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Properties

Label 837.2.o.a.440.19
Level $837$
Weight $2$
Character 837.440
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(440,837)] 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("837.440"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 837 = 3^{3} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 837.o (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 440.19
Character \(\chi\) \(=\) 837.440
Dual form 837.2.o.a.584.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.709222 + 0.409470i) q^{2} +(-0.664669 - 1.15124i) q^{4} +4.31120i q^{5} -3.40596 q^{7} -2.72653i q^{8} +(-1.76530 + 3.05760i) q^{10} +(-1.40132 - 2.42715i) q^{11} +0.971533i q^{13} +(-2.41558 - 1.39464i) q^{14} +(-0.212908 + 0.368768i) q^{16} +(1.13384 + 1.96387i) q^{17} +(-2.11869 - 3.66968i) q^{19} +(4.96322 - 2.86552i) q^{20} -2.29519i q^{22} +(-4.23384 - 7.33323i) q^{23} -13.5864 q^{25} +(-0.397813 + 0.689033i) q^{26} +(2.26383 + 3.92108i) q^{28} +(0.335786 - 0.581598i) q^{29} +(-2.77764 - 4.82543i) q^{31} +(-5.02448 + 2.90089i) q^{32} +1.85709i q^{34} -14.6837i q^{35} +(-2.20494 + 1.27302i) q^{37} -3.47016i q^{38} +11.7546 q^{40} +6.59167i q^{41} +7.06710i q^{43} +(-1.86282 + 3.22650i) q^{44} -6.93452i q^{46} +(2.71943 + 1.57007i) q^{47} +4.60055 q^{49} +(-9.63578 - 5.56322i) q^{50} +(1.11847 - 0.645748i) q^{52} +(-0.281383 + 0.487369i) q^{53} +(10.4639 - 6.04135i) q^{55} +9.28643i q^{56} +(0.476293 - 0.274988i) q^{58} +(-5.29385 - 3.05640i) q^{59} +(3.65555 + 2.11054i) q^{61} +(0.00590265 - 4.55966i) q^{62} -3.89966 q^{64} -4.18847 q^{65} -11.9830 q^{67} +(1.50726 - 2.61065i) q^{68} +(6.01255 - 10.4140i) q^{70} +(10.5264 + 6.07741i) q^{71} +(-8.11830 - 4.68710i) q^{73} -2.08505 q^{74} +(-2.81646 + 4.87824i) q^{76} +(4.77282 + 8.26677i) q^{77} -0.510908i q^{79} +(-1.58983 - 0.917890i) q^{80} +(-2.69909 + 4.67496i) q^{82} +(4.01485 + 6.95392i) q^{83} +(-8.46662 + 4.88821i) q^{85} +(-2.89376 + 5.01215i) q^{86} +(-6.61769 + 3.82072i) q^{88} +11.6759 q^{89} -3.30900i q^{91} +(-5.62821 + 9.74834i) q^{92} +(1.28579 + 2.22705i) q^{94} +(15.8207 - 9.13409i) q^{95} +(-3.18229 + 5.51190i) q^{97} +(3.26281 + 1.88378i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 6 q^{2} + 26 q^{4} - 4 q^{10} + 3 q^{11} + 3 q^{14} - 22 q^{16} - 4 q^{19} + 21 q^{20} + 9 q^{23} - 52 q^{25} - 18 q^{26} + 4 q^{28} - 8 q^{31} - 6 q^{32} - 15 q^{37} + 16 q^{40} - 18 q^{44} + 9 q^{47}+ \cdots + 177 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(218\) \(406\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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 0.729331 − 0.684161i \(-0.239831\pi\)
−0.227835 + 0.973700i \(0.573165\pi\)
\(3\) 0 0
\(4\) −0.664669 − 1.15124i −0.332335 − 0.575620i
\(5\) 4.31120i 1.92803i 0.265856 + 0.964013i \(0.414345\pi\)
−0.265856 + 0.964013i \(0.585655\pi\)
\(6\) 0 0
\(7\) −3.40596 −1.28733 −0.643665 − 0.765307i \(-0.722587\pi\)
−0.643665 + 0.765307i \(0.722587\pi\)
\(8\) − 2.72653i − 0.963973i
\(9\) 0 0
\(10\) −1.76530 + 3.05760i −0.558238 + 0.966897i
\(11\) −1.40132 − 2.42715i −0.422513 − 0.731813i 0.573672 − 0.819085i \(-0.305518\pi\)
−0.996185 + 0.0872718i \(0.972185\pi\)
\(12\) 0 0
\(13\) 0.971533i 0.269455i 0.990883 + 0.134727i \(0.0430159\pi\)
−0.990883 + 0.134727i \(0.956984\pi\)
\(14\) −2.41558 − 1.39464i −0.645591 − 0.372732i
\(15\) 0 0
\(16\) −0.212908 + 0.368768i −0.0532271 + 0.0921921i
\(17\) 1.13384 + 1.96387i 0.274997 + 0.476308i 0.970134 − 0.242568i \(-0.0779899\pi\)
−0.695138 + 0.718877i \(0.744657\pi\)
\(18\) 0 0
\(19\) −2.11869 − 3.66968i −0.486061 − 0.841882i 0.513811 − 0.857904i \(-0.328233\pi\)
−0.999872 + 0.0160215i \(0.994900\pi\)
\(20\) 4.96322 − 2.86552i 1.10981 − 0.640750i
\(21\) 0 0
\(22\) − 2.29519i − 0.489335i
\(23\) −4.23384 − 7.33323i −0.882817 − 1.52908i −0.848196 − 0.529683i \(-0.822311\pi\)
−0.0346213 − 0.999401i \(-0.511022\pi\)
\(24\) 0 0
\(25\) −13.5864 −2.71728
\(26\) −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 \(-0.813473\pi\)
0.895517 + 0.445027i \(0.146806\pi\)
\(30\) 0 0
\(31\) −2.77764 − 4.82543i −0.498878 − 0.866672i
\(32\) −5.02448 + 2.90089i −0.888211 + 0.512809i
\(33\) 0 0
\(34\) 1.85709i 0.318489i
\(35\) − 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 \(0.400446\pi\)
\(38\) − 3.47016i − 0.562934i
\(39\) 0 0
\(40\) 11.7546 1.85856
\(41\) 6.59167i 1.02945i 0.857357 + 0.514723i \(0.172105\pi\)
−0.857357 + 0.514723i \(0.827895\pi\)
\(42\) 0 0
\(43\) 7.06710i 1.07772i 0.842394 + 0.538861i \(0.181145\pi\)
−0.842394 + 0.538861i \(0.818855\pi\)
\(44\) −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 0.685046 − 0.728500i \(-0.259782\pi\)
−0.288376 + 0.957517i \(0.593115\pi\)
\(48\) 0 0
\(49\) 4.60055 0.657221
\(50\) −9.63578 − 5.56322i −1.36271 − 0.786759i
\(51\) 0 0
\(52\) 1.11847 − 0.645748i 0.155104 − 0.0895492i
\(53\) −0.281383 + 0.487369i −0.0386509 + 0.0669453i −0.884704 − 0.466154i \(-0.845639\pi\)
0.846053 + 0.533099i \(0.178973\pi\)
\(54\) 0 0
\(55\) 10.4639 − 6.04135i 1.41095 − 0.814615i
\(56\) 9.28643i 1.24095i
\(57\) 0 0
\(58\) 0.476293 − 0.274988i 0.0625404 − 0.0361077i
\(59\) −5.29385 − 3.05640i −0.689200 − 0.397910i 0.114112 − 0.993468i \(-0.463598\pi\)
−0.803312 + 0.595558i \(0.796931\pi\)
\(60\) 0 0
\(61\) 3.65555 + 2.11054i 0.468046 + 0.270226i 0.715421 − 0.698693i \(-0.246235\pi\)
−0.247375 + 0.968920i \(0.579568\pi\)
\(62\) 0.00590265 − 4.55966i 0.000749637 − 0.579077i
\(63\) 0 0
\(64\) −3.89966 −0.487458
\(65\) −4.18847 −0.519516
\(66\) 0 0
\(67\) −11.9830 −1.46396 −0.731980 − 0.681326i \(-0.761404\pi\)
−0.731980 + 0.681326i \(0.761404\pi\)
\(68\) 1.50726 − 2.61065i 0.182782 − 0.316587i
\(69\) 0 0
\(70\) 6.01255 − 10.4140i 0.718637 − 1.24472i
\(71\) 10.5264 + 6.07741i 1.24925 + 0.721256i 0.970960 − 0.239241i \(-0.0768985\pi\)
0.278292 + 0.960497i \(0.410232\pi\)
\(72\) 0 0
\(73\) −8.11830 − 4.68710i −0.950175 − 0.548584i −0.0570399 − 0.998372i \(-0.518166\pi\)
−0.893135 + 0.449788i \(0.851500\pi\)
\(74\) −2.08505 −0.242382
\(75\) 0 0
\(76\) −2.81646 + 4.87824i −0.323070 + 0.559573i
\(77\) 4.77282 + 8.26677i 0.543914 + 0.942086i
\(78\) 0 0
\(79\) − 0.510908i − 0.0574816i −0.999587 − 0.0287408i \(-0.990850\pi\)
0.999587 − 0.0287408i \(-0.00914974\pi\)
\(80\) −1.58983 − 0.917890i −0.177749 − 0.102623i
\(81\) 0 0
\(82\) −2.69909 + 4.67496i −0.298064 + 0.516263i
\(83\) 4.01485 + 6.95392i 0.440687 + 0.763291i 0.997741 − 0.0671848i \(-0.0214017\pi\)
−0.557054 + 0.830476i \(0.688068\pi\)
\(84\) 0 0
\(85\) −8.46662 + 4.88821i −0.918334 + 0.530201i
\(86\) −2.89376 + 5.01215i −0.312043 + 0.540474i
\(87\) 0 0
\(88\) −6.61769 + 3.82072i −0.705448 + 0.407291i
\(89\) 11.6759 1.23764 0.618821 − 0.785532i \(-0.287611\pi\)
0.618821 + 0.785532i \(0.287611\pi\)
\(90\) 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 −0.981129 − 0.193356i \(-0.938063\pi\)
0.658016 + 0.753004i \(0.271396\pi\)
\(98\) 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.o.a.440.19 60
3.2 odd 2 279.2.o.a.254.12 yes 60
9.4 even 3 279.2.r.a.68.12 yes 60
9.5 odd 6 837.2.r.a.719.19 60
31.26 odd 6 837.2.r.a.305.19 60
93.26 even 6 279.2.r.a.119.12 yes 60
279.212 even 6 inner 837.2.o.a.584.19 60
279.274 odd 6 279.2.o.a.212.12 ✓ 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.12 ✓ 60 279.274 odd 6
279.2.o.a.254.12 yes 60 3.2 odd 2
279.2.r.a.68.12 yes 60 9.4 even 3
279.2.r.a.119.12 yes 60 93.26 even 6
837.2.o.a.440.19 60 1.1 even 1 trivial
837.2.o.a.584.19 60 279.212 even 6 inner
837.2.r.a.305.19 60 31.26 odd 6
837.2.r.a.719.19 60 9.5 odd 6