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Properties

Label 1075.6.a.i
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 −10.8086 30.0700 84.8250 0 −325.014 54.3959 −570.963 661.206 0
1.2 −10.6831 −9.08102 82.1291 0 97.0136 −138.506 −535.536 −160.535 0
1.3 −10.3914 10.7135 75.9819 0 −111.328 32.3758 −457.035 −128.222 0
1.4 −10.0422 −25.6441 68.8458 0 257.523 −86.6514 −370.013 414.621 0
1.5 −9.45595 20.2754 57.4150 0 −191.723 137.133 −240.323 168.090 0
1.6 −8.36207 −24.1411 37.9243 0 201.870 224.337 −49.5390 339.794 0
1.7 −7.54165 15.8216 24.8765 0 −119.321 −147.614 53.7230 7.32245 0
1.8 −7.36705 −9.50108 22.2734 0 69.9949 66.6986 71.6560 −152.730 0
1.9 −7.20077 5.47224 19.8511 0 −39.4044 −128.650 87.4815 −213.055 0
1.10 −6.87442 5.75421 15.2577 0 −39.5569 159.977 115.094 −209.889 0
1.11 −5.38312 −10.6908 −3.02204 0 57.5496 193.079 188.528 −128.708 0
1.12 −5.09650 −28.9150 −6.02567 0 147.365 −141.657 193.798 593.078 0
1.13 −4.74820 −11.4452 −9.45457 0 54.3444 −225.148 196.835 −112.006 0
1.14 −3.95362 3.98544 −16.3689 0 −15.7569 −105.027 191.232 −227.116 0
1.15 −2.14099 23.1734 −27.4162 0 −49.6140 −101.342 127.210 294.004 0
1.16 −2.12910 16.4702 −27.4669 0 −35.0667 −114.364 126.611 28.2671 0
1.17 −1.48662 −16.7365 −29.7900 0 24.8807 109.207 91.8580 37.1088 0
1.18 −0.156002 −15.3198 −31.9757 0 2.38992 −16.1051 9.98030 −8.30330 0
1.19 0.171620 23.4398 −31.9705 0 4.02273 167.691 −10.9786 306.423 0
1.20 0.682970 −26.0895 −31.5336 0 −17.8183 −212.476 −43.3915 437.662 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.i ✓ 37
5.b even 2 1 1075.6.a.j yes 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1075.6.a.i ✓ 37 1.a even 1 1 trivial
1075.6.a.j yes 37 5.b even 2 1

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