Properties

Label 546.8.a.g
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 51426 x^{2} + 704960 x + 421951600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( -82 + \beta_{2} ) q^{5} + 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( -82 + \beta_{2} ) q^{5} + 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} + ( -656 + 8 \beta_{2} ) q^{10} + ( -1987 + \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{11} + 1728 q^{12} -2197 q^{13} + 2744 q^{14} + ( -2214 + 27 \beta_{2} ) q^{15} + 4096 q^{16} + ( -6789 - 111 \beta_{1} - 31 \beta_{2} - 9 \beta_{3} ) q^{17} + 5832 q^{18} + ( -8717 + 72 \beta_{1} + 31 \beta_{2} - 3 \beta_{3} ) q^{19} + ( -5248 + 64 \beta_{2} ) q^{20} + 9261 q^{21} + ( -15896 + 8 \beta_{1} - 72 \beta_{2} + 8 \beta_{3} ) q^{22} + ( -5696 + 392 \beta_{1} - 201 \beta_{2} + 38 \beta_{3} ) q^{23} + 13824 q^{24} + ( -16142 + 12 \beta_{1} - 325 \beta_{2} - 47 \beta_{3} ) q^{25} -17576 q^{26} + 19683 q^{27} + 21952 q^{28} + ( -39551 - 556 \beta_{1} + 27 \beta_{2} + 145 \beta_{3} ) q^{29} + ( -17712 + 216 \beta_{2} ) q^{30} + ( -68737 + 543 \beta_{1} - 138 \beta_{2} - 121 \beta_{3} ) q^{31} + 32768 q^{32} + ( -53649 + 27 \beta_{1} - 243 \beta_{2} + 27 \beta_{3} ) q^{33} + ( -54312 - 888 \beta_{1} - 248 \beta_{2} - 72 \beta_{3} ) q^{34} + ( -28126 + 343 \beta_{2} ) q^{35} + 46656 q^{36} + ( -79718 + 371 \beta_{1} - 31 \beta_{2} - 334 \beta_{3} ) q^{37} + ( -69736 + 576 \beta_{1} + 248 \beta_{2} - 24 \beta_{3} ) q^{38} -59319 q^{39} + ( -41984 + 512 \beta_{2} ) q^{40} + ( -177595 - 1372 \beta_{1} - 64 \beta_{2} + 249 \beta_{3} ) q^{41} + 74088 q^{42} + ( -222977 - 1560 \beta_{1} + 805 \beta_{2} + 217 \beta_{3} ) q^{43} + ( -127168 + 64 \beta_{1} - 576 \beta_{2} + 64 \beta_{3} ) q^{44} + ( -59778 + 729 \beta_{2} ) q^{45} + ( -45568 + 3136 \beta_{1} - 1608 \beta_{2} + 304 \beta_{3} ) q^{46} + ( -140079 - 2283 \beta_{1} - 1200 \beta_{2} - 265 \beta_{3} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( -129136 + 96 \beta_{1} - 2600 \beta_{2} - 376 \beta_{3} ) q^{50} + ( -183303 - 2997 \beta_{1} - 837 \beta_{2} - 243 \beta_{3} ) q^{51} -140608 q^{52} + ( -546425 + 1335 \beta_{1} + 1062 \beta_{2} - 851 \beta_{3} ) q^{53} + 157464 q^{54} + ( -310696 - 407 \beta_{1} - 501 \beta_{2} + 402 \beta_{3} ) q^{55} + 175616 q^{56} + ( -235359 + 1944 \beta_{1} + 837 \beta_{2} - 81 \beta_{3} ) q^{57} + ( -316408 - 4448 \beta_{1} + 216 \beta_{2} + 1160 \beta_{3} ) q^{58} + ( 11048 + 6876 \beta_{1} - 10098 \beta_{2} + 1066 \beta_{3} ) q^{59} + ( -141696 + 1728 \beta_{2} ) q^{60} + ( -607924 + 7391 \beta_{1} - 85 \beta_{2} + 2404 \beta_{3} ) q^{61} + ( -549896 + 4344 \beta_{1} - 1104 \beta_{2} - 968 \beta_{3} ) q^{62} + 250047 q^{63} + 262144 q^{64} + ( 180154 - 2197 \beta_{2} ) q^{65} + ( -429192 + 216 \beta_{1} - 1944 \beta_{2} + 216 \beta_{3} ) q^{66} + ( -1227178 + 1928 \beta_{1} + 7058 \beta_{2} - 1066 \beta_{3} ) q^{67} + ( -434496 - 7104 \beta_{1} - 1984 \beta_{2} - 576 \beta_{3} ) q^{68} + ( -153792 + 10584 \beta_{1} - 5427 \beta_{2} + 1026 \beta_{3} ) q^{69} + ( -225008 + 2744 \beta_{2} ) q^{70} + ( -1382337 - 19326 \beta_{1} + 12762 \beta_{2} + 1113 \beta_{3} ) q^{71} + 373248 q^{72} + ( 1528231 + 8398 \beta_{1} - 8251 \beta_{2} - 3393 \beta_{3} ) q^{73} + ( -637744 + 2968 \beta_{1} - 248 \beta_{2} - 2672 \beta_{3} ) q^{74} + ( -435834 + 324 \beta_{1} - 8775 \beta_{2} - 1269 \beta_{3} ) q^{75} + ( -557888 + 4608 \beta_{1} + 1984 \beta_{2} - 192 \beta_{3} ) q^{76} + ( -681541 + 343 \beta_{1} - 3087 \beta_{2} + 343 \beta_{3} ) q^{77} -474552 q^{78} + ( 1637639 - 14917 \beta_{1} + 5300 \beta_{2} - 6089 \beta_{3} ) q^{79} + ( -335872 + 4096 \beta_{2} ) q^{80} + 531441 q^{81} + ( -1420760 - 10976 \beta_{1} - 512 \beta_{2} + 1992 \beta_{3} ) q^{82} + ( 1313827 + 587 \beta_{1} - 3210 \beta_{2} + 4585 \beta_{3} ) q^{83} + 592704 q^{84} + ( -2589254 + 2421 \beta_{1} + 11949 \beta_{2} + 3584 \beta_{3} ) q^{85} + ( -1783816 - 12480 \beta_{1} + 6440 \beta_{2} + 1736 \beta_{3} ) q^{86} + ( -1067877 - 15012 \beta_{1} + 729 \beta_{2} + 3915 \beta_{3} ) q^{87} + ( -1017344 + 512 \beta_{1} - 4608 \beta_{2} + 512 \beta_{3} ) q^{88} + ( -1163832 - 22879 \beta_{1} + 16228 \beta_{2} - 472 \beta_{3} ) q^{89} + ( -478224 + 5832 \beta_{2} ) q^{90} -753571 q^{91} + ( -364544 + 25088 \beta_{1} - 12864 \beta_{2} + 2432 \beta_{3} ) q^{92} + ( -1855899 + 14661 \beta_{1} - 3726 \beta_{2} - 3267 \beta_{3} ) q^{93} + ( -1120632 - 18264 \beta_{1} - 9600 \beta_{2} - 2120 \beta_{3} ) q^{94} + ( 3253495 + 1194 \beta_{1} - 17747 \beta_{2} - 2819 \beta_{3} ) q^{95} + 884736 q^{96} + ( 3135121 + 18311 \beta_{1} + 33906 \beta_{2} + 3671 \beta_{3} ) q^{97} + 941192 q^{98} + ( -1448523 + 729 \beta_{1} - 6561 \beta_{2} + 729 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9} + O(q^{10}) \) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9} - 2624 q^{10} - 7945 q^{11} + 6912 q^{12} - 8788 q^{13} + 10976 q^{14} - 8856 q^{15} + 16384 q^{16} - 27285 q^{17} + 23328 q^{18} - 34802 q^{19} - 20992 q^{20} + 37044 q^{21} - 63560 q^{22} - 22316 q^{23} + 55296 q^{24} - 64650 q^{25} - 70304 q^{26} + 78732 q^{27} + 87808 q^{28} - 158470 q^{29} - 70848 q^{30} - 274647 q^{31} + 131072 q^{32} - 214515 q^{33} - 218280 q^{34} - 112504 q^{35} + 186624 q^{36} - 319169 q^{37} - 278416 q^{38} - 237276 q^{39} - 167936 q^{40} - 711254 q^{41} + 296352 q^{42} - 893034 q^{43} - 508480 q^{44} - 239112 q^{45} - 178528 q^{46} - 563129 q^{47} + 442368 q^{48} + 470596 q^{49} - 517200 q^{50} - 736695 q^{51} - 562432 q^{52} - 2186067 q^{53} + 629856 q^{54} - 1242387 q^{55} + 702464 q^{56} - 939654 q^{57} - 1267760 q^{58} + 53200 q^{59} - 566784 q^{60} - 2419497 q^{61} - 2197176 q^{62} + 1000188 q^{63} + 1048576 q^{64} + 720616 q^{65} - 1716120 q^{66} - 4908916 q^{67} - 1746240 q^{68} - 602532 q^{69} - 900032 q^{70} - 5546448 q^{71} + 1492992 q^{72} + 6114536 q^{73} - 2553352 q^{74} - 1745550 q^{75} - 2227328 q^{76} - 2725135 q^{77} - 1898208 q^{78} + 6523461 q^{79} - 1343488 q^{80} + 2125764 q^{81} - 5690032 q^{82} + 5265065 q^{83} + 2370816 q^{84} - 10347427 q^{85} - 7144272 q^{86} - 4278690 q^{87} - 4067840 q^{88} - 4679151 q^{89} - 1912896 q^{90} - 3014284 q^{91} - 1428224 q^{92} - 7415469 q^{93} - 4505032 q^{94} + 13009536 q^{95} + 3538944 q^{96} + 12566137 q^{97} + 3764768 q^{98} - 5791905 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 51426 x^{2} + 704960 x + 421951600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 11 \nu^{2} + 39536 \nu - 782880 \)\()/9300\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 299 \nu^{2} - 28066 \nu - 7188770 \)\()/4650\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(15 \beta_{3} + 30 \beta_{2} - 37 \beta_{1} + 25715\)
\(\nu^{3}\)\(=\)\(165 \beta_{3} - 8970 \beta_{2} + 39129 \beta_{1} - 500015\)

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
−91.1808
190.494
−212.586
114.273
8.00000 27.0000 64.0000 −462.460 216.000 343.000 512.000 729.000 −3699.68
1.2 8.00000 27.0000 64.0000 −56.7282 216.000 343.000 512.000 729.000 −453.826
1.3 8.00000 27.0000 64.0000 16.5813 216.000 343.000 512.000 729.000 132.650
1.4 8.00000 27.0000 64.0000 174.607 216.000 343.000 512.000 729.000 1396.86
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 546.8.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.g 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 328 T_{5}^{3} - 70133 T_{5}^{2} - 3512580 T_{5} + 75954600 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{4} \)
$3$ \( ( -27 + T )^{4} \)
$5$ \( 75954600 - 3512580 T - 70133 T^{2} + 328 T^{3} + T^{4} \)
$7$ \( ( -343 + T )^{4} \)
$11$ \( -8570492349000 - 2698199340 T + 13926874 T^{2} + 7945 T^{3} + T^{4} \)
$13$ \( ( 2197 + T )^{4} \)
$17$ \( 122398253429295232 - 10154289570444 T - 791176964 T^{2} + 27285 T^{3} + T^{4} \)
$19$ \( 14735909932415040 - 6260561827152 T - 29685251 T^{2} + 34802 T^{3} + T^{4} \)
$23$ \( 5594509032007764360 - 377460921888036 T - 10698441281 T^{2} + 22316 T^{3} + T^{4} \)
$29$ \( 67916412726096641700 - 3110701509633468 T - 36997723403 T^{2} + 158470 T^{3} + T^{4} \)
$31$ \( 24104186386534251136 - 2421810165601488 T - 7498731728 T^{2} + 274647 T^{3} + T^{4} \)
$37$ \( \)\(45\!\cdots\!00\)\( - 32650140708902620 T - 141011957232 T^{2} + 319169 T^{3} + T^{4} \)
$41$ \( -\)\(45\!\cdots\!00\)\( - 42701476091132208 T + 311552920 T^{2} + 711254 T^{3} + T^{4} \)
$43$ \( -\)\(17\!\cdots\!00\)\( - 88990000734288096 T + 88924953801 T^{2} + 893034 T^{3} + T^{4} \)
$47$ \( \)\(19\!\cdots\!04\)\( - 224781402533158092 T - 580027730792 T^{2} + 563129 T^{3} + T^{4} \)
$53$ \( -\)\(63\!\cdots\!00\)\( - 1422438311669395980 T + 441558742702 T^{2} + 2186067 T^{3} + T^{4} \)
$59$ \( \)\(31\!\cdots\!72\)\( + 1685787274291102608 T - 11840052686156 T^{2} - 53200 T^{3} + T^{4} \)
$61$ \( -\)\(18\!\cdots\!40\)\( - 31938302098707914436 T - 10108346140370 T^{2} + 2419497 T^{3} + T^{4} \)
$67$ \( -\)\(19\!\cdots\!40\)\( - 18632007486698911136 T + 1310147929044 T^{2} + 4908916 T^{3} + T^{4} \)
$71$ \( -\)\(12\!\cdots\!00\)\( - \)\(11\!\cdots\!40\)\( T - 15919783881228 T^{2} + 5546448 T^{3} + T^{4} \)
$73$ \( \)\(65\!\cdots\!00\)\( + 67689342187971182772 T - 12480664649993 T^{2} - 6114536 T^{3} + T^{4} \)
$79$ \( -\)\(53\!\cdots\!00\)\( + \)\(41\!\cdots\!20\)\( T - 53859475162766 T^{2} - 6523461 T^{3} + T^{4} \)
$83$ \( \)\(37\!\cdots\!36\)\( + 92016367338579525060 T - 23076631779938 T^{2} - 5265065 T^{3} + T^{4} \)
$89$ \( \)\(12\!\cdots\!92\)\( - \)\(11\!\cdots\!16\)\( T - 30296026373678 T^{2} + 4679151 T^{3} + T^{4} \)
$97$ \( -\)\(74\!\cdots\!48\)\( + \)\(24\!\cdots\!48\)\( T - 143472906281942 T^{2} - 12566137 T^{3} + T^{4} \)
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