Properties

Label 80.2.s.b
Level $80$
Weight $2$
Character orbit 80.s
Analytic conductor $0.639$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,2,Mod(3,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.s (of order \(4\), degree \(2\), 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: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + \beta_{13} q^{4} + \beta_{17} q^{5} + (\beta_{12} + \beta_{9} - 1) q^{6} + (\beta_{16} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{17} - \beta_{16} - \beta_{13} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + \beta_{13} q^{4} + \beta_{17} q^{5} + (\beta_{12} + \beta_{9} - 1) q^{6} + (\beta_{16} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{15} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{11} - 12 q^{14} - 20 q^{15} - 6 q^{17} - 24 q^{18} - 2 q^{19} - 12 q^{20} - 16 q^{21} + 12 q^{22} - 2 q^{23} - 4 q^{24} - 6 q^{25} - 16 q^{26} - 24 q^{27} + 40 q^{28} + 14 q^{29} + 40 q^{30} + 20 q^{32} - 8 q^{33} + 28 q^{34} + 2 q^{35} - 4 q^{36} + 24 q^{38} + 44 q^{40} + 8 q^{42} - 44 q^{44} - 14 q^{45} + 12 q^{46} + 38 q^{47} + 4 q^{48} - 8 q^{50} + 8 q^{51} + 8 q^{52} + 12 q^{53} + 4 q^{54} - 6 q^{55} + 20 q^{56} - 24 q^{57} + 20 q^{58} + 10 q^{59} + 8 q^{60} + 14 q^{61} - 40 q^{62} - 6 q^{63} + 16 q^{64} + 4 q^{66} - 60 q^{68} - 32 q^{69} - 28 q^{70} + 24 q^{71} - 68 q^{72} - 14 q^{73} - 48 q^{74} + 16 q^{75} - 16 q^{76} - 44 q^{77} - 36 q^{78} - 16 q^{79} - 92 q^{80} + 2 q^{81} + 48 q^{82} + 40 q^{83} + 24 q^{84} + 14 q^{85} - 36 q^{86} + 24 q^{87} - 8 q^{88} + 12 q^{89} - 8 q^{90} - 8 q^{92} - 28 q^{94} + 34 q^{95} - 40 q^{96} + 18 q^{97} - 56 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 129 \nu^{17} + 124 \nu^{16} + 398 \nu^{15} - 116 \nu^{14} - 797 \nu^{13} - 2778 \nu^{12} + \cdots - 58624 ) / 1280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + \cdots + 6048 ) / 160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + \cdots + 28416 ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + \cdots + 89600 ) / 1280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + \cdots + 75264 ) / 640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} + \cdots - 222976 ) / 1280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} + \cdots - 351232 ) / 1280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 228 \nu^{17} + 359 \nu^{16} + 1013 \nu^{15} + 672 \nu^{14} - 134 \nu^{13} - 3459 \nu^{12} + \cdots - 71232 ) / 320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1017 \nu^{17} - 1536 \nu^{16} - 4342 \nu^{15} - 2508 \nu^{14} + 1381 \nu^{13} + 16526 \nu^{12} + \cdots + 345088 ) / 1280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 129 \nu^{17} + 186 \nu^{16} + 524 \nu^{15} + 232 \nu^{14} - 321 \nu^{13} - 2280 \nu^{12} + \cdots - 46208 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 799 \nu^{17} - 1073 \nu^{16} - 3066 \nu^{15} - 930 \nu^{14} + 2727 \nu^{13} + 14975 \nu^{12} + \cdots + 305280 ) / 640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1799 \nu^{17} + 2608 \nu^{16} + 7346 \nu^{15} + 3380 \nu^{14} - 4267 \nu^{13} - 31490 \nu^{12} + \cdots - 636160 ) / 1280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} + \cdots - 33024 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + \cdots + 363008 ) / 640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1999 \nu^{17} + 2830 \nu^{16} + 8030 \nu^{15} + 3408 \nu^{14} - 5187 \nu^{13} - 35416 \nu^{12} + \cdots - 718848 ) / 1280 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + \cdots + 372864 ) / 640 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 593 \nu^{17} - 812 \nu^{16} - 2294 \nu^{15} - 774 \nu^{14} + 1929 \nu^{13} + 10958 \nu^{12} + \cdots + 218944 ) / 320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{16} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - 2\beta_{11} + \beta_{8} + \beta_{4} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 2\beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{17} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{17} + \beta_{16} - 5 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{17} + \beta_{16} - 2 \beta_{14} - 7 \beta_{13} - \beta_{12} - 3 \beta_{11} + 11 \beta_{10} + \cdots - 6 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3 \beta_{16} - 11 \beta_{15} - \beta_{14} + 7 \beta_{13} + 12 \beta_{12} - 8 \beta_{10} + \cdots + 10 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8 \beta_{17} + 4 \beta_{16} - 13 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - \beta_{12} + 3 \beta_{11} + \cdots + 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - \beta_{17} - 6 \beta_{16} - 6 \beta_{15} + 7 \beta_{14} + 4 \beta_{12} + 9 \beta_{11} - 3 \beta_{10} + \cdots + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18 \beta_{17} + 17 \beta_{16} + 5 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} + 2 \beta_{10} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 9 \beta_{17} - 21 \beta_{16} - 47 \beta_{15} - 20 \beta_{14} - 3 \beta_{13} + 34 \beta_{12} + \cdots - 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14 \beta_{17} + 3 \beta_{16} - 24 \beta_{15} - 8 \beta_{14} - 3 \beta_{13} - 25 \beta_{12} + \cdots + 114 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 42 \beta_{17} + 25 \beta_{16} - 47 \beta_{15} - 7 \beta_{14} - 49 \beta_{13} + 72 \beta_{12} + \cdots + 90 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 32 \beta_{17} + 160 \beta_{16} - 75 \beta_{15} + 13 \beta_{14} + 104 \beta_{13} - 33 \beta_{12} + \cdots + 116 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 59 \beta_{17} - 196 \beta_{16} - 80 \beta_{15} - 103 \beta_{14} - 58 \beta_{13} + 82 \beta_{12} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 54 \beta_{17} - 61 \beta_{16} - 63 \beta_{15} - 5 \beta_{14} - 121 \beta_{13} + 52 \beta_{12} + \cdots + 228 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{10}\) \(\beta_{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.480367 + 1.33013i
0.0376504 + 1.41371i
−1.37691 0.322680i
1.41323 0.0526497i
0.235136 1.39453i
1.41303 + 0.0578659i
−0.635486 1.26339i
0.482716 + 1.32928i
−1.08900 + 0.902261i
−0.480367 1.33013i
0.0376504 1.41371i
−1.37691 + 0.322680i
1.41323 + 0.0526497i
0.235136 + 1.39453i
1.41303 0.0578659i
−0.635486 + 1.26339i
0.482716 1.32928i
−1.08900 0.902261i
−1.38031 + 0.307817i 2.85601 1.81050 0.849763i −1.71489 1.43498i −3.94217 + 0.879127i −0.458895 + 0.458895i −2.23747 + 1.73024i 5.15678 2.80878 + 1.45284i
3.2 −1.29924 + 0.558542i −2.55161 1.37606 1.45136i 1.49107 1.66635i 3.31516 1.42518i 2.40368 2.40368i −0.977191 + 2.65426i 3.51070 −1.00653 + 2.99782i
3.3 −1.23576 0.687667i 0.614566 1.05423 + 1.69959i 0.832020 + 2.07551i −0.759459 0.422617i 2.83610 2.83610i −0.134028 2.82525i −2.62231 0.399079 3.13699i
3.4 −0.516777 + 1.31641i 1.28110 −1.46588 1.36058i 2.07160 + 0.841703i −0.662041 + 1.68645i −1.13975 + 1.13975i 2.54862 1.22659i −1.35879 −2.17858 + 2.29211i
3.5 0.430311 + 1.34716i −2.96561 −1.62967 + 1.15939i −0.177336 + 2.22902i −1.27613 3.99515i −0.115101 + 0.115101i −2.26315 1.69652i 5.79486 −3.07916 + 0.720273i
3.6 0.567819 1.29521i 1.96251 −1.35516 1.47090i −1.42182 + 1.72581i 1.11435 2.54187i −1.60205 + 1.60205i −2.67461 + 0.920026i 0.851447 1.42795 + 2.82151i
3.7 0.828280 + 1.14628i 0.692712 −0.627905 + 1.89888i −0.245325 2.22257i 0.573759 + 0.794040i −0.343872 + 0.343872i −2.69672 + 0.853049i −2.52015 2.34448 2.12212i
3.8 1.19301 0.759419i −1.39319 0.846564 1.81200i 2.17104 0.535339i −1.66209 + 1.05801i −2.13436 + 2.13436i −0.366101 2.80463i −1.05903 2.18353 2.28740i
3.9 1.41267 + 0.0660953i −0.496487 1.99126 + 0.186742i −2.00635 + 0.987189i −0.701372 0.0328155i 1.55426 1.55426i 2.80065 + 0.395417i −2.75350 −2.89956 + 1.26196i
27.1 −1.38031 0.307817i 2.85601 1.81050 + 0.849763i −1.71489 + 1.43498i −3.94217 0.879127i −0.458895 0.458895i −2.23747 1.73024i 5.15678 2.80878 1.45284i
27.2 −1.29924 0.558542i −2.55161 1.37606 + 1.45136i 1.49107 + 1.66635i 3.31516 + 1.42518i 2.40368 + 2.40368i −0.977191 2.65426i 3.51070 −1.00653 2.99782i
27.3 −1.23576 + 0.687667i 0.614566 1.05423 1.69959i 0.832020 2.07551i −0.759459 + 0.422617i 2.83610 + 2.83610i −0.134028 + 2.82525i −2.62231 0.399079 + 3.13699i
27.4 −0.516777 1.31641i 1.28110 −1.46588 + 1.36058i 2.07160 0.841703i −0.662041 1.68645i −1.13975 1.13975i 2.54862 + 1.22659i −1.35879 −2.17858 2.29211i
27.5 0.430311 1.34716i −2.96561 −1.62967 1.15939i −0.177336 2.22902i −1.27613 + 3.99515i −0.115101 0.115101i −2.26315 + 1.69652i 5.79486 −3.07916 0.720273i
27.6 0.567819 + 1.29521i 1.96251 −1.35516 + 1.47090i −1.42182 1.72581i 1.11435 + 2.54187i −1.60205 1.60205i −2.67461 0.920026i 0.851447 1.42795 2.82151i
27.7 0.828280 1.14628i 0.692712 −0.627905 1.89888i −0.245325 + 2.22257i 0.573759 0.794040i −0.343872 0.343872i −2.69672 0.853049i −2.52015 2.34448 + 2.12212i
27.8 1.19301 + 0.759419i −1.39319 0.846564 + 1.81200i 2.17104 + 0.535339i −1.66209 1.05801i −2.13436 2.13436i −0.366101 + 2.80463i −1.05903 2.18353 + 2.28740i
27.9 1.41267 0.0660953i −0.496487 1.99126 0.186742i −2.00635 0.987189i −0.701372 + 0.0328155i 1.55426 + 1.55426i 2.80065 0.395417i −2.75350 −2.89956 1.26196i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.2.s.b yes 18
3.b odd 2 1 720.2.z.g 18
4.b odd 2 1 320.2.s.b 18
5.b even 2 1 400.2.s.d 18
5.c odd 4 1 80.2.j.b 18
5.c odd 4 1 400.2.j.d 18
8.b even 2 1 640.2.s.d 18
8.d odd 2 1 640.2.s.c 18
15.e even 4 1 720.2.bd.g 18
16.e even 4 1 320.2.j.b 18
16.e even 4 1 640.2.j.c 18
16.f odd 4 1 80.2.j.b 18
16.f odd 4 1 640.2.j.d 18
20.d odd 2 1 1600.2.s.d 18
20.e even 4 1 320.2.j.b 18
20.e even 4 1 1600.2.j.d 18
40.i odd 4 1 640.2.j.d 18
40.k even 4 1 640.2.j.c 18
48.k even 4 1 720.2.bd.g 18
80.i odd 4 1 320.2.s.b 18
80.j even 4 1 400.2.s.d 18
80.j even 4 1 640.2.s.d 18
80.k odd 4 1 400.2.j.d 18
80.q even 4 1 1600.2.j.d 18
80.s even 4 1 inner 80.2.s.b yes 18
80.t odd 4 1 640.2.s.c 18
80.t odd 4 1 1600.2.s.d 18
240.z odd 4 1 720.2.z.g 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 5.c odd 4 1
80.2.j.b 18 16.f odd 4 1
80.2.s.b yes 18 1.a even 1 1 trivial
80.2.s.b yes 18 80.s even 4 1 inner
320.2.j.b 18 16.e even 4 1
320.2.j.b 18 20.e even 4 1
320.2.s.b 18 4.b odd 2 1
320.2.s.b 18 80.i odd 4 1
400.2.j.d 18 5.c odd 4 1
400.2.j.d 18 80.k odd 4 1
400.2.s.d 18 5.b even 2 1
400.2.s.d 18 80.j even 4 1
640.2.j.c 18 16.e even 4 1
640.2.j.c 18 40.k even 4 1
640.2.j.d 18 16.f odd 4 1
640.2.j.d 18 40.i odd 4 1
640.2.s.c 18 8.d odd 2 1
640.2.s.c 18 80.t odd 4 1
640.2.s.d 18 8.b even 2 1
640.2.s.d 18 80.j even 4 1
720.2.z.g 18 3.b odd 2 1
720.2.z.g 18 240.z odd 4 1
720.2.bd.g 18 15.e even 4 1
720.2.bd.g 18 48.k even 4 1
1600.2.j.d 18 20.e even 4 1
1600.2.j.d 18 80.q even 4 1
1600.2.s.d 18 20.d odd 2 1
1600.2.s.d 18 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 16T_{3}^{7} + 4T_{3}^{6} + 76T_{3}^{5} - 40T_{3}^{4} - 104T_{3}^{3} + 72T_{3}^{2} + 20T_{3} - 16 \) acting on \(S_{2}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{16} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( (T^{9} - 16 T^{7} + \cdots - 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} - 2 T^{17} + \cdots + 288 \) Copy content Toggle raw display
$11$ \( T^{18} + 2 T^{17} + \cdots + 5431808 \) Copy content Toggle raw display
$13$ \( T^{18} + 112 T^{16} + \cdots + 67108864 \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + \cdots + 512 \) Copy content Toggle raw display
$19$ \( T^{18} + 2 T^{17} + \cdots + 4608 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 17700587552 \) Copy content Toggle raw display
$29$ \( T^{18} - 14 T^{17} + \cdots + 82330112 \) Copy content Toggle raw display
$31$ \( T^{18} + 196 T^{16} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 574297214976 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 242788765696 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 337207844416 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16870640672 \) Copy content Toggle raw display
$53$ \( (T^{9} - 6 T^{8} + \cdots + 220832)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 144166720393728 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 121236758528 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 555525752896 \) Copy content Toggle raw display
$71$ \( (T^{9} - 12 T^{8} + \cdots - 27648)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 35535647232 \) Copy content Toggle raw display
$79$ \( (T^{9} + 8 T^{8} + \cdots + 45002752)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} - 20 T^{8} + \cdots - 8413744)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} - 6 T^{8} + \cdots - 251904)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 380349381734912 \) Copy content Toggle raw display
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