Properties

Label 133.2.e.c
Level $133$
Weight $2$
Character orbit 133.e
Analytic conductor $1.062$
Analytic rank $0$
Dimension $10$
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] = [133,2,Mod(64,133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(133, base_ring=CyclotomicField(6))
 
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("133.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.e (of order \(3\), 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: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{6}) q^{3} + (\beta_{9} + \beta_{6} - 1) q^{4} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{5} + (\beta_{9} - \beta_{8}) q^{6} - q^{7} + (\beta_{3} - 1) q^{8} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{6}) q^{3} + (\beta_{9} + \beta_{6} - 1) q^{4} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{5} + (\beta_{9} - \beta_{8}) q^{6} - q^{7} + (\beta_{3} - 1) q^{8} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{9} + (\beta_{8} - 2 \beta_{7} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{10} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4} - 1) q^{12} + ( - \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{13} - \beta_1 q^{14} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + (\beta_{9} + \beta_{8} + 2 \beta_{6} - \beta_{5} + \beta_{4}) q^{16} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{17} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{18} + (2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_1 - 1) q^{19} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{20} + ( - \beta_{7} - \beta_{6}) q^{21} + (2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{22} + ( - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{9} - \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_1) q^{24} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - \beta_{2} + \beta_1) q^{25} + (\beta_{5} - \beta_{3} + \beta_{2} - 2) q^{26} + (2 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{27} + ( - \beta_{9} - \beta_{6} + 1) q^{28} + ( - 4 \beta_{9} + \beta_{8} - \beta_{6} - 2 \beta_{2} + 2 \beta_1 + 1) q^{29} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} + 4) q^{30} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{31} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 1) q^{32} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{33} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{34} + (\beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1) q^{35} + (\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_1) q^{36} + (\beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 2) q^{37} + ( - 2 \beta_{9} + \beta_{7} - 4 \beta_{6} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{38} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 3) q^{39} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1) q^{40} + ( - 3 \beta_{7} + 3 \beta_{6} - 2 \beta_1) q^{41} + ( - \beta_{9} + \beta_{8}) q^{42} + (\beta_{9} - 2 \beta_{8} - 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{43} + (\beta_{7} - 2 \beta_{6} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{44} + (\beta_{3} + 2 \beta_{2} - 4) q^{45} + ( - 4 \beta_{5} - 5 \beta_{4} + \beta_{3} + 4 \beta_{2} - 6) q^{46} + (\beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{47} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{48} + q^{49} + (2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2) q^{50} + (\beta_{9} + \beta_{8} - 4 \beta_{7} - 3 \beta_{6} - 4 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{51} + (\beta_{9} - 2 \beta_{8} - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{52} + ( - \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - \beta_{6} + 3 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{53} + ( - \beta_{8} + 5 \beta_{7} + 4 \beta_{6} + \beta_{5} + \beta_1) q^{54} + (\beta_{9} + 2 \beta_{6} + \beta_{4} - 3 \beta_1) q^{55} + ( - \beta_{3} + 1) q^{56} + (\beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2) q^{57} + (\beta_{5} - \beta_{4} - 5 \beta_{3} + 4 \beta_{2}) q^{58} + ( - \beta_{9} + 2 \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_1) q^{59} + (\beta_{9} + \beta_{7} + 5 \beta_{6} + \beta_{4} + 2 \beta_1) q^{60} + (4 \beta_{9} + 7 \beta_{6} - 7) q^{61} + ( - 2 \beta_{9} + \beta_{8} - \beta_{7} - 6 \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{62} + (\beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{63} + ( - 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 5) q^{64} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3) q^{65} + (2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{3}) q^{66} + (3 \beta_{9} - \beta_{8} - \beta_{7} + 3 \beta_{6} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{67} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{68} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} - 5) q^{69} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{70} + (\beta_{9} - 2 \beta_{8} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_1) q^{71} + (3 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 4 \beta_{6} - 4 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{72} + ( - 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 6 \beta_1) q^{73}+ \cdots + (2 \beta_{8} + 3 \beta_{7} + 6 \beta_{6} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + 3 q^{3} - 3 q^{4} - q^{5} + 3 q^{6} - 10 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} + 3 q^{3} - 3 q^{4} - q^{5} + 3 q^{6} - 10 q^{7} - 6 q^{8} - 4 q^{10} + 2 q^{11} - 4 q^{12} - q^{14} + 6 q^{15} + 9 q^{16} - 7 q^{17} + 6 q^{18} - 7 q^{19} + 14 q^{20} - 3 q^{21} + 5 q^{22} - 2 q^{23} - 15 q^{24} - 2 q^{25} - 24 q^{26} - 24 q^{27} + 3 q^{28} - 6 q^{29} + 44 q^{30} - 6 q^{31} - 8 q^{32} + 10 q^{33} + 15 q^{34} + q^{35} + 6 q^{36} - 10 q^{37} + q^{38} + 34 q^{39} + 6 q^{40} + 19 q^{41} - 3 q^{42} - 16 q^{43} + 10 q^{44} - 32 q^{45} - 20 q^{46} + 10 q^{47} - 4 q^{48} + 10 q^{49} - 34 q^{50} + 7 q^{51} + 11 q^{52} + 7 q^{53} + 10 q^{54} + 5 q^{55} + 6 q^{56} + 6 q^{57} - 10 q^{58} + 16 q^{59} + 23 q^{60} - 27 q^{61} - 21 q^{62} - 38 q^{64} - 32 q^{65} + q^{66} - 11 q^{67} + 44 q^{68} - 54 q^{69} + 4 q^{70} + 21 q^{72} + 5 q^{73} - 8 q^{74} + 34 q^{75} + 3 q^{76} - 2 q^{77} + 4 q^{78} + 21 q^{79} - 8 q^{80} - 13 q^{81} + 11 q^{82} - 16 q^{83} + 4 q^{84} + 34 q^{85} - 48 q^{86} + 12 q^{87} - 20 q^{88} + 22 q^{89} + 26 q^{90} + 18 q^{92} + 12 q^{93} + 42 q^{94} + 44 q^{95} - 30 q^{96} + 28 q^{97} + q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1315 \nu^{9} + 10212 \nu^{8} - 29785 \nu^{7} + 62655 \nu^{6} - 116587 \nu^{5} + 246790 \nu^{4} - 580195 \nu^{3} + 353165 \nu^{2} - 41699 \nu + 94555 ) / 759446 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2630 \nu^{9} + 20424 \nu^{8} - 59570 \nu^{7} + 125310 \nu^{6} - 233174 \nu^{5} + 493580 \nu^{4} - 780667 \nu^{3} + 706330 \nu^{2} - 83398 \nu + 568833 ) / 379723 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8897 \nu^{9} - 20580 \nu^{8} + 60025 \nu^{7} - 74507 \nu^{6} + 234955 \nu^{5} - 497350 \nu^{4} + 379465 \nu^{3} - 711725 \nu^{2} + 84035 \nu - 2277023 ) / 759446 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15274 \nu^{9} + 38916 \nu^{8} - 113505 \nu^{7} + 207978 \nu^{6} - 444291 \nu^{5} + 940470 \nu^{4} - 805012 \nu^{3} + 1345845 \nu^{2} - 158907 \nu + 1130040 ) / 379723 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 94555 \nu^{9} - 93240 \nu^{8} + 651673 \nu^{7} - 159325 \nu^{6} + 2963105 \nu^{5} - 734408 \nu^{4} + 5710175 \nu^{3} + 2471295 \nu^{2} + 6076575 \nu + 44705 ) / 759446 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53119 \nu^{9} + 45493 \nu^{8} - 354193 \nu^{7} + 54788 \nu^{6} - 1690191 \nu^{5} + 276681 \nu^{4} - 3299920 \nu^{3} - 1496128 \nu^{2} - 4520934 \nu - 26801 ) / 379723 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 134864 \nu^{9} + 127326 \nu^{8} - 940952 \nu^{7} + 260698 \nu^{6} - 4301561 \nu^{5} + 1178430 \nu^{4} - 8421612 \nu^{3} - 2241795 \nu^{2} + \cdots + 1066270 ) / 379723 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 283665 \nu^{9} + 279720 \nu^{8} - 1955019 \nu^{7} + 477975 \nu^{6} - 8889315 \nu^{5} + 2203224 \nu^{4} - 17130525 \nu^{3} - 6654439 \nu^{2} + \cdots + 2144223 ) / 759446 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + \beta_{8} - 12\beta_{6} - \beta_{5} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + \beta_{8} - 6\beta_{7} - 7\beta_{6} - 6\beta_{3} + 16\beta_{2} - 16\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} + 23\beta_{4} - 2\beta_{3} + \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{9} - 9\beta_{8} + 30\beta_{7} + 40\beta_{6} + 9\beta_{5} + 10\beta_{4} + 65\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 104\beta_{9} - 39\beta_{8} + 19\beta_{7} + 223\beta_{6} + 19\beta_{3} - 11\beta_{2} + 11\beta _1 - 223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -58\beta_{5} - 69\beta_{4} + 143\beta_{3} - 269\beta_{2} - 215 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\)

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
−0.983281 1.70309i
−0.552442 0.956858i
0.0595946 + 0.103221i
0.883493 + 1.53025i
1.09264 + 1.89250i
−0.983281 + 1.70309i
−0.552442 + 0.956858i
0.0595946 0.103221i
0.883493 1.53025i
1.09264 1.89250i
−0.983281 1.70309i 0.130414 + 0.225884i −0.933684 + 1.61719i −1.22681 2.12490i 0.256468 0.444216i −1.00000 −0.260829 1.46598 2.53916i −2.41260 + 4.17875i
64.2 −0.552442 0.956858i 1.53536 + 2.65933i 0.389615 0.674832i 0.643958 + 1.11537i 1.69640 2.93825i −1.00000 −3.07073 −3.21469 + 5.56800i 0.711500 1.23235i
64.3 0.0595946 + 0.103221i −0.237532 0.411417i 0.992897 1.71975i −0.412094 0.713768i 0.0283112 0.0490364i −1.00000 0.475063 1.38716 2.40263i 0.0491171 0.0850734i
64.4 0.883493 + 1.53025i −0.775497 1.34320i −0.561119 + 0.971886i 1.75378 + 3.03764i 1.37029 2.37341i −1.00000 1.55099 0.297209 0.514782i −3.09891 + 5.36747i
64.5 1.09264 + 1.89250i 0.847250 + 1.46748i −1.38771 + 2.40358i −1.25884 2.18037i −1.85147 + 3.20684i −1.00000 −1.69450 0.0643360 0.111433i 2.75090 4.76470i
106.1 −0.983281 + 1.70309i 0.130414 0.225884i −0.933684 1.61719i −1.22681 + 2.12490i 0.256468 + 0.444216i −1.00000 −0.260829 1.46598 + 2.53916i −2.41260 4.17875i
106.2 −0.552442 + 0.956858i 1.53536 2.65933i 0.389615 + 0.674832i 0.643958 1.11537i 1.69640 + 2.93825i −1.00000 −3.07073 −3.21469 5.56800i 0.711500 + 1.23235i
106.3 0.0595946 0.103221i −0.237532 + 0.411417i 0.992897 + 1.71975i −0.412094 + 0.713768i 0.0283112 + 0.0490364i −1.00000 0.475063 1.38716 + 2.40263i 0.0491171 + 0.0850734i
106.4 0.883493 1.53025i −0.775497 + 1.34320i −0.561119 0.971886i 1.75378 3.03764i 1.37029 + 2.37341i −1.00000 1.55099 0.297209 + 0.514782i −3.09891 5.36747i
106.5 1.09264 1.89250i 0.847250 1.46748i −1.38771 2.40358i −1.25884 + 2.18037i −1.85147 3.20684i −1.00000 −1.69450 0.0643360 + 0.111433i 2.75090 + 4.76470i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.e.c 10
3.b odd 2 1 1197.2.k.g 10
7.b odd 2 1 931.2.e.c 10
7.c even 3 1 931.2.g.e 10
7.c even 3 1 931.2.h.f 10
7.d odd 6 1 931.2.g.f 10
7.d odd 6 1 931.2.h.e 10
19.c even 3 1 inner 133.2.e.c 10
19.c even 3 1 2527.2.a.j 5
19.d odd 6 1 2527.2.a.k 5
57.h odd 6 1 1197.2.k.g 10
133.g even 3 1 931.2.h.f 10
133.h even 3 1 931.2.g.e 10
133.k odd 6 1 931.2.h.e 10
133.m odd 6 1 931.2.e.c 10
133.t odd 6 1 931.2.g.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.e.c 10 1.a even 1 1 trivial
133.2.e.c 10 19.c even 3 1 inner
931.2.e.c 10 7.b odd 2 1
931.2.e.c 10 133.m odd 6 1
931.2.g.e 10 7.c even 3 1
931.2.g.e 10 133.h even 3 1
931.2.g.f 10 7.d odd 6 1
931.2.g.f 10 133.t odd 6 1
931.2.h.e 10 7.d odd 6 1
931.2.h.e 10 133.k odd 6 1
931.2.h.f 10 7.c even 3 1
931.2.h.f 10 133.g even 3 1
1197.2.k.g 10 3.b odd 2 1
1197.2.k.g 10 57.h odd 6 1
2527.2.a.j 5 19.c even 3 1
2527.2.a.k 5 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} + 7T_{2}^{8} - 2T_{2}^{7} + 32T_{2}^{6} - 9T_{2}^{5} + 63T_{2}^{4} + 20T_{2}^{3} + 68T_{2}^{2} - 8T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + 7 T^{8} - 2 T^{7} + 32 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + 12 T^{8} - 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + 14 T^{8} + 23 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( (T + 1)^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} - T^{4} - 12 T^{3} + 19 T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 20 T^{8} - 22 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( T^{10} + 7 T^{9} + 84 T^{8} + \cdots + 149769 \) Copy content Toggle raw display
$19$ \( T^{10} + 7 T^{9} + 30 T^{8} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 2 T^{9} + 104 T^{8} + \cdots + 36036009 \) Copy content Toggle raw display
$29$ \( T^{10} + 6 T^{9} + 120 T^{8} + \cdots + 78943225 \) Copy content Toggle raw display
$31$ \( (T^{5} + 3 T^{4} - 80 T^{3} + 73 T^{2} + \cdots - 1280)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 5 T^{4} - 90 T^{3} - 175 T^{2} + \cdots - 2924)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 19 T^{9} + 280 T^{8} + \cdots + 2627641 \) Copy content Toggle raw display
$43$ \( T^{10} + 16 T^{9} + 298 T^{8} + \cdots + 727609 \) Copy content Toggle raw display
$47$ \( T^{10} - 10 T^{9} + 149 T^{8} + \cdots + 3717184 \) Copy content Toggle raw display
$53$ \( T^{10} - 7 T^{9} + 214 T^{8} + \cdots + 25230529 \) Copy content Toggle raw display
$59$ \( T^{10} - 16 T^{9} + 228 T^{8} + \cdots + 3143529 \) Copy content Toggle raw display
$61$ \( T^{10} + 27 T^{9} + 559 T^{8} + \cdots + 408321 \) Copy content Toggle raw display
$67$ \( T^{10} + 11 T^{9} + 172 T^{8} + \cdots + 3003289 \) Copy content Toggle raw display
$71$ \( T^{10} + 42 T^{8} - 94 T^{7} + \cdots + 109561 \) Copy content Toggle raw display
$73$ \( T^{10} - 5 T^{9} + 291 T^{8} + \cdots + 685444761 \) Copy content Toggle raw display
$79$ \( T^{10} - 21 T^{9} + \cdots + 5068158481 \) Copy content Toggle raw display
$83$ \( (T^{5} + 8 T^{4} - 29 T^{3} - 167 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 22 T^{9} + \cdots + 380718144 \) Copy content Toggle raw display
$97$ \( T^{10} - 28 T^{9} + \cdots + 201554809 \) Copy content Toggle raw display
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