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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 339.6
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.6

$q$-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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

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

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 −0.897288 − 0.441445i \(-0.854466\pi\)
0.985396 + 0.170276i \(0.0544660\pi\)
\(4\) −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 0.922967 + 0.384878i \(0.125757\pi\)
0.0808284 + 0.996728i \(0.474243\pi\)
\(12\) 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
\(17\) −3.84862 − 1.47923i −0.933428 − 0.358766i
\(18\) − 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 −0.260128 − 0.965574i \(-0.583765\pi\)
0.837931 + 0.545775i \(0.183765\pi\)
\(24\) − 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 −0.0878715 − 0.996132i \(-0.528006\pi\)
−0.656601 + 0.754238i \(0.728006\pi\)
\(30\) 2.58179 + 0.446180i 0.471368 + 0.0814610i
\(31\) 3.14374 − 1.02146i 0.564633 − 0.183460i −0.0127717 − 0.999918i \(-0.504065\pi\)
0.577405 + 0.816458i \(0.304065\pi\)
\(32\) − 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 0.764279 − 0.644886i \(-0.223095\pi\)
−0.377148 + 0.926153i \(0.623095\pi\)
\(38\) 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 −0.735655 − 0.677356i \(-0.763126\pi\)
0.871534 + 0.490335i \(0.163126\pi\)
\(42\) −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 0.845865 − 0.533397i \(-0.179085\pi\)
−0.768677 + 0.639637i \(0.779085\pi\)
\(62\) −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 −0.585420 − 0.810730i \(-0.699070\pi\)
−0.950150 + 0.311793i \(0.899070\pi\)
\(72\) 10.1337 + 3.29265i 1.19427 + 0.388042i
\(73\) −10.7422 + 7.80465i −1.25728 + 0.913466i −0.998621 − 0.0525037i \(-0.983280\pi\)
−0.258657 + 0.965969i \(0.583280\pi\)
\(74\) − 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 0.820977 − 0.570961i \(-0.193429\pi\)
0.328582 + 0.944476i \(0.393429\pi\)
\(80\) 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
\(85\) 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 −0.475201 − 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 −0.958478 − 0.285166i \(-0.907951\pi\)
0.943041 + 0.332675i \(0.107951\pi\)
\(98\) 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.339.6 yes 176
17.16 even 2 inner 425.2.q.a.339.5 yes 176
25.9 even 10 inner 425.2.q.a.84.5 ✓ 176
425.84 even 10 inner 425.2.q.a.84.6 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.5 ✓ 176 25.9 even 10 inner
425.2.q.a.84.6 yes 176 425.84 even 10 inner
425.2.q.a.339.5 yes 176 17.16 even 2 inner
425.2.q.a.339.6 yes 176 1.1 even 1 trivial