Properties

Label 4026.2.a.r
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: 4.4.7537.1
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.

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

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

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

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
2.15976
−2.04717
1.37933
−0.491918
−1.00000 1.00000 1.00000 −1.66454 −1.00000 −2.43525 −1.00000 1.00000 1.66454
1.2 −1.00000 1.00000 1.00000 −1.19091 −1.00000 −0.609175 −1.00000 1.00000 1.19091
1.3 −1.00000 1.00000 1.00000 1.09744 −1.00000 1.89307 −1.00000 1.00000 −1.09744
1.4 −1.00000 1.00000 1.00000 2.75802 −1.00000 −2.84864 −1.00000 1.00000 −2.75802
\(n\): 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.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{4} - T_{5}^{3} - 6 T_{5}^{2} + T_{5} + 6 \)
\( T_{7}^{4} + 4 T_{7}^{3} - T_{7}^{2} - 15 T_{7} - 8 \)
\( T_{13}^{4} + 3 T_{13}^{3} - 32 T_{13}^{2} - 155 T_{13} - 178 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 - T + 14 T^{2} - 14 T^{3} + 96 T^{4} - 70 T^{5} + 350 T^{6} - 125 T^{7} + 625 T^{8} \)
$7$ \( 1 + 4 T + 27 T^{2} + 69 T^{3} + 272 T^{4} + 483 T^{5} + 1323 T^{6} + 1372 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 + 3 T + 20 T^{2} - 38 T^{3} + 4 T^{4} - 494 T^{5} + 3380 T^{6} + 6591 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 3 T + 37 T^{2} + 122 T^{3} + 674 T^{4} + 2074 T^{5} + 10693 T^{6} + 14739 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 4 T + 69 T^{2} + 213 T^{3} + 1898 T^{4} + 4047 T^{5} + 24909 T^{6} + 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 10 T + 89 T^{2} - 413 T^{3} + 2352 T^{4} - 9499 T^{5} + 47081 T^{6} - 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 10 T + 127 T^{2} + 793 T^{3} + 5546 T^{4} + 22997 T^{5} + 106807 T^{6} + 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 9 T + 88 T^{2} + 326 T^{3} + 2380 T^{4} + 10106 T^{5} + 84568 T^{6} + 268119 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 69 T^{2} + 397 T^{3} + 3722 T^{4} + 14689 T^{5} + 94461 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 3 T + 115 T^{2} - 286 T^{3} + 6044 T^{4} - 11726 T^{5} + 193315 T^{6} - 206763 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 18 T + 225 T^{2} + 2065 T^{3} + 15776 T^{4} + 88795 T^{5} + 416025 T^{6} + 1431126 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 21 T + 304 T^{2} - 3062 T^{3} + 24182 T^{4} - 143914 T^{5} + 671536 T^{6} - 2180283 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 20 T + 295 T^{2} + 3005 T^{3} + 25640 T^{4} + 159265 T^{5} + 828655 T^{6} + 2977540 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 5 T + 18 T^{2} + 2 T^{3} + 3718 T^{4} + 118 T^{5} + 62658 T^{6} - 1026895 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 - T )^{4} \)
$67$ \( 1 + 3 T + 266 T^{2} + 598 T^{3} + 26668 T^{4} + 40066 T^{5} + 1194074 T^{6} + 902289 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 9 T + 244 T^{2} + 1588 T^{3} + 24338 T^{4} + 112748 T^{5} + 1230004 T^{6} + 3221199 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 51 T^{2} - 377 T^{3} + 7094 T^{4} - 27521 T^{5} + 271779 T^{6} + 28398241 T^{8} \)
$79$ \( 1 + 31 T + 594 T^{2} + 7924 T^{3} + 81606 T^{4} + 625996 T^{5} + 3707154 T^{6} + 15284209 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 8 T + 267 T^{2} + 1711 T^{3} + 30448 T^{4} + 142013 T^{5} + 1839363 T^{6} + 4574296 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 5 T + 242 T^{2} - 1162 T^{3} + 28506 T^{4} - 103418 T^{5} + 1916882 T^{6} - 3524845 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 25 T + 412 T^{2} + 5726 T^{3} + 65012 T^{4} + 555422 T^{5} + 3876508 T^{6} + 22816825 T^{7} + 88529281 T^{8} \)
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