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Properties

Label 6045.2.a.x
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $1$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 7 x^{10} + 43 x^{9} - 5 x^{8} - 141 x^{7} + 90 x^{6} + 165 x^{5} - 141 x^{4} + \cdots - 2 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 7 x^{10} + 43 x^{9} - 5 x^{8} - 141 x^{7} + 90 x^{6} + 165 x^{5} - 141 x^{4} + \cdots - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10 \nu^{11} + 72 \nu^{10} - 7 \nu^{9} - 738 \nu^{8} + 795 \nu^{7} + 2170 \nu^{6} - 2829 \nu^{5} + \cdots + 198 ) / 59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32 \nu^{11} - 77 \nu^{10} - 308 \nu^{9} + 745 \nu^{8} + 760 \nu^{7} - 1929 \nu^{6} - 281 \nu^{5} + \cdots + 39 ) / 59 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14 \nu^{11} + 30 \nu^{10} + 179 \nu^{9} - 396 \nu^{8} - 716 \nu^{7} + 1799 \nu^{6} + 783 \nu^{5} + \cdots - 65 ) / 59 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24 \nu^{11} - 102 \nu^{10} - 172 \nu^{9} + 1075 \nu^{8} + 39 \nu^{7} - 3379 \nu^{6} + 925 \nu^{5} + \cdots + 44 ) / 59 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29 \nu^{11} + 79 \nu^{10} + 316 \nu^{9} - 913 \nu^{8} - 1028 \nu^{7} + 3402 \nu^{6} + 994 \nu^{5} + \cdots + 114 ) / 59 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24 \nu^{11} + 43 \nu^{10} + 349 \nu^{9} - 603 \nu^{8} - 1750 \nu^{7} + 2907 \nu^{6} + 3500 \nu^{5} + \cdots - 103 ) / 59 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 58 \nu^{11} - 217 \nu^{10} - 514 \nu^{9} + 2475 \nu^{8} + 876 \nu^{7} - 9046 \nu^{6} + 1375 \nu^{5} + \cdots + 126 ) / 59 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72 \nu^{11} - 247 \nu^{10} - 634 \nu^{9} + 2694 \nu^{8} + 1120 \nu^{7} - 9134 \nu^{6} + 1064 \nu^{5} + \cdots + 309 ) / 59 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 84 \nu^{11} + 298 \nu^{10} + 720 \nu^{9} - 3261 \nu^{8} - 1051 \nu^{7} + 11089 \nu^{6} - 2382 \nu^{5} + \cdots - 154 ) / 59 \) 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
\(\nu^{3}\)\(=\) \( -2\beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{3} + 2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{5} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{11} - 8 \beta_{10} + 3 \beta_{9} - 7 \beta_{8} + 16 \beta_{7} - 7 \beta_{6} + \cdots + 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} - 2 \beta_{10} + 11 \beta_{9} + 8 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 14 \beta_{5} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 78 \beta_{11} - 54 \beta_{10} + 33 \beta_{9} - 44 \beta_{8} + 110 \beta_{7} - 46 \beta_{6} + 103 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 25 \beta_{11} - 24 \beta_{10} + 92 \beta_{9} + 50 \beta_{8} + 32 \beta_{7} - 28 \beta_{6} + \cdots + 507 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 475 \beta_{11} - 350 \beta_{10} + 271 \beta_{9} - 272 \beta_{8} + 728 \beta_{7} - 307 \beta_{6} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 230 \beta_{11} - 214 \beta_{10} + 698 \beta_{9} + 285 \beta_{8} + 342 \beta_{7} - 275 \beta_{6} + \cdots + 3112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2953 \beta_{11} - 2247 \beta_{10} + 2015 \beta_{9} - 1681 \beta_{8} + 4761 \beta_{7} - 2068 \beta_{6} + \cdots + 861 \) 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
2.61671
2.11867
1.86550
1.73030
0.896827
0.697642
0.329040
−0.132530
−0.942831
−1.05349
−1.70019
−2.42565
−2.61671 1.00000 4.84717 −1.00000 −2.61671 4.81574 −7.45022 1.00000 2.61671
1.2 −2.11867 1.00000 2.48877 −1.00000 −2.11867 −2.45447 −1.03555 1.00000 2.11867
1.3 −1.86550 1.00000 1.48010 −1.00000 −1.86550 3.58361 0.969878 1.00000 1.86550
1.4 −1.73030 1.00000 0.993928 −1.00000 −1.73030 0.0923132 1.74080 1.00000 1.73030
1.5 −0.896827 1.00000 −1.19570 −1.00000 −0.896827 −1.45932 2.86599 1.00000 0.896827
1.6 −0.697642 1.00000 −1.51330 −1.00000 −0.697642 −2.51812 2.45102 1.00000 0.697642
1.7 −0.329040 1.00000 −1.89173 −1.00000 −0.329040 1.18922 1.28053 1.00000 0.329040
1.8 0.132530 1.00000 −1.98244 −1.00000 0.132530 3.30524 −0.527794 1.00000 −0.132530
1.9 0.942831 1.00000 −1.11107 −1.00000 0.942831 −3.20677 −2.93321 1.00000 −0.942831
1.10 1.05349 1.00000 −0.890165 −1.00000 1.05349 3.27468 −3.04475 1.00000 −1.05349
1.11 1.70019 1.00000 0.890648 −1.00000 1.70019 −0.680300 −1.88611 1.00000 −1.70019
1.12 2.42565 1.00000 3.88378 −1.00000 2.42565 −2.94183 4.56940 1.00000 −2.42565
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.x ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.x ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 7 T_{2}^{10} - 43 T_{2}^{9} - 5 T_{2}^{8} + 141 T_{2}^{7} + 90 T_{2}^{6} + \cdots - 2 \) Copy content Toggle raw display
\( T_{7}^{12} - 3 T_{7}^{11} - 42 T_{7}^{10} + 89 T_{7}^{9} + 713 T_{7}^{8} - 843 T_{7}^{7} - 5896 T_{7}^{6} + \cdots + 1187 \) Copy content Toggle raw display
\( T_{11}^{12} + 10 T_{11}^{11} - 11 T_{11}^{10} - 326 T_{11}^{9} - 418 T_{11}^{8} + 2320 T_{11}^{7} + \cdots - 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 1187 \) Copy content Toggle raw display
$11$ \( T^{12} + 10 T^{11} + \cdots - 80 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 7 T^{11} + \cdots + 28568 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots - 790982 \) Copy content Toggle raw display
$23$ \( T^{12} + 3 T^{11} + \cdots - 134096 \) Copy content Toggle raw display
$29$ \( T^{12} + 23 T^{11} + \cdots - 2524763 \) Copy content Toggle raw display
$31$ \( (T - 1)^{12} \) Copy content Toggle raw display
$37$ \( T^{12} - 9 T^{11} + \cdots + 45079448 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 3497228953 \) Copy content Toggle raw display
$43$ \( T^{12} + 20 T^{11} + \cdots + 13150751 \) Copy content Toggle raw display
$47$ \( T^{12} + 9 T^{11} + \cdots - 92276750 \) Copy content Toggle raw display
$53$ \( T^{12} + 20 T^{11} + \cdots + 1793830 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1352998585 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 46022475430 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 20887641239 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 564970112 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 2271587282 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 988021858 \) Copy content Toggle raw display
$83$ \( T^{12} - 3 T^{11} + \cdots + 29449009 \) Copy content Toggle raw display
$89$ \( T^{12} + 39 T^{11} + \cdots - 65034248 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 2550577691 \) Copy content Toggle raw display
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