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Properties

Label 425.2.c.b.424.2
Level $425$
Weight $2$
Character 425.424
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(424,425)] 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("425.424"); 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([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 424.2
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 425.424
Dual form 425.2.c.b.424.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.48119i q^{2} -1.67513 q^{3} -0.193937 q^{4} +2.48119i q^{6} -1.28726 q^{7} -2.67513i q^{8} -0.193937 q^{9} -0.481194i q^{11} +0.324869 q^{12} -2.15633i q^{13} +1.90668i q^{14} -4.35026 q^{16} +(-3.67513 + 1.86907i) q^{17} +0.287258i q^{18} -3.35026 q^{19} +2.15633 q^{21} -0.712742 q^{22} -8.24965 q^{23} +4.48119i q^{24} -3.19394 q^{26} +5.35026 q^{27} +0.249646 q^{28} -0.649738i q^{29} +1.83146i q^{31} +1.09332i q^{32} +0.806063i q^{33} +(2.76845 + 5.44358i) q^{34} +0.0376114 q^{36} +4.31265 q^{37} +4.96239i q^{38} +3.61213i q^{39} -11.2750i q^{41} -3.19394i q^{42} +8.15633i q^{43} +0.0933212i q^{44} +12.2193i q^{46} +6.54420i q^{47} +7.28726 q^{48} -5.34297 q^{49} +(6.15633 - 3.13093i) q^{51} +0.418190i q^{52} -8.57452i q^{53} -7.92478i q^{54} +3.44358i q^{56} +5.61213 q^{57} -0.962389 q^{58} -4.96239 q^{59} +2.83638i q^{61} +2.71274 q^{62} +0.249646 q^{63} -7.08110 q^{64} +1.19394 q^{66} -4.93207i q^{67} +(0.712742 - 0.362481i) q^{68} +13.8192 q^{69} -14.5320i q^{71} +0.518806i q^{72} +13.3503 q^{73} -6.38787i q^{74} +0.649738 q^{76} +0.619421i q^{77} +5.35026 q^{78} -9.05571i q^{79} -8.38058 q^{81} -16.7005 q^{82} -13.4314i q^{83} -0.418190 q^{84} +12.0811 q^{86} +1.08840i q^{87} -1.28726 q^{88} +16.7816 q^{89} +2.77575i q^{91} +1.59991 q^{92} -3.06793i q^{93} +9.69323 q^{94} -1.83146i q^{96} -3.66291 q^{97} +7.91397i q^{98} +0.0933212i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} + 4 q^{7} - 2 q^{9} + 12 q^{12} - 6 q^{16} - 12 q^{17} - 8 q^{21} - 16 q^{22} - 16 q^{23} - 20 q^{26} + 12 q^{27} - 32 q^{28} - 6 q^{34} + 22 q^{36} - 16 q^{37} + 32 q^{48} + 14 q^{49} + 16 q^{51}+ \cdots + 40 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.48119i − 1.04736i −0.851914 − 0.523681i \(-0.824558\pi\)
0.851914 − 0.523681i \(-0.175442\pi\)
\(3\) −1.67513 −0.967137 −0.483569 − 0.875306i \(-0.660660\pi\)
−0.483569 + 0.875306i \(0.660660\pi\)
\(4\) −0.193937 −0.0969683
\(5\) 0 0
\(6\) 2.48119i 1.01294i
\(7\) −1.28726 −0.486538 −0.243269 − 0.969959i \(-0.578220\pi\)
−0.243269 + 0.969959i \(0.578220\pi\)
\(8\) − 2.67513i − 0.945802i
\(9\) −0.193937 −0.0646455
\(10\) 0 0
\(11\) − 0.481194i − 0.145086i −0.997365 − 0.0725428i \(-0.976889\pi\)
0.997365 − 0.0725428i \(-0.0231114\pi\)
\(12\) 0.324869 0.0937816
\(13\) − 2.15633i − 0.598057i −0.954244 − 0.299028i \(-0.903337\pi\)
0.954244 − 0.299028i \(-0.0966626\pi\)
\(14\) 1.90668i 0.509581i
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) −3.67513 + 1.86907i −0.891350 + 0.453315i
\(18\) 0.287258i 0.0677073i
\(19\) −3.35026 −0.768603 −0.384301 − 0.923208i \(-0.625558\pi\)
−0.384301 + 0.923208i \(0.625558\pi\)
\(20\) 0 0
\(21\) 2.15633 0.470549
\(22\) −0.712742 −0.151957
\(23\) −8.24965 −1.72017 −0.860085 − 0.510151i \(-0.829590\pi\)
−0.860085 + 0.510151i \(0.829590\pi\)
\(24\) 4.48119i 0.914720i
\(25\) 0 0
\(26\) −3.19394 −0.626382
\(27\) 5.35026 1.02966
\(28\) 0.249646 0.0471787
\(29\) − 0.649738i − 0.120653i −0.998179 − 0.0603267i \(-0.980786\pi\)
0.998179 − 0.0603267i \(-0.0192142\pi\)
\(30\) 0 0
\(31\) 1.83146i 0.328939i 0.986382 + 0.164470i \(0.0525912\pi\)
−0.986382 + 0.164470i \(0.947409\pi\)
\(32\) 1.09332i 0.193274i
\(33\) 0.806063i 0.140318i
\(34\) 2.76845 + 5.44358i 0.474786 + 0.933567i
\(35\) 0 0
\(36\) 0.0376114 0.00626857
\(37\) 4.31265 0.708995 0.354498 − 0.935057i \(-0.384652\pi\)
0.354498 + 0.935057i \(0.384652\pi\)
\(38\) 4.96239i 0.805006i
\(39\) 3.61213i 0.578403i
\(40\) 0 0
\(41\) − 11.2750i − 1.76087i −0.474171 − 0.880433i \(-0.657252\pi\)
0.474171 − 0.880433i \(-0.342748\pi\)
\(42\) − 3.19394i − 0.492835i
\(43\) 8.15633i 1.24383i 0.783086 + 0.621914i \(0.213645\pi\)
−0.783086 + 0.621914i \(0.786355\pi\)
\(44\) 0.0933212i 0.0140687i
\(45\) 0 0
\(46\) 12.2193i 1.80164i
\(47\) 6.54420i 0.954569i 0.878749 + 0.477285i \(0.158379\pi\)
−0.878749 + 0.477285i \(0.841621\pi\)
\(48\) 7.28726 1.05183
\(49\) −5.34297 −0.763281
\(50\) 0 0
\(51\) 6.15633 − 3.13093i 0.862058 − 0.438418i
\(52\) 0.418190i 0.0579926i
\(53\) − 8.57452i − 1.17780i −0.808206 − 0.588900i \(-0.799561\pi\)
0.808206 − 0.588900i \(-0.200439\pi\)
\(54\) − 7.92478i − 1.07843i
\(55\) 0 0
\(56\) 3.44358i 0.460168i
\(57\) 5.61213 0.743344
\(58\) −0.962389 −0.126368
\(59\) −4.96239 −0.646048 −0.323024 − 0.946391i \(-0.604699\pi\)
−0.323024 + 0.946391i \(0.604699\pi\)
\(60\) 0 0
\(61\) 2.83638i 0.363161i 0.983376 + 0.181581i \(0.0581213\pi\)
−0.983376 + 0.181581i \(0.941879\pi\)
\(62\) 2.71274 0.344519
\(63\) 0.249646 0.0314525
\(64\) −7.08110 −0.885138
\(65\) 0 0
\(66\) 1.19394 0.146963
\(67\) − 4.93207i − 0.602548i −0.953538 − 0.301274i \(-0.902588\pi\)
0.953538 − 0.301274i \(-0.0974120\pi\)
\(68\) 0.712742 − 0.362481i 0.0864327 − 0.0439572i
\(69\) 13.8192 1.66364
\(70\) 0 0
\(71\) − 14.5320i − 1.72463i −0.506373 − 0.862314i \(-0.669014\pi\)
0.506373 − 0.862314i \(-0.330986\pi\)
\(72\) 0.518806i 0.0611418i
\(73\) 13.3503 1.56253 0.781265 − 0.624200i \(-0.214575\pi\)
0.781265 + 0.624200i \(0.214575\pi\)
\(74\) − 6.38787i − 0.742575i
\(75\) 0 0
\(76\) 0.649738 0.0745301
\(77\) 0.619421i 0.0705896i
\(78\) 5.35026 0.605798
\(79\) − 9.05571i − 1.01885i −0.860516 − 0.509423i \(-0.829859\pi\)
0.860516 − 0.509423i \(-0.170141\pi\)
\(80\) 0 0
\(81\) −8.38058 −0.931175
\(82\) −16.7005 −1.84426
\(83\) − 13.4314i − 1.47428i −0.675738 − 0.737142i \(-0.736175\pi\)
0.675738 − 0.737142i \(-0.263825\pi\)
\(84\) −0.418190 −0.0456283
\(85\) 0 0
\(86\) 12.0811 1.30274
\(87\) 1.08840i 0.116688i
\(88\) −1.28726 −0.137222
\(89\) 16.7816 1.77885 0.889424 − 0.457082i \(-0.151106\pi\)
0.889424 + 0.457082i \(0.151106\pi\)
\(90\) 0 0
\(91\) 2.77575i 0.290977i
\(92\) 1.59991 0.166802
\(93\) − 3.06793i − 0.318129i
\(94\) 9.69323 0.999780
\(95\) 0 0
\(96\) − 1.83146i − 0.186922i
\(97\) −3.66291 −0.371912 −0.185956 − 0.982558i \(-0.559538\pi\)
−0.185956 + 0.982558i \(0.559538\pi\)
\(98\) 7.91397i 0.799432i
\(99\) 0.0933212i 0.00937913i
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.c.b.424.2 6
5.2 odd 4 85.2.d.a.16.6 yes 6
5.3 odd 4 425.2.d.c.101.1 6
5.4 even 2 425.2.c.a.424.5 6
15.2 even 4 765.2.g.b.271.2 6
17.16 even 2 425.2.c.a.424.2 6
20.7 even 4 1360.2.c.f.1121.2 6
85.13 odd 4 7225.2.a.r.1.3 3
85.33 odd 4 425.2.d.c.101.2 6
85.38 odd 4 7225.2.a.q.1.3 3
85.47 odd 4 1445.2.a.k.1.1 3
85.67 odd 4 85.2.d.a.16.5 ✓ 6
85.72 odd 4 1445.2.a.j.1.1 3
85.84 even 2 inner 425.2.c.b.424.5 6
255.152 even 4 765.2.g.b.271.1 6
340.67 even 4 1360.2.c.f.1121.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 85.67 odd 4
85.2.d.a.16.6 yes 6 5.2 odd 4
425.2.c.a.424.2 6 17.16 even 2
425.2.c.a.424.5 6 5.4 even 2
425.2.c.b.424.2 6 1.1 even 1 trivial
425.2.c.b.424.5 6 85.84 even 2 inner
425.2.d.c.101.1 6 5.3 odd 4
425.2.d.c.101.2 6 85.33 odd 4
765.2.g.b.271.1 6 255.152 even 4
765.2.g.b.271.2 6 15.2 even 4
1360.2.c.f.1121.2 6 20.7 even 4
1360.2.c.f.1121.5 6 340.67 even 4
1445.2.a.j.1.1 3 85.72 odd 4
1445.2.a.k.1.1 3 85.47 odd 4
7225.2.a.q.1.3 3 85.38 odd 4
7225.2.a.r.1.3 3 85.13 odd 4