Properties

Label 8024.2.a.v
Level $8024$
Weight $2$
Character orbit 8024.a
Self dual yes
Analytic conductor $64.072$
Analytic rank $0$
Dimension $18$
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] = [8024,2,Mod(1,8024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8024.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: \(64.0719625819\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} - 23166 x^{10} - 36429 x^{9} + 72042 x^{8} + 69272 x^{7} - 119982 x^{6} + \cdots + 1136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + \beta_{13} q^{11} + ( - \beta_{12} - \beta_{6} + 1) q^{13} + ( - \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + 2 \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{15}+ \cdots + ( - 2 \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} - 23166 x^{10} - 36429 x^{9} + 72042 x^{8} + 69272 x^{7} - 119982 x^{6} + \cdots + 1136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 122592670237151 \nu^{17} + \cdots + 68\!\cdots\!72 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 609319771140861 \nu^{17} + \cdots + 20\!\cdots\!72 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!09 \nu^{17} + \cdots - 69\!\cdots\!84 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!17 \nu^{17} + \cdots - 12\!\cdots\!72 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!40 \nu^{17} + \cdots - 55\!\cdots\!84 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!89 \nu^{17} + \cdots - 64\!\cdots\!12 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42\!\cdots\!47 \nu^{17} + \cdots - 81\!\cdots\!28 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 47\!\cdots\!33 \nu^{17} + \cdots + 12\!\cdots\!76 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54\!\cdots\!22 \nu^{17} + \cdots - 36\!\cdots\!64 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 58\!\cdots\!93 \nu^{17} + \cdots + 17\!\cdots\!96 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 64\!\cdots\!10 \nu^{17} + \cdots - 79\!\cdots\!36 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!54 \nu^{17} + \cdots - 26\!\cdots\!76 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 93\!\cdots\!21 \nu^{17} + \cdots - 30\!\cdots\!04 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!62 \nu^{17} + \cdots - 27\!\cdots\!44 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 13\!\cdots\!56 \nu^{17} + \cdots - 29\!\cdots\!20 ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{12} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + 2\beta_{2} + 8\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + \beta_{16} - 3 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 3 \beta_{8} + 4 \beta_{7} + \beta_{6} - 5 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 17 \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} + 4 \beta_{16} - 2 \beta_{15} - 21 \beta_{14} + 2 \beta_{13} - 16 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - \beta_{9} + 20 \beta_{8} + 23 \beta_{7} + 2 \beta_{5} - 26 \beta_{4} + 20 \beta_{3} + 32 \beta_{2} + 87 \beta _1 + 66 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{17} + 21 \beta_{16} - 7 \beta_{15} - 74 \beta_{14} + 21 \beta_{13} - 35 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + \beta_{9} + 70 \beta_{8} + 94 \beta_{7} + 13 \beta_{6} + 6 \beta_{5} - 109 \beta_{4} + 62 \beta_{3} + 134 \beta_{2} + \cdots + 316 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 77 \beta_{17} + 78 \beta_{16} - 49 \beta_{15} - 353 \beta_{14} + 70 \beta_{13} - 227 \beta_{12} + 96 \beta_{11} - 87 \beta_{10} + 2 \beta_{9} + 324 \beta_{8} + 409 \beta_{7} + 22 \beta_{6} + 35 \beta_{5} - 465 \beta_{4} + 329 \beta_{3} + \cdots + 916 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 384 \beta_{17} + 320 \beta_{16} - 187 \beta_{15} - 1317 \beta_{14} + 408 \beta_{13} - 683 \beta_{12} + 437 \beta_{11} - 364 \beta_{10} + 88 \beta_{9} + 1198 \beta_{8} + 1637 \beta_{7} + 212 \beta_{6} + 111 \beta_{5} + \cdots + 3725 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1523 \beta_{17} + 1165 \beta_{16} - 905 \beta_{15} - 5522 \beta_{14} + 1579 \beta_{13} - 3238 \beta_{12} + 1747 \beta_{11} - 1756 \beta_{10} + 429 \beta_{9} + 4942 \beta_{8} + 6572 \beta_{7} + 750 \beta_{6} + \cdots + 12357 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6744 \beta_{17} + 4428 \beta_{16} - 3514 \beta_{15} - 21002 \beta_{14} + 7417 \beta_{13} - 11166 \beta_{12} + 7309 \beta_{11} - 7154 \beta_{10} + 2642 \beta_{9} + 18701 \beta_{8} + 25757 \beta_{7} + \cdots + 47338 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 27344 \beta_{17} + 16049 \beta_{16} - 14889 \beta_{15} - 83662 \beta_{14} + 29711 \beta_{13} - 46630 \beta_{12} + 29110 \beta_{11} - 30672 \beta_{10} + 12123 \beta_{9} + 74002 \beta_{8} + \cdots + 166841 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 115062 \beta_{17} + 59365 \beta_{16} - 57635 \beta_{15} - 320138 \beta_{14} + 127306 \beta_{13} - 170294 \beta_{12} + 117748 \beta_{11} - 122547 \beta_{10} + 58242 \beta_{9} + 283099 \beta_{8} + \cdots + 625793 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 466947 \beta_{17} + 215469 \beta_{16} - 230415 \beta_{15} - 1248025 \beta_{14} + 510804 \beta_{13} - 674114 \beta_{12} + 466502 \beta_{11} - 498405 \beta_{10} + 254217 \beta_{9} + \cdots + 2269038 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1915590 \beta_{17} + 790426 \beta_{16} - 885654 \beta_{15} - 4784387 \beta_{14} + 2095904 \beta_{13} - 2521552 \beta_{12} + 1856393 \beta_{11} - 1964242 \beta_{10} + 1116217 \beta_{9} + \cdots + 8457084 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7731717 \beta_{17} + 2880251 \beta_{16} - 3443726 \beta_{15} - 18480803 \beta_{14} + 8357266 \beta_{13} - 9757789 \beta_{12} + 7316432 \beta_{11} - 7783058 \beta_{10} + \cdots + 31101868 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 31261528 \beta_{17} + 10562222 \beta_{16} - 13155222 \beta_{15} - 70893352 \beta_{14} + 33537363 \beta_{13} - 36872896 \beta_{12} + 28849339 \beta_{11} - 30390186 \beta_{10} + \cdots + 115834959 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 125371317 \beta_{17} + 38679802 \beta_{16} - 50447645 \beta_{15} - 272821665 \beta_{14} + 132816430 \beta_{13} - 141360258 \beta_{12} + 113195450 \beta_{11} + \cdots + 429304259 \) 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
3.87400
3.69527
3.16635
2.83830
2.44623
2.26378
1.28813
1.27925
0.379802
0.365311
−0.164030
−0.947778
−1.45822
−1.57328
−1.86517
−2.05229
−2.08137
−2.45430
0 −2.87400 0 −0.461304 0 −2.48202 0 5.25986 0
1.2 0 −2.69527 0 −0.822149 0 4.27633 0 4.26450 0
1.3 0 −2.16635 0 3.18635 0 1.65768 0 1.69308 0
1.4 0 −1.83830 0 −1.17026 0 −0.774148 0 0.379331 0
1.5 0 −1.44623 0 3.72024 0 0.836543 0 −0.908408 0
1.6 0 −1.26378 0 −4.20424 0 4.08685 0 −1.40287 0
1.7 0 −0.288129 0 2.13990 0 1.97111 0 −2.91698 0
1.8 0 −0.279252 0 −0.263460 0 −1.45782 0 −2.92202 0
1.9 0 0.620198 0 −3.37733 0 −0.896557 0 −2.61535 0
1.10 0 0.634689 0 −0.976475 0 3.68579 0 −2.59717 0
1.11 0 1.16403 0 3.89012 0 −0.552876 0 −1.64503 0
1.12 0 1.94778 0 −1.72233 0 −3.84469 0 0.793839 0
1.13 0 2.45822 0 −0.323529 0 −2.56007 0 3.04282 0
1.14 0 2.57328 0 3.36856 0 0.305606 0 3.62177 0
1.15 0 2.86517 0 −3.00205 0 3.89738 0 5.20917 0
1.16 0 3.05229 0 −3.77561 0 −1.23812 0 6.31647 0
1.17 0 3.08137 0 3.75209 0 1.50295 0 6.49484 0
1.18 0 3.45430 0 2.04147 0 2.58609 0 8.93216 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)
\(59\) \(1\)

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 8024.2.a.v 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8024.2.a.v 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\( T_{3}^{18} - 9 T_{3}^{17} - 2 T_{3}^{16} + 228 T_{3}^{15} - 409 T_{3}^{14} - 2148 T_{3}^{13} + 6104 T_{3}^{12} + 8922 T_{3}^{11} - 37829 T_{3}^{10} - 12323 T_{3}^{9} + 117939 T_{3}^{8} - 19740 T_{3}^{7} - 185041 T_{3}^{6} + \cdots + 2384 \) Copy content Toggle raw display
\( T_{5}^{18} - 2 T_{5}^{17} - 64 T_{5}^{16} + 114 T_{5}^{15} + 1673 T_{5}^{14} - 2469 T_{5}^{13} - 23140 T_{5}^{12} + 24640 T_{5}^{11} + 183203 T_{5}^{10} - 99212 T_{5}^{9} - 831655 T_{5}^{8} - 55166 T_{5}^{7} + \cdots - 26072 \) Copy content Toggle raw display
\( T_{7}^{18} - 11 T_{7}^{17} + 4 T_{7}^{16} + 335 T_{7}^{15} - 853 T_{7}^{14} - 3402 T_{7}^{13} + 13020 T_{7}^{12} + 13804 T_{7}^{11} - 82433 T_{7}^{10} - 16487 T_{7}^{9} + 259709 T_{7}^{8} - 18031 T_{7}^{7} - 434687 T_{7}^{6} + \cdots + 13792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 9 T^{17} - 2 T^{16} + 228 T^{15} + \cdots + 2384 \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} - 64 T^{16} + \cdots - 26072 \) Copy content Toggle raw display
$7$ \( T^{18} - 11 T^{17} + 4 T^{16} + \cdots + 13792 \) Copy content Toggle raw display
$11$ \( T^{18} + T^{17} - 82 T^{16} - 60 T^{15} + \cdots + 51712 \) Copy content Toggle raw display
$13$ \( T^{18} - 15 T^{17} - 3 T^{16} + \cdots - 283136 \) Copy content Toggle raw display
$17$ \( (T - 1)^{18} \) Copy content Toggle raw display
$19$ \( T^{18} - 26 T^{17} + 199 T^{16} + \cdots + 32194400 \) Copy content Toggle raw display
$23$ \( T^{18} - 22 T^{17} + \cdots + 1857379648 \) Copy content Toggle raw display
$29$ \( T^{18} - 6 T^{17} - 259 T^{16} + \cdots - 979345592 \) Copy content Toggle raw display
$31$ \( T^{18} - 13 T^{17} + \cdots - 30838012416 \) Copy content Toggle raw display
$37$ \( T^{18} - 4 T^{17} + \cdots + 1151655804928 \) Copy content Toggle raw display
$41$ \( T^{18} + 15 T^{17} + \cdots - 301477579552 \) Copy content Toggle raw display
$43$ \( T^{18} - 12 T^{17} + \cdots + 1052269797376 \) Copy content Toggle raw display
$47$ \( T^{18} - 8 T^{17} - 498 T^{16} + \cdots + 559464448 \) Copy content Toggle raw display
$53$ \( T^{18} + 11 T^{17} + \cdots + 29233484264 \) Copy content Toggle raw display
$59$ \( (T + 1)^{18} \) Copy content Toggle raw display
$61$ \( T^{18} - 53 T^{17} + \cdots - 47709267279488 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 129931149953536 \) Copy content Toggle raw display
$71$ \( T^{18} - 8 T^{17} + \cdots + 19288096538624 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 596309081595328 \) Copy content Toggle raw display
$79$ \( T^{18} + 9 T^{17} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 112738221185024 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 235101122713856 \) Copy content Toggle raw display
$97$ \( T^{18} + 56 T^{17} + \cdots + 81\!\cdots\!56 \) Copy content Toggle raw display
show more
show less