Properties

Label 12.20.a.b
Level 12
Weight 20
Character orbit 12.a
Self dual Yes
Analytic conductor 27.458
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(27.4580035868\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3456\sqrt{3295681}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 19683 q^{3} \) \( + ( -1633554 - \beta ) q^{5} \) \( + ( -13511992 - 15 \beta ) q^{7} \) \( + 387420489 q^{9} \) \(+O(q^{10})\) \( q\) \( + 19683 q^{3} \) \( + ( -1633554 - \beta ) q^{5} \) \( + ( -13511992 - 15 \beta ) q^{7} \) \( + 387420489 q^{9} \) \( + ( 5340983076 + 770 \beta ) q^{11} \) \( + ( 16833984950 + 690 \beta ) q^{13} \) \( + ( -32153243382 - 19683 \beta ) q^{15} \) \( + ( -131363072622 - 2210 \beta ) q^{17} \) \( + ( 217658914460 + 280410 \beta ) q^{19} \) \( + ( -265956538536 - 295245 \beta ) q^{21} \) \( + ( 3006982281672 - 1665950 \beta ) q^{23} \) \( + ( 22958415283207 + 3267108 \beta ) q^{25} \) \( + 7625597484987 q^{27} \) \( + ( 75240133574166 + 8284445 \beta ) q^{29} \) \( + ( 43209083628464 - 35729235 \beta ) q^{31} \) \( + ( 105126569884908 + 15155910 \beta ) q^{33} \) \( + ( 612523612685808 + 38015302 \beta ) q^{35} \) \( + ( 1278719697331934 + 22582920 \beta ) q^{37} \) \( + ( 331343325770850 + 13581270 \beta ) q^{39} \) \( + ( 1260634510817370 - 321756070 \beta ) q^{41} \) \( + ( 229561335491012 + 355903470 \beta ) q^{43} \) \( + ( -632872289487906 - 387420489 \beta ) q^{45} \) \( + ( -4868758017993168 + 562522470 \beta ) q^{47} \) \( + ( -2359555595971479 + 405359760 \beta ) q^{49} \) \( + ( -2585619358418826 - 43499430 \beta ) q^{51} \) \( + ( -13425591676152546 - 180732175 \beta ) q^{53} \) \( + ( -39034604531852424 - 6598819656 \beta ) q^{55} \) \( + ( 4284180413316180 + 5519310030 \beta ) q^{57} \) \( + ( -56607040781321004 + 3602567320 \beta ) q^{59} \) \( + ( -30644254986893338 + 8986950180 \beta ) q^{61} \) \( + ( -5234822548004088 - 5811307335 \beta ) q^{63} \) \( + ( -54659971479899340 - 17961137210 \beta ) q^{65} \) \( + ( 228771592640757452 + 2223489240 \beta ) q^{67} \) \( + ( 59186432250149976 - 32790893850 \beta ) q^{69} \) \( + ( 375985005885267480 + 39513342970 \beta ) q^{71} \) \( + ( 509613452893284506 + 60240971520 \beta ) q^{73} \) \( + ( 451890488019363381 + 64306486764 \beta ) q^{75} \) \( + ( -526814624556852192 - 90518979980 \beta ) q^{77} \) \( + ( 300546956350576832 - 214563427455 \beta ) q^{79} \) \( + 150094635296999121 q^{81} \) \( + ( -983052122898690468 + 133864255050 \beta ) q^{83} \) \( + ( 301581793232277948 + 134973226962 \beta ) q^{85} \) \( + ( 1480951549140309378 + 163062730935 \beta ) q^{87} \) \( + ( -4245850157877660150 + 206914558900 \beta ) q^{89} \) \( + ( -634871890405826000 - 261833048730 \beta ) q^{91} \) \( + ( 850484393059056912 - 703258532505 \beta ) q^{93} \) \( + ( -11393449408873841400 - 675723791600 \beta ) q^{95} \) \( + ( 373021345833756386 + 1320228180420 \beta ) q^{97} \) \( + ( 2069206275044644164 + 298313776530 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 39366q^{3} \) \(\mathstrut -\mathstrut 3267108q^{5} \) \(\mathstrut -\mathstrut 27023984q^{7} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 39366q^{3} \) \(\mathstrut -\mathstrut 3267108q^{5} \) \(\mathstrut -\mathstrut 27023984q^{7} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut 10681966152q^{11} \) \(\mathstrut +\mathstrut 33667969900q^{13} \) \(\mathstrut -\mathstrut 64306486764q^{15} \) \(\mathstrut -\mathstrut 262726145244q^{17} \) \(\mathstrut +\mathstrut 435317828920q^{19} \) \(\mathstrut -\mathstrut 531913077072q^{21} \) \(\mathstrut +\mathstrut 6013964563344q^{23} \) \(\mathstrut +\mathstrut 45916830566414q^{25} \) \(\mathstrut +\mathstrut 15251194969974q^{27} \) \(\mathstrut +\mathstrut 150480267148332q^{29} \) \(\mathstrut +\mathstrut 86418167256928q^{31} \) \(\mathstrut +\mathstrut 210253139769816q^{33} \) \(\mathstrut +\mathstrut 1225047225371616q^{35} \) \(\mathstrut +\mathstrut 2557439394663868q^{37} \) \(\mathstrut +\mathstrut 662686651541700q^{39} \) \(\mathstrut +\mathstrut 2521269021634740q^{41} \) \(\mathstrut +\mathstrut 459122670982024q^{43} \) \(\mathstrut -\mathstrut 1265744578975812q^{45} \) \(\mathstrut -\mathstrut 9737516035986336q^{47} \) \(\mathstrut -\mathstrut 4719111191942958q^{49} \) \(\mathstrut -\mathstrut 5171238716837652q^{51} \) \(\mathstrut -\mathstrut 26851183352305092q^{53} \) \(\mathstrut -\mathstrut 78069209063704848q^{55} \) \(\mathstrut +\mathstrut 8568360826632360q^{57} \) \(\mathstrut -\mathstrut 113214081562642008q^{59} \) \(\mathstrut -\mathstrut 61288509973786676q^{61} \) \(\mathstrut -\mathstrut 10469645096008176q^{63} \) \(\mathstrut -\mathstrut 109319942959798680q^{65} \) \(\mathstrut +\mathstrut 457543185281514904q^{67} \) \(\mathstrut +\mathstrut 118372864500299952q^{69} \) \(\mathstrut +\mathstrut 751970011770534960q^{71} \) \(\mathstrut +\mathstrut 1019226905786569012q^{73} \) \(\mathstrut +\mathstrut 903780976038726762q^{75} \) \(\mathstrut -\mathstrut 1053629249113704384q^{77} \) \(\mathstrut +\mathstrut 601093912701153664q^{79} \) \(\mathstrut +\mathstrut 300189270593998242q^{81} \) \(\mathstrut -\mathstrut 1966104245797380936q^{83} \) \(\mathstrut +\mathstrut 603163586464555896q^{85} \) \(\mathstrut +\mathstrut 2961903098280618756q^{87} \) \(\mathstrut -\mathstrut 8491700315755320300q^{89} \) \(\mathstrut -\mathstrut 1269743780811652000q^{91} \) \(\mathstrut +\mathstrut 1700968786118113824q^{93} \) \(\mathstrut -\mathstrut 22786898817747682800q^{95} \) \(\mathstrut +\mathstrut 746042691667512772q^{97} \) \(\mathstrut +\mathstrut 4138412550089288328q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
908.201
−907.201
0 19683.0 0 −7.90758e6 0 −1.07622e8 0 3.87420e8 0
1.2 0 19683.0 0 4.64047e6 0 8.05984e7 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 \(T_{5}^{2} \) \(\mathstrut +\mathstrut 3267108 T_{5} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\( \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(12))\).