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Properties

Label 396.4.j.d.289.1
Level $396$
Weight $4$
Character 396.289
Analytic conductor $23.365$
Analytic rank $0$
Dimension $12$
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] = [396,4,Mod(37,396)] 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("396.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 396.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.3647563623\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 70 x^{10} - 84 x^{9} + 2459 x^{8} - 8514 x^{7} + 54995 x^{6} - 432951 x^{5} + \cdots + 40896025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 289.1
Root \(1.50918 + 1.09648i\) of defining polynomial
Character \(\chi\) \(=\) 396.289
Dual form 396.4.j.d.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-14.9139 - 10.8356i) q^{5} +(9.02047 - 27.7622i) q^{7} +(-5.54276 - 36.0594i) q^{11} +(-63.1228 + 45.8614i) q^{13} +(23.1093 + 16.7899i) q^{17} +(-13.4717 - 41.4615i) q^{19} +104.563 q^{23} +(66.3872 + 204.319i) q^{25} +(-13.1435 + 40.4515i) q^{29} +(51.0435 - 37.0853i) q^{31} +(-435.349 + 316.300i) q^{35} +(-98.0877 + 301.883i) q^{37} +(21.1199 + 65.0003i) q^{41} -227.116 q^{43} +(3.32926 + 10.2464i) q^{47} +(-411.875 - 299.245i) q^{49} +(-402.111 + 292.151i) q^{53} +(-308.060 + 597.844i) q^{55} +(130.834 - 402.665i) q^{59} +(154.010 + 111.895i) q^{61} +1438.34 q^{65} -386.223 q^{67} +(-559.414 - 406.438i) q^{71} +(98.3036 - 302.547i) q^{73} +(-1051.08 - 171.393i) q^{77} +(-130.828 + 95.0523i) q^{79} +(318.845 + 231.654i) q^{83} +(-162.721 - 500.804i) q^{85} -108.129 q^{89} +(703.814 + 2166.12i) q^{91} +(-248.344 + 764.325i) q^{95} +(549.111 - 398.952i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} + 6 q^{7} - 39 q^{11} - 10 q^{13} + 56 q^{17} - 141 q^{19} + 388 q^{23} - 203 q^{25} - 772 q^{29} + 882 q^{31} - 412 q^{35} - 192 q^{37} + 180 q^{41} - 2330 q^{43} - 196 q^{47} - 973 q^{49}+ \cdots + 2651 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −14.9139 − 10.8356i −1.33394 − 0.969163i −0.999644 − 0.0266942i \(-0.991502\pi\)
−0.334295 − 0.942469i \(-0.608498\pi\)
\(6\) 0 0
\(7\) 9.02047 − 27.7622i 0.487060 − 1.49902i −0.341916 − 0.939731i \(-0.611076\pi\)
0.828976 − 0.559285i \(-0.188924\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.54276 − 36.0594i −0.151928 − 0.988392i
\(12\) 0 0
\(13\) −63.1228 + 45.8614i −1.34670 + 0.978435i −0.347532 + 0.937668i \(0.612980\pi\)
−0.999169 + 0.0407672i \(0.987020\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.1093 + 16.7899i 0.329695 + 0.239538i 0.740301 − 0.672275i \(-0.234683\pi\)
−0.410606 + 0.911813i \(0.634683\pi\)
\(18\) 0 0
\(19\) −13.4717 − 41.4615i −0.162664 − 0.500628i 0.836193 − 0.548436i \(-0.184776\pi\)
−0.998857 + 0.0478081i \(0.984776\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 104.563 0.947952 0.473976 − 0.880538i \(-0.342818\pi\)
0.473976 + 0.880538i \(0.342818\pi\)
\(24\) 0 0
\(25\) 66.3872 + 204.319i 0.531098 + 1.63455i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −13.1435 + 40.4515i −0.0841615 + 0.259022i −0.984278 − 0.176627i \(-0.943481\pi\)
0.900116 + 0.435650i \(0.143481\pi\)
\(30\) 0 0
\(31\) 51.0435 − 37.0853i 0.295732 − 0.214862i −0.430018 − 0.902820i \(-0.641493\pi\)
0.725750 + 0.687958i \(0.241493\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −435.349 + 316.300i −2.10250 + 1.52755i
\(36\) 0 0
\(37\) −98.0877 + 301.883i −0.435825 + 1.34133i 0.456414 + 0.889768i \(0.349134\pi\)
−0.892239 + 0.451564i \(0.850866\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 21.1199 + 65.0003i 0.0804481 + 0.247594i 0.983189 − 0.182591i \(-0.0584484\pi\)
−0.902741 + 0.430185i \(0.858448\pi\)
\(42\) 0 0
\(43\) −227.116 −0.805463 −0.402731 − 0.915318i \(-0.631939\pi\)
−0.402731 + 0.915318i \(0.631939\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.32926 + 10.2464i 0.0103324 + 0.0317999i 0.956090 − 0.293074i \(-0.0946782\pi\)
−0.945757 + 0.324874i \(0.894678\pi\)
\(48\) 0 0
\(49\) −411.875 − 299.245i −1.20080 − 0.872434i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −402.111 + 292.151i −1.04215 + 0.757170i −0.970705 − 0.240274i \(-0.922763\pi\)
−0.0714497 + 0.997444i \(0.522763\pi\)
\(54\) 0 0
\(55\) −308.060 + 597.844i −0.755250 + 1.46570i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 130.834 − 402.665i 0.288697 − 0.888517i −0.696570 − 0.717489i \(-0.745291\pi\)
0.985266 − 0.171028i \(-0.0547088\pi\)
\(60\) 0 0
\(61\) 154.010 + 111.895i 0.323262 + 0.234864i 0.737566 − 0.675275i \(-0.235975\pi\)
−0.414304 + 0.910139i \(0.635975\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1438.34 2.74468
\(66\) 0 0
\(67\) −386.223 −0.704248 −0.352124 − 0.935953i \(-0.614541\pi\)
−0.352124 + 0.935953i \(0.614541\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −559.414 − 406.438i −0.935074 − 0.679371i 0.0121559 − 0.999926i \(-0.496131\pi\)
−0.947230 + 0.320555i \(0.896131\pi\)
\(72\) 0 0
\(73\) 98.3036 − 302.547i 0.157610 − 0.485075i −0.840806 − 0.541337i \(-0.817918\pi\)
0.998416 + 0.0562620i \(0.0179182\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1051.08 − 171.393i −1.55561 − 0.253664i
\(78\) 0 0
\(79\) −130.828 + 95.0523i −0.186321 + 0.135370i −0.677035 − 0.735950i \(-0.736736\pi\)
0.490715 + 0.871320i \(0.336736\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 318.845 + 231.654i 0.421660 + 0.306354i 0.778305 − 0.627886i \(-0.216080\pi\)
−0.356645 + 0.934240i \(0.616080\pi\)
\(84\) 0 0
\(85\) −162.721 − 500.804i −0.207642 − 0.639057i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −108.129 −0.128782 −0.0643912 − 0.997925i \(-0.520511\pi\)
−0.0643912 + 0.997925i \(0.520511\pi\)
\(90\) 0 0
\(91\) 703.814 + 2166.12i 0.810766 + 2.49528i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −248.344 + 764.325i −0.268206 + 0.825454i
\(96\) 0 0
\(97\) 549.111 − 398.952i 0.574781 − 0.417603i −0.262058 − 0.965052i \(-0.584401\pi\)
0.836839 + 0.547449i \(0.184401\pi\)
\(98\) 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 396.4.j.d.289.1 12
3.2 odd 2 44.4.e.a.25.2 ✓ 12
11.4 even 5 inner 396.4.j.d.37.1 12
12.11 even 2 176.4.m.d.113.2 12
33.2 even 10 484.4.a.h.1.3 6
33.20 odd 10 484.4.a.i.1.3 6
33.26 odd 10 44.4.e.a.37.2 yes 12
132.35 odd 10 1936.4.a.bs.1.4 6
132.59 even 10 176.4.m.d.81.2 12
132.119 even 10 1936.4.a.br.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.e.a.25.2 ✓ 12 3.2 odd 2
44.4.e.a.37.2 yes 12 33.26 odd 10
176.4.m.d.81.2 12 132.59 even 10
176.4.m.d.113.2 12 12.11 even 2
396.4.j.d.37.1 12 11.4 even 5 inner
396.4.j.d.289.1 12 1.1 even 1 trivial
484.4.a.h.1.3 6 33.2 even 10
484.4.a.i.1.3 6 33.20 odd 10
1936.4.a.br.1.4 6 132.119 even 10
1936.4.a.bs.1.4 6 132.35 odd 10