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Properties

Label 230.4.g.d.31.4
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 31.4
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.4

$q$-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+(-0.284630 + 1.97964i) q^{2} +(0.779667 + 1.70723i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-3.60163 + 1.05753i) q^{6} +(-17.1073 - 10.9942i) q^{7} +(3.32332 - 7.27706i) q^{8} +(15.3745 - 17.7431i) q^{9} +(8.41254 - 5.40641i) q^{10} +(8.49461 + 59.0813i) q^{11} +(-1.06841 - 7.43095i) q^{12} +(33.1290 - 21.2907i) q^{13} +(26.6338 - 30.7371i) q^{14} +(3.89834 - 8.53617i) q^{15} +(13.4601 + 8.65025i) q^{16} +(17.7741 - 5.21895i) q^{17} +(30.7489 + 35.4862i) q^{18} +(132.493 + 38.9035i) q^{19} +(8.30830 + 18.1926i) q^{20} +(5.43165 - 37.7780i) q^{21} -119.378 q^{22} +(109.346 - 14.5103i) q^{23} +15.0147 q^{24} +(-3.55787 + 24.7455i) q^{25} +(32.7186 + 71.6437i) q^{26} +(90.9006 + 26.6908i) q^{27} +(53.2676 + 61.4741i) q^{28} +(-206.351 + 60.5901i) q^{29} +(15.7890 + 10.1470i) q^{30} +(46.8109 - 102.502i) q^{31} +(-20.9555 + 24.1840i) q^{32} +(-94.2426 + 60.5661i) q^{33} +(5.27262 + 36.6719i) q^{34} +(14.4702 + 100.642i) q^{35} +(-79.0020 + 50.7715i) q^{36} +(203.732 - 235.119i) q^{37} +(-114.726 + 251.216i) q^{38} +(62.1779 + 39.9593i) q^{39} +(-38.3797 + 11.2693i) q^{40} +(-11.5063 - 13.2790i) q^{41} +(73.2409 + 21.5055i) q^{42} +(21.6703 + 47.4514i) q^{43} +(33.9784 - 236.325i) q^{44} -117.387 q^{45} +(-2.39774 + 220.595i) q^{46} +494.741 q^{47} +(-4.27363 + 29.7238i) q^{48} +(29.2999 + 64.1578i) q^{49} +(-47.9746 - 14.0866i) q^{50} +(22.7679 + 26.2755i) q^{51} +(-151.142 + 44.3792i) q^{52} +(415.232 + 266.853i) q^{53} +(-78.7113 + 172.354i) q^{54} +(195.439 - 225.549i) q^{55} +(-136.858 + 87.9535i) q^{56} +(36.8832 + 256.528i) q^{57} +(-61.2132 - 425.747i) q^{58} +(-675.615 + 434.192i) q^{59} +(-24.5814 + 28.3684i) q^{60} +(-129.883 + 284.405i) q^{61} +(189.593 + 121.844i) q^{62} +(-458.087 + 134.506i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-188.927 - 55.4740i) q^{65} +(-93.0749 - 203.806i) q^{66} +(8.78267 - 61.0848i) q^{67} -74.0979 q^{68} +(110.026 + 175.365i) q^{69} -203.355 q^{70} +(117.894 - 819.969i) q^{71} +(-78.0232 - 170.847i) q^{72} +(822.997 + 241.654i) q^{73} +(407.464 + 470.239i) q^{74} +(-45.0204 + 13.2192i) q^{75} +(-464.663 - 298.621i) q^{76} +(504.231 - 1104.11i) q^{77} +(-96.8029 + 111.716i) q^{78} +(880.102 - 565.607i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(-64.9074 - 451.441i) q^{81} +(29.5627 - 18.9988i) q^{82} +(-365.154 + 421.410i) q^{83} +(-63.4196 + 138.870i) q^{84} +(-77.9189 - 50.0754i) q^{85} +(-100.105 + 29.3935i) q^{86} +(-264.327 - 305.049i) q^{87} +(458.168 + 134.530i) q^{88} +(-413.075 - 904.508i) q^{89} +(33.4119 - 232.385i) q^{90} -800.823 q^{91} +(-436.017 - 67.5346i) q^{92} +211.491 q^{93} +(-140.818 + 979.411i) q^{94} +(-286.816 - 628.040i) q^{95} +(-57.6261 - 16.9205i) q^{96} +(-365.566 - 421.885i) q^{97} +(-135.349 + 39.7421i) q^{98} +(1178.89 + 757.623i) 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
\(3\) 0.779667 + 1.70723i 0.150047 + 0.328557i 0.969698 − 0.244306i \(-0.0785600\pi\)
−0.819651 + 0.572863i \(0.805833\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) −3.60163 + 1.05753i −0.245060 + 0.0719561i
\(7\) −17.1073 − 10.9942i −0.923707 − 0.593630i −0.00997602 − 0.999950i \(-0.503176\pi\)
−0.913731 + 0.406320i \(0.866812\pi\)
\(8\) 3.32332 − 7.27706i 0.146871 − 0.321603i
\(9\) 15.3745 − 17.7431i 0.569425 − 0.657151i
\(10\) 8.41254 − 5.40641i 0.266028 − 0.170966i
\(11\) 8.49461 + 59.0813i 0.232838 + 1.61943i 0.685727 + 0.727859i \(0.259484\pi\)
−0.452889 + 0.891567i \(0.649607\pi\)
\(12\) −1.06841 − 7.43095i −0.0257019 − 0.178761i
\(13\) 33.1290 − 21.2907i 0.706796 − 0.454230i −0.137225 − 0.990540i \(-0.543818\pi\)
0.844021 + 0.536310i \(0.180182\pi\)
\(14\) 26.6338 − 30.7371i 0.508442 − 0.586773i
\(15\) 3.89834 − 8.53617i 0.0671031 − 0.146935i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) 17.7741 − 5.21895i 0.253580 − 0.0744577i −0.152472 − 0.988308i \(-0.548723\pi\)
0.406052 + 0.913850i \(0.366905\pi\)
\(18\) 30.7489 + 35.4862i 0.402644 + 0.464676i
\(19\) 132.493 + 38.9035i 1.59979 + 0.469740i 0.955488 − 0.295031i \(-0.0953301\pi\)
0.644301 + 0.764772i \(0.277148\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) 5.43165 − 37.7780i 0.0564421 − 0.392563i
\(22\) −119.378 −1.15688
\(23\) 109.346 − 14.5103i 0.991310 − 0.131548i
\(24\) 15.0147 0.127703
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) 32.7186 + 71.6437i 0.246794 + 0.540403i
\(27\) 90.9006 + 26.6908i 0.647919 + 0.190246i
\(28\) 53.2676 + 61.4741i 0.359523 + 0.414911i
\(29\) −206.351 + 60.5901i −1.32132 + 0.387976i −0.864971 − 0.501822i \(-0.832663\pi\)
−0.456354 + 0.889798i \(0.650845\pi\)
\(30\) 15.7890 + 10.1470i 0.0960887 + 0.0617524i
\(31\) 46.8109 − 102.502i 0.271209 − 0.593866i −0.724198 − 0.689592i \(-0.757790\pi\)
0.995408 + 0.0957261i \(0.0305173\pi\)
\(32\) −20.9555 + 24.1840i −0.115764 + 0.133599i
\(33\) −94.2426 + 60.5661i −0.497137 + 0.319491i
\(34\) 5.27262 + 36.6719i 0.0265955 + 0.184976i
\(35\) 14.4702 + 100.642i 0.0698832 + 0.486048i
\(36\) −79.0020 + 50.7715i −0.365750 + 0.235053i
\(37\) 203.732 − 235.119i 0.905226 − 1.04469i −0.0935691 − 0.995613i \(-0.529828\pi\)
0.998795 − 0.0490737i \(-0.0156269\pi\)
\(38\) −114.726 + 251.216i −0.489765 + 1.07244i
\(39\) 62.1779 + 39.9593i 0.255293 + 0.164067i
\(40\) −38.3797 + 11.2693i −0.151709 + 0.0445458i
\(41\) −11.5063 − 13.2790i −0.0438289 − 0.0505812i 0.733412 − 0.679785i \(-0.237927\pi\)
−0.777241 + 0.629203i \(0.783381\pi\)
\(42\) 73.2409 + 21.5055i 0.269079 + 0.0790087i
\(43\) 21.6703 + 47.4514i 0.0768534 + 0.168286i 0.944159 − 0.329491i \(-0.106877\pi\)
−0.867305 + 0.497777i \(0.834150\pi\)
\(44\) 33.9784 − 236.325i 0.116419 − 0.809713i
\(45\) −117.387 −0.388868
\(46\) −2.39774 + 220.595i −0.00768539 + 0.707065i
\(47\) 494.741 1.53543 0.767717 − 0.640789i \(-0.221393\pi\)
0.767717 + 0.640789i \(0.221393\pi\)
\(48\) −4.27363 + 29.7238i −0.0128510 + 0.0893804i
\(49\) 29.2999 + 64.1578i 0.0854224 + 0.187049i
\(50\) −47.9746 − 14.0866i −0.135693 − 0.0398430i
\(51\) 22.7679 + 26.2755i 0.0625125 + 0.0721433i
\(52\) −151.142 + 44.3792i −0.403068 + 0.118352i
\(53\) 415.232 + 266.853i 1.07616 + 0.691606i 0.953667 − 0.300864i \(-0.0972749\pi\)
0.122492 + 0.992469i \(0.460911\pi\)
\(54\) −78.7113 + 172.354i −0.198356 + 0.434340i
\(55\) 195.439 − 225.549i 0.479146 − 0.552964i
\(56\) −136.858 + 87.9535i −0.326580 + 0.209880i
\(57\) 36.8832 + 256.528i 0.0857070 + 0.596105i
\(58\) −61.2132 − 425.747i −0.138581 − 0.963850i
\(59\) −675.615 + 434.192i −1.49081 + 0.958084i −0.494782 + 0.869017i \(0.664752\pi\)
−0.996026 + 0.0890663i \(0.971612\pi\)
\(60\) −24.5814 + 28.3684i −0.0528907 + 0.0610391i
\(61\) −129.883 + 284.405i −0.272621 + 0.596956i −0.995578 − 0.0939365i \(-0.970055\pi\)
0.722958 + 0.690892i \(0.242782\pi\)
\(62\) 189.593 + 121.844i 0.388360 + 0.249584i
\(63\) −458.087 + 134.506i −0.916087 + 0.268987i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) −188.927 − 55.4740i −0.360515 − 0.105857i
\(66\) −93.0749 − 203.806i −0.173587 − 0.380102i
\(67\) 8.78267 − 61.0848i 0.0160145 − 0.111384i −0.980246 − 0.197781i \(-0.936626\pi\)
0.996261 + 0.0863976i \(0.0275355\pi\)
\(68\) −74.0979 −0.132143
\(69\) 110.026 + 175.365i 0.191964 + 0.305964i
\(70\) −203.355 −0.347222
\(71\) 117.894 − 819.969i 0.197062 − 1.37060i −0.615690 − 0.787988i \(-0.711123\pi\)
0.812752 − 0.582609i \(-0.197968\pi\)
\(72\) −78.0232 − 170.847i −0.127710 − 0.279646i
\(73\) 822.997 + 241.654i 1.31951 + 0.387444i 0.864317 − 0.502947i \(-0.167751\pi\)
0.455197 + 0.890391i \(0.349569\pi\)
\(74\) 407.464 + 470.239i 0.640091 + 0.738705i
\(75\) −45.0204 + 13.2192i −0.0693134 + 0.0203522i
\(76\) −464.663 − 298.621i −0.701322 − 0.450713i
\(77\) 504.231 − 1104.11i 0.746266 − 1.63409i
\(78\) −96.8029 + 111.716i −0.140523 + 0.162172i
\(79\) 880.102 − 565.607i 1.25341 − 0.805516i 0.266040 − 0.963962i \(-0.414285\pi\)
0.987368 + 0.158446i \(0.0506484\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) −64.9074 − 451.441i −0.0890363 − 0.619261i
\(82\) 29.5627 − 18.9988i 0.0398128 − 0.0255862i
\(83\) −365.154 + 421.410i −0.482902 + 0.557299i −0.943955 − 0.330074i \(-0.892926\pi\)
0.461053 + 0.887373i \(0.347472\pi\)
\(84\) −63.4196 + 138.870i −0.0823768 + 0.180380i
\(85\) −77.9189 − 50.0754i −0.0994293 − 0.0638994i
\(86\) −100.105 + 29.3935i −0.125519 + 0.0368556i
\(87\) −264.327 − 305.049i −0.325733 − 0.375916i
\(88\) 458.168 + 134.530i 0.555010 + 0.162966i
\(89\) −413.075 − 904.508i −0.491976 − 1.07728i −0.978995 − 0.203885i \(-0.934643\pi\)
0.487019 − 0.873391i \(-0.338084\pi\)
\(90\) 33.4119 − 232.385i 0.0391325 − 0.272173i
\(91\) −800.823 −0.922517
\(92\) −436.017 − 67.5346i −0.494108 − 0.0765323i
\(93\) 211.491 0.235813
\(94\) −140.818 + 979.411i −0.154513 + 1.07466i
\(95\) −286.816 − 628.040i −0.309755 − 0.678268i
\(96\) −57.6261 − 16.9205i −0.0612650 − 0.0179890i
\(97\) −365.566 − 421.885i −0.382655 − 0.441608i 0.531447 − 0.847092i \(-0.321649\pi\)
−0.914102 + 0.405484i \(0.867103\pi\)
\(98\) −135.349 + 39.7421i −0.139514 + 0.0409649i
\(99\) 1178.89 + 757.623i 1.19679 + 0.769132i
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.31.4 ✓ 70
23.3 even 11 inner 230.4.g.d.141.4 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.4 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.4 yes 70 23.3 even 11 inner