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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 339.8
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.8

$q$-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+(-1.27397 - 1.75347i) q^{2} +(0.347412 - 1.06922i) q^{3} +(-0.833625 + 2.56564i) q^{4} +(2.16832 - 0.546269i) q^{5} +(-2.31745 + 0.752985i) q^{6} +2.57428 q^{7} +(1.43812 - 0.467275i) q^{8} +(1.40451 + 1.02043i) q^{9} +(-3.72024 - 3.10615i) q^{10} +(1.51857 + 2.09014i) q^{11} +(2.45363 + 1.78267i) q^{12} +(1.80492 - 2.48426i) q^{13} +(-3.27956 - 4.51393i) q^{14} +(0.169215 - 2.50820i) q^{15} +(1.71343 + 1.24488i) q^{16} +(4.12067 + 0.141819i) q^{17} -3.76276i q^{18} +(0.831872 + 2.56024i) q^{19} +(-0.406037 + 6.01849i) q^{20} +(0.894336 - 2.75248i) q^{21} +(1.73038 - 5.32555i) q^{22} +(-4.46069 + 3.24088i) q^{23} -1.70002i q^{24} +(4.40318 - 2.36897i) q^{25} -6.65551 q^{26} +(4.30762 - 3.12967i) q^{27} +(-2.14599 + 6.60466i) q^{28} +(-7.26390 - 2.36018i) q^{29} +(-4.61363 + 2.89866i) q^{30} +(-5.00591 + 1.62652i) q^{31} -7.61467i q^{32} +(2.76240 - 0.897557i) q^{33} +(-5.00094 - 7.40614i) q^{34} +(5.58185 - 1.40625i) q^{35} +(-3.78889 + 2.75279i) q^{36} +(-0.672839 - 0.488846i) q^{37} +(3.42952 - 4.72034i) q^{38} +(-2.02918 - 2.79293i) q^{39} +(2.86305 - 1.79880i) q^{40} +(-4.01713 + 5.52911i) q^{41} +(-5.96576 + 1.93839i) q^{42} +9.25232i q^{43} +(-6.62845 + 2.15371i) q^{44} +(3.60284 + 1.44538i) q^{45} +(11.3656 + 3.69290i) q^{46} +(-8.37265 - 2.72044i) q^{47} +(1.92633 - 1.39956i) q^{48} -0.373079 q^{49} +(-9.76344 - 4.70286i) q^{50} +(1.58320 - 4.35665i) q^{51} +(4.86908 + 6.70172i) q^{52} +(-12.7937 - 4.15692i) q^{53} +(-10.9756 - 3.56618i) q^{54} +(4.43452 + 3.70253i) q^{55} +(3.70214 - 1.20290i) q^{56} +3.02647 q^{57} +(5.11548 + 15.7438i) q^{58} +(-4.88543 - 3.54947i) q^{59} +(6.29405 + 2.52504i) q^{60} +(-6.99628 - 9.62955i) q^{61} +(9.22944 + 6.70558i) q^{62} +(3.61559 + 2.62688i) q^{63} +(-9.92525 + 7.21111i) q^{64} +(2.55657 - 6.37264i) q^{65} +(-5.09305 - 3.70032i) q^{66} +(13.1306 - 4.26637i) q^{67} +(-3.79895 + 10.4539i) q^{68} +(1.91553 + 5.89540i) q^{69} +(-9.57694 - 7.99610i) q^{70} +(-0.636317 - 0.206752i) q^{71} +(2.49668 + 0.811219i) q^{72} +(-6.34682 + 4.61124i) q^{73} +1.80258i q^{74} +(-1.00324 - 5.53100i) q^{75} -7.26211 q^{76} +(3.90923 + 5.38060i) q^{77} +(-2.31220 + 7.11623i) q^{78} +(4.25088 + 1.38120i) q^{79} +(4.39531 + 1.76330i) q^{80} +(-0.240381 - 0.739818i) q^{81} +14.8128 q^{82} +(6.25556 - 2.03256i) q^{83} +(6.31633 + 4.58908i) q^{84} +(9.01237 - 1.94348i) q^{85} +(16.2237 - 11.7872i) q^{86} +(-5.04713 + 6.94678i) q^{87} +(3.16057 + 2.29629i) q^{88} +(9.87913 - 7.17761i) q^{89} +(-2.05548 - 8.15886i) q^{90} +(4.64638 - 6.39519i) q^{91} +(-4.59637 - 14.1462i) q^{92} +5.91751i q^{93} +(5.89631 + 18.1470i) q^{94} +(3.20234 + 5.09698i) q^{95} +(-8.14180 - 2.64543i) q^{96} +(2.29897 - 7.07549i) q^{97} +(0.475293 + 0.654184i) q^{98} +4.48521i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27397 − 1.75347i −0.900834 − 1.23989i −0.970201 − 0.242301i \(-0.922098\pi\)
0.0693670 − 0.997591i \(-0.477902\pi\)
\(3\) 0.347412 − 1.06922i 0.200578 − 0.617317i −0.799288 − 0.600949i \(-0.794790\pi\)
0.999866 − 0.0163683i \(-0.00521043\pi\)
\(4\) −0.833625 + 2.56564i −0.416813 + 1.28282i
\(5\) 2.16832 − 0.546269i 0.969700 − 0.244299i
\(6\) −2.31745 + 0.752985i −0.946094 + 0.307405i
\(7\) 2.57428 0.972987 0.486493 − 0.873684i \(-0.338276\pi\)
0.486493 + 0.873684i \(0.338276\pi\)
\(8\) 1.43812 − 0.467275i 0.508454 − 0.165207i
\(9\) 1.40451 + 1.02043i 0.468168 + 0.340144i
\(10\) −3.72024 − 3.10615i −1.17644 − 0.982251i
\(11\) 1.51857 + 2.09014i 0.457867 + 0.630200i 0.974065 − 0.226270i \(-0.0726531\pi\)
−0.516198 + 0.856469i \(0.672653\pi\)
\(12\) 2.45363 + 1.78267i 0.708302 + 0.514611i
\(13\) 1.80492 − 2.48426i 0.500595 − 0.689010i −0.481703 − 0.876335i \(-0.659981\pi\)
0.982298 + 0.187324i \(0.0599815\pi\)
\(14\) −3.27956 − 4.51393i −0.876500 − 1.20640i
\(15\) 0.169215 − 2.50820i 0.0436912 − 0.647613i
\(16\) 1.71343 + 1.24488i 0.428359 + 0.311221i
\(17\) 4.12067 + 0.141819i 0.999408 + 0.0343960i
\(18\) − 3.76276i − 0.886892i
\(19\) 0.831872 + 2.56024i 0.190844 + 0.587359i 1.00000 0.000235666i \(-7.50149e-5\pi\)
−0.809155 + 0.587595i \(0.800075\pi\)
\(20\) −0.406037 + 6.01849i −0.0907926 + 1.34578i
\(21\) 0.894336 − 2.75248i 0.195160 − 0.600641i
\(22\) 1.73038 − 5.32555i 0.368917 − 1.13541i
\(23\) −4.46069 + 3.24088i −0.930117 + 0.675770i −0.946022 − 0.324104i \(-0.894937\pi\)
0.0159041 + 0.999874i \(0.494937\pi\)
\(24\) − 1.70002i − 0.347014i
\(25\) 4.40318 − 2.36897i 0.880636 − 0.473793i
\(26\) −6.65551 −1.30525
\(27\) 4.30762 − 3.12967i 0.829003 − 0.602306i
\(28\) −2.14599 + 6.60466i −0.405553 + 1.24816i
\(29\) −7.26390 − 2.36018i −1.34887 − 0.438275i −0.456559 − 0.889693i \(-0.650918\pi\)
−0.892312 + 0.451418i \(0.850918\pi\)
\(30\) −4.61363 + 2.89866i −0.842329 + 0.529220i
\(31\) −5.00591 + 1.62652i −0.899088 + 0.292131i −0.721860 − 0.692039i \(-0.756713\pi\)
−0.177227 + 0.984170i \(0.556713\pi\)
\(32\) − 7.61467i − 1.34610i
\(33\) 2.76240 − 0.897557i 0.480871 − 0.156245i
\(34\) −5.00094 − 7.40614i −0.857654 − 1.27014i
\(35\) 5.58185 − 1.40625i 0.943505 − 0.237699i
\(36\) −3.78889 + 2.75279i −0.631482 + 0.458798i
\(37\) −0.672839 − 0.488846i −0.110614 − 0.0803658i 0.531103 − 0.847307i \(-0.321778\pi\)
−0.641717 + 0.766941i \(0.721778\pi\)
\(38\) 3.42952 − 4.72034i 0.556342 − 0.765740i
\(39\) −2.02918 − 2.79293i −0.324929 − 0.447227i
\(40\) 2.86305 − 1.79880i 0.452688 − 0.284416i
\(41\) −4.01713 + 5.52911i −0.627371 + 0.863501i −0.997863 − 0.0653338i \(-0.979189\pi\)
0.370493 + 0.928835i \(0.379189\pi\)
\(42\) −5.96576 + 1.93839i −0.920537 + 0.299101i
\(43\) 9.25232i 1.41097i 0.708727 + 0.705483i \(0.249270\pi\)
−0.708727 + 0.705483i \(0.750730\pi\)
\(44\) −6.62845 + 2.15371i −0.999276 + 0.324684i
\(45\) 3.60284 + 1.44538i 0.537080 + 0.215465i
\(46\) 11.3656 + 3.69290i 1.67576 + 0.544488i
\(47\) −8.37265 − 2.72044i −1.22128 − 0.396817i −0.373730 − 0.927538i \(-0.621921\pi\)
−0.847547 + 0.530721i \(0.821921\pi\)
\(48\) 1.92633 − 1.39956i 0.278041 − 0.202009i
\(49\) −0.373079 −0.0532971
\(50\) −9.76344 − 4.70286i −1.38076 − 0.665085i
\(51\) 1.58320 − 4.35665i 0.221693 − 0.610053i
\(52\) 4.86908 + 6.70172i 0.675220 + 0.929361i
\(53\) −12.7937 − 4.15692i −1.75735 − 0.570996i −0.760426 − 0.649425i \(-0.775010\pi\)
−0.996920 + 0.0784284i \(0.975010\pi\)
\(54\) −10.9756 − 3.56618i −1.49359 − 0.485296i
\(55\) 4.43452 + 3.70253i 0.597951 + 0.499248i
\(56\) 3.70214 − 1.20290i 0.494719 − 0.160744i
\(57\) 3.02647 0.400866
\(58\) 5.11548 + 15.7438i 0.671696 + 2.06727i
\(59\) −4.88543 − 3.54947i −0.636029 − 0.462102i 0.222455 − 0.974943i \(-0.428593\pi\)
−0.858484 + 0.512841i \(0.828593\pi\)
\(60\) 6.29405 + 2.52504i 0.812559 + 0.325981i
\(61\) −6.99628 − 9.62955i −0.895782 − 1.23294i −0.971794 − 0.235831i \(-0.924219\pi\)
0.0760125 − 0.997107i \(-0.475781\pi\)
\(62\) 9.22944 + 6.70558i 1.17214 + 0.851610i
\(63\) 3.61559 + 2.62688i 0.455522 + 0.330956i
\(64\) −9.92525 + 7.21111i −1.24066 + 0.901389i
\(65\) 2.55657 − 6.37264i 0.317103 − 0.790428i
\(66\) −5.09305 − 3.70032i −0.626912 − 0.455478i
\(67\) 13.1306 − 4.26637i 1.60415 − 0.521221i 0.636022 − 0.771671i \(-0.280579\pi\)
0.968130 + 0.250450i \(0.0805785\pi\)
\(68\) −3.79895 + 10.4539i −0.460690 + 1.26772i
\(69\) 1.91553 + 5.89540i 0.230603 + 0.709722i
\(70\) −9.57694 − 7.99610i −1.14466 − 0.955717i
\(71\) −0.636317 − 0.206752i −0.0755170 − 0.0245369i 0.271015 − 0.962575i \(-0.412641\pi\)
−0.346532 + 0.938038i \(0.612641\pi\)
\(72\) 2.49668 + 0.811219i 0.294236 + 0.0956031i
\(73\) −6.34682 + 4.61124i −0.742839 + 0.539704i −0.893599 − 0.448866i \(-0.851828\pi\)
0.150760 + 0.988570i \(0.451828\pi\)
\(74\) 1.80258i 0.209546i
\(75\) −1.00324 − 5.53100i −0.115844 − 0.638664i
\(76\) −7.26211 −0.833021
\(77\) 3.90923 + 5.38060i 0.445498 + 0.613176i
\(78\) −2.31220 + 7.11623i −0.261805 + 0.805754i
\(79\) 4.25088 + 1.38120i 0.478262 + 0.155397i 0.538220 − 0.842804i \(-0.319097\pi\)
−0.0599586 + 0.998201i \(0.519097\pi\)
\(80\) 4.39531 + 1.76330i 0.491410 + 0.197143i
\(81\) −0.240381 − 0.739818i −0.0267091 − 0.0822020i
\(82\) 14.8128 1.63581
\(83\) 6.25556 − 2.03256i 0.686637 − 0.223102i 0.0551383 − 0.998479i \(-0.482440\pi\)
0.631499 + 0.775377i \(0.282440\pi\)
\(84\) 6.31633 + 4.58908i 0.689168 + 0.500710i
\(85\) 9.01237 − 1.94348i 0.977529 − 0.210800i
\(86\) 16.2237 − 11.7872i 1.74944 − 1.27105i
\(87\) −5.04713 + 6.94678i −0.541109 + 0.744773i
\(88\) 3.16057 + 2.29629i 0.336917 + 0.244785i
\(89\) 9.87913 − 7.17761i 1.04719 − 0.760825i 0.0755106 − 0.997145i \(-0.475941\pi\)
0.971675 + 0.236320i \(0.0759413\pi\)
\(90\) −2.05548 − 8.15886i −0.216667 − 0.860019i
\(91\) 4.64638 − 6.39519i 0.487073 − 0.670398i
\(92\) −4.59637 − 14.1462i −0.479205 − 1.47484i
\(93\) 5.91751i 0.613617i
\(94\) 5.89631 + 18.1470i 0.608158 + 1.87172i
\(95\) 3.20234 + 5.09698i 0.328553 + 0.522939i
\(96\) −8.14180 − 2.64543i −0.830969 − 0.269998i
\(97\) 2.29897 − 7.07549i 0.233425 − 0.718407i −0.763902 − 0.645332i \(-0.776719\pi\)
0.997326 − 0.0730747i \(-0.0232811\pi\)
\(98\) 0.475293 + 0.654184i 0.0480118 + 0.0660826i
\(99\) 4.48521i 0.450780i
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.339.8 yes 176
17.16 even 2 inner 425.2.q.a.339.7 yes 176
25.9 even 10 inner 425.2.q.a.84.7 ✓ 176
425.84 even 10 inner 425.2.q.a.84.8 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.7 ✓ 176 25.9 even 10 inner
425.2.q.a.84.8 yes 176 425.84 even 10 inner
425.2.q.a.339.7 yes 176 17.16 even 2 inner
425.2.q.a.339.8 yes 176 1.1 even 1 trivial