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Properties

Label 230.2.l.a.67.2
Level $230$
Weight $2$
Character 230.67
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 67.2
Character \(\chi\) \(=\) 230.67
Dual form 230.2.l.a.103.2

$q$-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+(-0.212565 - 0.977147i) q^{2} +(-0.409159 - 0.306293i) q^{3} +(-0.909632 + 0.415415i) q^{4} +(-0.0488687 + 2.23553i) q^{5} +(-0.212320 + 0.464916i) q^{6} +(-0.318744 - 4.45663i) q^{7} +(0.599278 + 0.800541i) q^{8} +(-0.771602 - 2.62783i) q^{9} +(2.19483 - 0.427445i) q^{10} +(-3.31624 - 5.16017i) q^{11} +(0.499423 + 0.108643i) q^{12} +(5.72063 + 0.409147i) q^{13} +(-4.28703 + 1.25878i) q^{14} +(0.704723 - 0.899722i) q^{15} +(0.654861 - 0.755750i) q^{16} +(0.199099 + 0.533806i) q^{17} +(-2.40376 + 1.31255i) q^{18} +(-1.30909 - 2.86651i) q^{19} +(-0.884222 - 2.05381i) q^{20} +(-1.23462 + 1.92110i) q^{21} +(-4.33733 + 4.33733i) q^{22} +(0.973320 + 4.69602i) q^{23} -0.511103i q^{24} +(-4.99522 - 0.218495i) q^{25} +(-0.816210 - 5.67686i) q^{26} +(-1.02502 + 2.74817i) q^{27} +(2.14129 + 3.92148i) q^{28} +(1.09164 + 0.498537i) q^{29} +(-1.02896 - 0.497368i) q^{30} +(-0.682538 + 4.74716i) q^{31} +(-0.877679 - 0.479249i) q^{32} +(-0.223653 + 3.12707i) q^{33} +(0.479285 - 0.308018i) q^{34} +(9.97852 - 0.494774i) q^{35} +(1.79352 + 2.06983i) q^{36} +(1.77395 - 3.24874i) q^{37} +(-2.52274 + 1.88850i) q^{38} +(-2.21533 - 1.91959i) q^{39} +(-1.81892 + 1.30058i) q^{40} +(3.17909 + 0.933464i) q^{41} +(2.13963 + 0.798042i) q^{42} +(3.83212 - 5.11911i) q^{43} +(5.16017 + 3.31624i) q^{44} +(5.91232 - 1.59652i) q^{45} +(4.38181 - 1.94929i) q^{46} +(2.83845 + 2.83845i) q^{47} +(-0.499423 + 0.108643i) q^{48} +(-12.8312 + 1.84484i) q^{49} +(0.848309 + 4.92751i) q^{50} +(0.0820375 - 0.279394i) q^{51} +(-5.37363 + 2.00426i) q^{52} +(-3.02013 + 0.216004i) q^{53} +(2.90325 + 0.417425i) q^{54} +(11.6978 - 7.16140i) q^{55} +(3.37670 - 2.92593i) q^{56} +(-0.342365 + 1.57383i) q^{57} +(0.255098 - 1.17267i) q^{58} +(-0.100712 + 0.0872672i) q^{59} +(-0.267281 + 1.11117i) q^{60} +(8.41993 + 1.21060i) q^{61} +(4.78375 - 0.342141i) q^{62} +(-11.4653 + 4.27635i) q^{63} +(-0.281733 + 0.959493i) q^{64} +(-1.19422 + 12.7687i) q^{65} +(3.10315 - 0.446166i) q^{66} +(3.29980 - 0.717828i) q^{67} +(-0.402858 - 0.402858i) q^{68} +(1.04012 - 2.21954i) q^{69} +(-2.60455 - 9.64531i) q^{70} +(-4.82259 - 3.09929i) q^{71} +(1.64129 - 2.19250i) q^{72} +(-0.0305242 - 0.0113849i) q^{73} +(-3.55158 - 1.04284i) q^{74} +(1.97692 + 1.61940i) q^{75} +(2.38159 + 2.06366i) q^{76} +(-21.9399 + 16.4240i) q^{77} +(-1.40482 + 2.57274i) q^{78} +(-2.54719 - 2.93961i) q^{79} +(1.65750 + 1.50090i) q^{80} +(-5.65087 + 3.63159i) q^{81} +(0.236368 - 3.30486i) q^{82} +(15.8196 + 8.63813i) q^{83} +(0.324992 - 2.26037i) q^{84} +(-1.20307 + 0.419007i) q^{85} +(-5.81670 - 2.65640i) q^{86} +(-0.293958 - 0.538344i) q^{87} +(2.14358 - 5.74716i) q^{88} +(0.587398 + 4.08544i) q^{89} +(-2.81679 - 5.43784i) q^{90} -25.6251i q^{91} +(-2.83616 - 3.86732i) q^{92} +(1.73329 - 1.73329i) q^{93} +(2.17022 - 3.37693i) q^{94} +(6.47216 - 2.78644i) q^{95} +(0.212320 + 0.464916i) q^{96} +(8.59298 - 4.69212i) q^{97} +(4.53015 + 12.1458i) q^{98} +(-11.0013 + 12.6961i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{13}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.212565 − 0.977147i −0.150306 − 0.690947i
\(3\) −0.409159 − 0.306293i −0.236228 − 0.176838i 0.474605 − 0.880199i \(-0.342591\pi\)
−0.710833 + 0.703361i \(0.751682\pi\)
\(4\) −0.909632 + 0.415415i −0.454816 + 0.207708i
\(5\) −0.0488687 + 2.23553i −0.0218547 + 0.999761i
\(6\) −0.212320 + 0.464916i −0.0866793 + 0.189801i
\(7\) −0.318744 − 4.45663i −0.120474 − 1.68445i −0.594625 − 0.804003i \(-0.702699\pi\)
0.474151 − 0.880444i \(-0.342755\pi\)
\(8\) 0.599278 + 0.800541i 0.211877 + 0.283034i
\(9\) −0.771602 − 2.62783i −0.257201 − 0.875945i
\(10\) 2.19483 − 0.427445i 0.694067 − 0.135170i
\(11\) −3.31624 − 5.16017i −0.999884 − 1.55585i −0.820161 − 0.572132i \(-0.806116\pi\)
−0.179722 − 0.983717i \(-0.557520\pi\)
\(12\) 0.499423 + 0.108643i 0.144171 + 0.0313625i
\(13\) 5.72063 + 0.409147i 1.58662 + 0.113477i 0.836552 − 0.547888i \(-0.184568\pi\)
0.750065 + 0.661365i \(0.230022\pi\)
\(14\) −4.28703 + 1.25878i −1.14576 + 0.336424i
\(15\) 0.704723 − 0.899722i 0.181959 − 0.232307i
\(16\) 0.654861 − 0.755750i 0.163715 − 0.188937i
\(17\) 0.199099 + 0.533806i 0.0482886 + 0.129467i 0.958855 − 0.283896i \(-0.0916269\pi\)
−0.910567 + 0.413363i \(0.864354\pi\)
\(18\) −2.40376 + 1.31255i −0.566573 + 0.309372i
\(19\) −1.30909 − 2.86651i −0.300326 − 0.657623i 0.697960 − 0.716137i \(-0.254091\pi\)
−0.998287 + 0.0585133i \(0.981364\pi\)
\(20\) −0.884222 − 2.05381i −0.197718 − 0.459247i
\(21\) −1.23462 + 1.92110i −0.269415 + 0.419218i
\(22\) −4.33733 + 4.33733i −0.924721 + 0.924721i
\(23\) 0.973320 + 4.69602i 0.202951 + 0.979189i
\(24\) − 0.511103i − 0.104329i
\(25\) −4.99522 − 0.218495i −0.999045 − 0.0436990i
\(26\) −0.816210 − 5.67686i −0.160072 − 1.11332i
\(27\) −1.02502 + 2.74817i −0.197264 + 0.528886i
\(28\) 2.14129 + 3.92148i 0.404666 + 0.741090i
\(29\) 1.09164 + 0.498537i 0.202713 + 0.0925760i 0.514185 − 0.857679i \(-0.328095\pi\)
−0.311472 + 0.950255i \(0.600822\pi\)
\(30\) −1.02896 − 0.497368i −0.187861 − 0.0908066i
\(31\) −0.682538 + 4.74716i −0.122587 + 0.852615i 0.832019 + 0.554746i \(0.187185\pi\)
−0.954607 + 0.297868i \(0.903724\pi\)
\(32\) −0.877679 − 0.479249i −0.155153 − 0.0847201i
\(33\) −0.223653 + 3.12707i −0.0389329 + 0.544353i
\(34\) 0.479285 − 0.308018i 0.0821967 − 0.0528246i
\(35\) 9.97852 − 0.494774i 1.68668 − 0.0836321i
\(36\) 1.79352 + 2.06983i 0.298919 + 0.344971i
\(37\) 1.77395 − 3.24874i 0.291635 − 0.534090i −0.690249 − 0.723572i \(-0.742499\pi\)
0.981884 + 0.189482i \(0.0606809\pi\)
\(38\) −2.52274 + 1.88850i −0.409242 + 0.306355i
\(39\) −2.21533 − 1.91959i −0.354737 − 0.307381i
\(40\) −1.81892 + 1.30058i −0.287597 + 0.205640i
\(41\) 3.17909 + 0.933464i 0.496490 + 0.145783i 0.520385 − 0.853932i \(-0.325789\pi\)
−0.0238950 + 0.999714i \(0.507607\pi\)
\(42\) 2.13963 + 0.798042i 0.330153 + 0.123141i
\(43\) 3.83212 − 5.11911i 0.584393 − 0.780657i −0.406739 − 0.913545i \(-0.633334\pi\)
0.991131 + 0.132887i \(0.0424249\pi\)
\(44\) 5.16017 + 3.31624i 0.777925 + 0.499942i
\(45\) 5.91232 − 1.59652i 0.881356 − 0.237996i
\(46\) 4.38181 − 1.94929i 0.646063 − 0.287407i
\(47\) 2.83845 + 2.83845i 0.414030 + 0.414030i 0.883140 − 0.469110i \(-0.155425\pi\)
−0.469110 + 0.883140i \(0.655425\pi\)
\(48\) −0.499423 + 0.108643i −0.0720855 + 0.0156812i
\(49\) −12.8312 + 1.84484i −1.83303 + 0.263549i
\(50\) 0.848309 + 4.92751i 0.119969 + 0.696855i
\(51\) 0.0820375 − 0.279394i 0.0114876 − 0.0391230i
\(52\) −5.37363 + 2.00426i −0.745189 + 0.277941i
\(53\) −3.02013 + 0.216004i −0.414847 + 0.0296704i −0.277203 − 0.960811i \(-0.589407\pi\)
−0.137644 + 0.990482i \(0.543953\pi\)
\(54\) 2.90325 + 0.417425i 0.395083 + 0.0568043i
\(55\) 11.6978 − 7.16140i 1.57733 − 0.965642i
\(56\) 3.37670 − 2.92593i 0.451230 − 0.390993i
\(57\) −0.342365 + 1.57383i −0.0453474 + 0.208458i
\(58\) 0.255098 − 1.17267i 0.0334961 − 0.153979i
\(59\) −0.100712 + 0.0872672i −0.0131115 + 0.0113612i −0.661392 − 0.750041i \(-0.730034\pi\)
0.648280 + 0.761402i \(0.275488\pi\)
\(60\) −0.267281 + 1.11117i −0.0345058 + 0.143451i
\(61\) 8.41993 + 1.21060i 1.07806 + 0.155002i 0.658392 − 0.752675i \(-0.271237\pi\)
0.419669 + 0.907677i \(0.362146\pi\)
\(62\) 4.78375 − 0.342141i 0.607537 − 0.0434519i
\(63\) −11.4653 + 4.27635i −1.44450 + 0.538769i
\(64\) −0.281733 + 0.959493i −0.0352166 + 0.119937i
\(65\) −1.19422 + 12.7687i −0.148125 + 1.58376i
\(66\) 3.10315 − 0.446166i 0.381971 − 0.0549192i
\(67\) 3.29980 − 0.717828i 0.403135 − 0.0876967i −0.00642607 − 0.999979i \(-0.502045\pi\)
0.409561 + 0.912283i \(0.365682\pi\)
\(68\) −0.402858 − 0.402858i −0.0488537 − 0.0488537i
\(69\) 1.04012 − 2.21954i 0.125215 − 0.267202i
\(70\) −2.60455 − 9.64531i −0.311304 − 1.15283i
\(71\) −4.82259 − 3.09929i −0.572336 − 0.367818i 0.222231 − 0.974994i \(-0.428666\pi\)
−0.794567 + 0.607176i \(0.792302\pi\)
\(72\) 1.64129 − 2.19250i 0.193427 − 0.258389i
\(73\) −0.0305242 − 0.0113849i −0.00357258 − 0.00133250i 0.347677 − 0.937614i \(-0.386970\pi\)
−0.351250 + 0.936282i \(0.614243\pi\)
\(74\) −3.55158 − 1.04284i −0.412862 − 0.121227i
\(75\) 1.97692 + 1.61940i 0.228275 + 0.186992i
\(76\) 2.38159 + 2.06366i 0.273187 + 0.236718i
\(77\) −21.9399 + 16.4240i −2.50029 + 1.87169i
\(78\) −1.40482 + 2.57274i −0.159065 + 0.291306i
\(79\) −2.54719 − 2.93961i −0.286581 − 0.330732i 0.594145 − 0.804358i \(-0.297490\pi\)
−0.880726 + 0.473626i \(0.842945\pi\)
\(80\) 1.65750 + 1.50090i 0.185314 + 0.167805i
\(81\) −5.65087 + 3.63159i −0.627874 + 0.403510i
\(82\) 0.236368 − 3.30486i 0.0261025 − 0.364960i
\(83\) 15.8196 + 8.63813i 1.73642 + 0.948158i 0.938416 + 0.345508i \(0.112293\pi\)
0.798006 + 0.602650i \(0.205889\pi\)
\(84\) 0.324992 − 2.26037i 0.0354596 − 0.246627i
\(85\) −1.20307 + 0.419007i −0.130491 + 0.0454476i
\(86\) −5.81670 − 2.65640i −0.627231 − 0.286447i
\(87\) −0.293958 − 0.538344i −0.0315156 − 0.0577165i
\(88\) 2.14358 − 5.74716i 0.228506 − 0.612649i
\(89\) 0.587398 + 4.08544i 0.0622640 + 0.433056i 0.996980 + 0.0776600i \(0.0247449\pi\)
−0.934716 + 0.355396i \(0.884346\pi\)
\(90\) −2.81679 − 5.43784i −0.296916 − 0.573198i
\(91\) − 25.6251i − 2.68624i
\(92\) −2.83616 − 3.86732i −0.295690 − 0.403196i
\(93\) 1.73329 − 1.73329i 0.179734 − 0.179734i
\(94\) 2.17022 − 3.37693i 0.223841 − 0.348304i
\(95\) 6.47216 − 2.78644i 0.664030 − 0.285883i
\(96\) 0.212320 + 0.464916i 0.0216698 + 0.0474503i
\(97\) 8.59298 − 4.69212i 0.872485 − 0.476413i 0.0203850 − 0.999792i \(-0.493511\pi\)
0.852100 + 0.523380i \(0.175329\pi\)
\(98\) 4.53015 + 12.1458i 0.457614 + 1.22691i
\(99\) −11.0013 + 12.6961i −1.10567 + 1.27601i
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.67.2 yes 240
5.3 odd 4 inner 230.2.l.a.113.5 yes 240
23.11 odd 22 inner 230.2.l.a.57.5 ✓ 240
115.103 even 44 inner 230.2.l.a.103.2 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.57.5 ✓ 240 23.11 odd 22 inner
230.2.l.a.67.2 yes 240 1.1 even 1 trivial
230.2.l.a.103.2 yes 240 115.103 even 44 inner
230.2.l.a.113.5 yes 240 5.3 odd 4 inner