Properties

Label 4026.2.a.s
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.26825.1
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 5 x^{2} + 5 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 3\)

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
−0.696894
2.92520
−1.68442
1.45612
1.00000 −1.00000 1.00000 −2.78091 −1.00000 −1.12055 1.00000 1.00000 −2.78091
1.2 1.00000 −1.00000 1.00000 −1.14111 −1.00000 −0.293619 1.00000 1.00000 −1.14111
1.3 1.00000 −1.00000 1.00000 3.40047 −1.00000 3.20612 1.00000 1.00000 3.40047
1.4 1.00000 −1.00000 1.00000 3.52154 −1.00000 −3.79195 1.00000 1.00000 3.52154
\(n\): 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.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{4} - 3 T_{5}^{3} - 12 T_{5}^{2} + 25 T_{5} + 38 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 11 T_{7}^{2} - 17 T_{7} - 4 \)
\( T_{13}^{4} - 13 T_{13}^{3} + 34 T_{13}^{2} + 133 T_{13} - 514 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 1 - 3 T + 8 T^{2} - 20 T^{3} + 68 T^{4} - 100 T^{5} + 200 T^{6} - 375 T^{7} + 625 T^{8} \)
$7$ \( 1 + 2 T + 17 T^{2} + 25 T^{3} + 136 T^{4} + 175 T^{5} + 833 T^{6} + 686 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 - 13 T + 86 T^{2} - 374 T^{3} + 1384 T^{4} - 4862 T^{5} + 14534 T^{6} - 28561 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 3 T + 33 T^{2} - 178 T^{3} + 614 T^{4} - 3026 T^{5} + 9537 T^{6} - 14739 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T + 71 T^{2} - 107 T^{3} + 1980 T^{4} - 2033 T^{5} + 25631 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 2 T + 87 T^{2} - 131 T^{3} + 2948 T^{4} - 3013 T^{5} + 46023 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 2 T + 99 T^{2} - 189 T^{3} + 4058 T^{4} - 5481 T^{5} + 83259 T^{6} - 48778 T^{7} + 707281 T^{8} \)
$31$ \( 1 + T + 66 T^{2} - 52 T^{3} + 2298 T^{4} - 1612 T^{5} + 63426 T^{6} + 29791 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 49 T^{2} - 173 T^{3} - 706 T^{4} - 6401 T^{5} + 67081 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 9 T + 169 T^{2} - 1026 T^{3} + 10410 T^{4} - 42066 T^{5} + 284089 T^{6} - 620289 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 20 T + 313 T^{2} + 2989 T^{3} + 23632 T^{4} + 128527 T^{5} + 578737 T^{6} + 1590140 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 19 T + 270 T^{2} - 2616 T^{3} + 20342 T^{4} - 122952 T^{5} + 596430 T^{6} - 1972637 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 8 T + 207 T^{2} - 1227 T^{3} + 16334 T^{4} - 65031 T^{5} + 581463 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 13 T + 110 T^{2} - 288 T^{3} + 134 T^{4} - 16992 T^{5} + 382910 T^{6} - 2669927 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + T )^{4} \)
$67$ \( 1 + 3 T + 188 T^{2} + 240 T^{3} + 16018 T^{4} + 16080 T^{5} + 843932 T^{6} + 902289 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + T + 200 T^{2} + 174 T^{3} + 18314 T^{4} + 12354 T^{5} + 1008200 T^{6} + 357911 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 2 T + 181 T^{2} + 461 T^{3} + 15934 T^{4} + 33653 T^{5} + 964549 T^{6} + 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 19 T + 192 T^{2} - 1652 T^{3} + 15770 T^{4} - 130508 T^{5} + 1198272 T^{6} - 9367741 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 12 T + 307 T^{2} - 2541 T^{3} + 36868 T^{4} - 210903 T^{5} + 2114923 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 19 T + 306 T^{2} - 3602 T^{3} + 41100 T^{4} - 320578 T^{5} + 2423826 T^{6} - 13394411 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + T + 288 T^{2} + 782 T^{3} + 36368 T^{4} + 75854 T^{5} + 2709792 T^{6} + 912673 T^{7} + 88529281 T^{8} \)
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