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Properties

Label 10.11.c.c.3.2
Level $10$
Weight $11$
Character 10.3
Analytic conductor $6.354$
Analytic rank $0$
Dimension $6$
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] = [10,11,Mod(3,10)] 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("10.3"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-96,128] 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: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 3.2
Root \(10.1043 - 10.1043i\) of defining polynomial
Character \(\chi\) \(=\) 10.3
Dual form 10.11.c.c.7.2

$q$-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+(-16.0000 + 16.0000i) q^{2} +(4.29207 + 4.29207i) q^{3} -512.000i q^{4} +(-2978.33 + 946.124i) q^{5} -137.346 q^{6} +(21284.7 - 21284.7i) q^{7} +(8192.00 + 8192.00i) q^{8} -59012.2i q^{9} +(32515.4 - 62791.3i) q^{10} +155649. q^{11} +(2197.54 - 2197.54i) q^{12} +(-358614. - 358614. i) q^{13} +681110. i q^{14} +(-16844.1 - 8722.40i) q^{15} -262144. q^{16} +(-609397. + 609397. i) q^{17} +(944194. + 944194. i) q^{18} -335226. i q^{19} +(484415. + 1.52491e6i) q^{20} +182711. q^{21} +(-2.49038e6 + 2.49038e6i) q^{22} +(-5.50132e6 - 5.50132e6i) q^{23} +70321.3i q^{24} +(7.97532e6 - 5.63575e6i) q^{25} +1.14757e7 q^{26} +(506727. - 506727. i) q^{27} +(-1.08978e7 - 1.08978e7i) q^{28} -1.33815e6i q^{29} +(409063. - 129947. i) q^{30} +2.59306e7 q^{31} +(4.19430e6 - 4.19430e6i) q^{32} +(668055. + 668055. i) q^{33} -1.95007e7i q^{34} +(-4.32550e7 + 8.35308e7i) q^{35} -3.02142e7 q^{36} +(-5.50890e7 + 5.50890e7i) q^{37} +(5.36362e6 + 5.36362e6i) q^{38} -3.07840e6i q^{39} +(-3.21492e7 - 1.66479e7i) q^{40} +1.37064e8 q^{41} +(-2.92337e6 + 2.92337e6i) q^{42} +(-9.34290e7 - 9.34290e7i) q^{43} -7.96921e7i q^{44} +(5.58328e7 + 1.75758e8i) q^{45} +1.76042e8 q^{46} +(1.14979e8 - 1.14979e8i) q^{47} +(-1.12514e6 - 1.12514e6i) q^{48} -6.23600e8i q^{49} +(-3.74333e7 + 2.17777e8i) q^{50} -5.23115e6 q^{51} +(-1.83611e8 + 1.83611e8i) q^{52} +(-2.29978e7 - 2.29978e7i) q^{53} +1.62153e7i q^{54} +(-4.63574e8 + 1.47263e8i) q^{55} +3.48728e8 q^{56} +(1.43882e6 - 1.43882e6i) q^{57} +(2.14104e7 + 2.14104e7i) q^{58} +8.73535e8i q^{59} +(-4.46587e6 + 8.62416e6i) q^{60} +5.87905e8 q^{61} +(-4.14890e8 + 4.14890e8i) q^{62} +(-1.25605e9 - 1.25605e9i) q^{63} +1.34218e8i q^{64} +(1.40737e9 + 7.28780e8i) q^{65} -2.13778e7 q^{66} +(-6.75197e8 + 6.75197e8i) q^{67} +(3.12011e8 + 3.12011e8i) q^{68} -4.72241e7i q^{69} +(-6.44414e8 - 2.02857e9i) q^{70} -5.54597e8 q^{71} +(4.83428e8 - 4.83428e8i) q^{72} +(8.91747e8 + 8.91747e8i) q^{73} -1.76285e9i q^{74} +(5.84197e7 + 1.00416e7i) q^{75} -1.71636e8 q^{76} +(3.31293e9 - 3.31293e9i) q^{77} +(4.92544e7 + 4.92544e7i) q^{78} +1.69149e9i q^{79} +(7.80752e8 - 2.48021e8i) q^{80} -3.48026e9 q^{81} +(-2.19302e9 + 2.19302e9i) q^{82} +(1.96163e9 + 1.96163e9i) q^{83} -9.35479e7i q^{84} +(1.23842e9 - 2.39155e9i) q^{85} +2.98973e9 q^{86} +(5.74345e6 - 5.74345e6i) q^{87} +(1.27507e9 + 1.27507e9i) q^{88} +7.73241e9i q^{89} +(-3.70545e9 - 1.91880e9i) q^{90} -1.52660e10 q^{91} +(-2.81667e9 + 2.81667e9i) q^{92} +(1.11296e8 + 1.11296e8i) q^{93} +3.67931e9i q^{94} +(3.17166e8 + 9.98416e8i) q^{95} +3.60045e7 q^{96} +(1.16280e10 - 1.16280e10i) q^{97} +(9.97760e9 + 9.97760e9i) q^{98} -9.18516e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8} - 173280 q^{10} + 647832 q^{11} + 65536 q^{12} - 742902 q^{13} + 1577720 q^{15} - 1572864 q^{16} - 755118 q^{17} - 5683744 q^{18}+ \cdots + 12874047264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 16.0000i −0.500000 + 0.500000i
\(3\) 4.29207 + 4.29207i 0.0176629 + 0.0176629i 0.715883 − 0.698220i \(-0.246024\pi\)
−0.698220 + 0.715883i \(0.746024\pi\)
\(4\) − 512.000i − 0.500000i
\(5\) −2978.33 + 946.124i −0.953067 + 0.302760i
\(6\) −137.346 −0.0176629
\(7\) 21284.7 − 21284.7i 1.26642 − 1.26642i 0.318492 − 0.947926i \(-0.396824\pi\)
0.947926 − 0.318492i \(-0.103176\pi\)
\(8\) 8192.00 + 8192.00i 0.250000 + 0.250000i
\(9\) − 59012.2i − 0.999376i
\(10\) 32515.4 − 62791.3i 0.325154 − 0.627913i
\(11\) 155649. 0.966456 0.483228 − 0.875495i \(-0.339464\pi\)
0.483228 + 0.875495i \(0.339464\pi\)
\(12\) 2197.54 − 2197.54i 0.00883143 − 0.00883143i
\(13\) −358614. − 358614.i −0.965853 − 0.965853i 0.0335834 − 0.999436i \(-0.489308\pi\)
−0.999436 + 0.0335834i \(0.989308\pi\)
\(14\) 681110.i 1.26642i
\(15\) −16844.1 − 8722.40i −0.0221815 − 0.0114863i
\(16\) −262144. −0.250000
\(17\) −609397. + 609397.i −0.429196 + 0.429196i −0.888354 − 0.459158i \(-0.848151\pi\)
0.459158 + 0.888354i \(0.348151\pi\)
\(18\) 944194. + 944194.i 0.499688 + 0.499688i
\(19\) − 335226.i − 0.135385i −0.997706 − 0.0676924i \(-0.978436\pi\)
0.997706 − 0.0676924i \(-0.0215637\pi\)
\(20\) 484415. + 1.52491e6i 0.151380 + 0.476533i
\(21\) 182711. 0.0447371
\(22\) −2.49038e6 + 2.49038e6i −0.483228 + 0.483228i
\(23\) −5.50132e6 − 5.50132e6i −0.854727 − 0.854727i 0.135984 − 0.990711i \(-0.456580\pi\)
−0.990711 + 0.135984i \(0.956580\pi\)
\(24\) 70321.3i 0.00883143i
\(25\) 7.97532e6 − 5.63575e6i 0.816673 − 0.577100i
\(26\) 1.14757e7 0.965853
\(27\) 506727. − 506727.i 0.0353147 − 0.0353147i
\(28\) −1.08978e7 − 1.08978e7i −0.633209 − 0.633209i
\(29\) − 1.33815e6i − 0.0652402i −0.999468 − 0.0326201i \(-0.989615\pi\)
0.999468 − 0.0326201i \(-0.0103851\pi\)
\(30\) 409063. − 129947.i 0.0168339 − 0.00534760i
\(31\) 2.59306e7 0.905743 0.452871 − 0.891576i \(-0.350400\pi\)
0.452871 + 0.891576i \(0.350400\pi\)
\(32\) 4.19430e6 − 4.19430e6i 0.125000 − 0.125000i
\(33\) 668055. + 668055.i 0.0170704 + 0.0170704i
\(34\) − 1.95007e7i − 0.429196i
\(35\) −4.32550e7 + 8.35308e7i −0.823561 + 1.59040i
\(36\) −3.02142e7 −0.499688
\(37\) −5.50890e7 + 5.50890e7i −0.794431 + 0.794431i −0.982211 − 0.187780i \(-0.939871\pi\)
0.187780 + 0.982211i \(0.439871\pi\)
\(38\) 5.36362e6 + 5.36362e6i 0.0676924 + 0.0676924i
\(39\) − 3.07840e6i − 0.0341194i
\(40\) −3.21492e7 − 1.66479e7i −0.313957 − 0.162577i
\(41\) 1.37064e8 1.18305 0.591525 − 0.806287i \(-0.298526\pi\)
0.591525 + 0.806287i \(0.298526\pi\)
\(42\) −2.92337e6 + 2.92337e6i −0.0223685 + 0.0223685i
\(43\) −9.34290e7 − 9.34290e7i −0.635535 − 0.635535i 0.313916 − 0.949451i \(-0.398359\pi\)
−0.949451 + 0.313916i \(0.898359\pi\)
\(44\) − 7.96921e7i − 0.483228i
\(45\) 5.58328e7 + 1.75758e8i 0.302571 + 0.952472i
\(46\) 1.76042e8 0.854727
\(47\) 1.14979e8 − 1.14979e8i 0.501334 − 0.501334i −0.410518 − 0.911852i \(-0.634652\pi\)
0.911852 + 0.410518i \(0.134652\pi\)
\(48\) −1.12514e6 − 1.12514e6i −0.00441571 − 0.00441571i
\(49\) − 6.23600e8i − 2.20763i
\(50\) −3.74333e7 + 2.17777e8i −0.119786 + 0.696887i
\(51\) −5.23115e6 −0.0151616
\(52\) −1.83611e8 + 1.83611e8i −0.482926 + 0.482926i
\(53\) −2.29978e7 − 2.29978e7i −0.0549930 − 0.0549930i 0.679075 − 0.734068i \(-0.262381\pi\)
−0.734068 + 0.679075i \(0.762381\pi\)
\(54\) 1.62153e7i 0.0353147i
\(55\) −4.63574e8 + 1.47263e8i −0.921097 + 0.292604i
\(56\) 3.48728e8 0.633209
\(57\) 1.43882e6 − 1.43882e6i 0.00239128 − 0.00239128i
\(58\) 2.14104e7 + 2.14104e7i 0.0326201 + 0.0326201i
\(59\) 8.73535e8i 1.22186i 0.791686 + 0.610929i \(0.209204\pi\)
−0.791686 + 0.610929i \(0.790796\pi\)
\(60\) −4.46587e6 + 8.62416e6i −0.00574314 + 0.0110907i
\(61\) 5.87905e8 0.696079 0.348039 − 0.937480i \(-0.386848\pi\)
0.348039 + 0.937480i \(0.386848\pi\)
\(62\) −4.14890e8 + 4.14890e8i −0.452871 + 0.452871i
\(63\) −1.25605e9 − 1.25605e9i −1.26563 − 1.26563i
\(64\) 1.34218e8i 0.125000i
\(65\) 1.40737e9 + 7.28780e8i 1.21294 + 0.628101i
\(66\) −2.13778e7 −0.0170704
\(67\) −6.75197e8 + 6.75197e8i −0.500100 + 0.500100i −0.911469 − 0.411369i \(-0.865051\pi\)
0.411369 + 0.911469i \(0.365051\pi\)
\(68\) 3.12011e8 + 3.12011e8i 0.214598 + 0.214598i
\(69\) − 4.72241e7i − 0.0301938i
\(70\) −6.44414e8 − 2.02857e9i −0.383420 − 1.20698i
\(71\) −5.54597e8 −0.307387 −0.153694 − 0.988119i \(-0.549117\pi\)
−0.153694 + 0.988119i \(0.549117\pi\)
\(72\) 4.83428e8 − 4.83428e8i 0.249844 − 0.249844i
\(73\) 8.91747e8 + 8.91747e8i 0.430158 + 0.430158i 0.888682 − 0.458524i \(-0.151622\pi\)
−0.458524 + 0.888682i \(0.651622\pi\)
\(74\) − 1.76285e9i − 0.794431i
\(75\) 5.84197e7 + 1.00416e7i 0.0246180 + 0.00423154i
\(76\) −1.71636e8 −0.0676924
\(77\) 3.31293e9 − 3.31293e9i 1.22394 − 1.22394i
\(78\) 4.92544e7 + 4.92544e7i 0.0170597 + 0.0170597i
\(79\) 1.69149e9i 0.549711i 0.961486 + 0.274856i \(0.0886300\pi\)
−0.961486 + 0.274856i \(0.911370\pi\)
\(80\) 7.80752e8 − 2.48021e8i 0.238267 − 0.0756899i
\(81\) −3.48026e9 −0.998129
\(82\) −2.19302e9 + 2.19302e9i −0.591525 + 0.591525i
\(83\) 1.96163e9 + 1.96163e9i 0.497996 + 0.497996i 0.910814 − 0.412818i \(-0.135455\pi\)
−0.412818 + 0.910814i \(0.635455\pi\)
\(84\) − 9.35479e7i − 0.0223685i
\(85\) 1.23842e9 − 2.39155e9i 0.279109 − 0.538996i
\(86\) 2.98973e9 0.635535
\(87\) 5.74345e6 − 5.74345e6i 0.00115233 − 0.00115233i
\(88\) 1.27507e9 + 1.27507e9i 0.241614 + 0.241614i
\(89\) 7.73241e9i 1.38473i 0.721548 + 0.692364i \(0.243431\pi\)
−0.721548 + 0.692364i \(0.756569\pi\)
\(90\) −3.70545e9 − 1.91880e9i −0.627521 − 0.324951i
\(91\) −1.52660e10 −2.44635
\(92\) −2.81667e9 + 2.81667e9i −0.427363 + 0.427363i
\(93\) 1.11296e8 + 1.11296e8i 0.0159980 + 0.0159980i
\(94\) 3.67931e9i 0.501334i
\(95\) 3.17166e8 + 9.98416e8i 0.0409891 + 0.129031i
\(96\) 3.60045e7 0.00441571
\(97\) 1.16280e10 − 1.16280e10i 1.35408 − 1.35408i 0.473039 − 0.881041i \(-0.343157\pi\)
0.881041 − 0.473039i \(-0.156843\pi\)
\(98\) 9.97760e9 + 9.97760e9i 1.10381 + 1.10381i
\(99\) − 9.18516e9i − 0.965853i
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 10.11.c.c.3.2 ✓ 6
3.2 odd 2 90.11.g.c.73.3 6
4.3 odd 2 80.11.p.c.33.2 6
5.2 odd 4 inner 10.11.c.c.7.2 yes 6
5.3 odd 4 50.11.c.e.7.2 6
5.4 even 2 50.11.c.e.43.2 6
15.2 even 4 90.11.g.c.37.3 6
20.7 even 4 80.11.p.c.17.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.c.3.2 ✓ 6 1.1 even 1 trivial
10.11.c.c.7.2 yes 6 5.2 odd 4 inner
50.11.c.e.7.2 6 5.3 odd 4
50.11.c.e.43.2 6 5.4 even 2
80.11.p.c.17.2 6 20.7 even 4
80.11.p.c.33.2 6 4.3 odd 2
90.11.g.c.37.3 6 15.2 even 4
90.11.g.c.73.3 6 3.2 odd 2