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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,2,Mod(2,135)] 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("135.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.q (of order \(36\), degree \(12\), 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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 38.13
Character \(\chi\) \(=\) 135.38
Dual form 135.2.q.a.32.13

$q$-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+(0.153697 + 1.75676i) q^{2} +(1.70488 + 0.305576i) q^{3} +(-1.09297 + 0.192720i) q^{4} +(-1.53583 + 1.62518i) q^{5} +(-0.274789 + 3.04204i) q^{6} +(-1.93217 - 2.75943i) q^{7} +(0.406292 + 1.51630i) q^{8} +(2.81325 + 1.04194i) q^{9} +(-3.09111 - 2.44830i) q^{10} +(-1.11880 - 3.07388i) q^{11} +(-1.92227 - 0.00542048i) q^{12} +(5.79013 + 0.506571i) q^{13} +(4.55068 - 3.81848i) q^{14} +(-3.11503 + 2.30143i) q^{15} +(-4.68713 + 1.70598i) q^{16} +(0.298295 - 1.11325i) q^{17} +(-1.39806 + 5.10234i) q^{18} +(-6.40749 - 3.69937i) q^{19} +(1.36541 - 2.07226i) q^{20} +(-2.45091 - 5.29492i) q^{21} +(5.22812 - 2.43791i) q^{22} +(1.56820 + 1.09807i) q^{23} +(0.229334 + 2.70927i) q^{24} +(-0.282447 - 4.99202i) q^{25} +10.2497i q^{26} +(4.47786 + 2.63605i) q^{27} +(2.64360 + 2.64360i) q^{28} +(-0.381851 - 0.320411i) q^{29} +(-4.52184 - 5.11864i) q^{30} +(0.573116 + 3.25030i) q^{31} +(-2.39054 - 5.12653i) q^{32} +(-0.968121 - 5.58249i) q^{33} +(2.00157 + 0.352930i) q^{34} +(7.45207 + 1.09788i) q^{35} +(-3.27559 - 0.596642i) q^{36} +(4.40348 + 1.17991i) q^{37} +(5.51409 - 11.8250i) q^{38} +(9.71669 + 2.63297i) q^{39} +(-3.08826 - 1.66848i) q^{40} +(-1.39279 - 1.65986i) q^{41} +(8.92522 - 5.11947i) q^{42} +(-3.62184 - 1.68889i) q^{43} +(1.81521 + 3.14404i) q^{44} +(-6.01402 + 2.97180i) q^{45} +(-1.68801 + 2.92372i) q^{46} +(-4.06724 + 2.84791i) q^{47} +(-8.51231 + 1.47621i) q^{48} +(-1.48701 + 4.08553i) q^{49} +(8.72636 - 1.26345i) q^{50} +(0.848742 - 1.80681i) q^{51} +(-6.42605 + 0.562207i) q^{52} +(-1.25033 + 1.25033i) q^{53} +(-3.94268 + 8.27168i) q^{54} +(6.71392 + 2.90271i) q^{55} +(3.39910 - 4.05089i) q^{56} +(-9.79358 - 8.26496i) q^{57} +(0.504196 - 0.720066i) q^{58} +(-9.92422 - 3.61212i) q^{59} +(2.96110 - 3.11572i) q^{60} +(-1.82504 + 10.3503i) q^{61} +(-5.62192 + 1.50639i) q^{62} +(-2.56051 - 9.77616i) q^{63} +(-0.000691655 + 0.000399327i) q^{64} +(-9.71593 + 8.63202i) q^{65} +(9.65830 - 2.55877i) q^{66} +(-0.509729 + 5.82623i) q^{67} +(-0.111481 + 1.27424i) q^{68} +(2.33806 + 2.35128i) q^{69} +(-0.783353 + 13.2602i) q^{70} +(-6.62671 + 3.82593i) q^{71} +(-0.436900 + 4.68906i) q^{72} +(6.89704 - 1.84806i) q^{73} +(-1.39602 + 7.91721i) q^{74} +(1.04390 - 8.59711i) q^{75} +(7.71612 + 2.80844i) q^{76} +(-6.32044 + 9.02653i) q^{77} +(-3.13207 + 17.4746i) q^{78} +(5.08874 - 6.06452i) q^{79} +(4.42611 - 10.2375i) q^{80} +(6.82871 + 5.86248i) q^{81} +(2.70191 - 2.70191i) q^{82} +(-6.22270 + 0.544416i) q^{83} +(3.69920 + 5.31485i) q^{84} +(1.35111 + 2.19455i) q^{85} +(2.41031 - 6.62228i) q^{86} +(-0.553100 - 0.662947i) q^{87} +(4.20637 - 2.94533i) q^{88} +(-0.260380 + 0.450992i) q^{89} +(-6.14507 - 10.1084i) q^{90} +(-9.78968 - 16.9562i) q^{91} +(-1.92562 - 0.897929i) q^{92} +(-0.0161196 + 5.71652i) q^{93} +(-5.62822 - 6.70745i) q^{94} +(15.8530 - 4.73175i) q^{95} +(-2.50905 - 9.47063i) q^{96} +(1.99563 - 4.27963i) q^{97} +(-7.40585 - 1.98439i) q^{98} +(0.0553442 - 9.81332i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 36 q^{6} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} - 12 q^{15} - 24 q^{16} - 18 q^{17} - 54 q^{18} + 36 q^{20} - 24 q^{21} - 12 q^{22}+ \cdots + 324 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.153697 + 1.75676i 0.108680 + 1.24222i 0.833203 + 0.552967i \(0.186504\pi\)
−0.724523 + 0.689250i \(0.757940\pi\)
\(3\) 1.70488 + 0.305576i 0.984314 + 0.176424i
\(4\) −1.09297 + 0.192720i −0.546484 + 0.0963599i
\(5\) −1.53583 + 1.62518i −0.686844 + 0.726804i
\(6\) −0.274789 + 3.04204i −0.112182 + 1.24191i
\(7\) −1.93217 − 2.75943i −0.730292 − 1.04297i −0.996863 − 0.0791502i \(-0.974779\pi\)
0.266570 − 0.963815i \(-0.414110\pi\)
\(8\) 0.406292 + 1.51630i 0.143646 + 0.536093i
\(9\) 2.81325 + 1.04194i 0.937749 + 0.347314i
\(10\) −3.09111 − 2.44830i −0.977495 − 0.774221i
\(11\) −1.11880 − 3.07388i −0.337332 − 0.926811i −0.986148 − 0.165866i \(-0.946958\pi\)
0.648817 − 0.760945i \(-0.275264\pi\)
\(12\) −1.92227 − 0.00542048i −0.554912 − 0.00156476i
\(13\) 5.79013 + 0.506571i 1.60589 + 0.140497i 0.854534 − 0.519396i \(-0.173843\pi\)
0.751359 + 0.659893i \(0.229399\pi\)
\(14\) 4.55068 − 3.81848i 1.21622 − 1.02053i
\(15\) −3.11503 + 2.30143i −0.804297 + 0.594228i
\(16\) −4.68713 + 1.70598i −1.17178 + 0.426494i
\(17\) 0.298295 − 1.11325i 0.0723472 − 0.270003i −0.920271 − 0.391280i \(-0.872032\pi\)
0.992619 + 0.121277i \(0.0386989\pi\)
\(18\) −1.39806 + 5.10234i −0.329525 + 1.20263i
\(19\) −6.40749 − 3.69937i −1.46998 − 0.848693i −0.470546 − 0.882375i \(-0.655943\pi\)
−0.999433 + 0.0336827i \(0.989276\pi\)
\(20\) 1.36541 − 2.07226i 0.305315 − 0.463371i
\(21\) −2.45091 − 5.29492i −0.534833 − 1.15545i
\(22\) 5.22812 − 2.43791i 1.11464 − 0.519765i
\(23\) 1.56820 + 1.09807i 0.326993 + 0.228963i 0.725532 − 0.688189i \(-0.241594\pi\)
−0.398539 + 0.917151i \(0.630483\pi\)
\(24\) 0.229334 + 2.70927i 0.0468126 + 0.553027i
\(25\) −0.282447 − 4.99202i −0.0564894 − 0.998403i
\(26\) 10.2497i 2.01014i
\(27\) 4.47786 + 2.63605i 0.861765 + 0.507308i
\(28\) 2.64360 + 2.64360i 0.499593 + 0.499593i
\(29\) −0.381851 − 0.320411i −0.0709079 − 0.0594988i 0.606645 − 0.794973i \(-0.292515\pi\)
−0.677553 + 0.735474i \(0.736959\pi\)
\(30\) −4.52184 − 5.11864i −0.825571 − 0.934531i
\(31\) 0.573116 + 3.25030i 0.102935 + 0.583772i 0.992025 + 0.126040i \(0.0402269\pi\)
−0.889090 + 0.457732i \(0.848662\pi\)
\(32\) −2.39054 − 5.12653i −0.422592 − 0.906251i
\(33\) −0.968121 − 5.58249i −0.168528 − 0.971787i
\(34\) 2.00157 + 0.352930i 0.343266 + 0.0605270i
\(35\) 7.45207 + 1.09788i 1.25963 + 0.185575i
\(36\) −3.27559 − 0.596642i −0.545932 − 0.0994403i
\(37\) 4.40348 + 1.17991i 0.723928 + 0.193976i 0.601924 − 0.798554i \(-0.294401\pi\)
0.122004 + 0.992530i \(0.461068\pi\)
\(38\) 5.51409 − 11.8250i 0.894503 − 1.91827i
\(39\) 9.71669 + 2.63297i 1.55592 + 0.421612i
\(40\) −3.08826 − 1.66848i −0.488297 − 0.263810i
\(41\) −1.39279 − 1.65986i −0.217517 − 0.259226i 0.646241 − 0.763133i \(-0.276340\pi\)
−0.863758 + 0.503907i \(0.831895\pi\)
\(42\) 8.92522 − 5.11947i 1.37719 − 0.789952i
\(43\) −3.62184 − 1.68889i −0.552325 − 0.257554i 0.126358 − 0.991985i \(-0.459671\pi\)
−0.678683 + 0.734431i \(0.737449\pi\)
\(44\) 1.81521 + 3.14404i 0.273654 + 0.473982i
\(45\) −6.01402 + 2.97180i −0.896517 + 0.443009i
\(46\) −1.68801 + 2.92372i −0.248884 + 0.431080i
\(47\) −4.06724 + 2.84791i −0.593268 + 0.415410i −0.831282 − 0.555851i \(-0.812392\pi\)
0.238014 + 0.971262i \(0.423504\pi\)
\(48\) −8.51231 + 1.47621i −1.22865 + 0.213073i
\(49\) −1.48701 + 4.08553i −0.212430 + 0.583647i
\(50\) 8.72636 − 1.26345i 1.23409 − 0.178678i
\(51\) 0.848742 − 1.80681i 0.118848 − 0.253004i
\(52\) −6.42605 + 0.562207i −0.891133 + 0.0779641i
\(53\) −1.25033 + 1.25033i −0.171746 + 0.171746i −0.787746 − 0.616000i \(-0.788752\pi\)
0.616000 + 0.787746i \(0.288752\pi\)
\(54\) −3.94268 + 8.27168i −0.536530 + 1.12563i
\(55\) 6.71392 + 2.90271i 0.905305 + 0.391401i
\(56\) 3.39910 − 4.05089i 0.454223 − 0.541322i
\(57\) −9.79358 − 8.26496i −1.29719 − 1.09472i
\(58\) 0.504196 − 0.720066i 0.0662041 − 0.0945493i
\(59\) −9.92422 − 3.61212i −1.29202 − 0.470258i −0.397633 − 0.917545i \(-0.630168\pi\)
−0.894391 + 0.447286i \(0.852391\pi\)
\(60\) 2.96110 − 3.11572i 0.382276 − 0.402238i
\(61\) −1.82504 + 10.3503i −0.233672 + 1.32522i 0.611720 + 0.791074i \(0.290478\pi\)
−0.845392 + 0.534146i \(0.820633\pi\)
\(62\) −5.62192 + 1.50639i −0.713985 + 0.191312i
\(63\) −2.56051 − 9.77616i −0.322594 − 1.23168i
\(64\) −0.000691655 0 0.000399327i −8.64569e−5 0 4.99159e-5i
\(65\) −9.71593 + 8.63202i −1.20511 + 1.07067i
\(66\) 9.65830 − 2.55877i 1.18885 − 0.314962i
\(67\) −0.509729 + 5.82623i −0.0622733 + 0.711787i 0.899317 + 0.437298i \(0.144064\pi\)
−0.961590 + 0.274490i \(0.911491\pi\)
\(68\) −0.111481 + 1.27424i −0.0135191 + 0.154524i
\(69\) 2.33806 + 2.35128i 0.281469 + 0.283061i
\(70\) −0.783353 + 13.2602i −0.0936285 + 1.58490i
\(71\) −6.62671 + 3.82593i −0.786446 + 0.454055i −0.838710 − 0.544579i \(-0.816690\pi\)
0.0522640 + 0.998633i \(0.483356\pi\)
\(72\) −0.436900 + 4.68906i −0.0514892 + 0.552611i
\(73\) 6.89704 − 1.84806i 0.807238 − 0.216299i 0.168478 − 0.985705i \(-0.446115\pi\)
0.638759 + 0.769407i \(0.279448\pi\)
\(74\) −1.39602 + 7.91721i −0.162284 + 0.920357i
\(75\) 1.04390 − 8.59711i 0.120539 − 0.992709i
\(76\) 7.71612 + 2.80844i 0.885100 + 0.322150i
\(77\) −6.32044 + 9.02653i −0.720281 + 1.02867i
\(78\) −3.13207 + 17.4746i −0.354637 + 1.97861i
\(79\) 5.08874 − 6.06452i 0.572528 − 0.682312i −0.399620 − 0.916681i \(-0.630858\pi\)
0.972148 + 0.234369i \(0.0753024\pi\)
\(80\) 4.42611 − 10.2375i 0.494854 − 1.14459i
\(81\) 6.82871 + 5.86248i 0.758746 + 0.651387i
\(82\) 2.70191 − 2.70191i 0.298375 − 0.298375i
\(83\) −6.22270 + 0.544416i −0.683031 + 0.0597574i −0.423389 − 0.905948i \(-0.639160\pi\)
−0.259641 + 0.965705i \(0.583604\pi\)
\(84\) 3.69920 + 5.31485i 0.403616 + 0.579897i
\(85\) 1.35111 + 2.19455i 0.146548 + 0.238033i
\(86\) 2.41031 − 6.62228i 0.259911 − 0.714099i
\(87\) −0.553100 − 0.662947i −0.0592986 − 0.0710754i
\(88\) 4.20637 − 2.94533i 0.448401 − 0.313974i
\(89\) −0.260380 + 0.450992i −0.0276002 + 0.0478050i −0.879496 − 0.475907i \(-0.842120\pi\)
0.851895 + 0.523712i \(0.175453\pi\)
\(90\) −6.14507 − 10.1084i −0.647747 − 1.06552i
\(91\) −9.78968 − 16.9562i −1.02624 − 1.77750i
\(92\) −1.92562 − 0.897929i −0.200759 − 0.0936156i
\(93\) −0.0161196 + 5.71652i −0.00167153 + 0.592775i
\(94\) −5.62822 − 6.70745i −0.580506 − 0.691820i
\(95\) 15.8530 − 4.73175i 1.62648 − 0.485467i
\(96\) −2.50905 − 9.47063i −0.256078 − 0.966592i
\(97\) 1.99563 − 4.27963i 0.202625 − 0.434531i −0.778568 − 0.627560i \(-0.784054\pi\)
0.981193 + 0.193029i \(0.0618313\pi\)
\(98\) −7.40585 − 1.98439i −0.748103 − 0.200454i
\(99\) 0.0553442 − 9.81332i 0.00556230 − 0.986276i
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 135.2.q.a.38.13 yes 192
3.2 odd 2 405.2.r.a.278.4 192
5.2 odd 4 inner 135.2.q.a.92.4 yes 192
5.3 odd 4 675.2.ba.b.632.13 192
5.4 even 2 675.2.ba.b.443.4 192
15.2 even 4 405.2.r.a.197.13 192
27.5 odd 18 inner 135.2.q.a.113.4 yes 192
27.22 even 9 405.2.r.a.368.13 192
135.22 odd 36 405.2.r.a.287.4 192
135.32 even 36 inner 135.2.q.a.32.13 ✓ 192
135.59 odd 18 675.2.ba.b.518.13 192
135.113 even 36 675.2.ba.b.32.4 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.32.13 ✓ 192 135.32 even 36 inner
135.2.q.a.38.13 yes 192 1.1 even 1 trivial
135.2.q.a.92.4 yes 192 5.2 odd 4 inner
135.2.q.a.113.4 yes 192 27.5 odd 18 inner
405.2.r.a.197.13 192 15.2 even 4
405.2.r.a.278.4 192 3.2 odd 2
405.2.r.a.287.4 192 135.22 odd 36
405.2.r.a.368.13 192 27.22 even 9
675.2.ba.b.32.4 192 135.113 even 36
675.2.ba.b.443.4 192 5.4 even 2
675.2.ba.b.518.13 192 135.59 odd 18
675.2.ba.b.632.13 192 5.3 odd 4