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Properties

Label 230.2.l.a.217.1
Level $230$
Weight $2$
Character 230.217
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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] = [230,2,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 217.1
Character \(\chi\) \(=\) 230.217
Dual form 230.2.l.a.53.1

$q$-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+(-0.0713392 + 0.997452i) q^{2} +(-3.05407 + 0.664372i) q^{3} +(-0.989821 - 0.142315i) q^{4} +(0.783298 + 2.09438i) q^{5} +(-0.444805 - 3.09368i) q^{6} +(-2.28419 + 1.24726i) q^{7} +(0.212565 - 0.977147i) q^{8} +(6.15705 - 2.81183i) q^{9} +(-2.14493 + 0.631890i) q^{10} +(-3.47691 - 3.01276i) q^{11} +(3.11753 - 0.222970i) q^{12} +(1.24765 - 2.28489i) q^{13} +(-1.08113 - 2.36735i) q^{14} +(-3.78370 - 5.87599i) q^{15} +(0.959493 + 0.281733i) q^{16} +(-2.99811 - 2.24436i) q^{17} +(2.36543 + 6.34196i) q^{18} +(0.215083 - 1.49593i) q^{19} +(-0.477263 - 2.18454i) q^{20} +(6.14742 - 5.32677i) q^{21} +(3.25312 - 3.25312i) q^{22} +(0.785478 + 4.73107i) q^{23} +3.12550i q^{24} +(-3.77289 + 3.28105i) q^{25} +(2.19007 + 1.40747i) q^{26} +(-9.42969 + 7.05898i) q^{27} +(2.43844 - 0.909491i) q^{28} +(-6.04466 + 0.869091i) q^{29} +(6.13095 - 3.35487i) q^{30} +(4.50690 - 2.89641i) q^{31} +(-0.349464 + 0.936950i) q^{32} +(12.6203 + 6.89121i) q^{33} +(2.45252 - 2.83036i) q^{34} +(-4.40144 - 3.80699i) q^{35} +(-6.49455 + 1.90697i) q^{36} +(-8.17698 - 3.04986i) q^{37} +(1.47678 + 0.321254i) q^{38} +(-2.29238 + 7.80713i) q^{39} +(2.21302 - 0.320204i) q^{40} +(-4.39226 + 9.61770i) q^{41} +(4.87465 + 6.51176i) q^{42} +(-0.143943 - 0.661697i) q^{43} +(3.01276 + 3.47691i) q^{44} +(10.7119 + 10.6927i) q^{45} +(-4.77505 + 0.445966i) q^{46} +(1.92809 + 1.92809i) q^{47} +(-3.11753 - 0.222970i) q^{48} +(-0.122635 + 0.190823i) q^{49} +(-3.00354 - 3.99734i) q^{50} +(10.6475 + 4.86256i) q^{51} +(-1.56012 + 2.08408i) q^{52} +(-3.01733 - 5.52584i) q^{53} +(-6.36829 - 9.90925i) q^{54} +(3.58642 - 9.64186i) q^{55} +(0.733218 + 2.49711i) q^{56} +(0.336979 + 4.71158i) q^{57} +(-0.435656 - 6.09126i) q^{58} +(1.80006 + 6.13045i) q^{59} +(2.90894 + 6.35466i) q^{60} +(-4.00554 - 6.23274i) q^{61} +(2.56751 + 4.70205i) q^{62} +(-10.5568 + 14.1022i) q^{63} +(-0.909632 - 0.415415i) q^{64} +(5.76272 + 0.823298i) q^{65} +(-7.77397 + 12.0965i) q^{66} +(-7.74682 - 0.554064i) q^{67} +(2.64819 + 2.64819i) q^{68} +(-5.54209 - 13.9272i) q^{69} +(4.11128 - 4.11864i) q^{70} +(-8.49053 - 9.79860i) q^{71} +(-1.43880 - 6.61404i) q^{72} +(-0.891553 - 1.19098i) q^{73} +(3.62542 - 7.93857i) q^{74} +(9.34282 - 12.5272i) q^{75} +(-0.425787 + 1.45010i) q^{76} +(11.6996 + 2.54509i) q^{77} +(-7.62370 - 2.84349i) q^{78} +(-4.18153 + 1.22781i) q^{79} +(0.161513 + 2.23023i) q^{80} +(10.8114 - 12.4770i) q^{81} +(-9.27986 - 5.06719i) q^{82} +(-0.858110 + 2.30068i) q^{83} +(-6.84293 + 4.39768i) q^{84} +(2.35213 - 8.03720i) q^{85} +(0.670280 - 0.0963717i) q^{86} +(17.8834 - 6.67017i) q^{87} +(-3.68298 + 2.75704i) q^{88} +(13.5970 + 8.73826i) q^{89} +(-11.4297 + 9.92176i) q^{90} +6.77526i q^{91} +(-0.104181 - 4.79470i) q^{92} +(-11.8401 + 11.8401i) q^{93} +(-2.06073 + 1.78563i) q^{94} +(3.30154 - 0.721296i) q^{95} +(0.444805 - 3.09368i) q^{96} +(-2.56949 - 6.88907i) q^{97} +(-0.181589 - 0.135936i) q^{98} +(-29.8789 - 8.77322i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0713392 + 0.997452i −0.0504444 + 0.705305i
\(3\) −3.05407 + 0.664372i −1.76327 + 0.383575i −0.973293 − 0.229565i \(-0.926270\pi\)
−0.789974 + 0.613140i \(0.789906\pi\)
\(4\) −0.989821 − 0.142315i −0.494911 − 0.0711574i
\(5\) 0.783298 + 2.09438i 0.350301 + 0.936637i
\(6\) −0.444805 − 3.09368i −0.181591 − 1.26299i
\(7\) −2.28419 + 1.24726i −0.863341 + 0.471420i −0.848948 − 0.528476i \(-0.822764\pi\)
−0.0143934 + 0.999896i \(0.504582\pi\)
\(8\) 0.212565 − 0.977147i 0.0751532 − 0.345474i
\(9\) 6.15705 − 2.81183i 2.05235 − 0.937277i
\(10\) −2.14493 + 0.631890i −0.678286 + 0.199821i
\(11\) −3.47691 − 3.01276i −1.04833 − 0.908380i −0.0524072 − 0.998626i \(-0.516689\pi\)
−0.995920 + 0.0902455i \(0.971235\pi\)
\(12\) 3.11753 − 0.222970i 0.899954 − 0.0643660i
\(13\) 1.24765 − 2.28489i 0.346035 − 0.633716i −0.645470 − 0.763786i \(-0.723338\pi\)
0.991505 + 0.130070i \(0.0415203\pi\)
\(14\) −1.08113 − 2.36735i −0.288944 − 0.632700i
\(15\) −3.78370 − 5.87599i −0.976946 − 1.51717i
\(16\) 0.959493 + 0.281733i 0.239873 + 0.0704331i
\(17\) −2.99811 − 2.24436i −0.727149 − 0.544337i 0.170334 − 0.985386i \(-0.445515\pi\)
−0.897483 + 0.441050i \(0.854606\pi\)
\(18\) 2.36543 + 6.34196i 0.557537 + 1.49481i
\(19\) 0.215083 − 1.49593i 0.0493434 − 0.343191i −0.950162 − 0.311755i \(-0.899083\pi\)
0.999506 − 0.0314355i \(-0.0100079\pi\)
\(20\) −0.477263 − 2.18454i −0.106719 − 0.488478i
\(21\) 6.14742 − 5.32677i 1.34148 − 1.16240i
\(22\) 3.25312 − 3.25312i 0.693568 − 0.693568i
\(23\) 0.785478 + 4.73107i 0.163783 + 0.986496i
\(24\) 3.12550i 0.637989i
\(25\) −3.77289 + 3.28105i −0.754578 + 0.656211i
\(26\) 2.19007 + 1.40747i 0.429507 + 0.276028i
\(27\) −9.42969 + 7.05898i −1.81475 + 1.35850i
\(28\) 2.43844 − 0.909491i 0.460822 − 0.171878i
\(29\) −6.04466 + 0.869091i −1.12247 + 0.161386i −0.678460 − 0.734638i \(-0.737352\pi\)
−0.444006 + 0.896024i \(0.646443\pi\)
\(30\) 6.13095 − 3.35487i 1.11935 − 0.612512i
\(31\) 4.50690 − 2.89641i 0.809464 − 0.520211i −0.0692275 − 0.997601i \(-0.522053\pi\)
0.878692 + 0.477390i \(0.158417\pi\)
\(32\) −0.349464 + 0.936950i −0.0617771 + 0.165631i
\(33\) 12.6203 + 6.89121i 2.19691 + 1.19961i
\(34\) 2.45252 − 2.83036i 0.420604 − 0.485403i
\(35\) −4.40144 − 3.80699i −0.743979 − 0.643498i
\(36\) −6.49455 + 1.90697i −1.08242 + 0.317829i
\(37\) −8.17698 − 3.04986i −1.34429 − 0.501393i −0.428676 − 0.903458i \(-0.641020\pi\)
−0.915611 + 0.402065i \(0.868293\pi\)
\(38\) 1.47678 + 0.321254i 0.239565 + 0.0521142i
\(39\) −2.29238 + 7.80713i −0.367074 + 1.25014i
\(40\) 2.21302 − 0.320204i 0.349910 − 0.0506286i
\(41\) −4.39226 + 9.61770i −0.685955 + 1.50203i 0.170252 + 0.985401i \(0.445542\pi\)
−0.856207 + 0.516632i \(0.827186\pi\)
\(42\) 4.87465 + 6.51176i 0.752174 + 1.00479i
\(43\) −0.143943 − 0.661697i −0.0219511 − 0.100908i 0.964868 − 0.262736i \(-0.0846247\pi\)
−0.986819 + 0.161828i \(0.948261\pi\)
\(44\) 3.01276 + 3.47691i 0.454190 + 0.524163i
\(45\) 10.7119 + 10.6927i 1.59683 + 1.59398i
\(46\) −4.77505 + 0.445966i −0.704043 + 0.0657540i
\(47\) 1.92809 + 1.92809i 0.281241 + 0.281241i 0.833604 − 0.552363i \(-0.186274\pi\)
−0.552363 + 0.833604i \(0.686274\pi\)
\(48\) −3.11753 − 0.222970i −0.449977 − 0.0321830i
\(49\) −0.122635 + 0.190823i −0.0175193 + 0.0272605i
\(50\) −3.00354 − 3.99734i −0.424764 − 0.565310i
\(51\) 10.6475 + 4.86256i 1.49095 + 0.680895i
\(52\) −1.56012 + 2.08408i −0.216350 + 0.289010i
\(53\) −3.01733 − 5.52584i −0.414463 − 0.759032i 0.584152 − 0.811644i \(-0.301427\pi\)
−0.998615 + 0.0526122i \(0.983245\pi\)
\(54\) −6.36829 − 9.90925i −0.866614 − 1.34848i
\(55\) 3.58642 − 9.64186i 0.483592 − 1.30011i
\(56\) 0.733218 + 2.49711i 0.0979803 + 0.333690i
\(57\) 0.336979 + 4.71158i 0.0446340 + 0.624064i
\(58\) −0.435656 − 6.09126i −0.0572044 − 0.799822i
\(59\) 1.80006 + 6.13045i 0.234348 + 0.798117i 0.989745 + 0.142849i \(0.0456263\pi\)
−0.755396 + 0.655268i \(0.772556\pi\)
\(60\) 2.90894 + 6.35466i 0.375543 + 0.820383i
\(61\) −4.00554 − 6.23274i −0.512856 − 0.798020i 0.484179 − 0.874969i \(-0.339118\pi\)
−0.997035 + 0.0769490i \(0.975482\pi\)
\(62\) 2.56751 + 4.70205i 0.326075 + 0.597161i
\(63\) −10.5568 + 14.1022i −1.33003 + 1.77671i
\(64\) −0.909632 − 0.415415i −0.113704 − 0.0519269i
\(65\) 5.76272 + 0.823298i 0.714778 + 0.102118i
\(66\) −7.77397 + 12.0965i −0.956910 + 1.48898i
\(67\) −7.74682 − 0.554064i −0.946425 − 0.0676897i −0.410413 − 0.911900i \(-0.634615\pi\)
−0.536012 + 0.844210i \(0.680070\pi\)
\(68\) 2.64819 + 2.64819i 0.321140 + 0.321140i
\(69\) −5.54209 − 13.9272i −0.667190 − 1.67663i
\(70\) 4.11128 − 4.11864i 0.491392 − 0.492272i
\(71\) −8.49053 − 9.79860i −1.00764 − 1.16288i −0.986609 − 0.163101i \(-0.947851\pi\)
−0.0210313 − 0.999779i \(-0.506695\pi\)
\(72\) −1.43880 − 6.61404i −0.169564 − 0.779472i
\(73\) −0.891553 − 1.19098i −0.104348 − 0.139393i 0.745327 − 0.666699i \(-0.232293\pi\)
−0.849676 + 0.527306i \(0.823202\pi\)
\(74\) 3.62542 − 7.93857i 0.421447 − 0.922840i
\(75\) 9.34282 − 12.5272i 1.07882 − 1.44651i
\(76\) −0.425787 + 1.45010i −0.0488412 + 0.166338i
\(77\) 11.6996 + 2.54509i 1.33329 + 0.290040i
\(78\) −7.62370 − 2.84349i −0.863214 − 0.321962i
\(79\) −4.18153 + 1.22781i −0.470459 + 0.138139i −0.508364 − 0.861142i \(-0.669750\pi\)
0.0379052 + 0.999281i \(0.487932\pi\)
\(80\) 0.161513 + 2.23023i 0.0180577 + 0.249347i
\(81\) 10.8114 − 12.4770i 1.20127 − 1.38634i
\(82\) −9.27986 − 5.06719i −1.02479 − 0.559577i
\(83\) −0.858110 + 2.30068i −0.0941898 + 0.252533i −0.975498 − 0.220010i \(-0.929391\pi\)
0.881308 + 0.472543i \(0.156664\pi\)
\(84\) −6.84293 + 4.39768i −0.746624 + 0.479826i
\(85\) 2.35213 − 8.03720i 0.255125 − 0.871757i
\(86\) 0.670280 − 0.0963717i 0.0722781 − 0.0103920i
\(87\) 17.8834 − 6.67017i 1.91730 − 0.715117i
\(88\) −3.68298 + 2.75704i −0.392606 + 0.293902i
\(89\) 13.5970 + 8.73826i 1.44128 + 0.926254i 0.999577 + 0.0290929i \(0.00926187\pi\)
0.441703 + 0.897161i \(0.354374\pi\)
\(90\) −11.4297 + 9.92176i −1.20479 + 1.04584i
\(91\) 6.77526i 0.710241i
\(92\) −0.104181 − 4.79470i −0.0108616 − 0.499882i
\(93\) −11.8401 + 11.8401i −1.22776 + 1.22776i
\(94\) −2.06073 + 1.78563i −0.212548 + 0.184174i
\(95\) 3.30154 − 0.721296i 0.338730 − 0.0740034i
\(96\) 0.444805 − 3.09368i 0.0453977 − 0.315748i
\(97\) −2.56949 − 6.88907i −0.260892 − 0.699479i −0.999674 − 0.0255342i \(-0.991871\pi\)
0.738782 − 0.673945i \(-0.235401\pi\)
\(98\) −0.181589 − 0.135936i −0.0183432 − 0.0137316i
\(99\) −29.8789 − 8.77322i −3.00294 − 0.881742i
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 230.2.l.a.217.1 yes 240
5.3 odd 4 inner 230.2.l.a.33.7 yes 240
23.7 odd 22 inner 230.2.l.a.7.7 ✓ 240
115.53 even 44 inner 230.2.l.a.53.1 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.7 ✓ 240 23.7 odd 22 inner
230.2.l.a.33.7 yes 240 5.3 odd 4 inner
230.2.l.a.53.1 yes 240 115.53 even 44 inner
230.2.l.a.217.1 yes 240 1.1 even 1 trivial