Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [289,4,Mod(1,289)] 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("289.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(289, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-7.36435\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} +7.36435 q^{3} +3.37228 q^{4} +10.1060 q^{5} +24.8347 q^{6} +17.4703 q^{7} -15.6060 q^{8} +27.2337 q^{9} +34.0802 q^{10} -51.5505 q^{11} +24.8347 q^{12} +75.2119 q^{13} +58.9148 q^{14} +74.4239 q^{15} -79.6060 q^{16} +91.8397 q^{18} +28.0000 q^{19} +34.0802 q^{20} +128.658 q^{21} -173.843 q^{22} -19.1913 q^{23} -114.928 q^{24} -22.8695 q^{25} +253.636 q^{26} +1.72096 q^{27} +58.9148 q^{28} -70.7417 q^{29} +250.978 q^{30} -41.4445 q^{31} -143.606 q^{32} -379.636 q^{33} +176.554 q^{35} +91.8397 q^{36} -135.460 q^{37} +94.4239 q^{38} +553.887 q^{39} -157.713 q^{40} +288.771 q^{41} +433.870 q^{42} -88.2934 q^{43} -173.843 q^{44} +275.223 q^{45} -64.7184 q^{46} +157.576 q^{47} -586.246 q^{48} -37.7881 q^{49} -77.1224 q^{50} +253.636 q^{52} -120.250 q^{53} +5.80356 q^{54} -520.967 q^{55} -272.641 q^{56} +206.202 q^{57} -238.561 q^{58} +696.119 q^{59} +250.978 q^{60} -683.544 q^{61} -139.763 q^{62} +475.781 q^{63} +152.568 q^{64} +760.089 q^{65} -1280.24 q^{66} +123.826 q^{67} -141.331 q^{69} +595.391 q^{70} -225.393 q^{71} -425.008 q^{72} -919.423 q^{73} -456.810 q^{74} -168.419 q^{75} +94.4239 q^{76} -900.603 q^{77} +1867.86 q^{78} -354.830 q^{79} -804.495 q^{80} -722.636 q^{81} +973.815 q^{82} +955.272 q^{83} +433.870 q^{84} -297.750 q^{86} -520.967 q^{87} +804.495 q^{88} +617.636 q^{89} +928.128 q^{90} +1313.98 q^{91} -64.7184 q^{92} -305.212 q^{93} +531.391 q^{94} +282.967 q^{95} -1057.56 q^{96} +428.533 q^{97} -127.432 q^{98} -1403.91 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} + 40 q^{9} + 140 q^{13} - 24 q^{15} - 238 q^{16} + 218 q^{18} + 112 q^{19} + 124 q^{21} + 460 q^{25} + 532 q^{26} + 912 q^{30} - 494 q^{32} - 1036 q^{33} + 936 q^{35}+ \cdots + 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37228 1.19228 0.596141 − 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(3\) 7.36435 1.41727 0.708635 − 0.705575i \(-0.249311\pi\)
0.708635 + 0.705575i \(0.249311\pi\)
\(4\) 3.37228 0.421535
\(5\) 10.1060 0.903905 0.451952 − 0.892042i \(-0.350728\pi\)
0.451952 + 0.892042i \(0.350728\pi\)
\(6\) 24.8347 1.68979
\(7\) 17.4703 0.943308 0.471654 − 0.881784i \(-0.343657\pi\)
0.471654 + 0.881784i \(0.343657\pi\)
\(8\) −15.6060 −0.689693
\(9\) 27.2337 1.00866
\(10\) 34.0802 1.07771
\(11\) −51.5505 −1.41300 −0.706502 − 0.707711i \(-0.749728\pi\)
−0.706502 + 0.707711i \(0.749728\pi\)
\(12\) 24.8347 0.597429
\(13\) 75.2119 1.60462 0.802309 − 0.596909i \(-0.203605\pi\)
0.802309 + 0.596909i \(0.203605\pi\)
\(14\) 58.9148 1.12469
\(15\) 74.4239 1.28108
\(16\) −79.6060 −1.24384
\(17\) 0 0
\(18\) 91.8397 1.20260
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 34.0802 0.381028
\(21\) 128.658 1.33692
\(22\) −173.843 −1.68470
\(23\) −19.1913 −0.173985 −0.0869926 − 0.996209i \(-0.527726\pi\)
−0.0869926 + 0.996209i \(0.527726\pi\)
\(24\) −114.928 −0.977481
\(25\) −22.8695 −0.182956
\(26\) 253.636 1.91316
\(27\) 1.72096 0.0122666
\(28\) 58.9148 0.397638
\(29\) −70.7417 −0.452980 −0.226490 − 0.974014i \(-0.572725\pi\)
−0.226490 + 0.974014i \(0.572725\pi\)
\(30\) 250.978 1.52740
\(31\) −41.4445 −0.240118 −0.120059 − 0.992767i \(-0.538308\pi\)
−0.120059 + 0.992767i \(0.538308\pi\)
\(32\) −143.606 −0.793318
\(33\) −379.636 −2.00261
\(34\) 0 0
\(35\) 176.554 0.852661
\(36\) 91.8397 0.425184
\(37\) −135.460 −0.601879 −0.300939 − 0.953643i \(-0.597300\pi\)
−0.300939 + 0.953643i \(0.597300\pi\)
\(38\) 94.4239 0.403094
\(39\) 553.887 2.27418
\(40\) −157.713 −0.623417
\(41\) 288.771 1.09996 0.549980 − 0.835178i \(-0.314635\pi\)
0.549980 + 0.835178i \(0.314635\pi\)
\(42\) 433.870 1.59399
\(43\) −88.2934 −0.313131 −0.156565 − 0.987668i \(-0.550042\pi\)
−0.156565 + 0.987668i \(0.550042\pi\)
\(44\) −173.843 −0.595631
\(45\) 275.223 0.911728
\(46\) −64.7184 −0.207439
\(47\) 157.576 0.489039 0.244520 − 0.969644i \(-0.421370\pi\)
0.244520 + 0.969644i \(0.421370\pi\)
\(48\) −586.246 −1.76286
\(49\) −37.7881 −0.110169
\(50\) −77.1224 −0.218135
\(51\) 0 0
\(52\) 253.636 0.676403
\(53\) −120.250 −0.311653 −0.155826 − 0.987784i \(-0.549804\pi\)
−0.155826 + 0.987784i \(0.549804\pi\)
\(54\) 5.80356 0.0146253
\(55\) −520.967 −1.27722
\(56\) −272.641 −0.650593
\(57\) 206.202 0.479160
\(58\) −238.561 −0.540079
\(59\) 696.119 1.53605 0.768026 − 0.640419i \(-0.221239\pi\)
0.768026 + 0.640419i \(0.221239\pi\)
\(60\) 250.978 0.540019
\(61\) −683.544 −1.43473 −0.717367 − 0.696695i \(-0.754653\pi\)
−0.717367 + 0.696695i \(0.754653\pi\)
\(62\) −139.763 −0.286288
\(63\) 475.781 0.951473
\(64\) 152.568 0.297984
\(65\) 760.089 1.45042
\(66\) −1280.24 −2.38767
\(67\) 123.826 0.225787 0.112894 − 0.993607i \(-0.463988\pi\)
0.112894 + 0.993607i \(0.463988\pi\)
\(68\) 0 0
\(69\) −141.331 −0.246584
\(70\) 595.391 1.01661
\(71\) −225.393 −0.376750 −0.188375 − 0.982097i \(-0.560322\pi\)
−0.188375 + 0.982097i \(0.560322\pi\)
\(72\) −425.008 −0.695662
\(73\) −919.423 −1.47411 −0.737057 − 0.675831i \(-0.763785\pi\)
−0.737057 + 0.675831i \(0.763785\pi\)
\(74\) −456.810 −0.717609
\(75\) −168.419 −0.259298
\(76\) 94.4239 0.142515
\(77\) −900.603 −1.33290
\(78\) 1867.86 2.71146
\(79\) −354.830 −0.505335 −0.252668 − 0.967553i \(-0.581308\pi\)
−0.252668 + 0.967553i \(0.581308\pi\)
\(80\) −804.495 −1.12432
\(81\) −722.636 −0.991270
\(82\) 973.815 1.31146
\(83\) 955.272 1.26331 0.631655 − 0.775250i \(-0.282376\pi\)
0.631655 + 0.775250i \(0.282376\pi\)
\(84\) 433.870 0.563560
\(85\) 0 0
\(86\) −297.750 −0.373340
\(87\) −520.967 −0.641995
\(88\) 804.495 0.974539
\(89\) 617.636 0.735610 0.367805 − 0.929903i \(-0.380109\pi\)
0.367805 + 0.929903i \(0.380109\pi\)
\(90\) 928.128 1.08704
\(91\) 1313.98 1.51365
\(92\) −64.7184 −0.0733408
\(93\) −305.212 −0.340312
\(94\) 531.391 0.583072
\(95\) 282.967 0.305598
\(96\) −1057.56 −1.12435
\(97\) 428.533 0.448566 0.224283 − 0.974524i \(-0.427996\pi\)
0.224283 + 0.974524i \(0.427996\pi\)
\(98\) −127.432 −0.131353
\(99\) −1403.91 −1.42523
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 289.4.a.e.1.4 4
17.4 even 4 17.4.b.a.16.1 ✓ 4
17.13 even 4 17.4.b.a.16.2 yes 4
17.16 even 2 inner 289.4.a.e.1.3 4
51.38 odd 4 153.4.d.b.118.4 4
51.47 odd 4 153.4.d.b.118.3 4
68.47 odd 4 272.4.b.d.33.1 4
68.55 odd 4 272.4.b.d.33.4 4
85.4 even 4 425.4.d.c.101.4 4
85.13 odd 4 425.4.c.c.424.7 8
85.38 odd 4 425.4.c.c.424.8 8
85.47 odd 4 425.4.c.c.424.2 8
85.64 even 4 425.4.d.c.101.3 4
85.72 odd 4 425.4.c.c.424.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 17.4 even 4
17.4.b.a.16.2 yes 4 17.13 even 4
153.4.d.b.118.3 4 51.47 odd 4
153.4.d.b.118.4 4 51.38 odd 4
272.4.b.d.33.1 4 68.47 odd 4
272.4.b.d.33.4 4 68.55 odd 4
289.4.a.e.1.3 4 17.16 even 2 inner
289.4.a.e.1.4 4 1.1 even 1 trivial
425.4.c.c.424.1 8 85.72 odd 4
425.4.c.c.424.2 8 85.47 odd 4
425.4.c.c.424.7 8 85.13 odd 4
425.4.c.c.424.8 8 85.38 odd 4
425.4.d.c.101.3 4 85.64 even 4
425.4.d.c.101.4 4 85.4 even 4