Properties

Label 4026.2.a.x
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.46101901.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 5 x^{4} + 12 x^{3} + 6 x^{2} - 5 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 12 \nu^{2} - 5 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 4 \nu^{4} + 2 \nu^{3} - 16 \nu^{2} + 4 \nu + 6 \)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 37 \nu^{2} - 3 \nu + 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 39 \nu^{2} - \nu + 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-13 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 5 \beta_{2} + 13 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(-42 \beta_{5} + 14 \beta_{4} + 17 \beta_{3} + 24 \beta_{2} + 52 \beta_{1} + 40\)

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
−0.654438
1.90193
3.29072
−1.86911
−0.371781
0.702675
1.00000 −1.00000 1.00000 −4.17718 −1.00000 2.87039 1.00000 1.00000 −4.17718
1.2 1.00000 −1.00000 1.00000 −4.05628 −1.00000 −1.59539 1.00000 1.00000 −4.05628
1.3 1.00000 −1.00000 1.00000 −0.656263 −1.00000 4.00423 1.00000 1.00000 −0.656263
1.4 1.00000 −1.00000 1.00000 0.199215 −1.00000 −0.938542 1.00000 1.00000 0.199215
1.5 1.00000 −1.00000 1.00000 1.28209 −1.00000 −0.306395 1.00000 1.00000 1.28209
1.6 1.00000 −1.00000 1.00000 1.40842 −1.00000 −3.03430 1.00000 1.00000 1.40842
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.x 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{6} + 6 T_{5}^{5} - T_{5}^{4} - 33 T_{5}^{3} + 17 T_{5}^{2} + 18 T_{5} - 4 \)
\( T_{7}^{6} - T_{7}^{5} - 18 T_{7}^{4} + 76 T_{7}^{2} + 75 T_{7} + 16 \)
\( T_{13}^{6} - 2 T_{13}^{5} - 18 T_{13}^{4} + 22 T_{13}^{3} + 69 T_{13}^{2} + 35 T_{13} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( 1 + 6 T + 29 T^{2} + 117 T^{3} + 372 T^{4} + 1023 T^{5} + 2516 T^{6} + 5115 T^{7} + 9300 T^{8} + 14625 T^{9} + 18125 T^{10} + 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 - T + 24 T^{2} - 35 T^{3} + 307 T^{4} - 415 T^{5} + 2648 T^{6} - 2905 T^{7} + 15043 T^{8} - 12005 T^{9} + 57624 T^{10} - 16807 T^{11} + 117649 T^{12} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( 1 - 2 T + 60 T^{2} - 108 T^{3} + 1668 T^{4} - 2487 T^{5} + 27484 T^{6} - 32331 T^{7} + 281892 T^{8} - 237276 T^{9} + 1713660 T^{10} - 742586 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 13 T + 101 T^{2} + 479 T^{3} + 1466 T^{4} + 1913 T^{5} + 885 T^{6} + 32521 T^{7} + 423674 T^{8} + 2353327 T^{9} + 8435621 T^{10} + 18458141 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - T + 20 T^{2} - 113 T^{3} + 113 T^{4} + 189 T^{5} + 7116 T^{6} + 3591 T^{7} + 40793 T^{8} - 775067 T^{9} + 2606420 T^{10} - 2476099 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 11 T + 113 T^{2} + 593 T^{3} + 3232 T^{4} + 9762 T^{5} + 53552 T^{6} + 224526 T^{7} + 1709728 T^{8} + 7215031 T^{9} + 31622033 T^{10} + 70799773 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 14 T + 169 T^{2} + 1529 T^{3} + 11755 T^{4} + 77809 T^{5} + 446222 T^{6} + 2256461 T^{7} + 9885955 T^{8} + 37290781 T^{9} + 119530489 T^{10} + 287156086 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 5 T + 94 T^{2} + 368 T^{3} + 4111 T^{4} + 12623 T^{5} + 133948 T^{6} + 391313 T^{7} + 3950671 T^{8} + 10963088 T^{9} + 86810974 T^{10} + 143145755 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 6 T + 157 T^{2} + 492 T^{3} + 9721 T^{4} + 15403 T^{5} + 396453 T^{6} + 569911 T^{7} + 13308049 T^{8} + 24921276 T^{9} + 294243277 T^{10} + 416063742 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 25 T + 371 T^{2} + 3752 T^{3} + 30437 T^{4} + 211679 T^{5} + 1388622 T^{6} + 8678839 T^{7} + 51164597 T^{8} + 258591592 T^{9} + 1048357331 T^{10} + 2896405025 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 19 T + 289 T^{2} - 2719 T^{3} + 23018 T^{4} - 149938 T^{5} + 1049156 T^{6} - 6447334 T^{7} + 42560282 T^{8} - 216179533 T^{9} + 988033489 T^{10} - 2793160417 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 10 T + 260 T^{2} + 2040 T^{3} + 29132 T^{4} + 178673 T^{5} + 1795972 T^{6} + 8397631 T^{7} + 64352588 T^{8} + 211798920 T^{9} + 1268717060 T^{10} + 2293450070 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 17 T + 370 T^{2} + 4081 T^{3} + 50917 T^{4} + 409681 T^{5} + 3620938 T^{6} + 21713093 T^{7} + 143025853 T^{8} + 607567037 T^{9} + 2919477970 T^{10} + 7109323381 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 14 T + 254 T^{2} + 2290 T^{3} + 25172 T^{4} + 172633 T^{5} + 1628952 T^{6} + 10185347 T^{7} + 87623732 T^{8} + 470317910 T^{9} + 3077809694 T^{10} + 10008940186 T^{11} + 42180533641 T^{12} \)
$61$ \( ( 1 - T )^{6} \)
$67$ \( 1 - 12 T + 260 T^{2} - 2692 T^{3} + 36308 T^{4} - 306087 T^{5} + 2989418 T^{6} - 20507829 T^{7} + 162986612 T^{8} - 809653996 T^{9} + 5239291460 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 6 T + 279 T^{2} - 1265 T^{3} + 36464 T^{4} - 129133 T^{5} + 3056036 T^{6} - 9168443 T^{7} + 183815024 T^{8} - 452757415 T^{9} + 7089858999 T^{10} - 10825376106 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 29 T + 649 T^{2} + 9813 T^{3} + 128754 T^{4} + 1353010 T^{5} + 12719488 T^{6} + 98769730 T^{7} + 686130066 T^{8} + 3817423821 T^{9} + 18430458409 T^{10} + 60119076197 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 24 T + 318 T^{2} + 2904 T^{3} + 29300 T^{4} + 312897 T^{5} + 3138590 T^{6} + 24718863 T^{7} + 182861300 T^{8} + 1431785256 T^{9} + 12386125758 T^{10} + 73849353576 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 9 T - 3 T^{2} - 625 T^{3} + 1196 T^{4} + 19774 T^{5} + 127172 T^{6} + 1641242 T^{7} + 8239244 T^{8} - 357366875 T^{9} - 142374963 T^{10} + 35451365787 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 4 T + 171 T^{2} + 1179 T^{3} + 28542 T^{4} + 151791 T^{5} + 2823324 T^{6} + 13509399 T^{7} + 226081182 T^{8} + 831158451 T^{9} + 10728923211 T^{10} + 22336237796 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 5 T + 334 T^{2} + 1183 T^{3} + 56850 T^{4} + 146998 T^{5} + 6383687 T^{6} + 14258806 T^{7} + 534901650 T^{8} + 1079692159 T^{9} + 29568779854 T^{10} + 42936701285 T^{11} + 832972004929 T^{12} \)
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