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Properties

Label 322.2.i.d.141.3
Level $322$
Weight $2$
Character 322.141
Analytic conductor $2.571$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [322,2,Mod(29,322)] 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("322.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 141.3
Character \(\chi\) \(=\) 322.141
Dual form 322.2.i.d.169.3

$q$-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+(-0.142315 - 0.989821i) q^{2} +(0.128569 - 0.281527i) q^{3} +(-0.959493 + 0.281733i) q^{4} +(-1.69136 + 1.95194i) q^{5} +(-0.296959 - 0.0871950i) q^{6} +(-0.841254 + 0.540641i) q^{7} +(0.415415 + 0.909632i) q^{8} +(1.90185 + 2.19486i) q^{9} +(2.17278 + 1.39636i) q^{10} +(0.223366 - 1.55354i) q^{11} +(-0.0440458 + 0.306345i) q^{12} +(5.87583 + 3.77617i) q^{13} +(0.654861 + 0.755750i) q^{14} +(0.332066 + 0.727124i) q^{15} +(0.841254 - 0.540641i) q^{16} +(3.24670 + 0.953318i) q^{17} +(1.90185 - 2.19486i) q^{18} +(-2.78645 + 0.818176i) q^{19} +(1.07293 - 2.34938i) q^{20} +(0.0440458 + 0.306345i) q^{21} -1.56952 q^{22} +(-3.55335 + 3.22082i) q^{23} +0.309496 q^{24} +(-0.237776 - 1.65377i) q^{25} +(2.90151 - 6.35343i) q^{26} +(1.75331 - 0.514818i) q^{27} +(0.654861 - 0.755750i) q^{28} +(-4.89082 - 1.43608i) q^{29} +(0.672465 - 0.432167i) q^{30} +(2.08555 + 4.56673i) q^{31} +(-0.654861 - 0.755750i) q^{32} +(-0.408647 - 0.262621i) q^{33} +(0.481561 - 3.34933i) q^{34} +(0.367568 - 2.55649i) q^{35} +(-2.44318 - 1.57014i) q^{36} +(-1.70330 - 1.96571i) q^{37} +(1.20640 + 2.64165i) q^{38} +(1.81854 - 1.16871i) q^{39} +(-2.47816 - 0.727654i) q^{40} +(2.81451 - 3.24811i) q^{41} +(0.296959 - 0.0871950i) q^{42} +(0.433199 - 0.948573i) q^{43} +(0.223366 + 1.55354i) q^{44} -7.50095 q^{45} +(3.69373 + 3.05881i) q^{46} -4.35535 q^{47} +(-0.0440458 - 0.306345i) q^{48} +(0.415415 - 0.909632i) q^{49} +(-1.60309 + 0.470711i) q^{50} +(0.685811 - 0.791468i) q^{51} +(-6.70169 - 1.96779i) q^{52} +(3.30701 - 2.12529i) q^{53} +(-0.759100 - 1.66220i) q^{54} +(2.65463 + 3.06360i) q^{55} +(-0.841254 - 0.540641i) q^{56} +(-0.127913 + 0.889654i) q^{57} +(-0.725421 + 5.04542i) q^{58} +(5.95911 + 3.82969i) q^{59} +(-0.523470 - 0.604116i) q^{60} +(3.26598 + 7.15149i) q^{61} +(4.22344 - 2.71424i) q^{62} +(-2.78657 - 0.818211i) q^{63} +(-0.654861 + 0.755750i) q^{64} +(-17.3090 + 5.08238i) q^{65} +(-0.201792 + 0.441862i) q^{66} +(-1.24794 - 8.67964i) q^{67} -3.38377 q^{68} +(0.449897 + 1.41446i) q^{69} -2.58278 q^{70} +(-1.40316 - 9.75920i) q^{71} +(-1.20645 + 2.64176i) q^{72} +(4.37444 - 1.28445i) q^{73} +(-1.70330 + 1.96571i) q^{74} +(-0.496151 - 0.145683i) q^{75} +(2.44307 - 1.57007i) q^{76} +(0.652002 + 1.42769i) q^{77} +(-1.41562 - 1.63371i) q^{78} +(-0.936623 - 0.601931i) q^{79} +(-0.367568 + 2.55649i) q^{80} +(-1.15945 + 8.06416i) q^{81} +(-3.61560 - 2.32360i) q^{82} +(-5.51811 - 6.36824i) q^{83} +(-0.128569 - 0.281527i) q^{84} +(-7.35218 + 4.72496i) q^{85} +(-1.00057 - 0.293794i) q^{86} +(-1.03310 + 1.19226i) q^{87} +(1.50594 - 0.442185i) q^{88} +(-5.26207 + 11.5223i) q^{89} +(1.06750 + 7.42460i) q^{90} -6.98462 q^{91} +(2.50201 - 4.09145i) q^{92} +1.55380 q^{93} +(0.619831 + 4.31102i) q^{94} +(3.11587 - 6.82281i) q^{95} +(-0.296959 + 0.0871950i) q^{96} +(7.73192 - 8.92312i) q^{97} +(-0.959493 - 0.281733i) q^{98} +(3.83462 - 2.46436i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} - 3 q^{15} - 4 q^{16} - 13 q^{17} - 8 q^{18} - 22 q^{19} - 2 q^{20} - 16 q^{22} - 9 q^{23} + 22 q^{24}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.142315 − 0.989821i −0.100632 − 0.699909i
\(3\) 0.128569 − 0.281527i 0.0742294 − 0.162540i −0.868880 − 0.495024i \(-0.835159\pi\)
0.943109 + 0.332484i \(0.107887\pi\)
\(4\) −0.959493 + 0.281733i −0.479746 + 0.140866i
\(5\) −1.69136 + 1.95194i −0.756401 + 0.872933i −0.995172 − 0.0981427i \(-0.968710\pi\)
0.238772 + 0.971076i \(0.423255\pi\)
\(6\) −0.296959 − 0.0871950i −0.121233 − 0.0355972i
\(7\) −0.841254 + 0.540641i −0.317964 + 0.204343i
\(8\) 0.415415 + 0.909632i 0.146871 + 0.321603i
\(9\) 1.90185 + 2.19486i 0.633952 + 0.731619i
\(10\) 2.17278 + 1.39636i 0.687092 + 0.441567i
\(11\) 0.223366 − 1.55354i 0.0673474 − 0.468411i −0.928041 − 0.372479i \(-0.878508\pi\)
0.995388 − 0.0959319i \(-0.0305831\pi\)
\(12\) −0.0440458 + 0.306345i −0.0127149 + 0.0884343i
\(13\) 5.87583 + 3.77617i 1.62966 + 1.04732i 0.949265 + 0.314478i \(0.101829\pi\)
0.680398 + 0.732843i \(0.261807\pi\)
\(14\) 0.654861 + 0.755750i 0.175019 + 0.201983i
\(15\) 0.332066 + 0.727124i 0.0857392 + 0.187743i
\(16\) 0.841254 − 0.540641i 0.210313 − 0.135160i
\(17\) 3.24670 + 0.953318i 0.787441 + 0.231214i 0.650642 − 0.759385i \(-0.274500\pi\)
0.136800 + 0.990599i \(0.456318\pi\)
\(18\) 1.90185 − 2.19486i 0.448271 − 0.517333i
\(19\) −2.78645 + 0.818176i −0.639256 + 0.187702i −0.585269 − 0.810839i \(-0.699011\pi\)
−0.0539868 + 0.998542i \(0.517193\pi\)
\(20\) 1.07293 − 2.34938i 0.239914 − 0.525338i
\(21\) 0.0440458 + 0.306345i 0.00961159 + 0.0668501i
\(22\) −1.56952 −0.334623
\(23\) −3.55335 + 3.22082i −0.740925 + 0.671588i
\(24\) 0.309496 0.0631755
\(25\) −0.237776 − 1.65377i −0.0475551 − 0.330753i
\(26\) 2.90151 − 6.35343i 0.569034 − 1.24601i
\(27\) 1.75331 − 0.514818i 0.337424 − 0.0990767i
\(28\) 0.654861 − 0.755750i 0.123757 − 0.142823i
\(29\) −4.89082 − 1.43608i −0.908203 − 0.266672i −0.205919 − 0.978569i \(-0.566018\pi\)
−0.702284 + 0.711897i \(0.747836\pi\)
\(30\) 0.672465 − 0.432167i 0.122775 − 0.0789025i
\(31\) 2.08555 + 4.56673i 0.374577 + 0.820208i 0.999227 + 0.0393040i \(0.0125141\pi\)
−0.624651 + 0.780904i \(0.714759\pi\)
\(32\) −0.654861 − 0.755750i −0.115764 − 0.133599i
\(33\) −0.408647 − 0.262621i −0.0711363 − 0.0457165i
\(34\) 0.481561 − 3.34933i 0.0825870 − 0.574405i
\(35\) 0.367568 − 2.55649i 0.0621304 − 0.432126i
\(36\) −2.44318 − 1.57014i −0.407196 − 0.261689i
\(37\) −1.70330 − 1.96571i −0.280021 − 0.323162i 0.598264 − 0.801299i \(-0.295857\pi\)
−0.878285 + 0.478138i \(0.841312\pi\)
\(38\) 1.20640 + 2.64165i 0.195704 + 0.428532i
\(39\) 1.81854 − 1.16871i 0.291200 − 0.187143i
\(40\) −2.47816 − 0.727654i −0.391832 − 0.115052i
\(41\) 2.81451 − 3.24811i 0.439552 − 0.507270i −0.492142 − 0.870515i \(-0.663786\pi\)
0.931694 + 0.363245i \(0.118331\pi\)
\(42\) 0.296959 − 0.0871950i 0.0458218 − 0.0134545i
\(43\) 0.433199 − 0.948573i 0.0660622 − 0.144656i −0.873721 − 0.486428i \(-0.838300\pi\)
0.939783 + 0.341772i \(0.111027\pi\)
\(44\) 0.223366 + 1.55354i 0.0336737 + 0.234206i
\(45\) −7.50095 −1.11818
\(46\) 3.69373 + 3.05881i 0.544611 + 0.450997i
\(47\) −4.35535 −0.635293 −0.317647 − 0.948209i \(-0.602893\pi\)
−0.317647 + 0.948209i \(0.602893\pi\)
\(48\) −0.0440458 − 0.306345i −0.00635747 − 0.0442172i
\(49\) 0.415415 − 0.909632i 0.0593450 − 0.129947i
\(50\) −1.60309 + 0.470711i −0.226712 + 0.0665685i
\(51\) 0.685811 − 0.791468i 0.0960328 − 0.110828i
\(52\) −6.70169 − 1.96779i −0.929357 − 0.272884i
\(53\) 3.30701 − 2.12529i 0.454253 − 0.291931i −0.293437 − 0.955978i \(-0.594799\pi\)
0.747690 + 0.664048i \(0.231163\pi\)
\(54\) −0.759100 − 1.66220i −0.103300 − 0.226196i
\(55\) 2.65463 + 3.06360i 0.357950 + 0.413096i
\(56\) −0.841254 − 0.540641i −0.112417 − 0.0722462i
\(57\) −0.127913 + 0.889654i −0.0169425 + 0.117838i
\(58\) −0.725421 + 5.04542i −0.0952525 + 0.662496i
\(59\) 5.95911 + 3.82969i 0.775810 + 0.498583i 0.867641 − 0.497192i \(-0.165636\pi\)
−0.0918308 + 0.995775i \(0.529272\pi\)
\(60\) −0.523470 − 0.604116i −0.0675797 − 0.0779911i
\(61\) 3.26598 + 7.15149i 0.418165 + 0.915654i 0.995101 + 0.0988646i \(0.0315211\pi\)
−0.576936 + 0.816790i \(0.695752\pi\)
\(62\) 4.22344 − 2.71424i 0.536377 − 0.344709i
\(63\) −2.78657 − 0.818211i −0.351075 − 0.103085i
\(64\) −0.654861 + 0.755750i −0.0818576 + 0.0944687i
\(65\) −17.3090 + 5.08238i −2.14692 + 0.630392i
\(66\) −0.201792 + 0.441862i −0.0248389 + 0.0543895i
\(67\) −1.24794 − 8.67964i −0.152461 − 1.06039i −0.912078 − 0.410017i \(-0.865523\pi\)
0.759617 − 0.650370i \(-0.225386\pi\)
\(68\) −3.38377 −0.410343
\(69\) 0.449897 + 1.41446i 0.0541612 + 0.170281i
\(70\) −2.58278 −0.308702
\(71\) −1.40316 − 9.75920i −0.166525 − 1.15820i −0.886000 − 0.463686i \(-0.846527\pi\)
0.719475 − 0.694518i \(-0.244382\pi\)
\(72\) −1.20645 + 2.64176i −0.142182 + 0.311335i
\(73\) 4.37444 − 1.28445i 0.511989 − 0.150334i −0.0155254 − 0.999879i \(-0.504942\pi\)
0.527515 + 0.849546i \(0.323124\pi\)
\(74\) −1.70330 + 1.96571i −0.198005 + 0.228510i
\(75\) −0.496151 − 0.145683i −0.0572905 − 0.0168220i
\(76\) 2.44307 − 1.57007i 0.280240 − 0.180099i
\(77\) 0.652002 + 1.42769i 0.0743025 + 0.162700i
\(78\) −1.41562 − 1.63371i −0.160287 − 0.184981i
\(79\) −0.936623 − 0.601931i −0.105378 − 0.0677225i 0.486890 − 0.873463i \(-0.338131\pi\)
−0.592269 + 0.805741i \(0.701768\pi\)
\(80\) −0.367568 + 2.55649i −0.0410954 + 0.285825i
\(81\) −1.15945 + 8.06416i −0.128828 + 0.896018i
\(82\) −3.61560 − 2.32360i −0.399276 − 0.256599i
\(83\) −5.51811 − 6.36824i −0.605692 − 0.699006i 0.367233 − 0.930129i \(-0.380305\pi\)
−0.972924 + 0.231124i \(0.925760\pi\)
\(84\) −0.128569 − 0.281527i −0.0140280 − 0.0307171i
\(85\) −7.35218 + 4.72496i −0.797455 + 0.512493i
\(86\) −1.00057 − 0.293794i −0.107894 − 0.0316806i
\(87\) −1.03310 + 1.19226i −0.110760 + 0.127824i
\(88\) 1.50594 − 0.442185i 0.160534 − 0.0471370i
\(89\) −5.26207 + 11.5223i −0.557779 + 1.22136i 0.395275 + 0.918563i \(0.370649\pi\)
−0.953054 + 0.302802i \(0.902078\pi\)
\(90\) 1.06750 + 7.42460i 0.112524 + 0.782622i
\(91\) −6.98462 −0.732187
\(92\) 2.50201 − 4.09145i 0.260852 − 0.426563i
\(93\) 1.55380 0.161121
\(94\) 0.619831 + 4.31102i 0.0639307 + 0.444648i
\(95\) 3.11587 − 6.82281i 0.319682 − 0.700006i
\(96\) −0.296959 + 0.0871950i −0.0303082 + 0.00889930i
\(97\) 7.73192 − 8.92312i 0.785058 − 0.906005i −0.212406 − 0.977182i \(-0.568130\pi\)
0.997464 + 0.0711763i \(0.0226753\pi\)
\(98\) −0.959493 − 0.281733i −0.0969234 − 0.0284593i
\(99\) 3.83462 − 2.46436i 0.385393 − 0.247677i
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 322.2.i.d.141.3 ✓ 40
23.8 even 11 inner 322.2.i.d.169.3 yes 40
23.10 odd 22 7406.2.a.bv.1.11 20
23.13 even 11 7406.2.a.bu.1.11 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.i.d.141.3 ✓ 40 1.1 even 1 trivial
322.2.i.d.169.3 yes 40 23.8 even 11 inner
7406.2.a.bu.1.11 20 23.13 even 11
7406.2.a.bv.1.11 20 23.10 odd 22