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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 76.9
Character \(\chi\) \(=\) 675.76
Dual form 675.2.l.f.151.9

$q$-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 \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\)

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 −0.203573 − 0.979060i \(-0.565255\pi\)
0.928836 + 0.370492i \(0.120811\pi\)
\(3\) 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 0.994058 − 0.108849i \(-0.0347166\pi\)
−0.971338 + 0.237703i \(0.923605\pi\)
\(8\) 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 0.284412 − 0.958702i \(-0.408202\pi\)
−0.398369 + 0.917225i \(0.630424\pi\)
\(12\) −0.341129 + 0.111921i −0.0984755 + 0.0323088i
\(13\) −0.575738 − 0.483102i −0.159681 − 0.133988i 0.559446 − 0.828867i \(-0.311014\pi\)
−0.719127 + 0.694878i \(0.755458\pi\)
\(14\) 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 −0.773129 − 0.634248i \(-0.781310\pi\)
0.935840 + 0.352426i \(0.114643\pi\)
\(18\) 0.442749 + 3.99230i 0.104357 + 0.940994i
\(19\) 1.87243 + 3.24314i 0.429565 + 0.744028i 0.996835 − 0.0795040i \(-0.0253336\pi\)
−0.567270 + 0.823532i \(0.692000\pi\)
\(20\) 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 0.876395 + 0.481594i \(0.159942\pi\)
−0.988256 + 0.152806i \(0.951169\pi\)
\(24\) −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 −0.218937 − 0.975739i \(-0.570259\pi\)
0.922897 + 0.385046i \(0.125815\pi\)
\(30\) 0 0
\(31\) −0.774172 + 4.39054i −0.139045 + 0.788565i 0.832912 + 0.553406i \(0.186672\pi\)
−0.971957 + 0.235159i \(0.924439\pi\)
\(32\) −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 −0.744696 − 0.667404i \(-0.767405\pi\)
0.950337 + 0.311223i \(0.100739\pi\)
\(38\) 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 0.768376 − 0.639999i \(-0.221065\pi\)
−0.496849 + 0.867837i \(0.665510\pi\)
\(42\) −0.164663 + 0.785740i −0.0254080 + 0.121242i
\(43\) −8.03293 − 2.92375i −1.22501 − 0.445867i −0.353124 − 0.935576i \(-0.614881\pi\)
−0.871886 + 0.489709i \(0.837103\pi\)
\(44\) 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 −0.590411 − 0.807103i \(-0.701034\pi\)
0.278760 − 0.960361i \(-0.410077\pi\)
\(48\) 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 0.670105 − 0.742266i \(-0.266249\pi\)
0.990450 + 0.137874i \(0.0440269\pi\)
\(60\) 0 0
\(61\) 2.31667 + 13.1385i 0.296619 + 1.68221i 0.660546 + 0.750785i \(0.270325\pi\)
−0.363927 + 0.931427i \(0.618564\pi\)
\(62\) 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 −0.149166 − 0.988812i \(-0.547659\pi\)
−0.999692 + 0.0248058i \(0.992103\pi\)
\(68\) 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 −0.925911 − 0.377741i \(-0.876701\pi\)
0.790089 + 0.612992i \(0.210034\pi\)
\(72\) −8.84795 − 0.567785i −1.04274 − 0.0669141i
\(73\) −6.23540 − 10.8000i −0.729799 − 1.26405i −0.956968 − 0.290194i \(-0.906280\pi\)
0.227169 − 0.973855i \(-0.427053\pi\)
\(74\) −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 0.0349328 − 0.999390i \(-0.488878\pi\)
0.990273 + 0.139140i \(0.0444338\pi\)
\(80\) 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 −0.415526 − 0.909581i \(-0.636402\pi\)
0.823608 + 0.567160i \(0.191958\pi\)
\(84\) −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 0.855373 − 0.518012i \(-0.173328\pi\)
−0.876298 + 0.481769i \(0.839995\pi\)
\(90\) 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 0.932531 − 0.361089i \(-0.117595\pi\)
0.482257 + 0.876030i \(0.339817\pi\)
\(98\) 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.76.9 ✓ 66
5.2 odd 4 675.2.u.e.49.17 132
5.3 odd 4 675.2.u.e.49.6 132
5.4 even 2 675.2.l.g.76.3 yes 66
27.16 even 9 inner 675.2.l.f.151.9 yes 66
135.43 odd 36 675.2.u.e.124.17 132
135.97 odd 36 675.2.u.e.124.6 132
135.124 even 18 675.2.l.g.151.3 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.2.l.f.76.9 ✓ 66 1.1 even 1 trivial
675.2.l.f.151.9 yes 66 27.16 even 9 inner
675.2.l.g.76.3 yes 66 5.4 even 2
675.2.l.g.151.3 yes 66 135.124 even 18
675.2.u.e.49.6 132 5.3 odd 4
675.2.u.e.49.17 132 5.2 odd 4
675.2.u.e.124.6 132 135.97 odd 36
675.2.u.e.124.17 132 135.43 odd 36