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Properties

Label 50.12.d.b.11.2
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 11.2
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.2

$q$-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+(-25.8885 + 18.8091i) q^{2} +(-200.084 + 615.795i) q^{3} +(316.433 - 973.882i) q^{4} +(-232.661 + 6983.84i) q^{5} +(-6402.68 - 19705.4i) q^{6} -50541.7 q^{7} +(10125.9 + 31164.2i) q^{8} +(-195855. - 142297. i) q^{9} +(-125337. - 185178. i) q^{10} +(596444. - 433342. i) q^{11} +(536398. + 389716. i) q^{12} +(1.63018e6 + 1.18440e6i) q^{13} +(1.30845e6 - 950646. i) q^{14} +(-4.25406e6 - 1.54062e6i) q^{15} +(-848316. - 616338. i) q^{16} +(-2.52616e6 - 7.77471e6i) q^{17} +7.74688e6 q^{18} +(-2.66164e6 - 8.19169e6i) q^{19} +(6.72781e6 + 2.43650e6i) q^{20} +(1.01126e7 - 3.11233e7i) q^{21} +(-7.29028e6 + 2.24372e7i) q^{22} +(3.29329e7 - 2.39271e7i) q^{23} -2.12168e7 q^{24} +(-4.87199e7 - 3.24973e6i) q^{25} -6.44806e7 q^{26} +(3.40188e7 - 2.47161e7i) q^{27} +(-1.59931e7 + 4.92217e7i) q^{28} +(1.84473e7 - 5.67748e7i) q^{29} +(1.39109e8 - 4.01306e7i) q^{30} +(-4.29868e7 - 1.32300e8i) q^{31} +3.35544e7 q^{32} +(1.47511e8 + 4.53992e8i) q^{33} +(2.11634e8 + 1.53761e8i) q^{34} +(1.17591e7 - 3.52975e8i) q^{35} +(-2.00555e8 + 1.45712e8i) q^{36} +(-1.81033e8 - 1.31528e8i) q^{37} +(2.22985e8 + 1.62008e8i) q^{38} +(-1.05552e9 + 7.66880e8i) q^{39} +(-2.20002e8 + 6.34667e7i) q^{40} +(6.26839e8 + 4.55425e8i) q^{41} +(3.23603e8 + 9.95947e8i) q^{42} +6.73513e8 q^{43} +(-2.33289e8 - 7.17989e8i) q^{44} +(1.03935e9 - 1.33471e9i) q^{45} +(-4.02536e8 + 1.23888e9i) q^{46} +(-4.24760e8 + 1.30728e9i) q^{47} +(5.49272e8 - 3.99069e8i) q^{48} +5.77139e8 q^{49} +(1.32241e9 - 8.32247e8i) q^{50} +5.29307e9 q^{51} +(1.66931e9 - 1.21282e9i) q^{52} +(4.33915e8 - 1.33545e9i) q^{53} +(-4.15809e8 + 1.27973e9i) q^{54} +(2.88762e9 + 4.26629e9i) q^{55} +(-5.11779e8 - 1.57509e9i) q^{56} +5.57695e9 q^{57} +(5.90312e8 + 1.81680e9i) q^{58} +(-7.73864e9 - 5.62245e9i) q^{59} +(-2.84651e9 + 3.65545e9i) q^{60} +(-1.55275e9 + 1.12814e9i) q^{61} +(3.60131e9 + 2.61650e9i) q^{62} +(9.89884e9 + 7.19193e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-8.65093e9 + 1.11094e10i) q^{65} +(-1.23580e10 - 8.97863e9i) q^{66} +(6.42443e9 + 1.97724e10i) q^{67} -8.37101e9 q^{68} +(8.14487e9 + 2.50673e10i) q^{69} +(6.33473e9 + 9.35919e9i) q^{70} +(1.97371e8 - 6.07444e8i) q^{71} +(2.45137e9 - 7.54454e9i) q^{72} +(1.23339e10 - 8.96110e9i) q^{73} +7.16060e9 q^{74} +(1.17492e10 - 2.93512e10i) q^{75} -8.81997e9 q^{76} +(-3.01453e10 + 2.19018e10i) q^{77} +(1.29015e10 - 3.97068e10i) q^{78} +(1.10087e10 - 3.38812e10i) q^{79} +(4.50177e9 - 5.78110e9i) q^{80} +(-4.83891e9 - 1.48926e10i) q^{81} -2.47941e10 q^{82} +(-1.16491e10 - 3.58524e10i) q^{83} +(-2.71105e10 - 1.96969e10i) q^{84} +(5.48851e10 - 1.58334e10i) q^{85} +(-1.74363e10 + 1.26682e10i) q^{86} +(3.12707e10 + 2.27195e10i) q^{87} +(1.95443e10 + 1.41997e10i) q^{88} +(-1.99289e10 + 1.44792e10i) q^{89} +(-1.80239e9 + 5.41029e10i) q^{90} +(-8.23923e10 - 5.98615e10i) q^{91} +(-1.28811e10 - 3.96441e10i) q^{92} +9.00704e10 q^{93} +(-1.35923e10 - 4.18328e10i) q^{94} +(5.78287e10 - 1.66826e10i) q^{95} +(-6.71370e9 + 2.06626e10i) q^{96} +(-1.32603e10 + 4.08110e10i) q^{97} +(-1.49413e10 + 1.08555e10i) q^{98} -1.78480e11 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
\(3\) −200.084 + 615.795i −0.475385 + 1.46308i 0.370054 + 0.929010i \(0.379339\pi\)
−0.845438 + 0.534073i \(0.820661\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) −232.661 + 6983.84i −0.0332957 + 0.999446i
\(6\) −6402.68 − 19705.4i −0.336148 − 1.03456i
\(7\) −50541.7 −1.13661 −0.568304 − 0.822819i \(-0.692400\pi\)
−0.568304 + 0.822819i \(0.692400\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) −195855. − 142297.i −1.10561 − 0.803270i
\(10\) −125337. − 185178.i −0.396349 − 0.585583i
\(11\) 596444. − 433342.i 1.11663 − 0.811280i 0.132936 − 0.991125i \(-0.457560\pi\)
0.983695 + 0.179845i \(0.0575596\pi\)
\(12\) 536398. + 389716.i 0.622287 + 0.452118i
\(13\) 1.63018e6 + 1.18440e6i 1.21772 + 0.884727i 0.995909 − 0.0903621i \(-0.0288024\pi\)
0.221813 + 0.975089i \(0.428802\pi\)
\(14\) 1.30845e6 − 950646.i 0.650210 − 0.472405i
\(15\) −4.25406e6 − 1.54062e6i −1.44644 − 0.523835i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −2.52616e6 − 7.77471e6i −0.431510 − 1.32805i −0.896621 − 0.442800i \(-0.853985\pi\)
0.465110 − 0.885253i \(-0.346015\pi\)
\(18\) 7.74688e6 0.966335
\(19\) −2.66164e6 − 8.19169e6i −0.246607 − 0.758977i −0.995368 − 0.0961374i \(-0.969351\pi\)
0.748762 − 0.662840i \(-0.230649\pi\)
\(20\) 6.72781e6 + 2.43650e6i 0.470120 + 0.170256i
\(21\) 1.01126e7 − 3.11233e7i 0.540326 − 1.66295i
\(22\) −7.29028e6 + 2.24372e7i −0.301592 + 0.928204i
\(23\) 3.29329e7 − 2.39271e7i 1.06691 − 0.775153i 0.0915533 − 0.995800i \(-0.470817\pi\)
0.975354 + 0.220647i \(0.0708168\pi\)
\(24\) −2.12168e7 −0.543898
\(25\) −4.87199e7 − 3.24973e6i −0.997783 − 0.0665544i
\(26\) −6.44806e7 −1.06433
\(27\) 3.40188e7 − 2.47161e7i 0.456266 − 0.331497i
\(28\) −1.59931e7 + 4.92217e7i −0.175616 + 0.540489i
\(29\) 1.84473e7 − 5.67748e7i 0.167010 − 0.514005i −0.832169 − 0.554523i \(-0.812901\pi\)
0.999179 + 0.0405181i \(0.0129009\pi\)
\(30\) 1.39109e8 − 4.01306e7i 1.04518 − 0.301515i
\(31\) −4.29868e7 − 1.32300e8i −0.269678 − 0.829983i −0.990579 − 0.136945i \(-0.956272\pi\)
0.720901 − 0.693038i \(-0.243728\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 1.47511e8 + 4.53992e8i 0.656141 + 2.01939i
\(34\) 2.11634e8 + 1.53761e8i 0.798825 + 0.580380i
\(35\) 1.17591e7 − 3.52975e8i 0.0378441 − 1.13598i
\(36\) −2.00555e8 + 1.45712e8i −0.552803 + 0.401635i
\(37\) −1.81033e8 − 1.31528e8i −0.429188 − 0.311823i 0.352136 − 0.935949i \(-0.385455\pi\)
−0.781324 + 0.624125i \(0.785455\pi\)
\(38\) 2.22985e8 + 1.62008e8i 0.456525 + 0.331685i
\(39\) −1.05552e9 + 7.66880e8i −1.87332 + 1.36104i
\(40\) −2.20002e8 + 6.34667e7i −0.339701 + 0.0979978i
\(41\) 6.26839e8 + 4.55425e8i 0.844977 + 0.613912i 0.923757 − 0.382980i \(-0.125102\pi\)
−0.0787792 + 0.996892i \(0.525102\pi\)
\(42\) 3.23603e8 + 9.95947e8i 0.382068 + 1.17589i
\(43\) 6.73513e8 0.698665 0.349333 − 0.936999i \(-0.386408\pi\)
0.349333 + 0.936999i \(0.386408\pi\)
\(44\) −2.33289e8 − 7.17989e8i −0.213258 − 0.656339i
\(45\) 1.03935e9 − 1.33471e9i 0.839637 − 1.07825i
\(46\) −4.02536e8 + 1.23888e9i −0.288162 + 0.886871i
\(47\) −4.24760e8 + 1.30728e9i −0.270150 + 0.831437i 0.720312 + 0.693650i \(0.243999\pi\)
−0.990462 + 0.137786i \(0.956001\pi\)
\(48\) 5.49272e8 − 3.99069e8i 0.311143 − 0.226059i
\(49\) 5.77139e8 0.291879
\(50\) 1.32241e9 − 8.32247e8i 0.598455 − 0.376632i
\(51\) 5.29307e9 2.14819
\(52\) 1.66931e9 − 1.21282e9i 0.608861 − 0.442364i
\(53\) 4.33915e8 − 1.33545e9i 0.142524 − 0.438643i −0.854160 − 0.520010i \(-0.825928\pi\)
0.996684 + 0.0813663i \(0.0259284\pi\)
\(54\) −4.15809e8 + 1.27973e9i −0.123233 + 0.379273i
\(55\) 2.88762e9 + 4.26629e9i 0.773651 + 1.14302i
\(56\) −5.11779e8 − 1.57509e9i −0.124179 − 0.382184i
\(57\) 5.57695e9 1.22768
\(58\) 5.90312e8 + 1.81680e9i 0.118094 + 0.363456i
\(59\) −7.73864e9 − 5.62245e9i −1.40922 − 1.02386i −0.993435 − 0.114395i \(-0.963507\pi\)
−0.415784 − 0.909463i \(-0.636493\pi\)
\(60\) −2.84651e9 + 3.65545e9i −0.472586 + 0.606888i
\(61\) −1.55275e9 + 1.12814e9i −0.235390 + 0.171021i −0.699227 − 0.714900i \(-0.746472\pi\)
0.463837 + 0.885920i \(0.346472\pi\)
\(62\) 3.60131e9 + 2.61650e9i 0.499236 + 0.362716i
\(63\) 9.89884e9 + 7.19193e9i 1.25664 + 0.913003i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −8.65093e9 + 1.11094e10i −0.924781 + 1.18759i
\(66\) −1.23580e10 − 8.97863e9i −1.21467 − 0.882508i
\(67\) 6.42443e9 + 1.97724e10i 0.581331 + 1.78915i 0.613530 + 0.789671i \(0.289749\pi\)
−0.0321990 + 0.999481i \(0.510251\pi\)
\(68\) −8.37101e9 −0.698198
\(69\) 8.14487e9 + 2.50673e10i 0.626923 + 1.92947i
\(70\) 6.33473e9 + 9.35919e9i 0.450494 + 0.665578i
\(71\) 1.97371e8 − 6.07444e8i 0.0129826 − 0.0399563i −0.944355 − 0.328927i \(-0.893313\pi\)
0.957338 + 0.288971i \(0.0933130\pi\)
\(72\) 2.45137e9 − 7.54454e9i 0.149307 − 0.459520i
\(73\) 1.23339e10 − 8.96110e9i 0.696345 − 0.505924i −0.182395 − 0.983225i \(-0.558385\pi\)
0.878740 + 0.477301i \(0.158385\pi\)
\(74\) 7.16060e9 0.375124
\(75\) 1.17492e10 − 2.93512e10i 0.571705 − 1.42820i
\(76\) −8.81997e9 −0.399018
\(77\) −3.01453e10 + 2.19018e10i −1.26917 + 0.922107i
\(78\) 1.29015e10 − 3.97068e10i 0.505965 − 1.55720i
\(79\) 1.10087e10 − 3.38812e10i 0.402518 − 1.23882i −0.520432 − 0.853903i \(-0.674229\pi\)
0.922950 − 0.384920i \(-0.125771\pi\)
\(80\) 4.50177e9 − 5.78110e9i 0.153599 − 0.197249i
\(81\) −4.83891e9 − 1.48926e10i −0.154198 − 0.474574i
\(82\) −2.47941e10 −0.738537
\(83\) −1.16491e10 − 3.58524e10i −0.324612 − 0.999052i −0.971616 − 0.236566i \(-0.923978\pi\)
0.647004 − 0.762487i \(-0.276022\pi\)
\(84\) −2.71105e10 − 1.96969e10i −0.707296 − 0.513881i
\(85\) 5.48851e10 − 1.58334e10i 1.34168 − 0.387053i
\(86\) −1.74363e10 + 1.26682e10i −0.399679 + 0.290384i
\(87\) 3.12707e10 + 2.27195e10i 0.672638 + 0.488700i
\(88\) 1.95443e10 + 1.41997e10i 0.394789 + 0.286831i
\(89\) −1.99289e10 + 1.44792e10i −0.378302 + 0.274852i −0.760645 − 0.649168i \(-0.775117\pi\)
0.382343 + 0.924020i \(0.375117\pi\)
\(90\) −1.80239e9 + 5.41029e10i −0.0321748 + 0.965800i
\(91\) −8.23923e10 − 5.98615e10i −1.38407 − 1.00559i
\(92\) −1.28811e10 − 3.96441e10i −0.203761 − 0.627112i
\(93\) 9.00704e10 1.34254
\(94\) −1.35923e10 − 4.18328e10i −0.191025 − 0.587915i
\(95\) 5.78287e10 − 1.66826e10i 0.766767 − 0.221199i
\(96\) −6.71370e9 + 2.06626e10i −0.0840369 + 0.258639i
\(97\) −1.32603e10 + 4.08110e10i −0.156786 + 0.482539i −0.998338 − 0.0576385i \(-0.981643\pi\)
0.841551 + 0.540178i \(0.181643\pi\)
\(98\) −1.49413e10 + 1.08555e10i −0.166972 + 0.121313i
\(99\) −1.78480e11 −1.88623
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.11.2 ✓ 56
25.16 even 5 inner 50.12.d.b.41.2 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.2 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.2 yes 56 25.16 even 5 inner