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Properties

Label 525.3.e.b.349.11
Level $525$
Weight $3$
Character 525.349
Analytic conductor $14.305$
Analytic rank $0$
Dimension $20$
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] = [525,3,Mod(349,525)] 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("525.349"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 25 x^{18} + 409 x^{16} - 3886 x^{14} + 26830 x^{12} - 118498 x^{10} + 377533 x^{8} + \cdots + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.11
Root \(-0.176161 + 0.101707i\) of defining polynomial
Character \(\chi\) \(=\) 525.349
Dual form 525.3.e.b.349.9

$q$-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)
 
\(f(q)\) \(=\) \(q+0.203414i q^{2} -1.73205 q^{3} +3.95862 q^{4} -0.352323i q^{6} +(-6.30811 - 3.03444i) q^{7} +1.61889i q^{8} +3.00000 q^{9} +3.28794 q^{11} -6.85654 q^{12} -4.69238 q^{13} +(0.617246 - 1.28316i) q^{14} +15.5052 q^{16} +16.0433 q^{17} +0.610241i q^{18} -17.3762i q^{19} +(10.9260 + 5.25580i) q^{21} +0.668812i q^{22} -38.9847i q^{23} -2.80401i q^{24} -0.954495i q^{26} -5.19615 q^{27} +(-24.9714 - 12.0122i) q^{28} +5.75378 q^{29} -1.73978i q^{31} +9.62954i q^{32} -5.69488 q^{33} +3.26342i q^{34} +11.8759 q^{36} +43.8846i q^{37} +3.53457 q^{38} +8.12744 q^{39} -39.0216i q^{41} +(-1.06910 + 2.22249i) q^{42} -44.8902i q^{43} +13.0157 q^{44} +7.93001 q^{46} +74.8886 q^{47} -26.8558 q^{48} +(30.5844 + 38.2831i) q^{49} -27.7878 q^{51} -18.5754 q^{52} -68.6484i q^{53} -1.05697i q^{54} +(4.91243 - 10.2121i) q^{56} +30.0965i q^{57} +1.17040i q^{58} -83.0734i q^{59} -5.95915i q^{61} +0.353895 q^{62} +(-18.9243 - 9.10331i) q^{63} +60.0620 q^{64} -1.15842i q^{66} +10.0470i q^{67} +63.5093 q^{68} +67.5234i q^{69} +70.6011 q^{71} +4.85668i q^{72} -66.8458 q^{73} -8.92674 q^{74} -68.7860i q^{76} +(-20.7407 - 9.97704i) q^{77} +1.65323i q^{78} -33.3562 q^{79} +9.00000 q^{81} +7.93753 q^{82} +26.8668 q^{83} +(43.2518 + 20.8057i) q^{84} +9.13128 q^{86} -9.96585 q^{87} +5.32282i q^{88} -146.421i q^{89} +(29.6000 + 14.2387i) q^{91} -154.326i q^{92} +3.01339i q^{93} +15.2334i q^{94} -16.6789i q^{96} +27.8988 q^{97} +(-7.78730 + 6.22129i) q^{98} +9.86382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 60 q^{9} + 64 q^{11} + 32 q^{14} - 108 q^{16} - 30 q^{21} + 80 q^{29} - 60 q^{36} - 60 q^{39} - 612 q^{44} - 452 q^{46} - 14 q^{49} - 96 q^{51} + 296 q^{56} + 336 q^{64} + 376 q^{71} + 28 q^{74}+ \cdots + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.203414i 0.101707i 0.998706 + 0.0508534i \(0.0161941\pi\)
−0.998706 + 0.0508534i \(0.983806\pi\)
\(3\) −1.73205 −0.577350
\(4\) 3.95862 0.989656
\(5\) 0 0
\(6\) − 0.352323i − 0.0587205i
\(7\) −6.30811 − 3.03444i −0.901158 − 0.433491i
\(8\) 1.61889i 0.202362i
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 3.28794 0.298904 0.149452 − 0.988769i \(-0.452249\pi\)
0.149452 + 0.988769i \(0.452249\pi\)
\(12\) −6.85654 −0.571378
\(13\) −4.69238 −0.360952 −0.180476 − 0.983579i \(-0.557764\pi\)
−0.180476 + 0.983579i \(0.557764\pi\)
\(14\) 0.617246 − 1.28316i 0.0440890 − 0.0916540i
\(15\) 0 0
\(16\) 15.5052 0.969074
\(17\) 16.0433 0.943722 0.471861 − 0.881673i \(-0.343582\pi\)
0.471861 + 0.881673i \(0.343582\pi\)
\(18\) 0.610241i 0.0339023i
\(19\) − 17.3762i − 0.914539i −0.889328 − 0.457270i \(-0.848827\pi\)
0.889328 − 0.457270i \(-0.151173\pi\)
\(20\) 0 0
\(21\) 10.9260 + 5.25580i 0.520284 + 0.250276i
\(22\) 0.668812i 0.0304006i
\(23\) − 38.9847i − 1.69498i −0.530807 − 0.847492i \(-0.678111\pi\)
0.530807 − 0.847492i \(-0.321889\pi\)
\(24\) − 2.80401i − 0.116834i
\(25\) 0 0
\(26\) − 0.954495i − 0.0367113i
\(27\) −5.19615 −0.192450
\(28\) −24.9714 − 12.0122i −0.891836 − 0.429007i
\(29\) 5.75378 0.198406 0.0992032 − 0.995067i \(-0.468371\pi\)
0.0992032 + 0.995067i \(0.468371\pi\)
\(30\) 0 0
\(31\) − 1.73978i − 0.0561219i −0.999606 − 0.0280610i \(-0.991067\pi\)
0.999606 − 0.0280610i \(-0.00893325\pi\)
\(32\) 9.62954i 0.300923i
\(33\) −5.69488 −0.172572
\(34\) 3.26342i 0.0959830i
\(35\) 0 0
\(36\) 11.8759 0.329885
\(37\) 43.8846i 1.18607i 0.805176 + 0.593036i \(0.202071\pi\)
−0.805176 + 0.593036i \(0.797929\pi\)
\(38\) 3.53457 0.0930149
\(39\) 8.12744 0.208396
\(40\) 0 0
\(41\) − 39.0216i − 0.951746i −0.879514 − 0.475873i \(-0.842132\pi\)
0.879514 − 0.475873i \(-0.157868\pi\)
\(42\) −1.06910 + 2.22249i −0.0254548 + 0.0529164i
\(43\) − 44.8902i − 1.04396i −0.852958 − 0.521979i \(-0.825194\pi\)
0.852958 − 0.521979i \(-0.174806\pi\)
\(44\) 13.0157 0.295812
\(45\) 0 0
\(46\) 7.93001 0.172392
\(47\) 74.8886 1.59337 0.796687 − 0.604392i \(-0.206584\pi\)
0.796687 + 0.604392i \(0.206584\pi\)
\(48\) −26.8558 −0.559495
\(49\) 30.5844 + 38.2831i 0.624172 + 0.781287i
\(50\) 0 0
\(51\) −27.7878 −0.544858
\(52\) −18.5754 −0.357219
\(53\) − 68.6484i − 1.29525i −0.761958 − 0.647626i \(-0.775762\pi\)
0.761958 − 0.647626i \(-0.224238\pi\)
\(54\) − 1.05697i − 0.0195735i
\(55\) 0 0
\(56\) 4.91243 − 10.2121i 0.0877219 − 0.182360i
\(57\) 30.0965i 0.528009i
\(58\) 1.17040i 0.0201793i
\(59\) − 83.0734i − 1.40802i −0.710189 − 0.704012i \(-0.751390\pi\)
0.710189 − 0.704012i \(-0.248610\pi\)
\(60\) 0 0
\(61\) − 5.95915i − 0.0976910i −0.998806 − 0.0488455i \(-0.984446\pi\)
0.998806 − 0.0488455i \(-0.0155542\pi\)
\(62\) 0.353895 0.00570799
\(63\) −18.9243 − 9.10331i −0.300386 − 0.144497i
\(64\) 60.0620 0.938468
\(65\) 0 0
\(66\) − 1.15842i − 0.0175518i
\(67\) 10.0470i 0.149955i 0.997185 + 0.0749775i \(0.0238885\pi\)
−0.997185 + 0.0749775i \(0.976112\pi\)
\(68\) 63.5093 0.933960
\(69\) 67.5234i 0.978600i
\(70\) 0 0
\(71\) 70.6011 0.994381 0.497191 − 0.867641i \(-0.334365\pi\)
0.497191 + 0.867641i \(0.334365\pi\)
\(72\) 4.85668i 0.0674539i
\(73\) −66.8458 −0.915696 −0.457848 − 0.889030i \(-0.651380\pi\)
−0.457848 + 0.889030i \(0.651380\pi\)
\(74\) −8.92674 −0.120632
\(75\) 0 0
\(76\) − 68.7860i − 0.905079i
\(77\) −20.7407 − 9.97704i −0.269359 − 0.129572i
\(78\) 1.65323i 0.0211953i
\(79\) −33.3562 −0.422230 −0.211115 − 0.977461i \(-0.567709\pi\)
−0.211115 + 0.977461i \(0.567709\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 7.93753 0.0967991
\(83\) 26.8668 0.323696 0.161848 − 0.986816i \(-0.448255\pi\)
0.161848 + 0.986816i \(0.448255\pi\)
\(84\) 43.2518 + 20.8057i 0.514902 + 0.247687i
\(85\) 0 0
\(86\) 9.13128 0.106178
\(87\) −9.96585 −0.114550
\(88\) 5.32282i 0.0604866i
\(89\) − 146.421i − 1.64519i −0.568631 − 0.822593i \(-0.692527\pi\)
0.568631 − 0.822593i \(-0.307473\pi\)
\(90\) 0 0
\(91\) 29.6000 + 14.2387i 0.325275 + 0.156470i
\(92\) − 154.326i − 1.67745i
\(93\) 3.01339i 0.0324020i
\(94\) 15.2334i 0.162057i
\(95\) 0 0
\(96\) − 16.6789i − 0.173738i
\(97\) 27.8988 0.287617 0.143808 − 0.989606i \(-0.454065\pi\)
0.143808 + 0.989606i \(0.454065\pi\)
\(98\) −7.78730 + 6.22129i −0.0794623 + 0.0634825i
\(99\) 9.86382 0.0996346
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 525.3.e.b.349.11 20
5.2 odd 4 525.3.h.c.76.6 yes 10
5.3 odd 4 525.3.h.b.76.5 ✓ 10
5.4 even 2 inner 525.3.e.b.349.10 20
7.6 odd 2 inner 525.3.e.b.349.12 20
35.13 even 4 525.3.h.b.76.6 yes 10
35.27 even 4 525.3.h.c.76.5 yes 10
35.34 odd 2 inner 525.3.e.b.349.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.3.e.b.349.9 20 35.34 odd 2 inner
525.3.e.b.349.10 20 5.4 even 2 inner
525.3.e.b.349.11 20 1.1 even 1 trivial
525.3.e.b.349.12 20 7.6 odd 2 inner
525.3.h.b.76.5 ✓ 10 5.3 odd 4
525.3.h.b.76.6 yes 10 35.13 even 4
525.3.h.c.76.5 yes 10 35.27 even 4
525.3.h.c.76.6 yes 10 5.2 odd 4