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Properties

Label 294.2.p.a.5.15
Level $294$
Weight $2$
Character 294.5
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 5.15
Character \(\chi\) \(=\) 294.5
Dual form 294.2.p.a.59.15

$q$-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} +(-0.133072 - 1.72693i) q^{3} +(-0.955573 + 0.294755i) q^{4} +(0.273625 - 3.65128i) q^{5} +(1.68781 - 0.388971i) q^{6} +(-1.75726 + 1.97789i) q^{7} +(-0.433884 - 0.900969i) q^{8} +(-2.96458 + 0.459611i) q^{9} +(3.65128 - 0.273625i) q^{10} +(-4.00468 - 1.57172i) q^{11} +(0.636182 + 1.61099i) q^{12} +(3.06793 - 2.44659i) q^{13} +(-2.21770 - 1.44284i) q^{14} +(-6.34192 + 0.0133490i) q^{15} +(0.826239 - 0.563320i) q^{16} +(-0.681534 + 0.632371i) q^{17} +(-0.896326 - 2.86297i) q^{18} +(-3.09569 - 1.78730i) q^{19} +(0.814764 + 3.56971i) q^{20} +(3.64952 + 2.77146i) q^{21} +(0.957300 - 4.19421i) q^{22} +(2.13217 - 2.29793i) q^{23} +(-1.49817 + 0.869181i) q^{24} +(-8.31280 - 1.25295i) q^{25} +(2.87652 + 2.66902i) q^{26} +(1.18822 + 5.05847i) q^{27} +(1.09620 - 2.40798i) q^{28} +(8.21256 - 1.87446i) q^{29} +(-0.958413 - 6.26909i) q^{30} +(6.51983 - 3.76423i) q^{31} +(0.680173 + 0.733052i) q^{32} +(-2.18135 + 7.12496i) q^{33} +(-0.726886 - 0.579672i) q^{34} +(6.74099 + 6.95744i) q^{35} +(2.69740 - 1.31302i) q^{36} +(-3.70685 - 1.14341i) q^{37} +(1.30595 - 3.32750i) q^{38} +(-4.63335 - 4.97253i) q^{39} +(-3.40841 + 1.33770i) q^{40} +(4.98847 - 2.40232i) q^{41} +(-2.19658 + 4.02182i) q^{42} +(7.93440 + 3.82101i) q^{43} +(4.29004 + 0.321494i) q^{44} +(0.866982 + 10.9503i) q^{45} +(2.59005 + 1.76586i) q^{46} +(-6.41280 + 0.966575i) q^{47} +(-1.08276 - 1.35190i) q^{48} +(-0.824086 - 6.95132i) q^{49} -8.40669i q^{50} +(1.18275 + 1.09281i) q^{51} +(-2.21048 + 3.24219i) q^{52} +(1.35622 + 4.39675i) q^{53} +(-4.82488 + 1.92887i) q^{54} +(-6.83458 + 14.1921i) q^{55} +(2.54446 + 0.725061i) q^{56} +(-2.67459 + 5.58389i) q^{57} +(3.07755 + 7.84146i) q^{58} +(0.717051 + 9.56838i) q^{59} +(6.05623 - 1.88207i) q^{60} +(0.314798 - 1.02055i) q^{61} +(4.69392 + 5.88598i) q^{62} +(4.30048 - 6.67127i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(-8.09372 - 11.8713i) q^{65} +(-7.37050 - 1.09506i) q^{66} +(-3.54872 - 6.14656i) q^{67} +(0.464861 - 0.805162i) q^{68} +(-4.25210 - 3.37632i) q^{69} +(-5.87504 + 7.70265i) q^{70} +(-7.23685 - 1.65176i) q^{71} +(1.70038 + 2.47158i) q^{72} +(2.51154 - 16.6630i) q^{73} +(0.578164 - 3.83587i) q^{74} +(-1.05757 + 14.5224i) q^{75} +(3.48497 + 0.795423i) q^{76} +(10.1460 - 5.15890i) q^{77} +(4.22643 - 5.32272i) q^{78} +(-6.63480 + 11.4918i) q^{79} +(-1.83076 - 3.17097i) q^{80} +(8.57752 - 2.72511i) q^{81} +(3.11898 + 4.57471i) q^{82} +(2.95637 - 3.70717i) q^{83} +(-4.30428 - 1.57262i) q^{84} +(2.12248 + 2.66150i) q^{85} +(-2.59577 + 8.41527i) q^{86} +(-4.32993 - 13.9331i) q^{87} +(0.321494 + 4.29004i) q^{88} +(3.06462 + 7.80853i) q^{89} +(-10.6988 + 2.48935i) q^{90} +(-0.552058 + 10.3673i) q^{91} +(-1.36011 + 2.82431i) q^{92} +(-7.36817 - 10.7584i) q^{93} +(-1.91156 - 6.19712i) q^{94} +(-7.37298 + 10.8142i) q^{95} +(1.17542 - 1.27216i) q^{96} -1.43166i q^{97} +(6.75086 - 1.85092i) q^{98} +(12.5946 + 2.81891i) 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 \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{29}{42}\right)\)

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
\(3\) −0.133072 − 1.72693i −0.0768289 − 0.997044i
\(4\) −0.955573 + 0.294755i −0.477786 + 0.147378i
\(5\) 0.273625 − 3.65128i 0.122369 − 1.63290i −0.511375 − 0.859358i \(-0.670864\pi\)
0.633744 − 0.773543i \(-0.281517\pi\)
\(6\) 1.68781 − 0.388971i 0.689045 − 0.158797i
\(7\) −1.75726 + 1.97789i −0.664181 + 0.747572i
\(8\) −0.433884 − 0.900969i −0.153401 − 0.318541i
\(9\) −2.96458 + 0.459611i −0.988195 + 0.153204i
\(10\) 3.65128 − 0.273625i 1.15464 − 0.0865279i
\(11\) −4.00468 − 1.57172i −1.20746 − 0.473892i −0.325645 − 0.945492i \(-0.605581\pi\)
−0.881813 + 0.471600i \(0.843677\pi\)
\(12\) 0.636182 + 1.61099i 0.183650 + 0.465051i
\(13\) 3.06793 − 2.44659i 0.850891 − 0.678563i −0.0976491 − 0.995221i \(-0.531132\pi\)
0.948540 + 0.316658i \(0.102561\pi\)
\(14\) −2.21770 − 1.44284i −0.592706 − 0.385616i
\(15\) −6.34192 + 0.0133490i −1.63748 + 0.00344669i
\(16\) 0.826239 − 0.563320i 0.206560 − 0.140830i
\(17\) −0.681534 + 0.632371i −0.165296 + 0.153373i −0.758484 − 0.651692i \(-0.774060\pi\)
0.593188 + 0.805064i \(0.297869\pi\)
\(18\) −0.896326 − 2.86297i −0.211266 − 0.674809i
\(19\) −3.09569 − 1.78730i −0.710200 − 0.410034i 0.100935 − 0.994893i \(-0.467817\pi\)
−0.811135 + 0.584859i \(0.801150\pi\)
\(20\) 0.814764 + 3.56971i 0.182187 + 0.798212i
\(21\) 3.64952 + 2.77146i 0.796390 + 0.604783i
\(22\) 0.957300 − 4.19421i 0.204097 − 0.894208i
\(23\) 2.13217 − 2.29793i 0.444587 − 0.479151i −0.470504 − 0.882398i \(-0.655928\pi\)
0.915091 + 0.403247i \(0.132118\pi\)
\(24\) −1.49817 + 0.869181i −0.305813 + 0.177421i
\(25\) −8.31280 − 1.25295i −1.66256 − 0.250591i
\(26\) 2.87652 + 2.66902i 0.564131 + 0.523437i
\(27\) 1.18822 + 5.05847i 0.228673 + 0.973503i
\(28\) 1.09620 − 2.40798i 0.207161 − 0.455065i
\(29\) 8.21256 − 1.87446i 1.52503 − 0.348079i 0.623860 − 0.781536i \(-0.285564\pi\)
0.901174 + 0.433457i \(0.142706\pi\)
\(30\) −0.958413 − 6.26909i −0.174982 − 1.14457i
\(31\) 6.51983 − 3.76423i 1.17100 − 0.676075i 0.217082 − 0.976153i \(-0.430346\pi\)
0.953915 + 0.300078i \(0.0970127\pi\)
\(32\) 0.680173 + 0.733052i 0.120239 + 0.129586i
\(33\) −2.18135 + 7.12496i −0.379724 + 1.24030i
\(34\) −0.726886 − 0.579672i −0.124660 − 0.0994129i
\(35\) 6.74099 + 6.95744i 1.13943 + 1.17602i
\(36\) 2.69740 − 1.31302i 0.449567 − 0.218836i
\(37\) −3.70685 − 1.14341i −0.609403 − 0.187976i −0.0253326 − 0.999679i \(-0.508064\pi\)
−0.584070 + 0.811703i \(0.698541\pi\)
\(38\) 1.30595 − 3.32750i 0.211853 − 0.539792i
\(39\) −4.63335 − 4.97253i −0.741930 − 0.796242i
\(40\) −3.40841 + 1.33770i −0.538917 + 0.211509i
\(41\) 4.98847 − 2.40232i 0.779068 − 0.375179i −0.00170139 − 0.999999i \(-0.500542\pi\)
0.780770 + 0.624819i \(0.214827\pi\)
\(42\) −2.19658 + 4.02182i −0.338939 + 0.620581i
\(43\) 7.93440 + 3.82101i 1.20999 + 0.582698i 0.926506 − 0.376280i \(-0.122797\pi\)
0.283479 + 0.958978i \(0.408511\pi\)
\(44\) 4.29004 + 0.321494i 0.646748 + 0.0484670i
\(45\) 0.866982 + 10.9503i 0.129242 + 1.63237i
\(46\) 2.59005 + 1.76586i 0.381881 + 0.260362i
\(47\) −6.41280 + 0.966575i −0.935404 + 0.140989i −0.599019 − 0.800735i \(-0.704442\pi\)
−0.336385 + 0.941725i \(0.609204\pi\)
\(48\) −1.08276 − 1.35190i −0.156284 − 0.195129i
\(49\) −0.824086 − 6.95132i −0.117727 − 0.993046i
\(50\) − 8.40669i − 1.18889i
\(51\) 1.18275 + 1.09281i 0.165619 + 0.153024i
\(52\) −2.21048 + 3.24219i −0.306539 + 0.449610i
\(53\) 1.35622 + 4.39675i 0.186291 + 0.603939i 0.999751 + 0.0223189i \(0.00710493\pi\)
−0.813460 + 0.581621i \(0.802419\pi\)
\(54\) −4.82488 + 1.92887i −0.656583 + 0.262486i
\(55\) −6.83458 + 14.1921i −0.921574 + 1.91367i
\(56\) 2.54446 + 0.725061i 0.340018 + 0.0968904i
\(57\) −2.67459 + 5.58389i −0.354259 + 0.739604i
\(58\) 3.07755 + 7.84146i 0.404102 + 1.02963i
\(59\) 0.717051 + 9.56838i 0.0933520 + 1.24570i 0.825441 + 0.564488i \(0.190926\pi\)
−0.732089 + 0.681209i \(0.761455\pi\)
\(60\) 6.05623 − 1.88207i 0.781856 − 0.242974i
\(61\) 0.314798 − 1.02055i 0.0403057 − 0.130668i −0.933184 − 0.359399i \(-0.882982\pi\)
0.973490 + 0.228731i \(0.0734578\pi\)
\(62\) 4.69392 + 5.88598i 0.596128 + 0.747521i
\(63\) 4.30048 − 6.67127i 0.541810 − 0.840501i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) −8.09372 − 11.8713i −1.00390 − 1.47245i
\(66\) −7.37050 − 1.09506i −0.907246 − 0.134793i
\(67\) −3.54872 − 6.14656i −0.433545 − 0.750922i 0.563631 − 0.826027i \(-0.309404\pi\)
−0.997176 + 0.0751053i \(0.976071\pi\)
\(68\) 0.464861 − 0.805162i 0.0563726 − 0.0976403i
\(69\) −4.25210 − 3.37632i −0.511892 − 0.406461i
\(70\) −5.87504 + 7.70265i −0.702201 + 0.920643i
\(71\) −7.23685 − 1.65176i −0.858856 − 0.196028i −0.229656 − 0.973272i \(-0.573760\pi\)
−0.629200 + 0.777243i \(0.716617\pi\)
\(72\) 1.70038 + 2.47158i 0.200392 + 0.291279i
\(73\) 2.51154 − 16.6630i 0.293954 − 1.95026i −0.0170684 − 0.999854i \(-0.505433\pi\)
0.311022 − 0.950403i \(-0.399329\pi\)
\(74\) 0.578164 − 3.83587i 0.0672102 − 0.445910i
\(75\) −1.05757 + 14.5224i −0.122117 + 1.67690i
\(76\) 3.48497 + 0.795423i 0.399754 + 0.0912412i
\(77\) 10.1460 − 5.15890i 1.15624 − 0.587911i
\(78\) 4.22643 − 5.32272i 0.478549 − 0.602679i
\(79\) −6.63480 + 11.4918i −0.746474 + 1.29293i 0.203029 + 0.979173i \(0.434921\pi\)
−0.949503 + 0.313758i \(0.898412\pi\)
\(80\) −1.83076 − 3.17097i −0.204685 − 0.354525i
\(81\) 8.57752 − 2.72511i 0.953057 − 0.302790i
\(82\) 3.11898 + 4.57471i 0.344434 + 0.505192i
\(83\) 2.95637 − 3.70717i 0.324503 − 0.406914i −0.592643 − 0.805465i \(-0.701915\pi\)
0.917146 + 0.398551i \(0.130487\pi\)
\(84\) −4.30428 − 1.57262i −0.469636 − 0.171587i
\(85\) 2.12248 + 2.66150i 0.230215 + 0.288680i
\(86\) −2.59577 + 8.41527i −0.279909 + 0.907442i
\(87\) −4.32993 − 13.9331i −0.464217 − 1.49378i
\(88\) 0.321494 + 4.29004i 0.0342714 + 0.457320i
\(89\) 3.06462 + 7.80853i 0.324849 + 0.827702i 0.996243 + 0.0866016i \(0.0276007\pi\)
−0.671394 + 0.741101i \(0.734304\pi\)
\(90\) −10.6988 + 2.48935i −1.12775 + 0.262401i
\(91\) −0.552058 + 10.3673i −0.0578714 + 1.08679i
\(92\) −1.36011 + 2.82431i −0.141802 + 0.294454i
\(93\) −7.36817 − 10.7584i −0.764043 − 1.11559i
\(94\) −1.91156 − 6.19712i −0.197162 − 0.639184i
\(95\) −7.37298 + 10.8142i −0.756452 + 1.10951i
\(96\) 1.17542 − 1.27216i 0.119966 − 0.129839i
\(97\) − 1.43166i − 0.145363i −0.997355 − 0.0726817i \(-0.976844\pi\)
0.997355 − 0.0726817i \(-0.0231557\pi\)
\(98\) 6.75086 − 1.85092i 0.681940 − 0.186971i
\(99\) 12.5946 + 2.81891i 1.26580 + 0.283311i
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.5.15 yes 216
3.2 odd 2 inner 294.2.p.a.5.8 ✓ 216
49.10 odd 42 inner 294.2.p.a.59.8 yes 216
147.59 even 42 inner 294.2.p.a.59.15 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.p.a.5.8 ✓ 216 3.2 odd 2 inner
294.2.p.a.5.15 yes 216 1.1 even 1 trivial
294.2.p.a.59.8 yes 216 49.10 odd 42 inner
294.2.p.a.59.15 yes 216 147.59 even 42 inner