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Properties

Label 6018.2.a.bc
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 49 x^{12} + 79 x^{11} + 956 x^{10} - 1179 x^{9} - 9396 x^{8} + 8315 x^{7} + \cdots - 43744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 49 x^{12} + 79 x^{11} + 956 x^{10} - 1179 x^{9} - 9396 x^{8} + 8315 x^{7} + \cdots - 43744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16618726754 \nu^{13} - 2880670567959 \nu^{12} - 19196372187770 \nu^{11} + \cdots - 12\!\cdots\!24 ) / 78\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 186379188393 \nu^{13} - 14081190935493 \nu^{12} + 28753457488965 \nu^{11} + \cdots - 14\!\cdots\!28 ) / 78\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 202997915147 \nu^{13} - 16961861503452 \nu^{12} + 9557085301195 \nu^{11} + \cdots - 32\!\cdots\!92 ) / 78\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 236276976171 \nu^{13} + 2329023024781 \nu^{12} + 3942662986183 \nu^{11} + \cdots - 13\!\cdots\!72 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 443751573941 \nu^{13} - 3008483462819 \nu^{12} - 15414700154749 \nu^{11} + \cdots + 13\!\cdots\!84 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2226419771871 \nu^{13} - 14073288977621 \nu^{12} - 88902889968565 \nu^{11} + \cdots - 79\!\cdots\!16 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5727717858944 \nu^{13} + 7373884997349 \nu^{12} + 260235001126690 \nu^{11} + \cdots - 90\!\cdots\!76 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3246512395886 \nu^{13} + 10334588586211 \nu^{12} + 126833659135550 \nu^{11} + \cdots + 17\!\cdots\!76 ) / 78\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10198052289283 \nu^{13} + 32343651313568 \nu^{12} + 416444703490635 \nu^{11} + \cdots - 25\!\cdots\!32 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12229460566967 \nu^{13} - 60219666399172 \nu^{12} - 443909983325205 \nu^{11} + \cdots - 26\!\cdots\!52 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17924326795869 \nu^{13} + 23455557245524 \nu^{12} + 879153061774715 \nu^{11} + \cdots + 16\!\cdots\!64 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21287880786177 \nu^{13} + 51088675909607 \nu^{12} + 889326633697785 \nu^{11} + \cdots + 18\!\cdots\!72 ) / 15\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} + 2\beta_{4} - 3\beta_{2} + 11\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + \cdots + 71 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{13} - 18 \beta_{12} - 23 \beta_{11} - 19 \beta_{10} - 8 \beta_{9} - 20 \beta_{8} - 19 \beta_{7} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 71 \beta_{13} - 25 \beta_{12} - 24 \beta_{11} - 46 \beta_{10} - 52 \beta_{9} - 91 \beta_{8} + 44 \beta_{7} + \cdots + 860 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 310 \beta_{13} - 281 \beta_{12} - 405 \beta_{11} - 292 \beta_{10} - 208 \beta_{9} - 346 \beta_{8} + \cdots + 1252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1320 \beta_{13} - 501 \beta_{12} - 473 \beta_{11} - 784 \beta_{10} - 1058 \beta_{9} - 1652 \beta_{8} + \cdots + 11634 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4995 \beta_{13} - 4282 \beta_{12} - 6505 \beta_{11} - 4184 \beta_{10} - 4174 \beta_{9} - 5947 \beta_{8} + \cdots + 24063 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22756 \beta_{13} - 9271 \beta_{12} - 8770 \beta_{11} - 11800 \beta_{10} - 19844 \beta_{9} + \cdots + 169394 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 81353 \beta_{13} - 65487 \beta_{12} - 100665 \beta_{11} - 57174 \beta_{10} - 77614 \beta_{9} + \cdots + 437040 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 380736 \beta_{13} - 164714 \beta_{12} - 156937 \beta_{11} - 162943 \beta_{10} - 358722 \beta_{9} + \cdots + 2589789 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1339890 \beta_{13} - 1013394 \beta_{12} - 1533614 \beta_{11} - 739948 \beta_{10} - 1402056 \beta_{9} + \cdots + 7704906 \) 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
−3.63246
−3.15319
−3.00862
−2.67453
−1.69357
−1.44259
−0.625391
0.800423
1.26062
2.33625
2.45834
3.24808
3.97677
4.14985
1.00000 −1.00000 1.00000 −3.63246 −1.00000 4.65920 1.00000 1.00000 −3.63246
1.2 1.00000 −1.00000 1.00000 −3.15319 −1.00000 −5.11963 1.00000 1.00000 −3.15319
1.3 1.00000 −1.00000 1.00000 −3.00862 −1.00000 −0.900066 1.00000 1.00000 −3.00862
1.4 1.00000 −1.00000 1.00000 −2.67453 −1.00000 −1.12425 1.00000 1.00000 −2.67453
1.5 1.00000 −1.00000 1.00000 −1.69357 −1.00000 3.25998 1.00000 1.00000 −1.69357
1.6 1.00000 −1.00000 1.00000 −1.44259 −1.00000 2.70884 1.00000 1.00000 −1.44259
1.7 1.00000 −1.00000 1.00000 −0.625391 −1.00000 −1.55533 1.00000 1.00000 −0.625391
1.8 1.00000 −1.00000 1.00000 0.800423 −1.00000 −4.52948 1.00000 1.00000 0.800423
1.9 1.00000 −1.00000 1.00000 1.26062 −1.00000 3.87711 1.00000 1.00000 1.26062
1.10 1.00000 −1.00000 1.00000 2.33625 −1.00000 −0.683869 1.00000 1.00000 2.33625
1.11 1.00000 −1.00000 1.00000 2.45834 −1.00000 −3.89101 1.00000 1.00000 2.45834
1.12 1.00000 −1.00000 1.00000 3.24808 −1.00000 4.67729 1.00000 1.00000 3.24808
1.13 1.00000 −1.00000 1.00000 3.97677 −1.00000 1.13692 1.00000 1.00000 3.97677
1.14 1.00000 −1.00000 1.00000 4.14985 −1.00000 −1.51570 1.00000 1.00000 4.14985
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.bc ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.bc ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{14} - 2 T_{5}^{13} - 49 T_{5}^{12} + 79 T_{5}^{11} + 956 T_{5}^{10} - 1179 T_{5}^{9} + \cdots - 43744 \) Copy content Toggle raw display
\( T_{7}^{14} - T_{7}^{13} - 73 T_{7}^{12} + 66 T_{7}^{11} + 1998 T_{7}^{10} - 1378 T_{7}^{9} + \cdots + 124864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{14} \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots - 43744 \) Copy content Toggle raw display
$7$ \( T^{14} - T^{13} + \cdots + 124864 \) Copy content Toggle raw display
$11$ \( T^{14} - 3 T^{13} + \cdots - 10734080 \) Copy content Toggle raw display
$13$ \( T^{14} - 16 T^{13} + \cdots - 2493184 \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 119325184 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 233633408 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 4977297248 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 169984000 \) Copy content Toggle raw display
$37$ \( T^{14} - 12 T^{13} + \cdots - 22410656 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 3981715600 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 1254891520 \) Copy content Toggle raw display
$47$ \( T^{14} - 338 T^{12} + \cdots + 16384000 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 4295955151168 \) Copy content Toggle raw display
$59$ \( (T - 1)^{14} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 352489546816 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 74154099712 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 75745018880 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 398142320624 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 5235919360 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 5519561541632 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 5815150000 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 133103753344 \) Copy content Toggle raw display
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