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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,10,Mod(2,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.2");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 2.16
Significant digits:
Format:

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 \(S_{10}^{\mathrm{new}}(15, [\chi])\).