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Properties

Label 425.2.n.c.349.2
Level $425$
Weight $2$
Character 425.349
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
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] = [425,2,Mod(49,425)] 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("425.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.n (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 349.2
Character \(\chi\) \(=\) 425.349
Dual form 425.2.n.c.274.2

$q$-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+(-1.01710 - 1.01710i) q^{2} +(0.0420595 - 0.101541i) q^{3} +0.0689897i q^{4} +(-0.146056 + 0.0604983i) q^{6} +(-0.642174 + 0.265997i) q^{7} +(-1.96403 + 1.96403i) q^{8} +(2.11278 + 2.11278i) q^{9} +(-4.48163 + 1.85635i) q^{11} +(0.00700526 + 0.00290167i) q^{12} -5.63906 q^{13} +(0.923703 + 0.382610i) q^{14} +4.13322 q^{16} +(-3.78674 - 1.63113i) q^{17} -4.29782i q^{18} +(1.64241 - 1.64241i) q^{19} +0.0763945i q^{21} +(6.44637 + 2.67018i) q^{22} +(1.77445 + 4.28390i) q^{23} +(0.116823 + 0.282035i) q^{24} +(5.73549 + 5.73549i) q^{26} +(0.608017 - 0.251849i) q^{27} +(-0.0183511 - 0.0443034i) q^{28} +(-2.48981 + 6.01093i) q^{29} +(-6.12711 - 2.53793i) q^{31} +(-0.275837 - 0.275837i) q^{32} +0.533145i q^{33} +(2.19248 + 5.51052i) q^{34} +(-0.145760 + 0.145760i) q^{36} +(0.0453958 - 0.109595i) q^{37} -3.34100 q^{38} +(-0.237176 + 0.572593i) q^{39} +(-0.412826 - 0.996650i) q^{41} +(0.0777010 - 0.0777010i) q^{42} +(0.453332 - 0.453332i) q^{43} +(-0.128069 - 0.309187i) q^{44} +(2.55237 - 6.16195i) q^{46} +4.93703 q^{47} +(0.173841 - 0.419690i) q^{48} +(-4.60811 + 4.60811i) q^{49} +(-0.324894 + 0.315904i) q^{51} -0.389037i q^{52} +(8.47565 + 8.47565i) q^{53} +(-0.874571 - 0.362259i) q^{54} +(0.738824 - 1.78368i) q^{56} +(-0.0976925 - 0.235850i) q^{57} +(8.64611 - 3.58134i) q^{58} +(-7.01329 - 7.01329i) q^{59} +(0.613413 + 1.48091i) q^{61} +(3.65056 + 8.81322i) q^{62} +(-1.91877 - 0.794779i) q^{63} -7.70533i q^{64} +(0.542263 - 0.542263i) q^{66} -2.99411i q^{67} +(0.112531 - 0.261246i) q^{68} +0.509622 q^{69} +(-4.33163 - 1.79422i) q^{71} -8.29913 q^{72} +(-5.08052 - 2.10442i) q^{73} +(-0.157641 + 0.0652972i) q^{74} +(0.113309 + 0.113309i) q^{76} +(2.38421 - 2.38421i) q^{77} +(0.823617 - 0.341153i) q^{78} +(13.7140 - 5.68053i) q^{79} +8.89143i q^{81} +(-0.593808 + 1.43358i) q^{82} +(-3.56033 - 3.56033i) q^{83} -0.00527044 q^{84} -0.922169 q^{86} +(0.505633 + 0.505633i) q^{87} +(5.15614 - 12.4480i) q^{88} -2.35657i q^{89} +(3.62126 - 1.49997i) q^{91} +(-0.295545 + 0.122419i) q^{92} +(-0.515406 + 0.515406i) q^{93} +(-5.02145 - 5.02145i) q^{94} +(-0.0396102 + 0.0164071i) q^{96} +(2.49522 + 1.03355i) q^{97} +9.37384 q^{98} +(-13.3908 - 5.54664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{3} - 8 q^{6} + 24 q^{9} - 8 q^{11} + 40 q^{12} + 16 q^{13} - 24 q^{16} + 8 q^{19} - 24 q^{22} + 8 q^{23} + 8 q^{24} + 16 q^{26} + 16 q^{27} - 40 q^{28} + 8 q^{29} - 16 q^{34} - 24 q^{36} - 16 q^{37}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.01710 − 1.01710i −0.719199 − 0.719199i 0.249242 − 0.968441i \(-0.419819\pi\)
−0.968441 + 0.249242i \(0.919819\pi\)
\(3\) 0.0420595 − 0.101541i 0.0242831 − 0.0586245i −0.911273 − 0.411803i \(-0.864899\pi\)
0.935556 + 0.353179i \(0.114899\pi\)
\(4\) 0.0689897i 0.0344949i
\(5\) 0 0
\(6\) −0.146056 + 0.0604983i −0.0596271 + 0.0246983i
\(7\) −0.642174 + 0.265997i −0.242719 + 0.100538i −0.500727 − 0.865605i \(-0.666934\pi\)
0.258008 + 0.966143i \(0.416934\pi\)
\(8\) −1.96403 + 1.96403i −0.694390 + 0.694390i
\(9\) 2.11278 + 2.11278i 0.704260 + 0.704260i
\(10\) 0 0
\(11\) −4.48163 + 1.85635i −1.35126 + 0.559712i −0.936644 − 0.350283i \(-0.886085\pi\)
−0.414620 + 0.909995i \(0.636085\pi\)
\(12\) 0.00700526 + 0.00290167i 0.00202224 + 0.000837641i
\(13\) −5.63906 −1.56399 −0.781996 − 0.623283i \(-0.785798\pi\)
−0.781996 + 0.623283i \(0.785798\pi\)
\(14\) 0.923703 + 0.382610i 0.246870 + 0.102257i
\(15\) 0 0
\(16\) 4.13322 1.03330
\(17\) −3.78674 − 1.63113i −0.918420 − 0.395606i
\(18\) − 4.29782i − 1.01301i
\(19\) 1.64241 − 1.64241i 0.376795 − 0.376795i −0.493150 − 0.869945i \(-0.664154\pi\)
0.869945 + 0.493150i \(0.164154\pi\)
\(20\) 0 0
\(21\) 0.0763945i 0.0166706i
\(22\) 6.44637 + 2.67018i 1.37437 + 0.569283i
\(23\) 1.77445 + 4.28390i 0.369998 + 0.893255i 0.993750 + 0.111632i \(0.0356078\pi\)
−0.623751 + 0.781623i \(0.714392\pi\)
\(24\) 0.116823 + 0.282035i 0.0238464 + 0.0575702i
\(25\) 0 0
\(26\) 5.73549 + 5.73549i 1.12482 + 1.12482i
\(27\) 0.608017 − 0.251849i 0.117013 − 0.0484684i
\(28\) −0.0183511 − 0.0443034i −0.00346803 − 0.00837256i
\(29\) −2.48981 + 6.01093i −0.462346 + 1.11620i 0.505086 + 0.863069i \(0.331461\pi\)
−0.967432 + 0.253132i \(0.918539\pi\)
\(30\) 0 0
\(31\) −6.12711 − 2.53793i −1.10046 − 0.455826i −0.242819 − 0.970072i \(-0.578072\pi\)
−0.857643 + 0.514246i \(0.828072\pi\)
\(32\) −0.275837 − 0.275837i −0.0487616 − 0.0487616i
\(33\) 0.533145i 0.0928087i
\(34\) 2.19248 + 5.51052i 0.376008 + 0.945047i
\(35\) 0 0
\(36\) −0.145760 + 0.145760i −0.0242933 + 0.0242933i
\(37\) 0.0453958 − 0.109595i 0.00746302 − 0.0180173i −0.920104 − 0.391674i \(-0.871896\pi\)
0.927567 + 0.373657i \(0.121896\pi\)
\(38\) −3.34100 −0.541981
\(39\) −0.237176 + 0.572593i −0.0379785 + 0.0916883i
\(40\) 0 0
\(41\) −0.412826 − 0.996650i −0.0644726 − 0.155651i 0.888359 − 0.459149i \(-0.151845\pi\)
−0.952832 + 0.303498i \(0.901845\pi\)
\(42\) 0.0777010 − 0.0777010i 0.0119895 − 0.0119895i
\(43\) 0.453332 − 0.453332i 0.0691325 − 0.0691325i −0.671695 − 0.740828i \(-0.734434\pi\)
0.740828 + 0.671695i \(0.234434\pi\)
\(44\) −0.128069 − 0.309187i −0.0193072 − 0.0466116i
\(45\) 0 0
\(46\) 2.55237 − 6.16195i 0.376326 − 0.908531i
\(47\) 4.93703 0.720139 0.360070 − 0.932925i \(-0.382753\pi\)
0.360070 + 0.932925i \(0.382753\pi\)
\(48\) 0.173841 − 0.419690i 0.0250918 − 0.0605770i
\(49\) −4.60811 + 4.60811i −0.658302 + 0.658302i
\(50\) 0 0
\(51\) −0.324894 + 0.315904i −0.0454943 + 0.0442354i
\(52\) − 0.389037i − 0.0539497i
\(53\) 8.47565 + 8.47565i 1.16422 + 1.16422i 0.983542 + 0.180678i \(0.0578292\pi\)
0.180678 + 0.983542i \(0.442171\pi\)
\(54\) −0.874571 − 0.362259i −0.119014 − 0.0492972i
\(55\) 0 0
\(56\) 0.738824 − 1.78368i 0.0987295 − 0.238354i
\(57\) −0.0976925 − 0.235850i −0.0129397 − 0.0312392i
\(58\) 8.64611 − 3.58134i 1.13529 − 0.470252i
\(59\) −7.01329 − 7.01329i −0.913053 − 0.913053i 0.0834587 − 0.996511i \(-0.473403\pi\)
−0.996511 + 0.0834587i \(0.973403\pi\)
\(60\) 0 0
\(61\) 0.613413 + 1.48091i 0.0785394 + 0.189611i 0.958272 − 0.285858i \(-0.0922785\pi\)
−0.879733 + 0.475469i \(0.842278\pi\)
\(62\) 3.65056 + 8.81322i 0.463621 + 1.11928i
\(63\) −1.91877 − 0.794779i −0.241742 − 0.100133i
\(64\) − 7.70533i − 0.963166i
\(65\) 0 0
\(66\) 0.542263 − 0.542263i 0.0667479 − 0.0667479i
\(67\) − 2.99411i − 0.365789i −0.983133 − 0.182894i \(-0.941453\pi\)
0.983133 − 0.182894i \(-0.0585467\pi\)
\(68\) 0.112531 − 0.261246i 0.0136464 − 0.0316808i
\(69\) 0.509622 0.0613513
\(70\) 0 0
\(71\) −4.33163 − 1.79422i −0.514070 − 0.212935i 0.110540 − 0.993872i \(-0.464742\pi\)
−0.624610 + 0.780937i \(0.714742\pi\)
\(72\) −8.29913 −0.978062
\(73\) −5.08052 − 2.10442i −0.594629 − 0.246304i 0.0650115 − 0.997885i \(-0.479292\pi\)
−0.659641 + 0.751581i \(0.729292\pi\)
\(74\) −0.157641 + 0.0652972i −0.0183254 + 0.00759065i
\(75\) 0 0
\(76\) 0.113309 + 0.113309i 0.0129975 + 0.0129975i
\(77\) 2.38421 − 2.38421i 0.271705 − 0.271705i
\(78\) 0.823617 − 0.341153i 0.0932563 − 0.0386280i
\(79\) 13.7140 − 5.68053i 1.54295 − 0.639110i 0.560924 − 0.827867i \(-0.310446\pi\)
0.982024 + 0.188757i \(0.0604460\pi\)
\(80\) 0 0
\(81\) 8.89143i 0.987937i
\(82\) −0.593808 + 1.43358i −0.0655752 + 0.158312i
\(83\) −3.56033 − 3.56033i −0.390797 − 0.390797i 0.484175 − 0.874971i \(-0.339120\pi\)
−0.874971 + 0.484175i \(0.839120\pi\)
\(84\) −0.00527044 −0.000575052
\(85\) 0 0
\(86\) −0.922169 −0.0994400
\(87\) 0.505633 + 0.505633i 0.0542096 + 0.0542096i
\(88\) 5.15614 − 12.4480i 0.549646 − 1.32696i
\(89\) − 2.35657i − 0.249796i −0.992170 − 0.124898i \(-0.960140\pi\)
0.992170 − 0.124898i \(-0.0398604\pi\)
\(90\) 0 0
\(91\) 3.62126 − 1.49997i 0.379611 − 0.157240i
\(92\) −0.295545 + 0.122419i −0.0308127 + 0.0127630i
\(93\) −0.515406 + 0.515406i −0.0534452 + 0.0534452i
\(94\) −5.02145 − 5.02145i −0.517924 − 0.517924i
\(95\) 0 0
\(96\) −0.0396102 + 0.0164071i −0.00404270 + 0.00167454i
\(97\) 2.49522 + 1.03355i 0.253351 + 0.104941i 0.505745 − 0.862683i \(-0.331218\pi\)
−0.252394 + 0.967625i \(0.581218\pi\)
\(98\) 9.37384 0.946900
\(99\) −13.3908 − 5.54664i −1.34582 − 0.557458i
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 425.2.n.c.349.2 24
5.2 odd 4 85.2.l.a.26.5 ✓ 24
5.3 odd 4 425.2.m.b.26.2 24
5.4 even 2 425.2.n.f.349.5 24
15.2 even 4 765.2.be.b.451.2 24
17.2 even 8 425.2.n.f.274.5 24
85.2 odd 8 85.2.l.a.36.5 yes 24
85.7 even 16 1445.2.d.j.866.6 24
85.19 even 8 inner 425.2.n.c.274.2 24
85.23 even 16 7225.2.a.bq.1.3 12
85.27 even 16 1445.2.d.j.866.5 24
85.28 even 16 7225.2.a.bs.1.3 12
85.53 odd 8 425.2.m.b.376.2 24
85.57 even 16 1445.2.a.q.1.10 12
85.62 even 16 1445.2.a.p.1.10 12
255.2 even 8 765.2.be.b.631.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.5 ✓ 24 5.2 odd 4
85.2.l.a.36.5 yes 24 85.2 odd 8
425.2.m.b.26.2 24 5.3 odd 4
425.2.m.b.376.2 24 85.53 odd 8
425.2.n.c.274.2 24 85.19 even 8 inner
425.2.n.c.349.2 24 1.1 even 1 trivial
425.2.n.f.274.5 24 17.2 even 8
425.2.n.f.349.5 24 5.4 even 2
765.2.be.b.451.2 24 15.2 even 4
765.2.be.b.631.2 24 255.2 even 8
1445.2.a.p.1.10 12 85.62 even 16
1445.2.a.q.1.10 12 85.57 even 16
1445.2.d.j.866.5 24 85.27 even 16
1445.2.d.j.866.6 24 85.7 even 16
7225.2.a.bq.1.3 12 85.23 even 16
7225.2.a.bs.1.3 12 85.28 even 16