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Properties

Label 15.8.e.a
Level $15$
Weight $8$
Character orbit 15.e
Analytic conductor $4.686$
Analytic rank $0$
Dimension $24$
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] = [15,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.2");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(4.68577538226\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7} - 3180 q^{10} - 2028 q^{12} + 16848 q^{13} + 37560 q^{15} - 68868 q^{16} - 106560 q^{18} + 84144 q^{21} + 166020 q^{22} - 275760 q^{25} - 264168 q^{27} + 602028 q^{28} + 892380 q^{30} - 208992 q^{31} - 877800 q^{33} - 256212 q^{36} - 635616 q^{37} - 1574040 q^{40} + 1219380 q^{42} + 1397328 q^{43} + 570240 q^{45} + 2787168 q^{46} - 476796 q^{48} - 4028016 q^{51} - 6426336 q^{52} + 2324880 q^{55} + 8457264 q^{57} + 986340 q^{58} + 2037900 q^{60} + 12157968 q^{61} - 1973952 q^{63} - 25406040 q^{66} - 16720176 q^{67} - 13165860 q^{70} + 36245880 q^{72} + 4059528 q^{73} + 11226120 q^{75} + 37968816 q^{76} - 26156280 q^{78} - 31289976 q^{81} - 53633520 q^{82} - 34181760 q^{85} + 44480160 q^{87} + 74717340 q^{88} + 71623080 q^{90} + 67599648 q^{91} - 58742592 q^{93} - 103105308 q^{96} - 75148056 q^{97}+O(q^{100}) \) 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 −14.4708 + 14.4708i −45.2107 − 11.9578i − 290.808i −140.740 + 241.490i 827.274 − 481.197i 196.040 + 196.040i 2355.96 + 2355.96i 1901.02 + 1081.24i −1457.93 − 5531.17i
2.2 −13.2671 + 13.2671i 42.4163 − 19.6942i − 224.030i 122.889 − 251.044i −301.455 + 824.023i 796.332 + 796.332i 1274.03 + 1274.03i 1411.28 − 1670.71i 1700.24 + 4961.00i
2.3 −9.36735 + 9.36735i 23.6872 + 40.3227i − 47.4943i 125.932 + 249.532i −599.602 − 155.830i −696.811 − 696.811i −754.124 − 754.124i −1064.84 + 1910.26i −3517.10 − 1157.80i
2.4 −6.70096 + 6.70096i −34.4749 + 31.5988i 38.1944i 8.44382 − 279.381i 19.2725 − 442.757i −68.7247 − 68.7247i −1113.66 − 1113.66i 190.033 − 2178.73i 1815.54 + 1928.70i
2.5 −5.42850 + 5.42850i 13.1156 − 44.8885i 69.0629i −275.850 + 45.0737i 172.479 + 314.875i −775.470 − 775.470i −1069.75 − 1069.75i −1842.96 − 1177.48i 1252.77 − 1742.13i
2.6 −1.72123 + 1.72123i −28.9107 − 36.7583i 122.075i 270.133 + 71.7849i 113.031 + 13.5074i 884.635 + 884.635i −430.436 − 430.436i −515.339 + 2125.42i −588.519 + 341.403i
2.7 1.72123 − 1.72123i 36.7583 + 28.9107i 122.075i −270.133 − 71.7849i 113.031 − 13.5074i 884.635 + 884.635i 430.436 + 430.436i 515.339 + 2125.42i −588.519 + 341.403i
2.8 5.42850 − 5.42850i 44.8885 − 13.1156i 69.0629i 275.850 − 45.0737i 172.479 − 314.875i −775.470 − 775.470i 1069.75 + 1069.75i 1842.96 − 1177.48i 1252.77 − 1742.13i
2.9 6.70096 − 6.70096i −31.5988 + 34.4749i 38.1944i −8.44382 + 279.381i 19.2725 + 442.757i −68.7247 − 68.7247i 1113.66 + 1113.66i −190.033 − 2178.73i 1815.54 + 1928.70i
2.10 9.36735 − 9.36735i −40.3227 − 23.6872i − 47.4943i −125.932 − 249.532i −599.602 + 155.830i −696.811 − 696.811i 754.124 + 754.124i 1064.84 + 1910.26i −3517.10 − 1157.80i
2.11 13.2671 − 13.2671i 19.6942 − 42.4163i − 224.030i −122.889 + 251.044i −301.455 − 824.023i 796.332 + 796.332i −1274.03 − 1274.03i −1411.28 − 1670.71i 1700.24 + 4961.00i
2.12 14.4708 − 14.4708i 11.9578 + 45.2107i − 290.808i 140.740 − 241.490i 827.274 + 481.197i 196.040 + 196.040i −2355.96 − 2355.96i −1901.02 + 1081.24i −1457.93 − 5531.17i
8.1 −14.4708 − 14.4708i −45.2107 + 11.9578i 290.808i −140.740 − 241.490i 827.274 + 481.197i 196.040 − 196.040i 2355.96 − 2355.96i 1901.02 − 1081.24i −1457.93 + 5531.17i
8.2 −13.2671 − 13.2671i 42.4163 + 19.6942i 224.030i 122.889 + 251.044i −301.455 − 824.023i 796.332 − 796.332i 1274.03 − 1274.03i 1411.28 + 1670.71i 1700.24 − 4961.00i
8.3 −9.36735 − 9.36735i 23.6872 − 40.3227i 47.4943i 125.932 − 249.532i −599.602 + 155.830i −696.811 + 696.811i −754.124 + 754.124i −1064.84 − 1910.26i −3517.10 + 1157.80i
8.4 −6.70096 − 6.70096i −34.4749 − 31.5988i − 38.1944i 8.44382 + 279.381i 19.2725 + 442.757i −68.7247 + 68.7247i −1113.66 + 1113.66i 190.033 + 2178.73i 1815.54 − 1928.70i
8.5 −5.42850 − 5.42850i 13.1156 + 44.8885i − 69.0629i −275.850 − 45.0737i 172.479 − 314.875i −775.470 + 775.470i −1069.75 + 1069.75i −1842.96 + 1177.48i 1252.77 + 1742.13i
8.6 −1.72123 − 1.72123i −28.9107 + 36.7583i − 122.075i 270.133 − 71.7849i 113.031 − 13.5074i 884.635 − 884.635i −430.436 + 430.436i −515.339 − 2125.42i −588.519 − 341.403i
8.7 1.72123 + 1.72123i 36.7583 − 28.9107i − 122.075i −270.133 + 71.7849i 113.031 + 13.5074i 884.635 − 884.635i 430.436 − 430.436i 515.339 − 2125.42i −588.519 − 341.403i
8.8 5.42850 + 5.42850i 44.8885 + 13.1156i − 69.0629i 275.850 + 45.0737i 172.479 + 314.875i −775.470 + 775.470i 1069.75 − 1069.75i 1842.96 + 1177.48i 1252.77 + 1742.13i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
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 15.8.e.a ✓ 24
3.b odd 2 1 inner 15.8.e.a ✓ 24
5.b even 2 1 75.8.e.d 24
5.c odd 4 1 inner 15.8.e.a ✓ 24
5.c odd 4 1 75.8.e.d 24
15.d odd 2 1 75.8.e.d 24
15.e even 4 1 inner 15.8.e.a ✓ 24
15.e even 4 1 75.8.e.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.e.a ✓ 24 1.a even 1 1 trivial
15.8.e.a ✓ 24 3.b odd 2 1 inner
15.8.e.a ✓ 24 5.c odd 4 1 inner
15.8.e.a ✓ 24 15.e even 4 1 inner
75.8.e.d 24 5.b even 2 1
75.8.e.d 24 5.c odd 4 1
75.8.e.d 24 15.d odd 2 1
75.8.e.d 24 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(15, [\chi])\).