Properties

Label 4018.2.a.bs
Level $4018$
Weight $2$
Character orbit 4018.a
Self dual yes
Analytic conductor $32.084$
Analytic rank $0$
Dimension $10$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 17x^{8} + 36x^{7} + 75x^{6} - 174x^{5} - 69x^{4} + 260x^{3} - 104x^{2} - 24x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 17x^{8} + 36x^{7} + 75x^{6} - 174x^{5} - 69x^{4} + 260x^{3} - 104x^{2} - 24x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13 \nu^{9} - 18 \nu^{8} - 234 \nu^{7} + 330 \nu^{6} + 1201 \nu^{5} - 1610 \nu^{4} - 1896 \nu^{3} + 2442 \nu^{2} - 18 \nu - 330 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 114 \nu^{9} + 167 \nu^{8} + 2023 \nu^{7} - 3005 \nu^{6} - 10109 \nu^{5} + 14188 \nu^{4} + 15490 \nu^{3} - 20714 \nu^{2} + 230 \nu + 2842 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 117 \nu^{9} - 169 \nu^{8} - 2086 \nu^{7} + 3064 \nu^{6} + 10513 \nu^{5} - 14673 \nu^{4} - 16220 \nu^{3} + 21804 \nu^{2} - 358 \nu - 2996 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24 \nu^{9} + 35 \nu^{8} + 426 \nu^{7} - 630 \nu^{6} - 2130 \nu^{5} + 2975 \nu^{4} + 3266 \nu^{3} - 4344 \nu^{2} + 52 \nu + 594 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 249 \nu^{9} + 362 \nu^{8} + 4424 \nu^{7} - 6528 \nu^{6} - 22157 \nu^{5} + 30938 \nu^{4} + 34004 \nu^{3} - 45404 \nu^{2} + 614 \nu + 6230 ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 261 \nu^{9} + 377 \nu^{8} + 4641 \nu^{7} - 6799 \nu^{6} - 23290 \nu^{5} + 32206 \nu^{4} + 35916 \nu^{3} - 47216 \nu^{2} + 356 \nu + 6496 ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 270 \nu^{9} + 390 \nu^{8} + 4809 \nu^{7} - 7053 \nu^{6} - 24201 \nu^{5} + 33605 \nu^{4} + 37378 \nu^{3} - 49632 \nu^{2} + 558 \nu + 6832 ) / 14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41 \nu^{9} - 60 \nu^{8} - 727 \nu^{7} + 1079 \nu^{6} + 3628 \nu^{5} - 5087 \nu^{4} - 5552 \nu^{3} + 7416 \nu^{2} - 84 \nu - 1018 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 319 \nu^{9} - 453 \nu^{8} - 5698 \nu^{7} + 8222 \nu^{6} + 28835 \nu^{5} - 39401 \nu^{4} - 44840 \nu^{3} + 58592 \nu^{2} - 474 \nu - 8050 ) / 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{6} + \beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{9} - \beta_{7} + 5\beta_{6} + 4\beta_{5} - 6\beta_{4} - \beta_{3} - 2\beta_{2} - 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - 9\beta_{8} + 3\beta_{7} - 13\beta_{6} + \beta_{5} + 14\beta_{3} + \beta_{2} - 7\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38 \beta_{9} + 6 \beta_{8} - 15 \beta_{7} + 54 \beta_{6} + 40 \beta_{5} - 60 \beta_{4} - 17 \beta_{3} - 11 \beta_{2} - 35 \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 33 \beta_{9} - 82 \beta_{8} + 44 \beta_{7} - 148 \beta_{6} + 10 \beta_{5} + 12 \beta_{4} + 159 \beta_{3} + 18 \beta_{2} - 52 \beta _1 + 265 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 382 \beta_{9} + 107 \beta_{8} - 174 \beta_{7} + 586 \beta_{6} + 406 \beta_{5} - 603 \beta_{4} - 220 \beta_{3} - 73 \beta_{2} - 323 \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 438 \beta_{9} - 789 \beta_{8} + 529 \beta_{7} - 1643 \beta_{6} + 45 \beta_{5} + 266 \beta_{4} + 1712 \beta_{3} + 245 \beta_{2} - 399 \beta _1 + 2520 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3933 \beta_{9} + 1434 \beta_{8} - 1905 \beta_{7} + 6349 \beta_{6} + 4129 \beta_{5} - 6123 \beta_{4} - 2655 \beta_{3} - 584 \beta_{2} - 3076 \beta _1 - 947 \) 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
−3.30199
−0.339114
0.835626
3.06975
1.92458
0.560789
−1.84058
2.39018
0.553933
−1.85317
−1.00000 −2.26016 1.00000 0.772879 2.26016 0 −1.00000 2.10833 −0.772879
1.2 −1.00000 −2.13421 1.00000 0.374303 2.13421 0 −1.00000 1.55487 −0.374303
1.3 −1.00000 −1.96764 1.00000 −1.63790 1.96764 0 −1.00000 0.871591 1.63790
1.4 −1.00000 −0.539617 1.00000 3.38661 0.539617 0 −1.00000 −2.70881 −3.38661
1.5 −1.00000 0.280978 1.00000 −0.181985 −0.280978 0 −1.00000 −2.92105 0.181985
1.6 −1.00000 0.748903 1.00000 −1.86270 −0.748903 0 −1.00000 −2.43914 1.86270
1.7 −1.00000 1.68800 1.00000 4.36119 −1.68800 0 −1.00000 −0.150660 −4.36119
1.8 −1.00000 2.25734 1.00000 −3.63252 −2.25734 0 −1.00000 2.09559 3.63252
1.9 −1.00000 2.84461 1.00000 0.100034 −2.84461 0 −1.00000 5.09179 −0.100034
1.10 −1.00000 3.08180 1.00000 2.32009 −3.08180 0 −1.00000 6.49750 −2.32009
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(41\) \(-1\)

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 4018.2.a.bs yes 10
7.b odd 2 1 4018.2.a.br 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.br 10 7.b odd 2 1
4018.2.a.bs yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{10} - 4T_{3}^{9} - 12T_{3}^{8} + 56T_{3}^{7} + 41T_{3}^{6} - 264T_{3}^{5} - 2T_{3}^{4} + 444T_{3}^{3} - 148T_{3}^{2} - 120T_{3} + 36 \) Copy content Toggle raw display
\( T_{5}^{10} - 4T_{5}^{9} - 20T_{5}^{8} + 76T_{5}^{7} + 101T_{5}^{6} - 340T_{5}^{5} - 160T_{5}^{4} + 380T_{5}^{3} - 76T_{5}^{2} - 16T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{10} - 4 T_{11}^{9} - 54 T_{11}^{8} + 184 T_{11}^{7} + 1010 T_{11}^{6} - 2468 T_{11}^{5} - 8350 T_{11}^{4} + 9456 T_{11}^{3} + 26288 T_{11}^{2} + 6960 T_{11} - 2104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 4 T^{9} - 12 T^{8} + 56 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} - 20 T^{8} + 76 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} - 54 T^{8} + \cdots - 2104 \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} - 46 T^{8} + 92 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{10} - 20 T^{9} + 100 T^{8} + \cdots + 196 \) Copy content Toggle raw display
$19$ \( T^{10} - 134 T^{8} + 220 T^{7} + \cdots + 17408 \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} - 76 T^{8} + 124 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{10} + 4 T^{9} - 136 T^{8} + \cdots + 53074 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} - 112 T^{8} + \cdots - 35064 \) Copy content Toggle raw display
$37$ \( T^{10} + 16 T^{9} - 12 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T - 1)^{10} \) Copy content Toggle raw display
$43$ \( T^{10} + 8 T^{9} - 274 T^{8} + \cdots + 11188592 \) Copy content Toggle raw display
$47$ \( T^{10} - 24 T^{9} - 54 T^{8} + \cdots + 14804368 \) Copy content Toggle raw display
$53$ \( T^{10} + 4 T^{9} - 262 T^{8} + \cdots + 5222914 \) Copy content Toggle raw display
$59$ \( T^{10} - 198 T^{8} - 124 T^{7} + \cdots + 32 \) Copy content Toggle raw display
$61$ \( T^{10} - 230 T^{8} - 36 T^{7} + \cdots - 4567934 \) Copy content Toggle raw display
$67$ \( T^{10} - 8 T^{9} - 200 T^{8} + \cdots - 1390336 \) Copy content Toggle raw display
$71$ \( T^{10} - 4 T^{9} - 226 T^{8} + \cdots - 192224 \) Copy content Toggle raw display
$73$ \( T^{10} + 24 T^{9} - 190 T^{8} + \cdots - 13190272 \) Copy content Toggle raw display
$79$ \( T^{10} - 24 T^{9} - 238 T^{8} + \cdots + 5358464 \) Copy content Toggle raw display
$83$ \( T^{10} - 48 T^{9} + 718 T^{8} + \cdots + 61592456 \) Copy content Toggle raw display
$89$ \( T^{10} - 20 T^{9} + \cdots - 4627391888 \) Copy content Toggle raw display
$97$ \( T^{10} - 4 T^{9} - 514 T^{8} + \cdots - 1767296 \) Copy content Toggle raw display
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