Properties

Label 4026.2.a.w
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: 6.6.30998405.1
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 9 x^{4} + 8 x^{3} + 16 x^{2} - 13 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 9 \nu^{3} - \nu^{2} - 15 \nu + 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 13 \nu^{2} + 39 \nu - 14 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} - 29 \nu^{3} + 13 \nu^{2} + 59 \nu - 14 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} - 29 \nu^{3} + 17 \nu^{2} + 59 \nu - 26 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} - 6 \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(-\beta_{5} - 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 30 \beta_{1} + 3\)

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
1.70935
0.215842
−1.58506
0.516938
2.64456
−2.50164
−1.00000 −1.00000 1.00000 −3.89759 1.00000 3.22249 −1.00000 1.00000 3.89759
1.2 −1.00000 −1.00000 1.00000 −1.40290 1.00000 2.64146 −1.00000 1.00000 1.40290
1.3 −1.00000 −1.00000 1.00000 −0.713913 1.00000 2.21640 −1.00000 1.00000 0.713913
1.4 −1.00000 −1.00000 1.00000 0.407480 1.00000 −3.39127 −1.00000 1.00000 −0.407480
1.5 −1.00000 −1.00000 1.00000 1.77758 1.00000 2.51943 −1.00000 1.00000 −1.77758
1.6 −1.00000 −1.00000 1.00000 2.82935 1.00000 −2.20851 −1.00000 1.00000 −2.82935
\(n\): 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.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{6} + T_{5}^{5} - 14 T_{5}^{4} - 3 T_{5}^{3} + 32 T_{5}^{2} + 8 T_{5} - 8 \)
\( T_{7}^{6} - 5 T_{7}^{5} - 10 T_{7}^{4} + 82 T_{7}^{3} - 48 T_{7}^{2} - 281 T_{7} + 356 \)
\( T_{13}^{6} - 2 T_{13}^{5} - 44 T_{13}^{4} + 104 T_{13}^{3} + 215 T_{13}^{2} - 457 T_{13} - 94 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( 1 + T + 16 T^{2} + 22 T^{3} + 127 T^{4} + 213 T^{5} + 712 T^{6} + 1065 T^{7} + 3175 T^{8} + 2750 T^{9} + 10000 T^{10} + 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 5 T + 32 T^{2} - 93 T^{3} + 407 T^{4} - 1009 T^{5} + 3604 T^{6} - 7063 T^{7} + 19943 T^{8} - 31899 T^{9} + 76832 T^{10} - 84035 T^{11} + 117649 T^{12} \)
$11$ \( ( 1 - T )^{6} \)
$13$ \( 1 - 2 T + 34 T^{2} - 26 T^{3} + 462 T^{4} + 219 T^{5} + 4820 T^{6} + 2847 T^{7} + 78078 T^{8} - 57122 T^{9} + 971074 T^{10} - 742586 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 4 T + 91 T^{2} + 279 T^{3} + 3523 T^{4} + 8442 T^{5} + 77012 T^{6} + 143514 T^{7} + 1018147 T^{8} + 1370727 T^{9} + 7600411 T^{10} + 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 5 T + 60 T^{2} + 225 T^{3} + 1851 T^{4} + 5521 T^{5} + 37764 T^{6} + 104899 T^{7} + 668211 T^{8} + 1543275 T^{9} + 7819260 T^{10} + 12380495 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 6 T + 79 T^{2} + 341 T^{3} + 3195 T^{4} + 11719 T^{5} + 88522 T^{6} + 269537 T^{7} + 1690155 T^{8} + 4148947 T^{9} + 22107439 T^{10} + 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 10 T + 153 T^{2} + 1149 T^{3} + 9903 T^{4} + 58129 T^{5} + 365798 T^{6} + 1685741 T^{7} + 8328423 T^{8} + 28022961 T^{9} + 108213993 T^{10} + 205111490 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 15 T + 180 T^{2} + 1096 T^{3} + 5509 T^{4} + 9033 T^{5} + 36380 T^{6} + 280023 T^{7} + 5294149 T^{8} + 32650936 T^{9} + 166233780 T^{10} + 429437265 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 3 T + 76 T^{2} + 119 T^{3} + 4159 T^{4} + 5613 T^{5} + 166518 T^{6} + 207681 T^{7} + 5693671 T^{8} + 6027707 T^{9} + 142436236 T^{10} + 208031871 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 9 T + 199 T^{2} + 1200 T^{3} + 16009 T^{4} + 73479 T^{5} + 787486 T^{6} + 3012639 T^{7} + 26911129 T^{8} + 82705200 T^{9} + 562326439 T^{10} + 1042705809 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 10 T + 199 T^{2} - 1345 T^{3} + 15837 T^{4} - 82771 T^{5} + 790822 T^{6} - 3559153 T^{7} + 29282613 T^{8} - 106936915 T^{9} + 680341399 T^{10} - 1470084430 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 14 T + 142 T^{2} + 948 T^{3} + 7946 T^{4} + 74593 T^{5} + 596918 T^{6} + 3505871 T^{7} + 17552714 T^{8} + 98424204 T^{9} + 692914702 T^{10} + 3210830098 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 11 T + 306 T^{2} + 2587 T^{3} + 39037 T^{4} + 257183 T^{5} + 2716878 T^{6} + 13630699 T^{7} + 109654933 T^{8} + 385144799 T^{9} + 2414487186 T^{10} + 4600150423 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 20 T + 388 T^{2} + 5090 T^{3} + 59534 T^{4} + 557963 T^{5} + 4732406 T^{6} + 32919817 T^{7} + 207237854 T^{8} + 1045379110 T^{9} + 4701536068 T^{10} + 14298485980 T^{11} + 42180533641 T^{12} \)
$61$ \( ( 1 - T )^{6} \)
$67$ \( 1 - 14 T + 298 T^{2} - 2778 T^{3} + 37102 T^{4} - 274023 T^{5} + 2934510 T^{6} - 18359541 T^{7} + 166550878 T^{8} - 835519614 T^{9} + 6005034058 T^{10} - 18901751498 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 21 T + 456 T^{2} + 6236 T^{3} + 82897 T^{4} + 822655 T^{5} + 7859980 T^{6} + 58408505 T^{7} + 417883777 T^{8} + 2231932996 T^{9} + 11587726536 T^{10} + 37888816371 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 16 T + 391 T^{2} - 5055 T^{3} + 69949 T^{4} - 676281 T^{5} + 6823446 T^{6} - 49368513 T^{7} + 372758221 T^{8} - 1966480935 T^{9} + 11103712231 T^{10} - 33169145488 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 6 T + 162 T^{2} + 484 T^{3} + 16196 T^{4} + 36935 T^{5} + 1488670 T^{6} + 2917865 T^{7} + 101079236 T^{8} + 238630876 T^{9} + 6309913122 T^{10} + 18462338394 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 10 T + 355 T^{2} + 2481 T^{3} + 55667 T^{4} + 305639 T^{5} + 5550770 T^{6} + 25368037 T^{7} + 383489963 T^{8} + 1418603547 T^{9} + 16847703955 T^{10} + 39390406430 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 11 T + 348 T^{2} + 3384 T^{3} + 51315 T^{4} + 477593 T^{5} + 5047832 T^{6} + 42505777 T^{7} + 406466115 T^{8} + 2385615096 T^{9} + 21834299868 T^{10} + 61424653939 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 2 T + 334 T^{2} - 368 T^{3} + 51532 T^{4} - 38707 T^{5} + 5558388 T^{6} - 3754579 T^{7} + 484864588 T^{8} - 335863664 T^{9} + 29568779854 T^{10} - 17174680514 T^{11} + 832972004929 T^{12} \)
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