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Properties

Label 578.2.d.b.423.1
Level $578$
Weight $2$
Character 578.423
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 = [4,0,0,0,8,0,0,0,-4,8,-4,-4,0,-8,8,-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 423.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 578.423
Dual form 578.2.d.b.399.1

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

Character values

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

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

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
\(3\) 0.707107 + 1.70711i 0.408248 + 0.985599i 0.985599 + 0.169102i \(0.0540867\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000i 0.500000i
\(5\) 3.41421 − 1.41421i 1.52688 − 0.632456i 0.547927 − 0.836526i \(-0.315417\pi\)
0.978956 + 0.204071i \(0.0654173\pi\)
\(6\) −0.707107 + 1.70711i −0.288675 + 0.696923i
\(7\) −1.41421 − 0.585786i −0.534522 − 0.221406i 0.0990602 − 0.995081i \(-0.468416\pi\)
−0.633583 + 0.773675i \(0.718416\pi\)
\(8\) −0.707107 + 0.707107i −0.250000 + 0.250000i
\(9\) −0.292893 + 0.292893i −0.0976311 + 0.0976311i
\(10\) 3.41421 + 1.41421i 1.07967 + 0.447214i
\(11\) −1.70711 + 4.12132i −0.514712 + 1.24262i 0.426401 + 0.904534i \(0.359781\pi\)
−0.941113 + 0.338091i \(0.890219\pi\)
\(12\) −1.70711 + 0.707107i −0.492799 + 0.204124i
\(13\) 0.828427i 0.229764i 0.993379 + 0.114882i \(0.0366490\pi\)
−0.993379 + 0.114882i \(0.963351\pi\)
\(14\) −0.585786 − 1.41421i −0.156558 − 0.377964i
\(15\) 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 \(-0.219706\pi\)
−0.636713 + 0.771101i \(0.719706\pi\)
\(20\) 1.41421 + 3.41421i 0.316228 + 0.763441i
\(21\) − 2.82843i − 0.617213i
\(22\) −4.12132 + 1.70711i −0.878668 + 0.363956i
\(23\) 1.17157 − 2.82843i 0.244290 − 0.589768i −0.753410 − 0.657551i \(-0.771593\pi\)
0.997700 + 0.0677829i \(0.0215925\pi\)
\(24\) −1.70711 − 0.707107i −0.348462 − 0.144338i
\(25\) 6.12132 − 6.12132i 1.22426 − 1.22426i
\(26\) −0.585786 + 0.585786i −0.114882 + 0.114882i
\(27\) 4.41421 + 1.82843i 0.849516 + 0.351881i
\(28\) 0.585786 − 1.41421i 0.110703 − 0.267261i
\(29\) 1.41421 − 0.585786i 0.262613 − 0.108778i −0.247492 − 0.968890i \(-0.579606\pi\)
0.510105 + 0.860112i \(0.329606\pi\)
\(30\) 6.82843i 1.24669i
\(31\) 1.41421 + 3.41421i 0.254000 + 0.613211i 0.998520 − 0.0543898i \(-0.0173214\pi\)
−0.744520 + 0.667601i \(0.767321\pi\)
\(32\) −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
\(37\) −0.828427 − 2.00000i −0.136193 − 0.328798i 0.841039 − 0.540975i \(-0.181945\pi\)
−0.977231 + 0.212177i \(0.931945\pi\)
\(38\) 0.828427i 0.134389i
\(39\) −1.41421 + 0.585786i −0.226455 + 0.0938009i
\(40\) −1.41421 + 3.41421i −0.223607 + 0.539835i
\(41\) −10.3640 − 4.29289i −1.61858 − 0.670437i −0.624695 − 0.780869i \(-0.714777\pi\)
−0.993884 + 0.110432i \(0.964777\pi\)
\(42\) 2.00000 − 2.00000i 0.308607 − 0.308607i
\(43\) −1.24264 + 1.24264i −0.189501 + 0.189501i −0.795480 − 0.605979i \(-0.792781\pi\)
0.605979 + 0.795480i \(0.292781\pi\)
\(44\) −4.12132 − 1.70711i −0.621312 − 0.257356i
\(45\) −0.585786 + 1.41421i −0.0873239 + 0.210819i
\(46\) 2.82843 − 1.17157i 0.417029 − 0.172739i
\(47\) − 9.65685i − 1.40860i −0.709904 − 0.704298i \(-0.751262\pi\)
0.709904 − 0.704298i \(-0.248738\pi\)
\(48\) −0.707107 − 1.70711i −0.102062 − 0.246400i
\(49\) −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 \(-0.231879\pi\)
−0.665729 + 0.746193i \(0.731879\pi\)
\(54\) 1.82843 + 4.41421i 0.248817 + 0.600698i
\(55\) 16.4853i 2.22287i
\(56\) 1.41421 − 0.585786i 0.188982 − 0.0782790i
\(57\) −0.585786 + 1.41421i −0.0775893 + 0.187317i
\(58\) 1.41421 + 0.585786i 0.185695 + 0.0769175i
\(59\) 0.414214 − 0.414214i 0.0539260 − 0.0539260i −0.679630 − 0.733556i \(-0.737859\pi\)
0.733556 + 0.679630i \(0.237859\pi\)
\(60\) −4.82843 + 4.82843i −0.623347 + 0.623347i
\(61\) −6.82843 − 2.82843i −0.874291 − 0.362143i −0.100011 − 0.994986i \(-0.531888\pi\)
−0.774280 + 0.632843i \(0.781888\pi\)
\(62\) −1.41421 + 3.41421i −0.179605 + 0.433606i
\(63\) 0.585786 − 0.242641i 0.0738022 − 0.0305699i
\(64\) − 1.00000i − 0.125000i
\(65\) 1.17157 + 2.82843i 0.145316 + 0.350823i
\(66\) −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
\(71\) −0.828427 − 2.00000i −0.0983162 − 0.237356i 0.867067 − 0.498191i \(-0.166002\pi\)
−0.965383 + 0.260835i \(0.916002\pi\)
\(72\) − 0.414214i − 0.0488155i
\(73\) 4.94975 − 2.05025i 0.579324 − 0.239964i −0.0737261 − 0.997279i \(-0.523489\pi\)
0.653050 + 0.757315i \(0.273489\pi\)
\(74\) 0.828427 − 2.00000i 0.0963027 − 0.232495i
\(75\) 14.7782 + 6.12132i 1.70644 + 0.706829i
\(76\) −0.585786 + 0.585786i −0.0671943 + 0.0671943i
\(77\) 4.82843 − 4.82843i 0.550250 − 0.550250i
\(78\) −1.41421 − 0.585786i −0.160128 − 0.0663273i
\(79\) 2.00000 − 4.82843i 0.225018 − 0.543240i −0.770540 − 0.637391i \(-0.780014\pi\)
0.995558 + 0.0941507i \(0.0300136\pi\)
\(80\) −3.41421 + 1.41421i −0.381721 + 0.158114i
\(81\) 10.0711i 1.11901i
\(82\) −4.29289 − 10.3640i −0.474071 − 1.14451i
\(83\) 4.41421 + 4.41421i 0.484523 + 0.484523i 0.906573 − 0.422050i \(-0.138689\pi\)
−0.422050 + 0.906573i \(0.638689\pi\)
\(84\) 2.82843 0.308607
\(85\) 0 0
\(86\) −1.75736 −0.189501
\(87\) 2.00000 + 2.00000i 0.214423 + 0.214423i
\(88\) −1.70711 − 4.12132i −0.181978 − 0.439334i
\(89\) − 15.0711i − 1.59753i −0.601643 − 0.798765i \(-0.705487\pi\)
0.601643 − 0.798765i \(-0.294513\pi\)
\(90\) −1.41421 + 0.585786i −0.149071 + 0.0617473i
\(91\) 0.485281 − 1.17157i 0.0508713 − 0.122814i
\(92\) 2.82843 + 1.17157i 0.294884 + 0.122145i
\(93\) −4.82843 + 4.82843i −0.500685 + 0.500685i
\(94\) 6.82843 − 6.82843i 0.704298 − 0.704298i
\(95\) 2.82843 + 1.17157i 0.290191 + 0.120201i
\(96\) 0.707107 − 1.70711i 0.0721688 − 0.174231i
\(97\) 5.12132 − 2.12132i 0.519991 − 0.215387i −0.107222 − 0.994235i \(-0.534196\pi\)
0.627213 + 0.778848i \(0.284196\pi\)
\(98\) − 4.65685i − 0.470413i
\(99\) −0.707107 − 1.70711i −0.0710669 − 0.171571i
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.b.423.1 4
17.2 even 8 578.2.d.c.179.1 4
17.3 odd 16 578.2.b.d.577.4 4
17.4 even 4 578.2.d.c.155.1 4
17.5 odd 16 578.2.a.i.1.1 4
17.6 odd 16 578.2.c.f.251.4 8
17.7 odd 16 578.2.c.f.327.4 8
17.8 even 8 inner 578.2.d.b.399.1 4
17.9 even 8 34.2.d.a.25.1 yes 4
17.10 odd 16 578.2.c.f.327.1 8
17.11 odd 16 578.2.c.f.251.1 8
17.12 odd 16 578.2.a.i.1.4 4
17.13 even 4 578.2.d.a.155.1 4
17.14 odd 16 578.2.b.d.577.1 4
17.15 even 8 578.2.d.a.179.1 4
17.16 even 2 34.2.d.a.15.1 ✓ 4
51.5 even 16 5202.2.a.bw.1.4 4
51.26 odd 8 306.2.l.c.127.1 4
51.29 even 16 5202.2.a.bw.1.1 4
51.50 odd 2 306.2.l.c.253.1 4
68.39 even 16 4624.2.a.bn.1.4 4
68.43 odd 8 272.2.v.b.161.1 4
68.63 even 16 4624.2.a.bn.1.1 4
68.67 odd 2 272.2.v.b.49.1 4
85.9 even 8 850.2.l.a.501.1 4
85.33 odd 4 850.2.o.b.49.1 4
85.43 odd 8 850.2.o.a.399.1 4
85.67 odd 4 850.2.o.a.49.1 4
85.77 odd 8 850.2.o.b.399.1 4
85.84 even 2 850.2.l.a.151.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.2.d.a.15.1 ✓ 4 17.16 even 2
34.2.d.a.25.1 yes 4 17.9 even 8
272.2.v.b.49.1 4 68.67 odd 2
272.2.v.b.161.1 4 68.43 odd 8
306.2.l.c.127.1 4 51.26 odd 8
306.2.l.c.253.1 4 51.50 odd 2
578.2.a.i.1.1 4 17.5 odd 16
578.2.a.i.1.4 4 17.12 odd 16
578.2.b.d.577.1 4 17.14 odd 16
578.2.b.d.577.4 4 17.3 odd 16
578.2.c.f.251.1 8 17.11 odd 16
578.2.c.f.251.4 8 17.6 odd 16
578.2.c.f.327.1 8 17.10 odd 16
578.2.c.f.327.4 8 17.7 odd 16
578.2.d.a.155.1 4 17.13 even 4
578.2.d.a.179.1 4 17.15 even 8
578.2.d.b.399.1 4 17.8 even 8 inner
578.2.d.b.423.1 4 1.1 even 1 trivial
578.2.d.c.155.1 4 17.4 even 4
578.2.d.c.179.1 4 17.2 even 8
850.2.l.a.151.1 4 85.84 even 2
850.2.l.a.501.1 4 85.9 even 8
850.2.o.a.49.1 4 85.67 odd 4
850.2.o.a.399.1 4 85.43 odd 8
850.2.o.b.49.1 4 85.33 odd 4
850.2.o.b.399.1 4 85.77 odd 8
4624.2.a.bn.1.1 4 68.63 even 16
4624.2.a.bn.1.4 4 68.39 even 16
5202.2.a.bw.1.1 4 51.29 even 16
5202.2.a.bw.1.4 4 51.5 even 16