Properties

Label 354.6.a.g
Level 354
Weight 6
Character orbit 354.a
Self dual Yes
Analytic conductor 56.776
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(56.7758722138\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + ( 1 - \beta_{5} ) q^{5} + 36 q^{6} + ( -9 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + ( 1 - \beta_{5} ) q^{5} + 36 q^{6} + ( -9 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} + ( -4 + 4 \beta_{5} ) q^{10} + ( 71 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{11} -144 q^{12} + ( -88 + 5 \beta_{1} + 10 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{13} + ( 36 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{14} + ( -9 + 9 \beta_{5} ) q^{15} + 256 q^{16} + ( 155 + 6 \beta_{1} - 22 \beta_{2} + 19 \beta_{3} - 3 \beta_{4} - 7 \beta_{5} ) q^{17} -324 q^{18} + ( 253 - 3 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 25 \beta_{5} ) q^{19} + ( 16 - 16 \beta_{5} ) q^{20} + ( 81 - 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{21} + ( -284 - 12 \beta_{1} - 28 \beta_{2} + 24 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} ) q^{22} + ( 279 + 12 \beta_{1} - 42 \beta_{2} + 45 \beta_{3} + 7 \beta_{4} - 27 \beta_{5} ) q^{23} + 576 q^{24} + ( -200 - 67 \beta_{2} - 31 \beta_{3} + 12 \beta_{4} - 5 \beta_{5} ) q^{25} + ( 352 - 20 \beta_{1} - 40 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 20 \beta_{5} ) q^{26} -729 q^{27} + ( -144 + 16 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{28} + ( -250 + 16 \beta_{1} + 130 \beta_{2} - 25 \beta_{3} - 37 \beta_{4} - 12 \beta_{5} ) q^{29} + ( 36 - 36 \beta_{5} ) q^{30} + ( -939 - 11 \beta_{1} + 250 \beta_{2} + 18 \beta_{3} + 22 \beta_{4} - 40 \beta_{5} ) q^{31} -1024 q^{32} + ( -639 - 27 \beta_{1} - 63 \beta_{2} + 54 \beta_{3} - 18 \beta_{4} - 27 \beta_{5} ) q^{33} + ( -620 - 24 \beta_{1} + 88 \beta_{2} - 76 \beta_{3} + 12 \beta_{4} + 28 \beta_{5} ) q^{34} + ( -1200 + 30 \beta_{1} - 33 \beta_{2} - 56 \beta_{3} - 31 \beta_{4} + 10 \beta_{5} ) q^{35} + 1296 q^{36} + ( -3431 + 26 \beta_{1} - 27 \beta_{2} - 88 \beta_{3} + 49 \beta_{4} + 15 \beta_{5} ) q^{37} + ( -1012 + 12 \beta_{1} + 40 \beta_{2} + 32 \beta_{3} + 12 \beta_{4} + 100 \beta_{5} ) q^{38} + ( 792 - 45 \beta_{1} - 90 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} + 45 \beta_{5} ) q^{39} + ( -64 + 64 \beta_{5} ) q^{40} + ( -22 + 5 \beta_{1} + 81 \beta_{2} + 79 \beta_{3} + 121 \beta_{4} + 76 \beta_{5} ) q^{41} + ( -324 + 36 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{42} + ( -5609 - 6 \beta_{1} - 215 \beta_{2} - 101 \beta_{3} - 102 \beta_{4} - 85 \beta_{5} ) q^{43} + ( 1136 + 48 \beta_{1} + 112 \beta_{2} - 96 \beta_{3} + 32 \beta_{4} + 48 \beta_{5} ) q^{44} + ( 81 - 81 \beta_{5} ) q^{45} + ( -1116 - 48 \beta_{1} + 168 \beta_{2} - 180 \beta_{3} - 28 \beta_{4} + 108 \beta_{5} ) q^{46} + ( -1928 - 29 \beta_{1} - 515 \beta_{2} + 155 \beta_{3} + 81 \beta_{4} + 106 \beta_{5} ) q^{47} -2304 q^{48} + ( -5413 - 98 \beta_{1} + 62 \beta_{2} + 77 \beta_{3} - 4 \beta_{4} - 83 \beta_{5} ) q^{49} + ( 800 + 268 \beta_{2} + 124 \beta_{3} - 48 \beta_{4} + 20 \beta_{5} ) q^{50} + ( -1395 - 54 \beta_{1} + 198 \beta_{2} - 171 \beta_{3} + 27 \beta_{4} + 63 \beta_{5} ) q^{51} + ( -1408 + 80 \beta_{1} + 160 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} - 80 \beta_{5} ) q^{52} + ( -2831 + 28 \beta_{1} + 164 \beta_{2} + 80 \beta_{3} - 150 \beta_{4} + 61 \beta_{5} ) q^{53} + 2916 q^{54} + ( -6039 + 84 \beta_{1} - 107 \beta_{2} + 35 \beta_{3} - 152 \beta_{4} - 251 \beta_{5} ) q^{55} + ( 576 - 64 \beta_{1} + 64 \beta_{2} - 64 \beta_{3} ) q^{56} + ( -2277 + 27 \beta_{1} + 90 \beta_{2} + 72 \beta_{3} + 27 \beta_{4} + 225 \beta_{5} ) q^{57} + ( 1000 - 64 \beta_{1} - 520 \beta_{2} + 100 \beta_{3} + 148 \beta_{4} + 48 \beta_{5} ) q^{58} + 3481 q^{59} + ( -144 + 144 \beta_{5} ) q^{60} + ( -7291 + 40 \beta_{1} + 182 \beta_{2} + 221 \beta_{3} + 20 \beta_{4} + 80 \beta_{5} ) q^{61} + ( 3756 + 44 \beta_{1} - 1000 \beta_{2} - 72 \beta_{3} - 88 \beta_{4} + 160 \beta_{5} ) q^{62} + ( -729 + 81 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( 10900 + 91 \beta_{1} - 701 \beta_{2} - 332 \beta_{3} - 55 \beta_{4} + 139 \beta_{5} ) q^{65} + ( 2556 + 108 \beta_{1} + 252 \beta_{2} - 216 \beta_{3} + 72 \beta_{4} + 108 \beta_{5} ) q^{66} + ( -8711 - 4 \beta_{1} + 903 \beta_{2} + 907 \beta_{3} - 253 \beta_{5} ) q^{67} + ( 2480 + 96 \beta_{1} - 352 \beta_{2} + 304 \beta_{3} - 48 \beta_{4} - 112 \beta_{5} ) q^{68} + ( -2511 - 108 \beta_{1} + 378 \beta_{2} - 405 \beta_{3} - 63 \beta_{4} + 243 \beta_{5} ) q^{69} + ( 4800 - 120 \beta_{1} + 132 \beta_{2} + 224 \beta_{3} + 124 \beta_{4} - 40 \beta_{5} ) q^{70} + ( 14046 - 264 \beta_{1} - 62 \beta_{2} - 240 \beta_{3} + 73 \beta_{4} + 159 \beta_{5} ) q^{71} -5184 q^{72} + ( -6203 + 189 \beta_{1} - 1921 \beta_{2} - 359 \beta_{3} + 252 \beta_{4} + 68 \beta_{5} ) q^{73} + ( 13724 - 104 \beta_{1} + 108 \beta_{2} + 352 \beta_{3} - 196 \beta_{4} - 60 \beta_{5} ) q^{74} + ( 1800 + 603 \beta_{2} + 279 \beta_{3} - 108 \beta_{4} + 45 \beta_{5} ) q^{75} + ( 4048 - 48 \beta_{1} - 160 \beta_{2} - 128 \beta_{3} - 48 \beta_{4} - 400 \beta_{5} ) q^{76} + ( 15084 - 124 \beta_{1} + 1009 \beta_{2} - 492 \beta_{3} + 203 \beta_{4} - 116 \beta_{5} ) q^{77} + ( -3168 + 180 \beta_{1} + 360 \beta_{2} + 36 \beta_{3} + 36 \beta_{4} - 180 \beta_{5} ) q^{78} + ( 14398 + 142 \beta_{1} + 199 \beta_{2} + 433 \beta_{3} + 123 \beta_{4} - 719 \beta_{5} ) q^{79} + ( 256 - 256 \beta_{5} ) q^{80} + 6561 q^{81} + ( 88 - 20 \beta_{1} - 324 \beta_{2} - 316 \beta_{3} - 484 \beta_{4} - 304 \beta_{5} ) q^{82} + ( 36502 + 283 \beta_{1} + 1727 \beta_{2} - 177 \beta_{3} - 73 \beta_{4} - 920 \beta_{5} ) q^{83} + ( 1296 - 144 \beta_{1} + 144 \beta_{2} - 144 \beta_{3} ) q^{84} + ( 4424 + 200 \beta_{1} - 356 \beta_{2} - 721 \beta_{3} - 74 \beta_{4} + 87 \beta_{5} ) q^{85} + ( 22436 + 24 \beta_{1} + 860 \beta_{2} + 404 \beta_{3} + 408 \beta_{4} + 340 \beta_{5} ) q^{86} + ( 2250 - 144 \beta_{1} - 1170 \beta_{2} + 225 \beta_{3} + 333 \beta_{4} + 108 \beta_{5} ) q^{87} + ( -4544 - 192 \beta_{1} - 448 \beta_{2} + 384 \beta_{3} - 128 \beta_{4} - 192 \beta_{5} ) q^{88} + ( 55568 - 42 \beta_{1} + 2457 \beta_{2} - 598 \beta_{3} - 55 \beta_{4} + 276 \beta_{5} ) q^{89} + ( -324 + 324 \beta_{5} ) q^{90} + ( 35913 - 230 \beta_{1} + 1085 \beta_{2} - 436 \beta_{3} - 76 \beta_{4} - 206 \beta_{5} ) q^{91} + ( 4464 + 192 \beta_{1} - 672 \beta_{2} + 720 \beta_{3} + 112 \beta_{4} - 432 \beta_{5} ) q^{92} + ( 8451 + 99 \beta_{1} - 2250 \beta_{2} - 162 \beta_{3} - 198 \beta_{4} + 360 \beta_{5} ) q^{93} + ( 7712 + 116 \beta_{1} + 2060 \beta_{2} - 620 \beta_{3} - 324 \beta_{4} - 424 \beta_{5} ) q^{94} + ( 76510 + 131 \beta_{1} - 1784 \beta_{2} - 688 \beta_{3} + 382 \beta_{4} - 533 \beta_{5} ) q^{95} + 9216 q^{96} + ( 28558 + 202 \beta_{1} + 2415 \beta_{2} + 1267 \beta_{3} - 193 \beta_{4} + 1093 \beta_{5} ) q^{97} + ( 21652 + 392 \beta_{1} - 248 \beta_{2} - 308 \beta_{3} + 16 \beta_{4} + 332 \beta_{5} ) q^{98} + ( 5751 + 243 \beta_{1} + 567 \beta_{2} - 486 \beta_{3} + 162 \beta_{4} + 243 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 24q^{2} - 54q^{3} + 96q^{4} + 4q^{5} + 216q^{6} - 54q^{7} - 384q^{8} + 486q^{9} + O(q^{10}) \) \( 6q - 24q^{2} - 54q^{3} + 96q^{4} + 4q^{5} + 216q^{6} - 54q^{7} - 384q^{8} + 486q^{9} - 16q^{10} + 436q^{11} - 864q^{12} - 536q^{13} + 216q^{14} - 36q^{15} + 1536q^{16} + 910q^{17} - 1944q^{18} + 1462q^{19} + 64q^{20} + 486q^{21} - 1744q^{22} + 1634q^{23} + 3456q^{24} - 1186q^{25} + 2144q^{26} - 4374q^{27} - 864q^{28} - 1598q^{29} + 144q^{30} - 5670q^{31} - 6144q^{32} - 3924q^{33} - 3640q^{34} - 7242q^{35} + 7776q^{36} - 20458q^{37} - 5848q^{38} + 4824q^{39} - 256q^{40} + 262q^{41} - 1944q^{42} - 34028q^{43} + 6976q^{44} + 324q^{45} - 6536q^{46} - 11194q^{47} - 13824q^{48} - 32652q^{49} + 4744q^{50} - 8190q^{51} - 8576q^{52} - 17164q^{53} + 17496q^{54} - 37040q^{55} + 3456q^{56} - 13158q^{57} + 6392q^{58} + 20886q^{59} - 576q^{60} - 43546q^{61} + 22680q^{62} - 4374q^{63} + 24576q^{64} + 65568q^{65} + 15696q^{66} - 52772q^{67} + 14560q^{68} - 14706q^{69} + 28968q^{70} + 84740q^{71} - 31104q^{72} - 36578q^{73} + 81832q^{74} + 10674q^{75} + 23392q^{76} + 90678q^{77} - 19296q^{78} + 85196q^{79} + 1024q^{80} + 39366q^{81} - 1048q^{82} + 217026q^{83} + 7776q^{84} + 26570q^{85} + 136112q^{86} + 14382q^{87} - 27904q^{88} + 333850q^{89} - 1296q^{90} + 214914q^{91} + 26144q^{92} + 51030q^{93} + 44776q^{94} + 458758q^{95} + 55296q^{96} + 173148q^{97} + 130608q^{98} + 35316q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 358 x^{4} - 404 x^{3} + 26492 x^{2} - 11664 x - 353376\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -34 \nu^{5} + 1148 \nu^{4} - 6156 \nu^{3} - 229231 \nu^{2} + 2808606 \nu + 4257918 \)\()/130266\)
\(\beta_{2}\)\(=\)\((\)\( -251 \nu^{5} - 1742 \nu^{4} + 82266 \nu^{3} + 616126 \nu^{2} - 3340824 \nu - 13795668 \)\()/260532\)
\(\beta_{3}\)\(=\)\((\)\( -134 \nu^{5} - 584 \nu^{4} + 39594 \nu^{3} + 299605 \nu^{2} - 1456758 \nu - 11662758 \)\()/130266\)
\(\beta_{4}\)\(=\)\((\)\( 598 \nu^{5} - 6143 \nu^{4} - 165030 \nu^{3} + 1407292 \nu^{2} + 8886672 \nu - 58414446 \)\()/390798\)
\(\beta_{5}\)\(=\)\((\)\( -1702 \nu^{5} - 2557 \nu^{4} + 539844 \nu^{3} + 1976846 \nu^{2} - 23388948 \nu - 58933332 \)\()/781596\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{1} - 1\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(10 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} - 24 \beta_{2} - \beta_{1} + 1069\)\()/9\)
\(\nu^{3}\)\(=\)\((\)\(314 \beta_{5} + 121 \beta_{4} - 336 \beta_{3} - 168 \beta_{2} + 34 \beta_{1} + 1673\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(1276 \beta_{5} + 488 \beta_{4} + 456 \beta_{3} - 2584 \beta_{2} - 44 \beta_{1} + 74592\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(87584 \beta_{5} + 35116 \beta_{4} - 77562 \beta_{3} - 69516 \beta_{2} + 2950 \beta_{1} + 1131320\)\()/9\)

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
17.2905
−3.64610
−14.3167
6.78105
−11.2438
5.13506
−4.00000 −9.00000 16.0000 −75.0224 36.0000 17.7979 −64.0000 81.0000 300.090
1.2 −4.00000 −9.00000 16.0000 −33.6768 36.0000 −116.721 −64.0000 81.0000 134.707
1.3 −4.00000 −9.00000 16.0000 −15.9371 36.0000 113.338 −64.0000 81.0000 63.7485
1.4 −4.00000 −9.00000 16.0000 −14.2062 36.0000 60.7262 −64.0000 81.0000 56.8247
1.5 −4.00000 −9.00000 16.0000 62.9440 36.0000 −185.768 −64.0000 81.0000 −251.776
1.6 −4.00000 −9.00000 16.0000 79.8984 36.0000 56.6271 −64.0000 81.0000 −319.594
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{6} - 4 T_{5}^{5} - 8774 T_{5}^{4} - 62240 T_{5}^{3} + 16501109 T_{5}^{2} + 425064268 T_{5} + 2876741688 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).