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Properties

Label 6275.2.a.n
Level $6275$
Weight $2$
Character orbit 6275.a
Self dual yes
Analytic conductor $50.106$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6275,2,Mod(1,6275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6275 = 5^{2} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6275.a (trivial)

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: yes
Analytic conductor: \(50.1061272684\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: no (minimal twist has level 1255)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 98 q^{4} + 16 q^{6} + 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 98 q^{4} + 16 q^{6} + 106 q^{9} + 16 q^{11} + 14 q^{14} + 134 q^{16} + 46 q^{19} + 74 q^{21} + 48 q^{24} + 16 q^{26} + 78 q^{29} + 50 q^{31} + 46 q^{34} + 162 q^{36} + 56 q^{39} + 38 q^{41} + 62 q^{44} + 98 q^{46} + 172 q^{49} + 72 q^{51} + 64 q^{54} + 60 q^{56} + 34 q^{59} + 146 q^{61} + 232 q^{64} + 36 q^{66} + 90 q^{69} + 50 q^{71} - 4 q^{74} + 116 q^{76} + 114 q^{79} + 204 q^{81} + 68 q^{84} + 26 q^{86} + 4 q^{89} + 78 q^{91} + 36 q^{94} + 112 q^{96} + 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80997 −2.07342 5.89594 0 5.82625 3.39566 −10.9475 1.29907 0
1.2 −2.75511 2.42688 5.59061 0 −6.68631 −4.34916 −9.89250 2.88974 0
1.3 −2.74031 −1.08983 5.50931 0 2.98648 −1.31839 −9.61660 −1.81226 0
1.4 −2.72323 1.19432 5.41598 0 −3.25241 3.47157 −9.30248 −1.57360 0
1.5 −2.64182 0.933266 4.97920 0 −2.46552 −3.65947 −7.87049 −2.12901 0
1.6 −2.60683 −3.13368 4.79556 0 8.16897 −0.693057 −7.28754 6.81996 0
1.7 −2.57536 1.56502 4.63246 0 −4.03049 5.13288 −6.77953 −0.550709 0
1.8 −2.55766 −2.15941 4.54163 0 5.52305 −0.783414 −6.50063 1.66307 0
1.9 −2.38893 −3.13141 3.70700 0 7.48074 −4.53028 −4.07791 6.80575 0
1.10 −2.34693 0.111270 3.50808 0 −0.261143 −2.19791 −3.53936 −2.98762 0
1.11 −2.32364 −1.68683 3.39932 0 3.91960 −2.58974 −3.25151 −0.154589 0
1.12 −2.21769 3.42922 2.91817 0 −7.60497 4.09929 −2.03622 8.75957 0
1.13 −2.01653 −2.47902 2.06640 0 4.99902 1.41290 −0.133907 3.14552 0
1.14 −1.97679 −2.05131 1.90770 0 4.05500 −4.80218 0.182460 1.20786 0
1.15 −1.97086 1.87035 1.88430 0 −3.68621 2.70307 0.228030 0.498219 0
1.16 −1.94294 0.765242 1.77501 0 −1.48682 −1.46751 0.437143 −2.41440 0
1.17 −1.90386 2.96578 1.62468 0 −5.64643 −1.97343 0.714552 5.79586 0
1.18 −1.89709 2.02619 1.59896 0 −3.84388 2.81498 0.760807 1.10547 0
1.19 −1.79244 2.90911 1.21286 0 −5.21441 −3.17729 1.41091 5.46290 0
1.20 −1.76707 −0.959398 1.12253 0 1.69532 3.39434 1.55055 −2.07956 0
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.76
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(251\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6275.2.a.n 76
5.b even 2 1 inner 6275.2.a.n 76
5.c odd 4 2 1255.2.b.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1255.2.b.b ✓ 76 5.c odd 4 2
6275.2.a.n 76 1.a even 1 1 trivial
6275.2.a.n 76 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6275))\):

\( T_{2}^{76} - 125 T_{2}^{74} + 7480 T_{2}^{72} - 285315 T_{2}^{70} + 7793055 T_{2}^{68} - 162337074 T_{2}^{66} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{3}^{76} - 167 T_{3}^{74} + 13313 T_{3}^{72} - 674530 T_{3}^{70} + 24400317 T_{3}^{68} + \cdots + 367856806144 \) Copy content Toggle raw display