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Properties

Label 8034.2.a.ba
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} + \cdots - 1492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - \beta_1 q^{5} + q^{6} - \beta_{7} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - \beta_1 q^{5} + q^{6} - \beta_{7} q^{7} - q^{8} + q^{9} + \beta_1 q^{10} + \beta_{11} q^{11} - q^{12} - q^{13} + \beta_{7} q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{17}+ \cdots + \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9} + q^{10} - q^{11} - 14 q^{12} - 14 q^{13} - 5 q^{14} + q^{15} + 14 q^{16} - 12 q^{17} - 14 q^{18} + 10 q^{19} - q^{20} - 5 q^{21} + q^{22} - q^{23} + 14 q^{24} + 19 q^{25} + 14 q^{26} - 14 q^{27} + 5 q^{28} + 6 q^{29} - q^{30} + 20 q^{31} - 14 q^{32} + q^{33} + 12 q^{34} - 16 q^{35} + 14 q^{36} - 3 q^{37} - 10 q^{38} + 14 q^{39} + q^{40} + q^{41} + 5 q^{42} + 6 q^{43} - q^{44} - q^{45} + q^{46} - 13 q^{47} - 14 q^{48} + 9 q^{49} - 19 q^{50} + 12 q^{51} - 14 q^{52} - 27 q^{53} + 14 q^{54} + 10 q^{55} - 5 q^{56} - 10 q^{57} - 6 q^{58} - 6 q^{59} + q^{60} - 4 q^{61} - 20 q^{62} + 5 q^{63} + 14 q^{64} + q^{65} - q^{66} + 13 q^{67} - 12 q^{68} + q^{69} + 16 q^{70} + 18 q^{71} - 14 q^{72} + 11 q^{73} + 3 q^{74} - 19 q^{75} + 10 q^{76} - 15 q^{77} - 14 q^{78} + 33 q^{79} - q^{80} + 14 q^{81} - q^{82} - 25 q^{83} - 5 q^{84} + 25 q^{85} - 6 q^{86} - 6 q^{87} + q^{88} + 3 q^{89} + q^{90} - 5 q^{91} - q^{92} - 20 q^{93} + 13 q^{94} + 30 q^{95} + 14 q^{96} + 11 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} + \cdots - 1492 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 266306853678383 \nu^{13} + 134001969718133 \nu^{12} + \cdots + 11\!\cdots\!40 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 276561632426969 \nu^{13} + 421045515071387 \nu^{12} + \cdots + 58\!\cdots\!32 ) / 21\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 281008689100883 \nu^{13} - 281957981628135 \nu^{12} + \cdots - 37\!\cdots\!57 ) / 21\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 563276920787713 \nu^{13} - 153049032331183 \nu^{12} + \cdots + 72\!\cdots\!90 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 587894636532799 \nu^{13} - 401515966880495 \nu^{12} + \cdots + 69\!\cdots\!80 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 600547475775077 \nu^{13} + 160676916331065 \nu^{12} + \cdots + 15\!\cdots\!26 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 323027709062960 \nu^{13} - 172381882120881 \nu^{12} + \cdots + 87\!\cdots\!91 ) / 21\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 682482407227411 \nu^{13} - 152764996823 \nu^{12} + \cdots - 16\!\cdots\!60 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 352781396910762 \nu^{13} - 166497424213521 \nu^{12} + \cdots + 91\!\cdots\!32 ) / 21\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 756784517059657 \nu^{13} - 144783307200521 \nu^{12} + \cdots + 14\!\cdots\!74 ) / 43\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 461379618788969 \nu^{13} + 101629522210970 \nu^{12} + \cdots - 64\!\cdots\!01 ) / 21\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!27 \nu^{13} - 773564887767339 \nu^{12} + \cdots + 81\!\cdots\!94 ) / 43\!\cdots\!62 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + 12\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{13} + 2 \beta_{12} + 18 \beta_{11} - 3 \beta_{10} - \beta_{9} + 14 \beta_{8} - 13 \beta_{7} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 23 \beta_{13} - 18 \beta_{12} - 10 \beta_{11} + 6 \beta_{10} - 19 \beta_{9} - 4 \beta_{8} - 28 \beta_{7} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 252 \beta_{13} + 35 \beta_{12} + 292 \beta_{11} - 69 \beta_{10} - 15 \beta_{9} + 186 \beta_{8} + \cdots + 1070 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 454 \beta_{13} - 291 \beta_{12} - 23 \beta_{11} + 114 \beta_{10} - 290 \beta_{9} - 91 \beta_{8} + \cdots + 493 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3992 \beta_{13} + 446 \beta_{12} + 4637 \beta_{11} - 1265 \beta_{10} - 143 \beta_{9} + 2477 \beta_{8} + \cdots + 14763 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8449 \beta_{13} - 4639 \beta_{12} + 1814 \beta_{11} + 1546 \beta_{10} - 4099 \beta_{9} - 1550 \beta_{8} + \cdots + 9432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 63639 \beta_{13} + 4544 \beta_{12} + 73258 \beta_{11} - 21545 \beta_{10} - 620 \beta_{9} + \cdots + 211587 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 152268 \beta_{13} - 73921 \beta_{12} + 60922 \beta_{11} + 17182 \beta_{10} - 56254 \beta_{9} + \cdots + 179584 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1020459 \beta_{13} + 30075 \beta_{12} + 1158028 \beta_{11} - 355420 \beta_{10} + 12256 \beta_{9} + \cdots + 3117311 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2688292 \beta_{13} - 1179924 \beta_{12} + 1428695 \beta_{11} + 142999 \beta_{10} - 765007 \beta_{9} + \cdots + 3383769 \) 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
4.07159
3.43788
3.09495
2.71343
0.924862
0.891518
0.503883
−0.522326
−0.643572
−1.05309
−2.46518
−2.56958
−3.59906
−3.78531
−1.00000 −1.00000 1.00000 −4.07159 1.00000 2.58120 −1.00000 1.00000 4.07159
1.2 −1.00000 −1.00000 1.00000 −3.43788 1.00000 −0.875663 −1.00000 1.00000 3.43788
1.3 −1.00000 −1.00000 1.00000 −3.09495 1.00000 3.97310 −1.00000 1.00000 3.09495
1.4 −1.00000 −1.00000 1.00000 −2.71343 1.00000 −2.34175 −1.00000 1.00000 2.71343
1.5 −1.00000 −1.00000 1.00000 −0.924862 1.00000 −4.24556 −1.00000 1.00000 0.924862
1.6 −1.00000 −1.00000 1.00000 −0.891518 1.00000 4.07845 −1.00000 1.00000 0.891518
1.7 −1.00000 −1.00000 1.00000 −0.503883 1.00000 2.75905 −1.00000 1.00000 0.503883
1.8 −1.00000 −1.00000 1.00000 0.522326 1.00000 −2.74868 −1.00000 1.00000 −0.522326
1.9 −1.00000 −1.00000 1.00000 0.643572 1.00000 1.17705 −1.00000 1.00000 −0.643572
1.10 −1.00000 −1.00000 1.00000 1.05309 1.00000 −0.189425 −1.00000 1.00000 −1.05309
1.11 −1.00000 −1.00000 1.00000 2.46518 1.00000 −0.952790 −1.00000 1.00000 −2.46518
1.12 −1.00000 −1.00000 1.00000 2.56958 1.00000 3.96287 −1.00000 1.00000 −2.56958
1.13 −1.00000 −1.00000 1.00000 3.59906 1.00000 0.919823 −1.00000 1.00000 −3.59906
1.14 −1.00000 −1.00000 1.00000 3.78531 1.00000 −3.09768 −1.00000 1.00000 −3.78531
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(-1\)

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 8034.2.a.ba ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.ba ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{14} + T_{5}^{13} - 44 T_{5}^{12} - 36 T_{5}^{11} + 722 T_{5}^{10} + 451 T_{5}^{9} - 5438 T_{5}^{8} + \cdots - 1492 \) Copy content Toggle raw display
\( T_{7}^{14} - 5 T_{7}^{13} - 41 T_{7}^{12} + 216 T_{7}^{11} + 596 T_{7}^{10} - 3377 T_{7}^{9} + \cdots - 6624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + T^{13} + \cdots - 1492 \) Copy content Toggle raw display
$7$ \( T^{14} - 5 T^{13} + \cdots - 6624 \) Copy content Toggle raw display
$11$ \( T^{14} + T^{13} + \cdots + 1214688 \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 12 T^{13} + \cdots - 19899904 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 558905344 \) Copy content Toggle raw display
$23$ \( T^{14} + T^{13} + \cdots + 43362048 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 204714696 \) Copy content Toggle raw display
$31$ \( T^{14} - 20 T^{13} + \cdots + 38404096 \) Copy content Toggle raw display
$37$ \( T^{14} + 3 T^{13} + \cdots - 1146304 \) Copy content Toggle raw display
$41$ \( T^{14} - T^{13} + \cdots - 2067248 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 52664932864 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 7354606408 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 231076800 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 3946534656 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 185165824 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 796056026 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 992054400 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 48509712950 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 41343457536 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 246609101456 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 244019584 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 175600576384 \) Copy content Toggle raw display
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