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Properties

Label 225.2.q.a.196.22
Level $225$
Weight $2$
Character 225.196
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 196.22
Character \(\chi\) \(=\) 225.196
Dual form 225.2.q.a.31.22

$q$-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+(0.956880 + 1.06272i) q^{2} +(0.819476 - 1.52593i) q^{3} +(-0.00470356 + 0.0447514i) q^{4} +(-2.07212 + 0.840433i) q^{5} +(2.40578 - 0.589254i) q^{6} +(1.92019 - 3.32586i) q^{7} +(2.26178 - 1.64328i) q^{8} +(-1.65692 - 2.50092i) q^{9} +(-2.87591 - 1.39789i) q^{10} +(3.49787 + 3.88478i) q^{11} +(0.0644329 + 0.0438500i) q^{12} +(-2.25011 + 2.49900i) q^{13} +(5.37185 - 1.14182i) q^{14} +(-0.415610 + 3.85062i) q^{15} +(3.99864 + 0.849937i) q^{16} +(-3.90998 + 2.84077i) q^{17} +(1.07232 - 4.15393i) q^{18} +(-1.90903 + 1.38699i) q^{19} +(-0.0278642 - 0.0966831i) q^{20} +(-3.50148 - 5.65553i) q^{21} +(-0.781401 + 7.43453i) q^{22} +(4.56147 - 0.969570i) q^{23} +(-0.654054 - 4.79795i) q^{24} +(3.58734 - 3.48295i) q^{25} -4.80884 q^{26} +(-5.17404 + 0.478889i) q^{27} +(0.139805 + 0.101574i) q^{28} +(-6.71254 + 2.98862i) q^{29} +(-4.48983 + 3.24290i) q^{30} +(0.593962 + 0.264449i) q^{31} +(0.127249 + 0.220402i) q^{32} +(8.79432 - 2.15402i) q^{33} +(-6.76033 - 1.43695i) q^{34} +(-1.18369 + 8.50536i) q^{35} +(0.119713 - 0.0623860i) q^{36} +(0.315262 + 0.970276i) q^{37} +(-3.30069 - 0.701584i) q^{38} +(1.96939 + 5.48139i) q^{39} +(-3.30561 + 5.30596i) q^{40} +(-2.50903 + 2.78656i) q^{41} +(2.65977 - 9.13276i) q^{42} +(-2.34851 + 4.06773i) q^{43} +(-0.190302 + 0.138262i) q^{44} +(5.53519 + 3.78968i) q^{45} +(5.39516 + 3.91981i) q^{46} +(10.9118 - 4.85827i) q^{47} +(4.57373 - 5.40514i) q^{48} +(-3.87423 - 6.71036i) q^{49} +(7.13407 + 0.479583i) q^{50} +(1.13067 + 8.29429i) q^{51} +(-0.101250 - 0.112450i) q^{52} +(2.92011 + 2.12158i) q^{53} +(-5.45986 - 5.04033i) q^{54} +(-10.5129 - 5.10999i) q^{55} +(-1.12228 - 10.6778i) q^{56} +(0.552044 + 4.04964i) q^{57} +(-9.59917 - 4.27383i) q^{58} +(-5.20768 + 5.78372i) q^{59} +(-0.170366 - 0.0367107i) q^{60} +(-4.40843 - 4.89605i) q^{61} +(0.287314 + 0.884263i) q^{62} +(-11.4993 + 0.708433i) q^{63} +(2.41404 - 7.42965i) q^{64} +(2.56225 - 7.06930i) q^{65} +(10.7042 + 7.28479i) q^{66} +(-0.930565 - 0.414314i) q^{67} +(-0.108737 - 0.188339i) q^{68} +(2.25852 - 7.75502i) q^{69} +(-10.1715 + 6.88068i) q^{70} +(-2.82082 - 2.04945i) q^{71} +(-7.85732 - 2.93377i) q^{72} +(-1.38140 + 4.25150i) q^{73} +(-0.729467 + 1.26347i) q^{74} +(-2.37500 - 8.32823i) q^{75} +(-0.0530904 - 0.0919553i) q^{76} +(19.6368 - 4.17393i) q^{77} +(-3.94073 + 7.33794i) q^{78} +(1.49826 - 0.667070i) q^{79} +(-8.99997 + 1.59942i) q^{80} +(-3.50925 + 8.28765i) q^{81} -5.36219 q^{82} +(-1.18152 - 11.2414i) q^{83} +(0.269562 - 0.130095i) q^{84} +(5.71446 - 9.17248i) q^{85} +(-6.57011 + 1.39652i) q^{86} +(-0.940354 + 12.6920i) q^{87} +(14.2952 + 3.03854i) q^{88} +(-0.935178 + 2.87818i) q^{89} +(1.26913 + 9.50864i) q^{90} +(3.99070 + 12.2821i) q^{91} +(0.0219345 + 0.208692i) q^{92} +(0.890268 - 0.689634i) q^{93} +(15.6043 + 6.94749i) q^{94} +(2.79006 - 4.47841i) q^{95} +(0.440596 - 0.0135589i) q^{96} +(14.8057 - 6.59193i) 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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.956880 + 1.06272i 0.676616 + 0.751458i 0.979472 − 0.201578i \(-0.0646069\pi\)
−0.302856 + 0.953036i \(0.597940\pi\)
\(3\) 0.819476 − 1.52593i 0.473125 − 0.880995i
\(4\) −0.00470356 + 0.0447514i −0.00235178 + 0.0223757i
\(5\) −2.07212 + 0.840433i −0.926679 + 0.375853i
\(6\) 2.40578 − 0.589254i 0.982155 − 0.240562i
\(7\) 1.92019 − 3.32586i 0.725762 − 1.25706i −0.232897 − 0.972501i \(-0.574821\pi\)
0.958660 − 0.284556i \(-0.0918460\pi\)
\(8\) 2.26178 − 1.64328i 0.799662 − 0.580988i
\(9\) −1.65692 − 2.50092i −0.552306 − 0.833642i
\(10\) −2.87591 − 1.39789i −0.909444 − 0.442052i
\(11\) 3.49787 + 3.88478i 1.05465 + 1.17131i 0.984790 + 0.173748i \(0.0555879\pi\)
0.0698578 + 0.997557i \(0.477745\pi\)
\(12\) 0.0644329 + 0.0438500i 0.0186002 + 0.0126584i
\(13\) −2.25011 + 2.49900i −0.624069 + 0.693099i −0.969429 − 0.245371i \(-0.921090\pi\)
0.345360 + 0.938470i \(0.387757\pi\)
\(14\) 5.37185 − 1.14182i 1.43569 − 0.305165i
\(15\) −0.415610 + 3.85062i −0.107310 + 0.994226i
\(16\) 3.99864 + 0.849937i 0.999660 + 0.212484i
\(17\) −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 \(0.387247\pi\)
\(20\) −0.0278642 − 0.0966831i −0.00623063 − 0.0216190i
\(21\) −3.50148 − 5.65553i −0.764085 − 1.23414i
\(22\) −0.781401 + 7.43453i −0.166595 + 1.58505i
\(23\) 4.56147 − 0.969570i 0.951132 − 0.202169i 0.293882 − 0.955842i \(-0.405053\pi\)
0.657250 + 0.753672i \(0.271719\pi\)
\(24\) −0.654054 − 4.79795i −0.133508 − 0.979378i
\(25\) 3.58734 − 3.48295i 0.717469 − 0.696591i
\(26\) −4.80884 −0.943090
\(27\) −5.17404 + 0.478889i −0.995744 + 0.0921622i
\(28\) 0.139805 + 0.101574i 0.0264207 + 0.0191957i
\(29\) −6.71254 + 2.98862i −1.24649 + 0.554972i −0.920626 − 0.390445i \(-0.872321\pi\)
−0.325862 + 0.945417i \(0.605654\pi\)
\(30\) −4.48983 + 3.24290i −0.819727 + 0.592070i
\(31\) 0.593962 + 0.264449i 0.106679 + 0.0474964i 0.459382 − 0.888239i \(-0.348071\pi\)
−0.352703 + 0.935735i \(0.614738\pi\)
\(32\) 0.127249 + 0.220402i 0.0224947 + 0.0389619i
\(33\) 8.79432 − 2.15402i 1.53089 − 0.374966i
\(34\) −6.76033 − 1.43695i −1.15939 − 0.246435i
\(35\) −1.18369 + 8.50536i −0.200080 + 1.43767i
\(36\) 0.119713 − 0.0623860i 0.0199522 − 0.0103977i
\(37\) 0.315262 + 0.970276i 0.0518287 + 0.159512i 0.973621 − 0.228172i \(-0.0732749\pi\)
−0.921792 + 0.387685i \(0.873275\pi\)
\(38\) −3.30069 − 0.701584i −0.535443 − 0.113812i
\(39\) 1.96939 + 5.48139i 0.315354 + 0.877724i
\(40\) −3.30561 + 5.30596i −0.522664 + 0.838945i
\(41\) −2.50903 + 2.78656i −0.391845 + 0.435188i −0.906498 − 0.422211i \(-0.861254\pi\)
0.514653 + 0.857399i \(0.327921\pi\)
\(42\) 2.65977 − 9.13276i 0.410411 − 1.40922i
\(43\) −2.34851 + 4.06773i −0.358144 + 0.620324i −0.987651 − 0.156671i \(-0.949924\pi\)
0.629507 + 0.776995i \(0.283257\pi\)
\(44\) −0.190302 + 0.138262i −0.0286890 + 0.0208438i
\(45\) 5.53519 + 3.78968i 0.825137 + 0.564932i
\(46\) 5.39516 + 3.91981i 0.795473 + 0.577945i
\(47\) 10.9118 − 4.85827i 1.59166 − 0.708651i 0.596108 − 0.802904i \(-0.296713\pi\)
0.995548 + 0.0942531i \(0.0300463\pi\)
\(48\) 4.57373 − 5.40514i 0.660161 − 0.780164i
\(49\) −3.87423 − 6.71036i −0.553461 − 0.958623i
\(50\) 7.13407 + 0.479583i 1.00891 + 0.0678233i
\(51\) 1.13067 + 8.29429i 0.158326 + 1.16143i
\(52\) −0.101250 − 0.112450i −0.0140409 − 0.0155940i
\(53\) 2.92011 + 2.12158i 0.401108 + 0.291422i 0.769992 − 0.638054i \(-0.220260\pi\)
−0.368884 + 0.929475i \(0.620260\pi\)
\(54\) −5.45986 − 5.04033i −0.742993 − 0.685902i
\(55\) −10.5129 − 5.10999i −1.41756 − 0.689031i
\(56\) −1.12228 − 10.6778i −0.149971 − 1.42688i
\(57\) 0.552044 + 4.04964i 0.0731200 + 0.536388i
\(58\) −9.59917 − 4.27383i −1.26043 − 0.561181i
\(59\) −5.20768 + 5.78372i −0.677982 + 0.752976i −0.979710 − 0.200418i \(-0.935770\pi\)
0.301728 + 0.953394i \(0.402437\pi\)
\(60\) −0.170366 − 0.0367107i −0.0219941 − 0.00473933i
\(61\) −4.40843 − 4.89605i −0.564441 − 0.626875i 0.391590 − 0.920140i \(-0.371925\pi\)
−0.956031 + 0.293264i \(0.905258\pi\)
\(62\) 0.287314 + 0.884263i 0.0364890 + 0.112301i
\(63\) −11.4993 + 0.708433i −1.44878 + 0.0892542i
\(64\) 2.41404 − 7.42965i 0.301755 − 0.928707i
\(65\) 2.56225 − 7.06930i 0.317808 − 0.876839i
\(66\) 10.7042 + 7.28479i 1.31760 + 0.896695i
\(67\) −0.930565 − 0.414314i −0.113687 − 0.0506165i 0.349105 − 0.937084i \(-0.386486\pi\)
−0.462792 + 0.886467i \(0.653152\pi\)
\(68\) −0.108737 − 0.188339i −0.0131863 − 0.0228394i
\(69\) 2.25852 − 7.75502i 0.271894 − 0.933594i
\(70\) −10.1715 + 6.88068i −1.21573 + 0.822398i
\(71\) −2.82082 − 2.04945i −0.334770 − 0.243225i 0.407682 − 0.913124i \(-0.366337\pi\)
−0.742452 + 0.669899i \(0.766337\pi\)
\(72\) −7.85732 − 2.93377i −0.925994 − 0.345748i
\(73\) −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
\(75\) −2.37500 − 8.32823i −0.274241 − 0.961661i
\(76\) −0.0530904 − 0.0919553i −0.00608989 − 0.0105480i
\(77\) 19.6368 − 4.17393i 2.23782 − 0.475663i
\(78\) −3.94073 + 7.33794i −0.446199 + 0.830858i
\(79\) 1.49826 − 0.667070i 0.168568 − 0.0750513i −0.320718 − 0.947175i \(-0.603924\pi\)
0.489286 + 0.872123i \(0.337257\pi\)
\(80\) −8.99997 + 1.59942i −1.00623 + 0.178821i
\(81\) −3.50925 + 8.28765i −0.389917 + 0.920850i
\(82\) −5.36219 −0.592154
\(83\) −1.18152 − 11.2414i −0.129688 − 1.23390i −0.844873 − 0.534967i \(-0.820324\pi\)
0.715185 − 0.698936i \(-0.246343\pi\)
\(84\) 0.269562 − 0.130095i 0.0294116 − 0.0141945i
\(85\) 5.71446 − 9.17248i 0.619820 − 0.994895i
\(86\) −6.57011 + 1.39652i −0.708473 + 0.150591i
\(87\) −0.940354 + 12.6920i −0.100817 + 1.36072i
\(88\) 14.2952 + 3.03854i 1.52388 + 0.323910i
\(89\) −0.935178 + 2.87818i −0.0991286 + 0.305087i −0.988308 − 0.152473i \(-0.951276\pi\)
0.889179 + 0.457559i \(0.151276\pi\)
\(90\) 1.26913 + 9.50864i 0.133778 + 1.00230i
\(91\) 3.99070 + 12.2821i 0.418339 + 1.28752i
\(92\) 0.0219345 + 0.208692i 0.00228682 + 0.0217577i
\(93\) 0.890268 − 0.689634i 0.0923165 − 0.0715117i
\(94\) 15.6043 + 6.94749i 1.60946 + 0.716579i
\(95\) 2.79006 − 4.47841i 0.286254 − 0.459475i
\(96\) 0.440596 − 0.0135589i 0.0449681 − 0.00138385i
\(97\) 14.8057 − 6.59193i 1.50329 − 0.669309i 0.520474 − 0.853878i \(-0.325755\pi\)
0.982820 + 0.184569i \(0.0590888\pi\)
\(98\) 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.196.22 yes 224
3.2 odd 2 675.2.r.a.46.7 224
9.4 even 3 inner 225.2.q.a.121.7 yes 224
9.5 odd 6 675.2.r.a.496.22 224
25.6 even 5 inner 225.2.q.a.106.7 yes 224
75.56 odd 10 675.2.r.a.181.22 224
225.31 even 15 inner 225.2.q.a.31.22 ✓ 224
225.131 odd 30 675.2.r.a.631.7 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.22 ✓ 224 225.31 even 15 inner
225.2.q.a.106.7 yes 224 25.6 even 5 inner
225.2.q.a.121.7 yes 224 9.4 even 3 inner
225.2.q.a.196.22 yes 224 1.1 even 1 trivial
675.2.r.a.46.7 224 3.2 odd 2
675.2.r.a.181.22 224 75.56 odd 10
675.2.r.a.496.22 224 9.5 odd 6
675.2.r.a.631.7 224 225.131 odd 30