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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 41.4
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.4

$q$-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} +(-1.37651 + 1.58858i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(3.53661 - 2.27284i) q^{6} +(-11.8356 - 25.9163i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(3.21370 + 22.3518i) q^{9} +(4.15415 - 9.09632i) q^{10} +(57.0776 - 16.7595i) q^{11} +(-8.06737 + 2.36879i) q^{12} +(-25.6650 + 56.1986i) q^{13} +(8.10938 + 56.4020i) q^{14} +(-6.88255 - 7.94289i) q^{15} +(6.64664 + 14.5541i) q^{16} +(20.3884 - 13.1028i) q^{17} +(6.42741 - 44.7036i) q^{18} +(-17.5998 - 11.3107i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(57.4619 + 16.8723i) q^{21} -118.974 q^{22} +(-109.153 + 15.8943i) q^{23} +16.8159 q^{24} +(-23.9873 - 7.04331i) q^{25} +(80.9168 - 93.3829i) q^{26} +(-87.6755 - 56.3456i) q^{27} +(16.2188 - 112.804i) q^{28} +(-157.841 + 101.438i) q^{29} +(8.73198 + 19.1204i) q^{30} +(-29.3795 - 33.9058i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(-51.9441 + 113.742i) q^{33} +(-46.5081 + 13.6560i) q^{34} +(136.684 - 40.1342i) q^{35} +(-37.5230 + 82.1639i) q^{36} +(34.6493 + 240.991i) q^{37} +(27.4006 + 31.6220i) q^{38} +(-53.9476 - 118.129i) q^{39} +(33.6501 - 21.6256i) q^{40} +(-68.8003 + 478.517i) q^{41} +(-100.762 - 64.7555i) q^{42} +(-280.562 + 323.785i) q^{43} +(228.310 + 67.0379i) q^{44} -112.908 q^{45} +(218.419 + 31.0030i) q^{46} -323.329 q^{47} +(-32.2695 - 9.47518i) q^{48} +(-306.957 + 354.247i) q^{49} +(42.0627 + 27.0320i) q^{50} +(-7.24999 + 50.4248i) q^{51} +(-207.896 + 133.607i) q^{52} +(-151.306 - 331.313i) q^{53} +(136.499 + 157.529i) q^{54} +(42.3296 + 294.409i) q^{55} +(-94.6847 + 207.330i) q^{56} +(42.1943 - 12.3894i) q^{57} +(360.051 - 105.720i) q^{58} +(95.4799 - 209.072i) q^{59} +(-5.98288 - 41.6119i) q^{60} +(18.3762 + 21.2073i) q^{61} +(37.2742 + 81.6191i) q^{62} +(541.240 - 347.834i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(-259.870 - 167.008i) q^{65} +(163.769 - 189.000i) q^{66} +(867.987 + 254.864i) q^{67} +96.9430 q^{68} +(125.001 - 195.277i) q^{69} -284.910 q^{70} +(-535.783 - 157.320i) q^{71} +(118.303 - 136.529i) q^{72} +(-292.068 - 187.701i) q^{73} +(69.2985 - 481.982i) q^{74} +(44.2076 - 28.4105i) q^{75} +(-34.7635 - 76.1215i) q^{76} +(-1109.89 - 1280.88i) q^{77} +(36.9633 + 257.085i) q^{78} +(-413.719 + 905.918i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(-374.811 + 110.055i) q^{81} +(401.654 - 879.500i) q^{82} +(98.5094 + 685.148i) q^{83} +(156.872 + 181.041i) q^{84} +(50.3395 + 110.228i) q^{85} +(720.836 - 463.253i) q^{86} +(56.1271 - 390.372i) q^{87} +(-400.351 - 257.290i) q^{88} +(-60.0227 + 69.2699i) q^{89} +(216.669 + 63.6198i) q^{90} +1760.22 q^{91} +(-401.674 - 182.566i) q^{92} +94.3032 q^{93} +(620.463 + 182.184i) q^{94} +(68.5016 - 79.0550i) q^{95} +(56.5858 + 36.3655i) q^{96} +(115.953 - 806.472i) q^{97} +(788.652 - 506.836i) q^{98} +(558.035 + 1221.93i) 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
\(3\) −1.37651 + 1.58858i −0.264909 + 0.305722i −0.872584 − 0.488465i \(-0.837557\pi\)
0.607674 + 0.794186i \(0.292103\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) 3.53661 − 2.27284i 0.240636 − 0.154647i
\(7\) −11.8356 − 25.9163i −0.639061 − 1.39935i −0.900810 − 0.434213i \(-0.857026\pi\)
0.261749 − 0.965136i \(-0.415701\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) 3.21370 + 22.3518i 0.119026 + 0.827844i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) 57.0776 − 16.7595i 1.56450 − 0.459379i 0.619108 − 0.785306i \(-0.287494\pi\)
0.945395 + 0.325927i \(0.105676\pi\)
\(12\) −8.06737 + 2.36879i −0.194071 + 0.0569844i
\(13\) −25.6650 + 56.1986i −0.547554 + 1.19898i 0.410361 + 0.911923i \(0.365403\pi\)
−0.957915 + 0.287052i \(0.907325\pi\)
\(14\) 8.10938 + 56.4020i 0.154809 + 1.07672i
\(15\) −6.88255 − 7.94289i −0.118471 − 0.136723i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) 20.3884 − 13.1028i 0.290877 − 0.186936i −0.387061 − 0.922054i \(-0.626510\pi\)
0.677939 + 0.735118i \(0.262873\pi\)
\(18\) 6.42741 − 44.7036i 0.0841641 − 0.585374i
\(19\) −17.5998 − 11.3107i −0.212509 − 0.136571i 0.430056 − 0.902802i \(-0.358494\pi\)
−0.642566 + 0.766231i \(0.722130\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) 57.4619 + 16.8723i 0.597105 + 0.175326i
\(22\) −118.974 −1.15297
\(23\) −109.153 + 15.8943i −0.989564 + 0.144095i
\(24\) 16.8159 0.143022
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) 80.9168 − 93.3829i 0.610349 − 0.704381i
\(27\) −87.6755 − 56.3456i −0.624932 − 0.401619i
\(28\) 16.2188 − 112.804i 0.109466 − 0.761355i
\(29\) −157.841 + 101.438i −1.01070 + 0.649537i −0.937574 − 0.347786i \(-0.886934\pi\)
−0.0731249 + 0.997323i \(0.523297\pi\)
\(30\) 8.73198 + 19.1204i 0.0531411 + 0.116363i
\(31\) −29.3795 − 33.9058i −0.170217 − 0.196441i 0.664231 − 0.747527i \(-0.268759\pi\)
−0.834448 + 0.551086i \(0.814214\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) −51.9441 + 113.742i −0.274009 + 0.599997i
\(34\) −46.5081 + 13.6560i −0.234590 + 0.0688819i
\(35\) 136.684 − 40.1342i 0.660111 − 0.193826i
\(36\) −37.5230 + 82.1639i −0.173718 + 0.380389i
\(37\) 34.6493 + 240.991i 0.153954 + 1.07077i 0.909508 + 0.415687i \(0.136459\pi\)
−0.755553 + 0.655087i \(0.772632\pi\)
\(38\) 27.4006 + 31.6220i 0.116973 + 0.134994i
\(39\) −53.9476 − 118.129i −0.221501 − 0.485019i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) −68.8003 + 478.517i −0.262068 + 1.82273i 0.255184 + 0.966892i \(0.417864\pi\)
−0.517253 + 0.855833i \(0.673045\pi\)
\(42\) −100.762 − 64.7555i −0.370187 − 0.237905i
\(43\) −280.562 + 323.785i −0.995006 + 1.14830i −0.00606580 + 0.999982i \(0.501931\pi\)
−0.988940 + 0.148316i \(0.952615\pi\)
\(44\) 228.310 + 67.0379i 0.782251 + 0.229690i
\(45\) −112.908 −0.374030
\(46\) 218.419 + 31.0030i 0.700089 + 0.0993726i
\(47\) −323.329 −1.00345 −0.501727 − 0.865026i \(-0.667302\pi\)
−0.501727 + 0.865026i \(0.667302\pi\)
\(48\) −32.2695 − 9.47518i −0.0970354 − 0.0284922i
\(49\) −306.957 + 354.247i −0.894918 + 1.03279i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) −7.24999 + 50.4248i −0.0199059 + 0.138449i
\(52\) −207.896 + 133.607i −0.554423 + 0.356306i
\(53\) −151.306 − 331.313i −0.392140 − 0.858668i −0.998007 − 0.0631015i \(-0.979901\pi\)
0.605867 − 0.795566i \(-0.292826\pi\)
\(54\) 136.499 + 157.529i 0.343985 + 0.396980i
\(55\) 42.3296 + 294.409i 0.103777 + 0.721783i
\(56\) −94.6847 + 207.330i −0.225942 + 0.494745i
\(57\) 42.1943 − 12.3894i 0.0980486 − 0.0287897i
\(58\) 360.051 − 105.720i 0.815120 − 0.239341i
\(59\) 95.4799 − 209.072i 0.210685 − 0.461336i −0.774557 − 0.632504i \(-0.782027\pi\)
0.985242 + 0.171169i \(0.0547543\pi\)
\(60\) −5.98288 − 41.6119i −0.0128731 − 0.0895345i
\(61\) 18.3762 + 21.2073i 0.0385711 + 0.0445134i 0.774707 − 0.632320i \(-0.217897\pi\)
−0.736136 + 0.676833i \(0.763352\pi\)
\(62\) 37.2742 + 81.6191i 0.0763521 + 0.167188i
\(63\) 541.240 − 347.834i 1.08238 − 0.695602i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) −259.870 − 167.008i −0.495891 − 0.318690i
\(66\) 163.769 − 189.000i 0.305434 − 0.352489i
\(67\) 867.987 + 254.864i 1.58271 + 0.464725i 0.950668 − 0.310210i \(-0.100399\pi\)
0.632040 + 0.774935i \(0.282218\pi\)
\(68\) 96.9430 0.172883
\(69\) 125.001 − 195.277i 0.218092 − 0.340703i
\(70\) −284.910 −0.486475
\(71\) −535.783 − 157.320i −0.895573 − 0.262964i −0.198617 − 0.980077i \(-0.563645\pi\)
−0.696957 + 0.717113i \(0.745463\pi\)
\(72\) 118.303 − 136.529i 0.193640 − 0.223473i
\(73\) −292.068 − 187.701i −0.468274 − 0.300942i 0.285142 − 0.958485i \(-0.407959\pi\)
−0.753416 + 0.657544i \(0.771596\pi\)
\(74\) 69.2985 − 481.982i 0.108862 − 0.757152i
\(75\) 44.2076 − 28.4105i 0.0680621 − 0.0437409i
\(76\) −34.7635 − 76.1215i −0.0524691 − 0.114891i
\(77\) −1109.89 − 1280.88i −1.64265 − 1.89571i
\(78\) 36.9633 + 257.085i 0.0536572 + 0.373194i
\(79\) −413.719 + 905.918i −0.589203 + 1.29017i 0.346720 + 0.937969i \(0.387295\pi\)
−0.935923 + 0.352205i \(0.885432\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) −374.811 + 110.055i −0.514145 + 0.150966i
\(82\) 401.654 − 879.500i 0.540918 − 1.18445i
\(83\) 98.5094 + 685.148i 0.130275 + 0.906082i 0.945195 + 0.326508i \(0.105872\pi\)
−0.814920 + 0.579574i \(0.803219\pi\)
\(84\) 156.872 + 181.041i 0.203764 + 0.235156i
\(85\) 50.3395 + 110.228i 0.0642363 + 0.140658i
\(86\) 720.836 − 463.253i 0.903834 − 0.580859i
\(87\) 56.1271 − 390.372i 0.0691661 − 0.481061i
\(88\) −400.351 − 257.290i −0.484972 − 0.311672i
\(89\) −60.0227 + 69.2699i −0.0714875 + 0.0825010i −0.790367 − 0.612634i \(-0.790110\pi\)
0.718879 + 0.695135i \(0.244655\pi\)
\(90\) 216.669 + 63.6198i 0.253766 + 0.0745124i
\(91\) 1760.22 2.02771
\(92\) −401.674 − 182.566i −0.455189 − 0.206889i
\(93\) 94.3032 0.105148
\(94\) 620.463 + 182.184i 0.680807 + 0.199903i
\(95\) 68.5016 − 79.0550i 0.0739801 − 0.0853776i
\(96\) 56.5858 + 36.3655i 0.0601590 + 0.0386618i
\(97\) 115.953 − 806.472i 0.121374 − 0.844174i −0.834628 − 0.550814i \(-0.814317\pi\)
0.956002 − 0.293360i \(-0.0947735\pi\)
\(98\) 788.652 − 506.836i 0.812917 − 0.522430i
\(99\) 558.035 + 1221.93i 0.566511 + 1.24049i
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.41.4 ✓ 70
23.9 even 11 inner 230.4.g.d.101.4 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.4 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.4 yes 70 23.9 even 11 inner