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Properties

Label 50.12.d.b.11.4
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 11.4
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.4

$q$-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} +(-170.771 + 525.580i) q^{3} +(316.433 - 973.882i) q^{4} +(6904.85 + 1072.90i) q^{5} +(-5464.68 - 16818.6i) q^{6} +1017.46 q^{7} +(10125.9 + 31164.2i) q^{8} +(-103757. - 75383.7i) q^{9} +(-198937. + 102098. i) q^{10} +(425173. - 308906. i) q^{11} +(457815. + 332622. i) q^{12} +(-1.69112e6 - 1.22867e6i) q^{13} +(-26340.5 + 19137.5i) q^{14} +(-1.74305e6 + 3.44583e6i) q^{15} +(-848316. - 616338. i) q^{16} +(3.32834e6 + 1.02436e7i) q^{17} +4.10401e6 q^{18} +(4.27085e6 + 1.31443e7i) q^{19} +(3.22981e6 - 6.38501e6i) q^{20} +(-173753. + 534756. i) q^{21} +(-5.19685e6 + 1.59943e7i) q^{22} +(-2.40545e7 + 1.74766e7i) q^{23} -1.81085e7 q^{24} +(4.65259e7 + 1.48164e7i) q^{25} +6.68910e7 q^{26} +(-2.18610e7 + 1.58829e7i) q^{27} +(321958. - 990885. i) q^{28} +(-2.94173e7 + 9.05372e7i) q^{29} +(-1.96882e7 - 1.21993e8i) q^{30} +(-5.81087e7 - 1.78840e8i) q^{31} +3.35544e7 q^{32} +(8.97475e7 + 2.76215e8i) q^{33} +(-2.78839e8 - 2.02588e8i) q^{34} +(7.02541e6 + 1.09163e6i) q^{35} +(-1.06247e8 + 7.71929e7i) q^{36} +(-5.39528e8 - 3.91990e8i) q^{37} +(-3.57799e8 - 2.59956e8i) q^{38} +(9.34562e8 - 6.78999e8i) q^{39} +(3.64815e7 + 2.26048e8i) q^{40} +(1.09664e9 + 7.96755e8i) q^{41} +(-5.56009e6 - 1.71122e7i) q^{42} -1.08600e9 q^{43} +(-1.66299e8 - 5.11816e8i) q^{44} +(-6.35546e8 - 6.31834e8i) q^{45} +(2.94016e8 - 9.04889e8i) q^{46} +(-2.67752e8 + 8.24055e8i) q^{47} +(4.68803e8 - 3.40605e8i) q^{48} -1.97629e9 q^{49} +(-1.48317e9 + 4.91535e8i) q^{50} -5.95221e9 q^{51} +(-1.73171e9 + 1.25816e9i) q^{52} +(-5.48014e8 + 1.68661e9i) q^{53} +(2.67205e8 - 8.22372e8i) q^{54} +(3.26718e9 - 1.67678e9i) q^{55} +(1.03027e7 + 3.17083e7i) q^{56} -7.63773e9 q^{57} +(-9.41354e8 - 2.89719e9i) q^{58} +(-5.21150e9 - 3.78637e9i) q^{59} +(2.80428e9 + 2.78790e9i) q^{60} +(7.86788e9 - 5.71635e9i) q^{61} +(4.86818e9 + 3.53694e9i) q^{62} +(-1.05568e8 - 7.66998e7i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-1.03587e10 - 1.02982e10i) q^{65} +(-7.51879e9 - 5.46272e9i) q^{66} +(-6.18414e8 - 1.90328e9i) q^{67} +1.10292e10 q^{68} +(-5.07755e9 - 1.56271e10i) q^{69} +(-2.02410e8 + 1.03881e8i) q^{70} +(-8.52765e8 + 2.62454e9i) q^{71} +(1.29865e9 - 3.99682e9i) q^{72} +(-8.61230e8 + 6.25720e8i) q^{73} +2.13406e10 q^{74} +(-1.57325e10 + 2.19229e10i) q^{75} +1.41525e10 q^{76} +(4.32596e8 - 3.14299e8i) q^{77} +(-1.14231e10 + 3.51566e10i) q^{78} +(-4.40179e9 + 1.35473e10i) q^{79} +(-5.19623e9 - 5.16588e9i) q^{80} +(-1.16351e10 - 3.58093e10i) q^{81} -4.33767e10 q^{82} +(1.61044e10 + 4.95644e10i) q^{83} +(4.65808e8 + 3.38429e8i) q^{84} +(1.19914e10 + 7.43014e10i) q^{85} +(2.81148e10 - 2.04266e10i) q^{86} +(-4.25609e10 - 3.09223e10i) q^{87} +(1.39321e10 + 1.01222e10i) q^{88} +(4.91868e10 - 3.57363e10i) q^{89} +(2.83376e10 + 4.40320e9i) q^{90} +(-1.72065e9 - 1.25013e9i) q^{91} +(9.40852e9 + 2.89565e10i) q^{92} +1.03918e11 q^{93} +(-8.56806e9 - 2.63698e10i) q^{94} +(1.53870e10 + 9.53418e10i) q^{95} +(-5.73014e9 + 1.76355e10i) q^{96} +(-1.28446e10 + 3.95317e10i) q^{97} +(5.11633e10 - 3.71723e10i) q^{98} -6.74010e10 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
\(3\) −170.771 + 525.580i −0.405740 + 1.24874i 0.514535 + 0.857469i \(0.327964\pi\)
−0.920276 + 0.391271i \(0.872036\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 6904.85 + 1072.90i 0.988142 + 0.153541i
\(6\) −5464.68 − 16818.6i −0.286902 − 0.882993i
\(7\) 1017.46 0.0228811 0.0114406 − 0.999935i \(-0.496358\pi\)
0.0114406 + 0.999935i \(0.496358\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) −103757. − 75383.7i −0.585710 − 0.425543i
\(10\) −198937. + 102098.i −0.629094 + 0.322864i
\(11\) 425173. − 308906.i 0.795986 − 0.578318i −0.113748 − 0.993510i \(-0.536286\pi\)
0.909734 + 0.415192i \(0.136286\pi\)
\(12\) 457815. + 332622.i 0.531121 + 0.385882i
\(13\) −1.69112e6 − 1.22867e6i −1.26324 − 0.917800i −0.264332 − 0.964432i \(-0.585151\pi\)
−0.998912 + 0.0466315i \(0.985151\pi\)
\(14\) −26340.5 + 19137.5i −0.0130894 + 0.00951002i
\(15\) −1.74305e6 + 3.44583e6i −0.592662 + 1.17164i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) 3.32834e6 + 1.02436e7i 0.568537 + 1.74978i 0.657201 + 0.753716i \(0.271740\pi\)
−0.0886638 + 0.996062i \(0.528260\pi\)
\(18\) 4.10401e6 0.511929
\(19\) 4.27085e6 + 1.31443e7i 0.395703 + 1.21785i 0.928413 + 0.371551i \(0.121174\pi\)
−0.532710 + 0.846298i \(0.678826\pi\)
\(20\) 3.22981e6 − 6.38501e6i 0.225689 − 0.446166i
\(21\) −173753. + 534756.i −0.00928380 + 0.0285726i
\(22\) −5.19685e6 + 1.59943e7i −0.214988 + 0.661667i
\(23\) −2.40545e7 + 1.74766e7i −0.779280 + 0.566180i −0.904763 − 0.425915i \(-0.859952\pi\)
0.125483 + 0.992096i \(0.459952\pi\)
\(24\) −1.81085e7 −0.464217
\(25\) 4.65259e7 + 1.48164e7i 0.952850 + 0.303441i
\(26\) 6.68910e7 1.10412
\(27\) −2.18610e7 + 1.58829e7i −0.293203 + 0.213025i
\(28\) 321958. − 990885.i 0.00353533 − 0.0108806i
\(29\) −2.94173e7 + 9.05372e7i −0.266326 + 0.819668i 0.725059 + 0.688687i \(0.241813\pi\)
−0.991385 + 0.130981i \(0.958187\pi\)
\(30\) −1.96882e7 − 1.21993e8i −0.147924 − 0.916573i
\(31\) −5.81087e7 − 1.78840e8i −0.364546 − 1.12196i −0.950265 − 0.311442i \(-0.899188\pi\)
0.585719 − 0.810514i \(-0.300812\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 8.97475e7 + 2.76215e8i 0.399205 + 1.22863i
\(34\) −2.78839e8 − 2.02588e8i −1.05249 − 0.764681i
\(35\) 7.02541e6 + 1.09163e6i 0.0226098 + 0.00351320i
\(36\) −1.06247e8 + 7.71929e7i −0.292855 + 0.212771i
\(37\) −5.39528e8 − 3.91990e8i −1.27910 − 0.929321i −0.279575 − 0.960124i \(-0.590193\pi\)
−0.999525 + 0.0308032i \(0.990193\pi\)
\(38\) −3.57799e8 − 2.59956e8i −0.732537 − 0.532219i
\(39\) 9.34562e8 − 6.78999e8i 1.65864 − 1.20507i
\(40\) 3.64815e7 + 2.26048e8i 0.0563304 + 0.349037i
\(41\) 1.09664e9 + 7.96755e8i 1.47827 + 1.07402i 0.978110 + 0.208088i \(0.0667242\pi\)
0.500156 + 0.865935i \(0.333276\pi\)
\(42\) −5.56009e6 − 1.71122e7i −0.00656464 − 0.0202039i
\(43\) −1.08600e9 −1.12655 −0.563276 − 0.826269i \(-0.690459\pi\)
−0.563276 + 0.826269i \(0.690459\pi\)
\(44\) −1.66299e8 − 5.11816e8i −0.152020 − 0.467869i
\(45\) −6.35546e8 − 6.31834e8i −0.513426 − 0.510427i
\(46\) 2.94016e8 − 9.04889e8i 0.210476 − 0.647780i
\(47\) −2.67752e8 + 8.24055e8i −0.170292 + 0.524105i −0.999387 − 0.0350026i \(-0.988856\pi\)
0.829095 + 0.559107i \(0.188856\pi\)
\(48\) 4.68803e8 − 3.40605e8i 0.265560 − 0.192941i
\(49\) −1.97629e9 −0.999476
\(50\) −1.48317e9 + 4.91535e8i −0.671207 + 0.222443i
\(51\) −5.95221e9 −2.41570
\(52\) −1.73171e9 + 1.25816e9i −0.631622 + 0.458900i
\(53\) −5.48014e8 + 1.68661e9i −0.180001 + 0.553986i −0.999826 − 0.0186276i \(-0.994070\pi\)
0.819826 + 0.572613i \(0.194070\pi\)
\(54\) 2.67205e8 − 8.22372e8i 0.0791915 − 0.243726i
\(55\) 3.26718e9 − 1.67678e9i 0.875343 − 0.449244i
\(56\) 1.03027e7 + 3.17083e7i 0.00249986 + 0.00769377i
\(57\) −7.63773e9 −1.68133
\(58\) −9.41354e8 − 2.89719e9i −0.188321 − 0.579593i
\(59\) −5.21150e9 − 3.78637e9i −0.949022 − 0.689505i 0.00155306 − 0.999999i \(-0.499506\pi\)
−0.950575 + 0.310494i \(0.899506\pi\)
\(60\) 2.80428e9 + 2.78790e9i 0.465574 + 0.462855i
\(61\) 7.86788e9 − 5.71635e9i 1.19273 − 0.866572i 0.199184 − 0.979962i \(-0.436171\pi\)
0.993551 + 0.113390i \(0.0361709\pi\)
\(62\) 4.86818e9 + 3.53694e9i 0.674858 + 0.490313i
\(63\) −1.05568e8 − 7.66998e7i −0.0134017 − 0.00973691i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −1.03587e10 − 1.02982e10i −1.10734 − 1.10088i
\(66\) −7.51879e9 − 5.46272e9i −0.739020 − 0.536929i
\(67\) −6.18414e8 − 1.90328e9i −0.0559588 − 0.172223i 0.919171 − 0.393859i \(-0.128860\pi\)
−0.975129 + 0.221636i \(0.928860\pi\)
\(68\) 1.10292e10 0.919912
\(69\) −5.07755e9 − 1.56271e10i −0.390827 − 1.20284i
\(70\) −2.02410e8 + 1.03881e8i −0.0143944 + 0.00738749i
\(71\) −8.52765e8 + 2.62454e9i −0.0560930 + 0.172636i −0.975178 − 0.221423i \(-0.928930\pi\)
0.919085 + 0.394060i \(0.128930\pi\)
\(72\) 1.29865e9 − 3.99682e9i 0.0790974 − 0.243437i
\(73\) −8.61230e8 + 6.25720e8i −0.0486232 + 0.0353268i −0.611831 − 0.790988i \(-0.709567\pi\)
0.563208 + 0.826315i \(0.309567\pi\)
\(74\) 2.13406e10 1.11797
\(75\) −1.57325e10 + 2.19229e10i −0.765528 + 1.06674i
\(76\) 1.41525e10 0.640261
\(77\) 4.32596e8 − 3.14299e8i 0.0182131 − 0.0132326i
\(78\) −1.14231e10 + 3.51566e10i −0.447984 + 1.37875i
\(79\) −4.40179e9 + 1.35473e10i −0.160946 + 0.495341i −0.998715 − 0.0506844i \(-0.983860\pi\)
0.837769 + 0.546025i \(0.183860\pi\)
\(80\) −5.19623e9 − 5.16588e9i −0.177294 − 0.176258i
\(81\) −1.16351e10 − 3.58093e10i −0.370769 − 1.14111i
\(82\) −4.33767e10 −1.29205
\(83\) 1.61044e10 + 4.95644e10i 0.448762 + 1.38115i 0.878305 + 0.478101i \(0.158675\pi\)
−0.429542 + 0.903047i \(0.641325\pi\)
\(84\) 4.65808e8 + 3.38429e8i 0.0121527 + 0.00882942i
\(85\) 1.19914e10 + 7.43014e10i 0.293133 + 1.81632i
\(86\) 2.81148e10 − 2.04266e10i 0.644457 − 0.468226i
\(87\) −4.25609e10 − 3.09223e10i −0.915493 − 0.665145i
\(88\) 1.39321e10 + 1.01222e10i 0.281424 + 0.204466i
\(89\) 4.91868e10 − 3.57363e10i 0.933692 − 0.678367i −0.0132023 − 0.999913i \(-0.504203\pi\)
0.946894 + 0.321546i \(0.104203\pi\)
\(90\) 2.83376e10 + 4.40320e9i 0.505859 + 0.0786021i
\(91\) −1.72065e9 − 1.25013e9i −0.0289045 − 0.0210003i
\(92\) 9.40852e9 + 2.89565e10i 0.148829 + 0.458050i
\(93\) 1.03918e11 1.54894
\(94\) −8.56806e9 − 2.63698e10i −0.120415 − 0.370598i
\(95\) 1.53870e10 + 9.53418e10i 0.204021 + 1.26416i
\(96\) −5.73014e9 + 1.76355e10i −0.0717254 + 0.220748i
\(97\) −1.28446e10 + 3.95317e10i −0.151872 + 0.467414i −0.997831 − 0.0658348i \(-0.979029\pi\)
0.845959 + 0.533249i \(0.179029\pi\)
\(98\) 5.11633e10 − 3.71723e10i 0.571762 − 0.415409i
\(99\) −6.74010e10 −0.712316
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.11.4 ✓ 56
25.16 even 5 inner 50.12.d.b.41.4 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.4 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.4 yes 56 25.16 even 5 inner