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Properties

Label 6069.2.a.bh
Level $6069$
Weight $2$
Character orbit 6069.a
Self dual yes
Analytic conductor $48.461$
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] = [6069,2,Mod(1,6069)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, 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("6069.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.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: \(48.4612089867\)
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: \( x^{15} - 24 x^{13} - 4 x^{12} + 225 x^{11} + 69 x^{10} - 1037 x^{9} - 432 x^{8} + 2439 x^{7} + \cdots - 171 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 24 x^{13} - 4 x^{12} + 225 x^{11} + 69 x^{10} - 1037 x^{9} - 432 x^{8} + 2439 x^{7} + \cdots - 171 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5624 \nu^{14} - 8155 \nu^{13} - 134610 \nu^{12} + 178998 \nu^{11} + 1262786 \nu^{10} + \cdots - 1878321 ) / 84023 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23674 \nu^{14} + 30960 \nu^{13} - 563229 \nu^{12} - 794128 \nu^{11} + 5042004 \nu^{10} + \cdots + 10192860 ) / 252069 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32953 \nu^{14} - 16872 \nu^{13} - 766407 \nu^{12} + 272018 \nu^{11} + 6877431 \nu^{10} + \cdots - 2477127 ) / 252069 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58313 \nu^{14} + 57111 \nu^{13} + 1332162 \nu^{12} - 1073188 \nu^{11} - 11801931 \nu^{10} + \cdots + 6191691 ) / 252069 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 88813 \nu^{14} + 108090 \nu^{13} + 2055723 \nu^{12} - 2145281 \nu^{11} - 18697824 \nu^{10} + \cdots + 25353876 ) / 252069 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36030 \nu^{14} + 25263 \nu^{13} + 833511 \nu^{12} - 428367 \nu^{11} - 7509193 \nu^{10} + \cdots + 5062341 ) / 84023 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55577 \nu^{14} - 25893 \nu^{13} - 1309823 \nu^{12} + 395676 \nu^{11} + 12021020 \nu^{10} + \cdots - 8811421 ) / 84023 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 182812 \nu^{14} - 70086 \nu^{13} - 4292046 \nu^{12} + 922052 \nu^{11} + 39152133 \nu^{10} + \cdots - 24788841 ) / 252069 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 184477 \nu^{14} + 147126 \nu^{13} + 4262247 \nu^{12} - 2649977 \nu^{11} - 38288856 \nu^{10} + \cdots + 25250397 ) / 252069 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 270143 \nu^{14} + 266295 \nu^{13} + 6227139 \nu^{12} - 5080999 \nu^{11} - 55885560 \nu^{10} + \cdots + 43194447 ) / 252069 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 373462 \nu^{14} - 300729 \nu^{13} - 8741754 \nu^{12} + 5565224 \nu^{11} + 79967421 \nu^{10} + \cdots - 82367079 ) / 252069 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 536669 \nu^{14} + 458337 \nu^{13} + 12492216 \nu^{12} - 8518717 \nu^{11} - 113556249 \nu^{10} + \cdots + 111258948 ) / 252069 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{5} - \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{14} + 8 \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - 8 \beta_{7} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} + \beta_{13} - 12 \beta_{10} + 12 \beta_{9} - \beta_{7} + 16 \beta_{5} - \beta_{4} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 57 \beta_{14} + 57 \beta_{13} - 12 \beta_{12} + 21 \beta_{11} - 15 \beta_{10} + 15 \beta_{9} - 10 \beta_{8} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 18 \beta_{14} + 16 \beta_{13} - 3 \beta_{12} - \beta_{11} - 110 \beta_{10} + 112 \beta_{9} - 18 \beta_{7} + \cdots + 536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 399 \beta_{14} + 396 \beta_{13} - 109 \beta_{12} + 161 \beta_{11} - 163 \beta_{10} + 168 \beta_{9} + \cdots + 590 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 218 \beta_{14} + 177 \beta_{13} - 59 \beta_{12} - 18 \beta_{11} - 918 \beta_{10} + 962 \beta_{9} + \cdots + 3443 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2796 \beta_{14} + 2726 \beta_{13} - 903 \beta_{12} + 1089 \beta_{11} - 1553 \beta_{10} + 1664 \beta_{9} + \cdots + 4630 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2228 \beta_{14} + 1678 \beta_{13} - 761 \beta_{12} - 227 \beta_{11} - 7345 \beta_{10} + 7965 \beta_{9} + \cdots + 22898 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19743 \beta_{14} + 18706 \beta_{13} - 7204 \beta_{12} + 6845 \beta_{11} - 13794 \beta_{10} + \cdots + 36055 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20773 \beta_{14} + 14651 \beta_{13} - 8177 \beta_{12} - 2503 \beta_{11} - 57581 \beta_{10} + \cdots + 156881 \) 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
2.80859
2.58176
2.28903
1.81029
1.08761
0.782309
0.730306
−0.620932
−0.725939
−0.878810
−0.954523
−1.99540
−2.07177
−2.36416
−2.47836
−2.80859 1.00000 5.88816 −0.821386 −2.80859 −1.00000 −10.9202 1.00000 2.30693
1.2 −2.58176 1.00000 4.66546 1.60063 −2.58176 −1.00000 −6.88156 1.00000 −4.13243
1.3 −2.28903 1.00000 3.23967 4.19587 −2.28903 −1.00000 −2.83764 1.00000 −9.60449
1.4 −1.81029 1.00000 1.27715 −3.48145 −1.81029 −1.00000 1.30857 1.00000 6.30243
1.5 −1.08761 1.00000 −0.817108 2.99193 −1.08761 −1.00000 3.06391 1.00000 −3.25405
1.6 −0.782309 1.00000 −1.38799 0.102983 −0.782309 −1.00000 2.65046 1.00000 −0.0805643
1.7 −0.730306 1.00000 −1.46665 0.415974 −0.730306 −1.00000 2.53172 1.00000 −0.303789
1.8 0.620932 1.00000 −1.61444 1.34353 0.620932 −1.00000 −2.24432 1.00000 0.834241
1.9 0.725939 1.00000 −1.47301 −3.81974 0.725939 −1.00000 −2.52119 1.00000 −2.77289
1.10 0.878810 1.00000 −1.22769 4.09522 0.878810 −1.00000 −2.83653 1.00000 3.59892
1.11 0.954523 1.00000 −1.08889 −0.787002 0.954523 −1.00000 −2.94841 1.00000 −0.751212
1.12 1.99540 1.00000 1.98163 −2.33326 1.99540 −1.00000 −0.0366648 1.00000 −4.65579
1.13 2.07177 1.00000 2.29222 3.05674 2.07177 −1.00000 0.605406 1.00000 6.33284
1.14 2.36416 1.00000 3.58923 −1.16247 2.36416 −1.00000 3.75719 1.00000 −2.74825
1.15 2.47836 1.00000 4.14227 3.60242 2.47836 −1.00000 5.30932 1.00000 8.92810
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(17\) \(1\)

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 6069.2.a.bh yes 15
17.b even 2 1 6069.2.a.bg ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6069.2.a.bg ✓ 15 17.b even 2 1
6069.2.a.bh yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):

\( T_{2}^{15} - 24 T_{2}^{13} + 4 T_{2}^{12} + 225 T_{2}^{11} - 69 T_{2}^{10} - 1037 T_{2}^{9} + 432 T_{2}^{8} + \cdots + 171 \) Copy content Toggle raw display
\( T_{5}^{15} - 9 T_{5}^{14} - 12 T_{5}^{13} + 305 T_{5}^{12} - 393 T_{5}^{11} - 3411 T_{5}^{10} + \cdots - 1216 \) Copy content Toggle raw display
\( T_{11}^{15} - 30 T_{11}^{14} + 333 T_{11}^{13} - 1281 T_{11}^{12} - 4908 T_{11}^{11} + 63309 T_{11}^{10} + \cdots + 317609 \) Copy content Toggle raw display
\( T_{23}^{15} - 12 T_{23}^{14} - 84 T_{23}^{13} + 1478 T_{23}^{12} + 603 T_{23}^{11} - 67980 T_{23}^{10} + \cdots - 1055048543 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 24 T^{13} + \cdots + 171 \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} - 9 T^{14} + \cdots - 1216 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} - 30 T^{14} + \cdots + 317609 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 466940224 \) Copy content Toggle raw display
$17$ \( T^{15} \) Copy content Toggle raw display
$19$ \( T^{15} - 3 T^{14} + \cdots + 39068864 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 1055048543 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 13574656077 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 656191296 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 763702470793 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 1572840262592 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 254339510563 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 8219166528 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 238296104817 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 384631453376 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 1237662912 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 2496190131 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 3015769381759 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 98501494464 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 9569745876609 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 47193598272 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 1189525678272 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 441315136 \) Copy content Toggle raw display
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