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Properties

Label 8022.2.a.p
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $1$
Dimension $9$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 17x^{7} - 12x^{6} + 72x^{5} + 81x^{4} - 67x^{3} - 105x^{2} - 17x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - \beta_{6} q^{5} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - \beta_{6} q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - \beta_{6} q^{10} + ( - \beta_{8} + \beta_{7} + \beta_{3}) q^{11} - q^{12} + ( - \beta_{8} - \beta_{7} + \beta_{6} - 1) q^{13} + q^{14} + \beta_{6} q^{15} + q^{16} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{8} + \beta_{7} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} + 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} + 9 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 3 q^{11} - 9 q^{12} - 9 q^{13} + 9 q^{14} + 4 q^{15} + 9 q^{16} - 12 q^{17} + 9 q^{18} - 18 q^{19} - 4 q^{20} - 9 q^{21} - 3 q^{22} + q^{23} - 9 q^{24} + 9 q^{25} - 9 q^{26} - 9 q^{27} + 9 q^{28} - 11 q^{29} + 4 q^{30} - 22 q^{31} + 9 q^{32} + 3 q^{33} - 12 q^{34} - 4 q^{35} + 9 q^{36} + 8 q^{37} - 18 q^{38} + 9 q^{39} - 4 q^{40} - 22 q^{41} - 9 q^{42} - 3 q^{43} - 3 q^{44} - 4 q^{45} + q^{46} - 49 q^{47} - 9 q^{48} + 9 q^{49} + 9 q^{50} + 12 q^{51} - 9 q^{52} + 8 q^{53} - 9 q^{54} - 19 q^{55} + 9 q^{56} + 18 q^{57} - 11 q^{58} - 18 q^{59} + 4 q^{60} + 3 q^{61} - 22 q^{62} + 9 q^{63} + 9 q^{64} - 32 q^{65} + 3 q^{66} - 11 q^{67} - 12 q^{68} - q^{69} - 4 q^{70} - 7 q^{71} + 9 q^{72} - 23 q^{73} + 8 q^{74} - 9 q^{75} - 18 q^{76} - 3 q^{77} + 9 q^{78} - 17 q^{79} - 4 q^{80} + 9 q^{81} - 22 q^{82} - 30 q^{83} - 9 q^{84} + 18 q^{85} - 3 q^{86} + 11 q^{87} - 3 q^{88} + 23 q^{89} - 4 q^{90} - 9 q^{91} + q^{92} + 22 q^{93} - 49 q^{94} - 30 q^{95} - 9 q^{96} - 46 q^{97} + 9 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 17x^{7} - 12x^{6} + 72x^{5} + 81x^{4} - 67x^{3} - 105x^{2} - 17x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -96\nu^{8} + 151\nu^{7} + 1380\nu^{6} - 1032\nu^{5} - 5048\nu^{4} + 494\nu^{3} + 4877\nu^{2} + 1233\nu - 177 ) / 107 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -143\nu^{8} + 177\nu^{7} + 2176\nu^{6} - 922\nu^{5} - 8634\nu^{4} - 1286\nu^{3} + 9252\nu^{2} + 3873\nu - 481 ) / 107 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 202 \nu^{8} + 262 \nu^{7} + 3091 \nu^{6} - 1583 \nu^{5} - 12423 \nu^{4} - 249 \nu^{3} + 13501 \nu^{2} + \cdots - 874 ) / 107 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 202 \nu^{8} - 262 \nu^{7} - 3091 \nu^{6} + 1583 \nu^{5} + 12423 \nu^{4} + 249 \nu^{3} - 13394 \nu^{2} + \cdots + 446 ) / 107 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 405 \nu^{8} - 520 \nu^{7} - 6183 \nu^{6} + 3043 \nu^{5} + 24747 \nu^{4} + 1146 \nu^{3} - 26831 \nu^{2} + \cdots + 1586 ) / 107 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 439 \nu^{8} + 598 \nu^{7} + 6645 \nu^{6} - 3783 \nu^{5} - 26410 \nu^{4} + 469 \nu^{3} + 28721 \nu^{2} + \cdots - 1963 ) / 107 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 616 \nu^{8} + 853 \nu^{7} + 9283 \nu^{6} - 5445 \nu^{5} - 36707 \nu^{4} + 798 \nu^{3} + 39756 \nu^{2} + \cdots - 2500 ) / 107 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 623 \nu^{8} + 825 \nu^{7} + 9504 \nu^{6} - 5119 \nu^{5} - 38154 \nu^{4} + 114 \nu^{3} + 41876 \nu^{2} + \cdots - 2864 ) / 107 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{8} + 7\beta_{7} - 6\beta_{6} - 11\beta_{5} + 11\beta_{4} + 15\beta_{3} - 11\beta_{2} - 4\beta _1 + 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{8} + 2\beta_{7} - \beta_{6} - 2\beta_{5} + 11\beta_{4} + 18\beta_{3} - 3\beta_{2} - \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 124 \beta_{8} + 81 \beta_{7} - 34 \beta_{6} - 113 \beta_{5} + 145 \beta_{4} + 241 \beta_{3} + \cdots + 357 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -81\beta_{8} + 44\beta_{7} - 15\beta_{6} - 43\beta_{5} + 141\beta_{4} + 260\beta_{3} - 59\beta_{2} - 32\beta _1 + 440 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1696 \beta_{8} + 1043 \beta_{7} - 326 \beta_{6} - 1275 \beta_{5} + 1999 \beta_{4} + 3607 \beta_{3} + \cdots + 5279 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1298 \beta_{8} + 732 \beta_{7} - 214 \beta_{6} - 728 \beta_{5} + 1913 \beta_{4} + 3635 \beta_{3} + \cdots + 5730 \) 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
−1.29242
−1.49428
−2.74795
−0.470396
3.74341
−1.37137
0.166868
2.17782
1.28832
1.00000 −1.00000 1.00000 −3.75450 −1.00000 1.00000 1.00000 1.00000 −3.75450
1.2 1.00000 −1.00000 1.00000 −3.72931 −1.00000 1.00000 1.00000 1.00000 −3.72931
1.3 1.00000 −1.00000 1.00000 −2.04654 −1.00000 1.00000 1.00000 1.00000 −2.04654
1.4 1.00000 −1.00000 1.00000 −1.03764 −1.00000 1.00000 1.00000 1.00000 −1.03764
1.5 1.00000 −1.00000 1.00000 −0.149044 −1.00000 1.00000 1.00000 1.00000 −0.149044
1.6 1.00000 −1.00000 1.00000 −0.137007 −1.00000 1.00000 1.00000 1.00000 −0.137007
1.7 1.00000 −1.00000 1.00000 0.787345 −1.00000 1.00000 1.00000 1.00000 0.787345
1.8 1.00000 −1.00000 1.00000 2.12019 −1.00000 1.00000 1.00000 1.00000 2.12019
1.9 1.00000 −1.00000 1.00000 3.94651 −1.00000 1.00000 1.00000 1.00000 3.94651
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(1\)

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 8022.2.a.p ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.p ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{9} + 4T_{5}^{8} - 19T_{5}^{7} - 83T_{5}^{6} + 48T_{5}^{5} + 333T_{5}^{4} + 93T_{5}^{3} - 189T_{5}^{2} - 56T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{9} + 3 T_{11}^{8} - 56 T_{11}^{7} - 166 T_{11}^{6} + 895 T_{11}^{5} + 2837 T_{11}^{4} + \cdots - 192 \) Copy content Toggle raw display
\( T_{13}^{9} + 9 T_{13}^{8} - 10 T_{13}^{7} - 260 T_{13}^{6} - 545 T_{13}^{5} + 593 T_{13}^{4} + 2365 T_{13}^{3} + \cdots - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 3 T^{8} + \cdots - 192 \) Copy content Toggle raw display
$13$ \( T^{9} + 9 T^{8} + \cdots - 72 \) Copy content Toggle raw display
$17$ \( T^{9} + 12 T^{8} + \cdots + 7284 \) Copy content Toggle raw display
$19$ \( T^{9} + 18 T^{8} + \cdots + 47744 \) Copy content Toggle raw display
$23$ \( T^{9} - T^{8} + \cdots - 11264 \) Copy content Toggle raw display
$29$ \( T^{9} + 11 T^{8} + \cdots + 415676 \) Copy content Toggle raw display
$31$ \( T^{9} + 22 T^{8} + \cdots + 1388416 \) Copy content Toggle raw display
$37$ \( T^{9} - 8 T^{8} + \cdots - 10170768 \) Copy content Toggle raw display
$41$ \( T^{9} + 22 T^{8} + \cdots - 53408 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} + \cdots + 51488 \) Copy content Toggle raw display
$47$ \( T^{9} + 49 T^{8} + \cdots + 27129792 \) Copy content Toggle raw display
$53$ \( T^{9} - 8 T^{8} + \cdots - 312484 \) Copy content Toggle raw display
$59$ \( T^{9} + 18 T^{8} + \cdots + 1933248 \) Copy content Toggle raw display
$61$ \( T^{9} - 3 T^{8} + \cdots + 204026668 \) Copy content Toggle raw display
$67$ \( T^{9} + 11 T^{8} + \cdots - 1728 \) Copy content Toggle raw display
$71$ \( T^{9} + 7 T^{8} + \cdots - 12358144 \) Copy content Toggle raw display
$73$ \( T^{9} + 23 T^{8} + \cdots - 27045748 \) Copy content Toggle raw display
$79$ \( T^{9} + 17 T^{8} + \cdots + 49418624 \) Copy content Toggle raw display
$83$ \( T^{9} + 30 T^{8} + \cdots + 22458624 \) Copy content Toggle raw display
$89$ \( T^{9} - 23 T^{8} + \cdots + 84869264 \) Copy content Toggle raw display
$97$ \( T^{9} + 46 T^{8} + \cdots - 1241768 \) Copy content Toggle raw display
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