Properties

Label 980.2.o.e
Level $980$
Weight $2$
Character orbit 980.o
Analytic conductor $7.825$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(31,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.o (of order \(6\), degree \(2\), not 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} - 3x^{12} + 4x^{10} - 4x^{8} + 16x^{6} - 48x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (\beta_{15} + \beta_{12} - \beta_{8}) q^{3} + \beta_{9} q^{4} - \beta_{6} q^{5} + (\beta_{14} - \beta_{11} - 2 \beta_{3}) q^{6} + ( - \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{13} + 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (\beta_{15} + \beta_{12} - \beta_{8}) q^{3} + \beta_{9} q^{4} - \beta_{6} q^{5} + (\beta_{14} - \beta_{11} - 2 \beta_{3}) q^{6} + ( - \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{13} + 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - \beta_1 - 3) q^{9} + ( - \beta_{15} + \beta_{8}) q^{10} + (\beta_{9} + 2 \beta_{4} - \beta_{2}) q^{11} + (2 \beta_{15} + 4 \beta_{6}) q^{12} - 2 \beta_{3} q^{13} + ( - \beta_{13} + \beta_{9} + \beta_{5}) q^{15} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_1 + 2) q^{16} + (2 \beta_{15} - \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{3}) q^{17} + ( - 2 \beta_{9} + 4 \beta_{7} - 3 \beta_{4}) q^{18} + (2 \beta_{15} - \beta_{14} + \beta_{12} + \beta_{10}) q^{19} - \beta_{14} q^{20} + ( - 2 \beta_{13} + 2 \beta_{9} + 4) q^{22} + ( - \beta_{13} + 3 \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{23} + (4 \beta_{15} + 2 \beta_{12} - 4 \beta_{8}) q^{24} + \beta_{7} q^{25} + 2 \beta_{15} q^{26} + ( - 2 \beta_{14} + 2 \beta_{11} + 4 \beta_{8}) q^{27} - 2 q^{29} + ( - \beta_{13} - 2 \beta_{7} + \beta_1 + 2) q^{30} + ( - 2 \beta_{15} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{8}) q^{31} + (2 \beta_{9} + 2 \beta_{7} + 2 \beta_{4} + \beta_{2}) q^{32} + (4 \beta_{15} + 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{10} + 8 \beta_{6}) q^{33} + (2 \beta_{14} + 2 \beta_{8} + 4 \beta_{3}) q^{34} + (3 \beta_{13} - 3 \beta_{9} - 4 \beta_{5} + 2 \beta_{2} - 2 \beta_1 - 4) q^{36} + (2 \beta_{7} - 2) q^{37} + (2 \beta_{12} - 4 \beta_{6} - 4 \beta_{3}) q^{38} + ( - 2 \beta_{9} - 2 \beta_{4}) q^{39} + (\beta_{10} - 2 \beta_{6}) q^{40} + ( - \beta_{14} + \beta_{11} - 2 \beta_{8} - 2 \beta_{3}) q^{41} + (\beta_{13} - \beta_{9} + \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{43} + ( - 4 \beta_{7} - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_1 + 4) q^{44} + (2 \beta_{15} - \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{8} + 3 \beta_{6} + 3 \beta_{3}) q^{45} + ( - 3 \beta_{9} - 6 \beta_{7} - \beta_{2}) q^{46} + ( - 3 \beta_{15} - 3 \beta_{10}) q^{47} + (4 \beta_{14} - 2 \beta_{11} - 4 \beta_{3}) q^{48} - \beta_{5} q^{50} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{51} + 2 \beta_{12} q^{52} - 2 \beta_{7} q^{53} + ( - 4 \beta_{14} + 4 \beta_{12} - 8 \beta_{6}) q^{54} + ( - \beta_{14} + \beta_{11} + 2 \beta_{8}) q^{55} + (2 \beta_{13} - 2 \beta_{9} + 4 \beta_{5} + 2 \beta_{2} - 2 \beta_1 - 8) q^{57} + (2 \beta_{5} - 2 \beta_{4}) q^{58} + ( - 2 \beta_{15} + \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 2 \beta_{8}) q^{59} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{60} - 2 \beta_{6} q^{61} + ( - 2 \beta_{14} - 2 \beta_{11} + 4 \beta_{3}) q^{62} + ( - 2 \beta_{13} + 2 \beta_{9} - 2 \beta_{5} - 3 \beta_{2} + 3 \beta_1 + 2) q^{64} + ( - 2 \beta_{7} + 2) q^{65} + (8 \beta_{15} + 4 \beta_{12} - 8 \beta_{8} + 8 \beta_{6} + 8 \beta_{3}) q^{66} + (\beta_{9} - 3 \beta_{4} + 4 \beta_{2}) q^{67} + ( - 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{12} - 2 \beta_{10} + 4 \beta_{6}) q^{68} + ( - 3 \beta_{14} + 3 \beta_{11} - 6 \beta_{8} + 10 \beta_{3}) q^{69} + ( - 3 \beta_{13} + 3 \beta_{9} + 2 \beta_{5} + \beta_{2} - \beta_1) q^{71} + (4 \beta_{13} + 10 \beta_{7} + 4 \beta_{5} - 4 \beta_{4} - \beta_1 - 10) q^{72} + (6 \beta_{15} - 3 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} - 6 \beta_{8} + 6 \beta_{6} + \cdots + 6 \beta_{3}) q^{73}+ \cdots + (9 \beta_{13} - 9 \beta_{9} - 10 \beta_{5} + \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 2 q^{4} + 28 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 2 q^{4} + 28 q^{8} - 16 q^{9} + 14 q^{16} + 30 q^{18} + 56 q^{22} + 8 q^{25} - 32 q^{29} + 12 q^{30} + 18 q^{32} - 60 q^{36} - 16 q^{37} + 20 q^{44} - 44 q^{46} - 4 q^{50} - 16 q^{53} - 96 q^{57} + 4 q^{58} - 28 q^{60} + 4 q^{64} + 16 q^{65} - 62 q^{72} - 4 q^{74} - 48 q^{78} - 48 q^{81} + 48 q^{85} + 20 q^{86} - 36 q^{88} - 24 q^{92} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} - 3x^{12} + 4x^{10} - 4x^{8} + 16x^{6} - 48x^{4} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{14} + 5\nu^{12} + 7\nu^{10} + 12\nu^{8} - 36\nu^{6} + 224\nu^{4} + 80\nu^{2} - 576 ) / 448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{14} - 3\nu^{12} + 7\nu^{10} + 36\nu^{8} - 12\nu^{6} + 464\nu^{2} + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{15} - 3\nu^{13} + 7\nu^{11} - \nu^{9} - 12\nu^{7} - 72\nu^{3} + 256\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{14} + 3\nu^{12} - 7\nu^{10} + \nu^{8} + 12\nu^{6} + 72\nu^{2} - 256 ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 2\nu^{12} - 3\nu^{8} + 34\nu^{6} + 28\nu^{4} - 48\nu^{2} + 96 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 2\nu^{13} - 3\nu^{9} + 34\nu^{7} + 28\nu^{5} - 48\nu^{3} + 96\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{14} - 3\nu^{12} + 7\nu^{10} + 12\nu^{8} - 12\nu^{6} - 144\nu^{2} + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{15} - 9\nu^{13} + 21\nu^{11} - 40\nu^{9} - 36\nu^{7} + 144\nu^{3} + 768\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{14} - 9\nu^{12} + 21\nu^{10} - 40\nu^{8} - 36\nu^{6} + 144\nu^{2} + 768 ) / 448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 3\nu^{9} + 3\nu^{7} - 28\nu^{5} + 48\nu^{3} + 216\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} - 9\nu^{13} + 21\nu^{11} + 60\nu^{9} - 36\nu^{7} + 176\nu^{3} + 768\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -3\nu^{15} + 11\nu^{13} - 7\nu^{11} - 12\nu^{9} - 12\nu^{7} + 224\nu^{5} + 368\nu^{3} - 192\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{14} + \nu^{12} + 3\nu^{10} - 4\nu^{8} + 4\nu^{6} - 16\nu^{4} + 48\nu^{2} + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -11\nu^{15} + 3\nu^{13} - 7\nu^{11} + 12\nu^{9} + 12\nu^{7} + 752\nu^{3} - 256\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} - 4\nu^{13} - 3\nu^{9} - 16\nu^{7} + 28\nu^{5} - 48\nu^{3} + 80\nu ) / 224 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 2\beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{11} + 2\beta_{8} - 4\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{13} - 4\beta_{7} + 2\beta_{5} - 2\beta_{4} + 3\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{15} + 5\beta_{12} + \beta_{11} - \beta_{10} - 2\beta_{8} + 4\beta_{6} + 4\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{13} - 3\beta_{9} + 10\beta_{5} + \beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{15} + 3\beta_{14} - 3\beta_{12} - \beta_{10} + 20\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -11\beta_{9} + 12\beta_{7} - 6\beta_{4} + 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3\beta_{14} + 15\beta_{11} - 22\beta_{8} + 12\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21\beta_{13} + 44\beta_{7} + 6\beta_{5} - 6\beta_{4} + 9\beta _1 - 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -42\beta_{15} - 13\beta_{12} + 31\beta_{11} - 31\beta_{10} + 42\beta_{8} + 12\beta_{6} + 12\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 37\beta_{13} - 37\beta_{9} - 26\beta_{5} - 25\beta_{2} + 25\beta _1 + 84 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -74\beta_{15} - 67\beta_{14} + 67\beta_{12} + 17\beta_{10} - 52\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 43\beta_{9} + 148\beta_{7} + 134\beta_{4} - 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -83\beta_{14} + 65\beta_{11} + 86\beta_{8} - 268\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1 - \beta_{7}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.40902 + 0.121053i
1.40902 0.121053i
−0.0865986 + 1.41156i
0.0865986 1.41156i
0.809347 1.15972i
−0.809347 + 1.15972i
1.26575 + 0.630783i
−1.26575 0.630783i
−1.40902 0.121053i
1.40902 + 0.121053i
−0.0865986 1.41156i
0.0865986 + 1.41156i
0.809347 + 1.15972i
−0.809347 1.15972i
1.26575 0.630783i
−1.26575 + 0.630783i
−1.15972 0.809347i −0.468213 + 0.810969i 0.689916 + 1.87724i 0.866025 0.500000i 1.19935 0.561553i 0 0.719224 2.73546i 1.06155 + 1.83866i −1.40902 0.121053i
31.2 −1.15972 0.809347i 0.468213 0.810969i 0.689916 + 1.87724i −0.866025 + 0.500000i −1.19935 + 0.561553i 0 0.719224 2.73546i 1.06155 + 1.83866i 1.40902 + 0.121053i
31.3 −0.630783 + 1.26575i −1.51022 + 2.61578i −1.20422 1.59682i −0.866025 + 0.500000i −2.35829 3.56155i 0 2.78078 0.516994i −3.06155 5.30277i −0.0865986 1.41156i
31.4 −0.630783 + 1.26575i 1.51022 2.61578i −1.20422 1.59682i 0.866025 0.500000i 2.35829 + 3.56155i 0 2.78078 0.516994i −3.06155 5.30277i 0.0865986 + 1.41156i
31.5 −0.121053 1.40902i −0.468213 + 0.810969i −1.97069 + 0.341134i −0.866025 + 0.500000i 1.19935 + 0.561553i 0 0.719224 + 2.73546i 1.06155 + 1.83866i 0.809347 + 1.15972i
31.6 −0.121053 1.40902i 0.468213 0.810969i −1.97069 + 0.341134i 0.866025 0.500000i −1.19935 0.561553i 0 0.719224 + 2.73546i 1.06155 + 1.83866i −0.809347 1.15972i
31.7 1.41156 + 0.0865986i −1.51022 + 2.61578i 1.98500 + 0.244478i 0.866025 0.500000i −2.35829 + 3.56155i 0 2.78078 + 0.516994i −3.06155 5.30277i 1.26575 0.630783i
31.8 1.41156 + 0.0865986i 1.51022 2.61578i 1.98500 + 0.244478i −0.866025 + 0.500000i 2.35829 3.56155i 0 2.78078 + 0.516994i −3.06155 5.30277i −1.26575 + 0.630783i
411.1 −1.15972 + 0.809347i −0.468213 0.810969i 0.689916 1.87724i 0.866025 + 0.500000i 1.19935 + 0.561553i 0 0.719224 + 2.73546i 1.06155 1.83866i −1.40902 + 0.121053i
411.2 −1.15972 + 0.809347i 0.468213 + 0.810969i 0.689916 1.87724i −0.866025 0.500000i −1.19935 0.561553i 0 0.719224 + 2.73546i 1.06155 1.83866i 1.40902 0.121053i
411.3 −0.630783 1.26575i −1.51022 2.61578i −1.20422 + 1.59682i −0.866025 0.500000i −2.35829 + 3.56155i 0 2.78078 + 0.516994i −3.06155 + 5.30277i −0.0865986 + 1.41156i
411.4 −0.630783 1.26575i 1.51022 + 2.61578i −1.20422 + 1.59682i 0.866025 + 0.500000i 2.35829 3.56155i 0 2.78078 + 0.516994i −3.06155 + 5.30277i 0.0865986 1.41156i
411.5 −0.121053 + 1.40902i −0.468213 0.810969i −1.97069 0.341134i −0.866025 0.500000i 1.19935 0.561553i 0 0.719224 2.73546i 1.06155 1.83866i 0.809347 1.15972i
411.6 −0.121053 + 1.40902i 0.468213 + 0.810969i −1.97069 0.341134i 0.866025 + 0.500000i −1.19935 + 0.561553i 0 0.719224 2.73546i 1.06155 1.83866i −0.809347 + 1.15972i
411.7 1.41156 0.0865986i −1.51022 2.61578i 1.98500 0.244478i 0.866025 + 0.500000i −2.35829 3.56155i 0 2.78078 0.516994i −3.06155 + 5.30277i 1.26575 + 0.630783i
411.8 1.41156 0.0865986i 1.51022 + 2.61578i 1.98500 0.244478i −0.866025 0.500000i 2.35829 + 3.56155i 0 2.78078 0.516994i −3.06155 + 5.30277i −1.26575 0.630783i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.o.e 16
4.b odd 2 1 inner 980.2.o.e 16
7.b odd 2 1 inner 980.2.o.e 16
7.c even 3 1 140.2.g.c 8
7.c even 3 1 inner 980.2.o.e 16
7.d odd 6 1 140.2.g.c 8
7.d odd 6 1 inner 980.2.o.e 16
21.g even 6 1 1260.2.c.c 8
21.h odd 6 1 1260.2.c.c 8
28.d even 2 1 inner 980.2.o.e 16
28.f even 6 1 140.2.g.c 8
28.f even 6 1 inner 980.2.o.e 16
28.g odd 6 1 140.2.g.c 8
28.g odd 6 1 inner 980.2.o.e 16
35.i odd 6 1 700.2.g.j 8
35.j even 6 1 700.2.g.j 8
35.k even 12 1 700.2.c.i 8
35.k even 12 1 700.2.c.j 8
35.l odd 12 1 700.2.c.i 8
35.l odd 12 1 700.2.c.j 8
56.j odd 6 1 2240.2.k.e 8
56.k odd 6 1 2240.2.k.e 8
56.m even 6 1 2240.2.k.e 8
56.p even 6 1 2240.2.k.e 8
84.j odd 6 1 1260.2.c.c 8
84.n even 6 1 1260.2.c.c 8
140.p odd 6 1 700.2.g.j 8
140.s even 6 1 700.2.g.j 8
140.w even 12 1 700.2.c.i 8
140.w even 12 1 700.2.c.j 8
140.x odd 12 1 700.2.c.i 8
140.x odd 12 1 700.2.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 7.c even 3 1
140.2.g.c 8 7.d odd 6 1
140.2.g.c 8 28.f even 6 1
140.2.g.c 8 28.g odd 6 1
700.2.c.i 8 35.k even 12 1
700.2.c.i 8 35.l odd 12 1
700.2.c.i 8 140.w even 12 1
700.2.c.i 8 140.x odd 12 1
700.2.c.j 8 35.k even 12 1
700.2.c.j 8 35.l odd 12 1
700.2.c.j 8 140.w even 12 1
700.2.c.j 8 140.x odd 12 1
700.2.g.j 8 35.i odd 6 1
700.2.g.j 8 35.j even 6 1
700.2.g.j 8 140.p odd 6 1
700.2.g.j 8 140.s even 6 1
980.2.o.e 16 1.a even 1 1 trivial
980.2.o.e 16 4.b odd 2 1 inner
980.2.o.e 16 7.b odd 2 1 inner
980.2.o.e 16 7.c even 3 1 inner
980.2.o.e 16 7.d odd 6 1 inner
980.2.o.e 16 28.d even 2 1 inner
980.2.o.e 16 28.f even 6 1 inner
980.2.o.e 16 28.g odd 6 1 inner
1260.2.c.c 8 21.g even 6 1
1260.2.c.c 8 21.h odd 6 1
1260.2.c.c 8 84.j odd 6 1
1260.2.c.c 8 84.n even 6 1
2240.2.k.e 8 56.j odd 6 1
2240.2.k.e 8 56.k odd 6 1
2240.2.k.e 8 56.m even 6 1
2240.2.k.e 8 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{8} + 10T_{3}^{6} + 92T_{3}^{4} + 80T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} - 28T_{11}^{6} + 656T_{11}^{4} - 3584T_{11}^{2} + 16384 \) Copy content Toggle raw display
\( T_{17}^{8} - 52T_{17}^{6} + 2640T_{17}^{4} - 3328T_{17}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{7} + T^{6} - 4 T^{5} - 6 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 10 T^{6} + 92 T^{4} + 80 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + 656 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} - 52 T^{6} + 2640 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 28 T^{6} + 656 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 74 T^{6} + 4124 T^{4} + \cdots + 1827904)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 56 T^{6} + 2624 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 4)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 58 T^{2} + 8)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 126 T^{6} + 13284 T^{4} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 4)^{8} \) Copy content Toggle raw display
$59$ \( (T^{8} + 124 T^{6} + 14864 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 218 T^{6} + 44636 T^{4} + \cdots + 8340544)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 92 T^{2} + 2048)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 324 T^{6} + 84240 T^{4} + \cdots + 429981696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 20 T^{6} + 368 T^{4} - 640 T^{2} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{2} + 8)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 144 T^{2} + 20736)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{2} + 64)^{4} \) Copy content Toggle raw display
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