↑

Properties

Label 675.2.r.a.46.3
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 46.3
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.64688 - 1.82905i) q^{2} +(-0.424138 + 4.03540i) q^{4} +(-0.681844 - 2.12957i) q^{5} +(2.55879 - 4.43196i) q^{7} +(4.09710 - 2.97672i) q^{8} +(-2.77218 + 4.75428i) q^{10} +(1.05935 + 1.17653i) q^{11} +(0.596899 - 0.662924i) q^{13} +(-12.3203 + 2.61876i) q^{14} +(-4.25406 - 0.904229i) q^{16} +(3.04259 - 2.21057i) q^{17} +(3.69404 - 2.68388i) q^{19} +(8.88289 - 1.84828i) q^{20} +(0.407301 - 3.87521i) q^{22} +(5.36085 - 1.13948i) q^{23} +(-4.07018 + 2.90407i) q^{25} -2.19554 q^{26} +(16.7994 + 12.2055i) q^{28} +(0.0385420 - 0.0171600i) q^{29} +(1.79980 + 0.801323i) q^{31} +(0.287768 + 0.498429i) q^{32} +(-9.05401 - 1.92449i) q^{34} +(-11.1829 - 2.42724i) q^{35} +(-1.43995 - 4.43171i) q^{37} +(-10.9926 - 2.33655i) q^{38} +(-9.13272 - 6.69542i) q^{40} +(-0.511798 + 0.568409i) q^{41} +(-4.11709 + 7.13101i) q^{43} +(-5.19708 + 3.77590i) q^{44} +(-10.9129 - 7.92865i) q^{46} +(-3.72722 + 1.65947i) q^{47} +(-9.59482 - 16.6187i) q^{49} +(12.0148 + 2.66188i) q^{50} +(2.42200 + 2.68990i) q^{52} +(2.00012 + 1.45317i) q^{53} +(1.78319 - 3.05818i) q^{55} +(-2.70906 - 25.7749i) q^{56} +(-0.0948606 - 0.0422347i) q^{58} +(-6.37293 + 7.07785i) q^{59} +(-4.84864 - 5.38496i) q^{61} +(-1.49840 - 4.61160i) q^{62} +(-2.25016 + 6.92529i) q^{64} +(-1.81874 - 0.819131i) q^{65} +(3.28565 + 1.46287i) q^{67} +(7.63006 + 13.2156i) q^{68} +(13.9773 + 24.4514i) q^{70} +(1.57130 + 1.14162i) q^{71} +(-4.42033 + 13.6044i) q^{73} +(-5.73438 + 9.93224i) q^{74} +(9.26375 + 16.0453i) q^{76} +(7.92498 - 1.68451i) q^{77} +(3.10949 - 1.38443i) q^{79} +(0.974982 + 9.67589i) q^{80} +1.88252 q^{82} +(-0.405949 - 3.86235i) q^{83} +(-6.78214 - 4.97215i) q^{85} +(19.8233 - 4.21357i) q^{86} +(7.84246 + 1.66697i) q^{88} +(0.935601 - 2.87948i) q^{89} +(-1.41071 - 4.34171i) q^{91} +(2.32454 + 22.1165i) q^{92} +(9.17353 + 4.08432i) q^{94} +(-8.23428 - 6.03676i) q^{95} +(-4.75481 + 2.11698i) q^{97} +(-14.5949 + 44.9184i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64688 − 1.82905i −1.16452 − 1.29333i −0.948440 − 0.316956i \(-0.897339\pi\)
−0.216081 − 0.976376i \(-0.569327\pi\)
\(3\) 0 0
\(4\) −0.424138 + 4.03540i −0.212069 + 2.01770i
\(5\) −0.681844 − 2.12957i −0.304930 − 0.952375i
\(6\) 0 0
\(7\) 2.55879 − 4.43196i 0.967132 − 1.67512i 0.263357 − 0.964698i \(-0.415170\pi\)
0.703775 − 0.710423i \(-0.251496\pi\)
\(8\) 4.09710 − 2.97672i 1.44854 − 1.05243i
\(9\) 0 0
\(10\) −2.77218 + 4.75428i −0.876639 + 1.50344i
\(11\) 1.05935 + 1.17653i 0.319406 + 0.354737i 0.881371 − 0.472424i \(-0.156621\pi\)
−0.561965 + 0.827161i \(0.689954\pi\)
\(12\) 0 0
\(13\) 0.596899 − 0.662924i 0.165550 − 0.183862i −0.654662 − 0.755922i \(-0.727189\pi\)
0.820212 + 0.572060i \(0.193856\pi\)
\(14\) −12.3203 + 2.61876i −3.29273 + 0.699892i
\(15\) 0 0
\(16\) −4.25406 − 0.904229i −1.06352 − 0.226057i
\(17\) 3.04259 − 2.21057i 0.737936 − 0.536142i −0.154128 − 0.988051i \(-0.549257\pi\)
0.892064 + 0.451909i \(0.149257\pi\)
\(18\) 0 0
\(19\) 3.69404 − 2.68388i 0.847472 − 0.615724i −0.0769761 − 0.997033i \(-0.524527\pi\)
0.924448 + 0.381309i \(0.124527\pi\)
\(20\) 8.88289 − 1.84828i 1.98627 − 0.413288i
\(21\) 0 0
\(22\) 0.407301 − 3.87521i 0.0868368 − 0.826197i
\(23\) 5.36085 − 1.13948i 1.11781 − 0.237599i 0.388279 − 0.921542i \(-0.373070\pi\)
0.729536 + 0.683943i \(0.239736\pi\)
\(24\) 0 0
\(25\) −4.07018 + 2.90407i −0.814036 + 0.580815i
\(26\) −2.19554 −0.430581
\(27\) 0 0
\(28\) 16.7994 + 12.2055i 3.17480 + 2.30662i
\(29\) 0.0385420 − 0.0171600i 0.00715708 − 0.00318654i −0.403155 − 0.915132i \(-0.632086\pi\)
0.410312 + 0.911945i \(0.365420\pi\)
\(30\) 0 0
\(31\) 1.79980 + 0.801323i 0.323254 + 0.143922i 0.561947 − 0.827173i \(-0.310052\pi\)
−0.238693 + 0.971095i \(0.576719\pi\)
\(32\) 0.287768 + 0.498429i 0.0508707 + 0.0881106i
\(33\) 0 0
\(34\) −9.05401 − 1.92449i −1.55275 − 0.330047i
\(35\) −11.1829 − 2.42724i −1.89025 − 0.410278i
\(36\) 0 0
\(37\) −1.43995 − 4.43171i −0.236726 − 0.728569i −0.996888 − 0.0788345i \(-0.974880\pi\)
0.760161 − 0.649734i \(-0.225120\pi\)
\(38\) −10.9926 − 2.33655i −1.78323 − 0.379038i
\(39\) 0 0
\(40\) −9.13272 − 6.69542i −1.44401 − 1.05864i
\(41\) −0.511798 + 0.568409i −0.0799294 + 0.0887706i −0.781783 − 0.623551i \(-0.785689\pi\)
0.701853 + 0.712321i \(0.252356\pi\)
\(42\) 0 0
\(43\) −4.11709 + 7.13101i −0.627850 + 1.08747i 0.360132 + 0.932901i \(0.382732\pi\)
−0.987982 + 0.154567i \(0.950602\pi\)
\(44\) −5.19708 + 3.77590i −0.783489 + 0.569238i
\(45\) 0 0
\(46\) −10.9129 − 7.92865i −1.60901 − 1.16902i
\(47\) −3.72722 + 1.65947i −0.543671 + 0.242058i −0.660148 − 0.751135i \(-0.729507\pi\)
0.116477 + 0.993193i \(0.462840\pi\)
\(48\) 0 0
\(49\) −9.59482 − 16.6187i −1.37069 − 2.37410i
\(50\) 12.0148 + 2.66188i 1.69915 + 0.376447i
\(51\) 0 0
\(52\) 2.42200 + 2.68990i 0.335870 + 0.373022i
\(53\) 2.00012 + 1.45317i 0.274737 + 0.199608i 0.716619 − 0.697465i \(-0.245689\pi\)
−0.441882 + 0.897073i \(0.645689\pi\)
\(54\) 0 0
\(55\) 1.78319 − 3.05818i 0.240446 − 0.412364i
\(56\) −2.70906 − 25.7749i −0.362013 − 3.44432i
\(57\) 0 0
\(58\) −0.0948606 − 0.0422347i −0.0124558 − 0.00554568i
\(59\) −6.37293 + 7.07785i −0.829684 + 0.921458i −0.997931 − 0.0642869i \(-0.979523\pi\)
0.168247 + 0.985745i \(0.446189\pi\)
\(60\) 0 0
\(61\) −4.84864 − 5.38496i −0.620805 − 0.689474i 0.347944 − 0.937515i \(-0.386880\pi\)
−0.968749 + 0.248041i \(0.920213\pi\)
\(62\) −1.49840 − 4.61160i −0.190297 − 0.585674i
\(63\) 0 0
\(64\) −2.25016 + 6.92529i −0.281270 + 0.865661i
\(65\) −1.81874 − 0.819131i −0.225587 − 0.101601i
\(66\) 0 0
\(67\) 3.28565 + 1.46287i 0.401406 + 0.178717i 0.597502 − 0.801867i \(-0.296160\pi\)
−0.196096 + 0.980585i \(0.562827\pi\)
\(68\) 7.63006 + 13.2156i 0.925280 + 1.60263i
\(69\) 0 0
\(70\) 13.9773 + 24.4514i 1.67061 + 2.92250i
\(71\) 1.57130 + 1.14162i 0.186479 + 0.135485i 0.677108 − 0.735884i \(-0.263233\pi\)
−0.490629 + 0.871369i \(0.663233\pi\)
\(72\) 0 0
\(73\) −4.42033 + 13.6044i −0.517361 + 1.59227i 0.261584 + 0.965181i \(0.415755\pi\)
−0.778945 + 0.627093i \(0.784245\pi\)
\(74\) −5.73438 + 9.93224i −0.666608 + 1.15460i
\(75\) 0 0
\(76\) 9.26375 + 16.0453i 1.06262 + 1.84052i
\(77\) 7.92498 − 1.68451i 0.903135 − 0.191967i
\(78\) 0 0
\(79\) 3.10949 − 1.38443i 0.349844 − 0.155761i −0.224289 − 0.974523i \(-0.572006\pi\)
0.574133 + 0.818762i \(0.305339\pi\)
\(80\) 0.974982 + 9.67589i 0.109006 + 1.08180i
\(81\) 0 0
\(82\) 1.88252 0.207889
\(83\) −0.405949 − 3.86235i −0.0445587 − 0.423948i −0.993948 − 0.109847i \(-0.964964\pi\)
0.949390 − 0.314100i \(-0.101703\pi\)
\(84\) 0 0
\(85\) −6.78214 − 4.97215i −0.735626 − 0.539306i
\(86\) 19.8233 − 4.21357i 2.13760 − 0.454361i
\(87\) 0 0
\(88\) 7.84246 + 1.66697i 0.836009 + 0.177699i
\(89\) 0.935601 − 2.87948i 0.0991735 − 0.305225i −0.889145 − 0.457625i \(-0.848700\pi\)
0.988319 + 0.152400i \(0.0487002\pi\)
\(90\) 0 0
\(91\) −1.41071 − 4.34171i −0.147882 − 0.455135i
\(92\) 2.32454 + 22.1165i 0.242350 + 2.30580i
\(93\) 0 0
\(94\) 9.17353 + 4.08432i 0.946178 + 0.421265i
\(95\) −8.23428 − 6.03676i −0.844819 − 0.619358i
\(96\) 0 0
\(97\) −4.75481 + 2.11698i −0.482777 + 0.214946i −0.633669 − 0.773604i \(-0.718452\pi\)
0.150892 + 0.988550i \(0.451785\pi\)
\(98\) −14.5949 + 44.9184i −1.47431 + 4.53744i
\(99\) 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.46.3 224
3.2 odd 2 225.2.q.a.196.26 yes 224
9.4 even 3 inner 675.2.r.a.496.26 224
9.5 odd 6 225.2.q.a.121.3 yes 224
25.6 even 5 inner 675.2.r.a.181.26 224
75.56 odd 10 225.2.q.a.106.3 yes 224
225.31 even 15 inner 675.2.r.a.631.3 224
225.131 odd 30 225.2.q.a.31.26 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.26 ✓ 224 225.131 odd 30
225.2.q.a.106.3 yes 224 75.56 odd 10
225.2.q.a.121.3 yes 224 9.5 odd 6
225.2.q.a.196.26 yes 224 3.2 odd 2
675.2.r.a.46.3 224 1.1 even 1 trivial
675.2.r.a.181.26 224 25.6 even 5 inner
675.2.r.a.496.26 224 9.4 even 3 inner
675.2.r.a.631.3 224 225.31 even 15 inner