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Properties

Label 425.2.n.f.349.5
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 349.5
Character \(\chi\) \(=\) 425.349
Dual form 425.2.n.f.274.5

$q$-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 0.968441 − 0.249242i \(-0.0801815\pi\)
−0.249242 + 0.968441i \(0.580181\pi\)
\(3\) −0.0420595 + 0.101541i −0.0242831 + 0.0586245i −0.935556 − 0.353179i \(-0.885101\pi\)
0.911273 + 0.411803i \(0.135101\pi\)
\(4\) 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 −0.258008 − 0.966143i \(-0.583066\pi\)
0.500727 + 0.865605i \(0.333066\pi\)
\(8\) 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 \(-0.214202\pi\)
0.781996 + 0.623283i \(0.214202\pi\)
\(14\) 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 \(-0.964392\pi\)
0.623751 − 0.781623i \(-0.285608\pi\)
\(24\) 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 −0.927567 − 0.373657i \(-0.878104\pi\)
0.920104 + 0.391674i \(0.128104\pi\)
\(38\) 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 −0.740828 − 0.671695i \(-0.765566\pi\)
0.671695 + 0.740828i \(0.265566\pi\)
\(44\) −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 \(-0.617247\pi\)
−0.360070 + 0.932925i \(0.617247\pi\)
\(48\) −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 \(-0.942171\pi\)
−0.180678 − 0.983542i \(-0.557829\pi\)
\(54\) −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 \(0.0585467\pi\)
−0.983133 + 0.182894i \(0.941453\pi\)
\(68\) −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 0.659641 − 0.751581i \(-0.270708\pi\)
−0.0650115 + 0.997885i \(0.520708\pi\)
\(74\) −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 0.874971 − 0.484175i \(-0.160880\pi\)
−0.484175 + 0.874971i \(0.660880\pi\)
\(84\) −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 0.252394 − 0.967625i \(-0.418782\pi\)
−0.505745 + 0.862683i \(0.668782\pi\)
\(98\) −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.f.349.5 24
5.2 odd 4 425.2.m.b.26.2 24
5.3 odd 4 85.2.l.a.26.5 ✓ 24
5.4 even 2 425.2.n.c.349.2 24
15.8 even 4 765.2.be.b.451.2 24
17.2 even 8 425.2.n.c.274.2 24
85.2 odd 8 425.2.m.b.376.2 24
85.19 even 8 inner 425.2.n.f.274.5 24
85.23 even 16 1445.2.a.q.1.10 12
85.28 even 16 1445.2.a.p.1.10 12
85.53 odd 8 85.2.l.a.36.5 yes 24
85.57 even 16 7225.2.a.bq.1.3 12
85.58 even 16 1445.2.d.j.866.6 24
85.62 even 16 7225.2.a.bs.1.3 12
85.78 even 16 1445.2.d.j.866.5 24
255.53 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 5.3 odd 4
85.2.l.a.36.5 yes 24 85.53 odd 8
425.2.m.b.26.2 24 5.2 odd 4
425.2.m.b.376.2 24 85.2 odd 8
425.2.n.c.274.2 24 17.2 even 8
425.2.n.c.349.2 24 5.4 even 2
425.2.n.f.274.5 24 85.19 even 8 inner
425.2.n.f.349.5 24 1.1 even 1 trivial
765.2.be.b.451.2 24 15.8 even 4
765.2.be.b.631.2 24 255.53 even 8
1445.2.a.p.1.10 12 85.28 even 16
1445.2.a.q.1.10 12 85.23 even 16
1445.2.d.j.866.5 24 85.78 even 16
1445.2.d.j.866.6 24 85.58 even 16
7225.2.a.bq.1.3 12 85.57 even 16
7225.2.a.bs.1.3 12 85.62 even 16