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Properties

Label 6045.2.a.bd
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
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: \( x^{14} - 2 x^{13} - 18 x^{12} + 35 x^{11} + 120 x^{10} - 226 x^{9} - 367 x^{8} + 658 x^{7} + 527 x^{6} + \cdots - 16 \) 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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + \beta_{6} q^{7} + (\beta_{12} - \beta_{11} - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + \beta_{6} q^{7} + (\beta_{12} - \beta_{11} - \beta_1 - 1) q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{9} - \beta_1) q^{11} + ( - \beta_{2} - 1) q^{12} - q^{13} + (\beta_{13} - \beta_{11} + \beta_{9} + \cdots - 1) q^{14}+ \cdots + (\beta_{9} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 14 q^{3} + 12 q^{4} + 14 q^{5} + 2 q^{6} + 5 q^{7} - 3 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 14 q^{3} + 12 q^{4} + 14 q^{5} + 2 q^{6} + 5 q^{7} - 3 q^{8} + 14 q^{9} - 2 q^{10} + 4 q^{11} - 12 q^{12} - 14 q^{13} - 7 q^{14} - 14 q^{15} + 8 q^{16} + 7 q^{17} - 2 q^{18} + 2 q^{19} + 12 q^{20} - 5 q^{21} + 21 q^{22} + 31 q^{23} + 3 q^{24} + 14 q^{25} + 2 q^{26} - 14 q^{27} + 8 q^{28} - 5 q^{29} + 2 q^{30} - 14 q^{31} - 8 q^{32} - 4 q^{33} - 11 q^{34} + 5 q^{35} + 12 q^{36} + 15 q^{37} + 2 q^{38} + 14 q^{39} - 3 q^{40} + 7 q^{41} + 7 q^{42} + 30 q^{43} + q^{44} + 14 q^{45} + 2 q^{46} + 23 q^{47} - 8 q^{48} + 11 q^{49} - 2 q^{50} - 7 q^{51} - 12 q^{52} + 14 q^{53} + 2 q^{54} + 4 q^{55} - 23 q^{56} - 2 q^{57} + 22 q^{58} + 10 q^{59} - 12 q^{60} + 2 q^{62} + 5 q^{63} - 3 q^{64} - 14 q^{65} - 21 q^{66} + 28 q^{67} + 23 q^{68} - 31 q^{69} - 7 q^{70} - 35 q^{71} - 3 q^{72} + 27 q^{73} - 12 q^{74} - 14 q^{75} + 26 q^{76} + 24 q^{77} - 2 q^{78} + 9 q^{79} + 8 q^{80} + 14 q^{81} + 45 q^{82} + 49 q^{83} - 8 q^{84} + 7 q^{85} + 4 q^{86} + 5 q^{87} + 49 q^{88} + 5 q^{89} - 2 q^{90} - 5 q^{91} + 107 q^{92} + 14 q^{93} + 26 q^{94} + 2 q^{95} + 8 q^{96} + 9 q^{97} - 6 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 18 x^{12} + 35 x^{11} + 120 x^{10} - 226 x^{9} - 367 x^{8} + 658 x^{7} + 527 x^{6} + \cdots - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 114 \nu^{13} + 1145 \nu^{12} + 1647 \nu^{11} - 20083 \nu^{10} - 9139 \nu^{9} + 127589 \nu^{8} + \cdots - 76062 ) / 15443 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 228 \nu^{13} - 2290 \nu^{12} - 3294 \nu^{11} + 40166 \nu^{10} + 18278 \nu^{9} - 255178 \nu^{8} + \cdots - 2306 ) / 15443 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1241 \nu^{13} - 4472 \nu^{12} - 16710 \nu^{11} + 77062 \nu^{10} + 50584 \nu^{9} - 481179 \nu^{8} + \cdots - 62809 ) / 15443 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1990 \nu^{13} + 5628 \nu^{12} + 33627 \nu^{11} - 98336 \nu^{10} - 200713 \nu^{9} + 623036 \nu^{8} + \cdots + 4413 ) / 15443 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2378 \nu^{13} - 5732 \nu^{12} - 37607 \nu^{11} + 99769 \nu^{10} + 195784 \nu^{9} - 641409 \nu^{8} + \cdots - 140551 ) / 15443 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2469 \nu^{13} - 821 \nu^{12} - 47456 \nu^{11} + 10273 \nu^{10} + 344640 \nu^{9} - 40058 \nu^{8} + \cdots + 25015 ) / 15443 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3346 \nu^{13} - 9494 \nu^{12} - 55656 \nu^{11} + 167345 \nu^{10} + 324862 \nu^{9} - 1088924 \nu^{8} + \cdots - 267654 ) / 15443 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3875 \nu^{13} + 9795 \nu^{12} + 67769 \nu^{11} - 170453 \nu^{10} - 431616 \nu^{9} + 1083446 \nu^{8} + \cdots + 148783 ) / 15443 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3938 \nu^{13} + 7583 \nu^{12} + 67460 \nu^{11} - 127501 \nu^{10} - 413502 \nu^{9} + 776415 \nu^{8} + \cdots + 9214 ) / 15443 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3938 \nu^{13} + 7583 \nu^{12} + 67460 \nu^{11} - 127501 \nu^{10} - 413502 \nu^{9} + 776415 \nu^{8} + \cdots + 24657 ) / 15443 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7691 \nu^{13} + 14798 \nu^{12} + 135092 \nu^{11} - 255871 \nu^{10} - 861076 \nu^{9} + \cdots + 261342 ) / 15443 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{11} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{3} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - 8\beta_{12} + 9\beta_{11} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 28\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{13} - \beta_{12} + 2 \beta_{11} - 11 \beta_{9} - 10 \beta_{8} + 11 \beta_{6} + \beta_{5} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15 \beta_{13} - 57 \beta_{12} + 67 \beta_{11} + 2 \beta_{10} - 13 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} + \cdots + 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 95 \beta_{13} - 19 \beta_{12} + 33 \beta_{11} + 3 \beta_{10} - 94 \beta_{9} - 80 \beta_{8} + \cdots + 469 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 156 \beta_{13} - 396 \beta_{12} + 475 \beta_{11} + 31 \beta_{10} - 128 \beta_{9} - 62 \beta_{8} + \cdots + 565 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 758 \beta_{13} - 227 \beta_{12} + 371 \beta_{11} + 52 \beta_{10} - 739 \beta_{9} - 599 \beta_{8} + \cdots + 3046 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1404 \beta_{13} - 2742 \beta_{12} + 3330 \beta_{11} + 329 \beta_{10} - 1132 \beta_{9} - 662 \beta_{8} + \cdots + 4411 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5838 \beta_{13} - 2248 \beta_{12} + 3548 \beta_{11} + 599 \beta_{10} - 5606 \beta_{9} - 4378 \beta_{8} + \cdots + 20595 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11758 \beta_{13} - 19064 \beta_{12} + 23382 \beta_{11} + 2989 \beta_{10} - 9463 \beta_{9} + \cdots + 34260 \) 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
2.72465
2.25713
1.95655
1.72815
1.45178
0.834130
0.496066
−0.283050
−0.467098
−0.646030
−1.44651
−1.78360
−2.38249
−2.43968
−2.72465 −1.00000 5.42372 1.00000 2.72465 4.13479 −9.32843 1.00000 −2.72465
1.2 −2.25713 −1.00000 3.09463 1.00000 2.25713 −1.78868 −2.47071 1.00000 −2.25713
1.3 −1.95655 −1.00000 1.82809 1.00000 1.95655 −1.02534 0.336358 1.00000 −1.95655
1.4 −1.72815 −1.00000 0.986492 1.00000 1.72815 −2.54226 1.75149 1.00000 −1.72815
1.5 −1.45178 −1.00000 0.107676 1.00000 1.45178 2.57523 2.74725 1.00000 −1.45178
1.6 −0.834130 −1.00000 −1.30423 1.00000 0.834130 4.45890 2.75615 1.00000 −0.834130
1.7 −0.496066 −1.00000 −1.75392 1.00000 0.496066 −1.84578 1.86219 1.00000 −0.496066
1.8 0.283050 −1.00000 −1.91988 1.00000 −0.283050 −1.33866 −1.10952 1.00000 0.283050
1.9 0.467098 −1.00000 −1.78182 1.00000 −0.467098 −0.825252 −1.76648 1.00000 0.467098
1.10 0.646030 −1.00000 −1.58264 1.00000 −0.646030 4.49127 −2.31450 1.00000 0.646030
1.11 1.44651 −1.00000 0.0923886 1.00000 −1.44651 0.649549 −2.75938 1.00000 1.44651
1.12 1.78360 −1.00000 1.18125 1.00000 −1.78360 −3.41120 −1.46033 1.00000 1.78360
1.13 2.38249 −1.00000 3.67624 1.00000 −2.38249 3.51669 3.99363 1.00000 2.38249
1.14 2.43968 −1.00000 3.95202 1.00000 −2.43968 −2.04926 4.76229 1.00000 2.43968
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(1\)

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 6045.2.a.bd ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 18 T_{2}^{12} - 35 T_{2}^{11} + 120 T_{2}^{10} + 226 T_{2}^{9} - 367 T_{2}^{8} + \cdots - 16 \) Copy content Toggle raw display
\( T_{7}^{14} - 5 T_{7}^{13} - 42 T_{7}^{12} + 175 T_{7}^{11} + 835 T_{7}^{10} - 2165 T_{7}^{9} + \cdots + 32372 \) Copy content Toggle raw display
\( T_{11}^{14} - 4 T_{11}^{13} - 60 T_{11}^{12} + 244 T_{11}^{11} + 1181 T_{11}^{10} - 5172 T_{11}^{9} + \cdots - 22480 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots - 16 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( (T - 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 5 T^{13} + \cdots + 32372 \) Copy content Toggle raw display
$11$ \( T^{14} - 4 T^{13} + \cdots - 22480 \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} - 7 T^{13} + \cdots + 2649920 \) Copy content Toggle raw display
$19$ \( T^{14} - 2 T^{13} + \cdots + 77158712 \) Copy content Toggle raw display
$23$ \( T^{14} - 31 T^{13} + \cdots - 13365904 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 2511497071 \) Copy content Toggle raw display
$31$ \( (T + 1)^{14} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 260663488 \) Copy content Toggle raw display
$41$ \( T^{14} - 7 T^{13} + \cdots + 1683743 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 1748375188 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 684556376 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11650062016 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 3274046588 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 201200561536 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 149132602844 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 513641811968 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 31407303040 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 468271478464 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 194051188612 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 6871425695936 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 1428416961184 \) Copy content Toggle raw display
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