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Properties

Label 8030.2.a.be
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: \( x^{15} - 3 x^{14} - 24 x^{13} + 64 x^{12} + 237 x^{11} - 524 x^{10} - 1225 x^{9} + 2074 x^{8} + \cdots - 52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 24 x^{13} + 64 x^{12} + 237 x^{11} - 524 x^{10} - 1225 x^{9} + 2074 x^{8} + \cdots - 52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6135455 \nu^{14} + 34883556 \nu^{13} + 73715727 \nu^{12} - 679696493 \nu^{11} + \cdots - 5186399506 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7950717 \nu^{14} + 11327530 \nu^{13} - 241656913 \nu^{12} - 452867377 \nu^{11} + \cdots + 16257697244 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24837945 \nu^{14} + 12188068 \nu^{13} + 748401619 \nu^{12} - 92125045 \nu^{11} + \cdots - 12450340512 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41839165 \nu^{14} - 97335759 \nu^{13} - 950444152 \nu^{12} + 1659753197 \nu^{11} + \cdots + 17646606370 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56714341 \nu^{14} - 129019881 \nu^{13} - 1338802868 \nu^{12} + 2167586419 \nu^{11} + \cdots + 8151943214 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 61936144 \nu^{14} - 225208367 \nu^{13} - 1288366493 \nu^{12} + 4591312948 \nu^{11} + \cdots - 18624471416 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65242380 \nu^{14} - 179003743 \nu^{13} - 1562396749 \nu^{12} + 3633751796 \nu^{11} + \cdots - 11100190102 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 71064649 \nu^{14} - 368116072 \nu^{13} - 1126751873 \nu^{12} + 7646764207 \nu^{11} + \cdots - 7735840732 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76022316 \nu^{14} + 248764393 \nu^{13} + 1603739133 \nu^{12} - 4818142064 \nu^{11} + \cdots + 8066141436 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 91585352 \nu^{14} - 318996982 \nu^{13} - 1872097191 \nu^{12} + 6110728165 \nu^{11} + \cdots - 6182535442 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 112912991 \nu^{14} + 678967023 \nu^{13} + 1226139356 \nu^{12} - 13031879625 \nu^{11} + \cdots - 5408793724 ) / 2177153354 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 129594252 \nu^{14} + 450924867 \nu^{13} + 2633495927 \nu^{12} - 8681786756 \nu^{11} + \cdots + 10284133320 ) / 2177153354 \) 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
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{4} - \beta_{3} + 2\beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 2\beta_{11} + \beta_{10} + 2\beta_{8} + \beta_{7} + 3\beta_{4} - \beta_{3} + 12\beta_{2} + 13\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 14 \beta_{11} + \beta_{10} + 3 \beta_{9} + 13 \beta_{8} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{14} - 3 \beta_{13} - 6 \beta_{12} + 33 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} + 37 \beta_{8} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 52 \beta_{14} - 21 \beta_{13} - 38 \beta_{12} + 162 \beta_{11} + 19 \beta_{10} + 55 \beta_{9} + \cdots + 523 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 270 \beta_{14} - 69 \beta_{13} - 122 \beta_{12} + 443 \beta_{11} + 124 \beta_{10} + 141 \beta_{9} + \cdots + 2293 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 881 \beta_{14} - 323 \beta_{13} - 544 \beta_{12} + 1836 \beta_{11} + 266 \beta_{10} + 771 \beta_{9} + \cdots + 6090 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3716 \beta_{14} - 1102 \beta_{13} - 1804 \beta_{12} + 5543 \beta_{11} + 1285 \beta_{10} + 2269 \beta_{9} + \cdots + 23669 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12635 \beta_{14} - 4418 \beta_{13} - 7062 \beta_{12} + 20997 \beta_{11} + 3346 \beta_{10} + 9943 \beta_{9} + \cdots + 70120 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 48467 \beta_{14} - 15279 \beta_{13} - 23800 \beta_{12} + 67189 \beta_{11} + 13729 \beta_{10} + \cdots + 255986 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 167007 \beta_{14} - 56988 \beta_{13} - 87682 \beta_{12} + 242649 \beta_{11} + 40201 \beta_{10} + \cdots + 807039 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 610525 \beta_{14} - 197701 \beta_{13} - 297946 \beta_{12} + 801830 \beta_{11} + 151022 \beta_{10} + \cdots + 2852415 \) 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
−2.53445
−2.37460
−1.85209
−1.72375
−1.51118
−0.907771
−0.424375
−0.0638672
0.823119
1.24368
1.38748
1.90080
2.77411
2.83620
3.42670
−1.00000 −2.53445 1.00000 1.00000 2.53445 −3.43636 −1.00000 3.42346 −1.00000
1.2 −1.00000 −2.37460 1.00000 1.00000 2.37460 −1.30659 −1.00000 2.63872 −1.00000
1.3 −1.00000 −1.85209 1.00000 1.00000 1.85209 1.81131 −1.00000 0.430244 −1.00000
1.4 −1.00000 −1.72375 1.00000 1.00000 1.72375 4.05342 −1.00000 −0.0286805 −1.00000
1.5 −1.00000 −1.51118 1.00000 1.00000 1.51118 −0.990203 −1.00000 −0.716341 −1.00000
1.6 −1.00000 −0.907771 1.00000 1.00000 0.907771 −0.750305 −1.00000 −2.17595 −1.00000
1.7 −1.00000 −0.424375 1.00000 1.00000 0.424375 −2.28477 −1.00000 −2.81991 −1.00000
1.8 −1.00000 −0.0638672 1.00000 1.00000 0.0638672 2.62362 −1.00000 −2.99592 −1.00000
1.9 −1.00000 0.823119 1.00000 1.00000 −0.823119 4.98726 −1.00000 −2.32248 −1.00000
1.10 −1.00000 1.24368 1.00000 1.00000 −1.24368 2.03579 −1.00000 −1.45326 −1.00000
1.11 −1.00000 1.38748 1.00000 1.00000 −1.38748 −5.08575 −1.00000 −1.07490 −1.00000
1.12 −1.00000 1.90080 1.00000 1.00000 −1.90080 −1.33242 −1.00000 0.613049 −1.00000
1.13 −1.00000 2.77411 1.00000 1.00000 −2.77411 0.220732 −1.00000 4.69567 −1.00000
1.14 −1.00000 2.83620 1.00000 1.00000 −2.83620 3.77087 −1.00000 5.04400 −1.00000
1.15 −1.00000 3.42670 1.00000 1.00000 −3.42670 2.68339 −1.00000 8.74228 −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.be ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.be ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{15} - 3 T_{3}^{14} - 24 T_{3}^{13} + 64 T_{3}^{12} + 237 T_{3}^{11} - 524 T_{3}^{10} - 1225 T_{3}^{9} + \cdots - 52 \) Copy content Toggle raw display
\( T_{7}^{15} - 7 T_{7}^{14} - 38 T_{7}^{13} + 345 T_{7}^{12} + 254 T_{7}^{11} - 5551 T_{7}^{10} + \cdots + 22560 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 3 T^{14} + \cdots - 52 \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 7 T^{14} + \cdots + 22560 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 13 T^{14} + \cdots + 7232400 \) Copy content Toggle raw display
$17$ \( T^{15} - 20 T^{14} + \cdots - 6524208 \) Copy content Toggle raw display
$19$ \( T^{15} - 3 T^{14} + \cdots + 97877884 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 262268032 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 19606528688 \) Copy content Toggle raw display
$31$ \( T^{15} + 3 T^{14} + \cdots - 438272 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 1714893968 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 4064576000 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 2589650688 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 3977039744 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 4714001664 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 2350970387328 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 10186331164800 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 26848829440 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 2169764212960 \) Copy content Toggle raw display
$73$ \( (T + 1)^{15} \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 789446144 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 54827655968 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 1829043563225 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 26355099797504 \) Copy content Toggle raw display
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