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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 algebraic \(q\)-expansion of this newform has not been computed, 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