Properties

Label 4.32.a.a
Level $4$
Weight $32$
Character orbit 4.a
Self dual yes
Analytic conductor $24.351$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.3508531276\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 648835980\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 576\sqrt{2595343921}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -15602580 - \beta ) q^{3} + ( 936152910 + 60 \beta ) q^{5} + ( 3045184714040 + 195174 \beta ) q^{7} + ( 486839931106149 + 31205160 \beta ) q^{9} +O(q^{10})\) \( q +(-15602580 - \beta) q^{3} +(936152910 + 60 \beta) q^{5} +(3045184714040 + 195174 \beta) q^{7} +(486839931106149 + 31205160 \beta) q^{9} +(5449482757404900 + 349933605 \beta) q^{11} +(-111978807839628730 - 7118697924 \beta) q^{13} +(-66270770154529560 - 1872307710 \beta) q^{15} +(5343860143768987890 + 396727474728 \beta) q^{17} +(-32753551660279010596 - 1317343114485 \beta) q^{19} +(-\)\(21\!\cdots\!04\)\( - 6090402662960 \beta) q^{21} +(-\)\(24\!\cdots\!80\)\( + 39506723420178 \beta) q^{23} +(-\)\(46\!\cdots\!25\)\( + 112338349200 \beta) q^{25} +(-\)\(24\!\cdots\!20\)\( - 356047540135002 \beta) q^{27} +(-\)\(63\!\cdots\!06\)\( - 124101321464820 \beta) q^{29} +(-\)\(62\!\cdots\!04\)\( + 5275159980514200 \beta) q^{31} +(-\)\(38\!\cdots\!80\)\( - 10909349824105800 \beta) q^{33} +(\)\(12\!\cdots\!40\)\( + 365423790898740 \beta) q^{35} +(\)\(82\!\cdots\!10\)\( - 25355709323724276 \beta) q^{37} +(\)\(78\!\cdots\!04\)\( + 223048861694672650 \beta) q^{39} +(\)\(38\!\cdots\!06\)\( - 356889936068955120 \beta) q^{41} +(\)\(10\!\cdots\!60\)\( - 125473602327915555 \beta) q^{43} +(\)\(20\!\cdots\!90\)\( + 58423197207384540 \beta) q^{45} +(-\)\(15\!\cdots\!00\)\( + 1777000944995426628 \beta) q^{47} +(-\)\(11\!\cdots\!47\)\( + 1188681762756085920 \beta) q^{49} +(-\)\(42\!\cdots\!88\)\( - 11533832306410586130 \beta) q^{51} +(-\)\(16\!\cdots\!10\)\( + 7048155549547205772 \beta) q^{53} +(\)\(23\!\cdots\!00\)\( + 654560328071834550 \beta) q^{55} +(\)\(16\!\cdots\!40\)\( + 53307502991480381896 \beta) q^{57} +(\)\(12\!\cdots\!88\)\( - 62478695796364019895 \beta) q^{59} +(\)\(14\!\cdots\!02\)\( - \)\(13\!\cdots\!60\)\( \beta) q^{61} +(\)\(67\!\cdots\!00\)\( + \)\(19\!\cdots\!26\)\( \beta) q^{63} +(-\)\(47\!\cdots\!40\)\( - 13382918247341282640 \beta) q^{65} +(-\)\(18\!\cdots\!40\)\( + \)\(41\!\cdots\!99\)\( \beta) q^{67} +(-\)\(30\!\cdots\!88\)\( - \)\(36\!\cdots\!60\)\( \beta) q^{69} +(-\)\(50\!\cdots\!12\)\( - \)\(10\!\cdots\!10\)\( \beta) q^{71} +(-\)\(57\!\cdots\!30\)\( - \)\(94\!\cdots\!24\)\( \beta) q^{73} +(\)\(72\!\cdots\!00\)\( + \)\(46\!\cdots\!25\)\( \beta) q^{75} +(\)\(75\!\cdots\!20\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{77} +(\)\(35\!\cdots\!92\)\( - \)\(10\!\cdots\!20\)\( \beta) q^{79} +(\)\(39\!\cdots\!89\)\( + \)\(11\!\cdots\!60\)\( \beta) q^{81} +(\)\(81\!\cdots\!00\)\( - \)\(21\!\cdots\!97\)\( \beta) q^{83} +(\)\(25\!\cdots\!80\)\( + \)\(69\!\cdots\!80\)\( \beta) q^{85} +(\)\(10\!\cdots\!00\)\( + \)\(65\!\cdots\!06\)\( \beta) q^{87} +(-\)\(18\!\cdots\!94\)\( - \)\(32\!\cdots\!60\)\( \beta) q^{89} +(-\)\(15\!\cdots\!96\)\( - \)\(43\!\cdots\!80\)\( \beta) q^{91} +(-\)\(35\!\cdots\!80\)\( - \)\(19\!\cdots\!96\)\( \beta) q^{93} +(-\)\(98\!\cdots\!60\)\( - \)\(31\!\cdots\!10\)\( \beta) q^{95} +(-\)\(33\!\cdots\!70\)\( - \)\(17\!\cdots\!76\)\( \beta) q^{97} +(\)\(12\!\cdots\!00\)\( + \)\(34\!\cdots\!45\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 31205160q^{3} + 1872305820q^{5} + 6090369428080q^{7} + 973679862212298q^{9} + O(q^{10}) \) \( 2q - 31205160q^{3} + 1872305820q^{5} + 6090369428080q^{7} + 973679862212298q^{9} + 10898965514809800q^{11} - 223957615679257460q^{13} - 132541540309059120q^{15} + 10687720287537975780q^{17} - 65507103320558021192q^{19} - \)\(43\!\cdots\!08\)\(q^{21} - \)\(49\!\cdots\!60\)\(q^{23} - \)\(93\!\cdots\!50\)\(q^{25} - \)\(49\!\cdots\!40\)\(q^{27} - \)\(12\!\cdots\!12\)\(q^{29} - \)\(12\!\cdots\!08\)\(q^{31} - \)\(77\!\cdots\!60\)\(q^{33} + \)\(25\!\cdots\!80\)\(q^{35} + \)\(16\!\cdots\!20\)\(q^{37} + \)\(15\!\cdots\!08\)\(q^{39} + \)\(77\!\cdots\!12\)\(q^{41} + \)\(20\!\cdots\!20\)\(q^{43} + \)\(41\!\cdots\!80\)\(q^{45} - \)\(30\!\cdots\!00\)\(q^{47} - \)\(23\!\cdots\!94\)\(q^{49} - \)\(84\!\cdots\!76\)\(q^{51} - \)\(32\!\cdots\!20\)\(q^{53} + \)\(46\!\cdots\!00\)\(q^{55} + \)\(32\!\cdots\!80\)\(q^{57} + \)\(25\!\cdots\!76\)\(q^{59} + \)\(28\!\cdots\!04\)\(q^{61} + \)\(13\!\cdots\!00\)\(q^{63} - \)\(94\!\cdots\!80\)\(q^{65} - \)\(37\!\cdots\!80\)\(q^{67} - \)\(60\!\cdots\!76\)\(q^{69} - \)\(10\!\cdots\!24\)\(q^{71} - \)\(11\!\cdots\!60\)\(q^{73} + \)\(14\!\cdots\!00\)\(q^{75} + \)\(15\!\cdots\!40\)\(q^{77} + \)\(70\!\cdots\!84\)\(q^{79} + \)\(78\!\cdots\!78\)\(q^{81} + \)\(16\!\cdots\!00\)\(q^{83} + \)\(50\!\cdots\!60\)\(q^{85} + \)\(21\!\cdots\!00\)\(q^{87} - \)\(36\!\cdots\!88\)\(q^{89} - \)\(30\!\cdots\!92\)\(q^{91} - \)\(71\!\cdots\!60\)\(q^{93} - \)\(19\!\cdots\!20\)\(q^{95} - \)\(67\!\cdots\!40\)\(q^{97} + \)\(24\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
25472.8
−25471.8
0 −4.49466e7 0 2.69680e9 0 8.77238e12 0 1.40253e15 0
1.2 0 1.37415e7 0 −8.24490e8 0 −2.68201e12 0 −4.28846e14 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.32.a.a 2
3.b odd 2 1 36.32.a.b 2
4.b odd 2 1 16.32.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.32.a.a 2 1.a even 1 1 trivial
16.32.a.d 2 4.b odd 2 1
36.32.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{32}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -617632322077296 + 31205160 T + T^{2} \)
$5$ \( -2223479898139837500 - 1872305820 T + T^{2} \)
$7$ \( -\)\(23\!\cdots\!96\)\( - 6090369428080 T + T^{2} \)
$11$ \( -\)\(75\!\cdots\!00\)\( - 10898965514809800 T + T^{2} \)
$13$ \( -\)\(31\!\cdots\!96\)\( + 223957615679257460 T + T^{2} \)
$17$ \( -\)\(10\!\cdots\!64\)\( - 10687720287537975780 T + T^{2} \)
$19$ \( -\)\(42\!\cdots\!84\)\( + 65507103320558021192 T + T^{2} \)
$23$ \( -\)\(12\!\cdots\!64\)\( + \)\(49\!\cdots\!60\)\( T + T^{2} \)
$29$ \( \)\(39\!\cdots\!36\)\( + \)\(12\!\cdots\!12\)\( T + T^{2} \)
$31$ \( -\)\(20\!\cdots\!84\)\( + \)\(12\!\cdots\!08\)\( T + T^{2} \)
$37$ \( \)\(13\!\cdots\!04\)\( - \)\(16\!\cdots\!20\)\( T + T^{2} \)
$41$ \( -\)\(94\!\cdots\!64\)\( - \)\(77\!\cdots\!12\)\( T + T^{2} \)
$43$ \( \)\(94\!\cdots\!00\)\( - \)\(20\!\cdots\!20\)\( T + T^{2} \)
$47$ \( -\)\(24\!\cdots\!64\)\( + \)\(30\!\cdots\!00\)\( T + T^{2} \)
$53$ \( -\)\(17\!\cdots\!64\)\( + \)\(32\!\cdots\!20\)\( T + T^{2} \)
$59$ \( -\)\(17\!\cdots\!56\)\( - \)\(25\!\cdots\!76\)\( T + T^{2} \)
$61$ \( -\)\(13\!\cdots\!96\)\( - \)\(28\!\cdots\!04\)\( T + T^{2} \)
$67$ \( -\)\(14\!\cdots\!96\)\( + \)\(37\!\cdots\!80\)\( T + T^{2} \)
$71$ \( \)\(15\!\cdots\!44\)\( + \)\(10\!\cdots\!24\)\( T + T^{2} \)
$73$ \( \)\(25\!\cdots\!04\)\( + \)\(11\!\cdots\!60\)\( T + T^{2} \)
$79$ \( -\)\(94\!\cdots\!36\)\( - \)\(70\!\cdots\!84\)\( T + T^{2} \)
$83$ \( -\)\(38\!\cdots\!64\)\( - \)\(16\!\cdots\!00\)\( T + T^{2} \)
$89$ \( \)\(23\!\cdots\!36\)\( + \)\(36\!\cdots\!88\)\( T + T^{2} \)
$97$ \( \)\(11\!\cdots\!04\)\( + \)\(67\!\cdots\!40\)\( T + T^{2} \)
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