Properties

Label 4034.2.a.c
Level 4034
Weight 2
Character orbit 4034.a
Self dual yes
Analytic conductor 32.212
Analytic rank 0
Dimension 49
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: \(49\)
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.

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

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 −3.15201 1.00000 −2.19549 3.15201 −1.72689 −1.00000 6.93518 2.19549
1.2 −1.00000 −3.09927 1.00000 −2.61970 3.09927 −2.12795 −1.00000 6.60550 2.61970
1.3 −1.00000 −2.82194 1.00000 −1.46398 2.82194 3.02991 −1.00000 4.96334 1.46398
1.4 −1.00000 −2.80221 1.00000 1.91358 2.80221 −1.55673 −1.00000 4.85238 −1.91358
1.5 −1.00000 −2.75660 1.00000 3.24124 2.75660 3.57367 −1.00000 4.59882 −3.24124
1.6 −1.00000 −2.73587 1.00000 −3.97663 2.73587 4.88134 −1.00000 4.48496 3.97663
1.7 −1.00000 −2.70268 1.00000 1.64140 2.70268 4.14098 −1.00000 4.30446 −1.64140
1.8 −1.00000 −2.62176 1.00000 1.44534 2.62176 −4.09354 −1.00000 3.87363 −1.44534
1.9 −1.00000 −2.38186 1.00000 1.49052 2.38186 −1.41584 −1.00000 2.67326 −1.49052
1.10 −1.00000 −1.96102 1.00000 1.55397 1.96102 −2.12307 −1.00000 0.845608 −1.55397
1.11 −1.00000 −1.82471 1.00000 −3.41474 1.82471 −0.804060 −1.00000 0.329557 3.41474
1.12 −1.00000 −1.80649 1.00000 0.991336 1.80649 3.46798 −1.00000 0.263400 −0.991336
1.13 −1.00000 −1.65034 1.00000 −3.69499 1.65034 0.484734 −1.00000 −0.276387 3.69499
1.14 −1.00000 −1.64240 1.00000 −1.67247 1.64240 −0.216059 −1.00000 −0.302515 1.67247
1.15 −1.00000 −1.33333 1.00000 2.98893 1.33333 1.91777 −1.00000 −1.22223 −2.98893
1.16 −1.00000 −1.09618 1.00000 −0.358556 1.09618 −2.93033 −1.00000 −1.79840 0.358556
1.17 −1.00000 −0.796272 1.00000 −4.15767 0.796272 2.72658 −1.00000 −2.36595 4.15767
1.18 −1.00000 −0.604236 1.00000 −1.43680 0.604236 3.64541 −1.00000 −2.63490 1.43680
1.19 −1.00000 −0.600384 1.00000 0.151444 0.600384 −2.68719 −1.00000 −2.63954 −0.151444
1.20 −1.00000 −0.372757 1.00000 0.124605 0.372757 −2.88406 −1.00000 −2.86105 −0.124605
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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.c 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4034.2.a.c 49 1.a even 1 1 trivial

Atkin-Lehner signs

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