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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,2,Mod(16,225)] 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("225.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.q (of order \(15\), degree \(8\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 106.20
Character \(\chi\) \(=\) 225.106
Dual form 225.2.q.a.121.20

$q$-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+(1.37843 + 0.292995i) q^{2} +(1.63456 - 0.572891i) q^{3} +(-0.0128554 - 0.00572357i) q^{4} +(1.56242 - 1.59963i) q^{5} +(2.42099 - 0.310774i) q^{6} +(-2.48652 + 4.30678i) q^{7} +(-2.29622 - 1.66830i) q^{8} +(2.34359 - 1.87285i) q^{9} +(2.62239 - 1.74721i) q^{10} +(3.07362 + 0.653319i) q^{11} +(-0.0242919 - 0.00199082i) q^{12} +(-1.44657 + 0.307478i) q^{13} +(-4.68937 + 5.20807i) q^{14} +(1.63747 - 3.50980i) q^{15} +(-2.65756 - 2.95152i) q^{16} +(-2.39081 - 1.73703i) q^{17} +(3.77922 - 1.89494i) q^{18} +(-1.51211 - 1.09861i) q^{19} +(-0.0292411 + 0.0116212i) q^{20} +(-1.59706 + 8.46420i) q^{21} +(4.04537 + 1.80111i) q^{22} +(-4.45542 + 4.94825i) q^{23} +(-4.70908 - 1.41146i) q^{24} +(-0.117657 - 4.99862i) q^{25} -2.08409 q^{26} +(2.75781 - 4.40392i) q^{27} +(0.0566152 - 0.0411334i) q^{28} +(0.427253 + 4.06504i) q^{29} +(3.28549 - 4.35826i) q^{30} +(-0.681027 + 6.47954i) q^{31} +(0.0398007 + 0.0689368i) q^{32} +(5.39831 - 0.692961i) q^{33} +(-2.78664 - 3.09487i) q^{34} +(3.00427 + 10.7065i) q^{35} +(-0.0408471 + 0.0106625i) q^{36} +(1.50355 - 4.62746i) q^{37} +(-1.76246 - 1.95741i) q^{38} +(-2.18836 + 1.33132i) q^{39} +(-6.25636 + 1.06652i) q^{40} +(2.38999 - 0.508007i) q^{41} +(-4.68141 + 11.1994i) q^{42} +(-1.92193 + 3.32888i) q^{43} +(-0.0357732 - 0.0259908i) q^{44} +(0.665807 - 6.67508i) q^{45} +(-7.59132 + 5.51542i) q^{46} +(-0.175217 - 1.66708i) q^{47} +(-6.03484 - 3.30195i) q^{48} +(-8.86554 - 15.3556i) q^{49} +(1.30239 - 6.92474i) q^{50} +(-4.90306 - 1.46960i) q^{51} +(0.0203561 + 0.00432682i) q^{52} +(0.228304 - 0.165873i) q^{53} +(5.09178 - 5.26249i) q^{54} +(5.84738 - 3.89591i) q^{55} +(12.8946 - 5.74105i) q^{56} +(-3.10103 - 0.929477i) q^{57} +(-0.602097 + 5.72857i) q^{58} +(2.95891 - 0.628937i) q^{59} +(-0.0411388 + 0.0357476i) q^{60} +(7.58789 + 1.61286i) q^{61} +(-2.83723 + 8.73208i) q^{62} +(2.23857 + 14.7502i) q^{63} +(2.48928 + 7.66123i) q^{64} +(-1.76831 + 2.79440i) q^{65} +(7.64425 + 0.626479i) q^{66} +(1.19291 - 11.3498i) q^{67} +(0.0207927 + 0.0360141i) q^{68} +(-4.44786 + 10.6407i) q^{69} +(1.00422 + 15.6385i) q^{70} +(13.0547 - 9.48483i) q^{71} +(-8.50590 + 0.390666i) q^{72} +(-2.03962 - 6.27730i) q^{73} +(3.42837 - 5.93811i) q^{74} +(-3.05598 - 8.10315i) q^{75} +(0.0131507 + 0.0227777i) q^{76} +(-10.4563 + 11.6129i) q^{77} +(-3.40658 + 1.19396i) q^{78} +(-0.847007 - 8.05873i) q^{79} +(-8.87358 - 0.360405i) q^{80} +(1.98484 - 8.77841i) q^{81} +3.44328 q^{82} +(-8.79712 + 3.91673i) q^{83} +(0.0689762 - 0.0996694i) q^{84} +(-6.51407 + 1.11045i) q^{85} +(-3.62460 + 4.02553i) q^{86} +(3.02720 + 6.39979i) q^{87} +(-5.96780 - 6.62791i) q^{88} +(0.845142 + 2.60108i) q^{89} +(2.87354 - 9.00608i) q^{90} +(2.27269 - 6.99461i) q^{91} +(0.0855977 - 0.0381106i) q^{92} +(2.59889 + 10.9814i) q^{93} +(0.246921 - 2.34930i) q^{94} +(-4.11994 + 0.702323i) q^{95} +(0.104550 + 0.0898801i) q^{96} +(1.02725 + 9.77359i) q^{97} +(-7.72146 - 23.7642i) q^{98} +(8.42689 - 4.22533i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q - 3 q^{2} - 8 q^{3} + 23 q^{4} - 8 q^{5} - 10 q^{6} - 8 q^{7} - 20 q^{8} - 8 q^{9} - 20 q^{10} - 11 q^{11} - 4 q^{12} - 3 q^{13} + q^{14} - 48 q^{15} + 23 q^{16} - 24 q^{17} - 12 q^{19} + q^{20}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37843 + 0.292995i 0.974700 + 0.207179i 0.667618 − 0.744504i \(-0.267314\pi\)
0.307082 + 0.951683i \(0.400647\pi\)
\(3\) 1.63456 − 0.572891i 0.943715 − 0.330759i
\(4\) −0.0128554 − 0.00572357i −0.00642768 − 0.00286179i
\(5\) 1.56242 − 1.59963i 0.698738 − 0.715378i
\(6\) 2.42099 − 0.310774i 0.988366 − 0.126873i
\(7\) −2.48652 + 4.30678i −0.939815 + 1.62781i −0.174002 + 0.984745i \(0.555670\pi\)
−0.765813 + 0.643063i \(0.777663\pi\)
\(8\) −2.29622 − 1.66830i −0.811838 − 0.589835i
\(9\) 2.34359 − 1.87285i 0.781197 − 0.624284i
\(10\) 2.62239 − 1.74721i 0.829271 − 0.552515i
\(11\) 3.07362 + 0.653319i 0.926732 + 0.196983i 0.646469 − 0.762940i \(-0.276245\pi\)
0.280263 + 0.959923i \(0.409578\pi\)
\(12\) −0.0242919 − 0.00199082i −0.00701246 − 0.000574701i
\(13\) −1.44657 + 0.307478i −0.401207 + 0.0852791i −0.404096 − 0.914717i \(-0.632414\pi\)
0.00288893 + 0.999996i \(0.499080\pi\)
\(14\) −4.68937 + 5.20807i −1.25329 + 1.39192i
\(15\) 1.63747 − 3.50980i 0.422792 − 0.906227i
\(16\) −2.65756 − 2.95152i −0.664389 − 0.737879i
\(17\) −2.39081 − 1.73703i −0.579857 − 0.421291i 0.258816 − 0.965927i \(-0.416668\pi\)
−0.838672 + 0.544636i \(0.816668\pi\)
\(18\) 3.77922 − 1.89494i 0.890772 − 0.446643i
\(19\) −1.51211 − 1.09861i −0.346902 − 0.252039i 0.400666 − 0.916224i \(-0.368779\pi\)
−0.747568 + 0.664185i \(0.768779\pi\)
\(20\) −0.0292411 + 0.0116212i −0.00653852 + 0.00259858i
\(21\) −1.59706 + 8.46420i −0.348506 + 1.84704i
\(22\) 4.04537 + 1.80111i 0.862476 + 0.383999i
\(23\) −4.45542 + 4.94825i −0.929020 + 1.03178i 0.0703930 + 0.997519i \(0.477575\pi\)
−0.999413 + 0.0342618i \(0.989092\pi\)
\(24\) −4.70908 − 1.41146i −0.961237 − 0.288114i
\(25\) −0.117657 − 4.99862i −0.0235313 − 0.999723i
\(26\) −2.08409 −0.408725
\(27\) 2.75781 − 4.40392i 0.530740 − 0.847535i
\(28\) 0.0566152 − 0.0411334i 0.0106993 − 0.00777347i
\(29\) 0.427253 + 4.06504i 0.0793388 + 0.754859i 0.959789 + 0.280721i \(0.0905735\pi\)
−0.880451 + 0.474138i \(0.842760\pi\)
\(30\) 3.28549 − 4.35826i 0.599847 − 0.795706i
\(31\) −0.681027 + 6.47954i −0.122316 + 1.16376i 0.745371 + 0.666649i \(0.232272\pi\)
−0.867687 + 0.497110i \(0.834394\pi\)
\(32\) 0.0398007 + 0.0689368i 0.00703584 + 0.0121864i
\(33\) 5.39831 − 0.692961i 0.939725 − 0.120629i
\(34\) −2.78664 − 3.09487i −0.477904 − 0.530766i
\(35\) 3.00427 + 10.7065i 0.507814 + 1.80973i
\(36\) −0.0408471 + 0.0106625i −0.00680785 + 0.00177708i
\(37\) 1.50355 − 4.62746i 0.247182 − 0.760749i −0.748088 − 0.663600i \(-0.769028\pi\)
0.995270 − 0.0971489i \(-0.0309723\pi\)
\(38\) −1.76246 − 1.95741i −0.285908 − 0.317533i
\(39\) −2.18836 + 1.33132i −0.350418 + 0.213182i
\(40\) −6.25636 + 1.06652i −0.989217 + 0.168631i
\(41\) 2.38999 − 0.508007i 0.373253 − 0.0793374i −0.0174658 − 0.999847i \(-0.505560\pi\)
0.390719 + 0.920510i \(0.372226\pi\)
\(42\) −4.68141 + 11.1994i −0.722357 + 1.72811i
\(43\) −1.92193 + 3.32888i −0.293091 + 0.507649i −0.974539 − 0.224218i \(-0.928017\pi\)
0.681448 + 0.731867i \(0.261351\pi\)
\(44\) −0.0357732 − 0.0259908i −0.00539301 − 0.00391825i
\(45\) 0.665807 − 6.67508i 0.0992527 − 0.995062i
\(46\) −7.59132 + 5.51542i −1.11928 + 0.813204i
\(47\) −0.175217 − 1.66708i −0.0255581 − 0.243169i −0.999841 − 0.0178399i \(-0.994321\pi\)
0.974283 − 0.225329i \(-0.0723456\pi\)
\(48\) −6.03484 − 3.30195i −0.871055 − 0.476595i
\(49\) −8.86554 − 15.3556i −1.26651 − 2.19365i
\(50\) 1.30239 − 6.92474i 0.184186 − 0.979306i
\(51\) −4.90306 − 1.46960i −0.686565 − 0.205786i
\(52\) 0.0203561 + 0.00432682i 0.00282288 + 0.000600021i
\(53\) 0.228304 − 0.165873i 0.0313600 − 0.0227844i −0.571995 − 0.820257i \(-0.693830\pi\)
0.603355 + 0.797473i \(0.293830\pi\)
\(54\) 5.09178 − 5.26249i 0.692904 − 0.716134i
\(55\) 5.84738 − 3.89591i 0.788460 − 0.525324i
\(56\) 12.8946 − 5.74105i 1.72312 − 0.767181i
\(57\) −3.10103 − 0.929477i −0.410741 − 0.123112i
\(58\) −0.602097 + 5.72857i −0.0790592 + 0.752198i
\(59\) 2.95891 − 0.628937i 0.385218 − 0.0818806i −0.0112321 − 0.999937i \(-0.503575\pi\)
0.396450 + 0.918056i \(0.370242\pi\)
\(60\) −0.0411388 + 0.0357476i −0.00531100 + 0.00461499i
\(61\) 7.58789 + 1.61286i 0.971530 + 0.206505i 0.666226 − 0.745750i \(-0.267909\pi\)
0.305304 + 0.952255i \(0.401242\pi\)
\(62\) −2.83723 + 8.73208i −0.360328 + 1.10898i
\(63\) 2.23857 + 14.7502i 0.282034 + 1.85835i
\(64\) 2.48928 + 7.66123i 0.311161 + 0.957654i
\(65\) −1.76831 + 2.79440i −0.219332 + 0.346602i
\(66\) 7.64425 + 0.626479i 0.940943 + 0.0771142i
\(67\) 1.19291 − 11.3498i 0.145738 − 1.38660i −0.640158 − 0.768243i \(-0.721131\pi\)
0.785896 − 0.618358i \(-0.212202\pi\)
\(68\) 0.0207927 + 0.0360141i 0.00252149 + 0.00436735i
\(69\) −4.44786 + 10.6407i −0.535460 + 1.28099i
\(70\) 1.00422 + 15.6385i 0.120027 + 1.86916i
\(71\) 13.0547 − 9.48483i 1.54931 − 1.12564i 0.605175 − 0.796092i \(-0.293103\pi\)
0.944138 − 0.329550i \(-0.106897\pi\)
\(72\) −8.50590 + 0.390666i −1.00243 + 0.0460404i
\(73\) −2.03962 − 6.27730i −0.238719 − 0.734703i −0.996606 − 0.0823167i \(-0.973768\pi\)
0.757887 − 0.652386i \(-0.226232\pi\)
\(74\) 3.42837 − 5.93811i 0.398540 − 0.690291i
\(75\) −3.05598 − 8.10315i −0.352874 − 0.935671i
\(76\) 0.0131507 + 0.0227777i 0.00150849 + 0.00261278i
\(77\) −10.4563 + 11.6129i −1.19161 + 1.32341i
\(78\) −3.40658 + 1.19396i −0.385720 + 0.135189i
\(79\) −0.847007 − 8.05873i −0.0952957 − 0.906678i −0.932836 − 0.360302i \(-0.882674\pi\)
0.837540 − 0.546376i \(-0.183993\pi\)
\(80\) −8.87358 − 0.360405i −0.992097 − 0.0402944i
\(81\) 1.98484 − 8.77841i 0.220538 − 0.975378i
\(82\) 3.44328 0.380247
\(83\) −8.79712 + 3.91673i −0.965609 + 0.429917i −0.828098 − 0.560583i \(-0.810577\pi\)
−0.137511 + 0.990500i \(0.543910\pi\)
\(84\) 0.0689762 − 0.0996694i 0.00752592 − 0.0108748i
\(85\) −6.51407 + 1.11045i −0.706550 + 0.120445i
\(86\) −3.62460 + 4.02553i −0.390851 + 0.434084i
\(87\) 3.02720 + 6.39979i 0.324549 + 0.686130i
\(88\) −5.96780 − 6.62791i −0.636169 − 0.706537i
\(89\) 0.845142 + 2.60108i 0.0895849 + 0.275714i 0.985805 − 0.167896i \(-0.0536974\pi\)
−0.896220 + 0.443610i \(0.853697\pi\)
\(90\) 2.87354 − 9.00608i 0.302898 − 0.949325i
\(91\) 2.27269 − 6.99461i 0.238242 − 0.733234i
\(92\) 0.0855977 − 0.0381106i 0.00892418 − 0.00397330i
\(93\) 2.59889 + 10.9814i 0.269492 + 1.13871i
\(94\) 0.246921 − 2.34930i 0.0254680 − 0.242312i
\(95\) −4.11994 + 0.702323i −0.422697 + 0.0720569i
\(96\) 0.104550 + 0.0898801i 0.0106706 + 0.00917335i
\(97\) 1.02725 + 9.77359i 0.104301 + 0.992358i 0.914056 + 0.405589i \(0.132934\pi\)
−0.809755 + 0.586769i \(0.800400\pi\)
\(98\) −7.72146 − 23.7642i −0.779985 − 2.40055i
\(99\) 8.42689 − 4.22533i 0.846934 − 0.424662i
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 225.2.q.a.106.20 yes 224
3.2 odd 2 675.2.r.a.181.9 224
9.4 even 3 inner 225.2.q.a.31.9 ✓ 224
9.5 odd 6 675.2.r.a.631.20 224
25.21 even 5 inner 225.2.q.a.196.9 yes 224
75.71 odd 10 675.2.r.a.46.20 224
225.121 even 15 inner 225.2.q.a.121.20 yes 224
225.221 odd 30 675.2.r.a.496.9 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.9 ✓ 224 9.4 even 3 inner
225.2.q.a.106.20 yes 224 1.1 even 1 trivial
225.2.q.a.121.20 yes 224 225.121 even 15 inner
225.2.q.a.196.9 yes 224 25.21 even 5 inner
675.2.r.a.46.20 224 75.71 odd 10
675.2.r.a.181.9 224 3.2 odd 2
675.2.r.a.496.9 224 225.221 odd 30
675.2.r.a.631.20 224 9.5 odd 6