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Properties

Label 8034.2.a.y
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + \cdots - 180 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + \cdots - 180 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29990952597 \nu^{12} - 43375192180 \nu^{11} - 1136475186639 \nu^{10} + 1007708200373 \nu^{9} + \cdots + 4629967598166 ) / 1742391215594 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65603934565 \nu^{12} - 181553890487 \nu^{11} - 2150123291131 \nu^{10} + \cdots + 10324372473579 ) / 2613586823391 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 162857412779 \nu^{12} - 677670898672 \nu^{11} - 4278067082381 \nu^{10} + \cdots + 13092136637310 ) / 5227173646782 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 102100965329 \nu^{12} - 304209647056 \nu^{11} - 3169887944810 \nu^{10} + \cdots - 14347124251584 ) / 2613586823391 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 217714208045 \nu^{12} + 1005449553016 \nu^{11} + 5411101570403 \nu^{10} + \cdots + 13131093814746 ) / 5227173646782 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 255095470579 \nu^{12} + 661810602824 \nu^{11} + 8285169875425 \nu^{10} + \cdots - 13275752967450 ) / 5227173646782 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 87548046093 \nu^{12} + 264959999010 \nu^{11} + 2727333681067 \nu^{10} + \cdots - 15396019776588 ) / 1742391215594 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 137156022856 \nu^{12} + 292035064997 \nu^{11} + 4686634469818 \nu^{10} + \cdots - 33755880545082 ) / 2613586823391 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48721679165 \nu^{12} - 146572031608 \nu^{11} - 1525594606540 \nu^{10} + 3583008630877 \nu^{9} + \cdots + 4959261111660 ) / 871195607797 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 389665509547 \nu^{12} + 1140098743058 \nu^{11} + 12323418899743 \nu^{10} + \cdots + 1172172013944 ) / 5227173646782 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 114093379497 \nu^{12} + 401458140693 \nu^{11} + 3322146417204 \nu^{10} + \cdots - 5083097540960 ) / 871195607797 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 3 \beta_{10} + \beta_{9} - 6 \beta_{8} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{12} + 9 \beta_{11} - \beta_{10} + 4 \beta_{9} - 22 \beta_{8} - 5 \beta_{7} - 14 \beta_{6} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + 37 \beta_{11} - 62 \beta_{10} + 34 \beta_{9} - 144 \beta_{8} - 13 \beta_{7} - 16 \beta_{6} + \cdots + 65 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 161 \beta_{12} + 243 \beta_{11} - 67 \beta_{10} + 127 \beta_{9} - 479 \beta_{8} - 145 \beta_{7} + \cdots + 900 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 35 \beta_{12} + 951 \beta_{11} - 1180 \beta_{10} + 801 \beta_{9} - 2920 \beta_{8} - 447 \beta_{7} + \cdots + 1468 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2393 \beta_{12} + 5389 \beta_{11} - 2073 \beta_{10} + 3071 \beta_{9} - 10213 \beta_{8} - 3254 \beta_{7} + \cdots + 15532 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1150 \beta_{12} + 21643 \beta_{11} - 22293 \beta_{10} + 17056 \beta_{9} - 57599 \beta_{8} - 11144 \beta_{7} + \cdots + 33244 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 38079 \beta_{12} + 113170 \beta_{11} - 52003 \beta_{10} + 67753 \beta_{9} - 214027 \beta_{8} + \cdots + 281896 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 33272 \beta_{12} + 467279 \beta_{11} - 424267 \beta_{10} + 350742 \beta_{9} - 1135805 \beta_{8} + \cdots + 737045 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 637279 \beta_{12} + 2327795 \beta_{11} - 1193552 \beta_{10} + 1436886 \beta_{9} - 4436546 \beta_{8} + \cdots + 5275921 \) 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
−2.14079
0.0852190
4.50145
−1.72356
1.94320
−1.46489
−0.723662
−3.83598
−0.892751
3.49310
2.89547
1.94844
0.914750
1.00000 −1.00000 1.00000 −4.27728 −1.00000 −2.14079 1.00000 1.00000 −4.27728
1.2 1.00000 −1.00000 1.00000 −4.16745 −1.00000 0.0852190 1.00000 1.00000 −4.16745
1.3 1.00000 −1.00000 1.00000 −1.68854 −1.00000 4.50145 1.00000 1.00000 −1.68854
1.4 1.00000 −1.00000 1.00000 −1.45246 −1.00000 −1.72356 1.00000 1.00000 −1.45246
1.5 1.00000 −1.00000 1.00000 −0.714571 −1.00000 1.94320 1.00000 1.00000 −0.714571
1.6 1.00000 −1.00000 1.00000 −0.556182 −1.00000 −1.46489 1.00000 1.00000 −0.556182
1.7 1.00000 −1.00000 1.00000 0.370604 −1.00000 −0.723662 1.00000 1.00000 0.370604
1.8 1.00000 −1.00000 1.00000 1.02710 −1.00000 −3.83598 1.00000 1.00000 1.02710
1.9 1.00000 −1.00000 1.00000 1.04935 −1.00000 −0.892751 1.00000 1.00000 1.04935
1.10 1.00000 −1.00000 1.00000 2.90785 −1.00000 3.49310 1.00000 1.00000 2.90785
1.11 1.00000 −1.00000 1.00000 3.35434 −1.00000 2.89547 1.00000 1.00000 3.35434
1.12 1.00000 −1.00000 1.00000 3.48931 −1.00000 1.94844 1.00000 1.00000 3.48931
1.13 1.00000 −1.00000 1.00000 3.65794 −1.00000 0.914750 1.00000 1.00000 3.65794
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.y ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.y ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{13} - 3 T_{5}^{12} - 40 T_{5}^{11} + 130 T_{5}^{10} + 496 T_{5}^{9} - 1749 T_{5}^{8} - 1960 T_{5}^{7} + \cdots - 864 \) Copy content Toggle raw display
\( T_{7}^{13} - 5 T_{7}^{12} - 25 T_{7}^{11} + 134 T_{7}^{10} + 196 T_{7}^{9} - 1151 T_{7}^{8} - 801 T_{7}^{7} + \cdots - 180 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{13} \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 3 T^{12} + \cdots - 864 \) Copy content Toggle raw display
$7$ \( T^{13} - 5 T^{12} + \cdots - 180 \) Copy content Toggle raw display
$11$ \( T^{13} - 8 T^{12} + \cdots - 587920 \) Copy content Toggle raw display
$13$ \( (T - 1)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} + T^{12} + \cdots + 5617408 \) Copy content Toggle raw display
$19$ \( T^{13} - 5 T^{12} + \cdots - 5142528 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 305262088 \) Copy content Toggle raw display
$29$ \( T^{13} + T^{12} + \cdots - 21563136 \) Copy content Toggle raw display
$31$ \( T^{13} - 5 T^{12} + \cdots - 5142528 \) Copy content Toggle raw display
$37$ \( T^{13} - 35 T^{12} + \cdots - 1583312 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 10405627608 \) Copy content Toggle raw display
$43$ \( T^{13} - 9 T^{12} + \cdots + 2236128 \) Copy content Toggle raw display
$47$ \( T^{13} + 8 T^{12} + \cdots + 3379806 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 2231257728 \) Copy content Toggle raw display
$59$ \( T^{13} - 12 T^{12} + \cdots - 2461440 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 11383480832 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 6772775080 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 552276992 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 445541544 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 1034356945408 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 27181773312 \) Copy content Toggle raw display
$89$ \( T^{13} - 14 T^{12} + \cdots - 1536000 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 4262466688 \) Copy content Toggle raw display
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