Properties

Label 4034.2.a.a
Level 4034
Weight 2
Character orbit 4034.a
Self dual yes
Analytic conductor 32.212
Analytic rank 1
Dimension 33
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: \(1\)
Dimension: \(33\)
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.

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

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 −3.41599 1.00000 1.09379 −3.41599 4.04655 1.00000 8.66901 1.09379
1.2 1.00000 −3.31212 1.00000 −3.97668 −3.31212 0.855437 1.00000 7.97011 −3.97668
1.3 1.00000 −2.82298 1.00000 0.779916 −2.82298 −3.19115 1.00000 4.96923 0.779916
1.4 1.00000 −2.73960 1.00000 −3.73363 −2.73960 −3.98248 1.00000 4.50543 −3.73363
1.5 1.00000 −2.69679 1.00000 −2.92996 −2.69679 3.32033 1.00000 4.27265 −2.92996
1.6 1.00000 −2.68746 1.00000 1.22075 −2.68746 −1.79428 1.00000 4.22244 1.22075
1.7 1.00000 −2.63161 1.00000 1.62454 −2.63161 2.40683 1.00000 3.92538 1.62454
1.8 1.00000 −2.03082 1.00000 −0.733478 −2.03082 −1.85114 1.00000 1.12424 −0.733478
1.9 1.00000 −1.80963 1.00000 0.856694 −1.80963 0.353536 1.00000 0.274766 0.856694
1.10 1.00000 −1.73259 1.00000 3.77683 −1.73259 0.621551 1.00000 0.00185264 3.77683
1.11 1.00000 −1.72838 1.00000 −2.88558 −1.72838 −0.799535 1.00000 −0.0126910 −2.88558
1.12 1.00000 −1.65119 1.00000 −2.90851 −1.65119 0.175106 1.00000 −0.273565 −2.90851
1.13 1.00000 −1.42842 1.00000 2.01766 −1.42842 0.289927 1.00000 −0.959611 2.01766
1.14 1.00000 −0.945997 1.00000 −2.44770 −0.945997 2.03235 1.00000 −2.10509 −2.44770
1.15 1.00000 −0.523973 1.00000 1.37573 −0.523973 −3.92552 1.00000 −2.72545 1.37573
1.16 1.00000 −0.423572 1.00000 −0.487122 −0.423572 2.40385 1.00000 −2.82059 −0.487122
1.17 1.00000 −0.366256 1.00000 0.145528 −0.366256 3.13381 1.00000 −2.86586 0.145528
1.18 1.00000 −0.289410 1.00000 −3.81688 −0.289410 −2.88122 1.00000 −2.91624 −3.81688
1.19 1.00000 −0.264302 1.00000 2.82441 −0.264302 −2.96778 1.00000 −2.93014 2.82441
1.20 1.00000 −0.138152 1.00000 −3.80380 −0.138152 3.68849 1.00000 −2.98091 −3.80380
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
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.a 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4034.2.a.a 33 1.a even 1 1 trivial

Atkin-Lehner signs

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