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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(49,675)] 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("675.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.u (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 124.6
Character \(\chi\) \(=\) 675.124
Dual form 675.2.u.e.49.6

$q$-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+(-0.860644 + 1.02568i) q^{2} +(-1.52697 - 0.817526i) q^{3} +(0.0359939 + 0.204131i) q^{4} +(2.15270 - 0.862581i) q^{6} +(0.340915 + 0.0601126i) q^{7} +(-2.55944 - 1.47769i) q^{8} +(1.66330 + 2.49668i) q^{9} +(-0.377953 + 0.137564i) q^{11} +(0.111921 - 0.341129i) q^{12} +(0.483102 + 0.575738i) q^{13} +(-0.355063 + 0.297933i) q^{14} +(3.32884 - 1.21160i) q^{16} +(-1.16198 + 0.670872i) q^{17} +(-3.99230 - 0.442749i) q^{18} +(-1.87243 + 3.24314i) q^{19} +(-0.471425 - 0.370498i) q^{21} +(0.184187 - 0.506050i) q^{22} +(-3.04247 + 0.536470i) q^{23} +(2.70014 + 4.34880i) q^{24} -1.00630 q^{26} +(-0.498713 - 5.17216i) q^{27} +0.0717552i q^{28} +(-3.79094 - 3.18098i) q^{29} +(-0.774172 - 4.39054i) q^{31} +(0.399363 - 1.09724i) q^{32} +(0.689586 + 0.0989303i) q^{33} +(0.311958 - 1.76920i) q^{34} +(-0.449783 + 0.429397i) q^{36} +(-2.16657 + 1.25087i) q^{37} +(-1.71492 - 4.71170i) q^{38} +(-0.267003 - 1.27409i) q^{39} +(1.73862 - 1.45887i) q^{41} +(0.785740 - 0.164663i) q^{42} +(2.92375 + 8.03293i) q^{43} +(-0.0416850 - 0.0722005i) q^{44} +(2.06824 - 3.58230i) q^{46} +(-12.1171 - 2.13658i) q^{47} +(-6.07356 - 0.871334i) q^{48} +(-6.46524 - 2.35315i) q^{49} +(2.32278 - 0.0744513i) q^{51} +(-0.100137 + 0.119339i) q^{52} -10.1571i q^{53} +(5.73418 + 3.93988i) q^{54} +(-0.783723 - 0.657622i) q^{56} +(5.51051 - 3.42144i) q^{57} +(6.52530 - 1.15059i) q^{58} +(-12.7550 - 4.64243i) q^{59} +(2.31667 - 13.1385i) q^{61} +(5.16956 + 2.98465i) q^{62} +(0.416963 + 0.951143i) q^{63} +(4.32418 + 7.48970i) q^{64} +(-0.694958 + 0.622148i) q^{66} +(-7.89073 - 9.40381i) q^{67} +(-0.178770 - 0.213050i) q^{68} +(5.08436 + 1.66813i) q^{69} +(-1.14446 - 1.98226i) q^{71} +(-0.567785 - 8.84795i) q^{72} +(10.8000 + 6.23540i) q^{73} +(0.581658 - 3.29875i) q^{74} +(-0.729423 - 0.265488i) q^{76} +(-0.137119 + 0.0241778i) q^{77} +(1.53659 + 0.822676i) q^{78} +(-9.11223 - 7.64607i) q^{79} +(-3.46686 + 8.30547i) q^{81} +3.03883i q^{82} +(3.11961 - 3.71780i) q^{83} +(0.0586617 - 0.109568i) q^{84} +(-10.7555 - 3.91468i) q^{86} +(3.18814 + 7.95646i) q^{87} +(1.17062 + 0.206412i) q^{88} +(0.197409 - 0.341923i) q^{89} +(0.130088 + 0.225318i) q^{91} +(-0.219021 - 0.601754i) q^{92} +(-2.40725 + 7.33715i) q^{93} +(12.6200 - 10.5894i) q^{94} +(-1.50684 + 1.34897i) q^{96} +(5.07158 + 13.9340i) q^{97} +(7.97784 - 4.60601i) q^{98} +(-0.972102 - 0.714819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 12 q^{6} + 12 q^{9} + 30 q^{11} - 30 q^{14} + 36 q^{16} - 24 q^{19} + 24 q^{21} + 78 q^{24} + 12 q^{26} + 30 q^{29} + 6 q^{31} + 60 q^{36} + 30 q^{39} + 78 q^{41} - 102 q^{44} + 18 q^{46} + 12 q^{49}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.860644 + 1.02568i −0.608567 + 0.725262i −0.979060 − 0.203573i \(-0.934745\pi\)
0.370492 + 0.928836i \(0.379189\pi\)
\(3\) −1.52697 − 0.817526i −0.881599 − 0.471999i
\(4\) 0.0359939 + 0.204131i 0.0179969 + 0.102066i
\(5\) 0 0
\(6\) 2.15270 − 0.862581i 0.878836 − 0.352147i
\(7\) 0.340915 + 0.0601126i 0.128854 + 0.0227204i 0.237703 − 0.971338i \(-0.423605\pi\)
−0.108849 + 0.994058i \(0.534717\pi\)
\(8\) −2.55944 − 1.47769i −0.904897 − 0.522443i
\(9\) 1.66330 + 2.49668i 0.554434 + 0.832228i
\(10\) 0 0
\(11\) −0.377953 + 0.137564i −0.113957 + 0.0414770i −0.398369 − 0.917225i \(-0.630424\pi\)
0.284412 + 0.958702i \(0.408202\pi\)
\(12\) 0.111921 − 0.341129i 0.0323088 − 0.0984755i
\(13\) 0.483102 + 0.575738i 0.133988 + 0.159681i 0.828867 − 0.559446i \(-0.188986\pi\)
−0.694878 + 0.719127i \(0.744542\pi\)
\(14\) −0.355063 + 0.297933i −0.0948945 + 0.0796260i
\(15\) 0 0
\(16\) 3.32884 − 1.21160i 0.832209 − 0.302899i
\(17\) −1.16198 + 0.670872i −0.281823 + 0.162710i −0.634248 − 0.773129i \(-0.718690\pi\)
0.352426 + 0.935840i \(0.385357\pi\)
\(18\) −3.99230 − 0.442749i −0.940994 − 0.104357i
\(19\) −1.87243 + 3.24314i −0.429565 + 0.744028i −0.996835 − 0.0795040i \(-0.974666\pi\)
0.567270 + 0.823532i \(0.308000\pi\)
\(20\) 0 0
\(21\) −0.471425 − 0.370498i −0.102873 − 0.0808492i
\(22\) 0.184187 − 0.506050i 0.0392688 − 0.107890i
\(23\) −3.04247 + 0.536470i −0.634399 + 0.111862i −0.481594 − 0.876395i \(-0.659942\pi\)
−0.152806 + 0.988256i \(0.548831\pi\)
\(24\) 2.70014 + 4.34880i 0.551164 + 0.887696i
\(25\) 0 0
\(26\) −1.00630 −0.197352
\(27\) −0.498713 − 5.17216i −0.0959773 − 0.995384i
\(28\) 0.0717552i 0.0135605i
\(29\) −3.79094 − 3.18098i −0.703960 − 0.590693i 0.218937 − 0.975739i \(-0.429741\pi\)
−0.922897 + 0.385046i \(0.874185\pi\)
\(30\) 0 0
\(31\) −0.774172 − 4.39054i −0.139045 − 0.788565i −0.971957 − 0.235159i \(-0.924439\pi\)
0.832912 − 0.553406i \(-0.186672\pi\)
\(32\) 0.399363 − 1.09724i 0.0705980 − 0.193966i
\(33\) 0.689586 + 0.0989303i 0.120041 + 0.0172216i
\(34\) 0.311958 − 1.76920i 0.0535003 − 0.303415i
\(35\) 0 0
\(36\) −0.449783 + 0.429397i −0.0749638 + 0.0715662i
\(37\) −2.16657 + 1.25087i −0.356181 + 0.205641i −0.667404 − 0.744696i \(-0.732595\pi\)
0.311223 + 0.950337i \(0.399261\pi\)
\(38\) −1.71492 − 4.71170i −0.278196 − 0.764338i
\(39\) −0.267003 − 1.27409i −0.0427546 − 0.204017i
\(40\) 0 0
\(41\) 1.73862 − 1.45887i 0.271526 − 0.227838i −0.496849 − 0.867837i \(-0.665510\pi\)
0.768376 + 0.639999i \(0.221065\pi\)
\(42\) 0.785740 − 0.164663i 0.121242 − 0.0254080i
\(43\) 2.92375 + 8.03293i 0.445867 + 1.22501i 0.935576 + 0.353124i \(0.114881\pi\)
−0.489709 + 0.871886i \(0.662897\pi\)
\(44\) −0.0416850 − 0.0722005i −0.00628425 − 0.0108846i
\(45\) 0 0
\(46\) 2.06824 − 3.58230i 0.304946 − 0.528181i
\(47\) −12.1171 − 2.13658i −1.76746 − 0.311652i −0.807103 − 0.590411i \(-0.798966\pi\)
−0.960361 + 0.278760i \(0.910077\pi\)
\(48\) −6.07356 − 0.871334i −0.876643 − 0.125766i
\(49\) −6.46524 − 2.35315i −0.923606 − 0.336165i
\(50\) 0 0
\(51\) 2.32278 − 0.0744513i 0.325254 − 0.0104253i
\(52\) −0.100137 + 0.119339i −0.0138866 + 0.0165494i
\(53\) − 10.1571i − 1.39519i −0.716494 − 0.697593i \(-0.754254\pi\)
0.716494 − 0.697593i \(-0.245746\pi\)
\(54\) 5.73418 + 3.93988i 0.780323 + 0.536149i
\(55\) 0 0
\(56\) −0.783723 − 0.657622i −0.104729 − 0.0878784i
\(57\) 5.51051 − 3.42144i 0.729884 − 0.453180i
\(58\) 6.52530 − 1.15059i 0.856814 − 0.151079i
\(59\) −12.7550 − 4.64243i −1.66055 − 0.604392i −0.670105 − 0.742266i \(-0.733751\pi\)
−0.990450 + 0.137874i \(0.955973\pi\)
\(60\) 0 0
\(61\) 2.31667 − 13.1385i 0.296619 − 1.68221i −0.363927 − 0.931427i \(-0.618564\pi\)
0.660546 − 0.750785i \(-0.270325\pi\)
\(62\) 5.16956 + 2.98465i 0.656535 + 0.379051i
\(63\) 0.416963 + 0.951143i 0.0525324 + 0.119833i
\(64\) 4.32418 + 7.48970i 0.540522 + 0.936212i
\(65\) 0 0
\(66\) −0.694958 + 0.622148i −0.0855435 + 0.0765811i
\(67\) −7.89073 − 9.40381i −0.964006 − 1.14886i −0.988812 − 0.149166i \(-0.952341\pi\)
0.0248058 − 0.999692i \(-0.492103\pi\)
\(68\) −0.178770 − 0.213050i −0.0216791 − 0.0258361i
\(69\) 5.08436 + 1.66813i 0.612085 + 0.200819i
\(70\) 0 0
\(71\) −1.14446 − 1.98226i −0.135822 − 0.235251i 0.790089 − 0.612992i \(-0.210034\pi\)
−0.925911 + 0.377741i \(0.876701\pi\)
\(72\) −0.567785 − 8.84795i −0.0669141 − 1.04274i
\(73\) 10.8000 + 6.23540i 1.26405 + 0.729799i 0.973855 − 0.227169i \(-0.0729470\pi\)
0.290194 + 0.956968i \(0.406280\pi\)
\(74\) 0.581658 − 3.29875i 0.0676164 − 0.383471i
\(75\) 0 0
\(76\) −0.729423 − 0.265488i −0.0836705 − 0.0304536i
\(77\) −0.137119 + 0.0241778i −0.0156262 + 0.00275532i
\(78\) 1.53659 + 0.822676i 0.173985 + 0.0931498i
\(79\) −9.11223 − 7.64607i −1.02521 − 0.860250i −0.0349328 − 0.999390i \(-0.511122\pi\)
−0.990273 + 0.139140i \(0.955566\pi\)
\(80\) 0 0
\(81\) −3.46686 + 8.30547i −0.385207 + 0.922830i
\(82\) 3.03883i 0.335583i
\(83\) 3.11961 − 3.71780i 0.342421 − 0.408082i −0.567160 − 0.823608i \(-0.691958\pi\)
0.909581 + 0.415526i \(0.136402\pi\)
\(84\) 0.0586617 − 0.109568i 0.00640052 − 0.0119549i
\(85\) 0 0
\(86\) −10.7555 − 3.91468i −1.15979 − 0.422130i
\(87\) 3.18814 + 7.95646i 0.341804 + 0.853022i
\(88\) 1.17062 + 0.206412i 0.124789 + 0.0220036i
\(89\) 0.197409 − 0.341923i 0.0209253 − 0.0362438i −0.855373 − 0.518012i \(-0.826672\pi\)
0.876298 + 0.481769i \(0.160005\pi\)
\(90\) 0 0
\(91\) 0.130088 + 0.225318i 0.0136369 + 0.0236198i
\(92\) −0.219021 − 0.601754i −0.0228345 − 0.0627372i
\(93\) −2.40725 + 7.33715i −0.249620 + 0.760827i
\(94\) 12.6200 − 10.5894i 1.30165 − 1.09221i
\(95\) 0 0
\(96\) −1.50684 + 1.34897i −0.153791 + 0.137678i
\(97\) 5.07158 + 13.9340i 0.514941 + 1.41479i 0.876030 + 0.482257i \(0.160183\pi\)
−0.361089 + 0.932531i \(0.617595\pi\)
\(98\) 7.97784 − 4.60601i 0.805884 − 0.465277i
\(99\) −0.972102 − 0.714819i −0.0976999 − 0.0718420i
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.u.e.124.6 132
5.2 odd 4 675.2.l.g.151.3 yes 66
5.3 odd 4 675.2.l.f.151.9 yes 66
5.4 even 2 inner 675.2.u.e.124.17 132
27.22 even 9 inner 675.2.u.e.49.17 132
135.22 odd 36 675.2.l.g.76.3 yes 66
135.49 even 18 inner 675.2.u.e.49.6 132
135.103 odd 36 675.2.l.f.76.9 ✓ 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.2.l.f.76.9 ✓ 66 135.103 odd 36
675.2.l.f.151.9 yes 66 5.3 odd 4
675.2.l.g.76.3 yes 66 135.22 odd 36
675.2.l.g.151.3 yes 66 5.2 odd 4
675.2.u.e.49.6 132 135.49 even 18 inner
675.2.u.e.49.17 132 27.22 even 9 inner
675.2.u.e.124.6 132 1.1 even 1 trivial
675.2.u.e.124.17 132 5.4 even 2 inner