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Properties

Label 425.2.r.a.69.12
Level $425$
Weight $2$
Character 425.69
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 69.12
Character \(\chi\) \(=\) 425.69
Dual form 425.2.r.a.154.12

$q$-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+(-1.38991 + 0.451609i) q^{2} +(1.86675 + 2.56936i) q^{3} +(0.109865 - 0.0798213i) q^{4} +(2.21373 - 0.315251i) q^{5} +(-3.75495 - 2.72813i) q^{6} +0.259005i q^{7} +(1.60137 - 2.20410i) q^{8} +(-2.18980 + 6.73951i) q^{9} +(-2.93452 + 1.43791i) q^{10} +(0.353460 + 1.08784i) q^{11} +(0.410179 + 0.133275i) q^{12} +(0.350179 + 0.113780i) q^{13} +(-0.116969 - 0.359993i) q^{14} +(4.94247 + 5.09938i) q^{15} +(-1.31430 + 4.04499i) q^{16} +(-0.587785 + 0.809017i) q^{17} -10.3562i q^{18} +(2.44657 + 1.77754i) q^{19} +(0.218047 - 0.211338i) q^{20} +(-0.665475 + 0.483496i) q^{21} +(-0.982555 - 1.35237i) q^{22} +(-2.23279 + 0.725476i) q^{23} +8.65246 q^{24} +(4.80123 - 1.39576i) q^{25} -0.538101 q^{26} +(-12.3426 + 4.01036i) q^{27} +(0.0206741 + 0.0284555i) q^{28} +(5.99815 - 4.35791i) q^{29} +(-9.17252 - 4.85561i) q^{30} +(-5.81450 - 4.22448i) q^{31} -0.766902i q^{32} +(-2.13522 + 2.93888i) q^{33} +(0.451609 - 1.38991i) q^{34} +(0.0816516 + 0.573367i) q^{35} +(0.297375 + 0.915226i) q^{36} +(-9.73573 - 3.16333i) q^{37} +(-4.20327 - 1.36572i) q^{38} +(0.361354 + 1.11213i) q^{39} +(2.85016 - 5.38411i) q^{40} +(0.123545 - 0.380232i) q^{41} +(0.706599 - 0.972551i) q^{42} -1.20039i q^{43} +(0.125665 + 0.0913012i) q^{44} +(-2.72299 + 15.6098i) q^{45} +(2.77574 - 2.01669i) q^{46} +(-4.59394 - 6.32301i) q^{47} +(-12.8465 + 4.17408i) q^{48} +6.93292 q^{49} +(-6.04294 + 4.10827i) q^{50} -3.17590 q^{51} +(0.0475543 - 0.0154513i) q^{52} +(5.64170 + 7.76513i) q^{53} +(15.3440 - 11.1481i) q^{54} +(1.12541 + 2.29675i) q^{55} +(0.570871 + 0.414762i) q^{56} +9.60433i q^{57} +(-6.36882 + 8.76593i) q^{58} +(-2.50095 + 7.69712i) q^{59} +(0.950042 + 0.165726i) q^{60} +(-0.665804 - 2.04913i) q^{61} +(9.98945 + 3.24577i) q^{62} +(-1.74556 - 0.567168i) q^{63} +(-2.28226 - 7.02406i) q^{64} +(0.811072 + 0.141484i) q^{65} +(1.64054 - 5.04907i) q^{66} +(7.79166 - 10.7243i) q^{67} +0.135800i q^{68} +(-6.03205 - 4.38254i) q^{69} +(-0.372426 - 0.760054i) q^{70} +(2.00137 - 1.45408i) q^{71} +(11.3478 + 15.6190i) q^{72} +(1.02055 - 0.331597i) q^{73} +14.9604 q^{74} +(12.5489 + 9.73054i) q^{75} +0.410677 q^{76} +(-0.281755 + 0.0915477i) q^{77} +(-1.00450 - 1.38257i) q^{78} +(1.43101 - 1.03969i) q^{79} +(-1.63432 + 9.36887i) q^{80} +(-16.1457 - 11.7305i) q^{81} +0.584282i q^{82} +(-7.49157 + 10.3113i) q^{83} +(-0.0345189 + 0.106238i) q^{84} +(-1.04616 + 1.97625i) q^{85} +(0.542109 + 1.66844i) q^{86} +(22.3941 + 7.27627i) q^{87} +(2.96372 + 0.962970i) q^{88} +(0.317468 + 0.977065i) q^{89} +(-3.26482 - 22.9260i) q^{90} +(-0.0294695 + 0.0906979i) q^{91} +(-0.187396 + 0.257928i) q^{92} -22.8256i q^{93} +(9.24069 + 6.71375i) q^{94} +(5.97643 + 3.16371i) q^{95} +(1.97045 - 1.43161i) q^{96} +(-8.37484 - 11.5270i) q^{97} +(-9.63613 + 3.13097i) q^{98} -8.10550 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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38991 + 0.451609i −0.982815 + 0.319336i −0.755978 − 0.654597i \(-0.772838\pi\)
−0.226837 + 0.973933i \(0.572838\pi\)
\(3\) 1.86675 + 2.56936i 1.07777 + 1.48342i 0.861956 + 0.506984i \(0.169240\pi\)
0.215811 + 0.976435i \(0.430760\pi\)
\(4\) 0.109865 − 0.0798213i 0.0549323 − 0.0399107i
\(5\) 2.21373 − 0.315251i 0.990012 − 0.140985i
\(6\) −3.75495 − 2.72813i −1.53295 − 1.11376i
\(7\) 0.259005i 0.0978946i 0.998801 + 0.0489473i \(0.0155866\pi\)
−0.998801 + 0.0489473i \(0.984413\pi\)
\(8\) 1.60137 − 2.20410i 0.566169 − 0.779265i
\(9\) −2.18980 + 6.73951i −0.729933 + 2.24650i
\(10\) −2.93452 + 1.43791i −0.927977 + 0.454708i
\(11\) 0.353460 + 1.08784i 0.106572 + 0.327995i 0.990096 − 0.140390i \(-0.0448358\pi\)
−0.883524 + 0.468386i \(0.844836\pi\)
\(12\) 0.410179 + 0.133275i 0.118408 + 0.0384732i
\(13\) 0.350179 + 0.113780i 0.0971221 + 0.0315569i 0.357175 − 0.934037i \(-0.383740\pi\)
−0.260053 + 0.965594i \(0.583740\pi\)
\(14\) −0.116969 − 0.359993i −0.0312612 − 0.0962122i
\(15\) 4.94247 + 5.09938i 1.27614 + 1.31665i
\(16\) −1.31430 + 4.04499i −0.328575 + 1.01125i
\(17\) −0.587785 + 0.809017i −0.142559 + 0.196215i
\(18\) − 10.3562i − 2.44099i
\(19\) 2.44657 + 1.77754i 0.561282 + 0.407795i 0.831928 − 0.554884i \(-0.187237\pi\)
−0.270646 + 0.962679i \(0.587237\pi\)
\(20\) 0.218047 − 0.211338i 0.0487568 − 0.0472566i
\(21\) −0.665475 + 0.483496i −0.145219 + 0.105508i
\(22\) −0.982555 − 1.35237i −0.209481 − 0.288326i
\(23\) −2.23279 + 0.725476i −0.465568 + 0.151272i −0.532401 − 0.846492i \(-0.678710\pi\)
0.0668336 + 0.997764i \(0.478710\pi\)
\(24\) 8.65246 1.76618
\(25\) 4.80123 − 1.39576i 0.960247 − 0.279153i
\(26\) −0.538101 −0.105530
\(27\) −12.3426 + 4.01036i −2.37534 + 0.771794i
\(28\) 0.0206741 + 0.0284555i 0.00390704 + 0.00537757i
\(29\) 5.99815 − 4.35791i 1.11383 − 0.809244i 0.130567 − 0.991440i \(-0.458320\pi\)
0.983262 + 0.182196i \(0.0583204\pi\)
\(30\) −9.17252 − 4.85561i −1.67466 − 0.886509i
\(31\) −5.81450 − 4.22448i −1.04432 − 0.758740i −0.0731924 − 0.997318i \(-0.523319\pi\)
−0.971123 + 0.238578i \(0.923319\pi\)
\(32\) − 0.766902i − 0.135570i
\(33\) −2.13522 + 2.93888i −0.371695 + 0.511594i
\(34\) 0.451609 − 1.38991i 0.0774503 − 0.238368i
\(35\) 0.0816516 + 0.573367i 0.0138016 + 0.0969168i
\(36\) 0.297375 + 0.915226i 0.0495625 + 0.152538i
\(37\) −9.73573 − 3.16333i −1.60054 − 0.520048i −0.633302 − 0.773905i \(-0.718301\pi\)
−0.967242 + 0.253857i \(0.918301\pi\)
\(38\) −4.20327 − 1.36572i −0.681860 − 0.221550i
\(39\) 0.361354 + 1.11213i 0.0578629 + 0.178084i
\(40\) 2.85016 − 5.38411i 0.450650 − 0.851303i
\(41\) 0.123545 − 0.380232i 0.0192945 − 0.0593823i −0.940946 − 0.338558i \(-0.890061\pi\)
0.960240 + 0.279175i \(0.0900610\pi\)
\(42\) 0.706599 − 0.972551i 0.109031 − 0.150068i
\(43\) − 1.20039i − 0.183058i −0.995802 − 0.0915292i \(-0.970825\pi\)
0.995802 − 0.0915292i \(-0.0291755\pi\)
\(44\) 0.125665 + 0.0913012i 0.0189448 + 0.0137642i
\(45\) −2.72299 + 15.6098i −0.405920 + 2.32697i
\(46\) 2.77574 − 2.01669i 0.409260 − 0.297345i
\(47\) −4.59394 − 6.32301i −0.670095 − 0.922306i 0.329668 − 0.944097i \(-0.393063\pi\)
−0.999763 + 0.0217908i \(0.993063\pi\)
\(48\) −12.8465 + 4.17408i −1.85423 + 0.602476i
\(49\) 6.93292 0.990417
\(50\) −6.04294 + 4.10827i −0.854601 + 0.580997i
\(51\) −3.17590 −0.444715
\(52\) 0.0475543 − 0.0154513i 0.00659460 − 0.00214272i
\(53\) 5.64170 + 7.76513i 0.774947 + 1.06662i 0.995822 + 0.0913210i \(0.0291089\pi\)
−0.220874 + 0.975302i \(0.570891\pi\)
\(54\) 15.3440 − 11.1481i 2.08806 − 1.51706i
\(55\) 1.12541 + 2.29675i 0.151750 + 0.309694i
\(56\) 0.570871 + 0.414762i 0.0762858 + 0.0554249i
\(57\) 9.60433i 1.27212i
\(58\) −6.36882 + 8.76593i −0.836267 + 1.15102i
\(59\) −2.50095 + 7.69712i −0.325595 + 1.00208i 0.645576 + 0.763696i \(0.276617\pi\)
−0.971171 + 0.238383i \(0.923383\pi\)
\(60\) 0.950042 + 0.165726i 0.122650 + 0.0213952i
\(61\) −0.665804 − 2.04913i −0.0852474 − 0.262365i 0.899342 − 0.437246i \(-0.144046\pi\)
−0.984590 + 0.174881i \(0.944046\pi\)
\(62\) 9.98945 + 3.24577i 1.26866 + 0.412213i
\(63\) −1.74556 − 0.567168i −0.219920 − 0.0714565i
\(64\) −2.28226 − 7.02406i −0.285282 − 0.878008i
\(65\) 0.811072 + 0.141484i 0.100601 + 0.0175490i
\(66\) 1.64054 − 5.04907i 0.201937 − 0.621497i
\(67\) 7.79166 − 10.7243i 0.951902 − 1.31018i 0.00122494 − 0.999999i \(-0.499610\pi\)
0.950677 − 0.310182i \(-0.100390\pi\)
\(68\) 0.135800i 0.0164682i
\(69\) −6.03205 − 4.38254i −0.726174 − 0.527596i
\(70\) −0.372426 − 0.760054i −0.0445134 − 0.0908439i
\(71\) 2.00137 − 1.45408i 0.237519 − 0.172568i −0.462658 − 0.886537i \(-0.653104\pi\)
0.700177 + 0.713969i \(0.253104\pi\)
\(72\) 11.3478 + 15.6190i 1.33736 + 1.84071i
\(73\) 1.02055 − 0.331597i 0.119446 − 0.0388104i −0.248684 − 0.968585i \(-0.579998\pi\)
0.368130 + 0.929774i \(0.379998\pi\)
\(74\) 14.9604 1.73911
\(75\) 12.5489 + 9.73054i 1.44902 + 1.12359i
\(76\) 0.410677 0.0471079
\(77\) −0.281755 + 0.0915477i −0.0321090 + 0.0104328i
\(78\) −1.00450 − 1.38257i −0.113737 − 0.156546i
\(79\) 1.43101 − 1.03969i 0.161002 − 0.116975i −0.504367 − 0.863489i \(-0.668274\pi\)
0.665369 + 0.746515i \(0.268274\pi\)
\(80\) −1.63432 + 9.36887i −0.182722 + 1.04747i
\(81\) −16.1457 − 11.7305i −1.79397 − 1.30339i
\(82\) 0.584282i 0.0645232i
\(83\) −7.49157 + 10.3113i −0.822307 + 1.13181i 0.166999 + 0.985957i \(0.446592\pi\)
−0.989306 + 0.145852i \(0.953408\pi\)
\(84\) −0.0345189 + 0.106238i −0.00376632 + 0.0115915i
\(85\) −1.04616 + 1.97625i −0.113472 + 0.214354i
\(86\) 0.542109 + 1.66844i 0.0584571 + 0.179912i
\(87\) 22.3941 + 7.27627i 2.40090 + 0.780098i
\(88\) 2.96372 + 0.962970i 0.315933 + 0.102653i
\(89\) 0.317468 + 0.977065i 0.0336515 + 0.103569i 0.966471 − 0.256774i \(-0.0826597\pi\)
−0.932820 + 0.360343i \(0.882660\pi\)
\(90\) −3.26482 − 22.9260i −0.344142 − 2.41661i
\(91\) −0.0294695 + 0.0906979i −0.00308925 + 0.00950773i
\(92\) −0.187396 + 0.257928i −0.0195373 + 0.0268909i
\(93\) − 22.8256i − 2.36690i
\(94\) 9.24069 + 6.71375i 0.953104 + 0.692471i
\(95\) 5.97643 + 3.16371i 0.613169 + 0.324590i
\(96\) 1.97045 − 1.43161i 0.201108 − 0.146113i
\(97\) −8.37484 − 11.5270i −0.850336 − 1.17039i −0.983789 − 0.179332i \(-0.942607\pi\)
0.133453 − 0.991055i \(-0.457393\pi\)
\(98\) −9.63613 + 3.13097i −0.973396 + 0.316276i
\(99\) −8.10550 −0.814633
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.69.12 ✓ 160
25.4 even 10 inner 425.2.r.a.154.12 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.r.a.69.12 ✓ 160 1.1 even 1 trivial
425.2.r.a.154.12 yes 160 25.4 even 10 inner