Properties

Label 1183.2.e.k
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - q^{2} - 23q^{4} + 13q^{5} - 28q^{6} - 3q^{7} - 26q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - q^{2} - 23q^{4} + 13q^{5} - 28q^{6} - 3q^{7} - 26q^{9} - 5q^{10} - q^{11} - 5q^{12} - 2q^{14} - 10q^{15} - 17q^{16} + 5q^{17} + 24q^{19} - 68q^{20} + q^{21} - 28q^{22} - 11q^{23} + 32q^{24} - 33q^{25} - 42q^{27} + 15q^{28} + 8q^{29} + 22q^{30} + 40q^{31} - 6q^{32} + 24q^{33} - 72q^{34} + 44q^{35} - 30q^{36} - 4q^{37} + 29q^{38} + 4q^{40} - 98q^{41} - 9q^{42} + 26q^{43} + 10q^{44} + 58q^{45} - 10q^{46} + 62q^{47} + 178q^{48} + 31q^{49} + 46q^{50} + 21q^{51} + 18q^{53} + 12q^{54} - 28q^{55} - 56q^{56} + 26q^{57} + 56q^{58} + 79q^{59} + 22q^{60} - 13q^{61} + 24q^{62} - 22q^{63} + 36q^{64} + 38q^{66} - 2q^{67} + 12q^{68} - 56q^{69} - 85q^{70} + 38q^{71} + 81q^{72} + 17q^{73} - 17q^{74} - 24q^{75} - 116q^{76} - 30q^{77} + 9q^{79} + 63q^{80} - 16q^{81} + 22q^{82} - 162q^{83} - 203q^{84} + 68q^{85} + 22q^{86} - 70q^{87} + 33q^{88} + 72q^{89} + 2q^{90} - 8q^{92} + 19q^{93} + 30q^{94} - 13q^{95} + 11q^{96} - 90q^{97} - 81q^{98} + 78q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1 −1.31989 + 2.28612i −0.745365 1.29101i −2.48423 4.30282i 1.40245 2.42911i 3.93520 −0.0241769 + 2.64564i 7.83610 0.388863 0.673530i 3.70216 + 6.41233i
170.2 −1.27052 + 2.20061i −0.156459 0.270995i −2.22846 3.85980i 1.54077 2.66869i 0.795139 −1.63233 2.08219i 6.24314 1.45104 2.51328i 3.91517 + 6.78127i
170.3 −1.18772 + 2.05719i 0.348245 + 0.603178i −1.82134 3.15466i −1.31283 + 2.27390i −1.65446 2.59846 0.497998i 3.90209 1.25745 2.17797i −3.11855 5.40149i
170.4 −1.13165 + 1.96008i 1.66347 + 2.88122i −1.56128 2.70422i −0.580975 + 1.00628i −7.52992 −2.63830 + 0.198459i 2.54072 −4.03430 + 6.98761i −1.31493 2.27752i
170.5 −0.973551 + 1.68624i 0.867274 + 1.50216i −0.895604 1.55123i 1.85966 3.22103i −3.37734 −1.95410 + 1.78368i −0.406541 −0.00432987 + 0.00749956i 3.62095 + 6.27168i
170.6 −0.875911 + 1.51712i −0.335909 0.581811i −0.534440 0.925678i −1.33875 + 2.31878i 1.17690 1.17520 2.37042i −1.63116 1.27433 2.20721i −2.34525 4.06209i
170.7 −0.680712 + 1.17903i 0.656465 + 1.13703i 0.0732621 + 0.126894i −1.52933 + 2.64889i −1.78745 −2.33513 1.24384i −2.92233 0.638108 1.10524i −2.08207 3.60626i
170.8 −0.593734 + 1.02838i 1.46725 + 2.54136i 0.294961 + 0.510887i 1.70195 2.94787i −3.48463 2.56390 + 0.653016i −3.07545 −2.80567 + 4.85955i 2.02101 + 3.50050i
170.9 −0.514418 + 0.890998i −1.33163 2.30645i 0.470749 + 0.815361i 0.745563 1.29135i 2.74006 1.93668 + 1.80257i −3.02632 −2.04647 + 3.54460i 0.767061 + 1.32859i
170.10 −0.308759 + 0.534786i −1.53664 2.66155i 0.809336 + 1.40181i 0.698024 1.20901i 1.89781 −2.36540 + 1.18527i −2.23459 −3.22255 + 5.58162i 0.431043 + 0.746588i
170.11 −0.294236 + 0.509631i −0.386839 0.670024i 0.826851 + 1.43215i −1.34801 + 2.33482i 0.455287 −2.52472 0.791073i −2.15010 1.20071 2.07969i −0.793264 1.37397i
170.12 −0.127376 + 0.220623i 0.787620 + 1.36420i 0.967550 + 1.67585i 0.259002 0.448605i −0.401297 1.92441 + 1.81567i −1.00248 0.259311 0.449140i 0.0659816 + 0.114283i
170.13 −0.0682269 + 0.118172i −1.48341 2.56934i 0.990690 + 1.71593i 1.64515 2.84948i 0.404833 1.56259 2.13502i −0.543274 −2.90099 + 5.02467i 0.224487 + 0.388822i
170.14 0.00118363 0.00205011i 1.54914 + 2.68320i 0.999997 + 1.73205i −0.565709 + 0.979836i 0.00733446 2.64547 0.0383280i 0.00946903 −3.29970 + 5.71524i 0.00133918 + 0.00231953i
170.15 0.438601 0.759680i 0.377512 + 0.653870i 0.615258 + 1.06566i −0.132920 + 0.230223i 0.662309 −0.588870 + 2.57939i 2.83382 1.21497 2.10439i 0.116597 + 0.201953i
170.16 0.545983 0.945670i −0.697705 1.20846i 0.403806 + 0.699412i −0.813252 + 1.40859i −1.52374 −2.51623 + 0.817685i 3.06581 0.526415 0.911778i 0.888042 + 1.53813i
170.17 0.603060 1.04453i −0.516035 0.893798i 0.272638 + 0.472223i 1.51464 2.62344i −1.24480 1.04407 2.43103i 3.06991 0.967416 1.67561i −1.82684 3.16418i
170.18 0.771913 1.33699i 0.676597 + 1.17190i −0.191698 0.332031i 0.170982 0.296150i 2.08910 −2.12749 + 1.57282i 2.49575 0.584432 1.01227i −0.263967 0.457204i
170.19 0.787207 1.36348i −1.10111 1.90717i −0.239388 0.414633i −1.88030 + 3.25678i −3.46719 0.609781 2.57452i 2.39503 −0.924870 + 1.60192i 2.96037 + 5.12751i
170.20 0.883662 1.53055i 1.16401 + 2.01613i −0.561718 0.972924i 2.03380 3.52265i 4.11437 −1.79914 1.93987i 1.54917 −1.20985 + 2.09552i −3.59438 6.22566i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 508.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.k 48
7.c even 3 1 inner 1183.2.e.k 48
7.c even 3 1 8281.2.a.cv 24
7.d odd 6 1 8281.2.a.cw 24
13.b even 2 1 1183.2.e.l yes 48
91.r even 6 1 1183.2.e.l yes 48
91.r even 6 1 8281.2.a.cu 24
91.s odd 6 1 8281.2.a.ct 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.e.k 48 1.a even 1 1 trivial
1183.2.e.k 48 7.c even 3 1 inner
1183.2.e.l yes 48 13.b even 2 1
1183.2.e.l yes 48 91.r even 6 1
8281.2.a.ct 24 91.s odd 6 1
8281.2.a.cu 24 91.r even 6 1
8281.2.a.cv 24 7.c even 3 1
8281.2.a.cw 24 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\(T_{2}^{48} + \cdots\)
\(10\!\cdots\!92\)\( T_{3}^{22} + 917156320984 T_{3}^{21} + \)\(21\!\cdots\!17\)\( T_{3}^{20} + \)\(18\!\cdots\!79\)\( T_{3}^{19} + \)\(36\!\cdots\!31\)\( T_{3}^{18} + \)\(29\!\cdots\!06\)\( T_{3}^{17} + \)\(49\!\cdots\!61\)\( T_{3}^{16} + \)\(37\!\cdots\!54\)\( T_{3}^{15} + \)\(54\!\cdots\!47\)\( T_{3}^{14} + \)\(38\!\cdots\!76\)\( T_{3}^{13} + \)\(46\!\cdots\!50\)\( T_{3}^{12} + \)\(30\!\cdots\!78\)\( T_{3}^{11} + \)\(31\!\cdots\!73\)\( T_{3}^{10} + \)\(18\!\cdots\!67\)\( T_{3}^{9} + \)\(15\!\cdots\!96\)\( T_{3}^{8} + 790250544491 T_{3}^{7} + 561525929980 T_{3}^{6} + 238793889789 T_{3}^{5} + 138029745763 T_{3}^{4} + 46857186096 T_{3}^{3} + 20032294513 T_{3}^{2} + 3978485812 T_{3} + 810483961 \)">\(T_{3}^{48} + \cdots\)