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Properties

Label 8036.2.a.r
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1148)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 2) q^{9} + ( - \beta_{12} + \beta_{5} - \beta_{3} + 1) q^{11} - \beta_{14} q^{13} + (\beta_{14} - \beta_{12} + \cdots + \beta_{2}) q^{15}+ \cdots + (\beta_{14} - 3 \beta_{13} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21568415171 \nu^{14} + 33366538546 \nu^{13} - 780431594905 \nu^{12} - 1076896328346 \nu^{11} + \cdots - 53909708702913 ) / 8954190428232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 207367915591 \nu^{14} + 75537732154 \nu^{13} - 6864511642829 \nu^{12} + \cdots + 964719687046731 ) / 53725142569392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 375675670889 \nu^{14} + 197307815606 \nu^{13} - 13458627125155 \nu^{12} + \cdots - 394220916659787 ) / 53725142569392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 563847518531 \nu^{14} + 271104011042 \nu^{13} - 20576643380593 \nu^{12} + \cdots - 14\!\cdots\!53 ) / 53725142569392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82318075711 \nu^{14} + 110890419958 \nu^{13} - 2903659812299 \nu^{12} + \cdots - 123529112322861 ) / 6715642821174 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57933555161 \nu^{14} + 19902931966 \nu^{13} - 2038445185159 \nu^{12} + \cdots - 132293240921499 ) / 4477095214116 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 263299916363 \nu^{14} + 284951510738 \nu^{13} - 9037992446569 \nu^{12} + \cdots - 369589303008177 ) / 17908380856464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 803771605457 \nu^{14} + 55554951370 \nu^{13} + 29873873602171 \nu^{12} + \cdots + 22\!\cdots\!23 ) / 53725142569392 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 527518007087 \nu^{14} - 247737754298 \nu^{13} + 19237736393941 \nu^{12} + \cdots + 13\!\cdots\!41 ) / 17908380856464 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1991383769819 \nu^{14} - 970794976610 \nu^{13} + 72904083529129 \nu^{12} + \cdots + 50\!\cdots\!45 ) / 53725142569392 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 154414062065 \nu^{14} + 69132074966 \nu^{13} - 5550942035752 \nu^{12} + \cdots - 312802565393409 ) / 3357821410587 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1366326190169 \nu^{14} - 650669688638 \nu^{13} + 49657832878051 \nu^{12} + \cdots + 34\!\cdots\!87 ) / 26862571284696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + 11\beta_{2} + \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{14} + 2 \beta_{12} + \beta_{10} + 14 \beta_{9} - 16 \beta_{8} - 14 \beta_{7} + 12 \beta_{6} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{14} + 14 \beta_{13} + \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 32 \beta_{8} + \cdots + 333 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 62 \beta_{14} - 10 \beta_{13} + 44 \beta_{12} - 7 \beta_{11} + 12 \beta_{10} + 170 \beta_{9} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 147 \beta_{14} + 152 \beta_{13} + 30 \beta_{12} + 60 \beta_{11} - 243 \beta_{10} + 234 \beta_{9} + \cdots + 3159 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 927 \beta_{14} - 256 \beta_{13} + 685 \beta_{12} - 156 \beta_{11} + 52 \beta_{10} + 2001 \beta_{9} + \cdots + 1342 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1327 \beta_{14} + 1492 \beta_{13} + 627 \beta_{12} + 1096 \beta_{11} - 3269 \beta_{10} + 3039 \beta_{9} + \cdots + 31539 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12261 \beta_{14} - 4464 \beta_{13} + 9349 \beta_{12} - 2401 \beta_{11} - 866 \beta_{10} + \cdots + 23366 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10196 \beta_{14} + 13647 \beta_{13} + 10931 \beta_{12} + 16261 \beta_{11} - 42318 \beta_{10} + \cdots + 326176 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 152923 \beta_{14} - 66240 \beta_{13} + 119702 \beta_{12} - 31483 \beta_{11} - 28360 \beta_{10} + \cdots + 337884 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 56795 \beta_{14} + 115353 \beta_{13} + 170227 \beta_{12} + 216472 \beta_{11} - 532792 \beta_{10} + \cdots + 3460210 \) 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
−3.24213
−2.90634
−2.73742
−2.47735
−2.11769
−0.741382
−0.449590
0.515082
1.27444
1.40418
1.72996
1.93225
2.29181
3.10047
3.42372
0 −3.24213 0 −1.49679 0 0 0 7.51143 0
1.2 0 −2.90634 0 −0.506103 0 0 0 5.44682 0
1.3 0 −2.73742 0 0.863054 0 0 0 4.49344 0
1.4 0 −2.47735 0 3.49844 0 0 0 3.13725 0
1.5 0 −2.11769 0 −3.93656 0 0 0 1.48461 0
1.6 0 −0.741382 0 2.48663 0 0 0 −2.45035 0
1.7 0 −0.449590 0 0.787803 0 0 0 −2.79787 0
1.8 0 0.515082 0 −2.19964 0 0 0 −2.73469 0
1.9 0 1.27444 0 −4.37429 0 0 0 −1.37580 0
1.10 0 1.40418 0 −3.61567 0 0 0 −1.02828 0
1.11 0 1.72996 0 0.138351 0 0 0 −0.00725198 0
1.12 0 1.93225 0 0.475465 0 0 0 0.733587 0
1.13 0 2.29181 0 3.92303 0 0 0 2.25239 0
1.14 0 3.10047 0 3.31369 0 0 0 6.61290 0
1.15 0 3.42372 0 −2.35740 0 0 0 8.72183 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
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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.r 15
7.b odd 2 1 8036.2.a.q 15
7.d odd 6 2 1148.2.i.e ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.e ✓ 30 7.d odd 6 2
8036.2.a.q 15 7.b odd 2 1
8036.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{15} - T_{3}^{14} - 37 T_{3}^{13} + 39 T_{3}^{12} + 537 T_{3}^{11} - 616 T_{3}^{10} - 3853 T_{3}^{9} + \cdots + 3381 \) Copy content Toggle raw display
\( T_{5}^{15} + 3 T_{5}^{14} - 49 T_{5}^{13} - 135 T_{5}^{12} + 911 T_{5}^{11} + 2258 T_{5}^{10} + \cdots + 1237 \) Copy content Toggle raw display
\( T_{11}^{15} - 9 T_{11}^{14} - 76 T_{11}^{13} + 888 T_{11}^{12} + 1161 T_{11}^{11} - 31027 T_{11}^{10} + \cdots + 470097 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - T^{14} + \cdots + 3381 \) Copy content Toggle raw display
$5$ \( T^{15} + 3 T^{14} + \cdots + 1237 \) Copy content Toggle raw display
$7$ \( T^{15} \) Copy content Toggle raw display
$11$ \( T^{15} - 9 T^{14} + \cdots + 470097 \) Copy content Toggle raw display
$13$ \( T^{15} + 7 T^{14} + \cdots + 32099328 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 1349910528 \) Copy content Toggle raw display
$19$ \( T^{15} + 7 T^{14} + \cdots - 8446623 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 3406092672 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 1176028416 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 131716224 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 19612697472 \) Copy content Toggle raw display
$41$ \( (T - 1)^{15} \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 3445653504 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 125194416 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 3933422208 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 406624858496 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 40737716604 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 22993294528 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 785361024 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 49810262688 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 656798733 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 40564783104 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 211923201536 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 9174340069632 \) Copy content Toggle raw display
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