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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 53.2
Character \(\chi\) \(=\) 230.53
Dual form 230.2.l.a.217.2

$q$-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+(-0.0713392 - 0.997452i) q^{2} +(-1.71120 - 0.372248i) q^{3} +(-0.989821 + 0.142315i) q^{4} +(-1.89205 + 1.19169i) q^{5} +(-0.249224 + 1.73339i) q^{6} +(3.16947 + 1.73066i) q^{7} +(0.212565 + 0.977147i) q^{8} +(0.0607343 + 0.0277364i) q^{9} +(1.32363 + 1.80222i) q^{10} +(-1.69797 + 1.47130i) q^{11} +(1.74676 + 0.124931i) q^{12} +(2.52851 + 4.63061i) q^{13} +(1.50014 - 3.28486i) q^{14} +(3.68129 - 1.33491i) q^{15} +(0.959493 - 0.281733i) q^{16} +(2.07167 - 1.55083i) q^{17} +(0.0233330 - 0.0625582i) q^{18} +(0.731070 + 5.08470i) q^{19} +(1.70320 - 1.44883i) q^{20} +(-4.77935 - 4.14133i) q^{21} +(1.58869 + 1.58869i) q^{22} +(-2.42036 + 4.14027i) q^{23} -1.75122i q^{24} +(2.15974 - 4.50949i) q^{25} +(4.43844 - 2.85241i) q^{26} +(4.11217 + 3.07833i) q^{27} +(-3.38351 - 1.26198i) q^{28} +(-4.62333 - 0.664735i) q^{29} +(-1.59413 - 3.57667i) q^{30} +(-7.65941 - 4.92241i) q^{31} +(-0.349464 - 0.936950i) q^{32} +(3.45326 - 1.88562i) q^{33} +(-1.69467 - 1.95575i) q^{34} +(-8.05922 + 0.502529i) q^{35} +(-0.0640634 - 0.0188107i) q^{36} +(-2.86759 + 1.06956i) q^{37} +(5.01960 - 1.09195i) q^{38} +(-2.60304 - 8.86513i) q^{39} +(-1.56664 - 1.59550i) q^{40} +(1.83808 + 4.02483i) q^{41} +(-3.78983 + 5.06262i) q^{42} +(-2.56386 + 11.7859i) 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} +(-1.74676 + 0.124931i) q^{48} +(3.26586 + 5.08178i) q^{49} +(-4.65208 - 1.83253i) q^{50} +(-4.12232 + 1.88260i) q^{51} +(-3.16178 - 4.22364i) q^{52} +(4.05640 - 7.42874i) q^{53} +(2.77713 - 4.32129i) q^{54} +(1.45932 - 4.80725i) q^{55} +(-1.01739 + 3.46492i) q^{56} +(0.641767 - 8.97308i) q^{57} +(-0.333216 + 4.65898i) q^{58} +(-1.72602 + 5.87829i) q^{59} +(-3.45384 + 1.84522i) q^{60} +(6.06094 - 9.43101i) q^{61} +(-4.36345 + 7.99106i) q^{62} +(0.144493 + 0.193020i) q^{63} +(-0.909632 + 0.415415i) q^{64} +(-10.3023 - 5.74817i) q^{65} +(-2.12717 - 3.30994i) q^{66} +(9.75197 - 0.697475i) 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} +(-0.0141926 + 0.0652421i) q^{72} +(4.01911 - 5.36891i) q^{73} +(1.27140 + 2.78398i) q^{74} +(-5.37439 + 6.91268i) q^{75} +(-1.44726 - 4.92891i) q^{76} +(-7.92800 + 1.72463i) q^{77} +(-8.65685 + 3.22884i) q^{78} +(2.52895 + 0.742567i) q^{79} +(-1.47967 + 1.67647i) q^{80} +(-6.02200 - 6.94976i) q^{81} +(3.88345 - 2.12053i) q^{82} +(-1.61492 - 4.32977i) q^{83} +(5.32008 + 3.41901i) q^{84} +(-2.07159 + 5.40304i) q^{85} +(11.9387 + 1.71653i) q^{86} +(7.66399 + 2.85852i) q^{87} +(-1.79861 - 1.34642i) q^{88} +(11.8000 - 7.58338i) q^{89} +(0.0304029 + 0.146169i) q^{90} +19.0526i q^{91} +(1.80650 - 4.44258i) q^{92} +(11.2744 + 11.2744i) q^{93} +(0.103316 + 0.0895236i) q^{94} +(-7.44263 - 8.74932i) q^{95} +(0.249224 + 1.73339i) q^{96} +(2.93367 - 7.86547i) q^{97} +(4.83585 - 3.62007i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{19}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0713392 − 0.997452i −0.0504444 − 0.705305i
\(3\) −1.71120 − 0.372248i −0.987961 − 0.214918i −0.310578 − 0.950548i \(-0.600522\pi\)
−0.677383 + 0.735630i \(0.736886\pi\)
\(4\) −0.989821 + 0.142315i −0.494911 + 0.0711574i
\(5\) −1.89205 + 1.19169i −0.846152 + 0.532941i
\(6\) −0.249224 + 1.73339i −0.101745 + 0.707655i
\(7\) 3.16947 + 1.73066i 1.19795 + 0.654129i 0.949271 − 0.314458i \(-0.101823\pi\)
0.248676 + 0.968587i \(0.420005\pi\)
\(8\) 0.212565 + 0.977147i 0.0751532 + 0.345474i
\(9\) 0.0607343 + 0.0277364i 0.0202448 + 0.00924547i
\(10\) 1.32363 + 1.80222i 0.418570 + 0.569912i
\(11\) −1.69797 + 1.47130i −0.511958 + 0.443614i −0.872132 − 0.489271i \(-0.837263\pi\)
0.360174 + 0.932885i \(0.382717\pi\)
\(12\) 1.74676 + 0.124931i 0.504245 + 0.0360643i
\(13\) 2.52851 + 4.63061i 0.701282 + 1.28430i 0.948941 + 0.315454i \(0.102157\pi\)
−0.247659 + 0.968847i \(0.579661\pi\)
\(14\) 1.50014 − 3.28486i 0.400930 − 0.877915i
\(15\) 3.68129 − 1.33491i 0.950504 − 0.344672i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) 2.07167 − 1.55083i 0.502453 − 0.376131i −0.317777 − 0.948165i \(-0.602936\pi\)
0.820230 + 0.572034i \(0.193845\pi\)
\(18\) 0.0233330 − 0.0625582i 0.00549964 − 0.0147451i
\(19\) 0.731070 + 5.08470i 0.167719 + 1.16651i 0.883584 + 0.468273i \(0.155123\pi\)
−0.715865 + 0.698239i \(0.753967\pi\)
\(20\) 1.70320 − 1.44883i 0.380847 − 0.323968i
\(21\) −4.77935 − 4.14133i −1.04294 − 0.903713i
\(22\) 1.58869 + 1.58869i 0.338709 + 0.338709i
\(23\) −2.42036 + 4.14027i −0.504680 + 0.863307i
\(24\) − 1.75122i − 0.357466i
\(25\) 2.15974 − 4.50949i 0.431947 − 0.901899i
\(26\) 4.43844 − 2.85241i 0.870449 − 0.559403i
\(27\) 4.11217 + 3.07833i 0.791387 + 0.592425i
\(28\) −3.38351 − 1.26198i −0.639423 − 0.238492i
\(29\) −4.62333 − 0.664735i −0.858531 − 0.123438i −0.301031 − 0.953615i \(-0.597331\pi\)
−0.557501 + 0.830176i \(0.688240\pi\)
\(30\) −1.59413 − 3.57667i −0.291046 − 0.653008i
\(31\) −7.65941 − 4.92241i −1.37567 − 0.884090i −0.376566 − 0.926390i \(-0.622895\pi\)
−0.999105 + 0.0422994i \(0.986532\pi\)
\(32\) −0.349464 − 0.936950i −0.0617771 − 0.165631i
\(33\) 3.45326 − 1.88562i 0.601135 − 0.328245i
\(34\) −1.69467 − 1.95575i −0.290633 − 0.335409i
\(35\) −8.05922 + 0.502529i −1.36226 + 0.0849429i
\(36\) −0.0640634 − 0.0188107i −0.0106772 − 0.00313512i
\(37\) −2.86759 + 1.06956i −0.471429 + 0.175834i −0.573939 − 0.818898i \(-0.694585\pi\)
0.102510 + 0.994732i \(0.467313\pi\)
\(38\) 5.01960 − 1.09195i 0.814286 − 0.177137i
\(39\) −2.60304 − 8.86513i −0.416820 − 1.41956i
\(40\) −1.56664 − 1.59550i −0.247708 − 0.252271i
\(41\) 1.83808 + 4.02483i 0.287060 + 0.628574i 0.997142 − 0.0755450i \(-0.0240696\pi\)
−0.710082 + 0.704119i \(0.751342\pi\)
\(42\) −3.78983 + 5.06262i −0.584783 + 0.781179i
\(43\) −2.56386 + 11.7859i −0.390985 + 1.79733i 0.188200 + 0.982131i \(0.439735\pi\)
−0.579184 + 0.815197i \(0.696629\pi\)
\(44\) 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 −0.714122 − 0.700022i \(-0.753174\pi\)
0.700022 + 0.714122i \(0.253174\pi\)
\(48\) −1.74676 + 0.124931i −0.252123 + 0.0180322i
\(49\) 3.26586 + 5.08178i 0.466552 + 0.725969i
\(50\) −4.65208 − 1.83253i −0.657903 − 0.259159i
\(51\) −4.12232 + 1.88260i −0.577241 + 0.263617i
\(52\) −3.16178 − 4.22364i −0.438459 − 0.585713i
\(53\) 4.05640 − 7.42874i 0.557189 − 1.02042i −0.435554 − 0.900163i \(-0.643448\pi\)
0.992743 − 0.120253i \(-0.0383706\pi\)
\(54\) 2.77713 − 4.32129i 0.377919 − 0.588054i
\(55\) 1.45932 − 4.80725i 0.196774 − 0.648209i
\(56\) −1.01739 + 3.46492i −0.135955 + 0.463019i
\(57\) 0.641767 − 8.97308i 0.0850041 − 1.18851i
\(58\) −0.333216 + 4.65898i −0.0437535 + 0.611753i
\(59\) −1.72602 + 5.87829i −0.224709 + 0.765288i 0.767538 + 0.641004i \(0.221482\pi\)
−0.992247 + 0.124284i \(0.960337\pi\)
\(60\) −3.45384 + 1.84522i −0.445889 + 0.238217i
\(61\) 6.06094 − 9.43101i 0.776024 − 1.20752i −0.197806 − 0.980241i \(-0.563382\pi\)
0.973830 − 0.227276i \(-0.0729820\pi\)
\(62\) −4.36345 + 7.99106i −0.554158 + 1.01487i
\(63\) 0.144493 + 0.193020i 0.0182044 + 0.0243183i
\(64\) −0.909632 + 0.415415i −0.113704 + 0.0519269i
\(65\) −10.3023 − 5.74817i −1.27785 − 0.712973i
\(66\) −2.12717 − 3.30994i −0.261837 − 0.407426i
\(67\) 9.75197 − 0.697475i 1.19139 − 0.0852101i 0.538477 − 0.842640i \(-0.319000\pi\)
0.652916 + 0.757430i \(0.273545\pi\)
\(68\) −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 \(0.343777\pi\)
\(72\) −0.0141926 + 0.0652421i −0.00167261 + 0.00768886i
\(73\) 4.01911 − 5.36891i 0.470402 − 0.628383i −0.501108 − 0.865385i \(-0.667074\pi\)
0.971509 + 0.237002i \(0.0761648\pi\)
\(74\) 1.27140 + 2.78398i 0.147798 + 0.323632i
\(75\) −5.37439 + 6.91268i −0.620581 + 0.798207i
\(76\) −1.44726 − 4.92891i −0.166012 − 0.565384i
\(77\) −7.92800 + 1.72463i −0.903480 + 0.196540i
\(78\) −8.65685 + 3.22884i −0.980195 + 0.365594i
\(79\) 2.52895 + 0.742567i 0.284529 + 0.0835454i 0.420883 − 0.907115i \(-0.361720\pi\)
−0.136354 + 0.990660i \(0.543538\pi\)
\(80\) −1.47967 + 1.67647i −0.165433 + 0.187435i
\(81\) −6.02200 − 6.94976i −0.669111 − 0.772195i
\(82\) 3.88345 − 2.12053i 0.428856 − 0.234173i
\(83\) −1.61492 − 4.32977i −0.177261 − 0.475254i 0.817685 − 0.575666i \(-0.195257\pi\)
−0.994946 + 0.100411i \(0.967984\pi\)
\(84\) 5.32008 + 3.41901i 0.580468 + 0.373044i
\(85\) −2.07159 + 5.40304i −0.224696 + 0.586042i
\(86\) 11.9387 + 1.71653i 1.28739 + 0.185098i
\(87\) 7.66399 + 2.85852i 0.821666 + 0.306466i
\(88\) −1.79861 − 1.34642i −0.191732 − 0.143529i
\(89\) 11.8000 − 7.58338i 1.25079 − 0.803836i 0.263797 − 0.964578i \(-0.415025\pi\)
0.986996 + 0.160742i \(0.0513887\pi\)
\(90\) 0.0304029 + 0.146169i 0.00320474 + 0.0154076i
\(91\) 19.0526i 1.99725i
\(92\) 1.80650 − 4.44258i 0.188341 − 0.463171i
\(93\) 11.2744 + 11.2744i 1.16910 + 1.16910i
\(94\) 0.103316 + 0.0895236i 0.0106562 + 0.00923365i
\(95\) −7.44263 − 8.74932i −0.763598 − 0.897662i
\(96\) 0.249224 + 1.73339i 0.0254364 + 0.176914i
\(97\) 2.93367 − 7.86547i 0.297869 − 0.798618i −0.698666 − 0.715448i \(-0.746223\pi\)
0.996535 − 0.0831701i \(-0.0265045\pi\)
\(98\) 4.83585 − 3.62007i 0.488495 − 0.365683i
\(99\) −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.53.2 yes 240
5.2 odd 4 inner 230.2.l.a.7.8 ✓ 240
23.10 odd 22 inner 230.2.l.a.33.8 yes 240
115.102 even 44 inner 230.2.l.a.217.2 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.8 ✓ 240 5.2 odd 4 inner
230.2.l.a.33.8 yes 240 23.10 odd 22 inner
230.2.l.a.53.2 yes 240 1.1 even 1 trivial
230.2.l.a.217.2 yes 240 115.102 even 44 inner