Properties

Label 12.16.a.b
Level 12
Weight 16
Character orbit 12.a
Self dual Yes
Analytic conductor 17.123
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(17.123220612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{8017}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2880\sqrt{8017}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 2187 q^{3} \) \( + ( 34830 - \beta ) q^{5} \) \( + ( 1245752 + 9 \beta ) q^{7} \) \( + 4782969 q^{9} \) \(+O(q^{10})\) \( q\) \( + 2187 q^{3} \) \( + ( 34830 - \beta ) q^{5} \) \( + ( 1245752 + 9 \beta ) q^{7} \) \( + 4782969 q^{9} \) \( + ( -7487964 - 286 \beta ) q^{11} \) \( + ( 5878790 + 1314 \beta ) q^{13} \) \( + ( 76173210 - 2187 \beta ) q^{15} \) \( + ( 2128038642 + 1198 \beta ) q^{17} \) \( + ( 4620844700 - 5382 \beta ) q^{19} \) \( + ( 2724459624 + 19683 \beta ) q^{21} \) \( + ( 12127411368 + 2002 \beta ) q^{23} \) \( + ( 37191755575 - 69660 \beta ) q^{25} \) \( + 10460353203 q^{27} \) \( + ( 73645129206 + 79565 \beta ) q^{29} \) \( + ( -38338765216 + 451413 \beta ) q^{31} \) \( + ( -16376177268 - 625482 \beta ) q^{33} \) \( + ( -555076301040 - 932282 \beta ) q^{35} \) \( + ( -485616578194 + 307512 \beta ) q^{37} \) \( + ( 12856913730 + 2873718 \beta ) q^{39} \) \( + ( -1005351710310 + 2742314 \beta ) q^{41} \) \( + ( 349483151108 - 8796114 \beta ) q^{43} \) \( + ( 166590810270 - 4782969 \beta ) q^{45} \) \( + ( -391922410032 + 3658902 \beta ) q^{47} \) \( + ( 2190529124361 + 22423536 \beta ) q^{49} \) \( + ( 4654020510054 + 2620026 \beta ) q^{51} \) \( + ( -2792937272994 - 36041935 \beta ) q^{53} \) \( + ( 18757108786680 - 2473416 \beta ) q^{55} \) \( + ( 10105787358900 - 11770434 \beta ) q^{57} \) \( + ( -7918081118604 + 56238424 \beta ) q^{59} \) \( + ( 908137030742 - 19865196 \beta ) q^{61} \) \( + ( 5958393197688 + 43046721 \beta ) q^{63} \) \( + ( -87171254851500 + 39887830 \beta ) q^{65} \) \( + ( -1129033705492 - 207154152 \beta ) q^{67} \) \( + ( 26522648661816 + 4378374 \beta ) q^{69} \) \( + ( -98497659855240 - 74494550 \beta ) q^{71} \) \( + ( -4012928613766 + 525698208 \beta ) q^{73} \) \( + ( 81338369442525 - 152346420 \beta ) q^{75} \) \( + ( -180489377284128 - 423676748 \beta ) q^{77} \) \( + ( 178193033072432 + 306386217 \beta ) q^{79} \) \( + 22876792454961 q^{81} \) \( + ( 180752178743388 + 638552202 \beta ) q^{83} \) \( + ( -5542867449540 - 2086312302 \beta ) q^{85} \) \( + ( 161061897573522 + 174008655 \beta ) q^{87} \) \( + ( -92228154240150 + 1605980116 \beta ) q^{89} \) \( + ( 793707632364880 + 1689827238 \beta ) q^{91} \) \( + ( -83846879527392 + 987240231 \beta ) q^{93} \) \( + ( 518826595134600 - 4808299760 \beta ) q^{95} \) \( + ( -907785730540126 - 421648668 \beta ) q^{97} \) \( + ( -35814699685116 - 1367929134 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4374q^{3} \) \(\mathstrut +\mathstrut 69660q^{5} \) \(\mathstrut +\mathstrut 2491504q^{7} \) \(\mathstrut +\mathstrut 9565938q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4374q^{3} \) \(\mathstrut +\mathstrut 69660q^{5} \) \(\mathstrut +\mathstrut 2491504q^{7} \) \(\mathstrut +\mathstrut 9565938q^{9} \) \(\mathstrut -\mathstrut 14975928q^{11} \) \(\mathstrut +\mathstrut 11757580q^{13} \) \(\mathstrut +\mathstrut 152346420q^{15} \) \(\mathstrut +\mathstrut 4256077284q^{17} \) \(\mathstrut +\mathstrut 9241689400q^{19} \) \(\mathstrut +\mathstrut 5448919248q^{21} \) \(\mathstrut +\mathstrut 24254822736q^{23} \) \(\mathstrut +\mathstrut 74383511150q^{25} \) \(\mathstrut +\mathstrut 20920706406q^{27} \) \(\mathstrut +\mathstrut 147290258412q^{29} \) \(\mathstrut -\mathstrut 76677530432q^{31} \) \(\mathstrut -\mathstrut 32752354536q^{33} \) \(\mathstrut -\mathstrut 1110152602080q^{35} \) \(\mathstrut -\mathstrut 971233156388q^{37} \) \(\mathstrut +\mathstrut 25713827460q^{39} \) \(\mathstrut -\mathstrut 2010703420620q^{41} \) \(\mathstrut +\mathstrut 698966302216q^{43} \) \(\mathstrut +\mathstrut 333181620540q^{45} \) \(\mathstrut -\mathstrut 783844820064q^{47} \) \(\mathstrut +\mathstrut 4381058248722q^{49} \) \(\mathstrut +\mathstrut 9308041020108q^{51} \) \(\mathstrut -\mathstrut 5585874545988q^{53} \) \(\mathstrut +\mathstrut 37514217573360q^{55} \) \(\mathstrut +\mathstrut 20211574717800q^{57} \) \(\mathstrut -\mathstrut 15836162237208q^{59} \) \(\mathstrut +\mathstrut 1816274061484q^{61} \) \(\mathstrut +\mathstrut 11916786395376q^{63} \) \(\mathstrut -\mathstrut 174342509703000q^{65} \) \(\mathstrut -\mathstrut 2258067410984q^{67} \) \(\mathstrut +\mathstrut 53045297323632q^{69} \) \(\mathstrut -\mathstrut 196995319710480q^{71} \) \(\mathstrut -\mathstrut 8025857227532q^{73} \) \(\mathstrut +\mathstrut 162676738885050q^{75} \) \(\mathstrut -\mathstrut 360978754568256q^{77} \) \(\mathstrut +\mathstrut 356386066144864q^{79} \) \(\mathstrut +\mathstrut 45753584909922q^{81} \) \(\mathstrut +\mathstrut 361504357486776q^{83} \) \(\mathstrut -\mathstrut 11085734899080q^{85} \) \(\mathstrut +\mathstrut 322123795147044q^{87} \) \(\mathstrut -\mathstrut 184456308480300q^{89} \) \(\mathstrut +\mathstrut 1587415264729760q^{91} \) \(\mathstrut -\mathstrut 167693759054784q^{93} \) \(\mathstrut +\mathstrut 1037653190269200q^{95} \) \(\mathstrut -\mathstrut 1815571461080252q^{97} \) \(\mathstrut -\mathstrut 71629399370232q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
45.2689
−44.2689
0 2187.00 0 −223039. 0 3.56657e6 0 4.78297e6 0
1.2 0 2187.00 0 292699. 0 −1.07507e6 0 4.78297e6 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 \(T_{5}^{2} \) \(\mathstrut -\mathstrut 69660 T_{5} \) \(\mathstrut -\mathstrut 65283075900 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(12))\).