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Properties

Label 294.2.m.b.37.2
Level $294$
Weight $2$
Character 294.37
Analytic conductor $2.348$
Analytic rank $0$
Dimension $24$
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 = [24,2] 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: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 37.2
Character \(\chi\) \(=\) 294.37
Dual form 294.2.m.b.151.2

$q$-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+(0.826239 - 0.563320i) q^{2} +(-0.733052 + 0.680173i) q^{3} +(0.365341 - 0.930874i) q^{4} +(1.32658 + 0.409195i) q^{5} +(-0.222521 + 0.974928i) q^{6} +(0.662652 - 2.56142i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(0.0747301 - 0.997204i) q^{9} +(1.32658 - 0.409195i) q^{10} +(-0.0332300 - 0.443423i) q^{11} +(0.365341 + 0.930874i) q^{12} +(4.28893 + 2.06544i) q^{13} +(-0.895392 - 2.48963i) q^{14} +(-1.25077 + 0.602340i) q^{15} +(-0.733052 - 0.680173i) q^{16} +(3.16037 + 0.476348i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-0.185455 + 0.321218i) q^{19} +(0.865562 - 1.08538i) q^{20} +(1.25645 + 2.32837i) q^{21} +(-0.277245 - 0.347654i) q^{22} +(1.06554 - 0.160605i) q^{23} +(0.826239 + 0.563320i) q^{24} +(-2.53883 - 1.73094i) q^{25} +(4.70719 - 0.709494i) q^{26} +(0.623490 + 0.781831i) q^{27} +(-2.14227 - 1.55264i) q^{28} +(-5.07096 + 6.35878i) q^{29} +(-0.694126 + 1.20226i) q^{30} +(-3.49647 - 6.05606i) q^{31} +(-0.988831 - 0.149042i) q^{32} +(0.325964 + 0.302450i) q^{33} +(2.87955 - 1.38672i) q^{34} +(1.92718 - 3.12677i) q^{35} +(-0.900969 - 0.433884i) q^{36} +(-2.63457 - 6.71279i) q^{37} +(0.0277182 + 0.369874i) q^{38} +(-4.54887 + 1.40314i) q^{39} +(0.103744 - 1.38437i) q^{40} +(1.57356 + 6.89423i) q^{41} +(2.34975 + 1.21601i) q^{42} +(-1.62663 + 7.12672i) q^{43} +(-0.424911 - 0.131068i) q^{44} +(0.507186 - 1.29229i) q^{45} +(0.789921 - 0.732940i) q^{46} +(-7.91100 + 5.39363i) q^{47} +1.00000 q^{48} +(-6.12178 - 3.39467i) q^{49} -3.07275 q^{50} +(-2.64071 + 1.80041i) q^{51} +(3.48959 - 3.23787i) q^{52} +(-3.32320 + 8.46739i) q^{53} +(0.955573 + 0.294755i) q^{54} +(0.137364 - 0.601833i) q^{55} +(-2.64466 - 0.0760677i) q^{56} +(-0.0825354 - 0.361611i) q^{57} +(-0.607794 + 8.11044i) q^{58} +(5.05775 - 1.56011i) q^{59} +(0.103744 + 1.38437i) q^{60} +(4.36117 + 11.1121i) q^{61} +(-6.30042 - 3.03412i) q^{62} +(-2.50474 - 0.852215i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(4.84443 + 4.49498i) q^{65} +(0.439700 + 0.0662741i) q^{66} +(-5.63206 - 9.75501i) q^{67} +(1.59803 - 2.76787i) q^{68} +(-0.671859 + 0.842485i) q^{69} +(-0.169062 - 3.66908i) q^{70} +(-1.77085 - 2.22058i) q^{71} +(-0.988831 + 0.149042i) q^{72} +(1.42932 + 0.974494i) q^{73} +(-5.95823 - 4.06226i) q^{74} +(3.03843 - 0.457970i) q^{75} +(0.231259 + 0.289990i) q^{76} +(-1.15781 - 0.208719i) q^{77} +(-2.96803 + 3.72180i) q^{78} +(3.04058 - 5.26644i) q^{79} +(-0.694126 - 1.20226i) q^{80} +(-0.988831 - 0.149042i) q^{81} +(5.18380 + 4.80986i) q^{82} +(-2.21046 + 1.06450i) q^{83} +(2.62646 - 0.318948i) q^{84} +(3.99755 + 1.92512i) q^{85} +(2.67064 + 6.80468i) q^{86} +(-0.607794 - 8.11044i) q^{87} +(-0.424911 + 0.131068i) q^{88} +(-0.300925 + 4.01556i) q^{89} +(-0.308915 - 1.35345i) q^{90} +(8.13254 - 9.61711i) q^{91} +(0.239784 - 1.05056i) q^{92} +(6.68226 + 2.06121i) q^{93} +(-3.49804 + 8.91285i) q^{94} +(-0.377462 + 0.350233i) q^{95} +(0.826239 - 0.563320i) q^{96} +4.66404 q^{97} +(-6.97034 + 0.643719i) q^{98} -0.444667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 4 q^{8} + 2 q^{9} + q^{10} + 8 q^{11} + 2 q^{12} - 2 q^{13} + 5 q^{15} + 2 q^{16} + 9 q^{17} - 12 q^{18} - 4 q^{19} + 5 q^{20} + 14 q^{21} + 5 q^{22}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(e\left(\frac{16}{21}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.826239 − 0.563320i 0.584239 − 0.398327i
\(3\) −0.733052 + 0.680173i −0.423228 + 0.392698i
\(4\) 0.365341 − 0.930874i 0.182671 − 0.465437i
\(5\) 1.32658 + 0.409195i 0.593263 + 0.182997i 0.576825 − 0.816867i \(-0.304291\pi\)
0.0164377 + 0.999865i \(0.494767\pi\)
\(6\) −0.222521 + 0.974928i −0.0908438 + 0.398013i
\(7\) 0.662652 − 2.56142i 0.250459 − 0.968127i
\(8\) −0.222521 − 0.974928i −0.0786730 − 0.344689i
\(9\) 0.0747301 − 0.997204i 0.0249100 − 0.332401i
\(10\) 1.32658 − 0.409195i 0.419500 − 0.129399i
\(11\) −0.0332300 − 0.443423i −0.0100192 − 0.133697i 0.989951 − 0.141411i \(-0.0451638\pi\)
−0.999970 + 0.00771341i \(0.997545\pi\)
\(12\) 0.365341 + 0.930874i 0.105465 + 0.268720i
\(13\) 4.28893 + 2.06544i 1.18954 + 0.572850i 0.920676 − 0.390329i \(-0.127639\pi\)
0.268861 + 0.963179i \(0.413353\pi\)
\(14\) −0.895392 − 2.48963i −0.239304 − 0.665382i
\(15\) −1.25077 + 0.602340i −0.322948 + 0.155524i
\(16\) −0.733052 − 0.680173i −0.183263 − 0.170043i
\(17\) 3.16037 + 0.476348i 0.766501 + 0.115531i 0.520643 − 0.853774i \(-0.325692\pi\)
0.245858 + 0.969306i \(0.420930\pi\)
\(18\) −0.500000 − 0.866025i −0.117851 − 0.204124i
\(19\) −0.185455 + 0.321218i −0.0425464 + 0.0736925i −0.886514 − 0.462701i \(-0.846880\pi\)
0.843968 + 0.536393i \(0.180214\pi\)
\(20\) 0.865562 − 1.08538i 0.193545 − 0.242698i
\(21\) 1.25645 + 2.32837i 0.274180 + 0.508093i
\(22\) −0.277245 − 0.347654i −0.0591089 − 0.0741202i
\(23\) 1.06554 − 0.160605i 0.222181 − 0.0334884i −0.0370081 − 0.999315i \(-0.511783\pi\)
0.259189 + 0.965827i \(0.416545\pi\)
\(24\) 0.826239 + 0.563320i 0.168655 + 0.114987i
\(25\) −2.53883 − 1.73094i −0.507766 − 0.346189i
\(26\) 4.70719 − 0.709494i 0.923155 − 0.139143i
\(27\) 0.623490 + 0.781831i 0.119991 + 0.150464i
\(28\) −2.14227 − 1.55264i −0.404851 − 0.293421i
\(29\) −5.07096 + 6.35878i −0.941653 + 1.18080i 0.0417071 + 0.999130i \(0.486720\pi\)
−0.983360 + 0.181666i \(0.941851\pi\)
\(30\) −0.694126 + 1.20226i −0.126730 + 0.219502i
\(31\) −3.49647 − 6.05606i −0.627985 − 1.08770i −0.987956 − 0.154738i \(-0.950547\pi\)
0.359971 − 0.932963i \(-0.382787\pi\)
\(32\) −0.988831 − 0.149042i −0.174802 − 0.0263472i
\(33\) 0.325964 + 0.302450i 0.0567430 + 0.0526498i
\(34\) 2.87955 − 1.38672i 0.493839 − 0.237820i
\(35\) 1.92718 − 3.12677i 0.325753 − 0.528521i
\(36\) −0.900969 − 0.433884i −0.150161 − 0.0723140i
\(37\) −2.63457 − 6.71279i −0.433121 − 1.10358i −0.966265 − 0.257550i \(-0.917085\pi\)
0.533143 − 0.846025i \(-0.321011\pi\)
\(38\) 0.0277182 + 0.369874i 0.00449649 + 0.0600014i
\(39\) −4.54887 + 1.40314i −0.728402 + 0.224682i
\(40\) 0.103744 − 1.38437i 0.0164034 − 0.218888i
\(41\) 1.57356 + 6.89423i 0.245749 + 1.07670i 0.935688 + 0.352830i \(0.114780\pi\)
−0.689938 + 0.723868i \(0.742362\pi\)
\(42\) 2.34975 + 1.21601i 0.362574 + 0.187634i
\(43\) −1.62663 + 7.12672i −0.248058 + 1.08681i 0.685411 + 0.728157i \(0.259623\pi\)
−0.933469 + 0.358658i \(0.883234\pi\)
\(44\) −0.424911 − 0.131068i −0.0640578 − 0.0197592i
\(45\) 0.507186 − 1.29229i 0.0756068 − 0.192643i
\(46\) 0.789921 − 0.732940i 0.116468 − 0.108066i
\(47\) −7.91100 + 5.39363i −1.15394 + 0.786742i −0.980192 − 0.198049i \(-0.936540\pi\)
−0.173746 + 0.984791i \(0.555587\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.12178 − 3.39467i −0.874541 − 0.484952i
\(50\) −3.07275 −0.434553
\(51\) −2.64071 + 1.80041i −0.369774 + 0.252107i
\(52\) 3.48959 − 3.23787i 0.483919 − 0.449011i
\(53\) −3.32320 + 8.46739i −0.456477 + 1.16308i 0.498627 + 0.866816i \(0.333838\pi\)
−0.955105 + 0.296269i \(0.904258\pi\)
\(54\) 0.955573 + 0.294755i 0.130037 + 0.0401111i
\(55\) 0.137364 − 0.601833i 0.0185222 − 0.0811511i
\(56\) −2.64466 − 0.0760677i −0.353407 − 0.0101650i
\(57\) −0.0825354 − 0.361611i −0.0109321 − 0.0478966i
\(58\) −0.607794 + 8.11044i −0.0798072 + 1.06495i
\(59\) 5.05775 − 1.56011i 0.658463 − 0.203109i 0.0525236 − 0.998620i \(-0.483274\pi\)
0.605939 + 0.795511i \(0.292797\pi\)
\(60\) 0.103744 + 1.38437i 0.0133933 + 0.178722i
\(61\) 4.36117 + 11.1121i 0.558391 + 1.42276i 0.879584 + 0.475743i \(0.157821\pi\)
−0.321193 + 0.947014i \(0.604084\pi\)
\(62\) −6.30042 − 3.03412i −0.800154 − 0.385334i
\(63\) −2.50474 − 0.852215i −0.315568 − 0.107369i
\(64\) −0.900969 + 0.433884i −0.112621 + 0.0542355i
\(65\) 4.84443 + 4.49498i 0.600878 + 0.557533i
\(66\) 0.439700 + 0.0662741i 0.0541233 + 0.00815778i
\(67\) −5.63206 − 9.75501i −0.688065 − 1.19176i −0.972463 − 0.233057i \(-0.925127\pi\)
0.284398 − 0.958706i \(-0.408206\pi\)
\(68\) 1.59803 − 2.76787i 0.193790 − 0.335654i
\(69\) −0.671859 + 0.842485i −0.0808824 + 0.101423i
\(70\) −0.169062 − 3.66908i −0.0202068 − 0.438539i
\(71\) −1.77085 − 2.22058i −0.210162 − 0.263534i 0.665567 − 0.746338i \(-0.268190\pi\)
−0.875729 + 0.482804i \(0.839618\pi\)
\(72\) −0.988831 + 0.149042i −0.116535 + 0.0175648i
\(73\) 1.42932 + 0.974494i 0.167289 + 0.114056i 0.644071 − 0.764966i \(-0.277244\pi\)
−0.476781 + 0.879022i \(0.658197\pi\)
\(74\) −5.95823 − 4.06226i −0.692631 − 0.472228i
\(75\) 3.03843 − 0.457970i 0.350848 − 0.0528818i
\(76\) 0.231259 + 0.289990i 0.0265272 + 0.0332641i
\(77\) −1.15781 − 0.208719i −0.131945 − 0.0237858i
\(78\) −2.96803 + 3.72180i −0.336064 + 0.421411i
\(79\) 3.04058 − 5.26644i 0.342092 − 0.592521i −0.642729 − 0.766094i \(-0.722198\pi\)
0.984821 + 0.173573i \(0.0555312\pi\)
\(80\) −0.694126 − 1.20226i −0.0776057 − 0.134417i
\(81\) −0.988831 − 0.149042i −0.109870 − 0.0165603i
\(82\) 5.18380 + 4.80986i 0.572455 + 0.531160i
\(83\) −2.21046 + 1.06450i −0.242629 + 0.116844i −0.551247 − 0.834342i \(-0.685848\pi\)
0.308618 + 0.951186i \(0.400134\pi\)
\(84\) 2.62646 − 0.318948i 0.286570 − 0.0348000i
\(85\) 3.99755 + 1.92512i 0.433595 + 0.208808i
\(86\) 2.67064 + 6.80468i 0.287983 + 0.733768i
\(87\) −0.607794 − 8.11044i −0.0651623 − 0.869531i
\(88\) −0.424911 + 0.131068i −0.0452957 + 0.0139719i
\(89\) −0.300925 + 4.01556i −0.0318979 + 0.425648i 0.958280 + 0.285833i \(0.0922701\pi\)
−0.990178 + 0.139816i \(0.955349\pi\)
\(90\) −0.308915 − 1.35345i −0.0325625 − 0.142666i
\(91\) 8.13254 − 9.61711i 0.852522 − 1.00815i
\(92\) 0.239784 − 1.05056i 0.0249992 − 0.109529i
\(93\) 6.68226 + 2.06121i 0.692918 + 0.213737i
\(94\) −3.49804 + 8.91285i −0.360795 + 0.919290i
\(95\) −0.377462 + 0.350233i −0.0387267 + 0.0359332i
\(96\) 0.826239 − 0.563320i 0.0843276 − 0.0574936i
\(97\) 4.66404 0.473562 0.236781 − 0.971563i \(-0.423908\pi\)
0.236781 + 0.971563i \(0.423908\pi\)
\(98\) −6.97034 + 0.643719i −0.704111 + 0.0650254i
\(99\) −0.444667 −0.0446907
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.b.37.2 ✓ 24
3.2 odd 2 882.2.z.a.37.1 24
49.4 even 21 inner 294.2.m.b.151.2 yes 24
147.53 odd 42 882.2.z.a.739.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.m.b.37.2 ✓ 24 1.1 even 1 trivial
294.2.m.b.151.2 yes 24 49.4 even 21 inner
882.2.z.a.37.1 24 3.2 odd 2
882.2.z.a.739.1 24 147.53 odd 42