Properties

Label 8034.2.a.bc
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} + \cdots + 2048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} + \cdots + 2048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!05 \nu^{14} + \cdots + 10\!\cdots\!44 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!53 \nu^{14} + \cdots + 31\!\cdots\!40 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\!\cdots\!89 \nu^{14} + \cdots + 15\!\cdots\!72 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48\!\cdots\!23 \nu^{14} + \cdots + 23\!\cdots\!72 ) / 55\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!53 \nu^{14} + \cdots - 17\!\cdots\!76 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!39 \nu^{14} + \cdots + 41\!\cdots\!80 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!99 \nu^{14} + \cdots - 10\!\cdots\!64 ) / 16\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!51 \nu^{14} + \cdots + 52\!\cdots\!40 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73\!\cdots\!63 \nu^{14} + \cdots - 20\!\cdots\!92 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 87\!\cdots\!33 \nu^{14} + \cdots + 54\!\cdots\!08 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 44\!\cdots\!09 \nu^{14} + \cdots - 11\!\cdots\!28 ) / 22\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 75\!\cdots\!63 \nu^{14} + \cdots - 22\!\cdots\!76 ) / 22\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 51\!\cdots\!33 \nu^{14} + \cdots + 97\!\cdots\!48 ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 2 \beta_{14} + 2 \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - 2\beta_{7} - \beta_{4} + 12\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 34 \beta_{14} + 32 \beta_{13} + 5 \beta_{12} + 19 \beta_{11} + 15 \beta_{10} - 10 \beta_{9} - 19 \beta_{8} + 18 \beta_{7} + 19 \beta_{6} + 18 \beta_{5} - 16 \beta_{4} - 3 \beta_{3} + 19 \beta_{2} - 16 \beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} + 8 \beta_{13} - 75 \beta_{12} - 20 \beta_{11} - 27 \beta_{10} - 22 \beta_{9} - 45 \beta_{7} - 8 \beta_{6} + \beta_{5} - 23 \beta_{4} - \beta_{3} + 4 \beta_{2} + 168 \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 556 \beta_{14} + 502 \beta_{13} + 115 \beta_{12} + 317 \beta_{11} + 229 \beta_{10} - 96 \beta_{9} - 333 \beta_{8} + 313 \beta_{7} + 337 \beta_{6} + 302 \beta_{5} - 252 \beta_{4} - 79 \beta_{3} + 310 \beta_{2} - 247 \beta _1 + 1198 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 31 \beta_{14} + 239 \beta_{13} - 1254 \beta_{12} - 346 \beta_{11} - 562 \beta_{10} - 432 \beta_{9} - 10 \beta_{8} - 849 \beta_{7} - 229 \beta_{6} + 18 \beta_{5} - 414 \beta_{4} - 35 \beta_{3} + 99 \beta_{2} + 2518 \beta _1 - 180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9097 \beta_{14} + 7967 \beta_{13} + 2088 \beta_{12} + 5193 \beta_{11} + 3576 \beta_{10} - 888 \beta_{9} - 5611 \beta_{8} + 5351 \beta_{7} + 5729 \beta_{6} + 5021 \beta_{5} - 3952 \beta_{4} - 1609 \beta_{3} + \cdots + 17955 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 619 \beta_{14} + 5095 \beta_{13} - 20805 \beta_{12} - 5891 \beta_{11} - 10625 \beta_{10} - 8055 \beta_{9} - 308 \beta_{8} - 15316 \beta_{7} - 5067 \beta_{6} + 61 \beta_{5} - 6877 \beta_{4} - 750 \beta_{3} + \cdots - 2196 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 149357 \beta_{14} + 127959 \beta_{13} + 35595 \beta_{12} + 84969 \beta_{11} + 56709 \beta_{10} - 7087 \beta_{9} - 93089 \beta_{8} + 90567 \beta_{7} + 95731 \beta_{6} + 83612 \beta_{5} + \cdots + 277897 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 10176 \beta_{14} + 95830 \beta_{13} - 346912 \beta_{12} - 100736 \beta_{11} - 192217 \beta_{10} - 145094 \beta_{9} - 6660 \beta_{8} - 270979 \beta_{7} - 101706 \beta_{6} - 5067 \beta_{5} + \cdots - 31788 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2460611 \beta_{14} + 2074863 \beta_{13} + 595649 \beta_{12} + 1393860 \beta_{11} + 908995 \beta_{10} - 28519 \beta_{9} - 1536904 \beta_{8} + 1524931 \beta_{7} + 1589921 \beta_{6} + \cdots + 4386294 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 143953 \beta_{14} + 1697102 \beta_{13} - 5809478 \beta_{12} - 1731649 \beta_{11} - 3399748 \beta_{10} - 2552103 \beta_{9} - 122438 \beta_{8} - 4738578 \beta_{7} - 1938971 \beta_{6} + \cdots - 564758 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 40658804 \beta_{14} + 33887265 \beta_{13} + 9929787 \beta_{12} + 22941355 \beta_{11} + 14689662 \beta_{10} + 623231 \beta_{9} - 25363596 \beta_{8} + 25613853 \beta_{7} + \cdots + 70152806 \) 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
4.06882
3.77963
2.75355
2.43337
2.05101
1.17201
0.676512
0.113262
−0.656329
−1.34829
−1.40985
−1.97610
−3.01123
−3.53778
−4.10857
1.00000 −1.00000 1.00000 −4.06882 −1.00000 4.31915 1.00000 1.00000 −4.06882
1.2 1.00000 −1.00000 1.00000 −3.77963 −1.00000 2.13919 1.00000 1.00000 −3.77963
1.3 1.00000 −1.00000 1.00000 −2.75355 −1.00000 0.824838 1.00000 1.00000 −2.75355
1.4 1.00000 −1.00000 1.00000 −2.43337 −1.00000 −2.53842 1.00000 1.00000 −2.43337
1.5 1.00000 −1.00000 1.00000 −2.05101 −1.00000 −1.63593 1.00000 1.00000 −2.05101
1.6 1.00000 −1.00000 1.00000 −1.17201 −1.00000 −4.40989 1.00000 1.00000 −1.17201
1.7 1.00000 −1.00000 1.00000 −0.676512 −1.00000 −2.72085 1.00000 1.00000 −0.676512
1.8 1.00000 −1.00000 1.00000 −0.113262 −1.00000 1.75034 1.00000 1.00000 −0.113262
1.9 1.00000 −1.00000 1.00000 0.656329 −1.00000 2.89208 1.00000 1.00000 0.656329
1.10 1.00000 −1.00000 1.00000 1.34829 −1.00000 −1.25683 1.00000 1.00000 1.34829
1.11 1.00000 −1.00000 1.00000 1.40985 −1.00000 4.86981 1.00000 1.00000 1.40985
1.12 1.00000 −1.00000 1.00000 1.97610 −1.00000 4.65524 1.00000 1.00000 1.97610
1.13 1.00000 −1.00000 1.00000 3.01123 −1.00000 −0.871929 1.00000 1.00000 3.01123
1.14 1.00000 −1.00000 1.00000 3.53778 −1.00000 −4.09818 1.00000 1.00000 3.53778
1.15 1.00000 −1.00000 1.00000 4.10857 −1.00000 1.08136 1.00000 1.00000 4.10857
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bc 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bc 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{15} + T_{5}^{14} - 48 T_{5}^{13} - 44 T_{5}^{12} + 872 T_{5}^{11} + 707 T_{5}^{10} - 7580 T_{5}^{9} - 5112 T_{5}^{8} + 33191 T_{5}^{7} + 16428 T_{5}^{6} - 71361 T_{5}^{5} - 21747 T_{5}^{4} + 65434 T_{5}^{3} + 11840 T_{5}^{2} + \cdots - 2048 \) Copy content Toggle raw display
\( T_{7}^{15} - 5 T_{7}^{14} - 56 T_{7}^{13} + 275 T_{7}^{12} + 1187 T_{7}^{11} - 5625 T_{7}^{10} - 12093 T_{7}^{9} + 53846 T_{7}^{8} + 63048 T_{7}^{7} - 255993 T_{7}^{6} - 164378 T_{7}^{5} + 590741 T_{7}^{4} + \cdots + 211634 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( (T + 1)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + T^{14} - 48 T^{13} - 44 T^{12} + \cdots - 2048 \) Copy content Toggle raw display
$7$ \( T^{15} - 5 T^{14} - 56 T^{13} + \cdots + 211634 \) Copy content Toggle raw display
$11$ \( T^{15} - 3 T^{14} - 100 T^{13} + \cdots + 12059584 \) Copy content Toggle raw display
$13$ \( (T + 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + 2 T^{14} - 140 T^{13} + \cdots - 19422208 \) Copy content Toggle raw display
$19$ \( T^{15} - 8 T^{14} - 160 T^{13} + \cdots - 186761216 \) Copy content Toggle raw display
$23$ \( T^{15} - 3 T^{14} - 227 T^{13} + \cdots - 98264192 \) Copy content Toggle raw display
$29$ \( T^{15} - 26 T^{14} + \cdots - 1382105088 \) Copy content Toggle raw display
$31$ \( T^{15} - 294 T^{13} + \cdots + 274726912 \) Copy content Toggle raw display
$37$ \( T^{15} - 25 T^{14} + 82 T^{13} + \cdots + 800768 \) Copy content Toggle raw display
$41$ \( T^{15} + T^{14} - 341 T^{13} + \cdots - 557822512 \) Copy content Toggle raw display
$43$ \( T^{15} - 10 T^{14} + \cdots + 777363968 \) Copy content Toggle raw display
$47$ \( T^{15} + 3 T^{14} + \cdots + 251671604384 \) Copy content Toggle raw display
$53$ \( T^{15} - 13 T^{14} + \cdots - 4553643008 \) Copy content Toggle raw display
$59$ \( T^{15} - 28 T^{14} + \cdots + 38526802432 \) Copy content Toggle raw display
$61$ \( T^{15} - 22 T^{14} + \cdots + 158418815744 \) Copy content Toggle raw display
$67$ \( T^{15} - 29 T^{14} + \cdots - 756505865536 \) Copy content Toggle raw display
$71$ \( T^{15} - 18 T^{14} + \cdots + 290069168128 \) Copy content Toggle raw display
$73$ \( T^{15} - 23 T^{14} + \cdots - 2146208123936 \) Copy content Toggle raw display
$79$ \( T^{15} - 27 T^{14} + \cdots + 4164592205824 \) Copy content Toggle raw display
$83$ \( T^{15} - 7 T^{14} - 442 T^{13} + \cdots - 96663104 \) Copy content Toggle raw display
$89$ \( T^{15} - 35 T^{14} + \cdots + 3996849127424 \) Copy content Toggle raw display
$97$ \( T^{15} - 19 T^{14} + \cdots + 15882900389888 \) Copy content Toggle raw display
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