Properties

Label 546.8.a.l
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 122890 x^{3} - 6160660 x^{2} + 3465881625 x + 278845474950\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 5 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( -50 + \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( -50 + \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} + ( -400 + 8 \beta_{1} ) q^{10} + ( 133 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{11} -1728 q^{12} -2197 q^{13} -2744 q^{14} + ( 1350 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 4911 + 7 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{17} + 5832 q^{18} + ( -1299 - 75 \beta_{1} + 47 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{19} + ( -3200 + 64 \beta_{1} ) q^{20} + 9261 q^{21} + ( 1064 + 32 \beta_{1} - 32 \beta_{2} - 8 \beta_{3} ) q^{22} + ( 6062 - 56 \beta_{1} - 58 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} ) q^{23} -13824 q^{24} + ( 7802 - 197 \beta_{1} - 35 \beta_{2} - 5 \beta_{3} + 14 \beta_{4} ) q^{25} -17576 q^{26} -19683 q^{27} -21952 q^{28} + ( 26185 - 147 \beta_{1} + 59 \beta_{2} - 17 \beta_{3} - 8 \beta_{4} ) q^{29} + ( 10800 - 216 \beta_{1} ) q^{30} + ( 52583 + 73 \beta_{1} + 38 \beta_{2} + 13 \beta_{3} + 14 \beta_{4} ) q^{31} + 32768 q^{32} + ( -3591 - 108 \beta_{1} + 108 \beta_{2} + 27 \beta_{3} ) q^{33} + ( 39288 + 56 \beta_{1} + 80 \beta_{2} + 40 \beta_{3} - 8 \beta_{4} ) q^{34} + ( 17150 - 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( -20324 - 50 \beta_{1} + 39 \beta_{2} + 16 \beta_{3} - 80 \beta_{4} ) q^{37} + ( -10392 - 600 \beta_{1} + 376 \beta_{2} + 8 \beta_{3} - 16 \beta_{4} ) q^{38} + 59319 q^{39} + ( -25600 + 512 \beta_{1} ) q^{40} + ( -49499 - 942 \beta_{1} + 129 \beta_{2} + 19 \beta_{3} + 28 \beta_{4} ) q^{41} + 74088 q^{42} + ( -3727 - 457 \beta_{1} - 555 \beta_{2} + 49 \beta_{3} - 38 \beta_{4} ) q^{43} + ( 8512 + 256 \beta_{1} - 256 \beta_{2} - 64 \beta_{3} ) q^{44} + ( -36450 + 729 \beta_{1} ) q^{45} + ( 48496 - 448 \beta_{1} - 464 \beta_{2} + 64 \beta_{3} + 40 \beta_{4} ) q^{46} + ( -25513 - 892 \beta_{1} + 1236 \beta_{2} - 45 \beta_{3} - 21 \beta_{4} ) q^{47} -110592 q^{48} + 117649 q^{49} + ( 62416 - 1576 \beta_{1} - 280 \beta_{2} - 40 \beta_{3} + 112 \beta_{4} ) q^{50} + ( -132597 - 189 \beta_{1} - 270 \beta_{2} - 135 \beta_{3} + 27 \beta_{4} ) q^{51} -140608 q^{52} + ( -60387 - 771 \beta_{1} - 332 \beta_{2} - 263 \beta_{3} + 92 \beta_{4} ) q^{53} -157464 q^{54} + ( 188726 + 588 \beta_{1} + 595 \beta_{2} - 120 \beta_{3} + 122 \beta_{4} ) q^{55} -175616 q^{56} + ( 35073 + 2025 \beta_{1} - 1269 \beta_{2} - 27 \beta_{3} + 54 \beta_{4} ) q^{57} + ( 209480 - 1176 \beta_{1} + 472 \beta_{2} - 136 \beta_{3} - 64 \beta_{4} ) q^{58} + ( -559278 + 1015 \beta_{1} - 2406 \beta_{2} + 80 \beta_{3} - 337 \beta_{4} ) q^{59} + ( 86400 - 1728 \beta_{1} ) q^{60} + ( -498278 + 606 \beta_{1} - 2661 \beta_{2} + 150 \beta_{3} + 66 \beta_{4} ) q^{61} + ( 420664 + 584 \beta_{1} + 304 \beta_{2} + 104 \beta_{3} + 112 \beta_{4} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( 109850 - 2197 \beta_{1} ) q^{65} + ( -28728 - 864 \beta_{1} + 864 \beta_{2} + 216 \beta_{3} ) q^{66} + ( 32698 + 752 \beta_{1} - 2194 \beta_{2} - 554 \beta_{3} + 130 \beta_{4} ) q^{67} + ( 314304 + 448 \beta_{1} + 640 \beta_{2} + 320 \beta_{3} - 64 \beta_{4} ) q^{68} + ( -163674 + 1512 \beta_{1} + 1566 \beta_{2} - 216 \beta_{3} - 135 \beta_{4} ) q^{69} + ( 137200 - 2744 \beta_{1} ) q^{70} + ( 768977 + 2854 \beta_{1} + 841 \beta_{2} + 181 \beta_{3} + 200 \beta_{4} ) q^{71} + 373248 q^{72} + ( 1130739 + 1183 \beta_{1} + 3015 \beta_{2} - 419 \beta_{3} + 250 \beta_{4} ) q^{73} + ( -162592 - 400 \beta_{1} + 312 \beta_{2} + 128 \beta_{3} - 640 \beta_{4} ) q^{74} + ( -210654 + 5319 \beta_{1} + 945 \beta_{2} + 135 \beta_{3} - 378 \beta_{4} ) q^{75} + ( -83136 - 4800 \beta_{1} + 3008 \beta_{2} + 64 \beta_{3} - 128 \beta_{4} ) q^{76} + ( -45619 - 1372 \beta_{1} + 1372 \beta_{2} + 343 \beta_{3} ) q^{77} + 474552 q^{78} + ( 1128879 - 6337 \beta_{1} + 3026 \beta_{2} + 285 \beta_{3} - 142 \beta_{4} ) q^{79} + ( -204800 + 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( -395992 - 7536 \beta_{1} + 1032 \beta_{2} + 152 \beta_{3} + 224 \beta_{4} ) q^{82} + ( -922431 - 1676 \beta_{1} + 4580 \beta_{2} - 47 \beta_{3} + 499 \beta_{4} ) q^{83} + 592704 q^{84} + ( 775208 - 2964 \beta_{1} - 3525 \beta_{2} + 710 \beta_{3} + 56 \beta_{4} ) q^{85} + ( -29816 - 3656 \beta_{1} - 4440 \beta_{2} + 392 \beta_{3} - 304 \beta_{4} ) q^{86} + ( -706995 + 3969 \beta_{1} - 1593 \beta_{2} + 459 \beta_{3} + 216 \beta_{4} ) q^{87} + ( 68096 + 2048 \beta_{1} - 2048 \beta_{2} - 512 \beta_{3} ) q^{88} + ( -3485522 - 8607 \beta_{1} + 2121 \beta_{2} + 730 \beta_{3} + 874 \beta_{4} ) q^{89} + ( -291600 + 5832 \beta_{1} ) q^{90} + 753571 q^{91} + ( 387968 - 3584 \beta_{1} - 3712 \beta_{2} + 512 \beta_{3} + 320 \beta_{4} ) q^{92} + ( -1419741 - 1971 \beta_{1} - 1026 \beta_{2} - 351 \beta_{3} - 378 \beta_{4} ) q^{93} + ( -204104 - 7136 \beta_{1} + 9888 \beta_{2} - 360 \beta_{3} - 168 \beta_{4} ) q^{94} + ( -4919351 - 4281 \beta_{1} + 3625 \beta_{2} - 295 \beta_{3} - 792 \beta_{4} ) q^{95} -884736 q^{96} + ( -3675177 - 4335 \beta_{1} - 5490 \beta_{2} + 399 \beta_{3} - 1184 \beta_{4} ) q^{97} + 941192 q^{98} + ( 96957 + 2916 \beta_{1} - 2916 \beta_{2} - 729 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} - 2000 q^{10} + 659 q^{11} - 8640 q^{12} - 10985 q^{13} - 13720 q^{14} + 6750 q^{15} + 20480 q^{16} + 24575 q^{17} + 29160 q^{18} - 6446 q^{19} - 16000 q^{20} + 46305 q^{21} + 5272 q^{22} + 30268 q^{23} - 69120 q^{24} + 38965 q^{25} - 87880 q^{26} - 98415 q^{27} - 109760 q^{28} + 130950 q^{29} + 54000 q^{30} + 262979 q^{31} + 163840 q^{32} - 17793 q^{33} + 196600 q^{34} + 85750 q^{35} + 233280 q^{36} - 101549 q^{37} - 51568 q^{38} + 296595 q^{39} - 128000 q^{40} - 247328 q^{41} + 370440 q^{42} - 19092 q^{43} + 42176 q^{44} - 182250 q^{45} + 242144 q^{46} - 126419 q^{47} - 552960 q^{48} + 588245 q^{49} + 311720 q^{50} - 663525 q^{51} - 703040 q^{52} - 302793 q^{53} - 787320 q^{54} + 943985 q^{55} - 878080 q^{56} + 174042 q^{57} + 1047600 q^{58} - 2798636 q^{59} + 432000 q^{60} - 2493751 q^{61} + 2103832 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 549250 q^{65} - 142344 q^{66} + 160188 q^{67} + 1572800 q^{68} - 817236 q^{69} + 686000 q^{70} + 3846088 q^{71} + 1866240 q^{72} + 5655872 q^{73} - 812392 q^{74} - 1052055 q^{75} - 412544 q^{76} - 226037 q^{77} + 2372760 q^{78} + 5647991 q^{79} - 1024000 q^{80} + 2657205 q^{81} - 1978624 q^{82} - 4607669 q^{83} + 2963520 q^{84} + 3873935 q^{85} - 152736 q^{86} - 3535650 q^{87} + 337408 q^{88} - 17424029 q^{89} - 1458000 q^{90} + 3767855 q^{91} + 1937152 q^{92} - 7100433 q^{93} - 1011352 q^{94} - 24593720 q^{95} - 4423680 q^{96} - 18380577 q^{97} + 4705960 q^{98} + 480411 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 122890 x^{3} - 6160660 x^{2} + 3465881625 x + 278845474950\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5819 \nu^{4} - 1178163 \nu^{3} - 550141385 \nu^{2} + 69816055635 \nu + 12467385733950 \)\()/ 10362009840 \)
\(\beta_{2}\)\(=\)\((\)\( -5819 \nu^{4} + 1178163 \nu^{3} + 550141385 \nu^{2} - 38730026115 \nu - 12477747743790 \)\()/ 10362009840 \)
\(\beta_{3}\)\(=\)\((\)\( 95 \nu^{4} + 26425 \nu^{3} - 12661669 \nu^{2} - 1766467065 \nu + 210119868966 \)\()/53138512\)
\(\beta_{4}\)\(=\)\((\)\( -140713 \nu^{4} + 29380281 \nu^{3} + 10122958435 \nu^{2} - 1309735057545 \nu - 148719135691770 \)\()/ 10362009840 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-10 \beta_{4} + \beta_{3} + 140 \beta_{2} - 105 \beta_{1} + 147442\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-806 \beta_{4} + 3572 \beta_{3} + 74935 \beta_{2} + 44073 \beta_{1} + 11515181\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-1108612 \beta_{4} + 817759 \beta_{3} + 16409930 \beta_{2} - 7659339 \beta_{1} + 9831368452\)\()/3\)

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
−201.135
−231.824
315.005
−90.2950
210.250
8.00000 −27.0000 64.0000 −505.603 −216.000 −343.000 512.000 729.000 −4044.82
1.2 8.00000 −27.0000 64.0000 −223.551 −216.000 −343.000 512.000 729.000 −1788.41
1.3 8.00000 −27.0000 64.0000 −17.2693 −216.000 −343.000 512.000 729.000 −138.154
1.4 8.00000 −27.0000 64.0000 232.967 −216.000 −343.000 512.000 729.000 1863.73
1.5 8.00000 −27.0000 64.0000 263.456 −216.000 −343.000 512.000 729.000 2107.65
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.l 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.l 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 250 T_{5}^{4} - 183545 T_{5}^{3} - 14595850 T_{5}^{2} + 6741142400 T_{5} + 119801402000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T )^{5} \)
$3$ \( ( 27 + T )^{5} \)
$5$ \( 119801402000 + 6741142400 T - 14595850 T^{2} - 183545 T^{3} + 250 T^{4} + T^{5} \)
$7$ \( ( 343 + T )^{5} \)
$11$ \( 164356746724271712 + 248876084787064 T + 41236262956 T^{2} - 43756206 T^{3} - 659 T^{4} + T^{5} \)
$13$ \( ( 2197 + T )^{5} \)
$17$ \( \)\(41\!\cdots\!76\)\( + 143175107384955000 T + 9383982990420 T^{2} - 673573230 T^{3} - 24575 T^{4} + T^{5} \)
$19$ \( \)\(56\!\cdots\!88\)\( + 654251293470820032 T - 12294669029916 T^{2} - 2522735967 T^{3} + 6446 T^{4} + T^{5} \)
$23$ \( \)\(72\!\cdots\!88\)\( + 11655712032270044248 T + 104556010962212 T^{2} - 6873478617 T^{3} - 30268 T^{4} + T^{5} \)
$29$ \( \)\(23\!\cdots\!16\)\( - \)\(14\!\cdots\!80\)\( T + 2814729805076390 T^{2} - 13264316365 T^{3} - 130950 T^{4} + T^{5} \)
$31$ \( \)\(58\!\cdots\!24\)\( - \)\(24\!\cdots\!72\)\( T + 2537726512578704 T^{2} + 9247332292 T^{3} - 262979 T^{4} + T^{5} \)
$37$ \( \)\(20\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T - 42375641710702260 T^{2} - 291033515550 T^{3} + 101549 T^{4} + T^{5} \)
$41$ \( \)\(24\!\cdots\!00\)\( - \)\(57\!\cdots\!00\)\( T - 78238352618592240 T^{2} - 206728243860 T^{3} + 247328 T^{4} + T^{5} \)
$43$ \( -\)\(27\!\cdots\!92\)\( + \)\(33\!\cdots\!28\)\( T + 37366883896088972 T^{2} - 463212154057 T^{3} + 19092 T^{4} + T^{5} \)
$47$ \( -\)\(85\!\cdots\!80\)\( + \)\(93\!\cdots\!36\)\( T + 120005194097902204 T^{2} - 1305267642644 T^{3} + 126419 T^{4} + T^{5} \)
$53$ \( -\)\(28\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T - 237037031459786880 T^{2} - 2712413229540 T^{3} + 302793 T^{4} + T^{5} \)
$59$ \( \)\(18\!\cdots\!32\)\( - \)\(41\!\cdots\!16\)\( T - 23701694727603442144 T^{2} - 6875467135836 T^{3} + 2798636 T^{4} + T^{5} \)
$61$ \( -\)\(30\!\cdots\!00\)\( - \)\(51\!\cdots\!00\)\( T - 9764296939204374880 T^{2} - 3966892938980 T^{3} + 2493751 T^{4} + T^{5} \)
$67$ \( -\)\(10\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T + 13320775028771852480 T^{2} - 13285169698540 T^{3} - 160188 T^{4} + T^{5} \)
$71$ \( \)\(41\!\cdots\!68\)\( - \)\(16\!\cdots\!32\)\( T + 13948237185812008192 T^{2} + 155502177028 T^{3} - 3846088 T^{4} + T^{5} \)
$73$ \( -\)\(42\!\cdots\!24\)\( + \)\(32\!\cdots\!84\)\( T + 31594634166079139662 T^{2} - 3744070691559 T^{3} - 5655872 T^{4} + T^{5} \)
$79$ \( -\)\(31\!\cdots\!40\)\( + \)\(54\!\cdots\!76\)\( T + 31405309985820738304 T^{2} - 2588657307054 T^{3} - 5647991 T^{4} + T^{5} \)
$83$ \( -\)\(90\!\cdots\!52\)\( + \)\(12\!\cdots\!04\)\( T - 27049562748360942156 T^{2} - 19508559810666 T^{3} + 4607669 T^{4} + T^{5} \)
$89$ \( -\)\(12\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T - \)\(35\!\cdots\!60\)\( T^{2} + 49036631517720 T^{3} + 17424029 T^{4} + T^{5} \)
$97$ \( -\)\(71\!\cdots\!60\)\( - \)\(51\!\cdots\!44\)\( T - \)\(89\!\cdots\!12\)\( T^{2} + 35252168542604 T^{3} + 18380577 T^{4} + T^{5} \)
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