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Properties

Label 6045.2.a.bc
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: \( x^{14} - 3 x^{13} - 17 x^{12} + 52 x^{11} + 104 x^{10} - 328 x^{9} - 275 x^{8} + 917 x^{7} + 296 x^{6} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 17 x^{12} + 52 x^{11} + 104 x^{10} - 328 x^{9} - 275 x^{8} + 917 x^{7} + 296 x^{6} + \cdots - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91 \nu^{13} + 131 \nu^{12} - 2181 \nu^{11} - 2588 \nu^{10} + 19444 \nu^{9} + 18826 \nu^{8} + \cdots - 3088 ) / 1558 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 169 \nu^{13} + 647 \nu^{12} + 2715 \nu^{11} - 11664 \nu^{10} - 14966 \nu^{9} + 77436 \nu^{8} + \cdots + 3064 ) / 1558 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 131 \nu^{13} + 1027 \nu^{12} + 777 \nu^{11} - 17744 \nu^{10} + 9658 \nu^{9} + 110876 \nu^{8} + \cdots + 2836 ) / 1558 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 373 \nu^{13} - 165 \nu^{12} - 7707 \nu^{11} + 1000 \nu^{10} + 59582 \nu^{9} + 10326 \nu^{8} + \cdots + 10062 ) / 1558 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 503 \nu^{13} + 423 \nu^{12} + 10155 \nu^{11} - 5538 \nu^{10} - 77566 \nu^{9} + 19758 \nu^{8} + \cdots - 5428 ) / 1558 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 571 \nu^{13} + 1301 \nu^{12} + 10261 \nu^{11} - 21718 \nu^{10} - 67510 \nu^{9} + 129166 \nu^{8} + \cdots + 64 ) / 1558 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 571 \nu^{13} - 1301 \nu^{12} - 10261 \nu^{11} + 21718 \nu^{10} + 67510 \nu^{9} - 129166 \nu^{8} + \cdots - 64 ) / 1558 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1065 \nu^{13} + 1035 \nu^{12} + 21433 \nu^{11} - 14346 \nu^{10} - 161712 \nu^{9} + 59018 \nu^{8} + \cdots - 1646 ) / 1558 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1123 \nu^{13} + 2013 \nu^{12} + 21111 \nu^{11} - 32454 \nu^{10} - 147636 \nu^{9} + 183130 \nu^{8} + \cdots + 10920 ) / 1558 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1123 \nu^{13} - 2013 \nu^{12} - 21111 \nu^{11} + 32454 \nu^{10} + 147636 \nu^{9} - 183130 \nu^{8} + \cdots - 3130 ) / 1558 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 728 \nu^{13} + 1289 \nu^{12} + 13553 \nu^{11} - 20583 \nu^{10} - 93232 \nu^{9} + 114252 \nu^{8} + \cdots + 2113 ) / 779 \) 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
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + 7\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + 8 \beta_{9} + 8 \beta_{8} - \beta_{5} - \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 11 \beta_{12} + 10 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{13} + 15 \beta_{12} + 13 \beta_{11} - 11 \beta_{10} + 53 \beta_{9} + 55 \beta_{8} + \beta_{7} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{13} + 96 \beta_{12} + 81 \beta_{11} - 14 \beta_{10} + 16 \beta_{9} + 14 \beta_{8} + 15 \beta_{7} + \cdots + 625 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 111 \beta_{13} + 158 \beta_{12} + 126 \beta_{11} - 94 \beta_{10} + 338 \beta_{9} + 366 \beta_{8} + \cdots + 459 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 172 \beta_{13} + 774 \beta_{12} + 619 \beta_{11} - 140 \beta_{10} + 174 \beta_{9} + 145 \beta_{8} + \cdots + 4166 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 925 \beta_{13} + 1439 \beta_{12} + 1095 \beta_{11} - 738 \beta_{10} + 2157 \beta_{9} + 2430 \beta_{8} + \cdots + 4180 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1572 \beta_{13} + 6012 \beta_{12} + 4637 \beta_{11} - 1225 \beta_{10} + 1615 \beta_{9} + 1337 \beta_{8} + \cdots + 28378 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7290 \beta_{13} + 12152 \beta_{12} + 9012 \beta_{11} - 5571 \beta_{10} + 13952 \beta_{9} + 16247 \beta_{8} + \cdots + 35481 \) 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
2.72778
2.71017
2.29320
1.56976
1.55900
0.851461
0.345538
−0.105610
−0.368324
−0.572635
−1.53015
−1.65115
−2.33050
−2.49856
−2.72778 −1.00000 5.44079 −1.00000 2.72778 −2.76621 −9.38572 1.00000 2.72778
1.2 −2.71017 −1.00000 5.34504 −1.00000 2.71017 4.40630 −9.06564 1.00000 2.71017
1.3 −2.29320 −1.00000 3.25878 −1.00000 2.29320 −0.293290 −2.88663 1.00000 2.29320
1.4 −1.56976 −1.00000 0.464155 −1.00000 1.56976 3.88992 2.41091 1.00000 1.56976
1.5 −1.55900 −1.00000 0.430481 −1.00000 1.55900 −4.23687 2.44688 1.00000 1.55900
1.6 −0.851461 −1.00000 −1.27501 −1.00000 0.851461 −3.99345 2.78855 1.00000 0.851461
1.7 −0.345538 −1.00000 −1.88060 −1.00000 0.345538 2.73141 1.34090 1.00000 0.345538
1.8 0.105610 −1.00000 −1.98885 −1.00000 −0.105610 −1.48171 −0.421262 1.00000 −0.105610
1.9 0.368324 −1.00000 −1.86434 −1.00000 −0.368324 −1.89187 −1.42333 1.00000 −0.368324
1.10 0.572635 −1.00000 −1.67209 −1.00000 −0.572635 1.91976 −2.10277 1.00000 −0.572635
1.11 1.53015 −1.00000 0.341349 −1.00000 −1.53015 −1.26884 −2.53798 1.00000 −1.53015
1.12 1.65115 −1.00000 0.726296 −1.00000 −1.65115 0.847545 −2.10308 1.00000 −1.65115
1.13 2.33050 −1.00000 3.43122 −1.00000 −2.33050 4.62579 3.33547 1.00000 −2.33050
1.14 2.49856 −1.00000 4.24278 −1.00000 −2.49856 −3.48849 5.60370 1.00000 −2.49856
\(n\): 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.bc ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.bc ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{14} + 3 T_{2}^{13} - 17 T_{2}^{12} - 52 T_{2}^{11} + 104 T_{2}^{10} + 328 T_{2}^{9} - 275 T_{2}^{8} + \cdots - 4 \) Copy content Toggle raw display
\( T_{7}^{14} + T_{7}^{13} - 64 T_{7}^{12} - 83 T_{7}^{11} + 1545 T_{7}^{10} + 2429 T_{7}^{9} + \cdots + 60016 \) Copy content Toggle raw display
\( T_{11}^{14} - 15 T_{11}^{13} + 35 T_{11}^{12} + 513 T_{11}^{11} - 3244 T_{11}^{10} + 1614 T_{11}^{9} + \cdots + 2752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 3 T^{13} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + T^{13} + \cdots + 60016 \) Copy content Toggle raw display
$11$ \( T^{14} - 15 T^{13} + \cdots + 2752 \) Copy content Toggle raw display
$13$ \( (T - 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 11 T^{13} + \cdots - 34296448 \) Copy content Toggle raw display
$19$ \( T^{14} - 12 T^{13} + \cdots - 1175152 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 110450944 \) Copy content Toggle raw display
$29$ \( T^{14} + 9 T^{13} + \cdots - 912824 \) Copy content Toggle raw display
$31$ \( (T + 1)^{14} \) Copy content Toggle raw display
$37$ \( T^{14} - 278 T^{12} + \cdots + 22746064 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 367201876 \) Copy content Toggle raw display
$43$ \( T^{14} + 8 T^{13} + \cdots + 34012616 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 130559770112 \) Copy content Toggle raw display
$53$ \( T^{14} + 28 T^{13} + \cdots - 859664 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 9354543136 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 4456517896072 \) Copy content Toggle raw display
$67$ \( T^{14} + 10 T^{13} + \cdots - 1181696 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 101292032 \) Copy content Toggle raw display
$73$ \( T^{14} + 8 T^{13} + \cdots + 76666748 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 135144064 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 64605149248 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 14724964144 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 6949242964 \) Copy content Toggle raw display
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