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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(97,112)] 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("112.97"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 112.97
Dual form 112.5.c.b.97.2

$q$-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)
 
\(f(q)\) \(=\) \(q-6.92820i q^{3} +20.7846i q^{5} +(7.00000 - 48.4974i) q^{7} +33.0000 q^{9} -18.0000 q^{11} -131.636i q^{13} +144.000 q^{15} -415.692i q^{17} -90.0666i q^{19} +(-336.000 - 48.4974i) q^{21} -738.000 q^{23} +193.000 q^{25} -789.815i q^{27} -846.000 q^{29} -1163.94i q^{31} +124.708i q^{33} +(1008.00 + 145.492i) q^{35} +2386.00 q^{37} -912.000 q^{39} -1704.34i q^{41} +2510.00 q^{43} +685.892i q^{45} +3408.68i q^{47} +(-2303.00 - 678.964i) q^{49} -2880.00 q^{51} -270.000 q^{53} -374.123i q^{55} -624.000 q^{57} +3138.48i q^{59} +6491.73i q^{61} +(231.000 - 1600.41i) q^{63} +2736.00 q^{65} -2450.00 q^{67} +5113.01i q^{69} +3150.00 q^{71} +235.559i q^{73} -1337.14i q^{75} +(-126.000 + 872.954i) q^{77} +3982.00 q^{79} -2799.00 q^{81} +5009.09i q^{83} +8640.00 q^{85} +5861.26i q^{87} -7607.17i q^{89} +(-6384.00 - 921.451i) q^{91} -8064.00 q^{93} +1872.00 q^{95} -12581.6i q^{97} -594.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{7} + 66 q^{9} - 36 q^{11} + 288 q^{15} - 672 q^{21} - 1476 q^{23} + 386 q^{25} - 1692 q^{29} + 2016 q^{35} + 4772 q^{37} - 1824 q^{39} + 5020 q^{43} - 4606 q^{49} - 5760 q^{51} - 540 q^{53}+ \cdots - 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 6.92820i − 0.769800i −0.922958 − 0.384900i \(-0.874236\pi\)
0.922958 − 0.384900i \(-0.125764\pi\)
\(4\) 0 0
\(5\) 20.7846i 0.831384i 0.909505 + 0.415692i \(0.136461\pi\)
−0.909505 + 0.415692i \(0.863539\pi\)
\(6\) 0 0
\(7\) 7.00000 − 48.4974i 0.142857 − 0.989743i
\(8\) 0 0
\(9\) 33.0000 0.407407
\(10\) 0 0
\(11\) −18.0000 −0.148760 −0.0743802 − 0.997230i \(-0.523698\pi\)
−0.0743802 + 0.997230i \(0.523698\pi\)
\(12\) 0 0
\(13\) − 131.636i − 0.778910i −0.921045 − 0.389455i \(-0.872663\pi\)
0.921045 − 0.389455i \(-0.127337\pi\)
\(14\) 0 0
\(15\) 144.000 0.640000
\(16\) 0 0
\(17\) − 415.692i − 1.43838i −0.694813 − 0.719191i \(-0.744513\pi\)
0.694813 − 0.719191i \(-0.255487\pi\)
\(18\) 0 0
\(19\) − 90.0666i − 0.249492i −0.992189 − 0.124746i \(-0.960188\pi\)
0.992189 − 0.124746i \(-0.0398116\pi\)
\(20\) 0 0
\(21\) −336.000 − 48.4974i −0.761905 − 0.109971i
\(22\) 0 0
\(23\) −738.000 −1.39509 −0.697543 − 0.716543i \(-0.745723\pi\)
−0.697543 + 0.716543i \(0.745723\pi\)
\(24\) 0 0
\(25\) 193.000 0.308800
\(26\) 0 0
\(27\) − 789.815i − 1.08342i
\(28\) 0 0
\(29\) −846.000 −1.00595 −0.502973 − 0.864302i \(-0.667760\pi\)
−0.502973 + 0.864302i \(0.667760\pi\)
\(30\) 0 0
\(31\) − 1163.94i − 1.21117i −0.795779 − 0.605587i \(-0.792938\pi\)
0.795779 − 0.605587i \(-0.207062\pi\)
\(32\) 0 0
\(33\) 124.708i 0.114516i
\(34\) 0 0
\(35\) 1008.00 + 145.492i 0.822857 + 0.118769i
\(36\) 0 0
\(37\) 2386.00 1.74288 0.871439 − 0.490504i \(-0.163187\pi\)
0.871439 + 0.490504i \(0.163187\pi\)
\(38\) 0 0
\(39\) −912.000 −0.599606
\(40\) 0 0
\(41\) − 1704.34i − 1.01388i −0.861980 − 0.506942i \(-0.830776\pi\)
0.861980 − 0.506942i \(-0.169224\pi\)
\(42\) 0 0
\(43\) 2510.00 1.35749 0.678745 − 0.734374i \(-0.262524\pi\)
0.678745 + 0.734374i \(0.262524\pi\)
\(44\) 0 0
\(45\) 685.892i 0.338712i
\(46\) 0 0
\(47\) 3408.68i 1.54309i 0.636177 + 0.771543i \(0.280515\pi\)
−0.636177 + 0.771543i \(0.719485\pi\)
\(48\) 0 0
\(49\) −2303.00 − 678.964i −0.959184 − 0.282784i
\(50\) 0 0
\(51\) −2880.00 −1.10727
\(52\) 0 0
\(53\) −270.000 −0.0961196 −0.0480598 − 0.998844i \(-0.515304\pi\)
−0.0480598 + 0.998844i \(0.515304\pi\)
\(54\) 0 0
\(55\) − 374.123i − 0.123677i
\(56\) 0 0
\(57\) −624.000 −0.192059
\(58\) 0 0
\(59\) 3138.48i 0.901602i 0.892625 + 0.450801i \(0.148862\pi\)
−0.892625 + 0.450801i \(0.851138\pi\)
\(60\) 0 0
\(61\) 6491.73i 1.74462i 0.488954 + 0.872309i \(0.337378\pi\)
−0.488954 + 0.872309i \(0.662622\pi\)
\(62\) 0 0
\(63\) 231.000 − 1600.41i 0.0582011 − 0.403229i
\(64\) 0 0
\(65\) 2736.00 0.647574
\(66\) 0 0
\(67\) −2450.00 −0.545779 −0.272889 − 0.962045i \(-0.587979\pi\)
−0.272889 + 0.962045i \(0.587979\pi\)
\(68\) 0 0
\(69\) 5113.01i 1.07394i
\(70\) 0 0
\(71\) 3150.00 0.624876 0.312438 − 0.949938i \(-0.398854\pi\)
0.312438 + 0.949938i \(0.398854\pi\)
\(72\) 0 0
\(73\) 235.559i 0.0442032i 0.999756 + 0.0221016i \(0.00703573\pi\)
−0.999756 + 0.0221016i \(0.992964\pi\)
\(74\) 0 0
\(75\) − 1337.14i − 0.237714i
\(76\) 0 0
\(77\) −126.000 + 872.954i −0.0212515 + 0.147235i
\(78\) 0 0
\(79\) 3982.00 0.638039 0.319019 − 0.947748i \(-0.396646\pi\)
0.319019 + 0.947748i \(0.396646\pi\)
\(80\) 0 0
\(81\) −2799.00 −0.426612
\(82\) 0 0
\(83\) 5009.09i 0.727114i 0.931572 + 0.363557i \(0.118438\pi\)
−0.931572 + 0.363557i \(0.881562\pi\)
\(84\) 0 0
\(85\) 8640.00 1.19585
\(86\) 0 0
\(87\) 5861.26i 0.774377i
\(88\) 0 0
\(89\) − 7607.17i − 0.960380i −0.877165 − 0.480190i \(-0.840568\pi\)
0.877165 − 0.480190i \(-0.159432\pi\)
\(90\) 0 0
\(91\) −6384.00 − 921.451i −0.770921 − 0.111273i
\(92\) 0 0
\(93\) −8064.00 −0.932362
\(94\) 0 0
\(95\) 1872.00 0.207424
\(96\) 0 0
\(97\) − 12581.6i − 1.33719i −0.743627 − 0.668595i \(-0.766896\pi\)
0.743627 − 0.668595i \(-0.233104\pi\)
\(98\) 0 0
\(99\) −594.000 −0.0606061
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 112.5.c.b.97.1 2
3.2 odd 2 1008.5.f.c.433.1 2
4.3 odd 2 28.5.b.a.13.2 yes 2
7.6 odd 2 inner 112.5.c.b.97.2 2
8.3 odd 2 448.5.c.c.321.1 2
8.5 even 2 448.5.c.d.321.2 2
12.11 even 2 252.5.d.a.181.1 2
20.3 even 4 700.5.h.a.349.2 4
20.7 even 4 700.5.h.a.349.3 4
20.19 odd 2 700.5.d.a.601.1 2
21.20 even 2 1008.5.f.c.433.2 2
28.3 even 6 196.5.h.a.117.1 2
28.11 odd 6 196.5.h.b.117.1 2
28.19 even 6 196.5.h.b.129.1 2
28.23 odd 6 196.5.h.a.129.1 2
28.27 even 2 28.5.b.a.13.1 ✓ 2
56.13 odd 2 448.5.c.d.321.1 2
56.27 even 2 448.5.c.c.321.2 2
84.83 odd 2 252.5.d.a.181.2 2
140.27 odd 4 700.5.h.a.349.1 4
140.83 odd 4 700.5.h.a.349.4 4
140.139 even 2 700.5.d.a.601.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.b.a.13.1 ✓ 2 28.27 even 2
28.5.b.a.13.2 yes 2 4.3 odd 2
112.5.c.b.97.1 2 1.1 even 1 trivial
112.5.c.b.97.2 2 7.6 odd 2 inner
196.5.h.a.117.1 2 28.3 even 6
196.5.h.a.129.1 2 28.23 odd 6
196.5.h.b.117.1 2 28.11 odd 6
196.5.h.b.129.1 2 28.19 even 6
252.5.d.a.181.1 2 12.11 even 2
252.5.d.a.181.2 2 84.83 odd 2
448.5.c.c.321.1 2 8.3 odd 2
448.5.c.c.321.2 2 56.27 even 2
448.5.c.d.321.1 2 56.13 odd 2
448.5.c.d.321.2 2 8.5 even 2
700.5.d.a.601.1 2 20.19 odd 2
700.5.d.a.601.2 2 140.139 even 2
700.5.h.a.349.1 4 140.27 odd 4
700.5.h.a.349.2 4 20.3 even 4
700.5.h.a.349.3 4 20.7 even 4
700.5.h.a.349.4 4 140.83 odd 4
1008.5.f.c.433.1 2 3.2 odd 2
1008.5.f.c.433.2 2 21.20 even 2