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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 339.5
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.5

$q$-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} +(-2.24413 + 3.45888i) 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} +(9.76902 - 0.516148i) 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} +(-8.04641 + 4.50058i) 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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39460 − 1.91951i −0.986133 − 1.35730i −0.933459 − 0.358684i \(-0.883225\pi\)
−0.0526745 − 0.998612i \(-0.516775\pi\)
\(3\) −0.152608 + 0.469678i −0.0881081 + 0.271169i −0.985396 − 0.170276i \(-0.945534\pi\)
0.897288 + 0.441445i \(0.145534\pi\)
\(4\) −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 \(-0.612852\pi\)
−0.347154 + 0.937808i \(0.612852\pi\)
\(8\) 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 −0.922967 − 0.384878i \(-0.874243\pi\)
−0.0808284 − 0.996728i \(-0.525757\pi\)
\(12\) −1.45007 − 1.05354i −0.418600 − 0.304131i
\(13\) −3.63764 + 5.00679i −1.00890 + 1.38863i −0.0891996 + 0.996014i \(0.528431\pi\)
−0.919701 + 0.392619i \(0.871569\pi\)
\(14\) 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
\(17\) −2.24413 + 3.45888i −0.544282 + 0.838903i
\(18\) − 6.53927i − 1.54132i
\(19\) 0.857473 + 2.63903i 0.196718 + 0.605435i 0.999952 + 0.00977227i \(0.00311066\pi\)
−0.803234 + 0.595663i \(0.796889\pi\)
\(20\) −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 \(-0.816235\pi\)
0.260128 + 0.965574i \(0.416235\pi\)
\(24\) 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 \(-0.271994\pi\)
0.0878715 + 0.996132i \(0.471994\pi\)
\(30\) 2.58179 + 0.446180i 0.471368 + 0.0814610i
\(31\) −3.14374 + 1.02146i −0.564633 + 0.183460i −0.577405 − 0.816458i \(-0.695935\pi\)
0.0127717 + 0.999918i \(0.495935\pi\)
\(32\) − 3.19115i − 0.564121i
\(33\) 2.66026 − 0.864370i 0.463091 − 0.150467i
\(34\) 9.76902 − 0.516148i 1.67537 − 0.0885187i
\(35\) −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 \(-0.376905\pi\)
−0.764279 + 0.644886i \(0.776905\pi\)
\(38\) 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 \(-0.836874\pi\)
0.735655 + 0.677356i \(0.236874\pi\)
\(42\) −2.04708 + 0.665136i −0.315871 + 0.102633i
\(43\) 1.83529i 0.279880i 0.990160 + 0.139940i \(0.0446909\pi\)
−0.990160 + 0.139940i \(0.955309\pi\)
\(44\) 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 0.616575 − 0.787296i \(-0.288520\pi\)
0.0360587 + 0.999350i \(0.488520\pi\)
\(48\) 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 0.809782 − 0.586731i \(-0.199586\pi\)
0.310255 + 0.950653i \(0.399586\pi\)
\(54\) 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 0.863187 − 0.504884i \(-0.168465\pi\)
−0.213434 + 0.976957i \(0.568465\pi\)
\(60\) −1.76920 − 3.59627i −0.228403 − 0.464277i
\(61\) −0.602854 − 0.829757i −0.0771875 − 0.106240i 0.768677 − 0.639637i \(-0.220915\pi\)
−0.845865 + 0.533397i \(0.820915\pi\)
\(62\) 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 −0.503964 − 0.863725i \(-0.668126\pi\)
0.0999692 + 0.994991i \(0.468126\pi\)
\(68\) −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 \(-0.100930\pi\)
0.585420 + 0.810730i \(0.300930\pi\)
\(72\) 10.1337 + 3.29265i 1.19427 + 0.388042i
\(73\) 10.7422 − 7.80465i 1.25728 − 0.913466i 0.258657 − 0.965969i \(-0.416720\pi\)
0.998621 + 0.0525037i \(0.0167201\pi\)
\(74\) 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 \(-0.606571\pi\)
−0.820977 + 0.570961i \(0.806571\pi\)
\(80\) −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 −0.732789 − 0.680456i \(-0.761782\pi\)
−0.192877 + 0.981223i \(0.561782\pi\)
\(84\) 2.66374 + 1.93532i 0.290638 + 0.211161i
\(85\) −8.04641 + 4.50058i −0.872756 + 0.488157i
\(86\) 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 −0.475201 − 0.879877i \(-0.657625\pi\)
0.689968 + 0.723840i \(0.257625\pi\)
\(90\) 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 \(-0.892049\pi\)
0.958478 + 0.285166i \(0.0920487\pi\)
\(98\) 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.339.5 yes 176
17.16 even 2 inner 425.2.q.a.339.6 yes 176
25.9 even 10 inner 425.2.q.a.84.6 yes 176
425.84 even 10 inner 425.2.q.a.84.5 ✓ 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.5 ✓ 176 425.84 even 10 inner
425.2.q.a.84.6 yes 176 25.9 even 10 inner
425.2.q.a.339.5 yes 176 1.1 even 1 trivial
425.2.q.a.339.6 yes 176 17.16 even 2 inner