Properties

Label 140.2.c.b
Level $140$
Weight $2$
Character orbit 140.c
Analytic conductor $1.118$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} + 28 x^{12} + 16 x^{10} - 40 x^{8} + 610 x^{6} + 1625 x^{4} - 524 x^{2} + 1444\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} + 28 x^{12} + 16 x^{10} - 40 x^{8} + 610 x^{6} + 1625 x^{4} - 524 x^{2} + 1444\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -160577 \nu^{15} - 19792868 \nu^{13} + 118635540 \nu^{11} - 640577256 \nu^{9} + 5860864 \nu^{7} + 1292054998 \nu^{5} - 13534107197 \nu^{3} - 3536177966 \nu \)\()/ 39980301296 \)
\(\beta_{2}\)\(=\)\((\)\( -701627 \nu^{15} - 5034292 \nu^{13} + 18194892 \nu^{11} - 122627780 \nu^{9} - 828065124 \nu^{7} + 442514098 \nu^{5} - 6218975035 \nu^{3} - 19599822798 \nu \)\()/ 9995075324 \)
\(\beta_{3}\)\(=\)\((\)\( -4622027 \nu^{14} + 18920224 \nu^{12} - 55700874 \nu^{10} - 428065102 \nu^{8} + 435253284 \nu^{6} - 2539415580 \nu^{4} - 9983737673 \nu^{2} - 9488500630 \)\()/ 9995075324 \)
\(\beta_{4}\)\(=\)\((\)\( -778741 \nu^{14} + 5214580 \nu^{12} - 27045620 \nu^{10} + 25125688 \nu^{8} - 6532248 \nu^{6} - 613266210 \nu^{4} + 311886279 \nu^{2} + 1386720738 \)\()/ 1537703896 \)
\(\beta_{5}\)\(=\)\((\)\( 10105411 \nu^{15} - 36502060 \nu^{13} + 157774044 \nu^{11} + 518633832 \nu^{9} + 1113094440 \nu^{7} + 506135110 \nu^{5} + 35184267703 \nu^{3} - 1520006582 \nu \)\()/ 39980301296 \)
\(\beta_{6}\)\(=\)\((\)\( 2006632 \nu^{14} - 7648781 \nu^{12} + 31976398 \nu^{10} + 145329823 \nu^{8} - 58607182 \nu^{6} + 1241411587 \nu^{4} + 5422581996 \nu^{2} + 2576062655 \)\()/ 2498768831 \)
\(\beta_{7}\)\(=\)\((\)\( -4095595 \nu^{14} + 28568136 \nu^{12} - 115147725 \nu^{10} - 60918915 \nu^{8} + 765952940 \nu^{6} - 2161769415 \nu^{4} - 4185433928 \nu^{2} + 16119618620 \)\()/ 4997537662 \)
\(\beta_{8}\)\(=\)\((\)\( 2918186 \nu^{14} - 20701507 \nu^{12} + 98570518 \nu^{10} - 52671282 \nu^{8} + 2447580 \nu^{6} + 929333314 \nu^{4} + 2868437416 \nu^{2} - 3350947293 \)\()/ 2498768831 \)
\(\beta_{9}\)\(=\)\((\)\( 5290343 \nu^{15} - 30566954 \nu^{13} + 139846998 \nu^{11} + 73573152 \nu^{9} + 8711350 \nu^{7} + 2721018508 \nu^{5} + 5600946403 \nu^{3} + 4522635990 \nu \)\()/ 9995075324 \)
\(\beta_{10}\)\(=\)\((\)\( 3003030 \nu^{14} - 18626943 \nu^{12} + 87707502 \nu^{10} + 56300402 \nu^{8} - 257526656 \nu^{6} + 2149887220 \nu^{4} + 4823331344 \nu^{2} - 3210930232 \)\()/ 2498768831 \)
\(\beta_{11}\)\(=\)\((\)\(-26047779 \nu^{15} + 212152652 \nu^{13} - 1049471708 \nu^{11} + 905836904 \nu^{9} + 2069617144 \nu^{7} - 15041745238 \nu^{5} - 25596257271 \nu^{3} + 94111892774 \nu\)\()/ 39980301296 \)
\(\beta_{12}\)\(=\)\((\)\( 16464459 \nu^{14} - 97577124 \nu^{12} + 428256914 \nu^{10} + 435323342 \nu^{8} - 945411436 \nu^{6} + 8697856648 \nu^{4} + 35362350517 \nu^{2} - 8632792082 \)\()/ 9995075324 \)
\(\beta_{13}\)\(=\)\((\)\(-50893351 \nu^{15} + 310852820 \nu^{13} - 1355898420 \nu^{11} - 1154887384 \nu^{9} + 4332540464 \nu^{7} - 31332033382 \nu^{5} - 74199704683 \nu^{3} + 95495899390 \nu\)\()/ 39980301296 \)
\(\beta_{14}\)\(=\)\((\)\(-54021013 \nu^{15} + 251129916 \nu^{13} - 1045847772 \nu^{11} - 2926553016 \nu^{9} + 1032001696 \nu^{7} - 26977866994 \nu^{5} - 126143819217 \nu^{3} - 69936350326 \nu\)\()/ 39980301296 \)
\(\beta_{15}\)\(=\)\((\)\(-57832737 \nu^{15} + 374118884 \nu^{13} - 1798365156 \nu^{11} - 138285752 \nu^{9} + 2995610328 \nu^{7} - 37488440626 \nu^{5} - 85289354813 \nu^{3} + 99947547522 \nu\)\()/ 39980301296 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} + \beta_{13} + \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{12} - \beta_{10} - \beta_{8} + 2 \beta_{3} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - 3 \beta_{9} + \beta_{5} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(8 \beta_{12} - \beta_{10} - 7 \beta_{8} + 4 \beta_{7} - 8 \beta_{6} - 8 \beta_{4} - 4 \beta_{3} - 5\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{15} + 3 \beta_{14} - 11 \beta_{13} + 6 \beta_{11} - 14 \beta_{9} - 10 \beta_{5} - 15 \beta_{2} + 12 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(9 \beta_{12} - \beta_{8} + 15 \beta_{7} - 17 \beta_{6} - 6 \beta_{4} - 20 \beta_{3} - 38\)
\(\nu^{7}\)\(=\)\((\)\(34 \beta_{15} + 35 \beta_{14} - 69 \beta_{13} - 36 \beta_{11} - 16 \beta_{9} - 58 \beta_{5} - 115 \beta_{2} - 60 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-96 \beta_{12} + 155 \beta_{10} + 67 \beta_{8} + 88 \beta_{7} - 78 \beta_{6} + 112 \beta_{4} - 184 \beta_{3} - 273\)\()/2\)
\(\nu^{9}\)\(=\)\(51 \beta_{15} + 41 \beta_{14} - 49 \beta_{13} - 114 \beta_{11} + 51 \beta_{9} - 180 \beta_{5} - 137 \beta_{2} - 351 \beta_{1}\)
\(\nu^{10}\)\(=\)\((\)\(-930 \beta_{12} + 881 \beta_{10} + 527 \beta_{8} + 48 \beta_{7} + 506 \beta_{6} + 792 \beta_{4} + 234 \beta_{3} - 133\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-674 \beta_{15} + 117 \beta_{14} + 1233 \beta_{13} - 410 \beta_{11} + 1330 \beta_{9} - 774 \beta_{5} + 531 \beta_{2} - 3614 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\(-2012 \beta_{12} + 566 \beta_{10} + 821 \beta_{8} - 1128 \beta_{7} + 3040 \beta_{6} + 920 \beta_{4} + 2648 \beta_{3} + 2279\)
\(\nu^{13}\)\(=\)\((\)\(-7262 \beta_{15} - 169 \beta_{14} + 9417 \beta_{13} + 4980 \beta_{11} + 7652 \beta_{9} + 4530 \beta_{5} + 10397 \beta_{2} - 8374 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-3394 \beta_{12} - 17629 \beta_{10} - 17 \beta_{8} - 17336 \beta_{7} + 27208 \beta_{6} - 9120 \beta_{4} + 25686 \beta_{3} + 34869\)\()/2\)
\(\nu^{15}\)\(=\)\(-12294 \beta_{15} - 1919 \beta_{14} + 14048 \beta_{13} + 19878 \beta_{11} + 12497 \beta_{9} + 27381 \beta_{5} + 28448 \beta_{2} + 20330 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
0.744612 + 0.556573i
−0.744612 0.556573i
−0.744612 + 0.556573i
0.744612 0.556573i
−0.328458 + 1.49331i
0.328458 1.49331i
0.328458 + 1.49331i
−0.328458 1.49331i
1.61596 1.02509i
−1.61596 + 1.02509i
−1.61596 1.02509i
1.61596 + 1.02509i
2.05580 + 0.953651i
−2.05580 0.953651i
−2.05580 + 0.953651i
2.05580 0.953651i
−1.06789 0.927153i 0.662153i 0.280776 + 1.98019i 1.31119 + 1.81129i −0.613917 + 0.707107i 1.19935 + 2.35829i 1.53610 2.37495i 2.56155 0.279135 3.14993i
139.2 −1.06789 0.927153i 0.662153i 0.280776 + 1.98019i −1.31119 1.81129i 0.613917 0.707107i 1.19935 2.35829i 1.53610 2.37495i 2.56155 −0.279135 + 3.14993i
139.3 −1.06789 + 0.927153i 0.662153i 0.280776 1.98019i −1.31119 + 1.81129i 0.613917 + 0.707107i 1.19935 + 2.35829i 1.53610 + 2.37495i 2.56155 −0.279135 3.14993i
139.4 −1.06789 + 0.927153i 0.662153i 0.280776 1.98019i 1.31119 1.81129i −0.613917 0.707107i 1.19935 2.35829i 1.53610 + 2.37495i 2.56155 0.279135 + 3.14993i
139.5 −0.331077 1.37491i 2.13578i −1.78078 + 0.910404i 1.94442 1.10418i −2.93651 + 0.707107i −2.35829 1.19935i 1.84130 + 2.14700i −1.56155 −2.16191 2.30784i
139.6 −0.331077 1.37491i 2.13578i −1.78078 + 0.910404i −1.94442 + 1.10418i 2.93651 0.707107i −2.35829 + 1.19935i 1.84130 + 2.14700i −1.56155 2.16191 + 2.30784i
139.7 −0.331077 + 1.37491i 2.13578i −1.78078 0.910404i −1.94442 1.10418i 2.93651 + 0.707107i −2.35829 1.19935i 1.84130 2.14700i −1.56155 2.16191 2.30784i
139.8 −0.331077 + 1.37491i 2.13578i −1.78078 0.910404i 1.94442 + 1.10418i −2.93651 0.707107i −2.35829 + 1.19935i 1.84130 2.14700i −1.56155 −2.16191 + 2.30784i
139.9 0.331077 1.37491i 2.13578i −1.78078 0.910404i −1.94442 + 1.10418i −2.93651 0.707107i 2.35829 1.19935i −1.84130 + 2.14700i −1.56155 0.874406 + 3.03898i
139.10 0.331077 1.37491i 2.13578i −1.78078 0.910404i 1.94442 1.10418i 2.93651 + 0.707107i 2.35829 + 1.19935i −1.84130 + 2.14700i −1.56155 −0.874406 3.03898i
139.11 0.331077 + 1.37491i 2.13578i −1.78078 + 0.910404i 1.94442 + 1.10418i 2.93651 0.707107i 2.35829 1.19935i −1.84130 2.14700i −1.56155 −0.874406 + 3.03898i
139.12 0.331077 + 1.37491i 2.13578i −1.78078 + 0.910404i −1.94442 1.10418i −2.93651 + 0.707107i 2.35829 + 1.19935i −1.84130 2.14700i −1.56155 0.874406 3.03898i
139.13 1.06789 0.927153i 0.662153i 0.280776 1.98019i −1.31119 1.81129i −0.613917 0.707107i −1.19935 + 2.35829i −1.53610 2.37495i 2.56155 −3.07955 0.718585i
139.14 1.06789 0.927153i 0.662153i 0.280776 1.98019i 1.31119 + 1.81129i 0.613917 + 0.707107i −1.19935 2.35829i −1.53610 2.37495i 2.56155 3.07955 + 0.718585i
139.15 1.06789 + 0.927153i 0.662153i 0.280776 + 1.98019i 1.31119 1.81129i 0.613917 0.707107i −1.19935 + 2.35829i −1.53610 + 2.37495i 2.56155 3.07955 0.718585i
139.16 1.06789 + 0.927153i 0.662153i 0.280776 + 1.98019i −1.31119 + 1.81129i −0.613917 + 0.707107i −1.19935 2.35829i −1.53610 + 2.37495i 2.56155 −3.07955 + 0.718585i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.c.b 16
4.b odd 2 1 inner 140.2.c.b 16
5.b even 2 1 inner 140.2.c.b 16
5.c odd 4 2 700.2.g.l 16
7.b odd 2 1 inner 140.2.c.b 16
7.c even 3 2 980.2.s.f 32
7.d odd 6 2 980.2.s.f 32
8.b even 2 1 2240.2.e.f 16
8.d odd 2 1 2240.2.e.f 16
20.d odd 2 1 inner 140.2.c.b 16
20.e even 4 2 700.2.g.l 16
28.d even 2 1 inner 140.2.c.b 16
28.f even 6 2 980.2.s.f 32
28.g odd 6 2 980.2.s.f 32
35.c odd 2 1 inner 140.2.c.b 16
35.f even 4 2 700.2.g.l 16
35.i odd 6 2 980.2.s.f 32
35.j even 6 2 980.2.s.f 32
40.e odd 2 1 2240.2.e.f 16
40.f even 2 1 2240.2.e.f 16
56.e even 2 1 2240.2.e.f 16
56.h odd 2 1 2240.2.e.f 16
140.c even 2 1 inner 140.2.c.b 16
140.j odd 4 2 700.2.g.l 16
140.p odd 6 2 980.2.s.f 32
140.s even 6 2 980.2.s.f 32
280.c odd 2 1 2240.2.e.f 16
280.n even 2 1 2240.2.e.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.b 16 1.a even 1 1 trivial
140.2.c.b 16 4.b odd 2 1 inner
140.2.c.b 16 5.b even 2 1 inner
140.2.c.b 16 7.b odd 2 1 inner
140.2.c.b 16 20.d odd 2 1 inner
140.2.c.b 16 28.d even 2 1 inner
140.2.c.b 16 35.c odd 2 1 inner
140.2.c.b 16 140.c even 2 1 inner
700.2.g.l 16 5.c odd 4 2
700.2.g.l 16 20.e even 4 2
700.2.g.l 16 35.f even 4 2
700.2.g.l 16 140.j odd 4 2
980.2.s.f 32 7.c even 3 2
980.2.s.f 32 7.d odd 6 2
980.2.s.f 32 28.f even 6 2
980.2.s.f 32 28.g odd 6 2
980.2.s.f 32 35.i odd 6 2
980.2.s.f 32 35.j even 6 2
980.2.s.f 32 140.p odd 6 2
980.2.s.f 32 140.s even 6 2
2240.2.e.f 16 8.b even 2 1
2240.2.e.f 16 8.d odd 2 1
2240.2.e.f 16 40.e odd 2 1
2240.2.e.f 16 40.f even 2 1
2240.2.e.f 16 56.e even 2 1
2240.2.e.f 16 56.h odd 2 1
2240.2.e.f 16 280.c odd 2 1
2240.2.e.f 16 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 5 T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + 12 T^{2} + 6 T^{4} + 3 T^{6} + T^{8} )^{2} \)
$3$ \( ( 2 + 5 T^{2} + T^{4} )^{4} \)
$5$ \( ( 625 - 50 T^{2} + 34 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$7$ \( ( 2401 + 30 T^{4} + T^{8} )^{2} \)
$11$ \( ( 52 + 15 T^{2} + T^{4} )^{4} \)
$13$ \( ( 26 - 23 T^{2} + T^{4} )^{4} \)
$17$ \( ( 104 - 29 T^{2} + T^{4} )^{4} \)
$19$ \( ( 208 - 38 T^{2} + T^{4} )^{4} \)
$23$ \( ( 128 - 40 T^{2} + T^{4} )^{4} \)
$29$ \( ( -2 + 3 T + T^{2} )^{8} \)
$31$ \( ( 3328 - 120 T^{2} + T^{4} )^{4} \)
$37$ \( ( 416 + 44 T^{2} + T^{4} )^{4} \)
$41$ \( ( 72 + T^{2} )^{8} \)
$43$ \( ( 32 - 20 T^{2} + T^{4} )^{4} \)
$47$ \( ( 8 + 95 T^{2} + T^{4} )^{4} \)
$53$ \( ( 6656 + 164 T^{2} + T^{4} )^{4} \)
$59$ \( ( 13312 - 270 T^{2} + T^{4} )^{4} \)
$61$ \( ( 16 + 42 T^{2} + T^{4} )^{4} \)
$67$ \( ( 128 - 28 T^{2} + T^{4} )^{4} \)
$71$ \( ( 13312 + 236 T^{2} + T^{4} )^{4} \)
$73$ \( ( 1664 - 116 T^{2} + T^{4} )^{4} \)
$79$ \( ( 208 + 115 T^{2} + T^{4} )^{4} \)
$83$ \( ( 2312 + 170 T^{2} + T^{4} )^{4} \)
$89$ \( ( 8 + T^{2} )^{8} \)
$97$ \( ( 8424 - 261 T^{2} + T^{4} )^{4} \)
show more
show less