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Properties

Label 17.14.b.a.16.5
Level $17$
Weight $14$
Character 17.16
Analytic conductor $18.229$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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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] = [17,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(18.2292579218\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 18192152 x^{16} + 135370932931108 x^{14} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{6}\cdot 17^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.5
Root \(-1711.77i\) of defining polynomial
Character \(\chi\) \(=\) 17.16
Dual form 17.14.b.a.16.6

$q$-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)
 
\(f(q)\) \(=\) \(q-66.6386 q^{2} -1711.77i q^{3} -3751.29 q^{4} -6896.67i q^{5} +114070. i q^{6} +473258. i q^{7} +795885. q^{8} -1.33583e6 q^{9} +459584. i q^{10} +1.79686e6i q^{11} +6.42135e6i q^{12} -6.66746e6 q^{13} -3.15373e7i q^{14} -1.18055e7 q^{15} -2.23060e7 q^{16} +(-8.59238e7 - 5.02163e7i) q^{17} +8.90178e7 q^{18} +3.43895e8 q^{19} +2.58714e7i q^{20} +8.10109e8 q^{21} -1.19740e8i q^{22} +2.84773e8i q^{23} -1.36237e9i q^{24} +1.17314e9 q^{25} +4.44310e8 q^{26} -4.42481e8i q^{27} -1.77533e9i q^{28} -3.28103e9i q^{29} +7.86702e8 q^{30} +5.23834e9i q^{31} -5.03344e9 q^{32} +3.07581e9 q^{33} +(5.72585e9 + 3.34634e9i) q^{34} +3.26390e9 q^{35} +5.01109e9 q^{36} +6.26884e9i q^{37} -2.29167e10 q^{38} +1.14132e10i q^{39} -5.48895e9i q^{40} -1.34900e10i q^{41} -5.39845e10 q^{42} +2.32863e10 q^{43} -6.74055e9i q^{44} +9.21277e9i q^{45} -1.89769e10i q^{46} +2.09569e10 q^{47} +3.81828e10i q^{48} -1.27084e11 q^{49} -7.81764e10 q^{50} +(-8.59586e10 + 1.47082e11i) q^{51} +2.50116e10 q^{52} +2.39547e11 q^{53} +2.94864e10i q^{54} +1.23923e10 q^{55} +3.76659e11i q^{56} -5.88668e11i q^{57} +2.18644e11i q^{58} +1.13745e11 q^{59} +4.42859e10 q^{60} -4.86158e11i q^{61} -3.49076e11i q^{62} -6.32192e11i q^{63} +5.18153e11 q^{64} +4.59833e10i q^{65} -2.04968e11 q^{66} +1.16382e12 q^{67} +(3.22326e11 + 1.88376e11i) q^{68} +4.87466e11 q^{69} -2.17502e11 q^{70} +1.37943e12i q^{71} -1.06317e12 q^{72} +1.88031e12i q^{73} -4.17747e11i q^{74} -2.00814e12i q^{75} -1.29005e12 q^{76} -8.50378e11 q^{77} -7.60557e11i q^{78} +1.60205e12i q^{79} +1.53837e11i q^{80} -2.88717e12 q^{81} +8.98956e11i q^{82} +2.63591e12 q^{83} -3.03896e12 q^{84} +(-3.46325e11 + 5.92588e11i) q^{85} -1.55177e12 q^{86} -5.61637e12 q^{87} +1.43009e12i q^{88} +3.75020e12 q^{89} -6.13926e11i q^{90} -3.15543e12i q^{91} -1.06827e12i q^{92} +8.96683e12 q^{93} -1.39654e12 q^{94} -2.37173e12i q^{95} +8.61609e12i q^{96} +6.91911e12i q^{97} +8.46872e12 q^{98} -2.40030e12i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 48786 q^{4} - 16386 q^{8} - 7686490 q^{9} + 16136148 q^{13} + 124323840 q^{15} + 65611842 q^{16} + 25023898 q^{17} - 123776710 q^{18} - 516719736 q^{19} - 665521472 q^{21} - 2475440166 q^{25}+ \cdots - 48142146348894 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −66.6386 −0.736260 −0.368130 − 0.929774i \(-0.620002\pi\)
−0.368130 + 0.929774i \(0.620002\pi\)
\(3\) − 1711.77i − 1.35568i −0.735210 − 0.677840i \(-0.762916\pi\)
0.735210 − 0.677840i \(-0.237084\pi\)
\(4\) −3751.29 −0.457922
\(5\) − 6896.67i − 0.197394i −0.995118 − 0.0986971i \(-0.968533\pi\)
0.995118 − 0.0986971i \(-0.0314675\pi\)
\(6\) 114070.i 0.998132i
\(7\) 473258.i 1.52041i 0.649683 + 0.760205i \(0.274902\pi\)
−0.649683 + 0.760205i \(0.725098\pi\)
\(8\) 795885. 1.07341
\(9\) −1.33583e6 −0.837866
\(10\) 459584.i 0.145333i
\(11\) 1.79686e6i 0.305817i 0.988240 + 0.152909i \(0.0488640\pi\)
−0.988240 + 0.152909i \(0.951136\pi\)
\(12\) 6.42135e6i 0.620795i
\(13\) −6.66746e6 −0.383115 −0.191557 − 0.981481i \(-0.561354\pi\)
−0.191557 + 0.981481i \(0.561354\pi\)
\(14\) − 3.15373e7i − 1.11942i
\(15\) −1.18055e7 −0.267603
\(16\) −2.23060e7 −0.332386
\(17\) −8.59238e7 − 5.02163e7i −0.863367 − 0.504576i
\(18\) 8.90178e7 0.616887
\(19\) 3.43895e8 1.67698 0.838488 − 0.544920i \(-0.183440\pi\)
0.838488 + 0.544920i \(0.183440\pi\)
\(20\) 2.58714e7i 0.0903911i
\(21\) 8.10109e8 2.06119
\(22\) − 1.19740e8i − 0.225161i
\(23\) 2.84773e8i 0.401115i 0.979682 + 0.200557i \(0.0642753\pi\)
−0.979682 + 0.200557i \(0.935725\pi\)
\(24\) − 1.36237e9i − 1.45520i
\(25\) 1.17314e9 0.961036
\(26\) 4.44310e8 0.282072
\(27\) − 4.42481e8i − 0.219801i
\(28\) − 1.77533e9i − 0.696229i
\(29\) − 3.28103e9i − 1.02429i −0.858898 − 0.512146i \(-0.828851\pi\)
0.858898 − 0.512146i \(-0.171149\pi\)
\(30\) 7.86702e8 0.197025
\(31\) 5.23834e9i 1.06009i 0.847969 + 0.530045i \(0.177825\pi\)
−0.847969 + 0.530045i \(0.822175\pi\)
\(32\) −5.03344e9 −0.828687
\(33\) 3.07581e9 0.414590
\(34\) 5.72585e9 + 3.34634e9i 0.635663 + 0.371499i
\(35\) 3.26390e9 0.300120
\(36\) 5.01109e9 0.383677
\(37\) 6.26884e9i 0.401676i 0.979624 + 0.200838i \(0.0643665\pi\)
−0.979624 + 0.200838i \(0.935633\pi\)
\(38\) −2.29167e10 −1.23469
\(39\) 1.14132e10i 0.519380i
\(40\) − 5.48895e9i − 0.211885i
\(41\) − 1.34900e10i − 0.443524i −0.975101 − 0.221762i \(-0.928819\pi\)
0.975101 − 0.221762i \(-0.0711808\pi\)
\(42\) −5.39845e10 −1.51757
\(43\) 2.32863e10 0.561766 0.280883 − 0.959742i \(-0.409373\pi\)
0.280883 + 0.959742i \(0.409373\pi\)
\(44\) − 6.74055e9i − 0.140040i
\(45\) 9.21277e9i 0.165390i
\(46\) − 1.89769e10i − 0.295324i
\(47\) 2.09569e10 0.283591 0.141795 − 0.989896i \(-0.454713\pi\)
0.141795 + 0.989896i \(0.454713\pi\)
\(48\) 3.81828e10i 0.450609i
\(49\) −1.27084e11 −1.31165
\(50\) −7.81764e10 −0.707572
\(51\) −8.59586e10 + 1.47082e11i −0.684043 + 1.17045i
\(52\) 2.50116e10 0.175436
\(53\) 2.39547e11 1.48456 0.742281 − 0.670089i \(-0.233744\pi\)
0.742281 + 0.670089i \(0.233744\pi\)
\(54\) 2.94864e10i 0.161831i
\(55\) 1.23923e10 0.0603665
\(56\) 3.76659e11i 1.63202i
\(57\) − 5.88668e11i − 2.27344i
\(58\) 2.18644e11i 0.754145i
\(59\) 1.13745e11 0.351071 0.175535 − 0.984473i \(-0.443834\pi\)
0.175535 + 0.984473i \(0.443834\pi\)
\(60\) 4.42859e10 0.122541
\(61\) − 4.86158e11i − 1.20819i −0.796914 − 0.604093i \(-0.793536\pi\)
0.796914 − 0.604093i \(-0.206464\pi\)
\(62\) − 3.49076e11i − 0.780502i
\(63\) − 6.32192e11i − 1.27390i
\(64\) 5.18153e11 0.942514
\(65\) 4.59833e10i 0.0756246i
\(66\) −2.04968e11 −0.305246
\(67\) 1.16382e12 1.57180 0.785902 − 0.618352i \(-0.212199\pi\)
0.785902 + 0.618352i \(0.212199\pi\)
\(68\) 3.22326e11 + 1.88376e11i 0.395355 + 0.231056i
\(69\) 4.87466e11 0.543783
\(70\) −2.17502e11 −0.220966
\(71\) 1.37943e12i 1.27797i 0.769219 + 0.638985i \(0.220645\pi\)
−0.769219 + 0.638985i \(0.779355\pi\)
\(72\) −1.06317e12 −0.899373
\(73\) 1.88031e12i 1.45423i 0.686518 + 0.727113i \(0.259138\pi\)
−0.686518 + 0.727113i \(0.740862\pi\)
\(74\) − 4.17747e11i − 0.295738i
\(75\) − 2.00814e12i − 1.30286i
\(76\) −1.29005e12 −0.767924
\(77\) −8.50378e11 −0.464967
\(78\) − 7.60557e11i − 0.382399i
\(79\) 1.60205e12i 0.741483i 0.928736 + 0.370742i \(0.120896\pi\)
−0.928736 + 0.370742i \(0.879104\pi\)
\(80\) 1.53837e11i 0.0656110i
\(81\) −2.88717e12 −1.13585
\(82\) 8.98956e11i 0.326549i
\(83\) 2.63591e12 0.884960 0.442480 − 0.896778i \(-0.354099\pi\)
0.442480 + 0.896778i \(0.354099\pi\)
\(84\) −3.03896e12 −0.943863
\(85\) −3.46325e11 + 5.92588e11i −0.0996003 + 0.170424i
\(86\) −1.55177e12 −0.413605
\(87\) −5.61637e12 −1.38861
\(88\) 1.43009e12i 0.328267i
\(89\) 3.75020e12 0.799870 0.399935 − 0.916543i \(-0.369033\pi\)
0.399935 + 0.916543i \(0.369033\pi\)
\(90\) − 6.13926e11i − 0.121770i
\(91\) − 3.15543e12i − 0.582491i
\(92\) − 1.06827e12i − 0.183679i
\(93\) 8.96683e12 1.43714
\(94\) −1.39654e12 −0.208796
\(95\) − 2.37173e12i − 0.331025i
\(96\) 8.61609e12i 1.12343i
\(97\) 6.91911e12i 0.843401i 0.906735 + 0.421700i \(0.138567\pi\)
−0.906735 + 0.421700i \(0.861433\pi\)
\(98\) 8.46872e12 0.965714
\(99\) − 2.40030e12i − 0.256234i
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 17.14.b.a.16.5 ✓ 18
3.2 odd 2 153.14.d.b.118.14 18
17.16 even 2 inner 17.14.b.a.16.6 yes 18
51.50 odd 2 153.14.d.b.118.13 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.14.b.a.16.5 ✓ 18 1.1 even 1 trivial
17.14.b.a.16.6 yes 18 17.16 even 2 inner
153.14.d.b.118.13 18 51.50 odd 2
153.14.d.b.118.14 18 3.2 odd 2