Properties

Label 8008.2.a.z
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + 11690 x^{7} - 7119 x^{6} - 22246 x^{5} + 11137 x^{4} + 20034 x^{3} - 8392 x^{2} + \cdots + 2560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{11} q^{5} - q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{11} q^{5} - q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + q^{13} + \beta_{3} q^{15} + (\beta_{9} + 1) q^{17} + ( - \beta_{11} + \beta_{6} - 1) q^{19} + \beta_1 q^{21} + (\beta_{13} + \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{11} - \beta_{10} + \beta_{4} + 2) q^{25} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_1 + 1) q^{27} + (\beta_{12} + \beta_{8} + \beta_1 + 1) q^{29} + ( - \beta_{12} - \beta_{11} + \beta_{5} - 1) q^{31} + \beta_1 q^{33} + \beta_{11} q^{35} + ( - \beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3}) q^{37} - \beta_1 q^{39} + ( - \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{4} + \beta_{3} - \beta_{2}) q^{41} + ( - \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{3} - 1) q^{43} + (\beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{45} + (\beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{47} + q^{49} + (2 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{51} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_{5} + \beta_{3} + 3) q^{53} + \beta_{11} q^{55} + (2 \beta_{14} - \beta_{13} - 2 \beta_{10} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{57} + ( - \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{3} - 1) q^{59} + (\beta_{12} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_{2} - 2) q^{63} - \beta_{11} q^{65} + (\beta_{11} + \beta_{10} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{67} + (\beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 3) q^{69}+ \cdots + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + 11690 x^{7} - 7119 x^{6} - 22246 x^{5} + 11137 x^{4} + 20034 x^{3} - 8392 x^{2} + \cdots + 2560 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1523579 \nu^{14} + 213661034 \nu^{13} - 179139600 \nu^{12} - 6986479725 \nu^{11} + 4541129757 \nu^{10} + 85561543273 \nu^{9} + \cdots - 253585799040 ) / 10828273438 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 204728997 \nu^{14} - 5797949397 \nu^{13} + 1200404289 \nu^{12} + 186669594656 \nu^{11} - 160969626719 \nu^{10} + \cdots + 11235942086336 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 157141431 \nu^{14} - 1230552919 \nu^{13} - 3155705101 \nu^{12} + 40387483536 \nu^{11} + 1702785579 \nu^{10} - 504234185256 \nu^{9} + \cdots + 3811771653184 ) / 86626187504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 413408383 \nu^{14} + 1544421129 \nu^{13} - 15746220005 \nu^{12} - 49653568744 \nu^{11} + 228419477571 \nu^{10} + 588397861744 \nu^{9} + \cdots + 545872471392 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 742937461 \nu^{14} - 1826318179 \nu^{13} + 30835167495 \nu^{12} + 59173290264 \nu^{11} - 502014930785 \nu^{10} + \cdots + 8430631658656 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1890437133 \nu^{14} - 1073055941 \nu^{13} - 66111397935 \nu^{12} + 31796527976 \nu^{11} + 904394485593 \nu^{10} + \cdots - 14280187531456 ) / 346504750016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1938443511 \nu^{14} + 1484217919 \nu^{13} + 68173131037 \nu^{12} - 44865881400 \nu^{11} - 931234396411 \nu^{10} + \cdots + 4190172143104 ) / 346504750016 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1184655527 \nu^{14} + 40599471 \nu^{13} + 42697917709 \nu^{12} + 530650360 \nu^{11} - 604962621531 \nu^{10} - 33239466432 \nu^{9} + \cdots + 9057739992992 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 396227811 \nu^{14} - 390133495 \nu^{13} - 13013329249 \nu^{12} + 11962731552 \nu^{11} + 161054746947 \nu^{10} - 137156782884 \nu^{9} + \cdots - 107982020560 ) / 43313093752 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2388875051 \nu^{14} - 8261584387 \nu^{13} - 71865979993 \nu^{12} + 259474132888 \nu^{11} + 775871452191 \nu^{10} + \cdots + 7139114332064 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2665656839 \nu^{14} + 4559868439 \nu^{13} + 84068282789 \nu^{12} - 139774279616 \nu^{11} - 980978775387 \nu^{10} + \cdots - 486592259136 ) / 173252375008 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3526798371 \nu^{14} + 4709151755 \nu^{13} + 117063051249 \nu^{12} - 143804690696 \nu^{11} - 1480208505991 \nu^{10} + \cdots + 7092433199008 ) / 173252375008 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{6} - 2\beta_{4} - \beta_{3} + 12\beta_{2} - \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} - \beta_{13} - \beta_{12} - 15 \beta_{11} - 14 \beta_{9} + 12 \beta_{8} + 14 \beta_{7} + 13 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 74 \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{14} - 3 \beta_{13} + 16 \beta_{12} + 31 \beta_{11} + 14 \beta_{10} + 3 \beta_{9} + \beta_{8} - 14 \beta_{6} - \beta_{5} - 33 \beta_{4} - 19 \beta_{3} + 129 \beta_{2} - 13 \beta _1 + 374 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 18 \beta_{14} - 21 \beta_{13} - 21 \beta_{12} - 184 \beta_{11} + 3 \beta_{10} - 167 \beta_{9} + 129 \beta_{8} + 161 \beta_{7} + 144 \beta_{6} + 18 \beta_{5} - 15 \beta_{4} - 17 \beta_{3} + 17 \beta_{2} + 733 \beta _1 - 157 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 252 \beta_{14} - 71 \beta_{13} + 202 \beta_{12} + 375 \beta_{11} + 148 \beta_{10} + 55 \beta_{9} + 17 \beta_{8} + 2 \beta_{7} - 154 \beta_{6} - 21 \beta_{5} - 423 \beta_{4} - 266 \beta_{3} + 1362 \beta_{2} - 123 \beta _1 + 3719 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 231 \beta_{14} - 310 \beta_{13} - 314 \beta_{12} - 2079 \beta_{11} + 70 \beta_{10} - 1908 \beta_{9} + 1362 \beta_{8} + 1762 \beta_{7} + 1520 \beta_{6} + 228 \beta_{5} - 172 \beta_{4} - 222 \beta_{3} + 222 \beta_{2} + \cdots - 1611 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3231 \beta_{14} - 1178 \beta_{13} + 2365 \beta_{12} + 4177 \beta_{11} + 1396 \beta_{10} + 696 \beta_{9} + 222 \beta_{8} + 61 \beta_{7} - 1578 \beta_{6} - 304 \beta_{5} - 4992 \beta_{4} - 3330 \beta_{3} + 14343 \beta_{2} + \cdots + 37990 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2559 \beta_{14} - 4056 \beta_{13} - 4140 \beta_{12} - 22536 \beta_{11} + 1102 \beta_{10} - 21413 \beta_{9} + 14343 \beta_{8} + 18970 \beta_{7} + 15716 \beta_{6} + 2557 \beta_{5} - 1838 \beta_{4} + \cdots - 15851 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 39686 \beta_{14} - 16983 \beta_{13} + 26769 \beta_{12} + 44927 \beta_{11} + 12188 \beta_{10} + 7551 \beta_{9} + 2681 \beta_{8} + 1224 \beta_{7} - 15767 \beta_{6} - 3816 \beta_{5} - 56875 \beta_{4} + \cdots + 393237 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 25891 \beta_{14} - 50465 \beta_{13} - 51380 \beta_{12} - 238942 \beta_{11} + 14779 \beta_{10} - 238140 \beta_{9} + 151136 \beta_{8} + 203010 \beta_{7} + 160838 \beta_{6} + 27223 \beta_{5} + \cdots - 152360 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 475419 \beta_{14} - 227599 \beta_{13} + 297511 \beta_{12} + 475676 \beta_{11} + 98135 \beta_{10} + 74828 \beta_{9} + 31400 \beta_{8} + 20288 \beta_{7} - 156157 \beta_{6} - 44832 \beta_{5} + \cdots + 4100536 \) 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
3.34164
3.21278
2.38300
2.04397
1.74947
1.40500
0.619952
0.455829
−0.804723
−1.32542
−1.58213
−1.69218
−2.42593
−3.11300
−3.26824
0 −3.34164 0 1.86211 0 −1.00000 0 8.16655 0
1.2 0 −3.21278 0 0.0641301 0 −1.00000 0 7.32193 0
1.3 0 −2.38300 0 −4.12953 0 −1.00000 0 2.67869 0
1.4 0 −2.04397 0 4.25466 0 −1.00000 0 1.17780 0
1.5 0 −1.74947 0 −0.876486 0 −1.00000 0 0.0606454 0
1.6 0 −1.40500 0 3.56716 0 −1.00000 0 −1.02598 0
1.7 0 −0.619952 0 −1.29231 0 −1.00000 0 −2.61566 0
1.8 0 −0.455829 0 0.836884 0 −1.00000 0 −2.79222 0
1.9 0 0.804723 0 −0.337817 0 −1.00000 0 −2.35242 0
1.10 0 1.32542 0 2.70237 0 −1.00000 0 −1.24326 0
1.11 0 1.58213 0 −3.58630 0 −1.00000 0 −0.496866 0
1.12 0 1.69218 0 −2.62665 0 −1.00000 0 −0.136527 0
1.13 0 2.42593 0 3.43967 0 −1.00000 0 2.88513 0
1.14 0 3.11300 0 −2.63092 0 −1.00000 0 6.69079 0
1.15 0 3.26824 0 2.75302 0 −1.00000 0 7.68141 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.z 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.z 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{15} + T_{3}^{14} - 35 T_{3}^{13} - 32 T_{3}^{12} + 477 T_{3}^{11} + 392 T_{3}^{10} - 3236 T_{3}^{9} - 2330 T_{3}^{8} + 11690 T_{3}^{7} + 7119 T_{3}^{6} - 22246 T_{3}^{5} - 11137 T_{3}^{4} + 20034 T_{3}^{3} + 8392 T_{3}^{2} + \cdots - 2560 \) Copy content Toggle raw display
\( T_{5}^{15} - 4 T_{5}^{14} - 46 T_{5}^{13} + 187 T_{5}^{12} + 783 T_{5}^{11} - 3259 T_{5}^{10} - 6166 T_{5}^{9} + 26402 T_{5}^{8} + 23304 T_{5}^{7} - 100657 T_{5}^{6} - 42529 T_{5}^{5} + 157237 T_{5}^{4} + 48979 T_{5}^{3} + \cdots + 1520 \) Copy content Toggle raw display
\( T_{17}^{15} - 8 T_{17}^{14} - 135 T_{17}^{13} + 1159 T_{17}^{12} + 6312 T_{17}^{11} - 61344 T_{17}^{10} - 117882 T_{17}^{9} + 1489316 T_{17}^{8} + 549286 T_{17}^{7} - 17081235 T_{17}^{6} + 6861491 T_{17}^{5} + \cdots + 63518720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + T^{14} - 35 T^{13} - 32 T^{12} + \cdots - 2560 \) Copy content Toggle raw display
$5$ \( T^{15} - 4 T^{14} - 46 T^{13} + \cdots + 1520 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( (T - 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 8 T^{14} - 135 T^{13} + \cdots + 63518720 \) Copy content Toggle raw display
$19$ \( T^{15} + 17 T^{14} + \cdots - 969411584 \) Copy content Toggle raw display
$23$ \( T^{15} - 7 T^{14} - 149 T^{13} + \cdots - 21274624 \) Copy content Toggle raw display
$29$ \( T^{15} - 14 T^{14} + \cdots + 8253251584 \) Copy content Toggle raw display
$31$ \( T^{15} + 4 T^{14} - 265 T^{13} + \cdots + 171913216 \) Copy content Toggle raw display
$37$ \( T^{15} - 3 T^{14} + \cdots + 953761153024 \) Copy content Toggle raw display
$41$ \( T^{15} - 324 T^{13} + 307 T^{12} + \cdots - 3473408 \) Copy content Toggle raw display
$43$ \( T^{15} + 13 T^{14} + \cdots + 1935535882240 \) Copy content Toggle raw display
$47$ \( T^{15} - 6 T^{14} + \cdots - 96164380672 \) Copy content Toggle raw display
$53$ \( T^{15} - 38 T^{14} + \cdots + 250000136560 \) Copy content Toggle raw display
$59$ \( T^{15} + 18 T^{14} + \cdots - 123374534656 \) Copy content Toggle raw display
$61$ \( T^{15} - 23 T^{14} + \cdots + 703905280 \) Copy content Toggle raw display
$67$ \( T^{15} + 8 T^{14} - 375 T^{13} + \cdots - 144646784 \) Copy content Toggle raw display
$71$ \( T^{15} + 12 T^{14} + \cdots - 43413893120 \) Copy content Toggle raw display
$73$ \( T^{15} - 11 T^{14} + \cdots - 1744469344256 \) Copy content Toggle raw display
$79$ \( T^{15} + T^{14} + \cdots + 37621321471744 \) Copy content Toggle raw display
$83$ \( T^{15} + 16 T^{14} + \cdots - 150160298752 \) Copy content Toggle raw display
$89$ \( T^{15} - 28 T^{14} + \cdots - 226733872 \) Copy content Toggle raw display
$97$ \( T^{15} - 30 T^{14} + \cdots + 10753709794304 \) Copy content Toggle raw display
show more
show less