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Properties

Label 425.2.n.e
Level $425$
Weight $2$
Character orbit 425.n
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
CM no
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 8 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 8 q^{6} - 12 q^{9} + 4 q^{11} - 20 q^{12} + 16 q^{13} + 24 q^{14} - 24 q^{16} + 20 q^{19} + 12 q^{22} + 8 q^{23} - 16 q^{24} + 16 q^{26} + 16 q^{27} + 20 q^{28} - 4 q^{29} + 24 q^{31} + 60 q^{32} - 16 q^{34} + 60 q^{36} - 16 q^{37} - 48 q^{38} - 8 q^{39} - 20 q^{41} + 12 q^{42} - 32 q^{43} - 64 q^{44} - 40 q^{46} - 88 q^{47} + 4 q^{48} - 24 q^{49} + 16 q^{51} - 12 q^{53} + 20 q^{54} - 32 q^{56} - 56 q^{57} - 28 q^{58} + 16 q^{59} - 64 q^{61} + 16 q^{62} - 40 q^{63} - 72 q^{66} + 48 q^{68} + 48 q^{69} - 24 q^{71} + 120 q^{72} + 20 q^{73} - 32 q^{74} + 52 q^{76} + 24 q^{77} - 100 q^{78} + 48 q^{79} + 8 q^{82} + 12 q^{83} + 40 q^{84} - 16 q^{86} - 24 q^{87} + 80 q^{88} + 24 q^{91} - 56 q^{92} + 32 q^{93} + 40 q^{94} + 132 q^{96} - 24 q^{97} + 48 q^{98} + 80 q^{99}+O(q^{100}) \) 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 −1.91681 + 1.91681i 0.977976 − 0.405091i − 5.34834i 0 −1.09811 + 2.65108i −0.544082 + 1.31353i 6.41814 + 6.41814i −1.32898 + 1.32898i 0
49.2 −0.921271 + 0.921271i −0.309218 + 0.128082i 0.302521i 0 0.166875 − 0.402872i 1.49424 − 3.60742i −2.12124 − 2.12124i −2.04211 + 2.04211i 0
49.3 −0.639117 + 0.639117i 2.66226 − 1.10274i 1.18306i 0 −0.996713 + 2.40628i −0.278207 + 0.671652i −2.03435 − 2.03435i 3.75027 − 3.75027i 0
49.4 −0.187572 + 0.187572i −1.67148 + 0.692349i 1.92963i 0 0.183657 − 0.443388i −1.88562 + 4.55230i −0.737090 − 0.737090i 0.193173 − 0.193173i 0
49.5 0.917051 − 0.917051i −1.69803 + 0.703348i 0.318036i 0 −0.912176 + 2.20219i 1.34200 − 3.23987i 2.12576 + 2.12576i 0.267297 − 0.267297i 0
49.6 1.33351 − 1.33351i 1.74560 − 0.723051i − 1.55648i 0 1.36358 − 3.29196i −0.128325 + 0.309803i 0.591431 + 0.591431i 0.402996 − 0.402996i 0
274.1 −1.71892 + 1.71892i 0.281370 + 0.679288i − 3.90934i 0 −1.65129 − 0.683987i −0.537508 − 0.222643i 3.28199 + 3.28199i 1.73906 − 1.73906i 0
274.2 −0.813283 + 0.813283i −0.786634 − 1.89910i 0.677141i 0 2.18426 + 0.904752i 0.346882 + 0.143683i −2.17727 − 2.17727i −0.866476 + 0.866476i 0
274.3 −0.392996 + 0.392996i 0.958007 + 2.31283i 1.69111i 0 −1.28543 − 0.532441i −3.72866 − 1.54446i −1.45059 − 1.45059i −2.31010 + 2.31010i 0
274.4 0.982785 − 0.982785i −0.0424107 − 0.102388i 0.0682683i 0 −0.142306 − 0.0589452i −1.58527 − 0.656642i 2.03266 + 2.03266i 2.11264 − 2.11264i 0
274.5 1.52941 − 1.52941i 1.06191 + 2.56367i − 2.67820i 0 5.54499 + 2.29681i 2.90308 + 1.20249i −1.03724 − 1.03724i −3.32343 + 3.32343i 0
274.6 1.82721 − 1.82721i −1.17935 − 2.84719i − 4.67741i 0 −7.35734 − 3.04751i 2.60148 + 1.07757i −4.89219 − 4.89219i −4.59433 + 4.59433i 0
349.1 −1.71892 − 1.71892i 0.281370 − 0.679288i 3.90934i 0 −1.65129 + 0.683987i −0.537508 + 0.222643i 3.28199 − 3.28199i 1.73906 + 1.73906i 0
349.2 −0.813283 − 0.813283i −0.786634 + 1.89910i − 0.677141i 0 2.18426 − 0.904752i 0.346882 − 0.143683i −2.17727 + 2.17727i −0.866476 − 0.866476i 0
349.3 −0.392996 − 0.392996i 0.958007 − 2.31283i − 1.69111i 0 −1.28543 + 0.532441i −3.72866 + 1.54446i −1.45059 + 1.45059i −2.31010 − 2.31010i 0
349.4 0.982785 + 0.982785i −0.0424107 + 0.102388i − 0.0682683i 0 −0.142306 + 0.0589452i −1.58527 + 0.656642i 2.03266 − 2.03266i 2.11264 + 2.11264i 0
349.5 1.52941 + 1.52941i 1.06191 − 2.56367i 2.67820i 0 5.54499 − 2.29681i 2.90308 − 1.20249i −1.03724 + 1.03724i −3.32343 − 3.32343i 0
349.6 1.82721 + 1.82721i −1.17935 + 2.84719i 4.67741i 0 −7.35734 + 3.04751i 2.60148 − 1.07757i −4.89219 + 4.89219i −4.59433 − 4.59433i 0
399.1 −1.91681 − 1.91681i 0.977976 + 0.405091i 5.34834i 0 −1.09811 − 2.65108i −0.544082 − 1.31353i 6.41814 − 6.41814i −1.32898 − 1.32898i 0
399.2 −0.921271 − 0.921271i −0.309218 − 0.128082i − 0.302521i 0 0.166875 + 0.402872i 1.49424 + 3.60742i −2.12124 + 2.12124i −2.04211 − 2.04211i 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.e 24
5.b even 2 1 425.2.n.d 24
5.c odd 4 1 425.2.m.c ✓ 24
5.c odd 4 1 425.2.m.d yes 24
17.d even 8 1 425.2.n.d 24
85.k odd 8 1 425.2.m.c ✓ 24
85.m even 8 1 inner 425.2.n.e 24
85.n odd 8 1 425.2.m.d yes 24
85.o even 16 2 7225.2.a.bx 24
85.r even 16 2 7225.2.a.cb 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.m.c ✓ 24 5.c odd 4 1
425.2.m.c ✓ 24 85.k odd 8 1
425.2.m.d yes 24 5.c odd 4 1
425.2.m.d yes 24 85.n odd 8 1
425.2.n.d 24 5.b even 2 1
425.2.n.d 24 17.d even 8 1
425.2.n.e 24 1.a even 1 1 trivial
425.2.n.e 24 85.m even 8 1 inner
7225.2.a.bx 24 85.o even 16 2
7225.2.a.cb 24 85.r even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} - 12 T_{2}^{19} + 100 T_{2}^{17} + 2351 T_{2}^{16} - 392 T_{2}^{15} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display