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Properties

Label 230.4.g.d.41.2
Level $230$
Weight $4$
Character 230.41
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 41.2
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.2

$q$-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+(-1.91899 - 0.563465i) q^{2} +(-5.54798 + 6.40271i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(14.2542 - 9.16062i) q^{6} +(-0.812937 - 1.78008i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(-6.37211 - 44.3190i) q^{9} +(4.15415 - 9.09632i) q^{10} +(-53.9271 + 15.8344i) q^{11} +(-32.5153 + 9.54735i) q^{12} +(-32.4740 + 71.1081i) q^{13} +(0.557000 + 3.87402i) q^{14} +(-27.7399 - 32.0135i) q^{15} +(6.64664 + 14.5541i) q^{16} +(24.1771 - 15.5377i) q^{17} +(-12.7442 + 88.6380i) q^{18} +(-71.0990 - 45.6925i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(15.9075 + 4.67087i) q^{21} +112.407 q^{22} +(107.941 + 22.7119i) q^{23} +67.7760 q^{24} +(-23.9873 - 7.04331i) q^{25} +(102.384 - 118.157i) q^{26} +(126.682 + 81.4138i) q^{27} +(1.11400 - 7.74804i) q^{28} +(144.210 - 92.6779i) q^{29} +(35.1940 + 77.0640i) q^{30} +(-118.830 - 137.137i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(197.803 - 433.128i) q^{33} +(-55.1505 + 16.1937i) q^{34} +(9.38829 - 2.75665i) q^{35} +(74.4004 - 162.914i) q^{36} +(9.72036 + 67.6066i) q^{37} +(110.692 + 127.745i) q^{38} +(-275.119 - 602.427i) q^{39} +(33.6501 - 21.6256i) q^{40} +(-41.5419 + 288.930i) q^{41} +(-27.8944 - 17.9267i) q^{42} +(53.1787 - 61.3715i) q^{43} +(-215.708 - 63.3377i) q^{44} +223.874 q^{45} +(-194.339 - 104.405i) q^{46} +349.955 q^{47} +(-130.061 - 38.1894i) q^{48} +(222.109 - 256.328i) q^{49} +(42.0627 + 27.0320i) q^{50} +(-34.6509 + 241.002i) q^{51} +(-263.051 + 169.053i) q^{52} +(97.1236 + 212.671i) q^{53} +(-197.228 - 227.613i) q^{54} +(-39.9931 - 278.158i) q^{55} +(-6.50350 + 14.2407i) q^{56} +(687.011 - 201.725i) q^{57} +(-328.957 + 96.5905i) q^{58} +(315.707 - 691.302i) q^{59} +(-24.1138 - 167.715i) q^{60} +(177.791 + 205.182i) q^{61} +(150.761 + 330.120i) q^{62} +(-73.7115 + 47.3715i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(-328.814 - 211.316i) q^{65} +(-623.634 + 719.712i) q^{66} +(-274.784 - 80.6840i) q^{67} +114.958 q^{68} +(-744.270 + 565.107i) q^{69} -19.5693 q^{70} +(-932.870 - 273.915i) q^{71} +(-234.570 + 270.708i) q^{72} +(-13.2791 - 8.53393i) q^{73} +(19.4407 - 135.213i) q^{74} +(178.177 - 114.508i) q^{75} +(-140.436 - 307.512i) q^{76} +(72.0259 + 83.1223i) q^{77} +(188.503 + 1311.07i) q^{78} +(17.6754 - 38.7038i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(-64.1519 + 18.8367i) q^{81} +(242.521 - 531.046i) q^{82} +(26.4515 + 183.974i) q^{83} +(43.4280 + 50.1186i) q^{84} +(59.6939 + 130.711i) q^{85} +(-136.630 + 87.8067i) q^{86} +(-206.682 + 1437.51i) q^{87} +(378.253 + 243.088i) q^{88} +(629.222 - 726.160i) q^{89} +(-429.611 - 126.145i) q^{90} +152.978 q^{91} +(314.106 + 309.854i) q^{92} +1537.31 q^{93} +(-671.558 - 197.187i) q^{94} +(276.729 - 319.363i) q^{95} +(228.067 + 146.570i) q^{96} +(144.812 - 1007.19i) q^{97} +(-570.657 + 366.739i) q^{98} +(1045.39 + 2289.10i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91899 − 0.563465i −0.678464 − 0.199215i
\(3\) −5.54798 + 6.40271i −1.06771 + 1.23220i −0.0961594 + 0.995366i \(0.530656\pi\)
−0.971550 + 0.236836i \(0.923890\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) 14.2542 − 9.16062i 0.969875 − 0.623301i
\(7\) −0.812937 − 1.78008i −0.0438945 − 0.0961155i 0.886414 − 0.462893i \(-0.153188\pi\)
−0.930309 + 0.366777i \(0.880461\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) −6.37211 − 44.3190i −0.236004 − 1.64144i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) −53.9271 + 15.8344i −1.47815 + 0.434023i −0.918738 − 0.394868i \(-0.870790\pi\)
−0.559410 + 0.828891i \(0.688972\pi\)
\(12\) −32.5153 + 9.54735i −0.782196 + 0.229674i
\(13\) −32.4740 + 71.1081i −0.692820 + 1.51706i 0.155646 + 0.987813i \(0.450254\pi\)
−0.848465 + 0.529251i \(0.822473\pi\)
\(14\) 0.557000 + 3.87402i 0.0106332 + 0.0739554i
\(15\) −27.7399 − 32.0135i −0.477494 − 0.551057i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) 24.1771 − 15.5377i 0.344930 − 0.221673i −0.356692 − 0.934222i \(-0.616096\pi\)
0.701623 + 0.712549i \(0.252459\pi\)
\(18\) −12.7442 + 88.6380i −0.166880 + 1.16068i
\(19\) −71.0990 − 45.6925i −0.858485 − 0.551715i 0.0357253 − 0.999362i \(-0.488626\pi\)
−0.894211 + 0.447647i \(0.852262\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) 15.9075 + 4.67087i 0.165300 + 0.0485365i
\(22\) 112.407 1.08933
\(23\) 107.941 + 22.7119i 0.978573 + 0.205902i
\(24\) 67.7760 0.576446
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) 102.384 − 118.157i 0.772275 − 0.891253i
\(27\) 126.682 + 81.4138i 0.902965 + 0.580300i
\(28\) 1.11400 − 7.74804i 0.00751879 − 0.0522943i
\(29\) 144.210 − 92.6779i 0.923415 − 0.593443i 0.00976875 − 0.999952i \(-0.496890\pi\)
0.913647 + 0.406509i \(0.133254\pi\)
\(30\) 35.1940 + 77.0640i 0.214184 + 0.468997i
\(31\) −118.830 − 137.137i −0.688465 − 0.794531i 0.298680 − 0.954353i \(-0.403454\pi\)
−0.987146 + 0.159822i \(0.948908\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) 197.803 − 433.128i 1.04343 − 2.28479i
\(34\) −55.1505 + 16.1937i −0.278183 + 0.0816820i
\(35\) 9.38829 − 2.75665i 0.0453403 − 0.0133131i
\(36\) 74.4004 − 162.914i 0.344446 − 0.754232i
\(37\) 9.72036 + 67.6066i 0.0431896 + 0.300391i 0.999955 + 0.00952838i \(0.00303302\pi\)
−0.956765 + 0.290862i \(0.906058\pi\)
\(38\) 110.692 + 127.745i 0.472541 + 0.545342i
\(39\) −275.119 − 602.427i −1.12960 − 2.47348i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) −41.5419 + 288.930i −0.158238 + 1.10057i 0.743641 + 0.668580i \(0.233097\pi\)
−0.901879 + 0.431990i \(0.857812\pi\)
\(42\) −27.8944 − 17.9267i −0.102481 − 0.0658606i
\(43\) 53.1787 − 61.3715i 0.188597 − 0.217653i −0.653575 − 0.756862i \(-0.726731\pi\)
0.842172 + 0.539209i \(0.181277\pi\)
\(44\) −215.708 − 63.3377i −0.739074 − 0.217012i
\(45\) 223.874 0.741625
\(46\) −194.339 − 104.405i −0.622907 − 0.334644i
\(47\) 349.955 1.08609 0.543044 − 0.839704i \(-0.317272\pi\)
0.543044 + 0.839704i \(0.317272\pi\)
\(48\) −130.061 − 38.1894i −0.391098 − 0.114837i
\(49\) 222.109 − 256.328i 0.647549 − 0.747312i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) −34.6509 + 241.002i −0.0951390 + 0.661706i
\(52\) −263.051 + 169.053i −0.701512 + 0.450834i
\(53\) 97.1236 + 212.671i 0.251716 + 0.551181i 0.992738 − 0.120300i \(-0.0383856\pi\)
−0.741022 + 0.671481i \(0.765658\pi\)
\(54\) −197.228 − 227.613i −0.497024 − 0.573597i
\(55\) −39.9931 − 278.158i −0.0980485 − 0.681943i
\(56\) −6.50350 + 14.2407i −0.0155190 + 0.0339820i
\(57\) 687.011 − 201.725i 1.59644 − 0.468756i
\(58\) −328.957 + 96.5905i −0.744727 + 0.218672i
\(59\) 315.707 − 691.302i 0.696637 − 1.52542i −0.147366 − 0.989082i \(-0.547080\pi\)
0.844003 − 0.536339i \(-0.180193\pi\)
\(60\) −24.1138 − 167.715i −0.0518847 − 0.360866i
\(61\) 177.791 + 205.182i 0.373178 + 0.430670i 0.911012 − 0.412381i \(-0.135303\pi\)
−0.537834 + 0.843051i \(0.680757\pi\)
\(62\) 150.761 + 330.120i 0.308816 + 0.676214i
\(63\) −73.7115 + 47.3715i −0.147409 + 0.0947340i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) −328.814 − 211.316i −0.627451 − 0.403238i
\(66\) −623.634 + 719.712i −1.16309 + 1.34228i
\(67\) −274.784 − 80.6840i −0.501048 − 0.147121i 0.0214355 − 0.999770i \(-0.493176\pi\)
−0.522484 + 0.852649i \(0.674995\pi\)
\(68\) 114.958 0.205010
\(69\) −744.270 + 565.107i −1.29854 + 0.985955i
\(70\) −19.5693 −0.0334139
\(71\) −932.870 − 273.915i −1.55931 − 0.457856i −0.615446 − 0.788179i \(-0.711024\pi\)
−0.943868 + 0.330323i \(0.892842\pi\)
\(72\) −234.570 + 270.708i −0.383949 + 0.443101i
\(73\) −13.2791 − 8.53393i −0.0212904 − 0.0136825i 0.529952 − 0.848027i \(-0.322210\pi\)
−0.551243 + 0.834345i \(0.685846\pi\)
\(74\) 19.4407 − 135.213i 0.0305397 − 0.212408i
\(75\) 178.177 − 114.508i 0.274322 − 0.176296i
\(76\) −140.436 − 307.512i −0.211962 − 0.464132i
\(77\) 72.0259 + 83.1223i 0.106599 + 0.123022i
\(78\) 188.503 + 1311.07i 0.273639 + 1.90320i
\(79\) 17.6754 − 38.7038i 0.0251727 − 0.0551204i −0.896628 − 0.442785i \(-0.853991\pi\)
0.921801 + 0.387664i \(0.126718\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) −64.1519 + 18.8367i −0.0879998 + 0.0258391i
\(82\) 242.521 − 531.046i 0.326609 − 0.715173i
\(83\) 26.4515 + 183.974i 0.0349810 + 0.243298i 0.999808 − 0.0195868i \(-0.00623506\pi\)
−0.964827 + 0.262885i \(0.915326\pi\)
\(84\) 43.4280 + 50.1186i 0.0564093 + 0.0650998i
\(85\) 59.6939 + 130.711i 0.0761731 + 0.166796i
\(86\) −136.630 + 87.8067i −0.171316 + 0.110098i
\(87\) −206.682 + 1437.51i −0.254697 + 1.77146i
\(88\) 378.253 + 243.088i 0.458203 + 0.294469i
\(89\) 629.222 − 726.160i 0.749408 − 0.864863i −0.245102 − 0.969497i \(-0.578822\pi\)
0.994511 + 0.104634i \(0.0333670\pi\)
\(90\) −429.611 − 126.145i −0.503166 − 0.147743i
\(91\) 152.978 0.176224
\(92\) 314.106 + 309.854i 0.355954 + 0.351136i
\(93\) 1537.31 1.71410
\(94\) −671.558 − 197.187i −0.736872 − 0.216365i
\(95\) 276.729 − 319.363i 0.298861 − 0.344904i
\(96\) 228.067 + 146.570i 0.242469 + 0.155825i
\(97\) 144.812 − 1007.19i 0.151582 − 1.05427i −0.761988 − 0.647591i \(-0.775776\pi\)
0.913570 − 0.406682i \(-0.133314\pi\)
\(98\) −570.657 + 366.739i −0.588215 + 0.378023i
\(99\) 1045.39 + 2289.10i 1.06127 + 2.32387i
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.41.2 ✓ 70
23.9 even 11 inner 230.4.g.d.101.2 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.2 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.2 yes 70 23.9 even 11 inner