Properties

Label 4.18.a.a
Level 4
Weight 18
Character orbit 4.a
Self dual Yes
Analytic conductor 7.329
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.32888349378\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9361}) \)
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 \(\beta = 192\sqrt{9361}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2940 - \beta ) q^{3} + ( 302022 + 36 \beta ) q^{5} + ( 12675080 - 594 \beta ) q^{7} + ( 224587341 + 5880 \beta ) q^{9} +O(q^{10})\) \( q + ( -2940 - \beta ) q^{3} + ( 302022 + 36 \beta ) q^{5} + ( 12675080 - 594 \beta ) q^{7} + ( 224587341 + 5880 \beta ) q^{9} + ( 629824140 - 38115 \beta ) q^{11} + ( -660026290 + 162756 \beta ) q^{13} + ( -13310965224 - 407862 \beta ) q^{15} + ( -13749113070 + 175032 \beta ) q^{17} + ( 50566816916 + 2861595 \beta ) q^{19} + ( 167715103776 - 10928720 \beta ) q^{21} + ( 67383745560 + 14187978 \beta ) q^{23} + ( -224493425057 + 21745584 \beta ) q^{25} + ( -2309708058840 - 112734378 \beta ) q^{27} + ( 1168577791326 + 136229940 \beta ) q^{29} + ( 139418056736 + 128935800 \beta ) q^{31} + ( 11301190029360 - 517766040 \beta ) q^{33} + ( -3551121191376 + 276901812 \beta ) q^{35} + ( 10464901444070 + 494092116 \beta ) q^{37} + ( -54223998586824 + 181523650 \beta ) q^{39} + ( 2083296157866 - 1902597840 \beta ) q^{41} + ( -55571574267220 - 1581547275 \beta ) q^{43} + ( 140877678702222 + 9861033636 \beta ) q^{45} + ( 98386325578800 - 5178600108 \beta ) q^{47} + ( 49785163370937 - 15057995040 \beta ) q^{49} + ( -19978333459128 + 13234518990 \beta ) q^{51} + ( -243982561368330 + 12154446612 \beta ) q^{53} + ( -283282681623480 + 11162100510 \beta ) q^{55} + ( -1136156815999920 - 58979906216 \beta ) q^{57} + ( 1417952098906812 - 9148192335 \beta ) q^{59} + ( -33772017497218 + 114760782180 \beta ) q^{61} + ( 1641381060983400 - 58875410154 \beta ) q^{63} + ( 1822578671500884 + 25394946192 \beta ) q^{65} + ( -497585660773180 - 109849087089 \beta ) q^{67} + ( -5094151050052512 - 109096400880 \beta ) q^{69} + ( 1941122746607688 + 347319240030 \beta ) q^{71} + ( -6373212940884790 - 6950534184 \beta ) q^{73} + ( -6844040351812356 + 160561408097 \beta ) q^{75} + ( 15795877923001440 - 857226213360 \beta ) q^{77} + ( -7492135767032752 - 360485585460 \beta ) q^{79} + ( 16690115143764729 + 1881802971720 \beta ) q^{81} + ( 21949708904946900 - 232562921877 \beta ) q^{83} + ( -1978108495770132 - 442104555816 \beta ) q^{85} + ( -50446378243384200 - 1569093814926 \beta ) q^{87} + ( 25954503979423194 - 224724957480 \beta ) q^{89} + ( -41727584700231056 + 2455000936740 \beta ) q^{91} + ( -44903558316167040 - 518489308736 \beta ) q^{93} + ( 50821944652211832 + 2684670054066 \beta ) q^{95} + ( -24640503894924190 - 3147743142504 \beta ) q^{97} + ( 64111635654566940 - 4856780559015 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + O(q^{10}) \) \( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + 1259648280q^{11} - 1320052580q^{13} - 26621930448q^{15} - 27498226140q^{17} + 101133633832q^{19} + 335430207552q^{21} + 134767491120q^{23} - 448986850114q^{25} - 4619416117680q^{27} + 2337155582652q^{29} + 278836113472q^{31} + 22602380058720q^{33} - 7102242382752q^{35} + 20929802888140q^{37} - 108447997173648q^{39} + 4166592315732q^{41} - 111143148534440q^{43} + 281755357404444q^{45} + 196772651157600q^{47} + 99570326741874q^{49} - 39956666918256q^{51} - 487965122736660q^{53} - 566565363246960q^{55} - 2272313631999840q^{57} + 2835904197813624q^{59} - 67544034994436q^{61} + 3282762121966800q^{63} + 3645157343001768q^{65} - 995171321546360q^{67} - 10188302100105024q^{69} + 3882245493215376q^{71} - 12746425881769580q^{73} - 13688080703624712q^{75} + 31591755846002880q^{77} - 14984271534065504q^{79} + 33380230287529458q^{81} + 43899417809893800q^{83} - 3956216991540264q^{85} - 100892756486768400q^{87} + 51909007958846388q^{89} - 83455169400462112q^{91} - 89807116632334080q^{93} + 101643889304423664q^{95} - 49281007789848380q^{97} + 128223271309133880q^{99} + O(q^{100}) \)

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
48.8761
−47.8761
0 −21516.4 0 970774. 0 1.64068e6 0 3.33817e8 0
1.2 0 15636.4 0 −366730. 0 2.37095e7 0 1.15358e8 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 \(S_{18}^{\mathrm{new}}(\Gamma_0(4))\).