Properties

Label 4018.2.a.bq
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.5163008.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 5 x^{2} - 6 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 7 \nu^{2} + 6 \nu - 3 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} + 5 \nu - 5 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{5} - 3 \nu^{4} - 11 \nu^{3} + 10 \nu^{2} + 13 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 5 \beta_{4} - 7 \beta_{3} + \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 8 \beta_{4} - 11 \beta_{3} + 7 \beta_{2} + 28 \beta_{1} + 10\)

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
−1.78566
1.60043
2.56754
0.545336
0.218114
−1.14577
1.00000 −2.71387 1.00000 3.56002 −2.71387 0 1.00000 4.36512 3.56002
1.2 1.00000 −1.82475 1.00000 2.37517 −1.82475 0 1.00000 0.329701 2.37517
1.3 1.00000 0.0387751 1.00000 2.61052 0.0387751 0 1.00000 −2.99850 2.61052
1.4 1.00000 1.93139 1.00000 1.16627 1.93139 0 1.00000 0.730254 1.16627
1.5 1.00000 3.26089 1.00000 −1.58475 3.26089 0 1.00000 7.63338 −1.58475
1.6 1.00000 3.30757 1.00000 3.87277 3.30757 0 1.00000 7.94005 3.87277
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bq yes 6
7.b odd 2 1 4018.2.a.bp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bp 6 7.b odd 2 1
4018.2.a.bq yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{6} - 4 T_{3}^{5} - 10 T_{3}^{4} + 44 T_{3}^{3} + 20 T_{3}^{2} - 104 T_{3} + 4 \)
\( T_{5}^{6} - 12 T_{5}^{5} + 50 T_{5}^{4} - 68 T_{5}^{3} - 68 T_{5}^{2} + 248 T_{5} - 158 \)
\( T_{11}^{6} - 50 T_{11}^{4} + 8 T_{11}^{3} + 498 T_{11}^{2} - 300 T_{11} - 398 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 35 T^{4} - 68 T^{5} + 124 T^{6} - 204 T^{7} + 315 T^{8} - 432 T^{9} + 648 T^{10} - 972 T^{11} + 729 T^{12} \)
$5$ \( 1 - 12 T + 80 T^{2} - 368 T^{3} + 1307 T^{4} - 3772 T^{5} + 9162 T^{6} - 18860 T^{7} + 32675 T^{8} - 46000 T^{9} + 50000 T^{10} - 37500 T^{11} + 15625 T^{12} \)
$7$ 1
$11$ \( 1 + 16 T^{2} + 8 T^{3} + 113 T^{4} - 36 T^{5} + 878 T^{6} - 396 T^{7} + 13673 T^{8} + 10648 T^{9} + 234256 T^{10} + 1771561 T^{12} \)
$13$ \( 1 - 8 T + 72 T^{2} - 404 T^{3} + 2275 T^{4} - 9484 T^{5} + 38732 T^{6} - 123292 T^{7} + 384475 T^{8} - 887588 T^{9} + 2056392 T^{10} - 2970344 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 16 T + 156 T^{2} - 1092 T^{3} + 6311 T^{4} - 31084 T^{5} + 136444 T^{6} - 528428 T^{7} + 1823879 T^{8} - 5364996 T^{9} + 13029276 T^{10} - 22717712 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 12 T + 128 T^{2} - 952 T^{3} + 6203 T^{4} - 32708 T^{5} + 157212 T^{6} - 621452 T^{7} + 2239283 T^{8} - 6529768 T^{9} + 16681088 T^{10} - 29713188 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 12 T + 134 T^{2} - 992 T^{3} + 7183 T^{4} - 40036 T^{5} + 214808 T^{6} - 920828 T^{7} + 3799807 T^{8} - 12069664 T^{9} + 37498694 T^{10} - 77236116 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 96 T^{2} + 48 T^{3} + 4853 T^{4} + 3300 T^{5} + 165406 T^{6} + 95700 T^{7} + 4081373 T^{8} + 1170672 T^{9} + 67898976 T^{10} + 594823321 T^{12} \)
$31$ \( 1 + 4 T + 96 T^{2} + 492 T^{3} + 5139 T^{4} + 23896 T^{5} + 194656 T^{6} + 740776 T^{7} + 4938579 T^{8} + 14657172 T^{9} + 88658016 T^{10} + 114516604 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 24 T + 398 T^{2} + 4600 T^{3} + 43895 T^{4} + 339152 T^{5} + 2260068 T^{6} + 12548624 T^{7} + 60092255 T^{8} + 233003800 T^{9} + 745916078 T^{10} + 1664254968 T^{11} + 2565726409 T^{12} \)
$41$ \( ( 1 + T )^{6} \)
$43$ \( 1 - 4 T + 150 T^{2} - 708 T^{3} + 11959 T^{4} - 55888 T^{5} + 614084 T^{6} - 2403184 T^{7} + 22112191 T^{8} - 56290956 T^{9} + 512820150 T^{10} - 588033772 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 16 T + 256 T^{2} + 2652 T^{3} + 27239 T^{4} + 215852 T^{5} + 1650068 T^{6} + 10145044 T^{7} + 60170951 T^{8} + 275338596 T^{9} + 1249198336 T^{10} + 3669520112 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 16 T + 234 T^{2} + 1596 T^{3} + 10025 T^{4} + 6328 T^{5} + 37970 T^{6} + 335384 T^{7} + 28160225 T^{8} + 237607692 T^{9} + 1846372554 T^{10} + 6691127888 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 16 T + 120 T^{2} - 1140 T^{3} + 13555 T^{4} - 89804 T^{5} + 500490 T^{6} - 5298436 T^{7} + 47184955 T^{8} - 234132060 T^{9} + 1454083320 T^{10} - 11438788784 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 32 T + 726 T^{2} - 11300 T^{3} + 143695 T^{4} - 1456668 T^{5} + 12552758 T^{6} - 88856748 T^{7} + 534689095 T^{8} - 2564885300 T^{9} + 10052080566 T^{10} - 27027081632 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 20 T + 362 T^{2} + 4304 T^{3} + 50513 T^{4} + 485936 T^{5} + 4399546 T^{6} + 32557712 T^{7} + 226752857 T^{8} + 1294483952 T^{9} + 7294705802 T^{10} + 27002502140 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 12 T + 334 T^{2} - 2596 T^{3} + 45983 T^{4} - 276920 T^{5} + 3969444 T^{6} - 19661320 T^{7} + 231800303 T^{8} - 929136956 T^{9} + 8487501454 T^{10} - 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 16 T + 512 T^{2} - 5856 T^{3} + 101027 T^{4} - 855728 T^{5} + 10072096 T^{6} - 62468144 T^{7} + 538372883 T^{8} - 2278083552 T^{9} + 14539899392 T^{10} - 33169145488 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 386 T^{2} - 288 T^{3} + 65775 T^{4} - 67872 T^{5} + 6560348 T^{6} - 5361888 T^{7} + 410501775 T^{8} - 141995232 T^{9} + 15034731266 T^{10} + 243087455521 T^{12} \)
$83$ \( 1 - 32 T + 632 T^{2} - 9580 T^{3} + 121787 T^{4} - 1345308 T^{5} + 13167754 T^{6} - 111660564 T^{7} + 838990643 T^{8} - 5477719460 T^{9} + 29993658872 T^{10} - 126049300576 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 8 T + 406 T^{2} - 3056 T^{3} + 76583 T^{4} - 508264 T^{5} + 8599604 T^{6} - 45235496 T^{7} + 606613943 T^{8} - 2154385264 T^{9} + 25473349846 T^{10} - 44672475592 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 8 T + 142 T^{2} - 584 T^{3} + 14639 T^{4} - 14000 T^{5} + 886340 T^{6} - 1358000 T^{7} + 137738351 T^{8} - 533001032 T^{9} + 12571157902 T^{10} - 68698722056 T^{11} + 832972004929 T^{12} \)
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