Properties

Label 354.8.a.d
Level 354
Weight 8
Character orbit 354.a
Self dual Yes
Analytic conductor 110.584
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{7}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( -74 + \beta_{3} ) q^{5} -216 q^{6} + ( -42 + \beta_{1} ) q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})\) \( q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( -74 + \beta_{3} ) q^{5} -216 q^{6} + ( -42 + \beta_{1} ) q^{7} + 512 q^{8} + 729 q^{9} + ( -592 + 8 \beta_{3} ) q^{10} + ( -358 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{11} -1728 q^{12} + ( 141 - 4 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{13} + ( -336 + 8 \beta_{1} ) q^{14} + ( 1998 - 27 \beta_{3} ) q^{15} + 4096 q^{16} + ( -2818 - 10 \beta_{1} + 11 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{17} + 5832 q^{18} + ( -4189 - 10 \beta_{1} - 5 \beta_{2} - 21 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} + 15 \beta_{6} - 14 \beta_{7} ) q^{19} + ( -4736 + 64 \beta_{3} ) q^{20} + ( 1134 - 27 \beta_{1} ) q^{21} + ( -2864 - 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} - 16 \beta_{6} + 24 \beta_{7} ) q^{22} + ( 5170 - 24 \beta_{1} + 31 \beta_{2} - 44 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} - 32 \beta_{7} ) q^{23} -13824 q^{24} + ( 26118 - 11 \beta_{1} + 12 \beta_{2} - 202 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} + 21 \beta_{6} + 34 \beta_{7} ) q^{25} + ( 1128 - 32 \beta_{1} - 56 \beta_{2} - 72 \beta_{3} + 24 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 32 \beta_{7} ) q^{26} -19683 q^{27} + ( -2688 + 64 \beta_{1} ) q^{28} + ( -6044 + 68 \beta_{1} - 65 \beta_{2} - 109 \beta_{3} + 33 \beta_{5} - 31 \beta_{6} + 16 \beta_{7} ) q^{29} + ( 15984 - 216 \beta_{3} ) q^{30} + ( 27157 - 56 \beta_{1} - 115 \beta_{2} - 159 \beta_{3} + 86 \beta_{4} - 35 \beta_{5} + 18 \beta_{6} - 26 \beta_{7} ) q^{31} + 32768 q^{32} + ( 9666 + 27 \beta_{1} - 54 \beta_{2} + 27 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} + 54 \beta_{6} - 81 \beta_{7} ) q^{33} + ( -22544 - 80 \beta_{1} + 88 \beta_{2} - 192 \beta_{3} + 48 \beta_{4} + 16 \beta_{5} + 16 \beta_{6} - 8 \beta_{7} ) q^{34} + ( -21553 - 79 \beta_{1} + 163 \beta_{2} - 328 \beta_{3} + 107 \beta_{4} + 28 \beta_{5} - 146 \beta_{6} + 186 \beta_{7} ) q^{35} + 46656 q^{36} + ( -9817 - 7 \beta_{1} - 218 \beta_{2} - 49 \beta_{3} - 194 \beta_{4} - 56 \beta_{5} - 47 \beta_{6} + 62 \beta_{7} ) q^{37} + ( -33512 - 80 \beta_{1} - 40 \beta_{2} - 168 \beta_{3} - 160 \beta_{4} - 32 \beta_{5} + 120 \beta_{6} - 112 \beta_{7} ) q^{38} + ( -3807 + 108 \beta_{1} + 189 \beta_{2} + 243 \beta_{3} - 81 \beta_{4} + 54 \beta_{5} - 54 \beta_{6} + 108 \beta_{7} ) q^{39} + ( -37888 + 512 \beta_{3} ) q^{40} + ( -55788 + 57 \beta_{1} + 411 \beta_{2} + 279 \beta_{3} + 45 \beta_{4} + 108 \beta_{5} - 149 \beta_{6} + 215 \beta_{7} ) q^{41} + ( 9072 - 216 \beta_{1} ) q^{42} + ( -58929 - 19 \beta_{1} - 426 \beta_{2} + 88 \beta_{3} - 59 \beta_{4} + 144 \beta_{5} - 22 \beta_{6} - 359 \beta_{7} ) q^{43} + ( -22912 - 64 \beta_{1} + 128 \beta_{2} - 64 \beta_{3} - 64 \beta_{4} + 64 \beta_{5} - 128 \beta_{6} + 192 \beta_{7} ) q^{44} + ( -53946 + 729 \beta_{3} ) q^{45} + ( 41360 - 192 \beta_{1} + 248 \beta_{2} - 352 \beta_{3} - 32 \beta_{4} - 72 \beta_{5} - 72 \beta_{6} - 256 \beta_{7} ) q^{46} + ( 234704 + 351 \beta_{1} + 115 \beta_{2} + 37 \beta_{3} - 137 \beta_{4} - 73 \beta_{5} + 210 \beta_{6} + 568 \beta_{7} ) q^{47} -110592 q^{48} + ( 208685 - 102 \beta_{1} + 341 \beta_{2} - 283 \beta_{3} + 300 \beta_{4} - 409 \beta_{5} + 157 \beta_{6} - 297 \beta_{7} ) q^{49} + ( 208944 - 88 \beta_{1} + 96 \beta_{2} - 1616 \beta_{3} + 24 \beta_{4} + 56 \beta_{5} + 168 \beta_{6} + 272 \beta_{7} ) q^{50} + ( 76086 + 270 \beta_{1} - 297 \beta_{2} + 648 \beta_{3} - 162 \beta_{4} - 54 \beta_{5} - 54 \beta_{6} + 27 \beta_{7} ) q^{51} + ( 9024 - 256 \beta_{1} - 448 \beta_{2} - 576 \beta_{3} + 192 \beta_{4} - 128 \beta_{5} + 128 \beta_{6} - 256 \beta_{7} ) q^{52} + ( -144348 - 10 \beta_{1} + 336 \beta_{2} - 803 \beta_{3} - 322 \beta_{4} + 334 \beta_{5} - 186 \beta_{6} - 648 \beta_{7} ) q^{53} -157464 q^{54} + ( 14593 + 919 \beta_{1} - 498 \beta_{2} + 658 \beta_{3} + 263 \beta_{4} + 77 \beta_{5} + 791 \beta_{6} - 286 \beta_{7} ) q^{55} + ( -21504 + 512 \beta_{1} ) q^{56} + ( 113103 + 270 \beta_{1} + 135 \beta_{2} + 567 \beta_{3} + 540 \beta_{4} + 108 \beta_{5} - 405 \beta_{6} + 378 \beta_{7} ) q^{57} + ( -48352 + 544 \beta_{1} - 520 \beta_{2} - 872 \beta_{3} + 264 \beta_{5} - 248 \beta_{6} + 128 \beta_{7} ) q^{58} -205379 q^{59} + ( 127872 - 1728 \beta_{3} ) q^{60} + ( -1136050 + 1296 \beta_{1} + 819 \beta_{2} + 430 \beta_{3} - 270 \beta_{4} - 96 \beta_{5} + 32 \beta_{6} + 258 \beta_{7} ) q^{61} + ( 217256 - 448 \beta_{1} - 920 \beta_{2} - 1272 \beta_{3} + 688 \beta_{4} - 280 \beta_{5} + 144 \beta_{6} - 208 \beta_{7} ) q^{62} + ( -30618 + 729 \beta_{1} ) q^{63} + 262144 q^{64} + ( -840808 - 247 \beta_{1} - 646 \beta_{2} + 622 \beta_{3} - 1449 \beta_{4} - 351 \beta_{5} - 328 \beta_{6} - 1002 \beta_{7} ) q^{65} + ( 77328 + 216 \beta_{1} - 432 \beta_{2} + 216 \beta_{3} + 216 \beta_{4} - 216 \beta_{5} + 432 \beta_{6} - 648 \beta_{7} ) q^{66} + ( -1068779 + 603 \beta_{1} + 226 \beta_{2} - 896 \beta_{3} - 47 \beta_{4} + 387 \beta_{5} + 363 \beta_{6} + 348 \beta_{7} ) q^{67} + ( -180352 - 640 \beta_{1} + 704 \beta_{2} - 1536 \beta_{3} + 384 \beta_{4} + 128 \beta_{5} + 128 \beta_{6} - 64 \beta_{7} ) q^{68} + ( -139590 + 648 \beta_{1} - 837 \beta_{2} + 1188 \beta_{3} + 108 \beta_{4} + 243 \beta_{5} + 243 \beta_{6} + 864 \beta_{7} ) q^{69} + ( -172424 - 632 \beta_{1} + 1304 \beta_{2} - 2624 \beta_{3} + 856 \beta_{4} + 224 \beta_{5} - 1168 \beta_{6} + 1488 \beta_{7} ) q^{70} + ( -729450 - 2152 \beta_{1} + 815 \beta_{2} + 532 \beta_{3} + 727 \beta_{4} + 39 \beta_{5} + 58 \beta_{6} - 248 \beta_{7} ) q^{71} + 373248 q^{72} + ( -953882 - 185 \beta_{1} + 989 \beta_{2} + 1666 \beta_{3} + 839 \beta_{4} - 1632 \beta_{5} + 719 \beta_{6} - 1198 \beta_{7} ) q^{73} + ( -78536 - 56 \beta_{1} - 1744 \beta_{2} - 392 \beta_{3} - 1552 \beta_{4} - 448 \beta_{5} - 376 \beta_{6} + 496 \beta_{7} ) q^{74} + ( -705186 + 297 \beta_{1} - 324 \beta_{2} + 5454 \beta_{3} - 81 \beta_{4} - 189 \beta_{5} - 567 \beta_{6} - 918 \beta_{7} ) q^{75} + ( -268096 - 640 \beta_{1} - 320 \beta_{2} - 1344 \beta_{3} - 1280 \beta_{4} - 256 \beta_{5} + 960 \beta_{6} - 896 \beta_{7} ) q^{76} + ( -2148407 - 1551 \beta_{1} - 734 \beta_{2} + 6552 \beta_{3} - 462 \beta_{4} + 194 \beta_{5} - 277 \beta_{6} + 186 \beta_{7} ) q^{77} + ( -30456 + 864 \beta_{1} + 1512 \beta_{2} + 1944 \beta_{3} - 648 \beta_{4} + 432 \beta_{5} - 432 \beta_{6} + 864 \beta_{7} ) q^{78} + ( -1578315 - 1175 \beta_{1} - 1807 \beta_{2} - 213 \beta_{3} + 2410 \beta_{4} + 420 \beta_{5} - 917 \beta_{6} + 322 \beta_{7} ) q^{79} + ( -303104 + 4096 \beta_{3} ) q^{80} + 531441 q^{81} + ( -446304 + 456 \beta_{1} + 3288 \beta_{2} + 2232 \beta_{3} + 360 \beta_{4} + 864 \beta_{5} - 1192 \beta_{6} + 1720 \beta_{7} ) q^{82} + ( -2394358 - 859 \beta_{1} - 1567 \beta_{2} + 12945 \beta_{3} + 1535 \beta_{4} - 506 \beta_{5} + 719 \beta_{6} + 2629 \beta_{7} ) q^{83} + ( 72576 - 1728 \beta_{1} ) q^{84} + ( -2515094 + 1980 \beta_{1} - 585 \beta_{2} + 5721 \beta_{3} - 3140 \beta_{4} + 920 \beta_{5} - 920 \beta_{6} - 2010 \beta_{7} ) q^{85} + ( -471432 - 152 \beta_{1} - 3408 \beta_{2} + 704 \beta_{3} - 472 \beta_{4} + 1152 \beta_{5} - 176 \beta_{6} - 2872 \beta_{7} ) q^{86} + ( 163188 - 1836 \beta_{1} + 1755 \beta_{2} + 2943 \beta_{3} - 891 \beta_{5} + 837 \beta_{6} - 432 \beta_{7} ) q^{87} + ( -183296 - 512 \beta_{1} + 1024 \beta_{2} - 512 \beta_{3} - 512 \beta_{4} + 512 \beta_{5} - 1024 \beta_{6} + 1536 \beta_{7} ) q^{88} + ( -2006665 - 1615 \beta_{1} + 671 \beta_{2} + 10320 \beta_{3} + 95 \beta_{4} + 1318 \beta_{5} + 2664 \beta_{6} + 414 \beta_{7} ) q^{89} + ( -431568 + 5832 \beta_{3} ) q^{90} + ( -2511525 + 1355 \beta_{1} - 4885 \beta_{2} - 831 \beta_{3} - 2419 \beta_{4} + 2370 \beta_{5} + 226 \beta_{6} - 1647 \beta_{7} ) q^{91} + ( 330880 - 1536 \beta_{1} + 1984 \beta_{2} - 2816 \beta_{3} - 256 \beta_{4} - 576 \beta_{5} - 576 \beta_{6} - 2048 \beta_{7} ) q^{92} + ( -733239 + 1512 \beta_{1} + 3105 \beta_{2} + 4293 \beta_{3} - 2322 \beta_{4} + 945 \beta_{5} - 486 \beta_{6} + 702 \beta_{7} ) q^{93} + ( 1877632 + 2808 \beta_{1} + 920 \beta_{2} + 296 \beta_{3} - 1096 \beta_{4} - 584 \beta_{5} + 1680 \beta_{6} + 4544 \beta_{7} ) q^{94} + ( -1019677 - 9458 \beta_{1} + 271 \beta_{2} + 2848 \beta_{3} + 4594 \beta_{4} - 3729 \beta_{5} - 942 \beta_{6} - 2218 \beta_{7} ) q^{95} -884736 q^{96} + ( -245071 + 3977 \beta_{1} + 2014 \beta_{2} - 15211 \beta_{3} - 2629 \beta_{4} + 994 \beta_{5} - 238 \beta_{6} + 8082 \beta_{7} ) q^{97} + ( 1669480 - 816 \beta_{1} + 2728 \beta_{2} - 2264 \beta_{3} + 2400 \beta_{4} - 3272 \beta_{5} + 1256 \beta_{6} - 2376 \beta_{7} ) q^{98} + ( -260982 - 729 \beta_{1} + 1458 \beta_{2} - 729 \beta_{3} - 729 \beta_{4} + 729 \beta_{5} - 1458 \beta_{6} + 2187 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 64q^{2} - 216q^{3} + 512q^{4} - 592q^{5} - 1728q^{6} - 340q^{7} + 4096q^{8} + 5832q^{9} + O(q^{10}) \) \( 8q + 64q^{2} - 216q^{3} + 512q^{4} - 592q^{5} - 1728q^{6} - 340q^{7} + 4096q^{8} + 5832q^{9} - 4736q^{10} - 2852q^{11} - 13824q^{12} + 1142q^{13} - 2720q^{14} + 15984q^{15} + 32768q^{16} - 22528q^{17} + 46656q^{18} - 33528q^{19} - 37888q^{20} + 9180q^{21} - 22816q^{22} + 41330q^{23} - 110592q^{24} + 209004q^{25} + 9136q^{26} - 157464q^{27} - 21760q^{28} - 48334q^{29} + 127872q^{30} + 217552q^{31} + 262144q^{32} + 77004q^{33} - 180224q^{34} - 171714q^{35} + 373248q^{36} - 77966q^{37} - 268224q^{38} - 30834q^{39} - 303104q^{40} - 446410q^{41} + 73440q^{42} - 470890q^{43} - 182528q^{44} - 431568q^{45} + 330640q^{46} + 1876568q^{47} - 884736q^{48} + 1667480q^{49} + 1672032q^{50} + 608256q^{51} + 73088q^{52} - 1155672q^{53} - 1259712q^{54} + 112064q^{55} - 174080q^{56} + 905256q^{57} - 386672q^{58} - 1643032q^{59} + 1022976q^{60} - 9094962q^{61} + 1740416q^{62} - 247860q^{63} + 2097152q^{64} - 6726234q^{65} + 616032q^{66} - 8552352q^{67} - 1441792q^{68} - 1115910q^{69} - 1373712q^{70} - 5829156q^{71} + 2985984q^{72} - 7639392q^{73} - 623728q^{74} - 5643108q^{75} - 2145792q^{76} - 17178270q^{77} - 246672q^{78} - 12614888q^{79} - 2424832q^{80} + 4251528q^{81} - 3571280q^{82} - 19145486q^{83} + 587520q^{84} - 20127842q^{85} - 3767120q^{86} + 1305018q^{87} - 1460224q^{88} - 16050066q^{89} - 3452544q^{90} - 20086856q^{91} + 2645120q^{92} - 5873904q^{93} + 15012544q^{94} - 8130136q^{95} - 7077888q^{96} - 1961876q^{97} + 13339840q^{98} - 2079108q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} - 17055507548256 x^{2} + 1662873307981032 x - 22179668074797840\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-403229322236707491211437818 \nu^{7} - 137967969456332414968902702557 \nu^{6} + 32272765742947592225785536661693 \nu^{5} + 9211362087496075255330491435177414 \nu^{4} - 1020553945814872731656206822126758163 \nu^{3} - 90014707113837876716775395861377686669 \nu^{2} + 9654151949266513610090445212328685886916 \nu - 356947010208639585264465995259988420923660\)\()/ \)\(18\!\cdots\!40\)\( \)
\(\beta_{2}\)\(=\)\((\)\(1097246569602563098146271901 \nu^{7} + 46107419100357503772032346707 \nu^{6} - 93887852792777244489516219220270 \nu^{5} + 1714050501852531549663212177849847 \nu^{4} + 1274385175313517237719367616634058697 \nu^{3} - 23880945938076077096670258052075750626 \nu^{2} - 611723712198081695672585787775968028320 \nu - 136300644436574650780228149573810545177760\)\()/ \)\(36\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(170108787082024923392891450 \nu^{7} + 14559238687147218040433102567 \nu^{6} - 15599687644076308419067191165577 \nu^{5} - 340057048169918736899478202025880 \nu^{4} + 340551898521674518832096786900065501 \nu^{3} - 1408668066647813515765449058104500175 \nu^{2} - 2099592624087832717873083692618986820238 \nu + 46495488789985667734566872416994794265256\)\()/ \)\(36\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-115498339258088308799016755 \nu^{7} - 18948739454581144940582807593 \nu^{6} + 12181131388669060994089749674350 \nu^{5} + 1136515713627213684759392232462679 \nu^{4} - 412117420072470122290629244631729171 \nu^{3} - 8973233723056551560865872788546296102 \nu^{2} + 4136104910966556997345478781124427369736 \nu - 95719710011998894723211837096027785717944\)\()/ \)\(24\!\cdots\!92\)\( \)
\(\beta_{5}\)\(=\)\((\)\(290990153212056172452479519 \nu^{7} + 79381498061657495434909325938 \nu^{6} - 26980938552899032571868666894055 \nu^{5} - 5611263382248124484752450453880977 \nu^{4} + 867188064622670732381583192381771398 \nu^{3} + 77769673790034588080518340869634168751 \nu^{2} - 7429229784256484293180525660533457104990 \nu - 68047200429759461243445824269030483673340\)\()/ \)\(61\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-712321044445775335554949789 \nu^{7} - 122501458831997685281243780521 \nu^{6} + 57185355372024056048534169134264 \nu^{5} + 6202885414457031247485712190533737 \nu^{4} - 1154950741736555661330307161449060759 \nu^{3} - 69537976025533733663762070617168056212 \nu^{2} + 7746165491601975485215549214247873591168 \nu + 10939378279915137457448417913880118449830\)\()/ \)\(92\!\cdots\!70\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-125957929380825480203012016 \nu^{7} - 10082856090107755136333445389 \nu^{6} + 11540722416899036972387132484165 \nu^{5} + 229570134912176440702599021209146 \nu^{4} - 243916127654407357669640540651106643 \nu^{3} - 842600631972009141505704822501743913 \nu^{2} + 1394845953596624549552085023962955881086 \nu - 11629851154660543572706887776244801561288\)\()/ \)\(12\!\cdots\!96\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-14 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 16 \beta_{3} - 12 \beta_{2} - 8 \beta_{1} + 9\)\()/90\)
\(\nu^{2}\)\(=\)\((\)\(6044 \beta_{7} - 1328 \beta_{6} + 2447 \beta_{5} + 566 \beta_{4} + 11711 \beta_{3} + 332 \beta_{2} + 5608 \beta_{1} + 6990631\)\()/270\)
\(\nu^{3}\)\(=\)\((\)\(-913904 \beta_{7} + 73783 \beta_{6} - 234227 \beta_{5} + 69334 \beta_{4} - 1407711 \beta_{3} - 623212 \beta_{2} - 451788 \beta_{1} - 192529676\)\()/90\)
\(\nu^{4}\)\(=\)\((\)\(651379252 \beta_{7} - 90115579 \beta_{6} + 214880806 \beta_{5} + 48289978 \beta_{4} + 1266219118 \beta_{3} + 114457786 \beta_{2} + 454356644 \beta_{1} + 377657536253\)\()/270\)
\(\nu^{5}\)\(=\)\((\)\(-223698357058 \beta_{7} + 14051070376 \beta_{6} - 66076732819 \beta_{5} + 2795744528 \beta_{4} - 381538523497 \beta_{3} - 119517756214 \beta_{2} - 117946120736 \beta_{1} - 66919763035637\)\()/270\)
\(\nu^{6}\)\(=\)\((\)\(12278943963646 \beta_{7} - 1362318816295 \beta_{6} + 3886489516477 \beta_{5} + 679818370000 \beta_{4} + 23193294403759 \beta_{3} + 3332439653182 \beta_{2} + 7689018218300 \beta_{1} + 5593246244731040\)\()/54\)
\(\nu^{7}\)\(=\)\((\)\(-19446099765027946 \beta_{7} + 1348338903513757 \beta_{6} - 5940184021424248 \beta_{5} - 201618974476444 \beta_{4} - 34365025060747174 \beta_{3} - 8856191398367488 \beta_{2} - 10734691494092252 \beta_{1} - 6634694907685724219\)\()/270\)

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
77.9355
137.528
−124.917
−135.103
211.499
116.839
16.5752
−299.356
8.00000 −27.0000 64.0000 −497.674 −216.000 −1339.98 512.000 729.000 −3981.39
1.2 8.00000 −27.0000 64.0000 −490.703 −216.000 975.190 512.000 729.000 −3925.63
1.3 8.00000 −27.0000 64.0000 −365.356 −216.000 −311.570 512.000 729.000 −2922.85
1.4 8.00000 −27.0000 64.0000 −11.9651 −216.000 1695.87 512.000 729.000 −95.7205
1.5 8.00000 −27.0000 64.0000 −6.85357 −216.000 554.074 512.000 729.000 −54.8286
1.6 8.00000 −27.0000 64.0000 188.524 −216.000 −784.370 512.000 729.000 1508.19
1.7 8.00000 −27.0000 64.0000 272.880 −216.000 −1263.34 512.000 729.000 2183.04
1.8 8.00000 −27.0000 64.0000 319.148 −216.000 134.129 512.000 729.000 2553.18
\(n\): 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\).