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Properties

Label 8030.2.a.bd
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 15 x^{12} + 143 x^{11} - 13 x^{10} - 1176 x^{9} + 1018 x^{8} + 4076 x^{7} + \cdots - 54 \) 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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 15 x^{12} + 143 x^{11} - 13 x^{10} - 1176 x^{9} + 1018 x^{8} + 4076 x^{7} + \cdots - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2914868715 \nu^{13} - 1481677836 \nu^{12} - 92329276273 \nu^{11} + 36500239287 \nu^{10} + \cdots + 1286475808555 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3013739076 \nu^{13} + 25971820874 \nu^{12} + 24044059112 \nu^{11} - 634643609480 \nu^{10} + \cdots - 1183349774126 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5956196641 \nu^{13} - 30375734869 \nu^{12} - 122300275334 \nu^{11} + 768938904475 \nu^{10} + \cdots - 2680171248537 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6000025764 \nu^{13} - 45403692907 \nu^{12} - 71158156672 \nu^{11} + 1109907562806 \nu^{10} + \cdots + 4161812783791 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6388173845 \nu^{13} - 29903795688 \nu^{12} - 123641888828 \nu^{11} + 713909753046 \nu^{10} + \cdots - 961873041978 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8203270317 \nu^{13} - 53522196132 \nu^{12} - 116123739560 \nu^{11} + 1309912732812 \nu^{10} + \cdots + 944421094086 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13083447036 \nu^{13} - 60182654009 \nu^{12} - 270235819099 \nu^{11} + 1478723382956 \nu^{10} + \cdots - 1416109772417 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14031049713 \nu^{13} - 93711317903 \nu^{12} - 204783187150 \nu^{11} + 2305886641289 \nu^{10} + \cdots + 402909869499 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28180525363 \nu^{13} - 154398148196 \nu^{12} - 490244344188 \nu^{11} + 3740214959036 \nu^{10} + \cdots + 6525355140337 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35738278950 \nu^{13} + 192127692799 \nu^{12} + 631908968296 \nu^{11} + \cdots - 4851763055327 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 36555709416 \nu^{13} + 213327294789 \nu^{12} + 597085618488 \nu^{11} + \cdots - 7547519788454 ) / 482819744623 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41694475591 \nu^{13} + 222503427668 \nu^{12} + 754209243630 \nu^{11} + \cdots - 4102870785282 ) / 482819744623 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{11} + \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + 2\beta_{3} - \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{13} - \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{6} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + 4 \beta_{9} + 18 \beta_{8} + 12 \beta_{7} - 13 \beta_{6} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 98 \beta_{13} - 17 \beta_{12} - 62 \beta_{11} + 38 \beta_{10} - 12 \beta_{9} + 35 \beta_{8} + \beta_{7} + \cdots + 278 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 45 \beta_{12} + 79 \beta_{11} + 19 \beta_{10} + 77 \beta_{9} + 243 \beta_{8} + 125 \beta_{7} + \cdots + 167 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 988 \beta_{13} - 241 \beta_{12} - 481 \beta_{11} + 533 \beta_{10} - 115 \beta_{9} + 474 \beta_{8} + \cdots + 2668 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4 \beta_{13} - 731 \beta_{12} + 1167 \beta_{11} + 271 \beta_{10} + 1073 \beta_{9} + 2976 \beta_{8} + \cdots + 1730 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10225 \beta_{13} - 3209 \beta_{12} - 3721 \beta_{11} + 6640 \beta_{10} - 1026 \beta_{9} + 5884 \beta_{8} + \cdots + 26703 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 43 \beta_{13} - 10370 \beta_{12} + 15404 \beta_{11} + 3503 \beta_{10} + 13282 \beta_{9} + \cdots + 17707 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 107883 \beta_{13} - 41167 \beta_{12} - 28344 \beta_{11} + 78038 \beta_{10} - 8955 \beta_{9} + \cdots + 274930 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 479 \beta_{13} - 136719 \beta_{12} + 191734 \beta_{11} + 43206 \beta_{10} + 155417 \beta_{9} + \cdots + 183262 \) 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
−3.28632
−2.47616
−1.98092
−1.51960
−0.856641
0.0877386
0.331408
0.454872
1.13475
2.25086
2.65302
2.86338
2.99130
3.35230
1.00000 −3.28632 1.00000 −1.00000 −3.28632 −0.411624 1.00000 7.79988 −1.00000
1.2 1.00000 −2.47616 1.00000 −1.00000 −2.47616 −1.89245 1.00000 3.13135 −1.00000
1.3 1.00000 −1.98092 1.00000 −1.00000 −1.98092 1.99444 1.00000 0.924028 −1.00000
1.4 1.00000 −1.51960 1.00000 −1.00000 −1.51960 −1.36556 1.00000 −0.690822 −1.00000
1.5 1.00000 −0.856641 1.00000 −1.00000 −0.856641 1.42002 1.00000 −2.26617 −1.00000
1.6 1.00000 0.0877386 1.00000 −1.00000 0.0877386 −3.09806 1.00000 −2.99230 −1.00000
1.7 1.00000 0.331408 1.00000 −1.00000 0.331408 4.90237 1.00000 −2.89017 −1.00000
1.8 1.00000 0.454872 1.00000 −1.00000 0.454872 −4.57770 1.00000 −2.79309 −1.00000
1.9 1.00000 1.13475 1.00000 −1.00000 1.13475 −2.24922 1.00000 −1.71235 −1.00000
1.10 1.00000 2.25086 1.00000 −1.00000 2.25086 4.10088 1.00000 2.06636 −1.00000
1.11 1.00000 2.65302 1.00000 −1.00000 2.65302 −2.26267 1.00000 4.03849 −1.00000
1.12 1.00000 2.86338 1.00000 −1.00000 2.86338 2.76153 1.00000 5.19896 −1.00000
1.13 1.00000 2.99130 1.00000 −1.00000 2.99130 4.10532 1.00000 5.94789 −1.00000
1.14 1.00000 3.35230 1.00000 −1.00000 3.35230 0.572732 1.00000 8.23794 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.bd ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bd ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{14} - 6 T_{3}^{13} - 15 T_{3}^{12} + 143 T_{3}^{11} - 13 T_{3}^{10} - 1176 T_{3}^{9} + 1018 T_{3}^{8} + \cdots - 54 \) Copy content Toggle raw display
\( T_{7}^{14} - 4 T_{7}^{13} - 51 T_{7}^{12} + 182 T_{7}^{11} + 997 T_{7}^{10} - 2892 T_{7}^{9} + \cdots - 28384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 6 T^{13} + \cdots - 54 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots - 28384 \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots - 128 \) Copy content Toggle raw display
$17$ \( T^{14} - 20 T^{13} + \cdots - 117936 \) Copy content Toggle raw display
$19$ \( T^{14} - 94 T^{12} + \cdots - 91601 \) Copy content Toggle raw display
$23$ \( T^{14} + 4 T^{13} + \cdots + 13408512 \) Copy content Toggle raw display
$29$ \( T^{14} - 7 T^{13} + \cdots - 726576 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 526235392 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 27183156112 \) Copy content Toggle raw display
$41$ \( T^{14} + 14 T^{13} + \cdots + 9145152 \) Copy content Toggle raw display
$43$ \( T^{14} - 12 T^{13} + \cdots - 6861156 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1744382208 \) Copy content Toggle raw display
$53$ \( T^{14} - 13 T^{13} + \cdots - 32299488 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 2843692477632 \) Copy content Toggle raw display
$61$ \( T^{14} - 25 T^{13} + \cdots - 53778568 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 2022266678112 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 924301501488 \) Copy content Toggle raw display
$73$ \( (T - 1)^{14} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 246542189184 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 39490688832 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 540987036507 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 1487589376 \) Copy content Toggle raw display
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