Properties

Label 4015.2.a.i
Level 4015
Weight 2
Character orbit 4015.a
Self dual Yes
Analytic conductor 32.060
Analytic rank 0
Dimension 38
CM No

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Newspace parameters

Level: \( N \) = \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \(38q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 50q^{4} \) \(\mathstrut +\mathstrut 38q^{5} \) \(\mathstrut +\mathstrut 11q^{6} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 63q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(38q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 50q^{4} \) \(\mathstrut +\mathstrut 38q^{5} \) \(\mathstrut +\mathstrut 11q^{6} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 63q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 38q^{11} \) \(\mathstrut +\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut 23q^{14} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 74q^{16} \) \(\mathstrut +\mathstrut 26q^{17} \) \(\mathstrut +\mathstrut 16q^{18} \) \(\mathstrut -\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut 50q^{20} \) \(\mathstrut +\mathstrut 21q^{21} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 10q^{23} \) \(\mathstrut +\mathstrut 41q^{24} \) \(\mathstrut +\mathstrut 38q^{25} \) \(\mathstrut +\mathstrut 25q^{26} \) \(\mathstrut +\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 28q^{29} \) \(\mathstrut +\mathstrut 11q^{30} \) \(\mathstrut +\mathstrut 24q^{31} \) \(\mathstrut +\mathstrut 39q^{32} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 38q^{34} \) \(\mathstrut +\mathstrut 111q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 19q^{38} \) \(\mathstrut -\mathstrut 18q^{39} \) \(\mathstrut +\mathstrut 15q^{40} \) \(\mathstrut +\mathstrut 62q^{41} \) \(\mathstrut -\mathstrut 17q^{42} \) \(\mathstrut -\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 50q^{44} \) \(\mathstrut +\mathstrut 63q^{45} \) \(\mathstrut -\mathstrut 9q^{46} \) \(\mathstrut +\mathstrut 31q^{47} \) \(\mathstrut +\mathstrut 53q^{48} \) \(\mathstrut +\mathstrut 88q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 21q^{52} \) \(\mathstrut +\mathstrut 30q^{53} \) \(\mathstrut +\mathstrut 49q^{54} \) \(\mathstrut -\mathstrut 38q^{55} \) \(\mathstrut +\mathstrut 32q^{56} \) \(\mathstrut +\mathstrut 49q^{57} \) \(\mathstrut +\mathstrut 12q^{58} \) \(\mathstrut +\mathstrut 31q^{59} \) \(\mathstrut +\mathstrut 12q^{60} \) \(\mathstrut +\mathstrut 25q^{61} \) \(\mathstrut +\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 15q^{63} \) \(\mathstrut +\mathstrut 137q^{64} \) \(\mathstrut -\mathstrut q^{65} \) \(\mathstrut -\mathstrut 11q^{66} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 75q^{68} \) \(\mathstrut +\mathstrut 92q^{69} \) \(\mathstrut +\mathstrut 23q^{70} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut -\mathstrut 38q^{73} \) \(\mathstrut +\mathstrut 55q^{74} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 57q^{76} \) \(\mathstrut -\mathstrut 17q^{78} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut 74q^{80} \) \(\mathstrut +\mathstrut 118q^{81} \) \(\mathstrut +\mathstrut 14q^{82} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut +\mathstrut 22q^{84} \) \(\mathstrut +\mathstrut 26q^{85} \) \(\mathstrut +\mathstrut 5q^{86} \) \(\mathstrut +\mathstrut 24q^{87} \) \(\mathstrut -\mathstrut 15q^{88} \) \(\mathstrut +\mathstrut 143q^{89} \) \(\mathstrut +\mathstrut 16q^{90} \) \(\mathstrut +\mathstrut 66q^{91} \) \(\mathstrut +\mathstrut 29q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut -\mathstrut 7q^{94} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut +\mathstrut 59q^{96} \) \(\mathstrut +\mathstrut 41q^{97} \) \(\mathstrut -\mathstrut 10q^{98} \) \(\mathstrut -\mathstrut 63q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.76981 1.77605 5.67182 1.00000 −4.91932 −0.953960 −10.1702 0.154367 −2.76981
1.2 −2.64676 −3.26170 5.00536 1.00000 8.63297 0.334745 −7.95449 7.63872 −2.64676
1.3 −2.64612 −1.80895 5.00197 1.00000 4.78670 −4.64214 −7.94360 0.272285 −2.64612
1.4 −2.40266 2.98645 3.77280 1.00000 −7.17544 4.58733 −4.25943 5.91889 −2.40266
1.5 −2.35033 0.502249 3.52405 1.00000 −1.18045 0.984719 −3.58202 −2.74775 −2.35033
1.6 −2.26509 0.846807 3.13061 1.00000 −1.91809 −3.17432 −2.56093 −2.28292 −2.26509
1.7 −1.83787 −3.02191 1.37778 1.00000 5.55388 4.06050 1.14356 6.13192 −1.83787
1.8 −1.73652 −1.80706 1.01551 1.00000 3.13799 −4.26912 1.70959 0.265453 −1.73652
1.9 −1.69384 0.407884 0.869088 1.00000 −0.690890 2.45284 1.91558 −2.83363 −1.69384
1.10 −1.61749 2.78278 0.616262 1.00000 −4.50111 −5.17171 2.23818 4.74386 −1.61749
1.11 −1.50911 −0.619433 0.277405 1.00000 0.934791 1.62740 2.59958 −2.61630 −1.50911
1.12 −1.49358 2.85677 0.230796 1.00000 −4.26684 0.0973246 2.64246 5.16116 −1.49358
1.13 −1.16964 −3.05105 −0.631953 1.00000 3.56861 −2.34807 3.07843 6.30888 −1.16964
1.14 −1.00909 −1.30445 −0.981746 1.00000 1.31630 1.28192 3.00884 −1.29840 −1.00909
1.15 −0.454453 2.93224 −1.79347 1.00000 −1.33257 4.01407 1.72395 5.59806 −0.454453
1.16 −0.445505 −0.895604 −1.80153 1.00000 0.398996 −1.53298 1.69360 −2.19789 −0.445505
1.17 −0.364998 1.30973 −1.86678 1.00000 −0.478050 −1.97253 1.41137 −1.28460 −0.364998
1.18 −0.0798923 0.0600931 −1.99362 1.00000 −0.00480098 −4.79075 0.319059 −2.99639 −0.0798923
1.19 0.0204713 2.32254 −1.99958 1.00000 0.0475455 2.78561 −0.0818767 2.39419 0.0204713
1.20 0.0631532 0.00106429 −1.99601 1.00000 6.72133e−5 0 3.26238 −0.252361 −3.00000 0.0631532
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{38} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).