Properties

Label 8001.2.a.n
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $12$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} - 71 x^{3} + 16 x^{2} + 7 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} - \beta_1 + 1) q^{4} + ( - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{7} + ( - \beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} - \beta_1 + 1) q^{4} + ( - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{7} + ( - \beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_1 + 1) q^{8} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{10} + (\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 3) q^{11} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{13} + ( - \beta_1 + 1) q^{14} + ( - 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 3 \beta_1) q^{16}+ \cdots + ( - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} - 71 x^{3} + 16 x^{2} + 7 x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 9 \nu^{10} - 78 \nu^{9} + 21 \nu^{8} + 551 \nu^{7} - 315 \nu^{6} - 1552 \nu^{5} + 474 \nu^{4} + 1974 \nu^{3} + 95 \nu^{2} - 664 \nu + 24 ) / 67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{11} + 97 \nu^{10} + 75 \nu^{9} - 868 \nu^{8} + 50 \nu^{7} + 2903 \nu^{6} - 193 \nu^{5} - 3981 \nu^{4} - 388 \nu^{3} + 1210 \nu^{2} - 114 \nu + 13 ) / 67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28 \nu^{11} + 150 \nu^{10} + 40 \nu^{9} - 1191 \nu^{8} + 652 \nu^{7} + 3393 \nu^{6} - 2037 \nu^{5} - 4093 \nu^{4} + 1544 \nu^{3} + 1628 \nu^{2} - 302 \nu - 69 ) / 67 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37 \nu^{11} + 203 \nu^{10} + 5 \nu^{9} - 1447 \nu^{8} + 986 \nu^{7} + 3615 \nu^{6} - 2340 \nu^{5} - 3669 \nu^{4} + 997 \nu^{3} + 840 \nu^{2} + 180 \nu + 117 ) / 67 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17 \nu^{11} + 48 \nu^{10} + 254 \nu^{9} - 692 \nu^{8} - 1260 \nu^{7} + 3077 \nu^{6} + 2867 \nu^{5} - 5043 \nu^{4} - 3006 \nu^{3} + 2271 \nu^{2} + 568 \nu - 207 ) / 67 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17 \nu^{11} - 48 \nu^{10} - 254 \nu^{9} + 692 \nu^{8} + 1260 \nu^{7} - 3077 \nu^{6} - 2867 \nu^{5} + 5043 \nu^{4} + 3006 \nu^{3} - 2204 \nu^{2} - 635 \nu + 73 ) / 67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36 \nu^{11} + 145 \nu^{10} + 262 \nu^{9} - 1292 \nu^{8} - 875 \nu^{7} + 4104 \nu^{6} + 2205 \nu^{5} - 5071 \nu^{4} - 3193 \nu^{3} + 1337 \nu^{2} + 856 \nu - 60 ) / 67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40 \nu^{11} - 176 \nu^{10} - 239 \nu^{9} + 1577 \nu^{8} + 466 \nu^{7} - 5163 \nu^{6} - 641 \nu^{5} + 7034 \nu^{4} + 905 \nu^{3} - 2967 \nu^{2} + 39 \nu + 223 ) / 67 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 50 \nu^{11} + 220 \nu^{10} + 282 \nu^{9} - 1921 \nu^{8} - 415 \nu^{7} + 5968 \nu^{6} + 282 \nu^{5} - 7419 \nu^{4} - 679 \nu^{3} + 2553 \nu^{2} + 102 \nu - 61 ) / 67 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 59 \nu^{11} + 273 \nu^{10} + 247 \nu^{9} - 2177 \nu^{8} - 81 \nu^{7} + 6190 \nu^{6} - 21 \nu^{5} - 7062 \nu^{4} - 1025 \nu^{3} + 1966 \nu^{2} - 19 \nu - 76 ) / 67 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + \beta_{10} + 3\beta_{9} - 3\beta_{8} + 6\beta_{7} + 9\beta_{6} + \beta_{5} - \beta_{4} + 9\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} + 2 \beta_{10} + 12 \beta_{9} - 13 \beta_{8} + 11 \beta_{7} + 24 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 31 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{11} + 11 \beta_{10} + 36 \beta_{9} - 42 \beta_{8} + 41 \beta_{7} + 81 \beta_{6} + 14 \beta_{5} - 12 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 72 \beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{11} + 30 \beta_{10} + 117 \beta_{9} - 144 \beta_{8} + 98 \beta_{7} + 232 \beta_{6} + 43 \beta_{5} - 33 \beta_{4} - 14 \beta_{3} - 21 \beta_{2} + 219 \beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 283 \beta_{11} + 110 \beta_{10} + 349 \beta_{9} - 458 \beta_{8} + 310 \beta_{7} + 720 \beta_{6} + 152 \beta_{5} - 128 \beta_{4} - 44 \beta_{3} - 77 \beta_{2} + 572 \beta _1 + 375 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 907 \beta_{11} + 327 \beta_{10} + 1069 \beta_{9} - 1466 \beta_{8} + 830 \beta_{7} + 2110 \beta_{6} + 469 \beta_{5} - 386 \beta_{4} - 154 \beta_{3} - 294 \beta_{2} + 1670 \beta _1 + 989 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2731 \beta_{11} + 1061 \beta_{10} + 3179 \beta_{9} - 4574 \beta_{8} + 2481 \beta_{7} + 6364 \beta_{6} + 1515 \beta_{5} - 1296 \beta_{4} - 481 \beta_{3} - 1006 \beta_{2} + 4609 \beta _1 + 2819 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8431 \beta_{11} + 3212 \beta_{10} + 9543 \beta_{9} - 14230 \beta_{8} + 6977 \beta_{7} + 18806 \beta_{6} + 4654 \beta_{5} - 3983 \beta_{4} - 1542 \beta_{3} - 3435 \beta_{2} + 13323 \beta _1 + 7783 \) Copy content Toggle raw display

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
2.98033
2.59078
1.94650
1.84951
0.649147
0.482477
0.127418
−0.329641
−0.920581
−1.17133
−1.42534
−1.77927
−1.98033 0 1.92172 1.61478 0 1.00000 0.155028 0 −3.19781
1.2 −1.59078 0 0.530574 1.30400 0 1.00000 2.33753 0 −2.07438
1.3 −0.946499 0 −1.10414 2.90523 0 1.00000 2.93806 0 −2.74979
1.4 −0.849509 0 −1.27834 −1.65890 0 1.00000 2.78497 0 1.40925
1.5 0.350853 0 −1.87690 0.112051 0 1.00000 −1.36022 0 0.0393133
1.6 0.517523 0 −1.73217 3.15891 0 1.00000 −1.93148 0 1.63481
1.7 0.872582 0 −1.23860 −3.32654 0 1.00000 −2.82594 0 −2.90267
1.8 1.32964 0 −0.232054 0.891044 0 1.00000 −2.96783 0 1.18477
1.9 1.92058 0 1.68863 −0.753423 0 1.00000 −0.598010 0 −1.44701
1.10 2.17133 0 2.71466 4.06798 0 1.00000 1.55176 0 8.83292
1.11 2.42534 0 3.88228 −2.61560 0 1.00000 4.56517 0 −6.34371
1.12 2.77927 0 5.72435 1.30046 0 1.00000 10.3510 0 3.61432
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

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 8001.2.a.n 12
3.b odd 2 1 889.2.a.a 12
21.c even 2 1 6223.2.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.a 12 3.b odd 2 1
6223.2.a.i 12 21.c even 2 1
8001.2.a.n 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{12} - 7 T_{2}^{11} + 8 T_{2}^{10} + 44 T_{2}^{9} - 107 T_{2}^{8} - 47 T_{2}^{7} + 303 T_{2}^{6} - 116 T_{2}^{5} - 278 T_{2}^{4} + 199 T_{2}^{3} + 60 T_{2}^{2} - 76 T_{2} + 15 \) Copy content Toggle raw display
\( T_{5}^{12} - 7 T_{5}^{11} - 7 T_{5}^{10} + 139 T_{5}^{9} - 164 T_{5}^{8} - 753 T_{5}^{7} + 1639 T_{5}^{6} + 633 T_{5}^{5} - 3700 T_{5}^{4} + 1888 T_{5}^{3} + 1390 T_{5}^{2} - 1165 T_{5} + 111 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7 T^{11} + 8 T^{10} + 44 T^{9} + \cdots + 15 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 7 T^{11} - 7 T^{10} + 139 T^{9} + \cdots + 111 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 22 T^{11} + 178 T^{10} + \cdots - 855 \) Copy content Toggle raw display
$13$ \( T^{12} - 78 T^{10} + 6 T^{9} + \cdots + 422935 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} - 54 T^{10} + \cdots + 3765 \) Copy content Toggle raw display
$19$ \( T^{12} + 7 T^{11} - 115 T^{10} + \cdots - 8619139 \) Copy content Toggle raw display
$23$ \( T^{12} - 29 T^{11} + 266 T^{10} + \cdots + 132471 \) Copy content Toggle raw display
$29$ \( T^{12} - 22 T^{11} + 96 T^{10} + \cdots + 2285685 \) Copy content Toggle raw display
$31$ \( T^{12} + 16 T^{11} - 85 T^{10} + \cdots + 69361385 \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} - 263 T^{10} + \cdots - 127475 \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots - 1174048215 \) Copy content Toggle raw display
$43$ \( T^{12} - 11 T^{11} - 83 T^{10} + \cdots + 4163911 \) Copy content Toggle raw display
$47$ \( T^{12} - 31 T^{11} + 287 T^{10} + \cdots - 7213725 \) Copy content Toggle raw display
$53$ \( T^{12} - 38 T^{11} + 478 T^{10} + \cdots + 3687897 \) Copy content Toggle raw display
$59$ \( T^{12} - 15 T^{11} - 151 T^{10} + \cdots - 5696385 \) Copy content Toggle raw display
$61$ \( T^{12} + 3 T^{11} - 300 T^{10} + \cdots - 3819875 \) Copy content Toggle raw display
$67$ \( T^{12} + T^{11} - 369 T^{10} + \cdots - 120068731 \) Copy content Toggle raw display
$71$ \( T^{12} - 57 T^{11} + \cdots - 84701830575 \) Copy content Toggle raw display
$73$ \( T^{12} + 7 T^{11} - 289 T^{10} + \cdots + 6839351 \) Copy content Toggle raw display
$79$ \( T^{12} + 18 T^{11} + \cdots + 116496565 \) Copy content Toggle raw display
$83$ \( T^{12} - 21 T^{11} + \cdots + 2914656237 \) Copy content Toggle raw display
$89$ \( T^{12} + 6 T^{11} + \cdots + 731582873037 \) Copy content Toggle raw display
$97$ \( T^{12} - 4 T^{11} + \cdots - 2242441657 \) Copy content Toggle raw display
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