Properties

Label 847.2.b.f
Level $847$
Weight $2$
Character orbit 847.b
Analytic conductor $6.763$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 20 x^{14} + 260 x^{12} + 2030 x^{10} + 11605 x^{8} + 42100 x^{6} + 106925 x^{4} + 113575 x^{2} + 87025\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{5} ) q^{2} -\beta_{14} q^{3} + ( 1 - \beta_{6} - \beta_{8} + 2 \beta_{13} ) q^{4} -\beta_{10} q^{5} -\beta_{7} q^{6} + ( -\beta_{2} + \beta_{3} + \beta_{12} ) q^{7} + ( -\beta_{3} - \beta_{5} + \beta_{12} ) q^{8} + ( -2 + 2 \beta_{8} - \beta_{13} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{5} ) q^{2} -\beta_{14} q^{3} + ( 1 - \beta_{6} - \beta_{8} + 2 \beta_{13} ) q^{4} -\beta_{10} q^{5} -\beta_{7} q^{6} + ( -\beta_{2} + \beta_{3} + \beta_{12} ) q^{7} + ( -\beta_{3} - \beta_{5} + \beta_{12} ) q^{8} + ( -2 + 2 \beta_{8} - \beta_{13} ) q^{9} + ( -\beta_{2} + \beta_{7} + \beta_{9} ) q^{10} + ( -\beta_{10} - \beta_{11} + \beta_{15} ) q^{12} -\beta_{2} q^{13} + ( \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{14} + ( -3 + 4 \beta_{6} + \beta_{8} - 2 \beta_{13} ) q^{15} + ( \beta_{6} - \beta_{8} - \beta_{13} ) q^{16} + ( -\beta_{2} + \beta_{9} ) q^{17} + ( -\beta_{1} + \beta_{5} + \beta_{12} ) q^{18} + ( -2 \beta_{4} - \beta_{9} ) q^{19} + ( \beta_{10} + 2 \beta_{11} - 3 \beta_{14} - \beta_{15} ) q^{20} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{9} - 3 \beta_{12} ) q^{21} + ( -2 - 3 \beta_{13} ) q^{23} + ( -2 \beta_{2} - \beta_{4} + \beta_{9} ) q^{24} + ( -1 + \beta_{6} - \beta_{8} + 4 \beta_{13} ) q^{25} + ( \beta_{10} + \beta_{11} - \beta_{14} ) q^{26} + ( \beta_{11} - 2 \beta_{15} ) q^{27} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{12} ) q^{28} + ( -2 \beta_{5} - \beta_{12} ) q^{29} + ( 3 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - \beta_{12} ) q^{30} + ( \beta_{10} - \beta_{11} + \beta_{15} ) q^{31} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{32} + ( 2 \beta_{10} + \beta_{11} - \beta_{14} ) q^{34} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 5 \beta_{5} - \beta_{9} - 2 \beta_{12} ) q^{35} + ( -4 + 3 \beta_{8} - 2 \beta_{13} ) q^{36} + ( -4 \beta_{6} + \beta_{8} + 2 \beta_{13} ) q^{37} + ( -\beta_{10} - 2 \beta_{15} ) q^{38} + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} - 3 \beta_{12} ) q^{39} + ( \beta_{2} + \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{40} + ( \beta_{2} + 2 \beta_{4} + 3 \beta_{7} + \beta_{9} ) q^{41} + ( 2 - 2 \beta_{6} - \beta_{8} + \beta_{11} + 6 \beta_{13} - \beta_{14} ) q^{42} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{12} ) q^{43} + ( \beta_{10} - 2 \beta_{11} + 2 \beta_{14} - \beta_{15} ) q^{45} + ( \beta_{1} + \beta_{5} - 3 \beta_{12} ) q^{46} + ( \beta_{10} - 2 \beta_{11} ) q^{47} + ( \beta_{10} - 2 \beta_{14} + \beta_{15} ) q^{48} + ( 3 + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{14} ) q^{49} + ( -4 \beta_{1} + \beta_{3} - 5 \beta_{5} + 3 \beta_{12} ) q^{50} + ( -\beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{51} + ( -2 \beta_{7} - \beta_{9} ) q^{52} + ( 1 + \beta_{6} - 2 \beta_{8} + \beta_{13} ) q^{53} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{54} + ( -2 \beta_{6} - \beta_{8} - \beta_{10} + 2 \beta_{14} + 2 \beta_{15} ) q^{56} + ( -4 \beta_{1} - 4 \beta_{3} + 5 \beta_{5} + 3 \beta_{12} ) q^{57} + ( 1 + \beta_{6} + 2 \beta_{8} - 2 \beta_{13} ) q^{58} + ( \beta_{11} - \beta_{14} - 3 \beta_{15} ) q^{59} + ( -4 - \beta_{6} - 2 \beta_{8} + 10 \beta_{13} ) q^{60} + ( 3 \beta_{4} + 2 \beta_{7} ) q^{61} + ( -\beta_{4} - \beta_{9} ) q^{62} + ( -\beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + \beta_{7} + 2 \beta_{9} - 3 \beta_{12} ) q^{63} + ( -3 + 4 \beta_{6} - \beta_{8} - 5 \beta_{13} ) q^{64} + ( -2 \beta_{1} - \beta_{3} - 5 \beta_{5} - 2 \beta_{12} ) q^{65} + ( 3 - 4 \beta_{6} + 3 \beta_{8} + 3 \beta_{13} ) q^{67} + ( \beta_{2} - 3 \beta_{7} ) q^{68} + ( 3 \beta_{11} - \beta_{14} ) q^{69} + ( 3 + 4 \beta_{6} + 5 \beta_{8} - \beta_{10} - 7 \beta_{13} + \beta_{15} ) q^{70} + ( \beta_{6} - 4 \beta_{8} - 3 \beta_{13} ) q^{71} + ( -4 \beta_{1} + 3 \beta_{5} + 3 \beta_{12} ) q^{72} + ( 2 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{73} + ( -6 \beta_{1} - 4 \beta_{3} - 5 \beta_{5} + 3 \beta_{12} ) q^{74} + ( \beta_{10} - 5 \beta_{11} + 4 \beta_{14} + \beta_{15} ) q^{75} + ( -\beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{76} + ( 2 - 2 \beta_{6} - \beta_{8} + 6 \beta_{13} ) q^{78} + ( -3 \beta_{1} + 4 \beta_{3} - \beta_{5} + 2 \beta_{12} ) q^{79} + ( -\beta_{10} + \beta_{14} + 2 \beta_{15} ) q^{80} + ( -2 - 2 \beta_{8} + 4 \beta_{13} ) q^{81} + ( 3 \beta_{10} + 2 \beta_{11} - 5 \beta_{14} - \beta_{15} ) q^{82} + ( -\beta_{2} - 3 \beta_{4} + \beta_{7} - \beta_{9} ) q^{83} + ( -8 \beta_{1} - \beta_{2} - 6 \beta_{3} - 5 \beta_{5} - \beta_{7} - 2 \beta_{9} - \beta_{12} ) q^{84} + ( -6 \beta_{1} + 5 \beta_{3} - 6 \beta_{5} + 2 \beta_{12} ) q^{85} + ( -1 - \beta_{6} - \beta_{8} + \beta_{13} ) q^{86} + ( -\beta_{2} - 2 \beta_{4} ) q^{87} + ( \beta_{10} - \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( -\beta_{2} + \beta_{4} - \beta_{9} ) q^{90} + ( 5 + \beta_{6} + \beta_{8} + \beta_{14} ) q^{91} + ( -2 - 4 \beta_{6} - \beta_{8} + 2 \beta_{13} ) q^{92} + ( -1 - 5 \beta_{6} + 4 \beta_{8} - 4 \beta_{13} ) q^{93} + ( -\beta_{2} - \beta_{7} - \beta_{9} ) q^{94} + ( 2 \beta_{1} + 5 \beta_{5} + 8 \beta_{12} ) q^{95} + ( -3 \beta_{2} - 3 \beta_{4} - 2 \beta_{7} + \beta_{9} ) q^{96} + ( -2 \beta_{10} + \beta_{11} + \beta_{14} + 3 \beta_{15} ) q^{97} + ( 5 \beta_{1} + 2 \beta_{3} + 7 \beta_{5} + 2 \beta_{7} + 2 \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{9} + O(q^{10}) \) \( 16q - 32q^{9} + 8q^{14} - 20q^{15} + 16q^{16} - 8q^{23} - 40q^{25} - 60q^{36} - 36q^{37} - 20q^{42} + 48q^{49} + 20q^{53} - 4q^{56} + 28q^{58} - 140q^{60} + 12q^{64} - 4q^{67} + 100q^{70} + 44q^{71} - 20q^{78} - 56q^{81} - 24q^{86} + 80q^{91} - 60q^{92} - 20q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 20 x^{14} + 260 x^{12} + 2030 x^{10} + 11605 x^{8} + 42100 x^{6} + 106925 x^{4} + 113575 x^{2} + 87025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1833998 \nu^{14} + 79007122 \nu^{12} + 1368289173 \nu^{10} + 15517814844 \nu^{8} + 88532447865 \nu^{6} + 352976217575 \nu^{4} + 604192602745 \nu^{2} + 487898960985\)\()/ 141563707035 \)
\(\beta_{2}\)\(=\)\((\)\(-2307412 \nu^{15} - 42954924 \nu^{13} - 546435360 \nu^{11} - 4022687807 \nu^{9} - 22443730260 \nu^{7} - 74163373020 \nu^{5} - 200532703805 \nu^{3} - 70674046800 \nu\)\()/ 141563707035 \)
\(\beta_{3}\)\(=\)\((\)\(14332206 \nu^{14} - 22738192 \nu^{12} - 1270611026 \nu^{10} - 21498650696 \nu^{8} - 125469295005 \nu^{6} - 521715656950 \nu^{4} - 866089585460 \nu^{2} - 712156894265\)\()/ 141563707035 \)
\(\beta_{4}\)\(=\)\((\)\( 2866677 \nu^{15} + 4906553 \nu^{13} + 62416920 \nu^{11} - 656915387 \nu^{9} + 2563649095 \nu^{7} + 8471357565 \nu^{5} + 126280408730 \nu^{3} + 8072787100 \nu \)\()/ 47187902345 \)
\(\beta_{5}\)\(=\)\((\)\(17550090 \nu^{14} + 15194037 \nu^{12} - 832677109 \nu^{10} - 20102189386 \nu^{8} - 117247734350 \nu^{6} - 516762324905 \nu^{4} - 703679870695 \nu^{2} - 622504551990\)\()/ 141563707035 \)
\(\beta_{6}\)\(=\)\((\)\( -312032 \nu^{14} - 6302671 \nu^{12} - 64624056 \nu^{10} - 423472000 \nu^{8} - 1648700690 \nu^{6} - 4513152840 \nu^{4} - 4868757880 \nu^{2} - 8556925565 \)\()/ 2399384865 \)
\(\beta_{7}\)\(=\)\((\)\(11651382 \nu^{15} + 531890783 \nu^{13} + 6766254120 \nu^{11} + 58247425041 \nu^{9} + 277910240545 \nu^{7} + 918330446715 \nu^{5} + 1215421813590 \nu^{3} + 875123748100 \nu\)\()/ 141563707035 \)
\(\beta_{8}\)\(=\)\((\)\( 366156 \nu^{14} + 7209311 \nu^{12} + 75833523 \nu^{10} + 496926000 \nu^{8} + 2038169710 \nu^{6} + 5295988845 \nu^{4} + 5713275915 \nu^{2} - 36566060 \)\()/ 2399384865 \)
\(\beta_{9}\)\(=\)\((\)\(-4614824 \nu^{15} - 85909848 \nu^{13} - 1092870720 \nu^{11} - 8045375614 \nu^{9} - 44887460520 \nu^{7} - 148326746040 \nu^{5} - 353877505265 \nu^{3} - 141348093600 \nu\)\()/ 47187902345 \)
\(\beta_{10}\)\(=\)\((\)\(-16997173 \nu^{15} - 259731426 \nu^{13} - 2577620454 \nu^{11} - 13172985771 \nu^{9} - 41205101320 \nu^{7} - 15301965200 \nu^{5} + 215720277125 \nu^{3} + 571605367040 \nu\)\()/ 141563707035 \)
\(\beta_{11}\)\(=\)\((\)\(19304585 \nu^{15} + 302686350 \nu^{13} + 3124055814 \nu^{11} + 17195673578 \nu^{9} + 63648831580 \nu^{7} + 89465338220 \nu^{5} - 15187573320 \nu^{3} - 217803906170 \nu\)\()/ 141563707035 \)
\(\beta_{12}\)\(=\)\((\)\(-34663122 \nu^{14} - 157372815 \nu^{12} - 357871758 \nu^{10} + 18037649624 \nu^{8} + 109297120970 \nu^{6} + 537175018240 \nu^{4} + 564511445800 \nu^{2} + 563850684635\)\()/ 141563707035 \)
\(\beta_{13}\)\(=\)\((\)\( 2380 \nu^{14} + 45553 \nu^{12} + 492915 \nu^{10} + 3230000 \nu^{8} + 13503405 \nu^{6} + 34423725 \nu^{4} + 37136075 \nu^{2} + 17923235 \)\()/6105305\)
\(\beta_{14}\)\(=\)\((\)\(36161330 \nu^{15} + 591428329 \nu^{13} + 6198172168 \nu^{11} + 37010612536 \nu^{9} + 153148476795 \nu^{7} + 368268112995 \nu^{5} + 556412873310 \nu^{3} + 677017839760 \nu\)\()/ 141563707035 \)
\(\beta_{15}\)\(=\)\((\)\(45560235 \nu^{15} + 729424284 \nu^{13} + 7316393468 \nu^{11} + 41356996652 \nu^{9} + 153943084810 \nu^{7} + 294465298240 \nu^{5} + 166033932275 \nu^{3} - 149364120930 \nu\)\()/ 141563707035 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{10} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{12} - \beta_{8} - \beta_{6} - 6 \beta_{5} - 3 \beta_{3} - 7 \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{9} - 6 \beta_{2}\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{13} + 23 \beta_{12} - 9 \beta_{8} - 7 \beta_{6} + 40 \beta_{5} + 13 \beta_{3} + 44 \beta_{1} - 30\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-10 \beta_{15} + 17 \beta_{14} - 44 \beta_{11} - 40 \beta_{10} - 9 \beta_{9} - \beta_{7} - 2 \beta_{4} + 37 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(-13 \beta_{13} + 69 \beta_{8} + 42 \beta_{6} + 189\)
\(\nu^{7}\)\(=\)\((\)\(83 \beta_{15} - 124 \beta_{14} + 273 \beta_{11} + 271 \beta_{10} - 69 \beta_{9} - 13 \beta_{7} - 27 \beta_{4} + 231 \beta_{2}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(120 \beta_{13} - 965 \beta_{12} - 505 \beta_{8} - 245 \beta_{6} - 1835 \beta_{5} - 320 \beta_{3} - 1700 \beta_{1} - 1210\)\()/2\)
\(\nu^{9}\)\(=\)\(505 \beta_{9} + 120 \beta_{7} + 260 \beta_{4} - 1455 \beta_{2}\)
\(\nu^{10}\)\(=\)\((\)\(965 \beta_{13} + 6395 \beta_{12} - 3610 \beta_{8} - 1420 \beta_{6} + 12390 \beta_{5} + 1560 \beta_{3} + 10655 \beta_{1} - 7815\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-4835 \beta_{15} + 5995 \beta_{14} - 10655 \beta_{11} - 12390 \beta_{10} - 3610 \beta_{9} - 965 \beta_{7} - 2190 \beta_{4} + 9235 \beta_{2}\)\()/2\)
\(\nu^{12}\)\(=\)\(-7210 \beta_{13} + 25410 \beta_{8} + 8205 \beta_{6} + 50815\)
\(\nu^{13}\)\(=\)\((\)\(35405 \beta_{15} - 40825 \beta_{14} + 67225 \beta_{11} + 83435 \beta_{10} - 25410 \beta_{9} - 7210 \beta_{7} - 17205 \beta_{4} + 59020 \beta_{2}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(51465 \beta_{13} - 285060 \beta_{12} - 176870 \beta_{8} - 47235 \beta_{6} - 560630 \beta_{5} - 30020 \beta_{3} - 426765 \beta_{1} - 332295\)\()/2\)
\(\nu^{15}\)\(=\)\(176870 \beta_{9} + 51465 \beta_{7} + 129635 \beta_{4} - 379530 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
−1.27939 2.21596i
1.27939 + 2.21596i
1.29877 + 2.24954i
−1.29877 2.24954i
−0.551501 + 0.955228i
0.551501 0.955228i
1.17141 + 2.02895i
−1.17141 2.02895i
−1.17141 + 2.02895i
1.17141 2.02895i
0.551501 + 0.955228i
−0.551501 0.955228i
−1.29877 + 2.24954i
1.29877 2.24954i
1.27939 2.21596i
−1.27939 + 2.21596i
2.40487i 1.72158i −3.78339 3.36793i −4.14018 2.55877 + 0.672816i 4.28881i 0.0361478 8.09942
846.2 2.40487i 1.72158i −3.78339 3.36793i 4.14018 −2.55877 + 0.672816i 4.28881i 0.0361478 −8.09942
846.3 1.22930i 1.30593i 0.488830 2.38605i −1.60537 −2.59754 0.502754i 3.05951i 1.29456 −2.93316
846.4 1.22930i 1.30593i 0.488830 2.38605i 1.60537 2.59754 0.502754i 3.05951i 1.29456 2.93316
846.5 0.672816i 2.65257i 1.54732 3.54984i −1.78470 1.10300 + 2.40487i 2.38669i −4.03615 −2.38839
846.6 0.672816i 2.65257i 1.54732 3.54984i 1.78470 −1.10300 + 2.40487i 2.38669i −4.03615 2.38839
846.7 0.502754i 2.88003i 1.74724 0.602090i −1.44795 −2.34283 1.22930i 1.88394i −5.29456 0.302703
846.8 0.502754i 2.88003i 1.74724 0.602090i 1.44795 2.34283 1.22930i 1.88394i −5.29456 −0.302703
846.9 0.502754i 2.88003i 1.74724 0.602090i 1.44795 2.34283 + 1.22930i 1.88394i −5.29456 −0.302703
846.10 0.502754i 2.88003i 1.74724 0.602090i −1.44795 −2.34283 + 1.22930i 1.88394i −5.29456 0.302703
846.11 0.672816i 2.65257i 1.54732 3.54984i 1.78470 −1.10300 2.40487i 2.38669i −4.03615 2.38839
846.12 0.672816i 2.65257i 1.54732 3.54984i −1.78470 1.10300 2.40487i 2.38669i −4.03615 −2.38839
846.13 1.22930i 1.30593i 0.488830 2.38605i 1.60537 2.59754 + 0.502754i 3.05951i 1.29456 2.93316
846.14 1.22930i 1.30593i 0.488830 2.38605i −1.60537 −2.59754 + 0.502754i 3.05951i 1.29456 −2.93316
846.15 2.40487i 1.72158i −3.78339 3.36793i 4.14018 −2.55877 0.672816i 4.28881i 0.0361478 −8.09942
846.16 2.40487i 1.72158i −3.78339 3.36793i −4.14018 2.55877 0.672816i 4.28881i 0.0361478 8.09942
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 846.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.b.f 16
7.b odd 2 1 inner 847.2.b.f 16
11.b odd 2 1 inner 847.2.b.f 16
11.c even 5 1 77.2.l.b 16
11.c even 5 1 847.2.l.e 16
11.c even 5 1 847.2.l.i 16
11.c even 5 1 847.2.l.j 16
11.d odd 10 1 77.2.l.b 16
11.d odd 10 1 847.2.l.e 16
11.d odd 10 1 847.2.l.i 16
11.d odd 10 1 847.2.l.j 16
33.f even 10 1 693.2.bu.d 16
33.h odd 10 1 693.2.bu.d 16
77.b even 2 1 inner 847.2.b.f 16
77.j odd 10 1 77.2.l.b 16
77.j odd 10 1 847.2.l.e 16
77.j odd 10 1 847.2.l.i 16
77.j odd 10 1 847.2.l.j 16
77.l even 10 1 77.2.l.b 16
77.l even 10 1 847.2.l.e 16
77.l even 10 1 847.2.l.i 16
77.l even 10 1 847.2.l.j 16
77.m even 15 1 539.2.s.b 16
77.m even 15 1 539.2.s.c 16
77.n even 30 1 539.2.s.b 16
77.n even 30 1 539.2.s.c 16
77.o odd 30 1 539.2.s.b 16
77.o odd 30 1 539.2.s.c 16
77.p odd 30 1 539.2.s.b 16
77.p odd 30 1 539.2.s.c 16
231.r odd 10 1 693.2.bu.d 16
231.u even 10 1 693.2.bu.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.b 16 11.c even 5 1
77.2.l.b 16 11.d odd 10 1
77.2.l.b 16 77.j odd 10 1
77.2.l.b 16 77.l even 10 1
539.2.s.b 16 77.m even 15 1
539.2.s.b 16 77.n even 30 1
539.2.s.b 16 77.o odd 30 1
539.2.s.b 16 77.p odd 30 1
539.2.s.c 16 77.m even 15 1
539.2.s.c 16 77.n even 30 1
539.2.s.c 16 77.o odd 30 1
539.2.s.c 16 77.p odd 30 1
693.2.bu.d 16 33.f even 10 1
693.2.bu.d 16 33.h odd 10 1
693.2.bu.d 16 231.r odd 10 1
693.2.bu.d 16 231.u even 10 1
847.2.b.f 16 1.a even 1 1 trivial
847.2.b.f 16 7.b odd 2 1 inner
847.2.b.f 16 11.b odd 2 1 inner
847.2.b.f 16 77.b even 2 1 inner
847.2.l.e 16 11.c even 5 1
847.2.l.e 16 11.d odd 10 1
847.2.l.e 16 77.j odd 10 1
847.2.l.e 16 77.l even 10 1
847.2.l.i 16 11.c even 5 1
847.2.l.i 16 11.d odd 10 1
847.2.l.i 16 77.j odd 10 1
847.2.l.i 16 77.l even 10 1
847.2.l.j 16 11.c even 5 1
847.2.l.j 16 11.d odd 10 1
847.2.l.j 16 77.j odd 10 1
847.2.l.j 16 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 8 T_{2}^{6} + 14 T_{2}^{4} + 7 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 7 T^{2} + 14 T^{4} + 8 T^{6} + T^{8} )^{2} \)
$3$ \( ( 295 + 350 T^{2} + 135 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$5$ \( ( 295 + 915 T^{2} + 290 T^{4} + 30 T^{6} + T^{8} )^{2} \)
$7$ \( 5764801 - 2823576 T^{2} + 605052 T^{4} - 74088 T^{6} + 8390 T^{8} - 1512 T^{10} + 252 T^{12} - 24 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 295 - 385 T^{2} + 140 T^{4} - 20 T^{6} + T^{8} )^{2} \)
$17$ \( ( 7375 - 6875 T^{2} + 1025 T^{4} - 55 T^{6} + T^{8} )^{2} \)
$19$ \( ( 295 - 2905 T^{2} + 1895 T^{4} - 95 T^{6} + T^{8} )^{2} \)
$23$ \( ( -11 + T + T^{2} )^{8} \)
$29$ \( ( 961 + 712 T^{2} + 194 T^{4} + 23 T^{6} + T^{8} )^{2} \)
$31$ \( ( 295 + 6020 T^{2} + 1170 T^{4} + 65 T^{6} + T^{8} )^{2} \)
$37$ \( ( 691 - 246 T - 34 T^{2} + 9 T^{3} + T^{4} )^{4} \)
$41$ \( ( 248095 - 392290 T^{2} + 19655 T^{4} - 260 T^{6} + T^{8} )^{2} \)
$43$ \( ( 3 + T^{2} )^{8} \)
$47$ \( ( 23895 + 23355 T^{2} + 5310 T^{4} + 150 T^{6} + T^{8} )^{2} \)
$53$ \( ( -5 + 20 T - 10 T^{2} - 5 T^{3} + T^{4} )^{4} \)
$59$ \( ( 1097695 + 281910 T^{2} + 14990 T^{4} + 225 T^{6} + T^{8} )^{2} \)
$61$ \( ( 16850695 - 1275110 T^{2} + 32780 T^{4} - 325 T^{6} + T^{8} )^{2} \)
$67$ \( ( 241 - 439 T - 109 T^{2} + T^{3} + T^{4} )^{4} \)
$71$ \( ( -2879 + 1069 T - 69 T^{2} - 11 T^{3} + T^{4} )^{4} \)
$73$ \( ( 1097695 - 253280 T^{2} + 13820 T^{4} - 235 T^{6} + T^{8} )^{2} \)
$79$ \( ( 44521 + 38707 T^{2} + 5414 T^{4} + 188 T^{6} + T^{8} )^{2} \)
$83$ \( ( 1097695 - 464360 T^{2} + 17930 T^{4} - 235 T^{6} + T^{8} )^{2} \)
$89$ \( ( 23895 + 16740 T^{2} + 3150 T^{4} + 165 T^{6} + T^{8} )^{2} \)
$97$ \( ( 105846295 + 5363390 T^{2} + 81390 T^{4} + 485 T^{6} + T^{8} )^{2} \)
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