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Properties

Label 225.2.q.a.121.7
Level $225$
Weight $2$
Character 225.121
Analytic conductor $1.797$
Analytic rank $0$
Dimension $224$
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(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 121.7
Character \(\chi\) \(=\) 225.121
Dual form 225.2.q.a.106.7

$q$-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+(-1.39878 + 0.297321i) q^{2} +(-1.19801 + 1.25091i) q^{3} +(0.0411076 - 0.0183023i) q^{4} +(1.76390 + 1.37429i) q^{5} +(1.30384 - 2.10594i) q^{6} +(1.92019 + 3.32586i) q^{7} +(2.26178 - 1.64328i) q^{8} +(-0.129533 - 2.99720i) q^{9} +(-2.87591 - 1.39789i) q^{10} +(-5.11325 + 1.08686i) q^{11} +(-0.0263530 + 0.0733481i) q^{12} +(3.28926 + 0.699153i) q^{13} +(-3.67477 - 4.08125i) q^{14} +(-3.83228 + 0.560051i) q^{15} +(-2.73539 + 3.03795i) q^{16} +(-3.90998 + 2.84077i) q^{17} +(1.07232 + 4.15393i) q^{18} +(-1.90903 + 1.38699i) q^{19} +(0.0976621 + 0.0242104i) q^{20} +(-6.46075 - 1.58245i) q^{21} +(6.82920 - 3.04055i) q^{22} +(-3.12041 - 3.46556i) q^{23} +(-0.654054 + 4.79795i) q^{24} +(1.22265 + 4.84821i) q^{25} -4.80884 q^{26} +(3.90440 + 3.42865i) q^{27} +(0.139805 + 0.101574i) q^{28} +(0.768054 - 7.30754i) q^{29} +(5.19401 - 1.92281i) q^{30} +(-0.0679615 - 0.646611i) q^{31} +(0.127249 - 0.220402i) q^{32} +(4.76619 - 7.69827i) q^{33} +(4.62460 - 5.13614i) q^{34} +(-1.18369 + 8.50536i) q^{35} +(-0.0601804 - 0.120837i) q^{36} +(0.315262 + 0.970276i) q^{37} +(2.25794 - 2.50769i) q^{38} +(-4.81515 + 3.27696i) q^{39} +(6.24790 + 0.209769i) q^{40} +(3.66775 + 0.779604i) q^{41} +(9.50769 + 0.292591i) q^{42} +(-2.34851 - 4.06773i) q^{43} +(-0.190302 + 0.138262i) q^{44} +(3.89054 - 5.46477i) q^{45} +(5.39516 + 3.91981i) q^{46} +(-1.24854 + 11.8791i) q^{47} +(-0.523169 - 7.06122i) q^{48} +(-3.87423 + 6.71036i) q^{49} +(-3.15170 - 6.41808i) q^{50} +(1.13067 - 8.29429i) q^{51} +(0.148010 - 0.0314604i) q^{52} +(2.92011 + 2.12158i) q^{53} +(-6.48082 - 3.63509i) q^{54} +(-10.5129 - 5.10999i) q^{55} +(9.80838 + 4.36697i) q^{56} +(0.552044 - 4.04964i) q^{57} +(1.09834 + 10.4500i) q^{58} +(7.61268 + 1.61813i) q^{59} +(-0.147285 + 0.0931618i) q^{60} +(6.44432 - 1.36978i) q^{61} +(0.287314 + 0.884263i) q^{62} +(9.71955 - 6.18599i) q^{63} +(2.41404 - 7.42965i) q^{64} +(4.84107 + 5.75363i) q^{65} +(-4.37801 + 12.1853i) q^{66} +(0.106476 + 1.01305i) q^{67} +(-0.108737 + 0.188339i) q^{68} +(8.07338 + 0.248451i) q^{69} +(-0.873097 - 12.2491i) q^{70} +(-2.82082 - 2.04945i) q^{71} +(-5.21823 - 6.56617i) q^{72} +(-1.38140 + 4.25150i) q^{73} +(-0.729467 - 1.26347i) q^{74} +(-7.52941 - 4.27879i) q^{75} +(-0.0530904 + 0.0919553i) q^{76} +(-13.4331 - 14.9190i) q^{77} +(5.76105 - 6.01540i) q^{78} +(-0.171432 + 1.63107i) q^{79} +(-8.99997 + 1.59942i) q^{80} +(-8.96644 + 0.776471i) q^{81} -5.36219 q^{82} +(10.3261 + 4.59747i) q^{83} +(-0.294548 + 0.0531957i) q^{84} +(-10.8008 - 0.362630i) q^{85} +(4.49448 + 4.99162i) q^{86} +(8.22091 + 9.71529i) q^{87} +(-9.77907 + 10.8608i) q^{88} +(-0.935178 + 2.87818i) q^{89} +(-3.81724 + 8.80077i) q^{90} +(3.99070 + 12.2821i) q^{91} +(-0.191700 - 0.0853504i) q^{92} +(0.890268 + 0.689634i) q^{93} +(-1.78546 - 16.9875i) q^{94} +(-5.27345 - 0.177052i) q^{95} +(0.123256 + 0.423221i) q^{96} +(-1.69408 + 16.1181i) q^{97} +(3.42408 - 10.5382i) q^{98} +(3.91986 + 15.1847i) 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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39878 + 0.297321i −0.989090 + 0.210238i −0.673925 − 0.738799i \(-0.735393\pi\)
−0.315165 + 0.949037i \(0.602060\pi\)
\(3\) −1.19801 + 1.25091i −0.691673 + 0.722211i
\(4\) 0.0411076 − 0.0183023i 0.0205538 − 0.00915114i
\(5\) 1.76390 + 1.37429i 0.788838 + 0.614601i
\(6\) 1.30384 − 2.10594i 0.532291 − 0.859747i
\(7\) 1.92019 + 3.32586i 0.725762 + 1.25706i 0.958660 + 0.284556i \(0.0918460\pi\)
−0.232897 + 0.972501i \(0.574821\pi\)
\(8\) 2.26178 − 1.64328i 0.799662 − 0.580988i
\(9\) −0.129533 − 2.99720i −0.0431775 − 0.999067i
\(10\) −2.87591 − 1.39789i −0.909444 − 0.442052i
\(11\) −5.11325 + 1.08686i −1.54170 + 0.327699i −0.898839 − 0.438280i \(-0.855588\pi\)
−0.642865 + 0.765979i \(0.722255\pi\)
\(12\) −0.0263530 + 0.0733481i −0.00760745 + 0.0211738i
\(13\) 3.28926 + 0.699153i 0.912276 + 0.193910i 0.640059 − 0.768326i \(-0.278910\pi\)
0.272217 + 0.962236i \(0.412243\pi\)
\(14\) −3.67477 − 4.08125i −0.982125 − 1.09076i
\(15\) −3.83228 + 0.560051i −0.989490 + 0.144605i
\(16\) −2.73539 + 3.03795i −0.683847 + 0.759489i
\(17\) −3.90998 + 2.84077i −0.948309 + 0.688987i −0.950406 − 0.311011i \(-0.899332\pi\)
0.00209712 + 0.999998i \(0.499332\pi\)
\(18\) 1.07232 + 4.15393i 0.252748 + 0.979090i
\(19\) −1.90903 + 1.38699i −0.437961 + 0.318197i −0.784824 − 0.619719i \(-0.787247\pi\)
0.346863 + 0.937916i \(0.387247\pi\)
\(20\) 0.0976621 + 0.0242104i 0.0218379 + 0.00541362i
\(21\) −6.46075 − 1.58245i −1.40985 − 0.345319i
\(22\) 6.82920 − 3.04055i 1.45599 − 0.648248i
\(23\) −3.12041 − 3.46556i −0.650650 − 0.722620i 0.324074 − 0.946032i \(-0.394947\pi\)
−0.974724 + 0.223412i \(0.928281\pi\)
\(24\) −0.654054 + 4.79795i −0.133508 + 0.979378i
\(25\) 1.22265 + 4.84821i 0.244531 + 0.969641i
\(26\) −4.80884 −0.943090
\(27\) 3.90440 + 3.42865i 0.751402 + 0.659844i
\(28\) 0.139805 + 0.101574i 0.0264207 + 0.0191957i
\(29\) 0.768054 − 7.30754i 0.142624 − 1.35698i −0.655824 − 0.754913i \(-0.727679\pi\)
0.798448 − 0.602063i \(-0.205654\pi\)
\(30\) 5.19401 − 1.92281i 0.948293 − 0.351055i
\(31\) −0.0679615 − 0.646611i −0.0122062 − 0.116135i 0.986722 − 0.162417i \(-0.0519291\pi\)
−0.998928 + 0.0462828i \(0.985262\pi\)
\(32\) 0.127249 − 0.220402i 0.0224947 − 0.0389619i
\(33\) 4.76619 − 7.69827i 0.829687 − 1.34010i
\(34\) 4.62460 − 5.13614i 0.793112 − 0.880840i
\(35\) −1.18369 + 8.50536i −0.200080 + 1.43767i
\(36\) −0.0601804 − 0.120837i −0.0100301 − 0.0201395i
\(37\) 0.315262 + 0.970276i 0.0518287 + 0.159512i 0.973621 − 0.228172i \(-0.0732749\pi\)
−0.921792 + 0.387685i \(0.873275\pi\)
\(38\) 2.25794 − 2.50769i 0.366286 − 0.406801i
\(39\) −4.81515 + 3.27696i −0.771041 + 0.524733i
\(40\) 6.24790 + 0.209769i 0.987880 + 0.0331674i
\(41\) 3.66775 + 0.779604i 0.572806 + 0.121754i 0.485203 − 0.874402i \(-0.338746\pi\)
0.0876036 + 0.996155i \(0.472079\pi\)
\(42\) 9.50769 + 0.292591i 1.46707 + 0.0451477i
\(43\) −2.34851 − 4.06773i −0.358144 − 0.620324i 0.629507 − 0.776995i \(-0.283257\pi\)
−0.987651 + 0.156671i \(0.949924\pi\)
\(44\) −0.190302 + 0.138262i −0.0286890 + 0.0208438i
\(45\) 3.89054 − 5.46477i 0.579968 − 0.814639i
\(46\) 5.39516 + 3.91981i 0.795473 + 0.577945i
\(47\) −1.24854 + 11.8791i −0.182118 + 1.73274i 0.397281 + 0.917697i \(0.369954\pi\)
−0.579400 + 0.815044i \(0.696713\pi\)
\(48\) −0.523169 − 7.06122i −0.0755130 − 1.01920i
\(49\) −3.87423 + 6.71036i −0.553461 + 0.958623i
\(50\) −3.15170 − 6.41808i −0.445718 − 0.907653i
\(51\) 1.13067 − 8.29429i 0.158326 − 1.16143i
\(52\) 0.148010 − 0.0314604i 0.0205252 − 0.00436277i
\(53\) 2.92011 + 2.12158i 0.401108 + 0.291422i 0.769992 − 0.638054i \(-0.220260\pi\)
−0.368884 + 0.929475i \(0.620260\pi\)
\(54\) −6.48082 − 3.63509i −0.881929 − 0.494673i
\(55\) −10.5129 − 5.10999i −1.41756 − 0.689031i
\(56\) 9.80838 + 4.36697i 1.31070 + 0.583561i
\(57\) 0.552044 − 4.04964i 0.0731200 − 0.536388i
\(58\) 1.09834 + 10.4500i 0.144220 + 1.37216i
\(59\) 7.61268 + 1.61813i 0.991087 + 0.210662i 0.674799 − 0.738001i \(-0.264230\pi\)
0.316288 + 0.948663i \(0.397563\pi\)
\(60\) −0.147285 + 0.0931618i −0.0190145 + 0.0120271i
\(61\) 6.44432 − 1.36978i 0.825110 − 0.175383i 0.224041 − 0.974580i \(-0.428075\pi\)
0.601069 + 0.799197i \(0.294742\pi\)
\(62\) 0.287314 + 0.884263i 0.0364890 + 0.112301i
\(63\) 9.71955 − 6.18599i 1.22455 − 0.779362i
\(64\) 2.41404 − 7.42965i 0.301755 − 0.928707i
\(65\) 4.84107 + 5.75363i 0.600461 + 0.713650i
\(66\) −4.37801 + 12.1853i −0.538896 + 1.49991i
\(67\) 0.106476 + 1.01305i 0.0130081 + 0.123764i 0.999099 − 0.0424441i \(-0.0135144\pi\)
−0.986091 + 0.166208i \(0.946848\pi\)
\(68\) −0.108737 + 0.188339i −0.0131863 + 0.0228394i
\(69\) 8.07338 + 0.248451i 0.971921 + 0.0299100i
\(70\) −0.873097 − 12.2491i −0.104355 − 1.46405i
\(71\) −2.82082 − 2.04945i −0.334770 − 0.243225i 0.407682 − 0.913124i \(-0.366337\pi\)
−0.742452 + 0.669899i \(0.766337\pi\)
\(72\) −5.21823 − 6.56617i −0.614974 − 0.773830i
\(73\) −1.38140 + 4.25150i −0.161680 + 0.497601i −0.998776 − 0.0494551i \(-0.984252\pi\)
0.837096 + 0.547056i \(0.184252\pi\)
\(74\) −0.729467 − 1.26347i −0.0847988 − 0.146876i
\(75\) −7.52941 − 4.27879i −0.869421 − 0.494072i
\(76\) −0.0530904 + 0.0919553i −0.00608989 + 0.0105480i
\(77\) −13.4331 − 14.9190i −1.53085 − 1.70018i
\(78\) 5.76105 − 6.01540i 0.652310 − 0.681110i
\(79\) −0.171432 + 1.63107i −0.0192877 + 0.183510i −0.999924 − 0.0123278i \(-0.996076\pi\)
0.980636 + 0.195838i \(0.0627425\pi\)
\(80\) −8.99997 + 1.59942i −1.00623 + 0.178821i
\(81\) −8.96644 + 0.776471i −0.996271 + 0.0862745i
\(82\) −5.36219 −0.592154
\(83\) 10.3261 + 4.59747i 1.13344 + 0.504638i 0.885731 − 0.464198i \(-0.153657\pi\)
0.247704 + 0.968836i \(0.420324\pi\)
\(84\) −0.294548 + 0.0531957i −0.0321378 + 0.00580413i
\(85\) −10.8008 − 0.362630i −1.17151 − 0.0393328i
\(86\) 4.49448 + 4.99162i 0.484652 + 0.538261i
\(87\) 8.22091 + 9.71529i 0.881374 + 1.04159i
\(88\) −9.77907 + 10.8608i −1.04245 + 1.15776i
\(89\) −0.935178 + 2.87818i −0.0991286 + 0.305087i −0.988308 − 0.152473i \(-0.951276\pi\)
0.889179 + 0.457559i \(0.151276\pi\)
\(90\) −3.81724 + 8.80077i −0.402373 + 0.927683i
\(91\) 3.99070 + 12.2821i 0.418339 + 1.28752i
\(92\) −0.191700 − 0.0853504i −0.0199861 − 0.00889839i
\(93\) 0.890268 + 0.689634i 0.0923165 + 0.0715117i
\(94\) −1.78546 − 16.9875i −0.184156 − 1.75212i
\(95\) −5.27345 − 0.177052i −0.541044 − 0.0181652i
\(96\) 0.123256 + 0.423221i 0.0125798 + 0.0431948i
\(97\) −1.69408 + 16.1181i −0.172008 + 1.63654i 0.479243 + 0.877682i \(0.340911\pi\)
−0.651251 + 0.758862i \(0.725755\pi\)
\(98\) 3.42408 − 10.5382i 0.345885 − 1.06452i
\(99\) 3.91986 + 15.1847i 0.393961 + 1.52612i
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.121.7 yes 224
3.2 odd 2 675.2.r.a.496.22 224
9.2 odd 6 675.2.r.a.46.7 224
9.7 even 3 inner 225.2.q.a.196.22 yes 224
25.6 even 5 inner 225.2.q.a.31.22 ✓ 224
75.56 odd 10 675.2.r.a.631.7 224
225.56 odd 30 675.2.r.a.181.22 224
225.106 even 15 inner 225.2.q.a.106.7 yes 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.22 ✓ 224 25.6 even 5 inner
225.2.q.a.106.7 yes 224 225.106 even 15 inner
225.2.q.a.121.7 yes 224 1.1 even 1 trivial
225.2.q.a.196.22 yes 224 9.7 even 3 inner
675.2.r.a.46.7 224 9.2 odd 6
675.2.r.a.181.22 224 225.56 odd 30
675.2.r.a.496.22 224 3.2 odd 2
675.2.r.a.631.7 224 75.56 odd 10