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Properties

Label 266.2.w.b.233.6
Level $266$
Weight $2$
Character 266.233
Analytic conductor $2.124$
Analytic rank $0$
Dimension $42$
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] = [266,2,Mod(25,266)] 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("266.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(266, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([12, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 233.6
Character \(\chi\) \(=\) 266.233
Dual form 266.2.w.b.137.6

$q$-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+(-0.173648 + 0.984808i) q^{2} +(2.18826 - 0.796462i) q^{3} +(-0.939693 - 0.342020i) q^{4} +(-1.48822 + 0.541667i) q^{5} +(0.404375 + 2.29332i) q^{6} +(2.02324 - 1.70485i) q^{7} +(0.500000 - 0.866025i) q^{8} +(1.85601 - 1.55737i) q^{9} +(-0.275011 - 1.55967i) q^{10} +5.15649 q^{11} -2.32870 q^{12} +(0.777469 + 4.40924i) q^{13} +(1.32761 + 2.28855i) q^{14} +(-2.82519 + 2.37062i) q^{15} +(0.766044 + 0.642788i) q^{16} +(-3.69063 - 3.09680i) q^{17} +(1.21142 + 2.09825i) q^{18} +(3.22968 + 2.92732i) q^{19} +1.58373 q^{20} +(3.06954 - 5.34208i) q^{21} +(-0.895416 + 5.07815i) q^{22} +(-1.27857 - 7.25113i) q^{23} +(0.404375 - 2.29332i) q^{24} +(-1.90883 + 1.60170i) q^{25} -4.47726 q^{26} +(-0.672010 + 1.16396i) q^{27} +(-2.48432 + 0.910041i) q^{28} +(-9.18792 - 3.34413i) q^{29} +(-1.84401 - 3.19393i) q^{30} +(2.37539 - 4.11429i) q^{31} +(-0.766044 + 0.642788i) q^{32} +(11.2838 - 4.10695i) q^{33} +(3.69063 - 3.09680i) q^{34} +(-2.08756 + 3.63310i) q^{35} +(-2.27673 + 0.828662i) q^{36} +(-5.29566 + 9.17236i) q^{37} +(-3.44367 + 2.67229i) q^{38} +(5.21310 + 9.02936i) q^{39} +(-0.275011 + 1.55967i) q^{40} +(-0.525350 + 2.97941i) q^{41} +(4.72791 + 3.95055i) q^{42} +(-1.67665 - 1.40687i) q^{43} +(-4.84552 - 1.76362i) q^{44} +(-1.91856 + 3.32305i) q^{45} +7.36299 q^{46} +(-4.33554 + 3.63795i) q^{47} +(2.18826 + 0.796462i) q^{48} +(1.18701 - 6.89862i) q^{49} +(-1.24590 - 2.15797i) q^{50} +(-10.5426 - 3.83717i) q^{51} +(0.777469 - 4.40924i) q^{52} +(-1.13332 - 0.412494i) q^{53} +(-1.02958 - 0.863920i) q^{54} +(-7.67398 + 2.79310i) q^{55} +(-0.464819 - 2.60460i) q^{56} +(9.39888 + 3.83342i) q^{57} +(4.88879 - 8.46763i) q^{58} +(5.89068 + 4.94286i) q^{59} +(3.46561 - 1.26138i) q^{60} +(0.385511 + 2.18634i) q^{61} +(3.63930 + 3.05374i) q^{62} +(1.10007 - 6.31515i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-3.54538 - 6.14078i) q^{65} +(2.08515 + 11.8255i) q^{66} +(-1.61433 - 9.15534i) q^{67} +(2.40889 + 4.17231i) q^{68} +(-8.57310 - 14.8490i) q^{69} +(-3.21541 - 2.68673i) q^{70} +(-8.87592 - 7.44778i) q^{71} +(-0.420723 - 2.38604i) q^{72} +(7.65354 - 2.78566i) q^{73} +(-8.11343 - 6.80797i) q^{74} +(-2.90133 + 5.02526i) q^{75} +(-2.03370 - 3.85539i) q^{76} +(10.4328 - 8.79102i) q^{77} +(-9.79743 + 3.56597i) q^{78} +(-1.44753 - 1.21463i) q^{79} +(-1.48822 - 0.541667i) q^{80} +(-1.80566 + 10.2404i) q^{81} +(-2.84292 - 1.03474i) q^{82} +(2.26739 + 3.92724i) q^{83} +(-4.71152 + 3.97007i) q^{84} +(7.16989 + 2.60963i) q^{85} +(1.67665 - 1.40687i) q^{86} -22.7690 q^{87} +(2.57825 - 4.46565i) q^{88} +(-2.62345 - 0.954858i) q^{89} +(-2.93941 - 2.46646i) q^{90} +(9.09008 + 7.59550i) q^{91} +(-1.27857 + 7.25113i) q^{92} +(1.92109 - 10.8951i) q^{93} +(-2.82982 - 4.90140i) q^{94} +(-6.39209 - 2.60707i) q^{95} +(-1.16435 + 2.01671i) q^{96} +(-10.0321 + 3.65139i) q^{97} +(6.58770 + 2.36691i) q^{98} +(9.57049 - 8.03059i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 12 q^{7} + 21 q^{8} - 6 q^{11} + 6 q^{12} - 9 q^{13} - 3 q^{15} - 3 q^{17} + 30 q^{18} - 3 q^{19} - 18 q^{20} - 6 q^{22} + 12 q^{23} - 12 q^{25} - 6 q^{26} - 21 q^{27} - 3 q^{28} - 18 q^{29} + 24 q^{33}+ \cdots + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 + 0.984808i −0.122788 + 0.696364i
\(3\) 2.18826 − 0.796462i 1.26339 − 0.459838i 0.378487 − 0.925606i \(-0.376444\pi\)
0.884907 + 0.465769i \(0.154222\pi\)
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) −1.48822 + 0.541667i −0.665551 + 0.242241i −0.652631 − 0.757676i \(-0.726335\pi\)
−0.0129199 + 0.999917i \(0.504113\pi\)
\(6\) 0.404375 + 2.29332i 0.165085 + 0.936245i
\(7\) 2.02324 − 1.70485i 0.764713 − 0.644371i
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 1.85601 − 1.55737i 0.618669 − 0.519125i
\(10\) −0.275011 − 1.55967i −0.0869663 − 0.493210i
\(11\) 5.15649 1.55474 0.777371 − 0.629043i \(-0.216553\pi\)
0.777371 + 0.629043i \(0.216553\pi\)
\(12\) −2.32870 −0.672238
\(13\) 0.777469 + 4.40924i 0.215631 + 1.22290i 0.879808 + 0.475329i \(0.157671\pi\)
−0.664177 + 0.747575i \(0.731218\pi\)
\(14\) 1.32761 + 2.28855i 0.354819 + 0.611640i
\(15\) −2.82519 + 2.37062i −0.729462 + 0.612091i
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) −3.69063 − 3.09680i −0.895109 − 0.751085i 0.0741195 − 0.997249i \(-0.476385\pi\)
−0.969228 + 0.246164i \(0.920830\pi\)
\(18\) 1.21142 + 2.09825i 0.285535 + 0.494561i
\(19\) 3.22968 + 2.92732i 0.740939 + 0.671573i
\(20\) 1.58373 0.354132
\(21\) 3.06954 − 5.34208i 0.669828 − 1.16574i
\(22\) −0.895416 + 5.07815i −0.190903 + 1.08267i
\(23\) −1.27857 − 7.25113i −0.266600 − 1.51196i −0.764440 − 0.644695i \(-0.776984\pi\)
0.497839 − 0.867269i \(-0.334127\pi\)
\(24\) 0.404375 − 2.29332i 0.0825426 − 0.468122i
\(25\) −1.90883 + 1.60170i −0.381767 + 0.320340i
\(26\) −4.47726 −0.878063
\(27\) −0.672010 + 1.16396i −0.129328 + 0.224003i
\(28\) −2.48432 + 0.910041i −0.469492 + 0.171982i
\(29\) −9.18792 − 3.34413i −1.70615 − 0.620989i −0.709651 − 0.704553i \(-0.751147\pi\)
−0.996502 + 0.0835644i \(0.973370\pi\)
\(30\) −1.84401 − 3.19393i −0.336669 − 0.583128i
\(31\) 2.37539 − 4.11429i 0.426632 − 0.738948i −0.569939 − 0.821687i \(-0.693033\pi\)
0.996571 + 0.0827386i \(0.0263667\pi\)
\(32\) −0.766044 + 0.642788i −0.135419 + 0.113630i
\(33\) 11.2838 − 4.10695i 1.96425 − 0.714929i
\(34\) 3.69063 − 3.09680i 0.632937 − 0.531098i
\(35\) −2.08756 + 3.63310i −0.352863 + 0.614106i
\(36\) −2.27673 + 0.828662i −0.379455 + 0.138110i
\(37\) −5.29566 + 9.17236i −0.870602 + 1.50793i −0.00922593 + 0.999957i \(0.502937\pi\)
−0.861376 + 0.507969i \(0.830397\pi\)
\(38\) −3.44367 + 2.67229i −0.558637 + 0.433502i
\(39\) 5.21310 + 9.02936i 0.834764 + 1.44585i
\(40\) −0.275011 + 1.55967i −0.0434831 + 0.246605i
\(41\) −0.525350 + 2.97941i −0.0820459 + 0.465306i 0.915909 + 0.401386i \(0.131471\pi\)
−0.997955 + 0.0639199i \(0.979640\pi\)
\(42\) 4.72791 + 3.95055i 0.729532 + 0.609583i
\(43\) −1.67665 − 1.40687i −0.255686 − 0.214546i 0.505930 − 0.862575i \(-0.331149\pi\)
−0.761616 + 0.648028i \(0.775594\pi\)
\(44\) −4.84552 − 1.76362i −0.730489 − 0.265876i
\(45\) −1.91856 + 3.32305i −0.286003 + 0.495371i
\(46\) 7.36299 1.08561
\(47\) −4.33554 + 3.63795i −0.632403 + 0.530649i −0.901675 − 0.432415i \(-0.857662\pi\)
0.269271 + 0.963064i \(0.413217\pi\)
\(48\) 2.18826 + 0.796462i 0.315848 + 0.114959i
\(49\) 1.18701 − 6.89862i 0.169572 − 0.985518i
\(50\) −1.24590 − 2.15797i −0.176197 − 0.305183i
\(51\) −10.5426 − 3.83717i −1.47625 − 0.537312i
\(52\) 0.777469 − 4.40924i 0.107815 − 0.611452i
\(53\) −1.13332 − 0.412494i −0.155673 − 0.0566605i 0.263008 − 0.964794i \(-0.415285\pi\)
−0.418681 + 0.908133i \(0.637508\pi\)
\(54\) −1.02958 − 0.863920i −0.140108 − 0.117565i
\(55\) −7.67398 + 2.79310i −1.03476 + 0.376622i
\(56\) −0.464819 − 2.60460i −0.0621140 − 0.348054i
\(57\) 9.39888 + 3.83342i 1.24491 + 0.507749i
\(58\) 4.88879 − 8.46763i 0.641929 − 1.11185i
\(59\) 5.89068 + 4.94286i 0.766901 + 0.643506i 0.939913 − 0.341414i \(-0.110906\pi\)
−0.173013 + 0.984920i \(0.555350\pi\)
\(60\) 3.46561 − 1.26138i 0.447409 − 0.162843i
\(61\) 0.385511 + 2.18634i 0.0493596 + 0.279932i 0.999490 − 0.0319192i \(-0.0101619\pi\)
−0.950131 + 0.311852i \(0.899051\pi\)
\(62\) 3.63930 + 3.05374i 0.462192 + 0.387825i
\(63\) 1.10007 − 6.31515i 0.138595 − 0.795634i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −3.54538 − 6.14078i −0.439751 − 0.761670i
\(66\) 2.08515 + 11.8255i 0.256665 + 1.45562i
\(67\) −1.61433 − 9.15534i −0.197222 − 1.11850i −0.909218 − 0.416320i \(-0.863320\pi\)
0.711996 − 0.702183i \(-0.247791\pi\)
\(68\) 2.40889 + 4.17231i 0.292120 + 0.505967i
\(69\) −8.57310 − 14.8490i −1.03208 − 1.78761i
\(70\) −3.21541 − 2.68673i −0.384314 − 0.321126i
\(71\) −8.87592 − 7.44778i −1.05338 − 0.883889i −0.0599331 − 0.998202i \(-0.519089\pi\)
−0.993445 + 0.114313i \(0.963533\pi\)
\(72\) −0.420723 − 2.38604i −0.0495826 − 0.281197i
\(73\) 7.65354 − 2.78566i 0.895779 − 0.326037i 0.147219 − 0.989104i \(-0.452968\pi\)
0.748560 + 0.663067i \(0.230746\pi\)
\(74\) −8.11343 − 6.80797i −0.943167 − 0.791411i
\(75\) −2.90133 + 5.02526i −0.335017 + 0.580267i
\(76\) −2.03370 − 3.85539i −0.233282 − 0.442244i
\(77\) 10.4328 − 8.79102i 1.18893 − 1.00183i
\(78\) −9.79743 + 3.56597i −1.10934 + 0.403767i
\(79\) −1.44753 − 1.21463i −0.162860 − 0.136656i 0.557715 − 0.830032i \(-0.311678\pi\)
−0.720576 + 0.693376i \(0.756122\pi\)
\(80\) −1.48822 − 0.541667i −0.166388 − 0.0605602i
\(81\) −1.80566 + 10.2404i −0.200628 + 1.13782i
\(82\) −2.84292 − 1.03474i −0.313948 − 0.114268i
\(83\) 2.26739 + 3.92724i 0.248878 + 0.431070i 0.963215 − 0.268732i \(-0.0866047\pi\)
−0.714336 + 0.699802i \(0.753271\pi\)
\(84\) −4.71152 + 3.97007i −0.514069 + 0.433170i
\(85\) 7.16989 + 2.60963i 0.777684 + 0.283054i
\(86\) 1.67665 − 1.40687i 0.180797 − 0.151707i
\(87\) −22.7690 −2.44110
\(88\) 2.57825 − 4.46565i 0.274842 − 0.476040i
\(89\) −2.62345 − 0.954858i −0.278085 − 0.101215i 0.199213 − 0.979956i \(-0.436162\pi\)
−0.477298 + 0.878742i \(0.658384\pi\)
\(90\) −2.93941 − 2.46646i −0.309841 − 0.259987i
\(91\) 9.09008 + 7.59550i 0.952900 + 0.796224i
\(92\) −1.27857 + 7.25113i −0.133300 + 0.755982i
\(93\) 1.92109 − 10.8951i 0.199208 − 1.12976i
\(94\) −2.82982 − 4.90140i −0.291874 − 0.505540i
\(95\) −6.39209 − 2.60707i −0.655815 − 0.267480i
\(96\) −1.16435 + 2.01671i −0.118836 + 0.205830i
\(97\) −10.0321 + 3.65139i −1.01861 + 0.370742i −0.796732 − 0.604333i \(-0.793440\pi\)
−0.221874 + 0.975075i \(0.571217\pi\)
\(98\) 6.58770 + 2.36691i 0.665458 + 0.239094i
\(99\) 9.57049 − 8.03059i 0.961870 − 0.807105i
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 266.2.w.b.233.6 yes 42
7.4 even 3 266.2.v.a.81.2 yes 42
19.4 even 9 266.2.v.a.23.2 ✓ 42
133.4 even 9 inner 266.2.w.b.137.6 yes 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.v.a.23.2 ✓ 42 19.4 even 9
266.2.v.a.81.2 yes 42 7.4 even 3
266.2.w.b.137.6 yes 42 133.4 even 9 inner
266.2.w.b.233.6 yes 42 1.1 even 1 trivial