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Properties

Label 225.2.q.a.196.23
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 196.23
Character \(\chi\) \(=\) 225.196
Dual form 225.2.q.a.31.23

$q$-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+(1.15183 + 1.27924i) q^{2} +(1.47785 - 0.903302i) q^{3} +(-0.100677 + 0.957875i) q^{4} +(-0.0644453 + 2.23514i) q^{5} +(2.85777 + 0.850072i) q^{6} +(-2.11497 + 3.66323i) q^{7} +(1.44394 - 1.04909i) q^{8} +(1.36809 - 2.66989i) q^{9} +(-2.93350 + 2.49206i) q^{10} +(-3.66270 - 4.06784i) q^{11} +(0.716464 + 1.50654i) q^{12} +(1.55472 - 1.72669i) q^{13} +(-7.12222 + 1.51387i) q^{14} +(1.92376 + 3.36142i) q^{15} +(4.88941 + 1.03928i) q^{16} +(-1.13466 + 0.824379i) q^{17} +(4.99123 - 1.32515i) q^{18} +(4.74872 - 3.45015i) q^{19} +(-2.13449 - 0.286757i) q^{20} +(0.183394 + 7.32417i) q^{21} +(0.984923 - 9.37092i) q^{22} +(-3.41812 + 0.726543i) q^{23} +(1.18629 - 2.85471i) q^{24} +(-4.99169 - 0.288088i) q^{25} +3.99961 q^{26} +(-0.389879 - 5.18150i) q^{27} +(-3.29599 - 2.39468i) q^{28} +(-3.94221 + 1.75518i) q^{29} +(-2.08420 + 6.33272i) q^{30} +(-2.89216 - 1.28767i) q^{31} +(2.51747 + 4.36039i) q^{32} +(-9.08742 - 2.70315i) q^{33} +(-2.36151 - 0.501954i) q^{34} +(-8.05154 - 4.96333i) q^{35} +(2.41969 + 1.57926i) q^{36} +(0.930443 + 2.86361i) q^{37} +(9.88326 + 2.10075i) q^{38} +(0.737922 - 3.95617i) q^{39} +(2.25180 + 3.29502i) q^{40} +(3.09385 - 3.43607i) q^{41} +(-9.15810 + 8.67079i) q^{42} +(-2.17619 + 3.76927i) q^{43} +(4.26523 - 3.09887i) q^{44} +(5.87941 + 3.22994i) q^{45} +(-4.86650 - 3.53572i) q^{46} +(-2.62209 + 1.16743i) q^{47} +(8.16460 - 2.88072i) q^{48} +(-5.44619 - 9.43308i) q^{49} +(-5.38104 - 6.71738i) q^{50} +(-0.932197 + 2.24325i) q^{51} +(1.49743 + 1.66306i) q^{52} +(4.61926 + 3.35609i) q^{53} +(6.17929 - 6.46695i) q^{54} +(9.32824 - 7.92450i) q^{55} +(0.789153 + 7.50829i) q^{56} +(3.90138 - 9.38833i) q^{57} +(-6.78604 - 3.02134i) q^{58} +(-7.75922 + 8.61749i) q^{59} +(-3.41350 + 1.50431i) q^{60} +(7.25373 + 8.05609i) q^{61} +(-1.68404 - 5.18293i) q^{62} +(6.88697 + 10.6584i) q^{63} +(0.411065 - 1.26513i) q^{64} +(3.75920 + 3.58629i) q^{65} +(-7.00919 - 14.7385i) q^{66} +(10.3055 + 4.58831i) q^{67} +(-0.675418 - 1.16986i) q^{68} +(-4.39518 + 4.16131i) q^{69} +(-2.92473 - 16.0167i) q^{70} +(-5.34318 - 3.88205i) q^{71} +(-0.825499 - 5.29042i) q^{72} +(1.20170 - 3.69845i) q^{73} +(-2.59152 + 4.48864i) q^{74} +(-7.63721 + 4.08325i) q^{75} +(2.82672 + 4.89603i) q^{76} +(22.6480 - 4.81398i) q^{77} +(5.91083 - 3.61286i) q^{78} +(-2.62186 + 1.16733i) q^{79} +(-2.63803 + 10.8615i) q^{80} +(-5.25665 - 7.30532i) q^{81} +7.95913 q^{82} +(-0.713599 - 6.78944i) q^{83} +(-7.03410 - 0.561705i) q^{84} +(-1.76948 - 2.58925i) q^{85} +(-7.32838 + 1.55770i) q^{86} +(-4.24054 + 6.15490i) q^{87} +(-9.55625 - 2.03124i) q^{88} +(-1.41471 + 4.35403i) q^{89} +(2.64022 + 11.2415i) q^{90} +(3.03709 + 9.34720i) q^{91} +(-0.351813 - 3.34727i) q^{92} +(-5.43734 + 0.709504i) q^{93} +(-4.51362 - 2.00959i) q^{94} +(7.40552 + 10.8364i) q^{95} +(7.65919 + 4.16997i) q^{96} +(12.6316 - 5.62394i) q^{97} +(5.79405 - 17.8322i) q^{98} +(-15.8716 + 4.21384i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.15183 + 1.27924i 0.814466 + 0.904556i 0.996901 − 0.0786620i \(-0.0250648\pi\)
−0.182436 + 0.983218i \(0.558398\pi\)
\(3\) 1.47785 − 0.903302i 0.853238 − 0.521521i
\(4\) −0.100677 + 0.957875i −0.0503383 + 0.478937i
\(5\) −0.0644453 + 2.23514i −0.0288208 + 0.999585i
\(6\) 2.85777 + 0.850072i 1.16668 + 0.347040i
\(7\) −2.11497 + 3.66323i −0.799383 + 1.38457i 0.120635 + 0.992697i \(0.461507\pi\)
−0.920018 + 0.391876i \(0.871826\pi\)
\(8\) 1.44394 − 1.04909i 0.510511 − 0.370908i
\(9\) 1.36809 − 2.66989i 0.456031 − 0.889964i
\(10\) −2.93350 + 2.49206i −0.927654 + 0.788057i
\(11\) −3.66270 − 4.06784i −1.10435 − 1.22650i −0.971921 − 0.235308i \(-0.924390\pi\)
−0.132426 − 0.991193i \(-0.542277\pi\)
\(12\) 0.716464 + 1.50654i 0.206825 + 0.434900i
\(13\) 1.55472 − 1.72669i 0.431201 − 0.478898i −0.487911 − 0.872893i \(-0.662241\pi\)
0.919112 + 0.393996i \(0.128908\pi\)
\(14\) −7.12222 + 1.51387i −1.90349 + 0.404600i
\(15\) 1.92376 + 3.36142i 0.496714 + 0.867914i
\(16\) 4.88941 + 1.03928i 1.22235 + 0.259819i
\(17\) −1.13466 + 0.824379i −0.275195 + 0.199941i −0.716819 − 0.697259i \(-0.754403\pi\)
0.441624 + 0.897200i \(0.354403\pi\)
\(18\) 4.99123 − 1.32515i 1.17644 − 0.312340i
\(19\) 4.74872 − 3.45015i 1.08943 − 0.791518i 0.110128 − 0.993917i \(-0.464874\pi\)
0.979303 + 0.202400i \(0.0648740\pi\)
\(20\) −2.13449 − 0.286757i −0.477288 − 0.0641208i
\(21\) 0.183394 + 7.32417i 0.0400198 + 1.59827i
\(22\) 0.984923 − 9.37092i 0.209986 − 1.99789i
\(23\) −3.41812 + 0.726543i −0.712727 + 0.151495i −0.549986 − 0.835174i \(-0.685367\pi\)
−0.162741 + 0.986669i \(0.552034\pi\)
\(24\) 1.18629 − 2.85471i 0.242151 − 0.582715i
\(25\) −4.99169 − 0.288088i −0.998339 − 0.0576177i
\(26\) 3.99961 0.784388
\(27\) −0.389879 − 5.18150i −0.0750323 − 0.997181i
\(28\) −3.29599 − 2.39468i −0.622884 − 0.452552i
\(29\) −3.94221 + 1.75518i −0.732050 + 0.325929i −0.738687 − 0.674049i \(-0.764554\pi\)
0.00663753 + 0.999978i \(0.497887\pi\)
\(30\) −2.08420 + 6.33272i −0.380521 + 1.15619i
\(31\) −2.89216 − 1.28767i −0.519448 − 0.231273i 0.130229 − 0.991484i \(-0.458429\pi\)
−0.649676 + 0.760211i \(0.725096\pi\)
\(32\) 2.51747 + 4.36039i 0.445030 + 0.770815i
\(33\) −9.08742 − 2.70315i −1.58192 − 0.470557i
\(34\) −2.36151 − 0.501954i −0.404995 − 0.0860844i
\(35\) −8.05154 − 4.96333i −1.36096 − 0.838956i
\(36\) 2.41969 + 1.57926i 0.403281 + 0.263209i
\(37\) 0.930443 + 2.86361i 0.152964 + 0.470775i 0.997949 − 0.0640154i \(-0.0203907\pi\)
−0.844985 + 0.534790i \(0.820391\pi\)
\(38\) 9.88326 + 2.10075i 1.60328 + 0.340787i
\(39\) 0.737922 − 3.95617i 0.118162 − 0.633495i
\(40\) 2.25180 + 3.29502i 0.356041 + 0.520989i
\(41\) 3.09385 − 3.43607i 0.483178 − 0.536624i −0.451428 − 0.892308i \(-0.649085\pi\)
0.934606 + 0.355684i \(0.115752\pi\)
\(42\) −9.15810 + 8.67079i −1.41313 + 1.33793i
\(43\) −2.17619 + 3.76927i −0.331866 + 0.574809i −0.982878 − 0.184259i \(-0.941011\pi\)
0.651012 + 0.759068i \(0.274345\pi\)
\(44\) 4.26523 − 3.09887i 0.643008 − 0.467173i
\(45\) 5.87941 + 3.22994i 0.876451 + 0.481491i
\(46\) −4.86650 − 3.53572i −0.717527 − 0.521314i
\(47\) −2.62209 + 1.16743i −0.382471 + 0.170287i −0.588961 − 0.808162i \(-0.700463\pi\)
0.206489 + 0.978449i \(0.433796\pi\)
\(48\) 8.16460 − 2.88072i 1.17846 − 0.415795i
\(49\) −5.44619 − 9.43308i −0.778027 − 1.34758i
\(50\) −5.38104 − 6.71738i −0.760994 − 0.949981i
\(51\) −0.932197 + 2.24325i −0.130534 + 0.314118i
\(52\) 1.49743 + 1.66306i 0.207656 + 0.230625i
\(53\) 4.61926 + 3.35609i 0.634504 + 0.460994i 0.857958 − 0.513720i \(-0.171733\pi\)
−0.223454 + 0.974715i \(0.571733\pi\)
\(54\) 6.17929 − 6.46695i 0.840895 − 0.880041i
\(55\) 9.32824 − 7.92450i 1.25782 − 1.06854i
\(56\) 0.789153 + 7.50829i 0.105455 + 1.00334i
\(57\) 3.90138 − 9.38833i 0.516750 − 1.24351i
\(58\) −6.78604 − 3.02134i −0.891051 − 0.396721i
\(59\) −7.75922 + 8.61749i −1.01016 + 1.12190i −0.0176411 + 0.999844i \(0.505616\pi\)
−0.992523 + 0.122057i \(0.961051\pi\)
\(60\) −3.41350 + 1.50431i −0.440680 + 0.194205i
\(61\) 7.25373 + 8.05609i 0.928745 + 1.03148i 0.999423 + 0.0339768i \(0.0108172\pi\)
−0.0706774 + 0.997499i \(0.522516\pi\)
\(62\) −1.68404 − 5.18293i −0.213873 − 0.658233i
\(63\) 6.88697 + 10.6584i 0.867676 + 1.34283i
\(64\) 0.411065 − 1.26513i 0.0513831 − 0.158141i
\(65\) 3.75920 + 3.58629i 0.466271 + 0.444824i
\(66\) −7.00919 − 14.7385i −0.862772 − 1.81419i
\(67\) 10.3055 + 4.58831i 1.25902 + 0.560551i 0.924264 − 0.381754i \(-0.124680\pi\)
0.334754 + 0.942305i \(0.391347\pi\)
\(68\) −0.675418 − 1.16986i −0.0819064 − 0.141866i
\(69\) −4.39518 + 4.16131i −0.529118 + 0.500963i
\(70\) −2.92473 − 16.0167i −0.349572 − 1.91436i
\(71\) −5.34318 − 3.88205i −0.634119 − 0.460714i 0.223706 − 0.974657i \(-0.428184\pi\)
−0.857825 + 0.513942i \(0.828184\pi\)
\(72\) −0.825499 − 5.29042i −0.0972860 − 0.623482i
\(73\) 1.20170 − 3.69845i 0.140648 − 0.432871i −0.855777 − 0.517344i \(-0.826921\pi\)
0.996426 + 0.0844731i \(0.0269207\pi\)
\(74\) −2.59152 + 4.48864i −0.301258 + 0.521794i
\(75\) −7.63721 + 4.08325i −0.881870 + 0.471493i
\(76\) 2.82672 + 4.89603i 0.324247 + 0.561613i
\(77\) 22.6480 − 4.81398i 2.58098 − 0.548603i
\(78\) 5.91083 − 3.61286i 0.669270 − 0.409075i
\(79\) −2.62186 + 1.16733i −0.294983 + 0.131335i −0.548892 − 0.835893i \(-0.684950\pi\)
0.253909 + 0.967228i \(0.418284\pi\)
\(80\) −2.63803 + 10.8615i −0.294940 + 1.21436i
\(81\) −5.25665 − 7.30532i −0.584072 − 0.811702i
\(82\) 7.95913 0.878938
\(83\) −0.713599 − 6.78944i −0.0783276 − 0.745237i −0.961242 − 0.275704i \(-0.911089\pi\)
0.882915 − 0.469533i \(-0.155578\pi\)
\(84\) −7.03410 − 0.561705i −0.767484 − 0.0612870i
\(85\) −1.76948 − 2.58925i −0.191927 − 0.280844i
\(86\) −7.32838 + 1.55770i −0.790240 + 0.167971i
\(87\) −4.24054 + 6.15490i −0.454633 + 0.659875i
\(88\) −9.55625 − 2.03124i −1.01870 − 0.216531i
\(89\) −1.41471 + 4.35403i −0.149959 + 0.461526i −0.997615 − 0.0690197i \(-0.978013\pi\)
0.847656 + 0.530545i \(0.178013\pi\)
\(90\) 2.64022 + 11.2415i 0.278304 + 1.18496i
\(91\) 3.03709 + 9.34720i 0.318373 + 0.979852i
\(92\) −0.351813 − 3.34727i −0.0366790 − 0.348977i
\(93\) −5.43734 + 0.709504i −0.563826 + 0.0735721i
\(94\) −4.51362 − 2.00959i −0.465544 − 0.207274i
\(95\) 7.40552 + 10.8364i 0.759791 + 1.11179i
\(96\) 7.65919 + 4.16997i 0.781713 + 0.425596i
\(97\) 12.6316 − 5.62394i 1.28254 − 0.571024i 0.351585 − 0.936156i \(-0.385643\pi\)
0.930956 + 0.365132i \(0.118976\pi\)
\(98\) 5.79405 − 17.8322i 0.585287 − 1.80133i
\(99\) −15.8716 + 4.21384i −1.59516 + 0.423506i
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.196.23 yes 224
3.2 odd 2 675.2.r.a.46.6 224
9.4 even 3 inner 225.2.q.a.121.6 yes 224
9.5 odd 6 675.2.r.a.496.23 224
25.6 even 5 inner 225.2.q.a.106.6 yes 224
75.56 odd 10 675.2.r.a.181.23 224
225.31 even 15 inner 225.2.q.a.31.23 ✓ 224
225.131 odd 30 675.2.r.a.631.6 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.23 ✓ 224 225.31 even 15 inner
225.2.q.a.106.6 yes 224 25.6 even 5 inner
225.2.q.a.121.6 yes 224 9.4 even 3 inner
225.2.q.a.196.23 yes 224 1.1 even 1 trivial
675.2.r.a.46.6 224 3.2 odd 2
675.2.r.a.181.23 224 75.56 odd 10
675.2.r.a.496.23 224 9.5 odd 6
675.2.r.a.631.6 224 225.131 odd 30