Properties

Label 8008.2.a.v
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
Dimension $11$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} - 295 x^{2} - 87 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{3} q^{5} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{3} q^{5} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{3} + 1) q^{9} + q^{11} - q^{13} + ( - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{15} + ( - \beta_{8} + \beta_{7} - 1) q^{17} + ( - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + \beta_1 q^{21} + (\beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{23} + (\beta_{10} - 2 \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{25} + ( - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{27} + (\beta_{9} - \beta_{8} + \beta_{6} - \beta_{4}) q^{29} + (2 \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 1) q^{31} - \beta_1 q^{33} - \beta_{3} q^{35} + ( - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{37}+ \cdots + (\beta_{10} + \beta_{9} + \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} - 295 x^{2} - 87 x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3471 \nu^{10} - 1645 \nu^{9} - 68837 \nu^{8} + 9923 \nu^{7} + 439510 \nu^{6} + 44393 \nu^{5} - 1009563 \nu^{4} - 166370 \nu^{3} + 809815 \nu^{2} + 91282 \nu - 165726 ) / 5542 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2324 \nu^{10} + 1582 \nu^{9} + 45542 \nu^{8} - 15875 \nu^{7} - 286486 \nu^{6} + 25969 \nu^{5} + 641822 \nu^{4} + 2582 \nu^{3} - 483424 \nu^{2} - 19195 \nu + 90446 ) / 2771 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6303 \nu^{10} + 2204 \nu^{9} + 125636 \nu^{8} - 3690 \nu^{7} - 803561 \nu^{6} - 162019 \nu^{5} + 1830536 \nu^{4} + 457295 \nu^{3} - 1424035 \nu^{2} + \cdots + 275704 ) / 5542 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7983 \nu^{10} - 3381 \nu^{9} - 158391 \nu^{8} + 15867 \nu^{7} + 1007688 \nu^{6} + 137771 \nu^{5} - 2279981 \nu^{4} - 438162 \nu^{3} + 1760411 \nu^{2} + 246972 \nu - 350902 ) / 5542 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8768 \nu^{10} - 4533 \nu^{9} - 173457 \nu^{8} + 32913 \nu^{7} + 1101741 \nu^{6} + 62246 \nu^{5} - 2494913 \nu^{4} - 328435 \nu^{3} + 1925042 \nu^{2} + 233075 \nu - 382328 ) / 5542 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16188 \nu^{10} - 8115 \nu^{9} - 320665 \nu^{8} + 55521 \nu^{7} + 2041865 \nu^{6} + 150968 \nu^{5} - 4651071 \nu^{4} - 700915 \nu^{3} + 3628688 \nu^{2} + \cdots - 715838 ) / 5542 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16691 \nu^{10} - 7773 \nu^{9} - 331173 \nu^{8} + 45871 \nu^{7} + 2112530 \nu^{6} + 223051 \nu^{5} - 4826435 \nu^{4} - 848530 \nu^{3} + 3813573 \nu^{2} + \cdots - 792496 ) / 5542 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1088 \nu^{10} + 503 \nu^{9} + 21531 \nu^{8} - 2861 \nu^{7} - 136641 \nu^{6} - 15636 \nu^{5} + 308417 \nu^{4} + 57859 \nu^{3} - 236944 \nu^{2} - 38459 \nu + 45924 ) / 326 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23144 \nu^{10} - 11715 \nu^{9} - 457111 \nu^{8} + 80387 \nu^{7} + 2895869 \nu^{6} + 213874 \nu^{5} - 6526733 \nu^{4} - 988767 \nu^{3} + 5000438 \nu^{2} + \cdots - 983768 ) / 5542 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{10} + 11\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} - \beta_{4} + 11\beta_{3} + \beta_{2} + \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{10} + 13 \beta_{9} - 11 \beta_{8} + 14 \beta_{7} - 4 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 25 \beta_{3} + 7 \beta_{2} + 58 \beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 107 \beta_{10} + 108 \beta_{9} - 25 \beta_{8} + 31 \beta_{7} - 24 \beta_{6} + 9 \beta_{5} - 6 \beta_{4} + 107 \beta_{3} + 10 \beta_{2} + 26 \beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 149 \beta_{10} + 153 \beta_{9} - 107 \beta_{8} + 162 \beta_{7} - 80 \beta_{6} + 170 \beta_{5} + 158 \beta_{4} + 266 \beta_{3} + 35 \beta_{2} + 521 \beta _1 + 269 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1039 \beta_{10} + 1056 \beta_{9} - 266 \beta_{8} + 391 \beta_{7} - 371 \beta_{6} + 204 \beta_{5} + 21 \beta_{4} + 1047 \beta_{3} + 63 \beta_{2} + 420 \beta _1 + 2128 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1691 \beta_{10} + 1771 \beta_{9} - 1047 \beta_{8} + 1801 \beta_{7} - 1178 \beta_{6} + 1956 \beta_{5} + 1689 \beta_{4} + 2789 \beta_{3} + 76 \beta_{2} + 4912 \beta _1 + 3393 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10280 \beta_{10} + 10517 \beta_{9} - 2789 \beta_{8} + 4670 \beta_{7} - 4888 \beta_{6} + 3216 \beta_{5} + 1103 \beta_{4} + 10519 \beta_{3} + 170 \beta_{2} + 5734 \beta _1 + 20944 \) 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
3.33407
2.73758
1.49423
1.48564
0.983398
0.447939
−0.674923
−0.979305
−1.62645
−2.29224
−2.90994
0 −3.33407 0 −0.131360 0 −1.00000 0 8.11603 0
1.2 0 −2.73758 0 4.10495 0 −1.00000 0 4.49435 0
1.3 0 −1.49423 0 −2.57953 0 −1.00000 0 −0.767273 0
1.4 0 −1.48564 0 −0.395242 0 −1.00000 0 −0.792869 0
1.5 0 −0.983398 0 −1.20732 0 −1.00000 0 −2.03293 0
1.6 0 −0.447939 0 3.28103 0 −1.00000 0 −2.79935 0
1.7 0 0.674923 0 −4.42380 0 −1.00000 0 −2.54448 0
1.8 0 0.979305 0 2.33286 0 −1.00000 0 −2.04096 0
1.9 0 1.62645 0 1.90839 0 −1.00000 0 −0.354663 0
1.10 0 2.29224 0 0.279184 0 −1.00000 0 2.25438 0
1.11 0 2.90994 0 −1.16917 0 −1.00000 0 5.46776 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

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 8008.2.a.v 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.v 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{11} + 2 T_{3}^{10} - 19 T_{3}^{9} - 33 T_{3}^{8} + 120 T_{3}^{7} + 178 T_{3}^{6} - 296 T_{3}^{5} - 380 T_{3}^{4} + 280 T_{3}^{3} + 295 T_{3}^{2} - 87 T_{3} - 64 \) Copy content Toggle raw display
\( T_{5}^{11} - 2 T_{5}^{10} - 31 T_{5}^{9} + 59 T_{5}^{8} + 272 T_{5}^{7} - 398 T_{5}^{6} - 908 T_{5}^{5} + 618 T_{5}^{4} + 1236 T_{5}^{3} + 177 T_{5}^{2} - 103 T_{5} - 14 \) Copy content Toggle raw display
\( T_{17}^{11} + 4 T_{17}^{10} - 75 T_{17}^{9} - 281 T_{17}^{8} + 2118 T_{17}^{7} + 7560 T_{17}^{6} - 27762 T_{17}^{5} - 96466 T_{17}^{4} + 165982 T_{17}^{3} + 573247 T_{17}^{2} - 353067 T_{17} - 1240534 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + 2 T^{10} - 19 T^{9} - 33 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{11} - 2 T^{10} - 31 T^{9} + 59 T^{8} + \cdots - 14 \) Copy content Toggle raw display
$7$ \( (T + 1)^{11} \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( (T + 1)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} + 4 T^{10} - 75 T^{9} + \cdots - 1240534 \) Copy content Toggle raw display
$19$ \( T^{11} + 16 T^{10} + 7 T^{9} + \cdots - 398384 \) Copy content Toggle raw display
$23$ \( T^{11} + 3 T^{10} - 108 T^{9} + \cdots + 433152 \) Copy content Toggle raw display
$29$ \( T^{11} - T^{10} - 182 T^{9} + \cdots - 9670272 \) Copy content Toggle raw display
$31$ \( T^{11} - 14 T^{10} - 178 T^{9} + \cdots - 17531904 \) Copy content Toggle raw display
$37$ \( T^{11} + 8 T^{10} - 210 T^{9} + \cdots - 2176128 \) Copy content Toggle raw display
$41$ \( T^{11} - 4 T^{10} - 176 T^{9} + \cdots - 331392 \) Copy content Toggle raw display
$43$ \( T^{11} + 30 T^{10} + 258 T^{9} + \cdots + 2453332 \) Copy content Toggle raw display
$47$ \( T^{11} + 3 T^{10} - 108 T^{9} + \cdots + 433152 \) Copy content Toggle raw display
$53$ \( T^{11} + 5 T^{10} - 307 T^{9} + \cdots - 52723918 \) Copy content Toggle raw display
$59$ \( T^{11} - 11 T^{10} - 137 T^{9} + \cdots - 2048 \) Copy content Toggle raw display
$61$ \( T^{11} - 15 T^{10} - 138 T^{9} + \cdots - 412898 \) Copy content Toggle raw display
$67$ \( T^{11} + 41 T^{10} + \cdots - 126938492 \) Copy content Toggle raw display
$71$ \( T^{11} - T^{10} - 546 T^{9} + \cdots + 1001186432 \) Copy content Toggle raw display
$73$ \( T^{11} + 8 T^{10} - 450 T^{9} + \cdots - 183384192 \) Copy content Toggle raw display
$79$ \( T^{11} + 26 T^{10} + \cdots - 110006064 \) Copy content Toggle raw display
$83$ \( T^{11} + 31 T^{10} + \cdots + 7126131504 \) Copy content Toggle raw display
$89$ \( T^{11} - 2 T^{10} - 383 T^{9} + \cdots + 147931482 \) Copy content Toggle raw display
$97$ \( T^{11} + 6 T^{10} + \cdots - 164065879168 \) Copy content Toggle raw display
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