Properties

Label 4026.2.a.o
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: 3.3.1129.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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
−2.39766
−0.440808
2.83847
1.00000 −1.00000 1.00000 −3.39766 −1.00000 −4.14644 1.00000 1.00000 −3.39766
1.2 1.00000 −1.00000 1.00000 −1.44081 −1.00000 3.36488 1.00000 1.00000 −1.44081
1.3 1.00000 −1.00000 1.00000 1.83847 −1.00000 −1.21844 1.00000 1.00000 1.83847
\(n\): 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.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.o 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{3} + 3 T_{5}^{2} - 4 T_{5} - 9 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 13 T_{7} - 17 \)
\( T_{13}^{3} + 7 T_{13}^{2} - 10 T_{13} - 97 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 + 3 T + 11 T^{2} + 21 T^{3} + 55 T^{4} + 75 T^{5} + 125 T^{6} \)
$7$ \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 56 T^{4} + 98 T^{5} + 343 T^{6} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 7 T + 29 T^{2} + 85 T^{3} + 377 T^{4} + 1183 T^{5} + 2197 T^{6} \)
$17$ \( ( 1 - T + 17 T^{2} )^{3} \)
$19$ \( 1 - 2 T + 32 T^{2} - 107 T^{3} + 608 T^{4} - 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 4 T + 48 T^{2} + 103 T^{3} + 1104 T^{4} + 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 8 T + 76 T^{2} + 437 T^{3} + 2204 T^{4} + 6728 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 17 T + 175 T^{2} + 1147 T^{3} + 5425 T^{4} + 16337 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 14 T + 160 T^{2} + 1045 T^{3} + 5920 T^{4} + 19166 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 5 T + 66 T^{2} + 437 T^{3} + 2706 T^{4} + 8405 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 122 T^{2} - 3 T^{3} + 5246 T^{4} + 79507 T^{6} \)
$47$ \( 1 + 5 T + 59 T^{2} + 303 T^{3} + 2773 T^{4} + 11045 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 8 T + 106 T^{2} - 545 T^{3} + 5618 T^{4} - 22472 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 13 T + 127 T^{2} - 733 T^{3} + 7493 T^{4} - 45253 T^{5} + 205379 T^{6} \)
$61$ \( ( 1 + T )^{3} \)
$67$ \( 1 - 17 T + 283 T^{2} - 2371 T^{3} + 18961 T^{4} - 76313 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 17 T + 295 T^{2} + 2507 T^{3} + 20945 T^{4} + 85697 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 10 T + 220 T^{2} - 1349 T^{3} + 16060 T^{4} - 53290 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 7 T + 227 T^{2} + 1009 T^{3} + 17933 T^{4} + 43687 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 14 T + 168 T^{2} + 2063 T^{3} + 13944 T^{4} + 96446 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 23 T + 261 T^{2} + 2357 T^{3} + 23229 T^{4} + 182183 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 7 T + 161 T^{2} + 709 T^{3} + 15617 T^{4} + 65863 T^{5} + 912673 T^{6} \)
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