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Properties

Label 230.4.g.d.31.3
Level $230$
Weight $4$
Character 230.31
Analytic conductor $13.570$
Analytic rank $0$
Dimension $70$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 31.3
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.3

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{11}\right)\)

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 0.848426 − 0.529314i \(-0.177551\pi\)
−0.955630 + 0.294570i \(0.904823\pi\)
\(4\) −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 0.647910 − 0.761717i \(-0.275643\pi\)
−0.423730 + 0.905788i \(0.639280\pi\)
\(8\) 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 0.936244 + 0.351350i \(0.114277\pi\)
−0.799333 + 0.600888i \(0.794814\pi\)
\(12\) 0.763345 + 5.30918i 0.0183632 + 0.127719i
\(13\) −70.5795 + 45.3587i −1.50579 + 0.967710i −0.511695 + 0.859167i \(0.670982\pi\)
−0.994092 + 0.108543i \(0.965381\pi\)
\(14\) −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 0.0600986 − 0.998192i \(-0.480859\pi\)
0.590222 + 0.807241i \(0.299040\pi\)
\(18\) 33.0074 + 38.0926i 0.432218 + 0.498806i
\(19\) −129.770 − 38.1040i −1.56691 − 0.460087i −0.620812 − 0.783959i \(-0.713197\pi\)
−0.946101 + 0.323872i \(0.895015\pi\)
\(20\) 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 −0.774627 − 0.632419i \(-0.782062\pi\)
−0.309746 + 0.950819i \(0.600244\pi\)
\(30\) −11.2807 − 7.24969i −0.0686524 − 0.0441202i
\(31\) 43.6430 − 95.5648i 0.252855 − 0.553675i −0.740055 − 0.672547i \(-0.765200\pi\)
0.992910 + 0.118871i \(0.0379276\pi\)
\(32\) −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 −0.997445 − 0.0714455i \(-0.977239\pi\)
0.212669 + 0.977124i \(0.431784\pi\)
\(38\) 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 0.999990 − 0.00440547i \(-0.00140231\pi\)
−0.146674 + 0.989185i \(0.546857\pi\)
\(42\) 12.6999 + 3.72901i 0.0466579 + 0.0137000i
\(43\) −40.7423 − 89.2131i −0.144492 − 0.316392i 0.823525 − 0.567281i \(-0.192005\pi\)
−0.968016 + 0.250888i \(0.919277\pi\)
\(44\) 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 −0.0983306 − 0.995154i \(-0.531350\pi\)
−0.946072 + 0.323957i \(0.894987\pi\)
\(54\) 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 −0.716903 − 0.697173i \(-0.754441\pi\)
0.336359 + 0.941734i \(0.390804\pi\)
\(60\) 17.5626 − 20.2684i 0.0377888 − 0.0436106i
\(61\) 109.590 − 239.969i 0.230026 − 0.503687i −0.759061 − 0.651020i \(-0.774341\pi\)
0.989087 + 0.147332i \(0.0470687\pi\)
\(62\) 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 0.824672 + 0.565612i \(0.191360\pi\)
−0.950618 + 0.310363i \(0.899549\pi\)
\(68\) −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 0.789819 + 0.613341i \(0.210175\pi\)
−0.930623 + 0.365978i \(0.880734\pi\)
\(72\) −83.7539 − 183.395i −0.137090 − 0.300186i
\(73\) 823.022 + 241.661i 1.31955 + 0.387456i 0.864332 − 0.502922i \(-0.167742\pi\)
0.455222 + 0.890378i \(0.349560\pi\)
\(74\) −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 −0.400450 − 0.916318i \(-0.631146\pi\)
0.667159 + 0.744915i \(0.267510\pi\)
\(80\) −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 −0.994649 − 0.103313i \(-0.967056\pi\)
0.243815 + 0.969822i \(0.421601\pi\)
\(84\) −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 −0.995431 − 0.0954820i \(-0.969561\pi\)
0.579708 − 0.814824i \(-0.303167\pi\)
\(90\) 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 0.965483 + 0.260467i \(0.0838765\pi\)
0.120413 + 0.992724i \(0.461578\pi\)
\(98\) 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.31.3 ✓ 70
23.3 even 11 inner 230.4.g.d.141.3 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.3 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.3 yes 70 23.3 even 11 inner