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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 3.51
Character \(\chi\) \(=\) 56.3
Dual form 56.10.m.a.19.51

$q$-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+(13.8786 - 17.8713i) q^{2} +(9.89269 + 5.71155i) q^{3} +(-126.768 - 496.058i) 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} +(32890.0 + 4524.34i) q^{10} +(20569.2 - 35626.9i) q^{11} +(1579.19 - 5631.39i) q^{12} +166990. q^{13} +(78945.2 + 120120. i) q^{14} +16760.4i q^{15} +(-230004. + 125768. i) q^{16} +(249085. + 143810. i) q^{17} +(-438295. - 60291.7i) 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} +(-78723.5 - 106378. i) q^{24} +(-99828.9 + 172909. i) q^{25} +(2.31759e6 - 2.98433e6i) q^{26} -448191. i q^{27} +(3.24234e6 + 256238. i) q^{28} -1.14929e6i q^{29} +(299530. + 232611. i) q^{30} +(828254. - 1.43458e6i) q^{31} +(-944487. + 5.85596e6i) 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} +(2.53821e6 - 1.84517e7i) q^{38} +(1.65198e6 + 953772. i) q^{39} +(-1.92506e6 - 1.68889e7i) q^{40} +5.84392e6i q^{41} +(94912.6 + 1.63920e6i) q^{42} -1.32179e7 q^{43} +(-2.02805e7 - 5.68718e6i) q^{44} +(1.43441e7 - 2.48447e7i) q^{45} +(220524. - 1.60312e6i) 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} +(-2.11690e7 - 8.28368e7i) q^{52} +(-3.47551e7 - 2.00659e7i) q^{53} +(-8.00976e6 - 6.22027e6i) q^{54} +6.03597e7 q^{55} +(4.95786e7 - 5.43887e7i) q^{56} +9.40278e6 q^{57} +(-2.05394e7 - 1.59506e7i) q^{58} +(2.80582e7 + 1.61994e7i) q^{59} +(8.31413e6 - 2.12468e6i) 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} +(1.44906e6 - 1.05341e7i) q^{66} +(-1.03080e7 + 1.78539e7i) q^{67} +(3.97619e7 - 1.41791e8i) q^{68} +816930. q^{69} +(-9.47159e7 + 1.88435e8i) q^{70} -2.44433e8i q^{71} +(2.56535e7 + 2.25063e8i) q^{72} +(-1.92553e8 - 1.11170e8i) q^{73} +(5.81972e7 - 4.23069e8i) 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} +(-3.28545e8 - 1.99992e8i) q^{80} +(-1.89866e8 + 3.28858e8i) q^{81} +(1.04439e8 + 8.11056e7i) q^{82} +4.24101e8i q^{83} +(3.06120e7 + 2.10537e7i) q^{84} +4.22005e8i q^{85} +(-1.83446e8 + 2.36221e8i) q^{86} +(6.56425e6 - 1.13696e7i) q^{87} +(-3.83103e8 + 2.83510e8i) 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} +(4.43538e8 + 6.10129e7i) q^{94} +(1.04593e9 + 6.03869e8i) q^{95} +(-4.27901e7 + 5.25368e7i) q^{96} +9.74074e8i q^{97} +(-8.84173e8 + 2.28005e8i) 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 \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.8786 − 17.8713i 0.613354 − 0.789808i
\(3\) 9.89269 + 5.71155i 0.0705129 + 0.0407107i 0.534842 − 0.844952i \(-0.320371\pi\)
−0.464329 + 0.885663i \(0.653704\pi\)
\(4\) −126.768 − 496.058i −0.247593 − 0.968864i
\(5\) 733.618 + 1270.66i 0.524934 + 0.909213i 0.999578 + 0.0290353i \(0.00924351\pi\)
−0.474644 + 0.880178i \(0.657423\pi\)
\(6\) 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
\(10\) 32890.0 + 4524.34i 1.04007 + 0.143072i
\(11\) 20569.2 − 35626.9i 0.423594 − 0.733687i −0.572694 − 0.819769i \(-0.694102\pi\)
0.996288 + 0.0860825i \(0.0274349\pi\)
\(12\) 1579.19 − 5631.39i 0.0219846 − 0.0783971i
\(13\) 166990. 1.62161 0.810804 − 0.585318i \(-0.199030\pi\)
0.810804 + 0.585318i \(0.199030\pi\)
\(14\) 78945.2 + 120120.i 0.549224 + 0.835675i
\(15\) 16760.4i 0.0854817i
\(16\) −230004. + 125768.i −0.877395 + 0.479769i
\(17\) 249085. + 143810.i 0.723316 + 0.417607i 0.815972 − 0.578091i \(-0.196202\pi\)
−0.0926559 + 0.995698i \(0.529536\pi\)
\(18\) −438295. − 60291.7i −0.984103 − 0.135373i
\(19\) 712858. − 411569.i 1.25491 − 0.724521i 0.282828 − 0.959171i \(-0.408728\pi\)
0.972080 + 0.234649i \(0.0753942\pi\)
\(20\) 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 \(-0.658185\pi\)
0.522897 + 0.852396i \(0.324851\pi\)
\(24\) −78723.5 − 106378.i −0.0484344 − 0.0654488i
\(25\) −99828.9 + 172909.i −0.0511124 + 0.0885293i
\(26\) 2.31759e6 − 2.98433e6i 0.994619 − 1.28076i
\(27\) − 448191.i − 0.162303i
\(28\) 3.24234e6 + 256238.i 0.996892 + 0.0787829i
\(29\) − 1.14929e6i − 0.301745i −0.988553 − 0.150873i \(-0.951792\pi\)
0.988553 − 0.150873i \(-0.0482083\pi\)
\(30\) 299530. + 232611.i 0.0675142 + 0.0524306i
\(31\) 828254. − 1.43458e6i 0.161078 − 0.278995i −0.774178 − 0.632968i \(-0.781836\pi\)
0.935256 + 0.353973i \(0.115170\pi\)
\(32\) −944487. + 5.85596e6i −0.159229 + 0.987242i
\(33\) 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 \(-0.356277\pi\)
0.997403 + 0.0720158i \(0.0229432\pi\)
\(38\) 2.53821e6 − 1.84517e7i 0.197470 − 1.43552i
\(39\) 1.65198e6 + 953772.i 0.114344 + 0.0660167i
\(40\) −1.92506e6 − 1.68889e7i −0.118898 − 1.04311i
\(41\) 5.84392e6i 0.322981i 0.986874 + 0.161491i \(0.0516301\pi\)
−0.986874 + 0.161491i \(0.948370\pi\)
\(42\) 94912.6 + 1.63920e6i 0.00470654 + 0.0812852i
\(43\) −1.32179e7 −0.589595 −0.294797 − 0.955560i \(-0.595252\pi\)
−0.294797 + 0.955560i \(0.595252\pi\)
\(44\) −2.02805e7 − 5.68718e6i −0.815722 − 0.228749i
\(45\) 1.43441e7 − 2.48447e7i 0.521454 − 0.903186i
\(46\) 220524. − 1.60312e6i 0.00726183 − 0.0527904i
\(47\) 9.89320e6 + 1.71355e7i 0.295731 + 0.512221i 0.975155 − 0.221525i \(-0.0711036\pi\)
−0.679424 + 0.733746i \(0.737770\pi\)
\(48\) −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
\(52\) −2.11690e7 − 8.28368e7i −0.401499 − 1.57112i
\(53\) −3.47551e7 − 2.00659e7i −0.605030 − 0.349314i 0.165988 − 0.986128i \(-0.446919\pi\)
−0.771018 + 0.636813i \(0.780252\pi\)
\(54\) −8.00976e6 − 6.22027e6i −0.128188 − 0.0995492i
\(55\) 6.03597e7 0.889437
\(56\) 4.95786e7 − 5.43887e7i 0.673671 − 0.739031i
\(57\) 9.40278e6 0.117983
\(58\) −2.05394e7 − 1.59506e7i −0.238321 − 0.185077i
\(59\) 2.80582e7 + 1.61994e7i 0.301457 + 0.174046i 0.643097 − 0.765784i \(-0.277649\pi\)
−0.341640 + 0.939831i \(0.610982\pi\)
\(60\) 8.31413e6 − 2.12468e6i 0.0828202 − 0.0211647i
\(61\) 7.68402e7 + 1.33091e8i 0.710565 + 1.23074i 0.964645 + 0.263552i \(0.0848940\pi\)
−0.254080 + 0.967183i \(0.581773\pi\)
\(62\) −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
\(66\) 1.44906e6 − 1.05341e7i 0.00940023 − 0.0683357i
\(67\) −1.03080e7 + 1.78539e7i −0.0624937 + 0.108242i −0.895579 − 0.444902i \(-0.853239\pi\)
0.833086 + 0.553144i \(0.186572\pi\)
\(68\) 3.97619e7 − 1.41791e8i 0.225516 − 0.804192i
\(69\) 816930. 0.00433875
\(70\) −9.47159e7 + 1.88435e8i −0.471500 + 0.938037i
\(71\) − 2.44433e8i − 1.14156i −0.821104 − 0.570779i \(-0.806641\pi\)
0.821104 − 0.570779i \(-0.193359\pi\)
\(72\) 2.56535e7 + 2.25063e8i 0.112500 + 0.986980i
\(73\) −1.92553e8 − 1.11170e8i −0.793592 − 0.458180i 0.0476337 − 0.998865i \(-0.484832\pi\)
−0.841226 + 0.540684i \(0.818165\pi\)
\(74\) 5.81972e7 − 4.23069e8i 0.225610 − 1.64009i
\(75\) −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 \(-0.883337\pi\)
−0.156447 + 0.987686i \(0.550004\pi\)
\(80\) −3.28545e8 − 1.99992e8i −0.896787 − 0.545892i
\(81\) −1.89866e8 + 3.28858e8i −0.490078 + 0.848840i
\(82\) 1.04439e8 + 8.11056e7i 0.255093 + 0.198102i
\(83\) 4.24101e8i 0.980884i 0.871474 + 0.490442i \(0.163165\pi\)
−0.871474 + 0.490442i \(0.836835\pi\)
\(84\) 3.06120e7 + 2.10537e7i 0.0670865 + 0.0461393i
\(85\) 4.22005e8i 0.876865i
\(86\) −1.83446e8 + 2.36221e8i −0.361630 + 0.465667i
\(87\) 6.56425e6 − 1.13696e7i 0.0122842 − 0.0212769i
\(88\) −3.83103e8 + 2.83510e8i −0.680995 + 0.503960i
\(89\) 5.10986e8 − 2.95018e8i 0.863284 − 0.498417i −0.00182677 − 0.999998i \(-0.500581\pi\)
0.865111 + 0.501581i \(0.167248\pi\)
\(90\) −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
\(94\) 4.43538e8 + 6.10129e7i 0.585944 + 0.0806021i
\(95\) 1.04593e9 + 6.03869e8i 1.31749 + 0.760653i
\(96\) −4.27901e7 + 5.25368e7i −0.0514190 + 0.0631310i
\(97\) 9.74074e8i 1.11717i 0.829448 + 0.558585i \(0.188655\pi\)
−0.829448 + 0.558585i \(0.811345\pi\)
\(98\) −8.84173e8 + 2.28005e8i −0.968322 + 0.249705i
\(99\) −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.3.51 yes 140
7.5 odd 6 inner 56.10.m.a.19.4 yes 140
8.3 odd 2 inner 56.10.m.a.3.4 ✓ 140
56.19 even 6 inner 56.10.m.a.19.51 yes 140
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.m.a.3.4 ✓ 140 8.3 odd 2 inner
56.10.m.a.3.51 yes 140 1.1 even 1 trivial
56.10.m.a.19.4 yes 140 7.5 odd 6 inner
56.10.m.a.19.51 yes 140 56.19 even 6 inner