Properties

Label 4026.2.a.v
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 5
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(5\)
Coefficient field: 5.5.11492689.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 9 x^{3} + 13 x^{2} + 18 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 9 \nu^{2} - \nu + 10 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 7 \nu^{2} - 11 \nu - 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{3} + 9 \beta_{2} + 10 \beta_{1} + 26\)

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
−2.42642
−1.26875
0.371979
2.34840
2.97479
1.00000 −1.00000 1.00000 −1.42642 −1.00000 −4.31394 1.00000 1.00000 −1.42642
1.2 1.00000 −1.00000 1.00000 −0.268751 −1.00000 1.12152 1.00000 1.00000 −0.268751
1.3 1.00000 −1.00000 1.00000 1.37198 −1.00000 4.23361 1.00000 1.00000 1.37198
1.4 1.00000 −1.00000 1.00000 3.34840 −1.00000 0.833399 1.00000 1.00000 3.34840
1.5 1.00000 −1.00000 1.00000 3.97479 −1.00000 −1.87458 1.00000 1.00000 3.97479
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.v 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{5} - 7 T_{5}^{4} + 9 T_{5}^{3} + 18 T_{5}^{2} - 22 T_{5} - 7 \)
\( T_{7}^{5} - 21 T_{7}^{3} + 3 T_{7}^{2} + 50 T_{7} - 32 \)
\( T_{13}^{5} + 2 T_{13}^{4} - 31 T_{13}^{3} - 23 T_{13}^{2} + 243 T_{13} - 158 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( 1 - 7 T + 34 T^{2} - 122 T^{3} + 363 T^{4} - 877 T^{5} + 1815 T^{6} - 3050 T^{7} + 4250 T^{8} - 4375 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 14 T^{2} + 3 T^{3} + 99 T^{4} + 10 T^{5} + 693 T^{6} + 147 T^{7} + 4802 T^{8} + 16807 T^{10} \)
$11$ \( ( 1 - T )^{5} \)
$13$ \( 1 + 2 T + 34 T^{2} + 81 T^{3} + 724 T^{4} + 1272 T^{5} + 9412 T^{6} + 13689 T^{7} + 74698 T^{8} + 57122 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 2 T + 53 T^{2} - 110 T^{3} + 1457 T^{4} - 2648 T^{5} + 24769 T^{6} - 31790 T^{7} + 260389 T^{8} - 167042 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 18 T + 200 T^{2} - 1571 T^{3} + 9599 T^{4} - 46478 T^{5} + 182381 T^{6} - 567131 T^{7} + 1371800 T^{8} - 2345778 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + T - 4 T^{2} + 200 T^{3} + 740 T^{4} - 1946 T^{5} + 17020 T^{6} + 105800 T^{7} - 48668 T^{8} + 279841 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 7 T + 130 T^{2} - 728 T^{3} + 7066 T^{4} - 30476 T^{5} + 204914 T^{6} - 612248 T^{7} + 3170570 T^{8} - 4950967 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 100 T^{2} - 67 T^{3} + 5228 T^{4} - 2620 T^{5} + 162068 T^{6} - 64387 T^{7} + 2979100 T^{8} + 28629151 T^{10} \)
$37$ \( 1 - 11 T + 124 T^{2} - 916 T^{3} + 7078 T^{4} - 39344 T^{5} + 261886 T^{6} - 1254004 T^{7} + 6280972 T^{8} - 20615771 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 8 T + 97 T^{2} - 558 T^{3} + 6329 T^{4} - 36766 T^{5} + 259489 T^{6} - 937998 T^{7} + 6685337 T^{8} - 22606088 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 7 T + 180 T^{2} + 1100 T^{3} + 14130 T^{4} + 69078 T^{5} + 607590 T^{6} + 2033900 T^{7} + 14311260 T^{8} + 23931607 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - T + 73 T^{2} + 141 T^{3} + 1206 T^{4} + 19128 T^{5} + 56682 T^{6} + 311469 T^{7} + 7579079 T^{8} - 4879681 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 10 T + 144 T^{2} - 1327 T^{3} + 13765 T^{4} - 89890 T^{5} + 729545 T^{6} - 3727543 T^{7} + 21438288 T^{8} - 78904810 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 8 T + 228 T^{2} - 1307 T^{3} + 23296 T^{4} - 104348 T^{5} + 1374464 T^{6} - 4549667 T^{7} + 46826412 T^{8} - 96938888 T^{9} + 714924299 T^{10} \)
$61$ \( ( 1 - T )^{5} \)
$67$ \( 1 + 9 T + 331 T^{2} + 2311 T^{3} + 44086 T^{4} + 229080 T^{5} + 2953762 T^{6} + 10374079 T^{7} + 99552553 T^{8} + 181360089 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 34 T + 622 T^{2} - 8033 T^{3} + 82242 T^{4} - 728020 T^{5} + 5839182 T^{6} - 40494353 T^{7} + 222620642 T^{8} - 863997154 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 13 T + 170 T^{2} - 2170 T^{3} + 22014 T^{4} - 162898 T^{5} + 1607022 T^{6} - 11563930 T^{7} + 66132890 T^{8} - 369177133 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 15 T + 349 T^{2} - 3743 T^{3} + 51768 T^{4} - 414980 T^{5} + 4089672 T^{6} - 23360063 T^{7} + 172070611 T^{8} - 584251215 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 27 T + 376 T^{2} + 3000 T^{3} + 16368 T^{4} + 90878 T^{5} + 1358544 T^{6} + 20667000 T^{7} + 214991912 T^{8} + 1281374667 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 11 T + 286 T^{2} - 2392 T^{3} + 41643 T^{4} - 296243 T^{5} + 3706227 T^{6} - 18947032 T^{7} + 201621134 T^{8} - 690164651 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 37 T + 838 T^{2} - 13892 T^{3} + 182723 T^{4} - 1965609 T^{5} + 17724131 T^{6} - 130709828 T^{7} + 764819974 T^{8} - 3275583397 T^{9} + 8587340257 T^{10} \)
show more
show less