Properties

Label 20.6.e.b
Level 20
Weight 6
Character orbit 20.e
Analytic conductor 3.208
Analytic rank 0
Dimension 24
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.20767639626\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(24q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut -\mathstrut 80q^{5} \) \(\mathstrut -\mathstrut 184q^{6} \) \(\mathstrut +\mathstrut 500q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(24q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut -\mathstrut 80q^{5} \) \(\mathstrut -\mathstrut 184q^{6} \) \(\mathstrut +\mathstrut 500q^{8} \) \(\mathstrut +\mathstrut 590q^{10} \) \(\mathstrut -\mathstrut 1280q^{12} \) \(\mathstrut -\mathstrut 1320q^{13} \) \(\mathstrut +\mathstrut 4024q^{16} \) \(\mathstrut +\mathstrut 2440q^{17} \) \(\mathstrut -\mathstrut 5190q^{18} \) \(\mathstrut +\mathstrut 1260q^{20} \) \(\mathstrut +\mathstrut 1632q^{21} \) \(\mathstrut -\mathstrut 2440q^{22} \) \(\mathstrut -\mathstrut 7320q^{25} \) \(\mathstrut -\mathstrut 4668q^{26} \) \(\mathstrut +\mathstrut 5920q^{28} \) \(\mathstrut +\mathstrut 8600q^{30} \) \(\mathstrut +\mathstrut 25800q^{32} \) \(\mathstrut +\mathstrut 10400q^{33} \) \(\mathstrut -\mathstrut 43460q^{36} \) \(\mathstrut +\mathstrut 14120q^{37} \) \(\mathstrut -\mathstrut 22160q^{38} \) \(\mathstrut -\mathstrut 11220q^{40} \) \(\mathstrut -\mathstrut 19552q^{41} \) \(\mathstrut -\mathstrut 39400q^{42} \) \(\mathstrut +\mathstrut 7960q^{45} \) \(\mathstrut +\mathstrut 29416q^{46} \) \(\mathstrut +\mathstrut 108160q^{48} \) \(\mathstrut +\mathstrut 141010q^{50} \) \(\mathstrut -\mathstrut 7540q^{52} \) \(\mathstrut -\mathstrut 106520q^{53} \) \(\mathstrut -\mathstrut 85984q^{56} \) \(\mathstrut +\mathstrut 40320q^{57} \) \(\mathstrut -\mathstrut 115160q^{58} \) \(\mathstrut -\mathstrut 263840q^{60} \) \(\mathstrut +\mathstrut 206048q^{61} \) \(\mathstrut -\mathstrut 109400q^{62} \) \(\mathstrut -\mathstrut 181800q^{65} \) \(\mathstrut +\mathstrut 186000q^{66} \) \(\mathstrut +\mathstrut 268100q^{68} \) \(\mathstrut +\mathstrut 364240q^{70} \) \(\mathstrut +\mathstrut 247620q^{72} \) \(\mathstrut +\mathstrut 107960q^{73} \) \(\mathstrut -\mathstrut 297600q^{76} \) \(\mathstrut -\mathstrut 53280q^{77} \) \(\mathstrut -\mathstrut 586200q^{78} \) \(\mathstrut -\mathstrut 390000q^{80} \) \(\mathstrut -\mathstrut 66744q^{81} \) \(\mathstrut -\mathstrut 498360q^{82} \) \(\mathstrut -\mathstrut 31880q^{85} \) \(\mathstrut +\mathstrut 545416q^{86} \) \(\mathstrut +\mathstrut 690080q^{88} \) \(\mathstrut +\mathstrut 791790q^{90} \) \(\mathstrut +\mathstrut 576800q^{92} \) \(\mathstrut +\mathstrut 182240q^{93} \) \(\mathstrut -\mathstrut 841984q^{96} \) \(\mathstrut +\mathstrut 60760q^{97} \) \(\mathstrut -\mathstrut 902450q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −5.45971 + 1.48039i 3.65704 3.65704i 27.6169 16.1650i −18.9750 + 52.5828i −14.5525 + 25.3802i 162.920 + 162.920i −126.850 + 129.140i 216.252i 25.7554 315.177i
3.2 −5.11607 + 2.41367i −16.4802 + 16.4802i 20.3484 24.6970i 21.8249 51.4653i 44.5361 124.092i −78.4623 78.4623i −44.4934 + 175.466i 300.193i 12.5624 + 315.978i
3.3 −5.05541 2.53828i 10.7203 10.7203i 19.1142 + 25.6641i −36.9698 41.9313i −81.4063 + 26.9842i −129.597 129.597i −31.4875 178.260i 13.1520i 80.4636 + 305.820i
3.4 −3.91187 4.08623i −6.89992 + 6.89992i −1.39448 + 31.9696i 53.4769 + 16.2857i 55.1863 + 1.20301i 57.8425 + 57.8425i 136.090 119.363i 147.782i −142.648 282.226i
3.5 −2.41367 + 5.11607i 16.4802 16.4802i −20.3484 24.6970i 21.8249 51.4653i 44.5361 + 124.092i 78.4623 + 78.4623i 175.466 44.4934i 300.193i 210.622 + 235.878i
3.6 −1.48039 + 5.45971i −3.65704 + 3.65704i −27.6169 16.1650i −18.9750 + 52.5828i −14.5525 25.3802i −162.920 162.920i 129.140 126.850i 216.252i −258.996 181.441i
3.7 0.216011 5.65273i −10.7585 + 10.7585i −31.9067 2.44211i −55.8903 + 1.12733i 58.4911 + 63.1390i −26.5759 26.5759i −20.6968 + 179.832i 11.5083i −5.70047 + 316.176i
3.8 0.294699 5.64917i 20.2176 20.2176i −31.8263 3.32961i 16.5333 + 53.4008i −108.255 120.171i −9.02060 9.02060i −28.1887 + 178.811i 574.502i 306.543 77.6625i
3.9 2.53828 + 5.05541i −10.7203 + 10.7203i −19.1142 + 25.6641i −36.9698 41.9313i −81.4063 26.9842i 129.597 + 129.597i −178.260 31.4875i 13.1520i 118.140 293.331i
3.10 4.08623 + 3.91187i 6.89992 6.89992i 1.39448 + 31.9696i 53.4769 + 16.2857i 55.1863 1.20301i −57.8425 57.8425i −119.363 + 136.090i 147.782i 154.811 + 275.742i
3.11 5.64917 0.294699i −20.2176 + 20.2176i 31.8263 3.32961i 16.5333 + 53.4008i −108.255 + 120.171i 9.02060 + 9.02060i 178.811 28.1887i 574.502i 109.137 + 296.798i
3.12 5.65273 0.216011i 10.7585 10.7585i 31.9067 2.44211i −55.8903 + 1.12733i 58.4911 63.1390i 26.5759 + 26.5759i 179.832 20.6968i 11.5083i −315.689 + 18.4454i
7.1 −5.45971 1.48039i 3.65704 + 3.65704i 27.6169 + 16.1650i −18.9750 52.5828i −14.5525 25.3802i 162.920 162.920i −126.850 129.140i 216.252i 25.7554 + 315.177i
7.2 −5.11607 2.41367i −16.4802 16.4802i 20.3484 + 24.6970i 21.8249 + 51.4653i 44.5361 + 124.092i −78.4623 + 78.4623i −44.4934 175.466i 300.193i 12.5624 315.978i
7.3 −5.05541 + 2.53828i 10.7203 + 10.7203i 19.1142 25.6641i −36.9698 + 41.9313i −81.4063 26.9842i −129.597 + 129.597i −31.4875 + 178.260i 13.1520i 80.4636 305.820i
7.4 −3.91187 + 4.08623i −6.89992 6.89992i −1.39448 31.9696i 53.4769 16.2857i 55.1863 1.20301i 57.8425 57.8425i 136.090 + 119.363i 147.782i −142.648 + 282.226i
7.5 −2.41367 5.11607i 16.4802 + 16.4802i −20.3484 + 24.6970i 21.8249 + 51.4653i 44.5361 124.092i 78.4623 78.4623i 175.466 + 44.4934i 300.193i 210.622 235.878i
7.6 −1.48039 5.45971i −3.65704 3.65704i −27.6169 + 16.1650i −18.9750 52.5828i −14.5525 + 25.3802i −162.920 + 162.920i 129.140 + 126.850i 216.252i −258.996 + 181.441i
7.7 0.216011 + 5.65273i −10.7585 10.7585i −31.9067 + 2.44211i −55.8903 1.12733i 58.4911 63.1390i −26.5759 + 26.5759i −20.6968 179.832i 11.5083i −5.70047 316.176i
7.8 0.294699 + 5.64917i 20.2176 + 20.2176i −31.8263 + 3.32961i 16.5333 53.4008i −108.255 + 120.171i −9.02060 + 9.02060i −28.1887 178.811i 574.502i 306.543 + 77.6625i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{24} \) \(\mathstrut +\mathstrut 1079568 T_{3}^{20} \) \(\mathstrut +\mathstrut 313012546656 T_{3}^{16} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\( T_{3}^{12} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!00\)\( T_{3}^{8} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\( T_{3}^{4} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\( \) acting on \(S_{6}^{\mathrm{new}}(20, [\chi])\).