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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(29,342)] 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("342.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 281.10
Character \(\chi\) \(=\) 342.281
Dual form 342.2.x.c.185.10

$q$-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+(0.766044 - 0.642788i) q^{2} +(1.63622 + 0.568143i) q^{3} +(0.173648 - 0.984808i) q^{4} +(0.440917 - 1.21141i) q^{5} +(1.61861 - 0.616519i) q^{6} +(-0.842522 - 1.45929i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(2.35443 + 1.85921i) q^{9} +(-0.440917 - 1.21141i) q^{10} -0.266344i q^{11} +(0.843638 - 1.51270i) q^{12} +(-0.317756 - 0.873027i) q^{13} +(-1.58342 - 0.576319i) q^{14} +(1.40969 - 1.73163i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-0.664089 + 1.82457i) q^{17} +(2.99867 - 0.0891572i) q^{18} +(1.37731 + 4.13558i) q^{19} +(-1.11644 - 0.644577i) q^{20} +(-0.549465 - 2.86639i) q^{21} +(-0.171202 - 0.204031i) q^{22} +(-4.79143 - 0.844859i) q^{23} +(-0.326084 - 1.70108i) q^{24} +(2.55712 + 2.14568i) q^{25} +(-0.804586 - 0.464528i) q^{26} +(2.79606 + 4.37973i) q^{27} +(-1.58342 + 0.576319i) q^{28} +(0.00952382 - 0.0540123i) q^{29} +(-0.0331832 - 2.23263i) q^{30} -2.02270i q^{31} +(-0.939693 + 0.342020i) q^{32} +(0.151321 - 0.435797i) q^{33} +(0.664089 + 1.82457i) q^{34} +(-2.13928 + 0.377212i) q^{35} +(2.23981 - 1.99581i) q^{36} +9.28190i q^{37} +(3.71338 + 2.28272i) q^{38} +(-0.0239142 - 1.60899i) q^{39} +(-1.26957 + 0.223859i) q^{40} +(-2.85929 + 2.39923i) q^{41} +(-2.26340 - 1.84259i) q^{42} +(-0.0688000 - 0.390184i) q^{43} +(-0.262297 - 0.0462501i) q^{44} +(3.29037 - 2.03241i) q^{45} +(-4.21351 + 2.43267i) q^{46} +(-2.32645 - 0.410216i) q^{47} +(-1.34323 - 1.09350i) q^{48} +(2.08031 - 3.60321i) q^{49} +3.33808 q^{50} +(-2.12321 + 2.60810i) q^{51} +(-0.914942 + 0.161329i) q^{52} +(-7.75233 - 6.50497i) q^{53} +(4.95714 + 1.55779i) q^{54} +(-0.322651 - 0.117435i) q^{55} +(-0.842522 + 1.45929i) q^{56} +(-0.0960103 + 7.54922i) q^{57} +(-0.0274227 - 0.0474976i) q^{58} +(-1.57060 - 8.90733i) q^{59} +(-1.46053 - 1.68897i) q^{60} +(-2.33717 + 0.850661i) q^{61} +(-1.30017 - 1.54948i) q^{62} +(0.729475 - 5.00222i) q^{63} +(-0.500000 + 0.866025i) q^{64} -1.19770 q^{65} +(-0.164206 - 0.431107i) q^{66} +(-0.522108 + 0.622225i) q^{67} +(1.68153 + 0.970833i) q^{68} +(-7.35983 - 4.10459i) q^{69} +(-1.39631 + 1.66406i) q^{70} +(-4.64075 + 3.89405i) q^{71} +(0.432912 - 2.96860i) q^{72} +(2.82620 + 16.0282i) q^{73} +(5.96629 + 7.11035i) q^{74} +(2.96496 + 4.96361i) q^{75} +(4.31192 - 0.638255i) q^{76} +(-0.388673 + 0.224400i) q^{77} +(-1.05256 - 1.21719i) q^{78} +(2.61242 - 7.17756i) q^{79} +(-0.828652 + 0.987549i) q^{80} +(2.08666 + 8.75476i) q^{81} +(-0.648148 + 3.67583i) q^{82} +(-3.91697 + 2.26147i) q^{83} +(-2.91826 + 0.0433735i) q^{84} +(1.91749 + 1.60897i) q^{85} +(-0.303509 - 0.254675i) q^{86} +(0.0462697 - 0.0829650i) q^{87} +(-0.230660 + 0.133172i) q^{88} +(1.61880 - 9.18068i) q^{89} +(1.21416 - 3.67193i) q^{90} +(-1.00628 + 1.19924i) q^{91} +(-1.66405 + 4.57193i) q^{92} +(1.14919 - 3.30959i) q^{93} +(-2.04585 + 1.18117i) q^{94} +(5.61715 + 0.154954i) q^{95} +(-1.73186 + 0.0257403i) q^{96} +(-1.43893 - 1.71485i) q^{97} +(-0.722486 - 4.09742i) q^{98} +(0.495189 - 0.627087i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 6 q^{3} - 30 q^{8} + 6 q^{9} + 3 q^{13} + 3 q^{14} + 27 q^{15} - 27 q^{17} - 18 q^{18} + 3 q^{19} + 18 q^{23} + 3 q^{24} - 6 q^{27} + 3 q^{28} + 57 q^{33} + 27 q^{34} + 9 q^{38} - 30 q^{39} + 9 q^{41}+ \cdots + 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.766044 − 0.642788i 0.541675 − 0.454519i
\(3\) 1.63622 + 0.568143i 0.944672 + 0.328017i
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) 0.440917 − 1.21141i 0.197184 − 0.541758i −0.801212 − 0.598381i \(-0.795811\pi\)
0.998396 + 0.0566227i \(0.0180332\pi\)
\(6\) 1.61861 − 0.616519i 0.660796 − 0.251693i
\(7\) −0.842522 − 1.45929i −0.318443 − 0.551560i 0.661720 − 0.749751i \(-0.269827\pi\)
−0.980163 + 0.198191i \(0.936493\pi\)
\(8\) −0.500000 − 0.866025i −0.176777 − 0.306186i
\(9\) 2.35443 + 1.85921i 0.784809 + 0.619738i
\(10\) −0.440917 − 1.21141i −0.139430 − 0.383081i
\(11\) − 0.266344i − 0.0803056i −0.999194 − 0.0401528i \(-0.987216\pi\)
0.999194 − 0.0401528i \(-0.0127845\pi\)
\(12\) 0.843638 − 1.51270i 0.243537 − 0.436680i
\(13\) −0.317756 − 0.873027i −0.0881296 − 0.242134i 0.887796 − 0.460237i \(-0.152235\pi\)
−0.975926 + 0.218103i \(0.930013\pi\)
\(14\) −1.58342 − 0.576319i −0.423188 − 0.154028i
\(15\) 1.40969 − 1.73163i 0.363980 − 0.447104i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −0.664089 + 1.82457i −0.161065 + 0.442523i −0.993805 − 0.111142i \(-0.964549\pi\)
0.832739 + 0.553665i \(0.186771\pi\)
\(18\) 2.99867 − 0.0891572i 0.706794 − 0.0210146i
\(19\) 1.37731 + 4.13558i 0.315978 + 0.948767i
\(20\) −1.11644 − 0.644577i −0.249644 − 0.144132i
\(21\) −0.549465 − 2.86639i −0.119903 − 0.625498i
\(22\) −0.171202 − 0.204031i −0.0365005 − 0.0434996i
\(23\) −4.79143 − 0.844859i −0.999083 − 0.176165i −0.349891 − 0.936790i \(-0.613781\pi\)
−0.649191 + 0.760625i \(0.724893\pi\)
\(24\) −0.326084 − 1.70108i −0.0665615 − 0.347231i
\(25\) 2.55712 + 2.14568i 0.511424 + 0.429136i
\(26\) −0.804586 − 0.464528i −0.157792 − 0.0911014i
\(27\) 2.79606 + 4.37973i 0.538102 + 0.842880i
\(28\) −1.58342 + 0.576319i −0.299239 + 0.108914i
\(29\) 0.00952382 − 0.0540123i 0.00176853 − 0.0100298i −0.983911 − 0.178661i \(-0.942824\pi\)
0.985679 + 0.168631i \(0.0539346\pi\)
\(30\) −0.0331832 − 2.23263i −0.00605840 − 0.407621i
\(31\) − 2.02270i − 0.363289i −0.983364 − 0.181644i \(-0.941858\pi\)
0.983364 − 0.181644i \(-0.0581419\pi\)
\(32\) −0.939693 + 0.342020i −0.166116 + 0.0604612i
\(33\) 0.151321 − 0.435797i 0.0263416 − 0.0758624i
\(34\) 0.664089 + 1.82457i 0.113890 + 0.312911i
\(35\) −2.13928 + 0.377212i −0.361604 + 0.0637605i
\(36\) 2.23981 − 1.99581i 0.373302 − 0.332635i
\(37\) 9.28190i 1.52594i 0.646437 + 0.762968i \(0.276258\pi\)
−0.646437 + 0.762968i \(0.723742\pi\)
\(38\) 3.71338 + 2.28272i 0.602390 + 0.370305i
\(39\) −0.0239142 − 1.60899i −0.00382934 − 0.257645i
\(40\) −1.26957 + 0.223859i −0.200736 + 0.0353952i
\(41\) −2.85929 + 2.39923i −0.446546 + 0.374696i −0.838152 − 0.545436i \(-0.816364\pi\)
0.391606 + 0.920133i \(0.371919\pi\)
\(42\) −2.26340 − 1.84259i −0.349249 − 0.284318i
\(43\) −0.0688000 − 0.390184i −0.0104919 − 0.0595025i 0.979112 − 0.203320i \(-0.0651732\pi\)
−0.989604 + 0.143818i \(0.954062\pi\)
\(44\) −0.262297 − 0.0462501i −0.0395428 − 0.00697246i
\(45\) 3.29037 − 2.03241i 0.490500 − 0.302975i
\(46\) −4.21351 + 2.43267i −0.621249 + 0.358678i
\(47\) −2.32645 − 0.410216i −0.339348 − 0.0598362i 0.00137745 − 0.999999i \(-0.499562\pi\)
−0.340725 + 0.940163i \(0.610673\pi\)
\(48\) −1.34323 − 1.09350i −0.193878 − 0.157833i
\(49\) 2.08031 − 3.60321i 0.297188 − 0.514744i
\(50\) 3.33808 0.472076
\(51\) −2.12321 + 2.60810i −0.297309 + 0.365207i
\(52\) −0.914942 + 0.161329i −0.126880 + 0.0223723i
\(53\) −7.75233 − 6.50497i −1.06486 − 0.893527i −0.0702865 − 0.997527i \(-0.522391\pi\)
−0.994577 + 0.104000i \(0.966836\pi\)
\(54\) 4.95714 + 1.55779i 0.674582 + 0.211989i
\(55\) −0.322651 − 0.117435i −0.0435062 − 0.0158350i
\(56\) −0.842522 + 1.45929i −0.112587 + 0.195006i
\(57\) −0.0960103 + 7.54922i −0.0127169 + 0.999919i
\(58\) −0.0274227 − 0.0474976i −0.00360078 − 0.00623674i
\(59\) −1.57060 − 8.90733i −0.204475 − 1.15964i −0.898264 − 0.439457i \(-0.855171\pi\)
0.693789 − 0.720178i \(-0.255940\pi\)
\(60\) −1.46053 − 1.68897i −0.188553 − 0.218045i
\(61\) −2.33717 + 0.850661i −0.299244 + 0.108916i −0.487279 − 0.873246i \(-0.662010\pi\)
0.188035 + 0.982162i \(0.439788\pi\)
\(62\) −1.30017 − 1.54948i −0.165122 − 0.196784i
\(63\) 0.729475 − 5.00222i 0.0919052 − 0.630220i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −1.19770 −0.148556
\(66\) −0.164206 − 0.431107i −0.0202123 − 0.0530656i
\(67\) −0.522108 + 0.622225i −0.0637857 + 0.0760168i −0.796994 − 0.603988i \(-0.793578\pi\)
0.733208 + 0.680004i \(0.238022\pi\)
\(68\) 1.68153 + 0.970833i 0.203916 + 0.117731i
\(69\) −7.35983 − 4.10459i −0.886020 − 0.494135i
\(70\) −1.39631 + 1.66406i −0.166891 + 0.198894i
\(71\) −4.64075 + 3.89405i −0.550756 + 0.462139i −0.875197 − 0.483767i \(-0.839268\pi\)
0.324441 + 0.945906i \(0.394824\pi\)
\(72\) 0.432912 − 2.96860i 0.0510191 − 0.349853i
\(73\) 2.82620 + 16.0282i 0.330782 + 1.87596i 0.465459 + 0.885069i \(0.345889\pi\)
−0.134677 + 0.990890i \(0.543000\pi\)
\(74\) 5.96629 + 7.11035i 0.693567 + 0.826561i
\(75\) 2.96496 + 4.96361i 0.342364 + 0.573148i
\(76\) 4.31192 − 0.638255i 0.494611 − 0.0732128i
\(77\) −0.388673 + 0.224400i −0.0442934 + 0.0255728i
\(78\) −1.05256 − 1.21719i −0.119179 − 0.137820i
\(79\) 2.61242 − 7.17756i 0.293920 − 0.807538i −0.701564 − 0.712607i \(-0.747514\pi\)
0.995484 − 0.0949317i \(-0.0302633\pi\)
\(80\) −0.828652 + 0.987549i −0.0926461 + 0.110411i
\(81\) 2.08666 + 8.75476i 0.231851 + 0.972751i
\(82\) −0.648148 + 3.67583i −0.0715760 + 0.405928i
\(83\) −3.91697 + 2.26147i −0.429944 + 0.248228i −0.699323 − 0.714806i \(-0.746515\pi\)
0.269379 + 0.963034i \(0.413182\pi\)
\(84\) −2.91826 + 0.0433735i −0.318408 + 0.00473244i
\(85\) 1.91749 + 1.60897i 0.207981 + 0.174517i
\(86\) −0.303509 − 0.254675i −0.0327283 − 0.0274623i
\(87\) 0.0462697 − 0.0829650i 0.00496064 − 0.00889478i
\(88\) −0.230660 + 0.133172i −0.0245885 + 0.0141962i
\(89\) 1.61880 − 9.18068i 0.171593 − 0.973150i −0.770411 − 0.637548i \(-0.779949\pi\)
0.942004 − 0.335603i \(-0.108940\pi\)
\(90\) 1.21416 − 3.67193i 0.127984 − 0.387055i
\(91\) −1.00628 + 1.19924i −0.105487 + 0.125715i
\(92\) −1.66405 + 4.57193i −0.173489 + 0.476657i
\(93\) 1.14919 − 3.30959i 0.119165 − 0.343188i
\(94\) −2.04585 + 1.18117i −0.211013 + 0.121828i
\(95\) 5.61715 + 0.154954i 0.576308 + 0.0158980i
\(96\) −1.73186 + 0.0257403i −0.176757 + 0.00262711i
\(97\) −1.43893 − 1.71485i −0.146101 − 0.174116i 0.688031 − 0.725681i \(-0.258475\pi\)
−0.834132 + 0.551565i \(0.814031\pi\)
\(98\) −0.722486 − 4.09742i −0.0729821 − 0.413902i
\(99\) 0.495189 − 0.627087i 0.0497684 − 0.0630246i
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 342.2.x.c.281.10 yes 60
9.5 odd 6 342.2.bf.c.167.8 yes 60
19.14 odd 18 342.2.bf.c.299.8 yes 60
171.14 even 18 inner 342.2.x.c.185.10 ✓ 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.x.c.185.10 ✓ 60 171.14 even 18 inner
342.2.x.c.281.10 yes 60 1.1 even 1 trivial
342.2.bf.c.167.8 yes 60 9.5 odd 6
342.2.bf.c.299.8 yes 60 19.14 odd 18