Properties

Label 354.8.a.b
Level 354
Weight 8
Character orbit 354.a
Self dual Yes
Analytic conductor 110.584
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(110.584299021\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( 33 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + 216 q^{6} + ( -16 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} - 4 \beta_{4} ) q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + ( 33 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + 216 q^{6} + ( -16 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} - 4 \beta_{4} ) q^{7} + 512 q^{8} + 729 q^{9} + ( 264 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} ) q^{10} + ( -3137 - 11 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} + 5 \beta_{4} ) q^{11} + 1728 q^{12} + ( -4382 - 18 \beta_{1} + 51 \beta_{2} + 4 \beta_{3} + 15 \beta_{4} ) q^{13} + ( -128 - 24 \beta_{1} - 8 \beta_{2} + 48 \beta_{3} - 32 \beta_{4} ) q^{14} + ( 891 + 27 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} + 27 \beta_{4} ) q^{15} + 4096 q^{16} + ( -6913 + 38 \beta_{1} + 116 \beta_{2} + 4 \beta_{3} - 145 \beta_{4} ) q^{17} + 5832 q^{18} + ( -8834 + 179 \beta_{1} + 20 \beta_{3} + 123 \beta_{4} ) q^{19} + ( 2112 + 64 \beta_{1} - 64 \beta_{2} - 64 \beta_{3} + 64 \beta_{4} ) q^{20} + ( -432 - 81 \beta_{1} - 27 \beta_{2} + 162 \beta_{3} - 108 \beta_{4} ) q^{21} + ( -25096 - 88 \beta_{1} - 72 \beta_{2} - 120 \beta_{3} + 40 \beta_{4} ) q^{22} + ( -22027 + 18 \beta_{1} + 10 \beta_{2} - 246 \beta_{3} + 17 \beta_{4} ) q^{23} + 13824 q^{24} + ( -34895 + 112 \beta_{1} + 101 \beta_{2} - 59 \beta_{3} - 197 \beta_{4} ) q^{25} + ( -35056 - 144 \beta_{1} + 408 \beta_{2} + 32 \beta_{3} + 120 \beta_{4} ) q^{26} + 19683 q^{27} + ( -1024 - 192 \beta_{1} - 64 \beta_{2} + 384 \beta_{3} - 256 \beta_{4} ) q^{28} + ( -61936 + 187 \beta_{1} + 325 \beta_{2} + 547 \beta_{3} + 516 \beta_{4} ) q^{29} + ( 7128 + 216 \beta_{1} - 216 \beta_{2} - 216 \beta_{3} + 216 \beta_{4} ) q^{30} + ( -55467 - 854 \beta_{1} + 1371 \beta_{2} - 312 \beta_{3} - 552 \beta_{4} ) q^{31} + 32768 q^{32} + ( -84699 - 297 \beta_{1} - 243 \beta_{2} - 405 \beta_{3} + 135 \beta_{4} ) q^{33} + ( -55304 + 304 \beta_{1} + 928 \beta_{2} + 32 \beta_{3} - 1160 \beta_{4} ) q^{34} + ( -152837 - 470 \beta_{1} - 725 \beta_{2} + 157 \beta_{3} + 728 \beta_{4} ) q^{35} + 46656 q^{36} + ( -169790 - 1133 \beta_{1} - 1416 \beta_{2} - 966 \beta_{3} - 347 \beta_{4} ) q^{37} + ( -70672 + 1432 \beta_{1} + 160 \beta_{3} + 984 \beta_{4} ) q^{38} + ( -118314 - 486 \beta_{1} + 1377 \beta_{2} + 108 \beta_{3} + 405 \beta_{4} ) q^{39} + ( 16896 + 512 \beta_{1} - 512 \beta_{2} - 512 \beta_{3} + 512 \beta_{4} ) q^{40} + ( 26422 - 1749 \beta_{1} + 1435 \beta_{2} - 829 \beta_{3} + 1204 \beta_{4} ) q^{41} + ( -3456 - 648 \beta_{1} - 216 \beta_{2} + 1296 \beta_{3} - 864 \beta_{4} ) q^{42} + ( -144192 + 264 \beta_{1} - 5465 \beta_{2} + 2217 \beta_{3} - 269 \beta_{4} ) q^{43} + ( -200768 - 704 \beta_{1} - 576 \beta_{2} - 960 \beta_{3} + 320 \beta_{4} ) q^{44} + ( 24057 + 729 \beta_{1} - 729 \beta_{2} - 729 \beta_{3} + 729 \beta_{4} ) q^{45} + ( -176216 + 144 \beta_{1} + 80 \beta_{2} - 1968 \beta_{3} + 136 \beta_{4} ) q^{46} + ( -72526 + 1381 \beta_{1} + 1523 \beta_{2} + 1277 \beta_{3} - 1660 \beta_{4} ) q^{47} + 110592 q^{48} + ( -41840 + 1408 \beta_{1} + 3446 \beta_{2} - 529 \beta_{3} - 1737 \beta_{4} ) q^{49} + ( -279160 + 896 \beta_{1} + 808 \beta_{2} - 472 \beta_{3} - 1576 \beta_{4} ) q^{50} + ( -186651 + 1026 \beta_{1} + 3132 \beta_{2} + 108 \beta_{3} - 3915 \beta_{4} ) q^{51} + ( -280448 - 1152 \beta_{1} + 3264 \beta_{2} + 256 \beta_{3} + 960 \beta_{4} ) q^{52} + ( 200573 + 877 \beta_{1} - 7139 \beta_{2} + 6481 \beta_{3} - 3385 \beta_{4} ) q^{53} + 157464 q^{54} + ( 16304 - 5814 \beta_{1} + 1971 \beta_{2} + 4281 \beta_{3} - 4295 \beta_{4} ) q^{55} + ( -8192 - 1536 \beta_{1} - 512 \beta_{2} + 3072 \beta_{3} - 2048 \beta_{4} ) q^{56} + ( -238518 + 4833 \beta_{1} + 540 \beta_{3} + 3321 \beta_{4} ) q^{57} + ( -495488 + 1496 \beta_{1} + 2600 \beta_{2} + 4376 \beta_{3} + 4128 \beta_{4} ) q^{58} + 205379 q^{59} + ( 57024 + 1728 \beta_{1} - 1728 \beta_{2} - 1728 \beta_{3} + 1728 \beta_{4} ) q^{60} + ( 20450 + 2379 \beta_{1} - 10441 \beta_{2} + 350 \beta_{3} + 6270 \beta_{4} ) q^{61} + ( -443736 - 6832 \beta_{1} + 10968 \beta_{2} - 2496 \beta_{3} - 4416 \beta_{4} ) q^{62} + ( -11664 - 2187 \beta_{1} - 729 \beta_{2} + 4374 \beta_{3} - 2916 \beta_{4} ) q^{63} + 262144 q^{64} + ( -629589 - 9119 \beta_{1} + 10081 \beta_{2} + 7274 \beta_{3} - 5943 \beta_{4} ) q^{65} + ( -677592 - 2376 \beta_{1} - 1944 \beta_{2} - 3240 \beta_{3} + 1080 \beta_{4} ) q^{66} + ( -1271094 + 8342 \beta_{1} - 6333 \beta_{2} - 4681 \beta_{3} + 311 \beta_{4} ) q^{67} + ( -442432 + 2432 \beta_{1} + 7424 \beta_{2} + 256 \beta_{3} - 9280 \beta_{4} ) q^{68} + ( -594729 + 486 \beta_{1} + 270 \beta_{2} - 6642 \beta_{3} + 459 \beta_{4} ) q^{69} + ( -1222696 - 3760 \beta_{1} - 5800 \beta_{2} + 1256 \beta_{3} + 5824 \beta_{4} ) q^{70} + ( -1783339 - 6497 \beta_{1} + 1397 \beta_{2} + 25666 \beta_{3} - 6951 \beta_{4} ) q^{71} + 373248 q^{72} + ( -968480 - 5387 \beta_{1} - 17897 \beta_{2} - 13610 \beta_{3} + 11042 \beta_{4} ) q^{73} + ( -1358320 - 9064 \beta_{1} - 11328 \beta_{2} - 7728 \beta_{3} - 2776 \beta_{4} ) q^{74} + ( -942165 + 3024 \beta_{1} + 2727 \beta_{2} - 1593 \beta_{3} - 5319 \beta_{4} ) q^{75} + ( -565376 + 11456 \beta_{1} + 1280 \beta_{3} + 7872 \beta_{4} ) q^{76} + ( -435331 + 17966 \beta_{1} - 5035 \beta_{2} - 28263 \beta_{3} + 14556 \beta_{4} ) q^{77} + ( -946512 - 3888 \beta_{1} + 11016 \beta_{2} + 864 \beta_{3} + 3240 \beta_{4} ) q^{78} + ( -547300 + 2968 \beta_{1} + 22783 \beta_{2} - 32918 \beta_{3} + 7153 \beta_{4} ) q^{79} + ( 135168 + 4096 \beta_{1} - 4096 \beta_{2} - 4096 \beta_{3} + 4096 \beta_{4} ) q^{80} + 531441 q^{81} + ( 211376 - 13992 \beta_{1} + 11480 \beta_{2} - 6632 \beta_{3} + 9632 \beta_{4} ) q^{82} + ( -288934 - 7305 \beta_{1} - 7461 \beta_{2} + 14603 \beta_{3} + 74 \beta_{4} ) q^{83} + ( -27648 - 5184 \beta_{1} - 1728 \beta_{2} + 10368 \beta_{3} - 6912 \beta_{4} ) q^{84} + ( -1517545 + 31866 \beta_{1} - 9818 \beta_{2} - 12951 \beta_{3} + 26765 \beta_{4} ) q^{85} + ( -1153536 + 2112 \beta_{1} - 43720 \beta_{2} + 17736 \beta_{3} - 2152 \beta_{4} ) q^{86} + ( -1672272 + 5049 \beta_{1} + 8775 \beta_{2} + 14769 \beta_{3} + 13932 \beta_{4} ) q^{87} + ( -1606144 - 5632 \beta_{1} - 4608 \beta_{2} - 7680 \beta_{3} + 2560 \beta_{4} ) q^{88} + ( -3440151 + 13928 \beta_{1} - 43367 \beta_{2} + 15341 \beta_{3} + 27336 \beta_{4} ) q^{89} + ( 192456 + 5832 \beta_{1} - 5832 \beta_{2} - 5832 \beta_{3} + 5832 \beta_{4} ) q^{90} + ( -187345 + 44228 \beta_{1} - 23395 \beta_{2} - 15290 \beta_{3} + 13998 \beta_{4} ) q^{91} + ( -1409728 + 1152 \beta_{1} + 640 \beta_{2} - 15744 \beta_{3} + 1088 \beta_{4} ) q^{92} + ( -1497609 - 23058 \beta_{1} + 37017 \beta_{2} - 8424 \beta_{3} - 14904 \beta_{4} ) q^{93} + ( -580208 + 11048 \beta_{1} + 12184 \beta_{2} + 10216 \beta_{3} - 13280 \beta_{4} ) q^{94} + ( 3187096 + 8655 \beta_{1} + 45208 \beta_{2} + 4081 \beta_{3} - 34797 \beta_{4} ) q^{95} + 884736 q^{96} + ( 4186272 - 4930 \beta_{1} + 77603 \beta_{2} - 68630 \beta_{3} - 2943 \beta_{4} ) q^{97} + ( -334720 + 11264 \beta_{1} + 27568 \beta_{2} - 4232 \beta_{3} - 13896 \beta_{4} ) q^{98} + ( -2286873 - 8019 \beta_{1} - 6561 \beta_{2} - 10935 \beta_{3} + 3645 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 40q^{2} + 135q^{3} + 320q^{4} + 164q^{5} + 1080q^{6} - 76q^{7} + 2560q^{8} + 3645q^{9} + O(q^{10}) \) \( 5q + 40q^{2} + 135q^{3} + 320q^{4} + 164q^{5} + 1080q^{6} - 76q^{7} + 2560q^{8} + 3645q^{9} + 1312q^{10} - 15730q^{11} + 8640q^{12} - 21854q^{13} - 608q^{14} + 4428q^{15} + 20480q^{16} - 34548q^{17} + 29160q^{18} - 43828q^{19} + 10496q^{20} - 2052q^{21} - 125840q^{22} - 110582q^{23} + 69120q^{24} - 174577q^{25} - 174832q^{26} + 98415q^{27} - 4864q^{28} - 307558q^{29} + 35424q^{30} - 277994q^{31} + 163840q^{32} - 424710q^{33} - 276384q^{34} - 764338q^{35} + 233280q^{36} - 853778q^{37} - 350624q^{38} - 590058q^{39} + 83968q^{40} + 131342q^{41} - 16416q^{42} - 721996q^{43} - 1006720q^{44} + 119556q^{45} - 884656q^{46} - 358832q^{47} + 552960q^{48} - 207141q^{49} - 1396616q^{50} - 932796q^{51} - 1398656q^{52} + 1006180q^{53} + 787320q^{54} + 81944q^{55} - 38912q^{56} - 1183356q^{57} - 2460464q^{58} + 1026895q^{59} + 283392q^{60} + 101158q^{61} - 2223952q^{62} - 55404q^{63} + 1310720q^{64} - 3138378q^{65} - 3397680q^{66} - 6362512q^{67} - 2211072q^{68} - 2985714q^{69} - 6114704q^{70} - 8877414q^{71} + 1866240q^{72} - 4881862q^{73} - 6830224q^{74} - 4713579q^{75} - 2804992q^{76} - 2205694q^{77} - 4720464q^{78} - 2769432q^{79} + 671744q^{80} + 2657205q^{81} + 1050736q^{82} - 1430156q^{83} - 131328q^{84} - 7564814q^{85} - 5775968q^{86} - 8304066q^{87} - 8053760q^{88} - 17172176q^{89} + 956448q^{90} - 932474q^{91} - 7077248q^{92} - 7505838q^{93} - 2870656q^{94} + 15962708q^{95} + 4423680q^{96} + 20863830q^{97} - 1657128q^{98} - 11467170q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6162 x^{3} - 12837 x^{2} + 3760259 x - 17264060\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3699 \nu^{4} + 275360 \nu^{3} - 19113038 \nu^{2} - 988069541 \nu - 897360330 \)\()/ 200090590 \)
\(\beta_{2}\)\(=\)\((\)\( -1091 \nu^{4} - 3940 \nu^{3} + 4818162 \nu^{2} + 136255669 \nu + 1457170680 \)\()/57168740\)
\(\beta_{3}\)\(=\)\((\)\( -1969 \nu^{4} + 123890 \nu^{3} + 7307048 \nu^{2} - 617411699 \nu + 4751462510 \)\()/ 100045295 \)
\(\beta_{4}\)\(=\)\((\)\( -58781 \nu^{4} - 264680 \nu^{3} + 340185062 \nu^{2} + 2644241339 \nu - 119148170200 \)\()/ 400181180 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(10 \beta_{4} - 20 \beta_{3} - 36 \beta_{2} + 21 \beta_{1} + 4939\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(159 \beta_{4} - 1322 \beta_{3} + 3655 \beta_{2} + 3629 \beta_{1} + 33239\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(43780 \beta_{4} - 126773 \beta_{3} - 189326 \beta_{2} + 125635 \beta_{1} + 24444871\)\()/2\)

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
4.85925
−30.0454
21.6805
−71.6338
76.1395
8.00000 27.0000 64.0000 −301.683 216.000 1135.68 512.000 729.000 −2413.47
1.2 8.00000 27.0000 64.0000 −27.7441 216.000 679.419 512.000 729.000 −221.952
1.3 8.00000 27.0000 64.0000 32.4075 216.000 −798.662 512.000 729.000 259.260
1.4 8.00000 27.0000 64.0000 138.462 216.000 133.721 512.000 729.000 1107.70
1.5 8.00000 27.0000 64.0000 322.557 216.000 −1226.16 512.000 729.000 2580.46
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \( T_{5}^{5} - 164 T_{5}^{4} - 94576 T_{5}^{3} + 14057394 T_{5}^{2} + 22059975 T_{5} - 12114512550 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\).