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Properties

Label 675.2.r.a.46.9
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 46.9
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.797750 - 0.885991i) q^{2} +(0.0604816 - 0.575444i) q^{4} +(-1.30610 - 1.81497i) q^{5} +(0.157578 - 0.272934i) q^{7} +(-2.48714 + 1.80701i) q^{8} +(-0.566103 + 2.60508i) q^{10} +(-1.93450 - 2.14848i) q^{11} +(4.36810 - 4.85127i) q^{13} +(-0.367525 + 0.0781199i) q^{14} +(2.45317 + 0.521438i) q^{16} +(-0.794350 + 0.577129i) q^{17} +(-5.88399 + 4.27497i) q^{19} +(-1.12341 + 0.641816i) q^{20} +(-0.360286 + 3.42790i) q^{22} +(-1.28104 + 0.272294i) q^{23} +(-1.58820 + 4.74106i) q^{25} -7.78284 q^{26} +(-0.147528 - 0.107185i) q^{28} +(3.94048 - 1.75442i) q^{29} +(-7.91741 - 3.52506i) q^{31} +(1.57924 + 2.73533i) q^{32} +(1.14502 + 0.243382i) q^{34} +(-0.701179 + 0.0704794i) q^{35} +(0.00738727 + 0.0227357i) q^{37} +(8.48155 + 1.80281i) q^{38} +(6.52812 + 2.15393i) q^{40} +(-3.06561 + 3.40470i) q^{41} +(-1.34599 + 2.33132i) q^{43} +(-1.35333 + 0.983252i) q^{44} +(1.26320 + 0.917772i) q^{46} +(4.76261 - 2.12045i) q^{47} +(3.45034 + 5.97616i) q^{49} +(5.46752 - 2.37504i) q^{50} +(-2.52745 - 2.80701i) q^{52} +(3.44104 + 2.50006i) q^{53} +(-1.37277 + 6.31717i) q^{55} +(0.101275 + 0.963571i) q^{56} +(-4.69792 - 2.09165i) q^{58} +(-3.46632 + 3.84974i) q^{59} +(-4.95936 - 5.50792i) q^{61} +(3.19295 + 9.82688i) q^{62} +(2.71365 - 8.35176i) q^{64} +(-14.5101 - 1.59172i) q^{65} +(-2.27694 - 1.01376i) q^{67} +(0.284062 + 0.492010i) q^{68} +(0.621810 + 0.565014i) q^{70} +(-9.63629 - 7.00118i) q^{71} +(-0.283594 + 0.872814i) q^{73} +(0.0142504 - 0.0246825i) q^{74} +(2.10413 + 3.64447i) q^{76} +(-0.891228 + 0.189436i) q^{77} +(-10.3980 + 4.62948i) q^{79} +(-2.25770 - 5.13348i) q^{80} +5.46212 q^{82} +(-1.03926 - 9.88793i) q^{83} +(2.08497 + 0.687930i) q^{85} +(3.13929 - 0.667278i) q^{86} +(8.69369 + 1.84790i) q^{88} +(-3.78718 + 11.6557i) q^{89} +(-0.635757 - 1.95666i) q^{91} +(0.0792106 + 0.753639i) q^{92} +(-5.67808 - 2.52804i) q^{94} +(15.4440 + 5.09571i) q^{95} +(-6.75362 + 3.00691i) q^{97} +(2.54232 - 7.82445i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.797750 − 0.885991i −0.564095 − 0.626491i 0.391853 − 0.920028i \(-0.371834\pi\)
−0.955948 + 0.293537i \(0.905168\pi\)
\(3\) 0 0
\(4\) 0.0604816 − 0.575444i 0.0302408 − 0.287722i
\(5\) −1.30610 − 1.81497i −0.584106 − 0.811678i
\(6\) 0 0
\(7\) 0.157578 − 0.272934i 0.0595591 − 0.103159i −0.834708 − 0.550692i \(-0.814364\pi\)
0.894268 + 0.447533i \(0.147697\pi\)
\(8\) −2.48714 + 1.80701i −0.879336 + 0.638875i
\(9\) 0 0
\(10\) −0.566103 + 2.60508i −0.179017 + 0.823800i
\(11\) −1.93450 − 2.14848i −0.583273 − 0.647790i 0.377210 − 0.926128i \(-0.376883\pi\)
−0.960483 + 0.278337i \(0.910217\pi\)
\(12\) 0 0
\(13\) 4.36810 − 4.85127i 1.21149 − 1.34550i 0.290039 − 0.957015i \(-0.406332\pi\)
0.921455 − 0.388486i \(-0.127002\pi\)
\(14\) −0.367525 + 0.0781199i −0.0982253 + 0.0208784i
\(15\) 0 0
\(16\) 2.45317 + 0.521438i 0.613293 + 0.130359i
\(17\) −0.794350 + 0.577129i −0.192658 + 0.139974i −0.679933 − 0.733274i \(-0.737991\pi\)
0.487275 + 0.873249i \(0.337991\pi\)
\(18\) 0 0
\(19\) −5.88399 + 4.27497i −1.34988 + 0.980746i −0.350863 + 0.936427i \(0.614112\pi\)
−0.999017 + 0.0443190i \(0.985888\pi\)
\(20\) −1.12341 + 0.641816i −0.251201 + 0.143514i
\(21\) 0 0
\(22\) −0.360286 + 3.42790i −0.0768133 + 0.730830i
\(23\) −1.28104 + 0.272294i −0.267116 + 0.0567773i −0.339523 − 0.940598i \(-0.610266\pi\)
0.0724063 + 0.997375i \(0.476932\pi\)
\(24\) 0 0
\(25\) −1.58820 + 4.74106i −0.317641 + 0.948211i
\(26\) −7.78284 −1.52634
\(27\) 0 0
\(28\) −0.147528 − 0.107185i −0.0278801 − 0.0202561i
\(29\) 3.94048 − 1.75442i 0.731729 − 0.325787i −0.00682892 − 0.999977i \(-0.502174\pi\)
0.738558 + 0.674190i \(0.235507\pi\)
\(30\) 0 0
\(31\) −7.91741 − 3.52506i −1.42201 − 0.633120i −0.455614 − 0.890178i \(-0.650580\pi\)
−0.966396 + 0.257058i \(0.917247\pi\)
\(32\) 1.57924 + 2.73533i 0.279173 + 0.483542i
\(33\) 0 0
\(34\) 1.14502 + 0.243382i 0.196370 + 0.0417397i
\(35\) −0.701179 + 0.0704794i −0.118521 + 0.0119132i
\(36\) 0 0
\(37\) 0.00738727 + 0.0227357i 0.00121446 + 0.00373772i 0.951662 − 0.307148i \(-0.0993745\pi\)
−0.950447 + 0.310885i \(0.899375\pi\)
\(38\) 8.48155 + 1.80281i 1.37589 + 0.292454i
\(39\) 0 0
\(40\) 6.52812 + 2.15393i 1.03219 + 0.340567i
\(41\) −3.06561 + 3.40470i −0.478767 + 0.531725i −0.933344 − 0.358983i \(-0.883124\pi\)
0.454577 + 0.890707i \(0.349790\pi\)
\(42\) 0 0
\(43\) −1.34599 + 2.33132i −0.205261 + 0.355523i −0.950216 − 0.311592i \(-0.899138\pi\)
0.744955 + 0.667115i \(0.232471\pi\)
\(44\) −1.35333 + 0.983252i −0.204022 + 0.148231i
\(45\) 0 0
\(46\) 1.26320 + 0.917772i 0.186249 + 0.135318i
\(47\) 4.76261 − 2.12045i 0.694698 − 0.309300i −0.0288398 − 0.999584i \(-0.509181\pi\)
0.723538 + 0.690284i \(0.242515\pi\)
\(48\) 0 0
\(49\) 3.45034 + 5.97616i 0.492905 + 0.853737i
\(50\) 5.46752 − 2.37504i 0.773225 − 0.335882i
\(51\) 0 0
\(52\) −2.52745 − 2.80701i −0.350494 − 0.389263i
\(53\) 3.44104 + 2.50006i 0.472663 + 0.343410i 0.798478 − 0.602024i \(-0.205639\pi\)
−0.325815 + 0.945433i \(0.605639\pi\)
\(54\) 0 0
\(55\) −1.37277 + 6.31717i −0.185104 + 0.851808i
\(56\) 0.101275 + 0.963571i 0.0135335 + 0.128763i
\(57\) 0 0
\(58\) −4.69792 − 2.09165i −0.616867 − 0.274647i
\(59\) −3.46632 + 3.84974i −0.451276 + 0.501193i −0.925256 − 0.379342i \(-0.876150\pi\)
0.473980 + 0.880536i \(0.342817\pi\)
\(60\) 0 0
\(61\) −4.95936 − 5.50792i −0.634980 − 0.705217i 0.336674 − 0.941621i \(-0.390698\pi\)
−0.971655 + 0.236404i \(0.924031\pi\)
\(62\) 3.19295 + 9.82688i 0.405505 + 1.24801i
\(63\) 0 0
\(64\) 2.71365 − 8.35176i 0.339207 − 1.04397i
\(65\) −14.5101 − 1.59172i −1.79975 − 0.197428i
\(66\) 0 0
\(67\) −2.27694 − 1.01376i −0.278173 − 0.123851i 0.262908 − 0.964821i \(-0.415318\pi\)
−0.541081 + 0.840970i \(0.681985\pi\)
\(68\) 0.284062 + 0.492010i 0.0344476 + 0.0596649i
\(69\) 0 0
\(70\) 0.621810 + 0.565014i 0.0743205 + 0.0675321i
\(71\) −9.63629 − 7.00118i −1.14362 − 0.830887i −0.155999 − 0.987757i \(-0.549860\pi\)
−0.987619 + 0.156870i \(0.949860\pi\)
\(72\) 0 0
\(73\) −0.283594 + 0.872814i −0.0331922 + 0.102155i −0.966280 − 0.257493i \(-0.917103\pi\)
0.933088 + 0.359649i \(0.117103\pi\)
\(74\) 0.0142504 − 0.0246825i 0.00165658 − 0.00286928i
\(75\) 0 0
\(76\) 2.10413 + 3.64447i 0.241361 + 0.418049i
\(77\) −0.891228 + 0.189436i −0.101565 + 0.0215883i
\(78\) 0 0
\(79\) −10.3980 + 4.62948i −1.16986 + 0.520857i −0.897361 − 0.441298i \(-0.854518\pi\)
−0.272503 + 0.962155i \(0.587851\pi\)
\(80\) −2.25770 − 5.13348i −0.252418 − 0.573940i
\(81\) 0 0
\(82\) 5.46212 0.603191
\(83\) −1.03926 − 9.88793i −0.114074 − 1.08534i −0.890453 − 0.455075i \(-0.849612\pi\)
0.776379 − 0.630266i \(-0.217054\pi\)
\(84\) 0 0
\(85\) 2.08497 + 0.687930i 0.226147 + 0.0746165i
\(86\) 3.13929 − 0.667278i 0.338519 − 0.0719544i
\(87\) 0 0
\(88\) 8.69369 + 1.84790i 0.926750 + 0.196987i
\(89\) −3.78718 + 11.6557i −0.401440 + 1.23551i 0.522391 + 0.852706i \(0.325040\pi\)
−0.923831 + 0.382800i \(0.874960\pi\)
\(90\) 0 0
\(91\) −0.635757 − 1.95666i −0.0666455 − 0.205114i
\(92\) 0.0792106 + 0.753639i 0.00825828 + 0.0785723i
\(93\) 0 0
\(94\) −5.67808 − 2.52804i −0.585649 − 0.260748i
\(95\) 15.4440 + 5.09571i 1.58452 + 0.522809i
\(96\) 0 0
\(97\) −6.75362 + 3.00691i −0.685726 + 0.305305i −0.719867 − 0.694112i \(-0.755797\pi\)
0.0341407 + 0.999417i \(0.489131\pi\)
\(98\) 2.54232 − 7.82445i 0.256813 − 0.790389i
\(99\) 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.46.9 224
3.2 odd 2 225.2.q.a.196.20 yes 224
9.4 even 3 inner 675.2.r.a.496.20 224
9.5 odd 6 225.2.q.a.121.9 yes 224
25.6 even 5 inner 675.2.r.a.181.20 224
75.56 odd 10 225.2.q.a.106.9 yes 224
225.31 even 15 inner 675.2.r.a.631.9 224
225.131 odd 30 225.2.q.a.31.20 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.20 ✓ 224 225.131 odd 30
225.2.q.a.106.9 yes 224 75.56 odd 10
225.2.q.a.121.9 yes 224 9.5 odd 6
225.2.q.a.196.20 yes 224 3.2 odd 2
675.2.r.a.46.9 224 1.1 even 1 trivial
675.2.r.a.181.20 224 25.6 even 5 inner
675.2.r.a.496.20 224 9.4 even 3 inner
675.2.r.a.631.9 224 225.31 even 15 inner