Properties

Label 354.6.a.f
Level 354
Weight 6
Character orbit 354.a
Self dual Yes
Analytic conductor 56.776
Analytic rank 1
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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + ( -8 + \beta_{5} ) q^{5} + 36 q^{6} + ( -18 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})\) \( q -4 q^{2} -9 q^{3} + 16 q^{4} + ( -8 + \beta_{5} ) q^{5} + 36 q^{6} + ( -18 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{7} -64 q^{8} + 81 q^{9} + ( 32 - 4 \beta_{5} ) q^{10} + ( -108 - 3 \beta_{4} + 2 \beta_{5} ) q^{11} -144 q^{12} + ( 106 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} ) q^{13} + ( 72 + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} ) q^{14} + ( 72 - 9 \beta_{5} ) q^{15} + 256 q^{16} + ( 115 + 5 \beta_{1} + \beta_{2} - 10 \beta_{3} - 7 \beta_{4} - 4 \beta_{5} ) q^{17} -324 q^{18} + ( -70 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - 22 \beta_{5} ) q^{19} + ( -128 + 16 \beta_{5} ) q^{20} + ( 162 + 9 \beta_{2} - 9 \beta_{3} - 9 \beta_{5} ) q^{21} + ( 432 + 12 \beta_{4} - 8 \beta_{5} ) q^{22} + ( -573 - 19 \beta_{1} + \beta_{2} + 10 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} ) q^{23} + 576 q^{24} + ( 656 + 10 \beta_{1} + 20 \beta_{2} - 5 \beta_{3} + 15 \beta_{4} - 40 \beta_{5} ) q^{25} + ( -424 + 4 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} - 28 \beta_{4} + 12 \beta_{5} ) q^{26} -729 q^{27} + ( -288 - 16 \beta_{2} + 16 \beta_{3} + 16 \beta_{5} ) q^{28} + ( 261 + 9 \beta_{1} + 23 \beta_{2} + 10 \beta_{3} - 16 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -288 + 36 \beta_{5} ) q^{30} + ( 995 - 24 \beta_{1} - 37 \beta_{2} + 14 \beta_{3} - 26 \beta_{4} - 3 \beta_{5} ) q^{31} -1024 q^{32} + ( 972 + 27 \beta_{4} - 18 \beta_{5} ) q^{33} + ( -460 - 20 \beta_{1} - 4 \beta_{2} + 40 \beta_{3} + 28 \beta_{4} + 16 \beta_{5} ) q^{34} + ( 1374 + 25 \beta_{1} - 31 \beta_{2} - 49 \beta_{3} + 55 \beta_{4} - 57 \beta_{5} ) q^{35} + 1296 q^{36} + ( 6218 + 21 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} - 68 \beta_{4} - 25 \beta_{5} ) q^{37} + ( 280 - 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} + 88 \beta_{5} ) q^{38} + ( -954 + 9 \beta_{1} - 36 \beta_{2} + 18 \beta_{3} - 63 \beta_{4} + 27 \beta_{5} ) q^{39} + ( 512 - 64 \beta_{5} ) q^{40} + ( 2319 + 29 \beta_{1} + 33 \beta_{2} - 52 \beta_{3} + 103 \beta_{4} - 13 \beta_{5} ) q^{41} + ( -648 - 36 \beta_{2} + 36 \beta_{3} + 36 \beta_{5} ) q^{42} + ( 1843 - 80 \beta_{1} - 58 \beta_{2} + 39 \beta_{3} + 4 \beta_{4} + 87 \beta_{5} ) q^{43} + ( -1728 - 48 \beta_{4} + 32 \beta_{5} ) q^{44} + ( -648 + 81 \beta_{5} ) q^{45} + ( 2292 + 76 \beta_{1} - 4 \beta_{2} - 40 \beta_{3} + 16 \beta_{4} + 60 \beta_{5} ) q^{46} + ( 2027 + 11 \beta_{1} + 81 \beta_{2} + 76 \beta_{3} - 26 \beta_{4} + 72 \beta_{5} ) q^{47} -2304 q^{48} + ( 9101 + 80 \beta_{1} + 49 \beta_{2} - 65 \beta_{3} - 26 \beta_{4} + 32 \beta_{5} ) q^{49} + ( -2624 - 40 \beta_{1} - 80 \beta_{2} + 20 \beta_{3} - 60 \beta_{4} + 160 \beta_{5} ) q^{50} + ( -1035 - 45 \beta_{1} - 9 \beta_{2} + 90 \beta_{3} + 63 \beta_{4} + 36 \beta_{5} ) q^{51} + ( 1696 - 16 \beta_{1} + 64 \beta_{2} - 32 \beta_{3} + 112 \beta_{4} - 48 \beta_{5} ) q^{52} + ( 3562 - 90 \beta_{1} + 56 \beta_{2} - 70 \beta_{3} + 54 \beta_{4} - 97 \beta_{5} ) q^{53} + 2916 q^{54} + ( 9759 + 80 \beta_{1} + 100 \beta_{2} - 55 \beta_{3} - 63 \beta_{4} - 286 \beta_{5} ) q^{55} + ( 1152 + 64 \beta_{2} - 64 \beta_{3} - 64 \beta_{5} ) q^{56} + ( 630 - 9 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} - 27 \beta_{4} + 198 \beta_{5} ) q^{57} + ( -1044 - 36 \beta_{1} - 92 \beta_{2} - 40 \beta_{3} + 64 \beta_{4} + 8 \beta_{5} ) q^{58} -3481 q^{59} + ( 1152 - 144 \beta_{5} ) q^{60} + ( -4020 - 226 \beta_{1} - 3 \beta_{2} - 23 \beta_{3} + 177 \beta_{4} + 88 \beta_{5} ) q^{61} + ( -3980 + 96 \beta_{1} + 148 \beta_{2} - 56 \beta_{3} + 104 \beta_{4} + 12 \beta_{5} ) q^{62} + ( -1458 - 81 \beta_{2} + 81 \beta_{3} + 81 \beta_{5} ) q^{63} + 4096 q^{64} + ( -6539 - 151 \beta_{1} + 24 \beta_{2} + 133 \beta_{3} + 77 \beta_{4} + 467 \beta_{5} ) q^{65} + ( -3888 - 108 \beta_{4} + 72 \beta_{5} ) q^{66} + ( 4397 + 376 \beta_{1} + 78 \beta_{2} + 7 \beta_{3} - 255 \beta_{4} - 272 \beta_{5} ) q^{67} + ( 1840 + 80 \beta_{1} + 16 \beta_{2} - 160 \beta_{3} - 112 \beta_{4} - 64 \beta_{5} ) q^{68} + ( 5157 + 171 \beta_{1} - 9 \beta_{2} - 90 \beta_{3} + 36 \beta_{4} + 135 \beta_{5} ) q^{69} + ( -5496 - 100 \beta_{1} + 124 \beta_{2} + 196 \beta_{3} - 220 \beta_{4} + 228 \beta_{5} ) q^{70} + ( -30863 - 337 \beta_{1} - 208 \beta_{2} + 195 \beta_{3} - 104 \beta_{4} - 72 \beta_{5} ) q^{71} -5184 q^{72} + ( -3742 + 54 \beta_{1} - 279 \beta_{2} - 333 \beta_{3} - 83 \beta_{4} - 352 \beta_{5} ) q^{73} + ( -24872 - 84 \beta_{1} + 52 \beta_{2} - 28 \beta_{3} + 272 \beta_{4} + 100 \beta_{5} ) q^{74} + ( -5904 - 90 \beta_{1} - 180 \beta_{2} + 45 \beta_{3} - 135 \beta_{4} + 360 \beta_{5} ) q^{75} + ( -1120 + 16 \beta_{1} - 48 \beta_{2} + 16 \beta_{3} + 48 \beta_{4} - 352 \beta_{5} ) q^{76} + ( -7284 + 239 \beta_{1} - 45 \beta_{2} - 223 \beta_{3} - 166 \beta_{4} - 590 \beta_{5} ) q^{77} + ( 3816 - 36 \beta_{1} + 144 \beta_{2} - 72 \beta_{3} + 252 \beta_{4} - 108 \beta_{5} ) q^{78} + ( -2780 + 159 \beta_{1} + 78 \beta_{2} + 80 \beta_{3} - 149 \beta_{4} - 391 \beta_{5} ) q^{79} + ( -2048 + 256 \beta_{5} ) q^{80} + 6561 q^{81} + ( -9276 - 116 \beta_{1} - 132 \beta_{2} + 208 \beta_{3} - 412 \beta_{4} + 52 \beta_{5} ) q^{82} + ( -7325 + 611 \beta_{1} + 151 \beta_{2} - 116 \beta_{3} - 263 \beta_{4} - 189 \beta_{5} ) q^{83} + ( 2592 + 144 \beta_{2} - 144 \beta_{3} - 144 \beta_{5} ) q^{84} + ( -5134 - 280 \beta_{1} + 11 \beta_{2} + 705 \beta_{3} - 617 \beta_{4} + 137 \beta_{5} ) q^{85} + ( -7372 + 320 \beta_{1} + 232 \beta_{2} - 156 \beta_{3} - 16 \beta_{4} - 348 \beta_{5} ) q^{86} + ( -2349 - 81 \beta_{1} - 207 \beta_{2} - 90 \beta_{3} + 144 \beta_{4} + 18 \beta_{5} ) q^{87} + ( 6912 + 192 \beta_{4} - 128 \beta_{5} ) q^{88} + ( -29870 + 175 \beta_{1} + 67 \beta_{2} - 375 \beta_{3} + 1021 \beta_{4} - 657 \beta_{5} ) q^{89} + ( 2592 - 324 \beta_{5} ) q^{90} + ( -57683 - 748 \beta_{1} + 3 \beta_{2} + 308 \beta_{3} + 929 \beta_{4} + 356 \beta_{5} ) q^{91} + ( -9168 - 304 \beta_{1} + 16 \beta_{2} + 160 \beta_{3} - 64 \beta_{4} - 240 \beta_{5} ) q^{92} + ( -8955 + 216 \beta_{1} + 333 \beta_{2} - 126 \beta_{3} + 234 \beta_{4} + 27 \beta_{5} ) q^{93} + ( -8108 - 44 \beta_{1} - 324 \beta_{2} - 304 \beta_{3} + 104 \beta_{4} - 288 \beta_{5} ) q^{94} + ( -89057 - 314 \beta_{1} - 673 \beta_{2} + 186 \beta_{3} - 182 \beta_{4} + 756 \beta_{5} ) q^{95} + 9216 q^{96} + ( -36114 - 201 \beta_{1} + 50 \beta_{2} - 218 \beta_{3} + 918 \beta_{4} - 228 \beta_{5} ) q^{97} + ( -36404 - 320 \beta_{1} - 196 \beta_{2} + 260 \beta_{3} + 104 \beta_{4} - 128 \beta_{5} ) q^{98} + ( -8748 - 243 \beta_{4} + 162 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 24q^{2} - 54q^{3} + 96q^{4} - 46q^{5} + 216q^{6} - 103q^{7} - 384q^{8} + 486q^{9} + O(q^{10}) \) \( 6q - 24q^{2} - 54q^{3} + 96q^{4} - 46q^{5} + 216q^{6} - 103q^{7} - 384q^{8} + 486q^{9} + 184q^{10} - 653q^{11} - 864q^{12} + 647q^{13} + 412q^{14} + 414q^{15} + 1536q^{16} + 621q^{17} - 1944q^{18} - 454q^{19} - 736q^{20} + 927q^{21} + 2612q^{22} - 3412q^{23} + 3456q^{24} + 3866q^{25} - 2588q^{26} - 4374q^{27} - 1648q^{28} + 1526q^{29} - 1656q^{30} + 5976q^{31} - 6144q^{32} + 5877q^{33} - 2484q^{34} + 8098q^{35} + 7776q^{36} + 37033q^{37} + 1816q^{38} - 5823q^{39} + 2944q^{40} + 13983q^{41} - 3708q^{42} + 11521q^{43} - 10448q^{44} - 3726q^{45} + 13648q^{46} + 12434q^{47} - 13824q^{48} + 54237q^{49} - 15464q^{50} - 5589q^{51} + 10352q^{52} + 21310q^{53} + 17496q^{54} + 57468q^{55} + 6592q^{56} + 4086q^{57} - 6104q^{58} - 20886q^{59} + 6624q^{60} - 23030q^{61} - 23904q^{62} - 8343q^{63} + 24576q^{64} - 37368q^{65} - 23508q^{66} + 24342q^{67} + 9936q^{68} + 30708q^{69} - 32392q^{70} - 184375q^{71} - 31104q^{72} - 24512q^{73} - 148132q^{74} - 34794q^{75} - 7264q^{76} - 46529q^{77} + 23292q^{78} - 17987q^{79} - 11776q^{80} + 39366q^{81} - 55932q^{82} - 46687q^{83} + 14832q^{84} - 29706q^{85} - 46084q^{86} - 13734q^{87} + 41792q^{88} - 178946q^{89} + 14904q^{90} - 340179q^{91} - 54592q^{92} - 53784q^{93} - 49736q^{94} - 532190q^{95} + 55296q^{96} - 214638q^{97} - 216948q^{98} - 52893q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 6296 x^{4} - 192180 x^{3} - 1919598 x^{2} - 7344954 x - 8433643\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -405091 \nu^{5} + 3994798 \nu^{4} + 2534385203 \nu^{3} + 57421225892 \nu^{2} + 247814195181 \nu + 163231989243 \)\()/ 4974382385 \)
\(\beta_{2}\)\(=\)\((\)\( -544836 \nu^{5} + 4297543 \nu^{4} + 3397439158 \nu^{3} + 84919990542 \nu^{2} + 595939730716 \nu + 1019381753668 \)\()/ 3553130275 \)
\(\beta_{3}\)\(=\)\((\)\( 20471 \nu^{5} - 243763 \nu^{4} - 126842723 \nu^{3} - 2676131527 \nu^{2} - 10304481901 \nu + 9856228127 \)\()/92979110\)
\(\beta_{4}\)\(=\)\((\)\( 13570971 \nu^{5} - 93564873 \nu^{4} - 85051805363 \nu^{3} - 2190963083437 \nu^{2} - 14910972914901 \nu - 21635683855423 \)\()/ 49743823850 \)
\(\beta_{5}\)\(=\)\((\)\( 9653159 \nu^{5} - 80139592 \nu^{4} - 60236074227 \nu^{3} - 1475985997048 \nu^{2} - 9432995734379 \nu - 15137248321417 \)\()/ 24871911925 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{2} + \beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(55 \beta_{5} - 33 \beta_{4} - 21 \beta_{3} + 53 \beta_{2} - 5 \beta_{1} + 6305\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(9677 \beta_{5} - 2925 \beta_{4} - 1527 \beta_{3} + 13804 \beta_{2} + 6200 \beta_{1} + 615391\)\()/6\)
\(\nu^{4}\)\(=\)\(154957 \beta_{5} - 73569 \beta_{4} - 47399 \beta_{3} + 180403 \beta_{2} + 24210 \beta_{1} + 14782871\)
\(\nu^{5}\)\(=\)\((\)\(85915328 \beta_{5} - 32008209 \beta_{4} - 18311427 \beta_{3} + 113285590 \beta_{2} + 39342341 \beta_{1} + 6515874271\)\()/6\)

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
−7.38516
−1.99174
−58.4572
−17.9445
−5.83848
93.6171
−4.00000 −9.00000 16.0000 −94.9298 36.0000 −186.983 −64.0000 81.0000 379.719
1.2 −4.00000 −9.00000 16.0000 −77.5601 36.0000 95.6136 −64.0000 81.0000 310.241
1.3 −4.00000 −9.00000 16.0000 −9.89453 36.0000 −127.898 −64.0000 81.0000 39.5781
1.4 −4.00000 −9.00000 16.0000 17.9373 36.0000 177.055 −64.0000 81.0000 −71.7490
1.5 −4.00000 −9.00000 16.0000 50.4423 36.0000 144.849 −64.0000 81.0000 −201.769
1.6 −4.00000 −9.00000 16.0000 68.0049 36.0000 −205.637 −64.0000 81.0000 −272.020
\(n\): 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 \( T_{5}^{6} + 46 T_{5}^{5} - 10250 T_{5}^{4} - 212480 T_{5}^{3} + 29222465 T_{5}^{2} - 153366726 T_{5} - 4482583056 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).