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Properties

Label 4232.2.a.ba.1.2
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 1.2
Root \(-2.59403\) 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)
 
\(f(q)\) \(=\) \(q-2.59403 q^{3} -3.58336 q^{5} -4.34127 q^{7} +3.72897 q^{9} -3.81113 q^{11} -2.60706 q^{13} +9.29532 q^{15} +3.12636 q^{17} -1.39333 q^{19} +11.2614 q^{21} +7.84043 q^{25} -1.89098 q^{27} +4.34076 q^{29} +5.76192 q^{31} +9.88618 q^{33} +15.5563 q^{35} -7.02696 q^{37} +6.76279 q^{39} -1.45431 q^{41} -5.28754 q^{43} -13.3622 q^{45} -0.409504 q^{47} +11.8466 q^{49} -8.10987 q^{51} +10.6502 q^{53} +13.6566 q^{55} +3.61433 q^{57} -2.14797 q^{59} +12.2175 q^{61} -16.1885 q^{63} +9.34203 q^{65} -6.75280 q^{67} -0.889430 q^{71} +8.92211 q^{73} -20.3383 q^{75} +16.5452 q^{77} +12.5955 q^{79} -6.28168 q^{81} -14.6789 q^{83} -11.2029 q^{85} -11.2601 q^{87} -6.69305 q^{89} +11.3180 q^{91} -14.9466 q^{93} +4.99279 q^{95} -2.05817 q^{97} -14.2116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} - q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} - 18 q^{33} - 18 q^{35} - 24 q^{37} - 19 q^{39} + 9 q^{41} - 48 q^{43} - 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 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
\(3\) −2.59403 −1.49766 −0.748831 − 0.662761i \(-0.769385\pi\)
−0.748831 + 0.662761i \(0.769385\pi\)
\(4\) 0 0
\(5\) −3.58336 −1.60253 −0.801263 − 0.598313i \(-0.795838\pi\)
−0.801263 + 0.598313i \(0.795838\pi\)
\(6\) 0 0
\(7\) −4.34127 −1.64085 −0.820423 − 0.571757i \(-0.806262\pi\)
−0.820423 + 0.571757i \(0.806262\pi\)
\(8\) 0 0
\(9\) 3.72897 1.24299
\(10\) 0 0
\(11\) −3.81113 −1.14910 −0.574550 − 0.818469i \(-0.694823\pi\)
−0.574550 + 0.818469i \(0.694823\pi\)
\(12\) 0 0
\(13\) −2.60706 −0.723069 −0.361534 − 0.932359i \(-0.617747\pi\)
−0.361534 + 0.932359i \(0.617747\pi\)
\(14\) 0 0
\(15\) 9.29532 2.40004
\(16\) 0 0
\(17\) 3.12636 0.758255 0.379127 − 0.925344i \(-0.376224\pi\)
0.379127 + 0.925344i \(0.376224\pi\)
\(18\) 0 0
\(19\) −1.39333 −0.319651 −0.159826 − 0.987145i \(-0.551093\pi\)
−0.159826 + 0.987145i \(0.551093\pi\)
\(20\) 0 0
\(21\) 11.2614 2.45743
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 7.84043 1.56809
\(26\) 0 0
\(27\) −1.89098 −0.363919
\(28\) 0 0
\(29\) 4.34076 0.806060 0.403030 − 0.915187i \(-0.367957\pi\)
0.403030 + 0.915187i \(0.367957\pi\)
\(30\) 0 0
\(31\) 5.76192 1.03487 0.517435 − 0.855722i \(-0.326887\pi\)
0.517435 + 0.855722i \(0.326887\pi\)
\(32\) 0 0
\(33\) 9.88618 1.72096
\(34\) 0 0
\(35\) 15.5563 2.62950
\(36\) 0 0
\(37\) −7.02696 −1.15523 −0.577613 − 0.816311i \(-0.696016\pi\)
−0.577613 + 0.816311i \(0.696016\pi\)
\(38\) 0 0
\(39\) 6.76279 1.08291
\(40\) 0 0
\(41\) −1.45431 −0.227126 −0.113563 − 0.993531i \(-0.536226\pi\)
−0.113563 + 0.993531i \(0.536226\pi\)
\(42\) 0 0
\(43\) −5.28754 −0.806342 −0.403171 − 0.915125i \(-0.632092\pi\)
−0.403171 + 0.915125i \(0.632092\pi\)
\(44\) 0 0
\(45\) −13.3622 −1.99192
\(46\) 0 0
\(47\) −0.409504 −0.0597322 −0.0298661 − 0.999554i \(-0.509508\pi\)
−0.0298661 + 0.999554i \(0.509508\pi\)
\(48\) 0 0
\(49\) 11.8466 1.69237
\(50\) 0 0
\(51\) −8.10987 −1.13561
\(52\) 0 0
\(53\) 10.6502 1.46292 0.731458 − 0.681887i \(-0.238840\pi\)
0.731458 + 0.681887i \(0.238840\pi\)
\(54\) 0 0
\(55\) 13.6566 1.84146
\(56\) 0 0
\(57\) 3.61433 0.478729
\(58\) 0 0
\(59\) −2.14797 −0.279642 −0.139821 − 0.990177i \(-0.544653\pi\)
−0.139821 + 0.990177i \(0.544653\pi\)
\(60\) 0 0
\(61\) 12.2175 1.56429 0.782143 − 0.623099i \(-0.214127\pi\)
0.782143 + 0.623099i \(0.214127\pi\)
\(62\) 0 0
\(63\) −16.1885 −2.03956
\(64\) 0 0
\(65\) 9.34203 1.15874
\(66\) 0 0
\(67\) −6.75280 −0.824986 −0.412493 − 0.910961i \(-0.635342\pi\)
−0.412493 + 0.910961i \(0.635342\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.889430 −0.105556 −0.0527779 − 0.998606i \(-0.516808\pi\)
−0.0527779 + 0.998606i \(0.516808\pi\)
\(72\) 0 0
\(73\) 8.92211 1.04425 0.522127 − 0.852868i \(-0.325139\pi\)
0.522127 + 0.852868i \(0.325139\pi\)
\(74\) 0 0
\(75\) −20.3383 −2.34846
\(76\) 0 0
\(77\) 16.5452 1.88550
\(78\) 0 0
\(79\) 12.5955 1.41710 0.708550 − 0.705661i \(-0.249350\pi\)
0.708550 + 0.705661i \(0.249350\pi\)
\(80\) 0 0
\(81\) −6.28168 −0.697964
\(82\) 0 0
\(83\) −14.6789 −1.61122 −0.805610 − 0.592446i \(-0.798162\pi\)
−0.805610 + 0.592446i \(0.798162\pi\)
\(84\) 0 0
\(85\) −11.2029 −1.21512
\(86\) 0 0
\(87\) −11.2601 −1.20720
\(88\) 0 0
\(89\) −6.69305 −0.709462 −0.354731 − 0.934968i \(-0.615428\pi\)
−0.354731 + 0.934968i \(0.615428\pi\)
\(90\) 0 0
\(91\) 11.3180 1.18644
\(92\) 0 0
\(93\) −14.9466 −1.54989
\(94\) 0 0
\(95\) 4.99279 0.512249
\(96\) 0 0
\(97\) −2.05817 −0.208975 −0.104488 − 0.994526i \(-0.533320\pi\)
−0.104488 + 0.994526i \(0.533320\pi\)
\(98\) 0 0
\(99\) −14.2116 −1.42832
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.1.2 15
4.3 odd 2 8464.2.a.ch.1.14 15
23.5 odd 22 184.2.i.b.25.3 ✓ 30
23.14 odd 22 184.2.i.b.81.3 yes 30
23.22 odd 2 4232.2.a.bb.1.2 15
92.51 even 22 368.2.m.e.209.1 30
92.83 even 22 368.2.m.e.81.1 30
92.91 even 2 8464.2.a.cg.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.3 ✓ 30 23.5 odd 22
184.2.i.b.81.3 yes 30 23.14 odd 22
368.2.m.e.81.1 30 92.83 even 22
368.2.m.e.209.1 30 92.51 even 22
4232.2.a.ba.1.2 15 1.1 even 1 trivial
4232.2.a.bb.1.2 15 23.22 odd 2
8464.2.a.cg.1.14 15 92.91 even 2
8464.2.a.ch.1.14 15 4.3 odd 2