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Properties

Label 294.2.j.a.41.3
Level $294$
Weight $2$
Character 294.41
Analytic conductor $2.348$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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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] = [294,2,Mod(41,294)] 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("294.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 41.3
Character \(\chi\) \(=\) 294.41
Dual form 294.2.j.a.251.3

$q$-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+(-0.781831 + 0.623490i) q^{2} +(-0.995627 + 1.41730i) q^{3} +(0.222521 - 0.974928i) q^{4} +(1.32694 - 0.639022i) q^{5} +(-0.105257 - 1.72885i) q^{6} +(1.06200 - 2.42325i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-1.01745 - 2.82220i) q^{9} +(-0.639022 + 1.32694i) q^{10} +(1.54863 - 1.23499i) q^{11} +(1.16021 + 1.28604i) q^{12} +(2.81675 - 2.24628i) q^{13} +(0.680565 + 2.55672i) q^{14} +(-0.415456 + 2.51690i) q^{15} +(-0.900969 - 0.433884i) q^{16} +(-0.364924 - 1.59884i) q^{17} +(2.55509 + 1.57211i) q^{18} +3.32283i q^{19} +(-0.327728 - 1.43587i) q^{20} +(2.37710 + 3.91783i) q^{21} +(-0.440762 + 1.93110i) q^{22} +(1.55008 + 0.353797i) q^{23} +(-1.70893 - 0.282087i) q^{24} +(-1.76502 + 2.21327i) q^{25} +(-0.801689 + 3.51243i) q^{26} +(5.01289 + 1.36782i) q^{27} +(-2.12618 - 1.57460i) q^{28} +(8.42627 - 1.92324i) q^{29} +(-1.24444 - 2.22682i) q^{30} +7.72115i q^{31} +(0.974928 - 0.222521i) q^{32} +(0.208490 + 3.42445i) q^{33} +(1.28217 + 1.02249i) q^{34} +(-0.139293 - 3.89416i) q^{35} +(-2.97784 + 0.363947i) q^{36} +(-1.59721 - 6.99784i) q^{37} +(-2.07175 - 2.59789i) q^{38} +(0.379216 + 6.22863i) q^{39} +(1.15148 + 0.918272i) q^{40} +(4.55013 - 2.19123i) q^{41} +(-4.30122 - 1.58098i) q^{42} +(-4.87660 - 2.34845i) q^{43} +(-0.859422 - 1.78461i) q^{44} +(-3.15355 - 3.09471i) q^{45} +(-1.43249 + 0.689853i) q^{46} +(2.86905 + 3.59768i) q^{47} +(1.51197 - 0.844953i) q^{48} +(-4.74430 - 5.14700i) q^{49} -2.83088i q^{50} +(2.62935 + 1.07464i) q^{51} +(-1.56318 - 3.24597i) q^{52} +(-13.0036 - 2.96798i) q^{53} +(-4.77206 + 2.05608i) q^{54} +(1.26575 - 2.62836i) q^{55} +(2.64406 - 0.0945772i) q^{56} +(-4.70943 - 3.30830i) q^{57} +(-5.38880 + 6.75734i) q^{58} +(-3.81131 - 1.83543i) q^{59} +(2.36134 + 0.965102i) q^{60} +(10.9005 - 2.48797i) q^{61} +(-4.81406 - 6.03664i) q^{62} +(-7.91943 - 0.531633i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(2.30224 - 4.78065i) q^{65} +(-2.29811 - 2.54735i) q^{66} -5.99565 q^{67} -1.63995 q^{68} +(-2.04474 + 1.84468i) q^{69} +(2.53687 + 2.95773i) q^{70} +(8.40837 + 1.91916i) q^{71} +(2.10125 - 2.14120i) q^{72} +(5.11254 + 4.07712i) q^{73} +(5.61183 + 4.47528i) q^{74} +(-1.37955 - 4.70515i) q^{75} +(3.23952 + 0.739399i) q^{76} +(-1.34804 - 5.06427i) q^{77} +(-4.17997 - 4.63330i) q^{78} -7.73563 q^{79} -1.47279 q^{80} +(-6.92957 + 5.74291i) q^{81} +(-2.19123 + 4.55013i) q^{82} +(-6.53297 + 8.19209i) q^{83} +(4.34855 - 1.44571i) q^{84} +(-1.50592 - 1.88837i) q^{85} +(5.27691 - 1.20442i) q^{86} +(-5.66362 + 13.8573i) q^{87} +(1.78461 + 0.859422i) q^{88} +(5.78822 - 7.25820i) q^{89} +(4.39507 + 0.453340i) q^{90} +(-2.45191 - 9.21126i) q^{91} +(0.689853 - 1.43249i) q^{92} +(-10.9432 - 7.68739i) q^{93} +(-4.48623 - 1.02395i) q^{94} +(2.12336 + 4.40920i) q^{95} +(-0.655286 + 1.60331i) q^{96} -7.64326i q^{97} +(6.91834 + 1.06607i) q^{98} +(-5.06103 - 3.11398i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 20 q^{4} - 14 q^{6} + 4 q^{7} - 10 q^{9} + 12 q^{15} - 20 q^{16} - 8 q^{18} - 12 q^{21} - 10 q^{22} - 24 q^{25} + 42 q^{27} - 4 q^{28} + 8 q^{30} + 10 q^{36} - 58 q^{37} + 8 q^{39} - 14 q^{40} - 2 q^{42}+ \cdots - 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.781831 + 0.623490i −0.552838 + 0.440874i
\(3\) −0.995627 + 1.41730i −0.574825 + 0.818276i
\(4\) 0.222521 − 0.974928i 0.111260 − 0.487464i
\(5\) 1.32694 − 0.639022i 0.593427 − 0.285779i −0.112967 − 0.993599i \(-0.536035\pi\)
0.706393 + 0.707820i \(0.250321\pi\)
\(6\) −0.105257 − 1.72885i −0.0429710 − 0.705800i
\(7\) 1.06200 − 2.42325i 0.401400 − 0.915903i
\(8\) 0.433884 + 0.900969i 0.153401 + 0.318541i
\(9\) −1.01745 − 2.82220i −0.339152 − 0.940732i
\(10\) −0.639022 + 1.32694i −0.202076 + 0.419616i
\(11\) 1.54863 − 1.23499i 0.466928 − 0.372363i −0.361579 − 0.932342i \(-0.617762\pi\)
0.828507 + 0.559979i \(0.189191\pi\)
\(12\) 1.16021 + 1.28604i 0.334925 + 0.371248i
\(13\) 2.81675 − 2.24628i 0.781226 − 0.623007i −0.149484 − 0.988764i \(-0.547761\pi\)
0.930710 + 0.365757i \(0.119190\pi\)
\(14\) 0.680565 + 2.55672i 0.181889 + 0.683313i
\(15\) −0.415456 + 2.51690i −0.107270 + 0.649860i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) −0.364924 − 1.59884i −0.0885071 − 0.387775i 0.911200 − 0.411963i \(-0.135157\pi\)
−0.999707 + 0.0241887i \(0.992300\pi\)
\(18\) 2.55509 + 1.57211i 0.602240 + 0.370550i
\(19\) 3.32283i 0.762309i 0.924511 + 0.381155i \(0.124473\pi\)
−0.924511 + 0.381155i \(0.875527\pi\)
\(20\) −0.327728 − 1.43587i −0.0732821 − 0.321070i
\(21\) 2.37710 + 3.91783i 0.518727 + 0.854940i
\(22\) −0.440762 + 1.93110i −0.0939707 + 0.411713i
\(23\) 1.55008 + 0.353797i 0.323215 + 0.0737717i 0.381050 − 0.924554i \(-0.375563\pi\)
−0.0578350 + 0.998326i \(0.518420\pi\)
\(24\) −1.70893 − 0.282087i −0.348833 − 0.0575808i
\(25\) −1.76502 + 2.21327i −0.353004 + 0.442654i
\(26\) −0.801689 + 3.51243i −0.157224 + 0.688844i
\(27\) 5.01289 + 1.36782i 0.964731 + 0.263237i
\(28\) −2.12618 − 1.57460i −0.401810 − 0.297572i
\(29\) 8.42627 − 1.92324i 1.56472 − 0.357137i 0.649585 − 0.760289i \(-0.274943\pi\)
0.915133 + 0.403152i \(0.132085\pi\)
\(30\) −1.24444 − 2.22682i −0.227203 − 0.406560i
\(31\) 7.72115i 1.38676i 0.720572 + 0.693380i \(0.243879\pi\)
−0.720572 + 0.693380i \(0.756121\pi\)
\(32\) 0.974928 − 0.222521i 0.172345 − 0.0393365i
\(33\) 0.208490 + 3.42445i 0.0362934 + 0.596120i
\(34\) 1.28217 + 1.02249i 0.219890 + 0.175356i
\(35\) −0.139293 − 3.89416i −0.0235448 − 0.658233i
\(36\) −2.97784 + 0.363947i −0.496307 + 0.0606579i
\(37\) −1.59721 − 6.99784i −0.262580 − 1.15044i −0.918442 − 0.395556i \(-0.870552\pi\)
0.655862 − 0.754881i \(-0.272305\pi\)
\(38\) −2.07175 − 2.59789i −0.336082 − 0.421434i
\(39\) 0.379216 + 6.22863i 0.0607232 + 0.997379i
\(40\) 1.15148 + 0.918272i 0.182065 + 0.145192i
\(41\) 4.55013 − 2.19123i 0.710611 − 0.342212i −0.0434147 − 0.999057i \(-0.513824\pi\)
0.754026 + 0.656845i \(0.228109\pi\)
\(42\) −4.30122 − 1.58098i −0.663693 − 0.243951i
\(43\) −4.87660 − 2.34845i −0.743674 − 0.358135i 0.0233718 − 0.999727i \(-0.492560\pi\)
−0.767046 + 0.641592i \(0.778274\pi\)
\(44\) −0.859422 − 1.78461i −0.129563 − 0.269040i
\(45\) −3.15355 − 3.09471i −0.470103 − 0.461333i
\(46\) −1.43249 + 0.689853i −0.211210 + 0.101713i
\(47\) 2.86905 + 3.59768i 0.418494 + 0.524775i 0.945734 − 0.324941i \(-0.105344\pi\)
−0.527240 + 0.849717i \(0.676773\pi\)
\(48\) 1.51197 − 0.844953i 0.218234 − 0.121958i
\(49\) −4.74430 − 5.14700i −0.677757 − 0.735286i
\(50\) − 2.83088i − 0.400346i
\(51\) 2.62935 + 1.07464i 0.368183 + 0.150480i
\(52\) −1.56318 − 3.24597i −0.216774 − 0.450136i
\(53\) −13.0036 − 2.96798i −1.78618 − 0.407684i −0.803849 − 0.594833i \(-0.797218\pi\)
−0.982331 + 0.187149i \(0.940075\pi\)
\(54\) −4.77206 + 2.05608i −0.649395 + 0.279797i
\(55\) 1.26575 − 2.62836i 0.170674 − 0.354408i
\(56\) 2.64406 − 0.0945772i 0.353327 − 0.0126384i
\(57\) −4.70943 − 3.30830i −0.623779 − 0.438195i
\(58\) −5.38880 + 6.75734i −0.707584 + 0.887282i
\(59\) −3.81131 − 1.83543i −0.496190 − 0.238952i 0.169015 − 0.985614i \(-0.445941\pi\)
−0.665205 + 0.746661i \(0.731656\pi\)
\(60\) 2.36134 + 0.965102i 0.304848 + 0.124594i
\(61\) 10.9005 − 2.48797i 1.39567 − 0.318552i 0.542441 − 0.840094i \(-0.317500\pi\)
0.853225 + 0.521542i \(0.174643\pi\)
\(62\) −4.81406 − 6.03664i −0.611386 − 0.766654i
\(63\) −7.91943 − 0.531633i −0.997754 − 0.0669795i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) 2.30224 − 4.78065i 0.285558 − 0.592967i
\(66\) −2.29811 − 2.54735i −0.282878 − 0.313557i
\(67\) −5.99565 −0.732485 −0.366243 − 0.930519i \(-0.619356\pi\)
−0.366243 + 0.930519i \(0.619356\pi\)
\(68\) −1.63995 −0.198874
\(69\) −2.04474 + 1.84468i −0.246158 + 0.222073i
\(70\) 2.53687 + 2.95773i 0.303214 + 0.353516i
\(71\) 8.40837 + 1.91916i 0.997890 + 0.227762i 0.690117 − 0.723698i \(-0.257559\pi\)
0.307773 + 0.951460i \(0.400416\pi\)
\(72\) 2.10125 − 2.14120i 0.247635 − 0.252343i
\(73\) 5.11254 + 4.07712i 0.598378 + 0.477191i 0.875220 − 0.483725i \(-0.160716\pi\)
−0.276842 + 0.960915i \(0.589288\pi\)
\(74\) 5.61183 + 4.47528i 0.652362 + 0.520241i
\(75\) −1.37955 − 4.70515i −0.159297 − 0.543304i
\(76\) 3.23952 + 0.739399i 0.371598 + 0.0848149i
\(77\) −1.34804 − 5.06427i −0.153623 − 0.577127i
\(78\) −4.17997 − 4.63330i −0.473288 − 0.524618i
\(79\) −7.73563 −0.870327 −0.435163 − 0.900351i \(-0.643309\pi\)
−0.435163 + 0.900351i \(0.643309\pi\)
\(80\) −1.47279 −0.164663
\(81\) −6.92957 + 5.74291i −0.769952 + 0.638101i
\(82\) −2.19123 + 4.55013i −0.241981 + 0.502478i
\(83\) −6.53297 + 8.19209i −0.717087 + 0.899199i −0.998169 − 0.0604852i \(-0.980735\pi\)
0.281082 + 0.959684i \(0.409307\pi\)
\(84\) 4.34855 − 1.44571i 0.474466 − 0.157740i
\(85\) −1.50592 − 1.88837i −0.163340 − 0.204822i
\(86\) 5.27691 − 1.20442i 0.569024 − 0.129876i
\(87\) −5.66362 + 13.8573i −0.607203 + 1.48566i
\(88\) 1.78461 + 0.859422i 0.190240 + 0.0916147i
\(89\) 5.78822 − 7.25820i 0.613550 − 0.769367i −0.373871 − 0.927481i \(-0.621970\pi\)
0.987421 + 0.158113i \(0.0505411\pi\)
\(90\) 4.39507 + 0.453340i 0.463281 + 0.0477863i
\(91\) −2.45191 − 9.21126i −0.257030 − 0.965602i
\(92\) 0.689853 − 1.43249i 0.0719221 − 0.149348i
\(93\) −10.9432 − 7.68739i −1.13475 − 0.797145i
\(94\) −4.48623 − 1.02395i −0.462719 − 0.105613i
\(95\) 2.12336 + 4.40920i 0.217852 + 0.452374i
\(96\) −0.655286 + 1.60331i −0.0668799 + 0.163637i
\(97\) − 7.64326i − 0.776056i −0.921648 − 0.388028i \(-0.873156\pi\)
0.921648 − 0.388028i \(-0.126844\pi\)
\(98\) 6.91834 + 1.06607i 0.698858 + 0.107689i
\(99\) −5.06103 − 3.11398i −0.508653 − 0.312967i
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 294.2.j.a.41.3 ✓ 120
3.2 odd 2 inner 294.2.j.a.41.20 yes 120
49.6 odd 14 inner 294.2.j.a.251.20 yes 120
147.104 even 14 inner 294.2.j.a.251.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.j.a.41.3 ✓ 120 1.1 even 1 trivial
294.2.j.a.41.20 yes 120 3.2 odd 2 inner
294.2.j.a.251.3 yes 120 147.104 even 14 inner
294.2.j.a.251.20 yes 120 49.6 odd 14 inner