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Properties

Label 560.3.p.f.209.1
Level $560$
Weight $3$
Character 560.209
Analytic conductor $15.259$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [560,3,Mod(209,560)] 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("560.209"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.p (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] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.1
Root \(-1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 560.209
Dual form 560.3.p.f.209.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.16228 q^{3} +(-1.58114 - 4.74342i) q^{5} +(6.32456 + 3.00000i) q^{7} +1.00000 q^{9} -14.0000 q^{11} -3.16228 q^{13} +(5.00000 + 15.0000i) q^{15} +6.32456 q^{17} -28.4605i q^{19} +(-20.0000 - 9.48683i) q^{21} +12.0000i q^{23} +(-20.0000 + 15.0000i) q^{25} +25.2982 q^{27} +14.0000 q^{29} +37.9473i q^{31} +44.2719 q^{33} +(4.23025 - 34.7434i) q^{35} +18.0000i q^{37} +10.0000 q^{39} +18.9737i q^{41} +42.0000i q^{43} +(-1.58114 - 4.74342i) q^{45} -44.2719 q^{47} +(31.0000 + 37.9473i) q^{49} -20.0000 q^{51} +54.0000i q^{53} +(22.1359 + 66.4078i) q^{55} +90.0000i q^{57} +9.48683i q^{59} +66.4078i q^{61} +(6.32456 + 3.00000i) q^{63} +(5.00000 + 15.0000i) q^{65} +102.000i q^{67} -37.9473i q^{69} +16.0000 q^{71} +63.2456 q^{73} +(63.2456 - 47.4342i) q^{75} +(-88.5438 - 42.0000i) q^{77} +76.0000 q^{79} -89.0000 q^{81} -72.7324 q^{83} +(-10.0000 - 30.0000i) q^{85} -44.2719 q^{87} -56.9210i q^{89} +(-20.0000 - 9.48683i) q^{91} -120.000i q^{93} +(-135.000 + 45.0000i) q^{95} -69.5701 q^{97} -14.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} - 56 q^{11} + 20 q^{15} - 80 q^{21} - 80 q^{25} + 56 q^{29} - 40 q^{35} + 40 q^{39} + 124 q^{49} - 80 q^{51} + 20 q^{65} + 64 q^{71} + 304 q^{79} - 356 q^{81} - 40 q^{85} - 80 q^{91} - 540 q^{95}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −3.16228 −1.05409 −0.527046 − 0.849837i \(-0.676701\pi\)
−0.527046 + 0.849837i \(0.676701\pi\)
\(4\) 0 0
\(5\) −1.58114 − 4.74342i −0.316228 − 0.948683i
\(6\) 0 0
\(7\) 6.32456 + 3.00000i 0.903508 + 0.428571i
\(8\) 0 0
\(9\) 1.00000 0.111111
\(10\) 0 0
\(11\) −14.0000 −1.27273 −0.636364 − 0.771389i \(-0.719562\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) −3.16228 −0.243252 −0.121626 − 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(14\) 0 0
\(15\) 5.00000 + 15.0000i 0.333333 + 1.00000i
\(16\) 0 0
\(17\) 6.32456 0.372033 0.186016 − 0.982547i \(-0.440442\pi\)
0.186016 + 0.982547i \(0.440442\pi\)
\(18\) 0 0
\(19\) − 28.4605i − 1.49792i −0.662615 − 0.748960i \(-0.730553\pi\)
0.662615 − 0.748960i \(-0.269447\pi\)
\(20\) 0 0
\(21\) −20.0000 − 9.48683i −0.952381 − 0.451754i
\(22\) 0 0
\(23\) 12.0000i 0.521739i 0.965374 + 0.260870i \(0.0840093\pi\)
−0.965374 + 0.260870i \(0.915991\pi\)
\(24\) 0 0
\(25\) −20.0000 + 15.0000i −0.800000 + 0.600000i
\(26\) 0 0
\(27\) 25.2982 0.936971
\(28\) 0 0
\(29\) 14.0000 0.482759 0.241379 − 0.970431i \(-0.422400\pi\)
0.241379 + 0.970431i \(0.422400\pi\)
\(30\) 0 0
\(31\) 37.9473i 1.22411i 0.790816 + 0.612054i \(0.209656\pi\)
−0.790816 + 0.612054i \(0.790344\pi\)
\(32\) 0 0
\(33\) 44.2719 1.34157
\(34\) 0 0
\(35\) 4.23025 − 34.7434i 0.120864 − 0.992669i
\(36\) 0 0
\(37\) 18.0000i 0.486486i 0.969965 + 0.243243i \(0.0782113\pi\)
−0.969965 + 0.243243i \(0.921789\pi\)
\(38\) 0 0
\(39\) 10.0000 0.256410
\(40\) 0 0
\(41\) 18.9737i 0.462772i 0.972862 + 0.231386i \(0.0743261\pi\)
−0.972862 + 0.231386i \(0.925674\pi\)
\(42\) 0 0
\(43\) 42.0000i 0.976744i 0.872635 + 0.488372i \(0.162409\pi\)
−0.872635 + 0.488372i \(0.837591\pi\)
\(44\) 0 0
\(45\) −1.58114 − 4.74342i −0.0351364 − 0.105409i
\(46\) 0 0
\(47\) −44.2719 −0.941955 −0.470978 − 0.882145i \(-0.656099\pi\)
−0.470978 + 0.882145i \(0.656099\pi\)
\(48\) 0 0
\(49\) 31.0000 + 37.9473i 0.632653 + 0.774435i
\(50\) 0 0
\(51\) −20.0000 −0.392157
\(52\) 0 0
\(53\) 54.0000i 1.01887i 0.860510 + 0.509434i \(0.170145\pi\)
−0.860510 + 0.509434i \(0.829855\pi\)
\(54\) 0 0
\(55\) 22.1359 + 66.4078i 0.402472 + 1.20742i
\(56\) 0 0
\(57\) 90.0000i 1.57895i
\(58\) 0 0
\(59\) 9.48683i 0.160794i 0.996763 + 0.0803969i \(0.0256188\pi\)
−0.996763 + 0.0803969i \(0.974381\pi\)
\(60\) 0 0
\(61\) 66.4078i 1.08865i 0.838873 + 0.544326i \(0.183215\pi\)
−0.838873 + 0.544326i \(0.816785\pi\)
\(62\) 0 0
\(63\) 6.32456 + 3.00000i 0.100390 + 0.0476190i
\(64\) 0 0
\(65\) 5.00000 + 15.0000i 0.0769231 + 0.230769i
\(66\) 0 0
\(67\) 102.000i 1.52239i 0.648524 + 0.761194i \(0.275387\pi\)
−0.648524 + 0.761194i \(0.724613\pi\)
\(68\) 0 0
\(69\) − 37.9473i − 0.549961i
\(70\) 0 0
\(71\) 16.0000 0.225352 0.112676 − 0.993632i \(-0.464058\pi\)
0.112676 + 0.993632i \(0.464058\pi\)
\(72\) 0 0
\(73\) 63.2456 0.866377 0.433189 − 0.901303i \(-0.357388\pi\)
0.433189 + 0.901303i \(0.357388\pi\)
\(74\) 0 0
\(75\) 63.2456 − 47.4342i 0.843274 − 0.632456i
\(76\) 0 0
\(77\) −88.5438 − 42.0000i −1.14992 − 0.545455i
\(78\) 0 0
\(79\) 76.0000 0.962025 0.481013 − 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(80\) 0 0
\(81\) −89.0000 −1.09877
\(82\) 0 0
\(83\) −72.7324 −0.876294 −0.438147 − 0.898903i \(-0.644365\pi\)
−0.438147 + 0.898903i \(0.644365\pi\)
\(84\) 0 0
\(85\) −10.0000 − 30.0000i −0.117647 − 0.352941i
\(86\) 0 0
\(87\) −44.2719 −0.508872
\(88\) 0 0
\(89\) − 56.9210i − 0.639562i −0.947492 − 0.319781i \(-0.896391\pi\)
0.947492 − 0.319781i \(-0.103609\pi\)
\(90\) 0 0
\(91\) −20.0000 − 9.48683i −0.219780 − 0.104251i
\(92\) 0 0
\(93\) − 120.000i − 1.29032i
\(94\) 0 0
\(95\) −135.000 + 45.0000i −1.42105 + 0.473684i
\(96\) 0 0
\(97\) −69.5701 −0.717218 −0.358609 − 0.933488i \(-0.616749\pi\)
−0.358609 + 0.933488i \(0.616749\pi\)
\(98\) 0 0
\(99\) −14.0000 −0.141414
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 560.3.p.f.209.1 4
4.3 odd 2 35.3.c.c.34.4 yes 4
5.4 even 2 inner 560.3.p.f.209.3 4
7.6 odd 2 inner 560.3.p.f.209.4 4
12.11 even 2 315.3.e.c.244.2 4
20.3 even 4 175.3.d.h.76.2 2
20.7 even 4 175.3.d.b.76.1 2
20.19 odd 2 35.3.c.c.34.1 ✓ 4
28.3 even 6 245.3.i.c.19.2 8
28.11 odd 6 245.3.i.c.19.1 8
28.19 even 6 245.3.i.c.129.4 8
28.23 odd 6 245.3.i.c.129.3 8
28.27 even 2 35.3.c.c.34.3 yes 4
35.34 odd 2 inner 560.3.p.f.209.2 4
60.59 even 2 315.3.e.c.244.3 4
84.83 odd 2 315.3.e.c.244.1 4
140.19 even 6 245.3.i.c.129.1 8
140.27 odd 4 175.3.d.b.76.2 2
140.39 odd 6 245.3.i.c.19.4 8
140.59 even 6 245.3.i.c.19.3 8
140.79 odd 6 245.3.i.c.129.2 8
140.83 odd 4 175.3.d.h.76.1 2
140.139 even 2 35.3.c.c.34.2 yes 4
420.419 odd 2 315.3.e.c.244.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.c.34.1 ✓ 4 20.19 odd 2
35.3.c.c.34.2 yes 4 140.139 even 2
35.3.c.c.34.3 yes 4 28.27 even 2
35.3.c.c.34.4 yes 4 4.3 odd 2
175.3.d.b.76.1 2 20.7 even 4
175.3.d.b.76.2 2 140.27 odd 4
175.3.d.h.76.1 2 140.83 odd 4
175.3.d.h.76.2 2 20.3 even 4
245.3.i.c.19.1 8 28.11 odd 6
245.3.i.c.19.2 8 28.3 even 6
245.3.i.c.19.3 8 140.59 even 6
245.3.i.c.19.4 8 140.39 odd 6
245.3.i.c.129.1 8 140.19 even 6
245.3.i.c.129.2 8 140.79 odd 6
245.3.i.c.129.3 8 28.23 odd 6
245.3.i.c.129.4 8 28.19 even 6
315.3.e.c.244.1 4 84.83 odd 2
315.3.e.c.244.2 4 12.11 even 2
315.3.e.c.244.3 4 60.59 even 2
315.3.e.c.244.4 4 420.419 odd 2
560.3.p.f.209.1 4 1.1 even 1 trivial
560.3.p.f.209.2 4 35.34 odd 2 inner
560.3.p.f.209.3 4 5.4 even 2 inner
560.3.p.f.209.4 4 7.6 odd 2 inner