Properties

Label 2009.2.a.u
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2009,2,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

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: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + 6449 x^{12} - 9852 x^{11} - 13797 x^{10} + 18080 x^{9} + 17721 x^{8} - 18446 x^{7} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{18} q^{5} + (\beta_{8} + 1) q^{6} + ( - \beta_{19} - \beta_{16} - \beta_{15} - \beta_{13} - \beta_{7} - \beta_1 + 1) q^{8} + (\beta_{15} + \beta_{9} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{18} q^{5} + (\beta_{8} + 1) q^{6} + ( - \beta_{19} - \beta_{16} - \beta_{15} - \beta_{13} - \beta_{7} - \beta_1 + 1) q^{8} + (\beta_{15} + \beta_{9} - \beta_{3} + \beta_1) q^{9} + ( - \beta_{13} - \beta_{7} + 1) q^{10} + (\beta_{18} - \beta_{11}) q^{11} + ( - \beta_{19} - \beta_{18} - \beta_{16} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{5} - \beta_{3} + \cdots + 2) q^{12}+ \cdots + (\beta_{19} + 2 \beta_{18} + \beta_{17} + \beta_{14} - \beta_{13} + \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + 6449 x^{12} - 9852 x^{11} - 13797 x^{10} + 18080 x^{9} + 17721 x^{8} - 18446 x^{7} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{19} + 125 \nu^{18} + 257 \nu^{17} - 3106 \nu^{16} - 5497 \nu^{15} + 30507 \nu^{14} + 53542 \nu^{13} - 148929 \nu^{12} - 277250 \nu^{11} + 366927 \nu^{10} + 789606 \nu^{9} + \cdots + 6860 ) / 844 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 974 \nu^{19} - 1689 \nu^{18} - 24289 \nu^{17} + 44215 \nu^{16} + 241298 \nu^{15} - 475649 \nu^{14} - 1207269 \nu^{13} + 2725642 \nu^{12} + 3116367 \nu^{11} - 9030010 \nu^{10} + \cdots + 207375 ) / 11816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 140 \nu^{19} + 277 \nu^{18} + 3905 \nu^{17} - 7023 \nu^{16} - 45386 \nu^{15} + 71783 \nu^{14} + 285307 \nu^{13} - 379058 \nu^{12} - 1051789 \nu^{11} + 1102030 \nu^{10} + \cdots - 1923 ) / 844 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1441 \nu^{19} - 4408 \nu^{18} - 36373 \nu^{17} + 115070 \nu^{16} + 370173 \nu^{15} - 1227388 \nu^{14} - 1944145 \nu^{13} + 6908584 \nu^{12} + 5578466 \nu^{11} + \cdots + 196049 ) / 2954 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4075 \nu^{19} - 10768 \nu^{18} - 106301 \nu^{17} + 279121 \nu^{16} + 1133945 \nu^{15} - 2948266 \nu^{14} - 6387514 \nu^{13} + 16368355 \nu^{12} + 20472398 \nu^{11} + \cdots + 397047 ) / 5908 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1871 \nu^{19} - 3616 \nu^{18} - 48804 \nu^{17} + 93461 \nu^{16} + 519731 \nu^{15} - 985066 \nu^{14} - 2915591 \nu^{13} + 5465175 \nu^{12} + 9276793 \nu^{11} - 17217267 \nu^{10} + \cdots + 136681 ) / 1688 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19767 \nu^{19} - 52631 \nu^{18} - 508397 \nu^{17} + 1369512 \nu^{16} + 5315407 \nu^{15} - 14549501 \nu^{14} - 29080478 \nu^{13} + 81489167 \nu^{12} + 89221158 \nu^{11} + \cdots + 2317504 ) / 11816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25318 \nu^{19} + 68773 \nu^{18} + 644715 \nu^{17} - 1793047 \nu^{16} - 6643398 \nu^{15} + 19103225 \nu^{14} + 35551495 \nu^{13} - 107435598 \nu^{12} + \cdots - 3155999 ) / 11816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4453 \nu^{19} - 10482 \nu^{18} - 116106 \nu^{17} + 271265 \nu^{16} + 1237033 \nu^{15} - 2860796 \nu^{14} - 6952139 \nu^{13} + 15862357 \nu^{12} + 22198929 \nu^{11} + \cdots + 385549 ) / 1688 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14341 \nu^{19} - 36774 \nu^{18} - 367103 \nu^{17} + 957937 \nu^{16} + 3811197 \nu^{15} - 10195866 \nu^{14} - 20628888 \nu^{13} + 57277865 \nu^{12} + 62243884 \nu^{11} + \cdots + 1715049 ) / 5908 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19699 \nu^{19} + 48813 \nu^{18} + 508339 \nu^{17} - 1268655 \nu^{16} - 5338649 \nu^{15} + 13459641 \nu^{14} + 29394218 \nu^{13} - 75265998 \nu^{12} + \cdots - 2112117 ) / 5908 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 55795 \nu^{19} + 137847 \nu^{18} + 1439923 \nu^{17} - 3582608 \nu^{16} - 15123407 \nu^{15} + 38010317 \nu^{14} + 83273652 \nu^{13} - 212578283 \nu^{12} + \cdots - 6105596 ) / 11816 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 57829 \nu^{19} - 148747 \nu^{18} - 1485881 \nu^{17} + 3870922 \nu^{16} + 15510665 \nu^{15} - 41139285 \nu^{14} - 84645344 \nu^{13} + 230601293 \nu^{12} + \cdots + 6810118 ) / 11816 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 63867 \nu^{19} - 147271 \nu^{18} - 1657237 \nu^{17} + 3819554 \nu^{16} + 17535335 \nu^{15} - 40414977 \nu^{14} - 97569654 \nu^{13} + 225221581 \nu^{12} + \cdots + 6091414 ) / 11816 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 65375 \nu^{19} + 157917 \nu^{18} + 1691191 \nu^{17} - 4101614 \nu^{16} - 17820791 \nu^{15} + 43481023 \nu^{14} + 98583464 \nu^{13} - 242911087 \nu^{12} + \cdots - 6892942 ) / 11816 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 35122 \nu^{19} - 85868 \nu^{18} - 910249 \nu^{17} + 2228382 \nu^{16} + 9617310 \nu^{15} - 23592048 \nu^{14} - 53410665 \nu^{13} + 131533684 \nu^{12} + 167604135 \nu^{11} + \cdots + 3591140 ) / 5908 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 45224 \nu^{19} + 109964 \nu^{18} + 1169521 \nu^{17} - 2855704 \nu^{16} - 12318296 \nu^{15} + 30265756 \nu^{14} + 68100795 \nu^{13} - 169013794 \nu^{12} + \cdots - 4729788 ) / 5908 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{16} + \beta_{15} + \beta_{13} + \beta_{7} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{15} + \beta_{10} + \beta_{9} + \beta_{8} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{19} + 2 \beta_{18} + 8 \beta_{16} + 7 \beta_{15} + \beta_{14} + 8 \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{9} - \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 27 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} + 9 \beta_{17} - \beta_{16} + 10 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + 9 \beta_{10} + 9 \beta_{9} + 11 \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 34 \beta_{2} + 9 \beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56 \beta_{19} + 24 \beta_{18} + \beta_{17} + 55 \beta_{16} + 44 \beta_{15} + 11 \beta_{14} + 55 \beta_{13} + 24 \beta_{12} - 12 \beta_{11} - 23 \beta_{9} - 10 \beta_{8} + 43 \beta_{7} + 11 \beta_{6} - 13 \beta_{5} - \beta_{3} + 10 \beta_{2} + 154 \beta _1 - 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{19} - 12 \beta_{18} + 67 \beta_{17} - 11 \beta_{16} + 81 \beta_{15} - 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 12 \beta_{11} + 65 \beta_{10} + 62 \beta_{9} + 91 \beta_{8} - 11 \beta_{7} + 12 \beta_{6} - 42 \beta_{5} - 12 \beta_{4} - \beta_{3} + \cdots + 493 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 380 \beta_{19} + 210 \beta_{18} + 16 \beta_{17} + 366 \beta_{16} + 279 \beta_{15} + 91 \beta_{14} + 368 \beta_{13} + 210 \beta_{12} - 106 \beta_{11} + \beta_{10} - 197 \beta_{9} - 73 \beta_{8} + 261 \beta_{7} + 91 \beta_{6} - 121 \beta_{5} + \cdots - 381 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22 \beta_{19} - 93 \beta_{18} + 478 \beta_{17} - 81 \beta_{16} + 611 \beta_{15} - 99 \beta_{14} + 155 \beta_{13} + 40 \beta_{12} + 98 \beta_{11} + 441 \beta_{10} + 389 \beta_{9} + 678 \beta_{8} - 87 \beta_{7} + 103 \beta_{6} - 411 \beta_{5} + \cdots + 2971 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2554 \beta_{19} + 1628 \beta_{18} + 176 \beta_{17} + 2423 \beta_{16} + 1816 \beta_{15} + 682 \beta_{14} + 2461 \beta_{13} + 1636 \beta_{12} - 832 \beta_{11} + 18 \beta_{10} - 1511 \beta_{9} - 472 \beta_{8} + 1606 \beta_{7} + \cdots - 2558 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 294 \beta_{19} - 577 \beta_{18} + 3367 \beta_{17} - 485 \beta_{16} + 4452 \beta_{15} - 702 \beta_{14} + 1377 \beta_{13} + 500 \beta_{12} + 675 \beta_{11} + 2924 \beta_{10} + 2325 \beta_{9} + 4806 \beta_{8} - 608 \beta_{7} + \cdots + 18231 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17120 \beta_{19} + 11891 \beta_{18} + 1652 \beta_{17} + 16078 \beta_{16} + 12110 \beta_{15} + 4892 \beta_{14} + 16540 \beta_{13} + 12060 \beta_{12} - 6153 \beta_{11} + 219 \beta_{10} - 10976 \beta_{9} + \cdots - 16993 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3149 \beta_{19} - 2968 \beta_{18} + 23582 \beta_{17} - 2396 \beta_{16} + 31813 \beta_{15} - 4601 \beta_{14} + 11318 \beta_{13} + 5070 \beta_{12} + 4186 \beta_{11} + 19207 \beta_{10} + 13429 \beta_{9} + \cdots + 113413 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 114763 \beta_{19} + 84042 \beta_{18} + 14221 \beta_{17} + 107111 \beta_{16} + 82238 \beta_{15} + 34321 \beta_{14} + 111719 \beta_{13} + 86284 \beta_{12} - 43982 \beta_{11} + 2260 \beta_{10} + \cdots - 111787 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 29855 \beta_{19} - 11483 \beta_{18} + 164528 \beta_{17} - 8368 \beta_{16} + 224695 \beta_{15} - 28732 \beta_{14} + 88914 \beta_{13} + 45818 \beta_{12} + 23838 \beta_{11} + 125728 \beta_{10} + \cdots + 713239 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 770182 \beta_{19} + 582625 \beta_{18} + 116045 \beta_{17} + 716308 \beta_{16} + 565412 \beta_{15} + 237841 \beta_{14} + 757526 \beta_{13} + 606656 \beta_{12} - 307904 \beta_{11} + 21286 \beta_{10} + \cdots - 728670 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 262512 \beta_{19} - 11693 \beta_{18} + 1144247 \beta_{17} + 3600 \beta_{16} + 1575715 \beta_{15} - 173241 \beta_{14} + 678888 \beta_{13} + 385908 \beta_{12} + 123712 \beta_{11} + 822592 \beta_{10} + \cdots + 4525211 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 5176782 \beta_{19} + 3992197 \beta_{18} + 913778 \beta_{17} + 4805830 \beta_{16} + 3917898 \beta_{15} + 1636097 \beta_{14} + 5150816 \beta_{13} + 4220190 \beta_{12} - 2127159 \beta_{11} + \cdots - 4709052 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.63622
2.34607
2.27905
2.27121
2.10538
1.41602
0.979921
0.928350
0.845808
0.212475
−0.328365
−0.463326
−0.466765
−0.874793
−1.30606
−1.50500
−1.89507
−2.11567
−2.51950
−2.54595
−2.63622 −2.63584 4.94964 1.42955 6.94865 0 −7.77590 3.94765 −3.76861
1.2 −2.34607 0.967441 3.50405 3.03222 −2.26969 0 −3.52861 −2.06406 −7.11381
1.3 −2.27905 −0.713649 3.19408 −3.36532 1.62644 0 −2.72136 −2.49071 7.66975
1.4 −2.27121 3.42520 3.15839 −0.765314 −7.77934 0 −2.63095 8.73199 1.73819
1.5 −2.10538 −1.20723 2.43261 1.24490 2.54167 0 −0.910797 −1.54260 −2.62098
1.6 −1.41602 −2.14541 0.00509860 −1.43123 3.03793 0 2.82481 1.60278 2.02665
1.7 −0.979921 1.94904 −1.03976 1.73965 −1.90990 0 2.97872 0.798739 −1.70472
1.8 −0.928350 2.79733 −1.13817 −3.35114 −2.59691 0 2.91332 4.82508 3.11103
1.9 −0.845808 0.0216736 −1.28461 −2.12752 −0.0183317 0 2.77815 −2.99953 1.79947
1.10 −0.212475 −2.41608 −1.95485 3.68342 0.513357 0 0.840308 2.83745 −0.782635
1.11 0.328365 −0.613482 −1.89218 0.220416 −0.201446 0 −1.27805 −2.62364 0.0723768
1.12 0.463326 0.981674 −1.78533 −3.31637 0.454835 0 −1.75384 −2.03632 −1.53656
1.13 0.466765 2.80691 −1.78213 2.27147 1.31017 0 −1.76537 4.87872 1.06024
1.14 0.874793 2.48225 −1.23474 4.06589 2.17145 0 −2.82973 3.16156 3.55681
1.15 1.30606 −1.82067 −0.294211 0.323122 −2.37790 0 −2.99637 0.314826 0.422016
1.16 1.50500 0.350711 0.265025 −3.19754 0.527821 0 −2.61114 −2.87700 −4.81229
1.17 1.89507 −1.57790 1.59127 3.12194 −2.99022 0 −0.774564 −0.510238 5.91628
1.18 2.11567 1.23352 2.47606 0.585449 2.60971 0 1.00718 −1.47844 1.23862
1.19 2.51950 2.78497 4.34790 1.47804 7.01674 0 5.91555 4.75604 3.72392
1.20 2.54595 1.32955 4.48185 2.35837 3.38496 0 6.31865 −1.23230 6.00428
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(41\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.2.a.u yes 20
7.b odd 2 1 2009.2.a.t 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.2.a.t 20 7.b odd 2 1
2009.2.a.u yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{20} + 2 T_{2}^{19} - 27 T_{2}^{18} - 52 T_{2}^{17} + 302 T_{2}^{16} + 552 T_{2}^{15} - 1820 T_{2}^{14} - 3090 T_{2}^{13} + 6449 T_{2}^{12} + 9852 T_{2}^{11} - 13797 T_{2}^{10} - 18080 T_{2}^{9} + 17721 T_{2}^{8} + 18446 T_{2}^{7} + \cdots + 49 \) Copy content Toggle raw display
\( T_{3}^{20} - 8 T_{3}^{19} - 6 T_{3}^{18} + 192 T_{3}^{17} - 236 T_{3}^{16} - 1804 T_{3}^{15} + 3770 T_{3}^{14} + 8248 T_{3}^{13} - 23916 T_{3}^{12} - 17616 T_{3}^{11} + 78924 T_{3}^{10} + 7624 T_{3}^{9} - 141680 T_{3}^{8} + \cdots + 89 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 2 T^{19} - 27 T^{18} - 52 T^{17} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{20} - 8 T^{19} - 6 T^{18} + 192 T^{17} + \cdots + 89 \) Copy content Toggle raw display
$5$ \( T^{20} - 8 T^{19} - 28 T^{18} + \cdots - 40384 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} - 102 T^{18} - 48 T^{17} + \cdots - 77312 \) Copy content Toggle raw display
$13$ \( T^{20} - 12 T^{19} + \cdots + 21455782873 \) Copy content Toggle raw display
$17$ \( T^{20} - 8 T^{19} - 152 T^{18} + \cdots + 202951601 \) Copy content Toggle raw display
$19$ \( T^{20} - 36 T^{19} + \cdots + 887123761 \) Copy content Toggle raw display
$23$ \( T^{20} + 12 T^{19} + \cdots + 1666581528929 \) Copy content Toggle raw display
$29$ \( T^{20} - 4 T^{19} + \cdots + 16555395476992 \) Copy content Toggle raw display
$31$ \( T^{20} - 80 T^{19} + \cdots + 350103980608 \) Copy content Toggle raw display
$37$ \( T^{20} - 4 T^{19} + \cdots - 196930033408 \) Copy content Toggle raw display
$41$ \( (T - 1)^{20} \) Copy content Toggle raw display
$43$ \( T^{20} - 354 T^{18} + \cdots - 23554608983 \) Copy content Toggle raw display
$47$ \( T^{20} - 32 T^{19} + \cdots + 37415328784 \) Copy content Toggle raw display
$53$ \( T^{20} - 4 T^{19} + \cdots + 4993537019392 \) Copy content Toggle raw display
$59$ \( T^{20} - 32 T^{19} + \cdots + 28531727785024 \) Copy content Toggle raw display
$61$ \( T^{20} - 44 T^{19} + \cdots + 1046452461632 \) Copy content Toggle raw display
$67$ \( T^{20} + 4 T^{19} + \cdots + 30547158444608 \) Copy content Toggle raw display
$71$ \( T^{20} - 8 T^{19} + \cdots + 31205527187392 \) Copy content Toggle raw display
$73$ \( T^{20} - 48 T^{19} + \cdots - 56\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 128543211633664 \) Copy content Toggle raw display
$83$ \( T^{20} - 8 T^{19} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{20} - 20 T^{19} + \cdots + 20\!\cdots\!93 \) Copy content Toggle raw display
$97$ \( T^{20} - 8 T^{19} + \cdots - 40\!\cdots\!11 \) Copy content Toggle raw display
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