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

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,2,Mod(139,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{9} q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{5} - \beta_1) q^{6} + ( - \beta_{14} - \beta_{9} + \beta_{4} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{3}) q^{8} + ( - \beta_{10} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{9} q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{5} - \beta_1) q^{6} + ( - \beta_{14} - \beta_{9} + \beta_{4} + \beta_1) q^{7} + ( - \beta_{12} - \beta_{3}) q^{8} + ( - \beta_{10} + \beta_{6}) q^{9} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_1) q^{10} + ( - \beta_{8} - \beta_{6}) q^{11} + (\beta_{14} + 2 \beta_{9} - \beta_{2} - \beta_1) q^{12} + ( - \beta_{15} + \beta_{9} + \beta_{5} + \beta_{2}) q^{13} + (\beta_{14} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6}) q^{14} + ( - \beta_{10} + \beta_{8} + \beta_{4}) q^{15} + (\beta_{10} + \beta_{8} - \beta_{6}) q^{16} + (\beta_{15} - \beta_{14} - \beta_{9} - \beta_{5} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{12} - \beta_{7}) q^{18} + (\beta_{15} + \beta_{13} + \beta_{5} + \beta_1) q^{19} + ( - 2 \beta_{14} - \beta_{11} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{20} + ( - \beta_{15} + \beta_{13} - \beta_{10} + \beta_{6} - \beta_{5} + \beta_1 - 1) q^{21} + (\beta_{12} - 2 \beta_{4}) q^{22} + (\beta_{7} - 2 \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{5} + \beta_1) q^{24} + (2 \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{25} + ( - \beta_{14} + \beta_{13} + \beta_{11} - \beta_1) q^{26} + (\beta_{14} - \beta_{9} - \beta_1) q^{27} + ( - \beta_{15} + \beta_{12} + 2 \beta_{9} + \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{28} + (\beta_{10} - \beta_{6} - 1) q^{29} + (\beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{30} + (\beta_{15} - \beta_{14} - \beta_{13} - 4 \beta_{11} + \beta_{5} + 2 \beta_1) q^{31} + (\beta_{12} + \beta_{7} + 2 \beta_{4}) q^{32} + ( - \beta_{15} + \beta_{14} + \beta_{9} + \beta_{5} - \beta_{2} - \beta_1) q^{33} + ( - \beta_{15} + \beta_{13} - \beta_{11} - \beta_{5} + 2 \beta_1) q^{34} + ( - \beta_{15} + 2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{35} + (\beta_{10} + \beta_{8} + \beta_{6} + 2) q^{36} + ( - \beta_{7} - 3 \beta_{3}) q^{37} + ( - \beta_{15} + \beta_{14} + \beta_{5} + 2 \beta_{2} - \beta_1) q^{38} + (\beta_{8} + \beta_{6}) q^{39} + (\beta_{15} + \beta_{13} - \beta_{11} - 2 \beta_{9} - 2 \beta_{2} + 3 \beta_1) q^{40} + (3 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 3 \beta_{5}) q^{41} + ( - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{7} + \beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{42} + ( - \beta_{7} + \beta_{3}) q^{43} + ( - 3 \beta_{10} + \beta_{8} + 2 \beta_{6} + 1) q^{44} + ( - 2 \beta_{14} + \beta_{13} - \beta_{9} - 2 \beta_{5}) q^{45} + ( - 2 \beta_{10} + 2 \beta_{8} - 2) q^{46} + (\beta_{14} + 5 \beta_{9} - \beta_1) q^{47} + (\beta_{15} - 3 \beta_{14} - 4 \beta_{9} - \beta_{5} + 2 \beta_{2} + 3 \beta_1) q^{48} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{10} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 1) q^{49}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 8 q^{9} - 4 q^{14} - 12 q^{16} - 8 q^{21} + 8 q^{25} - 24 q^{29} - 4 q^{30} + 28 q^{36} + 32 q^{44} - 32 q^{46} - 12 q^{50} - 20 q^{56} + 44 q^{60} - 36 q^{64} - 32 q^{65} + 40 q^{70} + 88 q^{74} - 48 q^{81} + 40 q^{84} - 56 q^{85} + 40 q^{86}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( - 160577 \nu^{15} - 19792868 \nu^{13} + 118635540 \nu^{11} - 640577256 \nu^{9} + 5860864 \nu^{7} + 1292054998 \nu^{5} - 13534107197 \nu^{3} + \cdots - 3536177966 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 701627 \nu^{15} - 5034292 \nu^{13} + 18194892 \nu^{11} - 122627780 \nu^{9} - 828065124 \nu^{7} + 442514098 \nu^{5} - 6218975035 \nu^{3} + \cdots - 19599822798 \nu ) / 9995075324 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4622027 \nu^{14} + 18920224 \nu^{12} - 55700874 \nu^{10} - 428065102 \nu^{8} + 435253284 \nu^{6} - 2539415580 \nu^{4} - 9983737673 \nu^{2} + \cdots - 9488500630 ) / 9995075324 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 778741 \nu^{14} + 5214580 \nu^{12} - 27045620 \nu^{10} + 25125688 \nu^{8} - 6532248 \nu^{6} - 613266210 \nu^{4} + 311886279 \nu^{2} + \cdots + 1386720738 ) / 1537703896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10105411 \nu^{15} - 36502060 \nu^{13} + 157774044 \nu^{11} + 518633832 \nu^{9} + 1113094440 \nu^{7} + 506135110 \nu^{5} + \cdots - 1520006582 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2006632 \nu^{14} - 7648781 \nu^{12} + 31976398 \nu^{10} + 145329823 \nu^{8} - 58607182 \nu^{6} + 1241411587 \nu^{4} + 5422581996 \nu^{2} + \cdots + 2576062655 ) / 2498768831 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4095595 \nu^{14} + 28568136 \nu^{12} - 115147725 \nu^{10} - 60918915 \nu^{8} + 765952940 \nu^{6} - 2161769415 \nu^{4} + \cdots + 16119618620 ) / 4997537662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2918186 \nu^{14} - 20701507 \nu^{12} + 98570518 \nu^{10} - 52671282 \nu^{8} + 2447580 \nu^{6} + 929333314 \nu^{4} + 2868437416 \nu^{2} + \cdots - 3350947293 ) / 2498768831 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5290343 \nu^{15} - 30566954 \nu^{13} + 139846998 \nu^{11} + 73573152 \nu^{9} + 8711350 \nu^{7} + 2721018508 \nu^{5} + 5600946403 \nu^{3} + \cdots + 4522635990 \nu ) / 9995075324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3003030 \nu^{14} - 18626943 \nu^{12} + 87707502 \nu^{10} + 56300402 \nu^{8} - 257526656 \nu^{6} + 2149887220 \nu^{4} + 4823331344 \nu^{2} + \cdots - 3210930232 ) / 2498768831 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26047779 \nu^{15} + 212152652 \nu^{13} - 1049471708 \nu^{11} + 905836904 \nu^{9} + 2069617144 \nu^{7} - 15041745238 \nu^{5} + \cdots + 94111892774 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16464459 \nu^{14} - 97577124 \nu^{12} + 428256914 \nu^{10} + 435323342 \nu^{8} - 945411436 \nu^{6} + 8697856648 \nu^{4} + 35362350517 \nu^{2} + \cdots - 8632792082 ) / 9995075324 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 50893351 \nu^{15} + 310852820 \nu^{13} - 1355898420 \nu^{11} - 1154887384 \nu^{9} + 4332540464 \nu^{7} - 31332033382 \nu^{5} + \cdots + 95495899390 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 54021013 \nu^{15} + 251129916 \nu^{13} - 1045847772 \nu^{11} - 2926553016 \nu^{9} + 1032001696 \nu^{7} - 26977866994 \nu^{5} + \cdots - 69936350326 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 57832737 \nu^{15} + 374118884 \nu^{13} - 1798365156 \nu^{11} - 138285752 \nu^{9} + 2995610328 \nu^{7} - 37488440626 \nu^{5} + \cdots + 99947547522 \nu ) / 39980301296 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{13} + \beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{12} - \beta_{10} - \beta_{8} + 2\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - 3\beta_{9} + \beta_{5} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{12} - \beta_{10} - 7\beta_{8} + 4\beta_{7} - 8\beta_{6} - 8\beta_{4} - 4\beta_{3} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{15} + 3\beta_{14} - 11\beta_{13} + 6\beta_{11} - 14\beta_{9} - 10\beta_{5} - 15\beta_{2} + 12\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{12} - \beta_{8} + 15\beta_{7} - 17\beta_{6} - 6\beta_{4} - 20\beta_{3} - 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34\beta_{15} + 35\beta_{14} - 69\beta_{13} - 36\beta_{11} - 16\beta_{9} - 58\beta_{5} - 115\beta_{2} - 60\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -96\beta_{12} + 155\beta_{10} + 67\beta_{8} + 88\beta_{7} - 78\beta_{6} + 112\beta_{4} - 184\beta_{3} - 273 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 51\beta_{15} + 41\beta_{14} - 49\beta_{13} - 114\beta_{11} + 51\beta_{9} - 180\beta_{5} - 137\beta_{2} - 351\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -930\beta_{12} + 881\beta_{10} + 527\beta_{8} + 48\beta_{7} + 506\beta_{6} + 792\beta_{4} + 234\beta_{3} - 133 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 674 \beta_{15} + 117 \beta_{14} + 1233 \beta_{13} - 410 \beta_{11} + 1330 \beta_{9} - 774 \beta_{5} + 531 \beta_{2} - 3614 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2012 \beta_{12} + 566 \beta_{10} + 821 \beta_{8} - 1128 \beta_{7} + 3040 \beta_{6} + 920 \beta_{4} + 2648 \beta_{3} + 2279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 7262 \beta_{15} - 169 \beta_{14} + 9417 \beta_{13} + 4980 \beta_{11} + 7652 \beta_{9} + 4530 \beta_{5} + 10397 \beta_{2} - 8374 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3394 \beta_{12} - 17629 \beta_{10} - 17 \beta_{8} - 17336 \beta_{7} + 27208 \beta_{6} - 9120 \beta_{4} + 25686 \beta_{3} + 34869 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12294 \beta_{15} - 1919 \beta_{14} + 14048 \beta_{13} + 19878 \beta_{11} + 12497 \beta_{9} + 27381 \beta_{5} + 28448 \beta_{2} + 20330 \beta_1 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{3}^{4} + 5T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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