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Properties

Label 675.2.r.a.46.2
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 46.2
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.69452 - 1.88195i) q^{2} +(-0.461298 + 4.38896i) q^{4} +(2.12223 - 0.704374i) q^{5} +(-1.03406 + 1.79105i) q^{7} +(4.94396 - 3.59200i) q^{8} +(-4.92175 - 2.80036i) q^{10} +(-1.63896 - 1.82025i) q^{11} +(-1.19441 + 1.32652i) q^{13} +(5.12291 - 1.08891i) q^{14} +(-6.50420 - 1.38251i) q^{16} +(-3.67466 + 2.66979i) q^{17} +(5.85004 - 4.25031i) q^{19} +(2.11249 + 9.63931i) q^{20} +(-0.648378 + 6.16890i) q^{22} +(2.31553 - 0.492181i) q^{23} +(4.00772 - 2.98968i) q^{25} +4.52040 q^{26} +(-7.38384 - 5.36467i) q^{28} +(7.47301 - 3.32720i) q^{29} +(6.20029 + 2.76055i) q^{31} +(2.30859 + 3.99860i) q^{32} +(11.2512 + 2.39152i) q^{34} +(-0.932951 + 4.52939i) q^{35} +(1.53985 + 4.73917i) q^{37} +(-17.9119 - 3.80729i) q^{38} +(7.96211 - 11.1054i) q^{40} +(3.20347 - 3.55781i) q^{41} +(4.14034 - 7.17127i) q^{43} +(8.74507 - 6.35367i) q^{44} +(-4.84997 - 3.52371i) q^{46} +(9.25434 - 4.12030i) q^{47} +(1.36143 + 2.35806i) q^{49} +(-12.4176 - 2.47626i) q^{50} +(-5.27108 - 5.85413i) q^{52} +(2.52356 + 1.83348i) q^{53} +(-4.76039 - 2.70855i) q^{55} +(1.32108 + 12.5692i) q^{56} +(-18.9248 - 8.42585i) q^{58} +(-6.99016 + 7.76336i) q^{59} +(-1.76446 - 1.95963i) q^{61} +(-5.31128 - 16.3465i) q^{62} +(-0.496397 + 1.52775i) q^{64} +(-1.60044 + 3.65649i) q^{65} +(-1.48939 - 0.663118i) q^{67} +(-10.0225 - 17.3595i) q^{68} +(10.1050 - 5.91936i) q^{70} +(-8.24169 - 5.98794i) q^{71} +(-0.608028 + 1.87132i) q^{73} +(6.30959 - 10.9285i) q^{74} +(15.9558 + 27.6363i) q^{76} +(4.95496 - 1.05321i) q^{77} +(13.6290 - 6.06801i) q^{79} +(-14.7772 + 1.64738i) q^{80} -12.1240 q^{82} +(0.424422 + 4.03811i) q^{83} +(-5.91793 + 8.25424i) q^{85} +(-20.5119 + 4.35993i) q^{86} +(-14.6413 - 3.11211i) q^{88} +(-0.105993 + 0.326212i) q^{89} +(-1.14078 - 3.51095i) q^{91} +(1.09201 + 10.3898i) q^{92} +(-23.4358 - 10.4343i) q^{94} +(9.42133 - 13.1407i) q^{95} +(-10.0640 + 4.48076i) q^{97} +(2.13079 - 6.55791i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.69452 − 1.88195i −1.19821 − 1.33074i −0.930080 − 0.367358i \(-0.880263\pi\)
−0.268125 − 0.963384i \(-0.586404\pi\)
\(3\) 0 0
\(4\) −0.461298 + 4.38896i −0.230649 + 2.19448i
\(5\) 2.12223 − 0.704374i 0.949090 − 0.315005i
\(6\) 0 0
\(7\) −1.03406 + 1.79105i −0.390839 + 0.676953i −0.992560 − 0.121753i \(-0.961148\pi\)
0.601721 + 0.798706i \(0.294482\pi\)
\(8\) 4.94396 − 3.59200i 1.74795 − 1.26996i
\(9\) 0 0
\(10\) −4.92175 − 2.80036i −1.55640 − 0.885552i
\(11\) −1.63896 − 1.82025i −0.494166 − 0.548827i 0.443541 − 0.896254i \(-0.353722\pi\)
−0.937707 + 0.347427i \(0.887055\pi\)
\(12\) 0 0
\(13\) −1.19441 + 1.32652i −0.331269 + 0.367911i −0.885651 − 0.464351i \(-0.846288\pi\)
0.554383 + 0.832262i \(0.312954\pi\)
\(14\) 5.12291 − 1.08891i 1.36916 − 0.291023i
\(15\) 0 0
\(16\) −6.50420 − 1.38251i −1.62605 − 0.345628i
\(17\) −3.67466 + 2.66979i −0.891235 + 0.647520i −0.936200 − 0.351469i \(-0.885682\pi\)
0.0449649 + 0.998989i \(0.485682\pi\)
\(18\) 0 0
\(19\) 5.85004 − 4.25031i 1.34209 − 0.975087i 0.342728 − 0.939435i \(-0.388649\pi\)
0.999364 − 0.0356524i \(-0.0113509\pi\)
\(20\) 2.11249 + 9.63931i 0.472367 + 2.15542i
\(21\) 0 0
\(22\) −0.648378 + 6.16890i −0.138235 + 1.31521i
\(23\) 2.31553 − 0.492181i 0.482821 − 0.102627i 0.0399305 − 0.999202i \(-0.487286\pi\)
0.442891 + 0.896576i \(0.353953\pi\)
\(24\) 0 0
\(25\) 4.00772 − 2.98968i 0.801543 − 0.597937i
\(26\) 4.52040 0.886523
\(27\) 0 0
\(28\) −7.38384 − 5.36467i −1.39541 − 1.01383i
\(29\) 7.47301 − 3.32720i 1.38770 − 0.617845i 0.429273 − 0.903175i \(-0.358770\pi\)
0.958429 + 0.285330i \(0.0921031\pi\)
\(30\) 0 0
\(31\) 6.20029 + 2.76055i 1.11361 + 0.495809i 0.879259 − 0.476344i \(-0.158038\pi\)
0.234346 + 0.972153i \(0.424705\pi\)
\(32\) 2.30859 + 3.99860i 0.408105 + 0.706859i
\(33\) 0 0
\(34\) 11.2512 + 2.39152i 1.92956 + 0.410141i
\(35\) −0.932951 + 4.52939i −0.157698 + 0.765606i
\(36\) 0 0
\(37\) 1.53985 + 4.73917i 0.253150 + 0.779115i 0.994189 + 0.107653i \(0.0343334\pi\)
−0.741039 + 0.671462i \(0.765667\pi\)
\(38\) −17.9119 − 3.80729i −2.90569 − 0.617624i
\(39\) 0 0
\(40\) 7.96211 − 11.1054i 1.25892 − 1.75592i
\(41\) 3.20347 − 3.55781i 0.500297 − 0.555636i −0.439113 − 0.898432i \(-0.644707\pi\)
0.939410 + 0.342796i \(0.111374\pi\)
\(42\) 0 0
\(43\) 4.14034 − 7.17127i 0.631395 − 1.09361i −0.355871 − 0.934535i \(-0.615816\pi\)
0.987267 − 0.159074i \(-0.0508508\pi\)
\(44\) 8.74507 − 6.35367i 1.31837 − 0.957851i
\(45\) 0 0
\(46\) −4.84997 − 3.52371i −0.715088 − 0.519542i
\(47\) 9.25434 − 4.12030i 1.34988 − 0.601007i 0.400839 − 0.916148i \(-0.368719\pi\)
0.949045 + 0.315141i \(0.102052\pi\)
\(48\) 0 0
\(49\) 1.36143 + 2.35806i 0.194489 + 0.336866i
\(50\) −12.4176 − 2.47626i −1.75611 − 0.350196i
\(51\) 0 0
\(52\) −5.27108 − 5.85413i −0.730967 − 0.811821i
\(53\) 2.52356 + 1.83348i 0.346638 + 0.251847i 0.747457 − 0.664310i \(-0.231274\pi\)
−0.400819 + 0.916157i \(0.631274\pi\)
\(54\) 0 0
\(55\) −4.76039 − 2.70855i −0.641891 − 0.365221i
\(56\) 1.32108 + 12.5692i 0.176537 + 1.67963i
\(57\) 0 0
\(58\) −18.9248 − 8.42585i −2.48494 − 1.10637i
\(59\) −6.99016 + 7.76336i −0.910041 + 1.01070i 0.0898499 + 0.995955i \(0.471361\pi\)
−0.999891 + 0.0147480i \(0.995305\pi\)
\(60\) 0 0
\(61\) −1.76446 − 1.95963i −0.225916 − 0.250905i 0.619521 − 0.784980i \(-0.287327\pi\)
−0.845437 + 0.534075i \(0.820660\pi\)
\(62\) −5.31128 − 16.3465i −0.674534 − 2.07600i
\(63\) 0 0
\(64\) −0.496397 + 1.52775i −0.0620496 + 0.190969i
\(65\) −1.60044 + 3.65649i −0.198510 + 0.453532i
\(66\) 0 0
\(67\) −1.48939 − 0.663118i −0.181958 − 0.0810127i 0.313736 − 0.949510i \(-0.398419\pi\)
−0.495693 + 0.868498i \(0.665086\pi\)
\(68\) −10.0225 − 17.3595i −1.21541 − 2.10515i
\(69\) 0 0
\(70\) 10.1050 − 5.91936i 1.20778 − 0.707498i
\(71\) −8.24169 − 5.98794i −0.978109 − 0.710638i −0.0208238 − 0.999783i \(-0.506629\pi\)
−0.957285 + 0.289145i \(0.906629\pi\)
\(72\) 0 0
\(73\) −0.608028 + 1.87132i −0.0711643 + 0.219021i −0.980313 − 0.197451i \(-0.936734\pi\)
0.909148 + 0.416472i \(0.136734\pi\)
\(74\) 6.30959 − 10.9285i 0.733475 − 1.27042i
\(75\) 0 0
\(76\) 15.9558 + 27.6363i 1.83026 + 3.17010i
\(77\) 4.95496 − 1.05321i 0.564670 − 0.120024i
\(78\) 0 0
\(79\) 13.6290 − 6.06801i 1.53338 − 0.682704i 0.545525 − 0.838094i \(-0.316330\pi\)
0.987853 + 0.155390i \(0.0496635\pi\)
\(80\) −14.7772 + 1.64738i −1.65214 + 0.184183i
\(81\) 0 0
\(82\) −12.1240 −1.33887
\(83\) 0.424422 + 4.03811i 0.0465864 + 0.443240i 0.992808 + 0.119720i \(0.0381996\pi\)
−0.946221 + 0.323520i \(0.895134\pi\)
\(84\) 0 0
\(85\) −5.91793 + 8.25424i −0.641890 + 0.895298i
\(86\) −20.5119 + 4.35993i −2.21185 + 0.470144i
\(87\) 0 0
\(88\) −14.6413 − 3.11211i −1.56077 − 0.331752i
\(89\) −0.105993 + 0.326212i −0.0112352 + 0.0345784i −0.956517 − 0.291677i \(-0.905787\pi\)
0.945282 + 0.326255i \(0.105787\pi\)
\(90\) 0 0
\(91\) −1.14078 − 3.51095i −0.119586 − 0.368048i
\(92\) 1.09201 + 10.3898i 0.113850 + 1.08321i
\(93\) 0 0
\(94\) −23.4358 − 10.4343i −2.41722 − 1.07622i
\(95\) 9.42133 − 13.1407i 0.966608 − 1.34821i
\(96\) 0 0
\(97\) −10.0640 + 4.48076i −1.02184 + 0.454953i −0.848098 − 0.529839i \(-0.822252\pi\)
−0.173742 + 0.984791i \(0.555586\pi\)
\(98\) 2.13079 − 6.55791i 0.215243 − 0.662449i
\(99\) 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.46.2 224
3.2 odd 2 225.2.q.a.196.27 yes 224
9.4 even 3 inner 675.2.r.a.496.27 224
9.5 odd 6 225.2.q.a.121.2 yes 224
25.6 even 5 inner 675.2.r.a.181.27 224
75.56 odd 10 225.2.q.a.106.2 yes 224
225.31 even 15 inner 675.2.r.a.631.2 224
225.131 odd 30 225.2.q.a.31.27 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.27 ✓ 224 225.131 odd 30
225.2.q.a.106.2 yes 224 75.56 odd 10
225.2.q.a.121.2 yes 224 9.5 odd 6
225.2.q.a.196.27 yes 224 3.2 odd 2
675.2.r.a.46.2 224 1.1 even 1 trivial
675.2.r.a.181.27 224 25.6 even 5 inner
675.2.r.a.496.27 224 9.4 even 3 inner
675.2.r.a.631.2 224 225.31 even 15 inner