Properties

Label 4013.2.a.c
Level 4013
Weight 2
Character orbit 4013.a
Self dual Yes
Analytic conductor 32.044
Analytic rank 0
Dimension 176
CM No

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Newspace parameters

Level: \( N \) = \( 4013 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4013.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0439663311\)
Analytic rank: \(0\)
Dimension: \(176\)
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.

\(\operatorname{Tr}(f)(q) = \) \(176q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 53q^{3} \) \(\mathstrut +\mathstrut 191q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut 46q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 193q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(176q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 53q^{3} \) \(\mathstrut +\mathstrut 191q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut 46q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 193q^{9} \) \(\mathstrut +\mathstrut 43q^{10} \) \(\mathstrut +\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 95q^{12} \) \(\mathstrut +\mathstrut 95q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 36q^{15} \) \(\mathstrut +\mathstrut 225q^{16} \) \(\mathstrut +\mathstrut 35q^{17} \) \(\mathstrut +\mathstrut 46q^{18} \) \(\mathstrut +\mathstrut 127q^{19} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut +\mathstrut 32q^{21} \) \(\mathstrut +\mathstrut 60q^{22} \) \(\mathstrut +\mathstrut 35q^{23} \) \(\mathstrut +\mathstrut 26q^{24} \) \(\mathstrut +\mathstrut 207q^{25} \) \(\mathstrut +\mathstrut 19q^{26} \) \(\mathstrut +\mathstrut 191q^{27} \) \(\mathstrut +\mathstrut 87q^{28} \) \(\mathstrut +\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 28q^{30} \) \(\mathstrut +\mathstrut 93q^{31} \) \(\mathstrut +\mathstrut 73q^{32} \) \(\mathstrut +\mathstrut 70q^{33} \) \(\mathstrut +\mathstrut 45q^{34} \) \(\mathstrut +\mathstrut 73q^{35} \) \(\mathstrut +\mathstrut 206q^{36} \) \(\mathstrut +\mathstrut 64q^{37} \) \(\mathstrut +\mathstrut 35q^{38} \) \(\mathstrut +\mathstrut 72q^{39} \) \(\mathstrut +\mathstrut 139q^{40} \) \(\mathstrut +\mathstrut 19q^{41} \) \(\mathstrut +\mathstrut 35q^{42} \) \(\mathstrut +\mathstrut 261q^{43} \) \(\mathstrut +\mathstrut 11q^{44} \) \(\mathstrut +\mathstrut 12q^{45} \) \(\mathstrut +\mathstrut 58q^{46} \) \(\mathstrut +\mathstrut 40q^{47} \) \(\mathstrut +\mathstrut 130q^{48} \) \(\mathstrut +\mathstrut 234q^{49} \) \(\mathstrut -\mathstrut 14q^{50} \) \(\mathstrut +\mathstrut 76q^{51} \) \(\mathstrut +\mathstrut 263q^{52} \) \(\mathstrut +\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut 28q^{54} \) \(\mathstrut +\mathstrut 170q^{55} \) \(\mathstrut -\mathstrut 10q^{56} \) \(\mathstrut +\mathstrut 60q^{57} \) \(\mathstrut +\mathstrut 52q^{58} \) \(\mathstrut +\mathstrut 69q^{59} \) \(\mathstrut +\mathstrut 37q^{60} \) \(\mathstrut +\mathstrut 110q^{61} \) \(\mathstrut +\mathstrut 71q^{62} \) \(\mathstrut +\mathstrut 101q^{63} \) \(\mathstrut +\mathstrut 250q^{64} \) \(\mathstrut -\mathstrut q^{65} \) \(\mathstrut +\mathstrut 43q^{66} \) \(\mathstrut +\mathstrut 190q^{67} \) \(\mathstrut +\mathstrut 48q^{68} \) \(\mathstrut +\mathstrut 45q^{69} \) \(\mathstrut +\mathstrut 14q^{70} \) \(\mathstrut +\mathstrut 9q^{71} \) \(\mathstrut +\mathstrut 98q^{72} \) \(\mathstrut +\mathstrut 182q^{73} \) \(\mathstrut -\mathstrut 23q^{74} \) \(\mathstrut +\mathstrut 219q^{75} \) \(\mathstrut +\mathstrut 197q^{76} \) \(\mathstrut +\mathstrut 25q^{77} \) \(\mathstrut -\mathstrut 26q^{78} \) \(\mathstrut +\mathstrut 105q^{79} \) \(\mathstrut +\mathstrut 20q^{80} \) \(\mathstrut +\mathstrut 236q^{81} \) \(\mathstrut +\mathstrut 107q^{82} \) \(\mathstrut +\mathstrut 130q^{83} \) \(\mathstrut +\mathstrut 38q^{84} \) \(\mathstrut +\mathstrut 73q^{85} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut +\mathstrut 171q^{87} \) \(\mathstrut +\mathstrut 165q^{88} \) \(\mathstrut +\mathstrut 40q^{89} \) \(\mathstrut +\mathstrut 45q^{90} \) \(\mathstrut +\mathstrut 182q^{91} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut +\mathstrut 23q^{93} \) \(\mathstrut +\mathstrut 98q^{94} \) \(\mathstrut +\mathstrut 30q^{95} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 168q^{97} \) \(\mathstrut +\mathstrut 82q^{98} \) \(\mathstrut +\mathstrut 50q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.79865 3.21303 5.83245 0.484406 −8.99216 2.55250 −10.7257 7.32359 −1.35568
1.2 −2.76303 −0.741544 5.63435 1.49530 2.04891 0.791604 −10.0418 −2.45011 −4.13156
1.3 −2.71973 2.24397 5.39695 −3.35299 −6.10299 −2.43693 −9.23880 2.03538 9.11924
1.4 −2.69276 −0.212283 5.25096 −3.83789 0.571627 2.56921 −8.75407 −2.95494 10.3345
1.5 −2.69142 −1.48564 5.24374 2.79908 3.99849 −0.286125 −8.73026 −0.792861 −7.53350
1.6 −2.65007 −1.98668 5.02285 −4.07833 5.26483 0.415430 −8.01074 0.946892 10.8079
1.7 −2.63729 1.99604 4.95530 −2.90741 −5.26413 3.48548 −7.79400 0.984162 7.66769
1.8 −2.62564 0.652884 4.89398 −2.04233 −1.71424 −2.37529 −7.59855 −2.57374 5.36241
1.9 −2.61853 0.324920 4.85671 3.20033 −0.850813 −1.73793 −7.48039 −2.89443 −8.38017
1.10 −2.61670 −2.37899 4.84710 −0.969862 6.22508 5.09491 −7.45000 2.65957 2.53783
1.11 −2.59980 1.10973 4.75898 2.90792 −2.88507 4.77007 −7.17280 −1.76851 −7.56002
1.12 −2.57761 −2.77945 4.64406 −1.52510 7.16432 −0.558594 −6.81534 4.72533 3.93111
1.13 −2.53190 1.30897 4.41050 −0.932957 −3.31419 0.385559 −6.10313 −1.28658 2.36215
1.14 −2.43054 3.05350 3.90754 −3.60333 −7.42166 −4.38981 −4.63635 6.32386 8.75804
1.15 −2.42541 2.84636 3.88263 −0.259504 −6.90359 3.45507 −4.56617 5.10174 0.629405
1.16 −2.39764 −0.316927 3.74866 −2.45861 0.759875 1.72271 −4.19265 −2.89956 5.89485
1.17 −2.39761 0.688358 3.74852 1.63631 −1.65041 −2.07745 −4.19226 −2.52616 −3.92323
1.18 −2.37116 2.70615 3.62242 4.34995 −6.41674 1.61584 −3.84703 4.32327 −10.3145
1.19 −2.28450 2.00074 3.21893 3.40444 −4.57068 0.591134 −2.78465 1.00294 −7.77744
1.20 −2.28329 3.29602 3.21343 1.63708 −7.52578 −3.76844 −2.77062 7.86375 −3.73794
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.176
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(4013\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{176} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\).