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Properties

Label 1075.6.a.j
Level $1075$
Weight $6$
Character orbit 1075.a
Self dual yes
Analytic conductor $172.413$
Analytic rank $0$
Dimension $37$
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] = [1075,6,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1075.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(172.412606299\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9} + 675 q^{11} - 4446 q^{12} + 1241 q^{13} + 2375 q^{14} + 10518 q^{16} + 1153 q^{17} - 6680 q^{18} + 4065 q^{19} + 9953 q^{21} + 9283 q^{22} - 360 q^{23} + 2265 q^{24} + 23695 q^{26} + 1323 q^{27} - 30375 q^{28} + 19290 q^{29} + 23291 q^{31} + 8166 q^{32} - 10388 q^{33} - 13153 q^{34} + 148705 q^{36} + 13501 q^{37} - 8127 q^{38} - 1327 q^{39} + 38345 q^{41} - 21835 q^{42} + 68413 q^{43} + 47768 q^{44} + 48755 q^{46} + 84859 q^{47} - 208720 q^{48} + 107255 q^{49} + 62027 q^{51} + 128320 q^{52} - 53559 q^{53} + 44158 q^{54} + 107538 q^{56} + 104239 q^{57} - 85186 q^{58} + 48186 q^{59} + 82364 q^{61} + 206506 q^{62} - 75269 q^{63} + 161467 q^{64} + 91969 q^{66} + 38168 q^{67} + 95991 q^{68} + 287103 q^{69} + 155302 q^{71} + 9979 q^{72} + 31927 q^{73} + 59946 q^{74} + 225407 q^{76} - 80007 q^{77} - 67815 q^{78} + 150174 q^{79} + 417489 q^{81} + 60603 q^{82} + 266568 q^{83} + 586273 q^{84} - 57554 q^{87} + 323054 q^{88} + 334356 q^{89} + 51747 q^{91} - 258529 q^{92} - 285287 q^{93} + 302744 q^{94} + 287282 q^{96} - 78640 q^{97} - 397117 q^{98} + 362152 q^{99}+O(q^{100}) \) 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 −10.9530 −11.8111 87.9689 0 129.367 32.4628 −613.029 −103.499 0
1.2 −10.8396 −29.0649 85.4968 0 315.051 −249.342 −579.884 601.767 0
1.3 −9.68160 5.21938 61.7333 0 −50.5319 −165.155 −287.866 −215.758 0
1.4 −9.39756 24.6425 56.3140 0 −231.579 45.1741 −228.493 364.253 0
1.5 −9.11541 −27.9963 51.0907 0 255.198 156.738 −174.020 540.793 0
1.6 −8.81448 28.7877 45.6951 0 −253.749 −98.9387 −120.716 585.731 0
1.7 −7.79265 −20.0614 28.7254 0 156.331 −205.945 25.5180 159.460 0
1.8 −7.68733 3.40773 27.0951 0 −26.1963 189.127 37.7059 −231.387 0
1.9 −7.65029 3.61004 26.5270 0 −27.6179 122.545 41.8704 −229.968 0
1.10 −6.05618 −23.8033 4.67727 0 144.157 −34.2246 165.471 323.599 0
1.11 −5.88541 18.4245 2.63800 0 −108.435 −37.9409 172.807 96.4607 0
1.12 −4.41252 −8.97599 −12.5297 0 39.6068 −156.734 196.488 −162.432 0
1.13 −4.20568 −17.5719 −14.3123 0 73.9018 253.679 194.775 65.7719 0
1.14 −3.78745 −2.98047 −17.6552 0 11.2884 −10.8305 188.067 −234.117 0
1.15 −3.70267 25.2071 −18.2902 0 −93.3338 50.9266 186.208 392.399 0
1.16 −1.57820 −0.410475 −29.5093 0 0.647811 −31.0376 97.0739 −242.832 0
1.17 −1.40676 10.9884 −30.0210 0 −15.4581 −133.152 87.2489 −122.255 0
1.18 −0.682970 26.0895 −31.5336 0 −17.8183 212.476 43.3915 437.662 0
1.19 −0.171620 −23.4398 −31.9705 0 4.02273 −167.691 10.9786 306.423 0
1.20 0.156002 15.3198 −31.9757 0 2.38992 16.1051 −9.98030 −8.30330 0
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(43\) \(-1\)

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 1075.6.a.j yes 37
5.b even 2 1 1075.6.a.i ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1075.6.a.i ✓ 37 5.b even 2 1
1075.6.a.j yes 37 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 907 T_{2}^{35} - 71 T_{2}^{34} + 374631 T_{2}^{33} + 57311 T_{2}^{32} + \cdots + 13\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1075))\). Copy content Toggle raw display