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Properties

Label 8034.2.a.t
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} + \cdots - 162 \) 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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} + \cdots - 162 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5953 \nu^{10} + 71878 \nu^{9} - 597285 \nu^{8} + 825299 \nu^{7} + 11387240 \nu^{6} + \cdots - 16896519 ) / 16688889 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32168 \nu^{10} - 265732 \nu^{9} + 2755953 \nu^{8} + 3489646 \nu^{7} - 42680480 \nu^{6} + \cdots + 42233265 ) / 16688889 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 53471 \nu^{10} - 338801 \nu^{9} - 262824 \nu^{8} + 4406315 \nu^{7} - 4924159 \nu^{6} + \cdots + 33384069 ) / 16688889 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35999 \nu^{10} - 355853 \nu^{9} + 249954 \nu^{8} + 6151373 \nu^{7} - 12119914 \nu^{6} + \cdots + 10853631 ) / 5562963 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57370 \nu^{10} + 225044 \nu^{9} + 1255382 \nu^{8} - 4321489 \nu^{7} - 10803735 \nu^{6} + \cdots + 18345483 ) / 5562963 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 176675 \nu^{10} + 631196 \nu^{9} + 4961220 \nu^{8} - 15848045 \nu^{7} - 51857339 \nu^{6} + \cdots + 184701816 ) / 16688889 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 75437 \nu^{10} + 263873 \nu^{9} + 1986210 \nu^{8} - 6019115 \nu^{7} - 19487438 \nu^{6} + \cdots + 22884885 ) / 5562963 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 168806 \nu^{10} + 844770 \nu^{9} + 2991638 \nu^{8} - 16491977 \nu^{7} - 18171259 \nu^{6} + \cdots + 58191552 ) / 5562963 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 77942 \nu^{10} - 390059 \nu^{9} - 1408296 \nu^{8} + 7828925 \nu^{7} + 8470799 \nu^{6} + \cdots - 31061394 ) / 2384127 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{6} - 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 9\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} - 13\beta_{9} + 9\beta_{8} + \beta_{7} + 14\beta_{6} - 13\beta_{5} + \beta_{4} + 2\beta_{3} + 17\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{10} - 6 \beta_{9} + 13 \beta_{8} - 2 \beta_{7} + 26 \beta_{6} - 33 \beta_{5} + 30 \beta_{4} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 24 \beta_{10} - 168 \beta_{9} + 85 \beta_{8} + 6 \beta_{7} + 194 \beta_{6} - 168 \beta_{5} + \cdots + 538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111 \beta_{10} - 166 \beta_{9} + 137 \beta_{8} - 48 \beta_{7} + 470 \beta_{6} - 489 \beta_{5} + \cdots + 1171 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 379 \beta_{10} - 2174 \beta_{9} + 823 \beta_{8} - 26 \beta_{7} + 2689 \beta_{6} - 2268 \beta_{5} + \cdots + 6803 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 852 \beta_{10} - 3280 \beta_{9} + 1337 \beta_{8} - 824 \beta_{7} + 7508 \beta_{6} - 7242 \beta_{5} + \cdots + 17861 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5248 \beta_{10} - 28433 \beta_{9} + 7861 \beta_{8} - 1367 \beta_{7} + 37184 \beta_{6} - 31807 \beta_{5} + \cdots + 91048 \) 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
−3.18653
−2.18067
−2.06747
−1.52965
−0.387444
−0.203778
1.30889
1.64076
3.03614
3.71131
3.85844
−1.00000 1.00000 1.00000 −3.18653 −1.00000 1.47944 −1.00000 1.00000 3.18653
1.2 −1.00000 1.00000 1.00000 −2.18067 −1.00000 −1.20699 −1.00000 1.00000 2.18067
1.3 −1.00000 1.00000 1.00000 −2.06747 −1.00000 −2.54251 −1.00000 1.00000 2.06747
1.4 −1.00000 1.00000 1.00000 −1.52965 −1.00000 −0.354974 −1.00000 1.00000 1.52965
1.5 −1.00000 1.00000 1.00000 −0.387444 −1.00000 5.18739 −1.00000 1.00000 0.387444
1.6 −1.00000 1.00000 1.00000 −0.203778 −1.00000 0.561930 −1.00000 1.00000 0.203778
1.7 −1.00000 1.00000 1.00000 1.30889 −1.00000 2.83911 −1.00000 1.00000 −1.30889
1.8 −1.00000 1.00000 1.00000 1.64076 −1.00000 −3.72013 −1.00000 1.00000 −1.64076
1.9 −1.00000 1.00000 1.00000 3.03614 −1.00000 −4.00525 −1.00000 1.00000 −3.03614
1.10 −1.00000 1.00000 1.00000 3.71131 −1.00000 1.88167 −1.00000 1.00000 −3.71131
1.11 −1.00000 1.00000 1.00000 3.85844 −1.00000 3.88029 −1.00000 1.00000 −3.85844
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.t ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.t ✓ 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{11} - 4 T_{5}^{10} - 24 T_{5}^{9} + 88 T_{5}^{8} + 220 T_{5}^{7} - 637 T_{5}^{6} - 977 T_{5}^{5} + \cdots - 162 \) Copy content Toggle raw display
\( T_{7}^{11} - 4 T_{7}^{10} - 39 T_{7}^{9} + 145 T_{7}^{8} + 487 T_{7}^{7} - 1738 T_{7}^{6} - 1985 T_{7}^{5} + \cdots + 1451 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{11} \) Copy content Toggle raw display
$3$ \( (T - 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} - 4 T^{10} + \cdots - 162 \) Copy content Toggle raw display
$7$ \( T^{11} - 4 T^{10} + \cdots + 1451 \) Copy content Toggle raw display
$11$ \( T^{11} - 5 T^{10} + \cdots + 3960 \) Copy content Toggle raw display
$13$ \( (T + 1)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} - 8 T^{10} + \cdots - 15928 \) Copy content Toggle raw display
$19$ \( T^{11} + 2 T^{10} + \cdots + 7472 \) Copy content Toggle raw display
$23$ \( T^{11} - 3 T^{10} + \cdots - 11088 \) Copy content Toggle raw display
$29$ \( T^{11} - 7 T^{10} + \cdots + 609222 \) Copy content Toggle raw display
$31$ \( T^{11} - 20 T^{10} + \cdots - 3919184 \) Copy content Toggle raw display
$37$ \( T^{11} - T^{10} + \cdots + 576400 \) Copy content Toggle raw display
$41$ \( T^{11} - 37 T^{10} + \cdots - 18250 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 107474032 \) Copy content Toggle raw display
$47$ \( T^{11} - 28 T^{10} + \cdots + 28302372 \) Copy content Toggle raw display
$53$ \( T^{11} + 5 T^{10} + \cdots - 2054968 \) Copy content Toggle raw display
$59$ \( T^{11} - 31 T^{10} + \cdots - 65166240 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 23303527296 \) Copy content Toggle raw display
$67$ \( T^{11} + 22 T^{10} + \cdots + 12509517 \) Copy content Toggle raw display
$71$ \( T^{11} - 42 T^{10} + \cdots - 8558660 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 2206561775 \) Copy content Toggle raw display
$79$ \( T^{11} - 33 T^{10} + \cdots + 95618960 \) Copy content Toggle raw display
$83$ \( T^{11} - 18 T^{10} + \cdots + 22143750 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 3961390960 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 2398891376 \) Copy content Toggle raw display
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