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Properties

Label 675.2.l.h.151.9
Level $675$
Weight $2$
Character 675.151
Analytic conductor $5.390$
Analytic rank $0$
Dimension $96$
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] = [675,2,Mod(76,675)] 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("675.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 151.9
Character \(\chi\) \(=\) 675.151
Dual form 675.2.l.h.76.9

$q$-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+(0.158986 + 0.133405i) q^{2} +(1.27412 - 1.17329i) q^{3} +(-0.339817 - 1.92720i) q^{4} +(0.359090 - 0.0165625i) q^{6} +(0.500029 - 2.83580i) q^{7} +(0.410612 - 0.711201i) q^{8} +(0.246774 - 2.98983i) q^{9} +(1.38169 - 0.502894i) q^{11} +(-2.69413 - 2.05678i) q^{12} +(-1.85496 + 1.55650i) q^{13} +(0.457807 - 0.384146i) q^{14} +(-3.51766 + 1.28032i) q^{16} +(0.704220 + 1.21975i) q^{17} +(0.438091 - 0.442420i) q^{18} +(-2.34516 + 4.06194i) q^{19} +(-2.69013 - 4.19984i) q^{21} +(0.286757 + 0.104371i) q^{22} +(-0.417535 - 2.36796i) q^{23} +(-0.311276 - 1.38792i) q^{24} -0.502557 q^{26} +(-3.19353 - 4.09895i) q^{27} -5.63507 q^{28} +(6.73596 + 5.65214i) q^{29} +(1.00865 + 5.72033i) q^{31} +(-2.27346 - 0.827470i) q^{32} +(1.17040 - 2.26187i) q^{33} +(-0.0507589 + 0.287868i) q^{34} +(-5.84585 + 0.540413i) q^{36} +(-4.37074 - 7.57034i) q^{37} +(-0.914728 + 0.332934i) q^{38} +(-0.537223 + 4.15958i) q^{39} +(8.32538 - 6.98582i) q^{41} +(0.132587 - 1.02659i) q^{42} +(7.23793 - 2.63439i) q^{43} +(-1.43870 - 2.49190i) q^{44} +(0.249515 - 0.432172i) q^{46} +(1.17948 - 6.68918i) q^{47} +(-2.97974 + 5.75853i) q^{48} +(-1.21391 - 0.441827i) q^{49} +(2.32838 + 0.727849i) q^{51} +(3.63003 + 3.04596i) q^{52} -5.43934 q^{53} +(0.0390949 - 1.07771i) q^{54} +(-1.81151 - 1.52004i) q^{56} +(1.77782 + 7.92696i) q^{57} +(0.316898 + 1.79722i) q^{58} +(-6.83272 - 2.48691i) q^{59} +(-1.03952 + 5.89541i) q^{61} +(-0.602759 + 1.04401i) q^{62} +(-8.35519 - 2.19481i) q^{63} +(3.49236 + 6.04894i) q^{64} +(0.487821 - 0.203468i) q^{66} +(-5.90251 + 4.95280i) q^{67} +(2.11138 - 1.77166i) q^{68} +(-3.31030 - 2.52718i) q^{69} +(4.51802 + 7.82544i) q^{71} +(-2.02504 - 1.40317i) q^{72} +(6.28791 - 10.8910i) q^{73} +(0.315035 - 1.78665i) q^{74} +(8.62507 + 3.13927i) q^{76} +(-0.735224 - 4.16966i) q^{77} +(-0.640319 + 0.589646i) q^{78} +(1.14861 + 0.963795i) q^{79} +(-8.87821 - 1.47563i) q^{81} +2.25556 q^{82} +(8.89751 + 7.46590i) q^{83} +(-7.17977 + 6.61158i) q^{84} +(1.50217 + 0.546744i) q^{86} +(15.2140 - 0.701727i) q^{87} +(0.209680 - 1.18915i) q^{88} +(5.96766 - 10.3363i) q^{89} +(3.48639 + 6.03861i) q^{91} +(-4.42164 + 1.60934i) q^{92} +(7.99676 + 6.10496i) q^{93} +(1.07989 - 0.906134i) q^{94} +(-3.86752 + 1.61313i) q^{96} +(5.28231 - 1.92260i) q^{97} +(-0.134052 - 0.232185i) q^{98} +(-1.16260 - 4.25512i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 12 q^{4} - 6 q^{6} + 18 q^{9} - 6 q^{11} + 18 q^{14} - 24 q^{16} + 6 q^{19} + 24 q^{21} + 30 q^{24} + 48 q^{26} + 30 q^{29} - 30 q^{31} + 24 q^{34} + 54 q^{36} + 6 q^{39} - 12 q^{41} - 78 q^{44}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.158986 + 0.133405i 0.112420 + 0.0943314i 0.697265 − 0.716814i \(-0.254400\pi\)
−0.584845 + 0.811145i \(0.698845\pi\)
\(3\) 1.27412 − 1.17329i 0.735615 − 0.677400i
\(4\) −0.339817 − 1.92720i −0.169908 − 0.963598i
\(5\) 0 0
\(6\) 0.359090 − 0.0165625i 0.146598 − 0.00676162i
\(7\) 0.500029 − 2.83580i 0.188993 − 1.07183i −0.731724 − 0.681602i \(-0.761284\pi\)
0.920717 − 0.390232i \(-0.127605\pi\)
\(8\) 0.410612 − 0.711201i 0.145173 − 0.251448i
\(9\) 0.246774 − 2.98983i 0.0822580 − 0.996611i
\(10\) 0 0
\(11\) 1.38169 − 0.502894i 0.416595 − 0.151628i −0.125214 − 0.992130i \(-0.539962\pi\)
0.541809 + 0.840502i \(0.317740\pi\)
\(12\) −2.69413 − 2.05678i −0.777729 − 0.593741i
\(13\) −1.85496 + 1.55650i −0.514474 + 0.431695i −0.862700 − 0.505715i \(-0.831229\pi\)
0.348226 + 0.937411i \(0.386784\pi\)
\(14\) 0.457807 − 0.384146i 0.122354 − 0.102667i
\(15\) 0 0
\(16\) −3.51766 + 1.28032i −0.879415 + 0.320081i
\(17\) 0.704220 + 1.21975i 0.170799 + 0.295832i 0.938699 − 0.344737i \(-0.112032\pi\)
−0.767901 + 0.640569i \(0.778699\pi\)
\(18\) 0.438091 − 0.442420i 0.103259 − 0.104279i
\(19\) −2.34516 + 4.06194i −0.538017 + 0.931872i 0.460994 + 0.887403i \(0.347493\pi\)
−0.999011 + 0.0444689i \(0.985840\pi\)
\(20\) 0 0
\(21\) −2.69013 − 4.19984i −0.587034 − 0.916480i
\(22\) 0.286757 + 0.104371i 0.0611368 + 0.0222520i
\(23\) −0.417535 − 2.36796i −0.0870620 − 0.493753i −0.996893 − 0.0787733i \(-0.974900\pi\)
0.909831 − 0.414980i \(-0.136211\pi\)
\(24\) −0.311276 − 1.38792i −0.0635390 − 0.283309i
\(25\) 0 0
\(26\) −0.502557 −0.0985595
\(27\) −3.19353 − 4.09895i −0.614594 − 0.788843i
\(28\) −5.63507 −1.06493
\(29\) 6.73596 + 5.65214i 1.25084 + 1.04958i 0.996596 + 0.0824353i \(0.0262698\pi\)
0.254240 + 0.967141i \(0.418175\pi\)
\(30\) 0 0
\(31\) 1.00865 + 5.72033i 0.181159 + 1.02740i 0.930792 + 0.365549i \(0.119119\pi\)
−0.749634 + 0.661853i \(0.769770\pi\)
\(32\) −2.27346 − 0.827470i −0.401894 − 0.146277i
\(33\) 1.17040 − 2.26187i 0.203740 − 0.393741i
\(34\) −0.0507589 + 0.287868i −0.00870509 + 0.0493690i
\(35\) 0 0
\(36\) −5.84585 + 0.540413i −0.974309 + 0.0900689i
\(37\) −4.37074 − 7.57034i −0.718545 − 1.24456i −0.961576 − 0.274538i \(-0.911475\pi\)
0.243031 − 0.970019i \(-0.421858\pi\)
\(38\) −0.914728 + 0.332934i −0.148388 + 0.0540090i
\(39\) −0.537223 + 4.15958i −0.0860245 + 0.666066i
\(40\) 0 0
\(41\) 8.32538 − 6.98582i 1.30021 − 1.09100i 0.310096 − 0.950705i \(-0.399639\pi\)
0.990110 − 0.140297i \(-0.0448056\pi\)
\(42\) 0.132587 − 1.02659i 0.0204586 − 0.158406i
\(43\) 7.23793 − 2.63439i 1.10377 − 0.401741i 0.275068 − 0.961425i \(-0.411300\pi\)
0.828706 + 0.559684i \(0.189077\pi\)
\(44\) −1.43870 − 2.49190i −0.216892 − 0.375667i
\(45\) 0 0
\(46\) 0.249515 − 0.432172i 0.0367889 − 0.0637203i
\(47\) 1.17948 − 6.68918i 0.172045 − 0.975717i −0.769454 − 0.638702i \(-0.779472\pi\)
0.941499 − 0.337015i \(-0.109417\pi\)
\(48\) −2.97974 + 5.75853i −0.430088 + 0.831172i
\(49\) −1.21391 − 0.441827i −0.173416 − 0.0631181i
\(50\) 0 0
\(51\) 2.32838 + 0.727849i 0.326038 + 0.101919i
\(52\) 3.63003 + 3.04596i 0.503394 + 0.422398i
\(53\) −5.43934 −0.747151 −0.373575 − 0.927600i \(-0.621868\pi\)
−0.373575 + 0.927600i \(0.621868\pi\)
\(54\) 0.0390949 − 1.07771i 0.00532014 − 0.146657i
\(55\) 0 0
\(56\) −1.81151 − 1.52004i −0.242073 − 0.203123i
\(57\) 1.77782 + 7.92696i 0.235477 + 1.04995i
\(58\) 0.316898 + 1.79722i 0.0416108 + 0.235986i
\(59\) −6.83272 − 2.48691i −0.889545 − 0.323768i −0.143489 − 0.989652i \(-0.545832\pi\)
−0.746055 + 0.665884i \(0.768055\pi\)
\(60\) 0 0
\(61\) −1.03952 + 5.89541i −0.133097 + 0.754830i 0.843069 + 0.537805i \(0.180746\pi\)
−0.976166 + 0.217025i \(0.930365\pi\)
\(62\) −0.602759 + 1.04401i −0.0765504 + 0.132589i
\(63\) −8.35519 − 2.19481i −1.05265 − 0.276520i
\(64\) 3.49236 + 6.04894i 0.436545 + 0.756118i
\(65\) 0 0
\(66\) 0.487821 − 0.203468i 0.0600466 − 0.0250452i
\(67\) −5.90251 + 4.95280i −0.721107 + 0.605081i −0.927691 − 0.373348i \(-0.878210\pi\)
0.206584 + 0.978429i \(0.433765\pi\)
\(68\) 2.11138 − 1.77166i 0.256043 − 0.214845i
\(69\) −3.31030 − 2.52718i −0.398513 − 0.304236i
\(70\) 0 0
\(71\) 4.51802 + 7.82544i 0.536190 + 0.928708i 0.999105 + 0.0423056i \(0.0134703\pi\)
−0.462915 + 0.886403i \(0.653196\pi\)
\(72\) −2.02504 − 1.40317i −0.238654 − 0.165365i
\(73\) 6.28791 − 10.8910i 0.735944 − 1.27469i −0.218364 − 0.975867i \(-0.570072\pi\)
0.954308 − 0.298825i \(-0.0965945\pi\)
\(74\) 0.315035 − 1.78665i 0.0366221 − 0.207694i
\(75\) 0 0
\(76\) 8.62507 + 3.13927i 0.989364 + 0.360099i
\(77\) −0.735224 − 4.16966i −0.0837865 − 0.475177i
\(78\) −0.640319 + 0.589646i −0.0725018 + 0.0667642i
\(79\) 1.14861 + 0.963795i 0.129228 + 0.108435i 0.705111 − 0.709097i \(-0.250897\pi\)
−0.575883 + 0.817532i \(0.695342\pi\)
\(80\) 0 0
\(81\) −8.87821 − 1.47563i −0.986467 − 0.163958i
\(82\) 2.25556 0.249085
\(83\) 8.89751 + 7.46590i 0.976629 + 0.819489i 0.983577 − 0.180487i \(-0.0577674\pi\)
−0.00694857 + 0.999976i \(0.502212\pi\)
\(84\) −7.17977 + 6.61158i −0.783377 + 0.721383i
\(85\) 0 0
\(86\) 1.50217 + 0.546744i 0.161983 + 0.0589569i
\(87\) 15.2140 − 0.701727i 1.63112 − 0.0752330i
\(88\) 0.209680 − 1.18915i 0.0223519 − 0.126764i
\(89\) 5.96766 − 10.3363i 0.632571 − 1.09564i −0.354454 − 0.935074i \(-0.615333\pi\)
0.987024 − 0.160571i \(-0.0513336\pi\)
\(90\) 0 0
\(91\) 3.48639 + 6.03861i 0.365473 + 0.633018i
\(92\) −4.42164 + 1.60934i −0.460987 + 0.167786i
\(93\) 7.99676 + 6.10496i 0.829225 + 0.633055i
\(94\) 1.07989 − 0.906134i 0.111382 − 0.0934606i
\(95\) 0 0
\(96\) −3.86752 + 1.61313i −0.394727 + 0.164639i
\(97\) 5.28231 − 1.92260i 0.536337 − 0.195211i −0.0596285 − 0.998221i \(-0.518992\pi\)
0.595966 + 0.803010i \(0.296769\pi\)
\(98\) −0.134052 − 0.232185i −0.0135413 − 0.0234543i
\(99\) −1.16260 − 4.25512i −0.116846 − 0.427656i
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 675.2.l.h.151.9 96
5.2 odd 4 135.2.p.a.124.8 yes 96
5.3 odd 4 135.2.p.a.124.9 yes 96
5.4 even 2 inner 675.2.l.h.151.8 96
15.2 even 4 405.2.p.a.289.9 96
15.8 even 4 405.2.p.a.289.8 96
27.22 even 9 inner 675.2.l.h.76.9 96
135.22 odd 36 135.2.p.a.49.9 yes 96
135.32 even 36 405.2.p.a.199.8 96
135.49 even 18 inner 675.2.l.h.76.8 96
135.103 odd 36 135.2.p.a.49.8 ✓ 96
135.113 even 36 405.2.p.a.199.9 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.p.a.49.8 ✓ 96 135.103 odd 36
135.2.p.a.49.9 yes 96 135.22 odd 36
135.2.p.a.124.8 yes 96 5.2 odd 4
135.2.p.a.124.9 yes 96 5.3 odd 4
405.2.p.a.199.8 96 135.32 even 36
405.2.p.a.199.9 96 135.113 even 36
405.2.p.a.289.8 96 15.8 even 4
405.2.p.a.289.9 96 15.2 even 4
675.2.l.h.76.8 96 135.49 even 18 inner
675.2.l.h.76.9 96 27.22 even 9 inner
675.2.l.h.151.8 96 5.4 even 2 inner
675.2.l.h.151.9 96 1.1 even 1 trivial