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Properties

Label 425.2.n.f
Level $425$
Weight $2$
Character orbit 425.n
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(49,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.n (of order \(8\), degree \(4\), not 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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} - 8 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} - 8 q^{6} + 24 q^{9} - 8 q^{11} - 40 q^{12} - 16 q^{13} - 24 q^{16} + 8 q^{19} + 24 q^{22} - 8 q^{23} + 8 q^{24} + 16 q^{26} - 16 q^{27} + 40 q^{28} + 8 q^{29} - 16 q^{34} - 24 q^{36} + 16 q^{37} + 48 q^{38} - 8 q^{39} + 16 q^{41} + 24 q^{42} + 8 q^{43} - 16 q^{44} + 8 q^{46} + 64 q^{47} + 8 q^{48} - 56 q^{51} - 24 q^{53} + 32 q^{54} + 64 q^{56} - 16 q^{57} - 56 q^{58} - 32 q^{59} + 32 q^{61} + 32 q^{62} - 80 q^{63} + 96 q^{66} - 24 q^{68} - 96 q^{69} - 24 q^{71} + 24 q^{72} + 64 q^{73} + 64 q^{74} - 8 q^{76} - 24 q^{77} - 8 q^{78} + 16 q^{82} + 96 q^{83} + 64 q^{84} - 16 q^{86} - 48 q^{87} - 8 q^{88} - 24 q^{91} - 112 q^{92} + 64 q^{93} - 56 q^{94} - 168 q^{96} - 48 q^{97} - 120 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.44607 + 1.44607i 2.95240 − 1.22292i − 2.18224i 0 −2.50094 + 6.03781i 0.450584 − 1.08781i 0.263530 + 0.263530i 5.09981 − 5.09981i 0
49.2 −0.528855 + 0.528855i −2.84096 + 1.17676i 1.44062i 0 0.880118 − 2.12479i 1.23707 − 2.98655i −1.81959 − 1.81959i 4.56494 − 4.56494i 0
49.3 −0.254738 + 0.254738i −0.0501087 + 0.0207557i 1.87022i 0 0.00747733 − 0.0180519i 0.114315 − 0.275980i −0.985893 − 0.985893i −2.11924 + 2.11924i 0
49.4 0.680853 − 0.680853i 2.44733 − 1.01372i 1.07288i 0 0.976080 − 2.35647i 1.18426 − 2.85906i 2.09218 + 2.09218i 2.84049 − 2.84049i 0
49.5 1.09631 − 1.09631i −1.05359 + 0.436412i − 0.403772i 0 −0.676617 + 1.63350i −1.43180 + 3.45666i 1.74995 + 1.74995i −1.20172 + 1.20172i 0
49.6 1.86672 − 1.86672i 1.95914 − 0.811501i − 4.96928i 0 2.14231 − 5.17200i −1.55444 + 3.75274i −5.54282 − 5.54282i 1.05836 − 1.05836i 0
274.1 −1.93083 + 1.93083i −0.591976 − 1.42916i − 5.45623i 0 3.90247 + 1.61646i 1.16820 + 0.483886i 6.67340 + 6.67340i 0.429267 − 0.429267i 0
274.2 −1.27691 + 1.27691i 0.263254 + 0.635552i − 1.26102i 0 −1.14770 − 0.475393i −4.01142 − 1.66158i −0.943613 − 0.943613i 1.78670 − 1.78670i 0
274.3 −1.09994 + 1.09994i 1.15110 + 2.77900i − 0.419729i 0 −4.32287 − 1.79059i 3.19170 + 1.32205i −1.73820 − 1.73820i −4.27649 + 4.27649i 0
274.4 0.213325 − 0.213325i 0.406032 + 0.980249i 1.90899i 0 0.295728 + 0.122495i 2.31879 + 0.960473i 0.833883 + 0.833883i 1.32529 − 1.32529i 0
274.5 1.01710 − 1.01710i −0.0420595 − 0.101541i − 0.0689897i 0 −0.146056 − 0.0604983i 0.642174 + 0.265997i 1.96403 + 1.96403i 2.11278 − 2.11278i 0
274.6 1.66305 − 1.66305i −0.600564 − 1.44989i − 3.53144i 0 −3.41000 − 1.41247i −3.30945 − 1.37082i −2.54686 − 2.54686i 0.379815 − 0.379815i 0
349.1 −1.93083 − 1.93083i −0.591976 + 1.42916i 5.45623i 0 3.90247 − 1.61646i 1.16820 − 0.483886i 6.67340 − 6.67340i 0.429267 + 0.429267i 0
349.2 −1.27691 − 1.27691i 0.263254 − 0.635552i 1.26102i 0 −1.14770 + 0.475393i −4.01142 + 1.66158i −0.943613 + 0.943613i 1.78670 + 1.78670i 0
349.3 −1.09994 − 1.09994i 1.15110 − 2.77900i 0.419729i 0 −4.32287 + 1.79059i 3.19170 − 1.32205i −1.73820 + 1.73820i −4.27649 − 4.27649i 0
349.4 0.213325 + 0.213325i 0.406032 − 0.980249i − 1.90899i 0 0.295728 − 0.122495i 2.31879 − 0.960473i 0.833883 − 0.833883i 1.32529 + 1.32529i 0
349.5 1.01710 + 1.01710i −0.0420595 + 0.101541i 0.0689897i 0 −0.146056 + 0.0604983i 0.642174 − 0.265997i 1.96403 − 1.96403i 2.11278 + 2.11278i 0
349.6 1.66305 + 1.66305i −0.600564 + 1.44989i 3.53144i 0 −3.41000 + 1.41247i −3.30945 + 1.37082i −2.54686 + 2.54686i 0.379815 + 0.379815i 0
399.1 −1.44607 − 1.44607i 2.95240 + 1.22292i 2.18224i 0 −2.50094 − 6.03781i 0.450584 + 1.08781i 0.263530 − 0.263530i 5.09981 + 5.09981i 0
399.2 −0.528855 − 0.528855i −2.84096 − 1.17676i − 1.44062i 0 0.880118 + 2.12479i 1.23707 + 2.98655i −1.81959 + 1.81959i 4.56494 + 4.56494i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.n.f 24
5.b even 2 1 425.2.n.c 24
5.c odd 4 1 85.2.l.a ✓ 24
5.c odd 4 1 425.2.m.b 24
15.e even 4 1 765.2.be.b 24
17.d even 8 1 425.2.n.c 24
85.k odd 8 1 425.2.m.b 24
85.m even 8 1 inner 425.2.n.f 24
85.n odd 8 1 85.2.l.a ✓ 24
85.o even 16 1 7225.2.a.bq 12
85.o even 16 1 7225.2.a.bs 12
85.r even 16 1 1445.2.a.p 12
85.r even 16 1 1445.2.a.q 12
85.r even 16 2 1445.2.d.j 24
255.v even 8 1 765.2.be.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.l.a ✓ 24 5.c odd 4 1
85.2.l.a ✓ 24 85.n odd 8 1
425.2.m.b 24 5.c odd 4 1
425.2.m.b 24 85.k odd 8 1
425.2.n.c 24 5.b even 2 1
425.2.n.c 24 17.d even 8 1
425.2.n.f 24 1.a even 1 1 trivial
425.2.n.f 24 85.m even 8 1 inner
765.2.be.b 24 15.e even 4 1
765.2.be.b 24 255.v even 8 1
1445.2.a.p 12 85.r even 16 1
1445.2.a.q 12 85.r even 16 1
1445.2.d.j 24 85.r even 16 2
7225.2.a.bq 12 85.o even 16 1
7225.2.a.bs 12 85.o even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} + 8 T_{2}^{17} + 2327 T_{2}^{16} + 128 T_{2}^{15} - 640 T_{2}^{13} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display