Properties

Label 171.2.g.c
Level $171$
Weight $2$
Character orbit 171.g
Analytic conductor $1.365$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} - 4 q^{13} - 2 q^{14} + q^{15} - 11 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20} + 11 q^{21} + 16 q^{22} + 5 q^{23} + 27 q^{24} + 18 q^{25} - 4 q^{26} - 5 q^{27} - 10 q^{28} - 20 q^{29} - 5 q^{30} - 10 q^{31} + 17 q^{32} + 34 q^{33} + 26 q^{34} - 3 q^{35} - 16 q^{36} + 2 q^{37} + 38 q^{38} - 24 q^{40} - 12 q^{41} + 25 q^{42} + 7 q^{43} + 20 q^{44} - 35 q^{45} + 18 q^{47} - 33 q^{48} - 13 q^{49} + q^{50} - 28 q^{51} + 19 q^{52} + 16 q^{53} + 35 q^{54} + 15 q^{55} - 6 q^{56} + 6 q^{57} - 74 q^{59} + 50 q^{60} + 24 q^{61} + 54 q^{62} - 30 q^{63} - 64 q^{64} + 54 q^{65} + 4 q^{66} - 11 q^{67} - 2 q^{68} + 3 q^{69} - 48 q^{70} + 9 q^{71} - 10 q^{73} + 6 q^{74} - 76 q^{75} + 29 q^{76} + 46 q^{77} - 82 q^{78} - 8 q^{79} - 24 q^{80} + 26 q^{81} + 7 q^{82} + 3 q^{83} + 12 q^{84} - 27 q^{85} + 17 q^{86} - 9 q^{87} + 9 q^{88} + 30 q^{89} - 74 q^{90} - q^{91} - 17 q^{92} - 24 q^{93} - 18 q^{94} - 6 q^{95} - 5 q^{96} + 18 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −1.23404 2.13742i 1.62292 + 0.605076i −2.04570 + 3.54325i 1.91524 −0.709450 4.21555i −0.324708 + 0.562412i 5.16173 2.26777 + 1.96399i −2.36348 4.09366i
106.2 −1.04884 1.81665i 0.154304 + 1.72516i −1.20014 + 2.07870i −2.89593 2.97217 2.08974i 0.116480 0.201749i 0.839660 −2.95238 + 0.532400i 3.03737 + 5.26088i
106.3 −0.978515 1.69484i −1.73022 0.0795352i −0.914982 + 1.58480i −0.196235 1.55825 + 3.01027i −2.23368 + 3.86885i −0.332766 2.98735 + 0.275227i 0.192018 + 0.332586i
106.4 −0.888985 1.53977i 1.15573 1.29008i −0.580589 + 1.00561i −1.27957 −3.01384 0.632688i 0.657761 1.13928i −1.49140 −0.328599 2.98195i 1.13752 + 1.97024i
106.5 −0.803309 1.39137i −1.24014 1.20915i −0.290611 + 0.503353i 3.75880 −0.686165 + 2.69682i 2.27973 3.94861i −2.27943 0.0758986 + 2.99904i −3.01948 5.22989i
106.6 −0.395929 0.685769i 0.659141 + 1.60173i 0.686481 1.18902i 2.59093 0.837442 1.08619i −0.373088 + 0.646207i −2.67091 −2.13107 + 2.11153i −1.02582 1.77678i
106.7 −0.269545 0.466866i −1.40907 + 1.00723i 0.854691 1.48037i −0.947325 0.850050 + 0.386354i 1.18430 2.05126i −1.99969 0.970970 2.83852i 0.255347 + 0.442274i
106.8 −0.185445 0.321199i −0.894876 1.48297i 0.931221 1.61292i −3.55521 −0.310379 + 0.562442i −0.124876 + 0.216291i −1.43254 −1.39839 + 2.65415i 0.659294 + 1.14193i
106.9 −0.0732670 0.126902i 0.157437 1.72488i 0.989264 1.71346i 2.57004 −0.230426 + 0.106398i −1.73898 + 3.01201i −0.582990 −2.95043 0.543121i −0.188299 0.326143i
106.10 0.0973467 + 0.168609i 1.55267 + 0.767609i 0.981047 1.69922i −1.90563 0.0217210 + 0.336519i 1.69446 2.93489i 0.771394 1.82155 + 2.38368i −0.185507 0.321308i
106.11 0.616796 + 1.06832i 0.506944 + 1.65620i 0.239126 0.414178i −0.551543 −1.45668 + 1.56312i −1.62156 + 2.80862i 3.05715 −2.48602 + 1.67920i −0.340189 0.589225i
106.12 0.847114 + 1.46725i 0.0521113 1.73127i −0.435206 + 0.753799i 0.0882176 2.58434 1.39012i 1.84695 3.19901i 1.91378 −2.99457 0.180437i 0.0747304 + 0.129437i
106.13 1.01016 + 1.74964i −1.70377 + 0.311691i −1.04083 + 1.80277i −4.18739 −2.26643 2.66614i −0.976107 + 1.69067i −0.164982 2.80570 1.06210i −4.22991 7.32643i
106.14 1.19971 + 2.07797i −0.340918 + 1.69817i −1.87863 + 3.25388i 0.719678 −3.93774 + 1.32890i 1.65862 2.87282i −4.21641 −2.76755 1.15787i 0.863408 + 1.49547i
106.15 1.30160 + 2.25443i −1.17382 1.27363i −2.38830 + 4.13666i 2.87794 1.34348 4.30405i −1.80240 + 3.12185i −7.22804 −0.244287 + 2.99004i 3.74592 + 6.48812i
106.16 1.30515 + 2.26059i 1.63157 0.581354i −2.40684 + 4.16877i −2.00201 3.44365 + 2.92956i 0.257107 0.445323i −7.34455 2.32405 1.89704i −2.61292 4.52572i
121.1 −1.23404 + 2.13742i 1.62292 0.605076i −2.04570 3.54325i 1.91524 −0.709450 + 4.21555i −0.324708 0.562412i 5.16173 2.26777 1.96399i −2.36348 + 4.09366i
121.2 −1.04884 + 1.81665i 0.154304 1.72516i −1.20014 2.07870i −2.89593 2.97217 + 2.08974i 0.116480 + 0.201749i 0.839660 −2.95238 0.532400i 3.03737 5.26088i
121.3 −0.978515 + 1.69484i −1.73022 + 0.0795352i −0.914982 1.58480i −0.196235 1.55825 3.01027i −2.23368 3.86885i −0.332766 2.98735 0.275227i 0.192018 0.332586i
121.4 −0.888985 + 1.53977i 1.15573 + 1.29008i −0.580589 1.00561i −1.27957 −3.01384 + 0.632688i 0.657761 + 1.13928i −1.49140 −0.328599 + 2.98195i 1.13752 1.97024i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.g.c 32
3.b odd 2 1 513.2.g.c 32
9.c even 3 1 171.2.h.c yes 32
9.d odd 6 1 513.2.h.c 32
19.c even 3 1 171.2.h.c yes 32
57.h odd 6 1 513.2.h.c 32
171.g even 3 1 inner 171.2.g.c 32
171.n odd 6 1 513.2.g.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.c 32 1.a even 1 1 trivial
171.2.g.c 32 171.g even 3 1 inner
171.2.h.c yes 32 9.c even 3 1
171.2.h.c yes 32 19.c even 3 1
513.2.g.c 32 3.b odd 2 1
513.2.g.c 32 171.n odd 6 1
513.2.h.c 32 9.d odd 6 1
513.2.h.c 32 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{32} - T_{2}^{31} + 25 T_{2}^{30} - 10 T_{2}^{29} + 358 T_{2}^{28} - 34 T_{2}^{27} + 3447 T_{2}^{26} + 717 T_{2}^{25} + 24681 T_{2}^{24} + 10382 T_{2}^{23} + 135298 T_{2}^{22} + 82109 T_{2}^{21} + 580960 T_{2}^{20} + 420887 T_{2}^{19} + \cdots + 81 \) Copy content Toggle raw display
\( T_{5}^{16} + 3 T_{5}^{15} - 40 T_{5}^{14} - 115 T_{5}^{13} + 600 T_{5}^{12} + 1690 T_{5}^{11} - 4152 T_{5}^{10} - 12111 T_{5}^{9} + 12405 T_{5}^{8} + 43371 T_{5}^{7} - 6663 T_{5}^{6} - 67191 T_{5}^{5} - 25815 T_{5}^{4} + \cdots - 189 \) Copy content Toggle raw display