Properties

Label 74.4.c.b
Level $74$
Weight $4$
Character orbit 74.c
Analytic conductor $4.366$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 84 x^{8} - 140 x^{7} + 6309 x^{6} - 5214 x^{5} + 67648 x^{4} + 164178 x^{3} + 511389 x^{2} + 497502 x + 443556\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} + ( -1 + \beta_{1} - \beta_{3} ) q^{3} + ( -4 - 4 \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{5} + ( 2 - 2 \beta_{2} ) q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{7} + 8 q^{8} + ( -5 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + 2 \beta_{3} q^{2} + ( -1 + \beta_{1} - \beta_{3} ) q^{3} + ( -4 - 4 \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{5} + ( 2 - 2 \beta_{2} ) q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{7} + 8 q^{8} + ( -5 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{9} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{10} + ( -8 + 2 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{11} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{12} + ( 14 - 4 \beta_{1} + 14 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{13} + ( 2 \beta_{4} - 2 \beta_{7} ) q^{14} + ( 4 \beta_{1} - 4 \beta_{2} - 22 \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{8} ) q^{15} + 16 \beta_{3} q^{16} + ( -4 \beta_{1} + 4 \beta_{2} - 13 \beta_{3} - 4 \beta_{6} + 4 \beta_{9} ) q^{17} + ( -16 + 10 \beta_{1} - 16 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{18} + ( 8 - 2 \beta_{1} + 8 \beta_{3} - 7 \beta_{4} - 7 \beta_{6} - 4 \beta_{7} - 4 \beta_{9} ) q^{19} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{6} ) q^{20} + ( -\beta_{1} + \beta_{2} + 23 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{21} + ( 4 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} + 6 \beta_{5} - 6 \beta_{8} - 2 \beta_{9} ) q^{22} + ( -52 + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{7} ) q^{23} + ( -8 + 8 \beta_{1} - 8 \beta_{3} ) q^{24} + ( 11 \beta_{1} - 11 \beta_{2} - 29 \beta_{3} - 3 \beta_{5} + 13 \beta_{6} + 3 \beta_{8} ) q^{25} + ( -28 + 8 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} - 6 \beta_{7} ) q^{26} + ( 90 - 26 \beta_{2} - \beta_{4} + 13 \beta_{5} + 5 \beta_{7} ) q^{27} + ( 4 \beta_{6} - 4 \beta_{9} ) q^{28} + ( 29 + 14 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} + 9 \beta_{7} ) q^{29} + ( 44 - 8 \beta_{1} + 44 \beta_{3} + 6 \beta_{4} + 6 \beta_{6} + 2 \beta_{8} ) q^{30} + ( -76 + 10 \beta_{2} - 12 \beta_{4} + 7 \beta_{5} + \beta_{7} ) q^{31} + ( -32 - 32 \beta_{3} ) q^{32} + ( 26 - 54 \beta_{1} + 26 \beta_{3} - 6 \beta_{4} - 6 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{33} + ( 26 + 8 \beta_{1} + 26 \beta_{3} + 8 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} - 8 \beta_{9} ) q^{34} + ( 15 \beta_{1} - 15 \beta_{2} + 51 \beta_{3} - 3 \beta_{5} + 8 \beta_{6} + 3 \beta_{8} + 3 \beta_{9} ) q^{35} + ( 32 - 20 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} + 4 \beta_{7} ) q^{36} + ( -5 + 17 \beta_{1} - \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} + 18 \beta_{6} - 8 \beta_{7} - 5 \beta_{8} - 11 \beta_{9} ) q^{37} + ( -16 + 4 \beta_{2} + 14 \beta_{4} + 8 \beta_{7} ) q^{38} + ( 40 \beta_{1} - 40 \beta_{2} - 140 \beta_{3} + 11 \beta_{5} + 21 \beta_{6} - 11 \beta_{8} - 13 \beta_{9} ) q^{39} + ( -8 \beta_{1} - 8 \beta_{4} - 8 \beta_{6} ) q^{40} + ( 72 + 3 \beta_{1} + 72 \beta_{3} - 15 \beta_{4} - 15 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} - 4 \beta_{9} ) q^{41} + ( -46 + 2 \beta_{1} - 46 \beta_{3} + 12 \beta_{4} + 12 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{42} + ( -49 + 43 \beta_{2} - \beta_{4} + 8 \beta_{5} - 4 \beta_{7} ) q^{43} + ( 32 - 8 \beta_{1} + 32 \beta_{3} + 4 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{44} + ( -181 + 24 \beta_{2} - 3 \beta_{4} - 14 \beta_{5} - 7 \beta_{7} ) q^{45} + ( 4 \beta_{1} - 4 \beta_{2} - 104 \beta_{3} + 8 \beta_{5} - 4 \beta_{6} - 8 \beta_{8} + 6 \beta_{9} ) q^{46} + ( -2 - 4 \beta_{2} - 6 \beta_{4} - 9 \beta_{5} - 13 \beta_{7} ) q^{47} + ( 16 - 16 \beta_{2} ) q^{48} + ( 13 \beta_{1} - 13 \beta_{2} - 30 \beta_{3} + 12 \beta_{5} - 12 \beta_{8} - 3 \beta_{9} ) q^{49} + ( 58 - 22 \beta_{1} + 58 \beta_{3} - 26 \beta_{4} - 26 \beta_{6} - 6 \beta_{8} ) q^{50} + ( 31 + \beta_{2} + 16 \beta_{4} + 8 \beta_{7} ) q^{51} + ( 16 \beta_{1} - 16 \beta_{2} - 56 \beta_{3} + 4 \beta_{5} - 12 \beta_{6} - 4 \beta_{8} - 12 \beta_{9} ) q^{52} + ( -43 \beta_{1} + 43 \beta_{2} + 11 \beta_{3} - 14 \beta_{5} - 36 \beta_{6} + 14 \beta_{8} + 27 \beta_{9} ) q^{53} + ( -52 \beta_{1} + 52 \beta_{2} + 180 \beta_{3} - 26 \beta_{5} - 2 \beta_{6} + 26 \beta_{8} + 10 \beta_{9} ) q^{54} + ( 60 + 58 \beta_{1} + 60 \beta_{3} + 40 \beta_{4} + 40 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} ) q^{55} + ( -8 \beta_{4} - 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{9} ) q^{56} + ( 20 \beta_{1} - 20 \beta_{2} - 36 \beta_{3} + \beta_{5} - 27 \beta_{6} - \beta_{8} + 13 \beta_{9} ) q^{57} + ( 28 \beta_{1} - 28 \beta_{2} + 58 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{8} + 18 \beta_{9} ) q^{58} + ( -8 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} + 3 \beta_{5} - 15 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} ) q^{59} + ( -88 + 16 \beta_{2} - 12 \beta_{4} - 4 \beta_{5} ) q^{60} + ( 39 + 58 \beta_{1} + 39 \beta_{3} + 24 \beta_{4} + 24 \beta_{6} - 2 \beta_{7} - 21 \beta_{8} - 2 \beta_{9} ) q^{61} + ( 20 \beta_{1} - 20 \beta_{2} - 152 \beta_{3} - 14 \beta_{5} - 24 \beta_{6} + 14 \beta_{8} + 2 \beta_{9} ) q^{62} + ( -30 - 54 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 20 \beta_{7} ) q^{63} + 64 q^{64} + ( -80 \beta_{1} + 80 \beta_{2} - 230 \beta_{3} - 3 \beta_{5} - 41 \beta_{6} + 3 \beta_{8} + 11 \beta_{9} ) q^{65} + ( -52 + 108 \beta_{2} + 12 \beta_{4} - 24 \beta_{5} - 8 \beta_{7} ) q^{66} + ( 150 - 60 \beta_{1} + 150 \beta_{3} + 23 \beta_{4} + 23 \beta_{6} - 28 \beta_{7} - 8 \beta_{8} - 28 \beta_{9} ) q^{67} + ( -52 - 16 \beta_{2} - 16 \beta_{4} + 16 \beta_{7} ) q^{68} + ( 125 - 115 \beta_{1} + 125 \beta_{3} - 21 \beta_{4} - 21 \beta_{6} - 2 \beta_{7} + 21 \beta_{8} - 2 \beta_{9} ) q^{69} + ( -102 - 30 \beta_{1} - 102 \beta_{3} - 16 \beta_{4} - 16 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} ) q^{70} + ( 108 + 12 \beta_{1} + 108 \beta_{3} + 15 \beta_{4} + 15 \beta_{6} + 7 \beta_{7} + 16 \beta_{8} + 7 \beta_{9} ) q^{71} + ( -40 \beta_{1} + 40 \beta_{2} + 64 \beta_{3} - 16 \beta_{5} - 16 \beta_{6} + 16 \beta_{8} + 8 \beta_{9} ) q^{72} + ( 31 - 39 \beta_{2} + 19 \beta_{4} - 15 \beta_{5} - 20 \beta_{7} ) q^{73} + ( 24 - 2 \beta_{1} - 32 \beta_{2} + 14 \beta_{3} - 36 \beta_{4} + 22 \beta_{5} - 4 \beta_{6} + 22 \beta_{7} - 12 \beta_{8} + 6 \beta_{9} ) q^{74} + ( -286 + 34 \beta_{2} - 46 \beta_{4} + 2 \beta_{7} ) q^{75} + ( 8 \beta_{1} - 8 \beta_{2} - 32 \beta_{3} + 28 \beta_{6} + 16 \beta_{9} ) q^{76} + ( 170 - 34 \beta_{1} + 170 \beta_{3} + 34 \beta_{4} + 34 \beta_{6} + 40 \beta_{7} + 40 \beta_{9} ) q^{77} + ( 280 - 80 \beta_{1} + 280 \beta_{3} - 42 \beta_{4} - 42 \beta_{6} + 26 \beta_{7} + 22 \beta_{8} + 26 \beta_{9} ) q^{78} + ( -156 - 6 \beta_{1} - 156 \beta_{3} + 53 \beta_{4} + 53 \beta_{6} - 6 \beta_{7} + 20 \beta_{8} - 6 \beta_{9} ) q^{79} + ( 16 \beta_{2} + 16 \beta_{4} ) q^{80} + ( -522 + 223 \beta_{1} - 522 \beta_{3} + \beta_{4} + \beta_{6} - 8 \beta_{7} - 31 \beta_{8} - 8 \beta_{9} ) q^{81} + ( -144 - 6 \beta_{2} + 30 \beta_{4} + 14 \beta_{5} + 8 \beta_{7} ) q^{82} + ( -107 \beta_{1} + 107 \beta_{2} - 441 \beta_{3} - 24 \beta_{6} - 24 \beta_{9} ) q^{83} + ( 92 - 4 \beta_{2} - 24 \beta_{4} + 8 \beta_{5} - 4 \beta_{7} ) q^{84} + ( -292 - 61 \beta_{2} - 61 \beta_{4} - 16 \beta_{5} - 12 \beta_{7} ) q^{85} + ( 86 \beta_{1} - 86 \beta_{2} - 98 \beta_{3} - 16 \beta_{5} - 2 \beta_{6} + 16 \beta_{8} - 8 \beta_{9} ) q^{86} + ( 536 - 10 \beta_{1} + 536 \beta_{3} - 20 \beta_{4} - 20 \beta_{6} - 7 \beta_{7} + 28 \beta_{8} - 7 \beta_{9} ) q^{87} + ( -64 + 16 \beta_{2} - 24 \beta_{5} - 8 \beta_{7} ) q^{88} + ( 109 \beta_{1} - 109 \beta_{2} - 180 \beta_{3} - 11 \beta_{5} - \beta_{6} + 11 \beta_{8} - 16 \beta_{9} ) q^{89} + ( 48 \beta_{1} - 48 \beta_{2} - 362 \beta_{3} + 28 \beta_{5} - 6 \beta_{6} - 28 \beta_{8} - 14 \beta_{9} ) q^{90} + ( -30 \beta_{1} + 30 \beta_{2} + 282 \beta_{3} + 67 \beta_{5} - 5 \beta_{6} - 67 \beta_{8} - 27 \beta_{9} ) q^{91} + ( 208 - 8 \beta_{1} + 208 \beta_{3} + 8 \beta_{4} + 8 \beta_{6} - 12 \beta_{7} + 16 \beta_{8} - 12 \beta_{9} ) q^{92} + ( 662 - 26 \beta_{1} + 662 \beta_{3} - 74 \beta_{4} - 74 \beta_{6} + 20 \beta_{7} + 12 \beta_{8} + 20 \beta_{9} ) q^{93} + ( -8 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 18 \beta_{5} - 12 \beta_{6} - 18 \beta_{8} - 26 \beta_{9} ) q^{94} + ( 80 \beta_{1} - 80 \beta_{2} + 654 \beta_{3} - 12 \beta_{5} + 67 \beta_{6} + 12 \beta_{8} - 12 \beta_{9} ) q^{95} + ( -32 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{96} + ( -480 + 69 \beta_{2} + 102 \beta_{4} + 34 \beta_{5} + 3 \beta_{7} ) q^{97} + ( 60 - 26 \beta_{1} + 60 \beta_{3} + 6 \beta_{7} + 24 \beta_{8} + 6 \beta_{9} ) q^{98} + ( 340 \beta_{1} - 340 \beta_{2} - 1374 \beta_{3} + 53 \beta_{5} + 86 \beta_{6} - 53 \beta_{8} - 29 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} - 5q^{3} - 20q^{4} - q^{5} + 20q^{6} - q^{7} + 80q^{8} - 38q^{9} + O(q^{10}) \) \( 10q - 10q^{2} - 5q^{3} - 20q^{4} - q^{5} + 20q^{6} - q^{7} + 80q^{8} - 38q^{9} + 4q^{10} - 80q^{11} - 20q^{12} + 73q^{13} + 4q^{14} + 113q^{15} - 80q^{16} + 69q^{17} - 76q^{18} + 33q^{19} - 4q^{20} - 109q^{21} + 80q^{22} - 524q^{23} - 40q^{24} + 132q^{25} - 292q^{26} + 898q^{27} - 4q^{28} + 296q^{29} + 226q^{30} - 784q^{31} - 160q^{32} + 124q^{33} + 138q^{34} - 263q^{35} + 304q^{36} + 24q^{37} - 132q^{38} + 679q^{39} - 8q^{40} + 345q^{41} - 218q^{42} - 492q^{43} + 160q^{44} - 1816q^{45} + 524q^{46} - 32q^{47} + 160q^{48} + 150q^{49} + 264q^{50} + 342q^{51} + 292q^{52} - 19q^{53} - 898q^{54} + 340q^{55} - 8q^{56} + 207q^{57} - 296q^{58} - 105q^{59} - 904q^{60} + 219q^{61} + 784q^{62} - 304q^{63} + 640q^{64} + 1191q^{65} - 496q^{66} + 773q^{67} - 552q^{68} + 604q^{69} - 526q^{70} + 555q^{71} - 304q^{72} + 348q^{73} + 102q^{74} - 2952q^{75} + 132q^{76} + 884q^{77} + 1358q^{78} - 727q^{79} + 32q^{80} - 2609q^{81} - 1380q^{82} + 2229q^{83} + 872q^{84} - 3042q^{85} + 492q^{86} + 2660q^{87} - 640q^{88} + 901q^{89} + 1816q^{90} - 1405q^{91} + 1048q^{92} + 3236q^{93} + 32q^{94} - 3337q^{95} - 160q^{96} - 4596q^{97} + 300q^{98} + 6784q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 84 x^{8} - 140 x^{7} + 6309 x^{6} - 5214 x^{5} + 67648 x^{4} + 164178 x^{3} + 511389 x^{2} + 497502 x + 443556\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2923369 \nu^{9} + 1291648158 \nu^{8} - 11318780244 \nu^{7} + 97444452524 \nu^{6} - 1041192911556 \nu^{5} + 8941322845902 \nu^{4} - 76851150344224 \nu^{3} + 73510030975680 \nu^{2} + 72259964551152 \nu - 620934447646350\)\()/ 703951748722773 \)
\(\beta_{3}\)\(=\)\((\)\(-103592667275 \nu^{9} + 216329306 \nu^{8} - 8797366014792 \nu^{7} + 15340563156556 \nu^{6} - 660777027324751 \nu^{5} + 617180442626994 \nu^{4} - 7669494646415948 \nu^{3} - 11320651802402374 \nu^{2} - 58415892817295295 \nu - 56884796531432298\)\()/ 52092429405485202 \)
\(\beta_{4}\)\(=\)\((\)\(82964449469 \nu^{9} - 780820708113 \nu^{8} + 6842372630934 \nu^{7} - 79298402920351 \nu^{6} + 629416750566366 \nu^{5} - 5405163931650297 \nu^{4} + 10255188571799297 \nu^{3} - 44437917620472480 \nu^{2} - 43682233694673672 \nu - 151653093189987762\)\()/ 21118552461683190 \)
\(\beta_{5}\)\(=\)\((\)\(-276538800347 \nu^{9} + 3364748327259 \nu^{8} - 29485465261362 \nu^{7} + 318765756856633 \nu^{6} - 2712311464862538 \nu^{5} + 23292179764972371 \nu^{4} - 72853561118616281 \nu^{3} + 191493908174780640 \nu^{2} + 188237480420177496 \nu + 260020960627171716\)\()/ 21118552461683190 \)
\(\beta_{6}\)\(=\)\((\)\(14502998162881 \nu^{9} - 622808271963 \nu^{8} + 1201327539595005 \nu^{7} - 1882111164964658 \nu^{6} + 90630484257083826 \nu^{5} - 61976010456722316 \nu^{4} + 863941772170989457 \nu^{3} + 3385101636576729159 \nu^{2} + 6453480882011100669 \nu + 6268457212298551206\)\()/ 781386441082278030 \)
\(\beta_{7}\)\(=\)\((\)\(378746420581 \nu^{9} - 4253543126982 \nu^{8} + 37274020494276 \nu^{7} - 409020400061429 \nu^{6} + 3428765740407924 \nu^{5} - 29444785022728758 \nu^{4} + 80760297682695928 \nu^{3} - 242076826482678720 \nu^{2} - 237960216695881008 \nu - 798612045641956998\)\()/ 10559276230841595 \)
\(\beta_{8}\)\(=\)\((\)\(-21187262572329 \nu^{9} + 33668105489074 \nu^{8} - 1770475510640473 \nu^{7} + 5662916693503905 \nu^{6} - 137232325886487043 \nu^{5} + 310299148572658375 \nu^{4} - 1502883505705851906 \nu^{3} - 2110652418918317709 \nu^{2} - 1851170184076462500 \nu - 802609209663182532\)\()/ 260462147027426010 \)
\(\beta_{9}\)\(=\)\((\)\(94253847831539 \nu^{9} + 20662805213298 \nu^{8} + 7683965959724250 \nu^{7} - 11150616860155192 \nu^{6} + 572203665882183294 \nu^{5} - 303331120279030014 \nu^{4} + 4760688828058942478 \nu^{3} + 19957398724377803151 \nu^{2} + 34919597698843104711 \nu + 33836539355339861034\)\()/ 390693220541139015 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + 34 \beta_{3} + 3 \beta_{2} - 3 \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} + 7 \beta_{5} + 5 \beta_{4} - 68 \beta_{2} + 41\)
\(\nu^{4}\)\(=\)\(-75 \beta_{9} - 153 \beta_{8} - 75 \beta_{7} + 171 \beta_{6} + 171 \beta_{4} - 2259 \beta_{3} + 334 \beta_{1} - 2259\)
\(\nu^{5}\)\(=\)\(238 \beta_{9} + 728 \beta_{8} + 280 \beta_{6} - 728 \beta_{5} + 6490 \beta_{3} + 5175 \beta_{2} - 5175 \beta_{1}\)
\(\nu^{6}\)\(=\)\(5693 \beta_{7} + 11848 \beta_{5} - 12520 \beta_{4} - 31241 \beta_{2} + 167228\)
\(\nu^{7}\)\(=\)\(-24414 \beta_{9} - 67965 \beta_{8} - 24414 \beta_{7} - 6483 \beta_{6} - 6483 \beta_{4} - 695307 \beta_{3} + 409282 \beta_{1} - 695307\)
\(\nu^{8}\)\(=\)\(440179 \beta_{9} + 936563 \beta_{8} - 901013 \beta_{6} - 936563 \beta_{5} + 12841285 \beta_{3} + 2782284 \beta_{2} - 2782284 \beta_{1}\)
\(\nu^{9}\)\(=\)\(2321450 \beta_{7} + 6096502 \beta_{5} - 654874 \beta_{4} - 32923277 \beta_{2} + 66768212\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−4.57142 7.91792i
−0.858393 1.48678i
−0.632114 1.09485i
2.13869 + 3.70432i
3.92323 + 6.79524i
−4.57142 + 7.91792i
−0.858393 + 1.48678i
−0.632114 + 1.09485i
2.13869 3.70432i
3.92323 6.79524i
−1.00000 + 1.73205i −5.07142 8.78395i −2.00000 3.46410i 3.82762 + 6.62963i 20.2857 −2.78651 4.82638i 8.00000 −37.9385 + 65.7114i −15.3105
47.2 −1.00000 + 1.73205i −1.35839 2.35281i −2.00000 3.46410i 0.388081 + 0.672176i 5.43357 11.9492 + 20.6965i 8.00000 9.80954 16.9906i −1.55232
47.3 −1.00000 + 1.73205i −1.13211 1.96088i −2.00000 3.46410i 2.10613 + 3.64793i 4.52846 −10.8960 18.8724i 8.00000 10.9366 18.9428i −8.42453
47.4 −1.00000 + 1.73205i 1.63869 + 2.83829i −2.00000 3.46410i −9.76482 16.9132i −6.55476 −7.16860 12.4164i 8.00000 8.12939 14.0805i 39.0593
47.5 −1.00000 + 1.73205i 3.42323 + 5.92921i −2.00000 3.46410i 2.94299 + 5.09740i −13.6929 8.40192 + 14.5526i 8.00000 −9.93705 + 17.2115i −11.7719
63.1 −1.00000 1.73205i −5.07142 + 8.78395i −2.00000 + 3.46410i 3.82762 6.62963i 20.2857 −2.78651 + 4.82638i 8.00000 −37.9385 65.7114i −15.3105
63.2 −1.00000 1.73205i −1.35839 + 2.35281i −2.00000 + 3.46410i 0.388081 0.672176i 5.43357 11.9492 20.6965i 8.00000 9.80954 + 16.9906i −1.55232
63.3 −1.00000 1.73205i −1.13211 + 1.96088i −2.00000 + 3.46410i 2.10613 3.64793i 4.52846 −10.8960 + 18.8724i 8.00000 10.9366 + 18.9428i −8.42453
63.4 −1.00000 1.73205i 1.63869 2.83829i −2.00000 + 3.46410i −9.76482 + 16.9132i −6.55476 −7.16860 + 12.4164i 8.00000 8.12939 + 14.0805i 39.0593
63.5 −1.00000 1.73205i 3.42323 5.92921i −2.00000 + 3.46410i 2.94299 5.09740i −13.6929 8.40192 14.5526i 8.00000 −9.93705 17.2115i −11.7719
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.c.b 10
3.b odd 2 1 666.4.f.d 10
37.c even 3 1 inner 74.4.c.b 10
111.i odd 6 1 666.4.f.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.c.b 10 1.a even 1 1 trivial
74.4.c.b 10 37.c even 3 1 inner
666.4.f.d 10 3.b odd 2 1
666.4.f.d 10 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{5} \)
$3$ \( 1960000 + 896000 T + 650400 T^{2} + 97120 T^{3} + 69944 T^{4} + 7728 T^{5} + 5696 T^{6} - 26 T^{7} + 99 T^{8} + 5 T^{9} + T^{10} \)
$5$ \( 8277129 - 14692839 T + 20344711 T^{2} - 8767874 T^{3} + 2716837 T^{4} - 477433 T^{5} + 63629 T^{6} - 4234 T^{7} + 247 T^{8} + T^{9} + T^{10} \)
$7$ \( 488936577600 + 94716253440 T + 20043285696 T^{2} + 765266016 T^{3} + 111103128 T^{4} + 2323896 T^{5} + 478492 T^{6} + 4066 T^{7} + 783 T^{8} + T^{9} + T^{10} \)
$11$ \( ( 147890304 + 5040208 T - 156272 T^{2} - 4728 T^{3} + 40 T^{4} + T^{5} )^{2} \)
$13$ \( 432040198066331904 - 4179151029795840 T + 376561940092416 T^{2} - 5251331127168 T^{3} + 254663110800 T^{4} - 3036701424 T^{5} + 72808560 T^{6} - 550620 T^{7} + 11797 T^{8} - 73 T^{9} + T^{10} \)
$17$ \( 472303425246849 + 384807294362043 T + 301606093334799 T^{2} + 10151599568994 T^{3} + 483175640493 T^{4} - 3142999539 T^{5} + 124773165 T^{6} - 390558 T^{7} + 14991 T^{8} - 69 T^{9} + T^{10} \)
$19$ \( 11208770916650465536 - 361678744752455424 T + 13615699607791360 T^{2} - 84300791903872 T^{3} + 2599875341584 T^{4} - 8979587728 T^{5} + 393561184 T^{6} - 437236 T^{7} + 23053 T^{8} - 33 T^{9} + T^{10} \)
$23$ \( ( -8739280512 - 379839816 T - 4333176 T^{2} + 2292 T^{3} + 262 T^{4} + T^{5} )^{2} \)
$29$ \( ( -103287670938 + 470419899 T + 10764630 T^{2} - 62316 T^{3} - 148 T^{4} + T^{5} )^{2} \)
$31$ \( ( -102244293760 - 2685067952 T - 20922064 T^{2} - 14456 T^{3} + 392 T^{4} + T^{5} )^{2} \)
$37$ \( \)\(33\!\cdots\!93\)\( - \)\(15\!\cdots\!44\)\( T + 8951894598824698837 T^{2} - 2302477747983782 T^{3} + 249222250801957 T^{4} - 306264956102 T^{5} + 4920187369 T^{6} - 897398 T^{7} + 68881 T^{8} - 24 T^{9} + T^{10} \)
$41$ \( \)\(21\!\cdots\!21\)\( + \)\(11\!\cdots\!47\)\( T + 3880032682527424287 T^{2} - 30618933769155222 T^{3} + 420410644432533 T^{4} - 1695951042975 T^{5} + 10568077821 T^{6} - 29424246 T^{7} + 164103 T^{8} - 345 T^{9} + T^{10} \)
$43$ \( ( 838671946912 + 10345859880 T - 37515784 T^{2} - 220192 T^{3} + 246 T^{4} + T^{5} )^{2} \)
$47$ \( ( -25825580928 - 1293354608 T - 21689168 T^{2} - 124248 T^{3} + 16 T^{4} + T^{5} )^{2} \)
$53$ \( \)\(29\!\cdots\!04\)\( + \)\(12\!\cdots\!56\)\( T + \)\(70\!\cdots\!00\)\( T^{2} + 12974197267706412672 T^{3} + 46472623275688848 T^{4} + 54348224056080 T^{5} + 222692066400 T^{6} + 137129460 T^{7} + 545485 T^{8} + 19 T^{9} + T^{10} \)
$59$ \( 18025318258417483776 + 1715094373027037184 T + 117988142675853312 T^{2} + 3779915301086208 T^{3} + 89011237394560 T^{4} + 733823801472 T^{5} + 5286425232 T^{6} + 14850932 T^{7} + 72381 T^{8} + 105 T^{9} + T^{10} \)
$61$ \( \)\(11\!\cdots\!25\)\( + \)\(73\!\cdots\!85\)\( T + \)\(24\!\cdots\!19\)\( T^{2} - 2288758310834088818 T^{3} + 30253357268681725 T^{4} - 92442365726453 T^{5} + 278539699357 T^{6} - 262008506 T^{7} + 529159 T^{8} - 219 T^{9} + T^{10} \)
$67$ \( \)\(12\!\cdots\!16\)\( - \)\(58\!\cdots\!48\)\( T + \)\(84\!\cdots\!08\)\( T^{2} + 10501233212823683328 T^{3} + 291116495237436144 T^{4} - 306659109181776 T^{5} + 887224470808 T^{6} - 450544952 T^{7} + 1335921 T^{8} - 773 T^{9} + T^{10} \)
$71$ \( \)\(17\!\cdots\!00\)\( - \)\(46\!\cdots\!60\)\( T + 84855390273014272704 T^{2} - 136108719160185312 T^{3} + 2802344636761272 T^{4} - 4441917699432 T^{5} + 43460476092 T^{6} - 40252098 T^{7} + 435363 T^{8} - 555 T^{9} + T^{10} \)
$73$ \( ( -551579403872 + 2362796688 T + 135541040 T^{2} - 520888 T^{3} - 174 T^{4} + T^{5} )^{2} \)
$79$ \( \)\(28\!\cdots\!16\)\( + \)\(30\!\cdots\!36\)\( T + \)\(37\!\cdots\!28\)\( T^{2} + 55169749231795479552 T^{3} + 428237841351752688 T^{4} + 524741872529712 T^{5} + 1144232537416 T^{6} + 673532920 T^{7} + 1382937 T^{8} + 727 T^{9} + T^{10} \)
$83$ \( \)\(61\!\cdots\!84\)\( + \)\(96\!\cdots\!36\)\( T + \)\(48\!\cdots\!96\)\( T^{2} - \)\(27\!\cdots\!64\)\( T^{3} + 1500574482403868632 T^{4} - 1290249202888608 T^{5} + 3680601937224 T^{6} - 3849525814 T^{7} + 4464867 T^{8} - 2229 T^{9} + T^{10} \)
$89$ \( \)\(20\!\cdots\!49\)\( - \)\(13\!\cdots\!65\)\( T + \)\(96\!\cdots\!71\)\( T^{2} + \)\(17\!\cdots\!22\)\( T^{3} + 659252356679779117 T^{4} - 50778872355043 T^{5} + 1243582608397 T^{6} - 282843214 T^{7} + 1818943 T^{8} - 901 T^{9} + T^{10} \)
$97$ \( ( -1508402031813726 - 6163097256435 T - 7371952788 T^{2} - 1416262 T^{3} + 2298 T^{4} + T^{5} )^{2} \)
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