Properties

Label 8046.2.a.o
Level $8046$
Weight $2$
Character orbit 8046.a
Self dual yes
Analytic conductor $64.248$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
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: \( x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} - 6927 x^{3} - 3639 x^{2} + 5508 x + 423 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{5} + (\beta_{6} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{5} + (\beta_{6} + 1) q^{7} + q^{8} + \beta_1 q^{10} + ( - \beta_{5} + 1) q^{11} + (\beta_{9} - \beta_{5} + 1) q^{13} + (\beta_{6} + 1) q^{14} + q^{16} + (\beta_{7} + 1) q^{17} - \beta_{11} q^{19} + \beta_1 q^{20} + ( - \beta_{5} + 1) q^{22} + (\beta_{10} + \beta_{8} - \beta_{7} - \beta_{2}) q^{23} + ( - \beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{25} + (\beta_{9} - \beta_{5} + 1) q^{26} + (\beta_{6} + 1) q^{28} + ( - \beta_{8} + 2) q^{29} + (\beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{31} + q^{32} + (\beta_{7} + 1) q^{34} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1 + 1) q^{35} + (\beta_{11} - \beta_{9} + \beta_{7} + 1) q^{37} - \beta_{11} q^{38} + \beta_1 q^{40} + ( - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{41} + (\beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + \beta_1) q^{43} + ( - \beta_{5} + 1) q^{44} + (\beta_{10} + \beta_{8} - \beta_{7} - \beta_{2}) q^{46} + ( - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{47} + (\beta_{11} - \beta_{10} - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{49} + ( - \beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{50} + (\beta_{9} - \beta_{5} + 1) q^{52} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 + 2) q^{53} + (\beta_{11} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{55} + (\beta_{6} + 1) q^{56} + ( - \beta_{8} + 2) q^{58} + (\beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2) q^{59} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{61} + (\beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{62} + q^{64} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 1) q^{65} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{67} + (\beta_{7} + 1) q^{68} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1 + 1) q^{70} + (2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 4) q^{71}+ \cdots + (\beta_{11} - \beta_{10} - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} - 6927 x^{3} - 3639 x^{2} + 5508 x + 423 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1293103 \nu^{11} + 6001945 \nu^{10} + 23580251 \nu^{9} - 131132508 \nu^{8} - 90139960 \nu^{7} + 1024446702 \nu^{6} - 573136895 \nu^{5} + \cdots - 735740835 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1915090 \nu^{11} - 1046362 \nu^{10} + 74238453 \nu^{9} + 35401589 \nu^{8} - 1005031345 \nu^{7} - 425190469 \nu^{6} + 5944816403 \nu^{5} + \cdots + 1318582776 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4181218 \nu^{11} + 5656554 \nu^{10} + 132467602 \nu^{9} - 109619327 \nu^{8} - 1543743823 \nu^{7} + 649333723 \nu^{6} + 8064151794 \nu^{5} + \cdots + 1376131821 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4389972 \nu^{11} + 4775951 \nu^{10} + 138192352 \nu^{9} - 72357264 \nu^{8} - 1596499871 \nu^{7} + 185639731 \nu^{6} + 8187073056 \nu^{5} + \cdots + 1169371344 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6533895 \nu^{11} - 11760004 \nu^{10} - 202559541 \nu^{9} + 251802455 \nu^{8} + 2345517923 \nu^{7} - 1816569674 \nu^{6} - 12292593347 \nu^{5} + \cdots - 1422840759 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6869077 \nu^{11} + 5839142 \nu^{10} + 230548446 \nu^{9} - 91270315 \nu^{8} - 2856540142 \nu^{7} + 274308218 \nu^{6} + 15985954088 \nu^{5} + \cdots + 2092764192 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9158236 \nu^{11} - 12464738 \nu^{10} - 290204756 \nu^{9} + 242859302 \nu^{8} + 3388516467 \nu^{7} - 1488120542 \nu^{6} - 17686820067 \nu^{5} + \cdots - 3556778949 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10768077 \nu^{11} + 13709156 \nu^{10} + 343745338 \nu^{9} - 248488722 \nu^{8} - 4077208325 \nu^{7} + 1283221540 \nu^{6} + \cdots + 4392624474 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11759125 \nu^{11} + 19783265 \nu^{10} + 355999031 \nu^{9} - 376860878 \nu^{8} - 4047108810 \nu^{7} + 2253401432 \nu^{6} + \cdots + 3637207113 ) / 174942609 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13139198 \nu^{11} - 18649271 \nu^{10} - 411865851 \nu^{9} + 355642798 \nu^{8} + 4791452746 \nu^{7} - 2137541739 \nu^{6} - 25035485466 \nu^{5} + \cdots - 2837900049 ) / 174942609 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 10 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 18 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 16 \beta_{8} - 19 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 18 \beta_{4} - 3 \beta_{3} - 17 \beta_{2} + 20 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 62 \beta_{11} - 20 \beta_{10} + 42 \beta_{9} + 47 \beta_{8} - 61 \beta_{7} + 23 \beta_{6} - 26 \beta_{5} + 53 \beta_{4} - 35 \beta_{3} - 56 \beta_{2} + 122 \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 296 \beta_{11} - 61 \beta_{10} + 153 \beta_{9} + 256 \beta_{8} - 304 \beta_{7} + 94 \beta_{6} - 119 \beta_{5} + 295 \beta_{4} - 81 \beta_{3} - 254 \beta_{2} + 336 \beta _1 + 248 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1096 \beta_{11} - 361 \beta_{10} + 776 \beta_{9} + 897 \beta_{8} - 1076 \beta_{7} + 428 \beta_{6} - 516 \beta_{5} + 1042 \beta_{4} - 538 \beta_{3} - 905 \beta_{2} + 1656 \beta _1 + 333 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4821 \beta_{11} - 1304 \beta_{10} + 3055 \beta_{9} + 4200 \beta_{8} - 4846 \beta_{7} + 1764 \beta_{6} - 2258 \beta_{5} + 4857 \beta_{4} - 1606 \beta_{3} - 3827 \beta_{2} + 5403 \beta _1 + 2803 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18627 \beta_{11} - 6354 \beta_{10} + 13806 \beta_{9} + 15966 \beta_{8} - 18280 \beta_{7} + 7495 \beta_{6} - 9404 \beta_{5} + 18623 \beta_{4} - 8326 \beta_{3} - 14361 \beta_{2} + 24022 \beta _1 + 6252 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 78834 \beta_{11} - 24603 \beta_{10} + 55891 \beta_{9} + 69732 \beta_{8} - 78265 \beta_{7} + 30911 \beta_{6} - 40046 \beta_{5} + 80628 \beta_{4} - 28744 \beta_{3} - 59098 \beta_{2} + \cdots + 36254 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 312025 \beta_{11} - 109982 \beta_{10} + 240193 \beta_{9} + 275200 \beta_{8} - 305996 \beta_{7} + 128162 \beta_{6} - 164919 \beta_{5} + 320329 \beta_{4} - 131955 \beta_{3} + \cdots + 106224 \) 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
−2.89416
−2.72416
−2.41916
−1.76721
−1.46524
−0.0737297
1.41673
1.51858
1.84210
2.15566
3.34249
4.06809
1.00000 0 1.00000 −2.89416 0 0.647278 1.00000 0 −2.89416
1.2 1.00000 0 1.00000 −2.72416 0 −2.55489 1.00000 0 −2.72416
1.3 1.00000 0 1.00000 −2.41916 0 0.521062 1.00000 0 −2.41916
1.4 1.00000 0 1.00000 −1.76721 0 2.53166 1.00000 0 −1.76721
1.5 1.00000 0 1.00000 −1.46524 0 −3.53999 1.00000 0 −1.46524
1.6 1.00000 0 1.00000 −0.0737297 0 2.82077 1.00000 0 −0.0737297
1.7 1.00000 0 1.00000 1.41673 0 −1.27536 1.00000 0 1.41673
1.8 1.00000 0 1.00000 1.51858 0 4.35237 1.00000 0 1.51858
1.9 1.00000 0 1.00000 1.84210 0 2.73011 1.00000 0 1.84210
1.10 1.00000 0 1.00000 2.15566 0 −3.96511 1.00000 0 2.15566
1.11 1.00000 0 1.00000 3.34249 0 1.09959 1.00000 0 3.34249
1.12 1.00000 0 1.00000 4.06809 0 2.63250 1.00000 0 4.06809
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(-1\)

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 8046.2.a.o yes 12
3.b odd 2 1 8046.2.a.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.j 12 3.b odd 2 1
8046.2.a.o yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{12} - 3 T_{5}^{11} - 29 T_{5}^{10} + 76 T_{5}^{9} + 320 T_{5}^{8} - 724 T_{5}^{7} - 1643 T_{5}^{6} + 3265 T_{5}^{5} + 3921 T_{5}^{4} - 6927 T_{5}^{3} - 3639 T_{5}^{2} + 5508 T_{5} + 423 \) Copy content Toggle raw display
\( T_{11}^{12} - 10 T_{11}^{11} - 13 T_{11}^{10} + 358 T_{11}^{9} - 377 T_{11}^{8} - 3849 T_{11}^{7} + 5783 T_{11}^{6} + 11799 T_{11}^{5} - 9084 T_{11}^{4} - 12831 T_{11}^{3} + 2745 T_{11}^{2} + 4500 T_{11} + 639 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} - 29 T^{10} + 76 T^{9} + \cdots + 423 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} - 25 T^{10} + \cdots + 3789 \) Copy content Toggle raw display
$11$ \( T^{12} - 10 T^{11} - 13 T^{10} + \cdots + 639 \) Copy content Toggle raw display
$13$ \( T^{12} - 5 T^{11} - 73 T^{10} + \cdots + 8883 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} - 41 T^{10} + 413 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{12} - 2 T^{11} - 85 T^{10} + \cdots - 46435 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} - 112 T^{10} + \cdots - 13061943 \) Copy content Toggle raw display
$29$ \( T^{12} - 19 T^{11} + 73 T^{10} + \cdots - 56433 \) Copy content Toggle raw display
$31$ \( T^{12} - 10 T^{11} - 134 T^{10} + \cdots + 7017 \) Copy content Toggle raw display
$37$ \( T^{12} - 11 T^{11} - 85 T^{10} + \cdots + 7275111 \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} - 212 T^{10} + \cdots + 13665672 \) Copy content Toggle raw display
$43$ \( T^{12} - 13 T^{11} - 202 T^{10} + \cdots - 1231641 \) Copy content Toggle raw display
$47$ \( T^{12} - 11 T^{11} - 184 T^{10} + \cdots - 54427221 \) Copy content Toggle raw display
$53$ \( T^{12} - 24 T^{11} - 33 T^{10} + \cdots + 51571617 \) Copy content Toggle raw display
$59$ \( T^{12} - 10 T^{11} - 237 T^{10} + \cdots + 67964739 \) Copy content Toggle raw display
$61$ \( T^{12} - 214 T^{10} - 300 T^{9} + \cdots - 607311 \) Copy content Toggle raw display
$67$ \( T^{12} - 21 T^{11} - 5 T^{10} + \cdots + 3133611 \) Copy content Toggle raw display
$71$ \( T^{12} - 37 T^{11} + \cdots + 4543751439 \) Copy content Toggle raw display
$73$ \( T^{12} + 2 T^{11} - 374 T^{10} + \cdots - 864833403 \) Copy content Toggle raw display
$79$ \( T^{12} - 7 T^{11} - 248 T^{10} + \cdots - 267344821 \) Copy content Toggle raw display
$83$ \( T^{12} - 22 T^{11} + \cdots + 1324074105 \) Copy content Toggle raw display
$89$ \( T^{12} - 40 T^{11} + \cdots + 236167035 \) Copy content Toggle raw display
$97$ \( T^{12} - 7 T^{11} + \cdots + 1975044843 \) Copy content Toggle raw display
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