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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [128,4,Mod(65,128)] 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("128.65"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-64,0,-74] 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: \(7.55224448073\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 128.65
Dual form 128.4.b.a.65.2

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000i q^{3} +12.0000i q^{5} -32.0000 q^{7} -37.0000 q^{9} -8.00000i q^{11} +20.0000i q^{13} +96.0000 q^{15} -98.0000 q^{17} +88.0000i q^{19} +256.000i q^{21} +32.0000 q^{23} -19.0000 q^{25} +80.0000i q^{27} -172.000i q^{29} -256.000 q^{31} -64.0000 q^{33} -384.000i q^{35} +92.0000i q^{37} +160.000 q^{39} -102.000 q^{41} -296.000i q^{43} -444.000i q^{45} -320.000 q^{47} +681.000 q^{49} +784.000i q^{51} +76.0000i q^{53} +96.0000 q^{55} +704.000 q^{57} +408.000i q^{59} -636.000i q^{61} +1184.00 q^{63} -240.000 q^{65} -552.000i q^{67} -256.000i q^{69} -416.000 q^{71} -138.000 q^{73} +152.000i q^{75} +256.000i q^{77} -64.0000 q^{79} -359.000 q^{81} -392.000i q^{83} -1176.00i q^{85} -1376.00 q^{87} +582.000 q^{89} -640.000i q^{91} +2048.00i q^{93} -1056.00 q^{95} +238.000 q^{97} +296.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{7} - 74 q^{9} + 192 q^{15} - 196 q^{17} + 64 q^{23} - 38 q^{25} - 512 q^{31} - 128 q^{33} + 320 q^{39} - 204 q^{41} - 640 q^{47} + 1362 q^{49} + 192 q^{55} + 1408 q^{57} + 2368 q^{63} - 480 q^{65}+ \cdots + 476 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 8.00000i − 1.53960i −0.638285 − 0.769800i \(-0.720356\pi\)
0.638285 − 0.769800i \(-0.279644\pi\)
\(4\) 0 0
\(5\) 12.0000i 1.07331i 0.843801 + 0.536656i \(0.180313\pi\)
−0.843801 + 0.536656i \(0.819687\pi\)
\(6\) 0 0
\(7\) −32.0000 −1.72784 −0.863919 − 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) −37.0000 −1.37037
\(10\) 0 0
\(11\) − 8.00000i − 0.219281i −0.993971 − 0.109640i \(-0.965030\pi\)
0.993971 − 0.109640i \(-0.0349700\pi\)
\(12\) 0 0
\(13\) 20.0000i 0.426692i 0.976977 + 0.213346i \(0.0684362\pi\)
−0.976977 + 0.213346i \(0.931564\pi\)
\(14\) 0 0
\(15\) 96.0000 1.65247
\(16\) 0 0
\(17\) −98.0000 −1.39815 −0.699073 − 0.715050i \(-0.746404\pi\)
−0.699073 + 0.715050i \(0.746404\pi\)
\(18\) 0 0
\(19\) 88.0000i 1.06256i 0.847197 + 0.531279i \(0.178288\pi\)
−0.847197 + 0.531279i \(0.821712\pi\)
\(20\) 0 0
\(21\) 256.000i 2.66018i
\(22\) 0 0
\(23\) 32.0000 0.290107 0.145054 − 0.989424i \(-0.453665\pi\)
0.145054 + 0.989424i \(0.453665\pi\)
\(24\) 0 0
\(25\) −19.0000 −0.152000
\(26\) 0 0
\(27\) 80.0000i 0.570222i
\(28\) 0 0
\(29\) − 172.000i − 1.10137i −0.834715 − 0.550683i \(-0.814367\pi\)
0.834715 − 0.550683i \(-0.185633\pi\)
\(30\) 0 0
\(31\) −256.000 −1.48319 −0.741596 − 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(32\) 0 0
\(33\) −64.0000 −0.337605
\(34\) 0 0
\(35\) − 384.000i − 1.85451i
\(36\) 0 0
\(37\) 92.0000i 0.408776i 0.978890 + 0.204388i \(0.0655204\pi\)
−0.978890 + 0.204388i \(0.934480\pi\)
\(38\) 0 0
\(39\) 160.000 0.656936
\(40\) 0 0
\(41\) −102.000 −0.388530 −0.194265 − 0.980949i \(-0.562232\pi\)
−0.194265 + 0.980949i \(0.562232\pi\)
\(42\) 0 0
\(43\) − 296.000i − 1.04976i −0.851177 − 0.524879i \(-0.824111\pi\)
0.851177 − 0.524879i \(-0.175889\pi\)
\(44\) 0 0
\(45\) − 444.000i − 1.47084i
\(46\) 0 0
\(47\) −320.000 −0.993123 −0.496562 − 0.868001i \(-0.665404\pi\)
−0.496562 + 0.868001i \(0.665404\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 784.000i 2.15259i
\(52\) 0 0
\(53\) 76.0000i 0.196970i 0.995139 + 0.0984849i \(0.0313996\pi\)
−0.995139 + 0.0984849i \(0.968600\pi\)
\(54\) 0 0
\(55\) 96.0000 0.235357
\(56\) 0 0
\(57\) 704.000 1.63591
\(58\) 0 0
\(59\) 408.000i 0.900289i 0.892956 + 0.450145i \(0.148628\pi\)
−0.892956 + 0.450145i \(0.851372\pi\)
\(60\) 0 0
\(61\) − 636.000i − 1.33494i −0.744636 − 0.667471i \(-0.767377\pi\)
0.744636 − 0.667471i \(-0.232623\pi\)
\(62\) 0 0
\(63\) 1184.00 2.36778
\(64\) 0 0
\(65\) −240.000 −0.457974
\(66\) 0 0
\(67\) − 552.000i − 1.00653i −0.864132 − 0.503265i \(-0.832132\pi\)
0.864132 − 0.503265i \(-0.167868\pi\)
\(68\) 0 0
\(69\) − 256.000i − 0.446649i
\(70\) 0 0
\(71\) −416.000 −0.695354 −0.347677 − 0.937614i \(-0.613029\pi\)
−0.347677 + 0.937614i \(0.613029\pi\)
\(72\) 0 0
\(73\) −138.000 −0.221256 −0.110628 − 0.993862i \(-0.535286\pi\)
−0.110628 + 0.993862i \(0.535286\pi\)
\(74\) 0 0
\(75\) 152.000i 0.234019i
\(76\) 0 0
\(77\) 256.000i 0.378882i
\(78\) 0 0
\(79\) −64.0000 −0.0911464 −0.0455732 − 0.998961i \(-0.514511\pi\)
−0.0455732 + 0.998961i \(0.514511\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 0 0
\(83\) − 392.000i − 0.518405i −0.965823 − 0.259202i \(-0.916540\pi\)
0.965823 − 0.259202i \(-0.0834597\pi\)
\(84\) 0 0
\(85\) − 1176.00i − 1.50065i
\(86\) 0 0
\(87\) −1376.00 −1.69566
\(88\) 0 0
\(89\) 582.000 0.693167 0.346584 − 0.938019i \(-0.387342\pi\)
0.346584 + 0.938019i \(0.387342\pi\)
\(90\) 0 0
\(91\) − 640.000i − 0.737255i
\(92\) 0 0
\(93\) 2048.00i 2.28352i
\(94\) 0 0
\(95\) −1056.00 −1.14046
\(96\) 0 0
\(97\) 238.000 0.249126 0.124563 − 0.992212i \(-0.460247\pi\)
0.124563 + 0.992212i \(0.460247\pi\)
\(98\) 0 0
\(99\) 296.000i 0.300496i
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 128.4.b.a.65.1 ✓ 2
3.2 odd 2 1152.4.d.a.577.1 2
4.3 odd 2 128.4.b.d.65.2 yes 2
8.3 odd 2 128.4.b.d.65.1 yes 2
8.5 even 2 inner 128.4.b.a.65.2 yes 2
12.11 even 2 1152.4.d.h.577.1 2
16.3 odd 4 256.4.a.c.1.1 1
16.5 even 4 256.4.a.b.1.1 1
16.11 odd 4 256.4.a.f.1.1 1
16.13 even 4 256.4.a.g.1.1 1
24.5 odd 2 1152.4.d.a.577.2 2
24.11 even 2 1152.4.d.h.577.2 2
48.5 odd 4 2304.4.a.n.1.1 1
48.11 even 4 2304.4.a.m.1.1 1
48.29 odd 4 2304.4.a.d.1.1 1
48.35 even 4 2304.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.a.65.1 ✓ 2 1.1 even 1 trivial
128.4.b.a.65.2 yes 2 8.5 even 2 inner
128.4.b.d.65.1 yes 2 8.3 odd 2
128.4.b.d.65.2 yes 2 4.3 odd 2
256.4.a.b.1.1 1 16.5 even 4
256.4.a.c.1.1 1 16.3 odd 4
256.4.a.f.1.1 1 16.11 odd 4
256.4.a.g.1.1 1 16.13 even 4
1152.4.d.a.577.1 2 3.2 odd 2
1152.4.d.a.577.2 2 24.5 odd 2
1152.4.d.h.577.1 2 12.11 even 2
1152.4.d.h.577.2 2 24.11 even 2
2304.4.a.c.1.1 1 48.35 even 4
2304.4.a.d.1.1 1 48.29 odd 4
2304.4.a.m.1.1 1 48.11 even 4
2304.4.a.n.1.1 1 48.5 odd 4