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Properties

Label 46.4.c.a.29.2
Level $46$
Weight $4$
Character 46.29
Analytic conductor $2.714$
Analytic rank $0$
Dimension $30$
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] = [46,4,Mod(3,46)] 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("46.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(46, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 46 = 2 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 46.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71408786026\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 29.2
Character \(\chi\) \(=\) 46.29
Dual form 46.4.c.a.27.2

$q$-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+(0.830830 - 1.81926i) q^{2} +(-0.858963 - 0.252214i) q^{3} +(-2.61944 - 3.02300i) q^{4} +(9.50458 - 6.10822i) q^{5} +(-1.17250 + 1.35313i) q^{6} +(3.59878 - 25.0301i) q^{7} +(-7.67594 + 2.25386i) q^{8} +(-22.0396 - 14.1640i) q^{9} +(-3.21578 - 22.3662i) q^{10} +(26.9613 + 59.0369i) q^{11} +(1.48756 + 3.25731i) q^{12} +(3.47253 + 24.1520i) q^{13} +(-42.5463 - 27.3429i) q^{14} +(-9.70467 + 2.84955i) q^{15} +(-2.27704 + 15.8371i) q^{16} +(52.5965 - 60.6996i) q^{17} +(-44.0793 + 28.3280i) q^{18} +(94.3889 + 108.931i) q^{19} +(-43.3618 - 12.7322i) q^{20} +(-9.40416 + 20.5922i) q^{21} +129.804 q^{22} +(-108.786 + 18.2353i) q^{23} +7.16181 q^{24} +(1.09976 - 2.40814i) q^{25} +(46.8239 + 13.7487i) q^{26} +(31.1876 + 35.9924i) q^{27} +(-85.0926 + 54.6857i) q^{28} +(-23.2981 + 26.8874i) q^{29} +(-2.87885 + 20.0228i) q^{30} +(-70.4810 + 20.6951i) q^{31} +(26.9201 + 17.3005i) q^{32} +(-8.26877 - 57.5106i) q^{33} +(-66.7299 - 146.118i) q^{34} +(-118.684 - 259.882i) q^{35} +(14.9138 + 103.728i) q^{36} +(-186.068 - 119.579i) q^{37} +(276.595 - 81.2156i) q^{38} +(3.10870 - 21.6215i) q^{39} +(-59.1895 + 68.3084i) q^{40} +(251.587 - 161.685i) q^{41} +(29.6495 + 34.2173i) q^{42} +(285.799 + 83.9181i) q^{43} +(107.845 - 236.148i) q^{44} -295.994 q^{45} +(-57.2082 + 213.062i) q^{46} -57.9788 q^{47} +(5.95025 - 13.0292i) q^{48} +(-284.447 - 83.5210i) q^{49} +(-3.46733 - 4.00151i) q^{50} +(-60.4878 + 38.8731i) q^{51} +(63.9153 - 73.7622i) q^{52} +(74.0991 - 515.370i) q^{53} +(91.3913 - 26.8349i) q^{54} +(616.866 + 396.436i) q^{55} +(28.7902 + 200.240i) q^{56} +(-53.6028 - 117.374i) q^{57} +(29.5586 + 64.7242i) q^{58} +(9.90286 + 68.8759i) q^{59} +(34.0350 + 21.8730i) q^{60} +(-332.769 + 97.7097i) q^{61} +(-20.9079 + 145.418i) q^{62} +(-433.842 + 500.680i) q^{63} +(53.8402 - 34.6010i) q^{64} +(180.531 + 208.343i) q^{65} +(-111.497 - 32.7384i) q^{66} +(-359.825 + 787.908i) q^{67} -321.268 q^{68} +(98.0427 + 11.7741i) q^{69} -571.401 q^{70} +(31.1068 - 68.1144i) q^{71} +(201.099 + 59.0479i) q^{72} +(206.853 + 238.721i) q^{73} +(-372.136 + 239.157i) q^{74} +(-1.55202 + 1.79113i) q^{75} +(82.0507 - 570.675i) q^{76} +(1574.73 - 462.381i) q^{77} +(-36.7524 - 23.6193i) q^{78} +(37.6144 + 261.614i) q^{79} +(75.0945 + 164.434i) q^{80} +(276.137 + 604.656i) q^{81} +(-85.1218 - 592.035i) q^{82} +(-682.344 - 438.516i) q^{83} +(86.8840 - 25.5114i) q^{84} +(129.141 - 898.195i) q^{85} +(390.120 - 450.222i) q^{86} +(26.7936 - 17.2192i) q^{87} +(-340.014 - 392.397i) q^{88} +(-1465.12 - 430.198i) q^{89} +(-245.921 + 538.492i) q^{90} +617.022 q^{91} +(340.085 + 281.095i) q^{92} +65.7602 q^{93} +(-48.1705 + 105.479i) q^{94} +(1562.50 + 458.791i) q^{95} +(-18.7600 - 21.6501i) q^{96} +(-29.3855 + 18.8849i) q^{97} +(-388.274 + 448.092i) q^{98} +(241.984 - 1683.03i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} + 6 q^{3} - 12 q^{4} + 10 q^{5} + 12 q^{6} + 107 q^{7} - 24 q^{8} - 63 q^{9} + 20 q^{10} + 64 q^{11} - 20 q^{12} + 8 q^{13} + 16 q^{14} + 252 q^{15} - 48 q^{16} - 126 q^{17} - 126 q^{18}+ \cdots + 1377 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/46\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.830830 − 1.81926i 0.293743 − 0.643207i
\(3\) −0.858963 − 0.252214i −0.165308 − 0.0485387i 0.198031 − 0.980196i \(-0.436545\pi\)
−0.363339 + 0.931657i \(0.618363\pi\)
\(4\) −2.61944 − 3.02300i −0.327430 − 0.377875i
\(5\) 9.50458 − 6.10822i 0.850115 − 0.546336i −0.0414954 − 0.999139i \(-0.513212\pi\)
0.891611 + 0.452803i \(0.149576\pi\)
\(6\) −1.17250 + 1.35313i −0.0797783 + 0.0920691i
\(7\) 3.59878 − 25.0301i 0.194316 − 1.35150i −0.626106 − 0.779738i \(-0.715352\pi\)
0.820422 − 0.571758i \(-0.193739\pi\)
\(8\) −7.67594 + 2.25386i −0.339232 + 0.0996075i
\(9\) −22.0396 − 14.1640i −0.816283 − 0.524593i
\(10\) −3.21578 − 22.3662i −0.101692 − 0.707282i
\(11\) 26.9613 + 59.0369i 0.739012 + 1.61821i 0.785174 + 0.619276i \(0.212574\pi\)
−0.0461621 + 0.998934i \(0.514699\pi\)
\(12\) 1.48756 + 3.25731i 0.0357852 + 0.0783586i
\(13\) 3.47253 + 24.1520i 0.0740851 + 0.515273i 0.992746 + 0.120227i \(0.0383623\pi\)
−0.918661 + 0.395046i \(0.870729\pi\)
\(14\) −42.5463 − 27.3429i −0.812213 − 0.521978i
\(15\) −9.70467 + 2.84955i −0.167049 + 0.0490500i
\(16\) −2.27704 + 15.8371i −0.0355787 + 0.247455i
\(17\) 52.5965 − 60.6996i 0.750384 − 0.865989i −0.244222 − 0.969719i \(-0.578532\pi\)
0.994605 + 0.103730i \(0.0330779\pi\)
\(18\) −44.0793 + 28.3280i −0.577199 + 0.370943i
\(19\) 94.3889 + 108.931i 1.13970 + 1.31528i 0.942227 + 0.334976i \(0.108728\pi\)
0.197474 + 0.980308i \(0.436726\pi\)
\(20\) −43.3618 − 12.7322i −0.484800 − 0.142350i
\(21\) −9.40416 + 20.5922i −0.0977217 + 0.213981i
\(22\) 129.804 1.25792
\(23\) −108.786 + 18.2353i −0.986240 + 0.165318i
\(24\) 7.16181 0.0609124
\(25\) 1.09976 − 2.40814i 0.00879809 − 0.0192651i
\(26\) 46.8239 + 13.7487i 0.353189 + 0.103706i
\(27\) 31.1876 + 35.9924i 0.222298 + 0.256546i
\(28\) −85.0926 + 54.6857i −0.574321 + 0.369094i
\(29\) −23.2981 + 26.8874i −0.149184 + 0.172168i −0.825423 − 0.564515i \(-0.809063\pi\)
0.676239 + 0.736683i \(0.263609\pi\)
\(30\) −2.87885 + 20.0228i −0.0175201 + 0.121855i
\(31\) −70.4810 + 20.6951i −0.408347 + 0.119901i −0.479454 − 0.877567i \(-0.659165\pi\)
0.0711067 + 0.997469i \(0.477347\pi\)
\(32\) 26.9201 + 17.3005i 0.148714 + 0.0955727i
\(33\) −8.26877 − 57.5106i −0.0436185 − 0.303373i
\(34\) −66.7299 − 146.118i −0.336590 − 0.737030i
\(35\) −118.684 − 259.882i −0.573180 − 1.25509i
\(36\) 14.9138 + 103.728i 0.0690453 + 0.480220i
\(37\) −186.068 − 119.579i −0.826740 − 0.531313i 0.0575007 − 0.998345i \(-0.481687\pi\)
−0.884240 + 0.467032i \(0.845323\pi\)
\(38\) 276.595 − 81.2156i 1.18078 − 0.346708i
\(39\) 3.10870 − 21.6215i 0.0127639 − 0.0887746i
\(40\) −59.1895 + 68.3084i −0.233967 + 0.270012i
\(41\) 251.587 − 161.685i 0.958322 − 0.615876i 0.0347892 − 0.999395i \(-0.488924\pi\)
0.923533 + 0.383518i \(0.125288\pi\)
\(42\) 29.6495 + 34.2173i 0.108929 + 0.125711i
\(43\) 285.799 + 83.9181i 1.01358 + 0.297614i 0.746017 − 0.665927i \(-0.231964\pi\)
0.267563 + 0.963541i \(0.413782\pi\)
\(44\) 107.845 − 236.148i 0.369506 − 0.809105i
\(45\) −295.994 −0.980539
\(46\) −57.2082 + 213.062i −0.183367 + 0.682918i
\(47\) −57.9788 −0.179938 −0.0899689 − 0.995945i \(-0.528677\pi\)
−0.0899689 + 0.995945i \(0.528677\pi\)
\(48\) 5.95025 − 13.0292i 0.0178926 − 0.0391793i
\(49\) −284.447 − 83.5210i −0.829290 − 0.243502i
\(50\) −3.46733 − 4.00151i −0.00980709 − 0.0113180i
\(51\) −60.4878 + 38.8731i −0.166078 + 0.106732i
\(52\) 63.9153 − 73.7622i 0.170451 − 0.196711i
\(53\) 74.0991 − 515.370i 0.192043 − 1.33569i −0.634546 − 0.772885i \(-0.718813\pi\)
0.826590 − 0.562805i \(-0.190278\pi\)
\(54\) 91.3913 − 26.8349i 0.230311 − 0.0676253i
\(55\) 616.866 + 396.436i 1.51233 + 0.971916i
\(56\) 28.7902 + 200.240i 0.0687010 + 0.477826i
\(57\) −53.6028 − 117.374i −0.124559 − 0.272746i
\(58\) 29.5586 + 64.7242i 0.0669178 + 0.146529i
\(59\) 9.90286 + 68.8759i 0.0218516 + 0.151981i 0.997826 − 0.0659043i \(-0.0209932\pi\)
−0.975974 + 0.217885i \(0.930084\pi\)
\(60\) 34.0350 + 21.8730i 0.0732317 + 0.0470631i
\(61\) −332.769 + 97.7097i −0.698470 + 0.205089i −0.611643 − 0.791134i \(-0.709491\pi\)
−0.0868272 + 0.996223i \(0.527673\pi\)
\(62\) −20.9079 + 145.418i −0.0428275 + 0.297872i
\(63\) −433.842 + 500.680i −0.867602 + 1.00127i
\(64\) 53.8402 − 34.6010i 0.105157 − 0.0675801i
\(65\) 180.531 + 208.343i 0.344493 + 0.397566i
\(66\) −111.497 − 32.7384i −0.207944 − 0.0610579i
\(67\) −359.825 + 787.908i −0.656115 + 1.43669i 0.229984 + 0.973194i \(0.426133\pi\)
−0.886099 + 0.463496i \(0.846595\pi\)
\(68\) −321.268 −0.572934
\(69\) 98.0427 + 11.7741i 0.171057 + 0.0205425i
\(70\) −571.401 −0.975650
\(71\) 31.1068 − 68.1144i 0.0519958 − 0.113855i −0.881851 − 0.471528i \(-0.843703\pi\)
0.933847 + 0.357673i \(0.116430\pi\)
\(72\) 201.099 + 59.0479i 0.329163 + 0.0966509i
\(73\) 206.853 + 238.721i 0.331648 + 0.382742i 0.896943 − 0.442147i \(-0.145783\pi\)
−0.565295 + 0.824889i \(0.691237\pi\)
\(74\) −372.136 + 239.157i −0.584593 + 0.375695i
\(75\) −1.55202 + 1.79113i −0.00238949 + 0.00275762i
\(76\) 82.0507 − 570.675i 0.123840 − 0.861328i
\(77\) 1574.73 − 462.381i 2.33061 − 0.684328i
\(78\) −36.7524 − 23.6193i −0.0533511 − 0.0342867i
\(79\) 37.6144 + 261.614i 0.0535690 + 0.372581i 0.998917 + 0.0465187i \(0.0148127\pi\)
−0.945348 + 0.326062i \(0.894278\pi\)
\(80\) 75.0945 + 164.434i 0.104948 + 0.229804i
\(81\) 276.137 + 604.656i 0.378789 + 0.829433i
\(82\) −85.1218 − 592.035i −0.114636 − 0.797309i
\(83\) −682.344 − 438.516i −0.902373 − 0.579920i 0.00512026 − 0.999987i \(-0.498370\pi\)
−0.907493 + 0.420067i \(0.862007\pi\)
\(84\) 86.8840 − 25.5114i 0.112855 − 0.0331372i
\(85\) 129.141 − 898.195i 0.164792 − 1.14615i
\(86\) 390.120 − 450.222i 0.489159 − 0.564520i
\(87\) 26.7936 − 17.2192i 0.0330181 − 0.0212194i
\(88\) −340.014 − 392.397i −0.411882 − 0.475337i
\(89\) −1465.12 − 430.198i −1.74497 − 0.512370i −0.755258 − 0.655428i \(-0.772488\pi\)
−0.989714 + 0.143058i \(0.954306\pi\)
\(90\) −245.921 + 538.492i −0.288026 + 0.630689i
\(91\) 617.022 0.710786
\(92\) 340.085 + 281.095i 0.385395 + 0.318545i
\(93\) 65.7602 0.0733227
\(94\) −48.1705 + 105.479i −0.0528554 + 0.115737i
\(95\) 1562.50 + 458.791i 1.68746 + 0.495484i
\(96\) −18.7600 − 21.6501i −0.0199446 − 0.0230173i
\(97\) −29.3855 + 18.8849i −0.0307593 + 0.0197678i −0.555930 − 0.831229i \(-0.687638\pi\)
0.525171 + 0.850997i \(0.324001\pi\)
\(98\) −388.274 + 448.092i −0.400220 + 0.461878i
\(99\) 241.984 − 1683.03i 0.245659 − 1.70860i
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 46.4.c.a.29.2 yes 30
23.2 even 11 1058.4.a.v.1.8 15
23.4 even 11 inner 46.4.c.a.27.2 ✓ 30
23.21 odd 22 1058.4.a.w.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.c.a.27.2 ✓ 30 23.4 even 11 inner
46.4.c.a.29.2 yes 30 1.1 even 1 trivial
1058.4.a.v.1.8 15 23.2 even 11
1058.4.a.w.1.8 15 23.21 odd 22