↑

Properties

Label 675.2.bd.a.602.21
Level $675$
Weight $2$
Character 675.602
Analytic conductor $5.390$
Analytic rank $0$
Dimension $448$
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(8,675)] 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("675.8"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([10, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.bd (of order \(60\), degree \(16\), 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: \(448\)
Relative dimension: \(28\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 602.21
Character \(\chi\) \(=\) 675.602
Dual form 675.2.bd.a.638.21

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.26732 + 1.02626i) q^{2} +(0.137078 + 0.644900i) q^{4} +(1.08873 - 1.95312i) q^{5} +(-3.51917 + 0.942959i) q^{7} +(0.992568 - 1.94802i) q^{8} +(3.38417 - 1.35792i) q^{10} +(4.73499 - 0.497667i) q^{11} +(-2.32419 - 2.87014i) q^{13} +(-5.42764 - 2.41654i) q^{14} +(4.46169 - 1.98647i) q^{16} +(2.54982 + 1.29920i) q^{17} +(6.31730 - 2.05262i) q^{19} +(1.40881 + 0.434391i) q^{20} +(6.51148 + 4.22861i) q^{22} +(0.919867 + 2.39633i) q^{23} +(-2.62935 - 4.25283i) q^{25} -6.02261i q^{26} +(-1.09051 - 2.14025i) q^{28} +(-2.82765 - 3.14043i) q^{29} +(-0.908934 + 1.00947i) q^{31} +(3.46938 + 0.929618i) q^{32} +(1.89813 + 4.26327i) q^{34} +(-1.98971 + 7.89999i) q^{35} +(0.242591 + 1.53166i) q^{37} +(10.1126 + 3.88185i) q^{38} +(-2.72409 - 4.05947i) q^{40} +(-0.830574 - 0.0872968i) q^{41} +(-1.23082 - 4.59349i) q^{43} +(0.970006 + 2.98537i) q^{44} +(-1.29349 + 3.98095i) q^{46} +(-4.08223 - 0.213941i) q^{47} +(5.43322 - 3.13687i) q^{49} +(1.03226 - 8.08808i) q^{50} +(1.53236 - 1.89230i) q^{52} +(-2.05548 + 1.04732i) q^{53} +(4.18310 - 9.78981i) q^{55} +(-1.65611 + 7.79139i) q^{56} +(-0.360661 - 6.88183i) q^{58} +(-1.22755 + 11.6793i) q^{59} +(0.806761 + 7.67582i) q^{61} +(-2.18789 + 0.346528i) q^{62} +(-2.29860 - 3.16376i) q^{64} +(-8.13614 + 1.41463i) q^{65} +(-10.4288 + 0.546548i) q^{67} +(-0.488329 + 1.82247i) q^{68} +(-10.6290 + 7.96987i) q^{70} +(10.9516 + 3.55840i) q^{71} +(-2.51103 + 15.8540i) q^{73} +(-1.26444 + 2.19007i) q^{74} +(2.18969 + 3.79266i) q^{76} +(-16.1940 + 6.21628i) q^{77} +(-6.41701 + 5.77790i) q^{79} +(0.977746 - 10.8769i) q^{80} +(-0.963015 - 0.963015i) q^{82} +(5.32431 + 8.19871i) q^{83} +(5.31355 - 3.56563i) q^{85} +(3.15426 - 7.08457i) q^{86} +(3.73033 - 9.71784i) q^{88} +(2.03131 - 1.47583i) q^{89} +(10.8857 + 7.90890i) q^{91} +(-1.41930 + 0.921706i) q^{92} +(-4.95394 - 4.46055i) q^{94} +(2.86882 - 14.5732i) q^{95} +(0.188909 - 3.60460i) q^{97} +(10.1049 + 1.60045i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 448 q + 24 q^{2} - 10 q^{4} + 24 q^{5} - 8 q^{7} - 32 q^{10} + 18 q^{11} - 8 q^{13} + 30 q^{14} - 50 q^{16} - 40 q^{19} + 48 q^{20} + 48 q^{23} + 16 q^{25} - 24 q^{28} + 30 q^{29} - 6 q^{31} + 60 q^{32}+ \cdots - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.26732 + 1.02626i 0.896132 + 0.725673i 0.962258 − 0.272138i \(-0.0877308\pi\)
−0.0661265 + 0.997811i \(0.521064\pi\)
\(3\) 0 0
\(4\) 0.137078 + 0.644900i 0.0685388 + 0.322450i
\(5\) 1.08873 − 1.95312i 0.486894 − 0.873461i
\(6\) 0 0
\(7\) −3.51917 + 0.942959i −1.33012 + 0.356405i −0.852761 − 0.522302i \(-0.825073\pi\)
−0.477361 + 0.878707i \(0.658407\pi\)
\(8\) 0.992568 − 1.94802i 0.350926 − 0.688731i
\(9\) 0 0
\(10\) 3.38417 − 1.35792i 1.07017 − 0.429411i
\(11\) 4.73499 − 0.497667i 1.42765 − 0.150052i 0.641044 − 0.767504i \(-0.278502\pi\)
0.786608 + 0.617452i \(0.211835\pi\)
\(12\) 0 0
\(13\) −2.32419 − 2.87014i −0.644616 − 0.796034i 0.345179 − 0.938537i \(-0.387818\pi\)
−0.989794 + 0.142503i \(0.954485\pi\)
\(14\) −5.42764 − 2.41654i −1.45060 − 0.645848i
\(15\) 0 0
\(16\) 4.46169 − 1.98647i 1.11542 − 0.496618i
\(17\) 2.54982 + 1.29920i 0.618422 + 0.315102i 0.734987 − 0.678081i \(-0.237188\pi\)
−0.116565 + 0.993183i \(0.537188\pi\)
\(18\) 0 0
\(19\) 6.31730 − 2.05262i 1.44929 − 0.470903i 0.524509 − 0.851405i \(-0.324249\pi\)
0.924780 + 0.380502i \(0.124249\pi\)
\(20\) 1.40881 + 0.434391i 0.315019 + 0.0971327i
\(21\) 0 0
\(22\) 6.51148 + 4.22861i 1.38825 + 0.901542i
\(23\) 0.919867 + 2.39633i 0.191805 + 0.499670i 0.995433 − 0.0954622i \(-0.0304329\pi\)
−0.803628 + 0.595133i \(0.797100\pi\)
\(24\) 0 0
\(25\) −2.62935 − 4.25283i −0.525869 − 0.850565i
\(26\) − 6.02261i − 1.18113i
\(27\) 0 0
\(28\) −1.09051 − 2.14025i −0.206088 − 0.404470i
\(29\) −2.82765 − 3.14043i −0.525082 − 0.583163i 0.421012 − 0.907055i \(-0.361675\pi\)
−0.946094 + 0.323892i \(0.895008\pi\)
\(30\) 0 0
\(31\) −0.908934 + 1.00947i −0.163249 + 0.181307i −0.819220 − 0.573480i \(-0.805593\pi\)
0.655970 + 0.754787i \(0.272260\pi\)
\(32\) 3.46938 + 0.929618i 0.613306 + 0.164335i
\(33\) 0 0
\(34\) 1.89813 + 4.26327i 0.325527 + 0.731145i
\(35\) −1.98971 + 7.89999i −0.336322 + 1.33534i
\(36\) 0 0
\(37\) 0.242591 + 1.53166i 0.0398818 + 0.251803i 0.999572 − 0.0292566i \(-0.00931398\pi\)
−0.959690 + 0.281060i \(0.909314\pi\)
\(38\) 10.1126 + 3.88185i 1.64048 + 0.629719i
\(39\) 0 0
\(40\) −2.72409 − 4.05947i −0.430716 − 0.641859i
\(41\) −0.830574 − 0.0872968i −0.129714 − 0.0136335i 0.0394490 − 0.999222i \(-0.487440\pi\)
−0.169163 + 0.985588i \(0.554106\pi\)
\(42\) 0 0
\(43\) −1.23082 − 4.59349i −0.187699 − 0.700501i −0.994037 − 0.109046i \(-0.965220\pi\)
0.806338 − 0.591455i \(-0.201446\pi\)
\(44\) 0.970006 + 2.98537i 0.146234 + 0.450062i
\(45\) 0 0
\(46\) −1.29349 + 3.98095i −0.190714 + 0.586959i
\(47\) −4.08223 − 0.213941i −0.595454 − 0.0312064i −0.247771 − 0.968819i \(-0.579698\pi\)
−0.347683 + 0.937612i \(0.613031\pi\)
\(48\) 0 0
\(49\) 5.43322 − 3.13687i 0.776174 − 0.448125i
\(50\) 1.03226 − 8.08808i 0.145984 − 1.14383i
\(51\) 0 0
\(52\) 1.53236 − 1.89230i 0.212500 − 0.262415i
\(53\) −2.05548 + 1.04732i −0.282342 + 0.143861i −0.589427 − 0.807822i \(-0.700646\pi\)
0.307085 + 0.951682i \(0.400646\pi\)
\(54\) 0 0
\(55\) 4.18310 − 9.78981i 0.564050 − 1.32006i
\(56\) −1.65611 + 7.79139i −0.221307 + 1.04117i
\(57\) 0 0
\(58\) −0.360661 − 6.88183i −0.0473572 − 0.903629i
\(59\) −1.22755 + 11.6793i −0.159813 + 1.52052i 0.561249 + 0.827647i \(0.310321\pi\)
−0.721062 + 0.692870i \(0.756346\pi\)
\(60\) 0 0
\(61\) 0.806761 + 7.67582i 0.103295 + 0.982788i 0.916290 + 0.400516i \(0.131169\pi\)
−0.812994 + 0.582271i \(0.802164\pi\)
\(62\) −2.18789 + 0.346528i −0.277863 + 0.0440091i
\(63\) 0 0
\(64\) −2.29860 − 3.16376i −0.287326 − 0.395470i
\(65\) −8.13614 + 1.41463i −1.00916 + 0.175463i
\(66\) 0 0
\(67\) −10.4288 + 0.546548i −1.27408 + 0.0667715i −0.677342 − 0.735668i \(-0.736868\pi\)
−0.596733 + 0.802440i \(0.703535\pi\)
\(68\) −0.488329 + 1.82247i −0.0592186 + 0.221007i
\(69\) 0 0
\(70\) −10.6290 + 7.96987i −1.27041 + 0.952582i
\(71\) 10.9516 + 3.55840i 1.29972 + 0.422304i 0.875484 − 0.483247i \(-0.160543\pi\)
0.424236 + 0.905552i \(0.360543\pi\)
\(72\) 0 0
\(73\) −2.51103 + 15.8540i −0.293894 + 1.85557i 0.191785 + 0.981437i \(0.438572\pi\)
−0.485679 + 0.874137i \(0.661428\pi\)
\(74\) −1.26444 + 2.19007i −0.146988 + 0.254590i
\(75\) 0 0
\(76\) 2.18969 + 3.79266i 0.251175 + 0.435048i
\(77\) −16.1940 + 6.21628i −1.84547 + 0.708410i
\(78\) 0 0
\(79\) −6.41701 + 5.77790i −0.721970 + 0.650065i −0.945869 − 0.324547i \(-0.894788\pi\)
0.223899 + 0.974612i \(0.428121\pi\)
\(80\) 0.977746 − 10.8769i 0.109315 − 1.21608i
\(81\) 0 0
\(82\) −0.963015 − 0.963015i −0.106347 − 0.106347i
\(83\) 5.32431 + 8.19871i 0.584418 + 0.899926i 0.999953 − 0.00965169i \(-0.00307228\pi\)
−0.415535 + 0.909577i \(0.636406\pi\)
\(84\) 0 0
\(85\) 5.31355 − 3.56563i 0.576335 − 0.386747i
\(86\) 3.15426 − 7.08457i 0.340132 − 0.763949i
\(87\) 0 0
\(88\) 3.73033 − 9.71784i 0.397654 − 1.03593i
\(89\) 2.03131 − 1.47583i 0.215318 − 0.156438i −0.474898 − 0.880041i \(-0.657515\pi\)
0.690217 + 0.723603i \(0.257515\pi\)
\(90\) 0 0
\(91\) 10.8857 + 7.90890i 1.14113 + 0.829078i
\(92\) −1.41930 + 0.921706i −0.147972 + 0.0960945i
\(93\) 0 0
\(94\) −4.95394 − 4.46055i −0.510960 − 0.460070i
\(95\) 2.86882 − 14.5732i 0.294334 − 1.49518i
\(96\) 0 0
\(97\) 0.188909 − 3.60460i 0.0191808 − 0.365992i −0.972129 − 0.234446i \(-0.924673\pi\)
0.991310 − 0.131546i \(-0.0419941\pi\)
\(98\) 10.1049 + 1.60045i 1.02075 + 0.161670i
\(99\) 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.bd.a.602.21 448
3.2 odd 2 225.2.w.a.2.8 ✓ 448
9.4 even 3 225.2.w.a.77.8 yes 448
9.5 odd 6 inner 675.2.bd.a.152.21 448
25.13 odd 20 inner 675.2.bd.a.413.21 448
75.38 even 20 225.2.w.a.38.8 yes 448
225.13 odd 60 225.2.w.a.113.8 yes 448
225.113 even 60 inner 675.2.bd.a.638.21 448
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.w.a.2.8 ✓ 448 3.2 odd 2
225.2.w.a.38.8 yes 448 75.38 even 20
225.2.w.a.77.8 yes 448 9.4 even 3
225.2.w.a.113.8 yes 448 225.13 odd 60
675.2.bd.a.152.21 448 9.5 odd 6 inner
675.2.bd.a.413.21 448 25.13 odd 20 inner
675.2.bd.a.602.21 448 1.1 even 1 trivial
675.2.bd.a.638.21 448 225.113 even 60 inner