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Properties

Label 110.3.d.a.21.5
Level $110$
Weight $3$
Character 110.21
Analytic conductor $2.997$
Analytic rank $0$
Dimension $8$
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] = [110,3,Mod(21,110)] 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("110.21"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 110.d (of order \(2\), degree \(1\), 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: \(2.99728290796\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4956160000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 21.5
Root \(1.51954 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 110.21
Dual form 110.3.d.a.21.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -5.11433 q^{3} -2.00000 q^{4} +2.23607 q^{5} -7.23275i q^{6} -6.78202i q^{7} -2.82843i q^{8} +17.1564 q^{9} +3.16228i q^{10} +(7.23901 - 8.28232i) q^{11} +10.2287 q^{12} +9.87597i q^{13} +9.59122 q^{14} -11.4360 q^{15} +4.00000 q^{16} -29.1178i q^{17} +24.2628i q^{18} -27.1845i q^{19} -4.47214 q^{20} +34.6855i q^{21} +(11.7130 + 10.2375i) q^{22} -35.0515 q^{23} +14.4655i q^{24} +5.00000 q^{25} -13.9667 q^{26} -41.7143 q^{27} +13.5640i q^{28} +21.3472i q^{29} -16.1729i q^{30} +21.1545 q^{31} +5.65685i q^{32} +(-37.0227 + 42.3585i) q^{33} +41.1788 q^{34} -15.1651i q^{35} -34.3127 q^{36} -7.43123 q^{37} +38.4447 q^{38} -50.5089i q^{39} -6.32456i q^{40} -44.2799i q^{41} -49.0527 q^{42} -15.2380i q^{43} +(-14.4780 + 16.5646i) q^{44} +38.3628 q^{45} -49.5704i q^{46} -0.490048 q^{47} -20.4573 q^{48} +3.00421 q^{49} +7.07107i q^{50} +148.918i q^{51} -19.7519i q^{52} +40.6167 q^{53} -58.9929i q^{54} +(16.1869 - 18.5198i) q^{55} -19.1824 q^{56} +139.031i q^{57} -30.1895 q^{58} -15.7748 q^{59} +22.8720 q^{60} +36.9599i q^{61} +29.9169i q^{62} -116.355i q^{63} -8.00000 q^{64} +22.0833i q^{65} +(-59.9039 - 52.3580i) q^{66} +98.6639 q^{67} +58.2356i q^{68} +179.265 q^{69} +21.4466 q^{70} -114.279 q^{71} -48.5255i q^{72} -106.453i q^{73} -10.5093i q^{74} -25.5716 q^{75} +54.3691i q^{76} +(-56.1708 - 49.0951i) q^{77} +71.4304 q^{78} +35.4008i q^{79} +8.94427 q^{80} +58.9334 q^{81} +62.6212 q^{82} +119.115i q^{83} -69.3710i q^{84} -65.1094i q^{85} +21.5498 q^{86} -109.177i q^{87} +(-23.4259 - 20.4750i) q^{88} -121.240 q^{89} +54.2532i q^{90} +66.9790 q^{91} +70.1031 q^{92} -108.191 q^{93} -0.693032i q^{94} -60.7865i q^{95} -28.9310i q^{96} +124.898 q^{97} +4.24860i q^{98} +(124.195 - 142.094i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 16 q^{4} + 40 q^{9} + 16 q^{12} - 32 q^{14} - 40 q^{15} + 32 q^{16} - 136 q^{23} + 40 q^{25} + 80 q^{26} + 64 q^{27} - 64 q^{31} + 88 q^{33} + 112 q^{34} - 80 q^{36} - 48 q^{37} - 208 q^{42}+ \cdots + 704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) −5.11433 −1.70478 −0.852388 − 0.522910i \(-0.824847\pi\)
−0.852388 + 0.522910i \(0.824847\pi\)
\(4\) −2.00000 −0.500000
\(5\) 2.23607 0.447214
\(6\) − 7.23275i − 1.20546i
\(7\) − 6.78202i − 0.968860i −0.874830 − 0.484430i \(-0.839027\pi\)
0.874830 − 0.484430i \(-0.160973\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 17.1564 1.90626
\(10\) 3.16228i 0.316228i
\(11\) 7.23901 − 8.28232i 0.658092 − 0.752938i
\(12\) 10.2287 0.852388
\(13\) 9.87597i 0.759690i 0.925050 + 0.379845i \(0.124023\pi\)
−0.925050 + 0.379845i \(0.875977\pi\)
\(14\) 9.59122 0.685087
\(15\) −11.4360 −0.762399
\(16\) 4.00000 0.250000
\(17\) − 29.1178i − 1.71281i −0.516302 − 0.856406i \(-0.672692\pi\)
0.516302 − 0.856406i \(-0.327308\pi\)
\(18\) 24.2628i 1.34793i
\(19\) − 27.1845i − 1.43077i −0.698733 − 0.715383i \(-0.746252\pi\)
0.698733 − 0.715383i \(-0.253748\pi\)
\(20\) −4.47214 −0.223607
\(21\) 34.6855i 1.65169i
\(22\) 11.7130 + 10.2375i 0.532407 + 0.465341i
\(23\) −35.0515 −1.52398 −0.761990 − 0.647589i \(-0.775778\pi\)
−0.761990 + 0.647589i \(0.775778\pi\)
\(24\) 14.4655i 0.602729i
\(25\) 5.00000 0.200000
\(26\) −13.9667 −0.537182
\(27\) −41.7143 −1.54497
\(28\) 13.5640i 0.484430i
\(29\) 21.3472i 0.736110i 0.929804 + 0.368055i \(0.119976\pi\)
−0.929804 + 0.368055i \(0.880024\pi\)
\(30\) − 16.1729i − 0.539098i
\(31\) 21.1545 0.682402 0.341201 − 0.939990i \(-0.389166\pi\)
0.341201 + 0.939990i \(0.389166\pi\)
\(32\) 5.65685i 0.176777i
\(33\) −37.0227 + 42.3585i −1.12190 + 1.28359i
\(34\) 41.1788 1.21114
\(35\) − 15.1651i − 0.433287i
\(36\) −34.3127 −0.953131
\(37\) −7.43123 −0.200844 −0.100422 − 0.994945i \(-0.532019\pi\)
−0.100422 + 0.994945i \(0.532019\pi\)
\(38\) 38.4447 1.01170
\(39\) − 50.5089i − 1.29510i
\(40\) − 6.32456i − 0.158114i
\(41\) − 44.2799i − 1.08000i −0.841666 − 0.539999i \(-0.818425\pi\)
0.841666 − 0.539999i \(-0.181575\pi\)
\(42\) −49.0527 −1.16792
\(43\) − 15.2380i − 0.354373i −0.984177 − 0.177186i \(-0.943300\pi\)
0.984177 − 0.177186i \(-0.0566995\pi\)
\(44\) −14.4780 + 16.5646i −0.329046 + 0.376469i
\(45\) 38.3628 0.852506
\(46\) − 49.5704i − 1.07762i
\(47\) −0.490048 −0.0104265 −0.00521327 − 0.999986i \(-0.501659\pi\)
−0.00521327 + 0.999986i \(0.501659\pi\)
\(48\) −20.4573 −0.426194
\(49\) 3.00421 0.0613105
\(50\) 7.07107i 0.141421i
\(51\) 148.918i 2.91996i
\(52\) − 19.7519i − 0.379845i
\(53\) 40.6167 0.766353 0.383177 − 0.923675i \(-0.374830\pi\)
0.383177 + 0.923675i \(0.374830\pi\)
\(54\) − 58.9929i − 1.09246i
\(55\) 16.1869 − 18.5198i 0.294308 − 0.336724i
\(56\) −19.1824 −0.342544
\(57\) 139.031i 2.43913i
\(58\) −30.1895 −0.520509
\(59\) −15.7748 −0.267370 −0.133685 − 0.991024i \(-0.542681\pi\)
−0.133685 + 0.991024i \(0.542681\pi\)
\(60\) 22.8720 0.381200
\(61\) 36.9599i 0.605901i 0.953006 + 0.302950i \(0.0979716\pi\)
−0.953006 + 0.302950i \(0.902028\pi\)
\(62\) 29.9169i 0.482531i
\(63\) − 116.355i − 1.84690i
\(64\) −8.00000 −0.125000
\(65\) 22.0833i 0.339744i
\(66\) −59.9039 − 52.3580i −0.907636 − 0.793303i
\(67\) 98.6639 1.47260 0.736298 − 0.676658i \(-0.236572\pi\)
0.736298 + 0.676658i \(0.236572\pi\)
\(68\) 58.2356i 0.856406i
\(69\) 179.265 2.59805
\(70\) 21.4466 0.306380
\(71\) −114.279 −1.60957 −0.804784 − 0.593568i \(-0.797719\pi\)
−0.804784 + 0.593568i \(0.797719\pi\)
\(72\) − 48.5255i − 0.673965i
\(73\) − 106.453i − 1.45826i −0.684374 − 0.729131i \(-0.739925\pi\)
0.684374 − 0.729131i \(-0.260075\pi\)
\(74\) − 10.5093i − 0.142018i
\(75\) −25.5716 −0.340955
\(76\) 54.3691i 0.715383i
\(77\) −56.1708 − 49.0951i −0.729491 − 0.637599i
\(78\) 71.4304 0.915775
\(79\) 35.4008i 0.448112i 0.974576 + 0.224056i \(0.0719298\pi\)
−0.974576 + 0.224056i \(0.928070\pi\)
\(80\) 8.94427 0.111803
\(81\) 58.9334 0.727573
\(82\) 62.6212 0.763673
\(83\) 119.115i 1.43512i 0.696499 + 0.717558i \(0.254740\pi\)
−0.696499 + 0.717558i \(0.745260\pi\)
\(84\) − 69.3710i − 0.825845i
\(85\) − 65.1094i − 0.765993i
\(86\) 21.5498 0.250579
\(87\) − 109.177i − 1.25490i
\(88\) −23.4259 − 20.4750i −0.266204 − 0.232671i
\(89\) −121.240 −1.36225 −0.681123 − 0.732169i \(-0.738508\pi\)
−0.681123 + 0.732169i \(0.738508\pi\)
\(90\) 54.2532i 0.602813i
\(91\) 66.9790 0.736033
\(92\) 70.1031 0.761990
\(93\) −108.191 −1.16334
\(94\) − 0.693032i − 0.00737268i
\(95\) − 60.7865i − 0.639858i
\(96\) − 28.9310i − 0.301365i
\(97\) 124.898 1.28761 0.643805 − 0.765190i \(-0.277355\pi\)
0.643805 + 0.765190i \(0.277355\pi\)
\(98\) 4.24860i 0.0433531i
\(99\) 124.195 − 142.094i 1.25450 − 1.43530i
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 110.3.d.a.21.5 yes 8
3.2 odd 2 990.3.b.b.901.1 8
4.3 odd 2 880.3.j.c.241.8 8
5.2 odd 4 550.3.c.b.549.8 16
5.3 odd 4 550.3.c.b.549.9 16
5.4 even 2 550.3.d.f.351.4 8
11.10 odd 2 inner 110.3.d.a.21.1 ✓ 8
33.32 even 2 990.3.b.b.901.6 8
44.43 even 2 880.3.j.c.241.7 8
55.32 even 4 550.3.c.b.549.16 16
55.43 even 4 550.3.c.b.549.1 16
55.54 odd 2 550.3.d.f.351.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.1 ✓ 8 11.10 odd 2 inner
110.3.d.a.21.5 yes 8 1.1 even 1 trivial
550.3.c.b.549.1 16 55.43 even 4
550.3.c.b.549.8 16 5.2 odd 4
550.3.c.b.549.9 16 5.3 odd 4
550.3.c.b.549.16 16 55.32 even 4
550.3.d.f.351.4 8 5.4 even 2
550.3.d.f.351.8 8 55.54 odd 2
880.3.j.c.241.7 8 44.43 even 2
880.3.j.c.241.8 8 4.3 odd 2
990.3.b.b.901.1 8 3.2 odd 2
990.3.b.b.901.6 8 33.32 even 2