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Properties

Label 675.2.r.a.46.15
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
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] = [675,2,Mod(46,675)] 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("675.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), degree \(8\), not 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 46.15
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.136671 + 0.151788i) q^{2} +(0.204696 - 1.94755i) q^{4} +(1.79725 - 1.33037i) q^{5} +(0.530047 - 0.918069i) q^{7} +(0.654078 - 0.475215i) q^{8} +(0.447567 + 0.0909796i) q^{10} +(3.92008 + 4.35369i) q^{11} +(-2.18992 + 2.43215i) q^{13} +(0.211794 - 0.0450182i) q^{14} +(-3.66945 - 0.779966i) q^{16} +(5.93589 - 4.31267i) q^{17} +(2.26837 - 1.64806i) q^{19} +(-2.22307 - 3.77257i) q^{20} +(-0.125079 + 1.19004i) q^{22} +(-0.593613 + 0.126176i) q^{23} +(1.46024 - 4.78202i) q^{25} -0.668470 q^{26} +(-1.67949 - 1.22022i) q^{28} +(-7.56074 + 3.36626i) q^{29} +(-3.84047 - 1.70989i) q^{31} +(-1.19160 - 2.06392i) q^{32} +(1.46588 + 0.311582i) q^{34} +(-0.268740 - 2.35516i) q^{35} +(-3.21237 - 9.88666i) q^{37} +(0.560177 + 0.119069i) q^{38} +(0.543332 - 1.72425i) q^{40} +(-4.32208 + 4.80016i) q^{41} +(1.32582 - 2.29639i) q^{43} +(9.28146 - 6.74338i) q^{44} +(-0.100282 - 0.0728589i) q^{46} +(0.606865 - 0.270194i) q^{47} +(2.93810 + 5.08894i) q^{49} +(0.925427 - 0.431915i) q^{50} +(4.28848 + 4.76284i) q^{52} +(5.03668 + 3.65936i) q^{53} +(12.8374 + 2.60953i) q^{55} +(-0.0895882 - 0.852375i) q^{56} +(-1.54429 - 0.687563i) q^{58} +(0.238569 - 0.264958i) q^{59} +(-4.71368 - 5.23507i) q^{61} +(-0.265339 - 0.816631i) q^{62} +(-2.16809 + 6.67269i) q^{64} +(-0.700182 + 7.28459i) q^{65} +(10.1955 + 4.53932i) q^{67} +(-7.18411 - 12.4432i) q^{68} +(0.320757 - 0.362673i) q^{70} +(3.31446 + 2.40810i) q^{71} +(-2.32007 + 7.14044i) q^{73} +(1.06164 - 1.83882i) q^{74} +(-2.74537 - 4.75512i) q^{76} +(6.07481 - 1.29124i) q^{77} +(-10.4539 + 4.65438i) q^{79} +(-7.63258 + 3.47992i) q^{80} -1.31931 q^{82} +(0.600982 + 5.71796i) q^{83} +(4.93085 - 15.6479i) q^{85} +(0.529768 - 0.112606i) q^{86} +(4.63297 + 0.984769i) q^{88} +(-1.63945 + 5.04569i) q^{89} +(1.07212 + 3.29965i) q^{91} +(0.124225 + 1.18192i) q^{92} +(0.123953 + 0.0551875i) q^{94} +(1.88430 - 5.97975i) q^{95} +(0.225928 - 0.100590i) q^{97} +(-0.370889 + 1.14148i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.136671 + 0.151788i 0.0966409 + 0.107331i 0.789525 − 0.613718i \(-0.210327\pi\)
−0.692884 + 0.721049i \(0.743660\pi\)
\(3\) 0 0
\(4\) 0.204696 − 1.94755i 0.102348 − 0.973777i
\(5\) 1.79725 − 1.33037i 0.803756 − 0.594959i
\(6\) 0 0
\(7\) 0.530047 − 0.918069i 0.200339 − 0.346997i −0.748299 − 0.663362i \(-0.769129\pi\)
0.948638 + 0.316365i \(0.102462\pi\)
\(8\) 0.654078 − 0.475215i 0.231251 − 0.168014i
\(9\) 0 0
\(10\) 0.447567 + 0.0909796i 0.141533 + 0.0287703i
\(11\) 3.92008 + 4.35369i 1.18195 + 1.31269i 0.939511 + 0.342518i \(0.111280\pi\)
0.242436 + 0.970167i \(0.422053\pi\)
\(12\) 0 0
\(13\) −2.18992 + 2.43215i −0.607374 + 0.674557i −0.965886 − 0.258968i \(-0.916618\pi\)
0.358512 + 0.933525i \(0.383284\pi\)
\(14\) 0.211794 − 0.0450182i 0.0566044 − 0.0120316i
\(15\) 0 0
\(16\) −3.66945 − 0.779966i −0.917363 − 0.194991i
\(17\) 5.93589 − 4.31267i 1.43966 − 1.04598i 0.451551 − 0.892245i \(-0.350871\pi\)
0.988113 − 0.153731i \(-0.0491291\pi\)
\(18\) 0 0
\(19\) 2.26837 − 1.64806i 0.520399 − 0.378092i −0.296355 − 0.955078i \(-0.595771\pi\)
0.816754 + 0.576986i \(0.195771\pi\)
\(20\) −2.22307 − 3.77257i −0.497094 − 0.843572i
\(21\) 0 0
\(22\) −0.125079 + 1.19004i −0.0266669 + 0.253718i
\(23\) −0.593613 + 0.126176i −0.123777 + 0.0263096i −0.269384 − 0.963033i \(-0.586820\pi\)
0.145607 + 0.989343i \(0.453487\pi\)
\(24\) 0 0
\(25\) 1.46024 − 4.78202i 0.292048 − 0.956404i
\(26\) −0.668470 −0.131098
\(27\) 0 0
\(28\) −1.67949 − 1.22022i −0.317394 − 0.230600i
\(29\) −7.56074 + 3.36626i −1.40399 + 0.625099i −0.962281 − 0.272058i \(-0.912296\pi\)
−0.441713 + 0.897156i \(0.645629\pi\)
\(30\) 0 0
\(31\) −3.84047 − 1.70989i −0.689769 − 0.307105i 0.0317538 − 0.999496i \(-0.489891\pi\)
−0.721523 + 0.692391i \(0.756557\pi\)
\(32\) −1.19160 − 2.06392i −0.210647 − 0.364852i
\(33\) 0 0
\(34\) 1.46588 + 0.311582i 0.251396 + 0.0534358i
\(35\) −0.268740 − 2.35516i −0.0454254 − 0.398095i
\(36\) 0 0
\(37\) −3.21237 − 9.88666i −0.528111 − 1.62536i −0.758082 − 0.652160i \(-0.773863\pi\)
0.229971 − 0.973197i \(-0.426137\pi\)
\(38\) 0.560177 + 0.119069i 0.0908726 + 0.0193156i
\(39\) 0 0
\(40\) 0.543332 − 1.72425i 0.0859084 − 0.272627i
\(41\) −4.32208 + 4.80016i −0.674996 + 0.749659i −0.979189 − 0.202951i \(-0.934947\pi\)
0.304193 + 0.952611i \(0.401613\pi\)
\(42\) 0 0
\(43\) 1.32582 − 2.29639i 0.202186 − 0.350197i −0.747046 − 0.664772i \(-0.768529\pi\)
0.949233 + 0.314575i \(0.101862\pi\)
\(44\) 9.28146 − 6.74338i 1.39923 − 1.01660i
\(45\) 0 0
\(46\) −0.100282 − 0.0728589i −0.0147857 − 0.0107425i
\(47\) 0.606865 − 0.270194i 0.0885204 − 0.0394118i −0.361999 − 0.932179i \(-0.617906\pi\)
0.450519 + 0.892767i \(0.351239\pi\)
\(48\) 0 0
\(49\) 2.93810 + 5.08894i 0.419729 + 0.726991i
\(50\) 0.925427 − 0.431915i 0.130875 − 0.0610820i
\(51\) 0 0
\(52\) 4.28848 + 4.76284i 0.594705 + 0.660486i
\(53\) 5.03668 + 3.65936i 0.691841 + 0.502652i 0.877265 − 0.480007i \(-0.159366\pi\)
−0.185424 + 0.982659i \(0.559366\pi\)
\(54\) 0 0
\(55\) 12.8374 + 2.60953i 1.73099 + 0.351869i
\(56\) −0.0895882 − 0.852375i −0.0119717 − 0.113903i
\(57\) 0 0
\(58\) −1.54429 − 0.687563i −0.202775 − 0.0902815i
\(59\) 0.238569 − 0.264958i 0.0310590 − 0.0344945i −0.727415 − 0.686198i \(-0.759279\pi\)
0.758474 + 0.651703i \(0.225945\pi\)
\(60\) 0 0
\(61\) −4.71368 − 5.23507i −0.603525 − 0.670282i 0.361521 − 0.932364i \(-0.382258\pi\)
−0.965046 + 0.262082i \(0.915591\pi\)
\(62\) −0.265339 − 0.816631i −0.0336981 − 0.103712i
\(63\) 0 0
\(64\) −2.16809 + 6.67269i −0.271011 + 0.834087i
\(65\) −0.700182 + 7.28459i −0.0868470 + 0.903542i
\(66\) 0 0
\(67\) 10.1955 + 4.53932i 1.24558 + 0.554566i 0.920360 − 0.391073i \(-0.127896\pi\)
0.325217 + 0.945639i \(0.394563\pi\)
\(68\) −7.18411 − 12.4432i −0.871201 − 1.50896i
\(69\) 0 0
\(70\) 0.320757 − 0.362673i 0.0383378 − 0.0433478i
\(71\) 3.31446 + 2.40810i 0.393354 + 0.285788i 0.766829 − 0.641852i \(-0.221834\pi\)
−0.373475 + 0.927640i \(0.621834\pi\)
\(72\) 0 0
\(73\) −2.32007 + 7.14044i −0.271544 + 0.835725i 0.718570 + 0.695455i \(0.244797\pi\)
−0.990113 + 0.140270i \(0.955203\pi\)
\(74\) 1.06164 − 1.83882i 0.123413 − 0.213758i
\(75\) 0 0
\(76\) −2.74537 − 4.75512i −0.314915 − 0.545449i
\(77\) 6.07481 − 1.29124i 0.692289 − 0.147150i
\(78\) 0 0
\(79\) −10.4539 + 4.65438i −1.17616 + 0.523659i −0.899334 − 0.437262i \(-0.855948\pi\)
−0.276823 + 0.960921i \(0.589282\pi\)
\(80\) −7.63258 + 3.47992i −0.853348 + 0.389067i
\(81\) 0 0
\(82\) −1.31931 −0.145694
\(83\) 0.600982 + 5.71796i 0.0659663 + 0.627628i 0.976696 + 0.214626i \(0.0688531\pi\)
−0.910730 + 0.413002i \(0.864480\pi\)
\(84\) 0 0
\(85\) 4.93085 − 15.6479i 0.534826 − 1.69725i
\(86\) 0.529768 − 0.112606i 0.0571263 − 0.0121426i
\(87\) 0 0
\(88\) 4.63297 + 0.984769i 0.493877 + 0.104977i
\(89\) −1.63945 + 5.04569i −0.173781 + 0.534842i −0.999576 − 0.0291282i \(-0.990727\pi\)
0.825795 + 0.563971i \(0.190727\pi\)
\(90\) 0 0
\(91\) 1.07212 + 3.29965i 0.112389 + 0.345897i
\(92\) 0.124225 + 1.18192i 0.0129513 + 0.123224i
\(93\) 0 0
\(94\) 0.123953 + 0.0551875i 0.0127848 + 0.00569215i
\(95\) 1.88430 − 5.97975i 0.193325 − 0.613510i
\(96\) 0 0
\(97\) 0.225928 − 0.100590i 0.0229395 − 0.0102133i −0.395235 − 0.918580i \(-0.629337\pi\)
0.418174 + 0.908367i \(0.362670\pi\)
\(98\) −0.370889 + 1.14148i −0.0374655 + 0.115307i
\(99\) 0 0
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.r.a.46.15 224
3.2 odd 2 225.2.q.a.196.14 yes 224
9.4 even 3 inner 675.2.r.a.496.14 224
9.5 odd 6 225.2.q.a.121.15 yes 224
25.6 even 5 inner 675.2.r.a.181.14 224
75.56 odd 10 225.2.q.a.106.15 yes 224
225.31 even 15 inner 675.2.r.a.631.15 224
225.131 odd 30 225.2.q.a.31.14 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.14 ✓ 224 225.131 odd 30
225.2.q.a.106.15 yes 224 75.56 odd 10
225.2.q.a.121.15 yes 224 9.5 odd 6
225.2.q.a.196.14 yes 224 3.2 odd 2
675.2.r.a.46.15 224 1.1 even 1 trivial
675.2.r.a.181.14 224 25.6 even 5 inner
675.2.r.a.496.14 224 9.4 even 3 inner
675.2.r.a.631.15 224 225.31 even 15 inner