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Properties

Label 425.2.m.b.26.2
Level $425$
Weight $2$
Character 425.26
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(26,425)] 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("425.26"); 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([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,-8,0,0,-24,0,-8,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == 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 26.2
Character \(\chi\) \(=\) 425.26
Dual form 425.2.m.b.376.2

$q$-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} +(0.101541 + 0.0420595i) q^{3} -0.0689897i q^{4} +(-0.146056 + 0.0604983i) q^{6} +(0.265997 + 0.642174i) q^{7} +(-1.96403 - 1.96403i) q^{8} +(-2.11278 - 2.11278i) q^{9} +(-4.48163 + 1.85635i) q^{11} +(0.00290167 - 0.00700526i) q^{12} -5.63906i q^{13} +(-0.923703 - 0.382610i) q^{14} +4.13322 q^{16} +(-1.63113 + 3.78674i) q^{17} +4.29782 q^{18} +(-1.64241 + 1.64241i) q^{19} +0.0763945i q^{21} +(2.67018 - 6.44637i) q^{22} +(-4.28390 + 1.77445i) q^{23} +(-0.116823 - 0.282035i) q^{24} +(5.73549 + 5.73549i) q^{26} +(-0.251849 - 0.608017i) q^{27} +(0.0443034 - 0.0183511i) q^{28} +(2.48981 - 6.01093i) q^{29} +(-6.12711 - 2.53793i) q^{31} +(-0.275837 + 0.275837i) q^{32} -0.533145 q^{33} +(-2.19248 - 5.51052i) q^{34} +(-0.145760 + 0.145760i) q^{36} +(-0.109595 - 0.0453958i) q^{37} -3.34100i 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} -4.93703i q^{47} +(0.419690 + 0.173841i) q^{48} +(4.60811 - 4.60811i) q^{49} +(-0.324894 + 0.315904i) q^{51} -0.389037 q^{52} +(-8.47565 + 8.47565i) q^{53} +(0.874571 + 0.362259i) q^{54} +(0.738824 - 1.78368i) q^{56} +(-0.235850 + 0.0976925i) q^{57} +(3.58134 + 8.64611i) q^{58} +(7.01329 + 7.01329i) q^{59} +(0.613413 + 1.48091i) q^{61} +(8.81322 - 3.65056i) q^{62} +(0.794779 - 1.91877i) q^{63} +7.70533i q^{64} +(0.542263 - 0.542263i) q^{66} -2.99411 q^{67} +(0.261246 + 0.112531i) q^{68} -0.509622 q^{69} +(-4.33163 - 1.79422i) q^{71} +8.29913i q^{72} +(2.10442 - 5.08052i) q^{73} +(0.157641 - 0.0652972i) q^{74} +(0.113309 + 0.113309i) q^{76} +(-2.38421 - 2.38421i) q^{77} +(0.341153 + 0.823617i) q^{78} +(-13.7140 + 5.68053i) q^{79} +8.89143i q^{81} +(1.43358 + 0.593808i) q^{82} +(3.56033 - 3.56033i) q^{83} +0.00527044 q^{84} -0.922169 q^{86} +(0.505633 - 0.505633i) q^{87} +(12.4480 + 5.15614i) q^{88} +2.35657i q^{89} +(3.62126 - 1.49997i) q^{91} +(0.122419 + 0.295545i) q^{92} +(-0.515406 - 0.515406i) q^{93} +(5.02145 + 5.02145i) q^{94} +(-0.0396102 + 0.0164071i) q^{96} +(1.03355 - 2.49522i) q^{97} +9.37384i q^{98} +(13.3908 + 5.54664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} - 24 q^{12} - 24 q^{16} + 8 q^{17} - 8 q^{18} - 8 q^{19} + 32 q^{22} + 16 q^{23} - 8 q^{24} + 16 q^{26} - 24 q^{27} - 48 q^{28} - 8 q^{29} + 16 q^{34} - 24 q^{36} - 24 q^{37}+ \cdots - 8 q^{99}+O(q^{100}) \) 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 \(-0.919819\pi\)
0.249242 + 0.968441i \(0.419819\pi\)
\(3\) 0.101541 + 0.0420595i 0.0586245 + 0.0242831i 0.411803 − 0.911273i \(-0.364899\pi\)
−0.353179 + 0.935556i \(0.614899\pi\)
\(4\) − 0.0689897i − 0.0344949i
\(5\) 0 0
\(6\) −0.146056 + 0.0604983i −0.0596271 + 0.0246983i
\(7\) 0.265997 + 0.642174i 0.100538 + 0.242719i 0.966143 − 0.258008i \(-0.0830661\pi\)
−0.865605 + 0.500727i \(0.833066\pi\)
\(8\) −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 \(0.636085\pi\)
\(12\) 0.00290167 − 0.00700526i 0.000837641 − 0.00202224i
\(13\) − 5.63906i − 1.56399i −0.623283 − 0.781996i \(-0.714202\pi\)
0.623283 − 0.781996i \(-0.285798\pi\)
\(14\) −0.923703 − 0.382610i −0.246870 − 0.102257i
\(15\) 0 0
\(16\) 4.13322 1.03330
\(17\) −1.63113 + 3.78674i −0.395606 + 0.918420i
\(18\) 4.29782 1.01301
\(19\) −1.64241 + 1.64241i −0.376795 + 0.376795i −0.869945 − 0.493150i \(-0.835846\pi\)
0.493150 + 0.869945i \(0.335846\pi\)
\(20\) 0 0
\(21\) 0.0763945i 0.0166706i
\(22\) 2.67018 − 6.44637i 0.569283 − 1.37437i
\(23\) −4.28390 + 1.77445i −0.893255 + 0.369998i −0.781623 − 0.623751i \(-0.785608\pi\)
−0.111632 + 0.993750i \(0.535608\pi\)
\(24\) −0.116823 − 0.282035i −0.0238464 − 0.0575702i
\(25\) 0 0
\(26\) 5.73549 + 5.73549i 1.12482 + 1.12482i
\(27\) −0.251849 − 0.608017i −0.0484684 − 0.117013i
\(28\) 0.0443034 − 0.0183511i 0.00837256 − 0.00346803i
\(29\) 2.48981 − 6.01093i 0.462346 − 1.11620i −0.505086 − 0.863069i \(-0.668539\pi\)
0.967432 − 0.253132i \(-0.0814608\pi\)
\(30\) 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
\(33\) −0.533145 −0.0928087
\(34\) −2.19248 − 5.51052i −0.376008 − 0.945047i
\(35\) 0 0
\(36\) −0.145760 + 0.145760i −0.0242933 + 0.0242933i
\(37\) −0.109595 − 0.0453958i −0.0180173 − 0.00746302i 0.373657 − 0.927567i \(-0.378104\pi\)
−0.391674 + 0.920104i \(0.628104\pi\)
\(38\) − 3.34100i − 0.541981i
\(39\) 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 \(-0.234434\pi\)
−0.671695 + 0.740828i \(0.734434\pi\)
\(44\) 0.128069 + 0.309187i 0.0193072 + 0.0466116i
\(45\) 0 0
\(46\) 2.55237 − 6.16195i 0.376326 − 0.908531i
\(47\) − 4.93703i − 0.720139i −0.932925 − 0.360070i \(-0.882753\pi\)
0.932925 − 0.360070i \(-0.117247\pi\)
\(48\) 0.419690 + 0.173841i 0.0605770 + 0.0250918i
\(49\) 4.60811 − 4.60811i 0.658302 − 0.658302i
\(50\) 0 0
\(51\) −0.324894 + 0.315904i −0.0454943 + 0.0442354i
\(52\) −0.389037 −0.0539497
\(53\) −8.47565 + 8.47565i −1.16422 + 1.16422i −0.180678 + 0.983542i \(0.557829\pi\)
−0.983542 + 0.180678i \(0.942171\pi\)
\(54\) 0.874571 + 0.362259i 0.119014 + 0.0492972i
\(55\) 0 0
\(56\) 0.738824 − 1.78368i 0.0987295 − 0.238354i
\(57\) −0.235850 + 0.0976925i −0.0312392 + 0.0129397i
\(58\) 3.58134 + 8.64611i 0.470252 + 1.13529i
\(59\) 7.01329 + 7.01329i 0.913053 + 0.913053i 0.996511 − 0.0834587i \(-0.0265967\pi\)
−0.0834587 + 0.996511i \(0.526597\pi\)
\(60\) 0 0
\(61\) 0.613413 + 1.48091i 0.0785394 + 0.189611i 0.958272 − 0.285858i \(-0.0922785\pi\)
−0.879733 + 0.475469i \(0.842278\pi\)
\(62\) 8.81322 − 3.65056i 1.11928 − 0.463621i
\(63\) 0.794779 − 1.91877i 0.100133 − 0.241742i
\(64\) 7.70533i 0.963166i
\(65\) 0 0
\(66\) 0.542263 − 0.542263i 0.0667479 − 0.0667479i
\(67\) −2.99411 −0.365789 −0.182894 − 0.983133i \(-0.558547\pi\)
−0.182894 + 0.983133i \(0.558547\pi\)
\(68\) 0.261246 + 0.112531i 0.0316808 + 0.0136464i
\(69\) −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 \(0.714742\pi\)
\(72\) 8.29913i 0.978062i
\(73\) 2.10442 − 5.08052i 0.246304 − 0.594629i −0.751581 − 0.659641i \(-0.770708\pi\)
0.997885 + 0.0650115i \(0.0207084\pi\)
\(74\) 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
\(78\) 0.341153 + 0.823617i 0.0386280 + 0.0932563i
\(79\) −13.7140 + 5.68053i −1.54295 + 0.639110i −0.982024 − 0.188757i \(-0.939554\pi\)
−0.560924 + 0.827867i \(0.689554\pi\)
\(80\) 0 0
\(81\) 8.89143i 0.987937i
\(82\) 1.43358 + 0.593808i 0.158312 + 0.0655752i
\(83\) 3.56033 − 3.56033i 0.390797 − 0.390797i −0.484175 − 0.874971i \(-0.660880\pi\)
0.874971 + 0.484175i \(0.160880\pi\)
\(84\) 0.00527044 0.000575052
\(85\) 0 0
\(86\) −0.922169 −0.0994400
\(87\) 0.505633 − 0.505633i 0.0542096 − 0.0542096i
\(88\) 12.4480 + 5.15614i 1.32696 + 0.549646i
\(89\) 2.35657i 0.249796i 0.992170 + 0.124898i \(0.0398604\pi\)
−0.992170 + 0.124898i \(0.960140\pi\)
\(90\) 0 0
\(91\) 3.62126 − 1.49997i 0.379611 − 0.157240i
\(92\) 0.122419 + 0.295545i 0.0127630 + 0.0308127i
\(93\) −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
\(97\) 1.03355 − 2.49522i 0.104941 − 0.253351i −0.862683 − 0.505745i \(-0.831218\pi\)
0.967625 + 0.252394i \(0.0812178\pi\)
\(98\) 9.37384i 0.946900i
\(99\) 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.m.b.26.2 24
5.2 odd 4 425.2.n.c.349.2 24
5.3 odd 4 425.2.n.f.349.5 24
5.4 even 2 85.2.l.a.26.5 ✓ 24
15.14 odd 2 765.2.be.b.451.2 24
17.2 even 8 inner 425.2.m.b.376.2 24
17.6 odd 16 7225.2.a.bq.1.3 12
17.11 odd 16 7225.2.a.bs.1.3 12
85.2 odd 8 425.2.n.f.274.5 24
85.19 even 8 85.2.l.a.36.5 yes 24
85.24 odd 16 1445.2.d.j.866.6 24
85.44 odd 16 1445.2.d.j.866.5 24
85.53 odd 8 425.2.n.c.274.2 24
85.74 odd 16 1445.2.a.q.1.10 12
85.79 odd 16 1445.2.a.p.1.10 12
255.104 odd 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 5.4 even 2
85.2.l.a.36.5 yes 24 85.19 even 8
425.2.m.b.26.2 24 1.1 even 1 trivial
425.2.m.b.376.2 24 17.2 even 8 inner
425.2.n.c.274.2 24 85.53 odd 8
425.2.n.c.349.2 24 5.2 odd 4
425.2.n.f.274.5 24 85.2 odd 8
425.2.n.f.349.5 24 5.3 odd 4
765.2.be.b.451.2 24 15.14 odd 2
765.2.be.b.631.2 24 255.104 odd 8
1445.2.a.p.1.10 12 85.79 odd 16
1445.2.a.q.1.10 12 85.74 odd 16
1445.2.d.j.866.5 24 85.44 odd 16
1445.2.d.j.866.6 24 85.24 odd 16
7225.2.a.bq.1.3 12 17.6 odd 16
7225.2.a.bs.1.3 12 17.11 odd 16