↑

Properties

Label 4232.2.a.w.1.1
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

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 1.1
Root \(-0.196290\) 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)
 
\(f(q)\) \(=\) \(q-3.11247 q^{3} -2.45507 q^{5} -1.48669 q^{7} +6.68750 q^{9} +4.66795 q^{11} +4.68750 q^{13} +7.64133 q^{15} +6.87766 q^{17} -2.20971 q^{19} +4.62729 q^{21} +1.02735 q^{25} -11.4772 q^{27} +5.97265 q^{29} +4.71485 q^{31} -14.5289 q^{33} +3.64993 q^{35} -7.60067 q^{37} -14.5897 q^{39} +4.25230 q^{41} +0.0505476 q^{43} -16.4183 q^{45} +8.36477 q^{47} -4.78975 q^{49} -21.4065 q^{51} +7.16368 q^{53} -11.4601 q^{55} +6.87766 q^{57} -11.8273 q^{59} -1.96435 q^{61} -9.94225 q^{63} -11.5081 q^{65} +0.723015 q^{67} -7.02492 q^{71} +15.2249 q^{73} -3.19760 q^{75} -6.93980 q^{77} -9.33590 q^{79} +15.6601 q^{81} +6.87766 q^{83} -16.8851 q^{85} -18.5897 q^{87} +4.19029 q^{89} -6.96886 q^{91} -14.6748 q^{93} +5.42498 q^{95} -6.12704 q^{97} +31.2169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 18 q^{9} + 2 q^{13} + 34 q^{25} - 10 q^{27} + 22 q^{29} + 28 q^{31} - 16 q^{35} - 8 q^{39} + 6 q^{41} + 12 q^{47} + 8 q^{49} + 32 q^{55} - 58 q^{59} + 60 q^{71} + 68 q^{73} + 54 q^{75} + 8 q^{77}+ \cdots + 52 q^{95}+O(q^{100}) \) 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
\(3\) −3.11247 −1.79699 −0.898494 − 0.438986i \(-0.855338\pi\)
−0.898494 + 0.438986i \(0.855338\pi\)
\(4\) 0 0
\(5\) −2.45507 −1.09794 −0.548969 − 0.835842i \(-0.684980\pi\)
−0.548969 + 0.835842i \(0.684980\pi\)
\(6\) 0 0
\(7\) −1.48669 −0.561916 −0.280958 − 0.959720i \(-0.590652\pi\)
−0.280958 + 0.959720i \(0.590652\pi\)
\(8\) 0 0
\(9\) 6.68750 2.22917
\(10\) 0 0
\(11\) 4.66795 1.40744 0.703720 − 0.710478i \(-0.251521\pi\)
0.703720 + 0.710478i \(0.251521\pi\)
\(12\) 0 0
\(13\) 4.68750 1.30008 0.650039 − 0.759901i \(-0.274752\pi\)
0.650039 + 0.759901i \(0.274752\pi\)
\(14\) 0 0
\(15\) 7.64133 1.97298
\(16\) 0 0
\(17\) 6.87766 1.66808 0.834038 − 0.551707i \(-0.186023\pi\)
0.834038 + 0.551707i \(0.186023\pi\)
\(18\) 0 0
\(19\) −2.20971 −0.506941 −0.253471 − 0.967343i \(-0.581572\pi\)
−0.253471 + 0.967343i \(0.581572\pi\)
\(20\) 0 0
\(21\) 4.62729 1.00976
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 1.02735 0.205470
\(26\) 0 0
\(27\) −11.4772 −2.20880
\(28\) 0 0
\(29\) 5.97265 1.10909 0.554547 − 0.832153i \(-0.312892\pi\)
0.554547 + 0.832153i \(0.312892\pi\)
\(30\) 0 0
\(31\) 4.71485 0.846812 0.423406 − 0.905940i \(-0.360834\pi\)
0.423406 + 0.905940i \(0.360834\pi\)
\(32\) 0 0
\(33\) −14.5289 −2.52915
\(34\) 0 0
\(35\) 3.64993 0.616950
\(36\) 0 0
\(37\) −7.60067 −1.24954 −0.624771 − 0.780808i \(-0.714808\pi\)
−0.624771 + 0.780808i \(0.714808\pi\)
\(38\) 0 0
\(39\) −14.5897 −2.33623
\(40\) 0 0
\(41\) 4.25230 0.664098 0.332049 − 0.943262i \(-0.392260\pi\)
0.332049 + 0.943262i \(0.392260\pi\)
\(42\) 0 0
\(43\) 0.0505476 0.00770844 0.00385422 − 0.999993i \(-0.498773\pi\)
0.00385422 + 0.999993i \(0.498773\pi\)
\(44\) 0 0
\(45\) −16.4183 −2.44749
\(46\) 0 0
\(47\) 8.36477 1.22013 0.610064 − 0.792352i \(-0.291144\pi\)
0.610064 + 0.792352i \(0.291144\pi\)
\(48\) 0 0
\(49\) −4.78975 −0.684250
\(50\) 0 0
\(51\) −21.4065 −2.99751
\(52\) 0 0
\(53\) 7.16368 0.984007 0.492003 − 0.870593i \(-0.336265\pi\)
0.492003 + 0.870593i \(0.336265\pi\)
\(54\) 0 0
\(55\) −11.4601 −1.54528
\(56\) 0 0
\(57\) 6.87766 0.910968
\(58\) 0 0
\(59\) −11.8273 −1.53979 −0.769893 − 0.638173i \(-0.779691\pi\)
−0.769893 + 0.638173i \(0.779691\pi\)
\(60\) 0 0
\(61\) −1.96435 −0.251509 −0.125754 − 0.992061i \(-0.540135\pi\)
−0.125754 + 0.992061i \(0.540135\pi\)
\(62\) 0 0
\(63\) −9.94225 −1.25261
\(64\) 0 0
\(65\) −11.5081 −1.42741
\(66\) 0 0
\(67\) 0.723015 0.0883304 0.0441652 − 0.999024i \(-0.485937\pi\)
0.0441652 + 0.999024i \(0.485937\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.02492 −0.833705 −0.416853 − 0.908974i \(-0.636867\pi\)
−0.416853 + 0.908974i \(0.636867\pi\)
\(72\) 0 0
\(73\) 15.2249 1.78195 0.890973 − 0.454057i \(-0.150024\pi\)
0.890973 + 0.454057i \(0.150024\pi\)
\(74\) 0 0
\(75\) −3.19760 −0.369227
\(76\) 0 0
\(77\) −6.93980 −0.790863
\(78\) 0 0
\(79\) −9.33590 −1.05037 −0.525185 − 0.850988i \(-0.676004\pi\)
−0.525185 + 0.850988i \(0.676004\pi\)
\(80\) 0 0
\(81\) 15.6601 1.74002
\(82\) 0 0
\(83\) 6.87766 0.754921 0.377460 − 0.926026i \(-0.376797\pi\)
0.377460 + 0.926026i \(0.376797\pi\)
\(84\) 0 0
\(85\) −16.8851 −1.83145
\(86\) 0 0
\(87\) −18.5897 −1.99303
\(88\) 0 0
\(89\) 4.19029 0.444170 0.222085 − 0.975027i \(-0.428714\pi\)
0.222085 + 0.975027i \(0.428714\pi\)
\(90\) 0 0
\(91\) −6.96886 −0.730535
\(92\) 0 0
\(93\) −14.6748 −1.52171
\(94\) 0 0
\(95\) 5.42498 0.556591
\(96\) 0 0
\(97\) −6.12704 −0.622107 −0.311054 − 0.950392i \(-0.600682\pi\)
−0.311054 + 0.950392i \(0.600682\pi\)
\(98\) 0 0
\(99\) 31.2169 3.13742
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.1.1 ✓ 8
4.3 odd 2 8464.2.a.cc.1.7 8
23.22 odd 2 inner 4232.2.a.w.1.2 yes 8
92.91 even 2 8464.2.a.cc.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.w.1.1 ✓ 8 1.1 even 1 trivial
4232.2.a.w.1.2 yes 8 23.22 odd 2 inner
8464.2.a.cc.1.7 8 4.3 odd 2
8464.2.a.cc.1.8 8 92.91 even 2