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Properties

Label 75.2.l.a
Level $75$
Weight $2$
Character orbit 75.l
Analytic conductor $0.599$
Analytic rank $0$
Dimension $64$
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] = [75,2,Mod(2,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), 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: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 10 q^{3} - 20 q^{4} - 6 q^{6} - 20 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 10 q^{3} - 20 q^{4} - 6 q^{6} - 20 q^{7} - 10 q^{9} - 20 q^{10} - 10 q^{12} - 20 q^{13} - 10 q^{15} - 8 q^{16} - 10 q^{18} - 6 q^{21} + 20 q^{22} + 40 q^{25} - 10 q^{27} + 40 q^{28} - 10 q^{30} - 12 q^{31} - 10 q^{33} + 20 q^{34} - 22 q^{36} - 20 q^{37} + 30 q^{39} - 20 q^{40} + 90 q^{42} - 20 q^{43} + 70 q^{45} - 12 q^{46} + 100 q^{48} - 16 q^{51} + 20 q^{52} + 120 q^{54} - 20 q^{55} + 70 q^{57} - 20 q^{58} + 50 q^{60} - 12 q^{61} - 20 q^{63} - 100 q^{64} - 30 q^{66} - 60 q^{67} - 80 q^{69} - 100 q^{70} - 150 q^{72} - 60 q^{73} - 90 q^{75} - 64 q^{76} - 80 q^{78} - 60 q^{79} + 14 q^{81} - 60 q^{82} - 130 q^{84} + 60 q^{85} - 60 q^{87} + 20 q^{88} - 70 q^{90} - 12 q^{91} - 20 q^{93} + 260 q^{94} + 42 q^{96} + 120 q^{97}+O(q^{100}) \) 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 −0.400841 + 2.53081i −0.633409 + 1.61208i −4.34220 − 1.41087i 1.84517 − 1.26307i −3.82596 − 2.24922i 1.29796 + 1.29796i 2.98460 − 5.85760i −2.19759 − 2.04221i 2.45697 + 5.17606i
2.2 −0.338090 + 2.13462i 1.18218 − 1.26588i −2.54018 − 0.825354i 0.256193 + 2.22134i 2.30249 + 2.95148i −0.0545283 − 0.0545283i 0.658272 − 1.29193i −0.204907 − 2.99299i −4.82834 − 0.204141i
2.3 −0.184928 + 1.16759i 1.52140 + 0.827854i 0.573046 + 0.186194i −1.19069 − 1.89269i −1.24794 + 1.62328i −3.04094 − 3.04094i −1.39673 + 2.74125i 1.62932 + 2.51899i 2.43007 − 1.04022i
2.4 −0.0809972 + 0.511396i −1.50036 − 0.865399i 1.64715 + 0.535191i 2.21509 − 0.305570i 0.564087 − 0.697184i 0.155466 + 0.155466i −0.877235 + 1.72167i 1.50217 + 2.59682i −0.0231486 + 1.15754i
2.5 0.0809972 − 0.511396i 0.181767 − 1.72249i 1.64715 + 0.535191i −2.21509 + 0.305570i −0.866150 − 0.232472i 0.155466 + 0.155466i 0.877235 − 1.72167i −2.93392 − 0.626184i −0.0231486 + 1.15754i
2.6 0.184928 − 1.16759i −0.224509 + 1.71744i 0.573046 + 0.186194i 1.19069 + 1.89269i 1.96375 + 0.579736i −3.04094 − 3.04094i 1.39673 − 2.74125i −2.89919 − 0.771160i 2.43007 − 1.04022i
2.7 0.338090 − 2.13462i −1.71899 + 0.212337i −2.54018 − 0.825354i −0.256193 − 2.22134i −0.127915 + 3.74117i −0.0545283 − 0.0545283i −0.658272 + 1.29193i 2.90983 − 0.730008i −4.82834 − 0.204141i
2.8 0.400841 − 2.53081i 1.67651 + 0.435117i −4.34220 − 1.41087i −1.84517 + 1.26307i 1.77321 − 4.06850i 1.29796 + 1.29796i −2.98460 + 5.85760i 2.62135 + 1.45895i 2.45697 + 5.17606i
8.1 −1.12503 + 2.20799i −1.64218 − 0.550675i −2.43396 − 3.35006i −2.18396 − 0.479921i 3.06338 − 3.00639i −1.56473 + 1.56473i 5.24001 − 0.829937i 2.39351 + 1.80862i 3.51667 − 4.28223i
8.2 −0.888111 + 1.74302i 1.12908 − 1.31346i −1.07379 − 1.47795i 1.74306 + 1.40062i 1.28664 + 3.13450i −0.551254 + 0.551254i −0.334566 + 0.0529901i −0.450377 − 2.96600i −3.98933 + 1.79428i
8.3 −0.674175 + 1.32314i −0.140417 + 1.72635i −0.120624 − 0.166024i 1.20181 − 1.88564i −2.18954 − 1.34965i −1.72218 + 1.72218i −2.63243 + 0.416937i −2.96057 − 0.484816i 1.68474 + 2.86142i
8.4 −0.0231951 + 0.0455229i −1.00313 − 1.41200i 1.17404 + 1.61592i 1.22804 − 1.86867i 0.0875458 − 0.0129140i 1.44136 − 1.44136i −0.201718 + 0.0319491i −0.987463 + 2.83283i 0.0565829 + 0.0992477i
8.5 0.0231951 − 0.0455229i −1.39036 + 1.03290i 1.17404 + 1.61592i −1.22804 + 1.86867i 0.0147712 + 0.0872517i 1.44136 − 1.44136i 0.201718 − 0.0319491i 0.866221 − 2.87222i 0.0565829 + 0.0992477i
8.6 0.674175 − 1.32314i 0.399927 − 1.68525i −0.120624 − 0.166024i −1.20181 + 1.88564i −1.96020 − 1.66531i −1.72218 + 1.72218i 2.63243 − 0.416937i −2.68012 − 1.34795i 1.68474 + 2.86142i
8.7 0.888111 − 1.74302i 0.667932 + 1.59808i −1.07379 − 1.47795i −1.74306 − 1.40062i 3.37868 + 0.255059i −0.551254 + 0.551254i 0.334566 − 0.0529901i −2.10773 + 2.13482i −3.98933 + 1.79428i
8.8 1.12503 − 2.20799i −1.73197 + 0.0162612i −2.43396 − 3.35006i 2.18396 + 0.479921i −1.91261 + 3.84248i −1.56473 + 1.56473i −5.24001 + 0.829937i 2.99947 − 0.0563281i 3.51667 − 4.28223i
17.1 −2.24299 − 1.14286i 1.52152 − 0.827629i 2.54931 + 3.50882i 2.21582 + 0.300268i −4.35863 + 0.117475i −1.80298 − 1.80298i −0.920375 − 5.81102i 1.63006 − 2.51851i −4.62689 − 3.20587i
17.2 −1.33917 − 0.682341i −0.310450 + 1.70400i 0.152214 + 0.209505i 1.00904 + 1.99546i 1.57845 − 2.07011i 1.98452 + 1.98452i 0.409350 + 2.58454i −2.80724 − 1.05801i 0.0103112 − 3.36076i
17.3 −0.710801 − 0.362171i −1.70683 + 0.294526i −0.801501 − 1.10317i 0.787273 − 2.09289i 1.31988 + 0.408813i −2.94096 − 2.94096i 0.419762 + 2.65027i 2.82651 − 1.00541i −1.31758 + 1.20250i
17.4 −0.685506 − 0.349283i 1.03848 − 1.38621i −0.827650 − 1.13916i −2.01837 − 0.962386i −1.19606 + 0.587530i 1.53819 + 1.53819i 0.410179 + 2.58977i −0.843130 − 2.87909i 1.04746 + 1.36470i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.l.a ✓ 64
3.b odd 2 1 inner 75.2.l.a ✓ 64
5.b even 2 1 375.2.l.c 64
5.c odd 4 1 375.2.l.a 64
5.c odd 4 1 375.2.l.b 64
15.d odd 2 1 375.2.l.c 64
15.e even 4 1 375.2.l.a 64
15.e even 4 1 375.2.l.b 64
25.d even 5 1 375.2.l.a 64
25.e even 10 1 375.2.l.b 64
25.f odd 20 1 inner 75.2.l.a ✓ 64
25.f odd 20 1 375.2.l.c 64
75.h odd 10 1 375.2.l.b 64
75.j odd 10 1 375.2.l.a 64
75.l even 20 1 inner 75.2.l.a ✓ 64
75.l even 20 1 375.2.l.c 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.l.a ✓ 64 1.a even 1 1 trivial
75.2.l.a ✓ 64 3.b odd 2 1 inner
75.2.l.a ✓ 64 25.f odd 20 1 inner
75.2.l.a ✓ 64 75.l even 20 1 inner
375.2.l.a 64 5.c odd 4 1
375.2.l.a 64 15.e even 4 1
375.2.l.a 64 25.d even 5 1
375.2.l.a 64 75.j odd 10 1
375.2.l.b 64 5.c odd 4 1
375.2.l.b 64 15.e even 4 1
375.2.l.b 64 25.e even 10 1
375.2.l.b 64 75.h odd 10 1
375.2.l.c 64 5.b even 2 1
375.2.l.c 64 15.d odd 2 1
375.2.l.c 64 25.f odd 20 1
375.2.l.c 64 75.l even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(75, [\chi])\).