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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 84.13
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.13

$q$-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+(-0.753629 + 1.03728i) q^{2} +(-0.0324420 - 0.0998462i) q^{3} +(0.110039 + 0.338664i) q^{4} +(0.863935 - 2.06243i) q^{5} +(0.128018 + 0.0415955i) q^{6} -3.13290 q^{7} +(-2.87301 - 0.933498i) q^{8} +(2.41813 - 1.75688i) q^{9} +(1.48823 + 2.45045i) q^{10} +(2.43082 - 3.34573i) q^{11} +(0.0302445 - 0.0219739i) q^{12} +(3.08588 + 4.24735i) q^{13} +(2.36104 - 3.24969i) q^{14} +(-0.233954 - 0.0193513i) q^{15} +(2.55731 - 1.85800i) q^{16} +(0.210024 - 4.11775i) q^{17} +3.83232i q^{18} +(1.19454 - 3.67643i) q^{19} +(0.793537 + 0.0656370i) q^{20} +(0.101637 + 0.312808i) q^{21} +(1.63853 + 5.04288i) q^{22} +(3.18703 + 2.31551i) q^{23} +0.317144i q^{24} +(-3.50723 - 3.56361i) q^{25} -6.73131 q^{26} +(-0.508669 - 0.369570i) q^{27} +(-0.344740 - 1.06100i) q^{28} +(-1.85513 + 0.602768i) q^{29} +(0.196387 - 0.228092i) q^{30} +(1.41366 + 0.459326i) q^{31} -1.98883i q^{32} +(-0.412919 - 0.134165i) q^{33} +(4.11299 + 3.32111i) q^{34} +(-2.70662 + 6.46138i) q^{35} +(0.861080 + 0.625611i) q^{36} +(8.02629 - 5.83144i) q^{37} +(2.91325 + 4.00974i) q^{38} +(0.323970 - 0.445906i) q^{39} +(-4.40737 + 5.11890i) q^{40} +(0.967857 + 1.33214i) q^{41} +(-0.401066 - 0.130314i) q^{42} +9.66183i q^{43} +(1.40056 + 0.455070i) q^{44} +(-1.53432 - 6.50506i) q^{45} +(-4.80368 + 1.56081i) q^{46} +(-3.26651 + 1.06135i) q^{47} +(-0.268478 - 0.195061i) q^{48} +2.81504 q^{49} +(6.33962 - 0.952344i) q^{50} +(-0.417956 + 0.112618i) q^{51} +(-1.09886 + 1.51245i) q^{52} +(11.4469 - 3.71933i) q^{53} +(0.766695 - 0.249114i) q^{54} +(-4.80027 - 7.90388i) q^{55} +(9.00085 + 2.92455i) q^{56} -0.405831 q^{57} +(0.772839 - 2.37855i) q^{58} +(5.52109 - 4.01131i) q^{59} +(-0.0191903 - 0.0813611i) q^{60} +(-3.31802 + 4.56686i) q^{61} +(-1.54182 + 1.12020i) q^{62} +(-7.57576 + 5.50411i) q^{63} +(7.17761 + 5.21484i) q^{64} +(11.4259 - 2.69498i) q^{65} +(0.450355 - 0.327202i) q^{66} +(-11.6806 - 3.79526i) q^{67} +(1.41765 - 0.381984i) q^{68} +(0.127802 - 0.393333i) q^{69} +(-4.66248 - 7.67700i) q^{70} +(-7.01490 + 2.27928i) q^{71} +(-8.58737 + 2.79021i) q^{72} +(-9.22500 - 6.70236i) q^{73} +12.7203i q^{74} +(-0.242031 + 0.465794i) q^{75} +1.37652 q^{76} +(-7.61549 + 10.4818i) q^{77} +(0.218377 + 0.672095i) q^{78} +(-10.1228 + 3.28910i) q^{79} +(-1.62263 - 6.87947i) q^{80} +(2.75054 - 8.46528i) q^{81} -2.11121 q^{82} +(4.45687 + 1.44813i) q^{83} +(-0.0947527 + 0.0688419i) q^{84} +(-8.31113 - 3.99063i) q^{85} +(-10.0220 - 7.28143i) q^{86} +(0.120368 + 0.165673i) q^{87} +(-10.1070 + 7.34316i) q^{88} +(5.78530 + 4.20327i) q^{89} +(7.90388 + 3.31087i) q^{90} +(-9.66775 - 13.3065i) q^{91} +(-0.433485 + 1.33413i) q^{92} -0.156050i q^{93} +(1.36081 - 4.18815i) q^{94} +(-6.55036 - 5.63986i) q^{95} +(-0.198578 + 0.0645218i) q^{96} +(-1.25686 - 3.86823i) q^{97} +(-2.12150 + 2.91999i) q^{98} -12.3611i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.753629 + 1.03728i −0.532896 + 0.733468i −0.987568 − 0.157190i \(-0.949756\pi\)
0.454672 + 0.890659i \(0.349756\pi\)
\(3\) −0.0324420 − 0.0998462i −0.0187304 − 0.0576462i 0.941254 − 0.337698i \(-0.109648\pi\)
−0.959985 + 0.280052i \(0.909648\pi\)
\(4\) 0.110039 + 0.338664i 0.0550193 + 0.169332i
\(5\) 0.863935 − 2.06243i 0.386364 − 0.922347i
\(6\) 0.128018 + 0.0415955i 0.0522630 + 0.0169813i
\(7\) −3.13290 −1.18412 −0.592062 − 0.805893i \(-0.701686\pi\)
−0.592062 + 0.805893i \(0.701686\pi\)
\(8\) −2.87301 − 0.933498i −1.01576 − 0.330041i
\(9\) 2.41813 − 1.75688i 0.806045 − 0.585626i
\(10\) 1.48823 + 2.45045i 0.470620 + 0.774900i
\(11\) 2.43082 − 3.34573i 0.732918 − 1.00878i −0.266076 − 0.963952i \(-0.585727\pi\)
0.998995 − 0.0448238i \(-0.0142726\pi\)
\(12\) 0.0302445 − 0.0219739i 0.00873082 − 0.00634331i
\(13\) 3.08588 + 4.24735i 0.855870 + 1.17800i 0.982539 + 0.186059i \(0.0595715\pi\)
−0.126669 + 0.991945i \(0.540429\pi\)
\(14\) 2.36104 − 3.24969i 0.631015 − 0.868517i
\(15\) −0.233954 − 0.0193513i −0.0604065 − 0.00499649i
\(16\) 2.55731 − 1.85800i 0.639328 − 0.464499i
\(17\) 0.210024 − 4.11775i 0.0509384 − 0.998702i
\(18\) 3.83232i 0.903286i
\(19\) 1.19454 − 3.67643i 0.274047 − 0.843430i −0.715423 − 0.698692i \(-0.753766\pi\)
0.989470 − 0.144739i \(-0.0462341\pi\)
\(20\) 0.793537 + 0.0656370i 0.177440 + 0.0146769i
\(21\) 0.101637 + 0.312808i 0.0221791 + 0.0682603i
\(22\) 1.63853 + 5.04288i 0.349336 + 1.07514i
\(23\) 3.18703 + 2.31551i 0.664542 + 0.482818i 0.868194 − 0.496225i \(-0.165281\pi\)
−0.203652 + 0.979043i \(0.565281\pi\)
\(24\) 0.317144i 0.0647367i
\(25\) −3.50723 − 3.56361i −0.701446 − 0.712722i
\(26\) −6.73131 −1.32012
\(27\) −0.508669 − 0.369570i −0.0978935 − 0.0711238i
\(28\) −0.344740 − 1.06100i −0.0651497 − 0.200510i
\(29\) −1.85513 + 0.602768i −0.344489 + 0.111931i −0.476151 − 0.879363i \(-0.657969\pi\)
0.131662 + 0.991295i \(0.457969\pi\)
\(30\) 0.196387 − 0.228092i 0.0358552 − 0.0416437i
\(31\) 1.41366 + 0.459326i 0.253901 + 0.0824973i 0.433202 − 0.901297i \(-0.357384\pi\)
−0.179301 + 0.983794i \(0.557384\pi\)
\(32\) − 1.98883i − 0.351580i
\(33\) −0.412919 − 0.134165i −0.0718800 − 0.0233552i
\(34\) 4.11299 + 3.32111i 0.705371 + 0.569566i
\(35\) −2.70662 + 6.46138i −0.457502 + 1.09217i
\(36\) 0.861080 + 0.625611i 0.143513 + 0.104268i
\(37\) 8.02629 − 5.83144i 1.31951 − 0.958683i 0.319575 − 0.947561i \(-0.396460\pi\)
0.999938 − 0.0111217i \(-0.00354021\pi\)
\(38\) 2.91325 + 4.00974i 0.472591 + 0.650465i
\(39\) 0.323970 − 0.445906i 0.0518767 − 0.0714021i
\(40\) −4.40737 + 5.11890i −0.696866 + 0.809369i
\(41\) 0.967857 + 1.33214i 0.151154 + 0.208046i 0.877879 − 0.478883i \(-0.158958\pi\)
−0.726725 + 0.686929i \(0.758958\pi\)
\(42\) −0.401066 − 0.130314i −0.0618859 − 0.0201079i
\(43\) 9.66183i 1.47342i 0.676211 + 0.736708i \(0.263621\pi\)
−0.676211 + 0.736708i \(0.736379\pi\)
\(44\) 1.40056 + 0.455070i 0.211143 + 0.0686044i
\(45\) −1.53432 − 6.50506i −0.228724 − 0.969717i
\(46\) −4.80368 + 1.56081i −0.708263 + 0.230129i
\(47\) −3.26651 + 1.06135i −0.476469 + 0.154814i −0.537399 − 0.843328i \(-0.680593\pi\)
0.0609304 + 0.998142i \(0.480593\pi\)
\(48\) −0.268478 − 0.195061i −0.0387515 − 0.0281546i
\(49\) 2.81504 0.402149
\(50\) 6.33962 − 0.952344i 0.896557 − 0.134682i
\(51\) −0.417956 + 0.112618i −0.0585255 + 0.0157697i
\(52\) −1.09886 + 1.51245i −0.152384 + 0.209739i
\(53\) 11.4469 − 3.71933i 1.57235 − 0.510889i 0.612281 − 0.790640i \(-0.290252\pi\)
0.960072 + 0.279751i \(0.0902521\pi\)
\(54\) 0.766695 − 0.249114i 0.104334 − 0.0339002i
\(55\) −4.80027 − 7.90388i −0.647268 − 1.06576i
\(56\) 9.00085 + 2.92455i 1.20279 + 0.390810i
\(57\) −0.405831 −0.0537536
\(58\) 0.772839 − 2.37855i 0.101479 − 0.312320i
\(59\) 5.52109 − 4.01131i 0.718784 − 0.522227i −0.167211 − 0.985921i \(-0.553476\pi\)
0.885996 + 0.463694i \(0.153476\pi\)
\(60\) −0.0191903 − 0.0813611i −0.00247746 − 0.0105037i
\(61\) −3.31802 + 4.56686i −0.424829 + 0.584726i −0.966757 − 0.255698i \(-0.917695\pi\)
0.541928 + 0.840425i \(0.317695\pi\)
\(62\) −1.54182 + 1.12020i −0.195812 + 0.142266i
\(63\) −7.57576 + 5.50411i −0.954457 + 0.693453i
\(64\) 7.17761 + 5.21484i 0.897201 + 0.651855i
\(65\) 11.4259 − 2.69498i 1.41720 − 0.334271i
\(66\) 0.450355 − 0.327202i 0.0554348 − 0.0402758i
\(67\) −11.6806 − 3.79526i −1.42701 − 0.463664i −0.509189 − 0.860655i \(-0.670055\pi\)
−0.917823 + 0.396990i \(0.870055\pi\)
\(68\) 1.41765 − 0.381984i 0.171915 − 0.0463224i
\(69\) 0.127802 − 0.393333i 0.0153855 − 0.0473517i
\(70\) −4.66248 − 7.67700i −0.557273 − 0.917578i
\(71\) −7.01490 + 2.27928i −0.832515 + 0.270501i −0.694104 − 0.719874i \(-0.744199\pi\)
−0.138411 + 0.990375i \(0.544199\pi\)
\(72\) −8.58737 + 2.79021i −1.01203 + 0.328829i
\(73\) −9.22500 − 6.70236i −1.07970 − 0.784451i −0.102073 − 0.994777i \(-0.532548\pi\)
−0.977631 + 0.210325i \(0.932548\pi\)
\(74\) 12.7203i 1.47870i
\(75\) −0.242031 + 0.465794i −0.0279474 + 0.0537853i
\(76\) 1.37652 0.157898
\(77\) −7.61549 + 10.4818i −0.867866 + 1.19452i
\(78\) 0.218377 + 0.672095i 0.0247263 + 0.0760998i
\(79\) −10.1228 + 3.28910i −1.13890 + 0.370052i −0.816953 − 0.576704i \(-0.804338\pi\)
−0.321950 + 0.946757i \(0.604338\pi\)
\(80\) −1.62263 − 6.87947i −0.181416 − 0.769148i
\(81\) 2.75054 − 8.46528i 0.305615 − 0.940587i
\(82\) −2.11121 −0.233144
\(83\) 4.45687 + 1.44813i 0.489205 + 0.158952i 0.543225 − 0.839587i \(-0.317203\pi\)
−0.0540192 + 0.998540i \(0.517203\pi\)
\(84\) −0.0947527 + 0.0688419i −0.0103384 + 0.00751127i
\(85\) −8.31113 − 3.99063i −0.901468 − 0.432845i
\(86\) −10.0220 − 7.28143i −1.08070 − 0.785177i
\(87\) 0.120368 + 0.165673i 0.0129048 + 0.0177620i
\(88\) −10.1070 + 7.34316i −1.07741 + 0.782784i
\(89\) 5.78530 + 4.20327i 0.613241 + 0.445545i 0.850554 − 0.525888i \(-0.176267\pi\)
−0.237313 + 0.971433i \(0.576267\pi\)
\(90\) 7.90388 + 3.31087i 0.833143 + 0.348997i
\(91\) −9.66775 − 13.3065i −1.01346 − 1.39490i
\(92\) −0.433485 + 1.33413i −0.0451939 + 0.139093i
\(93\) − 0.156050i − 0.0161816i
\(94\) 1.36081 − 4.18815i 0.140357 − 0.431975i
\(95\) −6.55036 − 5.63986i −0.672053 − 0.578637i
\(96\) −0.198578 + 0.0645218i −0.0202672 + 0.00658522i
\(97\) −1.25686 − 3.86823i −0.127615 − 0.392759i 0.866753 − 0.498737i \(-0.166203\pi\)
−0.994368 + 0.105978i \(0.966203\pi\)
\(98\) −2.12150 + 2.91999i −0.214303 + 0.294963i
\(99\) − 12.3611i − 1.24233i
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.84.13 ✓ 176
17.16 even 2 inner 425.2.q.a.84.14 yes 176
25.14 even 10 inner 425.2.q.a.339.14 yes 176
425.339 even 10 inner 425.2.q.a.339.13 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.13 ✓ 176 1.1 even 1 trivial
425.2.q.a.84.14 yes 176 17.16 even 2 inner
425.2.q.a.339.13 yes 176 425.339 even 10 inner
425.2.q.a.339.14 yes 176 25.14 even 10 inner