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Properties

Label 8036.2.a.o
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{3} + \beta_{2} + 1) q^{9} + \beta_{6} q^{11} + ( - \beta_{7} - 1) q^{13} + (\beta_{9} - \beta_{8} - \beta_{5} + \cdots - 1) q^{15}+ \cdots + ( - 2 \beta_{9} - \beta_{5} + \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9} - 6 q^{13} - 4 q^{15} - 12 q^{17} - 8 q^{19} + 4 q^{23} + 16 q^{25} - 8 q^{27} + 2 q^{29} - 8 q^{31} + 6 q^{33} - 2 q^{37} - 2 q^{39} - 10 q^{41} - 2 q^{43} - 44 q^{45} + 14 q^{47} + 14 q^{51} + 8 q^{53} - 8 q^{55} - 10 q^{57} - 24 q^{59} - 14 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 10 q^{71} - 44 q^{73} + 50 q^{75} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 8 q^{85} - 20 q^{87} - 6 q^{89} + 8 q^{93} + 4 q^{95} - 46 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{9} - 3\nu^{8} + 36\nu^{7} + 41\nu^{6} - 213\nu^{5} - 154\nu^{4} + 445\nu^{3} + 132\nu^{2} - 275\nu - 28 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{9} + 3\nu^{8} - 36\nu^{7} - 41\nu^{6} + 213\nu^{5} + 154\nu^{4} - 445\nu^{3} - 111\nu^{2} + 275\nu - 56 ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{9} - \nu^{8} + 47\nu^{7} + 16\nu^{6} - 218\nu^{5} - 84\nu^{4} + 293\nu^{3} + 72\nu^{2} - 73\nu - 7 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13 \nu^{9} + 9 \nu^{8} - 213 \nu^{7} - 151 \nu^{6} + 1080 \nu^{5} + 812 \nu^{4} - 1811 \nu^{3} + \cdots + 371 ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22 \nu^{9} - 12 \nu^{8} + 354 \nu^{7} + 199 \nu^{6} - 1734 \nu^{5} - 1043 \nu^{4} + 2711 \nu^{3} + \cdots - 266 ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29 \nu^{9} - 12 \nu^{8} + 459 \nu^{7} + 206 \nu^{6} - 2196 \nu^{5} - 1162 \nu^{4} + 3334 \nu^{3} + \cdots - 343 ) / 21 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4\nu^{9} + 3\nu^{8} - 63\nu^{7} - 49\nu^{6} + 297\nu^{5} + 251\nu^{4} - 425\nu^{3} - 309\nu^{2} + 175\nu + 74 ) / 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12 \nu^{9} + 11 \nu^{8} - 195 \nu^{7} - 176 \nu^{6} + 970 \nu^{5} + 868 \nu^{4} - 1529 \nu^{3} + \cdots + 252 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{5} - \beta_{4} + 9\beta_{3} + 8\beta_{2} - \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + 10 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} - 2 \beta_{6} + 25 \beta_{5} - 13 \beta_{4} + 77 \beta_{3} + \cdots + 194 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 87 \beta_{9} - 86 \beta_{8} - 101 \beta_{7} + 88 \beta_{6} - 102 \beta_{5} + 27 \beta_{4} - 108 \beta_{3} + \cdots - 50 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 149 \beta_{9} + 151 \beta_{8} + 150 \beta_{7} - 33 \beta_{6} + 249 \beta_{5} - 129 \beta_{4} + \cdots + 1522 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 737 \beta_{9} - 722 \beta_{8} - 884 \beta_{7} + 750 \beta_{6} - 902 \beta_{5} + 277 \beta_{4} + \cdots - 745 \) 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
2.71781
2.70664
2.58799
1.20067
0.729469
−0.284914
−1.05099
−1.73395
−1.91097
−2.96176
0 −2.71781 0 0.521406 0 0 0 4.38650 0
1.2 0 −2.70664 0 −0.469317 0 0 0 4.32591 0
1.3 0 −2.58799 0 −3.78481 0 0 0 3.69770 0
1.4 0 −1.20067 0 0.960855 0 0 0 −1.55839 0
1.5 0 −0.729469 0 3.02160 0 0 0 −2.46788 0
1.6 0 0.284914 0 −2.38934 0 0 0 −2.91882 0
1.7 0 1.05099 0 1.02904 0 0 0 −1.89542 0
1.8 0 1.73395 0 3.41343 0 0 0 0.00658743 0
1.9 0 1.91097 0 −1.95825 0 0 0 0.651806 0
1.10 0 2.96176 0 −4.34462 0 0 0 5.77201 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.o ✓ 10
7.b odd 2 1 8036.2.a.p yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8036.2.a.o ✓ 10 1.a even 1 1 trivial
8036.2.a.p yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{10} + 2 T_{3}^{9} - 18 T_{3}^{8} - 32 T_{3}^{7} + 110 T_{3}^{6} + 154 T_{3}^{5} - 282 T_{3}^{4} + \cdots - 49 \) Copy content Toggle raw display
\( T_{5}^{10} + 4 T_{5}^{9} - 25 T_{5}^{8} - 92 T_{5}^{7} + 201 T_{5}^{6} + 614 T_{5}^{5} - 592 T_{5}^{4} + \cdots - 192 \) Copy content Toggle raw display
\( T_{11}^{10} - 50 T_{11}^{8} - 48 T_{11}^{7} + 705 T_{11}^{6} + 1184 T_{11}^{5} - 1804 T_{11}^{4} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots - 49 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots - 192 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 50 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 1559 \) Copy content Toggle raw display
$17$ \( T^{10} + 12 T^{9} + \cdots - 18693 \) Copy content Toggle raw display
$19$ \( T^{10} + 8 T^{9} + \cdots + 5067 \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} + \cdots + 3737 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots + 192 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots + 3471296 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots - 11649008 \) Copy content Toggle raw display
$41$ \( (T + 1)^{10} \) Copy content Toggle raw display
$43$ \( T^{10} + 2 T^{9} + \cdots - 980127 \) Copy content Toggle raw display
$47$ \( T^{10} - 14 T^{9} + \cdots - 286272 \) Copy content Toggle raw display
$53$ \( T^{10} - 8 T^{9} + \cdots - 2454336 \) Copy content Toggle raw display
$59$ \( T^{10} + 24 T^{9} + \cdots - 7947072 \) Copy content Toggle raw display
$61$ \( T^{10} + 14 T^{9} + \cdots + 523584 \) Copy content Toggle raw display
$67$ \( T^{10} + 8 T^{9} + \cdots + 228928 \) Copy content Toggle raw display
$71$ \( T^{10} - 10 T^{9} + \cdots + 59328 \) Copy content Toggle raw display
$73$ \( T^{10} + 44 T^{9} + \cdots + 22060224 \) Copy content Toggle raw display
$79$ \( T^{10} - 10 T^{9} + \cdots + 6294528 \) Copy content Toggle raw display
$83$ \( T^{10} + 20 T^{9} + \cdots - 20754496 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 3358621053 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 109176709 \) Copy content Toggle raw display
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