Properties

Label 56.2.b.a.29.1
Level 5656
Weight 22
Character 56.29
Analytic conductor 0.4470.447
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,2,Mod(29,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 56=237 56 = 2^{3} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 56.b (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.4471622513190.447162251319
Analytic rank: 00
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{-2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+2 x^{2} + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 29.1
Root 1.41421i-1.41421i of defining polynomial
Character χ\chi == 56.29
Dual form 56.2.b.a.29.2

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q1.41421iq21.41421iq32.00000q4+1.41421iq52.00000q6+1.00000q7+2.82843iq8+1.00000q9+2.00000q102.82843iq11+2.82843iq12+4.24264iq131.41421iq14+2.00000q15+4.00000q166.00000q171.41421iq18+4.24264iq192.82843iq201.41421iq214.00000q226.00000q23+4.00000q24+3.00000q25+6.00000q265.65685iq272.00000q282.82843iq292.82843iq304.00000q315.65685iq324.00000q33+8.48528iq34+1.41421iq352.00000q368.48528iq37+6.00000q38+6.00000q394.00000q40+6.00000q412.00000q42+8.48528iq43+5.65685iq44+1.41421iq45+8.48528iq465.65685iq48+1.00000q494.24264iq50+8.48528iq518.48528iq52+5.65685iq538.00000q54+4.00000q55+2.82843iq56+6.00000q574.00000q58+1.41421iq594.00000q6012.7279iq61+5.65685iq62+1.00000q638.00000q646.00000q65+5.65685iq66+12.0000q68+8.48528iq69+2.00000q70+2.82843iq72+2.00000q7312.0000q744.24264iq758.48528iq762.82843iq778.48528iq78+8.00000q79+5.65685iq805.00000q818.48528iq8215.5563iq83+2.82843iq848.48528iq85+12.0000q864.00000q87+8.00000q88+6.00000q89+2.00000q90+4.24264iq91+12.0000q92+5.65685iq936.00000q958.00000q9610.0000q971.41421iq982.82843iq99+O(q100)q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} -2.00000 q^{6} +1.00000 q^{7} +2.82843i q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.82843i q^{11} +2.82843i q^{12} +4.24264i q^{13} -1.41421i q^{14} +2.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -1.41421i q^{18} +4.24264i q^{19} -2.82843i q^{20} -1.41421i q^{21} -4.00000 q^{22} -6.00000 q^{23} +4.00000 q^{24} +3.00000 q^{25} +6.00000 q^{26} -5.65685i q^{27} -2.00000 q^{28} -2.82843i q^{29} -2.82843i q^{30} -4.00000 q^{31} -5.65685i q^{32} -4.00000 q^{33} +8.48528i q^{34} +1.41421i q^{35} -2.00000 q^{36} -8.48528i q^{37} +6.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} +8.48528i q^{43} +5.65685i q^{44} +1.41421i q^{45} +8.48528i q^{46} -5.65685i q^{48} +1.00000 q^{49} -4.24264i q^{50} +8.48528i q^{51} -8.48528i q^{52} +5.65685i q^{53} -8.00000 q^{54} +4.00000 q^{55} +2.82843i q^{56} +6.00000 q^{57} -4.00000 q^{58} +1.41421i q^{59} -4.00000 q^{60} -12.7279i q^{61} +5.65685i q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +5.65685i q^{66} +12.0000 q^{68} +8.48528i q^{69} +2.00000 q^{70} +2.82843i q^{72} +2.00000 q^{73} -12.0000 q^{74} -4.24264i q^{75} -8.48528i q^{76} -2.82843i q^{77} -8.48528i q^{78} +8.00000 q^{79} +5.65685i q^{80} -5.00000 q^{81} -8.48528i q^{82} -15.5563i q^{83} +2.82843i q^{84} -8.48528i q^{85} +12.0000 q^{86} -4.00000 q^{87} +8.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} +4.24264i q^{91} +12.0000 q^{92} +5.65685i q^{93} -6.00000 q^{95} -8.00000 q^{96} -10.0000 q^{97} -1.41421i q^{98} -2.82843i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q4q44q6+2q7+2q9+4q10+4q15+8q1612q178q2212q23+8q24+6q25+12q264q288q318q334q36+12q38+20q97+O(q100) 2 q - 4 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9} + 4 q^{10} + 4 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{22} - 12 q^{23} + 8 q^{24} + 6 q^{25} + 12 q^{26} - 4 q^{28} - 8 q^{31} - 8 q^{33} - 4 q^{36} + 12 q^{38}+ \cdots - 20 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 1515 1717 2929
χ(n)\chi(n) 11 11 1-1

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 − 1.41421i − 1.00000i
33 − 1.41421i − 0.816497i −0.912871 0.408248i 0.866140π-0.866140\pi
0.912871 0.408248i 0.133860π-0.133860\pi
44 −2.00000 −1.00000
55 1.41421i 0.632456i 0.948683 + 0.316228i 0.102416π0.102416\pi
−0.948683 + 0.316228i 0.897584π0.897584\pi
66 −2.00000 −0.816497
77 1.00000 0.377964
88 2.82843i 1.00000i
99 1.00000 0.333333
1010 2.00000 0.632456
1111 − 2.82843i − 0.852803i −0.904534 0.426401i 0.859781π-0.859781\pi
0.904534 0.426401i 0.140219π-0.140219\pi
1212 2.82843i 0.816497i
1313 4.24264i 1.17670i 0.808608 + 0.588348i 0.200222π0.200222\pi
−0.808608 + 0.588348i 0.799778π0.799778\pi
1414 − 1.41421i − 0.377964i
1515 2.00000 0.516398
1616 4.00000 1.00000
1717 −6.00000 −1.45521 −0.727607 0.685994i 0.759367π-0.759367\pi
−0.727607 + 0.685994i 0.759367π0.759367\pi
1818 − 1.41421i − 0.333333i
1919 4.24264i 0.973329i 0.873589 + 0.486664i 0.161786π0.161786\pi
−0.873589 + 0.486664i 0.838214π0.838214\pi
2020 − 2.82843i − 0.632456i
2121 − 1.41421i − 0.308607i
2222 −4.00000 −0.852803
2323 −6.00000 −1.25109 −0.625543 0.780189i 0.715123π-0.715123\pi
−0.625543 + 0.780189i 0.715123π0.715123\pi
2424 4.00000 0.816497
2525 3.00000 0.600000
2626 6.00000 1.17670
2727 − 5.65685i − 1.08866i
2828 −2.00000 −0.377964
2929 − 2.82843i − 0.525226i −0.964901 0.262613i 0.915416π-0.915416\pi
0.964901 0.262613i 0.0845842π-0.0845842\pi
3030 − 2.82843i − 0.516398i
3131 −4.00000 −0.718421 −0.359211 0.933257i 0.616954π-0.616954\pi
−0.359211 + 0.933257i 0.616954π0.616954\pi
3232 − 5.65685i − 1.00000i
3333 −4.00000 −0.696311
3434 8.48528i 1.45521i
3535 1.41421i 0.239046i
3636 −2.00000 −0.333333
3737 − 8.48528i − 1.39497i −0.716599 0.697486i 0.754302π-0.754302\pi
0.716599 0.697486i 0.245698π-0.245698\pi
3838 6.00000 0.973329
3939 6.00000 0.960769
4040 −4.00000 −0.632456
4141 6.00000 0.937043 0.468521 0.883452i 0.344787π-0.344787\pi
0.468521 + 0.883452i 0.344787π0.344787\pi
4242 −2.00000 −0.308607
4343 8.48528i 1.29399i 0.762493 + 0.646997i 0.223975π0.223975\pi
−0.762493 + 0.646997i 0.776025π0.776025\pi
4444 5.65685i 0.852803i
4545 1.41421i 0.210819i
4646 8.48528i 1.25109i
4747 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4848 − 5.65685i − 0.816497i
4949 1.00000 0.142857
5050 − 4.24264i − 0.600000i
5151 8.48528i 1.18818i
5252 − 8.48528i − 1.17670i
5353 5.65685i 0.777029i 0.921443 + 0.388514i 0.127012π0.127012\pi
−0.921443 + 0.388514i 0.872988π0.872988\pi
5454 −8.00000 −1.08866
5555 4.00000 0.539360
5656 2.82843i 0.377964i
5757 6.00000 0.794719
5858 −4.00000 −0.525226
5959 1.41421i 0.184115i 0.995754 + 0.0920575i 0.0293443π0.0293443\pi
−0.995754 + 0.0920575i 0.970656π0.970656\pi
6060 −4.00000 −0.516398
6161 − 12.7279i − 1.62964i −0.579712 0.814822i 0.696835π-0.696835\pi
0.579712 0.814822i 0.303165π-0.303165\pi
6262 5.65685i 0.718421i
6363 1.00000 0.125988
6464 −8.00000 −1.00000
6565 −6.00000 −0.744208
6666 5.65685i 0.696311i
6767 0 0 1.00000 00
−1.00000 π\pi
6868 12.0000 1.45521
6969 8.48528i 1.02151i
7070 2.00000 0.239046
7171 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
7272 2.82843i 0.333333i
7373 2.00000 0.234082 0.117041 0.993127i 0.462659π-0.462659\pi
0.117041 + 0.993127i 0.462659π0.462659\pi
7474 −12.0000 −1.39497
7575 − 4.24264i − 0.489898i
7676 − 8.48528i − 0.973329i
7777 − 2.82843i − 0.322329i
7878 − 8.48528i − 0.960769i
7979 8.00000 0.900070 0.450035 0.893011i 0.351411π-0.351411\pi
0.450035 + 0.893011i 0.351411π0.351411\pi
8080 5.65685i 0.632456i
8181 −5.00000 −0.555556
8282 − 8.48528i − 0.937043i
8383 − 15.5563i − 1.70753i −0.520658 0.853766i 0.674313π-0.674313\pi
0.520658 0.853766i 0.325687π-0.325687\pi
8484 2.82843i 0.308607i
8585 − 8.48528i − 0.920358i
8686 12.0000 1.29399
8787 −4.00000 −0.428845
8888 8.00000 0.852803
8989 6.00000 0.635999 0.317999 0.948091i 0.396989π-0.396989\pi
0.317999 + 0.948091i 0.396989π0.396989\pi
9090 2.00000 0.210819
9191 4.24264i 0.444750i
9292 12.0000 1.25109
9393 5.65685i 0.586588i
9494 0 0
9595 −6.00000 −0.615587
9696 −8.00000 −0.816497
9797 −10.0000 −1.01535 −0.507673 0.861550i 0.669494π-0.669494\pi
−0.507673 + 0.861550i 0.669494π0.669494\pi
9898 − 1.41421i − 0.142857i
9999 − 2.82843i − 0.284268i
100100 −6.00000 −0.600000
101101 9.89949i 0.985037i 0.870302 + 0.492518i 0.163924π0.163924\pi
−0.870302 + 0.492518i 0.836076π0.836076\pi
102102 12.0000 1.18818
103103 −4.00000 −0.394132 −0.197066 0.980390i 0.563141π-0.563141\pi
−0.197066 + 0.980390i 0.563141π0.563141\pi
104104 −12.0000 −1.17670
105105 2.00000 0.195180
106106 8.00000 0.777029
107107 5.65685i 0.546869i 0.961891 + 0.273434i 0.0881596π0.0881596\pi
−0.961891 + 0.273434i 0.911840π0.911840\pi
108108 11.3137i 1.08866i
109109 8.48528i 0.812743i 0.913708 + 0.406371i 0.133206π0.133206\pi
−0.913708 + 0.406371i 0.866794π0.866794\pi
110110 − 5.65685i − 0.539360i
111111 −12.0000 −1.13899
112112 4.00000 0.377964
113113 −12.0000 −1.12887 −0.564433 0.825479i 0.690905π-0.690905\pi
−0.564433 + 0.825479i 0.690905π0.690905\pi
114114 − 8.48528i − 0.794719i
115115 − 8.48528i − 0.791257i
116116 5.65685i 0.525226i
117117 4.24264i 0.392232i
118118 2.00000 0.184115
119119 −6.00000 −0.550019
120120 5.65685i 0.516398i
121121 3.00000 0.272727
122122 −18.0000 −1.62964
123123 − 8.48528i − 0.765092i
124124 8.00000 0.718421
125125 11.3137i 1.01193i
126126 − 1.41421i − 0.125988i
127127 2.00000 0.177471 0.0887357 0.996055i 0.471717π-0.471717\pi
0.0887357 + 0.996055i 0.471717π0.471717\pi
128128 11.3137i 1.00000i
129129 12.0000 1.05654
130130 8.48528i 0.744208i
131131 1.41421i 0.123560i 0.998090 + 0.0617802i 0.0196778π0.0196778\pi
−0.998090 + 0.0617802i 0.980322π0.980322\pi
132132 8.00000 0.696311
133133 4.24264i 0.367884i
134134 0 0
135135 8.00000 0.688530
136136 − 16.9706i − 1.45521i
137137 6.00000 0.512615 0.256307 0.966595i 0.417494π-0.417494\pi
0.256307 + 0.966595i 0.417494π0.417494\pi
138138 12.0000 1.02151
139139 4.24264i 0.359856i 0.983680 + 0.179928i 0.0575865π0.0575865\pi
−0.983680 + 0.179928i 0.942414π0.942414\pi
140140 − 2.82843i − 0.239046i
141141 0 0
142142 0 0
143143 12.0000 1.00349
144144 4.00000 0.333333
145145 4.00000 0.332182
146146 − 2.82843i − 0.234082i
147147 − 1.41421i − 0.116642i
148148 16.9706i 1.39497i
149149 − 11.3137i − 0.926855i −0.886135 0.463428i 0.846619π-0.846619\pi
0.886135 0.463428i 0.153381π-0.153381\pi
150150 −6.00000 −0.489898
151151 −10.0000 −0.813788 −0.406894 0.913475i 0.633388π-0.633388\pi
−0.406894 + 0.913475i 0.633388π0.633388\pi
152152 −12.0000 −0.973329
153153 −6.00000 −0.485071
154154 −4.00000 −0.322329
155155 − 5.65685i − 0.454369i
156156 −12.0000 −0.960769
157157 − 12.7279i − 1.01580i −0.861416 0.507899i 0.830422π-0.830422\pi
0.861416 0.507899i 0.169578π-0.169578\pi
158158 − 11.3137i − 0.900070i
159159 8.00000 0.634441
160160 8.00000 0.632456
161161 −6.00000 −0.472866
162162 7.07107i 0.555556i
163163 − 8.48528i − 0.664619i −0.943170 0.332309i 0.892172π-0.892172\pi
0.943170 0.332309i 0.107828π-0.107828\pi
164164 −12.0000 −0.937043
165165 − 5.65685i − 0.440386i
166166 −22.0000 −1.70753
167167 24.0000 1.85718 0.928588 0.371113i 0.121024π-0.121024\pi
0.928588 + 0.371113i 0.121024π0.121024\pi
168168 4.00000 0.308607
169169 −5.00000 −0.384615
170170 −12.0000 −0.920358
171171 4.24264i 0.324443i
172172 − 16.9706i − 1.29399i
173173 9.89949i 0.752645i 0.926489 + 0.376322i 0.122811π0.122811\pi
−0.926489 + 0.376322i 0.877189π0.877189\pi
174174 5.65685i 0.428845i
175175 3.00000 0.226779
176176 − 11.3137i − 0.852803i
177177 2.00000 0.150329
178178 − 8.48528i − 0.635999i
179179 5.65685i 0.422813i 0.977398 + 0.211407i 0.0678044π0.0678044\pi
−0.977398 + 0.211407i 0.932196π0.932196\pi
180180 − 2.82843i − 0.210819i
181181 − 12.7279i − 0.946059i −0.881047 0.473029i 0.843160π-0.843160\pi
0.881047 0.473029i 0.156840π-0.156840\pi
182182 6.00000 0.444750
183183 −18.0000 −1.33060
184184 − 16.9706i − 1.25109i
185185 12.0000 0.882258
186186 8.00000 0.586588
187187 16.9706i 1.24101i
188188 0 0
189189 − 5.65685i − 0.411476i
190190 8.48528i 0.615587i
191191 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
192192 11.3137i 0.816497i
193193 −4.00000 −0.287926 −0.143963 0.989583i 0.545985π-0.545985\pi
−0.143963 + 0.989583i 0.545985π0.545985\pi
194194 14.1421i 1.01535i
195195 8.48528i 0.607644i
196196 −2.00000 −0.142857
197197 5.65685i 0.403034i 0.979485 + 0.201517i 0.0645872π0.0645872\pi
−0.979485 + 0.201517i 0.935413π0.935413\pi
198198 −4.00000 −0.284268
199199 20.0000 1.41776 0.708881 0.705328i 0.249200π-0.249200\pi
0.708881 + 0.705328i 0.249200π0.249200\pi
200200 8.48528i 0.600000i
201201 0 0
202202 14.0000 0.985037
203203 − 2.82843i − 0.198517i
204204 − 16.9706i − 1.18818i
205205 8.48528i 0.592638i
206206 5.65685i 0.394132i
207207 −6.00000 −0.417029
208208 16.9706i 1.17670i
209209 12.0000 0.830057
210210 − 2.82843i − 0.195180i
211211 16.9706i 1.16830i 0.811645 + 0.584151i 0.198572π0.198572\pi
−0.811645 + 0.584151i 0.801428π0.801428\pi
212212 − 11.3137i − 0.777029i
213213 0 0
214214 8.00000 0.546869
215215 −12.0000 −0.818393
216216 16.0000 1.08866
217217 −4.00000 −0.271538
218218 12.0000 0.812743
219219 − 2.82843i − 0.191127i
220220 −8.00000 −0.539360
221221 − 25.4558i − 1.71235i
222222 16.9706i 1.13899i
223223 −28.0000 −1.87502 −0.937509 0.347960i 0.886874π-0.886874\pi
−0.937509 + 0.347960i 0.886874π0.886874\pi
224224 − 5.65685i − 0.377964i
225225 3.00000 0.200000
226226 16.9706i 1.12887i
227227 9.89949i 0.657053i 0.944495 + 0.328526i 0.106552π0.106552\pi
−0.944495 + 0.328526i 0.893448π0.893448\pi
228228 −12.0000 −0.794719
229229 − 4.24264i − 0.280362i −0.990126 0.140181i 0.955232π-0.955232\pi
0.990126 0.140181i 0.0447684π-0.0447684\pi
230230 −12.0000 −0.791257
231231 −4.00000 −0.263181
232232 8.00000 0.525226
233233 6.00000 0.393073 0.196537 0.980497i 0.437031π-0.437031\pi
0.196537 + 0.980497i 0.437031π0.437031\pi
234234 6.00000 0.392232
235235 0 0
236236 − 2.82843i − 0.184115i
237237 − 11.3137i − 0.734904i
238238 8.48528i 0.550019i
239239 6.00000 0.388108 0.194054 0.980991i 0.437836π-0.437836\pi
0.194054 + 0.980991i 0.437836π0.437836\pi
240240 8.00000 0.516398
241241 −10.0000 −0.644157 −0.322078 0.946713i 0.604381π-0.604381\pi
−0.322078 + 0.946713i 0.604381π0.604381\pi
242242 − 4.24264i − 0.272727i
243243 − 9.89949i − 0.635053i
244244 25.4558i 1.62964i
245245 1.41421i 0.0903508i
246246 −12.0000 −0.765092
247247 −18.0000 −1.14531
248248 − 11.3137i − 0.718421i
249249 −22.0000 −1.39419
250250 16.0000 1.01193
251251 18.3848i 1.16044i 0.814461 + 0.580218i 0.197033π0.197033\pi
−0.814461 + 0.580218i 0.802967π0.802967\pi
252252 −2.00000 −0.125988
253253 16.9706i 1.06693i
254254 − 2.82843i − 0.177471i
255255 −12.0000 −0.751469
256256 16.0000 1.00000
257257 −6.00000 −0.374270 −0.187135 0.982334i 0.559920π-0.559920\pi
−0.187135 + 0.982334i 0.559920π0.559920\pi
258258 − 16.9706i − 1.05654i
259259 − 8.48528i − 0.527250i
260260 12.0000 0.744208
261261 − 2.82843i − 0.175075i
262262 2.00000 0.123560
263263 24.0000 1.47990 0.739952 0.672660i 0.234848π-0.234848\pi
0.739952 + 0.672660i 0.234848π0.234848\pi
264264 − 11.3137i − 0.696311i
265265 −8.00000 −0.491436
266266 6.00000 0.367884
267267 − 8.48528i − 0.519291i
268268 0 0
269269 − 7.07107i − 0.431131i −0.976489 0.215565i 0.930841π-0.930841\pi
0.976489 0.215565i 0.0691594π-0.0691594\pi
270270 − 11.3137i − 0.688530i
271271 20.0000 1.21491 0.607457 0.794353i 0.292190π-0.292190\pi
0.607457 + 0.794353i 0.292190π0.292190\pi
272272 −24.0000 −1.45521
273273 6.00000 0.363137
274274 − 8.48528i − 0.512615i
275275 − 8.48528i − 0.511682i
276276 − 16.9706i − 1.02151i
277277 16.9706i 1.01966i 0.860274 + 0.509831i 0.170292π0.170292\pi
−0.860274 + 0.509831i 0.829708π0.829708\pi
278278 6.00000 0.359856
279279 −4.00000 −0.239474
280280 −4.00000 −0.239046
281281 −6.00000 −0.357930 −0.178965 0.983855i 0.557275π-0.557275\pi
−0.178965 + 0.983855i 0.557275π0.557275\pi
282282 0 0
283283 − 12.7279i − 0.756596i −0.925684 0.378298i 0.876509π-0.876509\pi
0.925684 0.378298i 0.123491π-0.123491\pi
284284 0 0
285285 8.48528i 0.502625i
286286 − 16.9706i − 1.00349i
287287 6.00000 0.354169
288288 − 5.65685i − 0.333333i
289289 19.0000 1.11765
290290 − 5.65685i − 0.332182i
291291 14.1421i 0.829027i
292292 −4.00000 −0.234082
293293 − 24.0416i − 1.40453i −0.711917 0.702264i 0.752173π-0.752173\pi
0.711917 0.702264i 0.247827π-0.247827\pi
294294 −2.00000 −0.116642
295295 −2.00000 −0.116445
296296 24.0000 1.39497
297297 −16.0000 −0.928414
298298 −16.0000 −0.926855
299299 − 25.4558i − 1.47215i
300300 8.48528i 0.489898i
301301 8.48528i 0.489083i
302302 14.1421i 0.813788i
303303 14.0000 0.804279
304304 16.9706i 0.973329i
305305 18.0000 1.03068
306306 8.48528i 0.485071i
307307 12.7279i 0.726421i 0.931707 + 0.363210i 0.118319π0.118319\pi
−0.931707 + 0.363210i 0.881681π0.881681\pi
308308 5.65685i 0.322329i
309309 5.65685i 0.321807i
310310 −8.00000 −0.454369
311311 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
312312 16.9706i 0.960769i
313313 −10.0000 −0.565233 −0.282617 0.959233i 0.591202π-0.591202\pi
−0.282617 + 0.959233i 0.591202π0.591202\pi
314314 −18.0000 −1.01580
315315 1.41421i 0.0796819i
316316 −16.0000 −0.900070
317317 22.6274i 1.27088i 0.772149 + 0.635441i 0.219182π0.219182\pi
−0.772149 + 0.635441i 0.780818π0.780818\pi
318318 − 11.3137i − 0.634441i
319319 −8.00000 −0.447914
320320 − 11.3137i − 0.632456i
321321 8.00000 0.446516
322322 8.48528i 0.472866i
323323 − 25.4558i − 1.41640i
324324 10.0000 0.555556
325325 12.7279i 0.706018i
326326 −12.0000 −0.664619
327327 12.0000 0.663602
328328 16.9706i 0.937043i
329329 0 0
330330 −8.00000 −0.440386
331331 − 25.4558i − 1.39918i −0.714545 0.699590i 0.753366π-0.753366\pi
0.714545 0.699590i 0.246634π-0.246634\pi
332332 31.1127i 1.70753i
333333 − 8.48528i − 0.464991i
334334 − 33.9411i − 1.85718i
335335 0 0
336336 − 5.65685i − 0.308607i
337337 32.0000 1.74315 0.871576 0.490261i 0.163099π-0.163099\pi
0.871576 + 0.490261i 0.163099π0.163099\pi
338338 7.07107i 0.384615i
339339 16.9706i 0.921714i
340340 16.9706i 0.920358i
341341 11.3137i 0.612672i
342342 6.00000 0.324443
343343 1.00000 0.0539949
344344 −24.0000 −1.29399
345345 −12.0000 −0.646058
346346 14.0000 0.752645
347347 14.1421i 0.759190i 0.925153 + 0.379595i 0.123937π0.123937\pi
−0.925153 + 0.379595i 0.876063π0.876063\pi
348348 8.00000 0.428845
349349 − 4.24264i − 0.227103i −0.993532 0.113552i 0.963777π-0.963777\pi
0.993532 0.113552i 0.0362227π-0.0362227\pi
350350 − 4.24264i − 0.226779i
351351 24.0000 1.28103
352352 −16.0000 −0.852803
353353 6.00000 0.319348 0.159674 0.987170i 0.448956π-0.448956\pi
0.159674 + 0.987170i 0.448956π0.448956\pi
354354 − 2.82843i − 0.150329i
355355 0 0
356356 −12.0000 −0.635999
357357 8.48528i 0.449089i
358358 8.00000 0.422813
359359 −30.0000 −1.58334 −0.791670 0.610949i 0.790788π-0.790788\pi
−0.791670 + 0.610949i 0.790788π0.790788\pi
360360 −4.00000 −0.210819
361361 1.00000 0.0526316
362362 −18.0000 −0.946059
363363 − 4.24264i − 0.222681i
364364 − 8.48528i − 0.444750i
365365 2.82843i 0.148047i
366366 25.4558i 1.33060i
367367 −28.0000 −1.46159 −0.730794 0.682598i 0.760850π-0.760850\pi
−0.730794 + 0.682598i 0.760850π0.760850\pi
368368 −24.0000 −1.25109
369369 6.00000 0.312348
370370 − 16.9706i − 0.882258i
371371 5.65685i 0.293689i
372372 − 11.3137i − 0.586588i
373373 33.9411i 1.75740i 0.477370 + 0.878702i 0.341590π0.341590\pi
−0.477370 + 0.878702i 0.658410π0.658410\pi
374374 24.0000 1.24101
375375 16.0000 0.826236
376376 0 0
377377 12.0000 0.618031
378378 −8.00000 −0.411476
379379 − 25.4558i − 1.30758i −0.756677 0.653789i 0.773178π-0.773178\pi
0.756677 0.653789i 0.226822π-0.226822\pi
380380 12.0000 0.615587
381381 − 2.82843i − 0.144905i
382382 0 0
383383 −24.0000 −1.22634 −0.613171 0.789950i 0.710106π-0.710106\pi
−0.613171 + 0.789950i 0.710106π0.710106\pi
384384 16.0000 0.816497
385385 4.00000 0.203859
386386 5.65685i 0.287926i
387387 8.48528i 0.431331i
388388 20.0000 1.01535
389389 − 2.82843i − 0.143407i −0.997426 0.0717035i 0.977156π-0.977156\pi
0.997426 0.0717035i 0.0228435π-0.0228435\pi
390390 12.0000 0.607644
391391 36.0000 1.82060
392392 2.82843i 0.142857i
393393 2.00000 0.100887
394394 8.00000 0.403034
395395 11.3137i 0.569254i
396396 5.65685i 0.284268i
397397 − 21.2132i − 1.06466i −0.846537 0.532330i 0.821317π-0.821317\pi
0.846537 0.532330i 0.178683π-0.178683\pi
398398 − 28.2843i − 1.41776i
399399 6.00000 0.300376
400400 12.0000 0.600000
401401 −24.0000 −1.19850 −0.599251 0.800561i 0.704535π-0.704535\pi
−0.599251 + 0.800561i 0.704535π0.704535\pi
402402 0 0
403403 − 16.9706i − 0.845364i
404404 − 19.7990i − 0.985037i
405405 − 7.07107i − 0.351364i
406406 −4.00000 −0.198517
407407 −24.0000 −1.18964
408408 −24.0000 −1.18818
409409 −22.0000 −1.08783 −0.543915 0.839140i 0.683059π-0.683059\pi
−0.543915 + 0.839140i 0.683059π0.683059\pi
410410 12.0000 0.592638
411411 − 8.48528i − 0.418548i
412412 8.00000 0.394132
413413 1.41421i 0.0695889i
414414 8.48528i 0.417029i
415415 22.0000 1.07994
416416 24.0000 1.17670
417417 6.00000 0.293821
418418 − 16.9706i − 0.830057i
419419 26.8701i 1.31269i 0.754462 + 0.656344i 0.227898π0.227898\pi
−0.754462 + 0.656344i 0.772102π0.772102\pi
420420 −4.00000 −0.195180
421421 − 16.9706i − 0.827095i −0.910483 0.413547i 0.864290π-0.864290\pi
0.910483 0.413547i 0.135710π-0.135710\pi
422422 24.0000 1.16830
423423 0 0
424424 −16.0000 −0.777029
425425 −18.0000 −0.873128
426426 0 0
427427 − 12.7279i − 0.615947i
428428 − 11.3137i − 0.546869i
429429 − 16.9706i − 0.819346i
430430 16.9706i 0.818393i
431431 −6.00000 −0.289010 −0.144505 0.989504i 0.546159π-0.546159\pi
−0.144505 + 0.989504i 0.546159π0.546159\pi
432432 − 22.6274i − 1.08866i
433433 2.00000 0.0961139 0.0480569 0.998845i 0.484697π-0.484697\pi
0.0480569 + 0.998845i 0.484697π0.484697\pi
434434 5.65685i 0.271538i
435435 − 5.65685i − 0.271225i
436436 − 16.9706i − 0.812743i
437437 − 25.4558i − 1.21772i
438438 −4.00000 −0.191127
439439 −28.0000 −1.33637 −0.668184 0.743996i 0.732928π-0.732928\pi
−0.668184 + 0.743996i 0.732928π0.732928\pi
440440 11.3137i 0.539360i
441441 1.00000 0.0476190
442442 −36.0000 −1.71235
443443 22.6274i 1.07506i 0.843244 + 0.537531i 0.180643π0.180643\pi
−0.843244 + 0.537531i 0.819357π0.819357\pi
444444 24.0000 1.13899
445445 8.48528i 0.402241i
446446 39.5980i 1.87502i
447447 −16.0000 −0.756774
448448 −8.00000 −0.377964
449449 −18.0000 −0.849473 −0.424736 0.905317i 0.639633π-0.639633\pi
−0.424736 + 0.905317i 0.639633π0.639633\pi
450450 − 4.24264i − 0.200000i
451451 − 16.9706i − 0.799113i
452452 24.0000 1.12887
453453 14.1421i 0.664455i
454454 14.0000 0.657053
455455 −6.00000 −0.281284
456456 16.9706i 0.794719i
457457 −28.0000 −1.30978 −0.654892 0.755722i 0.727286π-0.727286\pi
−0.654892 + 0.755722i 0.727286π0.727286\pi
458458 −6.00000 −0.280362
459459 33.9411i 1.58424i
460460 16.9706i 0.791257i
461461 − 15.5563i − 0.724531i −0.932075 0.362266i 0.882003π-0.882003\pi
0.932075 0.362266i 0.117997π-0.117997\pi
462462 5.65685i 0.263181i
463463 32.0000 1.48717 0.743583 0.668644i 0.233125π-0.233125\pi
0.743583 + 0.668644i 0.233125π0.233125\pi
464464 − 11.3137i − 0.525226i
465465 −8.00000 −0.370991
466466 − 8.48528i − 0.393073i
467467 − 7.07107i − 0.327210i −0.986526 0.163605i 0.947688π-0.947688\pi
0.986526 0.163605i 0.0523123π-0.0523123\pi
468468 − 8.48528i − 0.392232i
469469 0 0
470470 0 0
471471 −18.0000 −0.829396
472472 −4.00000 −0.184115
473473 24.0000 1.10352
474474 −16.0000 −0.734904
475475 12.7279i 0.583997i
476476 12.0000 0.550019
477477 5.65685i 0.259010i
478478 − 8.48528i − 0.388108i
479479 −12.0000 −0.548294 −0.274147 0.961688i 0.588395π-0.588395\pi
−0.274147 + 0.961688i 0.588395π0.588395\pi
480480 − 11.3137i − 0.516398i
481481 36.0000 1.64146
482482 14.1421i 0.644157i
483483 8.48528i 0.386094i
484484 −6.00000 −0.272727
485485 − 14.1421i − 0.642161i
486486 −14.0000 −0.635053
487487 2.00000 0.0906287 0.0453143 0.998973i 0.485571π-0.485571\pi
0.0453143 + 0.998973i 0.485571π0.485571\pi
488488 36.0000 1.62964
489489 −12.0000 −0.542659
490490 2.00000 0.0903508
491491 39.5980i 1.78703i 0.449032 + 0.893516i 0.351769π0.351769\pi
−0.449032 + 0.893516i 0.648231π0.648231\pi
492492 16.9706i 0.765092i
493493 16.9706i 0.764316i
494494 25.4558i 1.14531i
495495 4.00000 0.179787
496496 −16.0000 −0.718421
497497 0 0
498498 31.1127i 1.39419i
499499 − 16.9706i − 0.759707i −0.925047 0.379853i 0.875974π-0.875974\pi
0.925047 0.379853i 0.124026π-0.124026\pi
500500 − 22.6274i − 1.01193i
501501 − 33.9411i − 1.51638i
502502 26.0000 1.16044
503503 36.0000 1.60516 0.802580 0.596544i 0.203460π-0.203460\pi
0.802580 + 0.596544i 0.203460π0.203460\pi
504504 2.82843i 0.125988i
505505 −14.0000 −0.622992
506506 24.0000 1.06693
507507 7.07107i 0.314037i
508508 −4.00000 −0.177471
509509 1.41421i 0.0626839i 0.999509 + 0.0313420i 0.00997809π0.00997809\pi
−0.999509 + 0.0313420i 0.990022π0.990022\pi
510510 16.9706i 0.751469i
511511 2.00000 0.0884748
512512 − 22.6274i − 1.00000i
513513 24.0000 1.05963
514514 8.48528i 0.374270i
515515 − 5.65685i − 0.249271i
516516 −24.0000 −1.05654
517517 0 0
518518 −12.0000 −0.527250
519519 14.0000 0.614532
520520 − 16.9706i − 0.744208i
521521 −6.00000 −0.262865 −0.131432 0.991325i 0.541958π-0.541958\pi
−0.131432 + 0.991325i 0.541958π0.541958\pi
522522 −4.00000 −0.175075
523523 29.6985i 1.29862i 0.760522 + 0.649312i 0.224943π0.224943\pi
−0.760522 + 0.649312i 0.775057π0.775057\pi
524524 − 2.82843i − 0.123560i
525525 − 4.24264i − 0.185164i
526526 − 33.9411i − 1.47990i
527527 24.0000 1.04546
528528 −16.0000 −0.696311
529529 13.0000 0.565217
530530 11.3137i 0.491436i
531531 1.41421i 0.0613716i
532532 − 8.48528i − 0.367884i
533533 25.4558i 1.10262i
534534 −12.0000 −0.519291
535535 −8.00000 −0.345870
536536 0 0
537537 8.00000 0.345225
538538 −10.0000 −0.431131
539539 − 2.82843i − 0.121829i
540540 −16.0000 −0.688530
541541 16.9706i 0.729621i 0.931082 + 0.364811i 0.118866π0.118866\pi
−0.931082 + 0.364811i 0.881134π0.881134\pi
542542 − 28.2843i − 1.21491i
543543 −18.0000 −0.772454
544544 33.9411i 1.45521i
545545 −12.0000 −0.514024
546546 − 8.48528i − 0.363137i
547547 8.48528i 0.362804i 0.983409 + 0.181402i 0.0580636π0.0580636\pi
−0.983409 + 0.181402i 0.941936π0.941936\pi
548548 −12.0000 −0.512615
549549 − 12.7279i − 0.543214i
550550 −12.0000 −0.511682
551551 12.0000 0.511217
552552 −24.0000 −1.02151
553553 8.00000 0.340195
554554 24.0000 1.01966
555555 − 16.9706i − 0.720360i
556556 − 8.48528i − 0.359856i
557557 5.65685i 0.239689i 0.992793 + 0.119844i 0.0382395π0.0382395\pi
−0.992793 + 0.119844i 0.961760π0.961760\pi
558558 5.65685i 0.239474i
559559 −36.0000 −1.52264
560560 5.65685i 0.239046i
561561 24.0000 1.01328
562562 8.48528i 0.357930i
563563 1.41421i 0.0596020i 0.999556 + 0.0298010i 0.00948736π0.00948736\pi
−0.999556 + 0.0298010i 0.990513π0.990513\pi
564564 0 0
565565 − 16.9706i − 0.713957i
566566 −18.0000 −0.756596
567567 −5.00000 −0.209980
568568 0 0
569569 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
570570 12.0000 0.502625
571571 − 25.4558i − 1.06529i −0.846338 0.532647i 0.821197π-0.821197\pi
0.846338 0.532647i 0.178803π-0.178803\pi
572572 −24.0000 −1.00349
573573 0 0
574574 − 8.48528i − 0.354169i
575575 −18.0000 −0.750652
576576 −8.00000 −0.333333
577577 38.0000 1.58196 0.790980 0.611842i 0.209571π-0.209571\pi
0.790980 + 0.611842i 0.209571π0.209571\pi
578578 − 26.8701i − 1.11765i
579579 5.65685i 0.235091i
580580 −8.00000 −0.332182
581581 − 15.5563i − 0.645386i
582582 20.0000 0.829027
583583 16.0000 0.662652
584584 5.65685i 0.234082i
585585 −6.00000 −0.248069
586586 −34.0000 −1.40453
587587 − 41.0122i − 1.69275i −0.532584 0.846377i 0.678779π-0.678779\pi
0.532584 0.846377i 0.321221π-0.321221\pi
588588 2.82843i 0.116642i
589589 − 16.9706i − 0.699260i
590590 2.82843i 0.116445i
591591 8.00000 0.329076
592592 − 33.9411i − 1.39497i
593593 −30.0000 −1.23195 −0.615976 0.787765i 0.711238π-0.711238\pi
−0.615976 + 0.787765i 0.711238π0.711238\pi
594594 22.6274i 0.928414i
595595 − 8.48528i − 0.347863i
596596 22.6274i 0.926855i
597597 − 28.2843i − 1.15760i
598598 −36.0000 −1.47215
599599 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
600600 12.0000 0.489898
601601 −10.0000 −0.407909 −0.203954 0.978980i 0.565379π-0.565379\pi
−0.203954 + 0.978980i 0.565379π0.565379\pi
602602 12.0000 0.489083
603603 0 0
604604 20.0000 0.813788
605605 4.24264i 0.172488i
606606 − 19.7990i − 0.804279i
607607 32.0000 1.29884 0.649420 0.760430i 0.275012π-0.275012\pi
0.649420 + 0.760430i 0.275012π0.275012\pi
608608 24.0000 0.973329
609609 −4.00000 −0.162088
610610 − 25.4558i − 1.03068i
611611 0 0
612612 12.0000 0.485071
613613 8.48528i 0.342717i 0.985209 + 0.171359i 0.0548157π0.0548157\pi
−0.985209 + 0.171359i 0.945184π0.945184\pi
614614 18.0000 0.726421
615615 12.0000 0.483887
616616 8.00000 0.322329
617617 24.0000 0.966204 0.483102 0.875564i 0.339510π-0.339510\pi
0.483102 + 0.875564i 0.339510π0.339510\pi
618618 8.00000 0.321807
619619 4.24264i 0.170526i 0.996358 + 0.0852631i 0.0271731π0.0271731\pi
−0.996358 + 0.0852631i 0.972827π0.972827\pi
620620 11.3137i 0.454369i
621621 33.9411i 1.36201i
622622 0 0
623623 6.00000 0.240385
624624 24.0000 0.960769
625625 −1.00000 −0.0400000
626626 14.1421i 0.565233i
627627 − 16.9706i − 0.677739i
628628 25.4558i 1.01580i
629629 50.9117i 2.02998i
630630 2.00000 0.0796819
631631 −16.0000 −0.636950 −0.318475 0.947931i 0.603171π-0.603171\pi
−0.318475 + 0.947931i 0.603171π0.603171\pi
632632 22.6274i 0.900070i
633633 24.0000 0.953914
634634 32.0000 1.27088
635635 2.82843i 0.112243i
636636 −16.0000 −0.634441
637637 4.24264i 0.168100i
638638 11.3137i 0.447914i
639639 0 0
640640 −16.0000 −0.632456
641641 −48.0000 −1.89589 −0.947943 0.318440i 0.896841π-0.896841\pi
−0.947943 + 0.318440i 0.896841π0.896841\pi
642642 − 11.3137i − 0.446516i
643643 − 21.2132i − 0.836567i −0.908317 0.418284i 0.862632π-0.862632\pi
0.908317 0.418284i 0.137368π-0.137368\pi
644644 12.0000 0.472866
645645 16.9706i 0.668215i
646646 −36.0000 −1.41640
647647 12.0000 0.471769 0.235884 0.971781i 0.424201π-0.424201\pi
0.235884 + 0.971781i 0.424201π0.424201\pi
648648 − 14.1421i − 0.555556i
649649 4.00000 0.157014
650650 18.0000 0.706018
651651 5.65685i 0.221710i
652652 16.9706i 0.664619i
653653 − 36.7696i − 1.43890i −0.694542 0.719452i 0.744393π-0.744393\pi
0.694542 0.719452i 0.255607π-0.255607\pi
654654 − 16.9706i − 0.663602i
655655 −2.00000 −0.0781465
656656 24.0000 0.937043
657657 2.00000 0.0780274
658658 0 0
659659 − 2.82843i − 0.110180i −0.998481 0.0550899i 0.982455π-0.982455\pi
0.998481 0.0550899i 0.0175446π-0.0175446\pi
660660 11.3137i 0.440386i
661661 38.1838i 1.48518i 0.669748 + 0.742588i 0.266402π0.266402\pi
−0.669748 + 0.742588i 0.733598π0.733598\pi
662662 −36.0000 −1.39918
663663 −36.0000 −1.39812
664664 44.0000 1.70753
665665 −6.00000 −0.232670
666666 −12.0000 −0.464991
667667 16.9706i 0.657103i
668668 −48.0000 −1.85718
669669 39.5980i 1.53095i
670670 0 0
671671 −36.0000 −1.38976
672672 −8.00000 −0.308607
673673 −46.0000 −1.77317 −0.886585 0.462566i 0.846929π-0.846929\pi
−0.886585 + 0.462566i 0.846929π0.846929\pi
674674 − 45.2548i − 1.74315i
675675 − 16.9706i − 0.653197i
676676 10.0000 0.384615
677677 9.89949i 0.380468i 0.981739 + 0.190234i 0.0609248π0.0609248\pi
−0.981739 + 0.190234i 0.939075π0.939075\pi
678678 24.0000 0.921714
679679 −10.0000 −0.383765
680680 24.0000 0.920358
681681 14.0000 0.536481
682682 16.0000 0.612672
683683 5.65685i 0.216454i 0.994126 + 0.108227i 0.0345173π0.0345173\pi
−0.994126 + 0.108227i 0.965483π0.965483\pi
684684 − 8.48528i − 0.324443i
685685 8.48528i 0.324206i
686686 − 1.41421i − 0.0539949i
687687 −6.00000 −0.228914
688688 33.9411i 1.29399i
689689 −24.0000 −0.914327
690690 16.9706i 0.646058i
691691 12.7279i 0.484193i 0.970252 + 0.242096i 0.0778351π0.0778351\pi
−0.970252 + 0.242096i 0.922165π0.922165\pi
692692 − 19.7990i − 0.752645i
693693 − 2.82843i − 0.107443i
694694 20.0000 0.759190
695695 −6.00000 −0.227593
696696 − 11.3137i − 0.428845i
697697 −36.0000 −1.36360
698698 −6.00000 −0.227103
699699 − 8.48528i − 0.320943i
700700 −6.00000 −0.226779
701701 − 19.7990i − 0.747798i −0.927470 0.373899i 0.878021π-0.878021\pi
0.927470 0.373899i 0.121979π-0.121979\pi
702702 − 33.9411i − 1.28103i
703703 36.0000 1.35777
704704 22.6274i 0.852803i
705705 0 0
706706 − 8.48528i − 0.319348i
707707 9.89949i 0.372309i
708708 −4.00000 −0.150329
709709 25.4558i 0.956014i 0.878356 + 0.478007i 0.158641π0.158641\pi
−0.878356 + 0.478007i 0.841359π0.841359\pi
710710 0 0
711711 8.00000 0.300023
712712 16.9706i 0.635999i
713713 24.0000 0.898807
714714 12.0000 0.449089
715715 16.9706i 0.634663i
716716 − 11.3137i − 0.422813i
717717 − 8.48528i − 0.316889i
718718 42.4264i 1.58334i
719719 24.0000 0.895049 0.447524 0.894272i 0.352306π-0.352306\pi
0.447524 + 0.894272i 0.352306π0.352306\pi
720720 5.65685i 0.210819i
721721 −4.00000 −0.148968
722722 − 1.41421i − 0.0526316i
723723 14.1421i 0.525952i
724724 25.4558i 0.946059i
725725 − 8.48528i − 0.315135i
726726 −6.00000 −0.222681
727727 8.00000 0.296704 0.148352 0.988935i 0.452603π-0.452603\pi
0.148352 + 0.988935i 0.452603π0.452603\pi
728728 −12.0000 −0.444750
729729 −29.0000 −1.07407
730730 4.00000 0.148047
731731 − 50.9117i − 1.88304i
732732 36.0000 1.33060
733733 − 29.6985i − 1.09694i −0.836171 0.548469i 0.815211π-0.815211\pi
0.836171 0.548469i 0.184789π-0.184789\pi
734734 39.5980i 1.46159i
735735 2.00000 0.0737711
736736 33.9411i 1.25109i
737737 0 0
738738 − 8.48528i − 0.312348i
739739 − 42.4264i − 1.56068i −0.625355 0.780340i 0.715046π-0.715046\pi
0.625355 0.780340i 0.284954π-0.284954\pi
740740 −24.0000 −0.882258
741741 25.4558i 0.935144i
742742 8.00000 0.293689
743743 −30.0000 −1.10059 −0.550297 0.834969i 0.685485π-0.685485\pi
−0.550297 + 0.834969i 0.685485π0.685485\pi
744744 −16.0000 −0.586588
745745 16.0000 0.586195
746746 48.0000 1.75740
747747 − 15.5563i − 0.569177i
748748 − 33.9411i − 1.24101i
749749 5.65685i 0.206697i
750750 − 22.6274i − 0.826236i
751751 −22.0000 −0.802791 −0.401396 0.915905i 0.631475π-0.631475\pi
−0.401396 + 0.915905i 0.631475π0.631475\pi
752752 0 0
753753 26.0000 0.947493
754754 − 16.9706i − 0.618031i
755755 − 14.1421i − 0.514685i
756756 11.3137i 0.411476i
757757 − 25.4558i − 0.925208i −0.886565 0.462604i 0.846915π-0.846915\pi
0.886565 0.462604i 0.153085π-0.153085\pi
758758 −36.0000 −1.30758
759759 24.0000 0.871145
760760 − 16.9706i − 0.615587i
761761 30.0000 1.08750 0.543750 0.839248i 0.317004π-0.317004\pi
0.543750 + 0.839248i 0.317004π0.317004\pi
762762 −4.00000 −0.144905
763763 8.48528i 0.307188i
764764 0 0
765765 − 8.48528i − 0.306786i
766766 33.9411i 1.22634i
767767 −6.00000 −0.216647
768768 − 22.6274i − 0.816497i
769769 14.0000 0.504853 0.252426 0.967616i 0.418771π-0.418771\pi
0.252426 + 0.967616i 0.418771π0.418771\pi
770770 − 5.65685i − 0.203859i
771771 8.48528i 0.305590i
772772 8.00000 0.287926
773773 − 32.5269i − 1.16991i −0.811065 0.584956i 0.801112π-0.801112\pi
0.811065 0.584956i 0.198888π-0.198888\pi
774774 12.0000 0.431331
775775 −12.0000 −0.431053
776776 − 28.2843i − 1.01535i
777777 −12.0000 −0.430498
778778 −4.00000 −0.143407
779779 25.4558i 0.912050i
780780 − 16.9706i − 0.607644i
781781 0 0
782782 − 50.9117i − 1.82060i
783783 −16.0000 −0.571793
784784 4.00000 0.142857
785785 18.0000 0.642448
786786 − 2.82843i − 0.100887i
787787 38.1838i 1.36110i 0.732700 + 0.680552i 0.238260π0.238260\pi
−0.732700 + 0.680552i 0.761740π0.761740\pi
788788 − 11.3137i − 0.403034i
789789 − 33.9411i − 1.20834i
790790 16.0000 0.569254
791791 −12.0000 −0.426671
792792 8.00000 0.284268
793793 54.0000 1.91760
794794 −30.0000 −1.06466
795795 11.3137i 0.401256i
796796 −40.0000 −1.41776
797797 26.8701i 0.951786i 0.879503 + 0.475893i 0.157875π0.157875\pi
−0.879503 + 0.475893i 0.842125π0.842125\pi
798798 − 8.48528i − 0.300376i
799799 0 0
800800 − 16.9706i − 0.600000i
801801 6.00000 0.212000
802802 33.9411i 1.19850i
803803 − 5.65685i − 0.199626i
804804 0 0
805805 − 8.48528i − 0.299067i
806806 −24.0000 −0.845364
807807 −10.0000 −0.352017
808808 −28.0000 −0.985037
809809 12.0000 0.421898 0.210949 0.977497i 0.432345π-0.432345\pi
0.210949 + 0.977497i 0.432345π0.432345\pi
810810 −10.0000 −0.351364
811811 − 12.7279i − 0.446938i −0.974711 0.223469i 0.928262π-0.928262\pi
0.974711 0.223469i 0.0717381π-0.0717381\pi
812812 5.65685i 0.198517i
813813 − 28.2843i − 0.991973i
814814 33.9411i 1.18964i
815815 12.0000 0.420342
816816 33.9411i 1.18818i
817817 −36.0000 −1.25948
818818 31.1127i 1.08783i
819819 4.24264i 0.148250i
820820 − 16.9706i − 0.592638i
821821 22.6274i 0.789702i 0.918745 + 0.394851i 0.129204π0.129204\pi
−0.918745 + 0.394851i 0.870796π0.870796\pi
822822 −12.0000 −0.418548
823823 32.0000 1.11545 0.557725 0.830026i 0.311674π-0.311674\pi
0.557725 + 0.830026i 0.311674π0.311674\pi
824824 − 11.3137i − 0.394132i
825825 −12.0000 −0.417786
826826 2.00000 0.0695889
827827 − 45.2548i − 1.57366i −0.617167 0.786832i 0.711720π-0.711720\pi
0.617167 0.786832i 0.288280π-0.288280\pi
828828 12.0000 0.417029
829829 21.2132i 0.736765i 0.929674 + 0.368383i 0.120088π0.120088\pi
−0.929674 + 0.368383i 0.879912π0.879912\pi
830830 − 31.1127i − 1.07994i
831831 24.0000 0.832551
832832 − 33.9411i − 1.17670i
833833 −6.00000 −0.207888
834834 − 8.48528i − 0.293821i
835835 33.9411i 1.17458i
836836 −24.0000 −0.830057
837837 22.6274i 0.782118i
838838 38.0000 1.31269
839839 12.0000 0.414286 0.207143 0.978311i 0.433583π-0.433583\pi
0.207143 + 0.978311i 0.433583π0.433583\pi
840840 5.65685i 0.195180i
841841 21.0000 0.724138
842842 −24.0000 −0.827095
843843 8.48528i 0.292249i
844844 − 33.9411i − 1.16830i
845845 − 7.07107i − 0.243252i
846846 0 0
847847 3.00000 0.103081
848848 22.6274i 0.777029i
849849 −18.0000 −0.617758
850850 25.4558i 0.873128i
851851 50.9117i 1.74523i
852852 0 0
853853 − 4.24264i − 0.145265i −0.997359 0.0726326i 0.976860π-0.976860\pi
0.997359 0.0726326i 0.0231401π-0.0231401\pi
854854 −18.0000 −0.615947
855855 −6.00000 −0.205196
856856 −16.0000 −0.546869
857857 42.0000 1.43469 0.717346 0.696717i 0.245357π-0.245357\pi
0.717346 + 0.696717i 0.245357π0.245357\pi
858858 −24.0000 −0.819346
859859 46.6690i 1.59233i 0.605081 + 0.796164i 0.293141π0.293141\pi
−0.605081 + 0.796164i 0.706859π0.706859\pi
860860 24.0000 0.818393
861861 − 8.48528i − 0.289178i
862862 8.48528i 0.289010i
863863 24.0000 0.816970 0.408485 0.912765i 0.366057π-0.366057\pi
0.408485 + 0.912765i 0.366057π0.366057\pi
864864 −32.0000 −1.08866
865865 −14.0000 −0.476014
866866 − 2.82843i − 0.0961139i
867867 − 26.8701i − 0.912555i
868868 8.00000 0.271538
869869 − 22.6274i − 0.767583i
870870 −8.00000 −0.271225
871871 0 0
872872 −24.0000 −0.812743
873873 −10.0000 −0.338449
874874 −36.0000 −1.21772
875875 11.3137i 0.382473i
876876 5.65685i 0.191127i
877877 − 42.4264i − 1.43264i −0.697773 0.716319i 0.745826π-0.745826\pi
0.697773 0.716319i 0.254174π-0.254174\pi
878878 39.5980i 1.33637i
879879 −34.0000 −1.14679
880880 16.0000 0.539360
881881 −18.0000 −0.606435 −0.303218 0.952921i 0.598061π-0.598061\pi
−0.303218 + 0.952921i 0.598061π0.598061\pi
882882 − 1.41421i − 0.0476190i
883883 50.9117i 1.71331i 0.515886 + 0.856657i 0.327463π0.327463\pi
−0.515886 + 0.856657i 0.672537π0.672537\pi
884884 50.9117i 1.71235i
885885 2.82843i 0.0950765i
886886 32.0000 1.07506
887887 48.0000 1.61168 0.805841 0.592132i 0.201714π-0.201714\pi
0.805841 + 0.592132i 0.201714π0.201714\pi
888888 − 33.9411i − 1.13899i
889889 2.00000 0.0670778
890890 12.0000 0.402241
891891 14.1421i 0.473779i
892892 56.0000 1.87502
893893 0 0
894894 22.6274i 0.756774i
895895 −8.00000 −0.267411
896896 11.3137i 0.377964i
897897 −36.0000 −1.20201
898898 25.4558i 0.849473i
899899 11.3137i 0.377333i
900900 −6.00000 −0.200000
901901 − 33.9411i − 1.13074i
902902 −24.0000 −0.799113
903903 12.0000 0.399335
904904 − 33.9411i − 1.12887i
905905 18.0000 0.598340
906906 20.0000 0.664455
907907 − 33.9411i − 1.12700i −0.826117 0.563498i 0.809455π-0.809455\pi
0.826117 0.563498i 0.190545π-0.190545\pi
908908 − 19.7990i − 0.657053i
909909 9.89949i 0.328346i
910910 8.48528i 0.281284i
911911 30.0000 0.993944 0.496972 0.867766i 0.334445π-0.334445\pi
0.496972 + 0.867766i 0.334445π0.334445\pi
912912 24.0000 0.794719
913913 −44.0000 −1.45619
914914 39.5980i 1.30978i
915915 − 25.4558i − 0.841544i
916916 8.48528i 0.280362i
917917 1.41421i 0.0467014i
918918 48.0000 1.58424
919919 −16.0000 −0.527791 −0.263896 0.964551i 0.585007π-0.585007\pi
−0.263896 + 0.964551i 0.585007π0.585007\pi
920920 24.0000 0.791257
921921 18.0000 0.593120
922922 −22.0000 −0.724531
923923 0 0
924924 8.00000 0.263181
925925 − 25.4558i − 0.836983i
926926 − 45.2548i − 1.48717i
927927 −4.00000 −0.131377
928928 −16.0000 −0.525226
929929 18.0000 0.590561 0.295280 0.955411i 0.404587π-0.404587\pi
0.295280 + 0.955411i 0.404587π0.404587\pi
930930 11.3137i 0.370991i
931931 4.24264i 0.139047i
932932 −12.0000 −0.393073
933933 0 0
934934 −10.0000 −0.327210
935935 −24.0000 −0.784884
936936 −12.0000 −0.392232
937937 2.00000 0.0653372 0.0326686 0.999466i 0.489599π-0.489599\pi
0.0326686 + 0.999466i 0.489599π0.489599\pi
938938 0 0
939939 14.1421i 0.461511i
940940 0 0
941941 52.3259i 1.70578i 0.522094 + 0.852888i 0.325151π0.325151\pi
−0.522094 + 0.852888i 0.674849π0.674849\pi
942942 25.4558i 0.829396i
943943 −36.0000 −1.17232
944944 5.65685i 0.184115i
945945 8.00000 0.260240
946946 − 33.9411i − 1.10352i
947947 − 2.82843i − 0.0919115i −0.998943 0.0459558i 0.985367π-0.985367\pi
0.998943 0.0459558i 0.0146333π-0.0146333\pi
948948 22.6274i 0.734904i
949949 8.48528i 0.275444i
950950 18.0000 0.583997
951951 32.0000 1.03767
952952 − 16.9706i − 0.550019i
953953 18.0000 0.583077 0.291539 0.956559i 0.405833π-0.405833\pi
0.291539 + 0.956559i 0.405833π0.405833\pi
954954 8.00000 0.259010
955955 0 0
956956 −12.0000 −0.388108
957957 11.3137i 0.365720i
958958 16.9706i 0.548294i
959959 6.00000 0.193750
960960 −16.0000 −0.516398
961961 −15.0000 −0.483871
962962 − 50.9117i − 1.64146i
963963 5.65685i 0.182290i
964964 20.0000 0.644157
965965 − 5.65685i − 0.182101i
966966 12.0000 0.386094
967967 −22.0000 −0.707472 −0.353736 0.935345i 0.615089π-0.615089\pi
−0.353736 + 0.935345i 0.615089π0.615089\pi
968968 8.48528i 0.272727i
969969 −36.0000 −1.15649
970970 −20.0000 −0.642161
971971 − 32.5269i − 1.04384i −0.852995 0.521919i 0.825216π-0.825216\pi
0.852995 0.521919i 0.174784π-0.174784\pi
972972 19.7990i 0.635053i
973973 4.24264i 0.136013i
974974 − 2.82843i − 0.0906287i
975975 18.0000 0.576461
976976 − 50.9117i − 1.62964i
977977 −18.0000 −0.575871 −0.287936 0.957650i 0.592969π-0.592969\pi
−0.287936 + 0.957650i 0.592969π0.592969\pi
978978 16.9706i 0.542659i
979979 − 16.9706i − 0.542382i
980980 − 2.82843i − 0.0903508i
981981 8.48528i 0.270914i
982982 56.0000 1.78703
983983 −12.0000 −0.382741 −0.191370 0.981518i 0.561293π-0.561293\pi
−0.191370 + 0.981518i 0.561293π0.561293\pi
984984 24.0000 0.765092
985985 −8.00000 −0.254901
986986 24.0000 0.764316
987987 0 0
988988 36.0000 1.14531
989989 − 50.9117i − 1.61890i
990990 − 5.65685i − 0.179787i
991991 −16.0000 −0.508257 −0.254128 0.967170i 0.581789π-0.581789\pi
−0.254128 + 0.967170i 0.581789π0.581789\pi
992992 22.6274i 0.718421i
993993 −36.0000 −1.14243
994994 0 0
995995 28.2843i 0.896672i
996996 44.0000 1.39419
997997 − 21.2132i − 0.671829i −0.941893 0.335914i 0.890955π-0.890955\pi
0.941893 0.335914i 0.109045π-0.109045\pi
998998 −24.0000 −0.759707
999999 −48.0000 −1.51865
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 56.2.b.a.29.1 2
3.2 odd 2 504.2.c.a.253.2 2
4.3 odd 2 224.2.b.a.113.2 2
7.2 even 3 392.2.p.a.165.1 4
7.3 odd 6 392.2.p.b.373.2 4
7.4 even 3 392.2.p.a.373.2 4
7.5 odd 6 392.2.p.b.165.1 4
7.6 odd 2 392.2.b.b.197.1 2
8.3 odd 2 224.2.b.a.113.1 2
8.5 even 2 inner 56.2.b.a.29.2 yes 2
12.11 even 2 2016.2.c.a.1009.1 2
16.3 odd 4 1792.2.a.p.1.1 2
16.5 even 4 1792.2.a.n.1.1 2
16.11 odd 4 1792.2.a.p.1.2 2
16.13 even 4 1792.2.a.n.1.2 2
24.5 odd 2 504.2.c.a.253.1 2
24.11 even 2 2016.2.c.a.1009.2 2
28.3 even 6 1568.2.t.b.177.1 4
28.11 odd 6 1568.2.t.c.177.2 4
28.19 even 6 1568.2.t.b.753.2 4
28.23 odd 6 1568.2.t.c.753.1 4
28.27 even 2 1568.2.b.a.785.1 2
56.3 even 6 1568.2.t.b.177.2 4
56.5 odd 6 392.2.p.b.165.2 4
56.11 odd 6 1568.2.t.c.177.1 4
56.13 odd 2 392.2.b.b.197.2 2
56.19 even 6 1568.2.t.b.753.1 4
56.27 even 2 1568.2.b.a.785.2 2
56.37 even 6 392.2.p.a.165.2 4
56.45 odd 6 392.2.p.b.373.1 4
56.51 odd 6 1568.2.t.c.753.2 4
56.53 even 6 392.2.p.a.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.a.29.1 2 1.1 even 1 trivial
56.2.b.a.29.2 yes 2 8.5 even 2 inner
224.2.b.a.113.1 2 8.3 odd 2
224.2.b.a.113.2 2 4.3 odd 2
392.2.b.b.197.1 2 7.6 odd 2
392.2.b.b.197.2 2 56.13 odd 2
392.2.p.a.165.1 4 7.2 even 3
392.2.p.a.165.2 4 56.37 even 6
392.2.p.a.373.1 4 56.53 even 6
392.2.p.a.373.2 4 7.4 even 3
392.2.p.b.165.1 4 7.5 odd 6
392.2.p.b.165.2 4 56.5 odd 6
392.2.p.b.373.1 4 56.45 odd 6
392.2.p.b.373.2 4 7.3 odd 6
504.2.c.a.253.1 2 24.5 odd 2
504.2.c.a.253.2 2 3.2 odd 2
1568.2.b.a.785.1 2 28.27 even 2
1568.2.b.a.785.2 2 56.27 even 2
1568.2.t.b.177.1 4 28.3 even 6
1568.2.t.b.177.2 4 56.3 even 6
1568.2.t.b.753.1 4 56.19 even 6
1568.2.t.b.753.2 4 28.19 even 6
1568.2.t.c.177.1 4 56.11 odd 6
1568.2.t.c.177.2 4 28.11 odd 6
1568.2.t.c.753.1 4 28.23 odd 6
1568.2.t.c.753.2 4 56.51 odd 6
1792.2.a.n.1.1 2 16.5 even 4
1792.2.a.n.1.2 2 16.13 even 4
1792.2.a.p.1.1 2 16.3 odd 4
1792.2.a.p.1.2 2 16.11 odd 4
2016.2.c.a.1009.1 2 12.11 even 2
2016.2.c.a.1009.2 2 24.11 even 2