# Properties

 Label 15.10.e.a Level 15 Weight 10 Character orbit 15.e Analytic conductor 7.726 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 15.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.72553754246$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 150q^{3} - 4548q^{6} - 9760q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 150q^{3} - 4548q^{6} - 9760q^{7} - 39820q^{10} + 191940q^{12} + 114260q^{13} - 40230q^{15} - 1494724q^{16} + 994800q^{18} + 592212q^{21} + 308260q^{22} + 5808020q^{25} + 4786830q^{27} - 12953300q^{28} - 17343060q^{30} + 16124104q^{31} + 19638120q^{33} - 48943956q^{36} - 35578660q^{37} + 144132840q^{40} + 46648260q^{42} - 118721020q^{43} - 129688380q^{45} + 3145024q^{46} + 275395140q^{48} - 64226868q^{51} - 157103680q^{52} + 144539740q^{55} + 48163500q^{57} - 209634300q^{58} - 140815620q^{60} + 63607744q^{61} + 89858580q^{63} + 358931400q^{66} + 153311540q^{67} + 420519420q^{70} - 1534046760q^{72} + 628070360q^{73} + 138090210q^{75} - 2457023472q^{76} - 403337880q^{78} + 2250542952q^{81} + 4138031440q^{82} + 108907340q^{85} - 2878139760q^{87} - 595851300q^{88} + 693868920q^{90} - 4369040216q^{91} - 293672640q^{93} + 7291125636q^{96} + 3987642200q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −30.5207 + 30.5207i −54.5047 + 129.276i 1351.03i 1261.11 602.260i −2282.07 5609.11i −2749.16 2749.16i 25607.7 + 25607.7i −13741.5 14092.3i −20108.7 + 56871.5i
2.2 −24.1261 + 24.1261i 21.4940 138.640i 652.142i 844.923 + 1113.21i 2826.28 + 3863.41i 1781.62 + 1781.62i 3381.08 + 3381.08i −18759.0 5959.84i −47242.1 6472.66i
2.3 −24.0812 + 24.0812i 139.239 + 17.1863i 647.806i −1397.54 + 2.62404i −3766.92 + 2939.18i −4153.03 4153.03i 3270.36 + 3270.36i 19092.3 + 4786.01i 33591.2 33717.6i
2.4 −20.3831 + 20.3831i −124.250 65.1534i 318.940i −840.534 1116.52i 3860.62 1204.57i 1024.24 + 1024.24i −3935.17 3935.17i 11193.1 + 16190.6i 39890.9 + 5625.54i
2.5 −13.8737 + 13.8737i −55.7208 + 128.756i 127.039i −919.994 + 1052.02i −1013.28 2559.39i 6797.28 + 6797.28i −8865.86 8865.86i −13473.4 14348.8i −1831.63 27359.1i
2.6 −10.0225 + 10.0225i 112.058 + 84.4162i 311.097i 1254.86 615.193i −1969.17 + 277.038i 2707.78 + 2707.78i −8249.53 8249.53i 5430.82 + 18919.0i −6411.06 + 18742.6i
2.7 −6.35582 + 6.35582i −140.296 0.342183i 431.207i 1052.08 + 919.921i 893.869 889.519i −7866.00 7866.00i −5994.85 5994.85i 19682.8 + 96.0135i −12533.7 + 839.974i
2.8 −4.02674 + 4.02674i 52.2930 130.186i 479.571i 216.015 1380.75i 313.655 + 734.796i 17.2800 + 17.2800i −3992.80 3992.80i −14213.9 13615.7i 4690.07 + 6429.74i
2.9 4.02674 4.02674i 130.186 52.2930i 479.571i −216.015 + 1380.75i 313.655 734.796i 17.2800 + 17.2800i 3992.80 + 3992.80i 14213.9 13615.7i 4690.07 + 6429.74i
2.10 6.35582 6.35582i 0.342183 + 140.296i 431.207i −1052.08 919.921i 893.869 + 889.519i −7866.00 7866.00i 5994.85 + 5994.85i −19682.8 + 96.0135i −12533.7 + 839.974i
2.11 10.0225 10.0225i −84.4162 112.058i 311.097i −1254.86 + 615.193i −1969.17 277.038i 2707.78 + 2707.78i 8249.53 + 8249.53i −5430.82 + 18919.0i −6411.06 + 18742.6i
2.12 13.8737 13.8737i −128.756 + 55.7208i 127.039i 919.994 1052.02i −1013.28 + 2559.39i 6797.28 + 6797.28i 8865.86 + 8865.86i 13473.4 14348.8i −1831.63 27359.1i
2.13 20.3831 20.3831i 65.1534 + 124.250i 318.940i 840.534 + 1116.52i 3860.62 + 1204.57i 1024.24 + 1024.24i 3935.17 + 3935.17i −11193.1 + 16190.6i 39890.9 + 5625.54i
2.14 24.0812 24.0812i −17.1863 139.239i 647.806i 1397.54 2.62404i −3766.92 2939.18i −4153.03 4153.03i −3270.36 3270.36i −19092.3 + 4786.01i 33591.2 33717.6i
2.15 24.1261 24.1261i 138.640 21.4940i 652.142i −844.923 1113.21i 2826.28 3863.41i 1781.62 + 1781.62i −3381.08 3381.08i 18759.0 5959.84i −47242.1 6472.66i
2.16 30.5207 30.5207i −129.276 + 54.5047i 1351.03i −1261.11 + 602.260i −2282.07 + 5609.11i −2749.16 2749.16i −25607.7 25607.7i 13741.5 14092.3i −20108.7 + 56871.5i
8.1 −30.5207 30.5207i −54.5047 129.276i 1351.03i 1261.11 + 602.260i −2282.07 + 5609.11i −2749.16 + 2749.16i 25607.7 25607.7i −13741.5 + 14092.3i −20108.7 56871.5i
8.2 −24.1261 24.1261i 21.4940 + 138.640i 652.142i 844.923 1113.21i 2826.28 3863.41i 1781.62 1781.62i 3381.08 3381.08i −18759.0 + 5959.84i −47242.1 + 6472.66i
8.3 −24.0812 24.0812i 139.239 17.1863i 647.806i −1397.54 2.62404i −3766.92 2939.18i −4153.03 + 4153.03i 3270.36 3270.36i 19092.3 4786.01i 33591.2 + 33717.6i
8.4 −20.3831 20.3831i −124.250 + 65.1534i 318.940i −840.534 + 1116.52i 3860.62 + 1204.57i 1024.24 1024.24i −3935.17 + 3935.17i 11193.1 16190.6i 39890.9 5625.54i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.10.e.a 32
3.b odd 2 1 inner 15.10.e.a 32
5.b even 2 1 75.10.e.f 32
5.c odd 4 1 inner 15.10.e.a 32
5.c odd 4 1 75.10.e.f 32
15.d odd 2 1 75.10.e.f 32
15.e even 4 1 inner 15.10.e.a 32
15.e even 4 1 75.10.e.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.e.a 32 1.a even 1 1 trivial
15.10.e.a 32 3.b odd 2 1 inner
15.10.e.a 32 5.c odd 4 1 inner
15.10.e.a 32 15.e even 4 1 inner
75.10.e.f 32 5.b even 2 1
75.10.e.f 32 5.c odd 4 1
75.10.e.f 32 15.d odd 2 1
75.10.e.f 32 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database