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Properties

Label 110.3.d.a.21.6
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 21.6
Root \(-1.09132 + 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 110.21
Dual form 110.3.d.a.21.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.29552 q^{3} -2.00000 q^{4} -2.23607 q^{5} -3.24636i q^{6} -4.07370i q^{7} -2.82843i q^{8} -3.73057 q^{9} -3.16228i q^{10} +(-10.0795 - 4.40491i) q^{11} +4.59105 q^{12} -22.8069i q^{13} +5.76108 q^{14} +5.13295 q^{15} +4.00000 q^{16} +5.68523i q^{17} -5.27583i q^{18} +29.9904i q^{19} +4.47214 q^{20} +9.35127i q^{21} +(6.22949 - 14.2546i) q^{22} -32.8493 q^{23} +6.49272i q^{24} +5.00000 q^{25} +32.2538 q^{26} +29.2233 q^{27} +8.14739i q^{28} +2.84548i q^{29} +7.25908i q^{30} -46.2613 q^{31} +5.65685i q^{32} +(23.1378 + 10.1116i) q^{33} -8.04013 q^{34} +9.10906i q^{35} +7.46115 q^{36} +17.0706 q^{37} -42.4129 q^{38} +52.3538i q^{39} +6.32456i q^{40} +35.5868i q^{41} -13.2247 q^{42} -45.5683i q^{43} +(20.1590 + 8.80982i) q^{44} +8.34182 q^{45} -46.4559i q^{46} -56.9032 q^{47} -9.18209 q^{48} +32.4050 q^{49} +7.07107i q^{50} -13.0506i q^{51} +45.6138i q^{52} +62.1711 q^{53} +41.3280i q^{54} +(22.5385 + 9.84968i) q^{55} -11.5222 q^{56} -68.8437i q^{57} -4.02412 q^{58} -11.4721 q^{59} -10.2659 q^{60} -85.6023i q^{61} -65.4233i q^{62} +15.1972i q^{63} -8.00000 q^{64} +50.9978i q^{65} +(-14.2999 + 32.7218i) q^{66} +5.61280 q^{67} -11.3705i q^{68} +75.4063 q^{69} -12.8822 q^{70} +25.2141 q^{71} +10.5517i q^{72} -125.646i q^{73} +24.1415i q^{74} -11.4776 q^{75} -59.9809i q^{76} +(-17.9443 + 41.0609i) q^{77} -74.0394 q^{78} -40.4075i q^{79} -8.94427 q^{80} -33.5077 q^{81} -50.3273 q^{82} +8.92477i q^{83} -18.7025i q^{84} -12.7126i q^{85} +64.4433 q^{86} -6.53187i q^{87} +(-12.4590 + 28.5092i) q^{88} -44.8711 q^{89} +11.7971i q^{90} -92.9084 q^{91} +65.6986 q^{92} +106.194 q^{93} -80.4733i q^{94} -67.0606i q^{95} -12.9854i q^{96} -141.919 q^{97} +45.8276i q^{98} +(37.6024 + 16.4328i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 16 q^{4} + 40 q^{9} + 16 q^{12} - 32 q^{14} - 40 q^{15} + 32 q^{16} - 136 q^{23} + 40 q^{25} + 80 q^{26} + 64 q^{27} - 64 q^{31} + 88 q^{33} + 112 q^{34} - 80 q^{36} - 48 q^{37} - 208 q^{42}+ \cdots + 704 q^{99}+O(q^{100}) \) 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
\(3\) −2.29552 −0.765174 −0.382587 − 0.923919i \(-0.624967\pi\)
−0.382587 + 0.923919i \(0.624967\pi\)
\(4\) −2.00000 −0.500000
\(5\) −2.23607 −0.447214
\(6\) − 3.24636i − 0.541060i
\(7\) − 4.07370i − 0.581957i −0.956730 − 0.290978i \(-0.906019\pi\)
0.956730 − 0.290978i \(-0.0939808\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) −3.73057 −0.414508
\(10\) − 3.16228i − 0.316228i
\(11\) −10.0795 − 4.40491i −0.916320 − 0.400447i
\(12\) 4.59105 0.382587
\(13\) − 22.8069i − 1.75438i −0.480146 − 0.877189i \(-0.659416\pi\)
0.480146 − 0.877189i \(-0.340584\pi\)
\(14\) 5.76108 0.411506
\(15\) 5.13295 0.342196
\(16\) 4.00000 0.250000
\(17\) 5.68523i 0.334425i 0.985921 + 0.167213i \(0.0534767\pi\)
−0.985921 + 0.167213i \(0.946523\pi\)
\(18\) − 5.27583i − 0.293102i
\(19\) 29.9904i 1.57844i 0.614108 + 0.789222i \(0.289516\pi\)
−0.614108 + 0.789222i \(0.710484\pi\)
\(20\) 4.47214 0.223607
\(21\) 9.35127i 0.445298i
\(22\) 6.22949 − 14.2546i 0.283158 − 0.647936i
\(23\) −32.8493 −1.42823 −0.714115 − 0.700029i \(-0.753171\pi\)
−0.714115 + 0.700029i \(0.753171\pi\)
\(24\) 6.49272i 0.270530i
\(25\) 5.00000 0.200000
\(26\) 32.2538 1.24053
\(27\) 29.2233 1.08235
\(28\) 8.14739i 0.290978i
\(29\) 2.84548i 0.0981201i 0.998796 + 0.0490601i \(0.0156226\pi\)
−0.998796 + 0.0490601i \(0.984377\pi\)
\(30\) 7.25908i 0.241969i
\(31\) −46.2613 −1.49230 −0.746150 − 0.665778i \(-0.768100\pi\)
−0.746150 + 0.665778i \(0.768100\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 23.1378 + 10.1116i 0.701145 + 0.306411i
\(34\) −8.04013 −0.236474
\(35\) 9.10906i 0.260259i
\(36\) 7.46115 0.207254
\(37\) 17.0706 0.461369 0.230684 − 0.973029i \(-0.425904\pi\)
0.230684 + 0.973029i \(0.425904\pi\)
\(38\) −42.4129 −1.11613
\(39\) 52.3538i 1.34240i
\(40\) 6.32456i 0.158114i
\(41\) 35.5868i 0.867970i 0.900920 + 0.433985i \(0.142893\pi\)
−0.900920 + 0.433985i \(0.857107\pi\)
\(42\) −13.2247 −0.314874
\(43\) − 45.5683i − 1.05973i −0.848082 − 0.529864i \(-0.822243\pi\)
0.848082 − 0.529864i \(-0.177757\pi\)
\(44\) 20.1590 + 8.80982i 0.458160 + 0.200223i
\(45\) 8.34182 0.185374
\(46\) − 46.4559i − 1.00991i
\(47\) −56.9032 −1.21071 −0.605353 − 0.795957i \(-0.706968\pi\)
−0.605353 + 0.795957i \(0.706968\pi\)
\(48\) −9.18209 −0.191294
\(49\) 32.4050 0.661326
\(50\) 7.07107i 0.141421i
\(51\) − 13.0506i − 0.255894i
\(52\) 45.6138i 0.877189i
\(53\) 62.1711 1.17304 0.586520 − 0.809935i \(-0.300498\pi\)
0.586520 + 0.809935i \(0.300498\pi\)
\(54\) 41.3280i 0.765334i
\(55\) 22.5385 + 9.84968i 0.409791 + 0.179085i
\(56\) −11.5222 −0.205753
\(57\) − 68.8437i − 1.20778i
\(58\) −4.02412 −0.0693814
\(59\) −11.4721 −0.194442 −0.0972212 − 0.995263i \(-0.530995\pi\)
−0.0972212 + 0.995263i \(0.530995\pi\)
\(60\) −10.2659 −0.171098
\(61\) − 85.6023i − 1.40332i −0.712513 − 0.701659i \(-0.752443\pi\)
0.712513 − 0.701659i \(-0.247557\pi\)
\(62\) − 65.4233i − 1.05522i
\(63\) 15.1972i 0.241226i
\(64\) −8.00000 −0.125000
\(65\) 50.9978i 0.784581i
\(66\) −14.2999 + 32.7218i −0.216666 + 0.495784i
\(67\) 5.61280 0.0837731 0.0418865 − 0.999122i \(-0.486663\pi\)
0.0418865 + 0.999122i \(0.486663\pi\)
\(68\) − 11.3705i − 0.167213i
\(69\) 75.4063 1.09284
\(70\) −12.8822 −0.184031
\(71\) 25.2141 0.355128 0.177564 − 0.984109i \(-0.443178\pi\)
0.177564 + 0.984109i \(0.443178\pi\)
\(72\) 10.5517i 0.146551i
\(73\) − 125.646i − 1.72117i −0.509303 − 0.860587i \(-0.670097\pi\)
0.509303 − 0.860587i \(-0.329903\pi\)
\(74\) 24.1415i 0.326237i
\(75\) −11.4776 −0.153035
\(76\) − 59.9809i − 0.789222i
\(77\) −17.9443 + 41.0609i −0.233043 + 0.533259i
\(78\) −74.0394 −0.949223
\(79\) − 40.4075i − 0.511487i −0.966745 − 0.255743i \(-0.917680\pi\)
0.966745 − 0.255743i \(-0.0823202\pi\)
\(80\) −8.94427 −0.111803
\(81\) −33.5077 −0.413675
\(82\) −50.3273 −0.613747
\(83\) 8.92477i 0.107527i 0.998554 + 0.0537637i \(0.0171218\pi\)
−0.998554 + 0.0537637i \(0.982878\pi\)
\(84\) − 18.7025i − 0.222649i
\(85\) − 12.7126i − 0.149560i
\(86\) 64.4433 0.749341
\(87\) − 6.53187i − 0.0750790i
\(88\) −12.4590 + 28.5092i −0.141579 + 0.323968i
\(89\) −44.8711 −0.504169 −0.252085 − 0.967705i \(-0.581116\pi\)
−0.252085 + 0.967705i \(0.581116\pi\)
\(90\) 11.7971i 0.131079i
\(91\) −92.9084 −1.02097
\(92\) 65.6986 0.714115
\(93\) 106.194 1.14187
\(94\) − 80.4733i − 0.856099i
\(95\) − 67.0606i − 0.705901i
\(96\) − 12.9854i − 0.135265i
\(97\) −141.919 −1.46308 −0.731542 − 0.681796i \(-0.761199\pi\)
−0.731542 + 0.681796i \(0.761199\pi\)
\(98\) 45.8276i 0.467628i
\(99\) 37.6024 + 16.4328i 0.379822 + 0.165988i
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.21.6 yes 8
3.2 odd 2 990.3.b.b.901.3 8
4.3 odd 2 880.3.j.c.241.6 8
5.2 odd 4 550.3.c.b.549.6 16
5.3 odd 4 550.3.c.b.549.11 16
5.4 even 2 550.3.d.f.351.3 8
11.10 odd 2 inner 110.3.d.a.21.2 ✓ 8
33.32 even 2 990.3.b.b.901.8 8
44.43 even 2 880.3.j.c.241.5 8
55.32 even 4 550.3.c.b.549.14 16
55.43 even 4 550.3.c.b.549.3 16
55.54 odd 2 550.3.d.f.351.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.2 ✓ 8 11.10 odd 2 inner
110.3.d.a.21.6 yes 8 1.1 even 1 trivial
550.3.c.b.549.3 16 55.43 even 4
550.3.c.b.549.6 16 5.2 odd 4
550.3.c.b.549.11 16 5.3 odd 4
550.3.c.b.549.14 16 55.32 even 4
550.3.d.f.351.3 8 5.4 even 2
550.3.d.f.351.7 8 55.54 odd 2
880.3.j.c.241.5 8 44.43 even 2
880.3.j.c.241.6 8 4.3 odd 2
990.3.b.b.901.3 8 3.2 odd 2
990.3.b.b.901.8 8 33.32 even 2