Properties

Label 354.6.a.i
Level 354
Weight 6
Character orbit 354.a
Self dual Yes
Analytic conductor 56.776
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( 12 - \beta_{1} ) q^{5} + 36 q^{6} + ( 23 - \beta_{1} - \beta_{2} ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( 12 - \beta_{1} ) q^{5} + 36 q^{6} + ( 23 - \beta_{1} - \beta_{2} ) q^{7} + 64 q^{8} + 81 q^{9} + ( 48 - 4 \beta_{1} ) q^{10} + ( 112 + \beta_{1} - \beta_{4} ) q^{11} + 144 q^{12} + ( 218 - \beta_{1} - \beta_{3} ) q^{13} + ( 92 - 4 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 108 - 9 \beta_{1} ) q^{15} + 256 q^{16} + ( 233 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{17} + 324 q^{18} + ( 393 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( 192 - 16 \beta_{1} ) q^{20} + ( 207 - 9 \beta_{1} - 9 \beta_{2} ) q^{21} + ( 448 + 4 \beta_{1} - 4 \beta_{4} ) q^{22} + ( 473 + 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{23} + 576 q^{24} + ( 1450 - 10 \beta_{1} + 10 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{25} + ( 872 - 4 \beta_{1} - 4 \beta_{3} ) q^{26} + 729 q^{27} + ( 368 - 16 \beta_{1} - 16 \beta_{2} ) q^{28} + ( 41 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 8 \beta_{7} ) q^{29} + ( 432 - 36 \beta_{1} ) q^{30} + ( 70 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + 7 \beta_{4} - 6 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{31} + 1024 q^{32} + ( 1008 + 9 \beta_{1} - 9 \beta_{4} ) q^{33} + ( 932 + 16 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{34} + ( 4503 + 29 \beta_{1} + 21 \beta_{2} - 5 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} ) q^{35} + 1296 q^{36} + ( 1597 + 33 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{37} + ( 1572 - 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{38} + ( 1962 - 9 \beta_{1} - 9 \beta_{3} ) q^{39} + ( 768 - 64 \beta_{1} ) q^{40} + ( 2506 + 84 \beta_{1} + 37 \beta_{2} - 10 \beta_{3} - 15 \beta_{4} - 3 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{41} + ( 828 - 36 \beta_{1} - 36 \beta_{2} ) q^{42} + ( 3067 + 77 \beta_{1} + 8 \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 9 \beta_{6} - \beta_{7} ) q^{43} + ( 1792 + 16 \beta_{1} - 16 \beta_{4} ) q^{44} + ( 972 - 81 \beta_{1} ) q^{45} + ( 1892 + 36 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 16 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{46} + ( 2534 + 91 \beta_{1} + 51 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} - 11 \beta_{6} - \beta_{7} ) q^{47} + 2304 q^{48} + ( 7447 + 167 \beta_{1} - 45 \beta_{2} - \beta_{3} - 16 \beta_{4} + 14 \beta_{5} + 18 \beta_{6} - 13 \beta_{7} ) q^{49} + ( 5800 - 40 \beta_{1} + 40 \beta_{2} - 4 \beta_{3} + 16 \beta_{4} + 12 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{50} + ( 2097 + 36 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} + 9 \beta_{7} ) q^{51} + ( 3488 - 16 \beta_{1} - 16 \beta_{3} ) q^{52} + ( 1652 + 113 \beta_{1} + 56 \beta_{2} + 14 \beta_{3} + 18 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} ) q^{53} + 2916 q^{54} + ( -4304 + 30 \beta_{1} - 94 \beta_{2} + 30 \beta_{3} - 30 \beta_{4} - 18 \beta_{5} - 11 \beta_{6} + 10 \beta_{7} ) q^{55} + ( 1472 - 64 \beta_{1} - 64 \beta_{2} ) q^{56} + ( 3537 - 27 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} ) q^{57} + ( 164 + 56 \beta_{1} + 28 \beta_{2} + 20 \beta_{3} + 24 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} + 32 \beta_{7} ) q^{58} + 3481 q^{59} + ( 1728 - 144 \beta_{1} ) q^{60} + ( 11926 + 193 \beta_{1} - 69 \beta_{2} - 47 \beta_{3} + 39 \beta_{4} + 43 \beta_{5} + 18 \beta_{6} + 6 \beta_{7} ) q^{61} + ( 280 - 20 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 28 \beta_{4} - 24 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} ) q^{62} + ( 1863 - 81 \beta_{1} - 81 \beta_{2} ) q^{63} + 4096 q^{64} + ( 6839 - 245 \beta_{1} + 30 \beta_{2} - 40 \beta_{3} + 65 \beta_{4} + 44 \beta_{5} - 5 \beta_{6} + 15 \beta_{7} ) q^{65} + ( 4032 + 36 \beta_{1} - 36 \beta_{4} ) q^{66} + ( 3633 + 206 \beta_{1} - 152 \beta_{2} + 23 \beta_{3} - 18 \beta_{4} - 41 \beta_{5} - 19 \beta_{6} - 28 \beta_{7} ) q^{67} + ( 3728 + 64 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} + 16 \beta_{6} + 16 \beta_{7} ) q^{68} + ( 4257 + 81 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} - 36 \beta_{5} + 9 \beta_{6} - 18 \beta_{7} ) q^{69} + ( 18012 + 116 \beta_{1} + 84 \beta_{2} - 20 \beta_{3} - 64 \beta_{4} + 16 \beta_{5} + 24 \beta_{6} - 48 \beta_{7} ) q^{70} + ( 1909 - 209 \beta_{1} - 118 \beta_{2} - 16 \beta_{3} - 54 \beta_{4} - 30 \beta_{5} + 43 \beta_{6} - 23 \beta_{7} ) q^{71} + 5184 q^{72} + ( 8703 + 268 \beta_{1} - 69 \beta_{2} - 4 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} ) q^{73} + ( 6388 + 132 \beta_{1} + 20 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} ) q^{74} + ( 13050 - 90 \beta_{1} + 90 \beta_{2} - 9 \beta_{3} + 36 \beta_{4} + 27 \beta_{5} + 9 \beta_{6} - 18 \beta_{7} ) q^{75} + ( 6288 - 48 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} + 32 \beta_{4} - 16 \beta_{5} - 32 \beta_{6} - 16 \beta_{7} ) q^{76} + ( -2753 - 448 \beta_{1} - 167 \beta_{2} - 29 \beta_{3} - 97 \beta_{4} - 39 \beta_{5} + 20 \beta_{6} - 21 \beta_{7} ) q^{77} + ( 7848 - 36 \beta_{1} - 36 \beta_{3} ) q^{78} + ( -6105 + 94 \beta_{1} - 86 \beta_{2} + 28 \beta_{3} + 48 \beta_{4} + 36 \beta_{5} - 50 \beta_{6} + 73 \beta_{7} ) q^{79} + ( 3072 - 256 \beta_{1} ) q^{80} + 6561 q^{81} + ( 10024 + 336 \beta_{1} + 148 \beta_{2} - 40 \beta_{3} - 60 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} ) q^{82} + ( 520 - 610 \beta_{1} - 75 \beta_{2} + 46 \beta_{3} - 43 \beta_{4} + 41 \beta_{5} - 39 \beta_{6} - 16 \beta_{7} ) q^{83} + ( 3312 - 144 \beta_{1} - 144 \beta_{2} ) q^{84} + ( -17870 - 320 \beta_{1} + 37 \beta_{2} + 41 \beta_{3} - 43 \beta_{4} - 157 \beta_{5} - 4 \beta_{7} ) q^{85} + ( 12268 + 308 \beta_{1} + 32 \beta_{2} + 4 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} - 36 \beta_{6} - 4 \beta_{7} ) q^{86} + ( 369 + 126 \beta_{1} + 63 \beta_{2} + 45 \beta_{3} + 54 \beta_{4} + 54 \beta_{5} - 27 \beta_{6} + 72 \beta_{7} ) q^{87} + ( 7168 + 64 \beta_{1} - 64 \beta_{4} ) q^{88} + ( 3557 - 545 \beta_{1} + 97 \beta_{2} + 41 \beta_{3} + 112 \beta_{4} + 6 \beta_{5} - 26 \beta_{6} - 12 \beta_{7} ) q^{89} + ( 3888 - 324 \beta_{1} ) q^{90} + ( 2672 - 53 \beta_{1} - 55 \beta_{2} - 121 \beta_{3} - 22 \beta_{4} + 58 \beta_{5} + 25 \beta_{6} + 73 \beta_{7} ) q^{91} + ( 7568 + 144 \beta_{1} + 48 \beta_{2} + 48 \beta_{3} - 64 \beta_{5} + 16 \beta_{6} - 32 \beta_{7} ) q^{92} + ( 630 - 45 \beta_{1} + 9 \beta_{2} - 27 \beta_{3} + 63 \beta_{4} - 54 \beta_{5} - 9 \beta_{6} + 27 \beta_{7} ) q^{93} + ( 10136 + 364 \beta_{1} + 204 \beta_{2} + 40 \beta_{3} - 8 \beta_{4} - 40 \beta_{5} - 44 \beta_{6} - 4 \beta_{7} ) q^{94} + ( 26015 - 784 \beta_{1} + 169 \beta_{2} - 64 \beta_{3} + 37 \beta_{4} + 177 \beta_{5} + 3 \beta_{6} - 21 \beta_{7} ) q^{95} + 9216 q^{96} + ( 25325 - 9 \beta_{1} + 530 \beta_{2} + 6 \beta_{3} + 15 \beta_{4} + 7 \beta_{5} + 42 \beta_{6} - 74 \beta_{7} ) q^{97} + ( 29788 + 668 \beta_{1} - 180 \beta_{2} - 4 \beta_{3} - 64 \beta_{4} + 56 \beta_{5} + 72 \beta_{6} - 52 \beta_{7} ) q^{98} + ( 9072 + 81 \beta_{1} - 81 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 32q^{2} + 72q^{3} + 128q^{4} + 96q^{5} + 288q^{6} + 181q^{7} + 512q^{8} + 648q^{9} + O(q^{10}) \) \( 8q + 32q^{2} + 72q^{3} + 128q^{4} + 96q^{5} + 288q^{6} + 181q^{7} + 512q^{8} + 648q^{9} + 384q^{10} + 897q^{11} + 1152q^{12} + 1743q^{13} + 724q^{14} + 864q^{15} + 2048q^{16} + 1861q^{17} + 2592q^{18} + 3154q^{19} + 1536q^{20} + 1629q^{21} + 3588q^{22} + 3808q^{23} + 4608q^{24} + 11616q^{25} + 6972q^{26} + 5832q^{27} + 2896q^{28} + 328q^{29} + 3456q^{30} + 570q^{31} + 8192q^{32} + 8073q^{33} + 7444q^{34} + 36086q^{35} + 10368q^{36} + 12777q^{37} + 12616q^{38} + 15687q^{39} + 6144q^{40} + 20167q^{41} + 6516q^{42} + 24579q^{43} + 14352q^{44} + 7776q^{45} + 15232q^{46} + 20490q^{47} + 18432q^{48} + 59391q^{49} + 46464q^{50} + 16749q^{51} + 27888q^{52} + 13404q^{53} + 23328q^{54} - 34588q^{55} + 11584q^{56} + 28386q^{57} + 1312q^{58} + 27848q^{59} + 13824q^{60} + 94944q^{61} + 2280q^{62} + 14661q^{63} + 32768q^{64} + 54560q^{65} + 32292q^{66} + 28838q^{67} + 29776q^{68} + 34272q^{69} + 144344q^{70} + 14983q^{71} + 41472q^{72} + 69384q^{73} + 51108q^{74} + 104544q^{75} + 50464q^{76} - 22359q^{77} + 62748q^{78} - 49199q^{79} + 24576q^{80} + 52488q^{81} + 80668q^{82} + 3995q^{83} + 26064q^{84} - 142290q^{85} + 98316q^{86} + 2952q^{87} + 57408q^{88} + 28722q^{89} + 31104q^{90} + 20815q^{91} + 60928q^{92} + 5130q^{93} + 81960q^{94} + 208010q^{95} + 73728q^{96} + 204150q^{97} + 237564q^{98} + 72657q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 17732 x^{6} - 152272 x^{5} + 93277609 x^{4} + 1554240404 x^{3} - 156444406614 x^{2} - 3720609623076 x + 6279664243680\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(34359867950801 \nu^{7} + 3427298800323263 \nu^{6} - 457847921556604998 \nu^{5} - 51962185507949140911 \nu^{4} + 687275948056748919781 \nu^{3} + 159617109746534403693282 \nu^{2} + 2812334227771541448543442 \nu - 11018649223780601340260160\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-213080717119766 \nu^{7} - 22925387385554253 \nu^{6} + 2715138572817323143 \nu^{5} + 342667369114587819791 \nu^{4} - 2052986798692591539011 \nu^{3} - 1010580638031545315476102 \nu^{2} - 25351147263972518444143062 \nu - 63008708021328146106184740\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-241173404580701 \nu^{7} - 6798393085467258 \nu^{6} + 3382114184691115738 \nu^{5} + 121007091560755332506 \nu^{4} - 9233087863201015175891 \nu^{3} - 394180630611175552345132 \nu^{2} - 2092655704737397057298322 \nu - 40994355136964891412995940\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-86456910449438 \nu^{7} - 11496777086524839 \nu^{6} + 1064444569459333609 \nu^{5} + 172964568397079736153 \nu^{4} - 167611146836501499873 \nu^{3} - 526722639138656562098046 \nu^{2} - 12437140292707217870372146 \nu + 8913845298144052102086080\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-230696835595757 \nu^{7} - 3440941111655301 \nu^{6} + 3434623525975861456 \nu^{5} + 70669592665446123887 \nu^{4} - 11638632472838981936447 \nu^{3} - 254126879321207440569574 \nu^{2} + 5464785054938800208966166 \nu + 18919097225926752618904320\)\()/ \)\(61\!\cdots\!50\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-480161363957732 \nu^{7} - 5901048474325521 \nu^{6} + 6763564348167357631 \nu^{5} + 140257866397284578027 \nu^{4} - 19523425739807739471437 \nu^{3} - 574515111577747804697014 \nu^{2} + 1031438218577234831783526 \nu + 114985354857148133724920520\)\()/ \)\(12\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{7} + \beta_{6} + 3 \beta_{5} + 4 \beta_{4} - \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 4431\)
\(\nu^{3}\)\(=\)\(37 \beta_{7} + 114 \beta_{6} + 122 \beta_{5} - 274 \beta_{4} + 141 \beta_{3} + 640 \beta_{2} + 7088 \beta_{1} + 56889\)
\(\nu^{4}\)\(=\)\(-16896 \beta_{7} + 14543 \beta_{6} + 45544 \beta_{5} + 28672 \beta_{4} - 22748 \beta_{3} + 121050 \beta_{2} + 161470 \beta_{1} + 31946788\)
\(\nu^{5}\)\(=\)\(188865 \beta_{7} + 1915400 \beta_{6} + 2055010 \beta_{5} - 3641590 \beta_{4} + 1402125 \beta_{3} + 9002360 \beta_{2} + 61299226 \beta_{1} + 713420925\)
\(\nu^{6}\)\(=\)\(-146841062 \beta_{7} + 173726981 \beta_{6} + 528049738 \beta_{5} + 209247184 \beta_{4} - 256834176 \beta_{3} + 1380954250 \beta_{2} + 1881124554 \beta_{1} + 279133727996\)
\(\nu^{7}\)\(=\)\(162735387 \beta_{7} + 23261647354 \beta_{6} + 27210948742 \beta_{5} - 39136997954 \beta_{4} + 11725390971 \beta_{3} + 107207780920 \beta_{2} + 584708175498 \beta_{1} + 8575274969179\)

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
104.152
65.5126
65.0121
1.58411
−31.4107
−50.6379
−59.0243
−95.1878
4.00000 9.00000 16.0000 −92.1518 36.0000 −250.043 64.0000 81.0000 −368.607
1.2 4.00000 9.00000 16.0000 −53.5126 36.0000 183.368 64.0000 81.0000 −214.050
1.3 4.00000 9.00000 16.0000 −53.0121 36.0000 −109.285 64.0000 81.0000 −212.049
1.4 4.00000 9.00000 16.0000 10.4159 36.0000 172.112 64.0000 81.0000 41.6636
1.5 4.00000 9.00000 16.0000 43.4107 36.0000 −6.75144 64.0000 81.0000 173.643
1.6 4.00000 9.00000 16.0000 62.6379 36.0000 −69.5352 64.0000 81.0000 250.552
1.7 4.00000 9.00000 16.0000 71.0243 36.0000 223.194 64.0000 81.0000 284.097
1.8 4.00000 9.00000 16.0000 107.188 36.0000 37.9412 64.0000 81.0000 428.751
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(T_{5}^{8} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).