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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 339.7
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.7

$q$-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} +(3.25033 - 2.53680i) 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} +(-8.58904 - 2.46755i) 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} +(-5.66197 + 7.27614i) 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 \(0.205210\pi\)
−0.999866 + 0.0163683i \(0.994790\pi\)
\(4\) −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 \(-0.661724\pi\)
−0.486493 + 0.873684i \(0.661724\pi\)
\(8\) 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 0.516198 − 0.856469i \(-0.327347\pi\)
−0.974065 + 0.226270i \(0.927347\pi\)
\(12\) −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
\(17\) 3.25033 − 2.53680i 0.788321 − 0.615264i
\(18\) − 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 −0.0159041 − 0.999874i \(-0.505063\pi\)
0.946022 + 0.324104i \(0.105063\pi\)
\(24\) 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 0.892312 − 0.451418i \(-0.149082\pi\)
0.456559 + 0.889693i \(0.349082\pi\)
\(30\) −4.61363 + 2.89866i −0.842329 + 0.529220i
\(31\) 5.00591 − 1.62652i 0.899088 − 0.292131i 0.177227 − 0.984170i \(-0.443287\pi\)
0.721860 + 0.692039i \(0.243287\pi\)
\(32\) − 7.61467i − 1.34610i
\(33\) 2.76240 − 0.897557i 0.480871 − 0.156245i
\(34\) −8.58904 − 2.46755i −1.47301 − 0.423181i
\(35\) 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 0.641717 − 0.766941i \(-0.278222\pi\)
−0.531103 + 0.847307i \(0.678222\pi\)
\(38\) 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 −0.370493 − 0.928835i \(-0.620811\pi\)
0.997863 + 0.0653338i \(0.0208112\pi\)
\(42\) −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 \(0.0757811\pi\)
−0.0760125 + 0.997107i \(0.524219\pi\)
\(62\) −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 0.346532 − 0.938038i \(-0.387359\pi\)
−0.271015 + 0.962575i \(0.587359\pi\)
\(72\) 2.49668 + 0.811219i 0.294236 + 0.0956031i
\(73\) 6.34682 − 4.61124i 0.742839 − 0.539704i −0.150760 − 0.988570i \(-0.548172\pi\)
0.893599 + 0.448866i \(0.148172\pi\)
\(74\) − 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 0.0599586 − 0.998201i \(-0.480903\pi\)
−0.538220 + 0.842804i \(0.680903\pi\)
\(80\) −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
\(85\) −5.66197 + 7.27614i −0.614126 + 0.789208i
\(86\) 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 \(0.223281\pi\)
−0.997326 + 0.0730747i \(0.976719\pi\)
\(98\) 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.339.7 yes 176
17.16 even 2 inner 425.2.q.a.339.8 yes 176
25.9 even 10 inner 425.2.q.a.84.8 yes 176
425.84 even 10 inner 425.2.q.a.84.7 ✓ 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.7 ✓ 176 425.84 even 10 inner
425.2.q.a.84.8 yes 176 25.9 even 10 inner
425.2.q.a.339.7 yes 176 1.1 even 1 trivial
425.2.q.a.339.8 yes 176 17.16 even 2 inner