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Properties

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

$q$-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 \(0.220742\pi\)
−0.769025 + 0.639219i \(0.779258\pi\)
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) 0.688892i 0.281239i
\(7\) 1.59210i 0.601759i 0.953662 + 0.300879i \(0.0972802\pi\)
−0.953662 + 0.300879i \(0.902720\pi\)
\(8\) −1.21432 −0.429327
\(9\) −1.90321 −0.634404
\(10\) 0 0
\(11\) 1.31111i 0.395314i 0.980271 + 0.197657i \(0.0633332\pi\)
−0.980271 + 0.197657i \(0.936667\pi\)
\(12\) − 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 \(-0.826622\pi\)
0.855292 − 0.518147i \(-0.173378\pi\)
\(24\) − 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 \(0.286096\pi\)
−0.622551 + 0.782580i \(0.713904\pi\)
\(30\) 0 0
\(31\) − 7.73975i − 1.39010i −0.718962 − 0.695050i \(-0.755382\pi\)
0.718962 − 0.695050i \(-0.244618\pi\)
\(32\) 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 \(0.196782\pi\)
−0.814918 + 0.579577i \(0.803218\pi\)
\(38\) −1.37778 −0.223506
\(39\) − 7.80642i − 1.25003i
\(40\) 0 0
\(41\) 3.67307i 0.573637i 0.957985 + 0.286819i \(0.0925977\pi\)
−0.957985 + 0.286819i \(0.907402\pi\)
\(42\) −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 \(0.448852\pi\)
−0.159996 + 0.987118i \(0.551148\pi\)
\(62\) − 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 \(0.216461\pi\)
−0.777553 + 0.628818i \(0.783539\pi\)
\(72\) 2.31111 0.272367
\(73\) 5.57136i 0.652078i 0.945356 + 0.326039i \(0.105714\pi\)
−0.945356 + 0.326039i \(0.894286\pi\)
\(74\) 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 \(0.146049\pi\)
−0.896573 + 0.442897i \(0.853951\pi\)
\(80\) 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 \(-0.712225\pi\)
0.618414 − 0.785852i \(-0.287775\pi\)
\(98\) 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.101.4 6
5.2 odd 4 425.2.c.b.424.4 6
5.3 odd 4 425.2.c.a.424.3 6
5.4 even 2 85.2.d.a.16.3 ✓ 6
15.14 odd 2 765.2.g.b.271.4 6
17.4 even 4 7225.2.a.q.1.2 3
17.13 even 4 7225.2.a.r.1.2 3
17.16 even 2 inner 425.2.d.c.101.3 6
20.19 odd 2 1360.2.c.f.1121.6 6
85.4 even 4 1445.2.a.j.1.2 3
85.33 odd 4 425.2.c.b.424.3 6
85.64 even 4 1445.2.a.k.1.2 3
85.67 odd 4 425.2.c.a.424.4 6
85.84 even 2 85.2.d.a.16.4 yes 6
255.254 odd 2 765.2.g.b.271.3 6
340.339 odd 2 1360.2.c.f.1121.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.3 ✓ 6 5.4 even 2
85.2.d.a.16.4 yes 6 85.84 even 2
425.2.c.a.424.3 6 5.3 odd 4
425.2.c.a.424.4 6 85.67 odd 4
425.2.c.b.424.3 6 85.33 odd 4
425.2.c.b.424.4 6 5.2 odd 4
425.2.d.c.101.3 6 17.16 even 2 inner
425.2.d.c.101.4 6 1.1 even 1 trivial
765.2.g.b.271.3 6 255.254 odd 2
765.2.g.b.271.4 6 15.14 odd 2
1360.2.c.f.1121.1 6 340.339 odd 2
1360.2.c.f.1121.6 6 20.19 odd 2
1445.2.a.j.1.2 3 85.4 even 4
1445.2.a.k.1.2 3 85.64 even 4
7225.2.a.q.1.2 3 17.4 even 4
7225.2.a.r.1.2 3 17.13 even 4