Properties

Label 12.20.a.a
Level 12
Weight 20
Character orbit 12.a
Self dual Yes
Analytic conductor 27.458
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(27.4580035868\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193153}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3}\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8640\sqrt{193153}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -19683 q^{3} \) \( + ( 312390 - \beta ) q^{5} \) \( + ( 2333552 + 39 \beta ) q^{7} \) \( + 387420489 q^{9} \) \(+O(q^{10})\) \( q\) \( -19683 q^{3} \) \( + ( 312390 - \beta ) q^{5} \) \( + ( 2333552 + 39 \beta ) q^{7} \) \( + 387420489 q^{9} \) \( + ( 4507891596 - 2146 \beta ) q^{11} \) \( + ( 12974075750 + 6954 \beta ) q^{13} \) \( + ( -6148772370 + 19683 \beta ) q^{15} \) \( + ( -165474934398 - 153842 \beta ) q^{17} \) \( + ( -777899556460 + 390678 \beta ) q^{19} \) \( + ( -45931304016 - 767637 \beta ) q^{21} \) \( + ( -1477897094232 + 1328782 \beta ) q^{23} \) \( + ( -4557104627225 - 624780 \beta ) q^{25} \) \( -7625597484987 q^{27} \) \( + ( -85556481109794 - 7636915 \beta ) q^{29} \) \( + ( -219843102023896 + 2126763 \beta ) q^{31} \) \( + ( -88728830284068 + 42239718 \beta ) q^{33} \) \( + ( -561603995053920 + 9849658 \beta ) q^{35} \) \( + ( -456420877658434 - 151286928 \beta ) q^{37} \) \( + ( -255368732987250 - 136875582 \beta ) q^{39} \) \( + ( -1019934442223190 + 349518794 \beta ) q^{41} \) \( + ( -2803561946993332 + 720464706 \beta ) q^{43} \) \( + ( 121026286558710 - 387420489 \beta ) q^{45} \) \( + ( -360043315643232 - 2420612598 \beta ) q^{47} \) \( + ( 10537536240728361 + 182017056 \beta ) q^{49} \) \( + ( 3257043133755834 + 3028072086 \beta ) q^{51} \) \( + ( 28491507940995846 + 4734419105 \beta ) q^{53} \) \( + ( 32350952584839240 - 5178280536 \beta ) q^{55} \) \( + ( 15311396969802180 - 7689715074 \beta ) q^{57} \) \( + ( 8979601995508716 + 4473996424 \beta ) q^{59} \) \( + ( 14622235580907302 - 3856945716 \beta ) q^{61} \) \( + ( 904065856946928 + 15109399071 \beta ) q^{63} \) \( + ( -96215323265372700 - 10801715690 \beta ) q^{65} \) \( + ( -18555297284623372 + 34561414728 \beta ) q^{67} \) \( + ( 29089448505768456 - 26154416106 \beta ) q^{69} \) \( + ( -302251237170952680 - 79972862570 \beta ) q^{71} \) \( + ( -329831906607582886 + 67829833008 \beta ) q^{73} \) \( + ( 89697490377669675 + 12297544740 \beta ) q^{75} \) \( + ( -1196247161387798208 + 170799969652 \beta ) q^{77} \) \( + ( -176020869989194888 - 354231591753 \beta ) q^{79} \) \( + 150094635296999121 q^{81} \) \( + ( -1246451794643165292 + 358109982582 \beta ) q^{83} \) \( + ( 2166523420836778380 + 117416232018 \beta ) q^{85} \) \( + ( 1684008217684075302 + 150317397945 \beta ) q^{87} \) \( + ( 1261864300035059370 - 1385364045164 \beta ) q^{89} \) \( + ( 3940739177182256800 + 522216474858 \beta ) q^{91} \) \( + ( 4327171777136344968 - 41861076129 \beta ) q^{93} \) \( + ( -5876113718534545800 + 899943456880 \beta ) q^{95} \) \( + ( 8623369287414429314 + 1139749767492 \beta ) q^{97} \) \( + ( 1746449566481310444 - 831404369394 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 39366q^{3} \) \(\mathstrut +\mathstrut 624780q^{5} \) \(\mathstrut +\mathstrut 4667104q^{7} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 39366q^{3} \) \(\mathstrut +\mathstrut 624780q^{5} \) \(\mathstrut +\mathstrut 4667104q^{7} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut 9015783192q^{11} \) \(\mathstrut +\mathstrut 25948151500q^{13} \) \(\mathstrut -\mathstrut 12297544740q^{15} \) \(\mathstrut -\mathstrut 330949868796q^{17} \) \(\mathstrut -\mathstrut 1555799112920q^{19} \) \(\mathstrut -\mathstrut 91862608032q^{21} \) \(\mathstrut -\mathstrut 2955794188464q^{23} \) \(\mathstrut -\mathstrut 9114209254450q^{25} \) \(\mathstrut -\mathstrut 15251194969974q^{27} \) \(\mathstrut -\mathstrut 171112962219588q^{29} \) \(\mathstrut -\mathstrut 439686204047792q^{31} \) \(\mathstrut -\mathstrut 177457660568136q^{33} \) \(\mathstrut -\mathstrut 1123207990107840q^{35} \) \(\mathstrut -\mathstrut 912841755316868q^{37} \) \(\mathstrut -\mathstrut 510737465974500q^{39} \) \(\mathstrut -\mathstrut 2039868884446380q^{41} \) \(\mathstrut -\mathstrut 5607123893986664q^{43} \) \(\mathstrut +\mathstrut 242052573117420q^{45} \) \(\mathstrut -\mathstrut 720086631286464q^{47} \) \(\mathstrut +\mathstrut 21075072481456722q^{49} \) \(\mathstrut +\mathstrut 6514086267511668q^{51} \) \(\mathstrut +\mathstrut 56983015881991692q^{53} \) \(\mathstrut +\mathstrut 64701905169678480q^{55} \) \(\mathstrut +\mathstrut 30622793939604360q^{57} \) \(\mathstrut +\mathstrut 17959203991017432q^{59} \) \(\mathstrut +\mathstrut 29244471161814604q^{61} \) \(\mathstrut +\mathstrut 1808131713893856q^{63} \) \(\mathstrut -\mathstrut 192430646530745400q^{65} \) \(\mathstrut -\mathstrut 37110594569246744q^{67} \) \(\mathstrut +\mathstrut 58178897011536912q^{69} \) \(\mathstrut -\mathstrut 604502474341905360q^{71} \) \(\mathstrut -\mathstrut 659663813215165772q^{73} \) \(\mathstrut +\mathstrut 179394980755339350q^{75} \) \(\mathstrut -\mathstrut 2392494322775596416q^{77} \) \(\mathstrut -\mathstrut 352041739978389776q^{79} \) \(\mathstrut +\mathstrut 300189270593998242q^{81} \) \(\mathstrut -\mathstrut 2492903589286330584q^{83} \) \(\mathstrut +\mathstrut 4333046841673556760q^{85} \) \(\mathstrut +\mathstrut 3368016435368150604q^{87} \) \(\mathstrut +\mathstrut 2523728600070118740q^{89} \) \(\mathstrut +\mathstrut 7881478354364513600q^{91} \) \(\mathstrut +\mathstrut 8654343554272689936q^{93} \) \(\mathstrut -\mathstrut 11752227437069091600q^{95} \) \(\mathstrut +\mathstrut 17246738574828858628q^{97} \) \(\mathstrut +\mathstrut 3492899132962620888q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
220.246
−219.246
0 −19683.0 0 −3.48482e6 0 1.50425e8 0 3.87420e8 0
1.2 0 −19683.0 0 4.10960e6 0 −1.45758e8 0 3.87420e8 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
\(2\) \(-1\)
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{2} \) \(\mathstrut -\mathstrut 624780 T_{5} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\( \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(12))\).