Properties

Label 354.8.a.c
Level 354
Weight 8
Character orbit 354.a
Self dual Yes
Analytic conductor 110.584
Analytic rank 1
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -23 - \beta_{2} ) q^{5} -216 q^{6} + ( -83 - \beta_{5} ) q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -23 - \beta_{2} ) q^{5} -216 q^{6} + ( -83 - \beta_{5} ) q^{7} -512 q^{8} + 729 q^{9} + ( 184 + 8 \beta_{2} ) q^{10} + ( -311 + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{11} + 1728 q^{12} + ( -1206 + 6 \beta_{1} + 5 \beta_{2} + \beta_{3} - 3 \beta_{5} - 9 \beta_{6} ) q^{13} + ( 664 + 8 \beta_{5} ) q^{14} + ( -621 - 27 \beta_{2} ) q^{15} + 4096 q^{16} + ( -344 - 15 \beta_{1} - \beta_{2} - 23 \beta_{3} - 10 \beta_{4} - 21 \beta_{5} + 13 \beta_{6} ) q^{17} -5832 q^{18} + ( -5292 - 18 \beta_{1} - 37 \beta_{2} - 29 \beta_{4} + 35 \beta_{5} + 19 \beta_{6} ) q^{19} + ( -1472 - 64 \beta_{2} ) q^{20} + ( -2241 - 27 \beta_{5} ) q^{21} + ( 2488 - 32 \beta_{2} - 24 \beta_{3} - 24 \beta_{4} - 24 \beta_{5} - 24 \beta_{6} ) q^{22} + ( 14213 - 7 \beta_{1} + 110 \beta_{2} + 19 \beta_{3} + 2 \beta_{4} + 27 \beta_{5} - 90 \beta_{6} ) q^{23} -13824 q^{24} + ( 14438 + 25 \beta_{1} + 36 \beta_{2} - 48 \beta_{3} + 148 \beta_{4} + 67 \beta_{5} + 29 \beta_{6} ) q^{25} + ( 9648 - 48 \beta_{1} - 40 \beta_{2} - 8 \beta_{3} + 24 \beta_{5} + 72 \beta_{6} ) q^{26} + 19683 q^{27} + ( -5312 - 64 \beta_{5} ) q^{28} + ( 418 + 235 \beta_{1} + 149 \beta_{2} + 55 \beta_{3} - 134 \beta_{4} + 91 \beta_{5} + 6 \beta_{6} ) q^{29} + ( 4968 + 216 \beta_{2} ) q^{30} + ( -8000 - 175 \beta_{1} + 190 \beta_{2} + 106 \beta_{3} - 272 \beta_{4} - 12 \beta_{5} + 60 \beta_{6} ) q^{31} -32768 q^{32} + ( -8397 + 108 \beta_{2} + 81 \beta_{3} + 81 \beta_{4} + 81 \beta_{5} + 81 \beta_{6} ) q^{33} + ( 2752 + 120 \beta_{1} + 8 \beta_{2} + 184 \beta_{3} + 80 \beta_{4} + 168 \beta_{5} - 104 \beta_{6} ) q^{34} + ( 27276 - 138 \beta_{1} + 360 \beta_{2} - 67 \beta_{3} + 154 \beta_{4} - 56 \beta_{5} - 167 \beta_{6} ) q^{35} + 46656 q^{36} + ( -728 + 80 \beta_{1} + 544 \beta_{2} - 107 \beta_{3} + 256 \beta_{4} + 226 \beta_{5} + 254 \beta_{6} ) q^{37} + ( 42336 + 144 \beta_{1} + 296 \beta_{2} + 232 \beta_{4} - 280 \beta_{5} - 152 \beta_{6} ) q^{38} + ( -32562 + 162 \beta_{1} + 135 \beta_{2} + 27 \beta_{3} - 81 \beta_{5} - 243 \beta_{6} ) q^{39} + ( 11776 + 512 \beta_{2} ) q^{40} + ( -45576 + 265 \beta_{1} + 928 \beta_{2} - 430 \beta_{3} + 171 \beta_{4} + 132 \beta_{5} - 163 \beta_{6} ) q^{41} + ( 17928 + 216 \beta_{5} ) q^{42} + ( -90332 - 635 \beta_{1} + 123 \beta_{2} - 8 \beta_{3} + 92 \beta_{4} - 63 \beta_{5} - 4 \beta_{6} ) q^{43} + ( -19904 + 256 \beta_{2} + 192 \beta_{3} + 192 \beta_{4} + 192 \beta_{5} + 192 \beta_{6} ) q^{44} + ( -16767 - 729 \beta_{2} ) q^{45} + ( -113704 + 56 \beta_{1} - 880 \beta_{2} - 152 \beta_{3} - 16 \beta_{4} - 216 \beta_{5} + 720 \beta_{6} ) q^{46} + ( -231299 - 579 \beta_{1} + 1105 \beta_{2} + 728 \beta_{3} - 723 \beta_{4} - 186 \beta_{5} - 212 \beta_{6} ) q^{47} + 110592 q^{48} + ( -549612 + 42 \beta_{1} + 223 \beta_{2} - 167 \beta_{3} + 309 \beta_{4} + 36 \beta_{5} - 132 \beta_{6} ) q^{49} + ( -115504 - 200 \beta_{1} - 288 \beta_{2} + 384 \beta_{3} - 1184 \beta_{4} - 536 \beta_{5} - 232 \beta_{6} ) q^{50} + ( -9288 - 405 \beta_{1} - 27 \beta_{2} - 621 \beta_{3} - 270 \beta_{4} - 567 \beta_{5} + 351 \beta_{6} ) q^{51} + ( -77184 + 384 \beta_{1} + 320 \beta_{2} + 64 \beta_{3} - 192 \beta_{5} - 576 \beta_{6} ) q^{52} + ( -173091 + 530 \beta_{1} + 1007 \beta_{2} - 294 \beta_{3} - 678 \beta_{4} - 1440 \beta_{5} + 510 \beta_{6} ) q^{53} -157464 q^{54} + ( -475817 + 329 \beta_{1} - 1868 \beta_{2} + 90 \beta_{3} - 946 \beta_{4} + 1007 \beta_{5} + 979 \beta_{6} ) q^{55} + ( 42496 + 512 \beta_{5} ) q^{56} + ( -142884 - 486 \beta_{1} - 999 \beta_{2} - 783 \beta_{4} + 945 \beta_{5} + 513 \beta_{6} ) q^{57} + ( -3344 - 1880 \beta_{1} - 1192 \beta_{2} - 440 \beta_{3} + 1072 \beta_{4} - 728 \beta_{5} - 48 \beta_{6} ) q^{58} -205379 q^{59} + ( -39744 - 1728 \beta_{2} ) q^{60} + ( -454953 + 414 \beta_{1} - 1477 \beta_{2} + 3 \beta_{3} + 1271 \beta_{4} - 1258 \beta_{5} + 1069 \beta_{6} ) q^{61} + ( 64000 + 1400 \beta_{1} - 1520 \beta_{2} - 848 \beta_{3} + 2176 \beta_{4} + 96 \beta_{5} - 480 \beta_{6} ) q^{62} + ( -60507 - 729 \beta_{5} ) q^{63} + 262144 q^{64} + ( 77461 + 551 \beta_{1} + 2361 \beta_{2} - 1331 \beta_{3} + 1232 \beta_{4} - 978 \beta_{5} - 461 \beta_{6} ) q^{65} + ( 67176 - 864 \beta_{2} - 648 \beta_{3} - 648 \beta_{4} - 648 \beta_{5} - 648 \beta_{6} ) q^{66} + ( -766799 + 3389 \beta_{1} - 1848 \beta_{2} + 642 \beta_{3} - 854 \beta_{4} + 5145 \beta_{5} - 3091 \beta_{6} ) q^{67} + ( -22016 - 960 \beta_{1} - 64 \beta_{2} - 1472 \beta_{3} - 640 \beta_{4} - 1344 \beta_{5} + 832 \beta_{6} ) q^{68} + ( 383751 - 189 \beta_{1} + 2970 \beta_{2} + 513 \beta_{3} + 54 \beta_{4} + 729 \beta_{5} - 2430 \beta_{6} ) q^{69} + ( -218208 + 1104 \beta_{1} - 2880 \beta_{2} + 536 \beta_{3} - 1232 \beta_{4} + 448 \beta_{5} + 1336 \beta_{6} ) q^{70} + ( 251083 - 2807 \beta_{1} + 3352 \beta_{2} + 2876 \beta_{3} + 3705 \beta_{4} - 4289 \beta_{5} - 3910 \beta_{6} ) q^{71} -373248 q^{72} + ( -266786 + 2408 \beta_{1} - 4327 \beta_{2} + 1012 \beta_{3} + 3532 \beta_{4} - 2215 \beta_{5} - 751 \beta_{6} ) q^{73} + ( 5824 - 640 \beta_{1} - 4352 \beta_{2} + 856 \beta_{3} - 2048 \beta_{4} - 1808 \beta_{5} - 2032 \beta_{6} ) q^{74} + ( 389826 + 675 \beta_{1} + 972 \beta_{2} - 1296 \beta_{3} + 3996 \beta_{4} + 1809 \beta_{5} + 783 \beta_{6} ) q^{75} + ( -338688 - 1152 \beta_{1} - 2368 \beta_{2} - 1856 \beta_{4} + 2240 \beta_{5} + 1216 \beta_{6} ) q^{76} + ( -552811 + 924 \beta_{1} + 1197 \beta_{2} + 1729 \beta_{3} - 4270 \beta_{4} + 1848 \beta_{5} - 28 \beta_{6} ) q^{77} + ( 260496 - 1296 \beta_{1} - 1080 \beta_{2} - 216 \beta_{3} + 648 \beta_{5} + 1944 \beta_{6} ) q^{78} + ( -678965 - 1420 \beta_{1} - 2931 \beta_{2} + 2944 \beta_{3} - 5759 \beta_{4} + 6054 \beta_{5} + 697 \beta_{6} ) q^{79} + ( -94208 - 4096 \beta_{2} ) q^{80} + 531441 q^{81} + ( 364608 - 2120 \beta_{1} - 7424 \beta_{2} + 3440 \beta_{3} - 1368 \beta_{4} - 1056 \beta_{5} + 1304 \beta_{6} ) q^{82} + ( 735618 - 597 \beta_{1} - 4112 \beta_{2} - 5476 \beta_{3} + 4959 \beta_{4} + 3432 \beta_{5} + 10791 \beta_{6} ) q^{83} + ( -143424 - 1728 \beta_{5} ) q^{84} + ( -651438 - 5184 \beta_{1} + 10148 \beta_{2} + 4813 \beta_{3} - 1677 \beta_{4} - 5534 \beta_{5} - 7923 \beta_{6} ) q^{85} + ( 722656 + 5080 \beta_{1} - 984 \beta_{2} + 64 \beta_{3} - 736 \beta_{4} + 504 \beta_{5} + 32 \beta_{6} ) q^{86} + ( 11286 + 6345 \beta_{1} + 4023 \beta_{2} + 1485 \beta_{3} - 3618 \beta_{4} + 2457 \beta_{5} + 162 \beta_{6} ) q^{87} + ( 159232 - 2048 \beta_{2} - 1536 \beta_{3} - 1536 \beta_{4} - 1536 \beta_{5} - 1536 \beta_{6} ) q^{88} + ( 2870376 - 326 \beta_{1} + 2090 \beta_{2} - 7675 \beta_{3} + 820 \beta_{4} + 2632 \beta_{5} + 7971 \beta_{6} ) q^{89} + ( 134136 + 5832 \beta_{2} ) q^{90} + ( 960836 - 1339 \beta_{1} - 741 \beta_{2} + 269 \beta_{3} + 3097 \beta_{4} - 851 \beta_{5} - 1021 \beta_{6} ) q^{91} + ( 909632 - 448 \beta_{1} + 7040 \beta_{2} + 1216 \beta_{3} + 128 \beta_{4} + 1728 \beta_{5} - 5760 \beta_{6} ) q^{92} + ( -216000 - 4725 \beta_{1} + 5130 \beta_{2} + 2862 \beta_{3} - 7344 \beta_{4} - 324 \beta_{5} + 1620 \beta_{6} ) q^{93} + ( 1850392 + 4632 \beta_{1} - 8840 \beta_{2} - 5824 \beta_{3} + 5784 \beta_{4} + 1488 \beta_{5} + 1696 \beta_{6} ) q^{94} + ( 1855133 - 403 \beta_{1} + 4899 \beta_{2} + 4184 \beta_{3} - 2672 \beta_{4} + 6180 \beta_{5} + 5440 \beta_{6} ) q^{95} -884736 q^{96} + ( 881734 + 2734 \beta_{1} - 8820 \beta_{2} - 14364 \beta_{3} + 8601 \beta_{4} - 5576 \beta_{5} + 6704 \beta_{6} ) q^{97} + ( 4396896 - 336 \beta_{1} - 1784 \beta_{2} + 1336 \beta_{3} - 2472 \beta_{4} - 288 \beta_{5} + 1056 \beta_{6} ) q^{98} + ( -226719 + 2916 \beta_{2} + 2187 \beta_{3} + 2187 \beta_{4} + 2187 \beta_{5} + 2187 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 56q^{2} + 189q^{3} + 448q^{4} - 158q^{5} - 1512q^{6} - 581q^{7} - 3584q^{8} + 5103q^{9} + O(q^{10}) \) \( 7q - 56q^{2} + 189q^{3} + 448q^{4} - 158q^{5} - 1512q^{6} - 581q^{7} - 3584q^{8} + 5103q^{9} + 1264q^{10} - 2201q^{11} + 12096q^{12} - 8421q^{13} + 4648q^{14} - 4266q^{15} + 28672q^{16} - 2425q^{17} - 40824q^{18} - 37084q^{19} - 10112q^{20} - 15687q^{21} + 17608q^{22} + 99364q^{23} - 96768q^{24} + 101361q^{25} + 67368q^{26} + 137781q^{27} - 37184q^{28} + 2498q^{29} + 34128q^{30} - 57962q^{31} - 229376q^{32} - 59427q^{33} + 19400q^{34} + 190586q^{35} + 326592q^{36} - 6497q^{37} + 296672q^{38} - 227367q^{39} + 80896q^{40} - 319165q^{41} + 125496q^{42} - 633743q^{43} - 140864q^{44} - 115182q^{45} - 794912q^{46} - 1626560q^{47} + 774144q^{48} - 3846354q^{49} - 810888q^{50} - 65475q^{51} - 538944q^{52} - 1215602q^{53} - 1102248q^{54} - 3329556q^{55} + 297472q^{56} - 1001268q^{57} - 19984q^{58} - 1437653q^{59} - 273024q^{60} - 3180086q^{61} + 463696q^{62} - 423549q^{63} + 1835008q^{64} + 544086q^{65} + 475416q^{66} - 5349632q^{67} - 155200q^{68} + 2682828q^{69} - 1524688q^{70} + 1752423q^{71} - 2612736q^{72} - 1843424q^{73} + 51976q^{74} + 2736747q^{75} - 2373376q^{76} - 3885063q^{77} + 1818936q^{78} - 4769243q^{79} - 647168q^{80} + 3720087q^{81} + 2553320q^{82} + 5154441q^{83} - 1003968q^{84} - 4594902q^{85} + 5069944q^{86} + 67446q^{87} + 1126912q^{88} + 20086462q^{89} + 921456q^{90} + 6733847q^{91} + 6359296q^{92} - 1564974q^{93} + 13012480q^{94} + 12936212q^{95} - 6193152q^{96} + 6244248q^{97} + 30770832q^{98} - 1604529q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 77333 x^{5} - 3585829 x^{4} + 1295511138 x^{3} + 69321224657 x^{2} - 5554636281450 x - 316178833801950\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-161432551438685229 \nu^{6} + 20545129292767223968 \nu^{5} + 11454773027949544661802 \nu^{4} - 786990046841790284277179 \nu^{3} - 224931978600293320201281072 \nu^{2} + 6020420522104038984958268422 \nu + 1163771283362632321438737905460\)\()/ \)\(42\!\cdots\!10\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-196733341242858923 \nu^{6} + 7620099148149065499 \nu^{5} + 13618372907037526621114 \nu^{4} + 232968603052876691892325 \nu^{3} - 183039671559799173947956645 \nu^{2} - 2857198520585030880975451368 \nu + 533622478992507034110558924114\)\()/ \)\(50\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-44580892663256924 \nu^{6} + 6516467633708020923 \nu^{5} + 2802424176203131216072 \nu^{4} - 237305880905667423262964 \nu^{3} - 43555838999588731537231777 \nu^{2} + 1859467132479937454078837682 \nu + 163352141040762875936084172150\)\()/ \)\(74\!\cdots\!90\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-1608855959713149461 \nu^{6} + 128282765918895956757 \nu^{5} + 113153749046774651175118 \nu^{4} - 3282639430892396950816181 \nu^{3} - 1745836926959058107144302723 \nu^{2} + 41146007543313945584331409728 \nu + 6187434472999785432172333302090\)\()/ \)\(25\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(723974399688116349 \nu^{6} - 53760889294787996823 \nu^{5} - 52866073192238994088982 \nu^{4} + 1198366716105223972311189 \nu^{3} + 889098221697592492300456457 \nu^{2} - 7883094568255414292671135092 \nu - 3347899047897842426233429784690\)\()/ \)\(84\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-1031046303477688759 \nu^{6} + 80663484845428165443 \nu^{5} + 73854680569900470835832 \nu^{4} - 1788049353525507540783754 \nu^{3} - 1233021223623383316962209757 \nu^{2} + 15188033092314525248281681632 \nu + 4922337116530064696253857771220\)\()/ \)\(63\!\cdots\!15\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} - \beta_{2} - \beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(32 \beta_{6} - 91 \beta_{5} - 42 \beta_{4} + 63 \beta_{3} - 137 \beta_{2} - 230 \beta_{1} + 132634\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(23678 \beta_{6} + 11264 \beta_{5} - 16473 \beta_{4} + 5889 \beta_{3} - 23687 \beta_{2} - 33359 \beta_{1} + 4825969\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(5893265 \beta_{6} - 4072456 \beta_{5} - 5847441 \beta_{4} + 3820905 \beta_{3} - 12676355 \beta_{2} - 17617799 \beta_{1} + 5841433591\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(2707604242 \beta_{6} + 560461729 \beta_{5} - 2359858638 \beta_{4} + 1060652073 \beta_{3} - 3300296605 \beta_{2} - 4660357876 \beta_{1} + 933776679830\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(262301462710 \beta_{6} - 81688548641 \beta_{5} - 248058376026 \beta_{4} + 130453708722 \beta_{3} - 467475925489 \beta_{2} - 625275361027 \beta_{1} + 172399926214322\)\()/3\)

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
115.894
−61.8652
−144.493
−75.1579
265.867
−182.113
83.8682
−8.00000 27.0000 64.0000 −506.565 −216.000 −836.150 −512.000 729.000 4052.52
1.2 −8.00000 27.0000 64.0000 −290.695 −216.000 419.510 −512.000 729.000 2325.56
1.3 −8.00000 27.0000 64.0000 −195.893 −216.000 306.908 −512.000 729.000 1567.14
1.4 −8.00000 27.0000 64.0000 −15.7693 −216.000 −472.852 −512.000 729.000 126.154
1.5 −8.00000 27.0000 64.0000 176.399 −216.000 −148.362 −512.000 729.000 −1411.19
1.6 −8.00000 27.0000 64.0000 266.924 −216.000 663.308 −512.000 729.000 −2135.39
1.7 −8.00000 27.0000 64.0000 407.599 −216.000 −513.363 −512.000 729.000 −3260.79
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \( T_{5}^{7} + 158 T_{5}^{6} - 311636 T_{5}^{5} - 27269830 T_{5}^{4} + 25158582515 T_{5}^{3} + \)\(11\!\cdots\!00\)\( T_{5}^{2} - \)\(54\!\cdots\!00\)\( T_{5} - \)\(87\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\).