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Properties

Label 725.2.q.b
Level $725$
Weight $2$
Character orbit 725.q
Analytic conductor $5.789$
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] = [725,2,Mod(51,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.q (of order \(14\), degree \(6\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{6} - 4 q^{7} + 21 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{6} - 4 q^{7} + 21 q^{8} - 4 q^{9} - 14 q^{11} + 8 q^{13} - 42 q^{14} - 10 q^{16} + 49 q^{21} - 3 q^{22} + 2 q^{23} + 38 q^{24} + 21 q^{27} - 44 q^{28} + 7 q^{29} + 7 q^{31} - 28 q^{32} + 54 q^{33} + 20 q^{34} - 28 q^{36} - 2 q^{38} + 7 q^{39} + 51 q^{42} - 7 q^{43} - 21 q^{44} + 21 q^{48} - 2 q^{49} + 6 q^{51} - 45 q^{52} + 7 q^{53} + 107 q^{54} - 42 q^{56} - 32 q^{57} + 10 q^{58} - 66 q^{59} + 56 q^{61} + 19 q^{62} + 45 q^{63} + 33 q^{64} - 70 q^{66} - 25 q^{67} - 28 q^{68} - 28 q^{69} + 53 q^{71} + 35 q^{72} - 14 q^{73} + 33 q^{74} - 42 q^{76} - 21 q^{77} - 18 q^{78} - 56 q^{79} - 29 q^{81} - 88 q^{82} - 47 q^{83} + 14 q^{84} + 30 q^{86} - 71 q^{87} + 104 q^{88} + 14 q^{89} - 31 q^{91} - 15 q^{92} - 46 q^{93} - 38 q^{94} - q^{96} + 49 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.82288 + 0.416060i −0.431723 + 0.896481i 1.34784 − 0.649083i 0 0.413987 − 1.81380i 0.817215 + 0.393550i 0.736786 − 0.587568i 1.25318 + 1.57143i 0
51.2 −0.951685 + 0.217216i 1.07945 − 2.24150i −0.943415 + 0.454325i 0 −0.540406 + 2.36767i 1.85273 + 0.892230i 2.32553 − 1.85455i −1.98863 − 2.49366i 0
51.3 −0.0919844 + 0.0209948i 0.100733 − 0.209175i −1.79392 + 0.863905i 0 −0.00487430 + 0.0213557i −3.10702 − 1.49626i 0.294407 − 0.234781i 1.83686 + 2.30335i 0
51.4 2.02054 − 0.461174i −1.42691 + 2.96300i 2.06795 − 0.995870i 0 −1.51666 + 6.64490i −3.84525 − 1.85178i 0.478400 − 0.381511i −4.87285 − 6.11036i 0
151.1 −1.52678 − 1.21757i −0.605332 − 0.138163i 0.403549 + 1.76806i 0 0.755986 + 0.947977i 0.0281333 − 0.123260i −0.157993 + 0.328076i −2.35557 − 1.13438i 0
151.2 0.0865386 + 0.0690123i −2.43800 − 0.556457i −0.442316 − 1.93791i 0 −0.172579 − 0.216407i −0.622806 + 2.72869i 0.191513 − 0.397681i 2.93129 + 1.41163i 0
151.3 0.992056 + 0.791138i 2.55523 + 0.583214i −0.0867667 − 0.380150i 0 2.07353 + 2.60012i −0.000528848 0.00231704i 1.31577 − 2.73223i 3.48615 + 1.67884i 0
151.4 1.97265 + 1.57313i −0.357909 − 0.0816903i 0.971544 + 4.25661i 0 −0.577517 − 0.724184i −1.14089 + 4.99858i −2.59023 + 5.37867i −2.58148 − 1.24318i 0
526.1 −1.82288 − 0.416060i −0.431723 − 0.896481i 1.34784 + 0.649083i 0 0.413987 + 1.81380i 0.817215 − 0.393550i 0.736786 + 0.587568i 1.25318 − 1.57143i 0
526.2 −0.951685 − 0.217216i 1.07945 + 2.24150i −0.943415 − 0.454325i 0 −0.540406 − 2.36767i 1.85273 − 0.892230i 2.32553 + 1.85455i −1.98863 + 2.49366i 0
526.3 −0.0919844 − 0.0209948i 0.100733 + 0.209175i −1.79392 − 0.863905i 0 −0.00487430 − 0.0213557i −3.10702 + 1.49626i 0.294407 + 0.234781i 1.83686 − 2.30335i 0
526.4 2.02054 + 0.461174i −1.42691 − 2.96300i 2.06795 + 0.995870i 0 −1.51666 − 6.64490i −3.84525 + 1.85178i 0.478400 + 0.381511i −4.87285 + 6.11036i 0
651.1 −1.06949 + 2.22082i 0.917356 + 0.731567i −2.54126 − 3.18664i 0 −2.60579 + 1.25488i 0.329257 − 0.412875i 4.98858 − 1.13861i −0.361211 − 1.58257i 0
651.2 −0.581499 + 1.20749i −0.968761 − 0.772561i 0.127077 + 0.159349i 0 1.49620 − 0.720531i 2.40911 − 3.02093i −2.87954 + 0.657236i −0.325916 − 1.42793i 0
651.3 0.410581 − 0.852581i −0.810963 − 0.646721i 0.688663 + 0.863556i 0 −0.884348 + 0.425880i −0.780924 + 0.979248i 2.86414 − 0.653721i −0.428151 − 1.87585i 0
651.4 0.561962 − 1.16693i 2.38683 + 1.90343i 0.201065 + 0.252127i 0 3.56247 − 1.71559i 2.06098 − 2.58438i 2.93264 − 0.669356i 1.40633 + 6.16153i 0
676.1 −1.06949 − 2.22082i 0.917356 − 0.731567i −2.54126 + 3.18664i 0 −2.60579 − 1.25488i 0.329257 + 0.412875i 4.98858 + 1.13861i −0.361211 + 1.58257i 0
676.2 −0.581499 − 1.20749i −0.968761 + 0.772561i 0.127077 − 0.159349i 0 1.49620 + 0.720531i 2.40911 + 3.02093i −2.87954 − 0.657236i −0.325916 + 1.42793i 0
676.3 0.410581 + 0.852581i −0.810963 + 0.646721i 0.688663 − 0.863556i 0 −0.884348 − 0.425880i −0.780924 − 0.979248i 2.86414 + 0.653721i −0.428151 + 1.87585i 0
676.4 0.561962 + 1.16693i 2.38683 − 1.90343i 0.201065 − 0.252127i 0 3.56247 + 1.71559i 2.06098 + 2.58438i 2.93264 + 0.669356i 1.40633 − 6.16153i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.q.b 24
5.b even 2 1 145.2.m.a ✓ 24
5.c odd 4 2 725.2.p.b 48
29.e even 14 1 inner 725.2.q.b 24
145.l even 14 1 145.2.m.a ✓ 24
145.q odd 28 2 725.2.p.b 48
145.s odd 28 2 4205.2.a.y 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.m.a ✓ 24 5.b even 2 1
145.2.m.a ✓ 24 145.l even 14 1
725.2.p.b 48 5.c odd 4 2
725.2.p.b 48 145.q odd 28 2
725.2.q.b 24 1.a even 1 1 trivial
725.2.q.b 24 29.e even 14 1 inner
4205.2.a.y 24 145.s odd 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 4 T_{2}^{22} - 7 T_{2}^{21} + 29 T_{2}^{20} + 84 T_{2}^{19} - 84 T_{2}^{18} - 427 T_{2}^{17} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display