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Properties

Label 112.2.m.c.29.4
Level $112$
Weight $2$
Character 112.29
Analytic conductor $0.894$
Analytic rank $0$
Dimension $8$
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] = [112,2,Mod(29,112)] 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("112.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.214798336.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 29.4
Root \(1.41216 - 0.0762223i\) of defining polynomial
Character \(\chi\) \(=\) 112.29
Dual form 112.2.m.c.85.4

$q$-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+(1.29150 + 0.576222i) q^{2} +(-0.715276 - 0.715276i) q^{3} +(1.33594 + 1.48838i) q^{4} +(0.867721 - 0.867721i) q^{5} +(-0.511620 - 1.33594i) q^{6} +1.00000i q^{7} +(0.867721 + 2.69204i) q^{8} -1.97676i q^{9} +(1.62066 - 0.620660i) q^{10} +(-2.97676 + 2.97676i) q^{11} +(0.109040 - 2.02017i) q^{12} +(-2.02017 - 2.02017i) q^{13} +(-0.576222 + 1.29150i) q^{14} -1.24132 q^{15} +(-0.430552 + 3.97676i) q^{16} +0.264559 q^{17} +(1.13905 - 2.55298i) q^{18} +(-4.53959 - 4.53959i) q^{19} +(2.45072 + 0.132279i) q^{20} +(0.715276 - 0.715276i) q^{21} +(-5.55976 + 2.12921i) q^{22} -1.54621i q^{23} +(1.30489 - 2.54621i) q^{24} +3.49412i q^{25} +(-1.44498 - 3.77310i) q^{26} +(-3.55976 + 3.55976i) q^{27} +(-1.48838 + 1.33594i) q^{28} +(-0.328129 - 0.328129i) q^{29} +(-1.60316 - 0.715276i) q^{30} +6.04033 q^{31} +(-2.84756 + 4.88789i) q^{32} +4.25841 q^{33} +(0.341677 + 0.152445i) q^{34} +(0.867721 + 0.867721i) q^{35} +(2.94217 - 2.64082i) q^{36} +(6.64863 - 6.64863i) q^{37} +(-3.24706 - 8.47869i) q^{38} +2.88995i q^{39} +(3.08887 + 1.58300i) q^{40} +11.0327i q^{41} +(1.33594 - 0.511620i) q^{42} +(3.38407 - 3.38407i) q^{43} +(-8.40731 - 0.453791i) q^{44} +(-1.71528 - 1.71528i) q^{45} +(0.890960 - 1.99693i) q^{46} +3.12566 q^{47} +(3.15244 - 2.53652i) q^{48} -1.00000 q^{49} +(-2.01339 + 4.51265i) q^{50} +(-0.189233 - 0.189233i) q^{51} +(0.307963 - 5.70559i) q^{52} +(0.430552 - 0.430552i) q^{53} +(-6.64863 + 2.54621i) q^{54} +5.16599i q^{55} +(-2.69204 + 0.867721i) q^{56} +6.49412i q^{57} +(-0.234703 - 0.612853i) q^{58} +(-4.62640 + 4.62640i) q^{59} +(-1.65832 - 1.84756i) q^{60} +(4.86772 + 4.86772i) q^{61} +(7.80108 + 3.48057i) q^{62} +1.97676 q^{63} +(-6.49412 + 4.67187i) q^{64} -3.50588 q^{65} +(5.49973 + 2.45379i) q^{66} +(3.34374 + 3.34374i) q^{67} +(0.353433 + 0.393764i) q^{68} +(-1.10597 + 1.10597i) q^{69} +(0.620660 + 1.62066i) q^{70} -9.03885i q^{71} +(5.32151 - 1.71528i) q^{72} -14.8146i q^{73} +(12.4178 - 4.75561i) q^{74} +(2.49926 - 2.49926i) q^{75} +(0.692037 - 12.8212i) q^{76} +(-2.97676 - 2.97676i) q^{77} +(-1.66525 + 3.73237i) q^{78} +12.5904 q^{79} +(3.07712 + 3.82432i) q^{80} -0.837864 q^{81} +(-6.35729 + 14.2487i) q^{82} +(-0.715276 - 0.715276i) q^{83} +(2.02017 + 0.109040i) q^{84} +(0.229563 - 0.229563i) q^{85} +(6.32050 - 2.42055i) q^{86} +0.469405i q^{87} +(-10.5965 - 5.43055i) q^{88} +10.9924i q^{89} +(-1.22690 - 3.20366i) q^{90} +(2.02017 - 2.02017i) q^{91} +(2.30135 - 2.06564i) q^{92} +(-4.32050 - 4.32050i) q^{93} +(4.03679 + 1.80108i) q^{94} -7.87820 q^{95} +(5.53298 - 1.45940i) q^{96} -14.2452 q^{97} +(-1.29150 - 0.576222i) q^{98} +(5.88434 + 5.88434i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{4} - 4 q^{5} - 16 q^{6} - 4 q^{8} + 12 q^{10} - 12 q^{12} - 2 q^{14} - 8 q^{15} + 8 q^{16} + 24 q^{17} + 6 q^{18} - 12 q^{19} - 8 q^{20} - 4 q^{22} + 8 q^{26} + 12 q^{27} - 16 q^{29}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29150 + 0.576222i 0.913227 + 0.407451i
\(3\) −0.715276 − 0.715276i −0.412965 − 0.412965i 0.469805 − 0.882770i \(-0.344324\pi\)
−0.882770 + 0.469805i \(0.844324\pi\)
\(4\) 1.33594 + 1.48838i 0.667968 + 0.744190i
\(5\) 0.867721 − 0.867721i 0.388056 − 0.388056i −0.485937 − 0.873994i \(-0.661522\pi\)
0.873994 + 0.485937i \(0.161522\pi\)
\(6\) −0.511620 − 1.33594i −0.208868 − 0.545393i
\(7\) 1.00000i 0.377964i
\(8\) 0.867721 + 2.69204i 0.306786 + 0.951779i
\(9\) − 1.97676i − 0.658920i
\(10\) 1.62066 − 0.620660i 0.512498 − 0.196270i
\(11\) −2.97676 + 2.97676i −0.897527 + 0.897527i −0.995217 − 0.0976898i \(-0.968855\pi\)
0.0976898 + 0.995217i \(0.468855\pi\)
\(12\) 0.109040 − 2.02017i 0.0314771 − 0.583171i
\(13\) −2.02017 − 2.02017i −0.560293 − 0.560293i 0.369098 − 0.929391i \(-0.379667\pi\)
−0.929391 + 0.369098i \(0.879667\pi\)
\(14\) −0.576222 + 1.29150i −0.154002 + 0.345167i
\(15\) −1.24132 −0.320507
\(16\) −0.430552 + 3.97676i −0.107638 + 0.994190i
\(17\) 0.264559 0.0641649 0.0320825 − 0.999485i \(-0.489786\pi\)
0.0320825 + 0.999485i \(0.489786\pi\)
\(18\) 1.13905 − 2.55298i 0.268478 − 0.601744i
\(19\) −4.53959 − 4.53959i −1.04145 − 1.04145i −0.999103 − 0.0423510i \(-0.986515\pi\)
−0.0423510 − 0.999103i \(-0.513485\pi\)
\(20\) 2.45072 + 0.132279i 0.547997 + 0.0295786i
\(21\) 0.715276 − 0.715276i 0.156086 − 0.156086i
\(22\) −5.55976 + 2.12921i −1.18534 + 0.453948i
\(23\) − 1.54621i − 0.322407i −0.986921 − 0.161203i \(-0.948462\pi\)
0.986921 − 0.161203i \(-0.0515375\pi\)
\(24\) 1.30489 − 2.54621i 0.266359 − 0.519743i
\(25\) 3.49412i 0.698824i
\(26\) −1.44498 − 3.77310i −0.283383 − 0.739967i
\(27\) −3.55976 + 3.55976i −0.685076 + 0.685076i
\(28\) −1.48838 + 1.33594i −0.281277 + 0.252468i
\(29\) −0.328129 − 0.328129i −0.0609320 − 0.0609320i 0.675984 − 0.736916i \(-0.263719\pi\)
−0.736916 + 0.675984i \(0.763719\pi\)
\(30\) −1.60316 − 0.715276i −0.292696 − 0.130591i
\(31\) 6.04033 1.08488 0.542438 − 0.840096i \(-0.317501\pi\)
0.542438 + 0.840096i \(0.317501\pi\)
\(32\) −2.84756 + 4.88789i −0.503381 + 0.864064i
\(33\) 4.25841 0.741294
\(34\) 0.341677 + 0.152445i 0.0585972 + 0.0261440i
\(35\) 0.867721 + 0.867721i 0.146672 + 0.146672i
\(36\) 2.94217 − 2.64082i 0.490362 − 0.440137i
\(37\) 6.64863 − 6.64863i 1.09303 − 1.09303i 0.0978247 − 0.995204i \(-0.468812\pi\)
0.995204 − 0.0978247i \(-0.0311884\pi\)
\(38\) −3.24706 − 8.47869i −0.526743 − 1.37543i
\(39\) 2.88995i 0.462763i
\(40\) 3.08887 + 1.58300i 0.488394 + 0.250294i
\(41\) 11.0327i 1.72302i 0.507741 + 0.861510i \(0.330481\pi\)
−0.507741 + 0.861510i \(0.669519\pi\)
\(42\) 1.33594 − 0.511620i 0.206139 − 0.0789446i
\(43\) 3.38407 − 3.38407i 0.516066 − 0.516066i −0.400312 − 0.916379i \(-0.631098\pi\)
0.916379 + 0.400312i \(0.131098\pi\)
\(44\) −8.40731 − 0.453791i −1.26745 − 0.0684116i
\(45\) −1.71528 − 1.71528i −0.255698 − 0.255698i
\(46\) 0.890960 − 1.99693i 0.131365 − 0.294431i
\(47\) 3.12566 0.455925 0.227962 − 0.973670i \(-0.426794\pi\)
0.227962 + 0.973670i \(0.426794\pi\)
\(48\) 3.15244 − 2.53652i 0.455016 − 0.366115i
\(49\) −1.00000 −0.142857
\(50\) −2.01339 + 4.51265i −0.284736 + 0.638185i
\(51\) −0.189233 − 0.189233i −0.0264979 − 0.0264979i
\(52\) 0.307963 − 5.70559i 0.0427068 − 0.791222i
\(53\) 0.430552 − 0.430552i 0.0591409 − 0.0591409i −0.676918 − 0.736059i \(-0.736685\pi\)
0.736059 + 0.676918i \(0.236685\pi\)
\(54\) −6.64863 + 2.54621i −0.904764 + 0.346495i
\(55\) 5.16599i 0.696582i
\(56\) −2.69204 + 0.867721i −0.359739 + 0.115954i
\(57\) 6.49412i 0.860167i
\(58\) −0.234703 − 0.612853i −0.0308180 − 0.0804715i
\(59\) −4.62640 + 4.62640i −0.602306 + 0.602306i −0.940924 − 0.338618i \(-0.890041\pi\)
0.338618 + 0.940924i \(0.390041\pi\)
\(60\) −1.65832 − 1.84756i −0.214089 − 0.238518i
\(61\) 4.86772 + 4.86772i 0.623248 + 0.623248i 0.946360 − 0.323113i \(-0.104729\pi\)
−0.323113 + 0.946360i \(0.604729\pi\)
\(62\) 7.80108 + 3.48057i 0.990738 + 0.442033i
\(63\) 1.97676 0.249048
\(64\) −6.49412 + 4.67187i −0.811765 + 0.583984i
\(65\) −3.50588 −0.434851
\(66\) 5.49973 + 2.45379i 0.676970 + 0.302041i
\(67\) 3.34374 + 3.34374i 0.408503 + 0.408503i 0.881216 − 0.472713i \(-0.156725\pi\)
−0.472713 + 0.881216i \(0.656725\pi\)
\(68\) 0.353433 + 0.393764i 0.0428601 + 0.0477509i
\(69\) −1.10597 + 1.10597i −0.133143 + 0.133143i
\(70\) 0.620660 + 1.62066i 0.0741830 + 0.193706i
\(71\) − 9.03885i − 1.07271i −0.843991 − 0.536357i \(-0.819800\pi\)
0.843991 − 0.536357i \(-0.180200\pi\)
\(72\) 5.32151 − 1.71528i 0.627146 − 0.202147i
\(73\) − 14.8146i − 1.73392i −0.498377 − 0.866960i \(-0.666070\pi\)
0.498377 − 0.866960i \(-0.333930\pi\)
\(74\) 12.4178 − 4.75561i 1.44354 − 0.552828i
\(75\) 2.49926 − 2.49926i 0.288590 − 0.288590i
\(76\) 0.692037 − 12.8212i 0.0793820 − 1.47070i
\(77\) −2.97676 − 2.97676i −0.339233 − 0.339233i
\(78\) −1.66525 + 3.73237i −0.188553 + 0.422607i
\(79\) 12.5904 1.41653 0.708265 − 0.705947i \(-0.249478\pi\)
0.708265 + 0.705947i \(0.249478\pi\)
\(80\) 3.07712 + 3.82432i 0.344032 + 0.427572i
\(81\) −0.837864 −0.0930960
\(82\) −6.35729 + 14.2487i −0.702045 + 1.57351i
\(83\) −0.715276 − 0.715276i −0.0785117 − 0.0785117i 0.666760 − 0.745272i \(-0.267680\pi\)
−0.745272 + 0.666760i \(0.767680\pi\)
\(84\) 2.02017 + 0.109040i 0.220418 + 0.0118972i
\(85\) 0.229563 − 0.229563i 0.0248996 − 0.0248996i
\(86\) 6.32050 − 2.42055i 0.681557 − 0.261014i
\(87\) 0.469405i 0.0503255i
\(88\) −10.5965 − 5.43055i −1.12960 − 0.578899i
\(89\) 10.9924i 1.16519i 0.812763 + 0.582595i \(0.197962\pi\)
−0.812763 + 0.582595i \(0.802038\pi\)
\(90\) −1.22690 − 3.20366i −0.129326 − 0.337695i
\(91\) 2.02017 − 2.02017i 0.211771 − 0.211771i
\(92\) 2.30135 − 2.06564i 0.239932 − 0.215357i
\(93\) −4.32050 − 4.32050i −0.448015 − 0.448015i
\(94\) 4.03679 + 1.80108i 0.416363 + 0.185767i
\(95\) −7.87820 −0.808286
\(96\) 5.53298 − 1.45940i 0.564707 − 0.148949i
\(97\) −14.2452 −1.44638 −0.723189 − 0.690650i \(-0.757325\pi\)
−0.723189 + 0.690650i \(0.757325\pi\)
\(98\) −1.29150 − 0.576222i −0.130461 − 0.0582072i
\(99\) 5.88434 + 5.88434i 0.591399 + 0.591399i
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 112.2.m.c.29.4 ✓ 8
4.3 odd 2 448.2.m.c.337.3 8
7.2 even 3 784.2.x.k.557.2 16
7.3 odd 6 784.2.x.j.765.2 16
7.4 even 3 784.2.x.k.765.2 16
7.5 odd 6 784.2.x.j.557.2 16
7.6 odd 2 784.2.m.g.589.4 8
8.3 odd 2 896.2.m.f.673.2 8
8.5 even 2 896.2.m.e.673.3 8
16.3 odd 4 896.2.m.f.225.2 8
16.5 even 4 inner 112.2.m.c.85.4 yes 8
16.11 odd 4 448.2.m.c.113.3 8
16.13 even 4 896.2.m.e.225.3 8
32.5 even 8 7168.2.a.bc.1.5 8
32.11 odd 8 7168.2.a.bd.1.5 8
32.21 even 8 7168.2.a.bc.1.4 8
32.27 odd 8 7168.2.a.bd.1.4 8
112.5 odd 12 784.2.x.j.165.2 16
112.37 even 12 784.2.x.k.165.2 16
112.53 even 12 784.2.x.k.373.2 16
112.69 odd 4 784.2.m.g.197.4 8
112.101 odd 12 784.2.x.j.373.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.4 ✓ 8 1.1 even 1 trivial
112.2.m.c.85.4 yes 8 16.5 even 4 inner
448.2.m.c.113.3 8 16.11 odd 4
448.2.m.c.337.3 8 4.3 odd 2
784.2.m.g.197.4 8 112.69 odd 4
784.2.m.g.589.4 8 7.6 odd 2
784.2.x.j.165.2 16 112.5 odd 12
784.2.x.j.373.2 16 112.101 odd 12
784.2.x.j.557.2 16 7.5 odd 6
784.2.x.j.765.2 16 7.3 odd 6
784.2.x.k.165.2 16 112.37 even 12
784.2.x.k.373.2 16 112.53 even 12
784.2.x.k.557.2 16 7.2 even 3
784.2.x.k.765.2 16 7.4 even 3
896.2.m.e.225.3 8 16.13 even 4
896.2.m.e.673.3 8 8.5 even 2
896.2.m.f.225.2 8 16.3 odd 4
896.2.m.f.673.2 8 8.3 odd 2
7168.2.a.bc.1.4 8 32.21 even 8
7168.2.a.bc.1.5 8 32.5 even 8
7168.2.a.bd.1.4 8 32.27 odd 8
7168.2.a.bd.1.5 8 32.11 odd 8