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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 339.17
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.17

$q$-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+(-0.568609 - 0.782624i) q^{2} +(-0.724084 + 2.22850i) q^{3} +(0.328851 - 1.01210i) q^{4} +(-1.46866 - 1.68613i) q^{5} +(2.15580 - 0.700461i) q^{6} -0.961359 q^{7} +(-2.81914 + 0.915994i) q^{8} +(-2.01486 - 1.46388i) q^{9} +(-0.484512 + 2.10816i) q^{10} +(3.16360 + 4.35432i) q^{11} +(2.01735 + 1.46569i) q^{12} +(-0.0411761 + 0.0566741i) q^{13} +(0.546637 + 0.752382i) q^{14} +(4.82098 - 2.05201i) q^{15} +(0.597981 + 0.434459i) q^{16} +(2.53031 + 3.25539i) q^{17} +2.40926i q^{18} +(0.700717 + 2.15658i) q^{19} +(-2.18950 + 0.931944i) q^{20} +(0.696104 - 2.14239i) q^{21} +(1.60894 - 4.95181i) q^{22} +(-4.45324 + 3.23547i) q^{23} -6.94571i q^{24} +(-0.686075 + 4.95271i) q^{25} +0.0677676 q^{26} +(-0.965828 + 0.701715i) q^{27} +(-0.316144 + 0.972990i) q^{28} +(5.43761 + 1.76679i) q^{29} +(-4.34720 - 2.60622i) q^{30} +(3.78449 - 1.22965i) q^{31} +5.21340i q^{32} +(-11.9943 + 3.89718i) q^{33} +(1.10898 - 3.83132i) q^{34} +(1.41191 + 1.62098i) q^{35} +(-2.14419 + 1.55784i) q^{36} +(-7.21286 - 5.24045i) q^{37} +(1.28936 - 1.77465i) q^{38} +(-0.0964832 - 0.132798i) q^{39} +(5.68484 + 3.40815i) q^{40} +(-3.96838 + 5.46201i) q^{41} +(-2.07249 + 0.673394i) q^{42} +7.97907i q^{43} +(5.44735 - 1.76995i) q^{44} +(0.490850 + 5.54728i) q^{45} +(5.06430 + 1.64549i) q^{46} +(10.1712 + 3.30482i) q^{47} +(-1.40118 + 1.01802i) q^{48} -6.07579 q^{49} +(4.26621 - 2.27922i) q^{50} +(-9.08678 + 3.28163i) q^{51} +(0.0438190 + 0.0603116i) q^{52} +(-1.39170 - 0.452190i) q^{53} +(1.09836 + 0.356878i) q^{54} +(2.69570 - 11.7292i) q^{55} +(2.71020 - 0.880598i) q^{56} -5.31333 q^{57} +(-1.70915 - 5.26021i) q^{58} +(2.12346 + 1.54279i) q^{59} +(-0.491455 - 5.55411i) q^{60} +(-4.11921 - 5.66960i) q^{61} +(-3.11425 - 2.26264i) q^{62} +(1.93701 + 1.40732i) q^{63} +(5.27610 - 3.83331i) q^{64} +(0.156034 - 0.0138066i) q^{65} +(9.87010 + 7.17105i) q^{66} +(13.5466 - 4.40156i) q^{67} +(4.12687 - 1.49039i) q^{68} +(-3.98572 - 12.2668i) q^{69} +(0.465790 - 2.02670i) q^{70} +(-3.77479 - 1.22651i) q^{71} +(7.02109 + 2.28129i) q^{72} +(8.90514 - 6.46996i) q^{73} +8.62472i q^{74} +(-10.5403 - 5.11509i) q^{75} +2.41311 q^{76} +(-3.04135 - 4.18606i) q^{77} +(-0.0490694 + 0.151020i) q^{78} +(-5.56411 - 1.80789i) q^{79} +(-0.145677 - 1.64635i) q^{80} +(-3.17326 - 9.76629i) q^{81} +6.53116 q^{82} +(-1.02862 + 0.334220i) q^{83} +(-1.93939 - 1.40905i) q^{84} +(1.77284 - 9.04749i) q^{85} +(6.24461 - 4.53698i) q^{86} +(-7.87456 + 10.8384i) q^{87} +(-12.9071 - 9.37758i) q^{88} +(-11.0652 + 8.03934i) q^{89} +(4.06233 - 3.53838i) q^{90} +(0.0395850 - 0.0544841i) q^{91} +(1.81016 + 5.57110i) q^{92} +9.32410i q^{93} +(-3.19700 - 9.83936i) q^{94} +(2.60717 - 4.34879i) q^{95} +(-11.6181 - 3.77494i) q^{96} +(-0.704269 + 2.16752i) q^{97} +(3.45475 + 4.75506i) q^{98} -13.4045i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.568609 − 0.782624i −0.402068 − 0.553398i 0.559194 − 0.829037i \(-0.311111\pi\)
−0.961261 + 0.275638i \(0.911111\pi\)
\(3\) −0.724084 + 2.22850i −0.418050 + 1.28663i 0.491445 + 0.870909i \(0.336469\pi\)
−0.909494 + 0.415716i \(0.863531\pi\)
\(4\) 0.328851 − 1.01210i 0.164425 − 0.506049i
\(5\) −1.46866 − 1.68613i −0.656805 − 0.754061i
\(6\) 2.15580 − 0.700461i 0.880101 − 0.285962i
\(7\) −0.961359 −0.363359 −0.181680 − 0.983358i \(-0.558153\pi\)
−0.181680 + 0.983358i \(0.558153\pi\)
\(8\) −2.81914 + 0.915994i −0.996716 + 0.323853i
\(9\) −2.01486 − 1.46388i −0.671622 − 0.487962i
\(10\) −0.484512 + 2.10816i −0.153216 + 0.666658i
\(11\) 3.16360 + 4.35432i 0.953860 + 1.31288i 0.949791 + 0.312884i \(0.101295\pi\)
0.00406835 + 0.999992i \(0.498705\pi\)
\(12\) 2.01735 + 1.46569i 0.582358 + 0.423108i
\(13\) −0.0411761 + 0.0566741i −0.0114202 + 0.0157186i −0.814689 − 0.579898i \(-0.803092\pi\)
0.803269 + 0.595617i \(0.203092\pi\)
\(14\) 0.546637 + 0.752382i 0.146095 + 0.201083i
\(15\) 4.82098 − 2.05201i 1.24477 − 0.529827i
\(16\) 0.597981 + 0.434459i 0.149495 + 0.108615i
\(17\) 2.53031 + 3.25539i 0.613690 + 0.789547i
\(18\) 2.40926i 0.567868i
\(19\) 0.700717 + 2.15658i 0.160755 + 0.494754i 0.998698 − 0.0510033i \(-0.0162419\pi\)
−0.837943 + 0.545758i \(0.816242\pi\)
\(20\) −2.18950 + 0.931944i −0.489587 + 0.208389i
\(21\) 0.696104 − 2.14239i 0.151902 − 0.467507i
\(22\) 1.60894 − 4.95181i 0.343027 − 1.05573i
\(23\) −4.45324 + 3.23547i −0.928564 + 0.674641i −0.945641 − 0.325213i \(-0.894564\pi\)
0.0170768 + 0.999854i \(0.494564\pi\)
\(24\) − 6.94571i − 1.41779i
\(25\) −0.686075 + 4.95271i −0.137215 + 0.990541i
\(26\) 0.0677676 0.0132903
\(27\) −0.965828 + 0.701715i −0.185874 + 0.135045i
\(28\) −0.316144 + 0.972990i −0.0597455 + 0.183878i
\(29\) 5.43761 + 1.76679i 1.00974 + 0.328084i 0.766750 − 0.641945i \(-0.221872\pi\)
0.242988 + 0.970029i \(0.421872\pi\)
\(30\) −4.34720 − 2.60622i −0.793687 − 0.475828i
\(31\) 3.78449 − 1.22965i 0.679714 − 0.220852i 0.0512438 − 0.998686i \(-0.483681\pi\)
0.628470 + 0.777834i \(0.283681\pi\)
\(32\) 5.21340i 0.921608i
\(33\) −11.9943 + 3.89718i −2.08794 + 0.678413i
\(34\) 1.10898 − 3.83132i 0.190189 − 0.657066i
\(35\) 1.41191 + 1.62098i 0.238656 + 0.273995i
\(36\) −2.14419 + 1.55784i −0.357364 + 0.259640i
\(37\) −7.21286 − 5.24045i −1.18579 − 0.861525i −0.192974 − 0.981204i \(-0.561813\pi\)
−0.992813 + 0.119679i \(0.961813\pi\)
\(38\) 1.28936 − 1.77465i 0.209162 − 0.287887i
\(39\) −0.0964832 − 0.132798i −0.0154497 − 0.0212647i
\(40\) 5.68484 + 3.40815i 0.898852 + 0.538876i
\(41\) −3.96838 + 5.46201i −0.619757 + 0.853022i −0.997335 − 0.0729553i \(-0.976757\pi\)
0.377578 + 0.925978i \(0.376757\pi\)
\(42\) −2.07249 + 0.673394i −0.319793 + 0.103907i
\(43\) 7.97907i 1.21680i 0.793632 + 0.608399i \(0.208188\pi\)
−0.793632 + 0.608399i \(0.791812\pi\)
\(44\) 5.44735 − 1.76995i 0.821219 − 0.266830i
\(45\) 0.490850 + 5.54728i 0.0731716 + 0.826939i
\(46\) 5.06430 + 1.64549i 0.746691 + 0.242615i
\(47\) 10.1712 + 3.30482i 1.48362 + 0.482058i 0.935192 − 0.354140i \(-0.115226\pi\)
0.548428 + 0.836198i \(0.315226\pi\)
\(48\) −1.40118 + 1.01802i −0.202243 + 0.146938i
\(49\) −6.07579 −0.867970
\(50\) 4.26621 − 2.27922i 0.603334 − 0.322330i
\(51\) −9.08678 + 3.28163i −1.27240 + 0.459519i
\(52\) 0.0438190 + 0.0603116i 0.00607660 + 0.00836372i
\(53\) −1.39170 − 0.452190i −0.191164 − 0.0621131i 0.211870 − 0.977298i \(-0.432045\pi\)
−0.403035 + 0.915185i \(0.632045\pi\)
\(54\) 1.09836 + 0.356878i 0.149468 + 0.0485649i
\(55\) 2.69570 − 11.7292i 0.363488 − 1.58157i
\(56\) 2.71020 − 0.880598i 0.362166 − 0.117675i
\(57\) −5.31333 −0.703767
\(58\) −1.70915 − 5.26021i −0.224422 − 0.690699i
\(59\) 2.12346 + 1.54279i 0.276451 + 0.200854i 0.717368 − 0.696694i \(-0.245347\pi\)
−0.440917 + 0.897548i \(0.645347\pi\)
\(60\) −0.491455 − 5.55411i −0.0634465 − 0.717032i
\(61\) −4.11921 − 5.66960i −0.527411 − 0.725918i 0.459322 − 0.888270i \(-0.348092\pi\)
−0.986733 + 0.162351i \(0.948092\pi\)
\(62\) −3.11425 − 2.26264i −0.395510 − 0.287355i
\(63\) 1.93701 + 1.40732i 0.244040 + 0.177305i
\(64\) 5.27610 − 3.83331i 0.659512 − 0.479163i
\(65\) 0.156034 − 0.0138066i 0.0193536 − 0.00171250i
\(66\) 9.87010 + 7.17105i 1.21493 + 0.882695i
\(67\) 13.5466 − 4.40156i 1.65498 − 0.537736i 0.675171 − 0.737662i \(-0.264070\pi\)
0.979811 + 0.199926i \(0.0640700\pi\)
\(68\) 4.12687 − 1.49039i 0.500456 − 0.180736i
\(69\) −3.98572 − 12.2668i −0.479824 − 1.47675i
\(70\) 0.465790 − 2.02670i 0.0556725 − 0.242236i
\(71\) −3.77479 − 1.22651i −0.447986 − 0.145559i 0.0763313 − 0.997083i \(-0.475679\pi\)
−0.524317 + 0.851523i \(0.675679\pi\)
\(72\) 7.02109 + 2.28129i 0.827444 + 0.268853i
\(73\) 8.90514 − 6.46996i 1.04227 − 0.757252i 0.0715406 − 0.997438i \(-0.477208\pi\)
0.970727 + 0.240186i \(0.0772085\pi\)
\(74\) 8.62472i 1.00260i
\(75\) −10.5403 − 5.11509i −1.21709 − 0.590640i
\(76\) 2.41311 0.276803
\(77\) −3.04135 − 4.18606i −0.346594 − 0.477046i
\(78\) −0.0490694 + 0.151020i −0.00555601 + 0.0170997i
\(79\) −5.56411 − 1.80789i −0.626012 − 0.203404i −0.0212042 − 0.999775i \(-0.506750\pi\)
−0.604808 + 0.796372i \(0.706750\pi\)
\(80\) −0.145677 − 1.64635i −0.0162872 − 0.184067i
\(81\) −3.17326 − 9.76629i −0.352585 − 1.08514i
\(82\) 6.53116 0.721245
\(83\) −1.02862 + 0.334220i −0.112906 + 0.0366854i −0.364925 − 0.931037i \(-0.618905\pi\)
0.252019 + 0.967722i \(0.418905\pi\)
\(84\) −1.93939 − 1.40905i −0.211605 − 0.153740i
\(85\) 1.77284 − 9.04749i 0.192292 − 0.981338i
\(86\) 6.24461 − 4.53698i 0.673374 − 0.489235i
\(87\) −7.87456 + 10.8384i −0.844242 + 1.16200i
\(88\) −12.9071 − 9.37758i −1.37591 − 0.999654i
\(89\) −11.0652 + 8.03934i −1.17291 + 0.852168i −0.991354 − 0.131213i \(-0.958113\pi\)
−0.181555 + 0.983381i \(0.558113\pi\)
\(90\) 4.06233 − 3.53838i 0.428207 − 0.372978i
\(91\) 0.0395850 − 0.0544841i 0.00414964 − 0.00571148i
\(92\) 1.81016 + 5.57110i 0.188722 + 0.580827i
\(93\) 9.32410i 0.966864i
\(94\) −3.19700 − 9.83936i −0.329746 − 1.01485i
\(95\) 2.60717 − 4.34879i 0.267490 − 0.446176i
\(96\) −11.6181 − 3.77494i −1.18576 − 0.385278i
\(97\) −0.704269 + 2.16752i −0.0715077 + 0.220078i −0.980423 − 0.196903i \(-0.936912\pi\)
0.908915 + 0.416981i \(0.136912\pi\)
\(98\) 3.45475 + 4.75506i 0.348983 + 0.480333i
\(99\) − 13.4045i − 1.34720i
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.339.17 yes 176
17.16 even 2 inner 425.2.q.a.339.18 yes 176
25.9 even 10 inner 425.2.q.a.84.18 yes 176
425.84 even 10 inner 425.2.q.a.84.17 ✓ 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.17 ✓ 176 425.84 even 10 inner
425.2.q.a.84.18 yes 176 25.9 even 10 inner
425.2.q.a.339.17 yes 176 1.1 even 1 trivial
425.2.q.a.339.18 yes 176 17.16 even 2 inner