Properties

Label 4026.2.a.bc
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 9
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 30 x^{7} + 7 x^{6} + 284 x^{5} + 100 x^{4} - 777 x^{3} - 250 x^{2} + 574 x - 68\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -99 \nu^{8} + 4436 \nu^{7} - 16215 \nu^{6} - 80028 \nu^{5} + 306876 \nu^{4} + 435761 \nu^{3} - 1379713 \nu^{2} - 800938 \nu + 1137130 \)\()/200458\)
\(\beta_{3}\)\(=\)\((\)\( -2961 \nu^{8} + 23336 \nu^{7} + 52616 \nu^{6} - 580331 \nu^{5} - 270479 \nu^{4} + 4358851 \nu^{3} + 711766 \nu^{2} - 8100922 \nu + 1153636 \)\()/400916\)
\(\beta_{4}\)\(=\)\((\)\( 3436 \nu^{8} - 13235 \nu^{7} - 78083 \nu^{6} + 253589 \nu^{5} + 562732 \nu^{4} - 1261001 \nu^{3} - 1100874 \nu^{2} + 1724498 \nu - 168584 \)\()/200458\)
\(\beta_{5}\)\(=\)\((\)\( 7105 \nu^{8} - 16662 \nu^{7} - 187860 \nu^{6} + 285499 \nu^{5} + 1499665 \nu^{4} - 976049 \nu^{3} - 2600248 \nu^{2} + 1017098 \nu - 403440 \)\()/400916\)
\(\beta_{6}\)\(=\)\((\)\( 8041 \nu^{8} - 3932 \nu^{7} - 253236 \nu^{6} - 14833 \nu^{5} + 2370547 \nu^{4} + 1045333 \nu^{3} - 5282530 \nu^{2} - 1141722 \nu + 1565484 \)\()/400916\)
\(\beta_{7}\)\(=\)\((\)\( 4471 \nu^{8} - 4941 \nu^{7} - 127244 \nu^{6} + 33285 \nu^{5} + 1126738 \nu^{4} + 525078 \nu^{3} - 2603901 \nu^{2} - 1838422 \nu + 663210 \)\()/200458\)
\(\beta_{8}\)\(=\)\((\)\( -7907 \nu^{8} + 18176 \nu^{7} + 205327 \nu^{6} - 286874 \nu^{5} - 1689470 \nu^{4} + 735923 \nu^{3} + 3905233 \nu^{2} - 86534 \nu - 1897832 \)\()/200458\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{4} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 12 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(14 \beta_{8} + 17 \beta_{7} - \beta_{6} - 4 \beta_{5} + 15 \beta_{4} - \beta_{3} + \beta_{2} + 24 \beta_{1} + 86\)
\(\nu^{5}\)\(=\)\(23 \beta_{8} + 48 \beta_{7} - 36 \beta_{6} + 33 \beta_{5} - 5 \beta_{4} - 9 \beta_{3} + 14 \beta_{2} + 181 \beta_{1} + 175\)
\(\nu^{6}\)\(=\)\(194 \beta_{8} + 290 \beta_{7} - 47 \beta_{6} - 57 \beta_{5} + 165 \beta_{4} - 42 \beta_{3} + 3 \beta_{2} + 498 \beta_{1} + 1246\)
\(\nu^{7}\)\(=\)\(430 \beta_{8} + 978 \beta_{7} - 583 \beta_{6} + 454 \beta_{5} - 184 \beta_{4} - 273 \beta_{3} + 136 \beta_{2} + 2953 \beta_{1} + 3428\)
\(\nu^{8}\)\(=\)\(2762 \beta_{8} + 5085 \beta_{7} - 1227 \beta_{6} - 593 \beta_{5} + 1332 \beta_{4} - 1178 \beta_{3} - 238 \beta_{2} + 9625 \beta_{1} + 19380\)

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
4.23292
3.88536
1.60053
0.731055
0.128459
−1.48002
−2.09192
−2.63661
−3.36977
1.00000 1.00000 1.00000 −3.23292 1.00000 4.29240 1.00000 1.00000 −3.23292
1.2 1.00000 1.00000 1.00000 −2.88536 1.00000 −2.96592 1.00000 1.00000 −2.88536
1.3 1.00000 1.00000 1.00000 −0.600528 1.00000 0.748033 1.00000 1.00000 −0.600528
1.4 1.00000 1.00000 1.00000 0.268945 1.00000 3.42672 1.00000 1.00000 0.268945
1.5 1.00000 1.00000 1.00000 0.871541 1.00000 1.79155 1.00000 1.00000 0.871541
1.6 1.00000 1.00000 1.00000 2.48002 1.00000 3.05465 1.00000 1.00000 2.48002
1.7 1.00000 1.00000 1.00000 3.09192 1.00000 −4.21340 1.00000 1.00000 3.09192
1.8 1.00000 1.00000 1.00000 3.63661 1.00000 −1.65542 1.00000 1.00000 3.63661
1.9 1.00000 1.00000 1.00000 4.36977 1.00000 4.52139 1.00000 1.00000 4.36977
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.bc 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.bc 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} - \cdots\)
\(T_{13}^{9} - \cdots\)