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Properties

Label 1001.4.a.g
Level $1001$
Weight $4$
Character orbit 1001.a
Self dual yes
Analytic conductor $59.061$
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] = [1001,4,Mod(1,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, 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("1001.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1001.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: \(59.0609119157\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 4 q^{2} + 4 q^{3} + 114 q^{4} - 2 q^{5} + 47 q^{6} + 175 q^{7} + 63 q^{8} + 251 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 4 q^{2} + 4 q^{3} + 114 q^{4} - 2 q^{5} + 47 q^{6} + 175 q^{7} + 63 q^{8} + 251 q^{9} + 105 q^{10} - 275 q^{11} + 21 q^{12} + 325 q^{13} + 28 q^{14} + 56 q^{15} + 678 q^{16} + 26 q^{17} + 6 q^{18} + 534 q^{19} + 186 q^{20} + 28 q^{21} - 44 q^{22} + 230 q^{23} + 365 q^{24} + 873 q^{25} + 52 q^{26} - 134 q^{27} + 798 q^{28} + 362 q^{29} + 404 q^{30} + 984 q^{31} + 568 q^{32} - 44 q^{33} + 1193 q^{34} - 14 q^{35} + 789 q^{36} + 306 q^{37} + 534 q^{38} + 52 q^{39} + 94 q^{40} + 534 q^{41} + 329 q^{42} + 1212 q^{43} - 1254 q^{44} + 600 q^{45} + 528 q^{46} + 534 q^{47} + 732 q^{48} + 1225 q^{49} + 1209 q^{50} + 998 q^{51} + 1482 q^{52} - 1670 q^{53} + 530 q^{54} + 22 q^{55} + 441 q^{56} + 1110 q^{57} + 1050 q^{58} + 1320 q^{59} - 3057 q^{60} + 970 q^{61} - 132 q^{62} + 1757 q^{63} + 5077 q^{64} - 26 q^{65} - 517 q^{66} + 1148 q^{67} + 270 q^{68} - 498 q^{69} + 735 q^{70} + 122 q^{71} + 1241 q^{72} + 3370 q^{73} - 2745 q^{74} - 674 q^{75} + 5998 q^{76} - 1925 q^{77} + 611 q^{78} + 3162 q^{79} + 1778 q^{80} + 969 q^{81} - 1080 q^{82} - 288 q^{83} + 147 q^{84} + 1240 q^{85} - 810 q^{86} + 1880 q^{87} - 693 q^{88} - 392 q^{89} + 4188 q^{90} + 2275 q^{91} + 2486 q^{92} + 2684 q^{93} + 2388 q^{94} - 2216 q^{95} + 2942 q^{96} + 2190 q^{97} + 196 q^{98} - 2761 q^{99}+O(q^{100}) \) 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 −5.44824 5.58606 21.6833 −0.337472 −30.4342 7.00000 −74.5498 4.20408 1.83863
1.2 −5.08324 −8.71995 17.8393 13.5683 44.3256 7.00000 −50.0156 49.0375 −68.9711
1.3 −4.87974 −4.00025 15.8118 −16.9325 19.5202 7.00000 −38.1197 −10.9980 82.6261
1.4 −4.56247 0.897479 12.8161 2.65401 −4.09472 7.00000 −21.9733 −26.1945 −12.1088
1.5 −3.43091 9.07752 3.77113 −8.12931 −31.1441 7.00000 14.5089 55.4014 27.8909
1.6 −3.39604 −5.78852 3.53311 3.92179 19.6580 7.00000 15.1698 6.50691 −13.3186
1.7 −2.94828 3.88655 0.692348 −19.6181 −11.4586 7.00000 21.5450 −11.8947 57.8396
1.8 −2.69367 1.13837 −0.744164 21.3002 −3.06640 7.00000 23.5539 −25.7041 −57.3755
1.9 −1.59523 −9.09895 −5.45526 −16.0024 14.5149 7.00000 21.4642 55.7909 25.5275
1.10 −1.23891 −8.01433 −6.46510 −7.22293 9.92902 7.00000 17.9209 37.2294 8.94856
1.11 −1.08458 0.216271 −6.82370 −2.11129 −0.234562 7.00000 16.0774 −26.9532 2.28985
1.12 −0.655624 9.63361 −7.57016 9.32356 −6.31602 7.00000 10.2082 65.8065 −6.11275
1.13 −0.226267 5.76745 −7.94880 4.05826 −1.30498 7.00000 3.60869 6.26349 −0.918249
1.14 1.12207 −4.87572 −6.74096 19.3366 −5.47090 7.00000 −16.5404 −3.22740 21.6971
1.15 1.19000 −2.91288 −6.58391 −4.30065 −3.46631 7.00000 −17.3548 −18.5152 −5.11775
1.16 2.17800 6.21083 −3.25630 −13.8336 13.5272 7.00000 −24.5163 11.5744 −30.1296
1.17 2.30192 2.62792 −2.70117 −18.9415 6.04927 7.00000 −24.6332 −20.0940 −43.6018
1.18 2.87128 0.202502 0.244254 10.4314 0.581440 7.00000 −22.2689 −26.9590 29.9515
1.19 3.18780 7.87263 2.16207 17.4367 25.0964 7.00000 −18.6102 34.9782 55.5846
1.20 3.37455 −9.05895 3.38759 −4.25940 −30.5699 7.00000 −15.5648 55.0645 −14.3736
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
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

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 1001.4.a.g ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.4.a.g ✓ 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 4 T_{2}^{24} - 149 T_{2}^{23} + 575 T_{2}^{22} + 9595 T_{2}^{21} - 35483 T_{2}^{20} + \cdots + 11470897152 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1001))\). Copy content Toggle raw display