Properties

Label 603.2.u.c
Level $603$
Weight $2$
Character orbit 603.u
Analytic conductor $4.815$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,2,Mod(64,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.u (of order \(11\), degree \(10\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + 2696 x^{12} + 9361 x^{11} - 20998 x^{10} + 10813 x^{9} + 44155 x^{8} + \cdots + 4489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 67)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + 1) q^{2} + ( - \beta_{14} + \beta_{13}) q^{4} + (\beta_{18} - \beta_{15} - \beta_{13} - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{2}) q^{5} + ( - \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{7} + (\beta_{16} - \beta_{15} + \beta_{8} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + 1) q^{2} + ( - \beta_{14} + \beta_{13}) q^{4} + (\beta_{18} - \beta_{15} - \beta_{13} - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{2}) q^{5} + ( - \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{7} + (\beta_{16} - \beta_{15} + \beta_{8} + 1) q^{8} + ( - \beta_{17} - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{8} - \beta_{6} + \beta_{2} - \beta_1) q^{10} + ( - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} - \beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{11} + \cdots - 2) q^{11}+ \cdots + ( - 2 \beta_{18} - \beta_{17} - 5 \beta_{16} - \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} - 4 q^{4} - q^{5} - 8 q^{7} + 22 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} - 4 q^{4} - q^{5} - 8 q^{7} + 22 q^{8} - 9 q^{10} + 12 q^{13} - 6 q^{14} + 8 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} + 31 q^{26} - 5 q^{28} + 6 q^{29} - 16 q^{32} - 8 q^{34} - 34 q^{35} + 24 q^{37} + 14 q^{38} - 11 q^{40} + 6 q^{41} - 22 q^{43} + 22 q^{44} + 15 q^{46} - 16 q^{47} + 42 q^{49} + 17 q^{50} + 2 q^{52} + q^{53} + 20 q^{55} - 11 q^{56} + 10 q^{58} + 26 q^{59} - 26 q^{61} - 11 q^{62} - 6 q^{64} - 9 q^{65} - 22 q^{67} + 52 q^{68} - 42 q^{70} - 20 q^{71} - 55 q^{73} - 37 q^{74} - 3 q^{76} + 70 q^{77} - 34 q^{79} - 40 q^{80} - 12 q^{82} - 56 q^{83} + 41 q^{85} + 33 q^{86} + 11 q^{88} + 3 q^{89} + 12 q^{91} + 18 q^{92} + 32 q^{94} - 74 q^{95} - 14 q^{97} - 95 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + 2696 x^{12} + 9361 x^{11} - 20998 x^{10} + 10813 x^{9} + 44155 x^{8} + \cdots + 4489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 46\!\cdots\!60 \nu^{19} + \cdots + 13\!\cdots\!10 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!64 \nu^{19} + \cdots - 10\!\cdots\!89 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\!\cdots\!18 \nu^{19} + \cdots - 45\!\cdots\!74 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\!\cdots\!99 \nu^{19} + \cdots + 15\!\cdots\!77 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67\!\cdots\!43 \nu^{19} + \cdots - 14\!\cdots\!22 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\!\cdots\!22 \nu^{19} + \cdots + 41\!\cdots\!11 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!67 \nu^{19} + \cdots - 52\!\cdots\!25 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!11 \nu^{19} + \cdots + 20\!\cdots\!59 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!36 \nu^{19} + \cdots - 56\!\cdots\!34 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!03 \nu^{19} + \cdots - 95\!\cdots\!77 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots + 17\!\cdots\!99 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21\!\cdots\!22 \nu^{19} + \cdots + 30\!\cdots\!98 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!31 \nu^{19} + \cdots - 52\!\cdots\!45 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!58 \nu^{19} + \cdots + 42\!\cdots\!38 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 65\!\cdots\!88 \nu^{19} + \cdots + 90\!\cdots\!00 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 68\!\cdots\!04 \nu^{19} + \cdots + 12\!\cdots\!28 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 69\!\cdots\!63 \nu^{19} + \cdots + 14\!\cdots\!60 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{16} - 4\beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6 \beta_{19} + \beta_{17} - \beta_{15} - 7 \beta_{14} + 8 \beta_{11} - 9 \beta_{10} - 8 \beta_{8} + \beta_{6} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} + \beta_{18} + 9 \beta_{17} - 7 \beta_{16} + 5 \beta_{15} + 10 \beta_{14} - 16 \beta_{13} + 29 \beta_{11} - \beta_{10} - 3 \beta_{8} - 18 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} - \beta_{4} + 7 \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} - 2 \beta_{17} + 8 \beta_{16} + 24 \beta_{15} - 8 \beta_{14} + 24 \beta_{13} + 15 \beta_{12} + 7 \beta_{11} - 14 \beta_{9} + 20 \beta_{8} + 7 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} - 3 \beta_{4} + 58 \beta_{3} + 3 \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 45 \beta_{19} - 77 \beta_{18} - 19 \beta_{17} - 104 \beta_{16} - 96 \beta_{15} + 51 \beta_{14} + 98 \beta_{13} - 45 \beta_{12} - 168 \beta_{11} + 12 \beta_{10} - 26 \beta_{9} + 24 \beta_{8} + 225 \beta_{7} + 77 \beta_{6} + 32 \beta_{5} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 325 \beta_{19} - 221 \beta_{18} + 122 \beta_{17} - 430 \beta_{16} - 355 \beta_{15} + 817 \beta_{14} - 474 \beta_{13} - 502 \beta_{12} - 576 \beta_{11} + 695 \beta_{10} + 654 \beta_{9} + 546 \beta_{8} - 209 \beta_{7} + 134 \beta_{6} + \cdots - 857 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 309 \beta_{19} + 680 \beta_{18} - 309 \beta_{17} + 1732 \beta_{16} + 585 \beta_{15} - 585 \beta_{14} - 136 \beta_{11} + 136 \beta_{10} + 1141 \beta_{9} + 490 \beta_{8} - 1141 \beta_{7} - 526 \beta_{6} - 386 \beta_{5} + \cdots - 490 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1200 \beta_{19} + 2398 \beta_{18} - 2534 \beta_{17} + 4853 \beta_{16} + 1827 \beta_{15} - 5725 \beta_{14} + 4737 \beta_{13} + 4451 \beta_{12} + 3027 \beta_{11} - 4498 \beta_{10} - 6906 \beta_{9} - 5725 \beta_{8} + \cdots + 9052 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2760 \beta_{19} - 4199 \beta_{18} - 1672 \beta_{17} - 14575 \beta_{16} - 7226 \beta_{15} + 14584 \beta_{14} - 7625 \beta_{13} + 1672 \beta_{12} - 7067 \beta_{11} + 19007 \beta_{10} - 10802 \beta_{9} + \cdots - 4199 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7068 \beta_{19} - 12830 \beta_{18} - 24282 \beta_{16} - 24282 \beta_{15} + 19898 \beta_{14} - 25959 \beta_{13} - 19898 \beta_{12} - 45857 \beta_{11} + 43444 \beta_{10} + 43444 \beta_{9} + 47434 \beta_{8} + \cdots - 79273 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 55107 \beta_{19} + 28866 \beta_{18} + 9300 \beta_{17} + 88243 \beta_{16} + 11392 \beta_{15} - 152170 \beta_{14} + 80072 \beta_{13} + 2363 \beta_{12} + 93273 \beta_{11} - 242876 \beta_{10} + 24411 \beta_{9} + \cdots + 108726 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 62564 \beta_{19} + 87418 \beta_{18} + 166296 \beta_{17} + 73420 \beta_{16} + 272576 \beta_{15} + 47963 \beta_{14} - 13998 \beta_{13} + 87418 \beta_{12} + 570642 \beta_{11} - 335140 \beta_{10} + \cdots + 595496 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 197672 \beta_{19} + 60285 \beta_{17} + 706584 \beta_{15} + 290566 \beta_{14} - 146365 \beta_{13} + 274855 \beta_{12} - 230281 \beta_{11} + 1106079 \beta_{10} - 128490 \beta_{9} + \cdots - 373004 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 433106 \beta_{19} - 900567 \beta_{18} - 1106079 \beta_{17} - 784612 \beta_{16} - 1733161 \beta_{15} - 2117640 \beta_{14} + 1637241 \beta_{13} - 108690 \beta_{12} - 3350956 \beta_{11} + \cdots - 1999381 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2854959 \beta_{18} + 2854959 \beta_{17} - 8164703 \beta_{16} - 7847972 \beta_{15} + 8164703 \beta_{14} - 7847972 \beta_{13} - 7313083 \beta_{12} + 279847 \beta_{11} + 6288794 \beta_{9} + \cdots - 6288794 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3628744 \beta_{19} + 11812158 \beta_{18} + 7961140 \beta_{17} + 20664705 \beta_{16} + 19773298 \beta_{15} + 6787974 \beta_{14} - 13567272 \beta_{13} - 3628744 \beta_{12} + \cdots + 285470 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 20352800 \beta_{19} + 42193117 \beta_{18} - 41404666 \beta_{17} + 117996676 \beta_{16} + 75682468 \beta_{15} - 144428287 \beta_{14} + 138958874 \beta_{13} + 89723120 \beta_{12} + \cdots + 165405588 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 42150258 \beta_{19} - 152375454 \beta_{18} - 42150258 \beta_{17} - 336873110 \beta_{16} - 213804591 \beta_{15} + 213804591 \beta_{14} - 249911180 \beta_{11} + \cdots - 62459923 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(470\)
\(\chi(n)\) \(-\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
2.28033 1.46548i
−0.165981 + 0.106669i
−1.27213 2.78557i
1.07319 + 2.34996i
−0.625078 + 0.721379i
1.66916 1.92631i
1.58638 + 0.465803i
−1.35948 0.399179i
−0.127556 + 0.887172i
0.441163 3.06836i
2.28033 + 1.46548i
−0.165981 0.106669i
−0.127556 0.887172i
0.441163 + 3.06836i
−1.27213 + 2.78557i
1.07319 2.34996i
−0.625078 0.721379i
1.66916 + 1.92631i
1.58638 0.465803i
−1.35948 + 0.399179i
−1.25667 0.368991i 0 −0.239446 0.153882i −0.681980 1.49333i 0 −3.00345 0.881891i 1.95949 + 2.26138i 0 0.305998 + 2.12826i
64.2 −1.25667 0.368991i 0 −0.239446 0.153882i 1.35049 + 2.95717i 0 4.00914 + 1.17719i 1.95949 + 2.26138i 0 −0.605953 4.21450i
82.1 0.239446 0.153882i 0 −0.797176 + 1.74557i −1.21919 1.40702i 0 −1.14774 + 0.737606i 0.158746 + 1.10411i 0 −0.508444 0.149293i
82.2 0.239446 0.153882i 0 −0.797176 + 1.74557i 1.85253 + 2.13793i 0 −3.42222 + 2.19933i 0.158746 + 1.10411i 0 0.772569 + 0.226847i
91.1 0.797176 1.74557i 0 −1.10181 1.27155i −0.201991 + 1.40488i 0 −0.918948 + 2.01222i 0.584585 0.171650i 0 2.29130 + 1.47253i
91.2 0.797176 1.74557i 0 −1.10181 1.27155i 0.451016 3.13689i 0 1.40141 3.06866i 0.584585 0.171650i 0 −5.11612 3.28793i
226.1 0.118239 + 0.822373i 0 1.25667 0.368991i −1.58839 + 1.02080i 0 −0.0441131 0.306813i 1.14231 + 2.50132i 0 −1.02729 1.18555i
226.2 0.118239 + 0.822373i 0 1.25667 0.368991i 3.36803 2.16450i 0 0.0592135 + 0.411839i 1.14231 + 2.50132i 0 2.17826 + 2.51385i
397.1 1.10181 1.27155i 0 −0.118239 0.822373i −2.46094 + 0.722597i 0 2.41674 2.78906i 1.65486 + 1.06351i 0 −1.79266 + 3.92538i
397.2 1.10181 1.27155i 0 −0.118239 0.822373i −1.36958 + 0.402144i 0 −3.35003 + 3.86614i 1.65486 + 1.06351i 0 −0.997662 + 2.18457i
424.1 −1.25667 + 0.368991i 0 −0.239446 + 0.153882i −0.681980 + 1.49333i 0 −3.00345 + 0.881891i 1.95949 2.26138i 0 0.305998 2.12826i
424.2 −1.25667 + 0.368991i 0 −0.239446 + 0.153882i 1.35049 2.95717i 0 4.00914 1.17719i 1.95949 2.26138i 0 −0.605953 + 4.21450i
442.1 1.10181 + 1.27155i 0 −0.118239 + 0.822373i −2.46094 0.722597i 0 2.41674 + 2.78906i 1.65486 1.06351i 0 −1.79266 3.92538i
442.2 1.10181 + 1.27155i 0 −0.118239 + 0.822373i −1.36958 0.402144i 0 −3.35003 3.86614i 1.65486 1.06351i 0 −0.997662 2.18457i
478.1 0.239446 + 0.153882i 0 −0.797176 1.74557i −1.21919 + 1.40702i 0 −1.14774 0.737606i 0.158746 1.10411i 0 −0.508444 + 0.149293i
478.2 0.239446 + 0.153882i 0 −0.797176 1.74557i 1.85253 2.13793i 0 −3.42222 2.19933i 0.158746 1.10411i 0 0.772569 0.226847i
550.1 0.797176 + 1.74557i 0 −1.10181 + 1.27155i −0.201991 1.40488i 0 −0.918948 2.01222i 0.584585 + 0.171650i 0 2.29130 1.47253i
550.2 0.797176 + 1.74557i 0 −1.10181 + 1.27155i 0.451016 + 3.13689i 0 1.40141 + 3.06866i 0.584585 + 0.171650i 0 −5.11612 + 3.28793i
595.1 0.118239 0.822373i 0 1.25667 + 0.368991i −1.58839 1.02080i 0 −0.0441131 + 0.306813i 1.14231 2.50132i 0 −1.02729 + 1.18555i
595.2 0.118239 0.822373i 0 1.25667 + 0.368991i 3.36803 + 2.16450i 0 0.0592135 0.411839i 1.14231 2.50132i 0 2.17826 2.51385i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.u.c 20
3.b odd 2 1 67.2.e.c 20
67.e even 11 1 inner 603.2.u.c 20
201.j even 22 1 4489.2.a.m 10
201.k odd 22 1 67.2.e.c 20
201.k odd 22 1 4489.2.a.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.e.c 20 3.b odd 2 1
67.2.e.c 20 201.k odd 22 1
603.2.u.c 20 1.a even 1 1 trivial
603.2.u.c 20 67.e even 11 1 inner
4489.2.a.l 10 201.k odd 22 1
4489.2.a.m 10 201.j even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2T_{2}^{9} + 4T_{2}^{8} + 3T_{2}^{7} - 6T_{2}^{6} + 12T_{2}^{5} + 9T_{2}^{4} - 7T_{2}^{3} + 14T_{2}^{2} - 6T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + 7 T^{18} + \cdots + 12243001 \) Copy content Toggle raw display
$7$ \( T^{20} + 8 T^{19} + 18 T^{18} + \cdots + 1739761 \) Copy content Toggle raw display
$11$ \( T^{20} + 5 T^{18} + 22 T^{17} + \cdots + 4190209 \) Copy content Toggle raw display
$13$ \( T^{20} - 12 T^{19} + 73 T^{18} + \cdots + 4489 \) Copy content Toggle raw display
$17$ \( T^{20} - 4 T^{19} + 57 T^{18} + \cdots + 876811321 \) Copy content Toggle raw display
$19$ \( T^{20} - 4 T^{19} - 61 T^{18} + \cdots + 23319241 \) Copy content Toggle raw display
$23$ \( T^{20} + 2 T^{19} + \cdots + 1098458449 \) Copy content Toggle raw display
$29$ \( (T^{10} - 3 T^{9} - 91 T^{8} + 70 T^{7} + \cdots + 737)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} - 21 T^{18} + \cdots + 867288026089 \) Copy content Toggle raw display
$37$ \( (T^{10} - 12 T^{9} - 54 T^{8} + \cdots - 279553)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 132185354901721 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 638971773407881 \) Copy content Toggle raw display
$47$ \( T^{20} + 16 T^{19} + \cdots + 2571253941169 \) Copy content Toggle raw display
$53$ \( T^{20} - T^{19} + 102 T^{18} + \cdots + 51457746649 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 279020843142889 \) Copy content Toggle raw display
$61$ \( T^{20} + 26 T^{19} + \cdots + 76356191669209 \) Copy content Toggle raw display
$67$ \( T^{20} + 22 T^{19} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{20} + 20 T^{19} + \cdots + 1590305111329 \) Copy content Toggle raw display
$73$ \( T^{20} + 55 T^{19} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{20} + 34 T^{19} + \cdots + 45\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{20} + 56 T^{19} + \cdots + 25849243729 \) Copy content Toggle raw display
$89$ \( T^{20} - 3 T^{19} + \cdots + 41\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{10} + 7 T^{9} - 405 T^{8} + \cdots - 294045257)^{2} \) Copy content Toggle raw display
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