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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [99,2,Mod(4,99)] 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("99.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.m (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 16.5
Character \(\chi\) \(=\) 99.16
Dual form 99.2.m.b.31.5

$q$-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+(-0.278917 - 0.0592857i) q^{2} +(-1.04736 + 1.37950i) q^{3} +(-1.75281 - 0.780402i) q^{4} +(-2.64954 + 0.563177i) q^{5} +(0.373912 - 0.322674i) q^{6} +(0.425779 + 4.05102i) q^{7} +(0.904002 + 0.656796i) q^{8} +(-0.806061 - 2.88968i) q^{9} +0.772391 q^{10} +(-3.31056 + 0.200500i) q^{11} +(2.91240 - 1.60065i) q^{12} +(2.57679 - 2.86182i) q^{13} +(0.121410 - 1.15514i) q^{14} +(1.99813 - 4.24490i) q^{15} +(2.35451 + 2.61494i) q^{16} +(-1.33502 + 4.10876i) q^{17} +(0.0535075 + 0.853770i) q^{18} +(1.31246 + 0.953561i) q^{19} +(5.08365 + 1.08056i) q^{20} +(-6.03434 - 3.65553i) q^{21} +(0.935259 + 0.140346i) q^{22} +(0.932117 + 1.61447i) q^{23} +(-1.85287 + 0.559170i) q^{24} +(2.13517 - 0.950641i) q^{25} +(-0.888377 + 0.645443i) q^{26} +(4.83057 + 1.91458i) q^{27} +(2.41511 - 7.43295i) q^{28} +(-0.371426 - 3.53388i) q^{29} +(-0.808974 + 1.06552i) q^{30} +(-3.25505 + 3.61509i) q^{31} +(-1.61909 - 2.80435i) q^{32} +(3.19077 - 4.77692i) q^{33} +(0.615950 - 1.06686i) q^{34} +(-3.40956 - 10.4936i) q^{35} +(-0.842241 + 5.69412i) q^{36} +(-6.26542 + 4.55210i) q^{37} +(-0.309536 - 0.343775i) q^{38} +(1.24905 + 6.55206i) q^{39} +(-2.76508 - 1.23109i) q^{40} +(-0.718335 + 6.83450i) q^{41} +(1.46636 + 1.37734i) q^{42} +(-0.492496 + 0.853027i) q^{43} +(5.95925 + 2.23213i) q^{44} +(3.76310 + 7.20238i) q^{45} +(-0.164268 - 0.505566i) q^{46} +(5.35918 - 2.38606i) q^{47} +(-6.07335 + 0.509254i) q^{48} +(-9.38245 + 1.99430i) q^{49} +(-0.651896 + 0.138565i) q^{50} +(-4.26980 - 6.14502i) q^{51} +(-6.75000 + 3.00529i) q^{52} +(-0.485721 - 1.49489i) q^{53} +(-1.23382 - 0.820393i) q^{54} +(8.65855 - 2.39567i) q^{55} +(-2.27579 + 3.94178i) q^{56} +(-2.69007 + 0.811824i) q^{57} +(-0.105911 + 1.00768i) q^{58} +(10.3802 + 4.62154i) q^{59} +(-6.81507 + 5.88117i) q^{60} +(-5.75953 - 6.39660i) q^{61} +(1.12221 - 0.815335i) q^{62} +(11.3630 - 4.49574i) q^{63} +(-1.88937 - 5.81490i) q^{64} +(-5.21561 + 9.03370i) q^{65} +(-1.17316 + 1.14320i) q^{66} +(0.870282 + 1.50737i) q^{67} +(5.54651 - 6.16003i) q^{68} +(-3.20344 - 0.405082i) q^{69} +(0.328868 + 3.12897i) q^{70} +(-1.77418 + 5.46036i) q^{71} +(1.16925 - 3.14170i) q^{72} +(3.27379 - 2.37855i) q^{73} +(2.01741 - 0.898208i) q^{74} +(-0.924890 + 3.94115i) q^{75} +(-1.55634 - 2.69566i) q^{76} +(-2.22180 - 13.3258i) q^{77} +(0.0400613 - 1.90153i) q^{78} +(4.18662 + 0.889893i) q^{79} +(-7.71104 - 5.60240i) q^{80} +(-7.70053 + 4.65852i) q^{81} +(0.605544 - 1.86367i) q^{82} +(-4.34947 - 4.83058i) q^{83} +(7.72429 + 11.1167i) q^{84} +(1.22322 - 11.6382i) q^{85} +(0.187938 - 0.208726i) q^{86} +(5.26402 + 3.18887i) q^{87} +(-3.12444 - 1.99311i) q^{88} +9.26243 q^{89} +(-0.622594 - 2.23196i) q^{90} +(12.6904 + 9.22014i) q^{91} +(-0.373887 - 3.55729i) q^{92} +(-1.57782 - 8.27667i) q^{93} +(-1.63623 + 0.347790i) q^{94} +(-4.01445 - 1.78735i) q^{95} +(5.56439 + 0.703629i) q^{96} +(7.25070 + 1.54118i) q^{97} +2.73516 q^{98} +(3.24790 + 9.40485i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - q^{2} - 12 q^{3} + 11 q^{4} - 8 q^{5} - 7 q^{6} - 2 q^{7} + 6 q^{8} - 22 q^{9} - 8 q^{10} - 2 q^{11} - 16 q^{12} - 11 q^{13} - 10 q^{14} - 20 q^{15} - 9 q^{16} - 20 q^{17} - 21 q^{18} + 8 q^{19}+ \cdots - 182 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.278917 − 0.0592857i −0.197224 − 0.0419213i 0.108240 − 0.994125i \(-0.465478\pi\)
−0.305465 + 0.952203i \(0.598812\pi\)
\(3\) −1.04736 + 1.37950i −0.604695 + 0.796457i
\(4\) −1.75281 − 0.780402i −0.876405 − 0.390201i
\(5\) −2.64954 + 0.563177i −1.18491 + 0.251861i −0.757891 − 0.652381i \(-0.773770\pi\)
−0.427020 + 0.904242i \(0.640437\pi\)
\(6\) 0.373912 − 0.322674i 0.152649 − 0.131731i
\(7\) 0.425779 + 4.05102i 0.160930 + 1.53114i 0.715268 + 0.698851i \(0.246305\pi\)
−0.554338 + 0.832291i \(0.687029\pi\)
\(8\) 0.904002 + 0.656796i 0.319613 + 0.232212i
\(9\) −0.806061 − 2.88968i −0.268687 − 0.963228i
\(10\) 0.772391 0.244251
\(11\) −3.31056 + 0.200500i −0.998171 + 0.0604531i
\(12\) 2.91240 − 1.60065i 0.840736 − 0.462066i
\(13\) 2.57679 − 2.86182i 0.714674 − 0.793726i −0.270966 − 0.962589i \(-0.587343\pi\)
0.985639 + 0.168863i \(0.0540097\pi\)
\(14\) 0.121410 − 1.15514i 0.0324483 − 0.308725i
\(15\) 1.99813 − 4.24490i 0.515914 − 1.09603i
\(16\) 2.35451 + 2.61494i 0.588626 + 0.653736i
\(17\) −1.33502 + 4.10876i −0.323789 + 0.996520i 0.648195 + 0.761474i \(0.275524\pi\)
−0.971984 + 0.235046i \(0.924476\pi\)
\(18\) 0.0535075 + 0.853770i 0.0126118 + 0.201236i
\(19\) 1.31246 + 0.953561i 0.301100 + 0.218762i 0.728068 − 0.685505i \(-0.240418\pi\)
−0.426968 + 0.904267i \(0.640418\pi\)
\(20\) 5.08365 + 1.08056i 1.13674 + 0.241621i
\(21\) −6.03434 − 3.65553i −1.31680 − 0.797701i
\(22\) 0.935259 + 0.140346i 0.199398 + 0.0299218i
\(23\) 0.932117 + 1.61447i 0.194360 + 0.336641i 0.946691 − 0.322145i \(-0.104404\pi\)
−0.752331 + 0.658786i \(0.771070\pi\)
\(24\) −1.85287 + 0.559170i −0.378216 + 0.114140i
\(25\) 2.13517 − 0.950641i 0.427035 − 0.190128i
\(26\) −0.888377 + 0.645443i −0.174225 + 0.126582i
\(27\) 4.83057 + 1.91458i 0.929643 + 0.368462i
\(28\) 2.41511 − 7.43295i 0.456413 − 1.40470i
\(29\) −0.371426 − 3.53388i −0.0689720 − 0.656225i −0.973324 − 0.229437i \(-0.926312\pi\)
0.904352 − 0.426788i \(-0.140355\pi\)
\(30\) −0.808974 + 1.06552i −0.147698 + 0.194536i
\(31\) −3.25505 + 3.61509i −0.584624 + 0.649290i −0.960795 − 0.277260i \(-0.910574\pi\)
0.376171 + 0.926550i \(0.377240\pi\)
\(32\) −1.61909 − 2.80435i −0.286218 − 0.495744i
\(33\) 3.19077 − 4.77692i 0.555441 − 0.831556i
\(34\) 0.615950 − 1.06686i 0.105634 − 0.182964i
\(35\) −3.40956 − 10.4936i −0.576321 − 1.77374i
\(36\) −0.842241 + 5.69412i −0.140373 + 0.949020i
\(37\) −6.26542 + 4.55210i −1.03003 + 0.748360i −0.968314 − 0.249734i \(-0.919657\pi\)
−0.0617145 + 0.998094i \(0.519657\pi\)
\(38\) −0.309536 − 0.343775i −0.0502134 − 0.0557676i
\(39\) 1.24905 + 6.55206i 0.200008 + 1.04917i
\(40\) −2.76508 − 1.23109i −0.437198 − 0.194653i
\(41\) −0.718335 + 6.83450i −0.112185 + 1.06737i 0.783106 + 0.621888i \(0.213634\pi\)
−0.895291 + 0.445481i \(0.853033\pi\)
\(42\) 1.46636 + 1.37734i 0.226265 + 0.212528i
\(43\) −0.492496 + 0.853027i −0.0751049 + 0.130085i −0.901132 − 0.433545i \(-0.857262\pi\)
0.826027 + 0.563631i \(0.190596\pi\)
\(44\) 5.95925 + 2.23213i 0.898391 + 0.336506i
\(45\) 3.76310 + 7.20238i 0.560969 + 1.07367i
\(46\) −0.164268 − 0.505566i −0.0242200 − 0.0745416i
\(47\) 5.35918 − 2.38606i 0.781716 − 0.348042i 0.0232288 − 0.999730i \(-0.492605\pi\)
0.758487 + 0.651688i \(0.225939\pi\)
\(48\) −6.07335 + 0.509254i −0.876612 + 0.0735045i
\(49\) −9.38245 + 1.99430i −1.34035 + 0.284900i
\(50\) −0.651896 + 0.138565i −0.0921920 + 0.0195960i
\(51\) −4.26980 − 6.14502i −0.597892 − 0.860475i
\(52\) −6.75000 + 3.00529i −0.936056 + 0.416759i
\(53\) −0.485721 − 1.49489i −0.0667189 − 0.205340i 0.912139 − 0.409881i \(-0.134430\pi\)
−0.978858 + 0.204541i \(0.934430\pi\)
\(54\) −1.23382 − 0.820393i −0.167902 − 0.111641i
\(55\) 8.65855 − 2.39567i 1.16752 − 0.323032i
\(56\) −2.27579 + 3.94178i −0.304115 + 0.526743i
\(57\) −2.69007 + 0.811824i −0.356308 + 0.107529i
\(58\) −0.105911 + 1.00768i −0.0139069 + 0.132315i
\(59\) 10.3802 + 4.62154i 1.35138 + 0.601674i 0.949427 − 0.313986i \(-0.101665\pi\)
0.401954 + 0.915660i \(0.368331\pi\)
\(60\) −6.81507 + 5.88117i −0.879822 + 0.759256i
\(61\) −5.75953 − 6.39660i −0.737432 − 0.819001i 0.251424 − 0.967877i \(-0.419101\pi\)
−0.988856 + 0.148876i \(0.952434\pi\)
\(62\) 1.12221 − 0.815335i 0.142521 − 0.103548i
\(63\) 11.3630 − 4.49574i 1.43160 − 0.566410i
\(64\) −1.88937 − 5.81490i −0.236172 − 0.726862i
\(65\) −5.21561 + 9.03370i −0.646916 + 1.12049i
\(66\) −1.17316 + 1.14320i −0.144406 + 0.140718i
\(67\) 0.870282 + 1.50737i 0.106322 + 0.184155i 0.914277 − 0.405088i \(-0.132759\pi\)
−0.807956 + 0.589243i \(0.799426\pi\)
\(68\) 5.54651 − 6.16003i 0.672614 − 0.747013i
\(69\) −3.20344 − 0.405082i −0.385649 − 0.0487661i
\(70\) 0.328868 + 3.12897i 0.0393073 + 0.373984i
\(71\) −1.77418 + 5.46036i −0.210556 + 0.648026i 0.788883 + 0.614543i \(0.210660\pi\)
−0.999439 + 0.0334822i \(0.989340\pi\)
\(72\) 1.16925 − 3.14170i 0.137798 − 0.370253i
\(73\) 3.27379 − 2.37855i 0.383168 − 0.278388i −0.379482 − 0.925199i \(-0.623898\pi\)
0.762650 + 0.646811i \(0.223898\pi\)
\(74\) 2.01741 − 0.898208i 0.234519 − 0.104415i
\(75\) −0.924890 + 3.94115i −0.106797 + 0.455084i
\(76\) −1.55634 − 2.69566i −0.178524 − 0.309213i
\(77\) −2.22180 − 13.3258i −0.253198 − 1.51861i
\(78\) 0.0400613 − 1.90153i 0.00453605 − 0.215306i
\(79\) 4.18662 + 0.889893i 0.471031 + 0.100121i 0.437311 − 0.899311i \(-0.355931\pi\)
0.0337205 + 0.999431i \(0.489264\pi\)
\(80\) −7.71104 − 5.60240i −0.862120 − 0.626367i
\(81\) −7.70053 + 4.65852i −0.855615 + 0.517613i
\(82\) 0.605544 − 1.86367i 0.0668711 − 0.205808i
\(83\) −4.34947 − 4.83058i −0.477417 − 0.530225i 0.455539 − 0.890216i \(-0.349446\pi\)
−0.932956 + 0.359991i \(0.882780\pi\)
\(84\) 7.72429 + 11.1167i 0.842789 + 1.21293i
\(85\) 1.22322 − 11.6382i 0.132677 − 1.26234i
\(86\) 0.187938 − 0.208726i 0.0202659 − 0.0225075i
\(87\) 5.26402 + 3.18887i 0.564362 + 0.341883i
\(88\) −3.12444 − 1.99311i −0.333066 − 0.212466i
\(89\) 9.26243 0.981816 0.490908 − 0.871211i \(-0.336665\pi\)
0.490908 + 0.871211i \(0.336665\pi\)
\(90\) −0.622594 − 2.23196i −0.0656272 − 0.235270i
\(91\) 12.6904 + 9.22014i 1.33032 + 0.966533i
\(92\) −0.373887 − 3.55729i −0.0389804 − 0.370874i
\(93\) −1.57782 − 8.27667i −0.163612 − 0.858250i
\(94\) −1.63623 + 0.347790i −0.168764 + 0.0358718i
\(95\) −4.01445 − 1.78735i −0.411874 − 0.183378i
\(96\) 5.56439 + 0.703629i 0.567913 + 0.0718138i
\(97\) 7.25070 + 1.54118i 0.736197 + 0.156483i 0.560726 − 0.828001i \(-0.310522\pi\)
0.175470 + 0.984485i \(0.443855\pi\)
\(98\) 2.73516 0.276293
\(99\) 3.24790 + 9.40485i 0.326426 + 0.945223i
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 99.2.m.b.16.5 ✓ 72
3.2 odd 2 297.2.n.b.181.5 72
9.2 odd 6 891.2.f.e.82.5 36
9.4 even 3 inner 99.2.m.b.49.5 yes 72
9.5 odd 6 297.2.n.b.280.5 72
9.7 even 3 891.2.f.f.82.5 36
11.3 even 5 1089.2.e.p.727.9 36
11.8 odd 10 1089.2.e.o.727.10 36
11.9 even 5 inner 99.2.m.b.97.5 yes 72
33.20 odd 10 297.2.n.b.262.5 72
99.20 odd 30 891.2.f.e.163.5 36
99.25 even 15 9801.2.a.cm.1.10 18
99.31 even 15 inner 99.2.m.b.31.5 yes 72
99.47 odd 30 9801.2.a.cp.1.9 18
99.52 odd 30 9801.2.a.co.1.9 18
99.58 even 15 1089.2.e.p.364.9 36
99.74 even 30 9801.2.a.cn.1.10 18
99.85 odd 30 1089.2.e.o.364.10 36
99.86 odd 30 297.2.n.b.64.5 72
99.97 even 15 891.2.f.f.163.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.b.16.5 ✓ 72 1.1 even 1 trivial
99.2.m.b.31.5 yes 72 99.31 even 15 inner
99.2.m.b.49.5 yes 72 9.4 even 3 inner
99.2.m.b.97.5 yes 72 11.9 even 5 inner
297.2.n.b.64.5 72 99.86 odd 30
297.2.n.b.181.5 72 3.2 odd 2
297.2.n.b.262.5 72 33.20 odd 10
297.2.n.b.280.5 72 9.5 odd 6
891.2.f.e.82.5 36 9.2 odd 6
891.2.f.e.163.5 36 99.20 odd 30
891.2.f.f.82.5 36 9.7 even 3
891.2.f.f.163.5 36 99.97 even 15
1089.2.e.o.364.10 36 99.85 odd 30
1089.2.e.o.727.10 36 11.8 odd 10
1089.2.e.p.364.9 36 99.58 even 15
1089.2.e.p.727.9 36 11.3 even 5
9801.2.a.cm.1.10 18 99.25 even 15
9801.2.a.cn.1.10 18 99.74 even 30
9801.2.a.co.1.9 18 99.52 odd 30
9801.2.a.cp.1.9 18 99.47 odd 30