Properties

Label 105.4.i.e
Level $105$
Weight $4$
Character orbit 105.i
Analytic conductor $6.195$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 31 x^{8} + 26 x^{7} + 738 x^{6} + 352 x^{5} + 5008 x^{4} + 5368 x^{3} + 26728 x^{2} + 13776 x + 7056\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -3 \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{8} ) q^{4} + ( 5 - 5 \beta_{4} ) q^{5} + 3 \beta_{3} q^{6} + ( 7 + \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{6} + \beta_{9} ) q^{7} + ( -12 + \beta_{1} - \beta_{2} + 5 \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{8} + ( -9 + 9 \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -3 \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{8} ) q^{4} + ( 5 - 5 \beta_{4} ) q^{5} + 3 \beta_{3} q^{6} + ( 7 + \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{6} + \beta_{9} ) q^{7} + ( -12 + \beta_{1} - \beta_{2} + 5 \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{8} + ( -9 + 9 \beta_{4} ) q^{9} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{10} + ( \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{8} + \beta_{9} ) q^{11} + ( -12 - 3 \beta_{1} + 12 \beta_{4} - 3 \beta_{8} ) q^{12} + ( -7 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{13} + ( 14 \beta_{1} - \beta_{2} + 3 \beta_{3} - 18 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{14} -15 q^{15} + ( -16 - 18 \beta_{1} + 3 \beta_{3} + 16 \beta_{4} + \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} ) q^{16} + ( 6 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 30 \beta_{4} + 2 \beta_{5} + \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{17} + ( -9 \beta_{1} - 9 \beta_{3} ) q^{18} + ( 13 - 16 \beta_{1} + 2 \beta_{3} - 13 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 6 \beta_{9} ) q^{19} + ( -20 - 5 \beta_{2} + 5 \beta_{3} ) q^{20} + ( -9 - 3 \beta_{2} + 3 \beta_{3} - 12 \beta_{4} + 3 \beta_{8} - 3 \beta_{9} ) q^{21} + ( -30 - 6 \beta_{1} + \beta_{2} - 39 \beta_{3} - 12 \beta_{5} + 6 \beta_{7} - 6 \beta_{9} ) q^{22} + ( 18 + 28 \beta_{1} - 3 \beta_{3} - 18 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} ) q^{23} + ( -15 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} + 36 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{24} -25 \beta_{4} q^{25} + ( 18 - 22 \beta_{1} - \beta_{3} - 18 \beta_{4} + \beta_{5} - \beta_{6} + 12 \beta_{8} - 3 \beta_{9} ) q^{26} + 27 q^{27} + ( -24 + 10 \beta_{1} - 11 \beta_{2} + 26 \beta_{3} - 100 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 15 \beta_{8} + 3 \beta_{9} ) q^{28} + ( 48 + 2 \beta_{1} + \beta_{2} - 40 \beta_{3} + 5 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{29} -15 \beta_{1} q^{30} + ( 29 \beta_{1} + 3 \beta_{2} + 30 \beta_{3} + 91 \beta_{4} - \beta_{5} - 3 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} ) q^{31} + ( -29 \beta_{1} + 23 \beta_{2} - 28 \beta_{3} + 108 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - 23 \beta_{8} + 2 \beta_{9} ) q^{32} + ( 18 - 6 \beta_{1} - 3 \beta_{3} - 18 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} + 12 \beta_{8} - 6 \beta_{9} ) q^{33} + ( -66 + 7 \beta_{1} - 30 \beta_{2} + 14 \beta_{3} + 4 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} ) q^{34} + ( 20 + 5 \beta_{1} + 5 \beta_{3} - 35 \beta_{4} + 5 \beta_{6} + 5 \beta_{8} ) q^{35} + ( 36 + 9 \beta_{2} - 9 \beta_{3} ) q^{36} + ( -5 + 12 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} - 15 \beta_{8} - \beta_{9} ) q^{37} + ( 11 \beta_{1} + 14 \beta_{2} + 13 \beta_{3} + 168 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 14 \beta_{8} - 4 \beta_{9} ) q^{38} + ( 12 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} + 21 \beta_{4} + 6 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} - 9 \beta_{8} - 3 \beta_{9} ) q^{39} + ( -60 - 20 \beta_{1} + 5 \beta_{3} + 60 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{40} + ( -114 + 7 \beta_{1} + 16 \beta_{2} + 29 \beta_{3} + 19 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} ) q^{41} + ( -54 - 9 \beta_{1} + 3 \beta_{2} + 33 \beta_{3} + 54 \beta_{4} + 6 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} ) q^{42} + ( 35 + 13 \beta_{1} + 10 \beta_{2} + 15 \beta_{3} + 23 \beta_{5} - 3 \beta_{6} - 13 \beta_{7} + 13 \beta_{9} ) q^{43} + ( 348 + 44 \beta_{1} - 5 \beta_{3} - 348 \beta_{4} - \beta_{5} - 5 \beta_{6} + 6 \beta_{7} + 43 \beta_{8} - 9 \beta_{9} ) q^{44} + 45 \beta_{4} q^{45} + ( -5 \beta_{1} - 45 \beta_{2} - 3 \beta_{3} - 330 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} + 45 \beta_{8} - 4 \beta_{9} ) q^{46} + ( 102 - 12 \beta_{1} - 11 \beta_{3} - 102 \beta_{4} - \beta_{5} - 11 \beta_{6} + 12 \beta_{7} - 3 \beta_{8} - 21 \beta_{9} ) q^{47} + ( 48 - 9 \beta_{1} + 9 \beta_{2} - 63 \beta_{3} - 12 \beta_{5} + 6 \beta_{6} + 9 \beta_{7} - 9 \beta_{9} ) q^{48} + ( -68 + 60 \beta_{1} + 30 \beta_{2} + 33 \beta_{3} + 57 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} ) q^{49} + 25 \beta_{3} q^{50} + ( 90 - 12 \beta_{1} + 6 \beta_{3} - 90 \beta_{4} + 15 \beta_{5} + 6 \beta_{6} - 21 \beta_{7} - 9 \beta_{8} - 3 \beta_{9} ) q^{51} + ( 61 \beta_{1} - \beta_{2} + 63 \beta_{3} + 220 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} + 8 \beta_{7} + \beta_{8} + 10 \beta_{9} ) q^{52} + ( -75 \beta_{1} + 11 \beta_{2} - 80 \beta_{3} + 216 \beta_{4} + 5 \beta_{5} - 13 \beta_{6} - 3 \beta_{7} - 11 \beta_{8} - 8 \beta_{9} ) q^{53} + 27 \beta_{1} q^{54} + ( 30 - 5 \beta_{1} + 20 \beta_{2} + 5 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - 5 \beta_{9} ) q^{55} + ( -120 + 25 \beta_{1} - 19 \beta_{2} + 143 \beta_{3} + 36 \beta_{4} + 16 \beta_{5} - 27 \beta_{7} - 16 \beta_{8} + 9 \beta_{9} ) q^{56} + ( -39 - 6 \beta_{1} - 6 \beta_{2} - 54 \beta_{3} + 12 \beta_{6} + 6 \beta_{7} - 6 \beta_{9} ) q^{57} + ( 498 + 89 \beta_{1} + 3 \beta_{3} - 498 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 24 \beta_{8} + 9 \beta_{9} ) q^{58} + ( 32 \beta_{1} - 33 \beta_{2} + 18 \beta_{3} + 84 \beta_{4} + 14 \beta_{5} - 31 \beta_{6} - 3 \beta_{7} + 33 \beta_{8} - 17 \beta_{9} ) q^{59} + ( -15 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} + 60 \beta_{4} - 15 \beta_{8} ) q^{60} + ( 124 - 66 \beta_{1} - 6 \beta_{3} - 124 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 10 \beta_{7} - 18 \beta_{8} - 8 \beta_{9} ) q^{61} + ( -336 - 5 \beta_{1} - 7 \beta_{2} - 98 \beta_{3} + 10 \beta_{6} + 5 \beta_{7} - 5 \beta_{9} ) q^{62} + ( -36 - 9 \beta_{1} - 9 \beta_{3} + 63 \beta_{4} - 9 \beta_{6} - 9 \beta_{8} ) q^{63} + ( 196 - \beta_{1} + 3 \beta_{2} - 153 \beta_{3} - 20 \beta_{5} - 18 \beta_{6} + \beta_{7} - \beta_{9} ) q^{64} + ( -35 + 10 \beta_{1} - 10 \beta_{3} + 35 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} + 15 \beta_{7} - 15 \beta_{8} - 15 \beta_{9} ) q^{65} + ( 117 \beta_{1} - 3 \beta_{2} + 99 \beta_{3} + 90 \beta_{4} + 18 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} + 3 \beta_{8} ) q^{66} + ( -127 \beta_{1} - 20 \beta_{2} - 128 \beta_{3} - 107 \beta_{4} + \beta_{5} + 12 \beta_{6} + 14 \beta_{7} + 20 \beta_{8} + 13 \beta_{9} ) q^{67} + ( -264 - 252 \beta_{1} + 28 \beta_{3} + 264 \beta_{4} + 10 \beta_{5} + 28 \beta_{6} - 38 \beta_{7} + 8 \beta_{8} + 46 \beta_{9} ) q^{68} + ( -54 + 9 \beta_{1} + 24 \beta_{2} + 93 \beta_{3} + 15 \beta_{5} - 3 \beta_{6} - 9 \beta_{7} + 9 \beta_{9} ) q^{69} + ( -90 + 55 \beta_{1} + 70 \beta_{3} - 5 \beta_{5} - 5 \beta_{6} + 15 \beta_{7} - 5 \beta_{8} ) q^{70} + ( -54 - 22 \beta_{1} - 9 \beta_{2} - 40 \beta_{3} - 45 \beta_{5} - \beta_{6} + 22 \beta_{7} - 22 \beta_{9} ) q^{71} + ( 108 + 36 \beta_{1} - 9 \beta_{3} - 108 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} + 18 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} ) q^{72} + ( -21 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 397 \beta_{4} - 17 \beta_{5} + 22 \beta_{6} - 12 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} ) q^{73} + ( -101 \beta_{1} - 30 \beta_{2} - 90 \beta_{3} - 150 \beta_{4} - 11 \beta_{5} + 5 \beta_{6} - 17 \beta_{7} + 30 \beta_{8} - 6 \beta_{9} ) q^{74} + ( -75 + 75 \beta_{4} ) q^{75} + ( -28 - 2 \beta_{1} + 33 \beta_{2} - 141 \beta_{3} - 24 \beta_{5} - 20 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{76} + ( -384 + 9 \beta_{1} + 60 \beta_{2} - 10 \beta_{3} + 576 \beta_{4} - 30 \beta_{5} + 6 \beta_{6} + 27 \beta_{7} - 47 \beta_{8} - 24 \beta_{9} ) q^{77} + ( -54 + 3 \beta_{1} - 36 \beta_{2} - 63 \beta_{3} - 6 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} ) q^{78} + ( 31 - 88 \beta_{1} + 19 \beta_{3} - 31 \beta_{4} + 17 \beta_{5} + 19 \beta_{6} - 36 \beta_{7} + 15 \beta_{8} + 21 \beta_{9} ) q^{79} + ( -105 \beta_{1} + 15 \beta_{2} - 90 \beta_{3} + 80 \beta_{4} - 15 \beta_{5} + 25 \beta_{6} - 5 \beta_{7} - 15 \beta_{8} + 10 \beta_{9} ) q^{80} -81 \beta_{4} q^{81} + ( -294 - 55 \beta_{1} - 2 \beta_{3} + 294 \beta_{4} - 26 \beta_{5} - 2 \beta_{6} + 28 \beta_{7} - 91 \beta_{8} + 22 \beta_{9} ) q^{82} + ( -192 + 25 \beta_{2} - 120 \beta_{3} - 27 \beta_{5} - 27 \beta_{6} ) q^{83} + ( -300 - 78 \beta_{1} - 12 \beta_{2} - 48 \beta_{3} + 372 \beta_{4} - 6 \beta_{5} + 18 \beta_{6} - 3 \beta_{7} - 33 \beta_{8} + 9 \beta_{9} ) q^{84} + ( 150 + 10 \beta_{1} - 15 \beta_{2} + 30 \beta_{3} + 35 \beta_{5} + 15 \beta_{6} - 10 \beta_{7} + 10 \beta_{9} ) q^{85} + ( -6 + 70 \beta_{1} + 16 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 16 \beta_{6} - 12 \beta_{7} - 81 \beta_{8} + 36 \beta_{9} ) q^{86} + ( 120 \beta_{1} - 3 \beta_{2} + 126 \beta_{3} - 144 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{87} + ( 441 \beta_{1} - 55 \beta_{2} + 436 \beta_{3} - 744 \beta_{4} + 5 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} + 55 \beta_{8} - 2 \beta_{9} ) q^{88} + ( 612 - 82 \beta_{1} - 612 \beta_{4} + 9 \beta_{5} - 9 \beta_{7} - 13 \beta_{8} - 9 \beta_{9} ) q^{89} -45 \beta_{3} q^{90} + ( -487 - 103 \beta_{1} - 55 \beta_{2} - 142 \beta_{3} + 375 \beta_{4} - 27 \beta_{5} + 7 \beta_{6} + 25 \beta_{7} + 69 \beta_{8} + 8 \beta_{9} ) q^{91} + ( 204 + 17 \beta_{1} - 31 \beta_{2} + 421 \beta_{3} + 54 \beta_{5} + 20 \beta_{6} - 17 \beta_{7} + 17 \beta_{9} ) q^{92} + ( 273 - 90 \beta_{1} - 3 \beta_{3} - 273 \beta_{4} - 15 \beta_{5} - 3 \beta_{6} + 18 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} ) q^{93} + ( 101 \beta_{1} - 25 \beta_{2} + 82 \beta_{3} + 204 \beta_{4} + 19 \beta_{5} - 21 \beta_{6} + 17 \beta_{7} + 25 \beta_{8} - 2 \beta_{9} ) q^{94} + ( -90 \beta_{1} - 10 \beta_{2} - 80 \beta_{3} - 65 \beta_{4} - 10 \beta_{5} + 30 \beta_{6} + 10 \beta_{7} + 10 \beta_{8} + 20 \beta_{9} ) q^{95} + ( 324 + 84 \beta_{1} - 3 \beta_{3} - 324 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 69 \beta_{8} - 9 \beta_{9} ) q^{96} + ( -424 - 26 \beta_{1} - 42 \beta_{2} + 152 \beta_{3} - 10 \beta_{5} + 42 \beta_{6} + 26 \beta_{7} - 26 \beta_{9} ) q^{97} + ( -396 + 143 \beta_{1} - 39 \beta_{2} - 61 \beta_{3} - 324 \beta_{4} - 31 \beta_{5} - 19 \beta_{6} + 65 \beta_{7} + 27 \beta_{8} - 32 \beta_{9} ) q^{98} + ( -54 + 9 \beta_{1} - 36 \beta_{2} - 9 \beta_{3} + 9 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 9 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + q^{2} - 15q^{3} - 21q^{4} + 25q^{5} - 6q^{6} + 56q^{7} - 138q^{8} - 45q^{9} + O(q^{10}) \) \( 10q + q^{2} - 15q^{3} - 21q^{4} + 25q^{5} - 6q^{6} + 56q^{7} - 138q^{8} - 45q^{9} - 5q^{10} + 33q^{11} - 63q^{12} - 46q^{13} - 73q^{14} - 150q^{15} - 113q^{16} + 136q^{17} + 9q^{18} + 39q^{19} - 210q^{20} - 147q^{21} - 174q^{22} + 133q^{23} + 207q^{24} - 125q^{25} + 73q^{26} + 270q^{27} - 809q^{28} + 544q^{29} - 15q^{30} + 430q^{31} + 573q^{32} + 99q^{33} - 744q^{34} + 35q^{35} + 378q^{36} - 3q^{37} + 837q^{38} + 69q^{39} - 345q^{40} - 1254q^{41} - 372q^{42} + 216q^{43} + 1809q^{44} + 225q^{45} - 1637q^{46} + 553q^{47} + 678q^{48} - 386q^{49} - 50q^{50} + 408q^{51} + 1047q^{52} + 1135q^{53} + 27q^{54} + 330q^{55} - 1356q^{56} - 234q^{57} + 2564q^{58} + 332q^{59} + 315q^{60} + 584q^{61} - 3124q^{62} - 63q^{63} + 2274q^{64} - 115q^{65} + 261q^{66} - 412q^{67} - 1712q^{68} - 798q^{69} - 985q^{70} - 284q^{71} + 621q^{72} + 2074q^{73} - 605q^{74} - 375q^{75} + 18q^{76} - 751q^{77} - 438q^{78} - 28q^{79} + 565q^{80} - 405q^{81} - 1515q^{82} - 1680q^{83} - 1077q^{84} + 1360q^{85} - 40q^{86} - 816q^{87} - 4181q^{88} + 2978q^{89} + 90q^{90} - 2736q^{91} + 1062q^{92} + 1290q^{93} + 843q^{94} - 195q^{95} + 1719q^{96} - 4336q^{97} - 5183q^{98} - 594q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 31 x^{8} + 26 x^{7} + 738 x^{6} + 352 x^{5} + 5008 x^{4} + 5368 x^{3} + 26728 x^{2} + 13776 x + 7056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(547053 \nu^{9} - 39677237 \nu^{8} + 74096045 \nu^{7} - 1011660144 \nu^{6} - 43979588 \nu^{5} - 28238719808 \nu^{4} + 17667877936 \nu^{3} - 186676141656 \nu^{2} - 97095457368 \nu - 1760861059296\)\()/ 152584070032 \)
\(\beta_{3}\)\(=\)\((\)\(-1229309 \nu^{9} - 547053 \nu^{8} + 1021605 \nu^{7} - 105036474 \nu^{6} - 606372 \nu^{5} - 389343552 \nu^{4} + 21692996784 \nu^{3} - 2573811864 \nu^{2} - 1338711192 \nu + 78821785392\)\()/ 152584070032 \)
\(\beta_{4}\)\(=\)\((\)\(234588647 \nu^{9} - 208773158 \nu^{8} + 7283736170 \nu^{7} + 6077851117 \nu^{6} + 175332187440 \nu^{5} + 82587937556 \nu^{4} + 1182996158768 \nu^{3} + 803718924632 \nu^{2} + 6324135406160 \nu + 3259806136104\)\()/ 3204265470672 \)
\(\beta_{5}\)\(=\)\((\)\(-407749371 \nu^{9} + 1645652265 \nu^{8} - 17320513781 \nu^{7} + 30310144474 \nu^{6} - 363550408560 \nu^{5} + 619719874560 \nu^{4} - 3633007303352 \nu^{3} + 1920482437352 \nu^{2} - 12733219802808 \nu + 4857556590096\)\()/ 1068088490224 \)
\(\beta_{6}\)\(=\)\((\)\(253478250 \nu^{9} + 501821041 \nu^{8} + 6186518193 \nu^{7} + 32231489069 \nu^{6} + 183243487894 \nu^{5} + 632905819744 \nu^{4} + 1091949805464 \nu^{3} + 5272048176992 \nu^{2} + 9608197090608 \nu + 14110741960056\)\()/ 534044245112 \)
\(\beta_{7}\)\(=\)\((\)\(408363607 \nu^{9} - 221600804 \nu^{8} + 11444006564 \nu^{7} + 19321193051 \nu^{6} + 265397721032 \nu^{5} + 303713239568 \nu^{4} + 1322180972588 \nu^{3} + 3447597478168 \nu^{2} + 5266938298784 \nu + 6396445900488\)\()/ 534044245112 \)
\(\beta_{8}\)\(=\)\((\)\(475394561 \nu^{9} - 554501960 \nu^{8} + 14823232880 \nu^{7} + 8982519389 \nu^{6} + 350512568624 \nu^{5} + 67703058216 \nu^{4} + 2351904401568 \nu^{3} + 1497123940104 \nu^{2} + 11779077955592 \nu + 80722315800\)\()/ 534044245112 \)
\(\beta_{9}\)\(=\)\((\)\(908381875 \nu^{9} - 1965915003 \nu^{8} + 28948768535 \nu^{7} - 6809581088 \nu^{6} + 628838769376 \nu^{5} - 386225403248 \nu^{4} + 3956634830204 \nu^{3} + 1062923717024 \nu^{2} + 17224675028304 \nu - 4008994682112\)\()/ 534044245112 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - 12 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{7} + 2 \beta_{5} + 21 \beta_{3} - \beta_{2} + \beta_{1} - 12\)
\(\nu^{4}\)\(=\)\(5 \beta_{9} - 27 \beta_{8} - 4 \beta_{7} + 3 \beta_{6} + \beta_{5} + 240 \beta_{4} + 3 \beta_{3} - 42 \beta_{1} - 240\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 55 \beta_{8} - 31 \beta_{7} + 35 \beta_{6} - 33 \beta_{5} + 492 \beta_{4} - 476 \beta_{3} + 55 \beta_{2} - 509 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-121 \beta_{9} + 121 \beta_{7} + 62 \beta_{6} - 180 \beta_{5} - 1569 \beta_{3} + 699 \beta_{2} - 121 \beta_{1} + 5700\)
\(\nu^{7}\)\(=\)\(-1059 \beta_{9} + 2047 \beta_{8} + 1764 \beta_{7} - 941 \beta_{6} - 823 \beta_{5} - 17004 \beta_{4} - 941 \beta_{3} + 12280 \beta_{1} + 17004\)
\(\nu^{8}\)\(=\)\(-1646 \beta_{9} + 18395 \beta_{8} + 2283 \beta_{7} - 5575 \beta_{6} + 3929 \beta_{5} - 146652 \beta_{4} + 46200 \beta_{3} - 18395 \beta_{2} + 50129 \beta_{1}\)
\(\nu^{9}\)\(=\)\(26253 \beta_{9} - 26253 \beta_{7} - 4566 \beta_{6} + 47940 \beta_{5} + 356365 \beta_{3} - 67119 \beta_{2} + 26253 \beta_{1} - 544524\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−2.13900 + 3.70486i
−1.18645 + 2.05499i
−0.272258 + 0.471565i
1.41747 2.45514i
2.68023 4.64230i
−2.13900 3.70486i
−1.18645 2.05499i
−0.272258 0.471565i
1.41747 + 2.45514i
2.68023 + 4.64230i
−2.13900 + 3.70486i −1.50000 2.59808i −5.15064 8.92117i 2.50000 4.33013i 12.8340 17.2464 + 6.74999i 9.84488 −4.50000 + 7.79423i 10.6950 + 18.5243i
16.2 −1.18645 + 2.05499i −1.50000 2.59808i 1.18467 + 2.05191i 2.50000 4.33013i 7.11870 −10.8596 15.0023i −24.6054 −4.50000 + 7.79423i 5.93225 + 10.2750i
16.3 −0.272258 + 0.471565i −1.50000 2.59808i 3.85175 + 6.67143i 2.50000 4.33013i 1.63355 −0.983423 + 18.4941i −8.55081 −4.50000 + 7.79423i 1.36129 + 2.35782i
16.4 1.41747 2.45514i −1.50000 2.59808i −0.0184684 0.0319883i 2.50000 4.33013i −8.50485 4.61394 17.9363i 22.5749 −4.50000 + 7.79423i −7.08737 12.2757i
16.5 2.68023 4.64230i −1.50000 2.59808i −10.3673 17.9567i 2.50000 4.33013i −16.0814 17.9827 4.42983i −68.2635 −4.50000 + 7.79423i −13.4012 23.2115i
46.1 −2.13900 3.70486i −1.50000 + 2.59808i −5.15064 + 8.92117i 2.50000 + 4.33013i 12.8340 17.2464 6.74999i 9.84488 −4.50000 7.79423i 10.6950 18.5243i
46.2 −1.18645 2.05499i −1.50000 + 2.59808i 1.18467 2.05191i 2.50000 + 4.33013i 7.11870 −10.8596 + 15.0023i −24.6054 −4.50000 7.79423i 5.93225 10.2750i
46.3 −0.272258 0.471565i −1.50000 + 2.59808i 3.85175 6.67143i 2.50000 + 4.33013i 1.63355 −0.983423 18.4941i −8.55081 −4.50000 7.79423i 1.36129 2.35782i
46.4 1.41747 + 2.45514i −1.50000 + 2.59808i −0.0184684 + 0.0319883i 2.50000 + 4.33013i −8.50485 4.61394 + 17.9363i 22.5749 −4.50000 7.79423i −7.08737 + 12.2757i
46.5 2.68023 + 4.64230i −1.50000 + 2.59808i −10.3673 + 17.9567i 2.50000 + 4.33013i −16.0814 17.9827 + 4.42983i −68.2635 −4.50000 7.79423i −13.4012 + 23.2115i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.i.e 10
3.b odd 2 1 315.4.j.f 10
7.c even 3 1 inner 105.4.i.e 10
7.c even 3 1 735.4.a.y 5
7.d odd 6 1 735.4.a.x 5
21.g even 6 1 2205.4.a.bv 5
21.h odd 6 1 315.4.j.f 10
21.h odd 6 1 2205.4.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.e 10 1.a even 1 1 trivial
105.4.i.e 10 7.c even 3 1 inner
315.4.j.f 10 3.b odd 2 1
315.4.j.f 10 21.h odd 6 1
735.4.a.x 5 7.d odd 6 1
735.4.a.y 5 7.c even 3 1
2205.4.a.bv 5 21.g even 6 1
2205.4.a.bw 5 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 7056 + 13776 T + 26728 T^{2} + 5368 T^{3} + 5008 T^{4} + 352 T^{5} + 738 T^{6} + 26 T^{7} + 31 T^{8} - T^{9} + T^{10} \)
$3$ \( ( 9 + 3 T + T^{2} )^{5} \)
$5$ \( ( 25 - 5 T + T^{2} )^{5} \)
$7$ \( 4747561509943 - 775112083256 T + 71062701927 T^{2} - 5517738100 T^{3} + 354471635 T^{4} - 19485732 T^{5} + 1033445 T^{6} - 46900 T^{7} + 1761 T^{8} - 56 T^{9} + T^{10} \)
$11$ \( 1986036295824 + 5499093388656 T + 15040569179920 T^{2} + 528126494592 T^{3} + 36524497140 T^{4} - 386384628 T^{5} + 24418388 T^{6} - 102048 T^{7} + 5985 T^{8} - 33 T^{9} + T^{10} \)
$13$ \( ( -25478281 + 4614873 T - 134666 T^{2} - 5870 T^{3} + 23 T^{4} + T^{5} )^{2} \)
$17$ \( \)\(34\!\cdots\!00\)\( - 2512440289366080000 T + 78587894771518864 T^{2} - 443293255910208 T^{3} + 11195798876608 T^{4} - 58733646272 T^{5} + 867632660 T^{6} - 3265696 T^{7} + 42192 T^{8} - 136 T^{9} + T^{10} \)
$19$ \( 2248247678314176841 - 113251636990501511 T + 6507647996707239 T^{2} - 28216064109234 T^{3} + 1957373559269 T^{4} - 7865529465 T^{5} + 469485405 T^{6} - 177938 T^{7} + 24415 T^{8} - 39 T^{9} + T^{10} \)
$23$ \( 13945631776351203600 - 1124927289361076880 T + 65138175671500624 T^{2} - 1796800684433664 T^{3} + 36674046634324 T^{4} - 322949104388 T^{5} + 2506253492 T^{6} - 8930056 T^{7} + 53649 T^{8} - 133 T^{9} + T^{10} \)
$29$ \( ( -12806983008 + 442539868 T + 7948768 T^{2} - 32232 T^{3} - 272 T^{4} + T^{5} )^{2} \)
$31$ \( 4081182415823420964 - 4058328655876698 T + 8100260675968545 T^{2} + 89012168959896 T^{3} + 15152305261486 T^{4} + 80012694626 T^{5} + 2122801203 T^{6} - 16631636 T^{7} + 164862 T^{8} - 430 T^{9} + T^{10} \)
$37$ \( \)\(16\!\cdots\!49\)\( - 2239673640559685253 T + 85471000055527311 T^{2} - 664854591272262 T^{3} + 28102507313401 T^{4} - 251132774399 T^{5} + 2879078833 T^{6} - 8735726 T^{7} + 55391 T^{8} + 3 T^{9} + T^{10} \)
$41$ \( ( 45007298820 - 9167437684 T - 72632464 T^{2} - 46144 T^{3} + 627 T^{4} + T^{5} )^{2} \)
$43$ \( ( 1217258521160 + 11616976573 T - 4723972 T^{2} - 219178 T^{3} - 108 T^{4} + T^{5} )^{2} \)
$47$ \( \)\(15\!\cdots\!76\)\( - \)\(65\!\cdots\!16\)\( T + 21108532315580033920 T^{2} - 231043904540262528 T^{3} + 1999408594849840 T^{4} - 8297878939184 T^{5} + 34488543800 T^{6} - 63008800 T^{7} + 362433 T^{8} - 553 T^{9} + T^{10} \)
$53$ \( \)\(10\!\cdots\!36\)\( + \)\(97\!\cdots\!56\)\( T + \)\(17\!\cdots\!64\)\( T^{2} - 400503198869264384 T^{3} + 8185807795063616 T^{4} - 5111172125120 T^{5} + 147382743040 T^{6} - 399432016 T^{7} + 1088753 T^{8} - 1135 T^{9} + T^{10} \)
$59$ \( \)\(84\!\cdots\!36\)\( + \)\(80\!\cdots\!44\)\( T + \)\(68\!\cdots\!32\)\( T^{2} + 17893448055423083776 T^{3} + 61320380672526368 T^{4} + 19848627769392 T^{5} + 294582123044 T^{6} + 39421152 T^{7} + 706424 T^{8} - 332 T^{9} + T^{10} \)
$61$ \( \)\(29\!\cdots\!00\)\( - \)\(46\!\cdots\!00\)\( T + \)\(14\!\cdots\!00\)\( T^{2} - 161599028160000000 T^{3} + 5484852826240000 T^{4} - 7179571712000 T^{5} + 55278737600 T^{6} - 93117440 T^{7} + 450896 T^{8} - 584 T^{9} + T^{10} \)
$67$ \( \)\(48\!\cdots\!00\)\( - \)\(31\!\cdots\!80\)\( T + \)\(18\!\cdots\!89\)\( T^{2} - 40710452921667468180 T^{3} + 113840936931665258 T^{4} - 90460280046032 T^{5} + 377305734303 T^{6} - 148977392 T^{7} + 870530 T^{8} + 412 T^{9} + T^{10} \)
$71$ \( ( -27239555642904 + 147258969164 T + 35683872 T^{2} - 769280 T^{3} + 142 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(38\!\cdots\!36\)\( - \)\(18\!\cdots\!02\)\( T + \)\(12\!\cdots\!93\)\( T^{2} - 36200024742546821760 T^{3} + 198536825323918070 T^{4} - 608629944296906 T^{5} + 1546069813175 T^{6} - 2443320492 T^{7} + 2951950 T^{8} - 2074 T^{9} + T^{10} \)
$79$ \( \)\(35\!\cdots\!64\)\( + \)\(17\!\cdots\!80\)\( T + \)\(10\!\cdots\!49\)\( T^{2} + 18675392318928123268 T^{3} + 80718492664113690 T^{4} + 88665849104232 T^{5} + 472267962083 T^{6} + 179251872 T^{7} + 751962 T^{8} + 28 T^{9} + T^{10} \)
$83$ \( ( 59475285699552 + 477301951676 T - 823734320 T^{2} - 1464184 T^{3} + 840 T^{4} + T^{5} )^{2} \)
$89$ \( \)\(17\!\cdots\!04\)\( + \)\(23\!\cdots\!08\)\( T + \)\(49\!\cdots\!24\)\( T^{2} - \)\(32\!\cdots\!20\)\( T^{3} + 1342913936197150256 T^{4} - 3230615570349480 T^{5} + 5690822944292 T^{6} - 6655748880 T^{7} + 5716460 T^{8} - 2978 T^{9} + T^{10} \)
$97$ \( ( 701198534208512 - 1769182097088 T - 5467925696 T^{2} - 1646480 T^{3} + 2168 T^{4} + T^{5} )^{2} \)
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