Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [17,4,Mod(16,17)] 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("17.16"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00303247010\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.1
Root \(-7.36435i\) of defining polynomial
Character \(\chi\) \(=\) 17.16
Dual form 17.4.b.a.16.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.37228 q^{2} -7.36435i q^{3} +3.37228 q^{4} -10.1060i q^{5} +24.8347i q^{6} +17.4703i q^{7} +15.6060 q^{8} -27.2337 q^{9} +34.0802i q^{10} -51.5505i q^{11} -24.8347i q^{12} +75.2119 q^{13} -58.9148i q^{14} -74.4239 q^{15} -79.6060 q^{16} +(-12.2119 + 69.0208i) q^{17} +91.8397 q^{18} -28.0000 q^{19} -34.0802i q^{20} +128.658 q^{21} +173.843i q^{22} -19.1913i q^{23} -114.928i q^{24} +22.8695 q^{25} -253.636 q^{26} +1.72096i q^{27} +58.9148i q^{28} +70.7417i q^{29} +250.978 q^{30} +41.4445i q^{31} +143.606 q^{32} -379.636 q^{33} +(41.1821 - 232.757i) q^{34} +176.554 q^{35} -91.8397 q^{36} +135.460i q^{37} +94.4239 q^{38} -553.887i q^{39} -157.713i q^{40} +288.771i q^{41} -433.870 q^{42} +88.2934 q^{43} -173.843i q^{44} +275.223i q^{45} +64.7184i q^{46} +157.576 q^{47} +586.246i q^{48} +37.7881 q^{49} -77.1224 q^{50} +(508.293 + 89.9330i) q^{51} +253.636 q^{52} +120.250 q^{53} -5.80356i q^{54} -520.967 q^{55} +272.641i q^{56} +206.202i q^{57} -238.561i q^{58} -696.119 q^{59} -250.978 q^{60} -683.544i q^{61} -139.763i q^{62} -475.781i q^{63} +152.568 q^{64} -760.089i q^{65} +1280.24 q^{66} +123.826 q^{67} +(-41.1821 + 232.757i) q^{68} -141.331 q^{69} -595.391 q^{70} +225.393i q^{71} -425.008 q^{72} +919.423i q^{73} -456.810i q^{74} -168.419i q^{75} -94.4239 q^{76} +900.603 q^{77} +1867.86i q^{78} -354.830i q^{79} +804.495i q^{80} -722.636 q^{81} -973.815i q^{82} -955.272 q^{83} +433.870 q^{84} +(697.522 + 123.413i) q^{85} -297.750 q^{86} +520.967 q^{87} -804.495i q^{88} +617.636 q^{89} -928.128i q^{90} +1313.98i q^{91} -64.7184i q^{92} +305.212 q^{93} -531.391 q^{94} +282.967i q^{95} -1057.56i q^{96} -428.533i q^{97} -127.432 q^{98} +1403.91i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9} + 140 q^{13} + 24 q^{15} - 238 q^{16} + 112 q^{17} + 218 q^{18} - 112 q^{19} + 124 q^{21} - 460 q^{25} - 532 q^{26} + 912 q^{30} + 494 q^{32} - 1036 q^{33}+ \cdots + 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.37228 −1.19228 −0.596141 − 0.802880i \(-0.703300\pi\)
−0.596141 + 0.802880i \(0.703300\pi\)
\(3\) − 7.36435i − 1.41727i −0.705575 − 0.708635i \(-0.749311\pi\)
0.705575 − 0.708635i \(-0.250689\pi\)
\(4\) 3.37228 0.421535
\(5\) − 10.1060i − 0.903905i −0.892042 − 0.451952i \(-0.850728\pi\)
0.892042 − 0.451952i \(-0.149272\pi\)
\(6\) 24.8347i 1.68979i
\(7\) 17.4703i 0.943308i 0.881784 + 0.471654i \(0.156343\pi\)
−0.881784 + 0.471654i \(0.843657\pi\)
\(8\) 15.6060 0.689693
\(9\) −27.2337 −1.00866
\(10\) 34.0802i 1.07771i
\(11\) − 51.5505i − 1.41300i −0.707711 − 0.706502i \(-0.750272\pi\)
0.707711 − 0.706502i \(-0.249728\pi\)
\(12\) − 24.8347i − 0.597429i
\(13\) 75.2119 1.60462 0.802309 − 0.596909i \(-0.203605\pi\)
0.802309 + 0.596909i \(0.203605\pi\)
\(14\) − 58.9148i − 1.12469i
\(15\) −74.4239 −1.28108
\(16\) −79.6060 −1.24384
\(17\) −12.2119 + 69.0208i −0.174225 + 0.984706i
\(18\) 91.8397 1.20260
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) − 34.0802i − 0.381028i
\(21\) 128.658 1.33692
\(22\) 173.843i 1.68470i
\(23\) − 19.1913i − 0.173985i −0.996209 − 0.0869926i \(-0.972274\pi\)
0.996209 − 0.0869926i \(-0.0277256\pi\)
\(24\) − 114.928i − 0.977481i
\(25\) 22.8695 0.182956
\(26\) −253.636 −1.91316
\(27\) 1.72096i 0.0122666i
\(28\) 58.9148i 0.397638i
\(29\) 70.7417i 0.452980i 0.974014 + 0.226490i \(0.0727250\pi\)
−0.974014 + 0.226490i \(0.927275\pi\)
\(30\) 250.978 1.52740
\(31\) 41.4445i 0.240118i 0.992767 + 0.120059i \(0.0383083\pi\)
−0.992767 + 0.120059i \(0.961692\pi\)
\(32\) 143.606 0.793318
\(33\) −379.636 −2.00261
\(34\) 41.1821 − 232.757i 0.207726 − 1.17405i
\(35\) 176.554 0.852661
\(36\) −91.8397 −0.425184
\(37\) 135.460i 0.601879i 0.953643 + 0.300939i \(0.0973002\pi\)
−0.953643 + 0.300939i \(0.902700\pi\)
\(38\) 94.4239 0.403094
\(39\) − 553.887i − 2.27418i
\(40\) − 157.713i − 0.623417i
\(41\) 288.771i 1.09996i 0.835178 + 0.549980i \(0.185365\pi\)
−0.835178 + 0.549980i \(0.814635\pi\)
\(42\) −433.870 −1.59399
\(43\) 88.2934 0.313131 0.156565 − 0.987668i \(-0.449958\pi\)
0.156565 + 0.987668i \(0.449958\pi\)
\(44\) − 173.843i − 0.595631i
\(45\) 275.223i 0.911728i
\(46\) 64.7184i 0.207439i
\(47\) 157.576 0.489039 0.244520 − 0.969644i \(-0.421370\pi\)
0.244520 + 0.969644i \(0.421370\pi\)
\(48\) 586.246i 1.76286i
\(49\) 37.7881 0.110169
\(50\) −77.1224 −0.218135
\(51\) 508.293 + 89.9330i 1.39559 + 0.246924i
\(52\) 253.636 0.676403
\(53\) 120.250 0.311653 0.155826 − 0.987784i \(-0.450196\pi\)
0.155826 + 0.987784i \(0.450196\pi\)
\(54\) − 5.80356i − 0.0146253i
\(55\) −520.967 −1.27722
\(56\) 272.641i 0.650593i
\(57\) 206.202i 0.479160i
\(58\) − 238.561i − 0.540079i
\(59\) −696.119 −1.53605 −0.768026 − 0.640419i \(-0.778761\pi\)
−0.768026 + 0.640419i \(0.778761\pi\)
\(60\) −250.978 −0.540019
\(61\) − 683.544i − 1.43473i −0.696695 − 0.717367i \(-0.745347\pi\)
0.696695 − 0.717367i \(-0.254653\pi\)
\(62\) − 139.763i − 0.286288i
\(63\) − 475.781i − 0.951473i
\(64\) 152.568 0.297984
\(65\) − 760.089i − 1.45042i
\(66\) 1280.24 2.38767
\(67\) 123.826 0.225787 0.112894 − 0.993607i \(-0.463988\pi\)
0.112894 + 0.993607i \(0.463988\pi\)
\(68\) −41.1821 + 232.757i −0.0734421 + 0.415088i
\(69\) −141.331 −0.246584
\(70\) −595.391 −1.01661
\(71\) 225.393i 0.376750i 0.982097 + 0.188375i \(0.0603220\pi\)
−0.982097 + 0.188375i \(0.939678\pi\)
\(72\) −425.008 −0.695662
\(73\) 919.423i 1.47411i 0.675831 + 0.737057i \(0.263785\pi\)
−0.675831 + 0.737057i \(0.736215\pi\)
\(74\) − 456.810i − 0.717609i
\(75\) − 168.419i − 0.259298i
\(76\) −94.4239 −0.142515
\(77\) 900.603 1.33290
\(78\) 1867.86i 2.71146i
\(79\) − 354.830i − 0.505335i −0.967553 − 0.252668i \(-0.918692\pi\)
0.967553 − 0.252668i \(-0.0813079\pi\)
\(80\) 804.495i 1.12432i
\(81\) −722.636 −0.991270
\(82\) − 973.815i − 1.31146i
\(83\) −955.272 −1.26331 −0.631655 − 0.775250i \(-0.717624\pi\)
−0.631655 + 0.775250i \(0.717624\pi\)
\(84\) 433.870 0.563560
\(85\) 697.522 + 123.413i 0.890080 + 0.157483i
\(86\) −297.750 −0.373340
\(87\) 520.967 0.641995
\(88\) − 804.495i − 0.974539i
\(89\) 617.636 0.735610 0.367805 − 0.929903i \(-0.380109\pi\)
0.367805 + 0.929903i \(0.380109\pi\)
\(90\) − 928.128i − 1.08704i
\(91\) 1313.98i 1.51365i
\(92\) − 64.7184i − 0.0733408i
\(93\) 305.212 0.340312
\(94\) −531.391 −0.583072
\(95\) 282.967i 0.305598i
\(96\) − 1057.56i − 1.12435i
\(97\) − 428.533i − 0.448566i −0.974524 − 0.224283i \(-0.927996\pi\)
0.974524 − 0.224283i \(-0.0720041\pi\)
\(98\) −127.432 −0.131353
\(99\) 1403.91i 1.42523i
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 17.4.b.a.16.1 ✓ 4
3.2 odd 2 153.4.d.b.118.4 4
4.3 odd 2 272.4.b.d.33.4 4
5.2 odd 4 425.4.c.c.424.1 8
5.3 odd 4 425.4.c.c.424.8 8
5.4 even 2 425.4.d.c.101.4 4
17.4 even 4 289.4.a.e.1.3 4
17.13 even 4 289.4.a.e.1.4 4
17.16 even 2 inner 17.4.b.a.16.2 yes 4
51.50 odd 2 153.4.d.b.118.3 4
68.67 odd 2 272.4.b.d.33.1 4
85.33 odd 4 425.4.c.c.424.7 8
85.67 odd 4 425.4.c.c.424.2 8
85.84 even 2 425.4.d.c.101.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 1.1 even 1 trivial
17.4.b.a.16.2 yes 4 17.16 even 2 inner
153.4.d.b.118.3 4 51.50 odd 2
153.4.d.b.118.4 4 3.2 odd 2
272.4.b.d.33.1 4 68.67 odd 2
272.4.b.d.33.4 4 4.3 odd 2
289.4.a.e.1.3 4 17.4 even 4
289.4.a.e.1.4 4 17.13 even 4
425.4.c.c.424.1 8 5.2 odd 4
425.4.c.c.424.2 8 85.67 odd 4
425.4.c.c.424.7 8 85.33 odd 4
425.4.c.c.424.8 8 5.3 odd 4
425.4.d.c.101.3 4 85.84 even 2
425.4.d.c.101.4 4 5.4 even 2