Properties

Label 8010.2.a.br
Level $8010$
Weight $2$
Character orbit 8010.a
Self dual yes
Analytic conductor $63.960$
Analytic rank $0$
Dimension $10$
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] = [8010,2,Mod(1,8010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8010.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: \(63.9601720190\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 44x^{8} + 68x^{7} + 707x^{6} - 704x^{5} - 5159x^{4} + 2172x^{3} + 16078x^{2} + 698x - 13023 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - \beta_{5} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} - \beta_{5} q^{7} + q^{8} + q^{10} + (\beta_{8} + 1) q^{11} + ( - \beta_{4} + 1) q^{13} - \beta_{5} q^{14} + q^{16} + (\beta_{3} + 1) q^{17} + \beta_1 q^{19} + q^{20} + (\beta_{8} + 1) q^{22} + ( - \beta_{6} + \beta_1 + 1) q^{23} + q^{25} + ( - \beta_{4} + 1) q^{26} - \beta_{5} q^{28} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3}) q^{29} + (\beta_{4} + 1) q^{31} + q^{32} + (\beta_{3} + 1) q^{34} - \beta_{5} q^{35} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2) q^{37} + \beta_1 q^{38} + q^{40} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 1) q^{41} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_1) q^{43} + (\beta_{8} + 1) q^{44} + ( - \beta_{6} + \beta_1 + 1) q^{46} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{47} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 3) q^{49} + q^{50} + ( - \beta_{4} + 1) q^{52} + ( - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{53} + (\beta_{8} + 1) q^{55} - \beta_{5} q^{56} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3}) q^{58} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_1 + 3) q^{59} + (\beta_{8} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 3) q^{61} + (\beta_{4} + 1) q^{62} + q^{64} + ( - \beta_{4} + 1) q^{65} + ( - 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_1) q^{67} + (\beta_{3} + 1) q^{68} - \beta_{5} q^{70} + ( - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{71} + (\beta_{9} - \beta_{7} + \beta_{5} + \beta_{3}) q^{73} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2) q^{74} + \beta_1 q^{76} + ( - 2 \beta_{9} + 2 \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_1 + 3) q^{77} + ( - \beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{79} + q^{80} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 1) q^{82} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 1) q^{83} + (\beta_{3} + 1) q^{85} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_1) q^{86} + (\beta_{8} + 1) q^{88} + q^{89} + (\beta_{9} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{91} + ( - \beta_{6} + \beta_1 + 1) q^{92} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{94} + \beta_1 q^{95} + (2 \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{97} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 3 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 3 q^{7} + 10 q^{8} + 10 q^{10} + 7 q^{11} + 6 q^{13} + 3 q^{14} + 10 q^{16} + 11 q^{17} + 3 q^{19} + 10 q^{20} + 7 q^{22} + 8 q^{23} + 10 q^{25} + 6 q^{26} + 3 q^{28} + 6 q^{29} + 14 q^{31} + 10 q^{32} + 11 q^{34} + 3 q^{35} + 8 q^{37} + 3 q^{38} + 10 q^{40} + 17 q^{41} + 4 q^{43} + 7 q^{44} + 8 q^{46} + 23 q^{47} + 29 q^{49} + 10 q^{50} + 6 q^{52} + 7 q^{55} + 3 q^{56} + 6 q^{58} + 22 q^{59} + 23 q^{61} + 14 q^{62} + 10 q^{64} + 6 q^{65} - q^{67} + 11 q^{68} + 3 q^{70} + 12 q^{71} - 4 q^{73} + 8 q^{74} + 3 q^{76} + 30 q^{77} + 23 q^{79} + 10 q^{80} + 17 q^{82} + 13 q^{83} + 11 q^{85} + 4 q^{86} + 7 q^{88} + 10 q^{89} - 4 q^{91} + 8 q^{92} + 23 q^{94} + 3 q^{95} + 8 q^{97} + 29 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 44x^{8} + 68x^{7} + 707x^{6} - 704x^{5} - 5159x^{4} + 2172x^{3} + 16078x^{2} + 698x - 13023 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6463441 \nu^{9} - 30748423 \nu^{8} - 243852693 \nu^{7} + 1119852401 \nu^{6} + 3258667102 \nu^{5} - 13343589826 \nu^{4} + \cdots - 65151289335 ) / 3525944964 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 783749 \nu^{9} - 382396 \nu^{8} + 29947282 \nu^{7} + 19019422 \nu^{6} - 360997650 \nu^{5} - 226446976 \nu^{4} + 1494327225 \nu^{3} + \cdots + 3116442839 ) / 293828747 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9674901 \nu^{9} + 94847045 \nu^{8} + 286409573 \nu^{7} - 3472513379 \nu^{6} - 2618680074 \nu^{5} + 40904353766 \nu^{4} + \cdots + 180667723089 ) / 3525944964 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3202032 \nu^{9} - 5384413 \nu^{8} + 115407416 \nu^{7} + 264446323 \nu^{6} - 1217193264 \nu^{5} - 4028586223 \nu^{4} + \cdots - 21570775716 ) / 881486241 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17794601 \nu^{9} + 70999113 \nu^{8} + 648156205 \nu^{7} - 2445777491 \nu^{6} - 7926350018 \nu^{5} + 26532186438 \nu^{4} + \cdots + 120161552865 ) / 3525944964 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18851467 \nu^{9} - 18141201 \nu^{8} - 783961571 \nu^{7} + 693285559 \nu^{6} + 11069022706 \nu^{5} - 9423421110 \nu^{4} + \cdots - 77827871049 ) / 3525944964 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11983343 \nu^{9} + 67605097 \nu^{8} + 440728973 \nu^{7} - 2459001957 \nu^{6} - 5560828694 \nu^{5} + 28711009990 \nu^{4} + \cdots + 124742446851 ) / 1762972482 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28686581 \nu^{9} - 84730599 \nu^{8} - 1055708761 \nu^{7} + 2935270121 \nu^{6} + 12773625782 \nu^{5} - 32524936086 \nu^{4} + \cdots - 154999829415 ) / 3525944964 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7610629 \nu^{9} - 19429777 \nu^{8} - 281112063 \nu^{7} + 654630254 \nu^{6} + 3427300018 \nu^{5} - 7055170084 \nu^{4} - 16519647282 \nu^{3} + \cdots - 35225039922 ) / 881486241 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{9} + 4\beta_{8} - 5\beta_{6} + 5\beta_{5} + 6\beta_{4} + 5\beta_{3} - 3\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27 \beta_{9} + 27 \beta_{8} + 26 \beta_{7} + 7 \beta_{6} - 45 \beta_{5} + 17 \beta_{4} - 5 \beta_{3} - 8 \beta_{2} - 26 \beta _1 + 233 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 228 \beta_{9} + 65 \beta_{8} + 28 \beta_{7} - 105 \beta_{6} + 104 \beta_{5} + 173 \beta_{4} + 104 \beta_{3} + 23 \beta_{2} - 134 \beta _1 + 310 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 289 \beta_{9} + 281 \beta_{8} + 285 \beta_{7} + 129 \beta_{6} - 439 \beta_{5} + 141 \beta_{4} - 105 \beta_{3} - 7 \beta_{2} - 302 \beta _1 + 1771 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3897 \beta_{9} + 527 \beta_{8} + 1036 \beta_{7} - 1079 \beta_{6} + 1097 \beta_{5} + 2743 \beta_{4} + 949 \beta_{3} + 784 \beta_{2} - 2688 \beta _1 + 4903 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11277 \beta_{9} + 10863 \beta_{8} + 11941 \beta_{7} + 6684 \beta_{6} - 16616 \beta_{5} + 4923 \beta_{4} - 5794 \beta_{3} + 1369 \beta_{2} - 12987 \beta _1 + 58646 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 34002 \beta_{9} + 2324 \beta_{8} + 13720 \beta_{7} - 4679 \beta_{6} + 5386 \beta_{5} + 23098 \beta_{4} + 2502 \beta_{3} + 9830 \beta_{2} - 26747 \beta _1 + 41846 \) 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
2.66874
−1.54229
−2.44774
0.959876
3.43493
−1.87128
−4.21976
−3.06974
3.62272
4.46455
1.00000 0 1.00000 1.00000 0 −4.49232 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −3.95936 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 −3.55821 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 −0.672856 1.00000 0 1.00000
1.5 1.00000 0 1.00000 1.00000 0 1.10935 1.00000 0 1.00000
1.6 1.00000 0 1.00000 1.00000 0 1.64343 1.00000 0 1.00000
1.7 1.00000 0 1.00000 1.00000 0 2.17320 1.00000 0 1.00000
1.8 1.00000 0 1.00000 1.00000 0 2.66853 1.00000 0 1.00000
1.9 1.00000 0 1.00000 1.00000 0 3.16554 1.00000 0 1.00000
1.10 1.00000 0 1.00000 1.00000 0 4.92270 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(89\) \(-1\)

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 8010.2.a.br yes 10
3.b odd 2 1 8010.2.a.bq 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8010.2.a.bq 10 3.b odd 2 1
8010.2.a.br yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\( T_{7}^{10} - 3 T_{7}^{9} - 45 T_{7}^{8} + 148 T_{7}^{7} + 598 T_{7}^{6} - 2454 T_{7}^{5} - 1426 T_{7}^{4} + 13716 T_{7}^{3} - 12948 T_{7}^{2} - 4360 T_{7} + 7016 \) Copy content Toggle raw display
\( T_{11}^{10} - 7 T_{11}^{9} - 45 T_{11}^{8} + 380 T_{11}^{7} + 548 T_{11}^{6} - 7236 T_{11}^{5} + 556 T_{11}^{4} + 57472 T_{11}^{3} - 41760 T_{11}^{2} - 159168 T_{11} + 157888 \) Copy content Toggle raw display
\( T_{13}^{10} - 6 T_{13}^{9} - 56 T_{13}^{8} + 252 T_{13}^{7} + 1142 T_{13}^{6} - 2484 T_{13}^{5} - 8908 T_{13}^{4} + 6856 T_{13}^{3} + 24392 T_{13}^{2} - 2640 T_{13} - 16416 \) Copy content Toggle raw display
\( T_{17}^{10} - 11 T_{17}^{9} - 37 T_{17}^{8} + 708 T_{17}^{7} - 860 T_{17}^{6} - 10916 T_{17}^{5} + 33596 T_{17}^{4} - 13376 T_{17}^{3} - 36192 T_{17}^{2} + 17664 T_{17} + 6912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} - 45 T^{8} + \cdots + 7016 \) Copy content Toggle raw display
$11$ \( T^{10} - 7 T^{9} - 45 T^{8} + \cdots + 157888 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} - 56 T^{8} + \cdots - 16416 \) Copy content Toggle raw display
$17$ \( T^{10} - 11 T^{9} - 37 T^{8} + \cdots + 6912 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} - 99 T^{8} + \cdots - 2592 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} - 104 T^{8} + \cdots + 491616 \) Copy content Toggle raw display
$29$ \( T^{10} - 6 T^{9} - 124 T^{8} + \cdots + 7776 \) Copy content Toggle raw display
$31$ \( T^{10} - 14 T^{9} + 16 T^{8} + \cdots - 2336 \) Copy content Toggle raw display
$37$ \( T^{10} - 8 T^{9} - 192 T^{8} + \cdots + 9117312 \) Copy content Toggle raw display
$41$ \( T^{10} - 17 T^{9} - 47 T^{8} + \cdots - 770304 \) Copy content Toggle raw display
$43$ \( T^{10} - 4 T^{9} - 244 T^{8} + \cdots - 147456 \) Copy content Toggle raw display
$47$ \( T^{10} - 23 T^{9} + 27 T^{8} + \cdots + 1455872 \) Copy content Toggle raw display
$53$ \( T^{10} - 392 T^{8} + \cdots + 123512832 \) Copy content Toggle raw display
$59$ \( T^{10} - 22 T^{9} - 92 T^{8} + \cdots + 19188864 \) Copy content Toggle raw display
$61$ \( T^{10} - 23 T^{9} + 5 T^{8} + \cdots - 16958232 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} - 567 T^{8} + \cdots - 6802752 \) Copy content Toggle raw display
$71$ \( T^{10} - 12 T^{9} - 376 T^{8} + \cdots - 14183424 \) Copy content Toggle raw display
$73$ \( T^{10} + 4 T^{9} - 216 T^{8} + \cdots + 4571136 \) Copy content Toggle raw display
$79$ \( T^{10} - 23 T^{9} - 123 T^{8} + \cdots + 45312 \) Copy content Toggle raw display
$83$ \( T^{10} - 13 T^{9} - 367 T^{8} + \cdots + 45840384 \) Copy content Toggle raw display
$89$ \( (T - 1)^{10} \) Copy content Toggle raw display
$97$ \( T^{10} - 8 T^{9} - 472 T^{8} + \cdots - 8912896 \) Copy content Toggle raw display
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