Properties

Label 1502.2.a.f
Level 1502
Weight 2
Character orbit 1502.a
Self dual yes
Analytic conductor 11.994
Analytic rank 1
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1502 = 2 \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1502.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.9935303836\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( -1 - \beta_{7} + \beta_{9} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -1 + \beta_{4} - \beta_{6} - \beta_{9} ) q^{7} + q^{8} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( -1 - \beta_{7} + \beta_{9} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -1 + \beta_{4} - \beta_{6} - \beta_{9} ) q^{7} + q^{8} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{9} + ( -1 - \beta_{7} + \beta_{9} ) q^{10} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{11} + ( -1 + \beta_{1} ) q^{12} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -1 + \beta_{4} - \beta_{6} - \beta_{9} ) q^{14} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{15} + q^{16} + ( -2 + \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{17} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{18} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{19} + ( -1 - \beta_{7} + \beta_{9} ) q^{20} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{21} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{22} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} + ( -1 + \beta_{1} ) q^{24} + ( 2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} ) q^{25} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{26} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} - 2 \beta_{10} ) q^{27} + ( -1 + \beta_{4} - \beta_{6} - \beta_{9} ) q^{28} + ( \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{29} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{30} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} + q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{33} + ( -2 + \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{34} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{35} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{36} + ( -1 + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{38} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{39} + ( -1 - \beta_{7} + \beta_{9} ) q^{40} + ( 1 + \beta_{1} - 2 \beta_{3} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{41} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{42} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{43} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{44} + ( -2 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} + 5 \beta_{10} ) q^{45} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{46} + ( -4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{47} + ( -1 + \beta_{1} ) q^{48} + ( -1 + \beta_{2} + \beta_{5} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{49} + ( 2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} ) q^{50} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{51} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{52} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{10} ) q^{53} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} - 2 \beta_{10} ) q^{54} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{55} + ( -1 + \beta_{4} - \beta_{6} - \beta_{9} ) q^{56} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{57} + ( \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{58} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{59} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{60} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{10} ) q^{61} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{62} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{63} + q^{64} + ( 3 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - 2 \beta_{10} ) q^{65} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{66} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{67} + ( -2 + \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{68} + ( 2 - 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + \beta_{10} ) q^{69} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{70} + ( -2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{71} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{72} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{73} + ( -1 + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{74} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + \beta_{8} + 4 \beta_{9} + 3 \beta_{10} ) q^{75} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{77} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{78} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{79} + ( -1 - \beta_{7} + \beta_{9} ) q^{80} + ( 6 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{81} + ( 1 + \beta_{1} - 2 \beta_{3} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{82} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{83} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{84} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{85} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{86} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{87} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{88} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{10} ) q^{89} + ( -2 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} + 5 \beta_{10} ) q^{90} + ( -1 + 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{92} + ( -3 - 5 \beta_{1} + 4 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 5 \beta_{10} ) q^{93} + ( -4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{94} + ( -5 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + 5 \beta_{10} ) q^{95} + ( -1 + \beta_{1} ) q^{96} + ( -6 - 4 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{97} + ( -1 + \beta_{2} + \beta_{5} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{98} + ( -2 - 6 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - 8 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{2} - 11q^{3} + 11q^{4} - 12q^{5} - 11q^{6} - 9q^{7} + 11q^{8} + 18q^{9} + O(q^{10}) \) \( 11q + 11q^{2} - 11q^{3} + 11q^{4} - 12q^{5} - 11q^{6} - 9q^{7} + 11q^{8} + 18q^{9} - 12q^{10} - 7q^{11} - 11q^{12} - 21q^{13} - 9q^{14} - 3q^{15} + 11q^{16} - 16q^{17} + 18q^{18} - 22q^{19} - 12q^{20} - q^{21} - 7q^{22} - 2q^{23} - 11q^{24} + 19q^{25} - 21q^{26} - 44q^{27} - 9q^{28} + 4q^{29} - 3q^{30} - 28q^{31} + 11q^{32} - 13q^{33} - 16q^{34} - 11q^{35} + 18q^{36} - 22q^{37} - 22q^{38} + 9q^{39} - 12q^{40} - 5q^{41} - q^{42} - 7q^{43} - 7q^{44} - 23q^{45} - 2q^{46} - 31q^{47} - 11q^{48} - 2q^{49} + 19q^{50} - 6q^{51} - 21q^{52} - 17q^{53} - 44q^{54} - 18q^{55} - 9q^{56} + 7q^{57} + 4q^{58} - 18q^{59} - 3q^{60} - 18q^{61} - 28q^{62} - 27q^{63} + 11q^{64} + 22q^{65} - 13q^{66} - 11q^{67} - 16q^{68} + 9q^{69} - 11q^{70} - 16q^{71} + 18q^{72} - 33q^{73} - 22q^{74} - 21q^{75} - 22q^{76} + 9q^{78} + 9q^{79} - 12q^{80} + 71q^{81} - 5q^{82} - 18q^{83} - q^{84} - 8q^{85} - 7q^{86} - 17q^{87} - 7q^{88} - 23q^{90} - 22q^{91} - 2q^{92} + 8q^{93} - 31q^{94} - 23q^{95} - 11q^{96} - 66q^{97} - 2q^{98} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 20 x^{9} - 7 x^{8} + 134 x^{7} + 70 x^{6} - 354 x^{5} - 193 x^{4} + 341 x^{3} + 163 x^{2} - 72 x - 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3377 \nu^{10} + 5788 \nu^{9} + 56308 \nu^{8} - 66273 \nu^{7} - 324370 \nu^{6} + 232522 \nu^{5} + 734978 \nu^{4} - 248639 \nu^{3} - 563689 \nu^{2} - 43087 \nu + 65684 \)\()/26056\)
\(\beta_{3}\)\(=\)\((\)\( -7719 \nu^{10} + 19580 \nu^{9} + 106732 \nu^{8} - 224575 \nu^{7} - 488494 \nu^{6} + 800590 \nu^{5} + 783622 \nu^{4} - 935697 \nu^{3} - 293791 \nu^{2} + 218599 \nu - 11908 \)\()/52112\)
\(\beta_{4}\)\(=\)\((\)\( -10331 \nu^{10} + 22668 \nu^{9} + 154204 \nu^{8} - 260419 \nu^{7} - 773302 \nu^{6} + 917910 \nu^{5} + 1447470 \nu^{4} - 1070205 \nu^{3} - 896739 \nu^{2} + 366731 \nu + 128476 \)\()/52112\)
\(\beta_{5}\)\(=\)\((\)\( -3309 \nu^{10} + 5548 \nu^{9} + 53496 \nu^{8} - 58357 \nu^{7} - 297982 \nu^{6} + 174802 \nu^{5} + 654830 \nu^{4} - 133691 \nu^{3} - 510145 \nu^{2} - 32459 \nu + 59296 \)\()/13028\)
\(\beta_{6}\)\(=\)\((\)\( 944 \nu^{10} - 2757 \nu^{9} - 12598 \nu^{8} + 33449 \nu^{7} + 55763 \nu^{6} - 130539 \nu^{5} - 90328 \nu^{4} + 180678 \nu^{3} + 48043 \nu^{2} - 66654 \nu - 1891 \)\()/3257\)
\(\beta_{7}\)\(=\)\((\)\( 16813 \nu^{10} - 33284 \nu^{9} - 255572 \nu^{8} + 361301 \nu^{7} + 1316490 \nu^{6} - 1152074 \nu^{5} - 2596834 \nu^{4} + 1055579 \nu^{3} + 1862053 \nu^{2} - 10397 \nu - 286404 \)\()/52112\)
\(\beta_{8}\)\(=\)\((\)\( 8321 \nu^{10} - 20172 \nu^{9} - 119748 \nu^{8} + 234113 \nu^{7} + 582246 \nu^{6} - 838742 \nu^{5} - 1075122 \nu^{4} + 967795 \nu^{3} + 687729 \nu^{2} - 218005 \nu - 54224 \)\()/13028\)
\(\beta_{9}\)\(=\)\((\)\( -17407 \nu^{10} + 43044 \nu^{9} + 248332 \nu^{8} - 498655 \nu^{7} - 1204054 \nu^{6} + 1792686 \nu^{5} + 2247430 \nu^{4} - 2127889 \nu^{3} - 1492535 \nu^{2} + 583519 \nu + 137972 \)\()/26056\)
\(\beta_{10}\)\(=\)\((\)\( -34863 \nu^{10} + 75532 \nu^{9} + 516508 \nu^{8} - 846295 \nu^{7} - 2578286 \nu^{6} + 2870574 \nu^{5} + 4827926 \nu^{4} - 3061481 \nu^{3} - 3000471 \nu^{2} + 543615 \nu + 194524 \)\()/52112\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{7} - \beta_{4} - \beta_{3} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{10} - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 7 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(9 \beta_{10} + \beta_{9} + \beta_{8} + 6 \beta_{7} - 3 \beta_{5} - 10 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(18 \beta_{10} + 5 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} + \beta_{6} - 17 \beta_{5} - 25 \beta_{4} - 20 \beta_{3} + 15 \beta_{2} + 64 \beta_{1} + 45\)
\(\nu^{6}\)\(=\)\(91 \beta_{10} + 23 \beta_{9} + 23 \beta_{8} + 43 \beta_{7} + 7 \beta_{6} - 58 \beta_{5} - 99 \beta_{4} - 113 \beta_{3} + 42 \beta_{2} + 168 \beta_{1} + 251\)
\(\nu^{7}\)\(=\)\(241 \beta_{10} + 98 \beta_{9} + 104 \beta_{8} + 55 \beta_{7} + 34 \beta_{6} - 236 \beta_{5} - 286 \beta_{4} - 278 \beta_{3} + 194 \beta_{2} + 650 \beta_{1} + 563\)
\(\nu^{8}\)\(=\)\(977 \beta_{10} + 358 \beta_{9} + 350 \beta_{8} + 366 \beta_{7} + 143 \beta_{6} - 812 \beta_{5} - 1032 \beta_{4} - 1203 \beta_{3} + 612 \beta_{2} + 1945 \beta_{1} + 2431\)
\(\nu^{9}\)\(=\)\(2939 \beta_{10} + 1380 \beta_{9} + 1379 \beta_{8} + 750 \beta_{7} + 565 \beta_{6} - 2977 \beta_{5} - 3231 \beta_{4} - 3460 \beta_{3} + 2373 \beta_{2} + 6983 \beta_{1} + 6654\)
\(\nu^{10}\)\(=\)\(10815 \beta_{10} + 4764 \beta_{9} + 4580 \beta_{8} + 3524 \beta_{7} + 2082 \beta_{6} - 10189 \beta_{5} - 11207 \beta_{4} - 13210 \beta_{3} + 7817 \beta_{2} + 22265 \beta_{1} + 25322\)

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
−2.37027
−2.36248
−2.14010
−1.29048
−0.799353
−0.178754
0.434885
1.21388
1.64282
2.47084
3.37900
1.00000 −3.37027 1.00000 −4.42460 −3.37027 1.01197 1.00000 8.35874 −4.42460
1.2 1.00000 −3.36248 1.00000 2.63222 −3.36248 −0.528777 1.00000 8.30627 2.63222
1.3 1.00000 −3.14010 1.00000 −0.286978 −3.14010 −4.72107 1.00000 6.86022 −0.286978
1.4 1.00000 −2.29048 1.00000 −2.28669 −2.29048 2.59902 1.00000 2.24628 −2.28669
1.5 1.00000 −1.79935 1.00000 −0.966216 −1.79935 −0.438762 1.00000 0.237673 −0.966216
1.6 1.00000 −1.17875 1.00000 3.48258 −1.17875 −3.41427 1.00000 −1.61054 3.48258
1.7 1.00000 −0.565115 1.00000 −0.615969 −0.565115 3.05802 1.00000 −2.68064 −0.615969
1.8 1.00000 0.213882 1.00000 −2.70660 0.213882 1.35942 1.00000 −2.95425 −2.70660
1.9 1.00000 0.642822 1.00000 −0.305614 0.642822 −3.12232 1.00000 −2.58678 −0.305614
1.10 1.00000 1.47084 1.00000 −3.42444 1.47084 −2.00214 1.00000 −0.836618 −3.42444
1.11 1.00000 2.37900 1.00000 −3.09771 2.37900 −2.80109 1.00000 2.65965 −3.09771
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1502.2.a.f 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.f 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{11} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).