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Properties

Label 8022.2.a.u
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $11$
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 13 x^{9} + 106 x^{8} - 85 x^{7} - 431 x^{6} + 844 x^{5} - 212 x^{4} - 385 x^{3} + \cdots - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 13 x^{9} + 106 x^{8} - 85 x^{7} - 431 x^{6} + 844 x^{5} - 212 x^{4} - 385 x^{3} + \cdots - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 381 \nu^{10} + 2960 \nu^{9} + 3801 \nu^{8} - 59377 \nu^{7} + 64572 \nu^{6} + 219527 \nu^{5} + \cdots + 22389 ) / 8084 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 461 \nu^{10} - 1534 \nu^{9} - 7559 \nu^{8} + 33313 \nu^{7} - 1778 \nu^{6} - 134485 \nu^{5} + \cdots + 12025 ) / 8084 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 391 \nu^{10} - 1266 \nu^{9} - 6797 \nu^{8} + 27825 \nu^{7} + 7444 \nu^{6} - 118457 \nu^{5} + \cdots - 3435 ) / 4042 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 230 \nu^{10} - 1458 \nu^{9} - 2215 \nu^{8} + 30158 \nu^{7} - 37230 \nu^{6} - 114737 \nu^{5} + \cdots + 7490 ) / 2021 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 479 \nu^{10} - 910 \nu^{9} - 11335 \nu^{8} + 24215 \nu^{7} + 73688 \nu^{6} - 166785 \nu^{5} + \cdots + 3297 ) / 4042 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 664 \nu^{10} + 1907 \nu^{9} + 14215 \nu^{8} - 44782 \nu^{7} - 66061 \nu^{6} + 251297 \nu^{5} + \cdots - 4313 ) / 4042 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1899 \nu^{10} + 6924 \nu^{9} + 32567 \nu^{8} - 151711 \nu^{7} - 17112 \nu^{6} + 673237 \nu^{5} + \cdots - 18961 ) / 8084 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 582 \nu^{10} + 2055 \nu^{9} + 10262 \nu^{8} - 46033 \nu^{7} - 11499 \nu^{6} + 219125 \nu^{5} + \cdots - 6124 ) / 2021 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3145 \nu^{10} + 10886 \nu^{9} + 55023 \nu^{8} - 240505 \nu^{7} - 58294 \nu^{6} + 1083029 \nu^{5} + \cdots + 13043 ) / 8084 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{7} - \beta_{5} - \beta_{3} - \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + 10\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{9} - 4\beta_{8} + 11\beta_{7} - \beta_{6} - 11\beta_{5} - \beta_{4} - 17\beta_{3} - 9\beta_{2} - 4\beta _1 + 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{10} + 4 \beta_{9} + 20 \beta_{8} - 35 \beta_{7} - 12 \beta_{6} + 21 \beta_{5} + 50 \beta_{4} + \cdots - 123 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{10} - 109 \beta_{9} - 82 \beta_{8} + 150 \beta_{7} - 8 \beta_{6} - 139 \beta_{5} + \cdots + 794 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 239 \beta_{10} + 111 \beta_{9} + 340 \beta_{8} - 544 \beta_{7} - 133 \beta_{6} + 363 \beta_{5} + \cdots - 2093 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 380 \beta_{10} - 1314 \beta_{9} - 1365 \beta_{8} + 2203 \beta_{7} - 9 \beta_{6} - 1911 \beta_{5} + \cdots + 10927 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3298 \beta_{10} + 2245 \beta_{9} + 5420 \beta_{8} - 8310 \beta_{7} - 1540 \beta_{6} + 5896 \beta_{5} + \cdots - 33799 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7433 \beta_{10} - 17111 \beta_{9} - 21450 \beta_{8} + 33093 \beta_{7} + 1058 \beta_{6} + \cdots + 157215 \) 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
2.07243
1.78626
−0.291427
−3.90193
0.0633955
1.61086
3.05783
−0.546656
0.545940
−2.48010
3.08340
1.00000 −1.00000 1.00000 −4.25396 −1.00000 −1.00000 1.00000 1.00000 −4.25396
1.2 1.00000 −1.00000 1.00000 −4.07687 −1.00000 −1.00000 1.00000 1.00000 −4.07687
1.3 1.00000 −1.00000 1.00000 −2.89683 −1.00000 −1.00000 1.00000 1.00000 −2.89683
1.4 1.00000 −1.00000 1.00000 −2.19323 −1.00000 −1.00000 1.00000 1.00000 −2.19323
1.5 1.00000 −1.00000 1.00000 −1.46237 −1.00000 −1.00000 1.00000 1.00000 −1.46237
1.6 1.00000 −1.00000 1.00000 −1.39983 −1.00000 −1.00000 1.00000 1.00000 −1.39983
1.7 1.00000 −1.00000 1.00000 −0.804532 −1.00000 −1.00000 1.00000 1.00000 −0.804532
1.8 1.00000 −1.00000 1.00000 0.995600 −1.00000 −1.00000 1.00000 1.00000 0.995600
1.9 1.00000 −1.00000 1.00000 2.15445 −1.00000 −1.00000 1.00000 1.00000 2.15445
1.10 1.00000 −1.00000 1.00000 2.77447 −1.00000 −1.00000 1.00000 1.00000 2.77447
1.11 1.00000 −1.00000 1.00000 3.16310 −1.00000 −1.00000 1.00000 1.00000 3.16310
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.u ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.u ✓ 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{11} + 8 T_{5}^{10} - 6 T_{5}^{9} - 178 T_{5}^{8} - 220 T_{5}^{7} + 1238 T_{5}^{6} + 2584 T_{5}^{5} + \cdots + 3416 \) Copy content Toggle raw display
\( T_{11}^{11} - 16 T_{11}^{10} + 50 T_{11}^{9} + 411 T_{11}^{8} - 2639 T_{11}^{7} - 350 T_{11}^{6} + \cdots - 93440 \) Copy content Toggle raw display
\( T_{13}^{11} + 2 T_{13}^{10} - 94 T_{13}^{9} - 193 T_{13}^{8} + 3267 T_{13}^{7} + 6600 T_{13}^{6} + \cdots - 2061528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{11} \) Copy content Toggle raw display
$3$ \( (T + 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + 8 T^{10} + \cdots + 3416 \) Copy content Toggle raw display
$7$ \( (T + 1)^{11} \) Copy content Toggle raw display
$11$ \( T^{11} - 16 T^{10} + \cdots - 93440 \) Copy content Toggle raw display
$13$ \( T^{11} + 2 T^{10} + \cdots - 2061528 \) Copy content Toggle raw display
$17$ \( T^{11} + T^{10} + \cdots - 1957806 \) Copy content Toggle raw display
$19$ \( T^{11} - 5 T^{10} + \cdots + 10752 \) Copy content Toggle raw display
$23$ \( T^{11} - 8 T^{10} + \cdots - 225024 \) Copy content Toggle raw display
$29$ \( T^{11} + 5 T^{10} + \cdots + 9632 \) Copy content Toggle raw display
$31$ \( T^{11} + 12 T^{10} + \cdots - 7764984 \) Copy content Toggle raw display
$37$ \( T^{11} - 3 T^{10} + \cdots - 39128 \) Copy content Toggle raw display
$41$ \( T^{11} + 4 T^{10} + \cdots - 96 \) Copy content Toggle raw display
$43$ \( T^{11} - 16 T^{10} + \cdots - 16 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 215488512 \) Copy content Toggle raw display
$53$ \( T^{11} + 3 T^{10} + \cdots - 77565600 \) Copy content Toggle raw display
$59$ \( T^{11} - 13 T^{10} + \cdots - 44629568 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 287025768 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 878132160 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 272733376 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 136630474 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 25843394376 \) Copy content Toggle raw display
$83$ \( T^{11} - 4 T^{10} + \cdots - 15765888 \) Copy content Toggle raw display
$89$ \( T^{11} + 7 T^{10} + \cdots - 14112 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 149922208 \) Copy content Toggle raw display
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