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Properties

Label 261.2.k.c.226.3
Level $261$
Weight $2$
Character 261.226
Analytic conductor $2.084$
Analytic rank $0$
Dimension $18$
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] = [261,2,Mod(82,261)] 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("261.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 226.3
Root \(0.491931 + 2.15529i\) of defining polynomial
Character \(\chi\) \(=\) 261.226
Dual form 261.2.k.c.82.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.754870 - 0.946578i) q^{2} +(0.118862 + 0.520769i) q^{4} +(-1.12131 + 1.40607i) q^{5} +(-0.951706 + 4.16970i) q^{7} +(2.76431 + 1.33122i) q^{8} +(0.484517 + 2.12281i) q^{10} +(0.951656 - 0.458293i) q^{11} +(4.75885 - 2.29174i) q^{13} +(3.22853 + 4.04845i) q^{14} +(2.38428 - 1.14821i) q^{16} -5.61769 q^{17} +(-1.42675 - 6.25099i) q^{19} +(-0.865520 - 0.416812i) q^{20} +(0.284567 - 1.24677i) q^{22} +(-1.27587 - 1.59988i) q^{23} +(0.392890 + 1.72136i) q^{25} +(1.42300 - 6.23459i) q^{26} -2.28457 q^{28} +(4.54927 + 2.88169i) q^{29} +(2.38467 - 2.99028i) q^{31} +(-0.652504 + 2.85881i) q^{32} +(-4.24063 + 5.31758i) q^{34} +(-4.79575 - 6.01368i) q^{35} +(0.315606 + 0.151988i) q^{37} +(-6.99406 - 3.36816i) q^{38} +(-4.97144 + 2.39412i) q^{40} +3.62240 q^{41} +(1.09049 + 1.36743i) q^{43} +(0.351781 + 0.441119i) q^{44} -2.47753 q^{46} +(6.28881 - 3.02853i) q^{47} +(-10.1739 - 4.89947i) q^{49} +(1.92598 + 0.927505i) q^{50} +(1.75911 + 2.20586i) q^{52} +(4.87488 - 6.11290i) q^{53} +(-0.422703 + 1.85198i) q^{55} +(-8.18161 + 10.2594i) q^{56} +(6.16185 - 2.13093i) q^{58} -0.382668 q^{59} +(1.21640 - 5.32939i) q^{61} +(-1.03041 - 4.51454i) q^{62} +(5.51347 + 6.91367i) q^{64} +(-2.11377 + 9.26104i) q^{65} +(-7.03806 - 3.38935i) q^{67} +(-0.667730 - 2.92552i) q^{68} -9.31258 q^{70} +(-14.0654 + 6.77356i) q^{71} +(-1.58547 - 1.98811i) q^{73} +(0.382110 - 0.184015i) q^{74} +(3.08574 - 1.48601i) q^{76} +(1.00525 + 4.40428i) q^{77} +(-9.25113 - 4.45511i) q^{79} +(-1.05904 + 4.63996i) q^{80} +(2.73445 - 3.42889i) q^{82} +(0.338541 + 1.48324i) q^{83} +(6.29915 - 7.89888i) q^{85} +2.11755 q^{86} +3.24076 q^{88} +(11.1881 - 14.0295i) q^{89} +(5.02684 + 22.0240i) q^{91} +(0.681518 - 0.854597i) q^{92} +(1.88050 - 8.23899i) q^{94} +(10.3892 + 5.00316i) q^{95} +(1.51530 + 6.63895i) q^{97} +(-12.3177 + 5.93188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} - 6 q^{4} + q^{5} - 4 q^{7} + 15 q^{8} - 14 q^{10} - 26 q^{11} + 9 q^{13} + 10 q^{14} - 14 q^{16} - 4 q^{17} - 10 q^{19} + q^{20} - 8 q^{22} + 8 q^{23} + 16 q^{25} - 5 q^{26} + 80 q^{28}+ \cdots - 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(e\left(\frac{5}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.754870 − 0.946578i 0.533774 − 0.669331i −0.439696 − 0.898147i \(-0.644914\pi\)
0.973470 + 0.228815i \(0.0734852\pi\)
\(3\) 0 0
\(4\) 0.118862 + 0.520769i 0.0594310 + 0.260384i
\(5\) −1.12131 + 1.40607i −0.501463 + 0.628815i −0.966559 − 0.256445i \(-0.917449\pi\)
0.465095 + 0.885261i \(0.346020\pi\)
\(6\) 0 0
\(7\) −0.951706 + 4.16970i −0.359711 + 1.57600i 0.394202 + 0.919024i \(0.371021\pi\)
−0.753913 + 0.656974i \(0.771836\pi\)
\(8\) 2.76431 + 1.33122i 0.977332 + 0.470658i
\(9\) 0 0
\(10\) 0.484517 + 2.12281i 0.153218 + 0.671290i
\(11\) 0.951656 − 0.458293i 0.286935 − 0.138181i −0.284880 − 0.958563i \(-0.591954\pi\)
0.571815 + 0.820383i \(0.306239\pi\)
\(12\) 0 0
\(13\) 4.75885 − 2.29174i 1.31987 − 0.635615i 0.364547 − 0.931185i \(-0.381224\pi\)
0.955321 + 0.295570i \(0.0955096\pi\)
\(14\) 3.22853 + 4.04845i 0.862860 + 1.08199i
\(15\) 0 0
\(16\) 2.38428 − 1.14821i 0.596069 − 0.287052i
\(17\) −5.61769 −1.36249 −0.681245 − 0.732056i \(-0.738561\pi\)
−0.681245 + 0.732056i \(0.738561\pi\)
\(18\) 0 0
\(19\) −1.42675 − 6.25099i −0.327319 − 1.43408i −0.824220 − 0.566269i \(-0.808386\pi\)
0.496902 − 0.867807i \(-0.334471\pi\)
\(20\) −0.865520 − 0.416812i −0.193536 − 0.0932021i
\(21\) 0 0
\(22\) 0.284567 − 1.24677i 0.0606698 − 0.265812i
\(23\) −1.27587 − 1.59988i −0.266036 − 0.333599i 0.630813 − 0.775935i \(-0.282721\pi\)
−0.896850 + 0.442336i \(0.854150\pi\)
\(24\) 0 0
\(25\) 0.392890 + 1.72136i 0.0785779 + 0.344272i
\(26\) 1.42300 − 6.23459i 0.279074 − 1.22270i
\(27\) 0 0
\(28\) −2.28457 −0.431743
\(29\) 4.54927 + 2.88169i 0.844778 + 0.535117i
\(30\) 0 0
\(31\) 2.38467 − 2.99028i 0.428299 − 0.537069i −0.520119 − 0.854094i \(-0.674112\pi\)
0.948417 + 0.317025i \(0.102684\pi\)
\(32\) −0.652504 + 2.85881i −0.115348 + 0.505371i
\(33\) 0 0
\(34\) −4.24063 + 5.31758i −0.727261 + 0.911957i
\(35\) −4.79575 − 6.01368i −0.810629 − 1.01650i
\(36\) 0 0
\(37\) 0.315606 + 0.151988i 0.0518854 + 0.0249867i 0.459647 − 0.888102i \(-0.347976\pi\)
−0.407761 + 0.913089i \(0.633690\pi\)
\(38\) −6.99406 − 3.36816i −1.13459 − 0.546388i
\(39\) 0 0
\(40\) −4.97144 + 2.39412i −0.786053 + 0.378543i
\(41\) 3.62240 0.565725 0.282862 − 0.959161i \(-0.408716\pi\)
0.282862 + 0.959161i \(0.408716\pi\)
\(42\) 0 0
\(43\) 1.09049 + 1.36743i 0.166298 + 0.208530i 0.857997 − 0.513655i \(-0.171709\pi\)
−0.691699 + 0.722186i \(0.743138\pi\)
\(44\) 0.351781 + 0.441119i 0.0530329 + 0.0665012i
\(45\) 0 0
\(46\) −2.47753 −0.365292
\(47\) 6.28881 − 3.02853i 0.917317 − 0.441756i 0.0852043 − 0.996363i \(-0.472846\pi\)
0.832112 + 0.554607i \(0.187131\pi\)
\(48\) 0 0
\(49\) −10.1739 − 4.89947i −1.45341 − 0.699924i
\(50\) 1.92598 + 0.927505i 0.272375 + 0.131169i
\(51\) 0 0
\(52\) 1.75911 + 2.20586i 0.243945 + 0.305898i
\(53\) 4.87488 − 6.11290i 0.669616 − 0.839672i −0.324736 − 0.945805i \(-0.605276\pi\)
0.994352 + 0.106133i \(0.0338469\pi\)
\(54\) 0 0
\(55\) −0.422703 + 1.85198i −0.0569973 + 0.249722i
\(56\) −8.18161 + 10.2594i −1.09331 + 1.37097i
\(57\) 0 0
\(58\) 6.16185 − 2.13093i 0.809091 − 0.279805i
\(59\) −0.382668 −0.0498191 −0.0249096 − 0.999690i \(-0.507930\pi\)
−0.0249096 + 0.999690i \(0.507930\pi\)
\(60\) 0 0
\(61\) 1.21640 − 5.32939i 0.155744 − 0.682358i −0.835409 − 0.549629i \(-0.814769\pi\)
0.991152 − 0.132729i \(-0.0423740\pi\)
\(62\) −1.03041 − 4.51454i −0.130863 − 0.573347i
\(63\) 0 0
\(64\) 5.51347 + 6.91367i 0.689184 + 0.864209i
\(65\) −2.11377 + 9.26104i −0.262181 + 1.14869i
\(66\) 0 0
\(67\) −7.03806 − 3.38935i −0.859836 − 0.414075i −0.0486169 − 0.998818i \(-0.515481\pi\)
−0.811219 + 0.584742i \(0.801196\pi\)
\(68\) −0.667730 − 2.92552i −0.0809742 − 0.354771i
\(69\) 0 0
\(70\) −9.31258 −1.11307
\(71\) −14.0654 + 6.77356i −1.66926 + 0.803874i −0.671224 + 0.741255i \(0.734231\pi\)
−0.998038 + 0.0626189i \(0.980055\pi\)
\(72\) 0 0
\(73\) −1.58547 − 1.98811i −0.185565 − 0.232691i 0.680344 − 0.732893i \(-0.261830\pi\)
−0.865909 + 0.500202i \(0.833259\pi\)
\(74\) 0.382110 − 0.184015i 0.0444194 − 0.0213913i
\(75\) 0 0
\(76\) 3.08574 − 1.48601i 0.353958 − 0.170457i
\(77\) 1.00525 + 4.40428i 0.114559 + 0.501914i
\(78\) 0 0
\(79\) −9.25113 − 4.45511i −1.04083 − 0.501239i −0.166235 − 0.986086i \(-0.553161\pi\)
−0.874599 + 0.484847i \(0.838875\pi\)
\(80\) −1.05904 + 4.63996i −0.118404 + 0.518764i
\(81\) 0 0
\(82\) 2.73445 − 3.42889i 0.301969 − 0.378657i
\(83\) 0.338541 + 1.48324i 0.0371597 + 0.162807i 0.990103 − 0.140340i \(-0.0448197\pi\)
−0.952944 + 0.303148i \(0.901963\pi\)
\(84\) 0 0
\(85\) 6.29915 − 7.89888i 0.683239 − 0.856754i
\(86\) 2.11755 0.228341
\(87\) 0 0
\(88\) 3.24076 0.345467
\(89\) 11.1881 − 14.0295i 1.18594 − 1.48712i 0.351351 − 0.936244i \(-0.385722\pi\)
0.834587 − 0.550876i \(-0.185706\pi\)
\(90\) 0 0
\(91\) 5.02684 + 22.0240i 0.526956 + 2.30875i
\(92\) 0.681518 − 0.854597i 0.0710532 − 0.0890979i
\(93\) 0 0
\(94\) 1.88050 − 8.23899i 0.193958 − 0.849787i
\(95\) 10.3892 + 5.00316i 1.06591 + 0.513314i
\(96\) 0 0
\(97\) 1.51530 + 6.63895i 0.153855 + 0.674083i 0.991743 + 0.128241i \(0.0409331\pi\)
−0.837888 + 0.545842i \(0.816210\pi\)
\(98\) −12.3177 + 5.93188i −1.24427 + 0.599210i
\(99\) 0 0
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 261.2.k.c.226.3 18
3.2 odd 2 87.2.g.a.52.1 ✓ 18
29.13 even 14 7569.2.a.bm.1.2 9
29.16 even 7 7569.2.a.bj.1.8 9
29.24 even 7 inner 261.2.k.c.82.3 18
87.53 odd 14 87.2.g.a.82.1 yes 18
87.71 odd 14 2523.2.a.o.1.8 9
87.74 odd 14 2523.2.a.r.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.52.1 ✓ 18 3.2 odd 2
87.2.g.a.82.1 yes 18 87.53 odd 14
261.2.k.c.82.3 18 29.24 even 7 inner
261.2.k.c.226.3 18 1.1 even 1 trivial
2523.2.a.o.1.8 9 87.71 odd 14
2523.2.a.r.1.2 9 87.74 odd 14
7569.2.a.bj.1.8 9 29.16 even 7
7569.2.a.bm.1.2 9 29.13 even 14