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Properties

Label 8036.2.a.t
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
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] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, 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("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.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: \(64.1677830643\)
Analytic rank: \(0\)
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: \( x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{11} q^{5} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{11} q^{5} + (\beta_{2} + 1) q^{9} + ( - \beta_{7} + \beta_1 - 1) q^{11} + (\beta_{14} + 1) q^{13} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_1) q^{15}+ \cdots + ( - \beta_{19} + \beta_{18} + \beta_{16} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 693771077180 \nu^{19} - 2778645197913 \nu^{18} - 20725264188092 \nu^{17} + \cdots + 59952362841513 ) / 8907432980628 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!34 \nu^{19} + \cdots + 19\!\cdots\!63 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!26 \nu^{19} + \cdots + 23\!\cdots\!47 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6661373649057 \nu^{19} - 27339265673408 \nu^{18} - 197062564273797 \nu^{17} + \cdots + 620836439810025 ) / 8907432980628 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!41 \nu^{19} + \cdots + 97\!\cdots\!54 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!21 \nu^{19} + \cdots + 60\!\cdots\!25 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!09 \nu^{19} + \cdots + 39\!\cdots\!73 ) / 35\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!75 \nu^{19} + \cdots - 22\!\cdots\!58 ) / 17\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 83\!\cdots\!21 \nu^{19} + \cdots - 66\!\cdots\!21 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!23 \nu^{19} + \cdots + 10\!\cdots\!69 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57\!\cdots\!19 \nu^{19} + \cdots - 50\!\cdots\!49 ) / 35\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!65 \nu^{19} + \cdots - 56\!\cdots\!20 ) / 35\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!23 \nu^{19} + \cdots + 12\!\cdots\!85 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!20 \nu^{19} + \cdots + 12\!\cdots\!01 ) / 70\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 74\!\cdots\!48 \nu^{19} + \cdots + 61\!\cdots\!26 ) / 35\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 81\!\cdots\!41 \nu^{19} + \cdots - 69\!\cdots\!71 ) / 35\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 90\!\cdots\!59 \nu^{19} + \cdots + 82\!\cdots\!10 ) / 35\!\cdots\!74 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{16} - \beta_{15} - \beta_{12} - 2\beta_{11} - \beta_{10} + \beta_{3} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{13} - \beta_{11} - 2 \beta_{10} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 10 \beta_{17} + 10 \beta_{16} - 14 \beta_{15} - \beta_{14} + \beta_{13} - 12 \beta_{12} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31 \beta_{19} - 16 \beta_{18} + 13 \beta_{17} - 18 \beta_{16} + 2 \beta_{15} - 4 \beta_{14} + \cdots + 211 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{19} + \beta_{18} - 87 \beta_{17} + 91 \beta_{16} - 165 \beta_{15} - 14 \beta_{14} + 14 \beta_{13} + \cdots + 151 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 378 \beta_{19} - 197 \beta_{18} + 141 \beta_{17} - 233 \beta_{16} + 29 \beta_{15} - 78 \beta_{14} + \cdots + 1806 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 38 \beta_{19} + 7 \beta_{18} - 748 \beta_{17} + 827 \beta_{16} - 1839 \beta_{15} - 152 \beta_{14} + \cdots + 1521 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4243 \beta_{19} - 2238 \beta_{18} + 1492 \beta_{17} - 2696 \beta_{16} + 233 \beta_{15} - 1046 \beta_{14} + \cdots + 16429 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 277 \beta_{19} - 133 \beta_{18} - 6518 \beta_{17} + 7570 \beta_{16} - 19975 \beta_{15} - 1542 \beta_{14} + \cdots + 15151 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 45923 \beta_{19} - 24655 \beta_{18} + 15782 \beta_{17} - 29677 \beta_{16} + 569 \beta_{15} + \cdots + 155705 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9828 \beta_{19} - 5006 \beta_{18} - 57756 \beta_{17} + 69586 \beta_{16} - 214052 \beta_{15} + \cdots + 152541 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 488333 \beta_{19} - 268379 \beta_{18} + 166948 \beta_{17} - 318472 \beta_{16} - 20717 \beta_{15} + \cdots + 1517375 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 171630 \beta_{19} - 101254 \beta_{18} - 518951 \beta_{17} + 639870 \beta_{16} - 2276953 \beta_{15} + \cdots + 1564378 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5147120 \beta_{19} - 2909362 \beta_{18} + 1762319 \beta_{17} - 3370083 \beta_{16} - 536726 \beta_{15} + \cdots + 15077038 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2477970 \beta_{19} - 1649754 \beta_{18} - 4708169 \beta_{17} + 5865640 \beta_{16} - 24125681 \beta_{15} + \cdots + 16359136 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 54018690 \beta_{19} - 31510453 \beta_{18} + 18543655 \beta_{17} - 35377845 \beta_{16} + \cdots + 151922809 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 32748466 \beta_{19} - 23979501 \beta_{18} - 42949033 \beta_{17} + 53432566 \beta_{16} + \cdots + 174055949 \) 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
−3.15720
−2.45978
−2.32542
−2.15199
−1.77641
−1.37724
−0.915292
−0.146508
−0.103746
−0.0954148
0.476040
0.538932
1.06664
1.15563
1.53485
2.15119
2.36764
2.83122
3.09326
3.29362
0 −3.15720 0 2.22019 0 0 0 6.96793 0
1.2 0 −2.45978 0 −2.10149 0 0 0 3.05053 0
1.3 0 −2.32542 0 1.16439 0 0 0 2.40759 0
1.4 0 −2.15199 0 −0.0591445 0 0 0 1.63107 0
1.5 0 −1.77641 0 −0.716761 0 0 0 0.155650 0
1.6 0 −1.37724 0 3.83294 0 0 0 −1.10321 0
1.7 0 −0.915292 0 −1.31004 0 0 0 −2.16224 0
1.8 0 −0.146508 0 3.56882 0 0 0 −2.97854 0
1.9 0 −0.103746 0 −3.71451 0 0 0 −2.98924 0
1.10 0 −0.0954148 0 −0.581124 0 0 0 −2.99090 0
1.11 0 0.476040 0 −2.17736 0 0 0 −2.77339 0
1.12 0 0.538932 0 0.810745 0 0 0 −2.70955 0
1.13 0 1.06664 0 −0.715473 0 0 0 −1.86229 0
1.14 0 1.15563 0 3.94572 0 0 0 −1.66452 0
1.15 0 1.53485 0 1.74990 0 0 0 −0.644239 0
1.16 0 2.15119 0 −3.75581 0 0 0 1.62763 0
1.17 0 2.36764 0 2.42438 0 0 0 2.60574 0
1.18 0 2.83122 0 −2.22134 0 0 0 5.01578 0
1.19 0 3.09326 0 2.53432 0 0 0 6.56828 0
1.20 0 3.29362 0 3.10165 0 0 0 7.84790 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
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 8036.2.a.t yes 20
7.b odd 2 1 8036.2.a.s ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8036.2.a.s ✓ 20 7.b odd 2 1
8036.2.a.t yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{20} - 4 T_{3}^{19} - 30 T_{3}^{18} + 128 T_{3}^{17} + 348 T_{3}^{16} - 1644 T_{3}^{15} - 1934 T_{3}^{14} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{20} - 8 T_{5}^{19} - 28 T_{5}^{18} + 360 T_{5}^{17} + 7 T_{5}^{16} - 6276 T_{5}^{15} + \cdots + 12352 \) Copy content Toggle raw display
\( T_{11}^{20} + 8 T_{11}^{19} - 78 T_{11}^{18} - 680 T_{11}^{17} + 2418 T_{11}^{16} + 22880 T_{11}^{15} + \cdots + 795136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{20} - 8 T^{19} + \cdots + 12352 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots + 795136 \) Copy content Toggle raw display
$13$ \( T^{20} - 12 T^{19} + \cdots + 288473 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 1152847367 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 5375323807 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 100720513921 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 189336292864 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 773521344 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 923069391104 \) Copy content Toggle raw display
$41$ \( (T + 1)^{20} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 517504990421151 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 844650777122928 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 22\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 27\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 104647296448 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 21503103552 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 543708917172288 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 633452493824 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 77\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 620195855401 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 92\!\cdots\!81 \) Copy content Toggle raw display
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