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Properties

Label 17.12.c.a.13.1
Level $17$
Weight $12$
Character 17.13
Analytic conductor $13.062$
Analytic rank $0$
Dimension $32$
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] = [17,12,Mod(4,17)] 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("17.4"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 17.c (of order \(4\), degree \(2\), 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: \(13.0618340695\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 17.13
Dual form 17.12.c.a.4.16

$q$-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-88.6653i q^{2} +(-145.298 - 145.298i) q^{3} -5813.53 q^{4} +(-6386.08 - 6386.08i) q^{5} +(-12882.9 + 12882.9i) q^{6} +(27657.5 - 27657.5i) q^{7} +333872. i q^{8} -134924. i q^{9} +(-566224. + 566224. i) q^{10} +(326010. - 326010. i) q^{11} +(844692. + 844692. i) q^{12} -942273. q^{13} +(-2.45226e6 - 2.45226e6i) q^{14} +1.85577e6i q^{15} +1.76967e7 q^{16} +(5.69555e6 + 1.35373e6i) q^{17} -1.19631e7 q^{18} -1.34809e7i q^{19} +(3.71257e7 + 3.71257e7i) q^{20} -8.03715e6 q^{21} +(-2.89058e7 - 2.89058e7i) q^{22} +(-1.73146e7 + 1.73146e7i) q^{23} +(4.85108e7 - 4.85108e7i) q^{24} +3.27360e7i q^{25} +8.35469e7i q^{26} +(-4.53432e7 + 4.53432e7i) q^{27} +(-1.60788e8 + 1.60788e8i) q^{28} +(5.58148e7 + 5.58148e7i) q^{29} +1.64542e8 q^{30} +(-5.79594e7 - 5.79594e7i) q^{31} -8.85314e8i q^{32} -9.47371e7 q^{33} +(1.20028e8 - 5.04998e8i) q^{34} -3.53246e8 q^{35} +7.84386e8i q^{36} +(2.77393e8 + 2.77393e8i) q^{37} -1.19529e9 q^{38} +(1.36910e8 + 1.36910e8i) q^{39} +(2.13213e9 - 2.13213e9i) q^{40} +(-2.25930e8 + 2.25930e8i) q^{41} +7.12616e8i q^{42} +5.89362e8i q^{43} +(-1.89527e9 + 1.89527e9i) q^{44} +(-8.61637e8 + 8.61637e8i) q^{45} +(1.53520e9 + 1.53520e9i) q^{46} -1.28340e9 q^{47} +(-2.57129e9 - 2.57129e9i) q^{48} +4.47450e8i q^{49} +2.90254e9 q^{50} +(-6.30857e8 - 1.02424e9i) q^{51} +5.47793e9 q^{52} -4.86248e9i q^{53} +(4.02037e9 + 4.02037e9i) q^{54} -4.16386e9 q^{55} +(9.23406e9 + 9.23406e9i) q^{56} +(-1.95875e9 + 1.95875e9i) q^{57} +(4.94883e9 - 4.94883e9i) q^{58} -1.51765e9i q^{59} -1.07885e10i q^{60} +(6.58302e9 - 6.58302e9i) q^{61} +(-5.13898e9 + 5.13898e9i) q^{62} +(-3.73167e9 - 3.73167e9i) q^{63} -4.22537e10 q^{64} +(6.01744e9 + 6.01744e9i) q^{65} +8.39989e9i q^{66} +1.16438e10 q^{67} +(-3.31113e10 - 7.86992e9i) q^{68} +5.03153e9 q^{69} +3.13207e10i q^{70} +(-1.63879e10 - 1.63879e10i) q^{71} +4.50473e10 q^{72} +(-3.10339e9 - 3.10339e9i) q^{73} +(2.45951e10 - 2.45951e10i) q^{74} +(4.75646e9 - 4.75646e9i) q^{75} +7.83719e10i q^{76} -1.80333e10i q^{77} +(1.21392e10 - 1.21392e10i) q^{78} +(1.85319e10 - 1.85319e10i) q^{79} +(-1.13013e11 - 1.13013e11i) q^{80} -1.07249e10 q^{81} +(2.00321e10 + 2.00321e10i) q^{82} +4.03852e10i q^{83} +4.67242e10 q^{84} +(-2.77273e10 - 4.50173e10i) q^{85} +5.22559e10 q^{86} -1.62195e10i q^{87} +(1.08846e11 + 1.08846e11i) q^{88} -6.44743e10 q^{89} +(7.63973e10 + 7.63973e10i) q^{90} +(-2.60609e10 + 2.60609e10i) q^{91} +(1.00659e11 - 1.00659e11i) q^{92} +1.68427e10i q^{93} +1.13793e11i q^{94} +(-8.60904e10 + 8.60904e10i) q^{95} +(-1.28634e11 + 1.28634e11i) q^{96} +(-3.87960e10 - 3.87960e10i) q^{97} +3.96733e10 q^{98} +(-4.39867e10 - 4.39867e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 528 q^{3} - 32772 q^{4} - 15496 q^{5} + 39058 q^{6} - 38368 q^{7} + 178138 q^{10} - 24816 q^{11} + 1322202 q^{12} - 1533752 q^{13} - 757692 q^{14} + 22082308 q^{16} + 17346416 q^{17} + 16775420 q^{18}+ \cdots + 229465269792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 88.6653i − 1.95924i −0.200850 − 0.979622i \(-0.564370\pi\)
0.200850 − 0.979622i \(-0.435630\pi\)
\(3\) −145.298 − 145.298i −0.345217 − 0.345217i 0.513108 − 0.858324i \(-0.328494\pi\)
−0.858324 + 0.513108i \(0.828494\pi\)
\(4\) −5813.53 −2.83864
\(5\) −6386.08 − 6386.08i −0.913902 − 0.913902i 0.0826748 − 0.996577i \(-0.473654\pi\)
−0.996577 + 0.0826748i \(0.973654\pi\)
\(6\) −12882.9 + 12882.9i −0.676364 + 0.676364i
\(7\) 27657.5 − 27657.5i 0.621977 − 0.621977i −0.324060 − 0.946037i \(-0.605048\pi\)
0.946037 + 0.324060i \(0.105048\pi\)
\(8\) 333872.i 3.60234i
\(9\) − 134924.i − 0.761651i
\(10\) −566224. + 566224.i −1.79056 + 1.79056i
\(11\) 326010. − 326010.i 0.610340 − 0.610340i −0.332695 − 0.943035i \(-0.607958\pi\)
0.943035 + 0.332695i \(0.107958\pi\)
\(12\) 844692. + 844692.i 0.979945 + 0.979945i
\(13\) −942273. −0.703864 −0.351932 − 0.936026i \(-0.614475\pi\)
−0.351932 + 0.936026i \(0.614475\pi\)
\(14\) −2.45226e6 − 2.45226e6i −1.21860 − 1.21860i
\(15\) 1.85577e6i 0.630988i
\(16\) 1.76967e7 4.21922
\(17\) 5.69555e6 + 1.35373e6i 0.972897 + 0.231239i
\(18\) −1.19631e7 −1.49226
\(19\) − 1.34809e7i − 1.24904i −0.781010 − 0.624519i \(-0.785295\pi\)
0.781010 − 0.624519i \(-0.214705\pi\)
\(20\) 3.71257e7 + 3.71257e7i 2.59424 + 2.59424i
\(21\) −8.03715e6 −0.429433
\(22\) −2.89058e7 − 2.89058e7i −1.19580 − 1.19580i
\(23\) −1.73146e7 + 1.73146e7i −0.560930 + 0.560930i −0.929572 − 0.368642i \(-0.879823\pi\)
0.368642 + 0.929572i \(0.379823\pi\)
\(24\) 4.85108e7 − 4.85108e7i 1.24359 − 1.24359i
\(25\) 3.27360e7i 0.670433i
\(26\) 8.35469e7i 1.37904i
\(27\) −4.53432e7 + 4.53432e7i −0.608151 + 0.608151i
\(28\) −1.60788e8 + 1.60788e8i −1.76557 + 1.76557i
\(29\) 5.58148e7 + 5.58148e7i 0.505313 + 0.505313i 0.913084 − 0.407771i \(-0.133694\pi\)
−0.407771 + 0.913084i \(0.633694\pi\)
\(30\) 1.64542e8 1.23626
\(31\) −5.79594e7 − 5.79594e7i −0.363609 − 0.363609i 0.501531 − 0.865140i \(-0.332770\pi\)
−0.865140 + 0.501531i \(0.832770\pi\)
\(32\) − 8.85314e8i − 4.66415i
\(33\) −9.47371e7 −0.421399
\(34\) 1.20028e8 − 5.04998e8i 0.453054 − 1.90614i
\(35\) −3.53246e8 −1.13685
\(36\) 7.84386e8i 2.16205i
\(37\) 2.77393e8 + 2.77393e8i 0.657637 + 0.657637i 0.954820 − 0.297184i \(-0.0960473\pi\)
−0.297184 + 0.954820i \(0.596047\pi\)
\(38\) −1.19529e9 −2.44717
\(39\) 1.36910e8 + 1.36910e8i 0.242985 + 0.242985i
\(40\) 2.13213e9 − 2.13213e9i 3.29218 − 3.29218i
\(41\) −2.25930e8 + 2.25930e8i −0.304553 + 0.304553i −0.842792 − 0.538239i \(-0.819090\pi\)
0.538239 + 0.842792i \(0.319090\pi\)
\(42\) 7.12616e8i 0.841365i
\(43\) 5.89362e8i 0.611372i 0.952132 + 0.305686i \(0.0988858\pi\)
−0.952132 + 0.305686i \(0.901114\pi\)
\(44\) −1.89527e9 + 1.89527e9i −1.73253 + 1.73253i
\(45\) −8.61637e8 + 8.61637e8i −0.696074 + 0.696074i
\(46\) 1.53520e9 + 1.53520e9i 1.09900 + 1.09900i
\(47\) −1.28340e9 −0.816252 −0.408126 − 0.912925i \(-0.633818\pi\)
−0.408126 + 0.912925i \(0.633818\pi\)
\(48\) −2.57129e9 − 2.57129e9i −1.45655 − 1.45655i
\(49\) 4.47450e8i 0.226290i
\(50\) 2.90254e9 1.31354
\(51\) −6.30857e8 − 1.02424e9i −0.256033 − 0.415688i
\(52\) 5.47793e9 1.99801
\(53\) − 4.86248e9i − 1.59713i −0.601908 − 0.798565i \(-0.705593\pi\)
0.601908 − 0.798565i \(-0.294407\pi\)
\(54\) 4.02037e9 + 4.02037e9i 1.19152 + 1.19152i
\(55\) −4.16386e9 −1.11558
\(56\) 9.23406e9 + 9.23406e9i 2.24057 + 2.24057i
\(57\) −1.95875e9 + 1.95875e9i −0.431189 + 0.431189i
\(58\) 4.94883e9 − 4.94883e9i 0.990031 − 0.990031i
\(59\) − 1.51765e9i − 0.276367i −0.990407 − 0.138183i \(-0.955874\pi\)
0.990407 − 0.138183i \(-0.0441263\pi\)
\(60\) − 1.07885e10i − 1.79115i
\(61\) 6.58302e9 − 6.58302e9i 0.997955 − 0.997955i −0.00204338 − 0.999998i \(-0.500650\pi\)
0.999998 + 0.00204338i \(0.000650428\pi\)
\(62\) −5.13898e9 + 5.13898e9i −0.712398 + 0.712398i
\(63\) −3.73167e9 − 3.73167e9i −0.473729 − 0.473729i
\(64\) −4.22537e10 −4.91898
\(65\) 6.01744e9 + 6.01744e9i 0.643262 + 0.643262i
\(66\) 8.39989e9i 0.825623i
\(67\) 1.16438e10 1.05362 0.526810 − 0.849983i \(-0.323388\pi\)
0.526810 + 0.849983i \(0.323388\pi\)
\(68\) −3.31113e10 − 7.86992e9i −2.76170 − 0.656405i
\(69\) 5.03153e9 0.387285
\(70\) 3.13207e10i 2.22737i
\(71\) −1.63879e10 − 1.63879e10i −1.07796 − 1.07796i −0.996692 − 0.0812664i \(-0.974104\pi\)
−0.0812664 − 0.996692i \(-0.525896\pi\)
\(72\) 4.50473e10 2.74372
\(73\) −3.10339e9 − 3.10339e9i −0.175211 − 0.175211i 0.614054 − 0.789264i \(-0.289538\pi\)
−0.789264 + 0.614054i \(0.789538\pi\)
\(74\) 2.45951e10 − 2.45951e10i 1.28847 − 1.28847i
\(75\) 4.75646e9 − 4.75646e9i 0.231445 − 0.231445i
\(76\) 7.83719e10i 3.54556i
\(77\) − 1.80333e10i − 0.759234i
\(78\) 1.21392e10 − 1.21392e10i 0.476068 − 0.476068i
\(79\) 1.85319e10 − 1.85319e10i 0.677595 − 0.677595i −0.281860 − 0.959455i \(-0.590951\pi\)
0.959455 + 0.281860i \(0.0909514\pi\)
\(80\) −1.13013e11 − 1.13013e11i −3.85596 − 3.85596i
\(81\) −1.07249e10 −0.341763
\(82\) 2.00321e10 + 2.00321e10i 0.596693 + 0.596693i
\(83\) 4.03852e10i 1.12536i 0.826673 + 0.562682i \(0.190230\pi\)
−0.826673 + 0.562682i \(0.809770\pi\)
\(84\) 4.67242e10 1.21901
\(85\) −2.77273e10 − 4.50173e10i −0.677802 − 1.10046i
\(86\) 5.22559e10 1.19783
\(87\) − 1.62195e10i − 0.348885i
\(88\) 1.08846e11 + 1.08846e11i 2.19865 + 2.19865i
\(89\) −6.44743e10 −1.22389 −0.611944 − 0.790901i \(-0.709612\pi\)
−0.611944 + 0.790901i \(0.709612\pi\)
\(90\) 7.63973e10 + 7.63973e10i 1.36378 + 1.36378i
\(91\) −2.60609e10 + 2.60609e10i −0.437787 + 0.437787i
\(92\) 1.00659e11 − 1.00659e11i 1.59228 − 1.59228i
\(93\) 1.68427e10i 0.251048i
\(94\) 1.13793e11i 1.59924i
\(95\) −8.60904e10 + 8.60904e10i −1.14150 + 1.14150i
\(96\) −1.28634e11 + 1.28634e11i −1.61014 + 1.61014i
\(97\) −3.87960e10 − 3.87960e10i −0.458715 − 0.458715i 0.439519 − 0.898233i \(-0.355149\pi\)
−0.898233 + 0.439519i \(0.855149\pi\)
\(98\) 3.96733e10 0.443358
\(99\) −4.39867e10 − 4.39867e10i −0.464866 − 0.464866i
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 17.12.c.a.13.1 yes 32
17.4 even 4 inner 17.12.c.a.4.16 ✓ 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.12.c.a.4.16 ✓ 32 17.4 even 4 inner
17.12.c.a.13.1 yes 32 1.1 even 1 trivial