Properties

Label 6018.2.a.bb
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $13$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 41 x^{11} + 179 x^{10} + 540 x^{9} - 2773 x^{8} - 2260 x^{7} + 17621 x^{6} - 838 x^{5} - 44478 x^{4} + 16472 x^{3} + 29944 x^{2} - 6856 x + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 41 x^{11} + 179 x^{10} + 540 x^{9} - 2773 x^{8} - 2260 x^{7} + 17621 x^{6} - 838 x^{5} - 44478 x^{4} + 16472 x^{3} + 29944 x^{2} - 6856 x + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1796388205553 \nu^{12} - 1489108919712 \nu^{11} + 81660319298773 \nu^{10} + 60295853640717 \nu^{9} + \cdots - 17\!\cdots\!84 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10253913982431 \nu^{12} + 12143449436408 \nu^{11} + 459309915620523 \nu^{10} - 550253809975141 \nu^{9} + \cdots - 25\!\cdots\!40 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2643124952245 \nu^{12} - 3844423118131 \nu^{11} - 118025134245614 \nu^{10} + 172279045954540 \nu^{9} + \cdots + 191018910747970 ) / 172262461018534 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2806630645981 \nu^{12} + 3962699821337 \nu^{11} + 124624713157090 \nu^{10} - 179172235132576 \nu^{9} + \cdots + 558668664048170 ) / 172262461018534 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11419786805749 \nu^{12} + 16698434980110 \nu^{11} + 504336575792465 \nu^{10} - 755378470856885 \nu^{9} + \cdots + 11\!\cdots\!96 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5836486040157 \nu^{12} - 6831410905550 \nu^{11} - 260080147076221 \nu^{10} + 311196222902321 \nu^{9} + \cdots + 990949223000480 ) / 344524922037068 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3800074510933 \nu^{12} + 4801952864360 \nu^{11} + 168177451801958 \nu^{10} - 219681987395845 \nu^{9} + \cdots + 228430125895582 ) / 172262461018534 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15713129420919 \nu^{12} + 20121588852528 \nu^{11} + 698648560224535 \nu^{10} - 915337629159613 \nu^{9} + \cdots + 927871747678256 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48613352793023 \nu^{12} - 64814272554394 \nu^{11} + \cdots - 72\!\cdots\!80 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 98658980574257 \nu^{12} - 121059374370454 \nu^{11} + \cdots + 497584607825056 ) / 689049844074136 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 114490409117923 \nu^{12} + 149510059488566 \nu^{11} + \cdots + 79\!\cdots\!64 ) / 689049844074136 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 3\beta_{9} + 3\beta_{8} + 3\beta_{7} - 3\beta_{6} - 2\beta_{5} - 3\beta_{2} + 9\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{12} + 2 \beta_{11} + 19 \beta_{10} + 20 \beta_{9} - 5 \beta_{8} + 14 \beta_{6} + 24 \beta_{5} + 9 \beta_{4} + 20 \beta_{3} + \beta _1 + 107 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{12} - 20 \beta_{11} + 3 \beta_{10} - 71 \beta_{9} + 61 \beta_{8} + 53 \beta_{7} - 73 \beta_{6} - 34 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} - 75 \beta_{2} + 108 \beta _1 - 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 83 \beta_{12} + 57 \beta_{11} + 343 \beta_{10} + 370 \beta_{9} - 121 \beta_{8} - 20 \beta_{7} + 197 \beta_{6} + 449 \beta_{5} + 220 \beta_{4} + 354 \beta_{3} - 4 \beta_{2} + 26 \beta _1 + 1611 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120 \beta_{12} - 324 \beta_{11} + 63 \beta_{10} - 1398 \beta_{9} + 1079 \beta_{8} + 814 \beta_{7} - 1413 \beta_{6} - 497 \beta_{5} + 434 \beta_{4} + 123 \beta_{3} - 1409 \beta_{2} + 1527 \beta _1 - 2028 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1747 \beta_{12} + 1234 \beta_{11} + 6054 \beta_{10} + 6709 \beta_{9} - 2374 \beta_{8} - 717 \beta_{7} + 2841 \beta_{6} + 7974 \beta_{5} + 4195 \beta_{4} + 5929 \beta_{3} - 42 \beta_{2} + 615 \beta _1 + 25577 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2658 \beta_{12} - 5011 \beta_{11} + 949 \beta_{10} - 26224 \beta_{9} + 18554 \beta_{8} + 12170 \beta_{7} - 25538 \beta_{6} - 7021 \beta_{5} + 8879 \beta_{4} + 2710 \beta_{3} - 24450 \beta_{2} + 23401 \beta _1 - 38176 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 33381 \beta_{12} + 24253 \beta_{11} + 105562 \beta_{10} + 121001 \beta_{9} - 43955 \beta_{8} - 17839 \beta_{7} + 42232 \beta_{6} + 138935 \beta_{5} + 73809 \beta_{4} + 97328 \beta_{3} + \cdots + 417089 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 52093 \beta_{12} - 77476 \beta_{11} + 10881 \beta_{10} - 483001 \beta_{9} + 317710 \beta_{8} + 182279 \beta_{7} - 449090 \beta_{6} - 99772 \beta_{5} + 165057 \beta_{4} + 53295 \beta_{3} + \cdots - 695315 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 610129 \beta_{12} + 454952 \beta_{11} + 1829309 \beta_{10} + 2177976 \beta_{9} - 797491 \beta_{8} - 382102 \beta_{7} + 647576 \beta_{6} + 2398243 \beta_{5} + 1257836 \beta_{4} + \cdots + 6906799 \) Copy content Toggle raw display

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
−4.21367
−3.91861
−2.41505
−2.08866
−0.877543
0.0205303
0.197152
1.47504
2.66604
2.71857
2.75551
3.59366
4.08702
1.00000 1.00000 1.00000 −4.21367 1.00000 1.46512 1.00000 1.00000 −4.21367
1.2 1.00000 1.00000 1.00000 −3.91861 1.00000 −4.71846 1.00000 1.00000 −3.91861
1.3 1.00000 1.00000 1.00000 −2.41505 1.00000 1.11075 1.00000 1.00000 −2.41505
1.4 1.00000 1.00000 1.00000 −2.08866 1.00000 3.96496 1.00000 1.00000 −2.08866
1.5 1.00000 1.00000 1.00000 −0.877543 1.00000 −4.30430 1.00000 1.00000 −0.877543
1.6 1.00000 1.00000 1.00000 0.0205303 1.00000 5.09020 1.00000 1.00000 0.0205303
1.7 1.00000 1.00000 1.00000 0.197152 1.00000 1.50880 1.00000 1.00000 0.197152
1.8 1.00000 1.00000 1.00000 1.47504 1.00000 −0.770392 1.00000 1.00000 1.47504
1.9 1.00000 1.00000 1.00000 2.66604 1.00000 4.24276 1.00000 1.00000 2.66604
1.10 1.00000 1.00000 1.00000 2.71857 1.00000 1.56988 1.00000 1.00000 2.71857
1.11 1.00000 1.00000 1.00000 2.75551 1.00000 1.74830 1.00000 1.00000 2.75551
1.12 1.00000 1.00000 1.00000 3.59366 1.00000 2.37015 1.00000 1.00000 3.59366
1.13 1.00000 1.00000 1.00000 4.08702 1.00000 −2.27776 1.00000 1.00000 4.08702
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

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 6018.2.a.bb 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.bb 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{13} - 4 T_{5}^{12} - 41 T_{5}^{11} + 179 T_{5}^{10} + 540 T_{5}^{9} - 2773 T_{5}^{8} - 2260 T_{5}^{7} + 17621 T_{5}^{6} - 838 T_{5}^{5} - 44478 T_{5}^{4} + 16472 T_{5}^{3} + 29944 T_{5}^{2} - 6856 T_{5} + 128 \) Copy content Toggle raw display
\( T_{7}^{13} - 11 T_{7}^{12} - T_{7}^{11} + 420 T_{7}^{10} - 1380 T_{7}^{9} - 2910 T_{7}^{8} + 22911 T_{7}^{7} - 30621 T_{7}^{6} - 53248 T_{7}^{5} + 192560 T_{7}^{4} - 173105 T_{7}^{3} - 21966 T_{7}^{2} + 115988 T_{7} - 48744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{13} \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 4 T^{12} - 41 T^{11} + 179 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$7$ \( T^{13} - 11 T^{12} - T^{11} + 420 T^{10} + \cdots - 48744 \) Copy content Toggle raw display
$11$ \( T^{13} - 7 T^{12} - 64 T^{11} + \cdots - 23744 \) Copy content Toggle raw display
$13$ \( T^{13} - 6 T^{12} - 83 T^{11} + \cdots - 801152 \) Copy content Toggle raw display
$17$ \( (T - 1)^{13} \) Copy content Toggle raw display
$19$ \( T^{13} - 9 T^{12} - 69 T^{11} + \cdots - 1815680 \) Copy content Toggle raw display
$23$ \( T^{13} - 2 T^{12} - 176 T^{11} + \cdots + 10405008 \) Copy content Toggle raw display
$29$ \( T^{13} - 14 T^{12} - 105 T^{11} + \cdots + 758240 \) Copy content Toggle raw display
$31$ \( T^{13} + 5 T^{12} - 156 T^{11} + \cdots - 3424256 \) Copy content Toggle raw display
$37$ \( T^{13} - 4 T^{12} - 213 T^{11} + \cdots - 69995728 \) Copy content Toggle raw display
$41$ \( T^{13} - 28 T^{12} + \cdots - 145363784 \) Copy content Toggle raw display
$43$ \( T^{13} - T^{12} - 314 T^{11} + \cdots - 190258176 \) Copy content Toggle raw display
$47$ \( T^{13} - 12 T^{12} + \cdots - 1017298944 \) Copy content Toggle raw display
$53$ \( T^{13} - 22 T^{12} + \cdots + 7993774176 \) Copy content Toggle raw display
$59$ \( (T + 1)^{13} \) Copy content Toggle raw display
$61$ \( T^{13} + 9 T^{12} + \cdots + 2166846944 \) Copy content Toggle raw display
$67$ \( T^{13} - 26 T^{12} + \cdots + 32421838848 \) Copy content Toggle raw display
$71$ \( T^{13} - 8 T^{12} - 216 T^{11} + \cdots + 41261184 \) Copy content Toggle raw display
$73$ \( T^{13} - 4 T^{12} - 338 T^{11} + \cdots + 994310528 \) Copy content Toggle raw display
$79$ \( T^{13} + 17 T^{12} + \cdots - 1040236960 \) Copy content Toggle raw display
$83$ \( T^{13} - 14 T^{12} + \cdots - 1193708032 \) Copy content Toggle raw display
$89$ \( T^{13} - 19 T^{12} + \cdots + 185798146080 \) Copy content Toggle raw display
$97$ \( T^{13} + 5 T^{12} - 284 T^{11} + \cdots - 39171584 \) Copy content Toggle raw display
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