Properties

Label 147.7.f.d
Level $147$
Weight $7$
Character orbit 147.f
Analytic conductor $33.818$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{2} + ( 18 + 9 \beta_{2} ) q^{3} + ( 42 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( 25 - 3 \beta_{1} - 25 \beta_{2} + 6 \beta_{3} - \beta_{6} + \beta_{7} ) q^{5} + ( -9 - 9 \beta_{1} - 18 \beta_{2} - 9 \beta_{3} ) q^{6} + ( 404 + 50 \beta_{1} - 50 \beta_{3} - 9 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{8} + ( 243 + 243 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{2} + ( 18 + 9 \beta_{2} ) q^{3} + ( 42 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( 25 - 3 \beta_{1} - 25 \beta_{2} + 6 \beta_{3} - \beta_{6} + \beta_{7} ) q^{5} + ( -9 - 9 \beta_{1} - 18 \beta_{2} - 9 \beta_{3} ) q^{6} + ( 404 + 50 \beta_{1} - 50 \beta_{3} - 9 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{8} + ( 243 + 243 \beta_{2} ) q^{9} + ( 580 - 64 \beta_{1} + 290 \beta_{2} + 32 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 4 \beta_{7} ) q^{10} + ( 55 \beta_{2} + 103 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{11} + ( -378 - 36 \beta_{1} + 378 \beta_{2} + 72 \beta_{3} + 18 \beta_{4} + 9 \beta_{5} ) q^{12} + ( -264 - 3 \beta_{1} - 528 \beta_{2} - 3 \beta_{3} - 20 \beta_{4} - 40 \beta_{5} + \beta_{6} ) q^{13} + ( 675 - 81 \beta_{1} + 81 \beta_{3} - 9 \beta_{6} + 18 \beta_{7} ) q^{15} + ( -3020 - 646 \beta_{1} - 3020 \beta_{2} - 55 \beta_{5} + 20 \beta_{6} + 20 \beta_{7} ) q^{16} + ( 884 + 164 \beta_{1} + 442 \beta_{2} - 82 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} + 18 \beta_{7} ) q^{17} + ( -243 \beta_{2} - 243 \beta_{3} ) q^{18} + ( 1550 - 85 \beta_{1} - 1550 \beta_{2} + 170 \beta_{3} - 88 \beta_{4} - 44 \beta_{5} + 81 \beta_{6} - 81 \beta_{7} ) q^{19} + ( 1440 - 366 \beta_{1} + 2880 \beta_{2} - 366 \beta_{3} + 15 \beta_{4} + 30 \beta_{5} + 12 \beta_{6} ) q^{20} + ( 10798 - 308 \beta_{1} + 308 \beta_{3} - 37 \beta_{4} - 44 \beta_{6} + 88 \beta_{7} ) q^{22} + ( 1088 - 280 \beta_{1} + 1088 \beta_{2} + 48 \beta_{5} - 64 \beta_{6} - 64 \beta_{7} ) q^{23} + ( 7272 + 900 \beta_{1} + 3636 \beta_{2} - 450 \beta_{3} - 81 \beta_{4} + 81 \beta_{5} - 108 \beta_{7} ) q^{24} + ( 4030 \beta_{2} + 1107 \beta_{3} - 60 \beta_{4} - 60 \beta_{5} - 86 \beta_{6} + 43 \beta_{7} ) q^{25} + ( -699 - 1457 \beta_{1} + 699 \beta_{2} + 2914 \beta_{3} + 210 \beta_{4} + 105 \beta_{5} - 236 \beta_{6} + 236 \beta_{7} ) q^{26} + ( 2187 + 4374 \beta_{2} ) q^{27} + ( -605 - 1247 \beta_{1} + 1247 \beta_{3} + 48 \beta_{4} + 145 \beta_{6} - 290 \beta_{7} ) q^{29} + ( 7830 - 864 \beta_{1} + 7830 \beta_{2} + 135 \beta_{5} + 36 \beta_{6} + 36 \beta_{7} ) q^{30} + ( 14662 + 1060 \beta_{1} + 7331 \beta_{2} - 530 \beta_{3} + 44 \beta_{4} - 44 \beta_{5} + 26 \beta_{7} ) q^{31} + ( 45324 \beta_{2} + 4098 \beta_{3} + 465 \beta_{4} + 465 \beta_{5} + 232 \beta_{6} - 116 \beta_{7} ) q^{32} + ( -495 - 927 \beta_{1} + 495 \beta_{2} + 1854 \beta_{3} - 216 \beta_{4} - 108 \beta_{5} + 27 \beta_{6} - 27 \beta_{7} ) q^{33} + ( -8908 + 464 \beta_{1} - 17816 \beta_{2} + 464 \beta_{3} + 74 \beta_{4} + 148 \beta_{5} - 216 \beta_{6} ) q^{34} + ( -10206 - 972 \beta_{1} + 972 \beta_{3} + 243 \beta_{4} ) q^{36} + ( 16510 - 1255 \beta_{1} + 16510 \beta_{2} + 260 \beta_{5} - 113 \beta_{6} - 113 \beta_{7} ) q^{37} + ( 14222 - 5278 \beta_{1} + 7111 \beta_{2} + 2639 \beta_{3} + 297 \beta_{4} - 297 \beta_{5} + 204 \beta_{7} ) q^{38} + ( -7128 \beta_{2} - 81 \beta_{3} - 540 \beta_{4} - 540 \beta_{5} + 18 \beta_{6} - 9 \beta_{7} ) q^{39} + ( -18340 - 614 \beta_{1} + 18340 \beta_{2} + 1228 \beta_{3} - 10 \beta_{4} - 5 \beta_{5} - 28 \beta_{6} + 28 \beta_{7} ) q^{40} + ( 7782 + 1052 \beta_{1} + 15564 \beta_{2} + 1052 \beta_{3} + 432 \beta_{4} + 864 \beta_{5} + 220 \beta_{6} ) q^{41} + ( 6182 - 1901 \beta_{1} + 1901 \beta_{3} + 620 \beta_{4} - 221 \beta_{6} + 442 \beta_{7} ) q^{43} + ( 24840 - 7614 \beta_{1} + 24840 \beta_{2} - 381 \beta_{5} + 388 \beta_{6} + 388 \beta_{7} ) q^{44} + ( 12150 - 1458 \beta_{1} + 6075 \beta_{2} + 729 \beta_{3} + 243 \beta_{7} ) q^{45} + ( 28024 \beta_{2} - 56 \beta_{3} - 344 \beta_{4} - 344 \beta_{5} - 128 \beta_{6} + 64 \beta_{7} ) q^{46} + ( -39992 - 82 \beta_{1} + 39992 \beta_{2} + 164 \beta_{3} - 192 \beta_{4} - 96 \beta_{5} + 1138 \beta_{6} - 1138 \beta_{7} ) q^{47} + ( -27180 - 5814 \beta_{1} - 54360 \beta_{2} - 5814 \beta_{3} - 495 \beta_{4} - 990 \beta_{5} + 540 \beta_{6} ) q^{48} + ( 119905 + 1687 \beta_{1} - 1687 \beta_{3} - 1065 \beta_{4} - 412 \beta_{6} + 824 \beta_{7} ) q^{50} + ( 11934 + 2214 \beta_{1} + 11934 \beta_{2} - 648 \beta_{5} + 162 \beta_{6} + 162 \beta_{7} ) q^{51} + ( 274836 + 14804 \beta_{1} + 137418 \beta_{2} - 7402 \beta_{3} - 1174 \beta_{4} + 1174 \beta_{5} - 252 \beta_{7} ) q^{52} + ( 137125 \beta_{2} - 6857 \beta_{3} + 864 \beta_{4} + 864 \beta_{5} - 1022 \beta_{6} + 511 \beta_{7} ) q^{53} + ( 2187 + 2187 \beta_{1} - 2187 \beta_{2} - 4374 \beta_{3} ) q^{54} + ( -43985 + 7433 \beta_{1} - 87970 \beta_{2} + 7433 \beta_{3} - 920 \beta_{4} - 1840 \beta_{5} - 331 \beta_{6} ) q^{55} + ( 41850 - 2295 \beta_{1} + 2295 \beta_{3} - 1188 \beta_{4} + 729 \beta_{6} - 1458 \beta_{7} ) q^{57} + ( 131828 + 14186 \beta_{1} + 131828 \beta_{2} + 617 \beta_{5} - 772 \beta_{6} - 772 \beta_{7} ) q^{58} + ( 31742 - 1142 \beta_{1} + 15871 \beta_{2} + 571 \beta_{3} + 756 \beta_{4} - 756 \beta_{5} - 1567 \beta_{7} ) q^{59} + ( 38880 \beta_{2} - 9882 \beta_{3} + 405 \beta_{4} + 405 \beta_{5} + 216 \beta_{6} - 108 \beta_{7} ) q^{60} + ( -38980 + 15208 \beta_{1} + 38980 \beta_{2} - 30416 \beta_{3} + 1440 \beta_{4} + 720 \beta_{5} - 2024 \beta_{6} + 2024 \beta_{7} ) q^{61} + ( -62717 - 6697 \beta_{1} - 125434 \beta_{2} - 6697 \beta_{3} - 258 \beta_{4} - 516 \beta_{5} - 424 \beta_{6} ) q^{62} + ( 285124 + 49742 \beta_{1} - 49742 \beta_{3} - 2207 \beta_{4} + 1044 \beta_{6} - 2088 \beta_{7} ) q^{64} + ( -109080 + 24294 \beta_{1} - 109080 \beta_{2} - 2040 \beta_{5} - 46 \beta_{6} - 46 \beta_{7} ) q^{65} + ( 194364 - 5544 \beta_{1} + 97182 \beta_{2} + 2772 \beta_{3} - 333 \beta_{4} + 333 \beta_{5} + 1188 \beta_{7} ) q^{66} + ( 291068 \beta_{2} - 6357 \beta_{3} - 2148 \beta_{4} - 2148 \beta_{5} + 2090 \beta_{6} - 1045 \beta_{7} ) q^{67} + ( 68544 - 1284 \beta_{1} - 68544 \beta_{2} + 2568 \beta_{3} + 540 \beta_{4} + 270 \beta_{5} - 1128 \beta_{6} + 1128 \beta_{7} ) q^{68} + ( 9792 - 2520 \beta_{1} + 19584 \beta_{2} - 2520 \beta_{3} + 432 \beta_{4} + 864 \beta_{5} - 1728 \beta_{6} ) q^{69} + ( 186172 - 22382 \beta_{1} + 22382 \beta_{3} + 3408 \beta_{4} - 758 \beta_{6} + 1516 \beta_{7} ) q^{71} + ( 98172 + 12150 \beta_{1} + 98172 \beta_{2} + 2187 \beta_{5} - 972 \beta_{6} - 972 \beta_{7} ) q^{72} + ( 70456 - 28634 \beta_{1} + 35228 \beta_{2} + 14317 \beta_{3} + 1924 \beta_{4} - 1924 \beta_{5} + 1551 \beta_{7} ) q^{73} + ( 113705 \beta_{2} - 22921 \beta_{3} - 723 \beta_{4} - 723 \beta_{5} + 1176 \beta_{6} - 588 \beta_{7} ) q^{74} + ( -36270 - 9963 \beta_{1} + 36270 \beta_{2} + 19926 \beta_{3} - 1080 \beta_{4} - 540 \beta_{5} - 1161 \beta_{6} + 1161 \beta_{7} ) q^{75} + ( 172566 + 20802 \beta_{1} + 345132 \beta_{2} + 20802 \beta_{3} + 1716 \beta_{4} + 3432 \beta_{5} + 2436 \beta_{6} ) q^{76} + ( -18873 - 39339 \beta_{1} + 39339 \beta_{3} + 2835 \beta_{4} - 2124 \beta_{6} + 4248 \beta_{7} ) q^{78} + ( 152983 - 7998 \beta_{1} + 152983 \beta_{2} - 732 \beta_{5} + 2750 \beta_{6} + 2750 \beta_{7} ) q^{79} + ( 349880 - 7980 \beta_{1} + 174940 \beta_{2} + 3990 \beta_{3} - 1605 \beta_{4} + 1605 \beta_{5} + 940 \beta_{7} ) q^{80} + 59049 \beta_{2} q^{81} + ( 120834 + 40378 \beta_{1} - 120834 \beta_{2} - 80756 \beta_{3} - 5544 \beta_{4} - 2772 \beta_{5} + 6064 \beta_{6} - 6064 \beta_{7} ) q^{82} + ( 95781 + 15973 \beta_{1} + 191562 \beta_{2} + 15973 \beta_{3} + 1092 \beta_{4} + 2184 \beta_{5} - 895 \beta_{6} ) q^{83} + ( -39210 + 21786 \beta_{1} - 21786 \beta_{3} + 3480 \beta_{4} - 326 \beta_{6} + 652 \beta_{7} ) q^{85} + ( 197143 + 26113 \beta_{1} + 197143 \beta_{2} + 6327 \beta_{5} - 1596 \beta_{6} - 1596 \beta_{7} ) q^{86} + ( -10890 - 22446 \beta_{1} - 5445 \beta_{2} + 11223 \beta_{3} + 432 \beta_{4} - 432 \beta_{5} - 3915 \beta_{7} ) q^{87} + ( 85844 \beta_{2} + 21950 \beta_{3} + 9479 \beta_{4} + 9479 \beta_{5} - 5576 \beta_{6} + 2788 \beta_{7} ) q^{88} + ( 81874 + 3154 \beta_{1} - 81874 \beta_{2} - 6308 \beta_{3} + 384 \beta_{4} + 192 \beta_{5} + 726 \beta_{6} - 726 \beta_{7} ) q^{89} + ( 70470 - 7776 \beta_{1} + 140940 \beta_{2} - 7776 \beta_{3} + 1215 \beta_{4} + 2430 \beta_{5} + 972 \beta_{6} ) q^{90} + ( 89712 - 13776 \beta_{1} + 13776 \beta_{3} - 1680 \beta_{4} + 2464 \beta_{6} - 4928 \beta_{7} ) q^{92} + ( 197937 + 14310 \beta_{1} + 197937 \beta_{2} - 1188 \beta_{5} + 234 \beta_{6} + 234 \beta_{7} ) q^{93} + ( 96052 + 105372 \beta_{1} + 48026 \beta_{2} - 52686 \beta_{3} + 2674 \beta_{4} - 2674 \beta_{5} - 3400 \beta_{7} ) q^{94} + ( 143310 \beta_{2} + 27108 \beta_{3} - 4680 \beta_{4} - 4680 \beta_{5} - 2584 \beta_{6} + 1292 \beta_{7} ) q^{95} + ( -407916 - 36882 \beta_{1} + 407916 \beta_{2} + 73764 \beta_{3} + 8370 \beta_{4} + 4185 \beta_{5} + 3132 \beta_{6} - 3132 \beta_{7} ) q^{96} + ( 114285 + 33555 \beta_{1} + 228570 \beta_{2} + 33555 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 4031 \beta_{6} ) q^{97} + ( -13365 - 25029 \beta_{1} + 25029 \beta_{3} - 2916 \beta_{4} + 243 \beta_{6} - 486 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 5q^{2} + 108q^{3} - 173q^{4} + 294q^{5} + 3326q^{8} + 972q^{9} + O(q^{10}) \) \( 8q - 5q^{2} + 108q^{3} - 173q^{4} + 294q^{5} + 3326q^{8} + 972q^{9} + 3411q^{10} - 314q^{11} - 4671q^{12} + 5292q^{15} - 12721q^{16} + 5532q^{17} + 1215q^{18} + 18234q^{19} + 86106q^{22} + 3928q^{23} + 44901q^{24} - 17038q^{25} - 12366q^{26} - 8300q^{29} + 30699q^{30} + 89508q^{31} - 186207q^{32} - 8478q^{33} - 84078q^{36} + 64706q^{37} + 77136q^{38} + 29106q^{39} - 221823q^{40} + 45740q^{43} + 92529q^{44} + 71442q^{45} - 111504q^{46} - 483276q^{47} + 967216q^{50} + 49788q^{51} + 1673988q^{52} - 540974q^{53} + 32805q^{54} + 328212q^{57} + 539799q^{58} + 181770q^{59} - 146367q^{60} - 418224q^{61} + 2378626q^{64} - 414204q^{65} + 1162431q^{66} - 1158902q^{67} + 821250q^{68} + 1442344q^{71} + 404109q^{72} + 378666q^{73} - 432940q^{74} - 460026q^{75} - 222588q^{78} + 611452q^{79} + 2094945q^{80} - 236196q^{81} + 1561266q^{82} - 275112q^{85} + 816224q^{86} - 112050q^{87} - 366441q^{88} + 989196q^{89} + 678720q^{92} + 805572q^{93} + 716148q^{94} - 591792q^{95} - 5027589q^{96} - 152604q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + 20860832470090 \nu^{3} + 529666993003657 \nu^{2} + 423382833624317 \nu + 1017283529220570\)\()/ 11540026222043130 \)
\(\beta_{3}\)\(=\)\((\)\(8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 8366760637851 \nu - 5583985297290\)\()/ 7542500798721 \)
\(\beta_{4}\)\(=\)\((\)\(-7295536 \nu^{7} - 50933099 \nu^{6} - 1392479248 \nu^{5} - 2649247696 \nu^{4} - 323422537422 \nu^{3} - 13289665936 \nu^{2} - 707660422560 \nu + 208034068231995\)\()/ 2514166932907 \)
\(\beta_{5}\)\(=\)\((\)\(-2339704235 \nu^{7} - 69853815577 \nu^{6} - 363164645146 \nu^{5} - 18532200031393 \nu^{4} - 115689918600610 \nu^{3} - 2937425023372453 \nu^{2} - 4453954916188463 \nu - 69640276582887300\)\()/ 769335081469542 \)
\(\beta_{6}\)\(=\)\((\)\(65881610383 \nu^{7} - 648926656693 \nu^{6} + 19199653600241 \nu^{5} - 98352920361781 \nu^{4} + 3069708627551885 \nu^{3} - 15721266306367057 \nu^{2} + 162140643786549358 \nu - 277910537962726260\)\()/ 7693350814695420 \)
\(\beta_{7}\)\(=\)\((\)\(-133418040331 \nu^{7} + 80927719096 \nu^{6} - 22152636639347 \nu^{5} - 109586797160888 \nu^{4} - 4010412443271095 \nu^{3} - 8163675106851176 \nu^{2} + 64933663679396204 \nu - 288352060383659040\)\()/ 7693350814695420 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} + 105 \beta_{2}\)
\(\nu^{3}\)\(=\)\(8 \beta_{7} - 4 \beta_{6} + 6 \beta_{4} + 169 \beta_{3} - 169 \beta_{1} - 216\)
\(\nu^{4}\)\(=\)\(4 \beta_{7} + 4 \beta_{6} - 217 \beta_{5} - 17781 \beta_{2} - 722 \beta_{1} - 17781\)
\(\nu^{5}\)\(=\)\(-848 \beta_{7} + 1696 \beta_{6} - 1614 \beta_{5} - 1614 \beta_{4} - 32053 \beta_{3} - 77112 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-4432 \beta_{7} + 2216 \beta_{6} - 43597 \beta_{4} - 197354 \beta_{3} + 197354 \beta_{1} + 3375249\)
\(\nu^{7}\)\(=\)\(-163308 \beta_{7} - 163308 \beta_{6} + 385038 \beta_{5} + 20983752 \beta_{2} + 6279409 \beta_{1} + 20983752\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
7.29767 12.6399i
2.26350 3.92050i
−2.75320 + 4.76869i
−6.30797 + 10.9257i
7.29767 + 12.6399i
2.26350 + 3.92050i
−2.75320 4.76869i
−6.30797 10.9257i
−7.79767 + 13.5060i 13.5000 7.79423i −89.6073 155.204i −22.3721 12.9165i 243.107i 0 1796.81 121.500 210.444i 348.901 201.438i
19.2 −2.76350 + 4.78652i 13.5000 7.79423i 16.7261 + 28.9705i 57.9943 + 33.4830i 86.1574i 0 −538.619 121.500 210.444i −320.535 + 185.061i
19.3 2.25320 3.90266i 13.5000 7.79423i 21.8461 + 37.8386i −53.9244 31.1333i 70.2479i 0 485.305 121.500 210.444i −243.005 + 140.299i
19.4 5.80797 10.0597i 13.5000 7.79423i −35.4650 61.4271i 165.302 + 95.4373i 181.074i 0 −80.4975 121.500 210.444i 1920.14 1108.59i
31.1 −7.79767 13.5060i 13.5000 + 7.79423i −89.6073 + 155.204i −22.3721 + 12.9165i 243.107i 0 1796.81 121.500 + 210.444i 348.901 + 201.438i
31.2 −2.76350 4.78652i 13.5000 + 7.79423i 16.7261 28.9705i 57.9943 33.4830i 86.1574i 0 −538.619 121.500 + 210.444i −320.535 185.061i
31.3 2.25320 + 3.90266i 13.5000 + 7.79423i 21.8461 37.8386i −53.9244 + 31.1333i 70.2479i 0 485.305 121.500 + 210.444i −243.005 140.299i
31.4 5.80797 + 10.0597i 13.5000 + 7.79423i −35.4650 + 61.4271i 165.302 95.4373i 181.074i 0 −80.4975 121.500 + 210.444i 1920.14 + 1108.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.f.d 8
7.b odd 2 1 21.7.f.a 8
7.c even 3 1 21.7.f.a 8
7.c even 3 1 147.7.d.b 8
7.d odd 6 1 147.7.d.b 8
7.d odd 6 1 inner 147.7.f.d 8
21.c even 2 1 63.7.m.d 8
21.g even 6 1 441.7.d.c 8
21.h odd 6 1 63.7.m.d 8
21.h odd 6 1 441.7.d.c 8
28.d even 2 1 336.7.bh.d 8
28.g odd 6 1 336.7.bh.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a 8 7.b odd 2 1
21.7.f.a 8 7.c even 3 1
63.7.m.d 8 21.c even 2 1
63.7.m.d 8 21.h odd 6 1
147.7.d.b 8 7.c even 3 1
147.7.d.b 8 7.d odd 6 1
147.7.f.d 8 1.a even 1 1 trivial
147.7.f.d 8 7.d odd 6 1 inner
336.7.bh.d 8 28.d even 2 1
336.7.bh.d 8 28.g odd 6 1
441.7.d.c 8 21.g even 6 1
441.7.d.c 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\):

\(T_{2}^{8} + \cdots\)
\(25\!\cdots\!00\)\( T_{5} + \)\(42\!\cdots\!00\)\( \)">\(T_{5}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 20358144 - 1281408 T + 992080 T^{2} + 12248 T^{3} + 37712 T^{4} - 442 T^{5} + 227 T^{6} + 5 T^{7} + T^{8} \)
$3$ \( ( 243 - 27 T + T^{2} )^{4} \)
$5$ \( 422734160250000 + 25332592050000 T + 334857307500 T^{2} - 10257232500 T^{3} - 72000675 T^{4} + 2447550 T^{5} + 20487 T^{6} - 294 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( \)\(18\!\cdots\!44\)\( + 98293722790383421872 T + 5063514049337341372 T^{2} + 8590189886857868 T^{3} + 14795841919805 T^{4} + 3395131754 T^{5} + 3844943 T^{6} + 314 T^{7} + T^{8} \)
$13$ \( \)\(21\!\cdots\!56\)\( + \)\(14\!\cdots\!08\)\( T^{2} + 331924469476329 T^{4} + 30553062 T^{6} + T^{8} \)
$17$ \( \)\(14\!\cdots\!04\)\( - \)\(32\!\cdots\!48\)\( T - 18954316226540153088 T^{2} + 49044433474028160 T^{3} + 325284835782096 T^{4} + 99181347120 T^{5} - 7727652 T^{6} - 5532 T^{7} + T^{8} \)
$19$ \( \)\(40\!\cdots\!56\)\( - \)\(29\!\cdots\!56\)\( T + \)\(80\!\cdots\!44\)\( T^{2} - 80019415606831775892 T^{3} - 3659130219753963 T^{4} + 1012664156058 T^{5} + 55289115 T^{6} - 18234 T^{7} + T^{8} \)
$23$ \( \)\(15\!\cdots\!56\)\( - \)\(35\!\cdots\!28\)\( T + \)\(83\!\cdots\!96\)\( T^{2} + 12921467685980241920 T^{3} + 20933343724396544 T^{4} + 749772681728 T^{5} + 161320832 T^{6} - 3928 T^{7} + T^{8} \)
$29$ \( ( -131918946476762880 - 26271861557728 T - 1346898667 T^{2} + 4150 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(16\!\cdots\!01\)\( + \)\(45\!\cdots\!96\)\( T - \)\(58\!\cdots\!94\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + 706793352941013399 T^{4} - 68718078342072 T^{5} + 3438291822 T^{6} - 89508 T^{7} + T^{8} \)
$37$ \( \)\(23\!\cdots\!36\)\( - \)\(16\!\cdots\!60\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(47\!\cdots\!28\)\( T^{3} + 2903640625281916109 T^{4} - 97788930340930 T^{5} + 3733577311 T^{6} - 64706 T^{7} + T^{8} \)
$41$ \( \)\(36\!\cdots\!44\)\( + \)\(43\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!32\)\( T^{4} + 21593029296 T^{6} + T^{8} \)
$43$ \( ( -293597400770976796 + 85125133057276 T - 4841210811 T^{2} - 22870 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(99\!\cdots\!96\)\( + \)\(26\!\cdots\!80\)\( T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} - \)\(15\!\cdots\!24\)\( T^{4} + 6508813429466640 T^{5} + 91320005532 T^{6} + 483276 T^{7} + T^{8} \)
$53$ \( \)\(73\!\cdots\!64\)\( - \)\(30\!\cdots\!60\)\( T + \)\(20\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!16\)\( T^{3} + \)\(60\!\cdots\!97\)\( T^{4} + 40803243733591190 T^{5} + 203978340551 T^{6} + 540974 T^{7} + T^{8} \)
$59$ \( \)\(20\!\cdots\!00\)\( - \)\(67\!\cdots\!60\)\( T - \)\(32\!\cdots\!28\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(51\!\cdots\!49\)\( T^{4} + 13446651706756290 T^{5} - 62962743777 T^{6} - 181770 T^{7} + T^{8} \)
$61$ \( \)\(68\!\cdots\!00\)\( - \)\(30\!\cdots\!00\)\( T - \)\(93\!\cdots\!00\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!00\)\( T^{4} - 71069068838315520 T^{5} - 111626861088 T^{6} + 418224 T^{7} + T^{8} \)
$67$ \( \)\(29\!\cdots\!04\)\( + \)\(10\!\cdots\!28\)\( T + \)\(26\!\cdots\!52\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} + 387875645278331366 T^{5} + 908045258107 T^{6} + 1158902 T^{7} + T^{8} \)
$71$ \( ( -\)\(19\!\cdots\!12\)\( + 115286062429846880 T - 36558200716 T^{2} - 721172 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(77\!\cdots\!24\)\( - \)\(13\!\cdots\!92\)\( T + \)\(10\!\cdots\!44\)\( T^{2} - \)\(36\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!01\)\( T^{4} + 88270688635072434 T^{5} - 185313643497 T^{6} - 378666 T^{7} + T^{8} \)
$79$ \( \)\(99\!\cdots\!01\)\( - \)\(96\!\cdots\!32\)\( T + \)\(92\!\cdots\!06\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!39\)\( T^{4} - 103324245868979728 T^{5} + 521211945622 T^{6} - 611452 T^{7} + T^{8} \)
$83$ \( \)\(52\!\cdots\!24\)\( + \)\(31\!\cdots\!16\)\( T^{2} + \)\(85\!\cdots\!29\)\( T^{4} + 524747194014 T^{6} + T^{8} \)
$89$ \( \)\(87\!\cdots\!56\)\( + \)\(92\!\cdots\!00\)\( T + \)\(22\!\cdots\!76\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!80\)\( T^{4} - 103856130641901456 T^{5} + 431160022908 T^{6} - 989196 T^{7} + T^{8} \)
$97$ \( \)\(90\!\cdots\!76\)\( + \)\(26\!\cdots\!08\)\( T^{2} + \)\(12\!\cdots\!73\)\( T^{4} + 2056709392566 T^{6} + T^{8} \)
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