Properties

Label 8021.2.a.c
Level 8021
Weight 2
Character orbit 8021.a
Self dual yes
Analytic conductor 64.048
Analytic rank 0
Dimension 169
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: \(0\)
Dimension: \(169\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 169q + 9q^{2} + 9q^{3} + 199q^{4} + 12q^{5} + 22q^{6} + 36q^{7} + 30q^{8} + 198q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 169q + 9q^{2} + 9q^{3} + 199q^{4} + 12q^{5} + 22q^{6} + 36q^{7} + 30q^{8} + 198q^{9} - 3q^{10} + 59q^{11} + 11q^{12} - 169q^{13} + 30q^{14} + 50q^{15} + 267q^{16} + q^{17} + 53q^{18} + 107q^{19} + 48q^{20} + 36q^{21} + 14q^{22} + 12q^{23} + 78q^{24} + 217q^{25} - 9q^{26} + 39q^{27} + 99q^{28} + 30q^{29} - q^{30} + 106q^{31} + 74q^{32} + 16q^{33} + 56q^{34} + 46q^{35} + 271q^{36} + 73q^{37} + 2q^{38} - 9q^{39} - 16q^{40} + 52q^{41} - 2q^{42} + 64q^{43} + 124q^{44} + 84q^{45} + 105q^{46} + 55q^{47} + 26q^{48} + 257q^{49} + 60q^{50} + 117q^{51} - 199q^{52} + 7q^{53} + 78q^{54} - 4q^{55} + 63q^{56} + 51q^{57} + 84q^{58} + 98q^{59} + 94q^{60} + 32q^{61} - 25q^{62} + 128q^{63} + 380q^{64} - 12q^{65} + 16q^{66} + 170q^{67} - 10q^{68} + 55q^{69} + 70q^{70} + 124q^{71} + 173q^{72} + 81q^{73} + 54q^{74} + 120q^{75} + 212q^{76} + 20q^{77} - 22q^{78} + 92q^{79} + 66q^{80} + 265q^{81} + 21q^{82} + 62q^{83} + 98q^{84} + 139q^{85} + 51q^{86} - 33q^{87} + 31q^{88} + 58q^{89} + 16q^{90} - 36q^{91} + 40q^{92} + 37q^{93} + 55q^{94} + 23q^{95} + 164q^{96} + 78q^{97} + 69q^{98} + 307q^{99} + O(q^{100}) \)

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 −2.81845 −2.05603 5.94365 −2.67963 5.79481 −3.26986 −11.1150 1.22725 7.55241
1.2 −2.79876 −0.249621 5.83306 3.64528 0.698630 4.03852 −10.7278 −2.93769 −10.2023
1.3 −2.78389 2.33732 5.75006 3.09975 −6.50684 −2.32296 −10.4398 2.46305 −8.62938
1.4 −2.77986 1.10062 5.72762 −3.35511 −3.05957 3.90263 −10.3623 −1.78864 9.32675
1.5 −2.75686 −1.90538 5.60028 −0.896577 5.25287 2.47327 −9.92547 0.630474 2.47174
1.6 −2.73464 1.05458 5.47825 1.58307 −2.88391 −1.67923 −9.51175 −1.88785 −4.32912
1.7 −2.66823 −2.44941 5.11946 1.79166 6.53561 −0.253558 −8.32344 2.99963 −4.78056
1.8 −2.61542 2.30971 4.84044 −1.40185 −6.04088 1.96303 −7.42897 2.33477 3.66644
1.9 −2.56283 −1.23829 4.56811 4.05265 3.17352 4.87248 −6.58165 −1.46665 −10.3863
1.10 −2.52538 0.561190 4.37756 2.21627 −1.41722 −4.18126 −6.00425 −2.68507 −5.59693
1.11 −2.51634 −2.90865 4.33198 −1.46069 7.31915 −4.08327 −5.86807 5.46023 3.67561
1.12 −2.49436 1.22166 4.22181 −2.33739 −3.04726 −2.97287 −5.54199 −1.50754 5.83029
1.13 −2.46202 0.861982 4.06154 2.09605 −2.12222 1.61995 −5.07555 −2.25699 −5.16051
1.14 −2.45702 −3.35722 4.03693 1.56470 8.24873 −2.41292 −5.00476 8.27090 −3.84450
1.15 −2.45553 −0.403768 4.02965 −2.39341 0.991465 2.40759 −4.98387 −2.83697 5.87709
1.16 −2.44727 −2.92033 3.98911 −3.99296 7.14681 3.05388 −4.86787 5.52831 9.77182
1.17 −2.42489 3.03758 3.88008 0.138091 −7.36579 2.88225 −4.55899 6.22689 −0.334855
1.18 −2.41610 −2.76712 3.83752 4.12238 6.68562 0.703742 −4.43962 4.65694 −9.96007
1.19 −2.33994 −2.01648 3.47532 0.773021 4.71844 −3.51939 −3.45216 1.06619 −1.80882
1.20 −2.32290 3.30690 3.39585 1.96073 −7.68158 2.65734 −3.24241 7.93557 −4.55456
See next 80 embeddings (of 169 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.169
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.c 169
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8021.2.a.c 169 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