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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 84.11
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.11

$q$-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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(-1\)

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 −0.996246 − 0.0865675i \(-0.972410\pi\)
0.390187 + 0.920735i \(0.372410\pi\)
\(3\) −0.707067 − 2.17613i −0.408225 − 1.25639i −0.918172 − 0.396182i \(-0.870335\pi\)
0.509947 − 0.860206i \(-0.329665\pi\)
\(4\) −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 −0.838701 − 0.544592i \(-0.816684\pi\)
0.777111 + 0.629364i \(0.216684\pi\)
\(12\) −0.233772 + 0.169845i −0.0674840 + 0.0490300i
\(13\) −0.627105 − 0.863136i −0.173928 − 0.239391i 0.713150 − 0.701012i \(-0.247268\pi\)
−0.887077 + 0.461621i \(0.847268\pi\)
\(14\) −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 0.670782 + 0.741655i \(0.265959\pi\)
−0.978608 + 0.205735i \(0.934041\pi\)
\(20\) −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 0.170773 − 0.985310i \(-0.445374\pi\)
−0.884314 + 0.466892i \(0.845374\pi\)
\(24\) 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 −0.706147 − 0.708066i \(-0.749568\pi\)
−0.155094 + 0.987900i \(0.549568\pi\)
\(30\) −3.38826 − 6.64683i −0.618609 − 1.21354i
\(31\) −9.91655 − 3.22208i −1.78106 − 0.578703i −0.782056 − 0.623209i \(-0.785829\pi\)
−0.999009 + 0.0445058i \(0.985829\pi\)
\(32\) 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 −0.124093 − 0.992271i \(-0.539602\pi\)
0.905359 + 0.424648i \(0.139602\pi\)
\(38\) −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 0.943330 − 0.331855i \(-0.107675\pi\)
−0.607118 + 0.794612i \(0.707675\pi\)
\(42\) 6.72844 + 2.18620i 1.03822 + 0.337338i
\(43\) − 3.26028i − 0.497188i −0.968608 − 0.248594i \(-0.920032\pi\)
0.968608 − 0.248594i \(-0.0799685\pi\)
\(44\) 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 −0.939050 − 0.343781i \(-0.888292\pi\)
−0.557638 + 0.830085i \(0.688292\pi\)
\(48\) −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 −0.106311 − 0.994333i \(-0.533904\pi\)
0.498447 + 0.866920i \(0.333904\pi\)
\(54\) 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 0.197871 − 0.980228i \(-0.436597\pi\)
0.993398 + 0.114721i \(0.0365973\pi\)
\(60\) 0.638192 + 0.100960i 0.0823903 + 0.0130339i
\(61\) −0.507392 + 0.698365i −0.0649649 + 0.0894165i −0.840263 − 0.542179i \(-0.817600\pi\)
0.775298 + 0.631595i \(0.217600\pi\)
\(62\) 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 0.667471 − 0.744636i \(-0.267377\pi\)
0.102309 + 0.994753i \(0.467377\pi\)
\(68\) 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 0.593465 − 0.804860i \(-0.297759\pi\)
0.953208 + 0.302315i \(0.0977594\pi\)
\(72\) 5.80886 − 1.88741i 0.684581 − 0.222434i
\(73\) −9.94242 − 7.22359i −1.16367 − 0.845458i −0.173435 − 0.984845i \(-0.555487\pi\)
−0.990238 + 0.139388i \(0.955487\pi\)
\(74\) 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 −0.739212 − 0.673473i \(-0.764802\pi\)
−0.202178 + 0.979349i \(0.564802\pi\)
\(80\) −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 0.640017 − 0.768361i \(-0.278927\pi\)
0.0661532 + 0.997809i \(0.478927\pi\)
\(84\) −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 0.361177 − 0.932497i \(-0.382375\pi\)
−0.775248 + 0.631657i \(0.782375\pi\)
\(90\) −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 0.931976 − 0.362519i \(-0.118083\pi\)
−0.967068 + 0.254518i \(0.918083\pi\)
\(98\) 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.84.11 ✓ 176
17.16 even 2 inner 425.2.q.a.84.12 yes 176
25.14 even 10 inner 425.2.q.a.339.12 yes 176
425.339 even 10 inner 425.2.q.a.339.11 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.11 ✓ 176 1.1 even 1 trivial
425.2.q.a.84.12 yes 176 17.16 even 2 inner
425.2.q.a.339.11 yes 176 425.339 even 10 inner
425.2.q.a.339.12 yes 176 25.14 even 10 inner