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Properties

Label 294.2.p.a.59.11
Level $294$
Weight $2$
Character 294.59
Analytic conductor $2.348$
Analytic rank $0$
Dimension $216$
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(5,294)] 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("294.5"); 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([21, 29])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.p (of order \(42\), degree \(12\), 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: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 59.11
Character \(\chi\) \(=\) 294.59
Dual form 294.2.p.a.5.11

$q$-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+(0.149042 - 0.988831i) q^{2} +(-1.54816 + 0.776659i) q^{3} +(-0.955573 - 0.294755i) q^{4} +(-0.0213134 - 0.284407i) q^{5} +(0.537243 + 1.64662i) q^{6} +(1.20478 + 2.35552i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(1.79360 - 2.40479i) q^{9} +(-0.284407 - 0.0213134i) q^{10} +(-1.59229 + 0.624927i) q^{11} +(1.70830 - 0.285826i) q^{12} +(4.60538 + 3.67267i) q^{13} +(2.50878 - 0.840255i) q^{14} +(0.253884 + 0.423755i) q^{15} +(0.826239 + 0.563320i) q^{16} +(3.84529 + 3.56791i) q^{17} +(-2.11060 - 2.13198i) q^{18} +(1.57607 - 0.909947i) q^{19} +(-0.0634640 + 0.278054i) q^{20} +(-3.69464 - 2.71102i) q^{21} +(0.380629 + 1.66765i) q^{22} +(-1.03152 - 1.11171i) q^{23} +(-0.0280243 - 1.73182i) q^{24} +(4.86372 - 0.733088i) q^{25} +(4.31805 - 4.00656i) q^{26} +(-0.909082 + 5.11601i) q^{27} +(-0.456956 - 2.60599i) q^{28} +(2.91215 + 0.664680i) q^{29} +(0.456861 - 0.187891i) q^{30} +(-1.59583 - 0.921355i) q^{31} +(0.680173 - 0.733052i) q^{32} +(1.97976 - 2.20415i) q^{33} +(4.10117 - 3.27058i) q^{34} +(0.644250 - 0.392853i) q^{35} +(-2.42274 + 1.76928i) q^{36} +(-9.69872 + 2.99166i) q^{37} +(-0.664882 - 1.69409i) q^{38} +(-9.98229 - 2.10907i) q^{39} +(0.265490 + 0.104197i) q^{40} +(-1.38570 - 0.667318i) q^{41} +(-3.23140 + 3.24932i) q^{42} +(-8.40003 + 4.04524i) q^{43} +(1.70575 - 0.127828i) q^{44} +(-0.722166 - 0.458859i) q^{45} +(-1.25304 + 0.854305i) q^{46} +(5.71314 + 0.861118i) q^{47} +(-1.71666 - 0.230404i) q^{48} +(-4.09699 + 5.67580i) q^{49} -4.91866i q^{50} +(-8.72418 - 2.53722i) q^{51} +(-3.31824 - 4.86697i) q^{52} +(3.30713 - 10.7214i) q^{53} +(4.92338 + 1.66143i) q^{54} +(0.211671 + 0.439539i) q^{55} +(-2.64499 + 0.0634495i) q^{56} +(-1.73330 + 2.63282i) q^{57} +(1.09129 - 2.78056i) q^{58} +(-0.890591 + 11.8841i) q^{59} +(-0.117701 - 0.479762i) q^{60} +(-1.31867 - 4.27502i) q^{61} +(-1.14891 + 1.44069i) q^{62} +(7.82543 + 1.32762i) q^{63} +(-0.623490 - 0.781831i) q^{64} +(0.946378 - 1.38808i) q^{65} +(-1.88447 - 2.28616i) q^{66} +(4.90509 - 8.49586i) q^{67} +(-2.62280 - 4.54282i) q^{68} +(2.46038 + 0.919971i) q^{69} +(-0.292445 - 0.695606i) q^{70} +(-2.21716 + 0.506053i) q^{71} +(1.38842 + 2.65938i) q^{72} +(-1.26593 - 8.39889i) q^{73} +(1.51273 + 10.0363i) q^{74} +(-6.96046 + 4.91239i) q^{75} +(-1.77427 + 0.404965i) q^{76} +(-3.39040 - 2.99777i) q^{77} +(-3.57330 + 9.55645i) q^{78} +(-0.786077 - 1.36152i) q^{79} +(0.142602 - 0.246994i) q^{80} +(-2.56599 - 8.62645i) q^{81} +(-0.866392 + 1.27076i) q^{82} +(5.94517 + 7.45501i) q^{83} +(2.73141 + 3.67959i) q^{84} +(0.932784 - 1.16967i) q^{85} +(2.74810 + 8.90912i) q^{86} +(-5.02471 + 1.23272i) q^{87} +(0.127828 - 1.70575i) q^{88} +(3.01624 - 7.68526i) q^{89} +(-0.561367 + 0.645711i) q^{90} +(-3.10257 + 15.2729i) q^{91} +(0.658008 + 1.36637i) q^{92} +(3.18618 + 0.186986i) q^{93} +(1.70300 - 5.52099i) q^{94} +(-0.292387 - 0.428853i) q^{95} +(-0.483685 + 1.66314i) q^{96} +7.01206i q^{97} +(5.00178 + 4.89716i) q^{98} +(-1.35311 + 4.94998i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q - 18 q^{4} + 14 q^{6} + 10 q^{7} + 16 q^{9} + 6 q^{10} - 12 q^{15} + 18 q^{16} - 4 q^{18} - 12 q^{19} + 10 q^{22} - 6 q^{24} + 16 q^{25} - 42 q^{27} + 2 q^{28} + 4 q^{30} + 6 q^{31} + 18 q^{33} - 10 q^{36}+ \cdots + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.149042 − 0.988831i 0.105389 − 0.699209i
\(3\) −1.54816 + 0.776659i −0.893831 + 0.448404i
\(4\) −0.955573 − 0.294755i −0.477786 − 0.147378i
\(5\) −0.0213134 − 0.284407i −0.00953163 − 0.127191i 0.990408 − 0.138173i \(-0.0441231\pi\)
−0.999940 + 0.0109826i \(0.996504\pi\)
\(6\) 0.537243 + 1.64662i 0.219329 + 0.672231i
\(7\) 1.20478 + 2.35552i 0.455366 + 0.890304i
\(8\) −0.433884 + 0.900969i −0.153401 + 0.318541i
\(9\) 1.79360 − 2.40479i 0.597867 − 0.801595i
\(10\) −0.284407 − 0.0213134i −0.0899374 − 0.00673988i
\(11\) −1.59229 + 0.624927i −0.480093 + 0.188423i −0.593031 − 0.805179i \(-0.702069\pi\)
0.112938 + 0.993602i \(0.463974\pi\)
\(12\) 1.70830 − 0.285826i 0.493145 − 0.0825109i
\(13\) 4.60538 + 3.67267i 1.27730 + 1.01862i 0.998298 + 0.0583267i \(0.0185765\pi\)
0.279006 + 0.960289i \(0.409995\pi\)
\(14\) 2.50878 − 0.840255i 0.670499 − 0.224568i
\(15\) 0.253884 + 0.423755i 0.0655526 + 0.109413i
\(16\) 0.826239 + 0.563320i 0.206560 + 0.140830i
\(17\) 3.84529 + 3.56791i 0.932621 + 0.865346i 0.991124 − 0.132941i \(-0.0424422\pi\)
−0.0585029 + 0.998287i \(0.518633\pi\)
\(18\) −2.11060 − 2.13198i −0.497474 − 0.502513i
\(19\) 1.57607 − 0.909947i 0.361576 − 0.208756i −0.308196 − 0.951323i \(-0.599725\pi\)
0.669772 + 0.742567i \(0.266392\pi\)
\(20\) −0.0634640 + 0.278054i −0.0141910 + 0.0621748i
\(21\) −3.69464 − 2.71102i −0.806236 − 0.591594i
\(22\) 0.380629 + 1.66765i 0.0811504 + 0.355543i
\(23\) −1.03152 − 1.11171i −0.215086 − 0.231808i 0.616288 − 0.787521i \(-0.288636\pi\)
−0.831374 + 0.555713i \(0.812445\pi\)
\(24\) −0.0280243 − 1.73182i −0.00572043 − 0.353507i
\(25\) 4.86372 − 0.733088i 0.972744 − 0.146618i
\(26\) 4.31805 − 4.00656i 0.846839 − 0.785752i
\(27\) −0.909082 + 5.11601i −0.174953 + 0.984577i
\(28\) −0.456956 − 2.60599i −0.0863566 − 0.492486i
\(29\) 2.91215 + 0.664680i 0.540773 + 0.123428i 0.484178 − 0.874969i \(-0.339119\pi\)
0.0565947 + 0.998397i \(0.481976\pi\)
\(30\) 0.456861 − 0.187891i 0.0834110 − 0.0343040i
\(31\) −1.59583 − 0.921355i −0.286620 − 0.165480i 0.349797 − 0.936826i \(-0.386251\pi\)
−0.636417 + 0.771346i \(0.719584\pi\)
\(32\) 0.680173 − 0.733052i 0.120239 − 0.129586i
\(33\) 1.97976 − 2.20415i 0.344632 − 0.383694i
\(34\) 4.10117 − 3.27058i 0.703345 − 0.560899i
\(35\) 0.644250 − 0.392853i 0.108898 − 0.0664044i
\(36\) −2.42274 + 1.76928i −0.403790 + 0.294879i
\(37\) −9.69872 + 2.99166i −1.59446 + 0.491826i −0.959734 − 0.280911i \(-0.909363\pi\)
−0.634726 + 0.772737i \(0.718887\pi\)
\(38\) −0.664882 − 1.69409i −0.107858 − 0.274818i
\(39\) −9.98229 − 2.10907i −1.59845 − 0.337721i
\(40\) 0.265490 + 0.104197i 0.0419776 + 0.0164750i
\(41\) −1.38570 − 0.667318i −0.216410 − 0.104218i 0.322538 − 0.946556i \(-0.395464\pi\)
−0.538948 + 0.842339i \(0.681178\pi\)
\(42\) −3.23140 + 3.24932i −0.498616 + 0.501380i
\(43\) −8.40003 + 4.04524i −1.28099 + 0.616894i −0.945645 − 0.325201i \(-0.894568\pi\)
−0.335348 + 0.942094i \(0.608854\pi\)
\(44\) 1.70575 − 0.127828i 0.257151 − 0.0192708i
\(45\) −0.722166 − 0.458859i −0.107654 − 0.0684026i
\(46\) −1.25304 + 0.854305i −0.184750 + 0.125960i
\(47\) 5.71314 + 0.861118i 0.833347 + 0.125607i 0.551838 − 0.833952i \(-0.313927\pi\)
0.281510 + 0.959558i \(0.409165\pi\)
\(48\) −1.71666 − 0.230404i −0.247778 − 0.0332559i
\(49\) −4.09699 + 5.67580i −0.585284 + 0.810828i
\(50\) − 4.91866i − 0.695603i
\(51\) −8.72418 − 2.53722i −1.22163 − 0.355281i
\(52\) −3.31824 − 4.86697i −0.460157 − 0.674927i
\(53\) 3.30713 − 10.7214i 0.454269 − 1.47270i −0.381885 − 0.924210i \(-0.624725\pi\)
0.836154 − 0.548494i \(-0.184799\pi\)
\(54\) 4.92338 + 1.66143i 0.669987 + 0.226092i
\(55\) 0.211671 + 0.439539i 0.0285417 + 0.0592674i
\(56\) −2.64499 + 0.0634495i −0.353452 + 0.00847880i
\(57\) −1.73330 + 2.63282i −0.229581 + 0.348725i
\(58\) 1.09129 − 2.78056i 0.143293 − 0.365105i
\(59\) −0.890591 + 11.8841i −0.115945 + 1.54718i 0.571293 + 0.820747i \(0.306442\pi\)
−0.687238 + 0.726433i \(0.741177\pi\)
\(60\) −0.117701 − 0.479762i −0.0151951 − 0.0619370i
\(61\) −1.31867 − 4.27502i −0.168838 − 0.547361i 0.831124 − 0.556088i \(-0.187698\pi\)
−0.999962 + 0.00872727i \(0.997222\pi\)
\(62\) −1.14891 + 1.44069i −0.145912 + 0.182968i
\(63\) 7.82543 + 1.32762i 0.985912 + 0.167265i
\(64\) −0.623490 − 0.781831i −0.0779362 − 0.0977289i
\(65\) 0.946378 − 1.38808i 0.117384 − 0.172170i
\(66\) −1.88447 − 2.28616i −0.231962 − 0.281407i
\(67\) 4.90509 − 8.49586i 0.599252 − 1.03793i −0.393680 − 0.919247i \(-0.628798\pi\)
0.992932 − 0.118687i \(-0.0378684\pi\)
\(68\) −2.62280 − 4.54282i −0.318061 − 0.550898i
\(69\) 2.46038 + 0.919971i 0.296195 + 0.110751i
\(70\) −0.292445 − 0.695606i −0.0349539 − 0.0831408i
\(71\) −2.21716 + 0.506053i −0.263129 + 0.0600574i −0.352050 − 0.935981i \(-0.614515\pi\)
0.0889210 + 0.996039i \(0.471658\pi\)
\(72\) 1.38842 + 2.65938i 0.163627 + 0.313410i
\(73\) −1.26593 − 8.39889i −0.148166 − 0.983016i −0.932108 − 0.362182i \(-0.882032\pi\)
0.783942 − 0.620834i \(-0.213206\pi\)
\(74\) 1.51273 + 10.0363i 0.175851 + 1.16669i
\(75\) −6.96046 + 4.91239i −0.803725 + 0.567234i
\(76\) −1.77427 + 0.404965i −0.203522 + 0.0464526i
\(77\) −3.39040 − 2.99777i −0.386372 − 0.341628i
\(78\) −3.57330 + 9.55645i −0.404596 + 1.08206i
\(79\) −0.786077 − 1.36152i −0.0884405 − 0.153183i 0.818412 − 0.574632i \(-0.194855\pi\)
−0.906852 + 0.421449i \(0.861522\pi\)
\(80\) 0.142602 − 0.246994i 0.0159434 − 0.0276148i
\(81\) −2.56599 − 8.62645i −0.285110 − 0.958495i
\(82\) −0.866392 + 1.27076i −0.0956770 + 0.140332i
\(83\) 5.94517 + 7.45501i 0.652567 + 0.818294i 0.992511 − 0.122152i \(-0.0389796\pi\)
−0.339944 + 0.940446i \(0.610408\pi\)
\(84\) 2.73141 + 3.67959i 0.298021 + 0.401477i
\(85\) 0.932784 − 1.16967i 0.101175 − 0.126869i
\(86\) 2.74810 + 8.90912i 0.296335 + 0.960695i
\(87\) −5.02471 + 1.23272i −0.538705 + 0.132161i
\(88\) 0.127828 − 1.70575i 0.0136265 − 0.181833i
\(89\) 3.01624 − 7.68526i 0.319721 − 0.814636i −0.677169 − 0.735828i \(-0.736793\pi\)
0.996890 − 0.0788083i \(-0.0251115\pi\)
\(90\) −0.561367 + 0.645711i −0.0591733 + 0.0680639i
\(91\) −3.10257 + 15.2729i −0.325238 + 1.60103i
\(92\) 0.658008 + 1.36637i 0.0686021 + 0.142454i
\(93\) 3.18618 + 0.186986i 0.330392 + 0.0193896i
\(94\) 1.70300 − 5.52099i 0.175651 − 0.569446i
\(95\) −0.292387 − 0.428853i −0.0299983 − 0.0439994i
\(96\) −0.483685 + 1.66314i −0.0493659 + 0.169744i
\(97\) 7.01206i 0.711967i 0.934492 + 0.355983i \(0.115854\pi\)
−0.934492 + 0.355983i \(0.884146\pi\)
\(98\) 5.00178 + 4.89716i 0.505256 + 0.494688i
\(99\) −1.35311 + 4.94998i −0.135993 + 0.497492i
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.p.a.59.11 yes 216
3.2 odd 2 inner 294.2.p.a.59.7 yes 216
49.5 odd 42 inner 294.2.p.a.5.7 ✓ 216
147.5 even 42 inner 294.2.p.a.5.11 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.p.a.5.7 ✓ 216 49.5 odd 42 inner
294.2.p.a.5.11 yes 216 147.5 even 42 inner
294.2.p.a.59.7 yes 216 3.2 odd 2 inner
294.2.p.a.59.11 yes 216 1.1 even 1 trivial