Properties

Label 4026.2.a.m
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} -2 \beta q^{5} - q^{6} + ( -1 + \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} -2 \beta q^{5} - q^{6} + ( -1 + \beta ) q^{7} + q^{8} + q^{9} -2 \beta q^{10} + q^{11} - q^{12} + 3 \beta q^{13} + ( -1 + \beta ) q^{14} + 2 \beta q^{15} + q^{16} + ( -6 - \beta ) q^{17} + q^{18} + ( -3 - \beta ) q^{19} -2 \beta q^{20} + ( 1 - \beta ) q^{21} + q^{22} + 2 q^{23} - q^{24} + ( -1 + 4 \beta ) q^{25} + 3 \beta q^{26} - q^{27} + ( -1 + \beta ) q^{28} + ( 6 - 6 \beta ) q^{29} + 2 \beta q^{30} + ( -4 + 6 \beta ) q^{31} + q^{32} - q^{33} + ( -6 - \beta ) q^{34} -2 q^{35} + q^{36} + ( 5 + 3 \beta ) q^{37} + ( -3 - \beta ) q^{38} -3 \beta q^{39} -2 \beta q^{40} + ( -6 + 4 \beta ) q^{41} + ( 1 - \beta ) q^{42} + ( -4 + 2 \beta ) q^{43} + q^{44} -2 \beta q^{45} + 2 q^{46} + ( 4 - 3 \beta ) q^{47} - q^{48} + ( -5 - \beta ) q^{49} + ( -1 + 4 \beta ) q^{50} + ( 6 + \beta ) q^{51} + 3 \beta q^{52} + ( -6 + 5 \beta ) q^{53} - q^{54} -2 \beta q^{55} + ( -1 + \beta ) q^{56} + ( 3 + \beta ) q^{57} + ( 6 - 6 \beta ) q^{58} + ( -7 - 3 \beta ) q^{59} + 2 \beta q^{60} - q^{61} + ( -4 + 6 \beta ) q^{62} + ( -1 + \beta ) q^{63} + q^{64} + ( -6 - 6 \beta ) q^{65} - q^{66} + ( -10 + 3 \beta ) q^{67} + ( -6 - \beta ) q^{68} -2 q^{69} -2 q^{70} + ( 6 - 2 \beta ) q^{71} + q^{72} + ( -6 + 4 \beta ) q^{73} + ( 5 + 3 \beta ) q^{74} + ( 1 - 4 \beta ) q^{75} + ( -3 - \beta ) q^{76} + ( -1 + \beta ) q^{77} -3 \beta q^{78} + ( 10 - 5 \beta ) q^{79} -2 \beta q^{80} + q^{81} + ( -6 + 4 \beta ) q^{82} + ( 8 - 12 \beta ) q^{83} + ( 1 - \beta ) q^{84} + ( 2 + 14 \beta ) q^{85} + ( -4 + 2 \beta ) q^{86} + ( -6 + 6 \beta ) q^{87} + q^{88} + ( -14 + 2 \beta ) q^{89} -2 \beta q^{90} + 3 q^{91} + 2 q^{92} + ( 4 - 6 \beta ) q^{93} + ( 4 - 3 \beta ) q^{94} + ( 2 + 8 \beta ) q^{95} - q^{96} + ( -5 - 7 \beta ) q^{97} + ( -5 - \beta ) q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 2q^{11} - 2q^{12} + 3q^{13} - q^{14} + 2q^{15} + 2q^{16} - 13q^{17} + 2q^{18} - 7q^{19} - 2q^{20} + q^{21} + 2q^{22} + 4q^{23} - 2q^{24} + 2q^{25} + 3q^{26} - 2q^{27} - q^{28} + 6q^{29} + 2q^{30} - 2q^{31} + 2q^{32} - 2q^{33} - 13q^{34} - 4q^{35} + 2q^{36} + 13q^{37} - 7q^{38} - 3q^{39} - 2q^{40} - 8q^{41} + q^{42} - 6q^{43} + 2q^{44} - 2q^{45} + 4q^{46} + 5q^{47} - 2q^{48} - 11q^{49} + 2q^{50} + 13q^{51} + 3q^{52} - 7q^{53} - 2q^{54} - 2q^{55} - q^{56} + 7q^{57} + 6q^{58} - 17q^{59} + 2q^{60} - 2q^{61} - 2q^{62} - q^{63} + 2q^{64} - 18q^{65} - 2q^{66} - 17q^{67} - 13q^{68} - 4q^{69} - 4q^{70} + 10q^{71} + 2q^{72} - 8q^{73} + 13q^{74} - 2q^{75} - 7q^{76} - q^{77} - 3q^{78} + 15q^{79} - 2q^{80} + 2q^{81} - 8q^{82} + 4q^{83} + q^{84} + 18q^{85} - 6q^{86} - 6q^{87} + 2q^{88} - 26q^{89} - 2q^{90} + 6q^{91} + 4q^{92} + 2q^{93} + 5q^{94} + 12q^{95} - 2q^{96} - 17q^{97} - 11q^{98} + 2q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −3.23607 −1.00000 0.618034 1.00000 1.00000 −3.23607
1.2 1.00000 −1.00000 1.00000 1.23607 −1.00000 −1.61803 1.00000 1.00000 1.23607
\(n\): 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 4026.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.m 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{7}^{2} + T_{7} - 1 \)
\( T_{13}^{2} - 3 T_{13} - 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + 13 T^{2} + 7 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 3 T + 17 T^{2} - 39 T^{3} + 169 T^{4} \)
$17$ \( 1 + 13 T + 75 T^{2} + 221 T^{3} + 289 T^{4} \)
$19$ \( 1 + 7 T + 49 T^{2} + 133 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 2 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 6 T + 22 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T + 18 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( 1 - 13 T + 105 T^{2} - 481 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 8 T + 78 T^{2} + 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 90 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 5 T + 89 T^{2} - 235 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 7 T + 87 T^{2} + 371 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 17 T + 179 T^{2} + 1003 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( 1 + 17 T + 195 T^{2} + 1139 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 10 T + 162 T^{2} - 710 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 8 T + 142 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 15 T + 183 T^{2} - 1185 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T - 10 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 26 T + 342 T^{2} + 2314 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 17 T + 205 T^{2} + 1649 T^{3} + 9409 T^{4} \)
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