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Properties

Label 784.2.x.k.165.2
Level $784$
Weight $2$
Character 784.165
Analytic conductor $6.260$
Analytic rank $0$
Dimension $16$
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] = [784,2,Mod(165,784)] 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("784.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} - 13 x^{12} + 32 x^{11} - 4 x^{10} - 34 x^{9} + 121 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 165.2
Root \(0.640069 + 1.26108i\) of defining polynomial
Character \(\chi\) \(=\) 784.165
Dual form 784.2.x.k.765.2

$q$-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+(-1.14477 - 0.830359i) q^{2} +(-0.261809 - 0.977085i) q^{3} +(0.621007 + 1.90114i) q^{4} +(0.317608 - 1.18533i) q^{5} +(-0.511620 + 1.33594i) q^{6} +(0.867721 - 2.69204i) q^{8} +(1.71192 - 0.988380i) q^{9} +(-1.34784 + 1.09320i) q^{10} +(4.06633 - 1.08957i) q^{11} +(1.69499 - 1.10451i) q^{12} +(-2.02017 + 2.02017i) q^{13} -1.24132 q^{15} +(-3.22870 + 2.36125i) q^{16} +(-0.132279 + 0.229115i) q^{17} +(-2.78047 - 0.290042i) q^{18} +(6.20120 + 1.66161i) q^{19} +(2.45072 - 0.132279i) q^{20} +(-5.55976 - 2.12921i) q^{22} +(1.33906 - 0.773104i) q^{23} +(-2.85753 - 0.143037i) q^{24} +(3.02600 + 1.74706i) q^{25} +(3.99009 - 0.635167i) q^{26} +(-3.55976 - 3.55976i) q^{27} +(-0.328129 + 0.328129i) q^{29} +(1.42103 + 1.03074i) q^{30} +(-3.02017 + 5.23108i) q^{31} +(5.65681 - 0.0221123i) q^{32} +(-2.12921 - 3.68789i) q^{33} +(0.341677 - 0.152445i) q^{34} +(2.94217 + 2.64082i) q^{36} +(2.43357 - 9.08220i) q^{37} +(-5.71923 - 7.05138i) q^{38} +(2.50277 + 1.44498i) q^{39} +(-2.91535 - 1.88355i) q^{40} -11.0327i q^{41} +(3.38407 + 3.38407i) q^{43} +(4.59665 + 7.05405i) q^{44} +(-0.627835 - 2.34311i) q^{45} +(-2.17487 - 0.226869i) q^{46} +(-1.56283 - 2.70690i) q^{47} +(3.15244 + 2.53652i) q^{48} +(-2.01339 - 4.51265i) q^{50} +(0.258496 + 0.0692639i) q^{51} +(-5.09516 - 2.58609i) q^{52} +(-0.588145 + 0.157593i) q^{53} +(1.11923 + 7.03099i) q^{54} -5.16599i q^{55} -6.49412i q^{57} +(0.648097 - 0.103168i) q^{58} +(6.31978 - 1.69338i) q^{59} +(-0.770868 - 2.35993i) q^{60} +(-6.64943 - 1.78171i) q^{61} +(7.80108 - 3.48057i) q^{62} +(-6.49412 - 4.67187i) q^{64} +(1.75294 + 3.03618i) q^{65} +(-0.624819 + 5.98980i) q^{66} +(1.22389 + 4.56764i) q^{67} +(-0.517726 - 0.109200i) q^{68} +(-1.10597 - 1.10597i) q^{69} +9.03885i q^{71} +(-1.17528 - 5.46620i) q^{72} +(-12.8298 - 7.40731i) q^{73} +(-10.3274 + 8.37632i) q^{74} +(0.914793 - 3.41405i) q^{75} +(0.692037 + 12.8212i) q^{76} +(-1.66525 - 3.73237i) q^{78} +(-6.29520 - 10.9036i) q^{79} +(1.77340 + 4.57702i) q^{80} +(0.418932 - 0.725612i) q^{81} +(-9.16111 + 12.6299i) q^{82} +(-0.715276 + 0.715276i) q^{83} +(0.229563 + 0.229563i) q^{85} +(-1.06400 - 6.68399i) q^{86} +(0.406517 + 0.234703i) q^{87} +(0.595277 - 11.8922i) q^{88} +(-9.51968 + 5.49619i) q^{89} +(-1.22690 + 3.20366i) q^{90} +(2.30135 + 2.06564i) q^{92} +(5.90192 + 1.58141i) q^{93} +(-0.458616 + 4.39650i) q^{94} +(3.93910 - 6.82272i) q^{95} +(-1.50261 - 5.52140i) q^{96} -14.2452 q^{97} +(5.88434 - 5.88434i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8} - 12 q^{10} + 12 q^{12} - 16 q^{15} - 8 q^{16} - 24 q^{17} - 6 q^{18} + 12 q^{19} - 16 q^{20} - 8 q^{22} - 8 q^{26} + 24 q^{27} - 32 q^{29} - 20 q^{30}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14477 − 0.830359i −0.809476 − 0.587153i
\(3\) −0.261809 − 0.977085i −0.151156 − 0.564120i −0.999404 − 0.0345217i \(-0.989009\pi\)
0.848248 − 0.529599i \(-0.177657\pi\)
\(4\) 0.621007 + 1.90114i 0.310504 + 0.950572i
\(5\) 0.317608 − 1.18533i 0.142039 − 0.530095i −0.857831 − 0.513932i \(-0.828188\pi\)
0.999869 − 0.0161629i \(-0.00514502\pi\)
\(6\) −0.511620 + 1.33594i −0.208868 + 0.545393i
\(7\) 0 0
\(8\) 0.867721 − 2.69204i 0.306786 − 0.951779i
\(9\) 1.71192 − 0.988380i 0.570642 − 0.329460i
\(10\) −1.34784 + 1.09320i −0.426223 + 0.345701i
\(11\) 4.06633 − 1.08957i 1.22604 − 0.328518i 0.413007 − 0.910728i \(-0.364479\pi\)
0.813038 + 0.582210i \(0.197812\pi\)
\(12\) 1.69499 − 1.10451i 0.489303 − 0.318846i
\(13\) −2.02017 + 2.02017i −0.560293 + 0.560293i −0.929391 − 0.369098i \(-0.879667\pi\)
0.369098 + 0.929391i \(0.379667\pi\)
\(14\) 0 0
\(15\) −1.24132 −0.320507
\(16\) −3.22870 + 2.36125i −0.807175 + 0.590312i
\(17\) −0.132279 + 0.229115i −0.0320825 + 0.0555685i −0.881621 − 0.471958i \(-0.843547\pi\)
0.849538 + 0.527527i \(0.176881\pi\)
\(18\) −2.78047 − 0.290042i −0.655364 − 0.0683636i
\(19\) 6.20120 + 1.66161i 1.42265 + 0.381199i 0.886424 − 0.462874i \(-0.153182\pi\)
0.536228 + 0.844073i \(0.319849\pi\)
\(20\) 2.45072 − 0.132279i 0.547997 − 0.0295786i
\(21\) 0 0
\(22\) −5.55976 − 2.12921i −1.18534 − 0.453948i
\(23\) 1.33906 − 0.773104i 0.279212 − 0.161203i −0.353854 − 0.935301i \(-0.615129\pi\)
0.633067 + 0.774097i \(0.281796\pi\)
\(24\) −2.85753 − 0.143037i −0.583290 − 0.0291973i
\(25\) 3.02600 + 1.74706i 0.605200 + 0.349412i
\(26\) 3.99009 − 0.635167i 0.782521 − 0.124566i
\(27\) −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 \(0.263719\pi\)
\(30\) 1.42103 + 1.03074i 0.259443 + 0.188187i
\(31\) −3.02017 + 5.23108i −0.542438 + 0.939530i 0.456326 + 0.889813i \(0.349165\pi\)
−0.998763 + 0.0497168i \(0.984168\pi\)
\(32\) 5.65681 − 0.0221123i 0.999992 − 0.00390894i
\(33\) −2.12921 − 3.68789i −0.370647 − 0.641980i
\(34\) 0.341677 − 0.152445i 0.0585972 − 0.0261440i
\(35\) 0 0
\(36\) 2.94217 + 2.64082i 0.490362 + 0.440137i
\(37\) 2.43357 − 9.08220i 0.400076 − 1.49310i −0.412883 − 0.910784i \(-0.635478\pi\)
0.812959 − 0.582320i \(-0.197855\pi\)
\(38\) −5.71923 − 7.05138i −0.927782 − 1.14389i
\(39\) 2.50277 + 1.44498i 0.400764 + 0.231381i
\(40\) −2.91535 − 1.88355i −0.460958 − 0.297815i
\(41\) − 11.0327i − 1.72302i −0.507741 − 0.861510i \(-0.669519\pi\)
0.507741 − 0.861510i \(-0.330481\pi\)
\(42\) 0 0
\(43\) 3.38407 + 3.38407i 0.516066 + 0.516066i 0.916379 − 0.400312i \(-0.131098\pi\)
−0.400312 + 0.916379i \(0.631098\pi\)
\(44\) 4.59665 + 7.05405i 0.692971 + 1.06344i
\(45\) −0.627835 − 2.34311i −0.0935921 − 0.349290i
\(46\) −2.17487 − 0.226869i −0.320667 − 0.0334500i
\(47\) −1.56283 − 2.70690i −0.227962 − 0.394842i 0.729242 − 0.684256i \(-0.239873\pi\)
−0.957204 + 0.289414i \(0.906540\pi\)
\(48\) 3.15244 + 2.53652i 0.455016 + 0.366115i
\(49\) 0 0
\(50\) −2.01339 − 4.51265i −0.284736 − 0.638185i
\(51\) 0.258496 + 0.0692639i 0.0361967 + 0.00969889i
\(52\) −5.09516 − 2.58609i −0.706572 − 0.358626i
\(53\) −0.588145 + 0.157593i −0.0807879 + 0.0216471i −0.298987 − 0.954257i \(-0.596649\pi\)
0.218199 + 0.975904i \(0.429982\pi\)
\(54\) 1.11923 + 7.03099i 0.152309 + 0.956796i
\(55\) − 5.16599i − 0.696582i
\(56\) 0 0
\(57\) − 6.49412i − 0.860167i
\(58\) 0.648097 − 0.103168i 0.0850994 − 0.0135466i
\(59\) 6.31978 − 1.69338i 0.822765 − 0.220459i 0.177210 − 0.984173i \(-0.443293\pi\)
0.645555 + 0.763714i \(0.276626\pi\)
\(60\) −0.770868 − 2.35993i −0.0995187 − 0.304665i
\(61\) −6.64943 − 1.78171i −0.851372 − 0.228125i −0.193356 − 0.981129i \(-0.561937\pi\)
−0.658016 + 0.753004i \(0.728604\pi\)
\(62\) 7.80108 − 3.48057i 0.990738 − 0.442033i
\(63\) 0 0
\(64\) −6.49412 − 4.67187i −0.811765 − 0.583984i
\(65\) 1.75294 + 3.03618i 0.217425 + 0.376592i
\(66\) −0.624819 + 5.98980i −0.0769100 + 0.737294i
\(67\) 1.22389 + 4.56764i 0.149523 + 0.558026i 0.999512 + 0.0312266i \(0.00994134\pi\)
−0.849990 + 0.526799i \(0.823392\pi\)
\(68\) −0.517726 − 0.109200i −0.0627835 − 0.0132425i
\(69\) −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 \(0.819800\pi\)
\(72\) −1.17528 − 5.46620i −0.138508 − 0.644198i
\(73\) −12.8298 − 7.40731i −1.50162 − 0.866960i −0.999998 − 0.00187294i \(-0.999404\pi\)
−0.501621 − 0.865087i \(-0.667263\pi\)
\(74\) −10.3274 + 8.37632i −1.20053 + 0.973727i
\(75\) 0.914793 − 3.41405i 0.105631 − 0.394221i
\(76\) 0.692037 + 12.8212i 0.0793820 + 1.47070i
\(77\) 0 0
\(78\) −1.66525 − 3.73237i −0.188553 − 0.422607i
\(79\) −6.29520 − 10.9036i −0.708265 − 1.22675i −0.965500 − 0.260402i \(-0.916145\pi\)
0.257235 − 0.966349i \(-0.417189\pi\)
\(80\) 1.77340 + 4.57702i 0.198272 + 0.511726i
\(81\) 0.418932 − 0.725612i 0.0465480 − 0.0806235i
\(82\) −9.16111 + 12.6299i −1.01168 + 1.39474i
\(83\) −0.715276 + 0.715276i −0.0785117 + 0.0785117i −0.745272 − 0.666760i \(-0.767680\pi\)
0.666760 + 0.745272i \(0.267680\pi\)
\(84\) 0 0
\(85\) 0.229563 + 0.229563i 0.0248996 + 0.0248996i
\(86\) −1.06400 − 6.68399i −0.114734 − 0.720753i
\(87\) 0.406517 + 0.234703i 0.0435832 + 0.0251628i
\(88\) 0.595277 − 11.8922i 0.0634567 − 1.26771i
\(89\) −9.51968 + 5.49619i −1.00908 + 0.582595i −0.910923 − 0.412576i \(-0.864629\pi\)
−0.0981604 + 0.995171i \(0.531296\pi\)
\(90\) −1.22690 + 3.20366i −0.129326 + 0.337695i
\(91\) 0 0
\(92\) 2.30135 + 2.06564i 0.239932 + 0.215357i
\(93\) 5.90192 + 1.58141i 0.612000 + 0.163985i
\(94\) −0.458616 + 4.39650i −0.0473026 + 0.453464i
\(95\) 3.93910 − 6.82272i 0.404143 − 0.699996i
\(96\) −1.50261 − 5.52140i −0.153360 − 0.563525i
\(97\) −14.2452 −1.44638 −0.723189 − 0.690650i \(-0.757325\pi\)
−0.723189 + 0.690650i \(0.757325\pi\)
\(98\) 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.x.k.165.2 16
7.2 even 3 inner 784.2.x.k.373.2 16
7.3 odd 6 784.2.m.g.197.4 8
7.4 even 3 112.2.m.c.85.4 yes 8
7.5 odd 6 784.2.x.j.373.2 16
7.6 odd 2 784.2.x.j.165.2 16
16.13 even 4 inner 784.2.x.k.557.2 16
28.11 odd 6 448.2.m.c.113.3 8
56.11 odd 6 896.2.m.f.225.2 8
56.53 even 6 896.2.m.e.225.3 8
112.11 odd 12 896.2.m.f.673.2 8
112.13 odd 4 784.2.x.j.557.2 16
112.45 odd 12 784.2.m.g.589.4 8
112.53 even 12 896.2.m.e.673.3 8
112.61 odd 12 784.2.x.j.765.2 16
112.67 odd 12 448.2.m.c.337.3 8
112.93 even 12 inner 784.2.x.k.765.2 16
112.109 even 12 112.2.m.c.29.4 ✓ 8
224.67 odd 24 7168.2.a.bd.1.5 8
224.109 even 24 7168.2.a.bc.1.5 8
224.179 odd 24 7168.2.a.bd.1.4 8
224.221 even 24 7168.2.a.bc.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.4 ✓ 8 112.109 even 12
112.2.m.c.85.4 yes 8 7.4 even 3
448.2.m.c.113.3 8 28.11 odd 6
448.2.m.c.337.3 8 112.67 odd 12
784.2.m.g.197.4 8 7.3 odd 6
784.2.m.g.589.4 8 112.45 odd 12
784.2.x.j.165.2 16 7.6 odd 2
784.2.x.j.373.2 16 7.5 odd 6
784.2.x.j.557.2 16 112.13 odd 4
784.2.x.j.765.2 16 112.61 odd 12
784.2.x.k.165.2 16 1.1 even 1 trivial
784.2.x.k.373.2 16 7.2 even 3 inner
784.2.x.k.557.2 16 16.13 even 4 inner
784.2.x.k.765.2 16 112.93 even 12 inner
896.2.m.e.225.3 8 56.53 even 6
896.2.m.e.673.3 8 112.53 even 12
896.2.m.f.225.2 8 56.11 odd 6
896.2.m.f.673.2 8 112.11 odd 12
7168.2.a.bc.1.4 8 224.221 even 24
7168.2.a.bc.1.5 8 224.109 even 24
7168.2.a.bd.1.4 8 224.179 odd 24
7168.2.a.bd.1.5 8 224.67 odd 24