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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 84.3
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.3

$q$-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+(-1.47509 + 2.03029i) q^{2} +(-0.532962 - 1.64029i) q^{3} +(-1.32814 - 4.08760i) q^{4} +(0.886990 + 2.05262i) q^{5} +(4.11642 + 1.33751i) q^{6} -3.49211 q^{7} +(5.48463 + 1.78207i) q^{8} +(0.0205526 - 0.0149323i) q^{9} +(-5.47580 - 1.22696i) q^{10} +(1.63185 - 2.24605i) q^{11} +(-5.99699 + 4.35707i) q^{12} +(2.93728 + 4.04282i) q^{13} +(5.15117 - 7.08998i) q^{14} +(2.89416 - 2.54889i) q^{15} +(-4.75420 + 3.45413i) q^{16} +(-4.11813 + 0.202457i) q^{17} +0.0637542i q^{18} +(-2.01401 + 6.19849i) q^{19} +(7.21224 - 6.35183i) q^{20} +(1.86116 + 5.72806i) q^{21} +(2.15300 + 6.62625i) q^{22} +(-1.41733 - 1.02975i) q^{23} -9.94616i q^{24} +(-3.42650 + 3.64131i) q^{25} -12.5408 q^{26} +(-4.22139 - 3.06702i) q^{27} +(4.63801 + 14.2743i) q^{28} +(-6.38667 + 2.07516i) q^{29} +(0.905831 + 9.63581i) q^{30} +(-2.96543 - 0.963528i) q^{31} -3.21376i q^{32} +(-4.55389 - 1.47965i) q^{33} +(5.66357 - 8.65963i) q^{34} +(-3.09746 - 7.16797i) q^{35} +(-0.0883342 - 0.0641785i) q^{36} +(-5.66078 + 4.11280i) q^{37} +(-9.61387 - 13.2324i) q^{38} +(5.06593 - 6.97265i) q^{39} +(1.20691 + 12.8385i) q^{40} +(2.49719 + 3.43708i) q^{41} +(-14.3750 - 4.67072i) q^{42} +8.26881i q^{43} +(-11.3483 - 3.68728i) q^{44} +(0.0488804 + 0.0289419i) q^{45} +(4.18139 - 1.35862i) q^{46} +(9.09369 - 2.95472i) q^{47} +(8.19957 + 5.95734i) q^{48} +5.19481 q^{49} +(-2.33850 - 12.3280i) q^{50} +(2.52690 + 6.64702i) q^{51} +(12.6243 - 17.3758i) q^{52} +(-7.02371 + 2.28214i) q^{53} +(12.4538 - 4.04650i) q^{54} +(6.05772 + 1.35735i) q^{55} +(-19.1529 - 6.22316i) q^{56} +11.2407 q^{57} +(5.20775 - 16.0278i) q^{58} +(-3.87604 + 2.81611i) q^{59} +(-14.2627 - 8.44487i) q^{60} +(1.80274 - 2.48126i) q^{61} +(6.33052 - 4.59939i) q^{62} +(-0.0717719 + 0.0521453i) q^{63} +(-2.98354 - 2.16767i) q^{64} +(-5.69303 + 9.61505i) q^{65} +(9.72150 - 7.06308i) q^{66} +(-5.84210 - 1.89821i) q^{67} +(6.29703 + 16.5644i) q^{68} +(-0.933707 + 2.87365i) q^{69} +(19.1221 + 4.28466i) q^{70} +(8.77051 - 2.84971i) q^{71} +(0.139334 - 0.0452723i) q^{72} +(-1.91096 - 1.38839i) q^{73} -17.5597i q^{74} +(7.79899 + 3.67977i) q^{75} +28.0118 q^{76} +(-5.69860 + 7.84345i) q^{77} +(6.68378 + 20.5706i) q^{78} +(-4.67642 + 1.51946i) q^{79} +(-11.3069 - 6.69478i) q^{80} +(-2.75740 + 8.48641i) q^{81} -10.6618 q^{82} +(4.64204 + 1.50829i) q^{83} +(20.9421 - 15.2154i) q^{84} +(-4.06831 - 8.27338i) q^{85} +(-16.7881 - 12.1972i) q^{86} +(6.80771 + 9.37001i) q^{87} +(12.9527 - 9.41070i) q^{88} +(-8.50595 - 6.17994i) q^{89} +(-0.130863 + 0.0565493i) q^{90} +(-10.2573 - 14.1179i) q^{91} +(-2.32680 + 7.16115i) q^{92} +5.37769i q^{93} +(-7.41508 + 22.8213i) q^{94} +(-14.5096 + 1.36400i) q^{95} +(-5.27149 + 1.71281i) q^{96} +(5.29550 + 16.2979i) q^{97} +(-7.66281 + 10.5470i) q^{98} -0.0705295i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.47509 + 2.03029i −1.04305 + 1.43563i −0.148360 + 0.988933i \(0.547400\pi\)
−0.894686 + 0.446696i \(0.852600\pi\)
\(3\) −0.532962 − 1.64029i −0.307706 − 0.947021i −0.978654 − 0.205517i \(-0.934113\pi\)
0.670948 − 0.741505i \(-0.265887\pi\)
\(4\) −1.32814 − 4.08760i −0.664071 − 2.04380i
\(5\) 0.886990 + 2.05262i 0.396674 + 0.917960i
\(6\) 4.11642 + 1.33751i 1.68052 + 0.546035i
\(7\) −3.49211 −1.31989 −0.659946 − 0.751313i \(-0.729421\pi\)
−0.659946 + 0.751313i \(0.729421\pi\)
\(8\) 5.48463 + 1.78207i 1.93911 + 0.630055i
\(9\) 0.0205526 − 0.0149323i 0.00685087 − 0.00497745i
\(10\) −5.47580 − 1.22696i −1.73160 − 0.387997i
\(11\) 1.63185 − 2.24605i 0.492022 − 0.677210i −0.488738 − 0.872431i \(-0.662542\pi\)
0.980759 + 0.195221i \(0.0625425\pi\)
\(12\) −5.99699 + 4.35707i −1.73118 + 1.25778i
\(13\) 2.93728 + 4.04282i 0.814654 + 1.12128i 0.990589 + 0.136873i \(0.0437054\pi\)
−0.175934 + 0.984402i \(0.556295\pi\)
\(14\) 5.15117 − 7.08998i 1.37671 − 1.89488i
\(15\) 2.89416 − 2.54889i 0.747268 − 0.658120i
\(16\) −4.75420 + 3.45413i −1.18855 + 0.863531i
\(17\) −4.11813 + 0.202457i −0.998794 + 0.0491031i
\(18\) 0.0637542i 0.0150270i
\(19\) −2.01401 + 6.19849i −0.462046 + 1.42203i 0.400614 + 0.916247i \(0.368797\pi\)
−0.862660 + 0.505785i \(0.831203\pi\)
\(20\) 7.21224 − 6.35183i 1.61271 − 1.42031i
\(21\) 1.86116 + 5.72806i 0.406139 + 1.24997i
\(22\) 2.15300 + 6.62625i 0.459021 + 1.41272i
\(23\) −1.41733 − 1.02975i −0.295534 − 0.214718i 0.430130 − 0.902767i \(-0.358468\pi\)
−0.725665 + 0.688049i \(0.758468\pi\)
\(24\) − 9.94616i − 2.03025i
\(25\) −3.42650 + 3.64131i −0.685300 + 0.728261i
\(26\) −12.5408 −2.45946
\(27\) −4.22139 − 3.06702i −0.812406 − 0.590248i
\(28\) 4.63801 + 14.2743i 0.876502 + 2.69760i
\(29\) −6.38667 + 2.07516i −1.18598 + 0.385347i −0.834583 − 0.550882i \(-0.814292\pi\)
−0.351392 + 0.936228i \(0.614292\pi\)
\(30\) 0.905831 + 9.63581i 0.165381 + 1.75925i
\(31\) −2.96543 − 0.963528i −0.532608 − 0.173055i 0.0303518 − 0.999539i \(-0.490337\pi\)
−0.562959 + 0.826485i \(0.690337\pi\)
\(32\) − 3.21376i − 0.568117i
\(33\) −4.55389 − 1.47965i −0.792730 − 0.257573i
\(34\) 5.66357 − 8.65963i 0.971294 − 1.48511i
\(35\) −3.09746 − 7.16797i −0.523567 − 1.21161i
\(36\) −0.0883342 − 0.0641785i −0.0147224 − 0.0106964i
\(37\) −5.66078 + 4.11280i −0.930626 + 0.676140i −0.946146 − 0.323740i \(-0.895060\pi\)
0.0155199 + 0.999880i \(0.495060\pi\)
\(38\) −9.61387 − 13.2324i −1.55958 − 2.14657i
\(39\) 5.06593 − 6.97265i 0.811198 − 1.11652i
\(40\) 1.20691 + 12.8385i 0.190829 + 2.02995i
\(41\) 2.49719 + 3.43708i 0.389995 + 0.536782i 0.958198 − 0.286107i \(-0.0923612\pi\)
−0.568203 + 0.822888i \(0.692361\pi\)
\(42\) −14.3750 − 4.67072i −2.21811 − 0.720707i
\(43\) 8.26881i 1.26098i 0.776197 + 0.630491i \(0.217146\pi\)
−0.776197 + 0.630491i \(0.782854\pi\)
\(44\) −11.3483 − 3.68728i −1.71082 − 0.555878i
\(45\) 0.0488804 + 0.0289419i 0.00728666 + 0.00431440i
\(46\) 4.18139 − 1.35862i 0.616512 − 0.200317i
\(47\) 9.09369 − 2.95472i 1.32645 − 0.430990i 0.441745 − 0.897141i \(-0.354359\pi\)
0.884705 + 0.466151i \(0.154359\pi\)
\(48\) 8.19957 + 5.95734i 1.18351 + 0.859867i
\(49\) 5.19481 0.742116
\(50\) −2.33850 − 12.3280i −0.330714 − 1.74345i
\(51\) 2.52690 + 6.64702i 0.353836 + 0.930769i
\(52\) 12.6243 − 17.3758i 1.75067 − 2.40960i
\(53\) −7.02371 + 2.28214i −0.964781 + 0.313476i −0.748707 − 0.662901i \(-0.769325\pi\)
−0.216074 + 0.976377i \(0.569325\pi\)
\(54\) 12.4538 − 4.04650i 1.69475 − 0.550659i
\(55\) 6.05772 + 1.35735i 0.816823 + 0.183025i
\(56\) −19.1529 − 6.22316i −2.55942 − 0.831605i
\(57\) 11.2407 1.48887
\(58\) 5.20775 − 16.0278i 0.683812 − 2.10456i
\(59\) −3.87604 + 2.81611i −0.504617 + 0.366626i −0.810778 − 0.585354i \(-0.800956\pi\)
0.306160 + 0.951980i \(0.400956\pi\)
\(60\) −14.2627 − 8.44487i −1.84130 − 1.09023i
\(61\) 1.80274 − 2.48126i 0.230817 − 0.317692i −0.677861 − 0.735190i \(-0.737093\pi\)
0.908678 + 0.417498i \(0.137093\pi\)
\(62\) 6.33052 − 4.59939i 0.803977 − 0.584123i
\(63\) −0.0717719 + 0.0521453i −0.00904241 + 0.00656969i
\(64\) −2.98354 − 2.16767i −0.372943 − 0.270959i
\(65\) −5.69303 + 9.61505i −0.706133 + 1.19260i
\(66\) 9.72150 − 7.06308i 1.19663 − 0.869405i
\(67\) −5.84210 − 1.89821i −0.713726 − 0.231904i −0.0704252 − 0.997517i \(-0.522436\pi\)
−0.643301 + 0.765613i \(0.722436\pi\)
\(68\) 6.29703 + 16.5644i 0.763627 + 2.00873i
\(69\) −0.933707 + 2.87365i −0.112405 + 0.345947i
\(70\) 19.1221 + 4.28466i 2.28552 + 0.512115i
\(71\) 8.77051 − 2.84971i 1.04087 − 0.338199i 0.261790 − 0.965125i \(-0.415687\pi\)
0.779079 + 0.626926i \(0.215687\pi\)
\(72\) 0.139334 − 0.0452723i 0.0164207 − 0.00533540i
\(73\) −1.91096 − 1.38839i −0.223660 − 0.162499i 0.470312 − 0.882500i \(-0.344141\pi\)
−0.693973 + 0.720001i \(0.744141\pi\)
\(74\) − 17.5597i − 2.04128i
\(75\) 7.79899 + 3.67977i 0.900549 + 0.424903i
\(76\) 28.0118 3.21318
\(77\) −5.69860 + 7.84345i −0.649415 + 0.893844i
\(78\) 6.68378 + 20.5706i 0.756790 + 2.32916i
\(79\) −4.67642 + 1.51946i −0.526138 + 0.170953i −0.560029 − 0.828473i \(-0.689210\pi\)
0.0338911 + 0.999426i \(0.489210\pi\)
\(80\) −11.3069 − 6.69478i −1.26415 − 0.748500i
\(81\) −2.75740 + 8.48641i −0.306378 + 0.942935i
\(82\) −10.6618 −1.17740
\(83\) 4.64204 + 1.50829i 0.509530 + 0.165556i 0.552489 − 0.833520i \(-0.313678\pi\)
−0.0429584 + 0.999077i \(0.513678\pi\)
\(84\) 20.9421 − 15.2154i 2.28497 − 1.66013i
\(85\) −4.06831 − 8.27338i −0.441270 − 0.897374i
\(86\) −16.7881 − 12.1972i −1.81030 − 1.31526i
\(87\) 6.80771 + 9.37001i 0.729863 + 1.00457i
\(88\) 12.9527 − 9.41070i 1.38076 − 1.00318i
\(89\) −8.50595 − 6.17994i −0.901629 − 0.655072i 0.0372548 − 0.999306i \(-0.488139\pi\)
−0.938884 + 0.344234i \(0.888139\pi\)
\(90\) −0.130863 + 0.0565493i −0.0137942 + 0.00596082i
\(91\) −10.2573 − 14.1179i −1.07526 − 1.47996i
\(92\) −2.32680 + 7.16115i −0.242585 + 0.746601i
\(93\) 5.37769i 0.557641i
\(94\) −7.41508 + 22.8213i −0.764807 + 2.35383i
\(95\) −14.5096 + 1.36400i −1.48865 + 0.139943i
\(96\) −5.27149 + 1.71281i −0.538019 + 0.174813i
\(97\) 5.29550 + 16.2979i 0.537677 + 1.65480i 0.737793 + 0.675027i \(0.235868\pi\)
−0.200116 + 0.979772i \(0.564132\pi\)
\(98\) −7.66281 + 10.5470i −0.774061 + 1.06540i
\(99\) − 0.0705295i − 0.00708848i
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.84.3 ✓ 176
17.16 even 2 inner 425.2.q.a.84.4 yes 176
25.14 even 10 inner 425.2.q.a.339.4 yes 176
425.339 even 10 inner 425.2.q.a.339.3 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.3 ✓ 176 1.1 even 1 trivial
425.2.q.a.84.4 yes 176 17.16 even 2 inner
425.2.q.a.339.3 yes 176 425.339 even 10 inner
425.2.q.a.339.4 yes 176 25.14 even 10 inner