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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 121.1
Root \(0.155554 + 0.269427i\) of defining polynomial
Character \(\chi\) \(=\) 296.121
Dual form 296.2.i.b.137.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.45161 - 2.51426i) q^{3} +(-0.951606 - 1.64823i) q^{5} +(-0.762714 - 1.32106i) q^{7} +(-2.71432 + 4.70134i) q^{9} +(1.00000 + 1.73205i) q^{13} +(-2.76271 + 4.78516i) q^{15} +(-3.02543 + 5.24019i) q^{17} +(-3.52543 - 6.10622i) q^{19} +(-2.21432 + 3.83531i) q^{21} +4.28100 q^{23} +(0.688892 - 1.19320i) q^{25} +7.05086 q^{27} -1.28100 q^{29} -9.80642 q^{31} +(-1.45161 + 2.51426i) q^{35} +(-0.854818 - 6.02240i) q^{37} +(2.90321 - 5.02851i) q^{39} +(-0.811108 - 1.40488i) q^{41} +7.76049 q^{43} +10.3319 q^{45} -5.80642 q^{47} +(2.33654 - 4.04700i) q^{49} +17.5669 q^{51} +(3.21432 - 5.56737i) q^{53} +(-10.2351 + 17.7276i) q^{57} +(3.52543 - 6.10622i) q^{59} +(-5.64050 - 9.76963i) q^{61} +8.28100 q^{63} +(1.90321 - 3.29646i) q^{65} +(3.80642 + 6.59292i) q^{67} +(-6.21432 - 10.7635i) q^{69} +(1.38493 + 2.39877i) q^{71} +11.2859 q^{73} -4.00000 q^{75} +(-7.95407 - 13.7768i) q^{79} +(-2.09210 - 3.62363i) q^{81} +(-6.78346 + 11.7493i) q^{83} +11.5161 q^{85} +(1.85950 + 3.22075i) q^{87} +(-4.46989 + 7.74207i) q^{89} +(1.52543 - 2.64212i) q^{91} +(14.2351 + 24.6559i) q^{93} +(-6.70964 + 11.6214i) q^{95} +11.9906 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 6 q^{13} - 10 q^{15} - 5 q^{17} - 8 q^{19} + 12 q^{23} + 4 q^{25} + 16 q^{27} + 6 q^{29} - 32 q^{31} - 2 q^{35} + 15 q^{37} + 4 q^{39} - 5 q^{41} - 20 q^{43}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.45161 − 2.51426i −0.838085 − 1.45161i −0.891494 − 0.453033i \(-0.850342\pi\)
0.0534085 − 0.998573i \(-0.482991\pi\)
\(4\) 0 0
\(5\) −0.951606 − 1.64823i −0.425571 − 0.737111i 0.570902 − 0.821018i \(-0.306593\pi\)
−0.996474 + 0.0839071i \(0.973260\pi\)
\(6\) 0 0
\(7\) −0.762714 − 1.32106i −0.288279 − 0.499313i 0.685120 − 0.728430i \(-0.259750\pi\)
−0.973399 + 0.229117i \(0.926416\pi\)
\(8\) 0 0
\(9\) −2.71432 + 4.70134i −0.904773 + 1.56711i
\(10\) 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 0.970725 − 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −2.76271 + 4.78516i −0.713330 + 1.23552i
\(16\) 0 0
\(17\) −3.02543 + 5.24019i −0.733774 + 1.27093i 0.221485 + 0.975164i \(0.428910\pi\)
−0.955259 + 0.295770i \(0.904424\pi\)
\(18\) 0 0
\(19\) −3.52543 − 6.10622i −0.808789 − 1.40086i −0.913703 − 0.406382i \(-0.866790\pi\)
0.104915 − 0.994481i \(-0.466543\pi\)
\(20\) 0 0
\(21\) −2.21432 + 3.83531i −0.483204 + 0.836934i
\(22\) 0 0
\(23\) 4.28100 0.892649 0.446325 − 0.894871i \(-0.352733\pi\)
0.446325 + 0.894871i \(0.352733\pi\)
\(24\) 0 0
\(25\) 0.688892 − 1.19320i 0.137778 − 0.238639i
\(26\) 0 0
\(27\) 7.05086 1.35694
\(28\) 0 0
\(29\) −1.28100 −0.237875 −0.118938 − 0.992902i \(-0.537949\pi\)
−0.118938 + 0.992902i \(0.537949\pi\)
\(30\) 0 0
\(31\) −9.80642 −1.76129 −0.880643 − 0.473781i \(-0.842889\pi\)
−0.880643 + 0.473781i \(0.842889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.45161 + 2.51426i −0.245366 + 0.424987i
\(36\) 0 0
\(37\) −0.854818 − 6.02240i −0.140531 − 0.990076i
\(38\) 0 0
\(39\) 2.90321 − 5.02851i 0.464886 − 0.805206i
\(40\) 0 0
\(41\) −0.811108 − 1.40488i −0.126674 − 0.219405i 0.795712 − 0.605675i \(-0.207097\pi\)
−0.922386 + 0.386270i \(0.873763\pi\)
\(42\) 0 0
\(43\) 7.76049 1.18346 0.591732 − 0.806135i \(-0.298444\pi\)
0.591732 + 0.806135i \(0.298444\pi\)
\(44\) 0 0
\(45\) 10.3319 1.54018
\(46\) 0 0
\(47\) −5.80642 −0.846954 −0.423477 − 0.905907i \(-0.639191\pi\)
−0.423477 + 0.905907i \(0.639191\pi\)
\(48\) 0 0
\(49\) 2.33654 − 4.04700i 0.333791 − 0.578143i
\(50\) 0 0
\(51\) 17.5669 2.45986
\(52\) 0 0
\(53\) 3.21432 − 5.56737i 0.441521 − 0.764736i −0.556282 − 0.830994i \(-0.687772\pi\)
0.997803 + 0.0662573i \(0.0211058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.2351 + 17.7276i −1.35567 + 2.34808i
\(58\) 0 0
\(59\) 3.52543 − 6.10622i 0.458972 − 0.794962i −0.539935 − 0.841707i \(-0.681551\pi\)
0.998907 + 0.0467444i \(0.0148846\pi\)
\(60\) 0 0
\(61\) −5.64050 − 9.76963i −0.722192 − 1.25087i −0.960119 − 0.279590i \(-0.909801\pi\)
0.237928 − 0.971283i \(-0.423532\pi\)
\(62\) 0 0
\(63\) 8.28100 1.04331
\(64\) 0 0
\(65\) 1.90321 − 3.29646i 0.236064 − 0.408876i
\(66\) 0 0
\(67\) 3.80642 + 6.59292i 0.465029 + 0.805453i 0.999203 − 0.0399211i \(-0.0127107\pi\)
−0.534174 + 0.845374i \(0.679377\pi\)
\(68\) 0 0
\(69\) −6.21432 − 10.7635i −0.748116 − 1.29578i
\(70\) 0 0
\(71\) 1.38493 + 2.39877i 0.164361 + 0.284681i 0.936428 − 0.350859i \(-0.114111\pi\)
−0.772067 + 0.635541i \(0.780777\pi\)
\(72\) 0 0
\(73\) 11.2859 1.32092 0.660458 − 0.750863i \(-0.270362\pi\)
0.660458 + 0.750863i \(0.270362\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.95407 − 13.7768i −0.894902 − 1.55002i −0.833925 − 0.551877i \(-0.813912\pi\)
−0.0609772 − 0.998139i \(-0.519422\pi\)
\(80\) 0 0
\(81\) −2.09210 − 3.62363i −0.232456 − 0.402626i
\(82\) 0 0
\(83\) −6.78346 + 11.7493i −0.744581 + 1.28965i 0.205809 + 0.978592i \(0.434017\pi\)
−0.950390 + 0.311060i \(0.899316\pi\)
\(84\) 0 0
\(85\) 11.5161 1.24909
\(86\) 0 0
\(87\) 1.85950 + 3.22075i 0.199360 + 0.345301i
\(88\) 0 0
\(89\) −4.46989 + 7.74207i −0.473807 + 0.820658i −0.999550 − 0.0299852i \(-0.990454\pi\)
0.525743 + 0.850643i \(0.323787\pi\)
\(90\) 0 0
\(91\) 1.52543 − 2.64212i 0.159908 − 0.276969i
\(92\) 0 0
\(93\) 14.2351 + 24.6559i 1.47611 + 2.55669i
\(94\) 0 0
\(95\) −6.70964 + 11.6214i −0.688394 + 1.19233i
\(96\) 0 0
\(97\) 11.9906 1.21746 0.608732 − 0.793376i \(-0.291678\pi\)
0.608732 + 0.793376i \(0.291678\pi\)
\(98\) 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.121.1 ✓ 6
3.2 odd 2 2664.2.r.i.1009.3 6
4.3 odd 2 592.2.i.g.417.3 6
37.26 even 3 inner 296.2.i.b.137.1 yes 6
111.26 odd 6 2664.2.r.i.433.3 6
148.63 odd 6 592.2.i.g.433.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.1 ✓ 6 1.1 even 1 trivial
296.2.i.b.137.1 yes 6 37.26 even 3 inner
592.2.i.g.417.3 6 4.3 odd 2
592.2.i.g.433.3 6 148.63 odd 6
2664.2.r.i.433.3 6 111.26 odd 6
2664.2.r.i.1009.3 6 3.2 odd 2