↑

Properties

Label 2366.2.d.r.337.7
Level $2366$
Weight $2$
Character 2366.337
Analytic conductor $18.893$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2366,2,Mod(337,2366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,4,-12,0,0,0,0,12,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.7
Root \(0.500000 - 1.69027i\) of defining polynomial
Character \(\chi\) \(=\) 2366.337
Dual form 2366.2.d.r.337.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.55629 q^{3} -1.00000 q^{4} -3.48754i q^{5} -2.55629i q^{6} -1.00000i q^{7} -1.00000i q^{8} +3.53463 q^{9} +3.48754 q^{10} -2.68172i q^{11} +2.55629 q^{12} +1.00000 q^{14} +8.91517i q^{15} +1.00000 q^{16} +5.91517 q^{17} +3.53463i q^{18} -5.19793i q^{19} +3.48754i q^{20} +2.55629i q^{21} +2.68172 q^{22} +7.05268 q^{23} +2.55629i q^{24} -7.16292 q^{25} -1.36668 q^{27} +1.00000i q^{28} +7.13278 q^{29} -8.91517 q^{30} +0.782383i q^{31} +1.00000i q^{32} +6.85526i q^{33} +5.91517i q^{34} -3.48754 q^{35} -3.53463 q^{36} +7.81450i q^{37} +5.19793 q^{38} -3.48754 q^{40} +0.157708i q^{41} -2.55629 q^{42} +0.330202 q^{43} +2.68172i q^{44} -12.3272i q^{45} +7.05268i q^{46} -1.60161i q^{47} -2.55629 q^{48} -1.00000 q^{49} -7.16292i q^{50} -15.1209 q^{51} +3.92601 q^{53} -1.36668i q^{54} -9.35259 q^{55} -1.00000 q^{56} +13.2874i q^{57} +7.13278i q^{58} -9.54021i q^{59} -8.91517i q^{60} +15.4105 q^{61} -0.782383 q^{62} -3.53463i q^{63} -1.00000 q^{64} -6.85526 q^{66} +0.966765i q^{67} -5.91517 q^{68} -18.0287 q^{69} -3.48754i q^{70} -4.18658i q^{71} -3.53463i q^{72} -15.0361i q^{73} -7.81450 q^{74} +18.3105 q^{75} +5.19793i q^{76} -2.68172 q^{77} -0.293356 q^{79} -3.48754i q^{80} -7.11027 q^{81} -0.157708 q^{82} -2.87495i q^{83} -2.55629i q^{84} -20.6294i q^{85} +0.330202i q^{86} -18.2335 q^{87} -2.68172 q^{88} +5.21325i q^{89} +12.3272 q^{90} -7.05268 q^{92} -2.00000i q^{93} +1.60161 q^{94} -18.1280 q^{95} -2.55629i q^{96} +3.15946i q^{97} -1.00000i q^{98} -9.47889i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 12 q^{4} + 12 q^{9} + 4 q^{10} - 4 q^{12} + 12 q^{14} + 12 q^{16} - 8 q^{17} + 4 q^{22} + 12 q^{23} - 24 q^{25} + 40 q^{27} + 20 q^{29} - 28 q^{30} - 4 q^{35} - 12 q^{36} + 8 q^{38} - 4 q^{40}+ \cdots - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −2.55629 −1.47588 −0.737938 − 0.674868i \(-0.764200\pi\)
−0.737938 + 0.674868i \(0.764200\pi\)
\(4\) −1.00000 −0.500000
\(5\) − 3.48754i − 1.55967i −0.625983 − 0.779837i \(-0.715302\pi\)
0.625983 − 0.779837i \(-0.284698\pi\)
\(6\) − 2.55629i − 1.04360i
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 3.53463 1.17821
\(10\) 3.48754 1.10286
\(11\) − 2.68172i − 0.808569i −0.914633 − 0.404284i \(-0.867521\pi\)
0.914633 − 0.404284i \(-0.132479\pi\)
\(12\) 2.55629 0.737938
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 8.91517i 2.30189i
\(16\) 1.00000 0.250000
\(17\) 5.91517 1.43464 0.717319 − 0.696745i \(-0.245369\pi\)
0.717319 + 0.696745i \(0.245369\pi\)
\(18\) 3.53463i 0.833121i
\(19\) − 5.19793i − 1.19249i −0.802804 − 0.596243i \(-0.796659\pi\)
0.802804 − 0.596243i \(-0.203341\pi\)
\(20\) 3.48754i 0.779837i
\(21\) 2.55629i 0.557829i
\(22\) 2.68172 0.571744
\(23\) 7.05268 1.47058 0.735292 − 0.677750i \(-0.237045\pi\)
0.735292 + 0.677750i \(0.237045\pi\)
\(24\) 2.55629i 0.521801i
\(25\) −7.16292 −1.43258
\(26\) 0 0
\(27\) −1.36668 −0.263017
\(28\) 1.00000i 0.188982i
\(29\) 7.13278 1.32452 0.662262 − 0.749272i \(-0.269596\pi\)
0.662262 + 0.749272i \(0.269596\pi\)
\(30\) −8.91517 −1.62768
\(31\) 0.782383i 0.140520i 0.997529 + 0.0702601i \(0.0223829\pi\)
−0.997529 + 0.0702601i \(0.977617\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.85526i 1.19335i
\(34\) 5.91517i 1.01444i
\(35\) −3.48754 −0.589501
\(36\) −3.53463 −0.589105
\(37\) 7.81450i 1.28470i 0.766413 + 0.642348i \(0.222040\pi\)
−0.766413 + 0.642348i \(0.777960\pi\)
\(38\) 5.19793 0.843215
\(39\) 0 0
\(40\) −3.48754 −0.551428
\(41\) 0.157708i 0.0246299i 0.999924 + 0.0123149i \(0.00392006\pi\)
−0.999924 + 0.0123149i \(0.996080\pi\)
\(42\) −2.55629 −0.394445
\(43\) 0.330202 0.0503553 0.0251777 − 0.999683i \(-0.491985\pi\)
0.0251777 + 0.999683i \(0.491985\pi\)
\(44\) 2.68172i 0.404284i
\(45\) − 12.3272i − 1.83762i
\(46\) 7.05268i 1.03986i
\(47\) − 1.60161i − 0.233619i −0.993154 − 0.116810i \(-0.962733\pi\)
0.993154 − 0.116810i \(-0.0372667\pi\)
\(48\) −2.55629 −0.368969
\(49\) −1.00000 −0.142857
\(50\) − 7.16292i − 1.01299i
\(51\) −15.1209 −2.11735
\(52\) 0 0
\(53\) 3.92601 0.539279 0.269639 − 0.962961i \(-0.413095\pi\)
0.269639 + 0.962961i \(0.413095\pi\)
\(54\) − 1.36668i − 0.185981i
\(55\) −9.35259 −1.26110
\(56\) −1.00000 −0.133631
\(57\) 13.2874i 1.75996i
\(58\) 7.13278i 0.936580i
\(59\) − 9.54021i − 1.24203i −0.783799 − 0.621015i \(-0.786721\pi\)
0.783799 − 0.621015i \(-0.213279\pi\)
\(60\) − 8.91517i − 1.15094i
\(61\) 15.4105 1.97311 0.986556 − 0.163421i \(-0.0522527\pi\)
0.986556 + 0.163421i \(0.0522527\pi\)
\(62\) −0.782383 −0.0993627
\(63\) − 3.53463i − 0.445322i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.85526 −0.843824
\(67\) 0.966765i 0.118109i 0.998255 + 0.0590546i \(0.0188086\pi\)
−0.998255 + 0.0590546i \(0.981191\pi\)
\(68\) −5.91517 −0.717319
\(69\) −18.0287 −2.17040
\(70\) − 3.48754i − 0.416840i
\(71\) − 4.18658i − 0.496855i −0.968650 − 0.248428i \(-0.920086\pi\)
0.968650 − 0.248428i \(-0.0799138\pi\)
\(72\) − 3.53463i − 0.416560i
\(73\) − 15.0361i − 1.75984i −0.475124 − 0.879919i \(-0.657597\pi\)
0.475124 − 0.879919i \(-0.342403\pi\)
\(74\) −7.81450 −0.908417
\(75\) 18.3105 2.11432
\(76\) 5.19793i 0.596243i
\(77\) −2.68172 −0.305610
\(78\) 0 0
\(79\) −0.293356 −0.0330052 −0.0165026 − 0.999864i \(-0.505253\pi\)
−0.0165026 + 0.999864i \(0.505253\pi\)
\(80\) − 3.48754i − 0.389919i
\(81\) −7.11027 −0.790030
\(82\) −0.157708 −0.0174159
\(83\) − 2.87495i − 0.315566i −0.987474 − 0.157783i \(-0.949565\pi\)
0.987474 − 0.157783i \(-0.0504347\pi\)
\(84\) − 2.55629i − 0.278914i
\(85\) − 20.6294i − 2.23757i
\(86\) 0.330202i 0.0356066i
\(87\) −18.2335 −1.95483
\(88\) −2.68172 −0.285872
\(89\) 5.21325i 0.552603i 0.961071 + 0.276302i \(0.0891089\pi\)
−0.961071 + 0.276302i \(0.910891\pi\)
\(90\) 12.3272 1.29940
\(91\) 0 0
\(92\) −7.05268 −0.735292
\(93\) − 2.00000i − 0.207390i
\(94\) 1.60161 0.165194
\(95\) −18.1280 −1.85989
\(96\) − 2.55629i − 0.260901i
\(97\) 3.15946i 0.320794i 0.987053 + 0.160397i \(0.0512775\pi\)
−0.987053 + 0.160397i \(0.948723\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) − 9.47889i − 0.952664i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2366.2.d.r.337.7 12
13.5 odd 4 2366.2.a.bh.1.1 6
13.8 odd 4 2366.2.a.bf.1.1 6
13.9 even 3 182.2.m.b.127.6 yes 12
13.10 even 6 182.2.m.b.43.6 ✓ 12
13.12 even 2 inner 2366.2.d.r.337.1 12
39.23 odd 6 1638.2.bj.g.1135.1 12
39.35 odd 6 1638.2.bj.g.127.3 12
52.23 odd 6 1456.2.cc.d.225.1 12
52.35 odd 6 1456.2.cc.d.673.1 12
91.9 even 3 1274.2.v.e.361.1 12
91.10 odd 6 1274.2.v.d.667.3 12
91.23 even 6 1274.2.o.d.459.3 12
91.48 odd 6 1274.2.m.c.491.4 12
91.61 odd 6 1274.2.v.d.361.3 12
91.62 odd 6 1274.2.m.c.589.4 12
91.74 even 3 1274.2.o.d.569.6 12
91.75 odd 6 1274.2.o.e.459.1 12
91.87 odd 6 1274.2.o.e.569.4 12
91.88 even 6 1274.2.v.e.667.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.m.b.43.6 ✓ 12 13.10 even 6
182.2.m.b.127.6 yes 12 13.9 even 3
1274.2.m.c.491.4 12 91.48 odd 6
1274.2.m.c.589.4 12 91.62 odd 6
1274.2.o.d.459.3 12 91.23 even 6
1274.2.o.d.569.6 12 91.74 even 3
1274.2.o.e.459.1 12 91.75 odd 6
1274.2.o.e.569.4 12 91.87 odd 6
1274.2.v.d.361.3 12 91.61 odd 6
1274.2.v.d.667.3 12 91.10 odd 6
1274.2.v.e.361.1 12 91.9 even 3
1274.2.v.e.667.1 12 91.88 even 6
1456.2.cc.d.225.1 12 52.23 odd 6
1456.2.cc.d.673.1 12 52.35 odd 6
1638.2.bj.g.127.3 12 39.35 odd 6
1638.2.bj.g.1135.1 12 39.23 odd 6
2366.2.a.bf.1.1 6 13.8 odd 4
2366.2.a.bh.1.1 6 13.5 odd 4
2366.2.d.r.337.1 12 13.12 even 2 inner
2366.2.d.r.337.7 12 1.1 even 1 trivial