Properties

Label 4029.2.a.c
Level 4029
Weight 2
Character orbit 4029.a
Self dual yes
Analytic conductor 32.172
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + ( -1 + \beta ) q^{5} + \beta q^{6} + q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + ( -1 + \beta ) q^{5} + \beta q^{6} + q^{7} -2 \beta q^{8} + q^{9} + ( 2 - \beta ) q^{10} -4 \beta q^{11} + \beta q^{14} + ( -1 + \beta ) q^{15} -4 q^{16} + q^{17} + \beta q^{18} + ( -5 + 2 \beta ) q^{19} + q^{21} -8 q^{22} + ( -3 + 3 \beta ) q^{23} -2 \beta q^{24} + ( -2 - 2 \beta ) q^{25} + q^{27} + ( 2 + 2 \beta ) q^{29} + ( 2 - \beta ) q^{30} + ( 4 - 4 \beta ) q^{31} -4 \beta q^{33} + \beta q^{34} + ( -1 + \beta ) q^{35} + ( -5 + 4 \beta ) q^{37} + ( 4 - 5 \beta ) q^{38} + ( -4 + 2 \beta ) q^{40} + ( 2 + 4 \beta ) q^{41} + \beta q^{42} + ( 2 - 6 \beta ) q^{43} + ( -1 + \beta ) q^{45} + ( 6 - 3 \beta ) q^{46} + ( -3 - 5 \beta ) q^{47} -4 q^{48} -6 q^{49} + ( -4 - 2 \beta ) q^{50} + q^{51} + ( -11 - \beta ) q^{53} + \beta q^{54} + ( -8 + 4 \beta ) q^{55} -2 \beta q^{56} + ( -5 + 2 \beta ) q^{57} + ( 4 + 2 \beta ) q^{58} + ( -5 + \beta ) q^{59} + ( 5 - 4 \beta ) q^{61} + ( -8 + 4 \beta ) q^{62} + q^{63} + 8 q^{64} -8 q^{66} -10 q^{67} + ( -3 + 3 \beta ) q^{69} + ( 2 - \beta ) q^{70} + ( -6 - 5 \beta ) q^{71} -2 \beta q^{72} + ( 2 - 9 \beta ) q^{73} + ( 8 - 5 \beta ) q^{74} + ( -2 - 2 \beta ) q^{75} -4 \beta q^{77} - q^{79} + ( 4 - 4 \beta ) q^{80} + q^{81} + ( 8 + 2 \beta ) q^{82} + ( 4 + \beta ) q^{83} + ( -1 + \beta ) q^{85} + ( -12 + 2 \beta ) q^{86} + ( 2 + 2 \beta ) q^{87} + 16 q^{88} + ( -12 + 4 \beta ) q^{89} + ( 2 - \beta ) q^{90} + ( 4 - 4 \beta ) q^{93} + ( -10 - 3 \beta ) q^{94} + ( 9 - 7 \beta ) q^{95} + ( 16 + \beta ) q^{97} -6 \beta q^{98} -4 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{10} - 2q^{15} - 8q^{16} + 2q^{17} - 10q^{19} + 2q^{21} - 16q^{22} - 6q^{23} - 4q^{25} + 2q^{27} + 4q^{29} + 4q^{30} + 8q^{31} - 2q^{35} - 10q^{37} + 8q^{38} - 8q^{40} + 4q^{41} + 4q^{43} - 2q^{45} + 12q^{46} - 6q^{47} - 8q^{48} - 12q^{49} - 8q^{50} + 2q^{51} - 22q^{53} - 16q^{55} - 10q^{57} + 8q^{58} - 10q^{59} + 10q^{61} - 16q^{62} + 2q^{63} + 16q^{64} - 16q^{66} - 20q^{67} - 6q^{69} + 4q^{70} - 12q^{71} + 4q^{73} + 16q^{74} - 4q^{75} - 2q^{79} + 8q^{80} + 2q^{81} + 16q^{82} + 8q^{83} - 2q^{85} - 24q^{86} + 4q^{87} + 32q^{88} - 24q^{89} + 4q^{90} + 8q^{93} - 20q^{94} + 18q^{95} + 32q^{97} + O(q^{100}) \)

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
−1.41421 1.00000 0 −2.41421 −1.41421 1.00000 2.82843 1.00000 3.41421
1.2 1.41421 1.00000 0 0.414214 1.41421 1.00000 −2.82843 1.00000 0.585786
\(n\): 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 4029.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.c 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\( T_{2}^{2} - 2 \)
\( T_{5}^{2} + 2 T_{5} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 2 T + 9 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 - 10 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( ( 1 - T )^{2} \)
$19$ \( 1 + 10 T + 55 T^{2} + 190 T^{3} + 361 T^{4} \)
$23$ \( 1 + 6 T + 37 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 54 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 46 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 10 T + 67 T^{2} + 370 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 18 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T + 53 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 22 T + 225 T^{2} + 1166 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 141 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 115 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 10 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 12 T + 128 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T - 12 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( 1 - 8 T + 180 T^{2} - 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 24 T + 290 T^{2} + 2136 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 32 T + 448 T^{2} - 3104 T^{3} + 9409 T^{4} \)
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