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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,10,Mod(3,56)] 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("56.3"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 56.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.8420068252\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(70\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.4
Character \(\chi\) \(=\) 56.19
Dual form 56.10.m.a.3.4

$q$-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+(-22.4163 - 3.08358i) q^{2} +(9.89269 - 5.71155i) q^{3} +(492.983 + 138.245i) q^{4} +(-733.618 + 1270.66i) q^{5} +(-239.370 + 97.5270i) q^{6} +(2052.82 + 6011.62i) q^{7} +(-10624.6 - 4619.10i) q^{8} +(-9776.26 + 16933.0i) q^{9} +(20363.2 - 26221.4i) q^{10} +(20569.2 + 35626.9i) q^{11} +(5666.52 - 1448.08i) q^{12} -166990. q^{13} +(-27479.3 - 141088. i) q^{14} +16760.4i q^{15} +(223921. + 136305. i) q^{16} +(249085. - 143810. i) q^{17} +(271362. - 349429. i) q^{18} +(712858. + 411569. i) q^{19} +(-537324. + 524997. i) q^{20} +(54643.5 + 47746.3i) q^{21} +(-351227. - 862051. i) q^{22} +(-61934.4 - 35757.8i) q^{23} +(-131488. + 14987.5i) q^{24} +(-99828.9 - 172909. i) q^{25} +(3.74330e6 + 514927. i) q^{26} +448191. i q^{27} +(180928. + 3.24742e6i) q^{28} -1.14929e6i q^{29} +(51682.0 - 375706. i) q^{30} +(-828254. - 1.43458e6i) q^{31} +(-4.59917e6 - 3.74593e6i) q^{32} +(406969. + 234964. i) q^{33} +(-6.02703e6 + 2.45561e6i) q^{34} +(-9.14473e6 - 1.80179e6i) q^{35} +(-7.16043e6 + 6.99615e6i) q^{36} +(-1.63447e7 - 9.43663e6i) q^{37} +(-1.47106e7 - 1.14240e7i) q^{38} +(-1.65198e6 + 953772. i) q^{39} +(1.36637e7 - 1.01116e7i) q^{40} -5.84392e6i q^{41} +(-1.07768e6 - 1.23879e6i) q^{42} -1.32179e7 q^{43} +(5.21502e6 + 2.04070e7i) q^{44} +(-1.43441e7 - 2.48447e7i) q^{45} +(1.27808e6 + 992539. i) q^{46} +(-9.89320e6 + 1.71355e7i) q^{47} +(2.99369e6 + 69489.1i) q^{48} +(-3.19255e7 + 2.46815e7i) q^{49} +(1.70462e6 + 4.18381e6i) q^{50} +(1.64275e6 - 2.84533e6i) q^{51} +(-8.23233e7 - 2.30856e7i) q^{52} +(3.47551e7 - 2.00659e7i) q^{53} +(1.38203e6 - 1.00468e7i) q^{54} -6.03597e7 q^{55} +(5.95793e6 - 7.33531e7i) q^{56} +9.40278e6 q^{57} +(-3.54394e6 + 2.57630e7i) q^{58} +(2.80582e7 - 1.61994e7i) q^{59} +(-2.31704e6 + 8.26259e6i) q^{60} +(-7.68402e7 + 1.33091e8i) q^{61} +(1.41428e7 + 3.47120e7i) q^{62} +(-1.21863e8 - 2.40108e7i) q^{63} +(9.15456e7 + 9.81519e7i) q^{64} +(1.22507e8 - 2.12188e8i) q^{65} +(-8.39823e6 - 6.52195e6i) q^{66} +(-1.03080e7 - 1.78539e7i) q^{67} +(1.42676e8 - 3.64608e7i) q^{68} -816930. q^{69} +(1.99435e8 + 6.85880e7i) q^{70} -2.44433e8i q^{71} +(1.82084e8 - 1.34748e8i) q^{72} +(-1.92553e8 + 1.11170e8i) q^{73} +(3.37290e8 + 2.61935e8i) q^{74} +(-1.97515e6 - 1.14036e6i) q^{75} +(2.94530e8 + 3.01446e8i) q^{76} +(-1.71950e8 + 1.96790e8i) q^{77} +(3.99724e7 - 1.62860e7i) q^{78} +(3.77365e8 + 2.17872e8i) q^{79} +(-3.37470e8 + 1.84532e8i) q^{80} +(-1.89866e8 - 3.28858e8i) q^{81} +(-1.80202e7 + 1.30999e8i) q^{82} -4.24101e8i q^{83} +(2.03376e7 + 3.10923e7i) q^{84} +4.22005e8i q^{85} +(2.96296e8 + 4.07584e7i) q^{86} +(-6.56425e6 - 1.13696e7i) q^{87} +(-5.39749e7 - 4.73532e8i) q^{88} +(5.10986e8 + 2.95018e8i) q^{89} +(2.44931e8 + 6.01157e8i) q^{90} +(-3.42800e8 - 1.00388e9i) q^{91} +(-2.55893e7 - 2.61901e7i) q^{92} +(-1.63873e7 - 9.46123e6i) q^{93} +(2.74608e8 - 3.53609e8i) q^{94} +(-1.04593e9 + 6.03869e8i) q^{95} +(-6.68932e7 - 1.07890e7i) q^{96} -9.74074e8i q^{97} +(7.91759e8 - 4.54824e8i) q^{98} -8.04359e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 140 q - 18 q^{2} - 6 q^{3} + 84 q^{4} + 17268 q^{8} + 433024 q^{9} - 26724 q^{10} - 65862 q^{11} - 82998 q^{12} - 214260 q^{14} + 410128 q^{16} - 6 q^{17} + 1110146 q^{18} - 6 q^{19} + 3455012 q^{22} - 5971242 q^{24}+ \cdots - 2771471384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.4163 − 3.08358i −0.990671 − 0.136276i
\(3\) 9.89269 − 5.71155i 0.0705129 − 0.0407107i −0.464329 − 0.885663i \(-0.653704\pi\)
0.534842 + 0.844952i \(0.320371\pi\)
\(4\) 492.983 + 138.245i 0.962858 + 0.270010i
\(5\) −733.618 + 1270.66i −0.524934 + 0.909213i 0.474644 + 0.880178i \(0.342577\pi\)
−0.999578 + 0.0290353i \(0.990756\pi\)
\(6\) −239.370 + 97.5270i −0.0754030 + 0.0307216i
\(7\) 2052.82 + 6011.62i 0.323154 + 0.946346i
\(8\) −10624.6 − 4619.10i −0.917079 − 0.398705i
\(9\) −9776.26 + 16933.0i −0.496685 + 0.860284i
\(10\) 20363.2 − 26221.4i 0.643941 − 0.829195i
\(11\) 20569.2 + 35626.9i 0.423594 + 0.733687i 0.996288 − 0.0860825i \(-0.0274349\pi\)
−0.572694 + 0.819769i \(0.694102\pi\)
\(12\) 5666.52 − 1448.08i 0.0788862 − 0.0201594i
\(13\) −166990. −1.62161 −0.810804 − 0.585318i \(-0.800970\pi\)
−0.810804 + 0.585318i \(0.800970\pi\)
\(14\) −27479.3 − 141088.i −0.191174 − 0.981556i
\(15\) 16760.4i 0.0854817i
\(16\) 223921. + 136305.i 0.854189 + 0.519962i
\(17\) 249085. − 143810.i 0.723316 − 0.417607i −0.0926559 − 0.995698i \(-0.529536\pi\)
0.815972 + 0.578091i \(0.196202\pi\)
\(18\) 271362. − 349429.i 0.609288 − 0.784572i
\(19\) 712858. + 411569.i 1.25491 + 0.724521i 0.972080 − 0.234649i \(-0.0753942\pi\)
0.282828 + 0.959171i \(0.408728\pi\)
\(20\) −537324. + 524997.i −0.750934 + 0.733705i
\(21\) 54643.5 + 47746.3i 0.0613129 + 0.0535739i
\(22\) −351227. − 862051.i −0.319658 − 0.784568i
\(23\) −61934.4 − 35757.8i −0.0461484 − 0.0266438i 0.476748 − 0.879040i \(-0.341815\pi\)
−0.522897 + 0.852396i \(0.675149\pi\)
\(24\) −131488. + 14987.5i −0.0808975 + 0.00922100i
\(25\) −99828.9 − 172909.i −0.0511124 − 0.0885293i
\(26\) 3.74330e6 + 514927.i 1.60648 + 0.220987i
\(27\) 448191.i 0.162303i
\(28\) 180928. + 3.24742e6i 0.0556282 + 0.998452i
\(29\) − 1.14929e6i − 0.301745i −0.988553 − 0.150873i \(-0.951792\pi\)
0.988553 − 0.150873i \(-0.0482083\pi\)
\(30\) 51682.0 − 375706.i 0.0116491 − 0.0846843i
\(31\) −828254. − 1.43458e6i −0.161078 − 0.278995i 0.774178 − 0.632968i \(-0.218164\pi\)
−0.935256 + 0.353973i \(0.884830\pi\)
\(32\) −4.59917e6 − 3.74593e6i −0.775362 − 0.631517i
\(33\) 406969. + 234964.i 0.0597378 + 0.0344896i
\(34\) −6.02703e6 + 2.45561e6i −0.773478 + 0.315140i
\(35\) −9.14473e6 − 1.80179e6i −1.03007 − 0.202954i
\(36\) −7.16043e6 + 6.99615e6i −0.710522 + 0.694221i
\(37\) −1.63447e7 − 9.43663e6i −1.43374 − 0.827769i −0.436334 − 0.899785i \(-0.643723\pi\)
−0.997403 + 0.0720158i \(0.977057\pi\)
\(38\) −1.47106e7 − 1.14240e7i −1.14447 − 0.888776i
\(39\) −1.65198e6 + 953772.i −0.114344 + 0.0660167i
\(40\) 1.36637e7 − 1.01116e7i 0.843915 − 0.624526i
\(41\) − 5.84392e6i − 0.322981i −0.986874 − 0.161491i \(-0.948370\pi\)
0.986874 − 0.161491i \(-0.0516301\pi\)
\(42\) −1.07768e6 − 1.23879e6i −0.0534401 − 0.0614296i
\(43\) −1.32179e7 −0.589595 −0.294797 − 0.955560i \(-0.595252\pi\)
−0.294797 + 0.955560i \(0.595252\pi\)
\(44\) 5.21502e6 + 2.04070e7i 0.209758 + 0.820811i
\(45\) −1.43441e7 − 2.48447e7i −0.521454 − 0.903186i
\(46\) 1.27808e6 + 992539.i 0.0420870 + 0.0326842i
\(47\) −9.89320e6 + 1.71355e7i −0.295731 + 0.512221i −0.975155 − 0.221525i \(-0.928896\pi\)
0.679424 + 0.733746i \(0.262230\pi\)
\(48\) 2.99369e6 + 69489.1i 0.0813994 + 0.00188943i
\(49\) −3.19255e7 + 2.46815e7i −0.791143 + 0.611631i
\(50\) 1.70462e6 + 4.18381e6i 0.0385711 + 0.0946688i
\(51\) 1.64275e6 − 2.84533e6i 0.0340021 − 0.0588934i
\(52\) −8.23233e7 − 2.30856e7i −1.56138 − 0.437850i
\(53\) 3.47551e7 − 2.00659e7i 0.605030 − 0.349314i −0.165988 − 0.986128i \(-0.553081\pi\)
0.771018 + 0.636813i \(0.219748\pi\)
\(54\) 1.38203e6 − 1.00468e7i 0.0221180 − 0.160789i
\(55\) −6.03597e7 −0.889437
\(56\) 5.95793e6 − 7.33531e7i 0.0809560 − 0.996718i
\(57\) 9.40278e6 0.117983
\(58\) −3.54394e6 + 2.57630e7i −0.0411207 + 0.298930i
\(59\) 2.80582e7 − 1.61994e7i 0.301457 − 0.174046i −0.341640 − 0.939831i \(-0.610982\pi\)
0.643097 + 0.765784i \(0.277649\pi\)
\(60\) −2.31704e6 + 8.26259e6i −0.0230809 + 0.0823067i
\(61\) −7.68402e7 + 1.33091e8i −0.710565 + 1.23074i 0.254080 + 0.967183i \(0.418227\pi\)
−0.964645 + 0.263552i \(0.915106\pi\)
\(62\) 1.41428e7 + 3.47120e7i 0.121555 + 0.298344i
\(63\) −1.21863e8 − 2.40108e7i −0.974633 − 0.192032i
\(64\) 9.15456e7 + 9.81519e7i 0.682068 + 0.731289i
\(65\) 1.22507e8 − 2.12188e8i 0.851237 − 1.47439i
\(66\) −8.39823e6 − 6.52195e6i −0.0544804 − 0.0423087i
\(67\) −1.03080e7 − 1.78539e7i −0.0624937 − 0.108242i 0.833086 − 0.553144i \(-0.186572\pi\)
−0.895579 + 0.444902i \(0.853239\pi\)
\(68\) 1.42676e8 − 3.64608e7i 0.809208 − 0.206793i
\(69\) −816930. −0.00433875
\(70\) 1.99435e8 + 6.85880e7i 0.992798 + 0.341434i
\(71\) − 2.44433e8i − 1.14156i −0.821104 − 0.570779i \(-0.806641\pi\)
0.821104 − 0.570779i \(-0.193359\pi\)
\(72\) 1.82084e8 − 1.34748e8i 0.798500 − 0.590917i
\(73\) −1.92553e8 + 1.11170e8i −0.793592 + 0.458180i −0.841226 − 0.540684i \(-0.818165\pi\)
0.0476337 + 0.998865i \(0.484832\pi\)
\(74\) 3.37290e8 + 2.61935e8i 1.30756 + 1.01543i
\(75\) −1.97515e6 − 1.14036e6i −0.00720817 − 0.00416164i
\(76\) 2.94530e8 + 3.01446e8i 1.01267 + 1.03645i
\(77\) −1.71950e8 + 1.96790e8i −0.557436 + 0.637961i
\(78\) 3.99724e7 − 1.62860e7i 0.122274 − 0.0498184i
\(79\) 3.77365e8 + 2.17872e8i 1.09003 + 0.629330i 0.933585 − 0.358356i \(-0.116663\pi\)
0.156447 + 0.987686i \(0.449996\pi\)
\(80\) −3.37470e8 + 1.84532e8i −0.921150 + 0.503694i
\(81\) −1.89866e8 − 3.28858e8i −0.490078 − 0.848840i
\(82\) −1.80202e7 + 1.30999e8i −0.0440146 + 0.319968i
\(83\) − 4.24101e8i − 0.980884i −0.871474 − 0.490442i \(-0.836835\pi\)
0.871474 − 0.490442i \(-0.163165\pi\)
\(84\) 2.03376e7 + 3.10923e7i 0.0445701 + 0.0681391i
\(85\) 4.22005e8i 0.876865i
\(86\) 2.96296e8 + 4.07584e7i 0.584094 + 0.0803478i
\(87\) −6.56425e6 − 1.13696e7i −0.0122842 − 0.0212769i
\(88\) −5.39749e7 − 4.73532e8i −0.0959445 − 0.841738i
\(89\) 5.10986e8 + 2.95018e8i 0.863284 + 0.498417i 0.865111 − 0.501581i \(-0.167248\pi\)
−0.00182677 + 0.999998i \(0.500581\pi\)
\(90\) 2.44931e8 + 6.01157e8i 0.393507 + 0.965822i
\(91\) −3.42800e8 − 1.00388e9i −0.524028 − 1.53460i
\(92\) −2.55893e7 − 2.61901e7i −0.0372402 − 0.0381147i
\(93\) −1.63873e7 − 9.46123e6i −0.0227162 − 0.0131152i
\(94\) 2.74608e8 − 3.53609e8i 0.362775 − 0.467141i
\(95\) −1.04593e9 + 6.03869e8i −1.31749 + 0.760653i
\(96\) −6.68932e7 − 1.07890e7i −0.0803825 − 0.0129646i
\(97\) − 9.74074e8i − 1.11717i −0.829448 − 0.558585i \(-0.811345\pi\)
0.829448 − 0.558585i \(-0.188655\pi\)
\(98\) 7.91759e8 − 4.54824e8i 0.867113 − 0.498111i
\(99\) −8.04359e8 −0.841572
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 56.10.m.a.19.4 yes 140
7.3 odd 6 inner 56.10.m.a.3.51 yes 140
8.3 odd 2 inner 56.10.m.a.19.51 yes 140
56.3 even 6 inner 56.10.m.a.3.4 ✓ 140
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.m.a.3.4 ✓ 140 56.3 even 6 inner
56.10.m.a.3.51 yes 140 7.3 odd 6 inner
56.10.m.a.19.4 yes 140 1.1 even 1 trivial
56.10.m.a.19.51 yes 140 8.3 odd 2 inner