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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2304,3,Mod(2177,2304)] 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("2304.2177"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.a.2177.2

$q$-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)
 
\(f(q)\) \(=\) \(q-7.07107 q^{5} -12.0000 q^{7} +5.65685 q^{11} -8.00000i q^{13} -9.89949i q^{17} -16.0000i q^{19} -39.5980i q^{23} +25.0000 q^{25} -29.6985 q^{29} -4.00000 q^{31} +84.8528 q^{35} -30.0000i q^{37} -21.2132i q^{41} +8.00000i q^{43} +16.9706i q^{47} +95.0000 q^{49} -49.4975 q^{53} -40.0000 q^{55} -79.1960 q^{59} -14.0000i q^{61} +56.5685i q^{65} -88.0000i q^{67} -28.2843i q^{71} +80.0000 q^{73} -67.8823 q^{77} +100.000 q^{79} -130.108 q^{83} +70.0000i q^{85} +148.492i q^{89} +96.0000i q^{91} +113.137i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{7} + 100 q^{25} - 16 q^{31} + 380 q^{49} - 160 q^{55} + 320 q^{73} + 400 q^{79} - 448 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −7.07107 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) −12.0000 −1.71429 −0.857143 − 0.515079i \(-0.827763\pi\)
−0.857143 + 0.515079i \(0.827763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 0.514259 0.257130 − 0.966377i \(-0.417223\pi\)
0.257130 + 0.966377i \(0.417223\pi\)
\(12\) 0 0
\(13\) − 8.00000i − 0.615385i −0.951486 − 0.307692i \(-0.900443\pi\)
0.951486 − 0.307692i \(-0.0995567\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 9.89949i − 0.582323i −0.956674 − 0.291162i \(-0.905958\pi\)
0.956674 − 0.291162i \(-0.0940417\pi\)
\(18\) 0 0
\(19\) − 16.0000i − 0.842105i −0.907036 − 0.421053i \(-0.861661\pi\)
0.907036 − 0.421053i \(-0.138339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 39.5980i − 1.72165i −0.508900 − 0.860826i \(-0.669948\pi\)
0.508900 − 0.860826i \(-0.330052\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −29.6985 −1.02409 −0.512043 − 0.858960i \(-0.671111\pi\)
−0.512043 + 0.858960i \(0.671111\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.129032 −0.0645161 − 0.997917i \(-0.520550\pi\)
−0.0645161 + 0.997917i \(0.520550\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 84.8528 2.42437
\(36\) 0 0
\(37\) − 30.0000i − 0.810811i −0.914137 − 0.405405i \(-0.867130\pi\)
0.914137 − 0.405405i \(-0.132870\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 21.2132i − 0.517395i −0.965958 − 0.258698i \(-0.916707\pi\)
0.965958 − 0.258698i \(-0.0832933\pi\)
\(42\) 0 0
\(43\) 8.00000i 0.186047i 0.995664 + 0.0930233i \(0.0296531\pi\)
−0.995664 + 0.0930233i \(0.970347\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 95.0000 1.93878
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −49.4975 −0.933915 −0.466957 − 0.884280i \(-0.654650\pi\)
−0.466957 + 0.884280i \(0.654650\pi\)
\(54\) 0 0
\(55\) −40.0000 −0.727273
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −79.1960 −1.34230 −0.671152 − 0.741320i \(-0.734200\pi\)
−0.671152 + 0.741320i \(0.734200\pi\)
\(60\) 0 0
\(61\) − 14.0000i − 0.229508i −0.993394 − 0.114754i \(-0.963392\pi\)
0.993394 − 0.114754i \(-0.0366080\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 56.5685i 0.870285i
\(66\) 0 0
\(67\) − 88.0000i − 1.31343i −0.754138 − 0.656716i \(-0.771945\pi\)
0.754138 − 0.656716i \(-0.228055\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 28.2843i − 0.398370i −0.979962 − 0.199185i \(-0.936171\pi\)
0.979962 − 0.199185i \(-0.0638295\pi\)
\(72\) 0 0
\(73\) 80.0000 1.09589 0.547945 − 0.836514i \(-0.315410\pi\)
0.547945 + 0.836514i \(0.315410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −67.8823 −0.881588
\(78\) 0 0
\(79\) 100.000 1.26582 0.632911 − 0.774224i \(-0.281860\pi\)
0.632911 + 0.774224i \(0.281860\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −130.108 −1.56756 −0.783781 − 0.621037i \(-0.786712\pi\)
−0.783781 + 0.621037i \(0.786712\pi\)
\(84\) 0 0
\(85\) 70.0000i 0.823529i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 148.492i 1.66845i 0.551421 + 0.834227i \(0.314086\pi\)
−0.551421 + 0.834227i \(0.685914\pi\)
\(90\) 0 0
\(91\) 96.0000i 1.05495i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 113.137i 1.19092i
\(96\) 0 0
\(97\) −112.000 −1.15464 −0.577320 − 0.816518i \(-0.695901\pi\)
−0.577320 + 0.816518i \(0.695901\pi\)
\(98\) 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 2304.3.h.a.2177.1 4
3.2 odd 2 inner 2304.3.h.a.2177.3 4
4.3 odd 2 2304.3.h.h.2177.1 4
8.3 odd 2 2304.3.h.h.2177.4 4
8.5 even 2 inner 2304.3.h.a.2177.4 4
12.11 even 2 2304.3.h.h.2177.3 4
16.3 odd 4 576.3.e.a.449.2 2
16.5 even 4 72.3.e.a.17.1 ✓ 2
16.11 odd 4 144.3.e.a.17.1 2
16.13 even 4 576.3.e.h.449.2 2
24.5 odd 2 inner 2304.3.h.a.2177.2 4
24.11 even 2 2304.3.h.h.2177.2 4
48.5 odd 4 72.3.e.a.17.2 yes 2
48.11 even 4 144.3.e.a.17.2 2
48.29 odd 4 576.3.e.h.449.1 2
48.35 even 4 576.3.e.a.449.1 2
80.27 even 4 3600.3.c.c.449.1 4
80.37 odd 4 1800.3.c.a.449.4 4
80.43 even 4 3600.3.c.c.449.3 4
80.53 odd 4 1800.3.c.a.449.2 4
80.59 odd 4 3600.3.l.l.1601.1 2
80.69 even 4 1800.3.l.a.1601.2 2
112.69 odd 4 3528.3.d.a.1961.2 2
144.5 odd 12 648.3.m.a.593.2 4
144.11 even 12 1296.3.q.k.1025.1 4
144.43 odd 12 1296.3.q.k.1025.2 4
144.59 even 12 1296.3.q.k.593.2 4
144.85 even 12 648.3.m.a.593.1 4
144.101 odd 12 648.3.m.a.377.1 4
144.133 even 12 648.3.m.a.377.2 4
144.139 odd 12 1296.3.q.k.593.1 4
240.53 even 4 1800.3.c.a.449.1 4
240.59 even 4 3600.3.l.l.1601.2 2
240.107 odd 4 3600.3.c.c.449.2 4
240.149 odd 4 1800.3.l.a.1601.1 2
240.197 even 4 1800.3.c.a.449.3 4
240.203 odd 4 3600.3.c.c.449.4 4
336.293 even 4 3528.3.d.a.1961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.e.a.17.1 ✓ 2 16.5 even 4
72.3.e.a.17.2 yes 2 48.5 odd 4
144.3.e.a.17.1 2 16.11 odd 4
144.3.e.a.17.2 2 48.11 even 4
576.3.e.a.449.1 2 48.35 even 4
576.3.e.a.449.2 2 16.3 odd 4
576.3.e.h.449.1 2 48.29 odd 4
576.3.e.h.449.2 2 16.13 even 4
648.3.m.a.377.1 4 144.101 odd 12
648.3.m.a.377.2 4 144.133 even 12
648.3.m.a.593.1 4 144.85 even 12
648.3.m.a.593.2 4 144.5 odd 12
1296.3.q.k.593.1 4 144.139 odd 12
1296.3.q.k.593.2 4 144.59 even 12
1296.3.q.k.1025.1 4 144.11 even 12
1296.3.q.k.1025.2 4 144.43 odd 12
1800.3.c.a.449.1 4 240.53 even 4
1800.3.c.a.449.2 4 80.53 odd 4
1800.3.c.a.449.3 4 240.197 even 4
1800.3.c.a.449.4 4 80.37 odd 4
1800.3.l.a.1601.1 2 240.149 odd 4
1800.3.l.a.1601.2 2 80.69 even 4
2304.3.h.a.2177.1 4 1.1 even 1 trivial
2304.3.h.a.2177.2 4 24.5 odd 2 inner
2304.3.h.a.2177.3 4 3.2 odd 2 inner
2304.3.h.a.2177.4 4 8.5 even 2 inner
2304.3.h.h.2177.1 4 4.3 odd 2
2304.3.h.h.2177.2 4 24.11 even 2
2304.3.h.h.2177.3 4 12.11 even 2
2304.3.h.h.2177.4 4 8.3 odd 2
3528.3.d.a.1961.1 2 336.293 even 4
3528.3.d.a.1961.2 2 112.69 odd 4
3600.3.c.c.449.1 4 80.27 even 4
3600.3.c.c.449.2 4 240.107 odd 4
3600.3.c.c.449.3 4 80.43 even 4
3600.3.c.c.449.4 4 240.203 odd 4
3600.3.l.l.1601.1 2 80.59 odd 4
3600.3.l.l.1601.2 2 240.59 even 4