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Properties

Label 50.12.d.b.21.6
Level $50$
Weight $12$
Character 50.21
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 21.6
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.6

$q$-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} +(-85.6566 - 62.2332i) q^{3} +(-828.433 - 601.892i) q^{4} +(4809.61 + 5069.10i) q^{5} +(-2741.01 + 1991.46i) q^{6} -45440.5 q^{7} +(-26509.9 + 19260.5i) q^{8} +(-51277.3 - 157815. i) q^{9} +(201832. - 96248.8i) q^{10} +(194871. - 599751. i) q^{11} +(33503.1 + 103112. i) q^{12} +(514494. + 1.58345e6i) q^{13} +(-449340. + 1.38293e6i) q^{14} +(-96509.1 - 733519. i) q^{15} +(324028. + 997255. i) q^{16} +(-3.70522e6 + 2.69200e6i) q^{17} -5.30998e6 q^{18} +(-760366. + 552438. i) q^{19} +(-933394. - 7.09428e6i) q^{20} +(3.89228e6 + 2.82790e6i) q^{21} +(-1.63257e7 - 1.18613e7i) q^{22} +(-8.60937e6 + 2.64969e7i) q^{23} +3.46939e6 q^{24} +(-2.56338e6 + 4.87608e7i) q^{25} +5.32780e7 q^{26} +(-1.12250e7 + 3.45470e7i) q^{27} +(3.76444e7 + 2.73503e7i) q^{28} +(6.67270e7 + 4.84800e7i) q^{29} +(-2.32781e7 - 4.31630e6i) q^{30} +(1.56359e8 - 1.13602e8i) q^{31} +3.35544e7 q^{32} +(-5.40164e7 + 3.92452e7i) q^{33} +(4.52885e7 + 1.39384e8i) q^{34} +(-2.18551e8 - 2.30342e8i) q^{35} +(-5.25080e7 + 1.61603e8i) q^{36} +(2.40887e8 + 7.41374e8i) q^{37} +(9.29389e6 + 2.86037e7i) q^{38} +(5.44733e7 - 1.67651e8i) q^{39} +(-2.25136e8 - 4.17453e7i) q^{40} +(-7.98292e7 - 2.45689e8i) q^{41} +(1.24553e8 - 9.04929e7i) q^{42} +4.68333e8 q^{43} +(-5.22423e8 + 3.79563e8i) q^{44} +(5.53358e8 - 1.01896e9i) q^{45} +(7.21268e8 + 5.24032e8i) q^{46} +(1.62459e9 + 1.18033e9i) q^{47} +(3.43072e7 - 1.05587e8i) q^{48} +8.75103e7 q^{49} +(1.45863e9 + 5.60187e8i) q^{50} +4.84908e8 q^{51} +(5.26842e8 - 1.62145e9i) q^{52} +(5.30640e8 + 3.85533e8i) q^{53} +(9.40397e8 + 6.83238e8i) q^{54} +(3.97745e9 - 1.89675e9i) q^{55} +(1.20462e9 - 8.75208e8i) q^{56} +9.95104e7 q^{57} +(2.13526e9 - 1.55136e9i) q^{58} +(1.52389e9 + 4.69005e9i) q^{59} +(-3.61548e8 + 6.65760e8i) q^{60} +(-6.75755e8 + 2.07976e9i) q^{61} +(-1.91117e9 - 5.88197e9i) q^{62} +(2.33007e9 + 7.17121e9i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(-5.55214e9 + 1.02238e10i) q^{65} +(6.60237e8 + 2.03200e9i) q^{66} +(-3.51314e9 + 2.55244e9i) q^{67} +4.68982e9 q^{68} +(2.38643e9 - 1.73385e9i) q^{69} +(-9.17134e9 + 4.37359e9i) q^{70} +(1.15514e10 + 8.39260e9i) q^{71} +(4.39897e9 + 3.19604e9i) q^{72} +(5.64792e9 - 1.73825e10i) q^{73} +2.49449e10 q^{74} +(3.25411e9 - 4.01716e9i) q^{75} +9.62421e8 q^{76} +(-8.85503e9 + 2.72530e10i) q^{77} +(-4.56361e9 - 3.31566e9i) q^{78} +(-4.88331e9 - 3.54793e9i) q^{79} +(-3.49673e9 + 6.43894e9i) q^{80} +(-2.06698e10 + 1.50175e10i) q^{81} -8.26665e9 q^{82} +(3.13567e10 - 2.27820e10i) q^{83} +(-1.52240e9 - 4.68546e9i) q^{84} +(-3.14667e10 - 5.83464e9i) q^{85} +(4.63113e9 - 1.42532e10i) q^{86} +(-2.69854e9 - 8.30526e9i) q^{87} +(6.38553e9 + 1.96526e10i) q^{88} +(-2.73154e10 + 8.40682e10i) q^{89} +(-2.55390e10 - 2.69168e10i) q^{90} +(-2.33789e10 - 7.19527e10i) q^{91} +(2.30806e10 - 1.67690e10i) q^{92} -2.04630e10 q^{93} +(5.19869e10 - 3.77707e10i) q^{94} +(-6.45743e9 - 1.19736e9i) q^{95} +(-2.87416e9 - 2.08820e9i) q^{96} +(-4.93984e10 - 3.58901e10i) q^{97} +(8.65349e8 - 2.66327e9i) q^{98} -1.04642e11 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
\(3\) −85.6566 − 62.2332i −0.203514 − 0.147861i 0.481360 − 0.876523i \(-0.340143\pi\)
−0.684874 + 0.728661i \(0.740143\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) 4809.61 + 5069.10i 0.688296 + 0.725430i
\(6\) −2741.01 + 1991.46i −0.143906 + 0.104554i
\(7\) −45440.5 −1.02189 −0.510944 − 0.859614i \(-0.670704\pi\)
−0.510944 + 0.859614i \(0.670704\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) −51277.3 − 157815.i −0.289462 − 0.890873i
\(10\) 201832. − 96248.8i 0.638249 − 0.304366i
\(11\) 194871. − 599751.i 0.364827 − 1.12282i −0.585262 − 0.810844i \(-0.699008\pi\)
0.950089 − 0.311979i \(-0.100992\pi\)
\(12\) 33503.1 + 103112.i 0.0388677 + 0.119622i
\(13\) 514494. + 1.58345e6i 0.384319 + 1.18281i 0.936973 + 0.349402i \(0.113615\pi\)
−0.552654 + 0.833411i \(0.686385\pi\)
\(14\) −449340. + 1.38293e6i −0.223291 + 0.687219i
\(15\) −96509.1 − 733519.i −0.0328145 − 0.249407i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) −3.70522e6 + 2.69200e6i −0.632914 + 0.459839i −0.857408 − 0.514637i \(-0.827927\pi\)
0.224495 + 0.974475i \(0.427927\pi\)
\(18\) −5.30998e6 −0.662360
\(19\) −760366. + 552438.i −0.0704495 + 0.0511846i −0.622453 − 0.782657i \(-0.713864\pi\)
0.552003 + 0.833842i \(0.313864\pi\)
\(20\) −933394. − 7.09428e6i −0.0652229 − 0.495728i
\(21\) 3.89228e6 + 2.82790e6i 0.207969 + 0.151098i
\(22\) −1.63257e7 − 1.18613e7i −0.675379 − 0.490692i
\(23\) −8.60937e6 + 2.64969e7i −0.278913 + 0.858405i 0.709245 + 0.704962i \(0.249036\pi\)
−0.988158 + 0.153443i \(0.950964\pi\)
\(24\) 3.46939e6 0.0889388
\(25\) −2.56338e6 + 4.87608e7i −0.0524980 + 0.998621i
\(26\) 5.32780e7 0.879416
\(27\) −1.12250e7 + 3.45470e7i −0.150552 + 0.463350i
\(28\) 3.76444e7 + 2.73503e7i 0.413363 + 0.300326i
\(29\) 6.67270e7 + 4.84800e7i 0.604105 + 0.438908i 0.847334 − 0.531061i \(-0.178207\pi\)
−0.243229 + 0.969969i \(0.578207\pi\)
\(30\) −2.32781e7 − 4.31630e6i −0.174896 − 0.0324298i
\(31\) 1.56359e8 − 1.13602e8i 0.980923 − 0.712682i 0.0230082 − 0.999735i \(-0.492676\pi\)
0.957915 + 0.287053i \(0.0926756\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −5.40164e7 + 3.92452e7i −0.240270 + 0.174566i
\(34\) 4.52885e7 + 1.39384e8i 0.170944 + 0.526112i
\(35\) −2.18551e8 − 2.30342e8i −0.703362 − 0.741309i
\(36\) −5.25080e7 + 1.61603e8i −0.144731 + 0.445436i
\(37\) 2.40887e8 + 7.41374e8i 0.571089 + 1.75763i 0.649125 + 0.760681i \(0.275135\pi\)
−0.0780363 + 0.996951i \(0.524865\pi\)
\(38\) 9.29389e6 + 2.86037e7i 0.0190278 + 0.0585614i
\(39\) 5.44733e7 − 1.67651e8i 0.0966781 − 0.297545i
\(40\) −2.25136e8 − 4.17453e7i −0.347628 − 0.0644582i
\(41\) −7.98292e7 − 2.45689e8i −0.107610 − 0.331188i 0.882725 − 0.469891i \(-0.155707\pi\)
−0.990334 + 0.138703i \(0.955707\pi\)
\(42\) 1.24553e8 − 9.04929e7i 0.147056 − 0.106842i
\(43\) 4.68333e8 0.485823 0.242912 − 0.970048i \(-0.421897\pi\)
0.242912 + 0.970048i \(0.421897\pi\)
\(44\) −5.22423e8 + 3.79563e8i −0.477565 + 0.346971i
\(45\) 5.53358e8 − 1.01896e9i 0.447031 − 0.823169i
\(46\) 7.21268e8 + 5.24032e8i 0.516331 + 0.375137i
\(47\) 1.62459e9 + 1.18033e9i 1.03325 + 0.750700i 0.968957 − 0.247231i \(-0.0795205\pi\)
0.0642938 + 0.997931i \(0.479521\pi\)
\(48\) 3.43072e7 − 1.05587e8i 0.0194338 − 0.0598112i
\(49\) 8.75103e7 0.0442569
\(50\) 1.45863e9 + 5.60187e8i 0.660100 + 0.253511i
\(51\) 4.84908e8 0.196799
\(52\) 5.26842e8 − 1.62145e9i 0.192160 − 0.591406i
\(53\) 5.30640e8 + 3.85533e8i 0.174294 + 0.126632i 0.671512 − 0.740994i \(-0.265645\pi\)
−0.497218 + 0.867626i \(0.665645\pi\)
\(54\) 9.40397e8 + 6.83238e8i 0.278705 + 0.202491i
\(55\) 3.97745e9 − 1.89675e9i 1.06564 − 0.508177i
\(56\) 1.20462e9 − 8.75208e8i 0.292292 − 0.212362i
\(57\) 9.95104e7 0.0219057
\(58\) 2.13526e9 − 1.55136e9i 0.427167 − 0.310355i
\(59\) 1.52389e9 + 4.69005e9i 0.277503 + 0.854066i 0.988546 + 0.150918i \(0.0482228\pi\)
−0.711044 + 0.703148i \(0.751777\pi\)
\(60\) −3.61548e8 + 6.65760e8i −0.0600253 + 0.110531i
\(61\) −6.75755e8 + 2.07976e9i −0.102441 + 0.315282i −0.989121 − 0.147101i \(-0.953006\pi\)
0.886680 + 0.462383i \(0.153006\pi\)
\(62\) −1.91117e9 − 5.88197e9i −0.264938 − 0.815396i
\(63\) 2.33007e9 + 7.17121e9i 0.295798 + 0.910373i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) −5.55214e9 + 1.02238e10i −0.593523 + 1.09292i
\(66\) 6.60237e8 + 2.03200e9i 0.0648946 + 0.199725i
\(67\) −3.51314e9 + 2.55244e9i −0.317895 + 0.230964i −0.735277 − 0.677767i \(-0.762948\pi\)
0.417382 + 0.908731i \(0.362948\pi\)
\(68\) 4.68982e9 0.391162
\(69\) 2.38643e9 − 1.73385e9i 0.183687 − 0.133457i
\(70\) −9.17134e9 + 4.37359e9i −0.652219 + 0.311028i
\(71\) 1.15514e10 + 8.39260e9i 0.759827 + 0.552046i 0.898857 − 0.438242i \(-0.144399\pi\)
−0.139031 + 0.990288i \(0.544399\pi\)
\(72\) 4.39897e9 + 3.19604e9i 0.267930 + 0.194663i
\(73\) 5.64792e9 − 1.73825e10i 0.318869 − 0.981379i −0.655263 − 0.755401i \(-0.727442\pi\)
0.974132 − 0.225978i \(-0.0725577\pi\)
\(74\) 2.49449e10 1.30679
\(75\) 3.25411e9 − 4.01716e9i 0.158342 − 0.195471i
\(76\) 9.62421e8 0.0435402
\(77\) −8.85503e9 + 2.72530e10i −0.372813 + 1.14740i
\(78\) −4.56361e9 − 3.31566e9i −0.178973 − 0.130032i
\(79\) −4.88331e9 − 3.54793e9i −0.178552 − 0.129726i 0.494919 − 0.868939i \(-0.335198\pi\)
−0.673471 + 0.739213i \(0.735198\pi\)
\(80\) −3.49673e9 + 6.43894e9i −0.119308 + 0.219695i
\(81\) −2.06698e10 + 1.50175e10i −0.658671 + 0.478552i
\(82\) −8.26665e9 −0.246237
\(83\) 3.13567e10 − 2.27820e10i 0.873778 − 0.634837i −0.0578203 − 0.998327i \(-0.518415\pi\)
0.931598 + 0.363490i \(0.118415\pi\)
\(84\) −1.52240e9 − 4.68546e9i −0.0397184 − 0.122241i
\(85\) −3.14667e10 − 5.83464e9i −0.769213 − 0.142630i
\(86\) 4.63113e9 − 1.42532e10i 0.106156 − 0.326716i
\(87\) −2.69854e9 − 8.30526e9i −0.0580461 − 0.178648i
\(88\) 6.38553e9 + 1.96526e10i 0.128986 + 0.396978i
\(89\) −2.73154e10 + 8.40682e10i −0.518517 + 1.59583i 0.258274 + 0.966072i \(0.416846\pi\)
−0.776791 + 0.629759i \(0.783154\pi\)
\(90\) −2.55390e10 − 2.69168e10i −0.455900 − 0.480496i
\(91\) −2.33789e10 − 7.19527e10i −0.392731 − 1.20870i
\(92\) 2.30806e10 − 1.67690e10i 0.365101 − 0.265262i
\(93\) −2.04630e10 −0.305010
\(94\) 5.19869e10 − 3.77707e10i 0.730618 − 0.530825i
\(95\) −6.45743e9 − 1.19736e9i −0.0856209 − 0.0158761i
\(96\) −2.87416e9 − 2.08820e9i −0.0359765 − 0.0261385i
\(97\) −4.93984e10 − 3.58901e10i −0.584075 − 0.424355i 0.256116 − 0.966646i \(-0.417557\pi\)
−0.840191 + 0.542291i \(0.817557\pi\)
\(98\) 8.65349e8 − 2.66327e9i 0.00967048 − 0.0297627i
\(99\) −1.04642e11 −1.10590
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.21.6 ✓ 56
25.6 even 5 inner 50.12.d.b.31.6 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.6 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.6 yes 56 25.6 even 5 inner