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Properties

Label 225.2.w.a.113.8
Level $225$
Weight $2$
Character 225.113
Analytic conductor $1.797$
Analytic rank $0$
Dimension $448$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,2,Mod(2,225)] 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("225.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.w (of order \(60\), degree \(16\), 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: \(448\)
Relative dimension: \(28\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 113.8
Character \(\chi\) \(=\) 225.113
Dual form 225.2.w.a.2.8

$q$-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+(-1.26732 + 1.02626i) q^{2} +(0.979850 + 1.42825i) q^{3} +(0.137078 - 0.644900i) q^{4} +(-1.08873 - 1.95312i) q^{5} +(-2.70753 - 0.804473i) q^{6} +(-3.51917 - 0.942959i) q^{7} +(-0.992568 - 1.94802i) q^{8} +(-1.07979 + 2.79894i) q^{9} +(3.38417 + 1.35792i) q^{10} +(-4.73499 - 0.497667i) q^{11} +(1.05539 - 0.436124i) q^{12} +(-2.32419 + 2.87014i) q^{13} +(5.42764 - 2.41654i) q^{14} +(1.72275 - 3.46874i) q^{15} +(4.46169 + 1.98647i) q^{16} +(-2.54982 + 1.29920i) q^{17} +(-1.50399 - 4.65530i) q^{18} +(6.31730 + 2.05262i) q^{19} +(-1.40881 + 0.434391i) q^{20} +(-2.10148 - 5.95021i) q^{21} +(6.51148 - 4.22861i) q^{22} +(-0.919867 + 2.39633i) q^{23} +(1.80970 - 3.32641i) q^{24} +(-2.62935 + 4.25283i) q^{25} -6.02261i q^{26} +(-5.05561 + 1.20033i) q^{27} +(-1.09051 + 2.14025i) q^{28} +(2.82765 - 3.14043i) q^{29} +(1.37653 + 6.16399i) q^{30} +(-0.908934 - 1.00947i) q^{31} +(-3.46938 + 0.929618i) q^{32} +(-3.92878 - 7.25038i) q^{33} +(1.89813 - 4.26327i) q^{34} +(1.98971 + 7.89999i) q^{35} +(1.65702 + 1.08003i) q^{36} +(0.242591 - 1.53166i) q^{37} +(-10.1126 + 3.88185i) q^{38} +(-6.37664 - 0.507221i) q^{39} +(-2.72409 + 4.05947i) q^{40} +(0.830574 - 0.0872968i) q^{41} +(8.76970 + 5.38417i) q^{42} +(-1.23082 + 4.59349i) q^{43} +(-0.970006 + 2.98537i) q^{44} +(6.64225 - 0.938324i) q^{45} +(-1.29349 - 3.98095i) q^{46} +(4.08223 - 0.213941i) q^{47} +(1.53461 + 8.31885i) q^{48} +(5.43322 + 3.13687i) q^{49} +(-1.03226 - 8.08808i) q^{50} +(-4.35402 - 2.36876i) q^{51} +(1.53236 + 1.89230i) q^{52} +(2.05548 + 1.04732i) q^{53} +(5.17524 - 6.70956i) q^{54} +(4.18310 + 9.78981i) q^{55} +(1.65611 + 7.79139i) q^{56} +(3.25836 + 11.0339i) q^{57} +(-0.360661 + 6.88183i) q^{58} +(1.22755 + 11.6793i) q^{59} +(-2.00084 - 1.58649i) q^{60} +(0.806761 - 7.67582i) q^{61} +(2.18789 + 0.346528i) q^{62} +(6.43925 - 8.83175i) q^{63} +(-2.29860 + 3.16376i) q^{64} +(8.13614 + 1.41463i) q^{65} +(12.4198 + 5.15662i) q^{66} +(-10.4288 - 0.546548i) q^{67} +(0.488329 + 1.82247i) q^{68} +(-4.32389 + 1.03425i) q^{69} +(-10.6290 - 7.96987i) q^{70} +(-10.9516 + 3.55840i) q^{71} +(6.52417 - 0.674682i) q^{72} +(-2.51103 - 15.8540i) q^{73} +(1.26444 + 2.19007i) q^{74} +(-8.65046 + 0.411770i) q^{75} +(2.18969 - 3.79266i) q^{76} +(16.1940 + 6.21628i) q^{77} +(8.60179 - 5.90126i) q^{78} +(-6.41701 - 5.77790i) q^{79} +(-0.977746 - 10.8769i) q^{80} +(-6.66811 - 6.04452i) q^{81} +(-0.963015 + 0.963015i) q^{82} +(-5.32431 + 8.19871i) q^{83} +(-4.12535 + 0.539602i) q^{84} +(5.31355 + 3.56563i) q^{85} +(-3.15426 - 7.08457i) q^{86} +(7.25599 + 0.961445i) q^{87} +(3.73033 + 9.71784i) q^{88} +(-2.03131 - 1.47583i) q^{89} +(-7.45491 + 8.00582i) q^{90} +(10.8857 - 7.90890i) q^{91} +(1.41930 + 0.921706i) q^{92} +(0.551161 - 2.28732i) q^{93} +(-4.95394 + 4.46055i) q^{94} +(-2.86882 - 14.5732i) q^{95} +(-4.72720 - 4.04426i) q^{96} +(0.188909 + 3.60460i) q^{97} +(-10.1049 + 1.60045i) q^{98} +(6.50572 - 12.7156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 448 q - 24 q^{2} - 14 q^{3} - 10 q^{4} - 24 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9} - 32 q^{10} - 18 q^{11} - 14 q^{12} - 8 q^{13} - 30 q^{14} - 14 q^{15} - 50 q^{16} - 56 q^{18} - 40 q^{19} - 48 q^{20} - 12 q^{21}+ \cdots - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26732 + 1.02626i −0.896132 + 0.725673i −0.962258 − 0.272138i \(-0.912269\pi\)
0.0661265 + 0.997811i \(0.478936\pi\)
\(3\) 0.979850 + 1.42825i 0.565717 + 0.824600i
\(4\) 0.137078 − 0.644900i 0.0685388 − 0.322450i
\(5\) −1.08873 − 1.95312i −0.486894 − 0.873461i
\(6\) −2.70753 − 0.804473i −1.10535 − 0.328425i
\(7\) −3.51917 − 0.942959i −1.33012 − 0.356405i −0.477361 − 0.878707i \(-0.658407\pi\)
−0.852761 + 0.522302i \(0.825073\pi\)
\(8\) −0.992568 − 1.94802i −0.350926 − 0.688731i
\(9\) −1.07979 + 2.79894i −0.359930 + 0.932979i
\(10\) 3.38417 + 1.35792i 1.07017 + 0.429411i
\(11\) −4.73499 − 0.497667i −1.42765 − 0.150052i −0.641044 − 0.767504i \(-0.721498\pi\)
−0.786608 + 0.617452i \(0.788165\pi\)
\(12\) 1.05539 − 0.436124i 0.304666 − 0.125898i
\(13\) −2.32419 + 2.87014i −0.644616 + 0.796034i −0.989794 − 0.142503i \(-0.954485\pi\)
0.345179 + 0.938537i \(0.387818\pi\)
\(14\) 5.42764 − 2.41654i 1.45060 − 0.645848i
\(15\) 1.72275 − 3.46874i 0.444812 − 0.895624i
\(16\) 4.46169 + 1.98647i 1.11542 + 0.496618i
\(17\) −2.54982 + 1.29920i −0.618422 + 0.315102i −0.734987 − 0.678081i \(-0.762812\pi\)
0.116565 + 0.993183i \(0.462812\pi\)
\(18\) −1.50399 − 4.65530i −0.354494 − 1.09726i
\(19\) 6.31730 + 2.05262i 1.44929 + 0.470903i 0.924780 − 0.380502i \(-0.124249\pi\)
0.524509 + 0.851405i \(0.324249\pi\)
\(20\) −1.40881 + 0.434391i −0.315019 + 0.0971327i
\(21\) −2.10148 − 5.95021i −0.458580 − 1.29844i
\(22\) 6.51148 − 4.22861i 1.38825 − 0.901542i
\(23\) −0.919867 + 2.39633i −0.191805 + 0.499670i −0.995433 − 0.0954622i \(-0.969567\pi\)
0.803628 + 0.595133i \(0.202900\pi\)
\(24\) 1.80970 − 3.32641i 0.369403 − 0.679000i
\(25\) −2.62935 + 4.25283i −0.525869 + 0.850565i
\(26\) − 6.02261i − 1.18113i
\(27\) −5.05561 + 1.20033i −0.972953 + 0.231004i
\(28\) −1.09051 + 2.14025i −0.206088 + 0.404470i
\(29\) 2.82765 − 3.14043i 0.525082 − 0.583163i −0.421012 − 0.907055i \(-0.638325\pi\)
0.946094 + 0.323892i \(0.104992\pi\)
\(30\) 1.37653 + 6.16399i 0.251320 + 1.12539i
\(31\) −0.908934 − 1.00947i −0.163249 − 0.181307i 0.655970 − 0.754787i \(-0.272260\pi\)
−0.819220 + 0.573480i \(0.805593\pi\)
\(32\) −3.46938 + 0.929618i −0.613306 + 0.164335i
\(33\) −3.92878 − 7.25038i −0.683913 − 1.26213i
\(34\) 1.89813 − 4.26327i 0.325527 − 0.731145i
\(35\) 1.98971 + 7.89999i 0.336322 + 1.33534i
\(36\) 1.65702 + 1.08003i 0.276170 + 0.180005i
\(37\) 0.242591 − 1.53166i 0.0398818 − 0.251803i −0.959690 − 0.281060i \(-0.909314\pi\)
0.999572 + 0.0292566i \(0.00931398\pi\)
\(38\) −10.1126 + 3.88185i −1.64048 + 0.629719i
\(39\) −6.37664 − 0.507221i −1.02108 − 0.0812203i
\(40\) −2.72409 + 4.05947i −0.430716 + 0.641859i
\(41\) 0.830574 − 0.0872968i 0.129714 − 0.0136335i −0.0394490 − 0.999222i \(-0.512560\pi\)
0.169163 + 0.985588i \(0.445894\pi\)
\(42\) 8.76970 + 5.38417i 1.35319 + 0.830796i
\(43\) −1.23082 + 4.59349i −0.187699 + 0.700501i 0.806338 + 0.591455i \(0.201446\pi\)
−0.994037 + 0.109046i \(0.965220\pi\)
\(44\) −0.970006 + 2.98537i −0.146234 + 0.450062i
\(45\) 6.64225 − 0.938324i 0.990169 − 0.139877i
\(46\) −1.29349 − 3.98095i −0.190714 − 0.586959i
\(47\) 4.08223 − 0.213941i 0.595454 − 0.0312064i 0.247771 − 0.968819i \(-0.420302\pi\)
0.347683 + 0.937612i \(0.386969\pi\)
\(48\) 1.53461 + 8.31885i 0.221502 + 1.20072i
\(49\) 5.43322 + 3.13687i 0.776174 + 0.448125i
\(50\) −1.03226 − 8.08808i −0.145984 − 1.14383i
\(51\) −4.35402 − 2.36876i −0.609685 − 0.331692i
\(52\) 1.53236 + 1.89230i 0.212500 + 0.262415i
\(53\) 2.05548 + 1.04732i 0.282342 + 0.143861i 0.589427 − 0.807822i \(-0.299354\pi\)
−0.307085 + 0.951682i \(0.599354\pi\)
\(54\) 5.17524 − 6.70956i 0.704260 − 0.913056i
\(55\) 4.18310 + 9.78981i 0.564050 + 1.32006i
\(56\) 1.65611 + 7.79139i 0.221307 + 1.04117i
\(57\) 3.25836 + 11.0339i 0.431581 + 1.46148i
\(58\) −0.360661 + 6.88183i −0.0473572 + 0.903629i
\(59\) 1.22755 + 11.6793i 0.159813 + 1.52052i 0.721062 + 0.692870i \(0.243654\pi\)
−0.561249 + 0.827647i \(0.689679\pi\)
\(60\) −2.00084 − 1.58649i −0.258307 − 0.204815i
\(61\) 0.806761 − 7.67582i 0.103295 − 0.982788i −0.812994 − 0.582271i \(-0.802164\pi\)
0.916290 − 0.400516i \(-0.131169\pi\)
\(62\) 2.18789 + 0.346528i 0.277863 + 0.0440091i
\(63\) 6.43925 − 8.83175i 0.811269 − 1.11270i
\(64\) −2.29860 + 3.16376i −0.287326 + 0.395470i
\(65\) 8.13614 + 1.41463i 1.00916 + 0.175463i
\(66\) 12.4198 + 5.15662i 1.52877 + 0.634736i
\(67\) −10.4288 − 0.546548i −1.27408 − 0.0667715i −0.596733 − 0.802440i \(-0.703535\pi\)
−0.677342 + 0.735668i \(0.736868\pi\)
\(68\) 0.488329 + 1.82247i 0.0592186 + 0.221007i
\(69\) −4.32389 + 1.03425i −0.520536 + 0.124509i
\(70\) −10.6290 − 7.96987i −1.27041 − 0.952582i
\(71\) −10.9516 + 3.55840i −1.29972 + 0.422304i −0.875484 − 0.483247i \(-0.839457\pi\)
−0.424236 + 0.905552i \(0.639457\pi\)
\(72\) 6.52417 − 0.674682i 0.768880 − 0.0795120i
\(73\) −2.51103 − 15.8540i −0.293894 − 1.85557i −0.485679 − 0.874137i \(-0.661428\pi\)
0.191785 − 0.981437i \(-0.438572\pi\)
\(74\) 1.26444 + 2.19007i 0.146988 + 0.254590i
\(75\) −8.65046 + 0.411770i −0.998869 + 0.0475471i
\(76\) 2.18969 − 3.79266i 0.251175 − 0.435048i
\(77\) 16.1940 + 6.21628i 1.84547 + 0.708410i
\(78\) 8.60179 − 5.90126i 0.973961 − 0.668186i
\(79\) −6.41701 − 5.77790i −0.721970 − 0.650065i 0.223899 − 0.974612i \(-0.428121\pi\)
−0.945869 + 0.324547i \(0.894788\pi\)
\(80\) −0.977746 − 10.8769i −0.109315 − 1.21608i
\(81\) −6.66811 − 6.04452i −0.740901 − 0.671614i
\(82\) −0.963015 + 0.963015i −0.106347 + 0.106347i
\(83\) −5.32431 + 8.19871i −0.584418 + 0.899926i −0.999953 − 0.00965169i \(-0.996928\pi\)
0.415535 + 0.909577i \(0.363594\pi\)
\(84\) −4.12535 + 0.539602i −0.450113 + 0.0588754i
\(85\) 5.31355 + 3.56563i 0.576335 + 0.386747i
\(86\) −3.15426 − 7.08457i −0.340132 − 0.763949i
\(87\) 7.25599 + 0.961445i 0.777923 + 0.103078i
\(88\) 3.73033 + 9.71784i 0.397654 + 1.03593i
\(89\) −2.03131 − 1.47583i −0.215318 − 0.156438i 0.474898 − 0.880041i \(-0.342485\pi\)
−0.690217 + 0.723603i \(0.742485\pi\)
\(90\) −7.45491 + 8.00582i −0.785817 + 0.843887i
\(91\) 10.8857 − 7.90890i 1.14113 − 0.829078i
\(92\) 1.41930 + 0.921706i 0.147972 + 0.0960945i
\(93\) 0.551161 − 2.28732i 0.0571527 − 0.237184i
\(94\) −4.95394 + 4.46055i −0.510960 + 0.460070i
\(95\) −2.86882 − 14.5732i −0.294334 − 1.49518i
\(96\) −4.72720 − 4.04426i −0.482468 − 0.412765i
\(97\) 0.188909 + 3.60460i 0.0191808 + 0.365992i 0.991310 + 0.131546i \(0.0419941\pi\)
−0.972129 + 0.234446i \(0.924673\pi\)
\(98\) −10.1049 + 1.60045i −1.02075 + 0.161670i
\(99\) 6.50572 − 12.7156i 0.653850 − 1.27796i
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.w.a.113.8 yes 448
3.2 odd 2 675.2.bd.a.638.21 448
9.2 odd 6 inner 225.2.w.a.38.8 yes 448
9.7 even 3 675.2.bd.a.413.21 448
25.2 odd 20 inner 225.2.w.a.77.8 yes 448
75.2 even 20 675.2.bd.a.152.21 448
225.2 even 60 inner 225.2.w.a.2.8 ✓ 448
225.52 odd 60 675.2.bd.a.602.21 448
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.w.a.2.8 ✓ 448 225.2 even 60 inner
225.2.w.a.38.8 yes 448 9.2 odd 6 inner
225.2.w.a.77.8 yes 448 25.2 odd 20 inner
225.2.w.a.113.8 yes 448 1.1 even 1 trivial
675.2.bd.a.152.21 448 75.2 even 20
675.2.bd.a.413.21 448 9.7 even 3
675.2.bd.a.602.21 448 225.52 odd 60
675.2.bd.a.638.21 448 3.2 odd 2