Properties

Label 4007.2.a.b
Level 4007
Weight 2
Character orbit 4007.a
Self dual Yes
Analytic conductor 31.996
Analytic rank 0
Dimension 195
CM No

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Newspace parameters

Level: \( N \) = \( 4007 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9960560899\)
Analytic rank: \(0\)
Dimension: \(195\)
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.

\(\operatorname{Tr}(f)(q) = \) \(195q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut +\mathstrut 22q^{3} \) \(\mathstrut +\mathstrut 220q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 48q^{7} \) \(\mathstrut +\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 245q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(195q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut +\mathstrut 22q^{3} \) \(\mathstrut +\mathstrut 220q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 48q^{7} \) \(\mathstrut +\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 245q^{9} \) \(\mathstrut +\mathstrut 40q^{10} \) \(\mathstrut +\mathstrut 13q^{11} \) \(\mathstrut +\mathstrut 57q^{12} \) \(\mathstrut +\mathstrut 97q^{13} \) \(\mathstrut +\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 23q^{15} \) \(\mathstrut +\mathstrut 270q^{16} \) \(\mathstrut +\mathstrut 66q^{17} \) \(\mathstrut +\mathstrut 60q^{18} \) \(\mathstrut +\mathstrut 33q^{19} \) \(\mathstrut +\mathstrut 24q^{20} \) \(\mathstrut +\mathstrut 27q^{21} \) \(\mathstrut +\mathstrut 127q^{22} \) \(\mathstrut +\mathstrut 42q^{23} \) \(\mathstrut +\mathstrut 33q^{24} \) \(\mathstrut +\mathstrut 357q^{25} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 79q^{27} \) \(\mathstrut +\mathstrut 131q^{28} \) \(\mathstrut +\mathstrut 57q^{29} \) \(\mathstrut +\mathstrut 51q^{30} \) \(\mathstrut +\mathstrut 41q^{31} \) \(\mathstrut +\mathstrut 67q^{32} \) \(\mathstrut +\mathstrut 74q^{33} \) \(\mathstrut +\mathstrut 26q^{34} \) \(\mathstrut +\mathstrut 14q^{35} \) \(\mathstrut +\mathstrut 279q^{36} \) \(\mathstrut +\mathstrut 133q^{37} \) \(\mathstrut +\mathstrut 7q^{38} \) \(\mathstrut +\mathstrut 28q^{39} \) \(\mathstrut +\mathstrut 97q^{40} \) \(\mathstrut +\mathstrut 64q^{41} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut +\mathstrut 123q^{43} \) \(\mathstrut +\mathstrut 21q^{44} \) \(\mathstrut +\mathstrut 40q^{45} \) \(\mathstrut +\mathstrut 84q^{46} \) \(\mathstrut +\mathstrut 14q^{47} \) \(\mathstrut +\mathstrut 122q^{48} \) \(\mathstrut +\mathstrut 335q^{49} \) \(\mathstrut +\mathstrut 35q^{50} \) \(\mathstrut +\mathstrut 37q^{51} \) \(\mathstrut +\mathstrut 220q^{52} \) \(\mathstrut +\mathstrut 83q^{53} \) \(\mathstrut +\mathstrut 21q^{54} \) \(\mathstrut +\mathstrut 47q^{55} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut +\mathstrut 235q^{57} \) \(\mathstrut +\mathstrut 138q^{58} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 81q^{61} \) \(\mathstrut +\mathstrut 39q^{62} \) \(\mathstrut +\mathstrut 102q^{63} \) \(\mathstrut +\mathstrut 343q^{64} \) \(\mathstrut +\mathstrut 165q^{65} \) \(\mathstrut -\mathstrut 54q^{66} \) \(\mathstrut +\mathstrut 147q^{67} \) \(\mathstrut +\mathstrut 74q^{68} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 13q^{70} \) \(\mathstrut +\mathstrut 31q^{71} \) \(\mathstrut +\mathstrut 112q^{72} \) \(\mathstrut +\mathstrut 300q^{73} \) \(\mathstrut +\mathstrut 5q^{74} \) \(\mathstrut +\mathstrut 84q^{75} \) \(\mathstrut +\mathstrut 64q^{76} \) \(\mathstrut +\mathstrut 67q^{77} \) \(\mathstrut +\mathstrut 61q^{78} \) \(\mathstrut +\mathstrut 144q^{79} \) \(\mathstrut -\mathstrut 8q^{80} \) \(\mathstrut +\mathstrut 359q^{81} \) \(\mathstrut +\mathstrut 85q^{82} \) \(\mathstrut +\mathstrut 26q^{83} \) \(\mathstrut +\mathstrut 12q^{84} \) \(\mathstrut +\mathstrut 201q^{85} \) \(\mathstrut -\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 80q^{87} \) \(\mathstrut +\mathstrut 347q^{88} \) \(\mathstrut +\mathstrut 29q^{89} \) \(\mathstrut +\mathstrut 62q^{90} \) \(\mathstrut +\mathstrut 70q^{91} \) \(\mathstrut +\mathstrut 79q^{92} \) \(\mathstrut +\mathstrut 76q^{93} \) \(\mathstrut +\mathstrut 72q^{94} \) \(\mathstrut +\mathstrut 71q^{95} \) \(\mathstrut +\mathstrut 31q^{96} \) \(\mathstrut +\mathstrut 264q^{97} \) \(\mathstrut +\mathstrut 30q^{98} \) \(\mathstrut +\mathstrut 19q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80768 0.834495 5.88309 4.09159 −2.34300 3.13441 −10.9025 −2.30362 −11.4879
1.2 −2.77582 −0.198171 5.70518 −0.442867 0.550088 0.719726 −10.2849 −2.96073 1.22932
1.3 −2.77052 −1.23937 5.67576 −3.30641 3.43369 0.994092 −10.1837 −1.46397 9.16047
1.4 −2.74088 3.29634 5.51241 −3.80383 −9.03487 −0.926403 −9.62710 7.86587 10.4258
1.5 −2.73169 2.16378 5.46210 −1.55160 −5.91076 4.97440 −9.45738 1.68193 4.23848
1.6 −2.71610 −2.73922 5.37720 −0.158886 7.44001 4.34515 −9.17282 4.50335 0.431551
1.7 −2.69070 −2.66626 5.23985 2.66315 7.17410 −3.38912 −8.71746 4.10894 −7.16574
1.8 −2.63607 1.85738 4.94884 −0.474491 −4.89618 −2.15797 −7.77333 0.449870 1.25079
1.9 −2.60343 1.83156 4.77782 −3.48115 −4.76834 −4.50968 −7.23185 0.354625 9.06293
1.10 −2.59764 −1.72984 4.74771 1.17522 4.49349 2.46697 −7.13754 −0.00766352 −3.05279
1.11 −2.59160 −1.40196 4.71637 −3.26317 3.63331 −1.83814 −7.03972 −1.03451 8.45682
1.12 −2.58803 0.217741 4.69789 −3.50376 −0.563520 3.48169 −6.98221 −2.95259 9.06783
1.13 −2.55145 −0.786154 4.50991 3.16082 2.00584 −3.53434 −6.40393 −2.38196 −8.06468
1.14 −2.54911 2.95187 4.49797 1.50040 −7.52466 0.00462218 −6.36761 5.71356 −3.82468
1.15 −2.50867 2.54308 4.29344 3.25074 −6.37975 2.35749 −5.75350 3.46723 −8.15505
1.16 −2.47249 0.496095 4.11323 2.63577 −1.22659 3.34809 −5.22495 −2.75389 −6.51692
1.17 −2.43669 0.943453 3.93744 −0.444210 −2.29890 −0.206330 −4.72093 −2.10990 1.08240
1.18 −2.42566 0.883893 3.88381 −3.90163 −2.14402 3.11093 −4.56949 −2.21873 9.46401
1.19 −2.41162 −0.257694 3.81590 2.26401 0.621460 −3.03056 −4.37927 −2.93359 −5.45992
1.20 −2.39027 −2.93715 3.71341 −2.06612 7.02059 −0.622431 −4.09551 5.62685 4.93859
See next 80 embeddings (of 195 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.195
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(4007\) \(-1\)