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Properties

Label 2312.2.b.i.577.3
Level $2312$
Weight $2$
Character 2312.577
Analytic conductor $18.461$
Analytic rank $0$
Dimension $4$
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] = [2312,2,Mod(577,2312)] 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("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2312.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,-16,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.4614129473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2 + \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.3
Root \(1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 2312.577
Dual form 2312.2.b.i.577.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.08239i q^{3} +3.37849i q^{5} +2.61313i q^{7} +1.82843 q^{9} +4.77791i q^{11} -5.41421 q^{13} -3.65685 q^{15} -3.65685 q^{19} -2.82843 q^{21} +0.448342i q^{23} -6.41421 q^{25} +5.22625i q^{27} -2.29610i q^{29} -2.61313i q^{31} -5.17157 q^{33} -8.82843 q^{35} +0.131316i q^{37} -5.86030i q^{39} -3.82683i q^{41} +11.6569 q^{43} +6.17733i q^{45} +6.82843 q^{47} +0.171573 q^{49} -0.242641 q^{53} -16.1421 q^{55} -3.95815i q^{57} +12.8284 q^{59} +0.317025i q^{61} +4.77791i q^{63} -18.2919i q^{65} -2.82843 q^{67} -0.485281 q^{69} -14.5964i q^{71} +14.0167i q^{73} -6.94269i q^{75} -12.4853 q^{77} +4.77791i q^{79} -0.171573 q^{81} +7.17157 q^{83} +2.48528 q^{87} -4.24264 q^{89} -14.1480i q^{91} +2.82843 q^{93} -12.3547i q^{95} +0.317025i q^{97} +8.73606i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 16 q^{13} + 8 q^{15} + 8 q^{19} - 20 q^{25} - 32 q^{33} - 24 q^{35} + 24 q^{43} + 16 q^{47} + 12 q^{49} + 16 q^{53} - 8 q^{55} + 40 q^{59} + 32 q^{69} - 16 q^{77} - 12 q^{81} + 40 q^{83}+ \cdots - 24 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.08239i 0.624919i 0.949931 + 0.312460i \(0.101153\pi\)
−0.949931 + 0.312460i \(0.898847\pi\)
\(4\) 0 0
\(5\) 3.37849i 1.51091i 0.655202 + 0.755454i \(0.272584\pi\)
−0.655202 + 0.755454i \(0.727416\pi\)
\(6\) 0 0
\(7\) 2.61313i 0.987669i 0.869556 + 0.493834i \(0.164405\pi\)
−0.869556 + 0.493834i \(0.835595\pi\)
\(8\) 0 0
\(9\) 1.82843 0.609476
\(10\) 0 0
\(11\) 4.77791i 1.44059i 0.693666 + 0.720297i \(0.255995\pi\)
−0.693666 + 0.720297i \(0.744005\pi\)
\(12\) 0 0
\(13\) −5.41421 −1.50163 −0.750816 − 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0 0
\(15\) −3.65685 −0.944196
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −3.65685 −0.838940 −0.419470 − 0.907769i \(-0.637784\pi\)
−0.419470 + 0.907769i \(0.637784\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) 0.448342i 0.0934857i 0.998907 + 0.0467428i \(0.0148841\pi\)
−0.998907 + 0.0467428i \(0.985116\pi\)
\(24\) 0 0
\(25\) −6.41421 −1.28284
\(26\) 0 0
\(27\) 5.22625i 1.00579i
\(28\) 0 0
\(29\) − 2.29610i − 0.426375i −0.977011 − 0.213188i \(-0.931615\pi\)
0.977011 − 0.213188i \(-0.0683845\pi\)
\(30\) 0 0
\(31\) − 2.61313i − 0.469331i −0.972076 − 0.234666i \(-0.924600\pi\)
0.972076 − 0.234666i \(-0.0753995\pi\)
\(32\) 0 0
\(33\) −5.17157 −0.900255
\(34\) 0 0
\(35\) −8.82843 −1.49228
\(36\) 0 0
\(37\) 0.131316i 0.0215882i 0.999942 + 0.0107941i \(0.00343594\pi\)
−0.999942 + 0.0107941i \(0.996564\pi\)
\(38\) 0 0
\(39\) − 5.86030i − 0.938399i
\(40\) 0 0
\(41\) − 3.82683i − 0.597651i −0.954308 − 0.298826i \(-0.903405\pi\)
0.954308 − 0.298826i \(-0.0965949\pi\)
\(42\) 0 0
\(43\) 11.6569 1.77765 0.888827 − 0.458243i \(-0.151521\pi\)
0.888827 + 0.458243i \(0.151521\pi\)
\(44\) 0 0
\(45\) 6.17733i 0.920862i
\(46\) 0 0
\(47\) 6.82843 0.996028 0.498014 − 0.867169i \(-0.334063\pi\)
0.498014 + 0.867169i \(0.334063\pi\)
\(48\) 0 0
\(49\) 0.171573 0.0245104
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.242641 −0.0333293 −0.0166646 − 0.999861i \(-0.505305\pi\)
−0.0166646 + 0.999861i \(0.505305\pi\)
\(54\) 0 0
\(55\) −16.1421 −2.17661
\(56\) 0 0
\(57\) − 3.95815i − 0.524270i
\(58\) 0 0
\(59\) 12.8284 1.67012 0.835059 − 0.550160i \(-0.185433\pi\)
0.835059 + 0.550160i \(0.185433\pi\)
\(60\) 0 0
\(61\) 0.317025i 0.0405909i 0.999794 + 0.0202955i \(0.00646069\pi\)
−0.999794 + 0.0202955i \(0.993539\pi\)
\(62\) 0 0
\(63\) 4.77791i 0.601960i
\(64\) 0 0
\(65\) − 18.2919i − 2.26883i
\(66\) 0 0
\(67\) −2.82843 −0.345547 −0.172774 − 0.984962i \(-0.555273\pi\)
−0.172774 + 0.984962i \(0.555273\pi\)
\(68\) 0 0
\(69\) −0.485281 −0.0584210
\(70\) 0 0
\(71\) − 14.5964i − 1.73227i −0.499810 − 0.866135i \(-0.666597\pi\)
0.499810 − 0.866135i \(-0.333403\pi\)
\(72\) 0 0
\(73\) 14.0167i 1.64053i 0.571983 + 0.820266i \(0.306174\pi\)
−0.571983 + 0.820266i \(0.693826\pi\)
\(74\) 0 0
\(75\) − 6.94269i − 0.801673i
\(76\) 0 0
\(77\) −12.4853 −1.42283
\(78\) 0 0
\(79\) 4.77791i 0.537557i 0.963202 + 0.268778i \(0.0866199\pi\)
−0.963202 + 0.268778i \(0.913380\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) 7.17157 0.787182 0.393591 − 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.48528 0.266450
\(88\) 0 0
\(89\) −4.24264 −0.449719 −0.224860 − 0.974391i \(-0.572192\pi\)
−0.224860 + 0.974391i \(0.572192\pi\)
\(90\) 0 0
\(91\) − 14.1480i − 1.48312i
\(92\) 0 0
\(93\) 2.82843 0.293294
\(94\) 0 0
\(95\) − 12.3547i − 1.26756i
\(96\) 0 0
\(97\) 0.317025i 0.0321890i 0.999870 + 0.0160945i \(0.00512327\pi\)
−0.999870 + 0.0160945i \(0.994877\pi\)
\(98\) 0 0
\(99\) 8.73606i 0.878007i
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 2312.2.b.i.577.3 4
17.4 even 4 2312.2.a.t.1.3 4
17.7 odd 16 136.2.n.b.121.1 yes 4
17.12 odd 16 136.2.n.b.9.1 ✓ 4
17.13 even 4 2312.2.a.t.1.2 4
17.16 even 2 inner 2312.2.b.i.577.2 4
51.29 even 16 1224.2.bq.b.145.1 4
51.41 even 16 1224.2.bq.b.937.1 4
68.7 even 16 272.2.v.a.257.1 4
68.47 odd 4 4624.2.a.bo.1.3 4
68.55 odd 4 4624.2.a.bo.1.2 4
68.63 even 16 272.2.v.a.145.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.n.b.9.1 ✓ 4 17.12 odd 16
136.2.n.b.121.1 yes 4 17.7 odd 16
272.2.v.a.145.1 4 68.63 even 16
272.2.v.a.257.1 4 68.7 even 16
1224.2.bq.b.145.1 4 51.29 even 16
1224.2.bq.b.937.1 4 51.41 even 16
2312.2.a.t.1.2 4 17.13 even 4
2312.2.a.t.1.3 4 17.4 even 4
2312.2.b.i.577.2 4 17.16 even 2 inner
2312.2.b.i.577.3 4 1.1 even 1 trivial
4624.2.a.bo.1.2 4 68.55 odd 4
4624.2.a.bo.1.3 4 68.47 odd 4