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Properties

Label 6275.2.a.j
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 6 q^{2} - 7 q^{3} + 42 q^{4} - q^{6} - 14 q^{7} - 18 q^{8} + 39 q^{9}+O(q^{10}) \) 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 −2.77118 −3.13084 5.67943 0 8.67613 1.36535 −10.1963 6.80219 0
1.2 −2.72684 2.18316 5.43566 0 −5.95313 −0.470377 −9.36850 1.76619 0
1.3 −2.66208 0.998140 5.08665 0 −2.65712 2.99748 −8.21689 −2.00372 0
1.4 −2.64488 −1.32765 4.99541 0 3.51149 −4.85995 −7.92251 −1.23733 0
1.5 −2.39346 −2.06207 3.72867 0 4.93549 −1.15201 −4.13750 1.25213 0
1.6 −2.29738 −2.23483 3.27796 0 5.13426 1.64599 −2.93597 1.99447 0
1.7 −2.22775 0.945470 2.96286 0 −2.10627 −2.68714 −2.14501 −2.10609 0
1.8 −2.09586 2.00037 2.39264 0 −4.19249 −4.91454 −0.822917 1.00146 0
1.9 −2.02953 −0.282387 2.11898 0 0.573113 4.65827 −0.241481 −2.92026 0
1.10 −1.88720 2.59964 1.56152 0 −4.90603 −2.39726 0.827497 3.75810 0
1.11 −1.65069 1.61490 0.724761 0 −2.66569 2.50348 2.10502 −0.392107 0
1.12 −1.51796 −2.60034 0.304203 0 3.94721 −3.44853 2.57415 3.76176 0
1.13 −1.48467 −0.138168 0.204259 0 0.205135 −0.606838 2.66609 −2.98091 0
1.14 −1.45929 −3.11649 0.129533 0 4.54787 −3.45295 2.72956 6.71250 0
1.15 −1.33037 3.12079 −0.230126 0 −4.15179 0.353952 2.96688 6.73934 0
1.16 −0.854297 −2.41732 −1.27018 0 2.06511 2.23704 2.79370 2.84343 0
1.17 −0.770250 −0.566050 −1.40671 0 0.436000 3.74924 2.62402 −2.67959 0
1.18 −0.750120 −1.18567 −1.43732 0 0.889393 0.0987624 2.57840 −1.59419 0
1.19 −0.661006 −0.526963 −1.56307 0 0.348326 3.34482 2.35521 −2.72231 0
1.20 −0.421665 3.05235 −1.82220 0 −1.28707 −3.53157 1.61169 6.31686 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.j ✓ 42
5.b even 2 1 6275.2.a.k yes 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6275.2.a.j ✓ 42 1.a even 1 1 trivial
6275.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6275))\):

\( T_{2}^{42} + 6 T_{2}^{41} - 45 T_{2}^{40} - 328 T_{2}^{39} + 830 T_{2}^{38} + 8186 T_{2}^{37} - 7083 T_{2}^{36} + \cdots - 4 \) 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