1.30296 −0.651480 − 0.758666i \(-0.725852\pi\)
−0.651480 + 0.758666i \(0.725852\pi\)
\(72\) 0 0
\(73\) 9.61722 1.12561 0.562805 − 0.826590i \(-0.309722\pi\)
0.562805 + 0.826590i \(0.309722\pi\)
\(74\) 0 0
\(75\) −0.326101 −0.0376549
\(76\) 0 0
\(77\) 10.3442 1.17883
\(78\) 0 0
\(79\) 8.50471 0.956854 0.478427 − 0.878127i \(-0.341207\pi\)
0.478427 + 0.878127i \(0.341207\pi\)
\(80\) 0 0
\(81\) 8.15192 0.905769
\(82\) 0 0
\(83\) 1.89807 0.208340 0.104170 − 0.994560i \(-0.466781\pi\)
0.104170 + 0.994560i \(0.466781\pi\)
\(84\) 0 0
\(85\) −3.76914 −0.408820
\(86\) 0 0
\(87\) 2.48635 0.266565
\(88\) 0 0
\(89\) 12.1348 1.28629 0.643144 − 0.765745i \(-0.277630\pi\)
0.643144 + 0.765745i \(0.277630\pi\)
\(90\) 0 0
\(91\) 11.9313 1.25074
\(92\) 0 0
\(93\) 2.14835 0.222774
\(94\) 0 0
\(95\) −12.2134 −1.25307
\(96\) 0 0
\(97\) −13.0284 −1.32284 −0.661419 − 0.750017i \(-0.730046\pi\)
−0.661419 + 0.750017i \(0.730046\pi\)
\(98\) 0 0
\(99\) 12.3521 1.24143
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 8464.2.a.cc.1.6 8
4.3 odd 2 4232.2.a.w.1.4 yes 8
23.22 odd 2 inner 8464.2.a.cc.1.5 8
92.91 even 2 4232.2.a.w.1.3 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.w.1.3 ✓ 8 92.91 even 2
4232.2.a.w.1.4 yes 8 4.3 odd 2
8464.2.a.cc.1.5 8 23.22 odd 2 inner
8464.2.a.cc.1.6 8 1.1 even 1 trivial