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Properties

Label 230.4.g.d.31.2
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 31.2
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.2

$q$-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} +(-2.58225 - 5.65433i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(11.9285 - 3.50254i) q^{6} +(8.08665 + 5.19698i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-7.62224 + 8.79653i) q^{9} +(8.41254 - 5.40641i) q^{10} +(-2.21658 - 15.4167i) q^{11} +(3.53855 + 24.6112i) q^{12} +(63.5622 - 40.8489i) q^{13} +(-12.5899 + 14.5295i) q^{14} +(-12.9112 + 28.2717i) q^{15} +(13.4601 + 8.65025i) q^{16} +(-112.732 + 33.1011i) q^{17} +(-15.2445 - 17.5931i) q^{18} +(-27.1474 - 7.97119i) q^{19} +(8.30830 + 18.1926i) q^{20} +(8.50370 - 59.1445i) q^{21} +31.1504 q^{22} +(-106.206 - 29.7863i) q^{23} -49.7285 q^{24} +(-3.55787 + 24.7455i) q^{25} +(62.7746 + 137.457i) q^{26} +(-91.6143 - 26.9004i) q^{27} +(-25.1797 - 29.0589i) q^{28} +(-189.907 + 55.7617i) q^{29} +(-52.2929 - 33.6066i) q^{30} +(-74.3622 + 162.831i) q^{31} +(-20.9555 + 24.1840i) q^{32} +(-81.4472 + 52.3429i) q^{33} +(-33.4415 - 232.591i) q^{34} +(-6.84009 - 47.5739i) q^{35} +(39.1670 - 25.1711i) q^{36} +(-130.410 + 150.501i) q^{37} +(23.5071 - 51.4733i) q^{38} +(-395.107 - 253.920i) q^{39} +(-38.3797 + 11.2693i) q^{40} +(-216.272 - 249.592i) q^{41} +(114.665 + 33.6686i) q^{42} +(22.9567 + 50.2682i) q^{43} +(-8.86632 + 61.6666i) q^{44} +58.1974 q^{45} +(89.1957 - 201.772i) q^{46} +472.897 q^{47} +(14.1542 - 98.4447i) q^{48} +(-104.102 - 227.951i) q^{49} +(-47.9746 - 14.0866i) q^{50} +(478.266 + 551.949i) q^{51} +(-289.984 + 85.1469i) q^{52} +(340.249 + 218.665i) q^{53} +(79.3293 - 173.707i) q^{54} +(-50.9979 + 58.8547i) q^{55} +(64.6932 - 41.5758i) q^{56} +(25.0295 + 174.084i) q^{57} +(-56.3351 - 391.819i) q^{58} +(-126.740 + 81.4509i) q^{59} +(81.4132 - 93.9558i) q^{60} +(330.451 - 723.586i) q^{61} +(-301.181 - 193.557i) q^{62} +(-107.354 + 31.5219i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-362.480 - 106.434i) q^{65} +(-80.4380 - 176.135i) q^{66} +(-80.8129 + 562.066i) q^{67} +469.965 q^{68} +(105.829 + 677.441i) q^{69} +96.1262 q^{70} +(154.703 - 1075.98i) q^{71} +(38.6817 + 84.7012i) q^{72} +(-503.623 - 147.877i) q^{73} +(-260.820 - 301.002i) q^{74} +(149.107 - 43.7817i) q^{75} +(95.2079 + 61.1864i) q^{76} +(62.1953 - 136.189i) q^{77} +(615.130 - 709.897i) q^{78} +(-675.283 + 433.978i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(129.192 + 898.549i) q^{81} +(555.660 - 357.101i) q^{82} +(-575.969 + 664.704i) q^{83} +(-99.2887 + 217.412i) q^{84} +(494.199 + 317.603i) q^{85} +(-106.047 + 31.1383i) q^{86} +(805.681 + 929.806i) q^{87} +(-119.554 - 35.1043i) q^{88} +(-333.474 - 730.206i) q^{89} +(-16.5647 + 115.210i) q^{90} +726.296 q^{91} +(374.050 + 234.006i) q^{92} +1112.72 q^{93} +(-134.600 + 936.167i) q^{94} +(58.7677 + 128.683i) q^{95} +(190.857 + 56.0406i) q^{96} +(-871.493 - 1005.76i) q^{97} +(480.893 - 141.203i) q^{98} +(152.508 + 98.0113i) 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
\(3\) −2.58225 − 5.65433i −0.496954 − 1.08818i −0.977447 − 0.211179i \(-0.932270\pi\)
0.480494 − 0.876998i \(-0.340457\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) 11.9285 − 3.50254i 0.811635 − 0.238317i
\(7\) 8.08665 + 5.19698i 0.436638 + 0.280610i 0.740437 − 0.672126i \(-0.234619\pi\)
−0.303799 + 0.952736i \(0.598255\pi\)
\(8\) 3.32332 − 7.27706i 0.146871 − 0.321603i
\(9\) −7.62224 + 8.79653i −0.282305 + 0.325798i
\(10\) 8.41254 − 5.40641i 0.266028 − 0.170966i
\(11\) −2.21658 − 15.4167i −0.0607568 − 0.422573i −0.997386 − 0.0722515i \(-0.976982\pi\)
0.936630 − 0.350321i \(-0.113928\pi\)
\(12\) 3.53855 + 24.6112i 0.0851244 + 0.592053i
\(13\) 63.5622 − 40.8489i 1.35608 − 0.871497i 0.358013 − 0.933717i \(-0.383454\pi\)
0.998062 + 0.0622197i \(0.0198179\pi\)
\(14\) −12.5899 + 14.5295i −0.240342 + 0.277369i
\(15\) −12.9112 + 28.2717i −0.222244 + 0.486648i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) −112.732 + 33.1011i −1.60832 + 0.472247i −0.957847 − 0.287279i \(-0.907249\pi\)
−0.650477 + 0.759526i \(0.725431\pi\)
\(18\) −15.2445 − 17.5931i −0.199620 − 0.230374i
\(19\) −27.1474 − 7.97119i −0.327791 − 0.0962483i 0.113696 − 0.993516i \(-0.463731\pi\)
−0.441488 + 0.897267i \(0.645549\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) 8.50370 − 59.1445i 0.0883647 − 0.614590i
\(22\) 31.1504 0.301877
\(23\) −106.206 − 29.7863i −0.962850 − 0.270038i
\(24\) −49.7285 −0.422950
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) 62.7746 + 137.457i 0.473505 + 1.03683i
\(27\) −91.6143 − 26.9004i −0.653007 − 0.191740i
\(28\) −25.1797 − 29.0589i −0.169947 − 0.196129i
\(29\) −189.907 + 55.7617i −1.21603 + 0.357058i −0.825960 − 0.563728i \(-0.809367\pi\)
−0.390067 + 0.920786i \(0.627548\pi\)
\(30\) −52.2929 − 33.6066i −0.318244 − 0.204523i
\(31\) −74.3622 + 162.831i −0.430834 + 0.943394i 0.562357 + 0.826895i \(0.309895\pi\)
−0.993191 + 0.116500i \(0.962833\pi\)
\(32\) −20.9555 + 24.1840i −0.115764 + 0.133599i
\(33\) −81.4472 + 52.3429i −0.429640 + 0.276113i
\(34\) −33.4415 − 232.591i −0.168681 − 1.17320i
\(35\) −6.84009 − 47.5739i −0.0330339 − 0.229756i
\(36\) 39.1670 − 25.1711i 0.181329 − 0.116533i
\(37\) −130.410 + 150.501i −0.579439 + 0.668708i −0.967484 − 0.252932i \(-0.918605\pi\)
0.388045 + 0.921640i \(0.373150\pi\)
\(38\) 23.5071 − 51.4733i 0.100351 − 0.219739i
\(39\) −395.107 − 253.920i −1.62225 − 1.04256i
\(40\) −38.3797 + 11.2693i −0.151709 + 0.0445458i
\(41\) −216.272 − 249.592i −0.823807 − 0.950724i 0.175624 − 0.984457i \(-0.443806\pi\)
−0.999431 + 0.0337336i \(0.989260\pi\)
\(42\) 114.665 + 33.6686i 0.421265 + 0.123695i
\(43\) 22.9567 + 50.2682i 0.0814154 + 0.178275i 0.945962 − 0.324278i \(-0.105121\pi\)
−0.864546 + 0.502553i \(0.832394\pi\)
\(44\) −8.86632 + 61.6666i −0.0303784 + 0.211286i
\(45\) 58.1974 0.192790
\(46\) 89.1957 − 201.772i 0.285895 − 0.646733i
\(47\) 472.897 1.46764 0.733820 − 0.679344i \(-0.237736\pi\)
0.733820 + 0.679344i \(0.237736\pi\)
\(48\) 14.1542 − 98.4447i 0.0425622 − 0.296027i
\(49\) −104.102 − 227.951i −0.303504 − 0.664582i
\(50\) −47.9746 − 14.0866i −0.135693 − 0.0398430i
\(51\) 478.266 + 551.949i 1.31315 + 1.51546i
\(52\) −289.984 + 85.1469i −0.773337 + 0.227072i
\(53\) 340.249 + 218.665i 0.881826 + 0.566715i 0.901349 − 0.433094i \(-0.142578\pi\)
−0.0195228 + 0.999809i \(0.506215\pi\)
\(54\) 79.3293 − 173.707i 0.199914 − 0.437750i
\(55\) −50.9979 + 58.8547i −0.125028 + 0.144290i
\(56\) 64.6932 − 41.5758i 0.154375 − 0.0992107i
\(57\) 25.0295 + 174.084i 0.0581621 + 0.404526i
\(58\) −56.3351 − 391.819i −0.127537 − 0.887041i
\(59\) −126.740 + 81.4509i −0.279664 + 0.179729i −0.672952 − 0.739686i \(-0.734974\pi\)
0.393288 + 0.919415i \(0.371337\pi\)
\(60\) 81.4132 − 93.9558i 0.175173 − 0.202161i
\(61\) 330.451 − 723.586i 0.693605 − 1.51878i −0.153953 − 0.988078i \(-0.549200\pi\)
0.847557 − 0.530704i \(-0.178072\pi\)
\(62\) −301.181 − 193.557i −0.616935 − 0.396480i
\(63\) −107.354 + 31.5219i −0.214687 + 0.0630379i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) −362.480 − 106.434i −0.691694 − 0.203100i
\(66\) −80.4380 − 176.135i −0.150019 − 0.328495i
\(67\) −80.8129 + 562.066i −0.147356 + 1.02488i 0.773169 + 0.634200i \(0.218671\pi\)
−0.920525 + 0.390684i \(0.872238\pi\)
\(68\) 469.965 0.838112
\(69\) 105.829 + 677.441i 0.184643 + 1.18195i
\(70\) 96.1262 0.164133
\(71\) 154.703 − 1075.98i 0.258590 − 1.79853i −0.284306 − 0.958733i \(-0.591763\pi\)
0.542896 − 0.839800i \(-0.317328\pi\)
\(72\) 38.6817 + 84.7012i 0.0633151 + 0.138641i
\(73\) −503.623 − 147.877i −0.807461 − 0.237092i −0.148152 − 0.988965i \(-0.547333\pi\)
−0.659308 + 0.751873i \(0.729151\pi\)
\(74\) −260.820 − 301.002i −0.409725 − 0.472848i
\(75\) 149.107 − 43.7817i 0.229565 − 0.0674063i
\(76\) 95.2079 + 61.1864i 0.143699 + 0.0923495i
\(77\) 62.1953 − 136.189i 0.0920495 − 0.201560i
\(78\) 615.130 − 709.897i 0.892945 − 1.03051i
\(79\) −675.283 + 433.978i −0.961713 + 0.618055i −0.924472 − 0.381251i \(-0.875493\pi\)
−0.0372413 + 0.999306i \(0.511857\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) 129.192 + 898.549i 0.177218 + 1.23258i
\(82\) 555.660 − 357.101i 0.748322 − 0.480917i
\(83\) −575.969 + 664.704i −0.761697 + 0.879045i −0.995647 − 0.0932035i \(-0.970289\pi\)
0.233950 + 0.972249i \(0.424835\pi\)
\(84\) −99.2887 + 217.412i −0.128968 + 0.282400i
\(85\) 494.199 + 317.603i 0.630629 + 0.405280i
\(86\) −106.047 + 31.1383i −0.132969 + 0.0390433i
\(87\) 805.681 + 929.806i 0.992852 + 1.14581i
\(88\) −119.554 − 35.1043i −0.144824 − 0.0425242i
\(89\) −333.474 − 730.206i −0.397171 − 0.869682i −0.997549 − 0.0699664i \(-0.977711\pi\)
0.600379 − 0.799716i \(-0.295016\pi\)
\(90\) −16.5647 + 115.210i −0.0194008 + 0.134936i
\(91\) 726.296 0.836665
\(92\) 374.050 + 234.006i 0.423885 + 0.265183i
\(93\) 1112.72 1.24068
\(94\) −134.600 + 936.167i −0.147691 + 1.02722i
\(95\) 58.7677 + 128.683i 0.0634677 + 0.138975i
\(96\) 190.857 + 56.0406i 0.202909 + 0.0595794i
\(97\) −871.493 − 1005.76i −0.912234 − 1.05277i −0.998403 − 0.0564933i \(-0.982008\pi\)
0.0861693 − 0.996281i \(-0.472537\pi\)
\(98\) 480.893 − 141.203i 0.495689 − 0.145547i
\(99\) 152.508 + 98.0113i 0.154825 + 0.0995000i
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.31.2 ✓ 70
23.3 even 11 inner 230.4.g.d.141.2 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.2 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.2 yes 70 23.3 even 11 inner