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Properties

Label 6045.2.a.bf
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 22 x^{13} + 21 x^{12} + 187 x^{11} - 166 x^{10} - 781 x^{9} + 617 x^{8} + 1697 x^{7} + \cdots + 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 22 x^{13} + 21 x^{12} + 187 x^{11} - 166 x^{10} - 781 x^{9} + 617 x^{8} + 1697 x^{7} + \cdots + 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57 \nu^{14} - 24 \nu^{13} - 1204 \nu^{12} - 139 \nu^{11} + 9684 \nu^{10} + 7390 \nu^{9} - 37555 \nu^{8} + \cdots + 3322 ) / 1214 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 337 \nu^{14} + 78 \nu^{13} + 6948 \nu^{12} - 1521 \nu^{11} - 53932 \nu^{10} + 10278 \nu^{9} + \cdots + 4682 ) / 1214 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73 \nu^{14} + 478 \nu^{13} + 1116 \nu^{12} - 9928 \nu^{11} - 4096 \nu^{10} + 77507 \nu^{9} + \cdots - 19323 ) / 607 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 120 \nu^{14} - 370 \nu^{13} - 2375 \nu^{12} + 7822 \nu^{11} + 16969 \nu^{10} - 62202 \nu^{9} + \cdots + 7409 ) / 607 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 160 \nu^{14} - 291 \nu^{13} - 3369 \nu^{12} + 5978 \nu^{11} + 26672 \nu^{10} - 45909 \nu^{9} + \cdots + 4011 ) / 607 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 329 \nu^{14} + 458 \nu^{13} + 6992 \nu^{12} - 10145 \nu^{11} - 55512 \nu^{10} + 85284 \nu^{9} + \cdots - 25862 ) / 1214 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 349 \nu^{14} - 722 \nu^{13} - 6882 \nu^{12} + 15293 \nu^{11} + 49134 \nu^{10} - 121752 \nu^{9} + \cdots + 18700 ) / 1214 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 523 \nu^{14} - 348 \nu^{13} - 11388 \nu^{12} + 7393 \nu^{11} + 94286 \nu^{10} - 58556 \nu^{9} + \cdots + 2644 ) / 1214 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 623 \nu^{14} + 454 \nu^{13} + 13266 \nu^{12} - 8853 \nu^{11} - 107314 \nu^{10} + 62438 \nu^{9} + \cdots + 2816 ) / 1214 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 381 \nu^{14} + 416 \nu^{13} + 7920 \nu^{12} - 8719 \nu^{11} - 61631 \nu^{10} + 68170 \nu^{9} + \cdots - 11247 ) / 607 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 623 \nu^{14} - 1061 \nu^{13} - 12659 \nu^{12} + 21600 \nu^{11} + 95174 \nu^{10} - 163200 \nu^{9} + \cdots + 22071 ) / 607 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - 2\beta_{13} + \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{4} + 7\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{13} + \beta_{12} - 2\beta_{11} - \beta_{10} + 2\beta_{8} + \beta_{6} + 8\beta_{3} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{14} - 22 \beta_{13} + 11 \beta_{12} - \beta_{11} + \beta_{10} + 11 \beta_{9} + 12 \beta_{8} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{14} - 26 \beta_{13} + 12 \beta_{12} - 24 \beta_{11} - 11 \beta_{10} + \beta_{9} + 24 \beta_{8} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 93 \beta_{14} - 189 \beta_{13} + 93 \beta_{12} - 16 \beta_{11} + 13 \beta_{10} + 93 \beta_{9} + \cdots + 541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 33 \beta_{14} - 250 \beta_{13} + 108 \beta_{12} - 214 \beta_{11} - 93 \beta_{10} + 18 \beta_{9} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 721 \beta_{14} - 1491 \beta_{13} + 719 \beta_{12} - 172 \beta_{11} + 123 \beta_{10} + 722 \beta_{9} + \cdots + 3510 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 369 \beta_{14} - 2144 \beta_{13} + 879 \beta_{12} - 1720 \beta_{11} - 723 \beta_{10} + 211 \beta_{9} + \cdots - 657 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5388 \beta_{14} - 11319 \beta_{13} + 5354 \beta_{12} - 1577 \beta_{11} + 1037 \beta_{10} + \cdots + 23411 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3515 \beta_{14} - 17381 \beta_{13} + 6849 \beta_{12} - 13195 \beta_{11} - 5427 \beta_{10} + \cdots - 3720 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 39566 \beta_{14} - 84277 \beta_{13} + 39207 \beta_{12} - 13353 \beta_{11} + 8260 \beta_{10} + \cdots + 159047 \) 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
2.71120
2.23960
2.05880
1.98360
1.14710
1.08493
0.694202
0.169033
−0.688624
−0.824705
−0.895769
−1.75324
−1.77259
−2.52826
−2.62528
−2.71120 1.00000 5.35062 −1.00000 −2.71120 −3.48234 −9.08421 1.00000 2.71120
1.2 −2.23960 1.00000 3.01581 −1.00000 −2.23960 0.493210 −2.27501 1.00000 2.23960
1.3 −2.05880 1.00000 2.23865 −1.00000 −2.05880 1.76520 −0.491342 1.00000 2.05880
1.4 −1.98360 1.00000 1.93466 −1.00000 −1.98360 −1.84365 0.129603 1.00000 1.98360
1.5 −1.14710 1.00000 −0.684165 −1.00000 −1.14710 −4.65242 3.07900 1.00000 1.14710
1.6 −1.08493 1.00000 −0.822921 −1.00000 −1.08493 3.47386 3.06268 1.00000 1.08493
1.7 −0.694202 1.00000 −1.51808 −1.00000 −0.694202 −2.18639 2.44226 1.00000 0.694202
1.8 −0.169033 1.00000 −1.97143 −1.00000 −0.169033 3.36699 0.671301 1.00000 0.169033
1.9 0.688624 1.00000 −1.52580 −1.00000 0.688624 1.81322 −2.42795 1.00000 −0.688624
1.10 0.824705 1.00000 −1.31986 −1.00000 0.824705 −4.26572 −2.73791 1.00000 −0.824705
1.11 0.895769 1.00000 −1.19760 −1.00000 0.895769 −0.597804 −2.86431 1.00000 −0.895769
1.12 1.75324 1.00000 1.07385 −1.00000 1.75324 −4.35158 −1.62376 1.00000 −1.75324
1.13 1.77259 1.00000 1.14207 −1.00000 1.77259 2.10158 −1.52076 1.00000 −1.77259
1.14 2.52826 1.00000 4.39212 −1.00000 2.52826 0.0357087 6.04790 1.00000 −2.52826
1.15 2.62528 1.00000 4.89208 −1.00000 2.62528 3.33013 7.59250 1.00000 −2.62528
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bf ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.bf ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{15} + T_{2}^{14} - 22 T_{2}^{13} - 21 T_{2}^{12} + 187 T_{2}^{11} + 166 T_{2}^{10} - 781 T_{2}^{9} + \cdots - 38 \) Copy content Toggle raw display
\( T_{7}^{15} + 5 T_{7}^{14} - 50 T_{7}^{13} - 231 T_{7}^{12} + 1057 T_{7}^{11} + 4097 T_{7}^{10} + \cdots + 3344 \) Copy content Toggle raw display
\( T_{11}^{15} - 6 T_{11}^{14} - 78 T_{11}^{13} + 530 T_{11}^{12} + 2139 T_{11}^{11} - 17986 T_{11}^{10} + \cdots - 1218368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + T^{14} + \cdots - 38 \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 5 T^{14} + \cdots + 3344 \) Copy content Toggle raw display
$11$ \( T^{15} - 6 T^{14} + \cdots - 1218368 \) Copy content Toggle raw display
$13$ \( (T + 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 13 T^{14} + \cdots + 390464 \) Copy content Toggle raw display
$19$ \( T^{15} - 4 T^{14} + \cdots + 3112 \) Copy content Toggle raw display
$23$ \( T^{15} + 3 T^{14} + \cdots - 5019776 \) Copy content Toggle raw display
$29$ \( T^{15} - 13 T^{14} + \cdots + 42042454 \) Copy content Toggle raw display
$31$ \( (T + 1)^{15} \) Copy content Toggle raw display
$37$ \( T^{15} + 9 T^{14} + \cdots + 1189312 \) Copy content Toggle raw display
$41$ \( T^{15} - 15 T^{14} + \cdots - 1120118 \) Copy content Toggle raw display
$43$ \( T^{15} - 22 T^{14} + \cdots - 66254896 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 42536950048 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 722216624 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 351255190000 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 4820186656 \) Copy content Toggle raw display
$67$ \( T^{15} + 2 T^{14} + \cdots + 14231104 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 338235392 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 122290208 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 1463319968 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 27217163719264 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 2296776512 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 1120146505024 \) Copy content Toggle raw display
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