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Properties

Label 50.12.d.b.11.9
Level $50$
Weight $12$
Character 50.11
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
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] = [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 11.9
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.9

$q$-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+(-25.8885 + 18.8091i) q^{2} +(42.9958 - 132.328i) q^{3} +(316.433 - 973.882i) q^{4} +(5249.33 - 4612.23i) q^{5} +(1375.87 + 4234.48i) q^{6} -79523.0 q^{7} +(10125.9 + 31164.2i) q^{8} +(127653. + 92745.3i) q^{9} +(-49145.4 + 218139. i) q^{10} +(533014. - 387257. i) q^{11} +(-115266. - 83745.8i) q^{12} +(1.10870e6 + 805516. i) q^{13} +(2.05874e6 - 1.49576e6i) q^{14} +(-384626. - 892938. i) q^{15} +(-848316. - 616338. i) q^{16} +(1.03081e6 + 3.17249e6i) q^{17} -5.04921e6 q^{18} +(167180. + 514529. i) q^{19} +(-2.83071e6 - 6.57169e6i) q^{20} +(-3.41916e6 + 1.05231e7i) q^{21} +(-6.51498e6 + 2.00511e7i) q^{22} +(-3.80431e7 + 2.76399e7i) q^{23} +4.55926e6 q^{24} +(6.28275e6 - 4.84222e7i) q^{25} -4.38536e7 q^{26} +(3.77018e7 - 2.73920e7i) q^{27} +(-2.51637e7 + 7.74460e7i) q^{28} +(6.26270e7 - 1.92746e8i) q^{29} +(2.67528e7 + 1.58824e7i) q^{30} +(-1.86002e7 - 5.72457e7i) q^{31} +3.35544e7 q^{32} +(-2.83275e7 - 8.71829e7i) q^{33} +(-8.63579e7 - 6.27427e7i) q^{34} +(-4.17442e8 + 3.66779e8i) q^{35} +(1.30717e8 - 9.49712e7i) q^{36} +(5.76631e8 + 4.18947e8i) q^{37} +(-1.40059e7 - 1.01759e7i) q^{38} +(1.54262e8 - 1.12078e8i) q^{39} +(1.96891e8 + 1.16888e8i) q^{40} +(-9.08243e8 - 6.59878e8i) q^{41} +(-1.09413e8 - 3.36739e8i) q^{42} +5.55671e6 q^{43} +(-2.08479e8 - 6.41634e8i) q^{44} +(1.09786e9 - 1.01915e8i) q^{45} +(4.64998e8 - 1.43112e9i) q^{46} +(5.83089e8 - 1.79456e9i) q^{47} +(-1.18033e8 + 8.57557e7i) q^{48} +4.34659e9 q^{49} +(7.48129e8 + 1.37175e9i) q^{50} +4.64129e8 q^{51} +(1.13531e9 - 8.24849e8i) q^{52} +(7.95176e8 - 2.44730e9i) q^{53} +(-4.60826e8 + 1.41828e9i) q^{54} +(1.01184e9 - 4.49123e9i) q^{55} +(-8.05240e8 - 2.47827e9i) q^{56} +7.52744e7 q^{57} +(2.00407e9 + 6.16788e9i) q^{58} +(-1.14570e9 - 8.32400e8i) q^{59} +(-9.91325e8 + 9.20254e7i) q^{60} +(5.05822e9 - 3.67501e9i) q^{61} +(1.55827e9 + 1.13215e9i) q^{62} +(-1.01514e10 - 7.37539e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(9.53515e9 - 8.85155e8i) q^{65} +(2.37319e9 + 1.72422e9i) q^{66} +(-3.19314e9 - 9.82747e9i) q^{67} +3.41582e9 q^{68} +(2.02183e9 + 6.22256e9i) q^{69} +(3.90819e9 - 1.73471e10i) q^{70} +(2.67734e7 - 8.24002e7i) q^{71} +(-1.59774e9 + 4.91733e9i) q^{72} +(1.62815e10 - 1.18292e10i) q^{73} -2.28082e10 q^{74} +(-6.13747e9 - 2.91334e9i) q^{75} +5.53991e8 q^{76} +(-4.23869e10 + 3.07959e10i) q^{77} +(-1.88552e9 + 5.80305e9i) q^{78} +(-3.15554e9 + 9.71175e9i) q^{79} +(-7.29578e9 + 6.77272e8i) q^{80} +(6.63384e9 + 2.04168e10i) q^{81} +3.59248e10 q^{82} +(-6.02264e9 - 1.85358e10i) q^{83} +(9.16631e9 + 6.65972e9i) q^{84} +(2.00433e10 + 1.18991e10i) q^{85} +(-1.43855e8 + 1.04517e8i) q^{86} +(-2.28129e10 - 1.65746e10i) q^{87} +(1.74658e10 + 1.26897e10i) q^{88} +(4.32486e10 - 3.14220e10i) q^{89} +(-2.65049e10 + 2.32881e10i) q^{90} +(-8.81671e10 - 6.40571e10i) q^{91} +(1.48799e10 + 4.57957e10i) q^{92} -8.37491e9 q^{93} +(1.86589e10 + 5.74261e10i) q^{94} +(3.25071e9 + 1.92985e9i) q^{95} +(1.44270e9 - 4.44018e9i) q^{96} +(3.72865e9 - 1.14756e10i) q^{97} +(-1.12527e11 + 8.17555e10i) q^{98} +1.03957e11 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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −25.8885 + 18.8091i −0.572061 + 0.415627i
\(3\) 42.9958 − 132.328i 0.102155 − 0.314401i −0.886897 − 0.461967i \(-0.847144\pi\)
0.989052 + 0.147566i \(0.0471439\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 5249.33 − 4612.23i 0.751223 − 0.660049i
\(6\) 1375.87 + 4234.48i 0.0722345 + 0.222315i
\(7\) −79523.0 −1.78836 −0.894178 − 0.447713i \(-0.852239\pi\)
−0.894178 + 0.447713i \(0.852239\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 127653. + 92745.3i 0.720605 + 0.523550i
\(10\) −49145.4 + 218139.i −0.155411 + 0.689817i
\(11\) 533014. − 387257.i 0.997881 − 0.725003i 0.0362485 − 0.999343i \(-0.488459\pi\)
0.961633 + 0.274340i \(0.0884592\pi\)
\(12\) −115266. − 83745.8i −0.133723 − 0.0971552i
\(13\) 1.10870e6 + 805516.i 0.828180 + 0.601708i 0.919044 − 0.394155i \(-0.128963\pi\)
−0.0908636 + 0.995863i \(0.528963\pi\)
\(14\) 2.05874e6 − 1.49576e6i 1.02305 − 0.743289i
\(15\) −384626. − 892938.i −0.130779 − 0.303612i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) 1.03081e6 + 3.17249e6i 0.176079 + 0.541916i 0.999681 − 0.0252525i \(-0.00803897\pi\)
−0.823602 + 0.567168i \(0.808039\pi\)
\(18\) −5.04921e6 −0.629832
\(19\) 167180. + 514529.i 0.0154896 + 0.0476721i 0.958503 − 0.285084i \(-0.0920215\pi\)
−0.943013 + 0.332756i \(0.892021\pi\)
\(20\) −2.83071e6 − 6.57169e6i −0.197802 − 0.459211i
\(21\) −3.41916e6 + 1.05231e7i −0.182689 + 0.562260i
\(22\) −6.51498e6 + 2.00511e7i −0.269519 + 0.829493i
\(23\) −3.80431e7 + 2.76399e7i −1.23246 + 0.895435i −0.997072 − 0.0764662i \(-0.975636\pi\)
−0.235389 + 0.971901i \(0.575636\pi\)
\(24\) 4.55926e6 0.116878
\(25\) 6.28275e6 − 4.84222e7i 0.128671 − 0.991687i
\(26\) −4.38536e7 −0.723856
\(27\) 3.77018e7 − 2.73920e7i 0.505663 − 0.367386i
\(28\) −2.51637e7 + 7.74460e7i −0.276316 + 0.850413i
\(29\) 6.26270e7 − 1.92746e8i 0.566987 − 1.74501i −0.0949870 − 0.995479i \(-0.530281\pi\)
0.661974 − 0.749527i \(-0.269719\pi\)
\(30\) 2.67528e7 + 1.58824e7i 0.201003 + 0.119330i
\(31\) −1.86002e7 − 5.72457e7i −0.116689 − 0.359131i 0.875607 − 0.483025i \(-0.160462\pi\)
−0.992296 + 0.123893i \(0.960462\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −2.83275e7 − 8.71829e7i −0.126003 − 0.387797i
\(34\) −8.63579e7 − 6.27427e7i −0.325963 − 0.236826i
\(35\) −4.17442e8 + 3.66779e8i −1.34345 + 1.18040i
\(36\) 1.30717e8 − 9.49712e7i 0.360302 − 0.261775i
\(37\) 5.76631e8 + 4.18947e8i 1.36706 + 0.993230i 0.997960 + 0.0638411i \(0.0203351\pi\)
0.369103 + 0.929388i \(0.379665\pi\)
\(38\) −1.40059e7 − 1.01759e7i −0.0286748 − 0.0208335i
\(39\) 1.54262e8 − 1.12078e8i 0.273780 − 0.198913i
\(40\) 1.96891e8 + 1.16888e8i 0.304015 + 0.180485i
\(41\) −9.08243e8 − 6.59878e8i −1.22431 − 0.889513i −0.227859 − 0.973694i \(-0.573172\pi\)
−0.996450 + 0.0841816i \(0.973172\pi\)
\(42\) −1.09413e8 − 3.36739e8i −0.129181 − 0.397578i
\(43\) 5.55671e6 0.00576423 0.00288211 − 0.999996i \(-0.499083\pi\)
0.00288211 + 0.999996i \(0.499083\pi\)
\(44\) −2.08479e8 − 6.41634e8i −0.190578 − 0.586540i
\(45\) 1.09786e9 − 1.01915e8i 0.886903 − 0.0823319i
\(46\) 4.64998e8 − 1.43112e9i 0.332876 − 1.02449i
\(47\) 5.83089e8 − 1.79456e9i 0.370849 − 1.14136i −0.575388 − 0.817881i \(-0.695149\pi\)
0.946237 − 0.323475i \(-0.104851\pi\)
\(48\) −1.18033e8 + 8.57557e7i −0.0668613 + 0.0485776i
\(49\) 4.34659e9 2.19821
\(50\) 7.48129e8 + 1.37175e9i 0.338564 + 0.620785i
\(51\) 4.64129e8 0.188366
\(52\) 1.13531e9 − 8.24849e8i 0.414090 − 0.300854i
\(53\) 7.95176e8 − 2.44730e9i 0.261184 − 0.803841i −0.731364 − 0.681987i \(-0.761116\pi\)
0.992548 − 0.121854i \(-0.0388839\pi\)
\(54\) −4.60826e8 + 1.41828e9i −0.136575 + 0.420334i
\(55\) 1.01184e9 − 4.49123e9i 0.271093 − 1.20329i
\(56\) −8.05240e8 − 2.47827e9i −0.195385 − 0.601333i
\(57\) 7.52744e7 0.0165705
\(58\) 2.00407e9 + 6.16788e9i 0.400920 + 1.23391i
\(59\) −1.14570e9 − 8.32400e8i −0.208634 − 0.151581i 0.478561 − 0.878054i \(-0.341158\pi\)
−0.687195 + 0.726473i \(0.741158\pi\)
\(60\) −9.91325e8 + 9.20254e7i −0.164583 + 0.0152783i
\(61\) 5.05822e9 − 3.67501e9i 0.766803 − 0.557115i −0.134187 − 0.990956i \(-0.542842\pi\)
0.900989 + 0.433841i \(0.142842\pi\)
\(62\) 1.55827e9 + 1.13215e9i 0.216018 + 0.156946i
\(63\) −1.01514e10 − 7.37539e9i −1.28870 − 0.936293i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) 9.53515e9 − 8.85155e8i 1.01930 − 0.0946228i
\(66\) 2.37319e9 + 1.72422e9i 0.233260 + 0.169474i
\(67\) −3.19314e9 − 9.82747e9i −0.288939 − 0.889263i −0.985190 − 0.171465i \(-0.945150\pi\)
0.696251 − 0.717798i \(-0.254850\pi\)
\(68\) 3.41582e9 0.284902
\(69\) 2.02183e9 + 6.22256e9i 0.155623 + 0.478960i
\(70\) 3.90819e9 − 1.73471e10i 0.277931 − 1.23364i
\(71\) 2.67734e7 − 8.24002e7i 0.00176110 − 0.00542010i −0.950172 − 0.311726i \(-0.899093\pi\)
0.951933 + 0.306306i \(0.0990930\pi\)
\(72\) −1.59774e9 + 4.91733e9i −0.0973143 + 0.299503i
\(73\) 1.62815e10 − 1.18292e10i 0.919218 − 0.667851i −0.0241112 − 0.999709i \(-0.507676\pi\)
0.943329 + 0.331858i \(0.107676\pi\)
\(74\) −2.28082e10 −1.19486
\(75\) −6.13747e9 − 2.91334e9i −0.298643 − 0.141760i
\(76\) 5.53991e8 0.0250627
\(77\) −4.23869e10 + 3.07959e10i −1.78457 + 1.29656i
\(78\) −1.88552e9 + 5.80305e9i −0.0739455 + 0.227581i
\(79\) −3.15554e9 + 9.71175e9i −0.115378 + 0.355098i −0.992026 − 0.126035i \(-0.959775\pi\)
0.876647 + 0.481133i \(0.159775\pi\)
\(80\) −7.29578e9 + 6.77272e8i −0.248930 + 0.0231083i
\(81\) 6.63384e9 + 2.04168e10i 0.211396 + 0.650611i
\(82\) 3.59248e10 1.07009
\(83\) −6.02264e9 − 1.85358e10i −0.167825 − 0.516513i 0.831408 − 0.555662i \(-0.187535\pi\)
−0.999233 + 0.0391493i \(0.987535\pi\)
\(84\) 9.16631e9 + 6.65972e9i 0.239144 + 0.173748i
\(85\) 2.00433e10 + 1.18991e10i 0.489965 + 0.290878i
\(86\) −1.43855e8 + 1.04517e8i −0.00329749 + 0.00239577i
\(87\) −2.28129e10 − 1.65746e10i −0.490711 − 0.356522i
\(88\) 1.74658e10 + 1.26897e10i 0.352804 + 0.256327i
\(89\) 4.32486e10 − 3.14220e10i 0.820970 − 0.596469i −0.0960205 − 0.995379i \(-0.530611\pi\)
0.916990 + 0.398910i \(0.130611\pi\)
\(90\) −2.65049e10 + 2.32881e10i −0.473144 + 0.415720i
\(91\) −8.81671e10 − 6.40571e10i −1.48108 − 1.07607i
\(92\) 1.48799e10 + 4.57957e10i 0.235379 + 0.724422i
\(93\) −8.37491e9 −0.124831
\(94\) 1.86589e10 + 5.74261e10i 0.262230 + 0.807060i
\(95\) 3.25071e9 + 1.92985e9i 0.0431021 + 0.0255885i
\(96\) 1.44270e9 − 4.44018e9i 0.0180586 − 0.0555787i
\(97\) 3.72865e9 − 1.14756e10i 0.0440867 − 0.135685i −0.926590 − 0.376072i \(-0.877274\pi\)
0.970677 + 0.240387i \(0.0772745\pi\)
\(98\) −1.12527e11 + 8.17555e10i −1.25751 + 0.913637i
\(99\) 1.03957e11 1.09865
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.11.9 ✓ 56
25.16 even 5 inner 50.12.d.b.41.9 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.9 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.9 yes 56 25.16 even 5 inner