Properties

Label 304.6.a.k
Level $304$
Weight $6$
Character orbit 304.a
Self dual yes
Analytic conductor $48.757$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.7566812231\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{3} ) q^{3} + ( -26 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{5} + ( -10 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{7} + ( 218 + \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{3} ) q^{3} + ( -26 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{5} + ( -10 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{7} + ( 218 + \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{9} + ( -176 + 3 \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{11} + ( 202 - 14 \beta_{1} + 15 \beta_{2} + 10 \beta_{3} ) q^{13} + ( -674 - 37 \beta_{1} + 6 \beta_{2} + 48 \beta_{3} ) q^{15} + ( 75 - 36 \beta_{1} + 48 \beta_{2} + 30 \beta_{3} ) q^{17} + 361 q^{19} + ( 1754 + 31 \beta_{1} - 75 \beta_{2} - 56 \beta_{3} ) q^{21} + ( -1450 - 4 \beta_{1} - 95 \beta_{2} + 42 \beta_{3} ) q^{23} + ( 2895 + 107 \beta_{1} - 132 \beta_{2} - 97 \beta_{3} ) q^{25} + ( -1157 + 137 \beta_{1} - 24 \beta_{2} - 101 \beta_{3} ) q^{27} + ( 1382 - 67 \beta_{1} + 19 \beta_{2} + 144 \beta_{3} ) q^{29} + ( 174 + 5 \beta_{1} - 150 \beta_{2} - 82 \beta_{3} ) q^{31} + ( 2800 - 151 \beta_{1} - 90 \beta_{2} + 226 \beta_{3} ) q^{33} + ( -4922 - 199 \beta_{1} + 186 \beta_{2} + 317 \beta_{3} ) q^{35} + ( -5004 + 133 \beta_{1} + 330 \beta_{2} + 358 \beta_{3} ) q^{37} + ( -3778 - 50 \beta_{1} + 411 \beta_{2} + 146 \beta_{3} ) q^{39} + ( -2580 + 13 \beta_{1} - 320 \beta_{2} - 58 \beta_{3} ) q^{41} + ( 3146 + 81 \beta_{1} - 378 \beta_{2} + 15 \beta_{3} ) q^{43} + ( -17464 + 7 \beta_{1} + 960 \beta_{2} + 725 \beta_{3} ) q^{45} + ( 1424 + 299 \beta_{1} - 32 \beta_{2} + 435 \beta_{3} ) q^{47} + ( -5602 + 144 \beta_{1} - 324 \beta_{2} - 516 \beta_{3} ) q^{49} + ( -8169 - 246 \beta_{1} + 1080 \beta_{2} + 921 \beta_{3} ) q^{51} + ( -19254 + 53 \beta_{1} + 251 \beta_{2} - 266 \beta_{3} ) q^{53} + ( 18200 - 9 \beta_{1} - 648 \beta_{2} - 57 \beta_{3} ) q^{55} + ( -1083 - 361 \beta_{3} ) q^{57} + ( -5779 + 661 \beta_{1} + 420 \beta_{2} + 391 \beta_{3} ) q^{59} + ( -8072 - 611 \beta_{1} - 570 \beta_{2} + 97 \beta_{3} ) q^{61} + ( 9968 + 403 \beta_{1} - 462 \beta_{2} - 1847 \beta_{3} ) q^{63} + ( -2178 + 538 \beta_{1} - 734 \beta_{2} - 1318 \beta_{3} ) q^{65} + ( -1667 - 1426 \beta_{1} + 660 \beta_{2} - 823 \beta_{3} ) q^{67} + ( -36886 + 1138 \beta_{1} + 885 \beta_{2} + 990 \beta_{3} ) q^{69} + ( -30112 - 376 \beta_{1} + 130 \beta_{2} + 426 \beta_{3} ) q^{71} + ( -26095 - 1382 \beta_{1} + 96 \beta_{2} + 160 \beta_{3} ) q^{73} + ( 20899 + 611 \beta_{1} - 3336 \beta_{2} - 5817 \beta_{3} ) q^{75} + ( -1990 - 467 \beta_{1} + 12 \beta_{2} + 1021 \beta_{3} ) q^{77} + ( -29016 - 17 \beta_{1} - 1326 \beta_{2} + 532 \beta_{3} ) q^{79} + ( 10953 - 1224 \beta_{1} - 1512 \beta_{2} - 556 \beta_{3} ) q^{81} + ( -23412 - 1344 \beta_{1} - 834 \beta_{2} - 408 \beta_{3} ) q^{83} + ( 17682 + 1179 \beta_{1} - 3186 \beta_{2} - 4797 \beta_{3} ) q^{85} + ( -74818 + 298 \beta_{1} + 3279 \beta_{2} + 438 \beta_{3} ) q^{87} + ( -742 - 953 \beta_{1} + 1494 \beta_{2} - 4478 \beta_{3} ) q^{89} + ( -40577 + 383 \beta_{1} + 444 \beta_{2} + 1655 \beta_{3} ) q^{91} + ( 3082 + 1832 \beta_{1} - 654 \beta_{2} - 1926 \beta_{3} ) q^{93} + ( -9386 + 361 \beta_{1} + 722 \beta_{2} + 1083 \beta_{3} ) q^{95} + ( 3510 - 908 \beta_{1} + 1104 \beta_{2} - 4466 \beta_{3} ) q^{97} + ( -110652 + 1635 \beta_{1} + 4176 \beta_{2} - 595 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{3} - 110q^{5} - 30q^{7} + 878q^{9} + O(q^{10}) \) \( 4q - 10q^{3} - 110q^{5} - 30q^{7} + 878q^{9} - 706q^{11} + 788q^{13} - 2792q^{15} + 240q^{17} + 1444q^{19} + 7128q^{21} - 5884q^{23} + 11774q^{25} - 4426q^{27} + 5240q^{29} + 860q^{31} + 10748q^{33} - 20322q^{35} - 20732q^{37} - 15404q^{39} - 10204q^{41} + 12554q^{43} - 71306q^{45} + 4826q^{47} - 21376q^{49} - 34518q^{51} - 76484q^{53} + 72914q^{55} - 3610q^{57} - 23898q^{59} - 32482q^{61} + 43566q^{63} - 6076q^{65} - 5022q^{67} - 149524q^{69} - 121300q^{71} - 104700q^{73} + 95230q^{75} - 10002q^{77} - 117128q^{79} + 44924q^{81} - 92832q^{83} + 80322q^{85} - 300148q^{87} + 5988q^{89} - 165618q^{91} + 16180q^{93} - 39710q^{95} + 22972q^{97} - 441418q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu^{2} - 79 \nu + 287 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{2} - 2 \nu - 71 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{3} + \beta_{1} + 142\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(40 \beta_{3} + 48 \beta_{2} + 89 \beta_{1} + 272\)\()/4\)

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
−10.0437
11.0547
−5.81615
4.80512
0 −27.9817 0 −64.0791 0 −65.8093 0 539.976 0
1.2 0 −17.5489 0 87.4671 0 −76.9277 0 64.9636 0
1.3 0 9.77006 0 −24.1428 0 −64.5622 0 −147.546 0
1.4 0 25.7605 0 −109.245 0 177.299 0 420.606 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 10 T_{3}^{3} - 875 T_{3}^{2} - 5988 T_{3} + 123588 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(304))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 123588 - 5988 T - 875 T^{2} + 10 T^{3} + T^{4} \)
$5$ \( -14782624 - 809300 T - 6087 T^{2} + 110 T^{3} + T^{4} \)
$7$ \( -57950181 - 2204622 T - 22476 T^{2} + 30 T^{3} + T^{4} \)
$11$ \( 842790380 - 17856160 T + 14469 T^{2} + 706 T^{3} + T^{4} \)
$13$ \( 15521087536 + 176779672 T - 298941 T^{2} - 788 T^{3} + T^{4} \)
$17$ \( 933010219809 + 260798832 T - 4066146 T^{2} - 240 T^{3} + T^{4} \)
$19$ \( ( -361 + T )^{4} \)
$23$ \( -69330313326016 - 58928167792 T - 4333677 T^{2} + 5884 T^{3} + T^{4} \)
$29$ \( -37858523058292 + 68597297852 T - 21188385 T^{2} - 5240 T^{3} + T^{4} \)
$31$ \( 129248370446848 + 11154181696 T - 23116560 T^{2} - 860 T^{3} + T^{4} \)
$37$ \( -4701007501700240 - 2332526971120 T - 49568352 T^{2} + 20732 T^{3} + T^{4} \)
$41$ \( 1168932051626240 - 507593819200 T - 74685948 T^{2} + 10204 T^{3} + T^{4} \)
$43$ \( -2367148348334144 + 1398101372356 T - 108385095 T^{2} - 12554 T^{3} + T^{4} \)
$47$ \( 1826425856030336 - 842949181576 T - 286986975 T^{2} - 4826 T^{3} + T^{4} \)
$53$ \( 75211347963000464 + 20504517033688 T + 1958495643 T^{2} + 76484 T^{3} + T^{4} \)
$59$ \( -129409346201894748 - 26521861709340 T - 1139036835 T^{2} + 23898 T^{3} + T^{4} \)
$61$ \( -385211274135945332 - 50337215709824 T - 1347049563 T^{2} + 32482 T^{3} + T^{4} \)
$67$ \( 508871346363957168 + 557781344136 T - 4416190023 T^{2} + 5022 T^{3} + T^{4} \)
$71$ \( 406332945301902704 + 81045731523248 T + 5020185504 T^{2} + 121300 T^{3} + T^{4} \)
$73$ \( 1025450251801515009 - 149622116529924 T - 174343674 T^{2} + 104700 T^{3} + T^{4} \)
$79$ \( -3407677020392545664 - 129375307451968 T + 1970108940 T^{2} + 117128 T^{3} + T^{4} \)
$83$ \( -2078981839792132416 - 202856324775840 T - 2488100004 T^{2} + 92832 T^{3} + T^{4} \)
$89$ \( \)\(14\!\cdots\!48\)\( + 117828675020736 T - 24926212848 T^{2} - 5988 T^{3} + T^{4} \)
$97$ \( \)\(12\!\cdots\!72\)\( + 281032633234496 T - 22191139356 T^{2} - 22972 T^{3} + T^{4} \)
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