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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(84,425)] 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("425.84"); 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([7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.q (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: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 339.11
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.11

$q$-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+(-0.857096 - 1.17969i) q^{2} +(-0.707067 + 2.17613i) q^{3} +(-0.0390246 + 0.120105i) q^{4} +(-1.58143 + 1.58085i) q^{5} +(3.17318 - 1.03103i) q^{6} +2.12040 q^{7} +(-2.59849 + 0.844299i) q^{8} +(-1.80854 - 1.31398i) q^{9} +(3.22035 + 0.510657i) q^{10} +(-0.204272 - 0.281156i) q^{11} +(-0.233772 - 0.169845i) q^{12} +(-0.627105 + 0.863136i) q^{13} +(-1.81739 - 2.50142i) q^{14} +(-2.32196 - 4.55916i) q^{15} +(3.42750 + 2.49023i) q^{16} +(-3.99763 + 1.00942i) q^{17} +3.25973i q^{18} +(-1.34178 - 4.12958i) q^{19} +(-0.128154 - 0.251630i) q^{20} +(-1.49927 + 4.61427i) q^{21} +(-0.156597 + 0.481955i) q^{22} +(-3.42202 + 2.48625i) q^{23} -6.25162i q^{24} +(0.00182737 - 5.00000i) q^{25} +1.55572 q^{26} +(-1.41523 + 1.02822i) q^{27} +(-0.0827479 + 0.254672i) q^{28} +(-4.63792 - 1.50695i) q^{29} +(-3.38826 + 6.64683i) q^{30} +(-9.91655 + 3.22208i) q^{31} -0.713336i q^{32} +(0.756265 - 0.245725i) q^{33} +(4.61716 + 3.85081i) q^{34} +(-3.35327 + 3.35204i) q^{35} +(0.228394 - 0.165938i) q^{36} +(4.75225 + 3.45271i) q^{37} +(-3.72159 + 5.12233i) q^{38} +(-1.43489 - 1.97496i) q^{39} +(2.77461 - 5.44302i) q^{40} +(2.15281 - 2.96309i) q^{41} +(6.72844 - 2.18620i) q^{42} +3.26028i q^{43} +(0.0417399 - 0.0135621i) q^{44} +(4.93728 - 0.781064i) q^{45} +(5.86601 + 1.90598i) q^{46} +(-10.2608 - 3.33393i) q^{47} +(-7.84252 + 5.69793i) q^{48} -2.50388 q^{49} +(-5.90002 + 4.28332i) q^{50} +(0.629968 - 9.41309i) q^{51} +(-0.0791947 - 0.109002i) q^{52} +(2.85480 + 0.927580i) q^{53} +(2.42597 + 0.788245i) q^{54} +(0.767506 + 0.121705i) q^{55} +(-5.50984 + 1.79026i) q^{56} +9.93522 q^{57} +(2.19741 + 6.76292i) q^{58} +(9.15031 + 6.64809i) q^{59} +(0.638192 - 0.100960i) q^{60} +(-0.507392 - 0.698365i) q^{61} +(12.3005 + 8.93684i) q^{62} +(-3.83484 - 2.78617i) q^{63} +(6.01349 - 4.36905i) q^{64} +(-0.372767 - 2.35635i) q^{65} +(-0.938072 - 0.681549i) q^{66} +(6.30092 - 2.04729i) q^{67} +(0.0347693 - 0.519529i) q^{68} +(-2.99079 - 9.20471i) q^{69} +(6.82845 + 1.08280i) q^{70} +(13.0325 + 4.23452i) q^{71} +(5.80886 + 1.88741i) q^{72} +(-9.94242 + 7.22359i) q^{73} -8.56550i q^{74} +(10.8793 + 3.53931i) q^{75} +0.548346 q^{76} +(-0.433139 - 0.596164i) q^{77} +(-1.10000 + 3.38545i) q^{78} +(-8.36725 - 2.71868i) q^{79} +(-9.35702 + 1.48025i) q^{80} +(-3.30929 - 10.1849i) q^{81} -5.34069 q^{82} +(6.43351 - 2.09038i) q^{83} +(-0.495690 - 0.360140i) q^{84} +(4.72623 - 7.91598i) q^{85} +(3.84613 - 2.79437i) q^{86} +(6.55865 - 9.02720i) q^{87} +(0.768177 + 0.558113i) q^{88} +(-3.90634 + 2.83812i) q^{89} +(-5.15314 - 5.15502i) q^{90} +(-1.32972 + 1.83020i) q^{91} +(-0.165068 - 0.508028i) q^{92} -23.8579i q^{93} +(4.86146 + 14.9620i) q^{94} +(8.65017 + 4.40947i) q^{95} +(1.55231 + 0.504376i) q^{96} +(-0.345614 + 1.06369i) q^{97} +(2.14607 + 2.95381i) q^{98} +0.776891i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 176 q - 10 q^{2} + 38 q^{4} - 10 q^{8} - 50 q^{9} - 10 q^{13} + 6 q^{15} - 58 q^{16} - 8 q^{19} - 44 q^{21} - 20 q^{25} - 64 q^{26} + 78 q^{30} - 20 q^{33} - 23 q^{34} - 46 q^{35} + 52 q^{36} - 10 q^{38}+ \cdots - 250 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.857096 − 1.17969i −0.606058 − 0.834168i 0.390187 − 0.920735i \(-0.372410\pi\)
−0.996246 + 0.0865675i \(0.972410\pi\)
\(3\) −0.707067 + 2.17613i −0.408225 + 1.25639i 0.509947 + 0.860206i \(0.329665\pi\)
−0.918172 + 0.396182i \(0.870335\pi\)
\(4\) −0.0390246 + 0.120105i −0.0195123 + 0.0600526i
\(5\) −1.58143 + 1.58085i −0.707236 + 0.706978i
\(6\) 3.17318 − 1.03103i 1.29545 − 0.420916i
\(7\) 2.12040 0.801438 0.400719 − 0.916201i \(-0.368760\pi\)
0.400719 + 0.916201i \(0.368760\pi\)
\(8\) −2.59849 + 0.844299i −0.918704 + 0.298505i
\(9\) −1.80854 − 1.31398i −0.602847 − 0.437994i
\(10\) 3.22035 + 0.510657i 1.01836 + 0.161484i
\(11\) −0.204272 − 0.281156i −0.0615902 − 0.0847717i 0.777111 − 0.629364i \(-0.216684\pi\)
−0.838701 + 0.544592i \(0.816684\pi\)
\(12\) −0.233772 − 0.169845i −0.0674840 − 0.0490300i
\(13\) −0.627105 + 0.863136i −0.173928 + 0.239391i −0.887077 − 0.461621i \(-0.847268\pi\)
0.713150 + 0.701012i \(0.247268\pi\)
\(14\) −1.81739 − 2.50142i −0.485718 − 0.668534i
\(15\) −2.32196 − 4.55916i −0.599527 − 1.17717i
\(16\) 3.42750 + 2.49023i 0.856875 + 0.622556i
\(17\) −3.99763 + 1.00942i −0.969569 + 0.244820i
\(18\) 3.25973i 0.768325i
\(19\) −1.34178 − 4.12958i −0.307826 − 0.947390i −0.978608 − 0.205735i \(-0.934041\pi\)
0.670782 − 0.741655i \(-0.265959\pi\)
\(20\) −0.128154 − 0.251630i −0.0286561 − 0.0562661i
\(21\) −1.49927 + 4.61427i −0.327167 + 1.00692i
\(22\) −0.156597 + 0.481955i −0.0333865 + 0.102753i
\(23\) −3.42202 + 2.48625i −0.713541 + 0.518418i −0.884314 − 0.466892i \(-0.845374\pi\)
0.170773 + 0.985310i \(0.445374\pi\)
\(24\) − 6.25162i − 1.27611i
\(25\) 0.00182737 − 5.00000i 0.000365475 − 1.00000i
\(26\) 1.55572 0.305102
\(27\) −1.41523 + 1.02822i −0.272360 + 0.197881i
\(28\) −0.0827479 + 0.254672i −0.0156379 + 0.0481285i
\(29\) −4.63792 − 1.50695i −0.861241 − 0.279834i −0.155094 − 0.987900i \(-0.549568\pi\)
−0.706147 + 0.708066i \(0.749568\pi\)
\(30\) −3.38826 + 6.64683i −0.618609 + 1.21354i
\(31\) −9.91655 + 3.22208i −1.78106 + 0.578703i −0.999009 − 0.0445058i \(-0.985829\pi\)
−0.782056 + 0.623209i \(0.785829\pi\)
\(32\) − 0.713336i − 0.126101i
\(33\) 0.756265 − 0.245725i 0.131649 − 0.0427753i
\(34\) 4.61716 + 3.85081i 0.791836 + 0.660408i
\(35\) −3.35327 + 3.35204i −0.566806 + 0.566598i
\(36\) 0.228394 − 0.165938i 0.0380656 − 0.0276563i
\(37\) 4.75225 + 3.45271i 0.781266 + 0.567623i 0.905359 − 0.424648i \(-0.139602\pi\)
−0.124093 + 0.992271i \(0.539602\pi\)
\(38\) −3.72159 + 5.12233i −0.603722 + 0.830952i
\(39\) −1.43489 − 1.97496i −0.229766 − 0.316246i
\(40\) 2.77461 − 5.44302i 0.438704 − 0.860616i
\(41\) 2.15281 − 2.96309i 0.336212 − 0.462756i −0.607118 − 0.794612i \(-0.707675\pi\)
0.943330 + 0.331855i \(0.107675\pi\)
\(42\) 6.72844 − 2.18620i 1.03822 − 0.337338i
\(43\) 3.26028i 0.497188i 0.968608 + 0.248594i \(0.0799685\pi\)
−0.968608 + 0.248594i \(0.920032\pi\)
\(44\) 0.0417399 − 0.0135621i 0.00629253 − 0.00204457i
\(45\) 4.93728 − 0.781064i 0.736007 − 0.116434i
\(46\) 5.86601 + 1.90598i 0.864896 + 0.281022i
\(47\) −10.2608 − 3.33393i −1.49669 − 0.486303i −0.557638 − 0.830085i \(-0.688292\pi\)
−0.939050 + 0.343781i \(0.888292\pi\)
\(48\) −7.84252 + 5.69793i −1.13197 + 0.822425i
\(49\) −2.50388 −0.357698
\(50\) −5.90002 + 4.28332i −0.834389 + 0.605754i
\(51\) 0.629968 − 9.41309i 0.0882132 − 1.31810i
\(52\) −0.0791947 − 0.109002i −0.0109823 − 0.0151159i
\(53\) 2.85480 + 0.927580i 0.392137 + 0.127413i 0.498447 − 0.866920i \(-0.333904\pi\)
−0.106311 + 0.994333i \(0.533904\pi\)
\(54\) 2.42597 + 0.788245i 0.330132 + 0.107267i
\(55\) 0.767506 + 0.121705i 0.103490 + 0.0164107i
\(56\) −5.50984 + 1.79026i −0.736284 + 0.239233i
\(57\) 9.93522 1.31595
\(58\) 2.19741 + 6.76292i 0.288534 + 0.888015i
\(59\) 9.15031 + 6.64809i 1.19127 + 0.865507i 0.993398 − 0.114721i \(-0.0365973\pi\)
0.197871 + 0.980228i \(0.436597\pi\)
\(60\) 0.638192 − 0.100960i 0.0823903 − 0.0130339i
\(61\) −0.507392 − 0.698365i −0.0649649 − 0.0894165i 0.775298 − 0.631595i \(-0.217600\pi\)
−0.840263 + 0.542179i \(0.817600\pi\)
\(62\) 12.3005 + 8.93684i 1.56216 + 1.13498i
\(63\) −3.83484 − 2.78617i −0.483144 − 0.351025i
\(64\) 6.01349 − 4.36905i 0.751686 − 0.546132i
\(65\) −0.372767 − 2.35635i −0.0462361 − 0.292269i
\(66\) −0.938072 − 0.681549i −0.115469 − 0.0838929i
\(67\) 6.30092 − 2.04729i 0.769780 − 0.250117i 0.102309 − 0.994753i \(-0.467377\pi\)
0.667471 + 0.744636i \(0.267377\pi\)
\(68\) 0.0347693 − 0.519529i 0.00421640 − 0.0630022i
\(69\) −2.99079 − 9.20471i −0.360049 − 1.10812i
\(70\) 6.82845 + 1.08280i 0.816156 + 0.129419i
\(71\) 13.0325 + 4.23452i 1.54667 + 0.502545i 0.953208 − 0.302315i \(-0.0977594\pi\)
0.593465 + 0.804860i \(0.297759\pi\)
\(72\) 5.80886 + 1.88741i 0.684581 + 0.222434i
\(73\) −9.94242 + 7.22359i −1.16367 + 0.845458i −0.990238 − 0.139388i \(-0.955487\pi\)
−0.173435 + 0.984845i \(0.555487\pi\)
\(74\) − 8.56550i − 0.995720i
\(75\) 10.8793 + 3.53931i 1.25624 + 0.408684i
\(76\) 0.548346 0.0628997
\(77\) −0.433139 − 0.596164i −0.0493607 − 0.0679392i
\(78\) −1.10000 + 3.38545i −0.124551 + 0.383327i
\(79\) −8.36725 − 2.71868i −0.941389 − 0.305876i −0.202178 − 0.979349i \(-0.564802\pi\)
−0.739212 + 0.673473i \(0.764802\pi\)
\(80\) −9.35702 + 1.48025i −1.04615 + 0.165497i
\(81\) −3.30929 − 10.1849i −0.367698 − 1.13166i
\(82\) −5.34069 −0.589781
\(83\) 6.43351 − 2.09038i 0.706170 − 0.229449i 0.0661532 − 0.997809i \(-0.478927\pi\)
0.640017 + 0.768361i \(0.278927\pi\)
\(84\) −0.495690 − 0.360140i −0.0540842 − 0.0392945i
\(85\) 4.72623 − 7.91598i 0.512631 − 0.858609i
\(86\) 3.84613 − 2.79437i 0.414738 − 0.301325i
\(87\) 6.55865 − 9.02720i 0.703161 − 0.967818i
\(88\) 0.768177 + 0.558113i 0.0818879 + 0.0594951i
\(89\) −3.90634 + 2.83812i −0.414071 + 0.300840i −0.775248 − 0.631657i \(-0.782375\pi\)
0.361177 + 0.932497i \(0.382375\pi\)
\(90\) −5.15314 − 5.15502i −0.543189 − 0.543387i
\(91\) −1.32972 + 1.83020i −0.139392 + 0.191857i
\(92\) −0.165068 − 0.508028i −0.0172096 − 0.0529656i
\(93\) − 23.8579i − 2.47395i
\(94\) 4.86146 + 14.9620i 0.501422 + 1.54322i
\(95\) 8.65017 + 4.40947i 0.887489 + 0.452402i
\(96\) 1.55231 + 0.504376i 0.158432 + 0.0514777i
\(97\) −0.345614 + 1.06369i −0.0350918 + 0.108002i −0.967068 − 0.254518i \(-0.918083\pi\)
0.931976 + 0.362519i \(0.118083\pi\)
\(98\) 2.14607 + 2.95381i 0.216786 + 0.298380i
\(99\) 0.776891i 0.0780805i
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.q.a.339.11 yes 176
17.16 even 2 inner 425.2.q.a.339.12 yes 176
25.9 even 10 inner 425.2.q.a.84.12 yes 176
425.84 even 10 inner 425.2.q.a.84.11 ✓ 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.11 ✓ 176 425.84 even 10 inner
425.2.q.a.84.12 yes 176 25.9 even 10 inner
425.2.q.a.339.11 yes 176 1.1 even 1 trivial
425.2.q.a.339.12 yes 176 17.16 even 2 inner