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Properties

Label 765.2.g.b.271.2
Level $765$
Weight $2$
Character 765.271
Analytic conductor $6.109$
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] = [765,2,Mod(271,765)] 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("765.271"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.10855575463\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 271.2
Root \(0.403032 + 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 765.271
Dual form 765.2.g.b.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.48119 q^{2} +0.193937 q^{4} +1.00000i q^{5} -1.28726i q^{7} +2.67513 q^{8} -1.48119i q^{10} +0.481194i q^{11} -2.15633 q^{13} +1.90668i q^{14} -4.35026 q^{16} +(1.86907 + 3.67513i) q^{17} +3.35026 q^{19} +0.193937i q^{20} -0.712742i q^{22} -8.24965i q^{23} -1.00000 q^{25} +3.19394 q^{26} -0.249646i q^{28} -0.649738i q^{29} +1.83146i q^{31} +1.09332 q^{32} +(-2.76845 - 5.44358i) q^{34} +1.28726 q^{35} +4.31265i q^{37} -4.96239 q^{38} +2.67513i q^{40} +11.2750i q^{41} +8.15633 q^{43} +0.0933212i q^{44} +12.2193i q^{46} +6.54420 q^{47} +5.34297 q^{49} +1.48119 q^{50} -0.418190 q^{52} +8.57452 q^{53} -0.481194 q^{55} -3.44358i q^{56} +0.962389i q^{58} -4.96239 q^{59} +2.83638i q^{61} -2.71274i q^{62} +7.08110 q^{64} -2.15633i q^{65} +4.93207 q^{67} +(0.362481 + 0.712742i) q^{68} -1.90668 q^{70} +14.5320i q^{71} -13.3503i q^{73} -6.38787i q^{74} +0.649738 q^{76} +0.619421 q^{77} +9.05571i q^{79} -4.35026i q^{80} -16.7005i q^{82} +13.4314 q^{83} +(-3.67513 + 1.86907i) q^{85} -12.0811 q^{86} +1.28726i q^{88} +16.7816 q^{89} +2.77575i q^{91} -1.59991i q^{92} -9.69323 q^{94} +3.35026i q^{95} -3.66291i q^{97} -7.91397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 8 q^{13} - 6 q^{16} + 2 q^{17} - 6 q^{25} + 20 q^{26} - 6 q^{32} + 6 q^{34} - 4 q^{35} - 8 q^{38} + 28 q^{43} + 20 q^{47} - 14 q^{49} - 2 q^{50} + 28 q^{53} + 8 q^{55}+ \cdots - 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/765\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.48119 −1.04736 −0.523681 − 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(3\) 0 0
\(4\) 0.193937 0.0969683
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) − 1.28726i − 0.486538i −0.969959 − 0.243269i \(-0.921780\pi\)
0.969959 − 0.243269i \(-0.0782197\pi\)
\(8\) 2.67513 0.945802
\(9\) 0 0
\(10\) − 1.48119i − 0.468395i
\(11\) 0.481194i 0.145086i 0.997365 + 0.0725428i \(0.0231114\pi\)
−0.997365 + 0.0725428i \(0.976889\pi\)
\(12\) 0 0
\(13\) −2.15633 −0.598057 −0.299028 − 0.954244i \(-0.596663\pi\)
−0.299028 + 0.954244i \(0.596663\pi\)
\(14\) 1.90668i 0.509581i
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) 1.86907 + 3.67513i 0.453315 + 0.891350i
\(18\) 0 0
\(19\) 3.35026 0.768603 0.384301 − 0.923208i \(-0.374442\pi\)
0.384301 + 0.923208i \(0.374442\pi\)
\(20\) 0.193937i 0.0433655i
\(21\) 0 0
\(22\) − 0.712742i − 0.151957i
\(23\) − 8.24965i − 1.72017i −0.510151 − 0.860085i \(-0.670410\pi\)
0.510151 − 0.860085i \(-0.329590\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 3.19394 0.626382
\(27\) 0 0
\(28\) − 0.249646i − 0.0471787i
\(29\) − 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 \(0.947409\pi\)
\(32\) 1.09332 0.193274
\(33\) 0 0
\(34\) −2.76845 − 5.44358i −0.474786 − 0.933567i
\(35\) 1.28726 0.217586
\(36\) 0 0
\(37\) 4.31265i 0.708995i 0.935057 + 0.354498i \(0.115348\pi\)
−0.935057 + 0.354498i \(0.884652\pi\)
\(38\) −4.96239 −0.805006
\(39\) 0 0
\(40\) 2.67513i 0.422975i
\(41\) 11.2750i 1.76087i 0.474171 + 0.880433i \(0.342748\pi\)
−0.474171 + 0.880433i \(0.657252\pi\)
\(42\) 0 0
\(43\) 8.15633 1.24383 0.621914 − 0.783086i \(-0.286355\pi\)
0.621914 + 0.783086i \(0.286355\pi\)
\(44\) 0.0933212i 0.0140687i
\(45\) 0 0
\(46\) 12.2193i 1.80164i
\(47\) 6.54420 0.954569 0.477285 − 0.878749i \(-0.341621\pi\)
0.477285 + 0.878749i \(0.341621\pi\)
\(48\) 0 0
\(49\) 5.34297 0.763281
\(50\) 1.48119 0.209473
\(51\) 0 0
\(52\) −0.418190 −0.0579926
\(53\) 8.57452 1.17780 0.588900 − 0.808206i \(-0.299561\pi\)
0.588900 + 0.808206i \(0.299561\pi\)
\(54\) 0 0
\(55\) −0.481194 −0.0648842
\(56\) − 3.44358i − 0.460168i
\(57\) 0 0
\(58\) 0.962389i 0.126368i
\(59\) −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 \(0.941879\pi\)
\(62\) − 2.71274i − 0.344519i
\(63\) 0 0
\(64\) 7.08110 0.885138
\(65\) − 2.15633i − 0.267459i
\(66\) 0 0
\(67\) 4.93207 0.602548 0.301274 − 0.953538i \(-0.402588\pi\)
0.301274 + 0.953538i \(0.402588\pi\)
\(68\) 0.362481 + 0.712742i 0.0439572 + 0.0864327i
\(69\) 0 0
\(70\) −1.90668 −0.227892
\(71\) 14.5320i 1.72463i 0.506373 + 0.862314i \(0.330986\pi\)
−0.506373 + 0.862314i \(0.669014\pi\)
\(72\) 0 0
\(73\) − 13.3503i − 1.56253i −0.624200 − 0.781265i \(-0.714575\pi\)
0.624200 − 0.781265i \(-0.285425\pi\)
\(74\) − 6.38787i − 0.742575i
\(75\) 0 0
\(76\) 0.649738 0.0745301
\(77\) 0.619421 0.0705896
\(78\) 0 0
\(79\) 9.05571i 1.01885i 0.860516 + 0.509423i \(0.170141\pi\)
−0.860516 + 0.509423i \(0.829859\pi\)
\(80\) − 4.35026i − 0.486374i
\(81\) 0 0
\(82\) − 16.7005i − 1.84426i
\(83\) 13.4314 1.47428 0.737142 − 0.675738i \(-0.236175\pi\)
0.737142 + 0.675738i \(0.236175\pi\)
\(84\) 0 0
\(85\) −3.67513 + 1.86907i −0.398624 + 0.202729i
\(86\) −12.0811 −1.30274
\(87\) 0 0
\(88\) 1.28726i 0.137222i
\(89\) 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
\(92\) − 1.59991i − 0.166802i
\(93\) 0 0
\(94\) −9.69323 −0.999780
\(95\) 3.35026i 0.343730i
\(96\) 0 0
\(97\) − 3.66291i − 0.371912i −0.982558 − 0.185956i \(-0.940462\pi\)
0.982558 − 0.185956i \(-0.0595383\pi\)
\(98\) −7.91397 −0.799432
\(99\) 0 0
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 765.2.g.b.271.2 6
3.2 odd 2 85.2.d.a.16.6 yes 6
12.11 even 2 1360.2.c.f.1121.2 6
15.2 even 4 425.2.c.a.424.5 6
15.8 even 4 425.2.c.b.424.2 6
15.14 odd 2 425.2.d.c.101.1 6
17.16 even 2 inner 765.2.g.b.271.1 6
51.38 odd 4 1445.2.a.j.1.1 3
51.47 odd 4 1445.2.a.k.1.1 3
51.50 odd 2 85.2.d.a.16.5 ✓ 6
204.203 even 2 1360.2.c.f.1121.5 6
255.89 odd 4 7225.2.a.q.1.3 3
255.149 odd 4 7225.2.a.r.1.3 3
255.152 even 4 425.2.c.b.424.5 6
255.203 even 4 425.2.c.a.424.2 6
255.254 odd 2 425.2.d.c.101.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 51.50 odd 2
85.2.d.a.16.6 yes 6 3.2 odd 2
425.2.c.a.424.2 6 255.203 even 4
425.2.c.a.424.5 6 15.2 even 4
425.2.c.b.424.2 6 15.8 even 4
425.2.c.b.424.5 6 255.152 even 4
425.2.d.c.101.1 6 15.14 odd 2
425.2.d.c.101.2 6 255.254 odd 2
765.2.g.b.271.1 6 17.16 even 2 inner
765.2.g.b.271.2 6 1.1 even 1 trivial
1360.2.c.f.1121.2 6 12.11 even 2
1360.2.c.f.1121.5 6 204.203 even 2
1445.2.a.j.1.1 3 51.38 odd 4
1445.2.a.k.1.1 3 51.47 odd 4
7225.2.a.q.1.3 3 255.89 odd 4
7225.2.a.r.1.3 3 255.149 odd 4