Properties

Label 4018.2.a.bm
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0838915322\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.37108.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 7 x^{2} + 5 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.836038
1.66364
−2.31364
2.48604
1.00000 −2.30104 1.00000 1.83604 −2.30104 0 1.00000 2.29479 1.83604
1.2 1.00000 −0.232297 1.00000 −0.663642 −0.232297 0 1.00000 −2.94604 −0.663642
1.3 1.00000 2.35295 1.00000 3.31364 2.35295 0 1.00000 2.53635 3.31364
1.4 1.00000 3.18039 1.00000 −1.48604 3.18039 0 1.00000 7.11489 −1.48604
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.bm yes 4
7.b odd 2 1 4018.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bl 4 7.b odd 2 1
4018.2.a.bm yes 4 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{4} - 3 T_{3}^{3} - 6 T_{3}^{2} + 16 T_{3} + 4 \)
\( T_{5}^{4} - 3 T_{5}^{3} - 4 T_{5}^{2} + 8 T_{5} + 6 \)
\( T_{11}^{4} - 20 T_{11}^{2} - 10 T_{11} + 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( 1 - 3 T + 6 T^{2} - 11 T^{3} + 22 T^{4} - 33 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8} \)
$5$ \( 1 - 3 T + 16 T^{2} - 37 T^{3} + 116 T^{4} - 185 T^{5} + 400 T^{6} - 375 T^{7} + 625 T^{8} \)
$7$ \( \)
$11$ \( 1 + 24 T^{2} - 10 T^{3} + 294 T^{4} - 110 T^{5} + 2904 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 10 T + 72 T^{2} - 354 T^{3} + 1478 T^{4} - 4602 T^{5} + 12168 T^{6} - 21970 T^{7} + 28561 T^{8} \)
$17$ \( 1 + T + 36 T^{2} + 115 T^{3} + 618 T^{4} + 1955 T^{5} + 10404 T^{6} + 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 14 T + 132 T^{2} - 858 T^{3} + 4310 T^{4} - 16302 T^{5} + 47652 T^{6} - 96026 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 4 T + 52 T^{2} + 208 T^{3} + 1366 T^{4} + 4784 T^{5} + 27508 T^{6} + 48668 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 3 T + 94 T^{2} + 259 T^{3} + 3776 T^{4} + 7511 T^{5} + 79054 T^{6} + 73167 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 3 T + 102 T^{2} - 251 T^{3} + 4394 T^{4} - 7781 T^{5} + 98022 T^{6} - 89373 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 14 T + 184 T^{2} + 1434 T^{3} + 10654 T^{4} + 53058 T^{5} + 251896 T^{6} + 709142 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - T )^{4} \)
$43$ \( 1 - T + 112 T^{2} + 127 T^{3} + 5646 T^{4} + 5461 T^{5} + 207088 T^{6} - 79507 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T + 96 T^{2} + 662 T^{3} + 4158 T^{4} + 31114 T^{5} + 212064 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 11 T + 112 T^{2} - 723 T^{3} + 6700 T^{4} - 38319 T^{5} + 314608 T^{6} - 1637647 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 2 T + 154 T^{2} + 560 T^{3} + 11126 T^{4} + 33040 T^{5} + 536074 T^{6} + 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 15 T + 262 T^{2} - 2421 T^{3} + 24360 T^{4} - 147681 T^{5} + 974902 T^{6} - 3404715 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 4 T + 192 T^{2} - 618 T^{3} + 17954 T^{4} - 41406 T^{5} + 861888 T^{6} - 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 15 T + 340 T^{2} - 3251 T^{3} + 38214 T^{4} - 230821 T^{5} + 1713940 T^{6} - 5368665 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 4 T + 228 T^{2} - 652 T^{3} + 23142 T^{4} - 47596 T^{5} + 1215012 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 15 T + 292 T^{2} - 3043 T^{3} + 34422 T^{4} - 240397 T^{5} + 1822372 T^{6} - 7395585 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 12 T + 286 T^{2} - 2126 T^{3} + 31506 T^{4} - 176458 T^{5} + 1970254 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 35 T + 790 T^{2} - 11589 T^{3} + 128770 T^{4} - 1031421 T^{5} + 6257590 T^{6} - 24673915 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 5 T + 202 T^{2} + 555 T^{3} + 24202 T^{4} + 53835 T^{5} + 1900618 T^{6} + 4563365 T^{7} + 88529281 T^{8} \)
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