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Properties

Label 837.2.o.a.440.23
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 440.23
Character \(\chi\) \(=\) 837.440
Dual form 837.2.o.a.584.23

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.45499 + 0.840038i) q^{2} +(0.411327 + 0.712440i) q^{4} +0.175209i q^{5} +4.05741 q^{7} -1.97803i q^{8} +(-0.147183 + 0.254928i) q^{10} +(-0.246063 - 0.426194i) q^{11} -2.38791i q^{13} +(5.90349 + 3.40838i) q^{14} +(2.48427 - 4.30289i) q^{16} +(0.0216897 + 0.0375676i) q^{17} +(-1.32655 - 2.29766i) q^{19} +(-0.124826 + 0.0720685i) q^{20} -0.826809i q^{22} +(1.81002 + 3.13505i) q^{23} +4.96930 q^{25} +(2.00594 - 3.47438i) q^{26} +(1.66893 + 2.89066i) q^{28} +(-3.85815 + 6.68252i) q^{29} +(2.67208 + 4.88467i) q^{31} +(3.80313 - 2.19574i) q^{32} +0.0728806i q^{34} +0.710897i q^{35} +(-3.97232 + 2.29342i) q^{37} -4.45742i q^{38} +0.346569 q^{40} -4.87910i q^{41} -9.09531i q^{43} +(0.202425 - 0.350610i) q^{44} +6.08196i q^{46} +(5.61428 + 3.24141i) q^{47} +9.46261 q^{49} +(7.23028 + 4.17440i) q^{50} +(1.70124 - 0.982213i) q^{52} +(-6.09442 + 10.5558i) q^{53} +(0.0746732 - 0.0431126i) q^{55} -8.02568i q^{56} +(-11.2271 + 6.48199i) q^{58} +(-8.67969 - 5.01122i) q^{59} +(7.20110 + 4.15756i) q^{61} +(-0.215453 + 9.35178i) q^{62} -2.55908 q^{64} +0.418385 q^{65} -3.24025 q^{67} +(-0.0178431 + 0.0309052i) q^{68} +(-0.597181 + 1.03435i) q^{70} +(1.70536 + 0.984588i) q^{71} +(3.33296 + 1.92429i) q^{73} -7.70623 q^{74} +(1.09130 - 1.89018i) q^{76} +(-0.998379 - 1.72924i) q^{77} +0.529369i q^{79} +(0.753907 + 0.435268i) q^{80} +(4.09863 - 7.09904i) q^{82} +(3.43452 + 5.94877i) q^{83} +(-0.00658220 + 0.00380024i) q^{85} +(7.64041 - 13.2336i) q^{86} +(-0.843023 + 0.486720i) q^{88} -4.98238 q^{89} -9.68875i q^{91} +(-1.48903 + 2.57907i) q^{92} +(5.44581 + 9.43242i) q^{94} +(0.402571 - 0.232425i) q^{95} +(1.29180 - 2.23747i) q^{97} +(13.7680 + 7.94895i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 6 q^{2} + 26 q^{4} - 4 q^{10} + 3 q^{11} + 3 q^{14} - 22 q^{16} - 4 q^{19} + 21 q^{20} + 9 q^{23} - 52 q^{25} - 18 q^{26} + 4 q^{28} - 8 q^{31} - 6 q^{32} - 15 q^{37} + 16 q^{40} - 18 q^{44} + 9 q^{47}+ \cdots + 177 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.45499 + 0.840038i 1.02883 + 0.593997i 0.916650 − 0.399691i \(-0.130883\pi\)
0.112182 + 0.993688i \(0.464216\pi\)
\(3\) 0 0
\(4\) 0.411327 + 0.712440i 0.205664 + 0.356220i
\(5\) 0.175209i 0.0783561i 0.999232 + 0.0391780i \(0.0124739\pi\)
−0.999232 + 0.0391780i \(0.987526\pi\)
\(6\) 0 0
\(7\) 4.05741 1.53356 0.766779 − 0.641911i \(-0.221858\pi\)
0.766779 + 0.641911i \(0.221858\pi\)
\(8\) − 1.97803i − 0.699339i
\(9\) 0 0
\(10\) −0.147183 + 0.254928i −0.0465432 + 0.0806152i
\(11\) −0.246063 − 0.426194i −0.0741908 − 0.128502i 0.826543 − 0.562873i \(-0.190304\pi\)
−0.900734 + 0.434371i \(0.856971\pi\)
\(12\) 0 0
\(13\) − 2.38791i − 0.662288i −0.943580 − 0.331144i \(-0.892565\pi\)
0.943580 − 0.331144i \(-0.107435\pi\)
\(14\) 5.90349 + 3.40838i 1.57777 + 0.910928i
\(15\) 0 0
\(16\) 2.48427 − 4.30289i 0.621069 − 1.07572i
\(17\) 0.0216897 + 0.0375676i 0.00526052 + 0.00911148i 0.868644 − 0.495437i \(-0.164992\pi\)
−0.863383 + 0.504549i \(0.831659\pi\)
\(18\) 0 0
\(19\) −1.32655 − 2.29766i −0.304332 − 0.527119i 0.672780 − 0.739842i \(-0.265100\pi\)
−0.977112 + 0.212724i \(0.931767\pi\)
\(20\) −0.124826 + 0.0720685i −0.0279120 + 0.0161150i
\(21\) 0 0
\(22\) − 0.826809i − 0.176276i
\(23\) 1.81002 + 3.13505i 0.377416 + 0.653704i 0.990686 − 0.136170i \(-0.0434793\pi\)
−0.613269 + 0.789874i \(0.710146\pi\)
\(24\) 0 0
\(25\) 4.96930 0.993860
\(26\) 2.00594 − 3.47438i 0.393396 − 0.681383i
\(27\) 0 0
\(28\) 1.66893 + 2.89066i 0.315397 + 0.546284i
\(29\) −3.85815 + 6.68252i −0.716441 + 1.24091i 0.245960 + 0.969280i \(0.420897\pi\)
−0.962401 + 0.271632i \(0.912437\pi\)
\(30\) 0 0
\(31\) 2.67208 + 4.88467i 0.479921 + 0.877312i
\(32\) 3.80313 − 2.19574i 0.672305 − 0.388156i
\(33\) 0 0
\(34\) 0.0728806i 0.0124989i
\(35\) 0.710897i 0.120164i
\(36\) 0 0
\(37\) −3.97232 + 2.29342i −0.653045 + 0.377036i −0.789622 − 0.613594i \(-0.789723\pi\)
0.136577 + 0.990629i \(0.456390\pi\)
\(38\) − 4.45742i − 0.723089i
\(39\) 0 0
\(40\) 0.346569 0.0547974
\(41\) − 4.87910i − 0.761988i −0.924578 − 0.380994i \(-0.875582\pi\)
0.924578 − 0.380994i \(-0.124418\pi\)
\(42\) 0 0
\(43\) − 9.09531i − 1.38702i −0.720446 − 0.693511i \(-0.756063\pi\)
0.720446 − 0.693511i \(-0.243937\pi\)
\(44\) 0.202425 − 0.350610i 0.0305167 − 0.0528565i
\(45\) 0 0
\(46\) 6.08196i 0.896736i
\(47\) 5.61428 + 3.24141i 0.818928 + 0.472808i 0.850047 − 0.526708i \(-0.176574\pi\)
−0.0311189 + 0.999516i \(0.509907\pi\)
\(48\) 0 0
\(49\) 9.46261 1.35180
\(50\) 7.23028 + 4.17440i 1.02252 + 0.590350i
\(51\) 0 0
\(52\) 1.70124 − 0.982213i 0.235920 − 0.136208i
\(53\) −6.09442 + 10.5558i −0.837133 + 1.44996i 0.0551497 + 0.998478i \(0.482436\pi\)
−0.892282 + 0.451478i \(0.850897\pi\)
\(54\) 0 0
\(55\) 0.0746732 − 0.0431126i 0.0100689 − 0.00581330i
\(56\) − 8.02568i − 1.07248i
\(57\) 0 0
\(58\) −11.2271 + 6.48199i −1.47420 + 0.851127i
\(59\) −8.67969 − 5.01122i −1.13000 − 0.652406i −0.186065 − 0.982537i \(-0.559574\pi\)
−0.943935 + 0.330132i \(0.892907\pi\)
\(60\) 0 0
\(61\) 7.20110 + 4.15756i 0.922007 + 0.532321i 0.884275 − 0.466967i \(-0.154653\pi\)
0.0377320 + 0.999288i \(0.487987\pi\)
\(62\) −0.215453 + 9.35178i −0.0273625 + 1.18768i
\(63\) 0 0
\(64\) −2.55908 −0.319885
\(65\) 0.418385 0.0518942
\(66\) 0 0
\(67\) −3.24025 −0.395860 −0.197930 − 0.980216i \(-0.563422\pi\)
−0.197930 + 0.980216i \(0.563422\pi\)
\(68\) −0.0178431 + 0.0309052i −0.00216380 + 0.00374780i
\(69\) 0 0
\(70\) −0.597181 + 1.03435i −0.0713768 + 0.123628i
\(71\) 1.70536 + 0.984588i 0.202388 + 0.116849i 0.597769 − 0.801668i \(-0.296054\pi\)
−0.395381 + 0.918517i \(0.629387\pi\)
\(72\) 0 0
\(73\) 3.33296 + 1.92429i 0.390094 + 0.225221i 0.682201 − 0.731165i \(-0.261023\pi\)
−0.292107 + 0.956386i \(0.594356\pi\)
\(74\) −7.70623 −0.895831
\(75\) 0 0
\(76\) 1.09130 − 1.89018i 0.125180 − 0.216818i
\(77\) −0.998379 − 1.72924i −0.113776 − 0.197066i
\(78\) 0 0
\(79\) 0.529369i 0.0595586i 0.999556 + 0.0297793i \(0.00948045\pi\)
−0.999556 + 0.0297793i \(0.990520\pi\)
\(80\) 0.753907 + 0.435268i 0.0842894 + 0.0486645i
\(81\) 0 0
\(82\) 4.09863 − 7.09904i 0.452618 − 0.783958i
\(83\) 3.43452 + 5.94877i 0.376988 + 0.652962i 0.990622 − 0.136628i \(-0.0436266\pi\)
−0.613635 + 0.789590i \(0.710293\pi\)
\(84\) 0 0
\(85\) −0.00658220 + 0.00380024i −0.000713940 + 0.000412193i
\(86\) 7.64041 − 13.2336i 0.823886 − 1.42701i
\(87\) 0 0
\(88\) −0.843023 + 0.486720i −0.0898666 + 0.0518845i
\(89\) −4.98238 −0.528131 −0.264066 − 0.964505i \(-0.585064\pi\)
−0.264066 + 0.964505i \(0.585064\pi\)
\(90\) 0 0
\(91\) − 9.68875i − 1.01566i
\(92\) −1.48903 + 2.57907i −0.155242 + 0.268886i
\(93\) 0 0
\(94\) 5.44581 + 9.43242i 0.561693 + 0.972880i
\(95\) 0.402571 − 0.232425i 0.0413030 − 0.0238463i
\(96\) 0 0
\(97\) 1.29180 − 2.23747i 0.131163 − 0.227180i −0.792962 − 0.609271i \(-0.791462\pi\)
0.924125 + 0.382090i \(0.124796\pi\)
\(98\) 13.7680 + 7.94895i 1.39078 + 0.802965i
\(99\) 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.440.23 60
3.2 odd 2 279.2.o.a.254.8 yes 60
9.4 even 3 279.2.r.a.68.8 yes 60
9.5 odd 6 837.2.r.a.719.23 60
31.26 odd 6 837.2.r.a.305.23 60
93.26 even 6 279.2.r.a.119.8 yes 60
279.212 even 6 inner 837.2.o.a.584.23 60
279.274 odd 6 279.2.o.a.212.8 ✓ 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.8 ✓ 60 279.274 odd 6
279.2.o.a.254.8 yes 60 3.2 odd 2
279.2.r.a.68.8 yes 60 9.4 even 3
279.2.r.a.119.8 yes 60 93.26 even 6
837.2.o.a.440.23 60 1.1 even 1 trivial
837.2.o.a.584.23 60 279.212 even 6 inner
837.2.r.a.305.23 60 31.26 odd 6
837.2.r.a.719.23 60 9.5 odd 6