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Properties

Label 230.2.l.a.7.8
Level $230$
Weight $2$
Character 230.7
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 7.8
Character \(\chi\) \(=\) 230.7
Dual form 230.2.l.a.33.8

$q$-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+(0.997452 - 0.0713392i) q^{2} +(-0.372248 + 1.71120i) q^{3} +(0.989821 - 0.142315i) q^{4} +(-2.13965 + 0.649526i) q^{5} +(-0.249224 + 1.73339i) q^{6} +(-1.73066 + 3.16947i) q^{7} +(0.977147 - 0.212565i) q^{8} +(-0.0607343 - 0.0277364i) q^{9} +(-2.08786 + 0.800513i) q^{10} +(-1.69797 + 1.47130i) q^{11} +(-0.124931 + 1.74676i) q^{12} +(4.63061 - 2.52851i) q^{13} +(-1.50014 + 3.28486i) q^{14} +(-0.314986 - 3.90316i) q^{15} +(0.959493 - 0.281733i) q^{16} +(1.55083 + 2.07167i) q^{17} +(-0.0625582 - 0.0233330i) q^{18} +(-0.731070 - 5.08470i) q^{19} +(-2.02544 + 0.947420i) q^{20} +(-4.77935 - 4.14133i) q^{21} +(-1.58869 + 1.58869i) q^{22} +(4.14027 + 2.42036i) q^{23} +1.75122i q^{24} +(4.15623 - 2.77952i) q^{25} +(4.43844 - 2.85241i) q^{26} +(-3.07833 + 4.11217i) q^{27} +(-1.26198 + 3.38351i) q^{28} +(4.62333 + 0.664735i) q^{29} +(-0.592631 - 3.87074i) q^{30} +(-7.65941 - 4.92241i) q^{31} +(0.936950 - 0.349464i) q^{32} +(-1.88562 - 3.45326i) q^{33} +(1.69467 + 1.95575i) q^{34} +(1.64436 - 7.90568i) q^{35} +(-0.0640634 - 0.0188107i) q^{36} +(-1.06956 - 2.86759i) q^{37} +(-1.09195 - 5.01960i) q^{38} +(2.60304 + 8.86513i) q^{39} +(-1.95269 + 1.08950i) q^{40} +(1.83808 + 4.02483i) q^{41} +(-5.06262 - 3.78983i) q^{42} +(11.7859 + 2.56386i) q^{43} +(-1.47130 + 1.69797i) q^{44} +(0.147966 + 0.0198978i) q^{45} +(4.30239 + 2.11883i) q^{46} +(-0.0966659 - 0.0966659i) q^{47} +(0.124931 + 1.74676i) q^{48} +(-3.26586 - 5.08178i) q^{49} +(3.94735 - 3.06894i) q^{50} +(-4.12232 + 1.88260i) q^{51} +(4.22364 - 3.16178i) q^{52} +(-7.42874 - 4.05640i) q^{53} +(-2.77713 + 4.32129i) q^{54} +(2.67742 - 4.25095i) q^{55} +(-1.01739 + 3.46492i) q^{56} +(8.97308 + 0.641767i) q^{57} +(4.65898 + 0.333216i) q^{58} +(1.72602 - 5.87829i) q^{59} +(-0.867257 - 3.81860i) q^{60} +(6.06094 - 9.43101i) q^{61} +(-7.99106 - 4.36345i) q^{62} +(0.193020 - 0.144493i) q^{63} +(0.909632 - 0.415415i) q^{64} +(-8.26558 + 8.41783i) q^{65} +(-2.12717 - 3.30994i) q^{66} +(0.697475 + 9.75197i) q^{67} +(1.82987 + 1.82987i) q^{68} +(-5.68292 + 6.18385i) q^{69} +(1.07619 - 8.00284i) q^{70} +(-3.94966 + 4.55815i) q^{71} +(-0.0652421 - 0.0141926i) q^{72} +(-5.36891 - 4.01911i) q^{73} +(-1.27140 - 2.78398i) q^{74} +(3.20916 + 8.14681i) q^{75} +(-1.44726 - 4.92891i) q^{76} +(-1.72463 - 7.92800i) q^{77} +(3.22884 + 8.65685i) q^{78} +(-2.52895 - 0.742567i) q^{79} +(-1.86999 + 1.22603i) q^{80} +(-6.02200 - 6.94976i) q^{81} +(2.12053 + 3.88345i) q^{82} +(-4.32977 + 1.61492i) q^{83} +(-5.32008 - 3.41901i) q^{84} +(-4.66384 - 3.42534i) q^{85} +(11.9387 + 1.71653i) q^{86} +(-2.85852 + 7.66399i) q^{87} +(-1.34642 + 1.79861i) q^{88} +(-11.8000 + 7.58338i) q^{89} +(0.149008 + 0.00929133i) q^{90} +19.0526i q^{91} +(4.44258 + 1.80650i) q^{92} +(11.2744 - 11.2744i) q^{93} +(-0.103316 - 0.0895236i) q^{94} +(4.86689 + 10.4047i) q^{95} +(0.249224 + 1.73339i) q^{96} +(7.86547 + 2.93367i) q^{97} +(-3.62007 - 4.83585i) q^{98} +(0.143934 - 0.0422628i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{19}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.997452 − 0.0713392i 0.705305 − 0.0504444i
\(3\) −0.372248 + 1.71120i −0.214918 + 0.987961i 0.735630 + 0.677383i \(0.236886\pi\)
−0.950548 + 0.310578i \(0.899478\pi\)
\(4\) 0.989821 − 0.142315i 0.494911 − 0.0711574i
\(5\) −2.13965 + 0.649526i −0.956882 + 0.290477i
\(6\) −0.249224 + 1.73339i −0.101745 + 0.707655i
\(7\) −1.73066 + 3.16947i −0.654129 + 1.19795i 0.314458 + 0.949271i \(0.398177\pi\)
−0.968587 + 0.248676i \(0.920005\pi\)
\(8\) 0.977147 − 0.212565i 0.345474 − 0.0751532i
\(9\) −0.0607343 − 0.0277364i −0.0202448 − 0.00924547i
\(10\) −2.08786 + 0.800513i −0.660241 + 0.253144i
\(11\) −1.69797 + 1.47130i −0.511958 + 0.443614i −0.872132 − 0.489271i \(-0.837263\pi\)
0.360174 + 0.932885i \(0.382717\pi\)
\(12\) −0.124931 + 1.74676i −0.0360643 + 0.504245i
\(13\) 4.63061 − 2.52851i 1.28430 − 0.701282i 0.315454 − 0.948941i \(-0.397843\pi\)
0.968847 + 0.247659i \(0.0796613\pi\)
\(14\) −1.50014 + 3.28486i −0.400930 + 0.877915i
\(15\) −0.314986 − 3.90316i −0.0813290 − 1.00779i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) 1.55083 + 2.07167i 0.376131 + 0.502453i 0.948165 − 0.317777i \(-0.102936\pi\)
−0.572034 + 0.820230i \(0.693845\pi\)
\(18\) −0.0625582 − 0.0233330i −0.0147451 − 0.00549964i
\(19\) −0.731070 − 5.08470i −0.167719 − 1.16651i −0.883584 − 0.468273i \(-0.844877\pi\)
0.715865 − 0.698239i \(-0.246033\pi\)
\(20\) −2.02544 + 0.947420i −0.452902 + 0.211849i
\(21\) −4.77935 − 4.14133i −1.04294 − 0.903713i
\(22\) −1.58869 + 1.58869i −0.338709 + 0.338709i
\(23\) 4.14027 + 2.42036i 0.863307 + 0.504680i
\(24\) 1.75122i 0.357466i
\(25\) 4.15623 − 2.77952i 0.831246 − 0.555904i
\(26\) 4.43844 − 2.85241i 0.870449 − 0.559403i
\(27\) −3.07833 + 4.11217i −0.592425 + 0.791387i
\(28\) −1.26198 + 3.38351i −0.238492 + 0.639423i
\(29\) 4.62333 + 0.664735i 0.858531 + 0.123438i 0.557501 − 0.830176i \(-0.311760\pi\)
0.301031 + 0.953615i \(0.402669\pi\)
\(30\) −0.592631 − 3.87074i −0.108199 − 0.706697i
\(31\) −7.65941 − 4.92241i −1.37567 − 0.884090i −0.376566 − 0.926390i \(-0.622895\pi\)
−0.999105 + 0.0422994i \(0.986532\pi\)
\(32\) 0.936950 − 0.349464i 0.165631 − 0.0617771i
\(33\) −1.88562 − 3.45326i −0.328245 − 0.601135i
\(34\) 1.69467 + 1.95575i 0.290633 + 0.335409i
\(35\) 1.64436 − 7.90568i 0.277948 − 1.33630i
\(36\) −0.0640634 − 0.0188107i −0.0106772 − 0.00313512i
\(37\) −1.06956 − 2.86759i −0.175834 − 0.471429i 0.818898 − 0.573939i \(-0.194585\pi\)
−0.994732 + 0.102510i \(0.967313\pi\)
\(38\) −1.09195 − 5.01960i −0.177137 − 0.814286i
\(39\) 2.60304 + 8.86513i 0.416820 + 1.41956i
\(40\) −1.95269 + 1.08950i −0.308747 + 0.172265i
\(41\) 1.83808 + 4.02483i 0.287060 + 0.628574i 0.997142 − 0.0755450i \(-0.0240696\pi\)
−0.710082 + 0.704119i \(0.751342\pi\)
\(42\) −5.06262 − 3.78983i −0.781179 − 0.584783i
\(43\) 11.7859 + 2.56386i 1.79733 + 0.390985i 0.982131 − 0.188200i \(-0.0602653\pi\)
0.815197 + 0.579184i \(0.196629\pi\)
\(44\) −1.47130 + 1.69797i −0.221807 + 0.255979i
\(45\) 0.147966 + 0.0198978i 0.0220574 + 0.00296619i
\(46\) 4.30239 + 2.11883i 0.634353 + 0.312404i
\(47\) −0.0966659 − 0.0966659i −0.0141002 − 0.0141002i 0.700022 − 0.714122i \(-0.253174\pi\)
−0.714122 + 0.700022i \(0.753174\pi\)
\(48\) 0.124931 + 1.74676i 0.0180322 + 0.252123i
\(49\) −3.26586 − 5.08178i −0.466552 − 0.725969i
\(50\) 3.94735 − 3.06894i 0.558240 − 0.434014i
\(51\) −4.12232 + 1.88260i −0.577241 + 0.263617i
\(52\) 4.22364 − 3.16178i 0.585713 − 0.438459i
\(53\) −7.42874 − 4.05640i −1.02042 − 0.557189i −0.120253 − 0.992743i \(-0.538371\pi\)
−0.900163 + 0.435554i \(0.856552\pi\)
\(54\) −2.77713 + 4.32129i −0.377919 + 0.588054i
\(55\) 2.67742 − 4.25095i 0.361024 − 0.573199i
\(56\) −1.01739 + 3.46492i −0.135955 + 0.463019i
\(57\) 8.97308 + 0.641767i 1.18851 + 0.0850041i
\(58\) 4.65898 + 0.333216i 0.611753 + 0.0437535i
\(59\) 1.72602 − 5.87829i 0.224709 − 0.765288i −0.767538 − 0.641004i \(-0.778518\pi\)
0.992247 − 0.124284i \(-0.0396635\pi\)
\(60\) −0.867257 − 3.81860i −0.111962 − 0.492979i
\(61\) 6.06094 − 9.43101i 0.776024 − 1.20752i −0.197806 − 0.980241i \(-0.563382\pi\)
0.973830 − 0.227276i \(-0.0729820\pi\)
\(62\) −7.99106 − 4.36345i −1.01487 − 0.554158i
\(63\) 0.193020 − 0.144493i 0.0243183 − 0.0182044i
\(64\) 0.909632 − 0.415415i 0.113704 − 0.0519269i
\(65\) −8.26558 + 8.41783i −1.02522 + 1.04410i
\(66\) −2.12717 − 3.30994i −0.261837 − 0.407426i
\(67\) 0.697475 + 9.75197i 0.0852101 + 1.19139i 0.842640 + 0.538477i \(0.181000\pi\)
−0.757430 + 0.652916i \(0.773545\pi\)
\(68\) 1.82987 + 1.82987i 0.221905 + 0.221905i
\(69\) −5.68292 + 6.18385i −0.684143 + 0.744449i
\(70\) 1.07619 − 8.00284i 0.128629 − 0.956522i
\(71\) −3.94966 + 4.55815i −0.468739 + 0.540953i −0.940060 − 0.341008i \(-0.889232\pi\)
0.471322 + 0.881961i \(0.343777\pi\)
\(72\) −0.0652421 − 0.0141926i −0.00768886 − 0.00167261i
\(73\) −5.36891 − 4.01911i −0.628383 − 0.470402i 0.237002 − 0.971509i \(-0.423835\pi\)
−0.865385 + 0.501108i \(0.832926\pi\)
\(74\) −1.27140 − 2.78398i −0.147798 − 0.323632i
\(75\) 3.20916 + 8.14681i 0.370562 + 0.940712i
\(76\) −1.44726 − 4.92891i −0.166012 − 0.565384i
\(77\) −1.72463 − 7.92800i −0.196540 − 0.903480i
\(78\) 3.22884 + 8.65685i 0.365594 + 0.980195i
\(79\) −2.52895 − 0.742567i −0.284529 − 0.0835454i 0.136354 − 0.990660i \(-0.456462\pi\)
−0.420883 + 0.907115i \(0.638280\pi\)
\(80\) −1.86999 + 1.22603i −0.209071 + 0.137074i
\(81\) −6.02200 − 6.94976i −0.669111 − 0.772195i
\(82\) 2.12053 + 3.88345i 0.234173 + 0.428856i
\(83\) −4.32977 + 1.61492i −0.475254 + 0.177261i −0.575666 − 0.817685i \(-0.695257\pi\)
0.100411 + 0.994946i \(0.467984\pi\)
\(84\) −5.32008 − 3.41901i −0.580468 − 0.373044i
\(85\) −4.66384 − 3.42534i −0.505864 − 0.371530i
\(86\) 11.9387 + 1.71653i 1.28739 + 0.185098i
\(87\) −2.85852 + 7.66399i −0.306466 + 0.821666i
\(88\) −1.34642 + 1.79861i −0.143529 + 0.191732i
\(89\) −11.8000 + 7.58338i −1.25079 + 0.803836i −0.986996 − 0.160742i \(-0.948611\pi\)
−0.263797 + 0.964578i \(0.584975\pi\)
\(90\) 0.149008 + 0.00929133i 0.0157069 + 0.000979393i
\(91\) 19.0526i 1.99725i
\(92\) 4.44258 + 1.80650i 0.463171 + 0.188341i
\(93\) 11.2744 − 11.2744i 1.16910 − 1.16910i
\(94\) −0.103316 − 0.0895236i −0.0106562 − 0.00923365i
\(95\) 4.86689 + 10.4047i 0.499332 + 1.06750i
\(96\) 0.249224 + 1.73339i 0.0254364 + 0.176914i
\(97\) 7.86547 + 2.93367i 0.798618 + 0.297869i 0.715448 − 0.698666i \(-0.246223\pi\)
0.0831701 + 0.996535i \(0.473496\pi\)
\(98\) −3.62007 − 4.83585i −0.365683 − 0.488495i
\(99\) 0.143934 − 0.0422628i 0.0144659 − 0.00424757i
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.7.8 ✓ 240
5.3 odd 4 inner 230.2.l.a.53.2 yes 240
23.10 odd 22 inner 230.2.l.a.217.2 yes 240
115.33 even 44 inner 230.2.l.a.33.8 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.8 ✓ 240 1.1 even 1 trivial
230.2.l.a.33.8 yes 240 115.33 even 44 inner
230.2.l.a.53.2 yes 240 5.3 odd 4 inner
230.2.l.a.217.2 yes 240 23.10 odd 22 inner