Properties

Label 4.26.a.a
Level $4$
Weight $26$
Character orbit 4.a
Self dual yes
Analytic conductor $15.840$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.8398711981\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{358121}) \)
Defining polynomial: \(x^{2} - x - 89530\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1728\sqrt{358121}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -449820 - \beta ) q^{3} + ( -199675098 - 444 \beta ) q^{5} + ( -20259231160 - 45234 \beta ) q^{7} + ( 424392999021 + 899640 \beta ) q^{9} +O(q^{10})\) \( q +(-449820 - \beta) q^{3} +(-199675098 - 444 \beta) q^{5} +(-20259231160 - 45234 \beta) q^{7} +(424392999021 + 899640 \beta) q^{9} +(-653144689620 - 4113795 \beta) q^{11} +(16364511746030 - 30041244 \beta) q^{13} +(564606400354776 + 399395178 \beta) q^{15} +(2312348505190290 - 1334625288 \beta) q^{17} +(12337010890387316 - 3100300965 \beta) q^{19} +(57483694680070176 + 40606389040 \beta) q^{21} +(-15521463939879720 - 137884422102 \beta) q^{23} +(-47346963904690817 + 177311487024 \beta) q^{25} +(-771797351290192920 + 18219545622 \beta) q^{27} +(-1664068692207202434 + 429107944020 \beta) q^{29} +(-3210901990635154144 - 3191043227400 \beta) q^{31} +(4692857800779071280 + 2503611956520 \beta) q^{33} +(25521849137215119024 + 18027202017972 \beta) q^{35} +(15843782904606461510 - 42932937760524 \beta) q^{37} +(24763326614771969016 - 2851359369950 \beta) q^{39} +(-8158404481979309334 + 74012242818480 \beta) q^{41} +(-\)\(17\!\cdots\!60\)\( + 111080724667605 \beta) q^{43} +(-\)\(51\!\cdots\!98\)\( - 368066196730044 \beta) q^{45} +(-\)\(55\!\cdots\!00\)\( - 362380229556588 \beta) q^{47} +(\)\(12\!\cdots\!77\)\( + 1832812124582880 \beta) q^{49} +(\)\(38\!\cdots\!32\)\( - 1712007358142130 \beta) q^{51} +(\)\(38\!\cdots\!10\)\( + 495922446512052 \beta) q^{53} +(\)\(20\!\cdots\!80\)\( + 1111418661968190 \beta) q^{55} +(-\)\(22\!\cdots\!60\)\( - 10942433510311016 \beta) q^{57} +(-\)\(22\!\cdots\!08\)\( + 21030367886292945 \beta) q^{59} +(-\)\(53\!\cdots\!18\)\( + 4201133961364740 \beta) q^{61} +(-\)\(52\!\cdots\!00\)\( - 37423007638498314 \beta) q^{63} +(\)\(10\!\cdots\!64\)\( - 1267354875495408 \beta) q^{65} +(-\)\(13\!\cdots\!40\)\( + 23118478364390031 \beta) q^{67} +(\)\(15\!\cdots\!28\)\( + 77544634689801360 \beta) q^{69} +(-\)\(15\!\cdots\!52\)\( - 56657217238532610 \beta) q^{71} +(\)\(19\!\cdots\!30\)\( - 107322367052387304 \beta) q^{73} +(-\)\(16\!\cdots\!96\)\( - 32411289188444863 \beta) q^{75} +(\)\(21\!\cdots\!20\)\( + 112886670740123280 \beta) q^{77} +(-\)\(57\!\cdots\!52\)\( + 269319129164539020 \beta) q^{79} +(-\)\(31\!\cdots\!11\)\( + 1347110679204360 \beta) q^{81} +(-\)\(12\!\cdots\!00\)\( - 431820919835393877 \beta) q^{83} +(\)\(17\!\cdots\!88\)\( - 760191301129810536 \beta) q^{85} +(\)\(28\!\cdots\!00\)\( + 1471047356828126034 \beta) q^{87} +(\)\(19\!\cdots\!34\)\( - 786203878756664040 \beta) q^{89} +(\)\(11\!\cdots\!44\)\( - 131619817789957980 \beta) q^{91} +(\)\(48\!\cdots\!80\)\( + 4646297055184222144 \beta) q^{93} +(-\)\(99\!\cdots\!28\)\( - 4858579936316098734 \beta) q^{95} +(-\)\(41\!\cdots\!70\)\( - 5367064275573580584 \beta) q^{97} +(-\)\(42\!\cdots\!20\)\( - 2333460885977331495 \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 899640q^{3} - 399350196q^{5} - 40518462320q^{7} + 848785998042q^{9} + O(q^{10}) \) \( 2q - 899640q^{3} - 399350196q^{5} - 40518462320q^{7} + 848785998042q^{9} - 1306289379240q^{11} + 32729023492060q^{13} + 1129212800709552q^{15} + 4624697010380580q^{17} + 24674021780774632q^{19} + 114967389360140352q^{21} - 31042927879759440q^{23} - 94693927809381634q^{25} - 1543594702580385840q^{27} - 3328137384414404868q^{29} - 6421803981270308288q^{31} + 9385715601558142560q^{33} + 51043698274430238048q^{35} + 31687565809212923020q^{37} + 49526653229543938032q^{39} - 16316808963958618668q^{41} - \)\(34\!\cdots\!20\)\(q^{43} - \)\(10\!\cdots\!96\)\(q^{45} - \)\(11\!\cdots\!00\)\(q^{47} + \)\(25\!\cdots\!54\)\(q^{49} + \)\(77\!\cdots\!64\)\(q^{51} + \)\(76\!\cdots\!20\)\(q^{53} + \)\(41\!\cdots\!60\)\(q^{55} - \)\(44\!\cdots\!20\)\(q^{57} - \)\(44\!\cdots\!16\)\(q^{59} - \)\(10\!\cdots\!36\)\(q^{61} - \)\(10\!\cdots\!00\)\(q^{63} + \)\(21\!\cdots\!28\)\(q^{65} - \)\(27\!\cdots\!80\)\(q^{67} + \)\(30\!\cdots\!56\)\(q^{69} - \)\(31\!\cdots\!04\)\(q^{71} + \)\(39\!\cdots\!60\)\(q^{73} - \)\(33\!\cdots\!92\)\(q^{75} + \)\(42\!\cdots\!40\)\(q^{77} - \)\(11\!\cdots\!04\)\(q^{79} - \)\(63\!\cdots\!22\)\(q^{81} - \)\(25\!\cdots\!00\)\(q^{83} + \)\(34\!\cdots\!76\)\(q^{85} + \)\(57\!\cdots\!00\)\(q^{87} + \)\(39\!\cdots\!68\)\(q^{89} + \)\(22\!\cdots\!88\)\(q^{91} + \)\(97\!\cdots\!60\)\(q^{93} - \)\(19\!\cdots\!56\)\(q^{95} - \)\(82\!\cdots\!40\)\(q^{97} - \)\(84\!\cdots\!40\)\(q^{99} + O(q^{100}) \)

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
299.716
−298.716
0 −1.48391e6 0 −6.58811e8 0 −6.70353e10 0 1.35470e12 0
1.2 0 584271. 0 2.59461e8 0 2.65168e10 0 −5.05916e11 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 4.26.a.a 2
3.b odd 2 1 36.26.a.d 2
4.b odd 2 1 16.26.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.26.a.a 2 1.a even 1 1 trivial
16.26.a.d 2 4.b odd 2 1
36.26.a.d 2 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -867005543664 + 899640 T + T^{2} \)
$5$ \( -170935970449643100 + 399350196 T + T^{2} \)
$7$ \( -\)\(17\!\cdots\!84\)\( + 40518462320 T + T^{2} \)
$11$ \( -\)\(17\!\cdots\!00\)\( + 1306289379240 T + T^{2} \)
$13$ \( -\)\(69\!\cdots\!04\)\( - 32729023492060 T + T^{2} \)
$17$ \( \)\(34\!\cdots\!84\)\( - 4624697010380580 T + T^{2} \)
$19$ \( \)\(14\!\cdots\!56\)\( - 24674021780774632 T + T^{2} \)
$23$ \( -\)\(20\!\cdots\!56\)\( + 31042927879759440 T + T^{2} \)
$29$ \( \)\(25\!\cdots\!56\)\( + 3328137384414404868 T + T^{2} \)
$31$ \( -\)\(57\!\cdots\!64\)\( + 6421803981270308288 T + T^{2} \)
$37$ \( -\)\(17\!\cdots\!64\)\( - 31687565809212923020 T + T^{2} \)
$41$ \( -\)\(57\!\cdots\!44\)\( + 16316808963958618668 T + T^{2} \)
$43$ \( \)\(15\!\cdots\!00\)\( + \)\(34\!\cdots\!20\)\( T + T^{2} \)
$47$ \( \)\(17\!\cdots\!84\)\( + \)\(11\!\cdots\!00\)\( T + T^{2} \)
$53$ \( \)\(14\!\cdots\!44\)\( - \)\(76\!\cdots\!20\)\( T + T^{2} \)
$59$ \( -\)\(46\!\cdots\!36\)\( + \)\(44\!\cdots\!16\)\( T + T^{2} \)
$61$ \( \)\(94\!\cdots\!24\)\( + \)\(10\!\cdots\!36\)\( T + T^{2} \)
$67$ \( -\)\(37\!\cdots\!04\)\( + \)\(27\!\cdots\!80\)\( T + T^{2} \)
$71$ \( -\)\(31\!\cdots\!96\)\( + \)\(31\!\cdots\!04\)\( T + T^{2} \)
$73$ \( \)\(25\!\cdots\!76\)\( - \)\(39\!\cdots\!60\)\( T + T^{2} \)
$79$ \( \)\(24\!\cdots\!04\)\( + \)\(11\!\cdots\!04\)\( T + T^{2} \)
$83$ \( \)\(14\!\cdots\!44\)\( + \)\(25\!\cdots\!00\)\( T + T^{2} \)
$89$ \( \)\(31\!\cdots\!56\)\( - \)\(39\!\cdots\!68\)\( T + T^{2} \)
$97$ \( -\)\(13\!\cdots\!84\)\( + \)\(82\!\cdots\!40\)\( T + T^{2} \)
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