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Properties

Label 1045.4.a.i
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $0$
Dimension $25$
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] = [1045,4,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(61.6569959560\)
Analytic rank: \(0\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100}) \) 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 −5.27434 4.83101 19.8186 −5.00000 −25.4803 −13.3199 −62.3354 −3.66139 26.3717
1.2 −5.23688 −8.48150 19.4249 −5.00000 44.4166 −12.7409 −59.8310 44.9358 26.1844
1.3 −4.81539 9.01446 15.1880 −5.00000 −43.4082 26.8150 −34.6132 54.2605 24.0770
1.4 −4.50858 −2.01806 12.3273 −5.00000 9.09858 15.1423 −19.5101 −22.9274 22.5429
1.5 −3.75267 −6.96028 6.08254 −5.00000 26.1196 −29.7997 7.19560 21.4455 18.7634
1.6 −3.65423 −3.03858 5.35338 −5.00000 11.1037 26.1615 9.67137 −17.7670 18.2711
1.7 −2.74955 4.78923 −0.439974 −5.00000 −13.1682 −9.83055 23.2061 −4.06331 13.7478
1.8 −2.74566 4.96249 −0.461347 −5.00000 −13.6253 −34.3482 23.2320 −2.37368 13.7283
1.9 −2.51141 8.82864 −1.69283 −5.00000 −22.1723 6.85719 24.3426 50.9449 12.5570
1.10 −1.72202 −1.27869 −5.03464 −5.00000 2.20194 3.57098 22.4459 −25.3649 8.61011
1.11 −1.64212 −7.74531 −5.30343 −5.00000 12.7188 −5.31024 21.8459 32.9899 8.21062
1.12 −0.333805 −8.69029 −7.88857 −5.00000 2.90086 18.9477 5.30368 48.5211 1.66902
1.13 0.109799 6.64848 −7.98794 −5.00000 0.729999 28.5341 −1.75547 17.2022 −0.548997
1.14 0.197347 −0.963676 −7.96105 −5.00000 −0.190178 0.480073 −3.14986 −26.0713 −0.986733
1.15 1.70323 9.22981 −5.09901 −5.00000 15.7205 −12.5263 −22.3106 58.1894 −8.51615
1.16 1.75191 3.91466 −4.93083 −5.00000 6.85812 −21.2936 −22.6536 −11.6754 −8.75953
1.17 2.33606 −1.07169 −2.54281 −5.00000 −2.50353 23.6079 −24.6287 −25.8515 −11.6803
1.18 3.11750 0.0197889 1.71881 −5.00000 0.0616921 −19.0525 −19.5816 −26.9996 −15.5875
1.19 3.49473 −5.86270 4.21313 −5.00000 −20.4885 −34.6786 −13.2341 7.37127 −17.4736
1.20 3.81190 −9.32002 6.53062 −5.00000 −35.5270 25.5875 −5.60115 59.8628 −19.0595
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(-1\)

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 1045.4.a.i ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.i ✓ 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 2 T_{2}^{24} - 159 T_{2}^{23} + 290 T_{2}^{22} + 11059 T_{2}^{21} - 18125 T_{2}^{20} + \cdots - 2492006400 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display