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Properties

Label 8034.2.a.z
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: \( x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + \cdots + 1552 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - \beta_1 q^{5} + q^{6} + \beta_{6} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - \beta_1 q^{5} + q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} + \beta_1 q^{10} + \beta_{7} q^{11} - q^{12} + q^{13} - \beta_{6} q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{9} - 1) q^{17} - q^{18} + ( - \beta_{13} + \beta_1 + 2) q^{19} - \beta_1 q^{20} - \beta_{6} q^{21} - \beta_{7} q^{22} + (\beta_{13} + \beta_{12} + \cdots + \beta_{4}) q^{23}+ \cdots + \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9} + 3 q^{10} + 5 q^{11} - 14 q^{12} + 14 q^{13} - 4 q^{14} + 3 q^{15} + 14 q^{16} - 7 q^{17} - 14 q^{18} + 31 q^{19} - 3 q^{20} - 4 q^{21} - 5 q^{22} + 14 q^{23} + 14 q^{24} + 11 q^{25} - 14 q^{26} - 14 q^{27} + 4 q^{28} + 9 q^{29} - 3 q^{30} + 23 q^{31} - 14 q^{32} - 5 q^{33} + 7 q^{34} - 2 q^{35} + 14 q^{36} + 12 q^{37} - 31 q^{38} - 14 q^{39} + 3 q^{40} - 5 q^{41} + 4 q^{42} - 2 q^{43} + 5 q^{44} - 3 q^{45} - 14 q^{46} - 11 q^{47} - 14 q^{48} + 52 q^{49} - 11 q^{50} + 7 q^{51} + 14 q^{52} + 6 q^{53} + 14 q^{54} + 30 q^{55} - 4 q^{56} - 31 q^{57} - 9 q^{58} - 4 q^{59} + 3 q^{60} + 12 q^{61} - 23 q^{62} + 4 q^{63} + 14 q^{64} - 3 q^{65} + 5 q^{66} + 24 q^{67} - 7 q^{68} - 14 q^{69} + 2 q^{70} + 20 q^{71} - 14 q^{72} + 2 q^{73} - 12 q^{74} - 11 q^{75} + 31 q^{76} - 28 q^{77} + 14 q^{78} + 59 q^{79} - 3 q^{80} + 14 q^{81} + 5 q^{82} + q^{83} - 4 q^{84} - 29 q^{85} + 2 q^{86} - 9 q^{87} - 5 q^{88} + 6 q^{89} + 3 q^{90} + 4 q^{91} + 14 q^{92} - 23 q^{93} + 11 q^{94} - 58 q^{95} + 14 q^{96} - 6 q^{97} - 52 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + \cdots + 1552 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3341759347 \nu^{13} + 518140367105 \nu^{12} - 1741789540612 \nu^{11} + \cdots - 592135085106752 ) / 30448521494580 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12670008395 \nu^{13} + 121522864993 \nu^{12} + 241219668796 \nu^{11} + \cdots - 7510828872730 ) / 15224260747290 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7437129421 \nu^{13} + 8250683722 \nu^{12} - 328982202812 \nu^{11} - 355914392252 \nu^{10} + \cdots + 41261138165888 ) / 7612130373645 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16735095548 \nu^{13} - 59503252771 \nu^{12} - 534709503235 \nu^{11} + 2013117619607 \nu^{10} + \cdots + 67234006456384 ) / 7612130373645 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6829796404 \nu^{13} - 24088984468 \nu^{12} - 220735533930 \nu^{11} + 818192200816 \nu^{10} + \cdots - 696096892188 ) / 2537376791215 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 101168043581 \nu^{13} - 428736219223 \nu^{12} - 3096334884004 \nu^{11} + \cdots + 321698456894260 ) / 30448521494580 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 132195076697 \nu^{13} - 378058949095 \nu^{12} - 4698358621144 \nu^{11} + \cdots + 93265012848748 ) / 30448521494580 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37454660035 \nu^{13} - 65377606061 \nu^{12} - 1543899870542 \nu^{11} + \cdots - 240724229362315 ) / 7612130373645 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 61385752497 \nu^{13} - 224068680263 \nu^{12} - 2016688298880 \nu^{11} + \cdots - 113491089496292 ) / 10149507164860 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61385752497 \nu^{13} - 224068680263 \nu^{12} - 2016688298880 \nu^{11} + \cdots - 52594046507132 ) / 10149507164860 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 47799908567 \nu^{13} - 114887884225 \nu^{12} - 1783176710104 \nu^{11} + \cdots - 91513305290492 ) / 7612130373645 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 120974609621 \nu^{13} + 303334971739 \nu^{12} + 4434352237832 \nu^{11} + \cdots + 280944669298364 ) / 10149507164860 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - 4 \beta_{12} - 19 \beta_{11} + 16 \beta_{10} + \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + \cdots + 71 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 60 \beta_{13} + 61 \beta_{12} + 53 \beta_{11} + 7 \beta_{10} + 21 \beta_{9} - 41 \beta_{8} + 41 \beta_{7} + \cdots - 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 59 \beta_{13} - 126 \beta_{12} - 354 \beta_{11} + 250 \beta_{10} + 15 \beta_{9} + 101 \beta_{8} + \cdots + 1064 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1048 \beta_{13} + 1103 \beta_{12} + 1135 \beta_{11} - 76 \beta_{10} + 365 \beta_{9} - 772 \beta_{8} + \cdots - 1613 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1728 \beta_{13} - 2966 \beta_{12} - 6601 \beta_{11} + 4024 \beta_{10} + 101 \beta_{9} + 2173 \beta_{8} + \cdots + 17581 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18247 \beta_{13} + 19912 \beta_{12} + 23085 \beta_{11} - 4313 \beta_{10} + 6128 \beta_{9} - 14338 \beta_{8} + \cdots - 38962 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 41245 \beta_{13} - 63259 \beta_{12} - 123644 \beta_{11} + 67177 \beta_{10} - 1642 \beta_{9} + \cdots + 305954 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 323677 \beta_{13} + 365063 \beta_{12} + 460293 \beta_{11} - 120204 \beta_{10} + 102976 \beta_{9} + \cdots - 854144 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 900094 \beta_{13} - 1291247 \beta_{12} - 2328995 \beta_{11} + 1159985 \beta_{10} - 89596 \beta_{9} + \cdots + 5498570 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5856451 \beta_{13} + 6791797 \beta_{12} + 9079144 \beta_{11} - 2797826 \beta_{10} + 1753751 \beta_{9} + \cdots - 17797008 \) 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
3.68342
3.65422
3.21497
2.05100
1.52247
0.937159
0.664239
0.343018
−1.02849
−1.34598
−1.59859
−2.28948
−2.41893
−4.38902
−1.00000 −1.00000 1.00000 −3.68342 1.00000 5.03243 −1.00000 1.00000 3.68342
1.2 −1.00000 −1.00000 1.00000 −3.65422 1.00000 −2.98447 −1.00000 1.00000 3.65422
1.3 −1.00000 −1.00000 1.00000 −3.21497 1.00000 −1.40491 −1.00000 1.00000 3.21497
1.4 −1.00000 −1.00000 1.00000 −2.05100 1.00000 5.03739 −1.00000 1.00000 2.05100
1.5 −1.00000 −1.00000 1.00000 −1.52247 1.00000 −3.40750 −1.00000 1.00000 1.52247
1.6 −1.00000 −1.00000 1.00000 −0.937159 1.00000 −0.792704 −1.00000 1.00000 0.937159
1.7 −1.00000 −1.00000 1.00000 −0.664239 1.00000 2.36162 −1.00000 1.00000 0.664239
1.8 −1.00000 −1.00000 1.00000 −0.343018 1.00000 2.39676 −1.00000 1.00000 0.343018
1.9 −1.00000 −1.00000 1.00000 1.02849 1.00000 −4.96874 −1.00000 1.00000 −1.02849
1.10 −1.00000 −1.00000 1.00000 1.34598 1.00000 −3.24510 −1.00000 1.00000 −1.34598
1.11 −1.00000 −1.00000 1.00000 1.59859 1.00000 4.24832 −1.00000 1.00000 −1.59859
1.12 −1.00000 −1.00000 1.00000 2.28948 1.00000 −1.19378 −1.00000 1.00000 −2.28948
1.13 −1.00000 −1.00000 1.00000 2.41893 1.00000 −0.257176 −1.00000 1.00000 −2.41893
1.14 −1.00000 −1.00000 1.00000 4.38902 1.00000 3.17787 −1.00000 1.00000 −4.38902
\(n\): 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.z ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.z ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{14} + 3 T_{5}^{13} - 36 T_{5}^{12} - 108 T_{5}^{11} + 434 T_{5}^{10} + 1239 T_{5}^{9} + \cdots + 1552 \) Copy content Toggle raw display
\( T_{7}^{14} - 4 T_{7}^{13} - 67 T_{7}^{12} + 235 T_{7}^{11} + 1761 T_{7}^{10} - 4948 T_{7}^{9} + \cdots + 108608 \) 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
$5$ \( T^{14} + 3 T^{13} + \cdots + 1552 \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots + 108608 \) Copy content Toggle raw display
$11$ \( T^{14} - 5 T^{13} + \cdots + 1849152 \) Copy content Toggle raw display
$13$ \( (T - 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 7 T^{13} + \cdots - 57600 \) Copy content Toggle raw display
$19$ \( T^{14} - 31 T^{13} + \cdots + 177664 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 2497011840 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 946837836 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 6181087360 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 106231104 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 228866240 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 157075968 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 159375400 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 9892473600 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 6276618880 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 672637440 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 12800870040 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 240723840 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 2427482624 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 3471863808 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 547859593152 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 22829917834624 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 9472848370240 \) Copy content Toggle raw display
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