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Properties

Label 6045.2.a.y
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: \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 78 x^{8} - 252 x^{7} - 149 x^{6} + 583 x^{5} + 18 x^{4} + \cdots - 26 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 78 x^{8} - 252 x^{7} - 149 x^{6} + 583 x^{5} + 18 x^{4} + \cdots - 26 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{11} + 5 \nu^{10} + 85 \nu^{9} - 84 \nu^{8} - 654 \nu^{7} + 481 \nu^{6} + 2169 \nu^{5} + \cdots - 208 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{11} - 30 \nu^{10} - 185 \nu^{9} + 426 \nu^{8} + 1181 \nu^{7} - 2054 \nu^{6} - 3381 \nu^{5} + \cdots + 247 ) / 65 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} - 50 \nu^{9} + 73 \nu^{8} + 303 \nu^{7} - 372 \nu^{6} - 788 \nu^{5} + 765 \nu^{4} + \cdots + 46 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38 \nu^{11} - 80 \nu^{10} - 645 \nu^{9} + 1188 \nu^{8} + 4068 \nu^{7} - 6162 \nu^{6} - 11343 \nu^{5} + \cdots + 1131 ) / 65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37 \nu^{11} + 95 \nu^{10} + 640 \nu^{9} - 1427 \nu^{8} - 4132 \nu^{7} + 7423 \nu^{6} + 11727 \nu^{5} + \cdots - 1534 ) / 65 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 74 \nu^{11} + 125 \nu^{10} + 1215 \nu^{9} - 1814 \nu^{8} - 7289 \nu^{7} + 9321 \nu^{6} + 18969 \nu^{5} + \cdots - 2028 ) / 65 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 88 \nu^{11} - 175 \nu^{10} - 1480 \nu^{9} + 2563 \nu^{8} + 9188 \nu^{7} - 13052 \nu^{6} - 24968 \nu^{5} + \cdots + 2431 ) / 65 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 82 \nu^{11} - 200 \nu^{10} - 1385 \nu^{9} + 2957 \nu^{8} + 8727 \nu^{7} - 15158 \nu^{6} - 24217 \nu^{5} + \cdots + 2704 ) / 65 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 8\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 8 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 11 \beta_{10} + \beta_{9} + 2 \beta_{8} - 10 \beta_{7} - 10 \beta_{6} - 12 \beta_{5} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{11} + 15 \beta_{10} + \beta_{9} + 14 \beta_{8} - 23 \beta_{7} - 15 \beta_{6} - 27 \beta_{5} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{11} + 94 \beta_{10} + 13 \beta_{9} + 28 \beta_{8} - 81 \beta_{7} - 82 \beta_{6} - 112 \beta_{5} + \cdots + 428 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 107 \beta_{11} + 157 \beta_{10} + 16 \beta_{9} + 137 \beta_{8} - 203 \beta_{7} - 155 \beta_{6} + \cdots + 261 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 159 \beta_{11} + 741 \beta_{10} + 121 \beta_{9} + 284 \beta_{8} - 623 \beta_{7} - 636 \beta_{6} + \cdots + 2539 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 858 \beta_{11} + 1421 \beta_{10} + 175 \beta_{9} + 1172 \beta_{8} - 1638 \beta_{7} - 1379 \beta_{6} + \cdots + 2216 \) 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
2.69492
2.69084
2.32897
1.40774
0.887639
0.847478
0.256555
−0.478938
−1.22780
−1.88956
−2.20326
−2.31460
−2.69492 1.00000 5.26259 −1.00000 −2.69492 2.26457 −8.79240 1.00000 2.69492
1.2 −2.69084 1.00000 5.24062 −1.00000 −2.69084 0.770536 −8.72000 1.00000 2.69084
1.3 −2.32897 1.00000 3.42411 −1.00000 −2.32897 −4.93080 −3.31671 1.00000 2.32897
1.4 −1.40774 1.00000 −0.0182582 −1.00000 −1.40774 4.37400 2.84119 1.00000 1.40774
1.5 −0.887639 1.00000 −1.21210 −1.00000 −0.887639 0.374138 2.85118 1.00000 0.887639
1.6 −0.847478 1.00000 −1.28178 −1.00000 −0.847478 1.40489 2.78124 1.00000 0.847478
1.7 −0.256555 1.00000 −1.93418 −1.00000 −0.256555 2.42431 1.00933 1.00000 0.256555
1.8 0.478938 1.00000 −1.77062 −1.00000 0.478938 −1.38011 −1.80589 1.00000 −0.478938
1.9 1.22780 1.00000 −0.492519 −1.00000 1.22780 −1.08279 −3.06030 1.00000 −1.22780
1.10 1.88956 1.00000 1.57045 −1.00000 1.88956 2.97241 −0.811668 1.00000 −1.88956
1.11 2.20326 1.00000 2.85433 −1.00000 2.20326 −3.44879 1.88232 1.00000 −2.20326
1.12 2.31460 1.00000 3.35735 −1.00000 2.31460 −1.74237 3.14172 1.00000 −2.31460
\(n\): 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.y ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.y ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 46 T_{2}^{9} + 78 T_{2}^{8} + 252 T_{2}^{7} - 149 T_{2}^{6} + \cdots - 26 \) Copy content Toggle raw display
\( T_{7}^{12} - 2 T_{7}^{11} - 40 T_{7}^{10} + 93 T_{7}^{9} + 464 T_{7}^{8} - 1205 T_{7}^{7} - 1743 T_{7}^{6} + \cdots - 1280 \) Copy content Toggle raw display
\( T_{11}^{12} + 24 T_{11}^{11} + 195 T_{11}^{10} + 266 T_{11}^{9} - 5130 T_{11}^{8} - 30740 T_{11}^{7} + \cdots + 40960 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots - 26 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots - 1280 \) Copy content Toggle raw display
$11$ \( T^{12} + 24 T^{11} + \cdots + 40960 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} + \cdots - 370816 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 52280 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots - 408896 \) Copy content Toggle raw display
$29$ \( T^{12} + 24 T^{11} + \cdots - 21973250 \) Copy content Toggle raw display
$31$ \( (T + 1)^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 13 T^{11} + \cdots + 5665600 \) Copy content Toggle raw display
$41$ \( T^{12} + 46 T^{11} + \cdots - 4633342 \) Copy content Toggle raw display
$43$ \( T^{12} + 5 T^{11} + \cdots + 14454508 \) Copy content Toggle raw display
$47$ \( T^{12} + 39 T^{11} + \cdots + 9027520 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 156621428 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 7112455492 \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots - 260 \) Copy content Toggle raw display
$67$ \( T^{12} - 5 T^{11} + \cdots - 6355900 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 3004622080 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 80217623980 \) Copy content Toggle raw display
$79$ \( T^{12} - 5 T^{11} + \cdots - 73929952 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 699208528 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1162726720 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 562269602 \) Copy content Toggle raw display
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