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Properties

Label 4232.2.a.ba.1.13
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 1.13
Root \(2.49141\) 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)
 
\(f(q)\) \(=\) \(q+2.49141 q^{3} -2.70623 q^{5} +3.48198 q^{7} +3.20713 q^{9} -4.95599 q^{11} -1.54369 q^{13} -6.74234 q^{15} +1.95771 q^{17} -4.64029 q^{19} +8.67504 q^{21} +2.32370 q^{25} +0.516035 q^{27} +0.0794550 q^{29} -6.97280 q^{31} -12.3474 q^{33} -9.42305 q^{35} -11.5259 q^{37} -3.84596 q^{39} +8.96232 q^{41} -9.79502 q^{43} -8.67923 q^{45} +7.58145 q^{47} +5.12418 q^{49} +4.87745 q^{51} -9.11026 q^{53} +13.4121 q^{55} -11.5609 q^{57} +8.19215 q^{59} +8.12359 q^{61} +11.1671 q^{63} +4.17758 q^{65} -3.24911 q^{67} -4.39028 q^{71} -14.8871 q^{73} +5.78929 q^{75} -17.2567 q^{77} +5.71432 q^{79} -8.33572 q^{81} -0.464051 q^{83} -5.29801 q^{85} +0.197955 q^{87} +1.93417 q^{89} -5.37509 q^{91} -17.3721 q^{93} +12.5577 q^{95} +1.69399 q^{97} -15.8945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} - q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} - 18 q^{33} - 18 q^{35} - 24 q^{37} - 19 q^{39} + 9 q^{41} - 48 q^{43} - 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 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
\(3\) 2.49141 1.43842 0.719208 − 0.694795i \(-0.244505\pi\)
0.719208 + 0.694795i \(0.244505\pi\)
\(4\) 0 0
\(5\) −2.70623 −1.21026 −0.605132 − 0.796125i \(-0.706880\pi\)
−0.605132 + 0.796125i \(0.706880\pi\)
\(6\) 0 0
\(7\) 3.48198 1.31606 0.658032 − 0.752990i \(-0.271389\pi\)
0.658032 + 0.752990i \(0.271389\pi\)
\(8\) 0 0
\(9\) 3.20713 1.06904
\(10\) 0 0
\(11\) −4.95599 −1.49429 −0.747144 − 0.664662i \(-0.768575\pi\)
−0.747144 + 0.664662i \(0.768575\pi\)
\(12\) 0 0
\(13\) −1.54369 −0.428142 −0.214071 − 0.976818i \(-0.568672\pi\)
−0.214071 + 0.976818i \(0.568672\pi\)
\(14\) 0 0
\(15\) −6.74234 −1.74086
\(16\) 0 0
\(17\) 1.95771 0.474813 0.237407 − 0.971410i \(-0.423703\pi\)
0.237407 + 0.971410i \(0.423703\pi\)
\(18\) 0 0
\(19\) −4.64029 −1.06455 −0.532277 − 0.846570i \(-0.678664\pi\)
−0.532277 + 0.846570i \(0.678664\pi\)
\(20\) 0 0
\(21\) 8.67504 1.89305
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 2.32370 0.464740
\(26\) 0 0
\(27\) 0.516035 0.0993111
\(28\) 0 0
\(29\) 0.0794550 0.0147544 0.00737722 − 0.999973i \(-0.497652\pi\)
0.00737722 + 0.999973i \(0.497652\pi\)
\(30\) 0 0
\(31\) −6.97280 −1.25235 −0.626176 − 0.779682i \(-0.715381\pi\)
−0.626176 + 0.779682i \(0.715381\pi\)
\(32\) 0 0
\(33\) −12.3474 −2.14941
\(34\) 0 0
\(35\) −9.42305 −1.59279
\(36\) 0 0
\(37\) −11.5259 −1.89484 −0.947420 − 0.319993i \(-0.896319\pi\)
−0.947420 + 0.319993i \(0.896319\pi\)
\(38\) 0 0
\(39\) −3.84596 −0.615846
\(40\) 0 0
\(41\) 8.96232 1.39968 0.699840 − 0.714300i \(-0.253255\pi\)
0.699840 + 0.714300i \(0.253255\pi\)
\(42\) 0 0
\(43\) −9.79502 −1.49373 −0.746863 − 0.664977i \(-0.768441\pi\)
−0.746863 + 0.664977i \(0.768441\pi\)
\(44\) 0 0
\(45\) −8.67923 −1.29382
\(46\) 0 0
\(47\) 7.58145 1.10587 0.552934 − 0.833225i \(-0.313508\pi\)
0.552934 + 0.833225i \(0.313508\pi\)
\(48\) 0 0
\(49\) 5.12418 0.732026
\(50\) 0 0
\(51\) 4.87745 0.682979
\(52\) 0 0
\(53\) −9.11026 −1.25139 −0.625695 − 0.780068i \(-0.715185\pi\)
−0.625695 + 0.780068i \(0.715185\pi\)
\(54\) 0 0
\(55\) 13.4121 1.80848
\(56\) 0 0
\(57\) −11.5609 −1.53127
\(58\) 0 0
\(59\) 8.19215 1.06653 0.533263 − 0.845949i \(-0.320965\pi\)
0.533263 + 0.845949i \(0.320965\pi\)
\(60\) 0 0
\(61\) 8.12359 1.04012 0.520060 − 0.854130i \(-0.325910\pi\)
0.520060 + 0.854130i \(0.325910\pi\)
\(62\) 0 0
\(63\) 11.1671 1.40693
\(64\) 0 0
\(65\) 4.17758 0.518165
\(66\) 0 0
\(67\) −3.24911 −0.396942 −0.198471 − 0.980107i \(-0.563597\pi\)
−0.198471 + 0.980107i \(0.563597\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.39028 −0.521030 −0.260515 − 0.965470i \(-0.583892\pi\)
−0.260515 + 0.965470i \(0.583892\pi\)
\(72\) 0 0
\(73\) −14.8871 −1.74240 −0.871201 − 0.490927i \(-0.836658\pi\)
−0.871201 + 0.490927i \(0.836658\pi\)
\(74\) 0 0
\(75\) 5.78929 0.668489
\(76\) 0 0
\(77\) −17.2567 −1.96658
\(78\) 0 0
\(79\) 5.71432 0.642911 0.321456 − 0.946925i \(-0.395828\pi\)
0.321456 + 0.946925i \(0.395828\pi\)
\(80\) 0 0
\(81\) −8.33572 −0.926191
\(82\) 0 0
\(83\) −0.464051 −0.0509363 −0.0254681 − 0.999676i \(-0.508108\pi\)
−0.0254681 + 0.999676i \(0.508108\pi\)
\(84\) 0 0
\(85\) −5.29801 −0.574650
\(86\) 0 0
\(87\) 0.197955 0.0212230
\(88\) 0 0
\(89\) 1.93417 0.205022 0.102511 − 0.994732i \(-0.467312\pi\)
0.102511 + 0.994732i \(0.467312\pi\)
\(90\) 0 0
\(91\) −5.37509 −0.563462
\(92\) 0 0
\(93\) −17.3721 −1.80140
\(94\) 0 0
\(95\) 12.5577 1.28839
\(96\) 0 0
\(97\) 1.69399 0.171999 0.0859993 − 0.996295i \(-0.472592\pi\)
0.0859993 + 0.996295i \(0.472592\pi\)
\(98\) 0 0
\(99\) −15.8945 −1.59746
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.1.13 15
4.3 odd 2 8464.2.a.ch.1.3 15
23.15 odd 22 184.2.i.b.41.1 yes 30
23.20 odd 22 184.2.i.b.9.1 ✓ 30
23.22 odd 2 4232.2.a.bb.1.13 15
92.15 even 22 368.2.m.e.225.3 30
92.43 even 22 368.2.m.e.193.3 30
92.91 even 2 8464.2.a.cg.1.3 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.9.1 ✓ 30 23.20 odd 22
184.2.i.b.41.1 yes 30 23.15 odd 22
368.2.m.e.193.3 30 92.43 even 22
368.2.m.e.225.3 30 92.15 even 22
4232.2.a.ba.1.13 15 1.1 even 1 trivial
4232.2.a.bb.1.13 15 23.22 odd 2
8464.2.a.cg.1.3 15 92.91 even 2
8464.2.a.ch.1.3 15 4.3 odd 2