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Properties

Label 4232.2.a.x.1.7
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
Dimension $12$
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 = [12,0,8,0,0,0,0] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.66287\) 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)
 
\(f(q)\) \(=\) \(q+1.23487 q^{3} -4.08929 q^{5} -5.04976 q^{7} -1.47508 q^{9} -1.54947 q^{11} -1.19661 q^{13} -5.04976 q^{15} -5.18225 q^{17} -0.101248 q^{19} -6.23583 q^{21} +11.7223 q^{25} -5.52617 q^{27} -5.58088 q^{29} -7.41918 q^{31} -1.91340 q^{33} +20.6500 q^{35} +2.86578 q^{37} -1.47766 q^{39} -7.75282 q^{41} -4.13899 q^{43} +6.03206 q^{45} +2.43323 q^{47} +18.5001 q^{49} -6.39943 q^{51} -8.63285 q^{53} +6.33622 q^{55} -0.125029 q^{57} -2.32823 q^{59} -2.43769 q^{61} +7.44883 q^{63} +4.89329 q^{65} +13.2843 q^{67} -6.33548 q^{71} -2.31339 q^{73} +14.4756 q^{75} +7.82444 q^{77} -1.53258 q^{79} -2.39887 q^{81} +0.719973 q^{83} +21.1918 q^{85} -6.89168 q^{87} +14.0774 q^{89} +6.04260 q^{91} -9.16175 q^{93} +0.414033 q^{95} +9.90518 q^{97} +2.28559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) \) 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
\(3\) 1.23487 0.712955 0.356478 − 0.934304i \(-0.383978\pi\)
0.356478 + 0.934304i \(0.383978\pi\)
\(4\) 0 0
\(5\) −4.08929 −1.82879 −0.914394 − 0.404826i \(-0.867332\pi\)
−0.914394 + 0.404826i \(0.867332\pi\)
\(6\) 0 0
\(7\) −5.04976 −1.90863 −0.954316 − 0.298800i \(-0.903414\pi\)
−0.954316 + 0.298800i \(0.903414\pi\)
\(8\) 0 0
\(9\) −1.47508 −0.491695
\(10\) 0 0
\(11\) −1.54947 −0.467181 −0.233591 − 0.972335i \(-0.575048\pi\)
−0.233591 + 0.972335i \(0.575048\pi\)
\(12\) 0 0
\(13\) −1.19661 −0.331880 −0.165940 − 0.986136i \(-0.553066\pi\)
−0.165940 + 0.986136i \(0.553066\pi\)
\(14\) 0 0
\(15\) −5.04976 −1.30384
\(16\) 0 0
\(17\) −5.18225 −1.25688 −0.628441 − 0.777858i \(-0.716306\pi\)
−0.628441 + 0.777858i \(0.716306\pi\)
\(18\) 0 0
\(19\) −0.101248 −0.0232279 −0.0116140 − 0.999933i \(-0.503697\pi\)
−0.0116140 + 0.999933i \(0.503697\pi\)
\(20\) 0 0
\(21\) −6.23583 −1.36077
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 11.7223 2.34446
\(26\) 0 0
\(27\) −5.52617 −1.06351
\(28\) 0 0
\(29\) −5.58088 −1.03634 −0.518171 − 0.855277i \(-0.673387\pi\)
−0.518171 + 0.855277i \(0.673387\pi\)
\(30\) 0 0
\(31\) −7.41918 −1.33252 −0.666262 − 0.745718i \(-0.732107\pi\)
−0.666262 + 0.745718i \(0.732107\pi\)
\(32\) 0 0
\(33\) −1.91340 −0.333079
\(34\) 0 0
\(35\) 20.6500 3.49048
\(36\) 0 0
\(37\) 2.86578 0.471131 0.235566 − 0.971858i \(-0.424306\pi\)
0.235566 + 0.971858i \(0.424306\pi\)
\(38\) 0 0
\(39\) −1.47766 −0.236616
\(40\) 0 0
\(41\) −7.75282 −1.21079 −0.605393 − 0.795927i \(-0.706984\pi\)
−0.605393 + 0.795927i \(0.706984\pi\)
\(42\) 0 0
\(43\) −4.13899 −0.631190 −0.315595 − 0.948894i \(-0.602204\pi\)
−0.315595 + 0.948894i \(0.602204\pi\)
\(44\) 0 0
\(45\) 6.03206 0.899206
\(46\) 0 0
\(47\) 2.43323 0.354923 0.177461 − 0.984128i \(-0.443212\pi\)
0.177461 + 0.984128i \(0.443212\pi\)
\(48\) 0 0
\(49\) 18.5001 2.64287
\(50\) 0 0
\(51\) −6.39943 −0.896100
\(52\) 0 0
\(53\) −8.63285 −1.18581 −0.592907 − 0.805271i \(-0.702020\pi\)
−0.592907 + 0.805271i \(0.702020\pi\)
\(54\) 0 0
\(55\) 6.33622 0.854376
\(56\) 0 0
\(57\) −0.125029 −0.0165605
\(58\) 0 0
\(59\) −2.32823 −0.303110 −0.151555 − 0.988449i \(-0.548428\pi\)
−0.151555 + 0.988449i \(0.548428\pi\)
\(60\) 0 0
\(61\) −2.43769 −0.312115 −0.156057 − 0.987748i \(-0.549878\pi\)
−0.156057 + 0.987748i \(0.549878\pi\)
\(62\) 0 0
\(63\) 7.44883 0.938465
\(64\) 0 0
\(65\) 4.89329 0.606938
\(66\) 0 0
\(67\) 13.2843 1.62293 0.811465 − 0.584401i \(-0.198670\pi\)
0.811465 + 0.584401i \(0.198670\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.33548 −0.751883 −0.375941 − 0.926643i \(-0.622681\pi\)
−0.375941 + 0.926643i \(0.622681\pi\)
\(72\) 0 0
\(73\) −2.31339 −0.270761 −0.135381 − 0.990794i \(-0.543226\pi\)
−0.135381 + 0.990794i \(0.543226\pi\)
\(74\) 0 0
\(75\) 14.4756 1.67150
\(76\) 0 0
\(77\) 7.82444 0.891677
\(78\) 0 0
\(79\) −1.53258 −0.172429 −0.0862145 − 0.996277i \(-0.527477\pi\)
−0.0862145 + 0.996277i \(0.527477\pi\)
\(80\) 0 0
\(81\) −2.39887 −0.266541
\(82\) 0 0
\(83\) 0.719973 0.0790273 0.0395137 − 0.999219i \(-0.487419\pi\)
0.0395137 + 0.999219i \(0.487419\pi\)
\(84\) 0 0
\(85\) 21.1918 2.29857
\(86\) 0 0
\(87\) −6.89168 −0.738866
\(88\) 0 0
\(89\) 14.0774 1.49220 0.746098 − 0.665836i \(-0.231925\pi\)
0.746098 + 0.665836i \(0.231925\pi\)
\(90\) 0 0
\(91\) 6.04260 0.633437
\(92\) 0 0
\(93\) −9.16175 −0.950029
\(94\) 0 0
\(95\) 0.414033 0.0424789
\(96\) 0 0
\(97\) 9.90518 1.00572 0.502859 − 0.864368i \(-0.332281\pi\)
0.502859 + 0.864368i \(0.332281\pi\)
\(98\) 0 0
\(99\) 2.28559 0.229711
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.x.1.7 ✓ 12
4.3 odd 2 8464.2.a.cf.1.5 12
23.22 odd 2 inner 4232.2.a.x.1.8 yes 12
92.91 even 2 8464.2.a.cf.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.x.1.7 ✓ 12 1.1 even 1 trivial
4232.2.a.x.1.8 yes 12 23.22 odd 2 inner
8464.2.a.cf.1.5 12 4.3 odd 2
8464.2.a.cf.1.6 12 92.91 even 2