Properties

Label 4.24.a.a
Level $4$
Weight $24$
Character orbit 4.a
Self dual yes
Analytic conductor $13.408$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4081614938\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 618312\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 192\sqrt{2473249}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 85260 - \beta ) q^{3} + ( -46133010 + 540 \beta ) q^{5} + ( 96041720 - 2106 \beta ) q^{7} + ( 4299939909 - 170520 \beta ) q^{9} +O(q^{10})\) \( q + ( 85260 - \beta ) q^{3} + ( -46133010 + 540 \beta ) q^{5} + ( 96041720 - 2106 \beta ) q^{7} + ( 4299939909 - 170520 \beta ) q^{9} + ( -623587347900 + 2983365 \beta ) q^{11} + ( -3730149990490 - 22978404 \beta ) q^{13} + ( -53167180046040 + 92173410 \beta ) q^{15} + ( -187448673951630 - 148205592 \beta ) q^{17} + ( -420180139606276 - 153252405 \beta ) q^{19} + ( 200200647539616 - 275599280 \beta ) q^{21} + ( 3216678961536360 + 11230584498 \beta ) q^{23} + ( 16793620647839575 - 49823650800 \beta ) q^{25} + ( 7886930545562040 + 75304703718 \beta ) q^{27} + ( -34027749623717226 + 111255541740 \beta ) q^{29} + ( -72792257273422624 - 638955815400 \beta ) q^{31} + ( -325171933676306640 + 877949047800 \beta ) q^{33} + ( -108117244095081840 + 149018647860 \beta ) q^{35} + ( 605947071775578830 - 1437407022996 \beta ) q^{37} + ( 1776996997449689544 + 1771011265450 \beta ) q^{39} + ( 2518389183567344346 - 6392398005360 \beta ) q^{41} + ( 467590472959967780 + 13445936110845 \beta ) q^{43} + ( -8593730322505084890 + 10188568416060 \beta ) q^{45} + ( -4215584448138798000 - 59591327627772 \beta ) q^{47} + ( -26955145781283329847 - 404527724640 \beta ) q^{49} + ( -2469399358585221288 + 174812665177710 \beta ) q^{51} + ( 1881568598685102270 - 42305364395028 \beta ) q^{53} + ( 175650594609494604600 - 474368775244650 \beta ) q^{55} + ( -21851946743127109680 + 407113839555976 \beta ) q^{57} + ( 174280803256450693908 + 418202976769065 \beta ) q^{59} + ( -291483296360355677578 - 440318419808580 \beta ) q^{61} + ( 33154882116323779800 - 25432707542754 \beta ) q^{63} + ( -959232929432213074860 - 954218053348560 \beta ) q^{65} + ( 105807634422707327180 + 1905529275465039 \beta ) q^{67} + ( -749681590930331236128 - 2259159327236880 \beta ) q^{69} + ( 1920023878566772005048 + 1924700485055670 \beta ) q^{71} + ( -340333125956797048390 + 8794324601147016 \beta ) q^{73} + ( 5974438217526049473300 - 21041585115047575 \beta ) q^{75} + ( -632732671149061047840 + 1599802460665200 \beta ) q^{77} + ( -321495697653361820368 + 16664050979942940 \beta ) q^{79} + ( -6598190160109239299991 + 14586843347014680 \beta ) q^{81} + ( -9399751316254358885700 + 4712530604907483 \beta ) q^{83} + ( 1350835275330679949820 - 94385113876088280 \beta ) q^{85} + ( -13044802133575925105400 + 43513397112469626 \beta ) q^{87} + ( 2500699389441625069866 + 67926459439942920 \beta ) q^{89} + ( 4053882286410810541264 + 5648810436957060 \beta ) q^{91} + ( 52049794540629083372160 + 18314884452418624 \beta ) q^{93} + ( 11838964124017576447560 - 219827280654999990 \beta ) q^{95} + ( 61011397130279910307490 + 119382738980386824 \beta ) q^{97} + ( -49063659646747689513900 + 119162404790521785 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 170520q^{3} - 92266020q^{5} + 192083440q^{7} + 8599879818q^{9} + O(q^{10}) \) \( 2q + 170520q^{3} - 92266020q^{5} + 192083440q^{7} + 8599879818q^{9} - 1247174695800q^{11} - 7460299980980q^{13} - 106334360092080q^{15} - 374897347903260q^{17} - 840360279212552q^{19} + 400401295079232q^{21} + 6433357923072720q^{23} + 33587241295679150q^{25} + 15773861091124080q^{27} - 68055499247434452q^{29} - 145584514546845248q^{31} - 650343867352613280q^{33} - 216234488190163680q^{35} + 1211894143551157660q^{37} + 3553993994899379088q^{39} + 5036778367134688692q^{41} + 935180945919935560q^{43} - 17187460645010169780q^{45} - 8431168896277596000q^{47} - 53910291562566659694q^{49} - 4938798717170442576q^{51} + 3763137197370204540q^{53} + 351301189218989209200q^{55} - 43703893486254219360q^{57} + 348561606512901387816q^{59} - 582966592720711355156q^{61} + 66309764232647559600q^{63} - 1918465858864426149720q^{65} + 211615268845414654360q^{67} - 1499363181860662472256q^{69} + 3840047757133544010096q^{71} - 680666251913594096780q^{73} + 11948876435052098946600q^{75} - 1265465342298122095680q^{77} - 642991395306723640736q^{79} - 13196380320218478599982q^{81} - 18799502632508717771400q^{83} + 2701670550661359899640q^{85} - 26089604267151850210800q^{87} + 5001398778883250139732q^{89} + 8107764572821621082528q^{91} + 104099589081258166744320q^{93} + 23677928248035152895120q^{95} + 122022794260559820614980q^{97} - 98127319293495379027800q^{99} + O(q^{100}) \)

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
786.828
−785.828
0 −216690. 0 1.16920e8 0 −5.39865e8 0 −4.71886e10 0
1.2 0 387210. 0 −2.09186e8 0 7.31949e8 0 5.57885e10 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 4.24.a.a 2
3.b odd 2 1 36.24.a.e 2
4.b odd 2 1 16.24.a.c 2
8.b even 2 1 64.24.a.e 2
8.d odd 2 1 64.24.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.24.a.a 2 1.a even 1 1 trivial
16.24.a.c 2 4.b odd 2 1
36.24.a.e 2 3.b odd 2 1
64.24.a.e 2 8.b even 2 1
64.24.a.f 2 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -83904583536 - 170520 T + T^{2} \)
$5$ \( -24458040379597500 + 92266020 T + T^{2} \)
$7$ \( -395153534836469696 - 192083440 T + T^{2} \)
$11$ \( -\)\(42\!\cdots\!00\)\( + 1247174695800 T + T^{2} \)
$13$ \( -\)\(34\!\cdots\!76\)\( + 7460299980980 T + T^{2} \)
$17$ \( \)\(33\!\cdots\!96\)\( + 374897347903260 T + T^{2} \)
$19$ \( \)\(17\!\cdots\!76\)\( + 840360279212552 T + T^{2} \)
$23$ \( -\)\(11\!\cdots\!44\)\( - 6433357923072720 T + T^{2} \)
$29$ \( \)\(29\!\cdots\!76\)\( + 68055499247434452 T + T^{2} \)
$31$ \( -\)\(31\!\cdots\!24\)\( + 145584514546845248 T + T^{2} \)
$37$ \( \)\(17\!\cdots\!24\)\( - 1211894143551157660 T + T^{2} \)
$41$ \( \)\(26\!\cdots\!16\)\( - 5036778367134688692 T + T^{2} \)
$43$ \( -\)\(16\!\cdots\!00\)\( - 935180945919935560 T + T^{2} \)
$47$ \( -\)\(30\!\cdots\!24\)\( + 8431168896277596000 T + T^{2} \)
$53$ \( -\)\(15\!\cdots\!24\)\( - 3763137197370204540 T + T^{2} \)
$59$ \( \)\(14\!\cdots\!64\)\( - \)\(34\!\cdots\!16\)\( T + T^{2} \)
$61$ \( \)\(67\!\cdots\!84\)\( + \)\(58\!\cdots\!56\)\( T + T^{2} \)
$67$ \( -\)\(31\!\cdots\!56\)\( - \)\(21\!\cdots\!60\)\( T + T^{2} \)
$71$ \( \)\(33\!\cdots\!04\)\( - \)\(38\!\cdots\!96\)\( T + T^{2} \)
$73$ \( -\)\(69\!\cdots\!16\)\( + \)\(68\!\cdots\!80\)\( T + T^{2} \)
$79$ \( -\)\(25\!\cdots\!76\)\( + \)\(64\!\cdots\!36\)\( T + T^{2} \)
$83$ \( \)\(86\!\cdots\!96\)\( + \)\(18\!\cdots\!00\)\( T + T^{2} \)
$89$ \( -\)\(41\!\cdots\!44\)\( - \)\(50\!\cdots\!32\)\( T + T^{2} \)
$97$ \( \)\(24\!\cdots\!64\)\( - \)\(12\!\cdots\!80\)\( T + T^{2} \)
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