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Properties

Label 578.2.d.c.179.1
Level $578$
Weight $2$
Character 578.179
Analytic conductor $4.615$
Analytic rank $0$
Dimension $4$
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] = [578,2,Mod(155,578)] 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("578.155"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.d (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,-4,8,0,4,0,-8,0,0,0,-8,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 34)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 179.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 578.179
Dual form 578.2.d.c.155.1

$q$-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} +(1.70711 + 0.707107i) q^{3} -1.00000i q^{4} +(-1.41421 + 3.41421i) q^{5} +(-1.70711 + 0.707107i) q^{6} +(0.585786 + 1.41421i) q^{7} +(0.707107 + 0.707107i) q^{8} +(0.292893 + 0.292893i) q^{9} +(-1.41421 - 3.41421i) q^{10} +(-4.12132 + 1.70711i) q^{11} +(0.707107 - 1.70711i) q^{12} -0.828427i q^{13} +(-1.41421 - 0.585786i) q^{14} +(-4.82843 + 4.82843i) q^{15} -1.00000 q^{16} -0.414214 q^{18} +(-0.585786 + 0.585786i) q^{19} +(3.41421 + 1.41421i) q^{20} +2.82843i q^{21} +(1.70711 - 4.12132i) q^{22} +(2.82843 - 1.17157i) q^{23} +(0.707107 + 1.70711i) q^{24} +(-6.12132 - 6.12132i) q^{25} +(0.585786 + 0.585786i) q^{26} +(-1.82843 - 4.41421i) q^{27} +(1.41421 - 0.585786i) q^{28} +(-0.585786 + 1.41421i) q^{29} -6.82843i q^{30} +(3.41421 + 1.41421i) q^{31} +(0.707107 - 0.707107i) q^{32} -8.24264 q^{33} -5.65685 q^{35} +(0.292893 - 0.292893i) q^{36} +(-2.00000 - 0.828427i) q^{37} -0.828427i q^{38} +(0.585786 - 1.41421i) q^{39} +(-3.41421 + 1.41421i) q^{40} +(4.29289 + 10.3640i) q^{41} +(-2.00000 - 2.00000i) q^{42} +(1.24264 + 1.24264i) q^{43} +(1.70711 + 4.12132i) q^{44} +(-1.41421 + 0.585786i) q^{45} +(-1.17157 + 2.82843i) q^{46} +9.65685i q^{47} +(-1.70711 - 0.707107i) q^{48} +(3.29289 - 3.29289i) q^{49} +8.65685 q^{50} -0.828427 q^{52} +(-0.585786 + 0.585786i) q^{53} +(4.41421 + 1.82843i) q^{54} -16.4853i q^{55} +(-0.585786 + 1.41421i) q^{56} +(-1.41421 + 0.585786i) q^{57} +(-0.585786 - 1.41421i) q^{58} +(-0.414214 - 0.414214i) q^{59} +(4.82843 + 4.82843i) q^{60} +(2.82843 + 6.82843i) q^{61} +(-3.41421 + 1.41421i) q^{62} +(-0.242641 + 0.585786i) q^{63} +1.00000i q^{64} +(2.82843 + 1.17157i) q^{65} +(5.82843 - 5.82843i) q^{66} -7.41421 q^{67} +5.65685 q^{69} +(4.00000 - 4.00000i) q^{70} +(-2.00000 - 0.828427i) q^{71} +0.414214i q^{72} +(-2.05025 + 4.94975i) q^{73} +(2.00000 - 0.828427i) q^{74} +(-6.12132 - 14.7782i) q^{75} +(0.585786 + 0.585786i) q^{76} +(-4.82843 - 4.82843i) q^{77} +(0.585786 + 1.41421i) q^{78} +(4.82843 - 2.00000i) q^{79} +(1.41421 - 3.41421i) q^{80} -10.0711i q^{81} +(-10.3640 - 4.29289i) q^{82} +(-4.41421 + 4.41421i) q^{83} +2.82843 q^{84} -1.75736 q^{86} +(-2.00000 + 2.00000i) q^{87} +(-4.12132 - 1.70711i) q^{88} +15.0711i q^{89} +(0.585786 - 1.41421i) q^{90} +(1.17157 - 0.485281i) q^{91} +(-1.17157 - 2.82843i) q^{92} +(4.82843 + 4.82843i) q^{93} +(-6.82843 - 6.82843i) q^{94} +(-1.17157 - 2.82843i) q^{95} +(1.70711 - 0.707107i) q^{96} +(-2.12132 + 5.12132i) q^{97} +4.65685i q^{98} +(-1.70711 - 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{6} + 8 q^{7} + 4 q^{9} - 8 q^{11} - 8 q^{15} - 4 q^{16} + 4 q^{18} - 8 q^{19} + 8 q^{20} + 4 q^{22} - 16 q^{25} + 8 q^{26} + 4 q^{27} - 8 q^{29} + 8 q^{31} - 16 q^{33} + 4 q^{36} - 8 q^{37}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\)

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
\(3\) 1.70711 + 0.707107i 0.985599 + 0.408248i 0.816497 − 0.577350i \(-0.195913\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) −1.41421 + 3.41421i −0.632456 + 1.52688i 0.204071 + 0.978956i \(0.434583\pi\)
−0.836526 + 0.547927i \(0.815417\pi\)
\(6\) −1.70711 + 0.707107i −0.696923 + 0.288675i
\(7\) 0.585786 + 1.41421i 0.221406 + 0.534522i 0.995081 − 0.0990602i \(-0.0315836\pi\)
−0.773675 + 0.633583i \(0.781584\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 0.292893 + 0.292893i 0.0976311 + 0.0976311i
\(10\) −1.41421 − 3.41421i −0.447214 − 1.07967i
\(11\) −4.12132 + 1.70711i −1.24262 + 0.514712i −0.904534 − 0.426401i \(-0.859781\pi\)
−0.338091 + 0.941113i \(0.609781\pi\)
\(12\) 0.707107 − 1.70711i 0.204124 − 0.492799i
\(13\) − 0.828427i − 0.229764i −0.993379 − 0.114882i \(-0.963351\pi\)
0.993379 − 0.114882i \(-0.0366490\pi\)
\(14\) −1.41421 − 0.585786i −0.377964 − 0.156558i
\(15\) −4.82843 + 4.82843i −1.24669 + 1.24669i
\(16\) −1.00000 −0.250000
\(17\) 0 0
\(18\) −0.414214 −0.0976311
\(19\) −0.585786 + 0.585786i −0.134389 + 0.134389i −0.771101 − 0.636713i \(-0.780294\pi\)
0.636713 + 0.771101i \(0.280294\pi\)
\(20\) 3.41421 + 1.41421i 0.763441 + 0.316228i
\(21\) 2.82843i 0.617213i
\(22\) 1.70711 − 4.12132i 0.363956 − 0.878668i
\(23\) 2.82843 − 1.17157i 0.589768 − 0.244290i −0.0677829 − 0.997700i \(-0.521593\pi\)
0.657551 + 0.753410i \(0.271593\pi\)
\(24\) 0.707107 + 1.70711i 0.144338 + 0.348462i
\(25\) −6.12132 − 6.12132i −1.22426 − 1.22426i
\(26\) 0.585786 + 0.585786i 0.114882 + 0.114882i
\(27\) −1.82843 − 4.41421i −0.351881 − 0.849516i
\(28\) 1.41421 − 0.585786i 0.267261 − 0.110703i
\(29\) −0.585786 + 1.41421i −0.108778 + 0.262613i −0.968890 − 0.247492i \(-0.920394\pi\)
0.860112 + 0.510105i \(0.170394\pi\)
\(30\) − 6.82843i − 1.24669i
\(31\) 3.41421 + 1.41421i 0.613211 + 0.254000i 0.667601 − 0.744520i \(-0.267321\pi\)
−0.0543898 + 0.998520i \(0.517321\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) −8.24264 −1.43486
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0.292893 − 0.292893i 0.0488155 − 0.0488155i
\(37\) −2.00000 − 0.828427i −0.328798 − 0.136193i 0.212177 − 0.977231i \(-0.431945\pi\)
−0.540975 + 0.841039i \(0.681945\pi\)
\(38\) − 0.828427i − 0.134389i
\(39\) 0.585786 − 1.41421i 0.0938009 − 0.226455i
\(40\) −3.41421 + 1.41421i −0.539835 + 0.223607i
\(41\) 4.29289 + 10.3640i 0.670437 + 1.61858i 0.780869 + 0.624695i \(0.214777\pi\)
−0.110432 + 0.993884i \(0.535223\pi\)
\(42\) −2.00000 − 2.00000i −0.308607 − 0.308607i
\(43\) 1.24264 + 1.24264i 0.189501 + 0.189501i 0.795480 − 0.605979i \(-0.207219\pi\)
−0.605979 + 0.795480i \(0.707219\pi\)
\(44\) 1.70711 + 4.12132i 0.257356 + 0.621312i
\(45\) −1.41421 + 0.585786i −0.210819 + 0.0873239i
\(46\) −1.17157 + 2.82843i −0.172739 + 0.417029i
\(47\) 9.65685i 1.40860i 0.709904 + 0.704298i \(0.248738\pi\)
−0.709904 + 0.704298i \(0.751262\pi\)
\(48\) −1.70711 − 0.707107i −0.246400 − 0.102062i
\(49\) 3.29289 − 3.29289i 0.470413 − 0.470413i
\(50\) 8.65685 1.22426
\(51\) 0 0
\(52\) −0.828427 −0.114882
\(53\) −0.585786 + 0.585786i −0.0804640 + 0.0804640i −0.746193 − 0.665729i \(-0.768121\pi\)
0.665729 + 0.746193i \(0.268121\pi\)
\(54\) 4.41421 + 1.82843i 0.600698 + 0.248817i
\(55\) − 16.4853i − 2.22287i
\(56\) −0.585786 + 1.41421i −0.0782790 + 0.188982i
\(57\) −1.41421 + 0.585786i −0.187317 + 0.0775893i
\(58\) −0.585786 − 1.41421i −0.0769175 − 0.185695i
\(59\) −0.414214 − 0.414214i −0.0539260 − 0.0539260i 0.679630 − 0.733556i \(-0.262141\pi\)
−0.733556 + 0.679630i \(0.762141\pi\)
\(60\) 4.82843 + 4.82843i 0.623347 + 0.623347i
\(61\) 2.82843 + 6.82843i 0.362143 + 0.874291i 0.994986 + 0.100011i \(0.0318877\pi\)
−0.632843 + 0.774280i \(0.718112\pi\)
\(62\) −3.41421 + 1.41421i −0.433606 + 0.179605i
\(63\) −0.242641 + 0.585786i −0.0305699 + 0.0738022i
\(64\) 1.00000i 0.125000i
\(65\) 2.82843 + 1.17157i 0.350823 + 0.145316i
\(66\) 5.82843 − 5.82843i 0.717430 − 0.717430i
\(67\) −7.41421 −0.905790 −0.452895 − 0.891564i \(-0.649609\pi\)
−0.452895 + 0.891564i \(0.649609\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 4.00000 − 4.00000i 0.478091 − 0.478091i
\(71\) −2.00000 − 0.828427i −0.237356 − 0.0983162i 0.260835 − 0.965383i \(-0.416002\pi\)
−0.498191 + 0.867067i \(0.666002\pi\)
\(72\) 0.414214i 0.0488155i
\(73\) −2.05025 + 4.94975i −0.239964 + 0.579324i −0.997279 − 0.0737261i \(-0.976511\pi\)
0.757315 + 0.653050i \(0.226511\pi\)
\(74\) 2.00000 − 0.828427i 0.232495 − 0.0963027i
\(75\) −6.12132 − 14.7782i −0.706829 − 1.70644i
\(76\) 0.585786 + 0.585786i 0.0671943 + 0.0671943i
\(77\) −4.82843 − 4.82843i −0.550250 − 0.550250i
\(78\) 0.585786 + 1.41421i 0.0663273 + 0.160128i
\(79\) 4.82843 − 2.00000i 0.543240 − 0.225018i −0.0941507 − 0.995558i \(-0.530014\pi\)
0.637391 + 0.770540i \(0.280014\pi\)
\(80\) 1.41421 − 3.41421i 0.158114 − 0.381721i
\(81\) − 10.0711i − 1.11901i
\(82\) −10.3640 − 4.29289i −1.14451 − 0.474071i
\(83\) −4.41421 + 4.41421i −0.484523 + 0.484523i −0.906573 − 0.422050i \(-0.861311\pi\)
0.422050 + 0.906573i \(0.361311\pi\)
\(84\) 2.82843 0.308607
\(85\) 0 0
\(86\) −1.75736 −0.189501
\(87\) −2.00000 + 2.00000i −0.214423 + 0.214423i
\(88\) −4.12132 − 1.70711i −0.439334 − 0.181978i
\(89\) 15.0711i 1.59753i 0.601643 + 0.798765i \(0.294513\pi\)
−0.601643 + 0.798765i \(0.705487\pi\)
\(90\) 0.585786 − 1.41421i 0.0617473 − 0.149071i
\(91\) 1.17157 − 0.485281i 0.122814 − 0.0508713i
\(92\) −1.17157 − 2.82843i −0.122145 − 0.294884i
\(93\) 4.82843 + 4.82843i 0.500685 + 0.500685i
\(94\) −6.82843 − 6.82843i −0.704298 − 0.704298i
\(95\) −1.17157 − 2.82843i −0.120201 − 0.290191i
\(96\) 1.70711 − 0.707107i 0.174231 − 0.0721688i
\(97\) −2.12132 + 5.12132i −0.215387 + 0.519991i −0.994235 − 0.107222i \(-0.965804\pi\)
0.778848 + 0.627213i \(0.215804\pi\)
\(98\) 4.65685i 0.470413i
\(99\) −1.70711 − 0.707107i −0.171571 − 0.0710669i
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 578.2.d.c.179.1 4
17.2 even 8 inner 578.2.d.c.155.1 4
17.3 odd 16 578.2.c.f.251.4 8
17.4 even 4 578.2.d.b.399.1 4
17.5 odd 16 578.2.c.f.327.1 8
17.6 odd 16 578.2.a.i.1.4 4
17.7 odd 16 578.2.b.d.577.1 4
17.8 even 8 34.2.d.a.15.1 ✓ 4
17.9 even 8 578.2.d.b.423.1 4
17.10 odd 16 578.2.b.d.577.4 4
17.11 odd 16 578.2.a.i.1.1 4
17.12 odd 16 578.2.c.f.327.4 8
17.13 even 4 34.2.d.a.25.1 yes 4
17.14 odd 16 578.2.c.f.251.1 8
17.15 even 8 578.2.d.a.155.1 4
17.16 even 2 578.2.d.a.179.1 4
51.8 odd 8 306.2.l.c.253.1 4
51.11 even 16 5202.2.a.bw.1.4 4
51.23 even 16 5202.2.a.bw.1.1 4
51.47 odd 4 306.2.l.c.127.1 4
68.11 even 16 4624.2.a.bn.1.4 4
68.23 even 16 4624.2.a.bn.1.1 4
68.47 odd 4 272.2.v.b.161.1 4
68.59 odd 8 272.2.v.b.49.1 4
85.8 odd 8 850.2.o.b.49.1 4
85.13 odd 4 850.2.o.a.399.1 4
85.42 odd 8 850.2.o.a.49.1 4
85.47 odd 4 850.2.o.b.399.1 4
85.59 even 8 850.2.l.a.151.1 4
85.64 even 4 850.2.l.a.501.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.2.d.a.15.1 ✓ 4 17.8 even 8
34.2.d.a.25.1 yes 4 17.13 even 4
272.2.v.b.49.1 4 68.59 odd 8
272.2.v.b.161.1 4 68.47 odd 4
306.2.l.c.127.1 4 51.47 odd 4
306.2.l.c.253.1 4 51.8 odd 8
578.2.a.i.1.1 4 17.11 odd 16
578.2.a.i.1.4 4 17.6 odd 16
578.2.b.d.577.1 4 17.7 odd 16
578.2.b.d.577.4 4 17.10 odd 16
578.2.c.f.251.1 8 17.14 odd 16
578.2.c.f.251.4 8 17.3 odd 16
578.2.c.f.327.1 8 17.5 odd 16
578.2.c.f.327.4 8 17.12 odd 16
578.2.d.a.155.1 4 17.15 even 8
578.2.d.a.179.1 4 17.16 even 2
578.2.d.b.399.1 4 17.4 even 4
578.2.d.b.423.1 4 17.9 even 8
578.2.d.c.155.1 4 17.2 even 8 inner
578.2.d.c.179.1 4 1.1 even 1 trivial
850.2.l.a.151.1 4 85.59 even 8
850.2.l.a.501.1 4 85.64 even 4
850.2.o.a.49.1 4 85.42 odd 8
850.2.o.a.399.1 4 85.13 odd 4
850.2.o.b.49.1 4 85.8 odd 8
850.2.o.b.399.1 4 85.47 odd 4
4624.2.a.bn.1.1 4 68.23 even 16
4624.2.a.bn.1.4 4 68.11 even 16
5202.2.a.bw.1.1 4 51.23 even 16
5202.2.a.bw.1.4 4 51.11 even 16