Properties

Label 684.6.a.e
Level $684$
Weight $6$
Character orbit 684.a
Self dual yes
Analytic conductor $109.703$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 684.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.702532752\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 140 x^{2} - 84 x + 3103\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 26 - 3 \beta_{1} - \beta_{2} ) q^{5} + ( 9 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 26 - 3 \beta_{1} - \beta_{2} ) q^{5} + ( 9 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -172 + 9 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{11} + ( 199 + 4 \beta_{1} - 14 \beta_{2} - 3 \beta_{3} ) q^{13} + ( -69 - 18 \beta_{1} + 36 \beta_{2} + 6 \beta_{3} ) q^{17} -361 q^{19} + ( -1497 - 52 \beta_{1} - 4 \beta_{2} - 47 \beta_{3} ) q^{23} + ( 2919 - 49 \beta_{1} + 107 \beta_{2} + 24 \beta_{3} ) q^{25} + ( -1318 - 16 \beta_{1} + 67 \beta_{2} + 64 \beta_{3} ) q^{29} + ( -153 + 124 \beta_{1} - 5 \beta_{2} + 21 \beta_{3} ) q^{31} + ( -5032 + 97 \beta_{1} - 199 \beta_{2} - 110 \beta_{3} ) q^{35} + ( -4969 + 428 \beta_{1} + 133 \beta_{2} + 35 \beta_{3} ) q^{37} + ( 2663 + 224 \beta_{1} - 13 \beta_{2} + 83 \beta_{3} ) q^{41} + ( -3042 + 193 \beta_{1} - 81 \beta_{2} + 104 \beta_{3} ) q^{43} + ( 1368 + 323 \beta_{1} + 299 \beta_{2} - 56 \beta_{3} ) q^{47} + ( -5490 - 292 \beta_{1} + 144 \beta_{2} + 112 \beta_{3} ) q^{49} + ( 19064 - 114 \beta_{1} - 53 \beta_{2} - 190 \beta_{3} ) q^{53} + ( -18000 + 457 \beta_{1} + 9 \beta_{2} + 200 \beta_{3} ) q^{55} + ( -5549 + 851 \beta_{1} + 661 \beta_{2} + 230 \beta_{3} ) q^{59} + ( -8498 - 755 \beta_{1} - 611 \beta_{2} - 426 \beta_{3} ) q^{61} + ( 1804 + 570 \beta_{1} - 538 \beta_{2} - 374 \beta_{3} ) q^{65} + ( 1648 + 785 \beta_{1} + 1426 \beta_{2} - 19 \beta_{3} ) q^{67} + ( -30336 - 22 \beta_{1} - 376 \beta_{2} - 224 \beta_{3} ) q^{71} + ( -26577 - 804 \beta_{1} - 1382 \beta_{2} - 482 \beta_{3} ) q^{73} + ( 2482 - 37 \beta_{1} + 467 \beta_{2} + 492 \beta_{3} ) q^{77} + ( 29641 + 718 \beta_{1} + 17 \beta_{2} + 625 \beta_{3} ) q^{79} + ( -24002 - 1588 \beta_{1} - 1344 \beta_{2} - 590 \beta_{3} ) q^{83} + ( 18612 - 2937 \beta_{1} + 1179 \beta_{2} + 930 \beta_{3} ) q^{85} + ( -931 + 1132 \beta_{1} + 953 \beta_{2} - 1673 \beta_{3} ) q^{89} + ( 40853 - 1103 \beta_{1} - 383 \beta_{2} + 276 \beta_{3} ) q^{91} + ( -9386 + 1083 \beta_{1} + 361 \beta_{2} ) q^{95} + ( 5064 - 1358 \beta_{1} - 908 \beta_{2} + 1554 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 110q^{5} + 30q^{7} + O(q^{10}) \) \( 4q + 110q^{5} + 30q^{7} - 706q^{11} + 788q^{13} - 240q^{17} - 1444q^{19} - 5884q^{23} + 11774q^{25} - 5240q^{29} - 860q^{31} - 20322q^{35} - 20732q^{37} + 10204q^{41} - 12554q^{43} + 4826q^{47} - 21376q^{49} + 76484q^{53} - 72914q^{55} - 23898q^{59} - 32482q^{61} + 6076q^{65} + 5022q^{67} - 121300q^{71} - 104700q^{73} + 10002q^{77} + 117128q^{79} - 92832q^{83} + 80322q^{85} - 5988q^{89} + 165618q^{91} - 39710q^{95} + 22972q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 140 x^{2} - 84 x + 3103\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 91 \nu - 139 \)\()/12\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} + 127 \nu - 287 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 17 \nu^{2} - 55 \nu + 1127 \)\()/12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 1\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 141\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(97 \beta_{3} + 267 \beta_{2} + 314 \beta_{1} + 913\)\()/12\)

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
11.0547
−5.81615
−10.0437
4.80512
0 0 0 −87.4671 0 76.9277 0 0 0
1.2 0 0 0 24.1428 0 64.5622 0 0 0
1.3 0 0 0 64.0791 0 65.8093 0 0 0
1.4 0 0 0 109.245 0 −177.299 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.6.a.e 4
3.b odd 2 1 76.6.a.b 4
12.b even 2 1 304.6.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.a.b 4 3.b odd 2 1
304.6.a.k 4 12.b even 2 1
684.6.a.e 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 110 T_{5}^{3} - 6087 T_{5}^{2} + 809300 T_{5} - 14782624 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(684))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( -14782624 + 809300 T - 6087 T^{2} - 110 T^{3} + T^{4} \)
$7$ \( -57950181 + 2204622 T - 22476 T^{2} - 30 T^{3} + T^{4} \)
$11$ \( 842790380 - 17856160 T + 14469 T^{2} + 706 T^{3} + T^{4} \)
$13$ \( 15521087536 + 176779672 T - 298941 T^{2} - 788 T^{3} + T^{4} \)
$17$ \( 933010219809 - 260798832 T - 4066146 T^{2} + 240 T^{3} + T^{4} \)
$19$ \( ( 361 + T )^{4} \)
$23$ \( -69330313326016 - 58928167792 T - 4333677 T^{2} + 5884 T^{3} + T^{4} \)
$29$ \( -37858523058292 - 68597297852 T - 21188385 T^{2} + 5240 T^{3} + T^{4} \)
$31$ \( 129248370446848 - 11154181696 T - 23116560 T^{2} + 860 T^{3} + T^{4} \)
$37$ \( -4701007501700240 - 2332526971120 T - 49568352 T^{2} + 20732 T^{3} + T^{4} \)
$41$ \( 1168932051626240 + 507593819200 T - 74685948 T^{2} - 10204 T^{3} + T^{4} \)
$43$ \( -2367148348334144 - 1398101372356 T - 108385095 T^{2} + 12554 T^{3} + T^{4} \)
$47$ \( 1826425856030336 - 842949181576 T - 286986975 T^{2} - 4826 T^{3} + T^{4} \)
$53$ \( 75211347963000464 - 20504517033688 T + 1958495643 T^{2} - 76484 T^{3} + T^{4} \)
$59$ \( -129409346201894748 - 26521861709340 T - 1139036835 T^{2} + 23898 T^{3} + T^{4} \)
$61$ \( -385211274135945332 - 50337215709824 T - 1347049563 T^{2} + 32482 T^{3} + T^{4} \)
$67$ \( 508871346363957168 - 557781344136 T - 4416190023 T^{2} - 5022 T^{3} + T^{4} \)
$71$ \( 406332945301902704 + 81045731523248 T + 5020185504 T^{2} + 121300 T^{3} + T^{4} \)
$73$ \( 1025450251801515009 - 149622116529924 T - 174343674 T^{2} + 104700 T^{3} + T^{4} \)
$79$ \( -3407677020392545664 + 129375307451968 T + 1970108940 T^{2} - 117128 T^{3} + T^{4} \)
$83$ \( -2078981839792132416 - 202856324775840 T - 2488100004 T^{2} + 92832 T^{3} + T^{4} \)
$89$ \( \)\(14\!\cdots\!48\)\( - 117828675020736 T - 24926212848 T^{2} + 5988 T^{3} + T^{4} \)
$97$ \( \)\(12\!\cdots\!72\)\( + 281032633234496 T - 22191139356 T^{2} - 22972 T^{3} + T^{4} \)
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