Properties

Label 4034.2.a.d
Level 4034
Weight 2
Character orbit 4034.a
Self dual yes
Analytic conductor 32.212
Analytic rank 0
Dimension 52
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4034 = 2 \cdot 2017 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2116521754\)
Analytic rank: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52q + 52q^{2} + 16q^{3} + 52q^{4} + 24q^{5} + 16q^{6} + 12q^{7} + 52q^{8} + 70q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 52q + 52q^{2} + 16q^{3} + 52q^{4} + 24q^{5} + 16q^{6} + 12q^{7} + 52q^{8} + 70q^{9} + 24q^{10} + 19q^{11} + 16q^{12} + 27q^{13} + 12q^{14} + 5q^{15} + 52q^{16} + 43q^{17} + 70q^{18} + 35q^{19} + 24q^{20} + 29q^{21} + 19q^{22} + 2q^{23} + 16q^{24} + 88q^{25} + 27q^{26} + 49q^{27} + 12q^{28} + 31q^{29} + 5q^{30} + 59q^{31} + 52q^{32} + 45q^{33} + 43q^{34} + 18q^{35} + 70q^{36} + 60q^{37} + 35q^{38} + 6q^{39} + 24q^{40} + 56q^{41} + 29q^{42} + 34q^{43} + 19q^{44} + 61q^{45} + 2q^{46} - 4q^{47} + 16q^{48} + 102q^{49} + 88q^{50} + 23q^{51} + 27q^{52} + 30q^{53} + 49q^{54} + 24q^{55} + 12q^{56} + 32q^{57} + 31q^{58} + 27q^{59} + 5q^{60} + 107q^{61} + 59q^{62} - 4q^{63} + 52q^{64} + 46q^{65} + 45q^{66} + 22q^{67} + 43q^{68} + 36q^{69} + 18q^{70} + 8q^{71} + 70q^{72} + 66q^{73} + 60q^{74} + 53q^{75} + 35q^{76} + 26q^{77} + 6q^{78} + 50q^{79} + 24q^{80} + 108q^{81} + 56q^{82} + 52q^{83} + 29q^{84} + 19q^{85} + 34q^{86} - 32q^{87} + 19q^{88} + 62q^{89} + 61q^{90} + 69q^{91} + 2q^{92} + 21q^{93} - 4q^{94} - 44q^{95} + 16q^{96} + 82q^{97} + 102q^{98} + 16q^{99} + O(q^{100}) \)

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 1.00000 −3.35325 1.00000 −0.0968667 −3.35325 −2.67437 1.00000 8.24428 −0.0968667
1.2 1.00000 −3.11260 1.00000 −0.160277 −3.11260 −3.29184 1.00000 6.68830 −0.160277
1.3 1.00000 −3.09514 1.00000 3.37243 −3.09514 1.17193 1.00000 6.57990 3.37243
1.4 1.00000 −2.86609 1.00000 4.19184 −2.86609 −2.33387 1.00000 5.21449 4.19184
1.5 1.00000 −2.59634 1.00000 −2.79905 −2.59634 1.81461 1.00000 3.74100 −2.79905
1.6 1.00000 −2.58772 1.00000 −1.78395 −2.58772 −0.188589 1.00000 3.69628 −1.78395
1.7 1.00000 −2.51773 1.00000 3.18361 −2.51773 3.34442 1.00000 3.33897 3.18361
1.8 1.00000 −2.49412 1.00000 2.01405 −2.49412 2.26153 1.00000 3.22062 2.01405
1.9 1.00000 −2.36933 1.00000 −2.34107 −2.36933 4.14004 1.00000 2.61374 −2.34107
1.10 1.00000 −1.89942 1.00000 0.311705 −1.89942 2.23532 1.00000 0.607790 0.311705
1.11 1.00000 −1.57271 1.00000 −1.60703 −1.57271 −4.83422 1.00000 −0.526597 −1.60703
1.12 1.00000 −1.53830 1.00000 3.94371 −1.53830 −3.54174 1.00000 −0.633629 3.94371
1.13 1.00000 −1.50477 1.00000 2.73541 −1.50477 −4.93109 1.00000 −0.735674 2.73541
1.14 1.00000 −1.45785 1.00000 0.272726 −1.45785 4.58744 1.00000 −0.874665 0.272726
1.15 1.00000 −1.22639 1.00000 −1.99280 −1.22639 −0.468701 1.00000 −1.49597 −1.99280
1.16 1.00000 −1.04358 1.00000 1.25349 −1.04358 0.828851 1.00000 −1.91094 1.25349
1.17 1.00000 −0.909483 1.00000 3.81385 −0.909483 4.56339 1.00000 −2.17284 3.81385
1.18 1.00000 −0.905567 1.00000 −1.11072 −0.905567 −1.83235 1.00000 −2.17995 −1.11072
1.19 1.00000 −0.570351 1.00000 −1.34990 −0.570351 −5.21632 1.00000 −2.67470 −1.34990
1.20 1.00000 −0.550547 1.00000 0.337407 −0.550547 1.25225 1.00000 −2.69690 0.337407
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
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 4034.2.a.d 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4034.2.a.d 52 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2017\) \(1\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database