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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 141.3
Character \(\chi\) \(=\) 230.141
Dual form 230.4.g.d.31.3

$q$-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+(-0.284630 - 1.97964i) q^{2} +(-0.557048 + 1.21977i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(-3.27430 + 3.77875i) q^{5} +(2.57325 + 0.755575i) q^{6} +(4.15187 - 2.66824i) q^{7} +(3.32332 + 7.27706i) q^{8} +(16.5037 + 19.0463i) q^{9} +(8.41254 + 5.40641i) q^{10} +(4.99491 - 34.7404i) q^{11} +(0.763345 - 5.30918i) q^{12} +(-70.5795 - 45.3587i) q^{13} +(-6.46391 - 7.45975i) q^{14} +(-2.78524 - 6.09883i) q^{15} +(13.4601 - 8.65025i) q^{16} +(45.5828 + 13.3843i) q^{17} +(33.0074 - 38.0926i) q^{18} +(-129.770 + 38.1040i) q^{19} +(8.30830 - 18.1926i) q^{20} +(0.941840 + 6.55064i) q^{21} -70.1953 q^{22} +(-106.509 + 28.6867i) q^{23} -10.7276 q^{24} +(-3.55787 - 24.7455i) q^{25} +(-69.7050 + 152.633i) q^{26} +(-67.1643 + 19.7212i) q^{27} +(-12.9278 + 14.9195i) q^{28} +(-169.346 - 49.7246i) q^{29} +(-11.2807 + 7.24969i) q^{30} +(43.6430 + 95.5648i) q^{31} +(-20.9555 - 24.1840i) q^{32} +(39.5927 + 25.4447i) q^{33} +(13.5219 - 94.0472i) q^{34} +(-3.51185 + 24.4255i) q^{35} +(-84.8046 - 54.5006i) q^{36} +(-176.623 - 203.834i) q^{37} +(112.369 + 246.053i) q^{38} +(94.6431 - 60.8234i) q^{39} +(-38.3797 - 11.2693i) q^{40} +(224.019 - 258.532i) q^{41} +(12.6999 - 3.72901i) q^{42} +(-40.7423 + 89.2131i) q^{43} +(19.9797 + 138.962i) q^{44} -126.009 q^{45} +(87.1049 + 202.684i) q^{46} -319.132 q^{47} +(3.05338 + 21.2367i) q^{48} +(-132.369 + 289.847i) q^{49} +(-47.9746 + 14.0866i) q^{50} +(-41.7175 + 48.1446i) q^{51} +(321.998 + 94.5472i) q^{52} +(-402.978 + 258.978i) q^{53} +(58.1579 + 127.348i) q^{54} +(114.920 + 132.625i) q^{55} +(33.2149 + 21.3459i) q^{56} +(25.8104 - 179.515i) q^{57} +(-50.2359 + 349.398i) q^{58} +(-172.458 - 110.832i) q^{59} +(17.5626 + 20.2684i) q^{60} +(109.590 + 239.969i) q^{61} +(176.762 - 113.598i) q^{62} +(119.341 + 35.0418i) q^{63} +(-41.9111 + 48.3680i) q^{64} +(402.498 - 118.184i) q^{65} +(39.1021 - 85.6217i) q^{66} +(-69.0712 - 480.401i) q^{67} -190.029 q^{68} +(24.3394 - 145.895i) q^{69} +49.3533 q^{70} +(-84.2374 - 585.884i) q^{71} +(-83.7539 + 183.395i) q^{72} +(823.022 - 241.661i) q^{73} +(-353.246 + 407.668i) q^{74} +(32.1657 + 9.44469i) q^{75} +(455.114 - 292.484i) q^{76} +(-71.9575 - 157.565i) q^{77} +(-147.347 - 170.047i) q^{78} +(187.274 + 120.354i) q^{79} +(-11.3852 + 79.1857i) q^{80} +(-83.4797 + 580.614i) q^{81} +(-575.564 - 369.893i) q^{82} +(-567.755 - 655.225i) q^{83} +(-10.9969 - 24.0798i) q^{84} +(-199.828 + 128.421i) q^{85} +(188.206 + 55.2624i) q^{86} +(154.986 - 178.864i) q^{87} +(269.407 - 79.1052i) q^{88} +(-349.051 + 764.315i) q^{89} +(35.8660 + 249.454i) q^{90} -414.064 q^{91} +(376.449 - 230.126i) q^{92} -140.878 q^{93} +(90.8345 + 631.768i) q^{94} +(280.922 - 615.133i) q^{95} +(41.1720 - 12.0892i) q^{96} +(1037.40 - 1197.22i) q^{97} +(611.470 + 179.544i) q^{98} +(744.110 - 478.211i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.284630 − 1.97964i −0.100632 − 0.699909i
\(3\) −0.557048 + 1.21977i −0.107204 + 0.234744i −0.955630 − 0.294570i \(-0.904823\pi\)
0.848426 + 0.529314i \(0.177551\pi\)
\(4\) −3.83797 + 1.12693i −0.479746 + 0.140866i
\(5\) −3.27430 + 3.77875i −0.292863 + 0.337981i
\(6\) 2.57325 + 0.755575i 0.175088 + 0.0514104i
\(7\) 4.15187 − 2.66824i 0.224180 − 0.144072i −0.423730 − 0.905788i \(-0.639280\pi\)
0.647910 + 0.761717i \(0.275643\pi\)
\(8\) 3.32332 + 7.27706i 0.146871 + 0.321603i
\(9\) 16.5037 + 19.0463i 0.611249 + 0.705419i
\(10\) 8.41254 + 5.40641i 0.266028 + 0.170966i
\(11\) 4.99491 − 34.7404i 0.136911 − 0.952238i −0.799333 − 0.600888i \(-0.794814\pi\)
0.936244 − 0.351350i \(-0.114277\pi\)
\(12\) 0.763345 − 5.30918i 0.0183632 − 0.127719i
\(13\) −70.5795 − 45.3587i −1.50579 − 0.967710i −0.994092 − 0.108543i \(-0.965381\pi\)
−0.511695 − 0.859167i \(-0.670982\pi\)
\(14\) −6.46391 − 7.45975i −0.123397 − 0.142407i
\(15\) −2.78524 − 6.09883i −0.0479431 − 0.104981i
\(16\) 13.4601 − 8.65025i 0.210313 − 0.135160i
\(17\) 45.5828 + 13.3843i 0.650320 + 0.190951i 0.590222 − 0.807241i \(-0.299040\pi\)
0.0600986 + 0.998192i \(0.480859\pi\)
\(18\) 33.0074 − 38.0926i 0.432218 − 0.498806i
\(19\) −129.770 + 38.1040i −1.56691 + 0.460087i −0.946101 − 0.323872i \(-0.895015\pi\)
−0.620812 + 0.783959i \(0.713197\pi\)
\(20\) 8.30830 − 18.1926i 0.0928896 − 0.203400i
\(21\) 0.941840 + 6.55064i 0.00978697 + 0.0680699i
\(22\) −70.1953 −0.680258
\(23\) −106.509 + 28.6867i −0.965590 + 0.260069i
\(24\) −10.7276 −0.0912397
\(25\) −3.55787 − 24.7455i −0.0284630 − 0.197964i
\(26\) −69.7050 + 152.633i −0.525780 + 1.15130i
\(27\) −67.1643 + 19.7212i −0.478732 + 0.140569i
\(28\) −12.9278 + 14.9195i −0.0872546 + 0.100697i
\(29\) −169.346 − 49.7246i −1.08437 − 0.318401i −0.309746 − 0.950819i \(-0.600244\pi\)
−0.774627 + 0.632419i \(0.782062\pi\)
\(30\) −11.2807 + 7.24969i −0.0686524 + 0.0441202i
\(31\) 43.6430 + 95.5648i 0.252855 + 0.553675i 0.992910 − 0.118871i \(-0.0379276\pi\)
−0.740055 + 0.672547i \(0.765200\pi\)
\(32\) −20.9555 − 24.1840i −0.115764 − 0.133599i
\(33\) 39.5927 + 25.4447i 0.208855 + 0.134223i
\(34\) 13.5219 − 94.0472i 0.0682057 − 0.474381i
\(35\) −3.51185 + 24.4255i −0.0169603 + 0.117962i
\(36\) −84.8046 − 54.5006i −0.392614 − 0.252318i
\(37\) −176.623 − 203.834i −0.784775 − 0.905679i 0.212669 − 0.977124i \(-0.431784\pi\)
−0.997445 + 0.0714455i \(0.977239\pi\)
\(38\) 112.369 + 246.053i 0.479701 + 1.05040i
\(39\) 94.6431 − 60.8234i 0.388590 − 0.249732i
\(40\) −38.3797 − 11.2693i −0.151709 − 0.0445458i
\(41\) 224.019 − 258.532i 0.853316 − 0.984779i −0.146674 − 0.989185i \(-0.546857\pi\)
0.999990 + 0.00440547i \(0.00140231\pi\)
\(42\) 12.6999 − 3.72901i 0.0466579 − 0.0137000i
\(43\) −40.7423 + 89.2131i −0.144492 + 0.316392i −0.968016 − 0.250888i \(-0.919277\pi\)
0.823525 + 0.567281i \(0.192005\pi\)
\(44\) 19.9797 + 138.962i 0.0684556 + 0.476119i
\(45\) −126.009 −0.417430
\(46\) 87.1049 + 202.684i 0.279194 + 0.649654i
\(47\) −319.132 −0.990431 −0.495215 − 0.868770i \(-0.664911\pi\)
−0.495215 + 0.868770i \(0.664911\pi\)
\(48\) 3.05338 + 21.2367i 0.00918161 + 0.0638595i
\(49\) −132.369 + 289.847i −0.385915 + 0.845036i
\(50\) −47.9746 + 14.0866i −0.135693 + 0.0398430i
\(51\) −41.7175 + 48.1446i −0.114542 + 0.132188i
\(52\) 321.998 + 94.5472i 0.858714 + 0.252141i
\(53\) −402.978 + 258.978i −1.04440 + 0.671197i −0.946072 − 0.323957i \(-0.894987\pi\)
−0.0983306 + 0.995154i \(0.531350\pi\)
\(54\) 58.1579 + 127.348i 0.146561 + 0.320924i
\(55\) 114.920 + 132.625i 0.281743 + 0.325148i
\(56\) 33.2149 + 21.3459i 0.0792595 + 0.0509370i
\(57\) 25.8104 − 179.515i 0.0599766 − 0.417147i
\(58\) −50.2359 + 349.398i −0.113729 + 0.791004i
\(59\) −172.458 − 110.832i −0.380544 − 0.244561i 0.336359 − 0.941734i \(-0.390804\pi\)
−0.716903 + 0.697173i \(0.754441\pi\)
\(60\) 17.5626 + 20.2684i 0.0377888 + 0.0436106i
\(61\) 109.590 + 239.969i 0.230026 + 0.503687i 0.989087 − 0.147332i \(-0.0470687\pi\)
−0.759061 + 0.651020i \(0.774341\pi\)
\(62\) 176.762 − 113.598i 0.362077 − 0.232693i
\(63\) 119.341 + 35.0418i 0.238660 + 0.0700770i
\(64\) −41.9111 + 48.3680i −0.0818576 + 0.0944687i
\(65\) 402.498 − 118.184i 0.768057 − 0.225522i
\(66\) 39.1021 − 85.6217i 0.0729264 − 0.159686i
\(67\) −69.0712 − 480.401i −0.125946 − 0.875975i −0.950618 − 0.310363i \(-0.899549\pi\)
0.824672 − 0.565612i \(-0.191360\pi\)
\(68\) −190.029 −0.338887
\(69\) 24.3394 − 145.895i 0.0424655 − 0.254547i
\(70\) 49.3533 0.0842693
\(71\) −84.2374 − 585.884i −0.140805 − 0.979319i −0.930623 − 0.365978i \(-0.880734\pi\)
0.789819 − 0.613341i \(-0.210175\pi\)
\(72\) −83.7539 + 183.395i −0.137090 + 0.300186i
\(73\) 823.022 − 241.661i 1.31955 − 0.387456i 0.455222 − 0.890378i \(-0.349560\pi\)
0.864332 + 0.502922i \(0.167742\pi\)
\(74\) −353.246 + 407.668i −0.554920 + 0.640412i
\(75\) 32.1657 + 9.44469i 0.0495223 + 0.0145410i
\(76\) 455.114 − 292.484i 0.686910 − 0.441450i
\(77\) −71.9575 − 157.565i −0.106498 − 0.233197i
\(78\) −147.347 − 170.047i −0.213894 − 0.246847i
\(79\) 187.274 + 120.354i 0.266709 + 0.171404i 0.667159 − 0.744915i \(-0.267510\pi\)
−0.400450 + 0.916318i \(0.631146\pi\)
\(80\) −11.3852 + 79.1857i −0.0159113 + 0.110665i
\(81\) −83.4797 + 580.614i −0.114513 + 0.796453i
\(82\) −575.564 − 369.893i −0.775127 − 0.498144i
\(83\) −567.755 − 655.225i −0.750834 − 0.866509i 0.243815 − 0.969822i \(-0.421601\pi\)
−0.994649 + 0.103313i \(0.967056\pi\)
\(84\) −10.9969 − 24.0798i −0.0142840 − 0.0312776i
\(85\) −199.828 + 128.421i −0.254992 + 0.163874i
\(86\) 188.206 + 55.2624i 0.235986 + 0.0692919i
\(87\) 154.986 − 178.864i 0.190992 − 0.220416i
\(88\) 269.407 − 79.1052i 0.326351 − 0.0958254i
\(89\) −349.051 + 764.315i −0.415723 + 0.910306i 0.579708 + 0.814824i \(0.303167\pi\)
−0.995431 + 0.0954820i \(0.969561\pi\)
\(90\) 35.8660 + 249.454i 0.0420068 + 0.292163i
\(91\) −414.064 −0.476986
\(92\) 376.449 − 230.126i 0.426604 − 0.260786i
\(93\) −140.878 −0.157079
\(94\) 90.8345 + 631.768i 0.0996688 + 0.693212i
\(95\) 280.922 − 615.133i 0.303389 − 0.664330i
\(96\) 41.1720 − 12.0892i 0.0437719 − 0.0128526i
\(97\) 1037.40 − 1197.22i 1.08590 − 1.25319i 0.120413 − 0.992724i \(-0.461578\pi\)
0.965483 − 0.260467i \(-0.0838765\pi\)
\(98\) 611.470 + 179.544i 0.630284 + 0.185068i
\(99\) 744.110 − 478.211i 0.755413 − 0.485475i
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 230.4.g.d.141.3 yes 70
23.8 even 11 inner 230.4.g.d.31.3 ✓ 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.3 ✓ 70 23.8 even 11 inner
230.4.g.d.141.3 yes 70 1.1 even 1 trivial