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Properties

Label 578.2.c.g.251.6
Level $578$
Weight $2$
Character 578.251
Analytic conductor $4.615$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(251,578)] 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("578.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.c (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.722204136308736.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{8} + 69x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 251.6
Root \(1.08335 + 1.08335i\) of defining polynomial
Character \(\chi\) \(=\) 578.251
Dual form 578.2.c.g.327.6

$q$-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+1.00000i q^{2} +(2.28161 + 2.28161i) q^{3} -1.00000 q^{4} +(-0.837775 - 0.837775i) q^{5} +(-2.28161 + 2.28161i) q^{6} +(-0.792394 + 0.792394i) q^{7} -1.00000i q^{8} +7.41147i q^{9} +(0.837775 - 0.837775i) q^{10} +(-2.41228 + 2.41228i) q^{11} +(-2.28161 - 2.28161i) q^{12} -0.347296 q^{13} +(-0.792394 - 0.792394i) q^{14} -3.82295i q^{15} +1.00000 q^{16} -7.41147 q^{18} -0.347296i q^{19} +(0.837775 + 0.837775i) q^{20} -3.61587 q^{21} +(-2.41228 - 2.41228i) q^{22} +(-0.290956 + 0.290956i) q^{23} +(2.28161 - 2.28161i) q^{24} -3.59627i q^{25} -0.347296i q^{26} +(-10.0653 + 10.0653i) q^{27} +(0.792394 - 0.792394i) q^{28} +(6.20915 + 6.20915i) q^{29} +3.82295 q^{30} +(6.16377 + 6.16377i) q^{31} +1.00000i q^{32} -11.0077 q^{33} +1.32770 q^{35} -7.41147i q^{36} +(-0.336337 - 0.336337i) q^{37} +0.347296 q^{38} +(-0.792394 - 0.792394i) q^{39} +(-0.837775 + 0.837775i) q^{40} +(-1.86546 + 1.86546i) q^{41} -3.61587i q^{42} -9.33275i q^{43} +(2.41228 - 2.41228i) q^{44} +(6.20915 - 6.20915i) q^{45} +(-0.290956 - 0.290956i) q^{46} +7.86484 q^{47} +(2.28161 + 2.28161i) q^{48} +5.74422i q^{49} +3.59627 q^{50} +0.347296 q^{52} -8.41921i q^{53} +(-10.0653 - 10.0653i) q^{54} +4.04189 q^{55} +(0.792394 + 0.792394i) q^{56} +(0.792394 - 0.792394i) q^{57} +(-6.20915 + 6.20915i) q^{58} -6.41147i q^{59} +3.82295i q^{60} +(4.03216 - 4.03216i) q^{61} +(-6.16377 + 6.16377i) q^{62} +(-5.87281 - 5.87281i) q^{63} -1.00000 q^{64} +(0.290956 + 0.290956i) q^{65} -11.0077i q^{66} +7.31315 q^{67} -1.32770 q^{69} +1.32770i q^{70} +(5.37137 + 5.37137i) q^{71} +7.41147 q^{72} +(-6.39358 - 6.39358i) q^{73} +(0.336337 - 0.336337i) q^{74} +(8.20527 - 8.20527i) q^{75} +0.347296i q^{76} -3.82295i q^{77} +(0.792394 - 0.792394i) q^{78} +(9.38777 - 9.38777i) q^{79} +(-0.837775 - 0.837775i) q^{80} -23.6955 q^{81} +(-1.86546 - 1.86546i) q^{82} +7.73917i q^{83} +3.61587 q^{84} +9.33275 q^{86} +28.3337i q^{87} +(2.41228 + 2.41228i) q^{88} -7.18479 q^{89} +(6.20915 + 6.20915i) q^{90} +(0.275196 - 0.275196i) q^{91} +(0.290956 - 0.290956i) q^{92} +28.1266i q^{93} +7.86484i q^{94} +(-0.290956 + 0.290956i) q^{95} +(-2.28161 + 2.28161i) q^{96} +(-0.156715 - 0.156715i) q^{97} -5.74422 q^{98} +(-17.8785 - 17.8785i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} + 12 q^{16} - 48 q^{18} - 36 q^{30} - 36 q^{33} - 12 q^{50} + 36 q^{55} - 12 q^{64} + 48 q^{72} - 12 q^{81} + 36 q^{86} - 72 q^{89} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 2.28161 + 2.28161i 1.31729 + 1.31729i 0.915915 + 0.401372i \(0.131467\pi\)
0.401372 + 0.915915i \(0.368533\pi\)
\(4\) −1.00000 −0.500000
\(5\) −0.837775 − 0.837775i −0.374664 − 0.374664i 0.494508 − 0.869173i \(-0.335348\pi\)
−0.869173 + 0.494508i \(0.835348\pi\)
\(6\) −2.28161 + 2.28161i −0.931463 + 0.931463i
\(7\) −0.792394 + 0.792394i −0.299497 + 0.299497i −0.840817 − 0.541320i \(-0.817925\pi\)
0.541320 + 0.840817i \(0.317925\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 7.41147i 2.47049i
\(10\) 0.837775 − 0.837775i 0.264928 − 0.264928i
\(11\) −2.41228 + 2.41228i −0.727329 + 0.727329i −0.970087 − 0.242758i \(-0.921948\pi\)
0.242758 + 0.970087i \(0.421948\pi\)
\(12\) −2.28161 − 2.28161i −0.658644 − 0.658644i
\(13\) −0.347296 −0.0963227 −0.0481613 − 0.998840i \(-0.515336\pi\)
−0.0481613 + 0.998840i \(0.515336\pi\)
\(14\) −0.792394 − 0.792394i −0.211776 − 0.211776i
\(15\) − 3.82295i − 0.987081i
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) −7.41147 −1.74690
\(19\) − 0.347296i − 0.0796752i −0.999206 − 0.0398376i \(-0.987316\pi\)
0.999206 − 0.0398376i \(-0.0126841\pi\)
\(20\) 0.837775 + 0.837775i 0.187332 + 0.187332i
\(21\) −3.61587 −0.789047
\(22\) −2.41228 − 2.41228i −0.514299 − 0.514299i
\(23\) −0.290956 + 0.290956i −0.0606686 + 0.0606686i −0.736790 − 0.676122i \(-0.763659\pi\)
0.676122 + 0.736790i \(0.263659\pi\)
\(24\) 2.28161 − 2.28161i 0.465731 − 0.465731i
\(25\) − 3.59627i − 0.719253i
\(26\) − 0.347296i − 0.0681104i
\(27\) −10.0653 + 10.0653i −1.93706 + 1.93706i
\(28\) 0.792394 − 0.792394i 0.149748 − 0.149748i
\(29\) 6.20915 + 6.20915i 1.15301 + 1.15301i 0.985946 + 0.167063i \(0.0534285\pi\)
0.167063 + 0.985946i \(0.446572\pi\)
\(30\) 3.82295 0.697972
\(31\) 6.16377 + 6.16377i 1.10705 + 1.10705i 0.993537 + 0.113508i \(0.0362087\pi\)
0.113508 + 0.993537i \(0.463791\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −11.0077 −1.91620
\(34\) 0 0
\(35\) 1.32770 0.224422
\(36\) − 7.41147i − 1.23525i
\(37\) −0.336337 − 0.336337i −0.0552934 − 0.0552934i 0.678919 − 0.734213i \(-0.262449\pi\)
−0.734213 + 0.678919i \(0.762449\pi\)
\(38\) 0.347296 0.0563389
\(39\) −0.792394 − 0.792394i −0.126885 − 0.126885i
\(40\) −0.837775 + 0.837775i −0.132464 + 0.132464i
\(41\) −1.86546 + 1.86546i −0.291336 + 0.291336i −0.837608 − 0.546272i \(-0.816046\pi\)
0.546272 + 0.837608i \(0.316046\pi\)
\(42\) − 3.61587i − 0.557940i
\(43\) − 9.33275i − 1.42323i −0.702569 − 0.711615i \(-0.747964\pi\)
0.702569 − 0.711615i \(-0.252036\pi\)
\(44\) 2.41228 − 2.41228i 0.363664 − 0.363664i
\(45\) 6.20915 − 6.20915i 0.925605 − 0.925605i
\(46\) −0.290956 − 0.290956i −0.0428991 − 0.0428991i
\(47\) 7.86484 1.14720 0.573602 − 0.819134i \(-0.305546\pi\)
0.573602 + 0.819134i \(0.305546\pi\)
\(48\) 2.28161 + 2.28161i 0.329322 + 0.329322i
\(49\) 5.74422i 0.820603i
\(50\) 3.59627 0.508589
\(51\) 0 0
\(52\) 0.347296 0.0481613
\(53\) − 8.41921i − 1.15647i −0.815871 − 0.578234i \(-0.803742\pi\)
0.815871 − 0.578234i \(-0.196258\pi\)
\(54\) −10.0653 − 10.0653i −1.36971 − 1.36971i
\(55\) 4.04189 0.545008
\(56\) 0.792394 + 0.792394i 0.105888 + 0.105888i
\(57\) 0.792394 − 0.792394i 0.104955 − 0.104955i
\(58\) −6.20915 + 6.20915i −0.815301 + 0.815301i
\(59\) − 6.41147i − 0.834703i −0.908745 − 0.417351i \(-0.862958\pi\)
0.908745 − 0.417351i \(-0.137042\pi\)
\(60\) 3.82295i 0.493540i
\(61\) 4.03216 − 4.03216i 0.516265 − 0.516265i −0.400174 − 0.916439i \(-0.631050\pi\)
0.916439 + 0.400174i \(0.131050\pi\)
\(62\) −6.16377 + 6.16377i −0.782799 + 0.782799i
\(63\) −5.87281 − 5.87281i −0.739904 − 0.739904i
\(64\) −1.00000 −0.125000
\(65\) 0.290956 + 0.290956i 0.0360887 + 0.0360887i
\(66\) − 11.0077i − 1.35496i
\(67\) 7.31315 0.893443 0.446722 − 0.894673i \(-0.352591\pi\)
0.446722 + 0.894673i \(0.352591\pi\)
\(68\) 0 0
\(69\) −1.32770 −0.159836
\(70\) 1.32770i 0.158690i
\(71\) 5.37137 + 5.37137i 0.637465 + 0.637465i 0.949929 − 0.312465i \(-0.101155\pi\)
−0.312465 + 0.949929i \(0.601155\pi\)
\(72\) 7.41147 0.873451
\(73\) −6.39358 − 6.39358i −0.748312 − 0.748312i 0.225850 − 0.974162i \(-0.427484\pi\)
−0.974162 + 0.225850i \(0.927484\pi\)
\(74\) 0.336337 − 0.336337i 0.0390983 − 0.0390983i
\(75\) 8.20527 − 8.20527i 0.947463 − 0.947463i
\(76\) 0.347296i 0.0398376i
\(77\) − 3.82295i − 0.435665i
\(78\) 0.792394 − 0.792394i 0.0897210 − 0.0897210i
\(79\) 9.38777 − 9.38777i 1.05621 − 1.05621i 0.0578833 − 0.998323i \(-0.481565\pi\)
0.998323 − 0.0578833i \(-0.0184351\pi\)
\(80\) −0.837775 − 0.837775i −0.0936661 − 0.0936661i
\(81\) −23.6955 −2.63284
\(82\) −1.86546 − 1.86546i −0.206005 − 0.206005i
\(83\) 7.73917i 0.849484i 0.905314 + 0.424742i \(0.139635\pi\)
−0.905314 + 0.424742i \(0.860365\pi\)
\(84\) 3.61587 0.394523
\(85\) 0 0
\(86\) 9.33275 1.00638
\(87\) 28.3337i 3.03769i
\(88\) 2.41228 + 2.41228i 0.257150 + 0.257150i
\(89\) −7.18479 −0.761586 −0.380793 − 0.924660i \(-0.624349\pi\)
−0.380793 + 0.924660i \(0.624349\pi\)
\(90\) 6.20915 + 6.20915i 0.654502 + 0.654502i
\(91\) 0.275196 − 0.275196i 0.0288483 − 0.0288483i
\(92\) 0.290956 − 0.290956i 0.0303343 − 0.0303343i
\(93\) 28.1266i 2.91659i
\(94\) 7.86484i 0.811196i
\(95\) −0.290956 + 0.290956i −0.0298515 + 0.0298515i
\(96\) −2.28161 + 2.28161i −0.232866 + 0.232866i
\(97\) −0.156715 − 0.156715i −0.0159120 − 0.0159120i 0.699106 − 0.715018i \(-0.253582\pi\)
−0.715018 + 0.699106i \(0.753582\pi\)
\(98\) −5.74422 −0.580254
\(99\) −17.8785 − 17.8785i −1.79686 − 1.79686i
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.c.g.251.6 12
17.2 even 8 578.2.a.f.1.3 yes 3
17.3 odd 16 578.2.d.h.423.6 24
17.4 even 4 inner 578.2.c.g.327.6 12
17.5 odd 16 578.2.d.h.155.1 24
17.6 odd 16 578.2.d.h.179.6 24
17.7 odd 16 578.2.d.h.399.6 24
17.8 even 8 578.2.b.f.577.1 6
17.9 even 8 578.2.b.f.577.6 6
17.10 odd 16 578.2.d.h.399.1 24
17.11 odd 16 578.2.d.h.179.1 24
17.12 odd 16 578.2.d.h.155.6 24
17.13 even 4 inner 578.2.c.g.327.1 12
17.14 odd 16 578.2.d.h.423.1 24
17.15 even 8 578.2.a.e.1.1 ✓ 3
17.16 even 2 inner 578.2.c.g.251.1 12
51.2 odd 8 5202.2.a.bo.1.2 3
51.32 odd 8 5202.2.a.bn.1.2 3
68.15 odd 8 4624.2.a.bj.1.3 3
68.19 odd 8 4624.2.a.ba.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.2.a.e.1.1 ✓ 3 17.15 even 8
578.2.a.f.1.3 yes 3 17.2 even 8
578.2.b.f.577.1 6 17.8 even 8
578.2.b.f.577.6 6 17.9 even 8
578.2.c.g.251.1 12 17.16 even 2 inner
578.2.c.g.251.6 12 1.1 even 1 trivial
578.2.c.g.327.1 12 17.13 even 4 inner
578.2.c.g.327.6 12 17.4 even 4 inner
578.2.d.h.155.1 24 17.5 odd 16
578.2.d.h.155.6 24 17.12 odd 16
578.2.d.h.179.1 24 17.11 odd 16
578.2.d.h.179.6 24 17.6 odd 16
578.2.d.h.399.1 24 17.10 odd 16
578.2.d.h.399.6 24 17.7 odd 16
578.2.d.h.423.1 24 17.14 odd 16
578.2.d.h.423.6 24 17.3 odd 16
4624.2.a.ba.1.1 3 68.19 odd 8
4624.2.a.bj.1.3 3 68.15 odd 8
5202.2.a.bn.1.2 3 51.32 odd 8
5202.2.a.bo.1.2 3 51.2 odd 8