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Properties

Label 6275.2.a.i
Level $6275$
Weight $2$
Character orbit 6275.a
Self dual yes
Analytic conductor $50.106$
Analytic rank $1$
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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 8 q^{2} - 5 q^{3} + 42 q^{4} - q^{6} - 14 q^{7} - 24 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 8 q^{2} - 5 q^{3} + 42 q^{4} - q^{6} - 14 q^{7} - 24 q^{8} + 39 q^{9} - 10 q^{12} - 47 q^{13} + q^{14} + 46 q^{16} - 15 q^{17} - 47 q^{18} - q^{19} - 27 q^{22} - 17 q^{23} - 6 q^{24} + 14 q^{26} - 20 q^{27} - 54 q^{28} - 3 q^{29} + 3 q^{31} - 46 q^{32} - 41 q^{33} - 10 q^{34} + 33 q^{36} - 76 q^{37} - 13 q^{38} + q^{39} - 5 q^{41} - 22 q^{42} - 63 q^{43} - 5 q^{44} - 8 q^{46} - 19 q^{47} - 29 q^{48} + 42 q^{49} - 17 q^{51} - 97 q^{52} - 52 q^{53} - q^{54} + 5 q^{56} - 54 q^{57} - 23 q^{58} + 8 q^{59} - 3 q^{61} - 20 q^{62} - 20 q^{63} + 44 q^{64} - 27 q^{66} - 59 q^{67} - 56 q^{68} - 13 q^{69} - 10 q^{71} - 81 q^{72} - 46 q^{73} + 47 q^{74} - 35 q^{76} - 67 q^{77} - 38 q^{78} + 11 q^{79} - 6 q^{81} - 5 q^{82} - 10 q^{83} + 70 q^{84} - 47 q^{86} - 60 q^{87} - 113 q^{88} + 17 q^{89} - 13 q^{91} + 17 q^{92} - 78 q^{93} - 3 q^{94} - 88 q^{96} - 61 q^{97} - 58 q^{98} + 27 q^{99}+O(q^{100}) \) 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 −2.75034 0.271267 5.56436 0 −0.746077 −3.92198 −9.80318 −2.92641 0
1.2 −2.70162 −2.31396 5.29874 0 6.25144 −1.24149 −8.91193 2.35442 0
1.3 −2.67653 3.20706 5.16383 0 −8.58380 −1.39662 −8.46811 7.28523 0
1.4 −2.59907 1.10778 4.75515 0 −2.87920 0.604603 −7.16081 −1.77282 0
1.5 −2.56883 −2.28380 4.59889 0 5.86670 −0.797930 −6.67611 2.21576 0
1.6 −2.29518 −3.14234 3.26785 0 7.21224 1.98331 −2.90996 6.87432 0
1.7 −2.24913 −0.156022 3.05857 0 0.350914 −3.03824 −2.38086 −2.97566 0
1.8 −2.24427 2.81957 3.03674 0 −6.32787 5.24849 −2.32673 4.94997 0
1.9 −2.00975 1.60834 2.03910 0 −3.23236 0.976858 −0.0785754 −0.413246 0
1.10 −1.97204 −3.01898 1.88895 0 5.95355 −3.70365 0.218999 6.11422 0
1.11 −1.69550 −1.17601 0.874730 0 1.99393 −4.25758 1.90790 −1.61700 0
1.12 −1.66453 −1.39734 0.770667 0 2.32592 4.79278 2.04626 −1.04743 0
1.13 −1.56511 0.407134 0.449555 0 −0.637208 1.72427 2.42661 −2.83424 0
1.14 −1.34809 1.67248 −0.182640 0 −2.25467 −1.62974 2.94241 −0.202801 0
1.15 −1.13298 2.20911 −0.716349 0 −2.50289 −0.239650 3.07758 1.88018 0
1.16 −1.02560 0.120228 −0.948148 0 −0.123306 −4.48819 3.02362 −2.98555 0
1.17 −0.998616 −2.76763 −1.00277 0 2.76380 0.162982 2.99861 4.65977 0
1.18 −0.828745 −1.95207 −1.31318 0 1.61776 3.86487 2.74578 0.810564 0
1.19 −0.826368 2.33425 −1.31712 0 −1.92895 −4.15366 2.74116 2.44871 0
1.20 −0.665723 2.50946 −1.55681 0 −1.67061 1.52100 2.36785 3.29740 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.i ✓ 42
5.b even 2 1 6275.2.a.l yes 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6275.2.a.i ✓ 42 1.a even 1 1 trivial
6275.2.a.l yes 42 5.b even 2 1

Hecke kernels

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

\( T_{2}^{42} + 8 T_{2}^{41} - 31 T_{2}^{40} - 400 T_{2}^{39} + 130 T_{2}^{38} + 9006 T_{2}^{37} + \cdots - 756 \) Copy content Toggle raw display
\( T_{3}^{42} + 5 T_{3}^{41} - 70 T_{3}^{40} - 375 T_{3}^{39} + 2206 T_{3}^{38} + 12888 T_{3}^{37} + \cdots + 198433 \) Copy content Toggle raw display