Properties

Label 8046.2.a.s
Level $8046$
Weight $2$
Character orbit 8046.a
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + 15173 x^{8} - 102466 x^{7} - 16612 x^{6} + 186120 x^{5} + 117 x^{4} + \cdots - 4260 \) 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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_1 q^{5} + \beta_{7} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_1 q^{5} + \beta_{7} q^{7} - q^{8} + \beta_1 q^{10} + \beta_{4} q^{11} + \beta_{9} q^{13} - \beta_{7} q^{14} + q^{16} + \beta_{2} q^{17} + (\beta_{14} + 1) q^{19} - \beta_1 q^{20} - \beta_{4} q^{22} + ( - \beta_{15} - 1) q^{23} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{25} - \beta_{9} q^{26} + \beta_{7} q^{28} + \beta_{5} q^{29} + (\beta_{14} - \beta_{11} + \beta_{6} + 2) q^{31} - q^{32} - \beta_{2} q^{34} + ( - \beta_{13} + \beta_{9} - \beta_{7} - \beta_{2} - \beta_1 - 1) q^{35} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{14} - 1) q^{38} + \beta_1 q^{40} + (\beta_{14} + \beta_{13} + \beta_{10} + \beta_{8} - \beta_{6} + \beta_{2} + \beta_1) q^{41} + ( - \beta_{15} - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{4}) q^{43} + \beta_{4} q^{44} + (\beta_{15} + 1) q^{46} + (\beta_{15} - \beta_{13} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{47} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 2) q^{49}+ \cdots + ( - \beta_{15} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + 15173 x^{8} - 102466 x^{7} - 16612 x^{6} + 186120 x^{5} + 117 x^{4} + \cdots - 4260 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!49 \nu^{15} + \cdots + 36\!\cdots\!18 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!42 \nu^{15} + \cdots - 18\!\cdots\!72 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!76 \nu^{15} + \cdots - 24\!\cdots\!82 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48\!\cdots\!13 \nu^{15} + \cdots + 67\!\cdots\!08 ) / 68\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18\!\cdots\!18 \nu^{15} + \cdots - 63\!\cdots\!52 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!22 \nu^{15} + \cdots - 81\!\cdots\!70 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!07 \nu^{15} + \cdots + 21\!\cdots\!78 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!56 \nu^{15} + \cdots + 16\!\cdots\!40 ) / 10\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41\!\cdots\!76 \nu^{15} + \cdots - 22\!\cdots\!88 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 48\!\cdots\!83 \nu^{15} + \cdots + 50\!\cdots\!00 ) / 22\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54\!\cdots\!06 \nu^{15} + \cdots - 51\!\cdots\!02 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 66\!\cdots\!95 \nu^{15} + \cdots - 58\!\cdots\!74 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 68\!\cdots\!06 \nu^{15} + \cdots + 10\!\cdots\!74 ) / 20\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26\!\cdots\!56 \nu^{15} + \cdots + 48\!\cdots\!86 ) / 68\!\cdots\!62 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} + \beta_{3} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + 2\beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 13\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} + 5 \beta_{14} + 9 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 5 \beta_{9} + 3 \beta_{8} - \beta_{7} - 18 \beta_{6} + 2 \beta_{5} + 19 \beta_{4} + 15 \beta_{3} + 4 \beta_{2} + 13 \beta _1 + 93 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 24 \beta_{15} + 3 \beta_{14} - 8 \beta_{13} + 57 \beta_{11} + 21 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 19 \beta_{6} + 23 \beta_{5} + 16 \beta_{4} + 21 \beta_{3} + 20 \beta_{2} + 200 \beta _1 - 53 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 122 \beta_{15} + 159 \beta_{14} + 281 \beta_{13} - 131 \beta_{12} + 45 \beta_{11} - 4 \beta_{10} - 157 \beta_{9} + 78 \beta_{8} - 37 \beta_{7} - 310 \beta_{6} + 47 \beta_{5} + 357 \beta_{4} + 234 \beta_{3} + 131 \beta_{2} + \cdots + 1472 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 489 \beta_{15} + 120 \beta_{14} - 245 \beta_{13} - 10 \beta_{12} + 1261 \beta_{11} + 382 \beta_{10} + 27 \beta_{9} - 63 \beta_{8} + 73 \beta_{7} + 341 \beta_{6} + 432 \beta_{5} + 532 \beta_{4} + 422 \beta_{3} + 389 \beta_{2} + \cdots - 1063 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2798 \beta_{15} + 3772 \beta_{14} + 6527 \beta_{13} - 3144 \beta_{12} + 847 \beta_{11} - 169 \beta_{10} - 3700 \beta_{9} + 1621 \beta_{8} - 861 \beta_{7} - 5452 \beta_{6} + 929 \beta_{5} + 6726 \beta_{4} + \cdots + 25073 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9469 \beta_{15} + 3260 \beta_{14} - 5513 \beta_{13} - 323 \beta_{12} + 25574 \beta_{11} + 6832 \beta_{10} + 550 \beta_{9} - 1446 \beta_{8} + 1901 \beta_{7} + 6147 \beta_{6} + 7835 \beta_{5} + 12914 \beta_{4} + \cdots - 19466 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57976 \beta_{15} + 80254 \beta_{14} + 136295 \beta_{13} - 67183 \beta_{12} + 15744 \beta_{11} - 4571 \beta_{10} - 78485 \beta_{9} + 31418 \beta_{8} - 17499 \beta_{7} - 98040 \beta_{6} + \cdots + 444808 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 179989 \beta_{15} + 76046 \beta_{14} - 110725 \beta_{13} - 7806 \beta_{12} + 499669 \beta_{11} + 123118 \beta_{10} + 9595 \beta_{9} - 29317 \beta_{8} + 42671 \beta_{7} + 111124 \beta_{6} + \cdots - 344176 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1148453 \beta_{15} + 1620472 \beta_{14} + 2709979 \beta_{13} - 1358174 \beta_{12} + 298719 \beta_{11} - 102964 \beta_{10} - 1582246 \beta_{9} + 593377 \beta_{8} - 338031 \beta_{7} + \cdots + 8080758 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3403966 \beta_{15} + 1646147 \beta_{14} - 2103514 \beta_{13} - 177729 \beta_{12} + 9591388 \beta_{11} + 2243295 \beta_{10} + 144851 \beta_{9} - 559047 \beta_{8} + 885346 \beta_{7} + \cdots - 5997125 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 22261980 \beta_{15} + 31813644 \beta_{14} + 52544664 \beta_{13} - 26656843 \beta_{12} + 5795859 \beta_{11} - 2115034 \beta_{10} - 31024473 \beta_{9} + 11102379 \beta_{8} + \cdots + 148849998 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 64406875 \beta_{15} + 34210426 \beta_{14} - 38709000 \beta_{13} - 4036107 \beta_{12} + 182490569 \beta_{11} + 41255743 \beta_{10} + 1760267 \beta_{9} - 10312043 \beta_{8} + \cdots - 103774387 \) 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
4.36260
3.39614
3.23488
3.09186
2.46314
1.31779
1.06521
0.939377
0.0881948
−0.859599
−1.16198
−1.23039
−1.68422
−3.23178
−3.47967
−4.31154
−1.00000 0 1.00000 −4.36260 0 2.57952 −1.00000 0 4.36260
1.2 −1.00000 0 1.00000 −3.39614 0 0.912273 −1.00000 0 3.39614
1.3 −1.00000 0 1.00000 −3.23488 0 −1.87212 −1.00000 0 3.23488
1.4 −1.00000 0 1.00000 −3.09186 0 4.92856 −1.00000 0 3.09186
1.5 −1.00000 0 1.00000 −2.46314 0 0.600028 −1.00000 0 2.46314
1.6 −1.00000 0 1.00000 −1.31779 0 −2.35088 −1.00000 0 1.31779
1.7 −1.00000 0 1.00000 −1.06521 0 −4.24065 −1.00000 0 1.06521
1.8 −1.00000 0 1.00000 −0.939377 0 3.42185 −1.00000 0 0.939377
1.9 −1.00000 0 1.00000 −0.0881948 0 4.92686 −1.00000 0 0.0881948
1.10 −1.00000 0 1.00000 0.859599 0 −4.94866 −1.00000 0 −0.859599
1.11 −1.00000 0 1.00000 1.16198 0 1.20200 −1.00000 0 −1.16198
1.12 −1.00000 0 1.00000 1.23039 0 0.0132875 −1.00000 0 −1.23039
1.13 −1.00000 0 1.00000 1.68422 0 −3.55865 −1.00000 0 −1.68422
1.14 −1.00000 0 1.00000 3.23178 0 3.10807 −1.00000 0 −3.23178
1.15 −1.00000 0 1.00000 3.47967 0 2.01581 −1.00000 0 −3.47967
1.16 −1.00000 0 1.00000 4.31154 0 −0.737290 −1.00000 0 −4.31154
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.s 16
3.b odd 2 1 8046.2.a.t yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.s 16 1.a even 1 1 trivial
8046.2.a.t yes 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{16} + 4 T_{5}^{15} - 46 T_{5}^{14} - 192 T_{5}^{13} + 752 T_{5}^{12} + 3378 T_{5}^{11} - 5277 T_{5}^{10} - 27132 T_{5}^{9} + 15173 T_{5}^{8} + 102466 T_{5}^{7} - 16612 T_{5}^{6} - 186120 T_{5}^{5} + 117 T_{5}^{4} + \cdots - 4260 \) Copy content Toggle raw display
\( T_{11}^{16} + 6 T_{11}^{15} - 119 T_{11}^{14} - 794 T_{11}^{13} + 4792 T_{11}^{12} + 38913 T_{11}^{11} - 58633 T_{11}^{10} - 829888 T_{11}^{9} - 617331 T_{11}^{8} + 6234164 T_{11}^{7} + 14164162 T_{11}^{6} + 6747208 T_{11}^{5} + \cdots + 73728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} - 46 T^{14} + \cdots - 4260 \) Copy content Toggle raw display
$7$ \( T^{16} - 6 T^{15} - 56 T^{14} + \cdots + 2845 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} - 119 T^{14} + \cdots + 73728 \) Copy content Toggle raw display
$13$ \( T^{16} - 6 T^{15} - 103 T^{14} + \cdots - 10771200 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} - 148 T^{14} + \cdots + 17280 \) Copy content Toggle raw display
$19$ \( T^{16} - 10 T^{15} - 71 T^{14} + \cdots + 304640 \) Copy content Toggle raw display
$23$ \( T^{16} + 10 T^{15} - 137 T^{14} + \cdots + 2533173 \) Copy content Toggle raw display
$29$ \( T^{16} + 6 T^{15} + \cdots + 8311916031 \) Copy content Toggle raw display
$31$ \( T^{16} - 21 T^{15} + \cdots + 1458191925 \) Copy content Toggle raw display
$37$ \( T^{16} - 17 T^{15} + \cdots - 805039488 \) Copy content Toggle raw display
$41$ \( T^{16} - 4 T^{15} + \cdots + 67170587895 \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{15} + \cdots + 5969792828 \) Copy content Toggle raw display
$47$ \( T^{16} + 25 T^{15} + \cdots - 91388839680 \) Copy content Toggle raw display
$53$ \( T^{16} + 14 T^{15} + \cdots + 91898661402501 \) Copy content Toggle raw display
$59$ \( T^{16} + 6 T^{15} + \cdots + 6060528000 \) Copy content Toggle raw display
$61$ \( T^{16} - 23 T^{15} + \cdots + 378983064960 \) Copy content Toggle raw display
$67$ \( T^{16} - 22 T^{15} + \cdots - 1370093000064 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 330049322557821 \) Copy content Toggle raw display
$73$ \( T^{16} - 16 T^{15} + \cdots - 218732400 \) Copy content Toggle raw display
$79$ \( T^{16} - 37 T^{15} + \cdots + 12099864448 \) Copy content Toggle raw display
$83$ \( T^{16} + 33 T^{15} + \cdots - 4064676660864 \) Copy content Toggle raw display
$89$ \( T^{16} - 3 T^{15} + \cdots + 5164890057 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 137150222775424 \) Copy content Toggle raw display
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