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Properties

Label 4232.2.a.w.1.3
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
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] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4232 = 2^{3} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.7926901354\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.13112\) of defining polynomial
Character \(\chi\) \(=\) 4232.1

$q$-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 \(-0.528395\pi\)
−0.0890873 + 0.996024i \(0.528395\pi\)
\(4\) 0 0
\(5\) −1.98578 −0.888067 −0.444034 − 0.896010i \(-0.646453\pi\)
−0.444034 + 0.896010i \(0.646453\pi\)
\(6\) 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 \(-0.426070\pi\)
0.230174 + 0.973149i \(0.426070\pi\)
\(18\) 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 \(-0.714965\pi\)
−0.625156 + 0.780500i \(0.714965\pi\)
\(32\) 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 \(-0.652733\pi\)
−0.461624 + 0.887076i \(0.652733\pi\)
\(38\) 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 \(-0.549670\pi\)
−0.155409 + 0.987850i \(0.549670\pi\)
\(48\) 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 \(-0.666403\pi\)
−0.499281 + 0.866440i \(0.666403\pi\)
\(54\) 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 \(-0.444756\pi\)
0.172685 + 0.984977i \(0.444756\pi\)
\(60\) 0 0
\(61\) −10.3151 −1.32071 −0.660354 − 0.750954i \(-0.729594\pi\)
−0.660354 + 0.750954i \(0.729594\pi\)
\(62\) 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 \(-0.274148\pi\)
0.651480 + 0.758666i \(0.274148\pi\)
\(72\) 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 \(-0.722370\pi\)
−0.643144 + 0.765745i \(0.722370\pi\)
\(90\) 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 \(-0.269954\pi\)
0.661419 + 0.750017i \(0.269954\pi\)
\(98\) 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 4232.2.a.w.1.3 ✓ 8
4.3 odd 2 8464.2.a.cc.1.5 8
23.22 odd 2 inner 4232.2.a.w.1.4 yes 8
92.91 even 2 8464.2.a.cc.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.w.1.3 ✓ 8 1.1 even 1 trivial
4232.2.a.w.1.4 yes 8 23.22 odd 2 inner
8464.2.a.cc.1.5 8 4.3 odd 2
8464.2.a.cc.1.6 8 92.91 even 2