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Properties

Label 512.2.b.c.257.3
Level $512$
Weight $2$
Character 512.257
Analytic conductor $4.088$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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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] = [512,2,Mod(257,512)] 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("512.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 257.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 512.257
Dual form 512.2.b.c.257.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.585786i q^{3} +2.65685 q^{9} -6.24264i q^{11} +5.65685 q^{17} +7.41421i q^{19} +5.00000 q^{25} +3.31371i q^{27} +3.65685 q^{33} +6.00000 q^{41} -13.0711i q^{43} -7.00000 q^{49} +3.31371i q^{51} -4.34315 q^{57} +14.2426i q^{59} -3.89949i q^{67} -16.9706 q^{73} +2.92893i q^{75} +6.02944 q^{81} -10.7279i q^{83} +5.65685 q^{89} -16.9706 q^{97} -16.5858i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} - 8 q^{33} + 24 q^{41} - 28 q^{49} - 40 q^{57} + 92 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(-1\) \(1\)

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 0
\(3\) 0.585786i 0.338204i 0.985599 + 0.169102i \(0.0540867\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.65685 0.885618
\(10\) 0 0
\(11\) − 6.24264i − 1.88223i −0.338091 − 0.941113i \(-0.609781\pi\)
0.338091 − 0.941113i \(-0.390219\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 − 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) 7.41421i 1.70094i 0.526026 + 0.850469i \(0.323682\pi\)
−0.526026 + 0.850469i \(0.676318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 3.31371i 0.637723i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.65685 0.636577
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 13.0711i − 1.99332i −0.0816682 − 0.996660i \(-0.526025\pi\)
0.0816682 − 0.996660i \(-0.473975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 3.31371i 0.464012i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.34315 −0.575264
\(58\) 0 0
\(59\) 14.2426i 1.85423i 0.374772 + 0.927117i \(0.377721\pi\)
−0.374772 + 0.927117i \(0.622279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.89949i − 0.476399i −0.971216 − 0.238200i \(-0.923443\pi\)
0.971216 − 0.238200i \(-0.0765572\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.9706 −1.98625 −0.993127 − 0.117041i \(-0.962659\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 2.92893i 0.338204i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 0 0
\(83\) − 10.7279i − 1.17754i −0.808300 − 0.588771i \(-0.799612\pi\)
0.808300 − 0.588771i \(-0.200388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.65685 0.599625 0.299813 − 0.953998i \(-0.403076\pi\)
0.299813 + 0.953998i \(0.403076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.9706 −1.72310 −0.861550 − 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) − 16.5858i − 1.66693i
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 512.2.b.c.257.3 4
3.2 odd 2 4608.2.d.k.2305.4 4
4.3 odd 2 inner 512.2.b.c.257.2 4
8.3 odd 2 CM 512.2.b.c.257.3 4
8.5 even 2 inner 512.2.b.c.257.2 4
12.11 even 2 4608.2.d.k.2305.1 4
16.3 odd 4 512.2.a.f.1.1 yes 2
16.5 even 4 512.2.a.f.1.1 yes 2
16.11 odd 4 512.2.a.a.1.2 ✓ 2
16.13 even 4 512.2.a.a.1.2 ✓ 2
24.5 odd 2 4608.2.d.k.2305.1 4
24.11 even 2 4608.2.d.k.2305.4 4
32.3 odd 8 1024.2.e.g.769.2 4
32.5 even 8 1024.2.e.o.257.1 4
32.11 odd 8 1024.2.e.o.257.1 4
32.13 even 8 1024.2.e.g.769.2 4
32.19 odd 8 1024.2.e.o.769.1 4
32.21 even 8 1024.2.e.g.257.2 4
32.27 odd 8 1024.2.e.g.257.2 4
32.29 even 8 1024.2.e.o.769.1 4
48.5 odd 4 4608.2.a.i.1.1 2
48.11 even 4 4608.2.a.k.1.2 2
48.29 odd 4 4608.2.a.k.1.2 2
48.35 even 4 4608.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.a.1.2 ✓ 2 16.11 odd 4
512.2.a.a.1.2 ✓ 2 16.13 even 4
512.2.a.f.1.1 yes 2 16.3 odd 4
512.2.a.f.1.1 yes 2 16.5 even 4
512.2.b.c.257.2 4 4.3 odd 2 inner
512.2.b.c.257.2 4 8.5 even 2 inner
512.2.b.c.257.3 4 1.1 even 1 trivial
512.2.b.c.257.3 4 8.3 odd 2 CM
1024.2.e.g.257.2 4 32.21 even 8
1024.2.e.g.257.2 4 32.27 odd 8
1024.2.e.g.769.2 4 32.3 odd 8
1024.2.e.g.769.2 4 32.13 even 8
1024.2.e.o.257.1 4 32.5 even 8
1024.2.e.o.257.1 4 32.11 odd 8
1024.2.e.o.769.1 4 32.19 odd 8
1024.2.e.o.769.1 4 32.29 even 8
4608.2.a.i.1.1 2 48.5 odd 4
4608.2.a.i.1.1 2 48.35 even 4
4608.2.a.k.1.2 2 48.11 even 4
4608.2.a.k.1.2 2 48.29 odd 4
4608.2.d.k.2305.1 4 12.11 even 2
4608.2.d.k.2305.1 4 24.5 odd 2
4608.2.d.k.2305.4 4 3.2 odd 2
4608.2.d.k.2305.4 4 24.11 even 2