Properties

Label 4026.2.a.t
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.2777.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)

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
−1.50848
2.36234
−0.679643
0.825785
1.00000 1.00000 1.00000 −2.78400 1.00000 0.966641 1.00000 1.00000 −2.78400
1.2 1.00000 1.00000 1.00000 −2.21831 1.00000 −1.29741 1.00000 1.00000 −2.21831
1.3 1.00000 1.00000 1.00000 −0.141558 1.00000 −4.12152 1.00000 1.00000 −0.141558
1.4 1.00000 1.00000 1.00000 1.14386 1.00000 −1.54771 1.00000 1.00000 1.14386
\(n\): 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.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.t 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{4} + 4 T_{5}^{3} + T_{5}^{2} - 7 T_{5} - 1 \)
\( T_{7}^{4} + 6 T_{7}^{3} + 7 T_{7}^{2} - 5 T_{7} - 8 \)
\( T_{13}^{4} + 5 T_{13}^{3} - 12 T_{13}^{2} - 39 T_{13} - 22 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 + 4 T + 21 T^{2} + 53 T^{3} + 159 T^{4} + 265 T^{5} + 525 T^{6} + 500 T^{7} + 625 T^{8} \)
$7$ \( 1 + 6 T + 35 T^{2} + 121 T^{3} + 384 T^{4} + 847 T^{5} + 1715 T^{6} + 2058 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 + 5 T + 40 T^{2} + 156 T^{3} + 680 T^{4} + 2028 T^{5} + 6760 T^{6} + 10985 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 10 T + 72 T^{2} + 332 T^{3} + 1521 T^{4} + 5644 T^{5} + 20808 T^{6} + 49130 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 8 T + 77 T^{2} + 373 T^{3} + 2080 T^{4} + 7087 T^{5} + 27797 T^{6} + 54872 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 7 T + 71 T^{2} + 397 T^{3} + 2311 T^{4} + 9131 T^{5} + 37559 T^{6} + 85169 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 4 T + 101 T^{2} + 309 T^{3} + 4178 T^{4} + 8961 T^{5} + 84941 T^{6} + 97556 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 9 T + 74 T^{2} + 220 T^{3} + 1320 T^{4} + 6820 T^{5} + 71114 T^{6} + 268119 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 11 T + 119 T^{2} + 827 T^{3} + 5511 T^{4} + 30599 T^{5} + 162911 T^{6} + 557183 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 17 T + 229 T^{2} + 2170 T^{3} + 15438 T^{4} + 88970 T^{5} + 384949 T^{6} + 1171657 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 11 T + 107 T^{2} + 921 T^{3} + 7617 T^{4} + 39603 T^{5} + 197843 T^{6} + 874577 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 156 T^{2} + 654 T^{3} + 9620 T^{4} + 30738 T^{5} + 344604 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 16 T + 227 T^{2} - 2271 T^{3} + 17942 T^{4} - 120363 T^{5} + 637643 T^{6} - 2382032 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 3 T + 116 T^{2} - 704 T^{3} + 8072 T^{4} - 41536 T^{5} + 403796 T^{6} - 616137 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + T )^{4} \)
$67$ \( 1 - 5 T + 236 T^{2} - 796 T^{3} + 22402 T^{4} - 53332 T^{5} + 1059404 T^{6} - 1503815 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 10 T + 199 T^{2} + 1427 T^{3} + 19545 T^{4} + 101317 T^{5} + 1003159 T^{6} + 3579110 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 9 T + 105 T^{2} + 741 T^{3} + 11403 T^{4} + 54093 T^{5} + 559545 T^{6} + 3501153 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 11 T + 278 T^{2} + 2478 T^{3} + 31770 T^{4} + 195762 T^{5} + 1734998 T^{6} + 5423429 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 5 T + 269 T^{2} - 1083 T^{3} + 32009 T^{4} - 89889 T^{5} + 1853141 T^{6} - 2858935 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 8 T + 309 T^{2} - 1663 T^{3} + 38487 T^{4} - 148007 T^{5} + 2447589 T^{6} - 5639752 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 12 T + 289 T^{2} + 2645 T^{3} + 38579 T^{4} + 256565 T^{5} + 2719201 T^{6} + 10952076 T^{7} + 88529281 T^{8} \)
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