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Properties

Label 578.2.d.e.423.1
Level $578$
Weight $2$
Character 578.423
Analytic conductor $4.615$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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 = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,8] 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 34)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 423.1
Root \(0.923880 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 578.423
Dual form 578.2.d.e.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)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{2} +(-0.765367 - 1.84776i) q^{3} +1.00000i q^{4} +(-0.765367 + 1.84776i) q^{6} +(-3.69552 - 1.53073i) q^{7} +(0.707107 - 0.707107i) q^{8} +(-0.707107 + 0.707107i) q^{9} +(-2.29610 + 5.54328i) q^{11} +(1.84776 - 0.765367i) q^{12} -2.00000i q^{13} +(1.53073 + 3.69552i) q^{14} -1.00000 q^{16} +1.00000 q^{18} +(2.82843 + 2.82843i) q^{19} +8.00000i q^{21} +(5.54328 - 2.29610i) q^{22} +(-1.84776 - 0.765367i) q^{24} +(-3.53553 + 3.53553i) q^{25} +(-1.41421 + 1.41421i) q^{26} +(-3.69552 - 1.53073i) q^{27} +(1.53073 - 3.69552i) q^{28} +(1.53073 + 3.69552i) q^{31} +(0.707107 + 0.707107i) q^{32} +12.0000 q^{33} +(-0.707107 - 0.707107i) q^{36} +(-1.53073 - 3.69552i) q^{37} -4.00000i q^{38} +(-3.69552 + 1.53073i) q^{39} +(5.54328 + 2.29610i) q^{41} +(5.65685 - 5.65685i) q^{42} +(-5.65685 + 5.65685i) q^{43} +(-5.54328 - 2.29610i) q^{44} +(0.765367 + 1.84776i) q^{48} +(6.36396 + 6.36396i) q^{49} +5.00000 q^{50} +2.00000 q^{52} +(4.24264 + 4.24264i) q^{53} +(1.53073 + 3.69552i) q^{54} +(-3.69552 + 1.53073i) q^{56} +(3.06147 - 7.39104i) q^{57} +(3.69552 + 1.53073i) q^{61} +(1.53073 - 3.69552i) q^{62} +(3.69552 - 1.53073i) q^{63} -1.00000i q^{64} +(-8.48528 - 8.48528i) q^{66} -8.00000 q^{67} +1.00000i q^{72} +(1.84776 - 0.765367i) q^{73} +(-1.53073 + 3.69552i) q^{74} +(9.23880 + 3.82683i) q^{75} +(-2.82843 + 2.82843i) q^{76} +(16.9706 - 16.9706i) q^{77} +(3.69552 + 1.53073i) q^{78} +(-3.06147 + 7.39104i) q^{79} +11.0000i q^{81} +(-2.29610 - 5.54328i) q^{82} -8.00000 q^{84} +8.00000 q^{86} +(2.29610 + 5.54328i) q^{88} -6.00000i q^{89} +(-3.06147 + 7.39104i) q^{91} +(5.65685 - 5.65685i) q^{93} +(0.765367 - 1.84776i) q^{96} +(-12.9343 + 5.35757i) q^{97} -9.00000i q^{98} +(-2.29610 - 5.54328i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} + 8 q^{18} + 96 q^{33} + 40 q^{50} + 16 q^{52} - 64 q^{67} - 64 q^{84} + 64 q^{86}+O(q^{100}) \) 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
\(3\) −0.765367 − 1.84776i −0.441885 − 1.06680i −0.975287 − 0.220942i \(-0.929087\pi\)
0.533402 − 0.845862i \(-0.320913\pi\)
\(4\) 1.00000i 0.500000i
\(5\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) −0.765367 + 1.84776i −0.312460 + 0.754344i
\(7\) −3.69552 − 1.53073i −1.39677 − 0.578563i −0.447862 − 0.894103i \(-0.647814\pi\)
−0.948912 + 0.315540i \(0.897814\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) −0.707107 + 0.707107i −0.235702 + 0.235702i
\(10\) 0 0
\(11\) −2.29610 + 5.54328i −0.692300 + 1.67136i 0.0477934 + 0.998857i \(0.484781\pi\)
−0.740094 + 0.672504i \(0.765219\pi\)
\(12\) 1.84776 − 0.765367i 0.533402 − 0.220942i
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 1.53073 + 3.69552i 0.409106 + 0.987669i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) 2.82843 + 2.82843i 0.648886 + 0.648886i 0.952724 − 0.303838i \(-0.0982682\pi\)
−0.303838 + 0.952724i \(0.598268\pi\)
\(20\) 0 0
\(21\) 8.00000i 1.74574i
\(22\) 5.54328 − 2.29610i 1.18183 − 0.489530i
\(23\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(24\) −1.84776 − 0.765367i −0.377172 − 0.156230i
\(25\) −3.53553 + 3.53553i −0.707107 + 0.707107i
\(26\) −1.41421 + 1.41421i −0.277350 + 0.277350i
\(27\) −3.69552 − 1.53073i −0.711203 − 0.294590i
\(28\) 1.53073 − 3.69552i 0.289281 − 0.698387i
\(29\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(30\) 0 0
\(31\) 1.53073 + 3.69552i 0.274928 + 0.663735i 0.999681 − 0.0252745i \(-0.00804598\pi\)
−0.724753 + 0.689009i \(0.758046\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) 12.0000 2.08893
\(34\) 0 0
\(35\) 0 0
\(36\) −0.707107 − 0.707107i −0.117851 − 0.117851i
\(37\) −1.53073 − 3.69552i −0.251651 − 0.607539i 0.746687 − 0.665176i \(-0.231644\pi\)
−0.998338 + 0.0576366i \(0.981644\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −3.69552 + 1.53073i −0.591756 + 0.245114i
\(40\) 0 0
\(41\) 5.54328 + 2.29610i 0.865714 + 0.358591i 0.770940 − 0.636908i \(-0.219787\pi\)
0.0947747 + 0.995499i \(0.469787\pi\)
\(42\) 5.65685 − 5.65685i 0.872872 − 0.872872i
\(43\) −5.65685 + 5.65685i −0.862662 + 0.862662i −0.991647 − 0.128984i \(-0.958828\pi\)
0.128984 + 0.991647i \(0.458828\pi\)
\(44\) −5.54328 − 2.29610i −0.835680 − 0.346150i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0.765367 + 1.84776i 0.110471 + 0.266701i
\(49\) 6.36396 + 6.36396i 0.909137 + 0.909137i
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 4.24264 + 4.24264i 0.582772 + 0.582772i 0.935664 − 0.352892i \(-0.114802\pi\)
−0.352892 + 0.935664i \(0.614802\pi\)
\(54\) 1.53073 + 3.69552i 0.208306 + 0.502896i
\(55\) 0 0
\(56\) −3.69552 + 1.53073i −0.493834 + 0.204553i
\(57\) 3.06147 − 7.39104i 0.405501 − 0.978967i
\(58\) 0 0
\(59\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) 3.69552 + 1.53073i 0.473163 + 0.195990i 0.606505 − 0.795080i \(-0.292571\pi\)
−0.133342 + 0.991070i \(0.542571\pi\)
\(62\) 1.53073 − 3.69552i 0.194403 − 0.469331i
\(63\) 3.69552 − 1.53073i 0.465592 − 0.192854i
\(64\) − 1.00000i − 0.125000i
\(65\) 0 0
\(66\) −8.48528 − 8.48528i −1.04447 − 1.04447i
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 1.84776 − 0.765367i 0.216264 − 0.0895794i −0.271921 − 0.962319i \(-0.587659\pi\)
0.488185 + 0.872740i \(0.337659\pi\)
\(74\) −1.53073 + 3.69552i −0.177944 + 0.429595i
\(75\) 9.23880 + 3.82683i 1.06680 + 0.441885i
\(76\) −2.82843 + 2.82843i −0.324443 + 0.324443i
\(77\) 16.9706 − 16.9706i 1.93398 − 1.93398i
\(78\) 3.69552 + 1.53073i 0.418435 + 0.173321i
\(79\) −3.06147 + 7.39104i −0.344442 + 0.831557i 0.652813 + 0.757519i \(0.273589\pi\)
−0.997255 + 0.0740378i \(0.976411\pi\)
\(80\) 0 0
\(81\) 11.0000i 1.22222i
\(82\) −2.29610 − 5.54328i −0.253562 − 0.612153i
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) −8.00000 −0.872872
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 2.29610 + 5.54328i 0.244765 + 0.590915i
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −3.06147 + 7.39104i −0.320929 + 0.774791i
\(92\) 0 0
\(93\) 5.65685 − 5.65685i 0.586588 − 0.586588i
\(94\) 0 0
\(95\) 0 0
\(96\) 0.765367 − 1.84776i 0.0781149 − 0.188586i
\(97\) −12.9343 + 5.35757i −1.31328 + 0.543979i −0.925840 − 0.377916i \(-0.876641\pi\)
−0.387441 + 0.921895i \(0.626641\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −2.29610 − 5.54328i −0.230767 − 0.557120i
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.e.423.1 8
17.2 even 8 inner 578.2.d.e.179.1 8
17.3 odd 16 578.2.b.a.577.1 2
17.4 even 4 inner 578.2.d.e.155.1 8
17.5 odd 16 578.2.a.a.1.1 1
17.6 odd 16 578.2.c.e.251.1 4
17.7 odd 16 578.2.c.e.327.1 4
17.8 even 8 inner 578.2.d.e.399.1 8
17.9 even 8 inner 578.2.d.e.399.2 8
17.10 odd 16 578.2.c.e.327.2 4
17.11 odd 16 578.2.c.e.251.2 4
17.12 odd 16 34.2.a.a.1.1 ✓ 1
17.13 even 4 inner 578.2.d.e.155.2 8
17.14 odd 16 578.2.b.a.577.2 2
17.15 even 8 inner 578.2.d.e.179.2 8
17.16 even 2 inner 578.2.d.e.423.2 8
51.5 even 16 5202.2.a.d.1.1 1
51.29 even 16 306.2.a.a.1.1 1
68.39 even 16 4624.2.a.a.1.1 1
68.63 even 16 272.2.a.d.1.1 1
85.12 even 16 850.2.c.b.749.2 2
85.29 odd 16 850.2.a.e.1.1 1
85.63 even 16 850.2.c.b.749.1 2
119.97 even 16 1666.2.a.m.1.1 1
136.29 odd 16 1088.2.a.l.1.1 1
136.131 even 16 1088.2.a.d.1.1 1
187.131 even 16 4114.2.a.a.1.1 1
204.131 odd 16 2448.2.a.k.1.1 1
221.12 odd 16 5746.2.a.b.1.1 1
255.29 even 16 7650.2.a.ci.1.1 1
340.199 even 16 6800.2.a.b.1.1 1
408.29 even 16 9792.2.a.y.1.1 1
408.131 odd 16 9792.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.2.a.a.1.1 ✓ 1 17.12 odd 16
272.2.a.d.1.1 1 68.63 even 16
306.2.a.a.1.1 1 51.29 even 16
578.2.a.a.1.1 1 17.5 odd 16
578.2.b.a.577.1 2 17.3 odd 16
578.2.b.a.577.2 2 17.14 odd 16
578.2.c.e.251.1 4 17.6 odd 16
578.2.c.e.251.2 4 17.11 odd 16
578.2.c.e.327.1 4 17.7 odd 16
578.2.c.e.327.2 4 17.10 odd 16
578.2.d.e.155.1 8 17.4 even 4 inner
578.2.d.e.155.2 8 17.13 even 4 inner
578.2.d.e.179.1 8 17.2 even 8 inner
578.2.d.e.179.2 8 17.15 even 8 inner
578.2.d.e.399.1 8 17.8 even 8 inner
578.2.d.e.399.2 8 17.9 even 8 inner
578.2.d.e.423.1 8 1.1 even 1 trivial
578.2.d.e.423.2 8 17.16 even 2 inner
850.2.a.e.1.1 1 85.29 odd 16
850.2.c.b.749.1 2 85.63 even 16
850.2.c.b.749.2 2 85.12 even 16
1088.2.a.d.1.1 1 136.131 even 16
1088.2.a.l.1.1 1 136.29 odd 16
1666.2.a.m.1.1 1 119.97 even 16
2448.2.a.k.1.1 1 204.131 odd 16
4114.2.a.a.1.1 1 187.131 even 16
4624.2.a.a.1.1 1 68.39 even 16
5202.2.a.d.1.1 1 51.5 even 16
5746.2.a.b.1.1 1 221.12 odd 16
6800.2.a.b.1.1 1 340.199 even 16
7650.2.a.ci.1.1 1 255.29 even 16
9792.2.a.y.1.1 1 408.29 even 16
9792.2.a.bj.1.1 1 408.131 odd 16