↑

Properties

Label 50.12.d.b.21.5
Level $50$
Weight $12$
Character 50.21
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,12,Mod(11,50)] 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("50.11"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 21.5
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.5

$q$-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+(9.88854 - 30.4338i) q^{2} +(-255.372 - 185.539i) q^{3} +(-828.433 - 601.892i) q^{4} +(5959.51 + 3648.62i) q^{5} +(-8171.90 + 5937.24i) q^{6} +53413.9 q^{7} +(-26509.9 + 19260.5i) q^{8} +(-23951.2 - 73714.1i) q^{9} +(169972. - 145291. i) q^{10} +(-265138. + 816011. i) q^{11} +(99884.5 + 307413. i) q^{12} +(-413767. - 1.27344e6i) q^{13} +(528186. - 1.62559e6i) q^{14} +(-844931. - 2.03747e6i) q^{15} +(324028. + 997255. i) q^{16} +(-7.78412e6 + 5.65549e6i) q^{17} -2.48024e6 q^{18} +(-1.24230e7 + 9.02583e6i) q^{19} +(-2.74098e6 - 6.60962e6i) q^{20} +(-1.36404e7 - 9.91035e6i) q^{21} +(2.22125e7 + 1.61383e7i) q^{22} +(9.64728e6 - 2.96913e7i) q^{23} +1.03435e7 q^{24} +(2.22033e7 + 4.34879e7i) q^{25} -4.28473e7 q^{26} +(-2.48399e7 + 7.64493e7i) q^{27} +(-4.42499e7 - 3.21494e7i) q^{28} +(1.12503e8 + 8.17380e7i) q^{29} +(-7.03632e7 + 5.56684e6i) q^{30} +(-1.14922e8 + 8.34955e7i) q^{31} +3.35544e7 q^{32} +(2.19110e8 - 1.59193e8i) q^{33} +(9.51446e7 + 2.92825e8i) q^{34} +(3.18321e8 + 1.94887e8i) q^{35} +(-2.45260e7 + 7.54832e7i) q^{36} +(-1.90493e8 - 5.86276e8i) q^{37} +(1.51845e8 + 4.67331e8i) q^{38} +(-1.30609e8 + 4.01972e8i) q^{39} +(-2.28260e8 + 1.80590e7i) q^{40} +(1.78310e8 + 5.48783e8i) q^{41} +(-4.36494e8 + 3.17131e8i) q^{42} -2.84652e8 q^{43} +(7.10800e8 - 5.16426e8i) q^{44} +(1.26217e8 - 5.26688e8i) q^{45} +(-8.08221e8 - 5.87207e8i) q^{46} +(4.86082e8 + 3.53160e8i) q^{47} +(1.02282e8 - 3.14791e8i) q^{48} +8.75722e8 q^{49} +(1.54306e9 - 2.45699e8i) q^{50} +3.03716e9 q^{51} +(-4.23698e8 + 1.30401e9i) q^{52} +(1.98547e9 + 1.44253e9i) q^{53} +(2.08101e9 + 1.51194e9i) q^{54} +(-4.55740e9 + 3.89564e9i) q^{55} +(-1.41600e9 + 1.02878e9i) q^{56} +4.84712e9 q^{57} +(3.60009e9 - 2.61562e9i) q^{58} +(8.07240e8 + 2.48443e9i) q^{59} +(-5.26370e8 + 2.19647e9i) q^{60} +(-3.68373e9 + 1.13373e10i) q^{61} +(1.40468e9 + 4.32315e9i) q^{62} +(-1.27933e9 - 3.93736e9i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(2.18046e9 - 9.09878e9i) q^{65} +(-2.67817e9 - 8.24255e9i) q^{66} +(-3.05271e9 + 2.21793e9i) q^{67} +9.85262e9 q^{68} +(-7.97253e9 + 5.79238e9i) q^{69} +(9.07888e9 - 7.76056e9i) q^{70} +(4.87450e9 + 3.54153e9i) q^{71} +(2.05472e9 + 1.49284e9i) q^{72} +(-9.30596e9 + 2.86408e10i) q^{73} -1.97263e10 q^{74} +(2.39859e9 - 1.52252e10i) q^{75} +1.57242e10 q^{76} +(-1.41621e10 + 4.35864e10i) q^{77} +(1.09420e10 + 7.94983e9i) q^{78} +(1.04787e10 + 7.61323e9i) q^{79} +(-1.70756e9 + 7.12540e9i) q^{80} +(9.41972e9 - 6.84382e9i) q^{81} +1.84648e10 q^{82} +(8.70731e9 - 6.32623e9i) q^{83} +(5.33522e9 + 1.64201e10i) q^{84} +(-6.70242e10 + 5.30267e9i) q^{85} +(-2.81480e9 + 8.66305e9i) q^{86} +(-1.35645e10 - 4.17472e10i) q^{87} +(-8.68804e9 - 2.67391e10i) q^{88} +(2.76812e10 - 8.51939e10i) q^{89} +(-1.47810e10 - 9.04945e9i) q^{90} +(-2.21009e10 - 6.80197e10i) q^{91} +(-2.58631e10 + 1.87906e10i) q^{92} +4.48394e10 q^{93} +(1.55546e10 - 1.13011e10i) q^{94} +(-1.06967e11 + 8.46274e9i) q^{95} +(-8.56886e9 - 6.22564e9i) q^{96} +(-9.48354e10 - 6.89020e10i) q^{97} +(8.65961e9 - 2.66515e10i) q^{98} +6.65019e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.88854 − 30.4338i 0.218508 − 0.672499i
\(3\) −255.372 − 185.539i −0.606745 − 0.440826i 0.241521 − 0.970395i \(-0.422354\pi\)
−0.848267 + 0.529569i \(0.822354\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) 5959.51 + 3648.62i 0.852855 + 0.522148i
\(6\) −8171.90 + 5937.24i −0.429034 + 0.311711i
\(7\) 53413.9 1.20120 0.600600 − 0.799550i \(-0.294928\pi\)
0.600600 + 0.799550i \(0.294928\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) −23951.2 − 73714.1i −0.135205 − 0.416118i
\(10\) 169972. − 145291.i 0.537499 − 0.459450i
\(11\) −265138. + 816011.i −0.496378 + 1.52769i 0.318421 + 0.947949i \(0.396848\pi\)
−0.814799 + 0.579744i \(0.803152\pi\)
\(12\) 99884.5 + 307413.i 0.115878 + 0.356636i
\(13\) −413767. − 1.27344e6i −0.309078 − 0.951243i −0.978124 − 0.208023i \(-0.933297\pi\)
0.669046 − 0.743221i \(-0.266703\pi\)
\(14\) 528186. − 1.62559e6i 0.262472 − 0.807805i
\(15\) −844931. − 2.03747e6i −0.287289 − 0.692772i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) −7.78412e6 + 5.65549e6i −1.32966 + 0.966054i −0.329902 + 0.944015i \(0.607016\pi\)
−0.999757 + 0.0220388i \(0.992984\pi\)
\(18\) −2.48024e6 −0.309382
\(19\) −1.24230e7 + 9.02583e6i −1.15102 + 0.836262i −0.988616 − 0.150463i \(-0.951923\pi\)
−0.162400 + 0.986725i \(0.551923\pi\)
\(20\) −2.74098e6 − 6.60962e6i −0.191532 − 0.461861i
\(21\) −1.36404e7 − 9.91035e6i −0.728822 − 0.529521i
\(22\) 2.22125e7 + 1.61383e7i 0.918909 + 0.667627i
\(23\) 9.64728e6 − 2.96913e7i 0.312537 − 0.961891i −0.664219 − 0.747538i \(-0.731236\pi\)
0.976756 − 0.214353i \(-0.0687643\pi\)
\(24\) 1.03435e7 0.265157
\(25\) 2.22033e7 + 4.34879e7i 0.454724 + 0.890633i
\(26\) −4.28473e7 −0.707246
\(27\) −2.48399e7 + 7.64493e7i −0.333157 + 1.02535i
\(28\) −4.42499e7 − 3.21494e7i −0.485896 − 0.353024i
\(29\) 1.12503e8 + 8.17380e7i 1.01853 + 0.740006i 0.965981 − 0.258613i \(-0.0832653\pi\)
0.0525494 + 0.998618i \(0.483265\pi\)
\(30\) −7.03632e7 + 5.56684e6i −0.528663 + 0.0418255i
\(31\) −1.14922e8 + 8.34955e7i −0.720962 + 0.523810i −0.886692 − 0.462362i \(-0.847002\pi\)
0.165729 + 0.986171i \(0.447002\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 2.19110e8 − 1.59193e8i 0.974622 − 0.708105i
\(34\) 9.51446e7 + 2.92825e8i 0.359129 + 1.10528i
\(35\) 3.18321e8 + 1.94887e8i 1.02445 + 0.627204i
\(36\) −2.45260e7 + 7.54832e7i −0.0676025 + 0.208059i
\(37\) −1.90493e8 − 5.86276e8i −0.451615 − 1.38993i −0.875063 − 0.484009i \(-0.839180\pi\)
0.423448 − 0.905920i \(-0.360820\pi\)
\(38\) 1.51845e8 + 4.67331e8i 0.310879 + 0.956786i
\(39\) −1.30609e8 + 4.01972e8i −0.231802 + 0.713412i
\(40\) −2.28260e8 + 1.80590e7i −0.352452 + 0.0278845i
\(41\) 1.78310e8 + 5.48783e8i 0.240362 + 0.739757i 0.996365 + 0.0851891i \(0.0271494\pi\)
−0.756003 + 0.654568i \(0.772851\pi\)
\(42\) −4.36494e8 + 3.17131e8i −0.515355 + 0.374428i
\(43\) −2.84652e8 −0.295283 −0.147641 − 0.989041i \(-0.547168\pi\)
−0.147641 + 0.989041i \(0.547168\pi\)
\(44\) 7.10800e8 − 5.16426e8i 0.649767 − 0.472083i
\(45\) 1.26217e8 − 5.26688e8i 0.101965 − 0.425485i
\(46\) −8.08221e8 − 5.87207e8i −0.578578 − 0.420362i
\(47\) 4.86082e8 + 3.53160e8i 0.309152 + 0.224612i 0.731532 − 0.681807i \(-0.238805\pi\)
−0.422381 + 0.906419i \(0.638805\pi\)
\(48\) 1.02282e8 − 3.14791e8i 0.0579390 − 0.178318i
\(49\) 8.75722e8 0.442882
\(50\) 1.54306e9 − 2.45699e8i 0.698310 − 0.111191i
\(51\) 3.03716e9 1.23263
\(52\) −4.23698e8 + 1.30401e9i −0.154539 + 0.475622i
\(53\) 1.98547e9 + 1.44253e9i 0.652149 + 0.473814i 0.864003 − 0.503487i \(-0.167950\pi\)
−0.211854 + 0.977301i \(0.567950\pi\)
\(54\) 2.08101e9 + 1.51194e9i 0.616750 + 0.448095i
\(55\) −4.55740e9 + 3.89564e9i −1.22102 + 1.04372i
\(56\) −1.41600e9 + 1.02878e9i −0.343580 + 0.249626i
\(57\) 4.84712e9 1.06702
\(58\) 3.60009e9 − 2.61562e9i 0.720210 − 0.523263i
\(59\) 8.07240e8 + 2.48443e9i 0.147000 + 0.452419i 0.997263 − 0.0739394i \(-0.0235571\pi\)
−0.850263 + 0.526358i \(0.823557\pi\)
\(60\) −5.26370e8 + 2.19647e9i −0.0873895 + 0.364664i
\(61\) −3.68373e9 + 1.13373e10i −0.558436 + 1.71869i 0.128257 + 0.991741i \(0.459062\pi\)
−0.686692 + 0.726948i \(0.740938\pi\)
\(62\) 1.40468e9 + 4.32315e9i 0.194725 + 0.599303i
\(63\) −1.27933e9 − 3.93736e9i −0.162408 − 0.499841i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) 2.18046e9 − 9.09878e9i 0.233091 − 0.972657i
\(66\) −2.67817e9 − 8.24255e9i −0.263236 − 0.810159i
\(67\) −3.05271e9 + 2.21793e9i −0.276232 + 0.200695i −0.717272 − 0.696793i \(-0.754610\pi\)
0.441040 + 0.897487i \(0.354610\pi\)
\(68\) 9.85262e9 0.821775
\(69\) −7.97253e9 + 5.79238e9i −0.613657 + 0.445848i
\(70\) 9.07888e9 − 7.76056e9i 0.645644 − 0.551892i
\(71\) 4.87450e9 + 3.54153e9i 0.320634 + 0.232954i 0.736446 − 0.676497i \(-0.236503\pi\)
−0.415812 + 0.909450i \(0.636503\pi\)
\(72\) 2.05472e9 + 1.49284e9i 0.125148 + 0.0909251i
\(73\) −9.30596e9 + 2.86408e10i −0.525395 + 1.61700i 0.238139 + 0.971231i \(0.423462\pi\)
−0.763534 + 0.645768i \(0.776538\pi\)
\(74\) −1.97263e10 −1.03341
\(75\) 2.39859e9 − 1.52252e10i 0.116713 − 0.740841i
\(76\) 1.57242e10 0.711367
\(77\) −1.41621e10 + 4.35864e10i −0.596249 + 1.83507i
\(78\) 1.09420e10 + 7.94983e9i 0.429118 + 0.311772i
\(79\) 1.04787e10 + 7.61323e9i 0.383141 + 0.278368i 0.762639 − 0.646824i \(-0.223903\pi\)
−0.379498 + 0.925193i \(0.623903\pi\)
\(80\) −1.70756e9 + 7.12540e9i −0.0582613 + 0.243116i
\(81\) 9.41972e9 − 6.84382e9i 0.300172 − 0.218088i
\(82\) 1.84648e10 0.550007
\(83\) 8.70731e9 − 6.32623e9i 0.242636 − 0.176285i −0.459821 − 0.888012i \(-0.652086\pi\)
0.702457 + 0.711726i \(0.252086\pi\)
\(84\) 5.33522e9 + 1.64201e10i 0.139193 + 0.428391i
\(85\) −6.70242e10 + 5.30267e9i −1.63843 + 0.129626i
\(86\) −2.81480e9 + 8.66305e9i −0.0645216 + 0.198577i
\(87\) −1.35645e10 − 4.17472e10i −0.291775 − 0.897990i
\(88\) −8.68804e9 − 2.67391e10i −0.175496 − 0.540121i
\(89\) 2.76812e10 − 8.51939e10i 0.525460 − 1.61720i −0.237944 − 0.971279i \(-0.576474\pi\)
0.763404 − 0.645921i \(-0.223526\pi\)
\(90\) −1.47810e10 − 9.04945e9i −0.263858 − 0.161543i
\(91\) −2.21009e10 − 6.80197e10i −0.371264 − 1.14263i
\(92\) −2.58631e10 + 1.87906e10i −0.409117 + 0.297241i
\(93\) 4.48394e10 0.668350
\(94\) 1.55546e10 − 1.13011e10i 0.218603 − 0.158825i
\(95\) −1.06967e11 + 8.46274e9i −1.41830 + 0.112210i
\(96\) −8.56886e9 − 6.22564e9i −0.107258 − 0.0779278i
\(97\) −9.48354e10 − 6.89020e10i −1.12131 − 0.814680i −0.136903 − 0.990584i \(-0.543715\pi\)
−0.984407 + 0.175905i \(0.943715\pi\)
\(98\) 8.65961e9 − 2.66515e10i 0.0967732 − 0.297837i
\(99\) 6.65019e10 0.702814
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 50.12.d.b.21.5 ✓ 56
25.6 even 5 inner 50.12.d.b.31.5 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.5 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.5 yes 56 25.6 even 5 inner