Properties

Label 4009.2.a.f
Level 4009
Weight 2
Character orbit 4009.a
Self dual yes
Analytic conductor 32.012
Analytic rank 0
Dimension 83
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 83q + 11q^{2} + 95q^{4} + 15q^{5} + 23q^{6} + 19q^{7} + 30q^{8} + 101q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 83q + 11q^{2} + 95q^{4} + 15q^{5} + 23q^{6} + 19q^{7} + 30q^{8} + 101q^{9} + 9q^{10} + 56q^{11} - 2q^{12} - 5q^{13} + 6q^{14} + 19q^{15} + 123q^{16} + 19q^{17} + 40q^{18} - 83q^{19} + 49q^{20} + 9q^{21} + 18q^{22} + 74q^{23} + 38q^{24} + 98q^{25} + 28q^{26} + 6q^{27} + 50q^{28} + 16q^{29} + 56q^{30} + 24q^{31} + 81q^{32} + 13q^{33} + 9q^{34} + 71q^{35} + 156q^{36} - 6q^{37} - 11q^{38} + 126q^{39} + q^{40} - q^{42} + 34q^{43} + 140q^{44} + 42q^{45} + 34q^{46} + 53q^{47} + 16q^{48} + 118q^{49} + 51q^{50} + 57q^{51} + 32q^{52} + q^{53} + 53q^{54} + 60q^{55} - 2q^{56} - 2q^{58} + 44q^{59} - 9q^{60} + 21q^{61} + 28q^{62} + 83q^{63} + 154q^{64} + 44q^{65} + 17q^{66} + 5q^{67} + 63q^{68} - 36q^{69} - 48q^{70} + 193q^{71} + 135q^{72} + 54q^{73} + 127q^{74} + 5q^{75} - 95q^{76} + 54q^{77} + 45q^{78} + 54q^{79} + 45q^{80} + 147q^{81} - 35q^{82} + 84q^{83} + 12q^{84} + 28q^{85} + 60q^{86} + 51q^{87} + 23q^{88} - 24q^{89} + 31q^{90} + 28q^{91} + 108q^{92} + 39q^{93} - 49q^{94} - 15q^{95} + 25q^{96} - 22q^{97} - 67q^{98} + 132q^{99} + O(q^{100}) \)

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 −2.76212 −1.26441 5.62933 −0.901770 3.49245 2.01058 −10.0247 −1.40127 2.49080
1.2 −2.70959 1.36959 5.34187 3.33123 −3.71102 3.43608 −9.05510 −1.12423 −9.02626
1.3 −2.67158 −3.12490 5.13731 1.64338 8.34840 −0.166545 −8.38157 6.76498 −4.39041
1.4 −2.57059 1.65554 4.60795 −0.615415 −4.25572 −3.21164 −6.70397 −0.259188 1.58198
1.5 −2.54651 2.15718 4.48471 0.595796 −5.49327 3.08751 −6.32733 1.65341 −1.51720
1.6 −2.52441 2.53119 4.37265 −2.23047 −6.38976 −2.32994 −5.98955 3.40692 5.63062
1.7 −2.39927 −1.08640 3.75651 4.25844 2.60656 3.20578 −4.21434 −1.81974 −10.2171
1.8 −2.36270 −3.38908 3.58234 0.417258 8.00738 4.46721 −3.73860 8.48588 −0.985855
1.9 −2.35956 −1.58207 3.56753 −1.28760 3.73299 −1.93582 −3.69869 −0.497055 3.03817
1.10 −2.30870 −1.28180 3.33008 1.74419 2.95929 −4.60684 −3.07074 −1.35699 −4.02679
1.11 −2.26307 −2.97799 3.12147 −3.73061 6.73939 −2.90027 −2.53796 5.86843 8.44263
1.12 −2.15681 −0.442776 2.65181 −2.84036 0.954981 0.461477 −1.40583 −2.80395 6.12610
1.13 −2.15273 1.11488 2.63426 −2.91632 −2.40003 1.19746 −1.36540 −1.75705 6.27807
1.14 −1.98014 0.389885 1.92094 2.34447 −0.772025 −3.40620 0.156544 −2.84799 −4.64237
1.15 −1.90192 0.698740 1.61729 0.858137 −1.32894 1.83968 0.727887 −2.51176 −1.63210
1.16 −1.77808 −1.50294 1.16155 1.25808 2.67234 4.62687 1.49083 −0.741168 −2.23696
1.17 −1.73978 2.70968 1.02684 3.17028 −4.71425 1.26105 1.69309 4.34237 −5.51560
1.18 −1.71565 −0.772476 0.943456 −1.16427 1.32530 −0.902518 1.81266 −2.40328 1.99748
1.19 −1.59861 3.13645 0.555538 2.23749 −5.01395 4.84396 2.30912 6.83733 −3.57686
1.20 −1.56314 0.463979 0.443399 3.91464 −0.725264 −3.83453 2.43318 −2.78472 −6.11913
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
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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 4009.2.a.f 83
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.f 83 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(19\) \(1\)
\(211\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{83} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).