Properties

Label 8021.2.a.b
Level 8021
Weight 2
Character orbit 8021.a
Self dual yes
Analytic conductor 64.048
Analytic rank 1
Dimension 140
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8021 = 13 \cdot 617 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8021.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0480074613\)
Analytic rank: \(1\)
Dimension: \(140\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 140q - 6q^{2} - 9q^{3} + 112q^{4} - 12q^{5} - 18q^{6} - 32q^{7} - 15q^{8} + 111q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 140q - 6q^{2} - 9q^{3} + 112q^{4} - 12q^{5} - 18q^{6} - 32q^{7} - 15q^{8} + 111q^{9} - q^{10} - 47q^{11} - 11q^{12} - 140q^{13} - 12q^{14} - 30q^{15} + 64q^{16} + 3q^{17} - 22q^{18} - 91q^{19} - 24q^{20} - 28q^{21} - 12q^{22} - 10q^{23} - 42q^{24} + 84q^{25} + 6q^{26} - 33q^{27} - 71q^{28} - 32q^{29} - 45q^{30} - 90q^{31} - 31q^{32} - 32q^{33} - 78q^{34} - 50q^{35} + 10q^{36} - 67q^{37} - 8q^{38} + 9q^{39} - 10q^{40} - 22q^{41} - 30q^{42} - 40q^{43} - 88q^{44} - 36q^{45} - 77q^{46} - 29q^{47} - 4q^{48} + 66q^{49} - 45q^{50} - 87q^{51} - 112q^{52} - 19q^{53} - 82q^{54} - 28q^{55} - 63q^{56} - 41q^{57} - 96q^{58} - 84q^{59} - 106q^{60} - 58q^{61} - 3q^{62} - 76q^{63} - 55q^{64} + 12q^{65} - 56q^{66} - 140q^{67} - 4q^{68} - 41q^{69} - 106q^{70} - 104q^{71} - 52q^{72} - 57q^{73} - 24q^{74} - 62q^{75} - 184q^{76} + 8q^{77} + 18q^{78} - 104q^{79} - 102q^{80} + 4q^{81} - 37q^{82} - 52q^{83} - 94q^{84} - 93q^{85} - 79q^{86} + 51q^{87} - 47q^{88} - 64q^{89} + 22q^{90} + 32q^{91} - 42q^{92} - 115q^{93} - 43q^{94} - 25q^{95} - 116q^{96} - 92q^{97} - 36q^{98} - 223q^{99} + O(q^{100}) \)

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 −2.74251 −0.873664 5.52139 −0.639723 2.39604 −2.37570 −9.65746 −2.23671 1.75445
1.2 −2.72092 2.93492 5.40338 −0.339682 −7.98568 −0.916249 −9.26032 5.61377 0.924245
1.3 −2.66546 1.20306 5.10470 −3.76588 −3.20671 −2.98135 −8.27547 −1.55265 10.0378
1.4 −2.62889 1.78605 4.91107 1.43411 −4.69533 3.37960 −7.65290 0.189969 −3.77012
1.5 −2.61802 −0.518785 4.85402 −1.93655 1.35819 1.44000 −7.47189 −2.73086 5.06994
1.6 −2.58382 −0.446323 4.67613 3.71919 1.15322 −4.18348 −6.91464 −2.80080 −9.60973
1.7 −2.55171 −0.135477 4.51121 −1.86622 0.345697 −2.01089 −6.40787 −2.98165 4.76206
1.8 −2.51611 −0.357802 4.33083 1.59960 0.900270 0.472398 −5.86464 −2.87198 −4.02477
1.9 −2.49455 −1.79114 4.22277 −0.0823474 4.46808 3.62956 −5.54480 0.208181 0.205419
1.10 −2.49397 −2.62257 4.21989 0.472078 6.54061 1.90995 −5.53635 3.87786 −1.17735
1.11 −2.47091 2.70447 4.10542 −1.18660 −6.68251 −3.35758 −5.20231 4.31416 2.93198
1.12 −2.46538 −1.17968 4.07807 2.88368 2.90835 −1.57366 −5.12323 −1.60835 −7.10935
1.13 −2.42591 −2.98941 3.88502 −2.94726 7.25203 −3.20190 −4.57289 5.93657 7.14978
1.14 −2.40289 0.734522 3.77388 0.776538 −1.76498 4.43052 −4.26243 −2.46048 −1.86594
1.15 −2.35872 2.44520 3.56358 2.52211 −5.76755 2.39189 −3.68804 2.97900 −5.94896
1.16 −2.33878 −3.04534 3.46990 −1.89662 7.12240 2.91891 −3.43778 6.27412 4.43579
1.17 −2.27005 2.30637 3.15311 −4.44364 −5.23557 0.498318 −2.61762 2.31934 10.0873
1.18 −2.16089 1.85142 2.66943 −2.55060 −4.00072 2.44468 −1.44657 0.427768 5.51156
1.19 −2.13500 1.79448 2.55821 1.65943 −3.83121 −2.70571 −1.19178 0.220161 −3.54288
1.20 −2.13485 −1.44998 2.55757 −1.78016 3.09549 −0.238675 −1.19032 −0.897552 3.80036
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.140
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 8021.2.a.b 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8021.2.a.b 140 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(617\) \(1\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database