↑

Properties

Label 70.2.g.a.13.1
Level $70$
Weight $2$
Character 70.13
Analytic conductor $0.559$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [70,2,Mod(13,70)] 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("70.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.g (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: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.1
Root \(-0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 70.13
Dual form 70.2.g.a.27.1

$q$-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+(-0.707107 + 0.707107i) q^{2} +(-1.30656 + 1.30656i) q^{3} -1.00000i q^{4} +(-0.158513 + 2.23044i) q^{5} -1.84776i q^{6} +(-2.47247 + 0.941740i) q^{7} +(0.707107 + 0.707107i) q^{8} -0.414214i q^{9} +(-1.46508 - 1.68925i) q^{10} +2.82843 q^{11} +(1.30656 + 1.30656i) q^{12} +(4.23671 - 4.23671i) q^{13} +(1.08239 - 2.41421i) q^{14} +(-2.70711 - 3.12132i) q^{15} -1.00000 q^{16} +(3.69552 + 3.69552i) q^{17} +(0.292893 + 0.292893i) q^{18} -1.39942 q^{19} +(2.23044 + 0.158513i) q^{20} +(2.00000 - 4.46088i) q^{21} +(-2.00000 + 2.00000i) q^{22} +(0.414214 + 0.414214i) q^{23} -1.84776 q^{24} +(-4.94975 - 0.707107i) q^{25} +5.99162i q^{26} +(-3.37849 - 3.37849i) q^{27} +(0.941740 + 2.47247i) q^{28} +0.828427i q^{29} +(4.12132 + 0.292893i) q^{30} +1.53073i q^{31} +(0.707107 - 0.707107i) q^{32} +(-3.69552 + 3.69552i) q^{33} -5.22625 q^{34} +(-1.70858 - 5.66399i) q^{35} -0.414214 q^{36} +(2.58579 - 2.58579i) q^{37} +(0.989538 - 0.989538i) q^{38} +11.0711i q^{39} +(-1.68925 + 1.46508i) q^{40} +3.69552i q^{41} +(1.74011 + 4.56854i) q^{42} +(4.00000 + 4.00000i) q^{43} -2.82843i q^{44} +(0.923880 + 0.0656581i) q^{45} -0.585786 q^{46} +(-1.08239 - 1.08239i) q^{47} +(1.30656 - 1.30656i) q^{48} +(5.22625 - 4.65685i) q^{49} +(4.00000 - 3.00000i) q^{50} -9.65685 q^{51} +(-4.23671 - 4.23671i) q^{52} +(-8.24264 - 8.24264i) q^{53} +4.77791 q^{54} +(-0.448342 + 6.30864i) q^{55} +(-2.41421 - 1.08239i) q^{56} +(1.82843 - 1.82843i) q^{57} +(-0.585786 - 0.585786i) q^{58} +9.23880 q^{59} +(-3.12132 + 2.70711i) q^{60} -6.43996i q^{61} +(-1.08239 - 1.08239i) q^{62} +(0.390081 + 1.02413i) q^{63} +1.00000i q^{64} +(8.77817 + 10.1213i) q^{65} -5.22625i q^{66} +(10.4853 - 10.4853i) q^{67} +(3.69552 - 3.69552i) q^{68} -1.08239 q^{69} +(5.21319 + 2.79690i) q^{70} -0.585786 q^{71} +(0.292893 - 0.292893i) q^{72} +(-4.14386 + 4.14386i) q^{73} +3.65685i q^{74} +(7.39104 - 5.54328i) q^{75} +1.39942i q^{76} +(-6.99321 + 2.66364i) q^{77} +(-7.82843 - 7.82843i) q^{78} -5.07107i q^{79} +(0.158513 - 2.23044i) q^{80} +10.0711 q^{81} +(-2.61313 - 2.61313i) q^{82} +(-5.31911 + 5.31911i) q^{83} +(-4.46088 - 2.00000i) q^{84} +(-8.82843 + 7.65685i) q^{85} -5.65685 q^{86} +(-1.08239 - 1.08239i) q^{87} +(2.00000 + 2.00000i) q^{88} -11.3492 q^{89} +(-0.699709 + 0.606854i) q^{90} +(-6.48528 + 14.4650i) q^{91} +(0.414214 - 0.414214i) q^{92} +(-2.00000 - 2.00000i) q^{93} +1.53073 q^{94} +(0.221825 - 3.12132i) q^{95} +1.84776i q^{96} +(4.59220 + 4.59220i) q^{97} +(-0.402625 + 6.98841i) q^{98} -1.17157i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} - 16 q^{15} - 8 q^{16} + 8 q^{18} + 16 q^{21} - 16 q^{22} - 8 q^{23} + 8 q^{28} + 16 q^{30} - 8 q^{35} + 8 q^{36} + 32 q^{37} + 32 q^{43} - 16 q^{46} + 32 q^{50} - 32 q^{51} - 32 q^{53} - 8 q^{56}+ \cdots + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.500000 + 0.500000i
\(3\) −1.30656 + 1.30656i −0.754344 + 0.754344i −0.975287 − 0.220942i \(-0.929087\pi\)
0.220942 + 0.975287i \(0.429087\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) −0.158513 + 2.23044i −0.0708890 + 0.997484i
\(6\) − 1.84776i − 0.754344i
\(7\) −2.47247 + 0.941740i −0.934507 + 0.355944i
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) − 0.414214i − 0.138071i
\(10\) −1.46508 − 1.68925i −0.463298 − 0.534187i
\(11\) 2.82843 0.852803 0.426401 − 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 1.30656 + 1.30656i 0.377172 + 0.377172i
\(13\) 4.23671 − 4.23671i 1.17505 − 1.17505i 0.194064 − 0.980989i \(-0.437833\pi\)
0.980989 − 0.194064i \(-0.0621670\pi\)
\(14\) 1.08239 − 2.41421i 0.289281 − 0.645226i
\(15\) −2.70711 − 3.12132i −0.698972 − 0.805921i
\(16\) −1.00000 −0.250000
\(17\) 3.69552 + 3.69552i 0.896295 + 0.896295i 0.995106 − 0.0988114i \(-0.0315040\pi\)
−0.0988114 + 0.995106i \(0.531504\pi\)
\(18\) 0.292893 + 0.292893i 0.0690356 + 0.0690356i
\(19\) −1.39942 −0.321048 −0.160524 − 0.987032i \(-0.551318\pi\)
−0.160524 + 0.987032i \(0.551318\pi\)
\(20\) 2.23044 + 0.158513i 0.498742 + 0.0354445i
\(21\) 2.00000 − 4.46088i 0.436436 − 0.973445i
\(22\) −2.00000 + 2.00000i −0.426401 + 0.426401i
\(23\) 0.414214 + 0.414214i 0.0863695 + 0.0863695i 0.748972 − 0.662602i \(-0.230548\pi\)
−0.662602 + 0.748972i \(0.730548\pi\)
\(24\) −1.84776 −0.377172
\(25\) −4.94975 − 0.707107i −0.989949 − 0.141421i
\(26\) 5.99162i 1.17505i
\(27\) −3.37849 − 3.37849i −0.650191 − 0.650191i
\(28\) 0.941740 + 2.47247i 0.177972 + 0.467254i
\(29\) 0.828427i 0.153835i 0.997037 + 0.0769175i \(0.0245078\pi\)
−0.997037 + 0.0769175i \(0.975492\pi\)
\(30\) 4.12132 + 0.292893i 0.752447 + 0.0534747i
\(31\) 1.53073i 0.274928i 0.990507 + 0.137464i \(0.0438951\pi\)
−0.990507 + 0.137464i \(0.956105\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) −3.69552 + 3.69552i −0.643307 + 0.643307i
\(34\) −5.22625 −0.896295
\(35\) −1.70858 − 5.66399i −0.288802 − 0.957389i
\(36\) −0.414214 −0.0690356
\(37\) 2.58579 − 2.58579i 0.425101 − 0.425101i −0.461855 − 0.886956i \(-0.652816\pi\)
0.886956 + 0.461855i \(0.152816\pi\)
\(38\) 0.989538 − 0.989538i 0.160524 − 0.160524i
\(39\) 11.0711i 1.77279i
\(40\) −1.68925 + 1.46508i −0.267093 + 0.231649i
\(41\) 3.69552i 0.577143i 0.957458 + 0.288571i \(0.0931803\pi\)
−0.957458 + 0.288571i \(0.906820\pi\)
\(42\) 1.74011 + 4.56854i 0.268505 + 0.704940i
\(43\) 4.00000 + 4.00000i 0.609994 + 0.609994i 0.942944 − 0.332950i \(-0.108044\pi\)
−0.332950 + 0.942944i \(0.608044\pi\)
\(44\) − 2.82843i − 0.426401i
\(45\) 0.923880 + 0.0656581i 0.137724 + 0.00978773i
\(46\) −0.585786 −0.0863695
\(47\) −1.08239 − 1.08239i −0.157883 − 0.157883i 0.623745 − 0.781628i \(-0.285610\pi\)
−0.781628 + 0.623745i \(0.785610\pi\)
\(48\) 1.30656 − 1.30656i 0.188586 − 0.188586i
\(49\) 5.22625 − 4.65685i 0.746607 − 0.665265i
\(50\) 4.00000 − 3.00000i 0.565685 − 0.424264i
\(51\) −9.65685 −1.35223
\(52\) −4.23671 − 4.23671i −0.587527 − 0.587527i
\(53\) −8.24264 − 8.24264i −1.13221 − 1.13221i −0.989808 − 0.142405i \(-0.954516\pi\)
−0.142405 − 0.989808i \(-0.545484\pi\)
\(54\) 4.77791 0.650191
\(55\) −0.448342 + 6.30864i −0.0604544 + 0.850657i
\(56\) −2.41421 − 1.08239i −0.322613 − 0.144641i
\(57\) 1.82843 − 1.82843i 0.242181 − 0.242181i
\(58\) −0.585786 − 0.585786i −0.0769175 − 0.0769175i
\(59\) 9.23880 1.20279 0.601394 − 0.798952i \(-0.294612\pi\)
0.601394 + 0.798952i \(0.294612\pi\)
\(60\) −3.12132 + 2.70711i −0.402961 + 0.349486i
\(61\) − 6.43996i − 0.824552i −0.911059 − 0.412276i \(-0.864734\pi\)
0.911059 − 0.412276i \(-0.135266\pi\)
\(62\) −1.08239 − 1.08239i −0.137464 − 0.137464i
\(63\) 0.390081 + 1.02413i 0.0491456 + 0.129029i
\(64\) 1.00000i 0.125000i
\(65\) 8.77817 + 10.1213i 1.08880 + 1.25540i
\(66\) − 5.22625i − 0.643307i
\(67\) 10.4853 − 10.4853i 1.28098 − 1.28098i 0.340871 − 0.940110i \(-0.389278\pi\)
0.940110 − 0.340871i \(-0.110722\pi\)
\(68\) 3.69552 − 3.69552i 0.448147 − 0.448147i
\(69\) −1.08239 −0.130305
\(70\) 5.21319 + 2.79690i 0.623096 + 0.334293i
\(71\) −0.585786 −0.0695201 −0.0347600 − 0.999396i \(-0.511067\pi\)
−0.0347600 + 0.999396i \(0.511067\pi\)
\(72\) 0.292893 − 0.292893i 0.0345178 − 0.0345178i
\(73\) −4.14386 + 4.14386i −0.485002 + 0.485002i −0.906725 − 0.421723i \(-0.861426\pi\)
0.421723 + 0.906725i \(0.361426\pi\)
\(74\) 3.65685i 0.425101i
\(75\) 7.39104 − 5.54328i 0.853443 − 0.640083i
\(76\) 1.39942i 0.160524i
\(77\) −6.99321 + 2.66364i −0.796950 + 0.303550i
\(78\) −7.82843 − 7.82843i −0.886395 − 0.886395i
\(79\) − 5.07107i − 0.570540i −0.958447 − 0.285270i \(-0.907917\pi\)
0.958447 − 0.285270i \(-0.0920832\pi\)
\(80\) 0.158513 − 2.23044i 0.0177223 − 0.249371i
\(81\) 10.0711 1.11901
\(82\) −2.61313 − 2.61313i −0.288571 − 0.288571i
\(83\) −5.31911 + 5.31911i −0.583848 + 0.583848i −0.935958 − 0.352111i \(-0.885464\pi\)
0.352111 + 0.935958i \(0.385464\pi\)
\(84\) −4.46088 − 2.00000i −0.486722 − 0.218218i
\(85\) −8.82843 + 7.65685i −0.957577 + 0.830502i
\(86\) −5.65685 −0.609994
\(87\) −1.08239 − 1.08239i −0.116045 − 0.116045i
\(88\) 2.00000 + 2.00000i 0.213201 + 0.213201i
\(89\) −11.3492 −1.20301 −0.601506 − 0.798869i \(-0.705432\pi\)
−0.601506 + 0.798869i \(0.705432\pi\)
\(90\) −0.699709 + 0.606854i −0.0737558 + 0.0639680i
\(91\) −6.48528 + 14.4650i −0.679842 + 1.51635i
\(92\) 0.414214 − 0.414214i 0.0431847 − 0.0431847i
\(93\) −2.00000 − 2.00000i −0.207390 − 0.207390i
\(94\) 1.53073 0.157883
\(95\) 0.221825 − 3.12132i 0.0227588 − 0.320241i
\(96\) 1.84776i 0.188586i
\(97\) 4.59220 + 4.59220i 0.466267 + 0.466267i 0.900703 − 0.434436i \(-0.143052\pi\)
−0.434436 + 0.900703i \(0.643052\pi\)
\(98\) −0.402625 + 6.98841i −0.0406713 + 0.705936i
\(99\) − 1.17157i − 0.117748i
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 70.2.g.a.13.1 ✓ 8
3.2 odd 2 630.2.p.a.433.4 8
4.3 odd 2 560.2.bj.c.433.4 8
5.2 odd 4 inner 70.2.g.a.27.2 yes 8
5.3 odd 4 350.2.g.a.307.3 8
5.4 even 2 350.2.g.a.293.4 8
7.2 even 3 490.2.l.a.423.3 16
7.3 odd 6 490.2.l.a.313.1 16
7.4 even 3 490.2.l.a.313.2 16
7.5 odd 6 490.2.l.a.423.4 16
7.6 odd 2 inner 70.2.g.a.13.2 yes 8
15.2 even 4 630.2.p.a.307.3 8
20.7 even 4 560.2.bj.c.97.1 8
21.20 even 2 630.2.p.a.433.3 8
28.27 even 2 560.2.bj.c.433.1 8
35.2 odd 12 490.2.l.a.227.1 16
35.12 even 12 490.2.l.a.227.2 16
35.13 even 4 350.2.g.a.307.4 8
35.17 even 12 490.2.l.a.117.3 16
35.27 even 4 inner 70.2.g.a.27.1 yes 8
35.32 odd 12 490.2.l.a.117.4 16
35.34 odd 2 350.2.g.a.293.3 8
105.62 odd 4 630.2.p.a.307.4 8
140.27 odd 4 560.2.bj.c.97.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.g.a.13.1 ✓ 8 1.1 even 1 trivial
70.2.g.a.13.2 yes 8 7.6 odd 2 inner
70.2.g.a.27.1 yes 8 35.27 even 4 inner
70.2.g.a.27.2 yes 8 5.2 odd 4 inner
350.2.g.a.293.3 8 35.34 odd 2
350.2.g.a.293.4 8 5.4 even 2
350.2.g.a.307.3 8 5.3 odd 4
350.2.g.a.307.4 8 35.13 even 4
490.2.l.a.117.3 16 35.17 even 12
490.2.l.a.117.4 16 35.32 odd 12
490.2.l.a.227.1 16 35.2 odd 12
490.2.l.a.227.2 16 35.12 even 12
490.2.l.a.313.1 16 7.3 odd 6
490.2.l.a.313.2 16 7.4 even 3
490.2.l.a.423.3 16 7.2 even 3
490.2.l.a.423.4 16 7.5 odd 6
560.2.bj.c.97.1 8 20.7 even 4
560.2.bj.c.97.4 8 140.27 odd 4
560.2.bj.c.433.1 8 28.27 even 2
560.2.bj.c.433.4 8 4.3 odd 2
630.2.p.a.307.3 8 15.2 even 4
630.2.p.a.307.4 8 105.62 odd 4
630.2.p.a.433.3 8 21.20 even 2
630.2.p.a.433.4 8 3.2 odd 2