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Properties

Label 6045.2.a.bg
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 401375 \nu^{15} + 5831475 \nu^{14} - 6105068 \nu^{13} - 127878505 \nu^{12} + 266535327 \nu^{11} + \cdots + 2656763704 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 675635 \nu^{15} - 166555 \nu^{14} + 24375904 \nu^{13} - 4276819 \nu^{12} - 340606193 \nu^{11} + \cdots - 2433705296 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2488534 \nu^{15} + 15164660 \nu^{14} + 28420427 \nu^{13} - 329766812 \nu^{12} + \cdots + 3930275208 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1368167 \nu^{15} + 5740111 \nu^{14} + 22545964 \nu^{13} - 124455102 \nu^{12} + \cdots + 1615863693 ) / 33320579 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2803945 \nu^{15} - 19153621 \nu^{14} - 38546041 \nu^{13} + 439852093 \nu^{12} - 11139427 \nu^{11} + \cdots - 1177545982 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2831059 \nu^{15} - 2064367 \nu^{14} - 89938815 \nu^{13} + 67211901 \nu^{12} + 1147216629 \nu^{11} + \cdots + 6127161078 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3037054 \nu^{15} + 3168600 \nu^{14} + 89382371 \nu^{13} - 82563440 \nu^{12} + \cdots - 4318069210 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6782934 \nu^{15} - 25215264 \nu^{14} - 135487993 \nu^{13} + 572705046 \nu^{12} + \cdots - 1296448684 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7250209 \nu^{15} + 23370511 \nu^{14} + 161207194 \nu^{13} - 539774147 \nu^{12} + \cdots - 2141597206 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9572188 \nu^{15} - 32068302 \nu^{14} - 208198863 \nu^{13} + 736125094 \nu^{12} + \cdots + 1789943440 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 9720317 \nu^{15} - 40614744 \nu^{14} - 177132377 \nu^{13} + 911381417 \nu^{12} + \cdots - 4303214968 ) / 66641158 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12686056 \nu^{15} + 41128413 \nu^{14} + 279181880 \nu^{13} - 947900454 \nu^{12} + \cdots - 2285385432 ) / 66641158 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - 2\beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + \beta_{5} + 10\beta_{3} + 40\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{13} + 2 \beta_{12} + 13 \beta_{11} - 14 \beta_{10} - \beta_{9} + 13 \beta_{7} - 11 \beta_{6} + \cdots + 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} - \beta_{14} - 12 \beta_{13} - 28 \beta_{12} - 11 \beta_{11} + 16 \beta_{10} + 13 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{15} + 3 \beta_{14} - 95 \beta_{13} + 36 \beta_{12} + 124 \beta_{11} - 143 \beta_{10} + \cdots + 1116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{15} - 20 \beta_{14} - 109 \beta_{13} - 286 \beta_{12} - 88 \beta_{11} + 179 \beta_{10} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20 \beta_{15} + 50 \beta_{14} - 759 \beta_{13} + 433 \beta_{12} + 1068 \beta_{11} - 1297 \beta_{10} + \cdots + 7978 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 222 \beta_{15} - 264 \beta_{14} - 894 \beta_{13} - 2602 \beta_{12} - 626 \beta_{11} + 1741 \beta_{10} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 256 \beta_{15} + 558 \beta_{14} - 5871 \beta_{13} + 4401 \beta_{12} + 8814 \beta_{11} - 11088 \beta_{10} + \cdots + 57910 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2329 \beta_{15} - 2883 \beta_{14} - 6973 \beta_{13} - 22371 \beta_{12} - 4217 \beta_{11} + 15779 \beta_{10} + \cdots - 3689 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2679 \beta_{15} + 5285 \beta_{14} - 44720 \beta_{13} + 40924 \beta_{12} + 71241 \beta_{11} + \cdots + 424747 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 22339 \beta_{15} - 28273 \beta_{14} - 52836 \beta_{13} - 186173 \beta_{12} - 27613 \beta_{11} + \cdots - 50618 \) 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
2.74907
2.67164
2.46449
2.34208
1.69426
1.17778
0.976688
0.399270
0.323092
−0.733028
−1.26786
−1.73595
−1.89108
−1.90524
−2.48293
−2.78228
−2.74907 −1.00000 5.55740 −1.00000 2.74907 −0.783225 −9.77954 1.00000 2.74907
1.2 −2.67164 −1.00000 5.13767 −1.00000 2.67164 1.81220 −8.38272 1.00000 2.67164
1.3 −2.46449 −1.00000 4.07370 −1.00000 2.46449 −4.85009 −5.11060 1.00000 2.46449
1.4 −2.34208 −1.00000 3.48532 −1.00000 2.34208 3.99026 −3.47875 1.00000 2.34208
1.5 −1.69426 −1.00000 0.870532 −1.00000 1.69426 0.135304 1.91362 1.00000 1.69426
1.6 −1.17778 −1.00000 −0.612829 −1.00000 1.17778 −0.648351 3.07734 1.00000 1.17778
1.7 −0.976688 −1.00000 −1.04608 −1.00000 0.976688 2.96131 2.97507 1.00000 0.976688
1.8 −0.399270 −1.00000 −1.84058 −1.00000 0.399270 −2.18061 1.53343 1.00000 0.399270
1.9 −0.323092 −1.00000 −1.89561 −1.00000 0.323092 1.90658 1.25864 1.00000 0.323092
1.10 0.733028 −1.00000 −1.46267 −1.00000 −0.733028 −2.61356 −2.53823 1.00000 −0.733028
1.11 1.26786 −1.00000 −0.392523 −1.00000 −1.26786 −1.11922 −3.03339 1.00000 −1.26786
1.12 1.73595 −1.00000 1.01354 −1.00000 −1.73595 −1.77896 −1.71245 1.00000 −1.73595
1.13 1.89108 −1.00000 1.57618 −1.00000 −1.89108 −4.66347 −0.801474 1.00000 −1.89108
1.14 1.90524 −1.00000 1.62994 −1.00000 −1.90524 4.91854 −0.705044 1.00000 −1.90524
1.15 2.48293 −1.00000 4.16494 −1.00000 −2.48293 −1.71749 5.37540 1.00000 −2.48293
1.16 2.78228 −1.00000 5.74107 −1.00000 −2.78228 2.63079 10.4087 1.00000 −2.78228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.bg ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.bg ✓ 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{16} + 2 T_{2}^{15} - 27 T_{2}^{14} - 51 T_{2}^{13} + 294 T_{2}^{12} + 517 T_{2}^{11} - 1657 T_{2}^{10} + \cdots - 428 \) Copy content Toggle raw display
\( T_{7}^{16} + 2 T_{7}^{15} - 62 T_{7}^{14} - 121 T_{7}^{13} + 1406 T_{7}^{12} + 2769 T_{7}^{11} + \cdots - 16000 \) Copy content Toggle raw display
\( T_{11}^{16} - 3 T_{11}^{15} - 112 T_{11}^{14} + 302 T_{11}^{13} + 4971 T_{11}^{12} - 11437 T_{11}^{11} + \cdots + 4757120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots - 428 \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots - 16000 \) Copy content Toggle raw display
$11$ \( T^{16} - 3 T^{15} + \cdots + 4757120 \) Copy content Toggle raw display
$13$ \( (T + 1)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 13 T^{15} + \cdots + 77365248 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 3865812800 \) Copy content Toggle raw display
$23$ \( T^{16} + 15 T^{15} + \cdots + 18739200 \) Copy content Toggle raw display
$29$ \( T^{16} + 4 T^{15} + \cdots - 12704624 \) Copy content Toggle raw display
$31$ \( (T - 1)^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 550632960 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 361478687256 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 248325248 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 1089903104 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 446291259520 \) Copy content Toggle raw display
$59$ \( T^{16} - 53 T^{15} + \cdots - 14893952 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 78424557696 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 569884655616 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 7295488819200 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 21054321229120 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 12746125834752 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 57132245826560 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 269845558784 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 34906853261440 \) Copy content Toggle raw display
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