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Properties

Label 837.2.r.a.719.19
Level $837$
Weight $2$
Character 837.719
Analytic conductor $6.683$
Analytic rank $0$
Dimension $60$
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] = [837,2,Mod(305,837)] 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("837.305"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(837, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 837 = 3^{3} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 837.r (of order \(6\), degree \(2\), not 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: \(6.68347864918\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 279)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 719.19
Character \(\chi\) \(=\) 837.719
Dual form 837.2.r.a.305.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.709222 - 0.409470i) q^{2} +(-0.664669 + 1.15124i) q^{4} +(-3.73361 + 2.15560i) q^{5} +(1.70298 - 2.94965i) q^{7} +2.72653i q^{8} +(-1.76530 + 3.05760i) q^{10} -2.80263 q^{11} +(0.841372 - 0.485767i) q^{13} -2.78927i q^{14} +(-0.212908 - 0.368768i) q^{16} +(-1.13384 - 1.96387i) q^{17} +(-2.11869 - 3.66968i) q^{19} -5.73104i q^{20} +(-1.98769 + 1.14759i) q^{22} +(4.23384 - 7.33323i) q^{23} +(6.79320 - 11.7662i) q^{25} +(0.397813 - 0.689033i) q^{26} +(2.26383 + 3.92108i) q^{28} +(-0.335786 - 0.581598i) q^{29} +(-2.79012 + 4.81822i) q^{31} +(-5.02448 - 2.90089i) q^{32} +(-1.60829 - 0.928547i) q^{34} +14.6837i q^{35} +(-2.20494 + 1.27302i) q^{37} +(-3.00524 - 1.73508i) q^{38} +(-5.87729 - 10.1798i) q^{40} +(-5.70855 + 3.29583i) q^{41} +(-6.12029 - 3.53355i) q^{43} +(1.86282 - 3.22650i) q^{44} -6.93452i q^{46} +(2.71943 - 1.57007i) q^{47} +(-2.30027 - 3.98419i) q^{49} -11.1264i q^{50} +1.29150i q^{52} +(0.281383 - 0.487369i) q^{53} +(10.4639 - 6.04135i) q^{55} +(8.04229 + 4.64322i) q^{56} +(-0.476293 - 0.274988i) q^{58} -6.11281i q^{59} +(-3.65555 + 2.11054i) q^{61} +(-0.00590265 + 4.55966i) q^{62} -3.89966 q^{64} +(-2.09424 + 3.62732i) q^{65} +(5.99151 + 10.3776i) q^{67} +3.01452 q^{68} +(6.01255 + 10.4140i) q^{70} +(-10.5264 - 6.07741i) q^{71} +(-8.11830 - 4.68710i) q^{73} +(-1.04253 + 1.80571i) q^{74} +5.63291 q^{76} +(-4.77282 + 8.26677i) q^{77} +(0.442459 + 0.255454i) q^{79} +(1.58983 + 0.917890i) q^{80} +(-2.69909 + 4.67496i) q^{82} +8.02969 q^{83} +(8.46662 + 4.88821i) q^{85} -5.78753 q^{86} -7.64145i q^{88} -11.6759 q^{89} -3.30900i q^{91} +(5.62821 + 9.74834i) q^{92} +(1.28579 - 2.22705i) q^{94} +(15.8207 + 9.13409i) q^{95} +(-3.18229 - 5.51190i) q^{97} +(-3.26281 - 1.88378i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 6 q^{2} + 26 q^{4} + 6 q^{5} - 4 q^{10} + 6 q^{11} - 3 q^{13} - 22 q^{16} - 4 q^{19} - 15 q^{22} - 9 q^{23} + 26 q^{25} + 18 q^{26} + 4 q^{28} + q^{31} - 6 q^{32} + 3 q^{34} - 15 q^{37} - 27 q^{38}+ \cdots - 177 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/837\mathbb{Z}\right)^\times\).

\(n\) \(218\) \(406\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.709222 − 0.409470i 0.501496 − 0.289539i −0.227835 − 0.973700i \(-0.573165\pi\)
0.729331 + 0.684161i \(0.239831\pi\)
\(3\) 0 0
\(4\) −0.664669 + 1.15124i −0.332335 + 0.575620i
\(5\) −3.73361 + 2.15560i −1.66972 + 0.964013i −0.701931 + 0.712245i \(0.747679\pi\)
−0.967788 + 0.251768i \(0.918988\pi\)
\(6\) 0 0
\(7\) 1.70298 − 2.94965i 0.643665 − 1.11486i −0.340943 − 0.940084i \(-0.610746\pi\)
0.984608 − 0.174777i \(-0.0559205\pi\)
\(8\) 2.72653i 0.963973i
\(9\) 0 0
\(10\) −1.76530 + 3.05760i −0.558238 + 0.966897i
\(11\) −2.80263 −0.845025 −0.422513 − 0.906357i \(-0.638852\pi\)
−0.422513 + 0.906357i \(0.638852\pi\)
\(12\) 0 0
\(13\) 0.841372 − 0.485767i 0.233355 − 0.134727i −0.378764 − 0.925493i \(-0.623651\pi\)
0.612119 + 0.790766i \(0.290317\pi\)
\(14\) − 2.78927i − 0.745464i
\(15\) 0 0
\(16\) −0.212908 − 0.368768i −0.0532271 − 0.0921921i
\(17\) −1.13384 − 1.96387i −0.274997 − 0.476308i 0.695138 − 0.718877i \(-0.255343\pi\)
−0.970134 + 0.242568i \(0.922010\pi\)
\(18\) 0 0
\(19\) −2.11869 − 3.66968i −0.486061 − 0.841882i 0.513811 − 0.857904i \(-0.328233\pi\)
−0.999872 + 0.0160215i \(0.994900\pi\)
\(20\) − 5.73104i − 1.28150i
\(21\) 0 0
\(22\) −1.98769 + 1.14759i −0.423777 + 0.244668i
\(23\) 4.23384 − 7.33323i 0.882817 − 1.52908i 0.0346213 − 0.999401i \(-0.488978\pi\)
0.848196 − 0.529683i \(-0.177689\pi\)
\(24\) 0 0
\(25\) 6.79320 − 11.7662i 1.35864 − 2.35324i
\(26\) 0.397813 − 0.689033i 0.0780176 − 0.135130i
\(27\) 0 0
\(28\) 2.26383 + 3.92108i 0.427825 + 0.741014i
\(29\) −0.335786 − 0.581598i −0.0623538 − 0.108000i 0.833163 − 0.553027i \(-0.186527\pi\)
−0.895517 + 0.445027i \(0.853194\pi\)
\(30\) 0 0
\(31\) −2.79012 + 4.81822i −0.501121 + 0.865377i
\(32\) −5.02448 − 2.90089i −0.888211 − 0.512809i
\(33\) 0 0
\(34\) −1.60829 − 0.928547i −0.275819 − 0.159244i
\(35\) 14.6837i 2.48201i
\(36\) 0 0
\(37\) −2.20494 + 1.27302i −0.362489 + 0.209283i −0.670172 − 0.742206i \(-0.733780\pi\)
0.307683 + 0.951489i \(0.400446\pi\)
\(38\) −3.00524 − 1.73508i −0.487515 − 0.281467i
\(39\) 0 0
\(40\) −5.87729 − 10.1798i −0.929282 − 1.60956i
\(41\) −5.70855 + 3.29583i −0.891526 + 0.514723i −0.874441 − 0.485131i \(-0.838772\pi\)
−0.0170846 + 0.999854i \(0.505438\pi\)
\(42\) 0 0
\(43\) −6.12029 − 3.53355i −0.933335 − 0.538861i −0.0454705 − 0.998966i \(-0.514479\pi\)
−0.887865 + 0.460104i \(0.847812\pi\)
\(44\) 1.86282 − 3.22650i 0.280831 − 0.486414i
\(45\) 0 0
\(46\) − 6.93452i − 1.02244i
\(47\) 2.71943 − 1.57007i 0.396670 − 0.229018i −0.288376 − 0.957517i \(-0.593115\pi\)
0.685046 + 0.728500i \(0.259782\pi\)
\(48\) 0 0
\(49\) −2.30027 − 3.98419i −0.328610 − 0.569170i
\(50\) − 11.1264i − 1.57352i
\(51\) 0 0
\(52\) 1.29150i 0.179098i
\(53\) 0.281383 − 0.487369i 0.0386509 − 0.0669453i −0.846053 − 0.533099i \(-0.821027\pi\)
0.884704 + 0.466154i \(0.154361\pi\)
\(54\) 0 0
\(55\) 10.4639 − 6.04135i 1.41095 − 0.814615i
\(56\) 8.04229 + 4.64322i 1.07470 + 0.620476i
\(57\) 0 0
\(58\) −0.476293 − 0.274988i −0.0625404 − 0.0361077i
\(59\) − 6.11281i − 0.795820i −0.917425 − 0.397910i \(-0.869736\pi\)
0.917425 − 0.397910i \(-0.130264\pi\)
\(60\) 0 0
\(61\) −3.65555 + 2.11054i −0.468046 + 0.270226i −0.715421 − 0.698693i \(-0.753765\pi\)
0.247375 + 0.968920i \(0.420432\pi\)
\(62\) −0.00590265 + 4.55966i −0.000749637 + 0.579077i
\(63\) 0 0
\(64\) −3.89966 −0.487458
\(65\) −2.09424 + 3.62732i −0.259758 + 0.449914i
\(66\) 0 0
\(67\) 5.99151 + 10.3776i 0.731980 + 1.26783i 0.956036 + 0.293250i \(0.0947369\pi\)
−0.224056 + 0.974576i \(0.571930\pi\)
\(68\) 3.01452 0.365564
\(69\) 0 0
\(70\) 6.01255 + 10.4140i 0.718637 + 1.24472i
\(71\) −10.5264 − 6.07741i −1.24925 − 0.721256i −0.278292 − 0.960497i \(-0.589768\pi\)
−0.970960 + 0.239241i \(0.923102\pi\)
\(72\) 0 0
\(73\) −8.11830 − 4.68710i −0.950175 − 0.548584i −0.0570399 − 0.998372i \(-0.518166\pi\)
−0.893135 + 0.449788i \(0.851500\pi\)
\(74\) −1.04253 + 1.80571i −0.121191 + 0.209909i
\(75\) 0 0
\(76\) 5.63291 0.646139
\(77\) −4.77282 + 8.26677i −0.543914 + 0.942086i
\(78\) 0 0
\(79\) 0.442459 + 0.255454i 0.0497805 + 0.0287408i 0.524684 − 0.851297i \(-0.324184\pi\)
−0.474903 + 0.880038i \(0.657517\pi\)
\(80\) 1.58983 + 0.917890i 0.177749 + 0.102623i
\(81\) 0 0
\(82\) −2.69909 + 4.67496i −0.298064 + 0.516263i
\(83\) 8.02969 0.881373 0.440687 − 0.897661i \(-0.354735\pi\)
0.440687 + 0.897661i \(0.354735\pi\)
\(84\) 0 0
\(85\) 8.46662 + 4.88821i 0.918334 + 0.530201i
\(86\) −5.78753 −0.624085
\(87\) 0 0
\(88\) − 7.64145i − 0.814581i
\(89\) −11.6759 −1.23764 −0.618821 − 0.785532i \(-0.712389\pi\)
−0.618821 + 0.785532i \(0.712389\pi\)
\(90\) 0 0
\(91\) − 3.30900i − 0.346878i
\(92\) 5.62821 + 9.74834i 0.586781 + 1.01633i
\(93\) 0 0
\(94\) 1.28579 − 2.22705i 0.132619 − 0.229703i
\(95\) 15.8207 + 9.13409i 1.62317 + 0.937138i
\(96\) 0 0
\(97\) −3.18229 − 5.51190i −0.323113 − 0.559648i 0.658016 − 0.753004i \(-0.271396\pi\)
−0.981129 + 0.193356i \(0.938063\pi\)
\(98\) −3.26281 − 1.88378i −0.329594 − 0.190291i
\(99\) 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 837.2.r.a.719.19 60
3.2 odd 2 279.2.r.a.68.12 yes 60
9.2 odd 6 837.2.o.a.440.19 60
9.7 even 3 279.2.o.a.254.12 yes 60
31.26 odd 6 837.2.o.a.584.19 60
93.26 even 6 279.2.o.a.212.12 ✓ 60
279.88 odd 6 279.2.r.a.119.12 yes 60
279.119 even 6 inner 837.2.r.a.305.19 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.12 ✓ 60 93.26 even 6
279.2.o.a.254.12 yes 60 9.7 even 3
279.2.r.a.68.12 yes 60 3.2 odd 2
279.2.r.a.119.12 yes 60 279.88 odd 6
837.2.o.a.440.19 60 9.2 odd 6
837.2.o.a.584.19 60 31.26 odd 6
837.2.r.a.305.19 60 279.119 even 6 inner
837.2.r.a.719.19 60 1.1 even 1 trivial