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Properties

Label 230.4.g.d.31.1
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 31.1
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.1

$q$-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} +(-4.10508 - 8.98887i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(18.9632 - 5.56809i) q^{6} +(-23.3667 - 15.0169i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-46.2669 + 53.3948i) q^{9} +(8.41254 - 5.40641i) q^{10} +(-7.53002 - 52.3725i) q^{11} +(5.62535 + 39.1252i) q^{12} +(-43.6255 + 28.0364i) q^{13} +(36.3789 - 41.9835i) q^{14} +(-20.5254 + 44.9444i) q^{15} +(13.4601 + 8.65025i) q^{16} +(122.137 - 35.8626i) q^{17} +(-92.5338 - 106.790i) q^{18} +(52.6747 + 15.4667i) q^{19} +(8.30830 + 18.1926i) q^{20} +(-39.0625 + 271.686i) q^{21} +105.822 q^{22} +(42.8174 - 101.655i) q^{23} -79.0550 q^{24} +(-3.55787 + 24.7455i) q^{25} +(-43.0849 - 94.3428i) q^{26} +(413.885 + 121.528i) q^{27} +(72.7578 + 83.9670i) q^{28} +(-189.773 + 55.7224i) q^{29} +(-83.1316 - 53.4255i) q^{30} +(-66.3220 + 145.225i) q^{31} +(-20.9555 + 24.1840i) q^{32} +(-439.858 + 282.679i) q^{33} +(36.2314 + 251.995i) q^{34} +(19.7647 + 137.467i) q^{35} +(237.743 - 152.788i) q^{36} +(40.7533 - 47.0319i) q^{37} +(-45.6113 + 99.8748i) q^{38} +(431.101 + 277.052i) q^{39} +(-38.3797 + 11.2693i) q^{40} +(46.3038 + 53.4374i) q^{41} +(-526.722 - 154.660i) q^{42} +(-100.566 - 220.209i) q^{43} +(-30.1201 + 209.490i) q^{44} +353.257 q^{45} +(189.053 + 113.697i) q^{46} -64.8219 q^{47} +(22.5014 - 156.501i) q^{48} +(178.009 + 389.786i) q^{49} +(-47.9746 - 14.0866i) q^{50} +(-823.746 - 950.654i) q^{51} +(199.028 - 58.4400i) q^{52} +(202.724 + 130.283i) q^{53} +(-358.386 + 784.755i) q^{54} +(-173.247 + 199.937i) q^{55} +(-186.934 + 120.135i) q^{56} +(-77.2058 - 536.978i) q^{57} +(-56.2955 - 391.543i) q^{58} +(-193.885 + 124.602i) q^{59} +(129.425 - 149.364i) q^{60} +(20.4733 - 44.8302i) q^{61} +(-268.616 - 172.629i) q^{62} +(1882.93 - 552.877i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(248.785 + 73.0500i) q^{65} +(-434.408 - 951.220i) q^{66} +(66.8041 - 464.632i) q^{67} -509.173 q^{68} +(-1089.53 + 32.4200i) q^{69} -277.761 q^{70} +(-87.7538 + 610.341i) q^{71} +(234.797 + 514.135i) q^{72} +(-460.317 - 135.161i) q^{73} +(81.5067 + 94.0637i) q^{74} +(237.040 - 69.6012i) q^{75} +(-184.734 - 118.721i) q^{76} +(-610.519 + 1336.85i) q^{77} +(-671.168 + 774.570i) q^{78} +(-671.707 + 431.680i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(-335.157 - 2331.07i) q^{81} +(-118.966 + 76.4550i) q^{82} +(136.608 - 157.654i) q^{83} +(456.092 - 998.702i) q^{84} +(-535.429 - 344.099i) q^{85} +(464.559 - 136.407i) q^{86} +(1279.92 + 1477.10i) q^{87} +(-406.142 - 119.254i) q^{88} +(-420.287 - 920.301i) q^{89} +(-100.548 + 699.323i) q^{90} +1440.40 q^{91} +(-278.890 + 341.895i) q^{92} +1577.66 q^{93} +(18.4502 - 128.324i) q^{94} +(-114.028 - 249.687i) q^{95} +(303.411 + 89.0895i) q^{96} +(646.296 + 745.865i) q^{97} +(-822.304 + 241.450i) q^{98} +(3144.81 + 2021.05i) 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
\(3\) −4.10508 − 8.98887i −0.790023 − 1.72991i −0.676586 − 0.736364i \(-0.736541\pi\)
−0.113437 − 0.993545i \(-0.536186\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) 18.9632 − 5.56809i 1.29028 − 0.378861i
\(7\) −23.3667 − 15.0169i −1.26168 − 0.810835i −0.273170 − 0.961966i \(-0.588072\pi\)
−0.988514 + 0.151131i \(0.951708\pi\)
\(8\) 3.32332 − 7.27706i 0.146871 − 0.321603i
\(9\) −46.2669 + 53.3948i −1.71359 + 1.97759i
\(10\) 8.41254 − 5.40641i 0.266028 − 0.170966i
\(11\) −7.53002 − 52.3725i −0.206399 − 1.43554i −0.784783 − 0.619771i \(-0.787226\pi\)
0.578384 − 0.815765i \(-0.303684\pi\)
\(12\) 5.62535 + 39.1252i 0.135325 + 0.941205i
\(13\) −43.6255 + 28.0364i −0.930733 + 0.598146i −0.915753 − 0.401742i \(-0.868405\pi\)
−0.0149800 + 0.999888i \(0.504768\pi\)
\(14\) 36.3789 − 41.9835i 0.694476 − 0.801469i
\(15\) −20.5254 + 44.9444i −0.353309 + 0.773639i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) 122.137 − 35.8626i 1.74250 − 0.511645i 0.753234 − 0.657752i \(-0.228493\pi\)
0.989269 + 0.146107i \(0.0466744\pi\)
\(18\) −92.5338 − 106.790i −1.21169 − 1.39836i
\(19\) 52.6747 + 15.4667i 0.636021 + 0.186753i 0.583819 − 0.811884i \(-0.301558\pi\)
0.0522023 + 0.998637i \(0.483376\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) −39.0625 + 271.686i −0.405911 + 2.82318i
\(22\) 105.822 1.02552
\(23\) 42.8174 − 101.655i 0.388176 − 0.921585i
\(24\) −79.0550 −0.672377
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) −43.0849 − 94.3428i −0.324987 − 0.711621i
\(27\) 413.885 + 121.528i 2.95008 + 0.866223i
\(28\) 72.7578 + 83.9670i 0.491069 + 0.566724i
\(29\) −189.773 + 55.7224i −1.21517 + 0.356807i −0.825635 − 0.564204i \(-0.809183\pi\)
−0.389537 + 0.921011i \(0.627365\pi\)
\(30\) −83.1316 − 53.4255i −0.505923 − 0.325137i
\(31\) −66.3220 + 145.225i −0.384251 + 0.841392i 0.614376 + 0.789013i \(0.289408\pi\)
−0.998627 + 0.0523790i \(0.983320\pi\)
\(32\) −20.9555 + 24.1840i −0.115764 + 0.133599i
\(33\) −439.858 + 282.679i −2.32029 + 1.49116i
\(34\) 36.2314 + 251.995i 0.182754 + 1.27108i
\(35\) 19.7647 + 137.467i 0.0954528 + 0.663889i
\(36\) 237.743 − 152.788i 1.10066 − 0.707353i
\(37\) 40.7533 − 47.0319i 0.181076 − 0.208973i −0.657954 − 0.753058i \(-0.728578\pi\)
0.839030 + 0.544085i \(0.183123\pi\)
\(38\) −45.6113 + 99.8748i −0.194714 + 0.426364i
\(39\) 431.101 + 277.052i 1.77004 + 1.13753i
\(40\) −38.3797 + 11.2693i −0.151709 + 0.0445458i
\(41\) 46.3038 + 53.4374i 0.176376 + 0.203549i 0.837054 − 0.547121i \(-0.184276\pi\)
−0.660677 + 0.750670i \(0.729731\pi\)
\(42\) −526.722 − 154.660i −1.93512 − 0.568202i
\(43\) −100.566 − 220.209i −0.356655 − 0.780966i −0.999883 − 0.0152914i \(-0.995132\pi\)
0.643228 − 0.765675i \(-0.277595\pi\)
\(44\) −30.1201 + 209.490i −0.103199 + 0.717768i
\(45\) 353.257 1.17023
\(46\) 189.053 + 113.697i 0.605963 + 0.364429i
\(47\) −64.8219 −0.201175 −0.100588 − 0.994928i \(-0.532072\pi\)
−0.100588 + 0.994928i \(0.532072\pi\)
\(48\) 22.5014 − 156.501i 0.0676624 − 0.470603i
\(49\) 178.009 + 389.786i 0.518977 + 1.13640i
\(50\) −47.9746 − 14.0866i −0.135693 − 0.0398430i
\(51\) −823.746 − 950.654i −2.26172 − 2.61016i
\(52\) 199.028 − 58.4400i 0.530774 − 0.155849i
\(53\) 202.724 + 130.283i 0.525400 + 0.337654i 0.776305 − 0.630358i \(-0.217092\pi\)
−0.250905 + 0.968012i \(0.580728\pi\)
\(54\) −358.386 + 784.755i −0.903150 + 1.97762i
\(55\) −173.247 + 199.937i −0.424738 + 0.490174i
\(56\) −186.934 + 120.135i −0.446073 + 0.286673i
\(57\) −77.2058 − 536.978i −0.179406 − 1.24780i
\(58\) −56.2955 − 391.543i −0.127448 − 0.886417i
\(59\) −193.885 + 124.602i −0.427825 + 0.274946i −0.736783 − 0.676129i \(-0.763656\pi\)
0.308958 + 0.951076i \(0.400020\pi\)
\(60\) 129.425 − 149.364i 0.278478 − 0.321381i
\(61\) 20.4733 − 44.8302i 0.0429727 − 0.0940971i −0.886928 − 0.461908i \(-0.847165\pi\)
0.929901 + 0.367810i \(0.119892\pi\)
\(62\) −268.616 − 172.629i −0.550230 − 0.353612i
\(63\) 1882.93 − 552.877i 3.76550 − 1.10565i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) 248.785 + 73.0500i 0.474739 + 0.139396i
\(66\) −434.408 − 951.220i −0.810180 − 1.77405i
\(67\) 66.8041 − 464.632i 0.121812 − 0.847222i −0.833689 − 0.552234i \(-0.813775\pi\)
0.955501 − 0.294988i \(-0.0953156\pi\)
\(68\) −509.173 −0.908033
\(69\) −1089.53 + 32.4200i −1.90093 + 0.0565639i
\(70\) −277.761 −0.474268
\(71\) −87.7538 + 610.341i −0.146683 + 1.02020i 0.774919 + 0.632061i \(0.217791\pi\)
−0.921601 + 0.388138i \(0.873118\pi\)
\(72\) 234.797 + 514.135i 0.384321 + 0.841547i
\(73\) −460.317 − 135.161i −0.738027 − 0.216704i −0.108950 − 0.994047i \(-0.534749\pi\)
−0.629077 + 0.777343i \(0.716567\pi\)
\(74\) 81.5067 + 94.0637i 0.128040 + 0.147766i
\(75\) 237.040 − 69.6012i 0.364947 − 0.107158i
\(76\) −184.734 − 118.721i −0.278822 − 0.179188i
\(77\) −610.519 + 1336.85i −0.903572 + 1.97855i
\(78\) −671.168 + 774.570i −0.974293 + 1.12439i
\(79\) −671.707 + 431.680i −0.956620 + 0.614782i −0.923061 − 0.384655i \(-0.874321\pi\)
−0.0335590 + 0.999437i \(0.510684\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) −335.157 − 2331.07i −0.459749 − 3.19763i
\(82\) −118.966 + 76.4550i −0.160215 + 0.102964i
\(83\) 136.608 − 157.654i 0.180659 − 0.208492i −0.658196 − 0.752847i \(-0.728680\pi\)
0.838855 + 0.544355i \(0.183226\pi\)
\(84\) 456.092 − 998.702i 0.592425 − 1.29723i
\(85\) −535.429 − 344.099i −0.683241 − 0.439092i
\(86\) 464.559 − 136.407i 0.582497 − 0.171036i
\(87\) 1279.92 + 1477.10i 1.57726 + 1.82025i
\(88\) −406.142 − 119.254i −0.491987 − 0.144460i
\(89\) −420.287 − 920.301i −0.500566 − 1.09609i −0.976285 − 0.216489i \(-0.930539\pi\)
0.475719 − 0.879597i \(-0.342188\pi\)
\(90\) −100.548 + 699.323i −0.117763 + 0.819057i
\(91\) 1440.40 1.65929
\(92\) −278.890 + 341.895i −0.316046 + 0.387446i
\(93\) 1577.66 1.75910
\(94\) 18.4502 − 128.324i 0.0202446 − 0.140805i
\(95\) −114.028 − 249.687i −0.123148 − 0.269656i
\(96\) 303.411 + 89.0895i 0.322570 + 0.0947152i
\(97\) 646.296 + 745.865i 0.676509 + 0.780733i 0.985380 − 0.170371i \(-0.0544966\pi\)
−0.308871 + 0.951104i \(0.599951\pi\)
\(98\) −822.304 + 241.450i −0.847604 + 0.248879i
\(99\) 3144.81 + 2021.05i 3.19258 + 2.05174i
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.31.1 ✓ 70
23.3 even 11 inner 230.4.g.d.141.1 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.1 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.1 yes 70 23.3 even 11 inner