Properties

Label 425.2.n.d
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: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} - 8 q^{6} - 12 q^{9}+O(q^{10}) \) 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

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 −1.33351 + 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
49.2 −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
49.3 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
49.4 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
49.5 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
49.6 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
274.1 −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
274.2 −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
274.3 −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
274.4 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
274.5 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
274.6 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
349.1 −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
349.2 −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
349.3 −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
349.4 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
349.5 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
349.6 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
399.1 −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
399.2 −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
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.d 24
5.b even 2 1 425.2.n.e 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.e 24
85.k odd 8 1 425.2.m.d yes 24
85.m even 8 1 inner 425.2.n.d 24
85.n odd 8 1 425.2.m.c 24
85.o even 16 2 7225.2.a.cb 24
85.r even 16 2 7225.2.a.bx 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.n odd 8 1
425.2.m.d yes 24 5.c odd 4 1
425.2.m.d yes 24 85.k odd 8 1
425.2.n.d 24 1.a even 1 1 trivial
425.2.n.d 24 85.m even 8 1 inner
425.2.n.e 24 5.b even 2 1
425.2.n.e 24 17.d even 8 1
7225.2.a.bx 24 85.r even 16 2
7225.2.a.cb 24 85.o 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