Properties

Label 4018.2.a.ba
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 82)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 −1.41421 1.00000 2.82843 −1.41421 0 1.00000 −1.00000 2.82843
1.2 1.00000 1.41421 1.00000 −2.82843 1.41421 0 1.00000 −1.00000 −2.82843
\(n\): 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.ba 2
7.b odd 2 1 82.2.a.b 2
21.c even 2 1 738.2.a.k 2
28.d even 2 1 656.2.a.e 2
35.c odd 2 1 2050.2.a.h 2
35.f even 4 2 2050.2.c.l 4
56.e even 2 1 2624.2.a.l 2
56.h odd 2 1 2624.2.a.j 2
77.b even 2 1 9922.2.a.i 2
84.h odd 2 1 5904.2.a.z 2
287.d odd 2 1 3362.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.2.a.b 2 7.b odd 2 1
656.2.a.e 2 28.d even 2 1
738.2.a.k 2 21.c even 2 1
2050.2.a.h 2 35.c odd 2 1
2050.2.c.l 4 35.f even 4 2
2624.2.a.j 2 56.h odd 2 1
2624.2.a.l 2 56.e even 2 1
3362.2.a.m 2 287.d odd 2 1
4018.2.a.ba 2 1.a even 1 1 trivial
5904.2.a.z 2 84.h odd 2 1
9922.2.a.i 2 77.b even 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 8 \)
\( T_{11}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 4 T^{2} + 9 T^{4} \)
$5$ \( 1 + 2 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 4 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 8 T + 52 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 42 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 48 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 8 T + 132 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 16 T + 178 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 12 T + 176 T^{2} + 948 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 24 T + 278 T^{2} + 1992 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 12 T + 182 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 4 T + 166 T^{2} - 388 T^{3} + 9409 T^{4} \)
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