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Properties

Label 4232.2.a.ba.1.9
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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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 = [15,0,1,0,0,0,-10] 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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.07060\) 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)
 
\(f(q)\) \(=\) \(q+1.07060 q^{3} +3.54589 q^{5} -3.82947 q^{7} -1.85382 q^{9} -4.38386 q^{11} +5.21152 q^{13} +3.79622 q^{15} +0.517206 q^{17} -3.99525 q^{19} -4.09982 q^{21} +7.57336 q^{25} -5.19649 q^{27} -9.23921 q^{29} -3.51763 q^{31} -4.69334 q^{33} -13.5789 q^{35} -3.44042 q^{37} +5.57944 q^{39} +10.8104 q^{41} -0.391013 q^{43} -6.57346 q^{45} +1.50879 q^{47} +7.66482 q^{49} +0.553720 q^{51} -0.746394 q^{53} -15.5447 q^{55} -4.27730 q^{57} -5.69987 q^{59} -5.94097 q^{61} +7.09915 q^{63} +18.4795 q^{65} -2.72082 q^{67} -3.85262 q^{71} -5.78124 q^{73} +8.10802 q^{75} +16.7878 q^{77} -2.75488 q^{79} -0.00187825 q^{81} -2.73234 q^{83} +1.83396 q^{85} -9.89147 q^{87} -7.82654 q^{89} -19.9574 q^{91} -3.76596 q^{93} -14.1667 q^{95} +1.93011 q^{97} +8.12689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} - q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} - 18 q^{33} - 18 q^{35} - 24 q^{37} - 19 q^{39} + 9 q^{41} - 48 q^{43} - 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 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
\(3\) 1.07060 0.618109 0.309055 − 0.951044i \(-0.399987\pi\)
0.309055 + 0.951044i \(0.399987\pi\)
\(4\) 0 0
\(5\) 3.54589 1.58577 0.792886 − 0.609370i \(-0.208578\pi\)
0.792886 + 0.609370i \(0.208578\pi\)
\(6\) 0 0
\(7\) −3.82947 −1.44740 −0.723701 − 0.690113i \(-0.757561\pi\)
−0.723701 + 0.690113i \(0.757561\pi\)
\(8\) 0 0
\(9\) −1.85382 −0.617941
\(10\) 0 0
\(11\) −4.38386 −1.32178 −0.660891 − 0.750482i \(-0.729822\pi\)
−0.660891 + 0.750482i \(0.729822\pi\)
\(12\) 0 0
\(13\) 5.21152 1.44542 0.722708 − 0.691153i \(-0.242897\pi\)
0.722708 + 0.691153i \(0.242897\pi\)
\(14\) 0 0
\(15\) 3.79622 0.980181
\(16\) 0 0
\(17\) 0.517206 0.125441 0.0627205 − 0.998031i \(-0.480022\pi\)
0.0627205 + 0.998031i \(0.480022\pi\)
\(18\) 0 0
\(19\) −3.99525 −0.916572 −0.458286 − 0.888805i \(-0.651536\pi\)
−0.458286 + 0.888805i \(0.651536\pi\)
\(20\) 0 0
\(21\) −4.09982 −0.894653
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 7.57336 1.51467
\(26\) 0 0
\(27\) −5.19649 −1.00006
\(28\) 0 0
\(29\) −9.23921 −1.71568 −0.857839 − 0.513918i \(-0.828194\pi\)
−0.857839 + 0.513918i \(0.828194\pi\)
\(30\) 0 0
\(31\) −3.51763 −0.631785 −0.315892 − 0.948795i \(-0.602304\pi\)
−0.315892 + 0.948795i \(0.602304\pi\)
\(32\) 0 0
\(33\) −4.69334 −0.817006
\(34\) 0 0
\(35\) −13.5789 −2.29525
\(36\) 0 0
\(37\) −3.44042 −0.565601 −0.282801 − 0.959179i \(-0.591263\pi\)
−0.282801 + 0.959179i \(0.591263\pi\)
\(38\) 0 0
\(39\) 5.57944 0.893426
\(40\) 0 0
\(41\) 10.8104 1.68830 0.844149 − 0.536108i \(-0.180106\pi\)
0.844149 + 0.536108i \(0.180106\pi\)
\(42\) 0 0
\(43\) −0.391013 −0.0596289 −0.0298145 − 0.999555i \(-0.509492\pi\)
−0.0298145 + 0.999555i \(0.509492\pi\)
\(44\) 0 0
\(45\) −6.57346 −0.979913
\(46\) 0 0
\(47\) 1.50879 0.220080 0.110040 − 0.993927i \(-0.464902\pi\)
0.110040 + 0.993927i \(0.464902\pi\)
\(48\) 0 0
\(49\) 7.66482 1.09497
\(50\) 0 0
\(51\) 0.553720 0.0775363
\(52\) 0 0
\(53\) −0.746394 −0.102525 −0.0512626 − 0.998685i \(-0.516325\pi\)
−0.0512626 + 0.998685i \(0.516325\pi\)
\(54\) 0 0
\(55\) −15.5447 −2.09605
\(56\) 0 0
\(57\) −4.27730 −0.566542
\(58\) 0 0
\(59\) −5.69987 −0.742060 −0.371030 − 0.928621i \(-0.620995\pi\)
−0.371030 + 0.928621i \(0.620995\pi\)
\(60\) 0 0
\(61\) −5.94097 −0.760663 −0.380332 − 0.924850i \(-0.624190\pi\)
−0.380332 + 0.924850i \(0.624190\pi\)
\(62\) 0 0
\(63\) 7.09915 0.894409
\(64\) 0 0
\(65\) 18.4795 2.29210
\(66\) 0 0
\(67\) −2.72082 −0.332401 −0.166201 − 0.986092i \(-0.553150\pi\)
−0.166201 + 0.986092i \(0.553150\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.85262 −0.457221 −0.228611 − 0.973518i \(-0.573418\pi\)
−0.228611 + 0.973518i \(0.573418\pi\)
\(72\) 0 0
\(73\) −5.78124 −0.676642 −0.338321 − 0.941031i \(-0.609859\pi\)
−0.338321 + 0.941031i \(0.609859\pi\)
\(74\) 0 0
\(75\) 8.10802 0.936233
\(76\) 0 0
\(77\) 16.7878 1.91315
\(78\) 0 0
\(79\) −2.75488 −0.309948 −0.154974 − 0.987919i \(-0.549529\pi\)
−0.154974 + 0.987919i \(0.549529\pi\)
\(80\) 0 0
\(81\) −0.00187825 −0.000208695 0
\(82\) 0 0
\(83\) −2.73234 −0.299913 −0.149957 − 0.988693i \(-0.547913\pi\)
−0.149957 + 0.988693i \(0.547913\pi\)
\(84\) 0 0
\(85\) 1.83396 0.198921
\(86\) 0 0
\(87\) −9.89147 −1.06048
\(88\) 0 0
\(89\) −7.82654 −0.829611 −0.414806 − 0.909910i \(-0.636150\pi\)
−0.414806 + 0.909910i \(0.636150\pi\)
\(90\) 0 0
\(91\) −19.9574 −2.09210
\(92\) 0 0
\(93\) −3.76596 −0.390512
\(94\) 0 0
\(95\) −14.1667 −1.45347
\(96\) 0 0
\(97\) 1.93011 0.195973 0.0979867 − 0.995188i \(-0.468760\pi\)
0.0979867 + 0.995188i \(0.468760\pi\)
\(98\) 0 0
\(99\) 8.12689 0.816783
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.ba.1.9 15
4.3 odd 2 8464.2.a.ch.1.7 15
23.5 odd 22 184.2.i.b.25.1 ✓ 30
23.14 odd 22 184.2.i.b.81.1 yes 30
23.22 odd 2 4232.2.a.bb.1.9 15
92.51 even 22 368.2.m.e.209.3 30
92.83 even 22 368.2.m.e.81.3 30
92.91 even 2 8464.2.a.cg.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.1 ✓ 30 23.5 odd 22
184.2.i.b.81.1 yes 30 23.14 odd 22
368.2.m.e.81.3 30 92.83 even 22
368.2.m.e.209.3 30 92.51 even 22
4232.2.a.ba.1.9 15 1.1 even 1 trivial
4232.2.a.bb.1.9 15 23.22 odd 2
8464.2.a.cg.1.7 15 92.91 even 2
8464.2.a.ch.1.7 15 4.3 odd 2