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Properties

Label 294.2.p.a.59.8
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 59.8
Character \(\chi\) \(=\) 294.59
Dual form 294.2.p.a.5.8

$q$-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} +(1.65617 + 0.507046i) q^{3} +(-0.955573 - 0.294755i) q^{4} +(-0.273625 - 3.65128i) q^{5} +(-0.748222 + 1.56210i) q^{6} +(-1.75726 - 1.97789i) q^{7} +(0.433884 - 0.900969i) q^{8} +(2.48581 + 1.67951i) q^{9} +(3.65128 + 0.273625i) q^{10} +(4.00468 - 1.57172i) q^{11} +(-1.43314 - 0.972685i) q^{12} +(3.06793 + 2.44659i) q^{13} +(2.21770 - 1.44284i) q^{14} +(1.39819 - 6.18588i) q^{15} +(0.826239 + 0.563320i) q^{16} +(0.681534 + 0.632371i) q^{17} +(-2.03124 + 2.20773i) q^{18} +(-3.09569 + 1.78730i) q^{19} +(-0.814764 + 3.56971i) q^{20} +(-1.90744 - 4.16673i) q^{21} +(0.957300 + 4.19421i) q^{22} +(-2.13217 - 2.29793i) q^{23} +(1.17542 - 1.27216i) q^{24} +(-8.31280 + 1.25295i) q^{25} +(-2.87652 + 2.66902i) q^{26} +(3.26534 + 4.04198i) q^{27} +(1.09620 + 2.40798i) q^{28} +(-8.21256 - 1.87446i) q^{29} +(5.90840 + 2.30454i) q^{30} +(6.51983 + 3.76423i) q^{31} +(-0.680173 + 0.733052i) q^{32} +(7.42938 - 0.572483i) q^{33} +(-0.726886 + 0.579672i) q^{34} +(-6.74099 + 6.95744i) q^{35} +(-1.88033 - 2.33760i) q^{36} +(-3.70685 + 1.14341i) q^{37} +(-1.30595 - 3.32750i) q^{38} +(3.84048 + 5.60756i) q^{39} +(-3.40841 - 1.33770i) q^{40} +(-4.98847 - 2.40232i) q^{41} +(4.40448 - 1.26512i) q^{42} +(7.93440 - 3.82101i) q^{43} +(-4.29004 + 0.321494i) q^{44} +(5.45218 - 9.53593i) 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} +(0.808096 + 1.39288i) q^{51} +(-2.21048 - 3.24219i) q^{52} +(-1.35622 + 4.39675i) q^{53} +(-4.48350 + 2.62644i) q^{54} +(-6.83458 - 14.1921i) q^{55} +(-2.54446 + 0.725061i) q^{56} +(-6.03324 + 1.39041i) q^{57} +(3.07755 - 7.84146i) q^{58} +(-0.717051 + 9.56838i) q^{59} +(-3.15940 + 5.49894i) q^{60} +(0.314798 + 1.02055i) q^{61} +(-4.69392 + 5.88598i) q^{62} +(-1.04632 - 7.86799i) q^{63} +(-0.623490 - 0.781831i) q^{64} +(8.09372 - 11.8713i) q^{65} +(-0.541202 + 7.43172i) q^{66} +(-3.54872 + 6.14656i) q^{67} +(-0.464861 - 0.805162i) q^{68} +(-2.36608 - 4.88687i) q^{69} +(-5.87504 - 7.70265i) q^{70} +(7.23685 - 1.65176i) q^{71} +(2.59174 - 1.51092i) q^{72} +(2.51154 + 16.6630i) q^{73} +(-0.578164 - 3.83587i) q^{74} +(-14.4027 - 2.13987i) q^{75} +(3.48497 - 0.795423i) q^{76} +(-10.1460 - 5.15890i) q^{77} +(-6.11732 + 2.96182i) q^{78} +(-6.63480 - 11.4918i) q^{79} +(1.83076 - 3.17097i) q^{80} +(3.35849 + 8.34988i) q^{81} +(3.11898 - 4.57471i) q^{82} +(-2.95637 - 3.70717i) q^{83} +(0.594532 + 4.54385i) q^{84} +(2.12248 - 2.66150i) q^{85} +(2.59577 + 8.41527i) q^{86} +(-12.6510 - 7.26858i) q^{87} +(0.321494 - 4.29004i) q^{88} +(-3.06462 + 7.80853i) q^{89} +(8.61682 + 6.81254i) q^{90} +(-0.552058 - 10.3673i) q^{91} +(1.36011 + 2.82431i) q^{92} +(8.88933 + 9.54006i) q^{93} +(-1.91156 + 6.19712i) q^{94} +(7.37298 + 10.8142i) q^{95} +(-1.49817 + 0.869181i) 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 \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.65617 + 0.507046i 0.956191 + 0.292743i
\(4\) −0.955573 − 0.294755i −0.477786 − 0.147378i
\(5\) −0.273625 − 3.65128i −0.122369 − 1.63290i −0.633744 − 0.773543i \(-0.718483\pi\)
0.511375 − 0.859358i \(-0.329136\pi\)
\(6\) −0.748222 + 1.56210i −0.305460 + 0.637726i
\(7\) −1.75726 − 1.97789i −0.664181 − 0.747572i
\(8\) 0.433884 − 0.900969i 0.153401 − 0.318541i
\(9\) 2.48581 + 1.67951i 0.828603 + 0.559837i
\(10\) 3.65128 + 0.273625i 1.15464 + 0.0865279i
\(11\) 4.00468 − 1.57172i 1.20746 − 0.473892i 0.325645 − 0.945492i \(-0.394419\pi\)
0.881813 + 0.471600i \(0.156323\pi\)
\(12\) −1.43314 − 0.972685i −0.413711 − 0.280790i
\(13\) 3.06793 + 2.44659i 0.850891 + 0.678563i 0.948540 − 0.316658i \(-0.102561\pi\)
−0.0976491 + 0.995221i \(0.531132\pi\)
\(14\) 2.21770 − 1.44284i 0.592706 − 0.385616i
\(15\) 1.39819 − 6.18588i 0.361012 − 1.59719i
\(16\) 0.826239 + 0.563320i 0.206560 + 0.140830i
\(17\) 0.681534 + 0.632371i 0.165296 + 0.153373i 0.758484 − 0.651692i \(-0.225940\pi\)
−0.593188 + 0.805064i \(0.702131\pi\)
\(18\) −2.03124 + 2.20773i −0.478768 + 0.520366i
\(19\) −3.09569 + 1.78730i −0.710200 + 0.410034i −0.811135 − 0.584859i \(-0.801150\pi\)
0.100935 + 0.994893i \(0.467817\pi\)
\(20\) −0.814764 + 3.56971i −0.182187 + 0.798212i
\(21\) −1.90744 − 4.16673i −0.416238 − 0.909256i
\(22\) 0.957300 + 4.19421i 0.204097 + 0.894208i
\(23\) −2.13217 − 2.29793i −0.444587 − 0.479151i 0.470504 − 0.882398i \(-0.344072\pi\)
−0.915091 + 0.403247i \(0.867882\pi\)
\(24\) 1.17542 − 1.27216i 0.239931 − 0.259679i
\(25\) −8.31280 + 1.25295i −1.66256 + 0.250591i
\(26\) −2.87652 + 2.66902i −0.564131 + 0.523437i
\(27\) 3.26534 + 4.04198i 0.628414 + 0.777879i
\(28\) 1.09620 + 2.40798i 0.207161 + 0.455065i
\(29\) −8.21256 − 1.87446i −1.52503 − 0.348079i −0.623860 − 0.781536i \(-0.714436\pi\)
−0.901174 + 0.433457i \(0.857294\pi\)
\(30\) 5.90840 + 2.30454i 1.07872 + 0.420749i
\(31\) 6.51983 + 3.76423i 1.17100 + 0.676075i 0.953915 − 0.300078i \(-0.0970127\pi\)
0.217082 + 0.976153i \(0.430346\pi\)
\(32\) −0.680173 + 0.733052i −0.120239 + 0.129586i
\(33\) 7.42938 − 0.572483i 1.29329 − 0.0996565i
\(34\) −0.726886 + 0.579672i −0.124660 + 0.0994129i
\(35\) −6.74099 + 6.95744i −1.13943 + 1.17602i
\(36\) −1.88033 − 2.33760i −0.313388 − 0.389600i
\(37\) −3.70685 + 1.14341i −0.609403 + 0.187976i −0.584070 − 0.811703i \(-0.698541\pi\)
−0.0253326 + 0.999679i \(0.508064\pi\)
\(38\) −1.30595 − 3.32750i −0.211853 − 0.539792i
\(39\) 3.84048 + 5.60756i 0.614969 + 0.897928i
\(40\) −3.40841 − 1.33770i −0.538917 − 0.211509i
\(41\) −4.98847 − 2.40232i −0.779068 − 0.375179i 0.00170139 − 0.999999i \(-0.499458\pi\)
−0.780770 + 0.624819i \(0.785173\pi\)
\(42\) 4.40448 − 1.26512i 0.679627 − 0.195212i
\(43\) 7.93440 − 3.82101i 1.20999 − 0.582698i 0.283479 − 0.958978i \(-0.408511\pi\)
0.926506 + 0.376280i \(0.122797\pi\)
\(44\) −4.29004 + 0.321494i −0.646748 + 0.0484670i
\(45\) 5.45218 − 9.53593i 0.812763 − 1.42153i
\(46\) 2.59005 − 1.76586i 0.381881 − 0.260362i
\(47\) 6.41280 + 0.966575i 0.935404 + 0.140989i 0.599019 − 0.800735i \(-0.295558\pi\)
0.336385 + 0.941725i \(0.390796\pi\)
\(48\) 1.08276 + 1.35190i 0.156284 + 0.195129i
\(49\) −0.824086 + 6.95132i −0.117727 + 0.993046i
\(50\) − 8.40669i − 1.18889i
\(51\) 0.808096 + 1.39288i 0.113156 + 0.195043i
\(52\) −2.21048 − 3.24219i −0.306539 − 0.449610i
\(53\) −1.35622 + 4.39675i −0.186291 + 0.603939i 0.813460 + 0.581621i \(0.197581\pi\)
−0.999751 + 0.0223189i \(0.992895\pi\)
\(54\) −4.48350 + 2.62644i −0.610128 + 0.357413i
\(55\) −6.83458 − 14.1921i −0.921574 − 1.91367i
\(56\) −2.54446 + 0.725061i −0.340018 + 0.0968904i
\(57\) −6.03324 + 1.39041i −0.799122 + 0.184165i
\(58\) 3.07755 − 7.84146i 0.404102 − 1.02963i
\(59\) −0.717051 + 9.56838i −0.0933520 + 1.24570i 0.732089 + 0.681209i \(0.238545\pi\)
−0.825441 + 0.564488i \(0.809074\pi\)
\(60\) −3.15940 + 5.49894i −0.407876 + 0.709909i
\(61\) 0.314798 + 1.02055i 0.0403057 + 0.130668i 0.973490 − 0.228731i \(-0.0734578\pi\)
−0.933184 + 0.359399i \(0.882982\pi\)
\(62\) −4.69392 + 5.88598i −0.596128 + 0.747521i
\(63\) −1.04632 − 7.86799i −0.131824 − 0.991273i
\(64\) −0.623490 − 0.781831i −0.0779362 − 0.0977289i
\(65\) 8.09372 − 11.8713i 1.00390 − 1.47245i
\(66\) −0.541202 + 7.43172i −0.0666174 + 0.914782i
\(67\) −3.54872 + 6.14656i −0.433545 + 0.750922i −0.997176 − 0.0751053i \(-0.976071\pi\)
0.563631 + 0.826027i \(0.309404\pi\)
\(68\) −0.464861 − 0.805162i −0.0563726 − 0.0976403i
\(69\) −2.36608 − 4.88687i −0.284842 − 0.588310i
\(70\) −5.87504 − 7.70265i −0.702201 − 0.920643i
\(71\) 7.23685 − 1.65176i 0.858856 − 0.196028i 0.229656 − 0.973272i \(-0.426240\pi\)
0.629200 + 0.777243i \(0.283383\pi\)
\(72\) 2.59174 − 1.51092i 0.305439 − 0.178064i
\(73\) 2.51154 + 16.6630i 0.293954 + 1.95026i 0.311022 + 0.950403i \(0.399329\pi\)
−0.0170684 + 0.999854i \(0.505433\pi\)
\(74\) −0.578164 − 3.83587i −0.0672102 − 0.445910i
\(75\) −14.4027 − 2.13987i −1.66308 − 0.247090i
\(76\) 3.48497 − 0.795423i 0.399754 − 0.0912412i
\(77\) −10.1460 − 5.15890i −1.15624 − 0.587911i
\(78\) −6.11732 + 2.96182i −0.692650 + 0.335361i
\(79\) −6.63480 − 11.4918i −0.746474 − 1.29293i −0.949503 − 0.313758i \(-0.898412\pi\)
0.203029 − 0.979173i \(-0.434921\pi\)
\(80\) 1.83076 − 3.17097i 0.204685 − 0.354525i
\(81\) 3.35849 + 8.34988i 0.373165 + 0.927765i
\(82\) 3.11898 − 4.57471i 0.344434 − 0.505192i
\(83\) −2.95637 − 3.70717i −0.324503 − 0.406914i 0.592643 − 0.805465i \(-0.298085\pi\)
−0.917146 + 0.398551i \(0.869513\pi\)
\(84\) 0.594532 + 4.54385i 0.0648688 + 0.495774i
\(85\) 2.12248 − 2.66150i 0.230215 − 0.288680i
\(86\) 2.59577 + 8.41527i 0.279909 + 0.907442i
\(87\) −12.6510 − 7.26858i −1.35633 − 0.779273i
\(88\) 0.321494 − 4.29004i 0.0342714 − 0.457320i
\(89\) −3.06462 + 7.80853i −0.324849 + 0.827702i 0.671394 + 0.741101i \(0.265696\pi\)
−0.996243 + 0.0866016i \(0.972399\pi\)
\(90\) 8.61682 + 6.81254i 0.908292 + 0.718105i
\(91\) −0.552058 − 10.3673i −0.0578714 − 1.08679i
\(92\) 1.36011 + 2.82431i 0.141802 + 0.294454i
\(93\) 8.88933 + 9.54006i 0.921780 + 0.989259i
\(94\) −1.91156 + 6.19712i −0.197162 + 0.639184i
\(95\) 7.37298 + 10.8142i 0.756452 + 1.10951i
\(96\) −1.49817 + 0.869181i −0.152907 + 0.0887104i
\(97\) 1.43166i 0.145363i 0.997355 + 0.0726817i \(0.0231557\pi\)
−0.997355 + 0.0726817i \(0.976844\pi\)
\(98\) −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.59.8 yes 216
3.2 odd 2 inner 294.2.p.a.59.15 yes 216
49.5 odd 42 inner 294.2.p.a.5.15 yes 216
147.5 even 42 inner 294.2.p.a.5.8 ✓ 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.p.a.5.8 ✓ 216 147.5 even 42 inner
294.2.p.a.5.15 yes 216 49.5 odd 42 inner
294.2.p.a.59.8 yes 216 1.1 even 1 trivial
294.2.p.a.59.15 yes 216 3.2 odd 2 inner