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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 84.6
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.6

$q$-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+(-1.39460 + 1.91951i) q^{2} +(0.152608 + 0.469678i) q^{3} +(-1.12155 - 3.45179i) q^{4} +(-1.97789 + 1.04304i) q^{5} +(-1.11438 - 0.362083i) q^{6} +1.83697 q^{7} +(3.67682 + 1.19467i) q^{8} +(2.22974 - 1.62000i) q^{9} +(0.756252 - 5.25121i) q^{10} +(3.32921 - 4.58227i) q^{11} +(1.45007 - 1.05354i) q^{12} +(-3.63764 - 5.00679i) q^{13} +(-2.56184 + 3.52607i) q^{14} +(-0.791736 - 0.769798i) q^{15} +(-1.54835 + 1.12494i) q^{16} +(-3.84862 + 1.47923i) q^{17} +6.53927i q^{18} +(0.857473 - 2.63903i) q^{19} +(5.81867 + 5.65744i) q^{20} +(0.280336 + 0.862784i) q^{21} +(4.15277 + 12.7809i) q^{22} +(2.77105 + 2.01328i) q^{23} +1.90924i q^{24} +(2.82413 - 4.12605i) q^{25} +14.6836 q^{26} +(2.29975 + 1.67087i) q^{27} +(-2.06026 - 6.34082i) q^{28} +(-4.00911 + 1.30264i) q^{29} +(2.58179 - 0.446180i) q^{30} +(3.14374 + 1.02146i) q^{31} +3.19115i q^{32} +(2.66026 + 0.864370i) q^{33} +(2.52791 - 9.45039i) q^{34} +(-3.63333 + 1.91603i) q^{35} +(-8.09268 - 5.87968i) q^{36} +(2.35482 - 1.71088i) q^{37} +(3.86980 + 5.32633i) q^{38} +(1.79645 - 2.47260i) q^{39} +(-8.51845 + 1.47214i) q^{40} +(0.870046 + 1.19752i) q^{41} +(-2.04708 - 0.665136i) q^{42} -1.83529i q^{43} +(-19.5509 - 6.35248i) q^{44} +(-2.72046 + 5.52991i) q^{45} +(-7.72902 + 2.51131i) q^{46} +(4.47423 - 1.45377i) q^{47} +(-0.764653 - 0.555553i) q^{48} -3.62555 q^{49} +(3.98144 + 11.1751i) q^{50} +(-1.28209 - 1.58187i) q^{51} +(-13.2026 + 18.1718i) q^{52} +(8.15399 - 2.64939i) q^{53} +(-6.41449 + 2.08419i) q^{54} +(-1.80533 + 12.5358i) q^{55} +(6.75420 + 2.19457i) q^{56} +1.37035 q^{57} +(3.09069 - 9.51217i) q^{58} +(4.99085 - 3.62606i) q^{59} +(-1.76920 + 3.59627i) q^{60} +(0.602854 - 0.829757i) q^{61} +(-6.34498 + 4.60990i) q^{62} +(4.09597 - 2.97589i) q^{63} +(-9.22214 - 6.70028i) q^{64} +(12.4172 + 6.10868i) q^{65} +(-5.36917 + 3.90093i) q^{66} +(-3.30684 - 1.07446i) q^{67} +(9.42242 + 11.6256i) q^{68} +(-0.522712 + 1.60874i) q^{69} +(1.38921 - 9.64631i) q^{70} +(-12.9389 + 4.20412i) q^{71} +(10.1337 - 3.29265i) q^{72} +(-10.7422 - 7.80465i) q^{73} +6.90610i q^{74} +(2.36890 + 0.696765i) q^{75} -10.0711 q^{76} +(6.11566 - 8.41749i) q^{77} +(2.24084 + 6.89658i) q^{78} +(10.2175 - 3.31987i) q^{79} +(1.88911 - 3.84002i) q^{80} +(2.12125 - 6.52853i) q^{81} -3.51201 q^{82} +(-8.43322 - 2.74012i) q^{83} +(2.66374 - 1.93532i) q^{84} +(6.06926 - 6.94003i) q^{85} +(3.52286 + 2.55951i) q^{86} +(-1.22364 - 1.68420i) q^{87} +(17.7152 - 12.8709i) q^{88} +(2.02611 + 1.47205i) q^{89} +(-6.82073 - 12.9340i) q^{90} +(-6.68224 - 9.19731i) q^{91} +(3.84155 - 11.8231i) q^{92} +1.63243i q^{93} +(-3.44926 + 10.6157i) q^{94} +(1.05663 + 6.11410i) q^{95} +(-1.49881 + 0.486994i) q^{96} +(-0.152034 - 0.467913i) q^{97} +(5.05620 - 6.95926i) q^{98} -15.6106i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39460 + 1.91951i −0.986133 + 1.35730i −0.0526745 + 0.998612i \(0.516775\pi\)
−0.933459 + 0.358684i \(0.883225\pi\)
\(3\) 0.152608 + 0.469678i 0.0881081 + 0.271169i 0.985396 − 0.170276i \(-0.0544660\pi\)
−0.897288 + 0.441445i \(0.854466\pi\)
\(4\) −1.12155 − 3.45179i −0.560777 − 1.72589i
\(5\) −1.97789 + 1.04304i −0.884541 + 0.466462i
\(6\) −1.11438 − 0.362083i −0.454943 − 0.147820i
\(7\) 1.83697 0.694309 0.347154 − 0.937808i \(-0.387148\pi\)
0.347154 + 0.937808i \(0.387148\pi\)
\(8\) 3.67682 + 1.19467i 1.29995 + 0.422380i
\(9\) 2.22974 − 1.62000i 0.743247 − 0.540001i
\(10\) 0.756252 − 5.25121i 0.239148 − 1.66058i
\(11\) 3.32921 − 4.58227i 1.00380 − 1.38161i 0.0808284 − 0.996728i \(-0.474243\pi\)
0.922967 − 0.384878i \(-0.125757\pi\)
\(12\) 1.45007 − 1.05354i 0.418600 − 0.304131i
\(13\) −3.63764 − 5.00679i −1.00890 − 1.38863i −0.919701 − 0.392619i \(-0.871569\pi\)
−0.0891996 − 0.996014i \(-0.528431\pi\)
\(14\) −2.56184 + 3.52607i −0.684681 + 0.942383i
\(15\) −0.791736 − 0.769798i −0.204425 − 0.198761i
\(16\) −1.54835 + 1.12494i −0.387088 + 0.281236i
\(17\) −3.84862 + 1.47923i −0.933428 + 0.358766i
\(18\) 6.53927i 1.54132i
\(19\) 0.857473 − 2.63903i 0.196718 − 0.605435i −0.803234 − 0.595663i \(-0.796889\pi\)
0.999952 − 0.00977227i \(-0.00311066\pi\)
\(20\) 5.81867 + 5.65744i 1.30109 + 1.26504i
\(21\) 0.280336 + 0.862784i 0.0611743 + 0.188275i
\(22\) 4.15277 + 12.7809i 0.885372 + 2.72490i
\(23\) 2.77105 + 2.01328i 0.577803 + 0.419799i 0.837931 − 0.545775i \(-0.183765\pi\)
−0.260128 + 0.965574i \(0.583765\pi\)
\(24\) 1.90924i 0.389722i
\(25\) 2.82413 − 4.12605i 0.564826 − 0.825210i
\(26\) 14.6836 2.87970
\(27\) 2.29975 + 1.67087i 0.442588 + 0.321559i
\(28\) −2.06026 − 6.34082i −0.389352 − 1.19830i
\(29\) −4.00911 + 1.30264i −0.744473 + 0.241894i −0.656601 − 0.754238i \(-0.728006\pi\)
−0.0878715 + 0.996132i \(0.528006\pi\)
\(30\) 2.58179 − 0.446180i 0.471368 − 0.0814610i
\(31\) 3.14374 + 1.02146i 0.564633 + 0.183460i 0.577405 − 0.816458i \(-0.304065\pi\)
−0.0127717 + 0.999918i \(0.504065\pi\)
\(32\) 3.19115i 0.564121i
\(33\) 2.66026 + 0.864370i 0.463091 + 0.150467i
\(34\) 2.52791 − 9.45039i 0.433533 − 1.62073i
\(35\) −3.63333 + 1.91603i −0.614145 + 0.323869i
\(36\) −8.09268 − 5.87968i −1.34878 − 0.979946i
\(37\) 2.35482 − 1.71088i 0.387130 − 0.281267i −0.377148 − 0.926153i \(-0.623095\pi\)
0.764279 + 0.644886i \(0.223095\pi\)
\(38\) 3.86980 + 5.32633i 0.627765 + 0.864044i
\(39\) 1.79645 − 2.47260i 0.287662 − 0.395932i
\(40\) −8.51845 + 1.47214i −1.34689 + 0.232766i
\(41\) 0.870046 + 1.19752i 0.135878 + 0.187020i 0.871534 − 0.490335i \(-0.163126\pi\)
−0.735655 + 0.677356i \(0.763126\pi\)
\(42\) −2.04708 − 0.665136i −0.315871 − 0.102633i
\(43\) − 1.83529i − 0.279880i −0.990160 − 0.139940i \(-0.955309\pi\)
0.990160 − 0.139940i \(-0.0446909\pi\)
\(44\) −19.5509 − 6.35248i −2.94741 − 0.957672i
\(45\) −2.72046 + 5.52991i −0.405543 + 0.824350i
\(46\) −7.72902 + 2.51131i −1.13958 + 0.370273i
\(47\) 4.47423 − 1.45377i 0.652634 − 0.212054i 0.0360587 − 0.999350i \(-0.488520\pi\)
0.616575 + 0.787296i \(0.288520\pi\)
\(48\) −0.764653 − 0.555553i −0.110368 − 0.0801872i
\(49\) −3.62555 −0.517935
\(50\) 3.98144 + 11.1751i 0.563061 + 1.58040i
\(51\) −1.28209 − 1.58187i −0.179529 − 0.221506i
\(52\) −13.2026 + 18.1718i −1.83086 + 2.51997i
\(53\) 8.15399 − 2.64939i 1.12004 − 0.363922i 0.310255 − 0.950653i \(-0.399586\pi\)
0.809782 + 0.586731i \(0.199586\pi\)
\(54\) −6.41449 + 2.08419i −0.872901 + 0.283623i
\(55\) −1.80533 + 12.5358i −0.243431 + 1.69032i
\(56\) 6.75420 + 2.19457i 0.902568 + 0.293262i
\(57\) 1.37035 0.181508
\(58\) 3.09069 − 9.51217i 0.405828 − 1.24901i
\(59\) 4.99085 − 3.62606i 0.649753 − 0.472073i −0.213434 − 0.976957i \(-0.568465\pi\)
0.863187 + 0.504884i \(0.168465\pi\)
\(60\) −1.76920 + 3.59627i −0.228403 + 0.464277i
\(61\) 0.602854 − 0.829757i 0.0771875 − 0.106240i −0.768677 − 0.639637i \(-0.779085\pi\)
0.845865 + 0.533397i \(0.179085\pi\)
\(62\) −6.34498 + 4.60990i −0.805814 + 0.585458i
\(63\) 4.09597 − 2.97589i 0.516043 − 0.374927i
\(64\) −9.22214 − 6.70028i −1.15277 − 0.837534i
\(65\) 12.4172 + 6.10868i 1.54016 + 0.757688i
\(66\) −5.36917 + 3.90093i −0.660899 + 0.480171i
\(67\) −3.30684 − 1.07446i −0.403995 − 0.131266i 0.0999692 − 0.994991i \(-0.468126\pi\)
−0.503964 + 0.863725i \(0.668126\pi\)
\(68\) 9.42242 + 11.6256i 1.14264 + 1.40981i
\(69\) −0.522712 + 1.60874i −0.0629272 + 0.193670i
\(70\) 1.38921 − 9.64631i 0.166042 − 1.15295i
\(71\) −12.9389 + 4.20412i −1.53557 + 0.498937i −0.950150 − 0.311793i \(-0.899070\pi\)
−0.585420 + 0.810730i \(0.699070\pi\)
\(72\) 10.1337 − 3.29265i 1.19427 − 0.388042i
\(73\) −10.7422 − 7.80465i −1.25728 − 0.913466i −0.258657 − 0.965969i \(-0.583280\pi\)
−0.998621 + 0.0525037i \(0.983280\pi\)
\(74\) 6.90610i 0.802817i
\(75\) 2.36890 + 0.696765i 0.273537 + 0.0804554i
\(76\) −10.0711 −1.15523
\(77\) 6.11566 − 8.41749i 0.696944 − 0.959261i
\(78\) 2.24084 + 6.89658i 0.253725 + 0.780884i
\(79\) 10.2175 − 3.31987i 1.14956 − 0.373514i 0.328582 − 0.944476i \(-0.393429\pi\)
0.820977 + 0.570961i \(0.193429\pi\)
\(80\) 1.88911 − 3.84002i 0.211209 − 0.429327i
\(81\) 2.12125 − 6.52853i 0.235694 − 0.725392i
\(82\) −3.51201 −0.387836
\(83\) −8.43322 − 2.74012i −0.925666 − 0.300767i −0.192877 − 0.981223i \(-0.561782\pi\)
−0.732789 + 0.680456i \(0.761782\pi\)
\(84\) 2.66374 − 1.93532i 0.290638 − 0.211161i
\(85\) 6.06926 − 6.94003i 0.658304 − 0.752752i
\(86\) 3.52286 + 2.55951i 0.379880 + 0.275999i
\(87\) −1.22364 − 1.68420i −0.131188 − 0.180565i
\(88\) 17.7152 − 12.8709i 1.88845 − 1.37204i
\(89\) 2.02611 + 1.47205i 0.214767 + 0.156037i 0.689968 − 0.723840i \(-0.257625\pi\)
−0.475201 + 0.879877i \(0.657625\pi\)
\(90\) −6.82073 − 12.9340i −0.718968 − 1.36336i
\(91\) −6.68224 − 9.19731i −0.700489 − 0.964140i
\(92\) 3.84155 − 11.8231i 0.400509 − 1.23264i
\(93\) 1.63243i 0.169275i
\(94\) −3.44926 + 10.6157i −0.355764 + 1.09493i
\(95\) 1.05663 + 6.11410i 0.108408 + 0.627294i
\(96\) −1.49881 + 0.486994i −0.152972 + 0.0497036i
\(97\) −0.152034 − 0.467913i −0.0154367 − 0.0475094i 0.943041 − 0.332675i \(-0.107951\pi\)
−0.958478 + 0.285166i \(0.907951\pi\)
\(98\) 5.05620 − 6.95926i 0.510753 − 0.702992i
\(99\) − 15.6106i − 1.56893i
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.84.6 yes 176
17.16 even 2 inner 425.2.q.a.84.5 ✓ 176
25.14 even 10 inner 425.2.q.a.339.5 yes 176
425.339 even 10 inner 425.2.q.a.339.6 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.5 ✓ 176 17.16 even 2 inner
425.2.q.a.84.6 yes 176 1.1 even 1 trivial
425.2.q.a.339.5 yes 176 25.14 even 10 inner
425.2.q.a.339.6 yes 176 425.339 even 10 inner