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Properties

Label 6275.2.a.k
Level $6275$
Weight $2$
Character orbit 6275.a
Self dual yes
Analytic conductor $50.106$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $1$

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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: \(42\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 6 q^{2} + 7 q^{3} + 42 q^{4} - q^{6} + 14 q^{7} + 18 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 6 q^{2} + 7 q^{3} + 42 q^{4} - q^{6} + 14 q^{7} + 18 q^{8} + 39 q^{9} + 26 q^{12} + 49 q^{13} + q^{14} + 46 q^{16} + 17 q^{17} + 23 q^{18} - q^{19} + 45 q^{22} + 19 q^{23} - 6 q^{24} + 14 q^{26} + 28 q^{27} + 30 q^{28} - 3 q^{29} + 3 q^{31} + 52 q^{32} + 23 q^{33} - 10 q^{34} + 33 q^{36} + 80 q^{37} + 19 q^{38} + q^{39} - 5 q^{41} + 26 q^{42} + 53 q^{43} - 5 q^{44} - 8 q^{46} + 13 q^{47} + 55 q^{48} + 42 q^{49} - 17 q^{51} + 95 q^{52} + 48 q^{53} - q^{54} + 5 q^{56} + 26 q^{57} + 53 q^{58} + 8 q^{59} - 3 q^{61} + 24 q^{62} + 64 q^{63} + 32 q^{64} + 21 q^{66} + 49 q^{67} + 24 q^{68} - 13 q^{69} + 6 q^{71} + 51 q^{72} + 82 q^{73} - 13 q^{74} + 15 q^{76} + 49 q^{77} + 42 q^{78} - 13 q^{79} + 26 q^{81} + 39 q^{82} + 34 q^{83} - 98 q^{84} + 41 q^{86} + 6 q^{87} + 103 q^{88} - 31 q^{89} + 15 q^{91} - 19 q^{92} + 98 q^{93} - 15 q^{94} + 72 q^{96} + 37 q^{97} + 40 q^{98} - 21 q^{99}+O(q^{100}) \) 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 −2.64705 −0.891631 5.00689 0 2.36020 3.75742 −7.95940 −2.20499 0
1.2 −2.58542 2.30319 4.68438 0 −5.95470 2.61626 −6.94024 2.30468 0
1.3 −2.51814 1.86335 4.34105 0 −4.69217 −3.82247 −5.89510 0.472059 0
1.4 −2.22523 −0.728286 2.95164 0 1.62060 0.371036 −2.11761 −2.46960 0
1.5 −2.17709 0.312204 2.73972 0 −0.679697 2.55746 −1.61045 −2.90253 0
1.6 −2.14114 2.37029 2.58448 0 −5.07512 −3.08484 −1.25145 2.61828 0
1.7 −2.02455 −1.80496 2.09881 0 3.65423 −0.867143 −0.200049 0.257864 0
1.8 −1.87136 −2.65091 1.50200 0 4.96081 4.25505 0.931945 4.02731 0
1.9 −1.71761 −1.20789 0.950191 0 2.07468 −2.73998 1.80316 −1.54101 0
1.10 −1.69577 3.40801 0.875645 0 −5.77922 −1.49608 1.90665 8.61456 0
1.11 −1.57922 1.62316 0.493948 0 −2.56334 −0.716702 2.37839 −0.365344 0
1.12 −1.26770 2.40188 −0.392926 0 −3.04487 4.72147 3.03352 2.76902 0
1.13 −1.02960 −1.38591 −0.939926 0 1.42693 0.351931 3.02694 −1.07927 0
1.14 −0.972810 −1.08437 −1.05364 0 1.05488 0.688190 2.97061 −1.82415 0
1.15 −0.914322 −2.94489 −1.16401 0 2.69257 0.205586 2.89293 5.67235 0
1.16 −0.551738 0.828199 −1.69558 0 −0.456949 0.550689 2.03900 −2.31409 0
1.17 −0.296499 −0.0750930 −1.91209 0 0.0222650 −0.792746 1.15993 −2.99436 0
1.18 −0.189518 0.866673 −1.96408 0 −0.164250 4.11214 0.751266 −2.24888 0
1.19 −0.0961754 1.30630 −1.99075 0 −0.125634 3.98228 0.383812 −1.29357 0
1.20 −0.0821513 −1.45084 −1.99325 0 0.119188 −1.42262 0.328051 −0.895072 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6275.2.a.k yes 42
5.b even 2 1 6275.2.a.j ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6275.2.a.j ✓ 42 5.b even 2 1
6275.2.a.k yes 42 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{42} - 6 T_{2}^{41} - 45 T_{2}^{40} + 328 T_{2}^{39} + 830 T_{2}^{38} - 8186 T_{2}^{37} - 7083 T_{2}^{36} + \cdots - 4 \) Copy content Toggle raw display
\( T_{3}^{42} - 7 T_{3}^{41} - 58 T_{3}^{40} + 497 T_{3}^{39} + 1362 T_{3}^{38} - 16056 T_{3}^{37} + \cdots - 468407 \) Copy content Toggle raw display