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Properties

Label 294.2.m.d.163.1
Level $294$
Weight $2$
Character 294.163
Analytic conductor $2.348$
Analytic rank $0$
Dimension $36$
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] = [294,2,Mod(25,294)] 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("294.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.m (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,3] 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: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 163.1
Character \(\chi\) \(=\) 294.163
Dual form 294.2.m.d.193.1

$q$-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+(0.365341 + 0.930874i) q^{2} +(-0.0747301 - 0.997204i) q^{3} +(-0.733052 + 0.680173i) q^{4} +(-2.53296 + 1.72694i) q^{5} +(0.900969 - 0.433884i) q^{6} +(-2.53425 + 0.759993i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(-0.988831 + 0.149042i) q^{9} +(-2.53296 - 1.72694i) q^{10} +(-1.75612 - 0.264692i) q^{11} +(0.733052 + 0.680173i) q^{12} +(1.16787 - 1.46447i) q^{13} +(-1.63332 - 2.08141i) q^{14} +(1.91140 + 2.39682i) q^{15} +(0.0747301 - 0.997204i) q^{16} +(-6.66741 + 2.05662i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-0.620889 + 1.07541i) q^{19} +(0.682172 - 2.98879i) q^{20} +(0.947253 + 2.47037i) q^{21} +(-0.395186 - 1.73142i) q^{22} +(5.02516 + 1.55006i) q^{23} +(-0.365341 + 0.930874i) q^{24} +(1.60685 - 4.09419i) q^{25} +(1.78991 + 0.552113i) q^{26} +(0.222521 + 0.974928i) q^{27} +(1.34081 - 2.28084i) q^{28} +(-2.17593 + 9.53336i) q^{29} +(-1.53283 + 2.65493i) q^{30} +(0.0797624 + 0.138153i) q^{31} +(0.955573 - 0.294755i) q^{32} +(-0.132717 + 1.77099i) q^{33} +(-4.35034 - 5.45515i) q^{34} +(5.10668 - 6.30153i) q^{35} +(0.623490 - 0.781831i) q^{36} +(0.411321 + 0.381650i) q^{37} +(-1.22791 - 0.185077i) q^{38} +(-1.54765 - 1.05517i) q^{39} +(3.03141 - 0.456912i) q^{40} +(-1.68711 - 0.812468i) q^{41} +(-1.95353 + 1.78430i) q^{42} +(8.14135 - 3.92067i) q^{43} +(1.46736 - 1.00043i) q^{44} +(2.24728 - 2.08517i) q^{45} +(0.392990 + 5.24409i) q^{46} +(-0.714453 - 1.82040i) q^{47} -1.00000 q^{48} +(5.84482 - 3.85202i) q^{49} +4.39822 q^{50} +(2.54913 + 6.49508i) q^{51} +(0.139979 + 1.86789i) q^{52} +(3.32007 - 3.08058i) q^{53} +(-0.826239 + 0.563320i) q^{54} +(4.90528 - 2.36226i) q^{55} +(2.61303 + 0.414839i) q^{56} +(1.11880 + 0.538787i) q^{57} +(-9.66931 + 1.45741i) q^{58} +(-4.65590 - 3.17434i) q^{59} +(-3.03141 - 0.456912i) q^{60} +(-7.22567 - 6.70445i) q^{61} +(-0.0994621 + 0.124722i) q^{62} +(2.39267 - 1.12921i) q^{63} +(0.623490 + 0.781831i) q^{64} +(-0.429126 + 5.72629i) q^{65} +(-1.69705 + 0.523471i) q^{66} +(-2.16143 - 3.74372i) q^{67} +(3.48870 - 6.04260i) q^{68} +(1.17019 - 5.12694i) q^{69} +(7.73161 + 2.45147i) q^{70} +(3.21023 + 14.0650i) q^{71} +(0.955573 + 0.294755i) q^{72} +(-4.51908 + 11.5144i) q^{73} +(-0.204995 + 0.522320i) q^{74} +(-4.20282 - 1.29640i) q^{75} +(-0.276322 - 1.21064i) q^{76} +(4.65160 - 0.663841i) q^{77} +(0.416809 - 1.82616i) q^{78} +(-4.09173 + 7.08709i) q^{79} +(1.53283 + 2.65493i) q^{80} +(0.955573 - 0.294755i) q^{81} +(0.139936 - 1.86731i) q^{82} +(6.15716 + 7.72083i) q^{83} +(-2.37466 - 1.16661i) q^{84} +(13.3366 - 16.7236i) q^{85} +(6.62401 + 6.14619i) q^{86} +(9.66931 + 1.45741i) q^{87} +(1.46736 + 1.00043i) q^{88} +(-12.6325 + 1.90405i) q^{89} +(2.76206 + 1.33014i) q^{90} +(-1.84670 + 4.59890i) q^{91} +(-4.73801 + 2.28170i) q^{92} +(0.131806 - 0.0898636i) q^{93} +(1.43354 - 1.33013i) q^{94} +(-0.284487 - 3.79621i) q^{95} +(-0.365341 - 0.930874i) q^{96} -15.3330 q^{97} +(5.72110 + 4.03349i) q^{98} +1.77595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 6 q^{6} - q^{7} - 6 q^{8} + 3 q^{9} + 13 q^{11} - 3 q^{12} + 2 q^{13} - 4 q^{14} + 3 q^{16} + q^{17} - 18 q^{18} - 2 q^{19} + 7 q^{20} - 12 q^{21} - 12 q^{22} + 42 q^{23}+ \cdots + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(e\left(\frac{10}{21}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.365341 + 0.930874i 0.258335 + 0.658227i
\(3\) −0.0747301 − 0.997204i −0.0431454 − 0.575736i
\(4\) −0.733052 + 0.680173i −0.366526 + 0.340086i
\(5\) −2.53296 + 1.72694i −1.13277 + 0.772313i −0.976525 − 0.215405i \(-0.930893\pi\)
−0.156250 + 0.987718i \(0.549940\pi\)
\(6\) 0.900969 − 0.433884i 0.367819 − 0.177132i
\(7\) −2.53425 + 0.759993i −0.957856 + 0.287250i
\(8\) −0.900969 − 0.433884i −0.318541 − 0.153401i
\(9\) −0.988831 + 0.149042i −0.329610 + 0.0496808i
\(10\) −2.53296 − 1.72694i −0.800993 − 0.546107i
\(11\) −1.75612 − 0.264692i −0.529489 − 0.0798076i −0.121143 − 0.992635i \(-0.538656\pi\)
−0.408346 + 0.912827i \(0.633894\pi\)
\(12\) 0.733052 + 0.680173i 0.211614 + 0.196349i
\(13\) 1.16787 − 1.46447i 0.323910 − 0.406170i −0.593040 − 0.805173i \(-0.702072\pi\)
0.916950 + 0.399003i \(0.130644\pi\)
\(14\) −1.63332 − 2.08141i −0.436524 − 0.556280i
\(15\) 1.91140 + 2.39682i 0.493522 + 0.618857i
\(16\) 0.0747301 − 0.997204i 0.0186825 − 0.249301i
\(17\) −6.66741 + 2.05662i −1.61708 + 0.498805i −0.965638 − 0.259890i \(-0.916314\pi\)
−0.651447 + 0.758694i \(0.725838\pi\)
\(18\) −0.500000 − 0.866025i −0.117851 − 0.204124i
\(19\) −0.620889 + 1.07541i −0.142442 + 0.246716i −0.928416 − 0.371543i \(-0.878829\pi\)
0.785974 + 0.618260i \(0.212162\pi\)
\(20\) 0.682172 − 2.98879i 0.152538 − 0.668314i
\(21\) 0.947253 + 2.47037i 0.206707 + 0.539078i
\(22\) −0.395186 − 1.73142i −0.0842540 − 0.369141i
\(23\) 5.02516 + 1.55006i 1.04782 + 0.323209i 0.770414 − 0.637544i \(-0.220050\pi\)
0.277404 + 0.960753i \(0.410526\pi\)
\(24\) −0.365341 + 0.930874i −0.0745749 + 0.190014i
\(25\) 1.60685 − 4.09419i 0.321370 − 0.818838i
\(26\) 1.78991 + 0.552113i 0.351029 + 0.108278i
\(27\) 0.222521 + 0.974928i 0.0428242 + 0.187625i
\(28\) 1.34081 − 2.28084i 0.253389 − 0.431038i
\(29\) −2.17593 + 9.53336i −0.404060 + 1.77030i 0.206618 + 0.978422i \(0.433754\pi\)
−0.610678 + 0.791879i \(0.709103\pi\)
\(30\) −1.53283 + 2.65493i −0.279854 + 0.484722i
\(31\) 0.0797624 + 0.138153i 0.0143258 + 0.0248129i 0.873099 − 0.487542i \(-0.162106\pi\)
−0.858774 + 0.512355i \(0.828773\pi\)
\(32\) 0.955573 − 0.294755i 0.168923 − 0.0521058i
\(33\) −0.132717 + 1.77099i −0.0231031 + 0.308289i
\(34\) −4.35034 − 5.45515i −0.746077 − 0.935550i
\(35\) 5.10668 − 6.30153i 0.863187 − 1.06515i
\(36\) 0.623490 − 0.781831i 0.103915 − 0.130305i
\(37\) 0.411321 + 0.381650i 0.0676207 + 0.0627428i 0.713259 − 0.700900i \(-0.247218\pi\)
−0.645639 + 0.763643i \(0.723409\pi\)
\(38\) −1.22791 − 0.185077i −0.199193 − 0.0300235i
\(39\) −1.54765 − 1.05517i −0.247822 − 0.168962i
\(40\) 3.03141 − 0.456912i 0.479308 − 0.0722441i
\(41\) −1.68711 − 0.812468i −0.263482 − 0.126886i 0.297479 − 0.954728i \(-0.403854\pi\)
−0.560961 + 0.827842i \(0.689568\pi\)
\(42\) −1.95353 + 1.78430i −0.301436 + 0.275323i
\(43\) 8.14135 − 3.92067i 1.24154 − 0.597896i 0.306312 − 0.951931i \(-0.400905\pi\)
0.935232 + 0.354035i \(0.115191\pi\)
\(44\) 1.46736 − 1.00043i 0.221213 − 0.150820i
\(45\) 2.24728 − 2.08517i 0.335005 − 0.310839i
\(46\) 0.392990 + 5.24409i 0.0579432 + 0.773198i
\(47\) −0.714453 − 1.82040i −0.104214 − 0.265532i 0.869250 − 0.494372i \(-0.164602\pi\)
−0.973464 + 0.228840i \(0.926507\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.84482 − 3.85202i 0.834974 − 0.550289i
\(50\) 4.39822 0.622003
\(51\) 2.54913 + 6.49508i 0.356950 + 0.909493i
\(52\) 0.139979 + 1.86789i 0.0194116 + 0.259029i
\(53\) 3.32007 − 3.08058i 0.456047 − 0.423150i −0.418406 − 0.908260i \(-0.637411\pi\)
0.874453 + 0.485110i \(0.161221\pi\)
\(54\) −0.826239 + 0.563320i −0.112437 + 0.0766582i
\(55\) 4.90528 − 2.36226i 0.661428 − 0.318527i
\(56\) 2.61303 + 0.414839i 0.349180 + 0.0554351i
\(57\) 1.11880 + 0.538787i 0.148189 + 0.0713641i
\(58\) −9.66931 + 1.45741i −1.26964 + 0.191368i
\(59\) −4.65590 − 3.17434i −0.606147 − 0.413264i 0.220981 − 0.975278i \(-0.429074\pi\)
−0.827128 + 0.562014i \(0.810027\pi\)
\(60\) −3.03141 − 0.456912i −0.391354 − 0.0589871i
\(61\) −7.22567 − 6.70445i −0.925153 − 0.858416i 0.0650743 − 0.997880i \(-0.479272\pi\)
−0.990227 + 0.139464i \(0.955462\pi\)
\(62\) −0.0994621 + 0.124722i −0.0126317 + 0.0158397i
\(63\) 2.39267 − 1.12921i 0.301448 − 0.142268i
\(64\) 0.623490 + 0.781831i 0.0779362 + 0.0977289i
\(65\) −0.429126 + 5.72629i −0.0532265 + 0.710259i
\(66\) −1.69705 + 0.523471i −0.208893 + 0.0644348i
\(67\) −2.16143 − 3.74372i −0.264061 − 0.457368i 0.703256 − 0.710937i \(-0.251729\pi\)
−0.967317 + 0.253569i \(0.918395\pi\)
\(68\) 3.48870 − 6.04260i 0.423067 − 0.732773i
\(69\) 1.17019 − 5.12694i 0.140874 − 0.617211i
\(70\) 7.73161 + 2.45147i 0.924105 + 0.293007i
\(71\) 3.21023 + 14.0650i 0.380985 + 1.66920i 0.694402 + 0.719587i \(0.255669\pi\)
−0.313418 + 0.949615i \(0.601474\pi\)
\(72\) 0.955573 + 0.294755i 0.112615 + 0.0347372i
\(73\) −4.51908 + 11.5144i −0.528918 + 1.34766i 0.377786 + 0.925893i \(0.376686\pi\)
−0.906704 + 0.421768i \(0.861410\pi\)
\(74\) −0.204995 + 0.522320i −0.0238302 + 0.0607184i
\(75\) −4.20282 − 1.29640i −0.485300 − 0.149695i
\(76\) −0.276322 − 1.21064i −0.0316963 − 0.138870i
\(77\) 4.65160 − 0.663841i 0.530098 − 0.0756517i
\(78\) 0.416809 − 1.82616i 0.0471943 − 0.206772i
\(79\) −4.09173 + 7.08709i −0.460356 + 0.797360i −0.998979 − 0.0451872i \(-0.985612\pi\)
0.538623 + 0.842547i \(0.318945\pi\)
\(80\) 1.53283 + 2.65493i 0.171375 + 0.296830i
\(81\) 0.955573 − 0.294755i 0.106175 − 0.0327506i
\(82\) 0.139936 − 1.86731i 0.0154533 − 0.206210i
\(83\) 6.15716 + 7.72083i 0.675836 + 0.847471i 0.994963 − 0.100240i \(-0.0319612\pi\)
−0.319127 + 0.947712i \(0.603390\pi\)
\(84\) −2.37466 − 1.16661i −0.259097 − 0.127288i
\(85\) 13.3366 − 16.7236i 1.44656 − 1.81393i
\(86\) 6.62401 + 6.14619i 0.714286 + 0.662760i
\(87\) 9.66931 + 1.45741i 1.03666 + 0.156251i
\(88\) 1.46736 + 1.00043i 0.156421 + 0.106646i
\(89\) −12.6325 + 1.90405i −1.33905 + 0.201829i −0.779204 − 0.626771i \(-0.784376\pi\)
−0.559842 + 0.828599i \(0.689138\pi\)
\(90\) 2.76206 + 1.33014i 0.291146 + 0.140209i
\(91\) −1.84670 + 4.59890i −0.193586 + 0.482095i
\(92\) −4.73801 + 2.28170i −0.493971 + 0.237884i
\(93\) 0.131806 − 0.0898636i 0.0136676 − 0.00931842i
\(94\) 1.43354 − 1.33013i 0.147858 − 0.137193i
\(95\) −0.284487 − 3.79621i −0.0291877 − 0.389483i
\(96\) −0.365341 − 0.930874i −0.0372875 − 0.0950069i
\(97\) −15.3330 −1.55683 −0.778413 − 0.627752i \(-0.783975\pi\)
−0.778413 + 0.627752i \(0.783975\pi\)
\(98\) 5.72110 + 4.03349i 0.577918 + 0.407444i
\(99\) 1.77595 0.178490
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.m.d.163.1 ✓ 36
3.2 odd 2 882.2.z.e.163.3 36
49.46 even 21 inner 294.2.m.d.193.1 yes 36
147.95 odd 42 882.2.z.e.487.3 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.m.d.163.1 ✓ 36 1.1 even 1 trivial
294.2.m.d.193.1 yes 36 49.46 even 21 inner
882.2.z.e.163.3 36 3.2 odd 2
882.2.z.e.487.3 36 147.95 odd 42