↑

Properties

Label 675.2.r.a.46.4
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
Inner twists $4$

Related objects

Downloads

Learn more

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 46.4
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.55906 - 1.73151i) q^{2} +(-0.358407 + 3.41002i) q^{4} +(-2.08227 + 0.814962i) q^{5} +(-0.0399999 + 0.0692819i) q^{7} +(2.69327 - 1.95678i) q^{8} +(4.65750 + 2.33489i) q^{10} +(0.0802290 + 0.0891034i) q^{11} +(-0.613486 + 0.681345i) q^{13} +(0.182325 - 0.0387543i) q^{14} +(-0.879417 - 0.186926i) q^{16} +(2.65368 - 1.92801i) q^{17} +(-2.09128 + 1.51940i) q^{19} +(-2.03273 - 7.39265i) q^{20} +(0.0292016 - 0.277835i) q^{22} +(7.02445 - 1.49309i) q^{23} +(3.67167 - 3.39394i) q^{25} +2.13622 q^{26} +(-0.221916 - 0.161231i) q^{28} +(-5.68401 + 2.53069i) q^{29} +(-5.63701 - 2.50976i) q^{31} +(-2.28167 - 3.95197i) q^{32} +(-7.47563 - 1.58899i) q^{34} +(0.0268284 - 0.176862i) q^{35} +(-3.64504 - 11.2183i) q^{37} +(5.89128 + 1.25223i) q^{38} +(-4.01342 + 6.26945i) q^{40} +(4.24828 - 4.71819i) q^{41} +(6.13162 - 10.6203i) q^{43} +(-0.332599 + 0.241647i) q^{44} +(-13.5368 - 9.83509i) q^{46} +(3.50843 - 1.56205i) q^{47} +(3.49680 + 6.05664i) q^{49} +(-11.6010 - 1.06619i) q^{50} +(-2.10352 - 2.33619i) q^{52} +(-6.52099 - 4.73778i) q^{53} +(-0.239674 - 0.120153i) q^{55} +(0.0278385 + 0.264866i) q^{56} +(13.2436 + 5.89644i) q^{58} +(-0.915873 + 1.01718i) q^{59} +(6.71123 + 7.45358i) q^{61} +(4.44276 + 13.6734i) q^{62} +(-3.84127 + 11.8222i) q^{64} +(0.722171 - 1.91871i) q^{65} +(-7.73960 - 3.44589i) q^{67} +(5.62346 + 9.74012i) q^{68} +(-0.348065 + 0.229284i) q^{70} +(-5.54404 - 4.02798i) q^{71} +(1.36759 - 4.20901i) q^{73} +(-13.7417 + 23.8014i) q^{74} +(-4.43165 - 7.67584i) q^{76} +(-0.00938240 + 0.00199429i) q^{77} +(1.95501 - 0.870427i) q^{79} +(1.98352 - 0.327462i) q^{80} -14.7929 q^{82} +(0.314936 + 2.99641i) q^{83} +(-3.95442 + 6.17729i) q^{85} +(-27.9487 + 5.94068i) q^{86} +(0.390434 + 0.0829894i) q^{88} +(4.33323 - 13.3363i) q^{89} +(-0.0226655 - 0.0697572i) q^{91} +(2.57386 + 24.4886i) q^{92} +(-8.17456 - 3.63955i) q^{94} +(3.11634 - 4.86811i) q^{95} +(11.1567 - 4.96728i) q^{97} +(5.03541 - 15.4974i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.55906 − 1.73151i −1.10242 − 1.22436i −0.972516 − 0.232837i \(-0.925199\pi\)
−0.129906 − 0.991526i \(-0.541468\pi\)
\(3\) 0 0
\(4\) −0.358407 + 3.41002i −0.179204 + 1.70501i
\(5\) −2.08227 + 0.814962i −0.931218 + 0.364462i
\(6\) 0 0
\(7\) −0.0399999 + 0.0692819i −0.0151185 + 0.0261861i −0.873486 − 0.486850i \(-0.838146\pi\)
0.858367 + 0.513036i \(0.171479\pi\)
\(8\) 2.69327 − 1.95678i 0.952216 − 0.691826i
\(9\) 0 0
\(10\) 4.65750 + 2.33489i 1.47283 + 0.738359i
\(11\) 0.0802290 + 0.0891034i 0.0241900 + 0.0268657i 0.755118 − 0.655588i \(-0.227579\pi\)
−0.730928 + 0.682454i \(0.760913\pi\)
\(12\) 0 0
\(13\) −0.613486 + 0.681345i −0.170150 + 0.188971i −0.822189 − 0.569215i \(-0.807247\pi\)
0.652039 + 0.758186i \(0.273914\pi\)
\(14\) 0.182325 − 0.0387543i 0.0487283 − 0.0103575i
\(15\) 0 0
\(16\) −0.879417 − 0.186926i −0.219854 − 0.0467315i
\(17\) 2.65368 − 1.92801i 0.643613 − 0.467612i −0.217477 − 0.976066i \(-0.569783\pi\)
0.861090 + 0.508453i \(0.169783\pi\)
\(18\) 0 0
\(19\) −2.09128 + 1.51940i −0.479771 + 0.348574i −0.801237 − 0.598347i \(-0.795824\pi\)
0.321466 + 0.946921i \(0.395824\pi\)
\(20\) −2.03273 − 7.39265i −0.454533 − 1.65305i
\(21\) 0 0
\(22\) 0.0292016 − 0.277835i 0.00622581 − 0.0592346i
\(23\) 7.02445 − 1.49309i 1.46470 − 0.311331i 0.594526 − 0.804077i \(-0.297340\pi\)
0.870174 + 0.492745i \(0.164007\pi\)
\(24\) 0 0
\(25\) 3.67167 − 3.39394i 0.734335 − 0.678787i
\(26\) 2.13622 0.418947
\(27\) 0 0
\(28\) −0.221916 − 0.161231i −0.0419382 − 0.0304699i
\(29\) −5.68401 + 2.53069i −1.05549 + 0.469937i −0.859748 − 0.510718i \(-0.829380\pi\)
−0.195747 + 0.980655i \(0.562713\pi\)
\(30\) 0 0
\(31\) −5.63701 − 2.50976i −1.01244 − 0.450766i −0.167637 − 0.985849i \(-0.553614\pi\)
−0.844799 + 0.535083i \(0.820280\pi\)
\(32\) −2.28167 − 3.95197i −0.403346 − 0.698616i
\(33\) 0 0
\(34\) −7.47563 − 1.58899i −1.28206 − 0.272510i
\(35\) 0.0268284 − 0.176862i 0.00453483 − 0.0298951i
\(36\) 0 0
\(37\) −3.64504 − 11.2183i −0.599241 − 1.84427i −0.532368 − 0.846513i \(-0.678698\pi\)
−0.0668736 − 0.997761i \(-0.521302\pi\)
\(38\) 5.89128 + 1.25223i 0.955692 + 0.203139i
\(39\) 0 0
\(40\) −4.01342 + 6.26945i −0.634577 + 0.991287i
\(41\) 4.24828 − 4.71819i 0.663470 − 0.736858i −0.313652 − 0.949538i \(-0.601553\pi\)
0.977122 + 0.212680i \(0.0682193\pi\)
\(42\) 0 0
\(43\) 6.13162 − 10.6203i 0.935064 − 1.61958i 0.160543 − 0.987029i \(-0.448675\pi\)
0.774520 − 0.632549i \(-0.217991\pi\)
\(44\) −0.332599 + 0.241647i −0.0501411 + 0.0364297i
\(45\) 0 0
\(46\) −13.5368 − 9.83509i −1.99590 − 1.45011i
\(47\) 3.50843 − 1.56205i 0.511757 − 0.227849i −0.134578 − 0.990903i \(-0.542968\pi\)
0.646334 + 0.763054i \(0.276301\pi\)
\(48\) 0 0
\(49\) 3.49680 + 6.05664i 0.499543 + 0.865234i
\(50\) −11.6010 − 1.06619i −1.64063 − 0.150783i
\(51\) 0 0
\(52\) −2.10352 − 2.33619i −0.291706 − 0.323972i
\(53\) −6.52099 − 4.73778i −0.895727 − 0.650784i 0.0416378 − 0.999133i \(-0.486742\pi\)
−0.937365 + 0.348349i \(0.886742\pi\)
\(54\) 0 0
\(55\) −0.239674 − 0.120153i −0.0323177 − 0.0162015i
\(56\) 0.0278385 + 0.264866i 0.00372008 + 0.0353942i
\(57\) 0 0
\(58\) 13.2436 + 5.89644i 1.73897 + 0.774241i
\(59\) −0.915873 + 1.01718i −0.119236 + 0.132425i −0.799807 − 0.600257i \(-0.795065\pi\)
0.680571 + 0.732682i \(0.261732\pi\)
\(60\) 0 0
\(61\) 6.71123 + 7.45358i 0.859285 + 0.954333i 0.999359 − 0.0358081i \(-0.0114005\pi\)
−0.140073 + 0.990141i \(0.544734\pi\)
\(62\) 4.44276 + 13.6734i 0.564231 + 1.73652i
\(63\) 0 0
\(64\) −3.84127 + 11.8222i −0.480159 + 1.47778i
\(65\) 0.722171 − 1.91871i 0.0895743 − 0.237987i
\(66\) 0 0
\(67\) −7.73960 − 3.44589i −0.945542 − 0.420982i −0.124736 − 0.992190i \(-0.539808\pi\)
−0.820806 + 0.571207i \(0.806475\pi\)
\(68\) 5.62346 + 9.74012i 0.681945 + 1.18116i
\(69\) 0 0
\(70\) −0.348065 + 0.229284i −0.0416018 + 0.0274047i
\(71\) −5.54404 − 4.02798i −0.657957 − 0.478034i 0.208016 − 0.978126i \(-0.433300\pi\)
−0.865972 + 0.500092i \(0.833300\pi\)
\(72\) 0 0
\(73\) 1.36759 − 4.20901i 0.160064 − 0.492627i −0.838575 − 0.544787i \(-0.816611\pi\)
0.998639 + 0.0521600i \(0.0166106\pi\)
\(74\) −13.7417 + 23.8014i −1.59745 + 2.76686i
\(75\) 0 0
\(76\) −4.43165 − 7.67584i −0.508345 − 0.880480i
\(77\) −0.00938240 + 0.00199429i −0.00106922 + 0.000227271i
\(78\) 0 0
\(79\) 1.95501 − 0.870427i 0.219956 − 0.0979307i −0.293801 − 0.955867i \(-0.594920\pi\)
0.513757 + 0.857936i \(0.328253\pi\)
\(80\) 1.98352 − 0.327462i 0.221764 − 0.0366113i
\(81\) 0 0
\(82\) −14.7929 −1.63360
\(83\) 0.314936 + 2.99641i 0.0345687 + 0.328899i 0.998116 + 0.0613599i \(0.0195437\pi\)
−0.963547 + 0.267539i \(0.913790\pi\)
\(84\) 0 0
\(85\) −3.95442 + 6.17729i −0.428917 + 0.670021i
\(86\) −27.9487 + 5.94068i −3.01379 + 0.640600i
\(87\) 0 0
\(88\) 0.390434 + 0.0829894i 0.0416204 + 0.00884670i
\(89\) 4.33323 − 13.3363i 0.459321 − 1.41364i −0.406665 − 0.913577i \(-0.633308\pi\)
0.865986 − 0.500068i \(-0.166692\pi\)
\(90\) 0 0
\(91\) −0.0226655 − 0.0697572i −0.00237599 − 0.00731254i
\(92\) 2.57386 + 24.4886i 0.268343 + 2.55312i
\(93\) 0 0
\(94\) −8.17456 − 3.63955i −0.843141 − 0.375391i
\(95\) 3.11634 − 4.86811i 0.319730 − 0.499457i
\(96\) 0 0
\(97\) 11.1567 − 4.96728i 1.13279 − 0.504350i 0.247267 − 0.968947i \(-0.420467\pi\)
0.885522 + 0.464597i \(0.153801\pi\)
\(98\) 5.03541 − 15.4974i 0.508653 − 1.56547i
\(99\) 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.46.4 224
3.2 odd 2 225.2.q.a.196.25 yes 224
9.4 even 3 inner 675.2.r.a.496.25 224
9.5 odd 6 225.2.q.a.121.4 yes 224
25.6 even 5 inner 675.2.r.a.181.25 224
75.56 odd 10 225.2.q.a.106.4 yes 224
225.31 even 15 inner 675.2.r.a.631.4 224
225.131 odd 30 225.2.q.a.31.25 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.25 ✓ 224 225.131 odd 30
225.2.q.a.106.4 yes 224 75.56 odd 10
225.2.q.a.121.4 yes 224 9.5 odd 6
225.2.q.a.196.25 yes 224 3.2 odd 2
675.2.r.a.46.4 224 1.1 even 1 trivial
675.2.r.a.181.25 224 25.6 even 5 inner
675.2.r.a.496.25 224 9.4 even 3 inner
675.2.r.a.631.4 224 225.31 even 15 inner