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

Related objects

Downloads

Learn more about

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: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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.

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

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:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

There are no other newforms in \(S_{24}^{\mathrm{new}}(\Gamma_0(4))\).