↑

Properties

Label 784.2.m.g.197.4
Level $784$
Weight $2$
Character 784.197
Analytic conductor $6.260$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,2,Mod(197,784)] 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("784.197"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,0,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
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: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 197.4
Root \(1.41216 + 0.0762223i\) of defining polynomial
Character \(\chi\) \(=\) 784.197
Dual form 784.2.m.g.589.4

$q$-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} +(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} -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} +(-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} +(-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} +(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} +(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} +(-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} -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} +(-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} +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} +(-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} +(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} +(2.30135 + 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} +(5.88434 - 5.88434i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{4} + 4 q^{5} + 16 q^{6} - 4 q^{8} - 12 q^{10} + 12 q^{12} - 8 q^{15} + 8 q^{16} - 24 q^{17} + 6 q^{18} + 12 q^{19} + 8 q^{20} - 4 q^{22} - 8 q^{26} - 12 q^{27} - 16 q^{29} + 20 q^{30}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29150 − 0.576222i 0.913227 − 0.407451i
\(3\) 0.715276 − 0.715276i 0.412965 − 0.412965i −0.469805 − 0.882770i \(-0.655676\pi\)
0.882770 + 0.469805i \(0.155676\pi\)
\(4\) 1.33594 − 1.48838i 0.667968 − 0.744190i
\(5\) −0.867721 − 0.867721i −0.388056 − 0.388056i 0.485937 − 0.873994i \(-0.338478\pi\)
−0.873994 + 0.485937i \(0.838478\pi\)
\(6\) 0.511620 − 1.33594i 0.208868 − 0.545393i
\(7\) 0 0
\(8\) 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 0.0976898 − 0.995217i \(-0.468855\pi\)
−0.995217 + 0.0976898i \(0.968855\pi\)
\(12\) −0.109040 − 2.02017i −0.0314771 − 0.583171i
\(13\) 2.02017 − 2.02017i 0.560293 − 0.560293i −0.369098 − 0.929391i \(-0.620333\pi\)
0.929391 + 0.369098i \(0.120333\pi\)
\(14\) 0 0
\(15\) −1.24132 −0.320507
\(16\) −0.430552 − 3.97676i −0.107638 − 0.994190i
\(17\) −0.264559 −0.0641649 −0.0320825 − 0.999485i \(-0.510214\pi\)
−0.0320825 + 0.999485i \(0.510214\pi\)
\(18\) 1.13905 + 2.55298i 0.268478 + 0.601744i
\(19\) 4.53959 − 4.53959i 1.04145 − 1.04145i 0.0423510 − 0.999103i \(-0.486515\pi\)
0.999103 − 0.0423510i \(-0.0134848\pi\)
\(20\) −2.45072 + 0.132279i −0.547997 + 0.0295786i
\(21\) 0 0
\(22\) −5.55976 − 2.12921i −1.18534 − 0.453948i
\(23\) 1.54621i 0.322407i 0.986921 + 0.161203i \(0.0515375\pi\)
−0.986921 + 0.161203i \(0.948462\pi\)
\(24\) −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
\(28\) 0 0
\(29\) −0.328129 + 0.328129i −0.0609320 + 0.0609320i −0.736916 − 0.675984i \(-0.763719\pi\)
0.675984 + 0.736916i \(0.263719\pi\)
\(30\) −1.60316 + 0.715276i −0.292696 + 0.130591i
\(31\) −6.04033 −1.08488 −0.542438 − 0.840096i \(-0.682499\pi\)
−0.542438 + 0.840096i \(0.682499\pi\)
\(32\) −2.84756 − 4.88789i −0.503381 − 0.864064i
\(33\) −4.25841 −0.741294
\(34\) −0.341677 + 0.152445i −0.0585972 + 0.0261440i
\(35\) 0 0
\(36\) 2.94217 + 2.64082i 0.490362 + 0.440137i
\(37\) 6.64863 + 6.64863i 1.09303 + 1.09303i 0.995204 + 0.0978247i \(0.0311884\pi\)
0.0978247 + 0.995204i \(0.468812\pi\)
\(38\) 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 \(0.669519\pi\)
\(42\) 0 0
\(43\) 3.38407 + 3.38407i 0.516066 + 0.516066i 0.916379 − 0.400312i \(-0.131098\pi\)
−0.400312 + 0.916379i \(0.631098\pi\)
\(44\) −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 \(-0.573206\pi\)
−0.227962 + 0.973670i \(0.573206\pi\)
\(48\) −3.15244 − 2.53652i −0.455016 − 0.366115i
\(49\) 0 0
\(50\) −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 0.736059 − 0.676918i \(-0.236685\pi\)
−0.676918 + 0.736059i \(0.736685\pi\)
\(54\) 6.64863 + 2.54621i 0.904764 + 0.346495i
\(55\) 5.16599i 0.696582i
\(56\) 0 0
\(57\) − 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 \(-0.109959\pi\)
−0.338618 + 0.940924i \(0.609959\pi\)
\(60\) −1.65832 + 1.84756i −0.214089 + 0.238518i
\(61\) −4.86772 + 4.86772i −0.623248 + 0.623248i −0.946360 − 0.323113i \(-0.895271\pi\)
0.323113 + 0.946360i \(0.395271\pi\)
\(62\) −7.80108 + 3.48057i −0.990738 + 0.442033i
\(63\) 0 0
\(64\) −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 −0.472713 − 0.881216i \(-0.656725\pi\)
0.881216 + 0.472713i \(0.156725\pi\)
\(68\) −0.353433 + 0.393764i −0.0428601 + 0.0477509i
\(69\) 1.10597 + 1.10597i 0.133143 + 0.133143i
\(70\) 0 0
\(71\) 9.03885i 1.07271i 0.843991 + 0.536357i \(0.180200\pi\)
−0.843991 + 0.536357i \(0.819800\pi\)
\(72\) 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
\(77\) 0 0
\(78\) −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 \(-0.732320\pi\)
0.745272 + 0.666760i \(0.232320\pi\)
\(84\) 0 0
\(85\) 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
\(91\) 0 0
\(92\) 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 \(-0.242675\pi\)
0.723189 + 0.690650i \(0.242675\pi\)
\(98\) 0 0
\(99\) 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 784.2.m.g.197.4 8
7.2 even 3 784.2.x.j.165.2 16
7.3 odd 6 784.2.x.k.373.2 16
7.4 even 3 784.2.x.j.373.2 16
7.5 odd 6 784.2.x.k.165.2 16
7.6 odd 2 112.2.m.c.85.4 yes 8
16.13 even 4 inner 784.2.m.g.589.4 8
28.27 even 2 448.2.m.c.113.3 8
56.13 odd 2 896.2.m.e.225.3 8
56.27 even 2 896.2.m.f.225.2 8
112.13 odd 4 112.2.m.c.29.4 ✓ 8
112.27 even 4 896.2.m.f.673.2 8
112.45 odd 12 784.2.x.k.765.2 16
112.61 odd 12 784.2.x.k.557.2 16
112.69 odd 4 896.2.m.e.673.3 8
112.83 even 4 448.2.m.c.337.3 8
112.93 even 12 784.2.x.j.557.2 16
112.109 even 12 784.2.x.j.765.2 16
224.13 odd 8 7168.2.a.bc.1.5 8
224.83 even 8 7168.2.a.bd.1.4 8
224.125 odd 8 7168.2.a.bc.1.4 8
224.195 even 8 7168.2.a.bd.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.4 ✓ 8 112.13 odd 4
112.2.m.c.85.4 yes 8 7.6 odd 2
448.2.m.c.113.3 8 28.27 even 2
448.2.m.c.337.3 8 112.83 even 4
784.2.m.g.197.4 8 1.1 even 1 trivial
784.2.m.g.589.4 8 16.13 even 4 inner
784.2.x.j.165.2 16 7.2 even 3
784.2.x.j.373.2 16 7.4 even 3
784.2.x.j.557.2 16 112.93 even 12
784.2.x.j.765.2 16 112.109 even 12
784.2.x.k.165.2 16 7.5 odd 6
784.2.x.k.373.2 16 7.3 odd 6
784.2.x.k.557.2 16 112.61 odd 12
784.2.x.k.765.2 16 112.45 odd 12
896.2.m.e.225.3 8 56.13 odd 2
896.2.m.e.673.3 8 112.69 odd 4
896.2.m.f.225.2 8 56.27 even 2
896.2.m.f.673.2 8 112.27 even 4
7168.2.a.bc.1.4 8 224.125 odd 8
7168.2.a.bc.1.5 8 224.13 odd 8
7168.2.a.bd.1.4 8 224.83 even 8
7168.2.a.bd.1.5 8 224.195 even 8