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Properties

Label 6045.2.a.ba
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $1$
Dimension $13$
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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - x^{12} - 17 x^{11} + 14 x^{10} + 106 x^{9} - 68 x^{8} - 299 x^{7} + 141 x^{6} + 380 x^{5} + \cdots - 4 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - x^{12} - 17 x^{11} + 14 x^{10} + 106 x^{9} - 68 x^{8} - 299 x^{7} + 141 x^{6} + 380 x^{5} + \cdots - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 236 \nu^{12} - 190 \nu^{11} - 3981 \nu^{10} + 2443 \nu^{9} + 24466 \nu^{8} - 10202 \nu^{7} + \cdots + 2681 ) / 223 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 508 \nu^{12} - 443 \nu^{11} - 8694 \nu^{10} + 5939 \nu^{9} + 54690 \nu^{8} - 26613 \nu^{7} + \cdots + 7846 ) / 223 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 567 \nu^{12} - 379 \nu^{11} - 9745 \nu^{10} + 4710 \nu^{9} + 61364 \nu^{8} - 18348 \nu^{7} + \cdots + 8572 ) / 223 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 632 \nu^{12} - 437 \nu^{11} - 10918 \nu^{10} + 5552 \nu^{9} + 69295 \nu^{8} - 22528 \nu^{7} + \cdots + 10604 ) / 223 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 726 \nu^{12} + 490 \nu^{11} + 12532 \nu^{10} - 6183 \nu^{9} - 79399 \nu^{8} + 24902 \nu^{7} + \cdots - 11961 ) / 223 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 962 \nu^{12} - 680 \nu^{11} - 16513 \nu^{10} + 8626 \nu^{9} + 103865 \nu^{8} - 35104 \nu^{7} + \cdots + 15088 ) / 223 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1059 \nu^{12} - 794 \nu^{11} - 18188 \nu^{10} + 10181 \nu^{9} + 114709 \nu^{8} - 42028 \nu^{7} + \cdots + 17098 ) / 223 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1170 \nu^{12} + 821 \nu^{11} + 20222 \nu^{10} - 10473 \nu^{9} - 128486 \nu^{8} + 42905 \nu^{7} + \cdots - 20074 ) / 223 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1576 \nu^{12} - 1197 \nu^{11} - 27065 \nu^{10} + 15547 \nu^{9} + 170504 \nu^{8} - 66012 \nu^{7} + \cdots + 24227 ) / 223 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2302 \nu^{12} - 1687 \nu^{11} - 39597 \nu^{10} + 21730 \nu^{9} + 249903 \nu^{8} - 90914 \nu^{7} + \cdots + 34850 ) / 223 \) 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
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{11} - \beta_{7} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} + 7 \beta_{8} + 8 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{12} + 10 \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - 10 \beta_{7} - \beta_{6} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{12} + 13 \beta_{11} - 23 \beta_{10} - 12 \beta_{9} + 43 \beta_{8} + 55 \beta_{7} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 82 \beta_{12} + 79 \beta_{11} - 17 \beta_{10} - 16 \beta_{9} - 25 \beta_{8} - 79 \beta_{7} + \cdots + 508 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 125 \beta_{12} + 122 \beta_{11} - 199 \beta_{10} - 111 \beta_{9} + 258 \beta_{8} + 358 \beta_{7} + \cdots + 380 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 631 \beta_{12} + 582 \beta_{11} - 188 \beta_{10} - 174 \beta_{9} - 222 \beta_{8} - 574 \beta_{7} + \cdots + 3124 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1067 \beta_{12} + 1008 \beta_{11} - 1558 \beta_{10} - 928 \beta_{9} + 1539 \beta_{8} + 2266 \beta_{7} + \cdots + 2560 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4706 \beta_{12} + 4174 \beta_{11} - 1748 \beta_{10} - 1611 \beta_{9} - 1736 \beta_{8} - 4007 \beta_{7} + \cdots + 19556 \) 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
2.62963
2.41287
1.87788
1.24594
1.08625
0.262347
0.250065
−0.433719
−0.751578
−1.37726
−1.61484
−2.07884
−2.50876
−2.62963 −1.00000 4.91498 1.00000 2.62963 −2.21047 −7.66533 1.00000 −2.62963
1.2 −2.41287 −1.00000 3.82195 1.00000 2.41287 −3.00058 −4.39612 1.00000 −2.41287
1.3 −1.87788 −1.00000 1.52644 1.00000 1.87788 1.19924 0.889290 1.00000 −1.87788
1.4 −1.24594 −1.00000 −0.447637 1.00000 1.24594 2.29536 3.04961 1.00000 −1.24594
1.5 −1.08625 −1.00000 −0.820055 1.00000 1.08625 −4.17568 3.06329 1.00000 −1.08625
1.6 −0.262347 −1.00000 −1.93117 1.00000 0.262347 0.235153 1.03133 1.00000 −0.262347
1.7 −0.250065 −1.00000 −1.93747 1.00000 0.250065 2.81469 0.984622 1.00000 −0.250065
1.8 0.433719 −1.00000 −1.81189 1.00000 −0.433719 −4.75584 −1.65329 1.00000 0.433719
1.9 0.751578 −1.00000 −1.43513 1.00000 −0.751578 −0.176283 −2.58177 1.00000 0.751578
1.10 1.37726 −1.00000 −0.103163 1.00000 −1.37726 −0.130011 −2.89660 1.00000 1.37726
1.11 1.61484 −1.00000 0.607701 1.00000 −1.61484 2.50670 −2.24834 1.00000 1.61484
1.12 2.07884 −1.00000 2.32156 1.00000 −2.07884 −3.88445 0.668471 1.00000 2.07884
1.13 2.50876 −1.00000 4.29389 1.00000 −2.50876 −1.71783 5.75483 1.00000 2.50876
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.ba ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.ba ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{13} + T_{2}^{12} - 17 T_{2}^{11} - 14 T_{2}^{10} + 106 T_{2}^{9} + 68 T_{2}^{8} - 299 T_{2}^{7} + \cdots + 4 \) Copy content Toggle raw display
\( T_{7}^{13} + 11 T_{7}^{12} + 14 T_{7}^{11} - 213 T_{7}^{10} - 601 T_{7}^{9} + 1359 T_{7}^{8} + 5308 T_{7}^{7} + \cdots - 92 \) Copy content Toggle raw display
\( T_{11}^{13} - 4 T_{11}^{12} - 53 T_{11}^{11} + 214 T_{11}^{10} + 958 T_{11}^{9} - 4022 T_{11}^{8} + \cdots - 18880 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + T^{12} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( (T - 1)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + 11 T^{12} + \cdots - 92 \) Copy content Toggle raw display
$11$ \( T^{13} - 4 T^{12} + \cdots - 18880 \) Copy content Toggle raw display
$13$ \( (T + 1)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} - 3 T^{12} + \cdots + 7863584 \) Copy content Toggle raw display
$19$ \( T^{13} + 2 T^{12} + \cdots + 582992 \) Copy content Toggle raw display
$23$ \( T^{13} + 9 T^{12} + \cdots + 372352 \) Copy content Toggle raw display
$29$ \( T^{13} - 7 T^{12} + \cdots - 6014750 \) Copy content Toggle raw display
$31$ \( (T - 1)^{13} \) Copy content Toggle raw display
$37$ \( T^{13} + 7 T^{12} + \cdots + 1548128 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 276718490 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 1223247620 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 4313432416 \) Copy content Toggle raw display
$53$ \( T^{13} + 18 T^{12} + \cdots + 2087656 \) Copy content Toggle raw display
$59$ \( T^{13} + 26 T^{12} + \cdots - 2320480 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 6416451944 \) Copy content Toggle raw display
$67$ \( T^{13} + 30 T^{12} + \cdots - 6404744 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 783680512 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 32861453032 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 119476345184 \) Copy content Toggle raw display
$83$ \( T^{13} + 35 T^{12} + \cdots + 2158012 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 132809120 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 210586823630 \) Copy content Toggle raw display
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