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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(8,405)] 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("405.8"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.r (of order \(36\), degree \(12\), 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 278.4
Character \(\chi\) \(=\) 405.278
Dual form 405.2.r.a.287.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.153697 - 1.75676i) q^{2} +(-1.09297 + 0.192720i) q^{4} +(1.53583 - 1.62518i) q^{5} +(-1.93217 - 2.75943i) q^{7} +(-0.406292 - 1.51630i) q^{8} +(-3.09111 - 2.44830i) q^{10} +(1.11880 + 3.07388i) q^{11} +(5.79013 + 0.506571i) q^{13} +(-4.55068 + 3.81848i) q^{14} +(-4.68713 + 1.70598i) q^{16} +(-0.298295 + 1.11325i) q^{17} +(-6.40749 - 3.69937i) q^{19} +(-1.36541 + 2.07226i) q^{20} +(5.22812 - 2.43791i) q^{22} +(-1.56820 - 1.09807i) q^{23} +(-0.282447 - 4.99202i) q^{25} -10.2497i q^{26} +(2.64360 + 2.64360i) q^{28} +(0.381851 + 0.320411i) q^{29} +(0.573116 + 3.25030i) q^{31} +(2.39054 + 5.12653i) q^{32} +(2.00157 + 0.352930i) q^{34} +(-7.45207 - 1.09788i) q^{35} +(4.40348 + 1.17991i) q^{37} +(-5.51409 + 11.8250i) q^{38} +(-3.08826 - 1.66848i) q^{40} +(1.39279 + 1.65986i) q^{41} +(-3.62184 - 1.68889i) q^{43} +(-1.81521 - 3.14404i) q^{44} +(-1.68801 + 2.92372i) q^{46} +(4.06724 - 2.84791i) q^{47} +(-1.48701 + 4.08553i) q^{49} +(-8.72636 + 1.26345i) q^{50} +(-6.42605 + 0.562207i) q^{52} +(1.25033 - 1.25033i) q^{53} +(6.71392 + 2.90271i) q^{55} +(-3.39910 + 4.05089i) q^{56} +(0.504196 - 0.720066i) q^{58} +(9.92422 + 3.61212i) q^{59} +(-1.82504 + 10.3503i) q^{61} +(5.62192 - 1.50639i) q^{62} +(-0.000691655 + 0.000399327i) q^{64} +(9.71593 - 8.63202i) q^{65} +(-0.509729 + 5.82623i) q^{67} +(0.111481 - 1.27424i) q^{68} +(-0.783353 + 13.2602i) q^{70} +(6.62671 - 3.82593i) q^{71} +(6.89704 - 1.84806i) q^{73} +(1.39602 - 7.91721i) q^{74} +(7.71612 + 2.80844i) q^{76} +(6.32044 - 9.02653i) q^{77} +(5.08874 - 6.06452i) q^{79} +(-4.42611 + 10.2375i) q^{80} +(2.70191 - 2.70191i) q^{82} +(6.22270 - 0.544416i) q^{83} +(1.35111 + 2.19455i) q^{85} +(-2.41031 + 6.62228i) q^{86} +(4.20637 - 2.94533i) q^{88} +(0.260380 - 0.450992i) q^{89} +(-9.78968 - 16.9562i) q^{91} +(1.92562 + 0.897929i) q^{92} +(-5.62822 - 6.70745i) q^{94} +(-15.8530 + 4.73175i) q^{95} +(1.99563 - 4.27963i) q^{97} +(7.40585 + 1.98439i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8} - 6 q^{10} + 36 q^{11} - 12 q^{13} - 24 q^{16} + 18 q^{17} - 36 q^{20} - 12 q^{22} + 36 q^{23} - 30 q^{25} - 24 q^{28} - 24 q^{31} + 48 q^{32} - 36 q^{35}+ \cdots - 324 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{13}{18}\right)\)

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 \(-0.813496\pi\)
0.724523 − 0.689250i \(-0.242060\pi\)
\(3\) 0 0
\(4\) −1.09297 + 0.192720i −0.546484 + 0.0963599i
\(5\) 1.53583 − 1.62518i 0.686844 − 0.726804i
\(6\) 0 0
\(7\) −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
\(9\) 0 0
\(10\) −3.09111 − 2.44830i −0.977495 − 0.774221i
\(11\) 1.11880 + 3.07388i 0.337332 + 0.926811i 0.986148 + 0.165866i \(0.0530420\pi\)
−0.648817 + 0.760945i \(0.724736\pi\)
\(12\) 0 0
\(13\) 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
\(15\) 0 0
\(16\) −4.68713 + 1.70598i −1.17178 + 0.426494i
\(17\) −0.298295 + 1.11325i −0.0723472 + 0.270003i −0.992619 − 0.121277i \(-0.961301\pi\)
0.920271 + 0.391280i \(0.127968\pi\)
\(18\) 0 0
\(19\) −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
\(21\) 0 0
\(22\) 5.22812 − 2.43791i 1.11464 − 0.519765i
\(23\) −1.56820 − 1.09807i −0.326993 − 0.228963i 0.398539 − 0.917151i \(-0.369517\pi\)
−0.725532 + 0.688189i \(0.758406\pi\)
\(24\) 0 0
\(25\) −0.282447 − 4.99202i −0.0564894 − 0.998403i
\(26\) − 10.2497i − 2.01014i
\(27\) 0 0
\(28\) 2.64360 + 2.64360i 0.499593 + 0.499593i
\(29\) 0.381851 + 0.320411i 0.0709079 + 0.0594988i 0.677553 − 0.735474i \(-0.263041\pi\)
−0.606645 + 0.794973i \(0.707485\pi\)
\(30\) 0 0
\(31\) 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 0
\(34\) 2.00157 + 0.352930i 0.343266 + 0.0605270i
\(35\) −7.45207 − 1.09788i −1.25963 − 0.185575i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) −3.08826 − 1.66848i −0.488297 − 0.263810i
\(41\) 1.39279 + 1.65986i 0.217517 + 0.259226i 0.863758 − 0.503907i \(-0.168105\pi\)
−0.646241 + 0.763133i \(0.723660\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −1.68801 + 2.92372i −0.248884 + 0.431080i
\(47\) 4.06724 − 2.84791i 0.593268 − 0.415410i −0.238014 − 0.971262i \(-0.576496\pi\)
0.831282 + 0.555851i \(0.187608\pi\)
\(48\) 0 0
\(49\) −1.48701 + 4.08553i −0.212430 + 0.583647i
\(50\) −8.72636 + 1.26345i −1.23409 + 0.178678i
\(51\) 0 0
\(52\) −6.42605 + 0.562207i −0.891133 + 0.0779641i
\(53\) 1.25033 − 1.25033i 0.171746 − 0.171746i −0.616000 − 0.787746i \(-0.711248\pi\)
0.787746 + 0.616000i \(0.211248\pi\)
\(54\) 0 0
\(55\) 6.71392 + 2.90271i 0.905305 + 0.391401i
\(56\) −3.39910 + 4.05089i −0.454223 + 0.541322i
\(57\) 0 0
\(58\) 0.504196 − 0.720066i 0.0662041 − 0.0945493i
\(59\) 9.92422 + 3.61212i 1.29202 + 0.470258i 0.894391 − 0.447286i \(-0.147609\pi\)
0.397633 + 0.917545i \(0.369832\pi\)
\(60\) 0 0
\(61\) −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
\(63\) 0 0
\(64\) −0.000691655 0 0.000399327i −8.64569e−5 0 4.99159e-5i
\(65\) 9.71593 − 8.63202i 1.20511 − 1.07067i
\(66\) 0 0
\(67\) −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
\(69\) 0 0
\(70\) −0.783353 + 13.2602i −0.0936285 + 1.58490i
\(71\) 6.62671 − 3.82593i 0.786446 − 0.454055i −0.0522640 − 0.998633i \(-0.516644\pi\)
0.838710 + 0.544579i \(0.183310\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) 7.71612 + 2.80844i 0.885100 + 0.322150i
\(77\) 6.32044 − 9.02653i 0.720281 − 1.02867i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) 2.70191 − 2.70191i 0.298375 − 0.298375i
\(83\) 6.22270 − 0.544416i 0.683031 − 0.0597574i 0.259641 − 0.965705i \(-0.416396\pi\)
0.423389 + 0.905948i \(0.360840\pi\)
\(84\) 0 0
\(85\) 1.35111 + 2.19455i 0.146548 + 0.238033i
\(86\) −2.41031 + 6.62228i −0.259911 + 0.714099i
\(87\) 0 0
\(88\) 4.20637 − 2.94533i 0.448401 − 0.313974i
\(89\) 0.260380 − 0.450992i 0.0276002 − 0.0478050i −0.851895 − 0.523712i \(-0.824547\pi\)
0.879496 + 0.475907i \(0.157880\pi\)
\(90\) 0 0
\(91\) −9.78968 − 16.9562i −1.02624 − 1.77750i
\(92\) 1.92562 + 0.897929i 0.200759 + 0.0936156i
\(93\) 0 0
\(94\) −5.62822 − 6.70745i −0.580506 − 0.691820i
\(95\) −15.8530 + 4.73175i −1.62648 + 0.485467i
\(96\) 0 0
\(97\) 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 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 405.2.r.a.278.4 192
3.2 odd 2 135.2.q.a.38.13 yes 192
5.2 odd 4 inner 405.2.r.a.197.13 192
15.2 even 4 135.2.q.a.92.4 yes 192
15.8 even 4 675.2.ba.b.632.13 192
15.14 odd 2 675.2.ba.b.443.4 192
27.5 odd 18 inner 405.2.r.a.368.13 192
27.22 even 9 135.2.q.a.113.4 yes 192
135.22 odd 36 135.2.q.a.32.13 ✓ 192
135.32 even 36 inner 405.2.r.a.287.4 192
135.49 even 18 675.2.ba.b.518.13 192
135.103 odd 36 675.2.ba.b.32.4 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.32.13 ✓ 192 135.22 odd 36
135.2.q.a.38.13 yes 192 3.2 odd 2
135.2.q.a.92.4 yes 192 15.2 even 4
135.2.q.a.113.4 yes 192 27.22 even 9
405.2.r.a.197.13 192 5.2 odd 4 inner
405.2.r.a.278.4 192 1.1 even 1 trivial
405.2.r.a.287.4 192 135.32 even 36 inner
405.2.r.a.368.13 192 27.5 odd 18 inner
675.2.ba.b.32.4 192 135.103 odd 36
675.2.ba.b.443.4 192 15.14 odd 2
675.2.ba.b.518.13 192 135.49 even 18
675.2.ba.b.632.13 192 15.8 even 4