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Properties

Label 425.2.r.a.154.6
Level $425$
Weight $2$
Character 425.154
Analytic conductor $3.394$
Analytic rank $0$
Dimension $160$
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] = [425,2,Mod(69,425)] 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("425.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([9, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 154.6
Character \(\chi\) \(=\) 425.154
Dual form 425.2.r.a.69.6

$q$-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+(-2.14124 - 0.695730i) q^{2} +(1.08145 - 1.48849i) q^{3} +(2.48282 + 1.80387i) q^{4} +(-0.133284 - 2.23209i) q^{5} +(-3.35124 + 2.43482i) q^{6} +4.38149i q^{7} +(-1.41458 - 1.94701i) q^{8} +(-0.119019 - 0.366304i) q^{9} +(-1.26754 + 4.87217i) q^{10} +(1.54742 - 4.76246i) q^{11} +(5.37011 - 1.74485i) q^{12} +(5.16707 - 1.67888i) q^{13} +(3.04833 - 9.38180i) q^{14} +(-3.46659 - 2.21551i) q^{15} +(-0.222341 - 0.684295i) q^{16} +(0.587785 + 0.809017i) q^{17} +0.867149i q^{18} +(2.84744 - 2.06879i) q^{19} +(3.69549 - 5.78231i) q^{20} +(6.52181 + 4.73838i) q^{21} +(-6.62677 + 9.12097i) q^{22} +(-3.72988 - 1.21191i) q^{23} -4.42791 q^{24} +(-4.96447 + 0.595005i) q^{25} -12.2320 q^{26} +(4.57553 + 1.48668i) q^{27} +(-7.90365 + 10.8784i) q^{28} +(-1.35616 - 0.985311i) q^{29} +(5.88140 + 7.15575i) q^{30} +(4.45495 - 3.23671i) q^{31} +6.43319i q^{32} +(-5.41543 - 7.45370i) q^{33} +(-0.695730 - 2.14124i) q^{34} +(9.77988 - 0.583982i) q^{35} +(0.365263 - 1.12416i) q^{36} +(-5.69639 + 1.85087i) q^{37} +(-7.53636 + 2.44871i) q^{38} +(3.08894 - 9.50678i) q^{39} +(-4.15735 + 3.41698i) q^{40} +(-1.38219 - 4.25393i) q^{41} +(-10.6681 - 14.6834i) q^{42} +2.73959i q^{43} +(12.4328 - 9.03298i) q^{44} +(-0.801761 + 0.314485i) q^{45} +(7.14340 + 5.18998i) q^{46} +(-2.75987 + 3.79863i) q^{47} +(-1.25902 - 0.409080i) q^{48} -12.1974 q^{49} +(11.0441 + 2.17989i) q^{50} +1.83988 q^{51} +(15.8574 + 5.15238i) q^{52} +(4.49551 - 6.18754i) q^{53} +(-8.76297 - 6.36667i) q^{54} +(-10.8365 - 2.81922i) q^{55} +(8.53078 - 6.19797i) q^{56} -6.47569i q^{57} +(2.21836 + 3.05331i) q^{58} +(-1.96156 - 6.03707i) q^{59} +(-4.61043 - 11.7540i) q^{60} +(-0.481029 + 1.48046i) q^{61} +(-11.7910 + 3.83112i) q^{62} +(1.60496 - 0.521482i) q^{63} +(4.03108 - 12.4064i) q^{64} +(-4.43611 - 11.3096i) q^{65} +(6.40995 + 19.7278i) q^{66} +(-6.97108 - 9.59486i) q^{67} +3.06893i q^{68} +(-5.83762 + 4.24128i) q^{69} +(-21.3473 - 5.55371i) q^{70} +(12.2357 + 8.88975i) q^{71} +(-0.544833 + 0.749898i) q^{72} +(1.39360 + 0.452808i) q^{73} +13.4850 q^{74} +(-4.48319 + 8.03305i) q^{75} +10.8015 q^{76} +(20.8666 + 6.77999i) q^{77} +(-13.2283 + 18.2072i) q^{78} +(-3.22393 - 2.34232i) q^{79} +(-1.49777 + 0.587491i) q^{80} +(8.09593 - 5.88204i) q^{81} +10.0703i q^{82} +(6.78571 + 9.33973i) q^{83} +(7.64506 + 23.5291i) q^{84} +(1.72746 - 1.41982i) q^{85} +(1.90601 - 5.86610i) q^{86} +(-2.93326 + 0.953074i) q^{87} +(-11.4615 + 3.72406i) q^{88} +(-4.22214 + 12.9944i) q^{89} +(1.93556 - 0.115577i) q^{90} +(7.35600 + 22.6394i) q^{91} +(-7.07449 - 9.73720i) q^{92} -10.1315i q^{93} +(8.55235 - 6.21364i) q^{94} +(-4.99724 - 6.08001i) q^{95} +(9.57576 + 6.95719i) q^{96} +(6.46389 - 8.89679i) q^{97} +(26.1176 + 8.48612i) q^{98} -1.92868 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 40 q^{4} - 8 q^{5} + 4 q^{6} - 30 q^{8} + 36 q^{9} - 6 q^{10} + 8 q^{11} - 40 q^{12} - 20 q^{14} - 40 q^{15} - 64 q^{16} + 6 q^{19} + 2 q^{20} - 50 q^{22} + 20 q^{23} + 20 q^{24} + 32 q^{25} + 20 q^{26}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{1}{10}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.14124 − 0.695730i −1.51408 − 0.491955i −0.569995 − 0.821648i \(-0.693055\pi\)
−0.944088 + 0.329693i \(0.893055\pi\)
\(3\) 1.08145 − 1.48849i 0.624378 − 0.859382i −0.373285 − 0.927717i \(-0.621769\pi\)
0.997662 + 0.0683347i \(0.0217686\pi\)
\(4\) 2.48282 + 1.80387i 1.24141 + 0.901937i
\(5\) −0.133284 − 2.23209i −0.0596064 − 0.998222i
\(6\) −3.35124 + 2.43482i −1.36814 + 0.994010i
\(7\) 4.38149i 1.65605i 0.560694 + 0.828023i \(0.310535\pi\)
−0.560694 + 0.828023i \(0.689465\pi\)
\(8\) −1.41458 − 1.94701i −0.500130 − 0.688370i
\(9\) −0.119019 − 0.366304i −0.0396731 − 0.122101i
\(10\) −1.26754 + 4.87217i −0.400832 + 1.54071i
\(11\) 1.54742 − 4.76246i 0.466564 − 1.43594i −0.390442 − 0.920628i \(-0.627678\pi\)
0.857005 − 0.515307i \(-0.172322\pi\)
\(12\) 5.37011 − 1.74485i 1.55022 − 0.503696i
\(13\) 5.16707 − 1.67888i 1.43309 − 0.465638i 0.513352 − 0.858178i \(-0.328404\pi\)
0.919735 + 0.392540i \(0.128404\pi\)
\(14\) 3.04833 − 9.38180i 0.814701 − 2.50739i
\(15\) −3.46659 − 2.21551i −0.895071 − 0.572043i
\(16\) −0.222341 − 0.684295i −0.0555852 − 0.171074i
\(17\) 0.587785 + 0.809017i 0.142559 + 0.196215i
\(18\) 0.867149i 0.204389i
\(19\) 2.84744 − 2.06879i 0.653248 − 0.474612i −0.211128 − 0.977458i \(-0.567714\pi\)
0.864376 + 0.502846i \(0.167714\pi\)
\(20\) 3.69549 − 5.78231i 0.826338 − 1.29296i
\(21\) 6.52181 + 4.73838i 1.42318 + 1.03400i
\(22\) −6.62677 + 9.12097i −1.41283 + 1.94460i
\(23\) −3.72988 − 1.21191i −0.777734 − 0.252701i −0.106862 − 0.994274i \(-0.534080\pi\)
−0.670872 + 0.741573i \(0.734080\pi\)
\(24\) −4.42791 −0.903843
\(25\) −4.96447 + 0.595005i −0.992894 + 0.119001i
\(26\) −12.2320 −2.39889
\(27\) 4.57553 + 1.48668i 0.880562 + 0.286112i
\(28\) −7.90365 + 10.8784i −1.49365 + 2.05583i
\(29\) −1.35616 − 0.985311i −0.251833 − 0.182968i 0.454706 − 0.890642i \(-0.349745\pi\)
−0.706539 + 0.707674i \(0.749745\pi\)
\(30\) 5.88140 + 7.15575i 1.07379 + 1.30646i
\(31\) 4.45495 − 3.23671i 0.800133 − 0.581331i −0.110820 − 0.993840i \(-0.535348\pi\)
0.910953 + 0.412510i \(0.135348\pi\)
\(32\) 6.43319i 1.13724i
\(33\) −5.41543 − 7.45370i −0.942705 − 1.29752i
\(34\) −0.695730 − 2.14124i −0.119317 − 0.367219i
\(35\) 9.77988 − 0.583982i 1.65310 − 0.0987110i
\(36\) 0.365263 − 1.12416i 0.0608771 − 0.187360i
\(37\) −5.69639 + 1.85087i −0.936481 + 0.304281i −0.737210 − 0.675663i \(-0.763857\pi\)
−0.199271 + 0.979944i \(0.563857\pi\)
\(38\) −7.53636 + 2.44871i −1.22256 + 0.397234i
\(39\) 3.08894 − 9.50678i 0.494626 − 1.52230i
\(40\) −4.15735 + 3.41698i −0.657335 + 0.540272i
\(41\) −1.38219 − 4.25393i −0.215861 − 0.664352i −0.999091 − 0.0426203i \(-0.986429\pi\)
0.783230 − 0.621732i \(-0.213571\pi\)
\(42\) −10.6681 − 14.6834i −1.64613 − 2.26570i
\(43\) 2.73959i 0.417783i 0.977939 + 0.208892i \(0.0669856\pi\)
−0.977939 + 0.208892i \(0.933014\pi\)
\(44\) 12.4328 − 9.03298i 1.87432 − 1.36177i
\(45\) −0.801761 + 0.314485i −0.119519 + 0.0468806i
\(46\) 7.14340 + 5.18998i 1.05324 + 0.765221i
\(47\) −2.75987 + 3.79863i −0.402568 + 0.554087i −0.961386 − 0.275203i \(-0.911255\pi\)
0.558818 + 0.829290i \(0.311255\pi\)
\(48\) −1.25902 − 0.409080i −0.181724 − 0.0590457i
\(49\) −12.1974 −1.74249
\(50\) 11.0441 + 2.17989i 1.56187 + 0.308282i
\(51\) 1.83988 0.257635
\(52\) 15.8574 + 5.15238i 2.19902 + 0.714507i
\(53\) 4.49551 − 6.18754i 0.617506 − 0.849925i −0.379662 − 0.925125i \(-0.623960\pi\)
0.997168 + 0.0752006i \(0.0239597\pi\)
\(54\) −8.76297 − 6.36667i −1.19249 − 0.866394i
\(55\) −10.8365 − 2.81922i −1.46119 − 0.380143i
\(56\) 8.53078 − 6.19797i 1.13997 − 0.828239i
\(57\) − 6.47569i − 0.857727i
\(58\) 2.21836 + 3.05331i 0.291285 + 0.400919i
\(59\) −1.96156 − 6.03707i −0.255374 − 0.785959i −0.993756 − 0.111577i \(-0.964410\pi\)
0.738382 − 0.674382i \(-0.235590\pi\)
\(60\) −4.61043 − 11.7540i −0.595203 − 1.51744i
\(61\) −0.481029 + 1.48046i −0.0615895 + 0.189553i −0.977117 − 0.212702i \(-0.931774\pi\)
0.915528 + 0.402255i \(0.131774\pi\)
\(62\) −11.7910 + 3.83112i −1.49746 + 0.486553i
\(63\) 1.60496 − 0.521482i 0.202205 − 0.0657005i
\(64\) 4.03108 − 12.4064i 0.503885 − 1.55080i
\(65\) −4.43611 − 11.3096i −0.550231 − 1.40278i
\(66\) 6.40995 + 19.7278i 0.789011 + 2.42832i
\(67\) −6.97108 − 9.59486i −0.851653 − 1.17220i −0.983496 − 0.180929i \(-0.942090\pi\)
0.131844 − 0.991271i \(-0.457910\pi\)
\(68\) 3.06893i 0.372163i
\(69\) −5.83762 + 4.24128i −0.702767 + 0.510590i
\(70\) −21.3473 − 5.55371i −2.55150 − 0.663796i
\(71\) 12.2357 + 8.88975i 1.45211 + 1.05502i 0.985333 + 0.170645i \(0.0545851\pi\)
0.466777 + 0.884375i \(0.345415\pi\)
\(72\) −0.544833 + 0.749898i −0.0642092 + 0.0883764i
\(73\) 1.39360 + 0.452808i 0.163108 + 0.0529972i 0.389433 − 0.921055i \(-0.372671\pi\)
−0.226325 + 0.974052i \(0.572671\pi\)
\(74\) 13.4850 1.56760
\(75\) −4.48319 + 8.03305i −0.517674 + 0.927577i
\(76\) 10.8015 1.23902
\(77\) 20.8666 + 6.77999i 2.37798 + 0.772651i
\(78\) −13.2283 + 18.2072i −1.49781 + 2.06156i
\(79\) −3.22393 − 2.34232i −0.362721 − 0.263532i 0.391465 − 0.920193i \(-0.371968\pi\)
−0.754186 + 0.656661i \(0.771968\pi\)
\(80\) −1.49777 + 0.587491i −0.167456 + 0.0656835i
\(81\) 8.09593 − 5.88204i 0.899548 − 0.653560i
\(82\) 10.0703i 1.11208i
\(83\) 6.78571 + 9.33973i 0.744829 + 1.02517i 0.998326 + 0.0578327i \(0.0184190\pi\)
−0.253498 + 0.967336i \(0.581581\pi\)
\(84\) 7.64506 + 23.5291i 0.834144 + 2.56723i
\(85\) 1.72746 − 1.41982i 0.187369 − 0.154001i
\(86\) 1.90601 − 5.86610i 0.205531 − 0.632558i
\(87\) −2.93326 + 0.953074i −0.314478 + 0.102180i
\(88\) −11.4615 + 3.72406i −1.22180 + 0.396986i
\(89\) −4.22214 + 12.9944i −0.447546 + 1.37740i 0.432121 + 0.901815i \(0.357765\pi\)
−0.879667 + 0.475589i \(0.842235\pi\)
\(90\) 1.93556 − 0.115577i 0.204026 − 0.0121829i
\(91\) 7.35600 + 22.6394i 0.771118 + 2.37326i
\(92\) −7.07449 − 9.73720i −0.737566 − 1.01517i
\(93\) − 10.1315i − 1.05059i
\(94\) 8.55235 − 6.21364i 0.882107 − 0.640888i
\(95\) −4.99724 − 6.08001i −0.512706 − 0.623796i
\(96\) 9.57576 + 6.95719i 0.977322 + 0.710066i
\(97\) 6.46389 − 8.89679i 0.656309 − 0.903332i −0.343043 − 0.939320i \(-0.611458\pi\)
0.999352 + 0.0359878i \(0.0114577\pi\)
\(98\) 26.1176 + 8.48612i 2.63827 + 0.857227i
\(99\) −1.92868 −0.193840
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 425.2.r.a.154.6 yes 160
25.19 even 10 inner 425.2.r.a.69.6 ✓ 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.r.a.69.6 ✓ 160 25.19 even 10 inner
425.2.r.a.154.6 yes 160 1.1 even 1 trivial