Properties

Label 6005.2.a.f
Level 6005
Weight 2
Character orbit 6005.a
Self dual Yes
Analytic conductor 47.950
Analytic rank 0
Dimension 111
CM No

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Newspace parameters

Level: \( N \) = \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9501664138\)
Analytic rank: \(0\)
Dimension: \(111\)
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.

\(\operatorname{Tr}(f)(q) = \) \(111q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 40q^{3} \) \(\mathstrut +\mathstrut 136q^{4} \) \(\mathstrut +\mathstrut 111q^{5} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 39q^{7} \) \(\mathstrut +\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 139q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(111q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 40q^{3} \) \(\mathstrut +\mathstrut 136q^{4} \) \(\mathstrut +\mathstrut 111q^{5} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 39q^{7} \) \(\mathstrut +\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 139q^{9} \) \(\mathstrut +\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 36q^{11} \) \(\mathstrut +\mathstrut 80q^{12} \) \(\mathstrut +\mathstrut 36q^{13} \) \(\mathstrut +\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut 40q^{15} \) \(\mathstrut +\mathstrut 190q^{16} \) \(\mathstrut +\mathstrut 38q^{17} \) \(\mathstrut +\mathstrut 48q^{18} \) \(\mathstrut +\mathstrut 77q^{19} \) \(\mathstrut +\mathstrut 136q^{20} \) \(\mathstrut +\mathstrut 11q^{21} \) \(\mathstrut +\mathstrut 39q^{22} \) \(\mathstrut +\mathstrut 82q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 111q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut +\mathstrut 130q^{27} \) \(\mathstrut +\mathstrut 87q^{28} \) \(\mathstrut +\mathstrut 20q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut +\mathstrut 41q^{31} \) \(\mathstrut +\mathstrut 85q^{32} \) \(\mathstrut +\mathstrut 33q^{33} \) \(\mathstrut +\mathstrut 7q^{34} \) \(\mathstrut +\mathstrut 39q^{35} \) \(\mathstrut +\mathstrut 191q^{36} \) \(\mathstrut +\mathstrut 80q^{37} \) \(\mathstrut +\mathstrut 42q^{38} \) \(\mathstrut +\mathstrut 21q^{39} \) \(\mathstrut +\mathstrut 45q^{40} \) \(\mathstrut +\mathstrut 16q^{41} \) \(\mathstrut +\mathstrut 33q^{42} \) \(\mathstrut +\mathstrut 164q^{43} \) \(\mathstrut +\mathstrut 37q^{44} \) \(\mathstrut +\mathstrut 139q^{45} \) \(\mathstrut +\mathstrut 32q^{46} \) \(\mathstrut +\mathstrut 148q^{47} \) \(\mathstrut +\mathstrut 149q^{48} \) \(\mathstrut +\mathstrut 160q^{49} \) \(\mathstrut +\mathstrut 20q^{50} \) \(\mathstrut +\mathstrut 51q^{51} \) \(\mathstrut +\mathstrut 87q^{52} \) \(\mathstrut +\mathstrut 83q^{53} \) \(\mathstrut -\mathstrut 6q^{54} \) \(\mathstrut +\mathstrut 36q^{55} \) \(\mathstrut -\mathstrut 10q^{56} \) \(\mathstrut +\mathstrut 28q^{57} \) \(\mathstrut +\mathstrut 47q^{58} \) \(\mathstrut +\mathstrut 14q^{59} \) \(\mathstrut +\mathstrut 80q^{60} \) \(\mathstrut +\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 120q^{63} \) \(\mathstrut +\mathstrut 231q^{64} \) \(\mathstrut +\mathstrut 36q^{65} \) \(\mathstrut -\mathstrut 4q^{66} \) \(\mathstrut +\mathstrut 253q^{67} \) \(\mathstrut +\mathstrut 80q^{68} \) \(\mathstrut +\mathstrut 6q^{69} \) \(\mathstrut +\mathstrut 7q^{70} \) \(\mathstrut +\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 124q^{72} \) \(\mathstrut +\mathstrut 64q^{73} \) \(\mathstrut -\mathstrut 37q^{74} \) \(\mathstrut +\mathstrut 40q^{75} \) \(\mathstrut +\mathstrut 92q^{76} \) \(\mathstrut +\mathstrut 63q^{77} \) \(\mathstrut +\mathstrut 29q^{78} \) \(\mathstrut +\mathstrut 91q^{79} \) \(\mathstrut +\mathstrut 190q^{80} \) \(\mathstrut +\mathstrut 187q^{81} \) \(\mathstrut -\mathstrut 7q^{82} \) \(\mathstrut +\mathstrut 63q^{83} \) \(\mathstrut -\mathstrut 69q^{84} \) \(\mathstrut +\mathstrut 38q^{85} \) \(\mathstrut -\mathstrut 22q^{86} \) \(\mathstrut +\mathstrut 57q^{87} \) \(\mathstrut +\mathstrut 121q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 48q^{90} \) \(\mathstrut +\mathstrut 119q^{91} \) \(\mathstrut +\mathstrut 104q^{92} \) \(\mathstrut +\mathstrut 14q^{93} \) \(\mathstrut -\mathstrut q^{94} \) \(\mathstrut +\mathstrut 77q^{95} \) \(\mathstrut -\mathstrut 38q^{96} \) \(\mathstrut +\mathstrut 96q^{97} \) \(\mathstrut +\mathstrut 81q^{98} \) \(\mathstrut +\mathstrut 106q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.74668 1.63661 5.54423 1.00000 −4.49523 −2.87647 −9.73484 −0.321512 −2.74668
1.2 −2.72268 3.38210 5.41301 1.00000 −9.20838 2.13000 −9.29255 8.43858 −2.72268
1.3 −2.71589 0.174668 5.37604 1.00000 −0.474377 2.58754 −9.16893 −2.96949 −2.71589
1.4 −2.65700 −0.218117 5.05966 1.00000 0.579538 4.50653 −8.12951 −2.95242 −2.65700
1.5 −2.63642 3.34656 4.95072 1.00000 −8.82296 −3.64864 −7.77935 8.19949 −2.63642
1.6 −2.62342 −2.19431 4.88231 1.00000 5.75659 1.89929 −7.56149 1.81500 −2.62342
1.7 −2.61069 −1.33217 4.81572 1.00000 3.47788 −0.680814 −7.35098 −1.22533 −2.61069
1.8 −2.56895 1.63205 4.59952 1.00000 −4.19265 3.45435 −6.67805 −0.336428 −2.56895
1.9 −2.47000 −2.23996 4.10090 1.00000 5.53271 3.40548 −5.18922 2.01743 −2.47000
1.10 −2.41916 1.95905 3.85233 1.00000 −4.73925 −1.88529 −4.48108 0.837875 −2.41916
1.11 −2.34362 −0.853258 3.49256 1.00000 1.99971 −1.48080 −3.49800 −2.27195 −2.34362
1.12 −2.32829 0.726641 3.42095 1.00000 −1.69183 −3.13935 −3.30840 −2.47199 −2.32829
1.13 −2.32774 1.75930 3.41838 1.00000 −4.09520 3.92990 −3.30162 0.0951392 −2.32774
1.14 −2.32622 −1.23055 3.41132 1.00000 2.86255 −0.720355 −3.28304 −1.48574 −2.32622
1.15 −2.23901 3.10819 3.01316 1.00000 −6.95927 3.97064 −2.26846 6.66087 −2.23901
1.16 −2.17647 1.64482 2.73702 1.00000 −3.57990 −1.13164 −1.60410 −0.294573 −2.17647
1.17 −2.03891 2.79275 2.15716 1.00000 −5.69418 −0.0959578 −0.320429 4.79947 −2.03891
1.18 −1.98616 −2.45604 1.94482 1.00000 4.87807 −0.764099 0.109603 3.03211 −1.98616
1.19 −1.98040 0.765234 1.92199 1.00000 −1.51547 −3.35858 0.154490 −2.41442 −1.98040
1.20 −1.91654 −2.98629 1.67314 1.00000 5.72336 0.938196 0.626446 5.91794 −1.91654
See next 80 embeddings (of 111 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.111
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1201\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{111} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).