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Properties

Label 296.2.i.b.137.3
Level $296$
Weight $2$
Character 296.137
Analytic conductor $2.364$
Analytic rank $0$
Dimension $6$
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] = [296,2,Mod(121,296)] 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("296.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 296.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.36357189983\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 137.3
Root \(1.08504 - 1.87935i\) of defining polynomial
Character \(\chi\) \(=\) 296.137
Dual form 296.2.i.b.121.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.854638 - 1.48028i) q^{3} +(1.35464 - 2.34630i) q^{5} +(-0.315449 + 0.546373i) q^{7} +(0.0391889 + 0.0678771i) q^{9} +(1.00000 - 1.73205i) q^{13} +(-2.31545 - 4.01047i) q^{15} +(-2.13090 - 3.69082i) q^{17} +(-2.63090 + 4.55685i) q^{19} +(0.539189 + 0.933903i) q^{21} -4.04945 q^{23} +(-1.17009 - 2.02665i) q^{25} +5.26180 q^{27} +7.04945 q^{29} -0.581449 q^{31} +(0.854638 + 1.48028i) q^{35} +(6.06391 + 0.478496i) q^{37} +(-1.70928 - 2.96055i) q^{39} +(-2.67009 + 4.62473i) q^{41} -7.86603 q^{43} +0.212347 q^{45} +3.41855 q^{47} +(3.30098 + 5.71747i) q^{49} -7.28458 q^{51} +(0.460811 + 0.798148i) q^{53} +(4.49693 + 7.78891i) q^{57} +(2.63090 + 4.55685i) q^{59} +(-1.47528 + 2.55525i) q^{61} -0.0494483 q^{63} +(-2.70928 - 4.69260i) q^{65} +(-5.41855 + 9.38521i) q^{67} +(-3.46081 + 5.99430i) q^{69} +(4.65562 - 8.06377i) q^{71} -5.23513 q^{73} -4.00000 q^{75} +(-1.55252 + 2.68904i) q^{79} +(4.37936 - 7.58528i) q^{81} +(5.64229 + 9.77273i) q^{83} -11.5464 q^{85} +(6.02472 - 10.4351i) q^{87} +(5.71953 + 9.90652i) q^{89} +(0.630898 + 1.09275i) q^{91} +(-0.496928 + 0.860705i) q^{93} +(7.12783 + 12.3458i) q^{95} -10.1773 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 6 q^{13} - 10 q^{15} - 5 q^{17} - 8 q^{19} + 12 q^{23} + 4 q^{25} + 16 q^{27} + 6 q^{29} - 32 q^{31} - 2 q^{35} + 15 q^{37} + 4 q^{39} - 5 q^{41} - 20 q^{43}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/296\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 0
\(3\) 0.854638 − 1.48028i 0.493425 − 0.854638i −0.506546 − 0.862213i \(-0.669078\pi\)
0.999971 + 0.00757525i \(0.00241130\pi\)
\(4\) 0 0
\(5\) 1.35464 − 2.34630i 0.605812 − 1.04930i −0.386110 − 0.922453i \(-0.626182\pi\)
0.991923 − 0.126845i \(-0.0404851\pi\)
\(6\) 0 0
\(7\) −0.315449 + 0.546373i −0.119228 + 0.206510i −0.919462 − 0.393179i \(-0.871375\pi\)
0.800234 + 0.599688i \(0.204709\pi\)
\(8\) 0 0
\(9\) 0.0391889 + 0.0678771i 0.0130630 + 0.0226257i
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 − 1.73205i 0.277350 − 0.480384i −0.693375 − 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) −2.31545 − 4.01047i −0.597846 − 1.03550i
\(16\) 0 0
\(17\) −2.13090 − 3.69082i −0.516819 − 0.895156i −0.999809 − 0.0195306i \(-0.993783\pi\)
0.482991 − 0.875625i \(-0.339550\pi\)
\(18\) 0 0
\(19\) −2.63090 + 4.55685i −0.603569 + 1.04541i 0.388707 + 0.921362i \(0.372922\pi\)
−0.992276 + 0.124051i \(0.960411\pi\)
\(20\) 0 0
\(21\) 0.539189 + 0.933903i 0.117661 + 0.203794i
\(22\) 0 0
\(23\) −4.04945 −0.844368 −0.422184 − 0.906510i \(-0.638736\pi\)
−0.422184 + 0.906510i \(0.638736\pi\)
\(24\) 0 0
\(25\) −1.17009 − 2.02665i −0.234017 − 0.405330i
\(26\) 0 0
\(27\) 5.26180 1.01263
\(28\) 0 0
\(29\) 7.04945 1.30905 0.654525 − 0.756040i \(-0.272869\pi\)
0.654525 + 0.756040i \(0.272869\pi\)
\(30\) 0 0
\(31\) −0.581449 −0.104431 −0.0522157 − 0.998636i \(-0.516628\pi\)
−0.0522157 + 0.998636i \(0.516628\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.854638 + 1.48028i 0.144460 + 0.250212i
\(36\) 0 0
\(37\) 6.06391 + 0.478496i 0.996901 + 0.0786643i
\(38\) 0 0
\(39\) −1.70928 − 2.96055i −0.273703 − 0.474068i
\(40\) 0 0
\(41\) −2.67009 + 4.62473i −0.416997 + 0.722261i −0.995636 − 0.0933234i \(-0.970251\pi\)
0.578638 + 0.815584i \(0.303584\pi\)
\(42\) 0 0
\(43\) −7.86603 −1.19956 −0.599779 − 0.800166i \(-0.704745\pi\)
−0.599779 + 0.800166i \(0.704745\pi\)
\(44\) 0 0
\(45\) 0.212347 0.0316548
\(46\) 0 0
\(47\) 3.41855 0.498647 0.249323 − 0.968420i \(-0.419792\pi\)
0.249323 + 0.968420i \(0.419792\pi\)
\(48\) 0 0
\(49\) 3.30098 + 5.71747i 0.471569 + 0.816782i
\(50\) 0 0
\(51\) −7.28458 −1.02005
\(52\) 0 0
\(53\) 0.460811 + 0.798148i 0.0632973 + 0.109634i 0.895937 − 0.444180i \(-0.146505\pi\)
−0.832640 + 0.553814i \(0.813172\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.49693 + 7.78891i 0.595633 + 1.03167i
\(58\) 0 0
\(59\) 2.63090 + 4.55685i 0.342514 + 0.593251i 0.984899 − 0.173131i \(-0.0553884\pi\)
−0.642385 + 0.766382i \(0.722055\pi\)
\(60\) 0 0
\(61\) −1.47528 + 2.55525i −0.188890 + 0.327167i −0.944880 − 0.327416i \(-0.893822\pi\)
0.755991 + 0.654582i \(0.227156\pi\)
\(62\) 0 0
\(63\) −0.0494483 −0.00622990
\(64\) 0 0
\(65\) −2.70928 − 4.69260i −0.336044 − 0.582046i
\(66\) 0 0
\(67\) −5.41855 + 9.38521i −0.661981 + 1.14659i 0.318113 + 0.948053i \(0.396951\pi\)
−0.980094 + 0.198532i \(0.936382\pi\)
\(68\) 0 0
\(69\) −3.46081 + 5.99430i −0.416633 + 0.721629i
\(70\) 0 0
\(71\) 4.65562 − 8.06377i 0.552521 − 0.956994i −0.445571 − 0.895247i \(-0.646999\pi\)
0.998092 − 0.0617473i \(-0.0196673\pi\)
\(72\) 0 0
\(73\) −5.23513 −0.612726 −0.306363 − 0.951915i \(-0.599112\pi\)
−0.306363 + 0.951915i \(0.599112\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.55252 + 2.68904i −0.174672 + 0.302541i −0.940048 − 0.341043i \(-0.889220\pi\)
0.765376 + 0.643584i \(0.222553\pi\)
\(80\) 0 0
\(81\) 4.37936 − 7.58528i 0.486596 − 0.842809i
\(82\) 0 0
\(83\) 5.64229 + 9.77273i 0.619322 + 1.07270i 0.989610 + 0.143780i \(0.0459257\pi\)
−0.370288 + 0.928917i \(0.620741\pi\)
\(84\) 0 0
\(85\) −11.5464 −1.25238
\(86\) 0 0
\(87\) 6.02472 − 10.4351i 0.645918 − 1.11876i
\(88\) 0 0
\(89\) 5.71953 + 9.90652i 0.606269 + 1.05009i 0.991849 + 0.127415i \(0.0406680\pi\)
−0.385580 + 0.922674i \(0.625999\pi\)
\(90\) 0 0
\(91\) 0.630898 + 1.09275i 0.0661360 + 0.114551i
\(92\) 0 0
\(93\) −0.496928 + 0.860705i −0.0515291 + 0.0892510i
\(94\) 0 0
\(95\) 7.12783 + 12.3458i 0.731300 + 1.26665i
\(96\) 0 0
\(97\) −10.1773 −1.03335 −0.516673 − 0.856183i \(-0.672830\pi\)
−0.516673 + 0.856183i \(0.672830\pi\)
\(98\) 0 0
\(99\) 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 296.2.i.b.137.3 yes 6
3.2 odd 2 2664.2.r.i.433.1 6
4.3 odd 2 592.2.i.g.433.1 6
37.10 even 3 inner 296.2.i.b.121.3 ✓ 6
111.47 odd 6 2664.2.r.i.1009.1 6
148.47 odd 6 592.2.i.g.417.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.3 ✓ 6 37.10 even 3 inner
296.2.i.b.137.3 yes 6 1.1 even 1 trivial
592.2.i.g.417.1 6 148.47 odd 6
592.2.i.g.433.1 6 4.3 odd 2
2664.2.r.i.433.1 6 3.2 odd 2
2664.2.r.i.1009.1 6 111.47 odd 6