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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(76,675)] 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("675.76"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [66,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 151.9
Character \(\chi\) \(=\) 675.151
Dual form 675.2.l.f.76.9

$q$-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+(1.02568 + 0.860644i) q^{2} +(0.817526 - 1.52697i) q^{3} +(-0.0359939 - 0.204131i) q^{4} +(2.15270 - 0.862581i) q^{6} +(0.0601126 - 0.340915i) q^{7} +(1.47769 - 2.55944i) q^{8} +(-1.66330 - 2.49668i) q^{9} +(-0.377953 + 0.137564i) q^{11} +(-0.341129 - 0.111921i) q^{12} +(-0.575738 + 0.483102i) q^{13} +(0.355063 - 0.297933i) q^{14} +(3.32884 - 1.21160i) q^{16} +(0.670872 + 1.16198i) q^{17} +(0.442749 - 3.99230i) q^{18} +(1.87243 - 3.24314i) q^{19} +(-0.471425 - 0.370498i) q^{21} +(-0.506050 - 0.184187i) q^{22} +(-0.536470 - 3.04247i) q^{23} +(-2.70014 - 4.34880i) q^{24} -1.00630 q^{26} +(-5.17216 + 0.498713i) q^{27} -0.0717552 q^{28} +(3.79094 + 3.18098i) q^{29} +(-0.774172 - 4.39054i) q^{31} +(-1.09724 - 0.399363i) q^{32} +(-0.0989303 + 0.689586i) q^{33} +(-0.311958 + 1.76920i) q^{34} +(-0.449783 + 0.429397i) q^{36} +(1.25087 + 2.16657i) q^{37} +(4.71170 - 1.71492i) q^{38} +(0.267003 + 1.27409i) q^{39} +(1.73862 - 1.45887i) q^{41} +(-0.164663 - 0.785740i) q^{42} +(-8.03293 + 2.92375i) q^{43} +(0.0416850 + 0.0722005i) q^{44} +(2.06824 - 3.58230i) q^{46} +(-2.13658 + 12.1171i) q^{47} +(0.871334 - 6.07356i) q^{48} +(6.46524 + 2.35315i) q^{49} +(2.32278 - 0.0744513i) q^{51} +(0.119339 + 0.100137i) q^{52} +10.1571 q^{53} +(-5.73418 - 3.93988i) q^{54} +(-0.783723 - 0.657622i) q^{56} +(-3.42144 - 5.51051i) q^{57} +(1.15059 + 6.52530i) q^{58} +(12.7550 + 4.64243i) q^{59} +(2.31667 - 13.1385i) q^{61} +(2.98465 - 5.16956i) q^{62} +(-0.951143 + 0.416963i) q^{63} +(-4.32418 - 7.48970i) q^{64} +(-0.694958 + 0.622148i) q^{66} +(-9.40381 + 7.89073i) q^{67} +(0.213050 - 0.178770i) q^{68} +(-5.08436 - 1.66813i) q^{69} +(-1.14446 - 1.98226i) q^{71} +(-8.84795 + 0.567785i) q^{72} +(-6.23540 + 10.8000i) q^{73} +(-0.581658 + 3.29875i) q^{74} +(-0.729423 - 0.265488i) q^{76} +(0.0241778 + 0.137119i) q^{77} +(-0.822676 + 1.53659i) q^{78} +(9.11223 + 7.64607i) q^{79} +(-3.46686 + 8.30547i) q^{81} +3.03883 q^{82} +(3.71780 + 3.11961i) q^{83} +(-0.0586617 + 0.109568i) q^{84} +(-10.7555 - 3.91468i) q^{86} +(7.95646 - 3.18814i) q^{87} +(-0.206412 + 1.17062i) q^{88} +(-0.197409 + 0.341923i) q^{89} +(0.130088 + 0.225318i) q^{91} +(-0.601754 + 0.219021i) q^{92} +(-7.33715 - 2.40725i) q^{93} +(-12.6200 + 10.5894i) q^{94} +(-1.50684 + 1.34897i) q^{96} +(13.9340 - 5.07158i) q^{97} +(4.60601 + 7.97784i) q^{98} +(0.972102 + 0.714819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q - 6 q^{2} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 6 q^{9} + 15 q^{11} - 18 q^{12} + 15 q^{14} + 18 q^{16} - 30 q^{17} + 12 q^{18} + 12 q^{19} + 12 q^{21} + 45 q^{22} - 36 q^{23} - 39 q^{24} + 6 q^{26} - 51 q^{27}+ \cdots - 48 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.02568 + 0.860644i 0.725262 + 0.608567i 0.928836 − 0.370492i \(-0.120811\pi\)
−0.203573 + 0.979060i \(0.565255\pi\)
\(3\) 0.817526 − 1.52697i 0.471999 − 0.881599i
\(4\) −0.0359939 − 0.204131i −0.0179969 − 0.102066i
\(5\) 0 0
\(6\) 2.15270 − 0.862581i 0.878836 − 0.352147i
\(7\) 0.0601126 − 0.340915i 0.0227204 − 0.128854i −0.971338 − 0.237703i \(-0.923605\pi\)
0.994058 + 0.108849i \(0.0347166\pi\)
\(8\) 1.47769 − 2.55944i 0.522443 − 0.904897i
\(9\) −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 \(0.408202\pi\)
\(12\) −0.341129 − 0.111921i −0.0984755 − 0.0323088i
\(13\) −0.575738 + 0.483102i −0.159681 + 0.133988i −0.719127 − 0.694878i \(-0.755458\pi\)
0.559446 + 0.828867i \(0.311014\pi\)
\(14\) 0.355063 − 0.297933i 0.0948945 − 0.0796260i
\(15\) 0 0
\(16\) 3.32884 − 1.21160i 0.832209 − 0.302899i
\(17\) 0.670872 + 1.16198i 0.162710 + 0.281823i 0.935840 − 0.352426i \(-0.114643\pi\)
−0.773129 + 0.634248i \(0.781310\pi\)
\(18\) 0.442749 − 3.99230i 0.104357 − 0.940994i
\(19\) 1.87243 − 3.24314i 0.429565 − 0.744028i −0.567270 − 0.823532i \(-0.692000\pi\)
0.996835 + 0.0795040i \(0.0253336\pi\)
\(20\) 0 0
\(21\) −0.471425 − 0.370498i −0.102873 − 0.0808492i
\(22\) −0.506050 − 0.184187i −0.107890 − 0.0392688i
\(23\) −0.536470 − 3.04247i −0.111862 − 0.634399i −0.988256 − 0.152806i \(-0.951169\pi\)
0.876395 − 0.481594i \(-0.159942\pi\)
\(24\) −2.70014 − 4.34880i −0.551164 − 0.887696i
\(25\) 0 0
\(26\) −1.00630 −0.197352
\(27\) −5.17216 + 0.498713i −0.995384 + 0.0959773i
\(28\) −0.0717552 −0.0135605
\(29\) 3.79094 + 3.18098i 0.703960 + 0.590693i 0.922897 − 0.385046i \(-0.125815\pi\)
−0.218937 + 0.975739i \(0.570259\pi\)
\(30\) 0 0
\(31\) −0.774172 − 4.39054i −0.139045 − 0.788565i −0.971957 − 0.235159i \(-0.924439\pi\)
0.832912 − 0.553406i \(-0.186672\pi\)
\(32\) −1.09724 − 0.399363i −0.193966 − 0.0705980i
\(33\) −0.0989303 + 0.689586i −0.0172216 + 0.120041i
\(34\) −0.311958 + 1.76920i −0.0535003 + 0.303415i
\(35\) 0 0
\(36\) −0.449783 + 0.429397i −0.0749638 + 0.0715662i
\(37\) 1.25087 + 2.16657i 0.205641 + 0.356181i 0.950337 − 0.311223i \(-0.100739\pi\)
−0.744696 + 0.667404i \(0.767405\pi\)
\(38\) 4.71170 − 1.71492i 0.764338 − 0.278196i
\(39\) 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 \(0.221065\pi\)
\(42\) −0.164663 − 0.785740i −0.0254080 − 0.121242i
\(43\) −8.03293 + 2.92375i −1.22501 + 0.445867i −0.871886 − 0.489709i \(-0.837103\pi\)
−0.353124 + 0.935576i \(0.614881\pi\)
\(44\) 0.0416850 + 0.0722005i 0.00628425 + 0.0108846i
\(45\) 0 0
\(46\) 2.06824 − 3.58230i 0.304946 − 0.528181i
\(47\) −2.13658 + 12.1171i −0.311652 + 1.76746i 0.278760 + 0.960361i \(0.410077\pi\)
−0.590411 + 0.807103i \(0.701034\pi\)
\(48\) 0.871334 − 6.07356i 0.125766 − 0.876643i
\(49\) 6.46524 + 2.35315i 0.923606 + 0.336165i
\(50\) 0 0
\(51\) 2.32278 − 0.0744513i 0.325254 − 0.0104253i
\(52\) 0.119339 + 0.100137i 0.0165494 + 0.0138866i
\(53\) 10.1571 1.39519 0.697593 − 0.716494i \(-0.254254\pi\)
0.697593 + 0.716494i \(0.254254\pi\)
\(54\) −5.73418 − 3.93988i −0.780323 − 0.536149i
\(55\) 0 0
\(56\) −0.783723 − 0.657622i −0.104729 − 0.0878784i
\(57\) −3.42144 − 5.51051i −0.453180 − 0.729884i
\(58\) 1.15059 + 6.52530i 0.151079 + 0.856814i
\(59\) 12.7550 + 4.64243i 1.66055 + 0.604392i 0.990450 − 0.137874i \(-0.0440269\pi\)
0.670105 + 0.742266i \(0.266249\pi\)
\(60\) 0 0
\(61\) 2.31667 − 13.1385i 0.296619 − 1.68221i −0.363927 − 0.931427i \(-0.618564\pi\)
0.660546 − 0.750785i \(-0.270325\pi\)
\(62\) 2.98465 − 5.16956i 0.379051 − 0.656535i
\(63\) −0.951143 + 0.416963i −0.119833 + 0.0525324i
\(64\) −4.32418 − 7.48970i −0.540522 − 0.936212i
\(65\) 0 0
\(66\) −0.694958 + 0.622148i −0.0855435 + 0.0765811i
\(67\) −9.40381 + 7.89073i −1.14886 + 0.964006i −0.999692 − 0.0248058i \(-0.992103\pi\)
−0.149166 + 0.988812i \(0.547659\pi\)
\(68\) 0.213050 − 0.178770i 0.0258361 − 0.0216791i
\(69\) −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 \(0.876701\pi\)
\(72\) −8.84795 + 0.567785i −1.04274 + 0.0669141i
\(73\) −6.23540 + 10.8000i −0.729799 + 1.26405i 0.227169 + 0.973855i \(0.427053\pi\)
−0.956968 + 0.290194i \(0.906280\pi\)
\(74\) −0.581658 + 3.29875i −0.0676164 + 0.383471i
\(75\) 0 0
\(76\) −0.729423 − 0.265488i −0.0836705 − 0.0304536i
\(77\) 0.0241778 + 0.137119i 0.00275532 + 0.0156262i
\(78\) −0.822676 + 1.53659i −0.0931498 + 0.173985i
\(79\) 9.11223 + 7.64607i 1.02521 + 0.860250i 0.990273 − 0.139140i \(-0.0444338\pi\)
0.0349328 + 0.999390i \(0.488878\pi\)
\(80\) 0 0
\(81\) −3.46686 + 8.30547i −0.385207 + 0.922830i
\(82\) 3.03883 0.335583
\(83\) 3.71780 + 3.11961i 0.408082 + 0.342421i 0.823608 − 0.567160i \(-0.191958\pi\)
−0.415526 + 0.909581i \(0.636402\pi\)
\(84\) −0.0586617 + 0.109568i −0.00640052 + 0.0119549i
\(85\) 0 0
\(86\) −10.7555 − 3.91468i −1.15979 − 0.422130i
\(87\) 7.95646 − 3.18814i 0.853022 − 0.341804i
\(88\) −0.206412 + 1.17062i −0.0220036 + 0.124789i
\(89\) −0.197409 + 0.341923i −0.0209253 + 0.0362438i −0.876298 − 0.481769i \(-0.839995\pi\)
0.855373 + 0.518012i \(0.173328\pi\)
\(90\) 0 0
\(91\) 0.130088 + 0.225318i 0.0136369 + 0.0236198i
\(92\) −0.601754 + 0.219021i −0.0627372 + 0.0228345i
\(93\) −7.33715 − 2.40725i −0.760827 − 0.249620i
\(94\) −12.6200 + 10.5894i −1.30165 + 1.09221i
\(95\) 0 0
\(96\) −1.50684 + 1.34897i −0.153791 + 0.137678i
\(97\) 13.9340 − 5.07158i 1.41479 − 0.514941i 0.482257 − 0.876030i \(-0.339817\pi\)
0.932531 + 0.361089i \(0.117595\pi\)
\(98\) 4.60601 + 7.97784i 0.465277 + 0.805884i
\(99\) 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.l.f.151.9 yes 66
5.2 odd 4 675.2.u.e.124.6 132
5.3 odd 4 675.2.u.e.124.17 132
5.4 even 2 675.2.l.g.151.3 yes 66
27.22 even 9 inner 675.2.l.f.76.9 ✓ 66
135.22 odd 36 675.2.u.e.49.17 132
135.49 even 18 675.2.l.g.76.3 yes 66
135.103 odd 36 675.2.u.e.49.6 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.2.l.f.76.9 ✓ 66 27.22 even 9 inner
675.2.l.f.151.9 yes 66 1.1 even 1 trivial
675.2.l.g.76.3 yes 66 135.49 even 18
675.2.l.g.151.3 yes 66 5.4 even 2
675.2.u.e.49.6 132 135.103 odd 36
675.2.u.e.49.17 132 135.22 odd 36
675.2.u.e.124.6 132 5.2 odd 4
675.2.u.e.124.17 132 5.3 odd 4