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Properties

Label 8024.2.a.w
Level $8024$
Weight $2$
Character orbit 8024.a
Self dual yes
Analytic conductor $64.072$
Analytic rank $1$
Dimension $20$
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] = [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: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{17} q^{5} + \beta_{15} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{17} q^{5} + \beta_{15} q^{7} + \beta_{2} q^{9} + \beta_{6} q^{11} + ( - \beta_{17} - \beta_{3}) q^{13} + \beta_{7} q^{15} + q^{17} + ( - \beta_{18} - \beta_{12} + \cdots - \beta_{6}) q^{19}+ \cdots + ( - 2 \beta_{18} - \beta_{16} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15008099526873 \nu^{19} + 60125553049071 \nu^{18} - 662031832725318 \nu^{17} + \cdots - 28\!\cdots\!88 ) / 324854300335192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12145529660002 \nu^{19} + 15707606525101 \nu^{18} + 366647734887630 \nu^{17} + \cdots + 12\!\cdots\!68 ) / 162427150167596 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16020819195294 \nu^{19} - 58870048375111 \nu^{18} - 398443278955476 \nu^{17} + \cdots + 992821189787352 ) / 162427150167596 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 76645812152769 \nu^{19} - 17936170631519 \nu^{18} + \cdots - 86\!\cdots\!28 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 133807711367713 \nu^{19} - 362894116240403 \nu^{18} + \cdots - 13\!\cdots\!28 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42993313897265 \nu^{19} + 67952978646472 \nu^{18} + \cdots - 219488687815120 ) / 162427150167596 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45331988284303 \nu^{19} - 66818051107591 \nu^{18} + \cdots - 13\!\cdots\!28 ) / 162427150167596 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 184101995107787 \nu^{19} + 343669186525913 \nu^{18} + \cdots - 15\!\cdots\!76 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 120925478470383 \nu^{19} + 250843014352425 \nu^{18} + \cdots - 435029148696828 ) / 324854300335192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 263860853334045 \nu^{19} + 343389161409139 \nu^{18} + \cdots + 50\!\cdots\!76 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 151331030049999 \nu^{19} + 328605339348677 \nu^{18} + \cdots + 11\!\cdots\!56 ) / 324854300335192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 80461065143945 \nu^{19} - 192949961716001 \nu^{18} + \cdots + 292932332005348 ) / 162427150167596 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 325730164389989 \nu^{19} - 530619240509251 \nu^{18} + \cdots - 58\!\cdots\!64 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 337802121038101 \nu^{19} + 685783496165623 \nu^{18} + \cdots + 10\!\cdots\!80 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 343398705436307 \nu^{19} + 552989699504901 \nu^{18} + \cdots + 16\!\cdots\!28 ) / 649708600670384 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 253562826274429 \nu^{19} - 412127320209451 \nu^{18} + \cdots - 60\!\cdots\!32 ) / 324854300335192 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 546278944556307 \nu^{19} + \cdots - 15\!\cdots\!84 ) / 649708600670384 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} - \beta_{18} + 2 \beta_{15} - 2 \beta_{12} - \beta_{10} + 2 \beta_{8} + \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{16} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{19} - 16 \beta_{18} - 2 \beta_{17} - 4 \beta_{16} + 22 \beta_{15} + \beta_{14} - 2 \beta_{13} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{19} - 14 \beta_{18} + 2 \beta_{17} - 35 \beta_{16} + 7 \beta_{15} - 12 \beta_{14} + \cdots + 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 119 \beta_{19} - 195 \beta_{18} - 21 \beta_{17} - 80 \beta_{16} + 217 \beta_{15} + 10 \beta_{14} + \cdots + 219 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 84 \beta_{19} - 296 \beta_{18} + 39 \beta_{17} - 454 \beta_{16} + 164 \beta_{15} - 113 \beta_{14} + \cdots + 1580 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1138 \beta_{19} - 2219 \beta_{18} - 143 \beta_{17} - 1172 \beta_{16} + 2132 \beta_{15} + 66 \beta_{14} + \cdots + 2822 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1264 \beta_{19} - 4473 \beta_{18} + 531 \beta_{17} - 5392 \beta_{16} + 2636 \beta_{15} - 1008 \beta_{14} + \cdots + 15866 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 10969 \beta_{19} - 24806 \beta_{18} - 568 \beta_{17} - 15248 \beta_{16} + 21386 \beta_{15} + \cdots + 34970 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16762 \beta_{19} - 59503 \beta_{18} + 6425 \beta_{17} - 62221 \beta_{16} + 36441 \beta_{15} + \cdots + 166222 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 108410 \beta_{19} - 276964 \beta_{18} + 2777 \beta_{17} - 187462 \beta_{16} + 220379 \beta_{15} + \cdots + 422041 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 209183 \beta_{19} - 744000 \beta_{18} + 74499 \beta_{17} - 711737 \beta_{16} + 467037 \beta_{15} + \cdots + 1792079 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1104331 \beta_{19} - 3104435 \beta_{18} + 105587 \beta_{17} - 2236833 \beta_{16} + 2330442 \beta_{15} + \cdots + 5004206 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2526122 \beta_{19} - 8992100 \beta_{18} + 851497 \beta_{17} - 8125784 \beta_{16} + 5737033 \beta_{15} + \cdots + 19702435 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 11579025 \beta_{19} - 34964785 \beta_{18} + 1772647 \beta_{17} - 26247096 \beta_{16} + \cdots + 58642246 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 29926581 \beta_{19} - 106533226 \beta_{18} + 9696174 \beta_{17} - 92787727 \beta_{16} + \cdots + 219512538 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 124401332 \beta_{19} - 395489373 \beta_{18} + 24516859 \beta_{17} - 304997153 \beta_{16} + \cdots + 681858800 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.38332
2.69870
2.25007
2.11393
1.80437
1.37441
1.06424
0.609415
0.283688
0.237078
−0.126023
−0.313711
−0.423017
−1.12918
−1.36564
−1.42163
−1.50131
−2.06684
−2.70148
−2.77040
0 −3.38332 0 −0.369303 0 0.374879 0 8.44685 0
1.2 0 −2.69870 0 −1.98742 0 −0.0153109 0 4.28301 0
1.3 0 −2.25007 0 1.77999 0 3.34275 0 2.06282 0
1.4 0 −2.11393 0 0.387846 0 2.46344 0 1.46870 0
1.5 0 −1.80437 0 3.22329 0 2.17270 0 0.255750 0
1.6 0 −1.37441 0 −2.22841 0 2.74311 0 −1.11098 0
1.7 0 −1.06424 0 −1.91818 0 −4.17854 0 −1.86738 0
1.8 0 −0.609415 0 0.676769 0 −1.71345 0 −2.62861 0
1.9 0 −0.283688 0 −2.93013 0 4.78235 0 −2.91952 0
1.10 0 −0.237078 0 2.74912 0 −2.88085 0 −2.94379 0
1.11 0 0.126023 0 −3.00184 0 1.14075 0 −2.98412 0
1.12 0 0.313711 0 3.13839 0 2.81435 0 −2.90159 0
1.13 0 0.423017 0 1.16659 0 0.651110 0 −2.82106 0
1.14 0 1.12918 0 −2.32377 0 1.49770 0 −1.72496 0
1.15 0 1.36564 0 2.75901 0 −0.969675 0 −1.13502 0
1.16 0 1.42163 0 −3.52594 0 −1.45144 0 −0.978971 0
1.17 0 1.50131 0 1.01639 0 −3.17746 0 −0.746069 0
1.18 0 2.06684 0 0.0661094 0 2.71031 0 1.27184 0
1.19 0 2.70148 0 −2.07497 0 −1.35978 0 4.29801 0
1.20 0 2.77040 0 1.39646 0 −3.94694 0 4.67511 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.w ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8024.2.a.w ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\( T_{3}^{20} + 2 T_{3}^{19} - 29 T_{3}^{18} - 51 T_{3}^{17} + 341 T_{3}^{16} + 514 T_{3}^{15} - 2114 T_{3}^{14} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{20} + 2 T_{5}^{19} - 46 T_{5}^{18} - 86 T_{5}^{17} + 887 T_{5}^{16} + 1513 T_{5}^{15} + \cdots - 1841 \) Copy content Toggle raw display
\( T_{7}^{20} - 5 T_{7}^{19} - 53 T_{7}^{18} + 294 T_{7}^{17} + 1010 T_{7}^{16} - 6840 T_{7}^{15} + \cdots - 5659 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{20} + 2 T^{19} + \cdots - 1841 \) Copy content Toggle raw display
$7$ \( T^{20} - 5 T^{19} + \cdots - 5659 \) Copy content Toggle raw display
$11$ \( T^{20} - 8 T^{19} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{20} + 3 T^{19} + \cdots - 1515328 \) Copy content Toggle raw display
$17$ \( (T - 1)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + 9 T^{19} + \cdots - 318097 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 300505856 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 776063249744 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 42453927104 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 3298103296 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 8126321129236 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 42959042240 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 27357449569216 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 729380668051415 \) Copy content Toggle raw display
$59$ \( (T + 1)^{20} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 61350342141376 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 459860060992 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 8103160451072 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 664573208187968 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 8662600986647 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 58\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
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