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Properties

Label 425.2.r.a
Level $425$
Weight $2$
Character orbit 425.r
Analytic conductor $3.394$
Analytic rank $0$
Dimension $160$
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(69,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.r (of order \(10\), degree \(4\), 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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 40 q^{4} - 8 q^{5} + 4 q^{6} - 30 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 40 q^{4} - 8 q^{5} + 4 q^{6} - 30 q^{8} + 36 q^{9} - 6 q^{10} + 8 q^{11} - 40 q^{12} - 20 q^{14} - 40 q^{15} - 64 q^{16} + 6 q^{19} + 2 q^{20} - 50 q^{22} + 20 q^{23} + 20 q^{24} + 32 q^{25} + 20 q^{26} + 30 q^{27} + 60 q^{28} + 16 q^{30} + 12 q^{31} - 40 q^{33} - 4 q^{34} - 54 q^{36} - 40 q^{38} - 12 q^{39} - 52 q^{40} - 2 q^{41} + 30 q^{42} - 20 q^{44} + 42 q^{45} + 48 q^{46} + 40 q^{47} - 100 q^{48} - 184 q^{49} + 46 q^{50} - 32 q^{51} - 10 q^{52} - 20 q^{53} - 40 q^{54} + 18 q^{55} + 24 q^{56} - 120 q^{58} + 6 q^{59} - 40 q^{60} + 12 q^{61} - 10 q^{62} + 90 q^{63} + 58 q^{64} - 20 q^{65} + 48 q^{66} - 90 q^{67} - 74 q^{69} - 120 q^{70} - 16 q^{71} - 10 q^{72} + 52 q^{74} + 76 q^{75} + 96 q^{76} + 80 q^{77} + 40 q^{78} - 60 q^{79} - 24 q^{80} - 118 q^{81} - 76 q^{84} + 2 q^{85} + 44 q^{86} + 40 q^{87} + 140 q^{88} + 50 q^{89} + 224 q^{90} + 28 q^{91} - 170 q^{92} + 96 q^{94} - 44 q^{95} - 116 q^{96} + 110 q^{97} + 70 q^{98} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −2.60604 + 0.846752i −1.86275 − 2.56385i 4.45640 − 3.23776i 0.0540158 + 2.23542i 7.02533 + 5.10420i − 2.47810i −5.65071 + 7.77753i −2.17645 + 6.69843i −2.03361 − 5.77983i
69.2 −2.55039 + 0.828673i −0.0154746 − 0.0212989i 4.19977 − 3.05131i 2.16953 + 0.541436i 0.0571160 + 0.0414972i 2.17034i −5.03006 + 6.92328i 0.926837 − 2.85251i −5.98182 + 0.416954i
69.3 −2.47862 + 0.805352i 1.68980 + 2.32581i 3.87693 − 2.81675i −1.39512 + 1.74746i −6.06145 − 4.40390i 3.88508i −4.27721 + 5.88707i −1.62691 + 5.00711i 2.05065 − 5.45486i
69.4 −2.42763 + 0.788786i −0.379903 − 0.522892i 3.65319 − 2.65420i −1.95346 − 1.08812i 1.33472 + 0.969728i 1.82616i −3.77429 + 5.19486i 0.797961 − 2.45587i 5.60057 + 1.10071i
69.5 −2.20587 + 0.716731i −0.551084 − 0.758502i 2.73413 − 1.98646i 1.11977 − 1.93549i 1.75926 + 1.27818i − 3.09233i −1.88077 + 2.58866i 0.655419 − 2.01717i −1.08286 + 5.07201i
69.6 −2.14124 + 0.695730i 1.08145 + 1.48849i 2.48282 − 1.80387i −0.133284 + 2.23209i −3.35124 − 2.43482i − 4.38149i −1.41458 + 1.94701i −0.119019 + 0.366304i −1.26754 − 4.87217i
69.7 −1.85902 + 0.604031i −1.84553 − 2.54016i 1.47306 − 1.07024i −1.33190 − 1.79612i 4.96521 + 3.60744i 3.19704i 0.205893 − 0.283388i −2.11936 + 6.52273i 3.56094 + 2.53450i
69.8 −1.80321 + 0.585900i 0.0268065 + 0.0368959i 1.29027 − 0.937434i 0.943879 + 2.02709i −0.0699551 − 0.0508253i − 0.426178i 0.451510 − 0.621450i 0.926408 − 2.85119i −2.88969 − 3.10226i
69.9 −1.63738 + 0.532018i −1.52304 − 2.09629i 0.779945 − 0.566663i 2.20989 − 0.341170i 3.60906 + 2.62214i − 0.996248i 1.04832 − 1.44289i −1.14771 + 3.53228i −3.43692 + 1.73433i
69.10 −1.50815 + 0.490028i 0.441680 + 0.607920i 0.416355 − 0.302500i 1.21824 − 1.87507i −0.964016 − 0.700399i 4.59462i 1.38448 − 1.90558i 0.752565 − 2.31616i −0.918447 + 3.42486i
69.11 −1.46986 + 0.477587i −1.10052 − 1.51474i 0.314369 − 0.228403i −2.19230 + 0.440256i 2.34104 + 1.70086i − 4.04029i 1.46385 − 2.01482i −0.156235 + 0.480841i 3.01211 − 1.69413i
69.12 −1.38991 + 0.451609i 1.86675 + 2.56936i 0.109865 − 0.0798213i 2.21373 − 0.315251i −3.75495 − 2.72813i 0.259005i 1.60137 − 2.20410i −2.18980 + 6.73951i −2.93452 + 1.43791i
69.13 −1.17389 + 0.381421i −1.02469 − 1.41037i −0.385489 + 0.280074i 0.897862 + 2.04789i 1.74082 + 1.26478i 2.70715i 1.79671 − 2.47296i −0.0120917 + 0.0372143i −1.83510 − 2.06154i
69.14 −1.07215 + 0.348362i 0.792633 + 1.09097i −0.589886 + 0.428578i −1.75044 + 1.39139i −1.22987 − 0.893554i 3.55811i 1.80840 − 2.48904i 0.365112 − 1.12370i 1.39202 − 2.10156i
69.15 −1.03358 + 0.335830i 0.604115 + 0.831493i −0.662529 + 0.481355i −2.14816 + 0.620799i −0.903642 − 0.656534i − 1.04210i 1.80070 − 2.47845i 0.600625 − 1.84853i 2.01182 − 1.36306i
69.16 −0.628349 + 0.204163i 0.482449 + 0.664034i −1.26489 + 0.918999i 2.18757 − 0.463165i −0.438718 − 0.318747i − 4.33633i 1.38385 − 1.90471i 0.718867 − 2.21244i −1.28000 + 0.737651i
69.17 −0.419745 + 0.136383i 1.18389 + 1.62948i −1.46045 + 1.06108i −0.544497 − 2.16876i −0.719166 − 0.522504i − 3.59659i 0.987136 − 1.35868i −0.326569 + 1.00508i 0.524333 + 0.836067i
69.18 −0.292236 + 0.0949534i −0.973705 − 1.34019i −1.54165 + 1.12007i 1.39711 − 1.74587i 0.411808 + 0.299196i 1.62592i 0.705395 − 0.970893i 0.0790435 − 0.243271i −0.242510 + 0.642869i
69.19 −0.256860 + 0.0834589i −1.58231 − 2.17787i −1.55902 + 1.13270i −2.05873 + 0.872706i 0.588195 + 0.427349i 2.44071i 0.623414 − 0.858055i −1.31234 + 4.03896i 0.455972 − 0.395983i
69.20 −0.248853 + 0.0808572i −0.408373 − 0.562077i −1.56264 + 1.13533i −1.61719 − 1.54424i 0.147073 + 0.106855i 1.76904i 0.604668 − 0.832255i 0.777889 − 2.39410i 0.527307 + 0.253527i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.r.a ✓ 160
25.e even 10 1 inner 425.2.r.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.r.a ✓ 160 1.a even 1 1 trivial
425.2.r.a ✓ 160 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(425, [\chi])\).