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Properties

Label 425.2.d.c.101.3
Level $425$
Weight $2$
Character 425.101
Analytic conductor $3.394$
Analytic rank $0$
Dimension $6$
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(101,425)] 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("425.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) 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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,2,0,2,0,0,0,6,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.3
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 425.101
Dual form 425.2.d.c.101.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.311108 q^{2} -2.21432i q^{3} -1.90321 q^{4} -0.688892i q^{6} -1.59210i q^{7} -1.21432 q^{8} -1.90321 q^{9} -1.31111i q^{11} +4.21432i q^{12} -3.52543 q^{13} -0.495316i q^{14} +3.42864 q^{16} +(-4.11753 + 0.214320i) q^{17} -0.592104 q^{18} -4.42864 q^{19} -3.52543 q^{21} -0.407896i q^{22} +4.96989i q^{23} +2.68889i q^{24} -1.09679 q^{26} -2.42864i q^{27} +3.03011i q^{28} -8.42864i q^{29} +7.73975i q^{31} +3.49532 q^{32} -2.90321 q^{33} +(-1.28100 + 0.0666765i) q^{34} +3.62222 q^{36} -7.05086i q^{37} -1.37778 q^{38} +7.80642i q^{39} -3.67307i q^{41} -1.09679 q^{42} -2.47457 q^{43} +2.49532i q^{44} +1.54617i q^{46} -3.33185 q^{47} -7.59210i q^{48} +4.46520 q^{49} +(0.474572 + 9.11753i) q^{51} +6.70964 q^{52} +9.18421 q^{53} -0.755569i q^{54} +1.93332i q^{56} +9.80642i q^{57} -2.62222i q^{58} +1.37778 q^{59} -15.4193i q^{61} +2.40790i q^{62} +3.03011i q^{63} -5.76986 q^{64} -0.903212 q^{66} +9.13828 q^{67} +(7.83654 - 0.407896i) q^{68} +11.0049 q^{69} -10.5970i q^{71} +2.31111 q^{72} -5.57136i q^{73} -2.19358i q^{74} +8.42864 q^{76} -2.08742 q^{77} +2.42864i q^{78} -7.87310i q^{79} -11.0874 q^{81} -1.14272i q^{82} -7.19850 q^{83} +6.70964 q^{84} -0.769859 q^{86} -18.6637 q^{87} +1.59210i q^{88} +11.6271 q^{89} +5.61285i q^{91} -9.45875i q^{92} +17.1383 q^{93} -1.03657 q^{94} -7.73975i q^{96} +15.4795i q^{97} +1.38916 q^{98} +2.49532i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 2 q^{9} - 8 q^{13} - 6 q^{16} + 2 q^{17} + 10 q^{18} - 8 q^{21} - 20 q^{26} - 6 q^{32} - 4 q^{33} + 6 q^{34} + 22 q^{36} - 8 q^{38} - 20 q^{42} - 28 q^{43} + 20 q^{47}+ \cdots - 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 − 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) − 2.21432i − 1.27844i −0.769025 − 0.639219i \(-0.779258\pi\)
0.769025 − 0.639219i \(-0.220742\pi\)
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) − 0.688892i − 0.281239i
\(7\) − 1.59210i − 0.601759i −0.953662 − 0.300879i \(-0.902720\pi\)
0.953662 − 0.300879i \(-0.0972802\pi\)
\(8\) −1.21432 −0.429327
\(9\) −1.90321 −0.634404
\(10\) 0 0
\(11\) − 1.31111i − 0.395314i −0.980271 − 0.197657i \(-0.936667\pi\)
0.980271 − 0.197657i \(-0.0633332\pi\)
\(12\) 4.21432i 1.21657i
\(13\) −3.52543 −0.977778 −0.488889 − 0.872346i \(-0.662598\pi\)
−0.488889 + 0.872346i \(0.662598\pi\)
\(14\) − 0.495316i − 0.132379i
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) −4.11753 + 0.214320i −0.998648 + 0.0519802i
\(18\) −0.592104 −0.139560
\(19\) −4.42864 −1.01600 −0.508000 − 0.861357i \(-0.669615\pi\)
−0.508000 + 0.861357i \(0.669615\pi\)
\(20\) 0 0
\(21\) −3.52543 −0.769311
\(22\) − 0.407896i − 0.0869637i
\(23\) 4.96989i 1.03629i 0.855292 + 0.518147i \(0.173378\pi\)
−0.855292 + 0.518147i \(0.826622\pi\)
\(24\) 2.68889i 0.548868i
\(25\) 0 0
\(26\) −1.09679 −0.215098
\(27\) − 2.42864i − 0.467392i
\(28\) 3.03011i 0.572637i
\(29\) − 8.42864i − 1.56516i −0.622551 − 0.782580i \(-0.713904\pi\)
0.622551 − 0.782580i \(-0.286096\pi\)
\(30\) 0 0
\(31\) 7.73975i 1.39010i 0.718962 + 0.695050i \(0.244618\pi\)
−0.718962 + 0.695050i \(0.755382\pi\)
\(32\) 3.49532 0.617890
\(33\) −2.90321 −0.505384
\(34\) −1.28100 + 0.0666765i −0.219689 + 0.0114349i
\(35\) 0 0
\(36\) 3.62222 0.603703
\(37\) − 7.05086i − 1.15915i −0.814918 − 0.579577i \(-0.803218\pi\)
0.814918 − 0.579577i \(-0.196782\pi\)
\(38\) −1.37778 −0.223506
\(39\) 7.80642i 1.25003i
\(40\) 0 0
\(41\) − 3.67307i − 0.573637i −0.957985 − 0.286819i \(-0.907402\pi\)
0.957985 − 0.286819i \(-0.0925977\pi\)
\(42\) −1.09679 −0.169238
\(43\) −2.47457 −0.377369 −0.188684 − 0.982038i \(-0.560422\pi\)
−0.188684 + 0.982038i \(0.560422\pi\)
\(44\) 2.49532i 0.376183i
\(45\) 0 0
\(46\) 1.54617i 0.227970i
\(47\) −3.33185 −0.486000 −0.243000 − 0.970026i \(-0.578132\pi\)
−0.243000 + 0.970026i \(0.578132\pi\)
\(48\) − 7.59210i − 1.09583i
\(49\) 4.46520 0.637886
\(50\) 0 0
\(51\) 0.474572 + 9.11753i 0.0664534 + 1.27671i
\(52\) 6.70964 0.930459
\(53\) 9.18421 1.26155 0.630774 − 0.775967i \(-0.282737\pi\)
0.630774 + 0.775967i \(0.282737\pi\)
\(54\) − 0.755569i − 0.102820i
\(55\) 0 0
\(56\) 1.93332i 0.258351i
\(57\) 9.80642i 1.29889i
\(58\) − 2.62222i − 0.344314i
\(59\) 1.37778 0.179372 0.0896861 − 0.995970i \(-0.471414\pi\)
0.0896861 + 0.995970i \(0.471414\pi\)
\(60\) 0 0
\(61\) − 15.4193i − 1.97424i −0.159996 − 0.987118i \(-0.551148\pi\)
0.159996 − 0.987118i \(-0.448852\pi\)
\(62\) 2.40790i 0.305803i
\(63\) 3.03011i 0.381758i
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) −0.903212 −0.111178
\(67\) 9.13828 1.11642 0.558209 − 0.829700i \(-0.311489\pi\)
0.558209 + 0.829700i \(0.311489\pi\)
\(68\) 7.83654 − 0.407896i 0.950320 − 0.0494646i
\(69\) 11.0049 1.32484
\(70\) 0 0
\(71\) − 10.5970i − 1.25764i −0.777553 − 0.628818i \(-0.783539\pi\)
0.777553 − 0.628818i \(-0.216461\pi\)
\(72\) 2.31111 0.272367
\(73\) − 5.57136i − 0.652078i −0.945356 − 0.326039i \(-0.894286\pi\)
0.945356 − 0.326039i \(-0.105714\pi\)
\(74\) − 2.19358i − 0.254998i
\(75\) 0 0
\(76\) 8.42864 0.966831
\(77\) −2.08742 −0.237884
\(78\) 2.42864i 0.274989i
\(79\) − 7.87310i − 0.885793i −0.896573 − 0.442897i \(-0.853951\pi\)
0.896573 − 0.442897i \(-0.146049\pi\)
\(80\) 0 0
\(81\) −11.0874 −1.23194
\(82\) − 1.14272i − 0.126192i
\(83\) −7.19850 −0.790138 −0.395069 − 0.918651i \(-0.629279\pi\)
−0.395069 + 0.918651i \(0.629279\pi\)
\(84\) 6.70964 0.732081
\(85\) 0 0
\(86\) −0.769859 −0.0830160
\(87\) −18.6637 −2.00096
\(88\) 1.59210i 0.169719i
\(89\) 11.6271 1.23247 0.616237 − 0.787561i \(-0.288656\pi\)
0.616237 + 0.787561i \(0.288656\pi\)
\(90\) 0 0
\(91\) 5.61285i 0.588386i
\(92\) − 9.45875i − 0.986143i
\(93\) 17.1383 1.77716
\(94\) −1.03657 −0.106914
\(95\) 0 0
\(96\) − 7.73975i − 0.789935i
\(97\) 15.4795i 1.57170i 0.618414 + 0.785852i \(0.287775\pi\)
−0.618414 + 0.785852i \(0.712225\pi\)
\(98\) 1.38916 0.140326
\(99\) 2.49532i 0.250789i
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.d.c.101.3 6
5.2 odd 4 425.2.c.a.424.4 6
5.3 odd 4 425.2.c.b.424.3 6
5.4 even 2 85.2.d.a.16.4 yes 6
15.14 odd 2 765.2.g.b.271.3 6
17.4 even 4 7225.2.a.r.1.2 3
17.13 even 4 7225.2.a.q.1.2 3
17.16 even 2 inner 425.2.d.c.101.4 6
20.19 odd 2 1360.2.c.f.1121.1 6
85.4 even 4 1445.2.a.k.1.2 3
85.33 odd 4 425.2.c.a.424.3 6
85.64 even 4 1445.2.a.j.1.2 3
85.67 odd 4 425.2.c.b.424.4 6
85.84 even 2 85.2.d.a.16.3 ✓ 6
255.254 odd 2 765.2.g.b.271.4 6
340.339 odd 2 1360.2.c.f.1121.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.3 ✓ 6 85.84 even 2
85.2.d.a.16.4 yes 6 5.4 even 2
425.2.c.a.424.3 6 85.33 odd 4
425.2.c.a.424.4 6 5.2 odd 4
425.2.c.b.424.3 6 5.3 odd 4
425.2.c.b.424.4 6 85.67 odd 4
425.2.d.c.101.3 6 1.1 even 1 trivial
425.2.d.c.101.4 6 17.16 even 2 inner
765.2.g.b.271.3 6 15.14 odd 2
765.2.g.b.271.4 6 255.254 odd 2
1360.2.c.f.1121.1 6 20.19 odd 2
1360.2.c.f.1121.6 6 340.339 odd 2
1445.2.a.j.1.2 3 85.64 even 4
1445.2.a.k.1.2 3 85.4 even 4
7225.2.a.q.1.2 3 17.13 even 4
7225.2.a.r.1.2 3 17.4 even 4