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.

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

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
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 \(S_{18}^{\mathrm{new}}(\Gamma_0(4))\).