Properties

Label 6001.2.a.b
Level $6001$
Weight $2$
Character orbit 6001.a
Self dual yes
Analytic conductor $47.918$
Analytic rank $1$
Dimension $114$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6001 = 17 \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9182262530\)
Analytic rank: \(1\)
Dimension: \(114\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 114q - 8q^{2} - 23q^{3} + 110q^{4} - 27q^{5} - 23q^{6} - 53q^{7} - 21q^{8} + 107q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 114q - 8q^{2} - 23q^{3} + 110q^{4} - 27q^{5} - 23q^{6} - 53q^{7} - 21q^{8} + 107q^{9} - 19q^{10} - 52q^{11} - 49q^{12} - 12q^{13} - 40q^{14} - 39q^{15} + 110q^{16} + 114q^{17} - 21q^{18} - 30q^{19} - 88q^{20} - 30q^{21} - 36q^{22} - 77q^{23} - 72q^{24} + 119q^{25} - 79q^{26} - 77q^{27} - 92q^{28} - 65q^{29} - 10q^{30} - 131q^{31} - 30q^{32} - 12q^{33} - 8q^{34} - 33q^{35} + 109q^{36} - 54q^{37} - 14q^{38} - 83q^{39} - 42q^{40} - 99q^{41} + 29q^{42} + 4q^{43} - 98q^{44} - 73q^{45} - 35q^{46} - 113q^{47} - 86q^{48} + 101q^{49} - 44q^{50} - 23q^{51} - 3q^{52} - 18q^{53} - 78q^{54} - 63q^{55} - 117q^{56} - 64q^{57} - 31q^{58} - 134q^{59} - 6q^{60} - 30q^{61} - 30q^{62} - 154q^{63} + 117q^{64} - 66q^{65} - 12q^{66} - 34q^{67} + 110q^{68} - 35q^{69} + 18q^{70} - 233q^{71} + 16q^{72} - 56q^{73} - 64q^{74} - 100q^{75} - 64q^{76} - 6q^{77} + 50q^{78} - 154q^{79} - 128q^{80} + 118q^{81} + 2q^{82} - 53q^{83} - 6q^{84} - 27q^{85} - 52q^{86} - 22q^{87} - 52q^{88} - 118q^{89} - 5q^{90} - 95q^{91} - 102q^{92} + 47q^{93} - 3q^{94} - 158q^{95} - 144q^{96} - 57q^{97} + 3q^{98} - 131q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.79736 1.48474 5.82522 0.639507 −4.15334 0.101329 −10.7005 −0.795561 −1.78893
1.2 −2.75915 3.04798 5.61289 −2.57270 −8.40983 −0.275912 −9.96848 6.29019 7.09847
1.3 −2.75487 −1.38395 5.58934 1.31563 3.81261 −4.81832 −9.88817 −1.08468 −3.62439
1.4 −2.73023 −3.10574 5.45415 −3.71411 8.47938 1.88828 −9.43061 6.64562 10.1404
1.5 −2.69250 0.578947 5.24958 4.35539 −1.55882 −1.96736 −8.74952 −2.66482 −11.7269
1.6 −2.68479 0.238039 5.20809 −3.86690 −0.639084 2.11314 −8.61305 −2.94334 10.3818
1.7 −2.61975 −2.31872 4.86309 0.216177 6.07447 3.38406 −7.50057 2.37647 −0.566328
1.8 −2.56595 1.01181 4.58410 2.09874 −2.59626 3.20271 −6.63067 −1.97623 −5.38525
1.9 −2.39336 0.882221 3.72817 0.123030 −2.11147 −1.77301 −4.13613 −2.22169 −0.294456
1.10 −2.39289 −2.96659 3.72594 3.31044 7.09873 0.664206 −4.12999 5.80065 −7.92153
1.11 −2.38632 −0.731332 3.69453 1.80073 1.74519 −2.93364 −4.04370 −2.46515 −4.29713
1.12 −2.36622 −1.16111 3.59897 −3.78541 2.74743 −1.24944 −3.78352 −1.65183 8.95709
1.13 −2.31496 −0.975130 3.35906 −3.23089 2.25739 5.09566 −3.14618 −2.04912 7.47940
1.14 −2.31236 3.26187 3.34699 −1.15202 −7.54261 −3.58394 −3.11471 7.63982 2.66388
1.15 −2.29894 0.676824 3.28511 −3.14176 −1.55598 −3.61058 −2.95439 −2.54191 7.22271
1.16 −2.28982 −3.10307 3.24328 2.16467 7.10548 −1.63956 −2.84688 6.62906 −4.95671
1.17 −2.18122 1.64876 2.75771 −2.64324 −3.59630 −4.67385 −1.65274 −0.281605 5.76549
1.18 −2.12375 2.77384 2.51031 −3.45627 −5.89093 3.25214 −1.08377 4.69417 7.34024
1.19 −2.10824 −2.66682 2.44469 −3.46040 5.62230 −1.09952 −0.937516 4.11192 7.29537
1.20 −2.09110 −1.81594 2.37271 1.01486 3.79732 2.98461 −0.779370 0.297648 −2.12218
See next 80 embeddings (of 114 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.114
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(353\) \(-1\)

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 6001.2.a.b 114
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.b 114 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database