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Properties

Label 1045.6.a.c
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
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] = [1045,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(167.601091705\)
Analytic rank: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 12 q^{2} - 27 q^{3} + 574 q^{4} - 925 q^{5} - 75 q^{6} + 337 q^{7} - 696 q^{8} + 3140 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 12 q^{2} - 27 q^{3} + 574 q^{4} - 925 q^{5} - 75 q^{6} + 337 q^{7} - 696 q^{8} + 3140 q^{9} + 300 q^{10} - 4477 q^{11} - 568 q^{12} + 719 q^{13} + 687 q^{14} + 675 q^{15} + 11494 q^{16} + 999 q^{17} - 595 q^{18} - 13357 q^{19} - 14350 q^{20} - 1077 q^{21} + 1452 q^{22} + 5096 q^{23} - 3154 q^{24} + 23125 q^{25} - 10395 q^{26} - 7578 q^{27} + 19863 q^{28} - 7969 q^{29} + 1875 q^{30} + 603 q^{31} - 27809 q^{32} + 3267 q^{33} - 24081 q^{34} - 8425 q^{35} + 59869 q^{36} + 7963 q^{37} + 4332 q^{38} + 86 q^{39} + 17400 q^{40} + 1475 q^{41} - 46542 q^{42} + 38059 q^{43} - 69454 q^{44} - 78500 q^{45} - 3413 q^{46} - 37658 q^{47} - 51317 q^{48} + 39188 q^{49} - 7500 q^{50} - 40262 q^{51} + 25358 q^{52} - 52545 q^{53} + 64732 q^{54} + 111925 q^{55} - 54173 q^{56} + 9747 q^{57} + 105808 q^{58} - 34039 q^{59} + 14200 q^{60} + 30023 q^{61} - 100198 q^{62} + 30376 q^{63} + 160888 q^{64} - 17975 q^{65} + 9075 q^{66} - 45284 q^{67} + 125176 q^{68} + 109244 q^{69} - 17175 q^{70} - 84020 q^{71} - 291176 q^{72} + 24542 q^{73} + 38795 q^{74} - 16875 q^{75} - 207214 q^{76} - 40777 q^{77} + 1042 q^{78} + 49303 q^{79} - 287350 q^{80} + 344453 q^{81} - 286030 q^{82} - 402155 q^{83} - 203270 q^{84} - 24975 q^{85} - 276426 q^{86} + 116994 q^{87} + 84216 q^{88} - 442930 q^{89} + 14875 q^{90} - 93040 q^{91} + 402160 q^{92} - 241950 q^{93} - 170720 q^{94} + 333925 q^{95} - 234384 q^{96} - 87732 q^{97} - 712662 q^{98} - 379940 q^{99}+O(q^{100}) \) 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 −10.9276 −26.0746 87.4116 −25.0000 284.932 122.691 −605.513 436.887 273.189
1.2 −10.5955 22.4642 80.2653 −25.0000 −238.020 228.020 −511.396 261.641 264.888
1.3 −10.5199 0.539776 78.6693 −25.0000 −5.67841 3.26226 −490.959 −242.709 262.999
1.4 −10.2487 19.1450 73.0349 −25.0000 −196.210 −226.466 −420.553 123.531 256.216
1.5 −9.84120 −9.26217 64.8493 −25.0000 91.1509 −35.6928 −323.277 −157.212 246.030
1.6 −8.73290 −22.7868 44.2636 −25.0000 198.995 −46.5268 −107.097 276.237 218.323
1.7 −8.15394 5.08384 34.4868 −25.0000 −41.4534 169.957 −20.2770 −217.155 203.849
1.8 −7.04813 7.74381 17.6762 −25.0000 −54.5794 −183.718 100.956 −183.033 176.203
1.9 −6.88533 20.3717 15.4078 −25.0000 −140.266 93.5387 114.243 172.007 172.133
1.10 −6.52420 −27.2044 10.5652 −25.0000 177.487 −162.861 139.845 497.081 163.105
1.11 −5.93303 −1.91423 3.20088 −25.0000 11.3572 107.575 170.866 −239.336 148.326
1.12 −5.50202 28.7846 −1.72782 −25.0000 −158.373 5.95765 185.571 585.551 137.550
1.13 −4.33468 −21.6700 −13.2105 −25.0000 93.9326 25.9276 195.973 226.588 108.367
1.14 −3.96708 6.17610 −16.2623 −25.0000 −24.5011 98.7920 191.460 −204.856 99.1771
1.15 −3.53022 −7.27933 −19.5375 −25.0000 25.6976 −171.265 181.939 −190.011 88.2555
1.16 −2.23923 −26.8282 −26.9859 −25.0000 60.0744 193.889 132.083 476.753 55.9807
1.17 −1.08270 23.1648 −30.8278 −25.0000 −25.0805 −27.0095 68.0236 293.607 27.0675
1.18 −1.05078 5.26669 −30.8959 −25.0000 −5.53415 −191.487 66.0899 −215.262 26.2696
1.19 −0.312283 −15.4864 −31.9025 −25.0000 4.83615 −30.9789 19.9557 −3.17065 7.80708
1.20 0.814963 −1.30743 −31.3358 −25.0000 −1.06551 233.682 −51.6164 −241.291 −20.3741
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
\(5\) \(1\)
\(11\) \(1\)
\(19\) \(1\)

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 1045.6.a.c ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.c ✓ 37 1.a even 1 1 trivial