Properties

Label 4.22.a.a
Level 4
Weight 22
Character orbit 4.a
Self dual Yes
Analytic conductor 11.179
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.1790937715\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2161}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2688\sqrt{2161}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 32820 - \beta ) q^{3} \) \( + ( 6844662 - 204 \beta ) q^{5} \) \( + ( -130254040 + 5886 \beta ) q^{7} \) \( + ( 6230767581 - 65640 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 32820 - \beta ) q^{3} \) \( + ( 6844662 - 204 \beta ) q^{5} \) \( + ( -130254040 + 5886 \beta ) q^{7} \) \( + ( 6230767581 - 65640 \beta ) q^{9} \) \( + ( 72717981660 + 314445 \beta ) q^{11} \) \( + ( 714450208670 + 324756 \beta ) q^{13} \) \( + ( 3409891357176 - 13539942 \beta ) q^{15} \) \( + ( 920310288210 + 84858072 \beta ) q^{17} \) \( + ( -8390371964284 - 265449285 \beta ) q^{19} \) \( + ( -96178755501024 + 323432560 \beta ) q^{21} \) \( + ( -159845962713480 + 627682938 \beta ) q^{23} \) \( + ( 219803147959663 - 2792622096 \beta ) q^{25} \) \( + ( 886085884611720 + 2075280822 \beta ) q^{27} \) \( + ( 1871055887883246 + 4723712580 \beta ) q^{29} \) \( + ( 56021176731296 + 4595859000 \beta ) q^{31} \) \( + ( -2523130130425680 - 62397896760 \beta ) q^{33} \) \( + ( -19639923731212176 + 66859504692 \beta ) q^{35} \) \( + ( -16681352818773610 + 245612021796 \beta ) q^{37} \) \( + ( 18377525932035096 - 703791716750 \beta ) q^{39} \) \( + ( 87564872161566666 + 265794737520 \beta ) q^{41} \) \( + ( 7673306708264060 + 1278047140965 \beta ) q^{43} \) \( + ( 251727278576557662 - 1720360200204 \beta ) q^{45} \) \( + ( -342424409644227600 + 553734430452 \beta ) q^{47} \) \( + ( -633876939155943 - 1533350558880 \beta ) q^{49} \) \( + ( -1294766669676143448 + 1864731634830 \beta ) q^{51} \) \( + ( 337652697122210790 + 7229502808932 \beta ) q^{53} \) \( + ( -503855789070504600 - 12682198516050 \beta ) q^{55} \) \( + ( 3869344735677604560 - 321673569416 \beta ) q^{57} \) \( + ( 521222625217717452 - 7341605116095 \beta ) q^{59} \) \( + ( 4532998914868234382 + 32044341559860 \beta ) q^{61} \) \( + ( -6844149257222100600 + 45224173167366 \beta ) q^{63} \) \( + ( 3855741291206701524 - 143524997516208 \beta ) q^{65} \) \( + ( -15232150523401387660 - 19726418252529 \beta ) q^{67} \) \( + ( -15046766045364645792 + 180446516738640 \beta ) q^{69} \) \( + ( -4099546751415259032 + 115464309246510 \beta ) q^{71} \) \( + ( 12707543329687396970 - 226051490503944 \beta ) q^{73} \) \( + ( 50817852431439952524 - 311457005150383 \beta ) q^{75} \) \( + ( 19426885130290589280 + 387060308442960 \beta ) q^{77} \) \( + ( 60602176612530082448 + 99610802523180 \beta ) q^{79} \) \( + ( -68498060032734793191 - 131357583788760 \beta ) q^{81} \) \( + ( 54371468378516904900 + 218924689501323 \beta ) q^{83} \) \( + ( -263994922822459877172 + 393081522016824 \beta ) q^{85} \) \( + ( -12347844638894937000 - 1716023641007646 \beta ) q^{87} \) \( + ( -92103999637482684486 - 1052618322986760 \beta ) q^{89} \) \( + ( -63213709769507333456 + 4162953147217380 \beta ) q^{91} \) \( + ( -69920982103000721280 + 94814915648704 \beta ) q^{93} \) \( + ( 788092955533462657752 - 105274753252734 \beta ) q^{95} \) \( + ( 369748892594238233570 - 5190622720371144 \beta ) q^{97} \) \( + ( 130813883985288961260 - 2813974604154855 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 65640q^{3} \) \(\mathstrut +\mathstrut 13689324q^{5} \) \(\mathstrut -\mathstrut 260508080q^{7} \) \(\mathstrut +\mathstrut 12461535162q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 65640q^{3} \) \(\mathstrut +\mathstrut 13689324q^{5} \) \(\mathstrut -\mathstrut 260508080q^{7} \) \(\mathstrut +\mathstrut 12461535162q^{9} \) \(\mathstrut +\mathstrut 145435963320q^{11} \) \(\mathstrut +\mathstrut 1428900417340q^{13} \) \(\mathstrut +\mathstrut 6819782714352q^{15} \) \(\mathstrut +\mathstrut 1840620576420q^{17} \) \(\mathstrut -\mathstrut 16780743928568q^{19} \) \(\mathstrut -\mathstrut 192357511002048q^{21} \) \(\mathstrut -\mathstrut 319691925426960q^{23} \) \(\mathstrut +\mathstrut 439606295919326q^{25} \) \(\mathstrut +\mathstrut 1772171769223440q^{27} \) \(\mathstrut +\mathstrut 3742111775766492q^{29} \) \(\mathstrut +\mathstrut 112042353462592q^{31} \) \(\mathstrut -\mathstrut 5046260260851360q^{33} \) \(\mathstrut -\mathstrut 39279847462424352q^{35} \) \(\mathstrut -\mathstrut 33362705637547220q^{37} \) \(\mathstrut +\mathstrut 36755051864070192q^{39} \) \(\mathstrut +\mathstrut 175129744323133332q^{41} \) \(\mathstrut +\mathstrut 15346613416528120q^{43} \) \(\mathstrut +\mathstrut 503454557153115324q^{45} \) \(\mathstrut -\mathstrut 684848819288455200q^{47} \) \(\mathstrut -\mathstrut 1267753878311886q^{49} \) \(\mathstrut -\mathstrut 2589533339352286896q^{51} \) \(\mathstrut +\mathstrut 675305394244421580q^{53} \) \(\mathstrut -\mathstrut 1007711578141009200q^{55} \) \(\mathstrut +\mathstrut 7738689471355209120q^{57} \) \(\mathstrut +\mathstrut 1042445250435434904q^{59} \) \(\mathstrut +\mathstrut 9065997829736468764q^{61} \) \(\mathstrut -\mathstrut 13688298514444201200q^{63} \) \(\mathstrut +\mathstrut 7711482582413403048q^{65} \) \(\mathstrut -\mathstrut 30464301046802775320q^{67} \) \(\mathstrut -\mathstrut 30093532090729291584q^{69} \) \(\mathstrut -\mathstrut 8199093502830518064q^{71} \) \(\mathstrut +\mathstrut 25415086659374793940q^{73} \) \(\mathstrut +\mathstrut 101635704862879905048q^{75} \) \(\mathstrut +\mathstrut 38853770260581178560q^{77} \) \(\mathstrut +\mathstrut 121204353225060164896q^{79} \) \(\mathstrut -\mathstrut 136996120065469586382q^{81} \) \(\mathstrut +\mathstrut 108742936757033809800q^{83} \) \(\mathstrut -\mathstrut 527989845644919754344q^{85} \) \(\mathstrut -\mathstrut 24695689277789874000q^{87} \) \(\mathstrut -\mathstrut 184207999274965368972q^{89} \) \(\mathstrut -\mathstrut 126427419539014666912q^{91} \) \(\mathstrut -\mathstrut 139841964206001442560q^{93} \) \(\mathstrut +\mathstrut 1576185911066925315504q^{95} \) \(\mathstrut +\mathstrut 739497785188476467140q^{97} \) \(\mathstrut +\mathstrut 261627767970577922520q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
23.7433
−22.7433
0 −92135.9 0 −1.86463e7 0 6.05236e8 0 −1.97134e9 0
1.2 0 157776. 0 3.23357e7 0 −8.65744e8 0 1.44329e10 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 \(S_{22}^{\mathrm{new}}(\Gamma_0(4))\).