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Properties

Label 6045.2.a.z
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: \( x^{12} - 15x^{10} - x^{9} + 81x^{8} + 9x^{7} - 192x^{6} - 27x^{5} + 197x^{4} + 28x^{3} - 82x^{2} - 10x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} - x^{9} + 81x^{8} + 9x^{7} - 192x^{6} - 27x^{5} + 197x^{4} + 28x^{3} - 82x^{2} - 10x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{11} - 14\nu^{9} - \nu^{8} + 68\nu^{7} + 7\nu^{6} - 134\nu^{5} - 12\nu^{4} + 93\nu^{3} - 4\nu^{2} - 17\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{11} - 14\nu^{9} - 2\nu^{8} + 69\nu^{7} + 17\nu^{6} - 141\nu^{5} - 43\nu^{4} + 105\nu^{3} + 29\nu^{2} - 22\nu - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{11} + 15\nu^{9} - 79\nu^{7} + \nu^{6} + 174\nu^{5} - 6\nu^{4} - 148\nu^{3} + 15\nu^{2} + 38\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{11} + \nu^{10} - 14 \nu^{9} - 15 \nu^{8} + 66 \nu^{7} + 76 \nu^{6} - 117 \nu^{5} - 153 \nu^{4} + \cdots - 15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{11} - \nu^{10} - 14 \nu^{9} + 12 \nu^{8} + 70 \nu^{7} - 51 \nu^{6} - 149 \nu^{5} + 92 \nu^{4} + \cdots + 15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{11} - \nu^{10} - 14 \nu^{9} + 12 \nu^{8} + 70 \nu^{7} - 51 \nu^{6} - 149 \nu^{5} + 92 \nu^{4} + \cdots + 14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\nu^{10} - \nu^{9} + 16\nu^{8} + 12\nu^{7} - 87\nu^{6} - 49\nu^{5} + 189\nu^{4} + 79\nu^{3} - 144\nu^{2} - 34\nu + 27 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 2\nu^{10} - 28\nu^{8} - 3\nu^{7} + 136\nu^{6} + 25\nu^{5} - 266\nu^{4} - 61\nu^{3} + 176\nu^{2} + 33\nu - 27 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 2 \nu^{11} - \nu^{10} - 29 \nu^{9} + 13 \nu^{8} + 148 \nu^{7} - 62 \nu^{6} - 315 \nu^{5} + 127 \nu^{4} + \cdots + 24 \) 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
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + 5\beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + 2\beta_{9} + 7\beta_{8} - 6\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 20\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{11} - \beta_{10} + 10 \beta_{9} + 10 \beta_{8} - \beta_{7} + 11 \beta_{6} + \beta_{4} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{11} - \beta_{10} + 22 \beta_{9} + 44 \beta_{8} - 33 \beta_{7} + 22 \beta_{6} + 9 \beta_{5} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 75 \beta_{11} - 11 \beta_{10} + 77 \beta_{9} + 76 \beta_{8} - 13 \beta_{7} + 87 \beta_{6} + \cdots + 337 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 116 \beta_{11} - 14 \beta_{10} + 177 \beta_{9} + 273 \beta_{8} - 183 \beta_{7} + 179 \beta_{6} + \cdots + 318 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 510 \beta_{11} - 87 \beta_{10} + 539 \beta_{9} + 526 \beta_{8} - 119 \beta_{7} + 611 \beta_{6} + \cdots + 1858 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 891 \beta_{11} - 132 \beta_{10} + 1269 \beta_{9} + 1693 \beta_{8} - 1035 \beta_{7} + 1300 \beta_{6} + \cdots + 2152 \) 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
2.51257
1.99226
1.81920
0.936465
0.873645
0.354579
−0.626598
−0.693206
−1.06039
−1.66487
−2.21701
−2.22665
−2.51257 −1.00000 4.31301 −1.00000 2.51257 −1.14239 −5.81161 1.00000 2.51257
1.2 −1.99226 −1.00000 1.96909 −1.00000 1.99226 −3.89117 0.0615801 1.00000 1.99226
1.3 −1.81920 −1.00000 1.30949 −1.00000 1.81920 0.377267 1.25617 1.00000 1.81920
1.4 −0.936465 −1.00000 −1.12303 −1.00000 0.936465 2.62789 2.92461 1.00000 0.936465
1.5 −0.873645 −1.00000 −1.23675 −1.00000 0.873645 0.198655 2.82776 1.00000 0.873645
1.6 −0.354579 −1.00000 −1.87427 −1.00000 0.354579 −2.36918 1.37374 1.00000 0.354579
1.7 0.626598 −1.00000 −1.60738 −1.00000 −0.626598 −3.54811 −2.26037 1.00000 −0.626598
1.8 0.693206 −1.00000 −1.51947 −1.00000 −0.693206 0.105900 −2.43971 1.00000 −0.693206
1.9 1.06039 −1.00000 −0.875570 −1.00000 −1.06039 4.02306 −3.04923 1.00000 −1.06039
1.10 1.66487 −1.00000 0.771781 −1.00000 −1.66487 −4.06912 −2.04482 1.00000 −1.66487
1.11 2.21701 −1.00000 2.91512 −1.00000 −2.21701 1.40989 2.02883 1.00000 −2.21701
1.12 2.22665 −1.00000 2.95796 −1.00000 −2.22665 −0.722704 2.13305 1.00000 −2.22665
\(n\): 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.z ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.z ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{12} - 15 T_{2}^{10} + T_{2}^{9} + 81 T_{2}^{8} - 9 T_{2}^{7} - 192 T_{2}^{6} + 27 T_{2}^{5} + \cdots + 10 \) Copy content Toggle raw display
\( T_{7}^{12} + 7 T_{7}^{11} - 14 T_{7}^{10} - 179 T_{7}^{9} - 121 T_{7}^{8} + 1187 T_{7}^{7} + 1554 T_{7}^{6} + \cdots + 13 \) Copy content Toggle raw display
\( T_{11}^{12} + 6 T_{11}^{11} - 35 T_{11}^{10} - 244 T_{11}^{9} + 90 T_{11}^{8} + 2574 T_{11}^{7} + \cdots + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 15 T^{10} + \cdots + 10 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} + \cdots + 13 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 80 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 5 T^{11} + \cdots - 689624 \) Copy content Toggle raw display
$19$ \( T^{12} + 24 T^{11} + \cdots - 2194 \) Copy content Toggle raw display
$23$ \( T^{12} - 13 T^{11} + \cdots - 1190800 \) Copy content Toggle raw display
$29$ \( T^{12} - 3 T^{11} + \cdots + 8158775 \) Copy content Toggle raw display
$31$ \( (T - 1)^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 9 T^{11} + \cdots + 1805464 \) Copy content Toggle raw display
$41$ \( T^{12} + 5 T^{11} + \cdots - 35939051 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 281033081 \) Copy content Toggle raw display
$47$ \( T^{12} - 17 T^{11} + \cdots + 26739790 \) Copy content Toggle raw display
$53$ \( T^{12} - 16 T^{11} + \cdots + 2598602 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 156440465 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 4147354810 \) Copy content Toggle raw display
$67$ \( T^{12} + 20 T^{11} + \cdots - 4613633 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 5363691904 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1413519446 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 751257034 \) Copy content Toggle raw display
$83$ \( T^{12} + 13 T^{11} + \cdots + 714661 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 77411012152 \) Copy content Toggle raw display
$97$ \( T^{12} + 11 T^{11} + \cdots + 36201847 \) Copy content Toggle raw display
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