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Properties

Label 8030.2.a.bg
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
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: \( x^{15} - 7 x^{14} - 6 x^{13} + 136 x^{12} - 149 x^{11} - 876 x^{10} + 1631 x^{9} + 2142 x^{8} + \cdots - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} - 6 x^{13} + 136 x^{12} - 149 x^{11} - 876 x^{10} + 1631 x^{9} + 2142 x^{8} + \cdots - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 43181499 \nu^{14} + 311865932 \nu^{13} + 175296873 \nu^{12} - 5835678177 \nu^{11} + \cdots - 1043521072 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 151113864 \nu^{14} - 856502009 \nu^{13} - 2141104338 \nu^{12} + 18174114793 \nu^{11} + \cdots + 10563926796 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 240146793 \nu^{14} + 1349405164 \nu^{13} + 3310464644 \nu^{12} - 28093591614 \nu^{11} + \cdots + 637595355 ) / 736025987 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 393129354 \nu^{14} + 2205793563 \nu^{13} + 5420958086 \nu^{12} - 45934140616 \nu^{11} + \cdots - 4867233006 ) / 736025987 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 795515309 \nu^{14} + 4429334644 \nu^{13} + 11257805713 \nu^{12} - 92926760279 \nu^{11} + \cdots - 13759637204 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 994410369 \nu^{14} + 5597120447 \nu^{13} + 13686695073 \nu^{12} - 116666528026 \nu^{11} + \cdots - 18366549708 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1069761995 \nu^{14} - 5991781794 \nu^{13} - 14929676803 \nu^{12} + 125312551991 \nu^{11} + \cdots + 12810965580 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 717823592 \nu^{14} + 4066794759 \nu^{13} + 9723141475 \nu^{12} - 84590402393 \nu^{11} + \cdots - 19927820364 ) / 736025987 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1615094007 \nu^{14} + 9152836614 \nu^{13} + 21956198693 \nu^{12} - 190633964393 \nu^{11} + \cdots - 31966415456 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1750079193 \nu^{14} - 9894268181 \nu^{13} - 23804879799 \nu^{12} + 205776820544 \nu^{11} + \cdots + 38701296324 ) / 1472051974 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 917646164 \nu^{14} - 5144440308 \nu^{13} - 12683340797 \nu^{12} + 107209799902 \nu^{11} + \cdots + 20954746371 ) / 736025987 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3971985085 \nu^{14} + 22439444825 \nu^{13} + 54093078921 \nu^{12} - 466825906562 \nu^{11} + \cdots - 92707558986 ) / 1472051974 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{12} + \beta_{10} + \beta_{6} + \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 10\beta_{2} + 10\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{14} - \beta_{13} - 15 \beta_{12} + 11 \beta_{10} + \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{14} + 10 \beta_{13} - 21 \beta_{12} - 14 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 144 \beta_{14} - 26 \beta_{13} - 174 \beta_{12} + 2 \beta_{11} + 103 \beta_{10} + 15 \beta_{9} + \cdots + 119 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 133 \beta_{14} + 60 \beta_{13} - 299 \beta_{12} - 148 \beta_{11} + 42 \beta_{10} - 96 \beta_{9} + \cdots + 2064 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1550 \beta_{14} - 436 \beta_{13} - 1884 \beta_{12} + 42 \beta_{11} + 954 \beta_{10} + 188 \beta_{9} + \cdots + 1316 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2067 \beta_{14} + 47 \beta_{13} - 3737 \beta_{12} - 1433 \beta_{11} + 641 \beta_{10} - 763 \beta_{9} + \cdots + 18300 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16565 \beta_{14} - 6067 \beta_{13} - 20005 \beta_{12} + 533 \beta_{11} + 8984 \beta_{10} + \cdots + 14655 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 27851 \beta_{14} - 5541 \beta_{13} - 44232 \beta_{12} - 13497 \beta_{11} + 8626 \beta_{10} + \cdots + 165526 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 176606 \beta_{14} - 76562 \beta_{13} - 211771 \beta_{12} + 5037 \beta_{11} + 86425 \beta_{10} + \cdots + 162718 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 348250 \beta_{14} - 115865 \beta_{13} - 509659 \beta_{12} - 126936 \beta_{11} + 108588 \beta_{10} + \cdots + 1523442 \) 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
−2.92483
−2.60943
−1.49337
−1.29384
−0.765121
−0.490398
0.0684817
0.348096
1.35747
1.35961
2.15389
2.24233
2.74529
3.03129
3.27053
1.00000 −2.92483 1.00000 −1.00000 −2.92483 0.830562 1.00000 5.55462 −1.00000
1.2 1.00000 −2.60943 1.00000 −1.00000 −2.60943 −1.48886 1.00000 3.80914 −1.00000
1.3 1.00000 −1.49337 1.00000 −1.00000 −1.49337 −0.532219 1.00000 −0.769851 −1.00000
1.4 1.00000 −1.29384 1.00000 −1.00000 −1.29384 −4.27001 1.00000 −1.32597 −1.00000
1.5 1.00000 −0.765121 1.00000 −1.00000 −0.765121 2.95797 1.00000 −2.41459 −1.00000
1.6 1.00000 −0.490398 1.00000 −1.00000 −0.490398 −1.29403 1.00000 −2.75951 −1.00000
1.7 1.00000 0.0684817 1.00000 −1.00000 0.0684817 −0.682969 1.00000 −2.99531 −1.00000
1.8 1.00000 0.348096 1.00000 −1.00000 0.348096 4.26660 1.00000 −2.87883 −1.00000
1.9 1.00000 1.35747 1.00000 −1.00000 1.35747 0.608019 1.00000 −1.15727 −1.00000
1.10 1.00000 1.35961 1.00000 −1.00000 1.35961 −4.65260 1.00000 −1.15147 −1.00000
1.11 1.00000 2.15389 1.00000 −1.00000 2.15389 1.68510 1.00000 1.63923 −1.00000
1.12 1.00000 2.24233 1.00000 −1.00000 2.24233 4.69629 1.00000 2.02806 −1.00000
1.13 1.00000 2.74529 1.00000 −1.00000 2.74529 −3.29124 1.00000 4.53661 −1.00000
1.14 1.00000 3.03129 1.00000 −1.00000 3.03129 3.50775 1.00000 6.18873 −1.00000
1.15 1.00000 3.27053 1.00000 −1.00000 3.27053 0.659654 1.00000 7.69640 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(1\)

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 8030.2.a.bg ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bg ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{15} - 7 T_{3}^{14} - 6 T_{3}^{13} + 136 T_{3}^{12} - 149 T_{3}^{11} - 876 T_{3}^{10} + 1631 T_{3}^{9} + \cdots - 32 \) Copy content Toggle raw display
\( T_{7}^{15} - 3 T_{7}^{14} - 56 T_{7}^{13} + 165 T_{7}^{12} + 1102 T_{7}^{11} - 3157 T_{7}^{10} + \cdots + 5344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 7 T^{14} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 3 T^{14} + \cdots + 5344 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + T^{14} + \cdots - 26800632 \) Copy content Toggle raw display
$17$ \( T^{15} - 2 T^{14} + \cdots - 17336496 \) Copy content Toggle raw display
$19$ \( T^{15} - 23 T^{14} + \cdots - 493692 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 911862208 \) Copy content Toggle raw display
$29$ \( T^{15} - 23 T^{14} + \cdots + 23359568 \) Copy content Toggle raw display
$31$ \( T^{15} - 9 T^{14} + \cdots + 8057856 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 205410224 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 821378048 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 1783042048 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 87521179008 \) Copy content Toggle raw display
$53$ \( T^{15} + 19 T^{14} + \cdots + 7024896 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 169992772128 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 407021605552 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 2966245522432 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 1576676904 \) Copy content Toggle raw display
$73$ \( (T + 1)^{15} \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 85128621568 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 1197555888 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 82532341518387 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 25510616072192 \) Copy content Toggle raw display
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