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Properties

Label 171.4.e.a
Level $171$
Weight $4$
Character orbit 171.e
Analytic conductor $10.089$
Analytic rank $0$
Dimension $54$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,4,Mod(58,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.58");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(10.0893266110\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(27\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{2} + 2 q^{3} - 108 q^{4} - 26 q^{5} + 59 q^{6} + 138 q^{8} - 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{2} + 2 q^{3} - 108 q^{4} - 26 q^{5} + 59 q^{6} + 138 q^{8} - 102 q^{9} - 70 q^{11} + 256 q^{12} - 121 q^{14} + 64 q^{15} - 468 q^{16} + 136 q^{17} - 102 q^{18} - 1026 q^{19} - 136 q^{20} + 133 q^{21} - 250 q^{23} - 762 q^{24} - 819 q^{25} + 1132 q^{26} + 1091 q^{27} - 920 q^{29} + 160 q^{30} - 18 q^{31} - 1368 q^{32} - 650 q^{33} + 528 q^{35} + 2043 q^{36} + 630 q^{37} + 114 q^{38} - 325 q^{39} - 854 q^{41} - 818 q^{42} + 144 q^{43} + 3496 q^{44} + 1984 q^{45} + 144 q^{46} - 81 q^{47} + 1060 q^{48} - 1521 q^{49} - 3776 q^{50} - 1234 q^{51} - 1251 q^{52} + 4532 q^{53} + 50 q^{54} - 1008 q^{55} - 3735 q^{56} - 38 q^{57} - 45 q^{58} - 1294 q^{59} - 1039 q^{60} + 18 q^{61} + 3116 q^{62} + 332 q^{63} + 4302 q^{64} - 2586 q^{65} - 635 q^{66} + 306 q^{67} - 3130 q^{68} - 1980 q^{69} + 2205 q^{70} + 5388 q^{71} + 4728 q^{72} - 72 q^{73} - 3202 q^{74} + 924 q^{75} + 2052 q^{76} - 2836 q^{77} - 85 q^{78} - 954 q^{79} + 9386 q^{80} + 5298 q^{81} - 2574 q^{82} - 1202 q^{83} - 10546 q^{84} + 468 q^{85} - 4029 q^{86} + 4797 q^{87} + 1044 q^{88} + 8812 q^{89} + 2101 q^{90} - 306 q^{91} - 7973 q^{92} - 3154 q^{93} + 225 q^{94} + 494 q^{95} - 434 q^{96} + 2142 q^{97} + 12128 q^{98} + 2690 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −2.71759 − 4.70700i −1.72680 − 4.90083i −10.7706 + 18.6552i −3.98845 + 6.90820i −18.3755 + 21.4465i 7.59826 + 13.1606i 73.5985 −21.0363 + 16.9255i 43.3559
58.2 −2.70529 − 4.68569i −1.87859 + 4.84468i −10.6371 + 18.4241i −8.74654 + 15.1494i 27.7828 − 4.30374i −3.11415 − 5.39387i 71.8214 −19.9418 − 18.2023i 94.6475
58.3 −2.63881 − 4.57055i −4.75458 + 2.09618i −9.92660 + 17.1934i 8.90090 − 15.4168i 22.1271 + 16.1996i −16.0088 − 27.7281i 62.5566 18.2121 − 19.9329i −93.9510
58.4 −2.25805 − 3.91105i 4.70555 − 2.20405i −6.19756 + 10.7345i −3.96830 + 6.87330i −19.2455 − 13.4268i −9.26005 − 16.0389i 19.8488 17.2843 − 20.7425i 35.8424
58.5 −2.17034 − 3.75914i −4.20970 − 3.04605i −5.42074 + 9.38901i 7.46889 − 12.9365i −2.31404 + 22.4358i 6.59214 + 11.4179i 12.3340 8.44317 + 25.6459i −64.8401
58.6 −1.78836 − 3.09754i 2.79226 − 4.38215i −2.39648 + 4.15083i 7.39022 − 12.8002i −18.5675 − 0.812253i 10.3127 + 17.8622i −11.4707 −11.4065 − 24.4722i −52.8656
58.7 −1.60398 − 2.77818i −4.39733 − 2.76831i −1.14551 + 1.98408i −9.96596 + 17.2615i −0.637611 + 16.6569i −16.3907 − 28.3896i −18.3142 11.6730 + 24.3463i 63.9408
58.8 −1.50144 − 2.60058i 4.94730 + 1.58879i −0.508662 + 0.881029i −0.754699 + 1.30718i −3.29631 − 15.2513i 15.8260 + 27.4114i −20.9682 21.9515 + 15.7205i 4.53255
58.9 −1.29457 − 2.24226i −0.0706213 + 5.19567i 0.648173 − 1.12267i 8.23332 − 14.2605i 11.7415 − 6.56781i −2.49245 − 4.31705i −24.0696 −26.9900 − 0.733850i −42.6345
58.10 −1.20745 − 2.09137i −1.10087 + 5.07820i 1.08411 − 1.87774i −10.4375 + 18.0783i 11.9496 − 3.82937i 14.9933 + 25.9691i −24.5553 −24.5762 − 11.1808i 50.4113
58.11 −1.04620 − 1.81207i −4.65752 + 2.30381i 1.81094 − 3.13665i −0.277634 + 0.480876i 9.04734 + 6.02949i −3.29263 − 5.70300i −24.3176 16.3849 − 21.4601i 1.16184
58.12 −0.668972 − 1.15869i 1.82065 − 4.86675i 3.10495 − 5.37794i 1.70839 − 2.95902i −6.85704 + 1.14614i −8.90834 − 15.4297i −19.0121 −20.3704 − 17.7213i −4.57146
58.13 −0.310405 − 0.537637i −4.36122 − 2.82484i 3.80730 − 6.59443i −2.88298 + 4.99346i −0.164993 + 3.22160i 11.8767 + 20.5710i −9.69370 11.0406 + 24.6395i 3.57956
58.14 −0.215194 − 0.372727i 0.915225 + 5.11492i 3.90738 − 6.76779i 2.35296 − 4.07544i 1.70952 − 1.44183i −11.2998 − 19.5718i −6.80648 −25.3247 + 9.36260i −2.02537
58.15 0.192549 + 0.333505i 4.52896 + 2.54726i 3.92585 − 6.79977i 5.55121 − 9.61497i 0.0225252 + 2.00090i 6.25466 + 10.8334i 6.10446 14.0230 + 23.0728i 4.27552
58.16 0.319091 + 0.552682i 0.778090 − 5.13757i 3.79636 − 6.57549i −9.44905 + 16.3662i 3.08772 − 1.20932i −6.84018 − 11.8475i 9.95101 −25.7892 − 7.99498i −12.0604
58.17 0.619031 + 1.07219i 5.12846 − 0.835986i 3.23360 − 5.60076i −9.52182 + 16.4923i 4.07102 + 4.98120i 3.48423 + 6.03487i 17.9113 25.6023 − 8.57465i −23.5772
58.18 0.813967 + 1.40983i −1.64161 − 4.93002i 2.67491 − 4.63309i 7.80212 − 13.5137i 5.61429 − 6.32728i 0.642561 + 1.11295i 21.7327 −21.6102 + 16.1864i 25.4027
58.19 0.989966 + 1.71467i 4.88583 + 1.76881i 2.03993 − 3.53327i 1.25650 − 2.17632i 1.80387 + 10.1287i −18.0950 − 31.3414i 23.9173 20.7426 + 17.2842i 4.97556
58.20 1.01953 + 1.76588i −3.01198 + 4.23415i 1.92111 − 3.32745i −1.62373 + 2.81238i −10.5478 − 1.00195i 11.8906 + 20.5951i 24.1470 −8.85599 − 25.5063i −6.62178
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.e.a ✓ 54
9.c even 3 1 inner 171.4.e.a ✓ 54
9.c even 3 1 1539.4.a.h 27
9.d odd 6 1 1539.4.a.g 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.4.e.a ✓ 54 1.a even 1 1 trivial
171.4.e.a ✓ 54 9.c even 3 1 inner
1539.4.a.g 27 9.d odd 6 1
1539.4.a.h 27 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} + 6 T_{2}^{53} + 180 T_{2}^{52} + 898 T_{2}^{51} + 17061 T_{2}^{50} + 75702 T_{2}^{49} + \cdots + 70\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display