Properties

Label 1920.2.m.d.959.1
Level 19201920
Weight 22
Character 1920.959
Analytic conductor 15.33115.331
Analytic rank 00
Dimension 44
Inner twists 22

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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] = [1920,2,Mod(959,1920)] 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("1920.959"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: N N == 1920=2735 1920 = 2^{7} \cdot 3 \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1920.m (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,-4,0,-4,0,0,0,0,0,8,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 15.331277188115.3312771881
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,5)\Q(i, \sqrt{5})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+3x2+1 x^{4} + 3x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 959.1
Root 1.61803i1.61803i of defining polynomial
Character χ\chi == 1920.959
Dual form 1920.2.m.d.959.2

qq-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)f(q) == q+(1.618030.618034i)q3+(1.00000+2.00000i)q53.23607q7+(2.23607+2.00000i)q94.00000iq11+6.47214q13+(2.854102.61803i)q156.47214q17+2.47214q19+(5.23607+2.00000i)q211.23607iq23+(3.000004.00000i)q25+(2.381974.61803i)q27+4.47214q292.47214iq31+(2.47214+6.47214i)q33+(3.236076.47214i)q351.52786q37+(10.47214.00000i)q394.00000iq41+9.23607iq43+(6.23607+2.47214i)q45+9.23607iq47+3.47214q49+(10.4721+4.00000i)q51+8.94427iq53+(8.00000+4.00000i)q55+(4.000001.52786i)q57+8.94427iq594.94427iq61+(7.236076.47214i)q63+(6.47214+12.9443i)q6511.7082iq67+(0.763932+2.00000i)q69+4.94427iq73+(2.38197+8.32624i)q75+12.9443iq77+10.4721iq79+(1.00000+8.94427i)q818.18034q83+(6.4721412.9443i)q85+(7.236072.76393i)q87+8.00000iq8920.9443q91+(1.52786+4.00000i)q93+(2.47214+4.94427i)q95+12.9443iq97+(8.000008.94427i)q99+O(q100)q+(-1.61803 - 0.618034i) q^{3} +(-1.00000 + 2.00000i) q^{5} -3.23607 q^{7} +(2.23607 + 2.00000i) q^{9} -4.00000i q^{11} +6.47214 q^{13} +(2.85410 - 2.61803i) q^{15} -6.47214 q^{17} +2.47214 q^{19} +(5.23607 + 2.00000i) q^{21} -1.23607i q^{23} +(-3.00000 - 4.00000i) q^{25} +(-2.38197 - 4.61803i) q^{27} +4.47214 q^{29} -2.47214i q^{31} +(-2.47214 + 6.47214i) q^{33} +(3.23607 - 6.47214i) q^{35} -1.52786 q^{37} +(-10.4721 - 4.00000i) q^{39} -4.00000i q^{41} +9.23607i q^{43} +(-6.23607 + 2.47214i) q^{45} +9.23607i q^{47} +3.47214 q^{49} +(10.4721 + 4.00000i) q^{51} +8.94427i q^{53} +(8.00000 + 4.00000i) q^{55} +(-4.00000 - 1.52786i) q^{57} +8.94427i q^{59} -4.94427i q^{61} +(-7.23607 - 6.47214i) q^{63} +(-6.47214 + 12.9443i) q^{65} -11.7082i q^{67} +(-0.763932 + 2.00000i) q^{69} +4.94427i q^{73} +(2.38197 + 8.32624i) q^{75} +12.9443i q^{77} +10.4721i q^{79} +(1.00000 + 8.94427i) q^{81} -8.18034 q^{83} +(6.47214 - 12.9443i) q^{85} +(-7.23607 - 2.76393i) q^{87} +8.00000i q^{89} -20.9443 q^{91} +(-1.52786 + 4.00000i) q^{93} +(-2.47214 + 4.94427i) q^{95} +12.9443i q^{97} +(8.00000 - 8.94427i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q2q34q54q7+8q132q158q178q19+12q2112q2514q27+8q33+4q3524q3724q3916q454q49+24q51+32q55++32q99+O(q100) 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{13} - 2 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{21} - 12 q^{25} - 14 q^{27} + 8 q^{33} + 4 q^{35} - 24 q^{37} - 24 q^{39} - 16 q^{45} - 4 q^{49} + 24 q^{51} + 32 q^{55}+ \cdots + 32 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1920Z)×\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times.

nn 511511 641641 901901 15371537
χ(n)\chi(n) 1-1 1-1 1-1 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 0 0
33 −1.61803 0.618034i −0.934172 0.356822i
44 0 0
55 −1.00000 + 2.00000i −0.447214 + 0.894427i
66 0 0
77 −3.23607 −1.22312 −0.611559 0.791199i 0.709457π-0.709457\pi
−0.611559 + 0.791199i 0.709457π0.709457\pi
88 0 0
99 2.23607 + 2.00000i 0.745356 + 0.666667i
1010 0 0
1111 4.00000i 1.20605i −0.797724 0.603023i 0.793963π-0.793963\pi
0.797724 0.603023i 0.206037π-0.206037\pi
1212 0 0
1313 6.47214 1.79505 0.897524 0.440966i 0.145364π-0.145364\pi
0.897524 + 0.440966i 0.145364π0.145364\pi
1414 0 0
1515 2.85410 2.61803i 0.736926 0.675973i
1616 0 0
1717 −6.47214 −1.56972 −0.784862 0.619671i 0.787266π-0.787266\pi
−0.784862 + 0.619671i 0.787266π0.787266\pi
1818 0 0
1919 2.47214 0.567147 0.283573 0.958951i 0.408480π-0.408480\pi
0.283573 + 0.958951i 0.408480π0.408480\pi
2020 0 0
2121 5.23607 + 2.00000i 1.14260 + 0.436436i
2222 0 0
2323 1.23607i 0.257738i −0.991662 0.128869i 0.958865π-0.958865\pi
0.991662 0.128869i 0.0411347π-0.0411347\pi
2424 0 0
2525 −3.00000 4.00000i −0.600000 0.800000i
2626 0 0
2727 −2.38197 4.61803i −0.458410 0.888741i
2828 0 0
2929 4.47214 0.830455 0.415227 0.909718i 0.363702π-0.363702\pi
0.415227 + 0.909718i 0.363702π0.363702\pi
3030 0 0
3131 2.47214i 0.444009i −0.975046 0.222004i 0.928740π-0.928740\pi
0.975046 0.222004i 0.0712599π-0.0712599\pi
3232 0 0
3333 −2.47214 + 6.47214i −0.430344 + 1.12665i
3434 0 0
3535 3.23607 6.47214i 0.546995 1.09399i
3636 0 0
3737 −1.52786 −0.251179 −0.125590 0.992082i 0.540082π-0.540082\pi
−0.125590 + 0.992082i 0.540082π0.540082\pi
3838 0 0
3939 −10.4721 4.00000i −1.67688 0.640513i
4040 0 0
4141 4.00000i 0.624695i −0.949968 0.312348i 0.898885π-0.898885\pi
0.949968 0.312348i 0.101115π-0.101115\pi
4242 0 0
4343 9.23607i 1.40849i 0.709958 + 0.704244i 0.248714π0.248714\pi
−0.709958 + 0.704244i 0.751286π0.751286\pi
4444 0 0
4545 −6.23607 + 2.47214i −0.929618 + 0.368524i
4646 0 0
4747 9.23607i 1.34722i 0.739087 + 0.673609i 0.235257π0.235257\pi
−0.739087 + 0.673609i 0.764743π0.764743\pi
4848 0 0
4949 3.47214 0.496019
5050 0 0
5151 10.4721 + 4.00000i 1.46639 + 0.560112i
5252 0 0
5353 8.94427i 1.22859i 0.789076 + 0.614295i 0.210560π0.210560\pi
−0.789076 + 0.614295i 0.789440π0.789440\pi
5454 0 0
5555 8.00000 + 4.00000i 1.07872 + 0.539360i
5656 0 0
5757 −4.00000 1.52786i −0.529813 0.202371i
5858 0 0
5959 8.94427i 1.16445i 0.813029 + 0.582223i 0.197817π0.197817\pi
−0.813029 + 0.582223i 0.802183π0.802183\pi
6060 0 0
6161 4.94427i 0.633049i −0.948584 0.316525i 0.897484π-0.897484\pi
0.948584 0.316525i 0.102516π-0.102516\pi
6262 0 0
6363 −7.23607 6.47214i −0.911659 0.815412i
6464 0 0
6565 −6.47214 + 12.9443i −0.802770 + 1.60554i
6666 0 0
6767 11.7082i 1.43038i −0.698928 0.715192i 0.746339π-0.746339\pi
0.698928 0.715192i 0.253661π-0.253661\pi
6868 0 0
6969 −0.763932 + 2.00000i −0.0919666 + 0.240772i
7070 0 0
7171 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
7272 0 0
7373 4.94427i 0.578683i 0.957226 + 0.289342i 0.0934364π0.0934364\pi
−0.957226 + 0.289342i 0.906564π0.906564\pi
7474 0 0
7575 2.38197 + 8.32624i 0.275046 + 0.961431i
7676 0 0
7777 12.9443i 1.47514i
7878 0 0
7979 10.4721i 1.17821i 0.808057 + 0.589104i 0.200519π0.200519\pi
−0.808057 + 0.589104i 0.799481π0.799481\pi
8080 0 0
8181 1.00000 + 8.94427i 0.111111 + 0.993808i
8282 0 0
8383 −8.18034 −0.897909 −0.448954 0.893555i 0.648203π-0.648203\pi
−0.448954 + 0.893555i 0.648203π0.648203\pi
8484 0 0
8585 6.47214 12.9443i 0.702002 1.40400i
8686 0 0
8787 −7.23607 2.76393i −0.775788 0.296325i
8888 0 0
8989 8.00000i 0.847998i 0.905663 + 0.423999i 0.139374π0.139374\pi
−0.905663 + 0.423999i 0.860626π0.860626\pi
9090 0 0
9191 −20.9443 −2.19556
9292 0 0
9393 −1.52786 + 4.00000i −0.158432 + 0.414781i
9494 0 0
9595 −2.47214 + 4.94427i −0.253636 + 0.507272i
9696 0 0
9797 12.9443i 1.31429i 0.753763 + 0.657146i 0.228236π0.228236\pi
−0.753763 + 0.657146i 0.771764π0.771764\pi
9898 0 0
9999 8.00000 8.94427i 0.804030 0.898933i
100100 0 0
101101 3.52786 0.351036 0.175518 0.984476i 0.443840π-0.443840\pi
0.175518 + 0.984476i 0.443840π0.443840\pi
102102 0 0
103103 1.70820 0.168314 0.0841572 0.996452i 0.473180π-0.473180\pi
0.0841572 + 0.996452i 0.473180π0.473180\pi
104104 0 0
105105 −9.23607 + 8.47214i −0.901348 + 0.826796i
106106 0 0
107107 9.70820 0.938527 0.469264 0.883058i 0.344519π-0.344519\pi
0.469264 + 0.883058i 0.344519π0.344519\pi
108108 0 0
109109 12.9443i 1.23984i 0.784666 + 0.619918i 0.212834π0.212834\pi
−0.784666 + 0.619918i 0.787166π0.787166\pi
110110 0 0
111111 2.47214 + 0.944272i 0.234645 + 0.0896263i
112112 0 0
113113 −14.4721 −1.36142 −0.680712 0.732551i 0.738329π-0.738329\pi
−0.680712 + 0.732551i 0.738329π0.738329\pi
114114 0 0
115115 2.47214 + 1.23607i 0.230528 + 0.115264i
116116 0 0
117117 14.4721 + 12.9443i 1.33795 + 1.19670i
118118 0 0
119119 20.9443 1.91996
120120 0 0
121121 −5.00000 −0.454545
122122 0 0
123123 −2.47214 + 6.47214i −0.222905 + 0.583573i
124124 0 0
125125 11.0000 2.00000i 0.983870 0.178885i
126126 0 0
127127 11.2361 0.997040 0.498520 0.866878i 0.333877π-0.333877\pi
0.498520 + 0.866878i 0.333877π0.333877\pi
128128 0 0
129129 5.70820 14.9443i 0.502579 1.31577i
130130 0 0
131131 0.944272i 0.0825014i −0.999149 0.0412507i 0.986866π-0.986866\pi
0.999149 0.0412507i 0.0131342π-0.0131342\pi
132132 0 0
133133 −8.00000 −0.693688
134134 0 0
135135 11.6180 0.145898i 0.999921 0.0125569i
136136 0 0
137137 1.52786 0.130534 0.0652671 0.997868i 0.479210π-0.479210\pi
0.0652671 + 0.997868i 0.479210π0.479210\pi
138138 0 0
139139 −23.4164 −1.98615 −0.993077 0.117466i 0.962523π-0.962523\pi
−0.993077 + 0.117466i 0.962523π0.962523\pi
140140 0 0
141141 5.70820 14.9443i 0.480717 1.25853i
142142 0 0
143143 25.8885i 2.16491i
144144 0 0
145145 −4.47214 + 8.94427i −0.371391 + 0.742781i
146146 0 0
147147 −5.61803 2.14590i −0.463368 0.176991i
148148 0 0
149149 11.8885 0.973947 0.486974 0.873417i 0.338101π-0.338101\pi
0.486974 + 0.873417i 0.338101π0.338101\pi
150150 0 0
151151 10.4721i 0.852210i 0.904674 + 0.426105i 0.140115π0.140115\pi
−0.904674 + 0.426105i 0.859885π0.859885\pi
152152 0 0
153153 −14.4721 12.9443i −1.17000 1.04648i
154154 0 0
155155 4.94427 + 2.47214i 0.397133 + 0.198567i
156156 0 0
157157 −1.52786 −0.121937 −0.0609684 0.998140i 0.519419π-0.519419\pi
−0.0609684 + 0.998140i 0.519419π0.519419\pi
158158 0 0
159159 5.52786 14.4721i 0.438388 1.14772i
160160 0 0
161161 4.00000i 0.315244i
162162 0 0
163163 14.7639i 1.15640i 0.815895 + 0.578200i 0.196245π0.196245\pi
−0.815895 + 0.578200i 0.803755π0.803755\pi
164164 0 0
165165 −10.4721 11.4164i −0.815255 0.888766i
166166 0 0
167167 1.23607i 0.0956498i 0.998856 + 0.0478249i 0.0152289π0.0152289\pi
−0.998856 + 0.0478249i 0.984771π0.984771\pi
168168 0 0
169169 28.8885 2.22220
170170 0 0
171171 5.52786 + 4.94427i 0.422726 + 0.378098i
172172 0 0
173173 4.00000i 0.304114i 0.988372 + 0.152057i 0.0485898π0.0485898\pi
−0.988372 + 0.152057i 0.951410π0.951410\pi
174174 0 0
175175 9.70820 + 12.9443i 0.733871 + 0.978495i
176176 0 0
177177 5.52786 14.4721i 0.415500 1.08779i
178178 0 0
179179 0.944272i 0.0705782i −0.999377 0.0352891i 0.988765π-0.988765\pi
0.999377 0.0352891i 0.0112352π-0.0112352\pi
180180 0 0
181181 0.944272i 0.0701872i 0.999384 + 0.0350936i 0.0111729π0.0111729\pi
−0.999384 + 0.0350936i 0.988827π0.988827\pi
182182 0 0
183183 −3.05573 + 8.00000i −0.225886 + 0.591377i
184184 0 0
185185 1.52786 3.05573i 0.112331 0.224662i
186186 0 0
187187 25.8885i 1.89316i
188188 0 0
189189 7.70820 + 14.9443i 0.560689 + 1.08704i
190190 0 0
191191 −4.94427 −0.357755 −0.178877 0.983871i 0.557247π-0.557247\pi
−0.178877 + 0.983871i 0.557247π0.557247\pi
192192 0 0
193193 20.9443i 1.50760i −0.657103 0.753801i 0.728218π-0.728218\pi
0.657103 0.753801i 0.271782π-0.271782\pi
194194 0 0
195195 18.4721 16.9443i 1.32282 1.21340i
196196 0 0
197197 8.94427i 0.637253i −0.947880 0.318626i 0.896778π-0.896778\pi
0.947880 0.318626i 0.103222π-0.103222\pi
198198 0 0
199199 13.5279i 0.958965i 0.877551 + 0.479482i 0.159176π0.159176\pi
−0.877551 + 0.479482i 0.840824π0.840824\pi
200200 0 0
201201 −7.23607 + 18.9443i −0.510393 + 1.33623i
202202 0 0
203203 −14.4721 −1.01574
204204 0 0
205205 8.00000 + 4.00000i 0.558744 + 0.279372i
206206 0 0
207207 2.47214 2.76393i 0.171825 0.192107i
208208 0 0
209209 9.88854i 0.684005i
210210 0 0
211211 7.41641 0.510567 0.255283 0.966866i 0.417831π-0.417831\pi
0.255283 + 0.966866i 0.417831π0.417831\pi
212212 0 0
213213 0 0
214214 0 0
215215 −18.4721 9.23607i −1.25979 0.629895i
216216 0 0
217217 8.00000i 0.543075i
218218 0 0
219219 3.05573 8.00000i 0.206487 0.540590i
220220 0 0
221221 −41.8885 −2.81773
222222 0 0
223223 −21.1246 −1.41461 −0.707304 0.706909i 0.750089π-0.750089\pi
−0.707304 + 0.706909i 0.750089π0.750089\pi
224224 0 0
225225 1.29180 14.9443i 0.0861197 0.996285i
226226 0 0
227227 12.7639 0.847172 0.423586 0.905856i 0.360771π-0.360771\pi
0.423586 + 0.905856i 0.360771π0.360771\pi
228228 0 0
229229 24.9443i 1.64836i 0.566325 + 0.824182i 0.308365π0.308365\pi
−0.566325 + 0.824182i 0.691635π0.691635\pi
230230 0 0
231231 8.00000 20.9443i 0.526361 1.37803i
232232 0 0
233233 9.52786 0.624191 0.312096 0.950051i 0.398969π-0.398969\pi
0.312096 + 0.950051i 0.398969π0.398969\pi
234234 0 0
235235 −18.4721 9.23607i −1.20499 0.602495i
236236 0 0
237237 6.47214 16.9443i 0.420410 1.10065i
238238 0 0
239239 −11.0557 −0.715136 −0.357568 0.933887i 0.616394π-0.616394\pi
−0.357568 + 0.933887i 0.616394π0.616394\pi
240240 0 0
241241 −12.4721 −0.803401 −0.401700 0.915771i 0.631581π-0.631581\pi
−0.401700 + 0.915771i 0.631581π0.631581\pi
242242 0 0
243243 3.90983 15.0902i 0.250816 0.968035i
244244 0 0
245245 −3.47214 + 6.94427i −0.221827 + 0.443653i
246246 0 0
247247 16.0000 1.01806
248248 0 0
249249 13.2361 + 5.05573i 0.838802 + 0.320394i
250250 0 0
251251 21.8885i 1.38159i 0.723049 + 0.690796i 0.242740π0.242740\pi
−0.723049 + 0.690796i 0.757260π0.757260\pi
252252 0 0
253253 −4.94427 −0.310844
254254 0 0
255255 −18.4721 + 16.9443i −1.15677 + 1.06109i
256256 0 0
257257 −4.58359 −0.285917 −0.142958 0.989729i 0.545661π-0.545661\pi
−0.142958 + 0.989729i 0.545661π0.545661\pi
258258 0 0
259259 4.94427 0.307222
260260 0 0
261261 10.0000 + 8.94427i 0.618984 + 0.553637i
262262 0 0
263263 14.1803i 0.874397i 0.899365 + 0.437199i 0.144029π0.144029\pi
−0.899365 + 0.437199i 0.855971π0.855971\pi
264264 0 0
265265 −17.8885 8.94427i −1.09888 0.549442i
266266 0 0
267267 4.94427 12.9443i 0.302585 0.792177i
268268 0 0
269269 19.8885 1.21263 0.606313 0.795226i 0.292648π-0.292648\pi
0.606313 + 0.795226i 0.292648π0.292648\pi
270270 0 0
271271 26.4721i 1.60807i 0.594583 + 0.804034i 0.297317π0.297317\pi
−0.594583 + 0.804034i 0.702683π0.702683\pi
272272 0 0
273273 33.8885 + 12.9443i 2.05103 + 0.783423i
274274 0 0
275275 −16.0000 + 12.0000i −0.964836 + 0.723627i
276276 0 0
277277 −11.4164 −0.685945 −0.342973 0.939345i 0.611434π-0.611434\pi
−0.342973 + 0.939345i 0.611434π0.611434\pi
278278 0 0
279279 4.94427 5.52786i 0.296006 0.330945i
280280 0 0
281281 12.0000i 0.715860i −0.933748 0.357930i 0.883483π-0.883483\pi
0.933748 0.357930i 0.116517π-0.116517\pi
282282 0 0
283283 3.70820i 0.220430i −0.993908 0.110215i 0.964846π-0.964846\pi
0.993908 0.110215i 0.0351539π-0.0351539\pi
284284 0 0
285285 7.05573 6.47214i 0.417945 0.383376i
286286 0 0
287287 12.9443i 0.764076i
288288 0 0
289289 24.8885 1.46403
290290 0 0
291291 8.00000 20.9443i 0.468968 1.22777i
292292 0 0
293293 13.8885i 0.811377i −0.914011 0.405689i 0.867032π-0.867032\pi
0.914011 0.405689i 0.132968π-0.132968\pi
294294 0 0
295295 −17.8885 8.94427i −1.04151 0.520756i
296296 0 0
297297 −18.4721 + 9.52786i −1.07186 + 0.552863i
298298 0 0
299299 8.00000i 0.462652i
300300 0 0
301301 29.8885i 1.72275i
302302 0 0
303303 −5.70820 2.18034i −0.327928 0.125257i
304304 0 0
305305 9.88854 + 4.94427i 0.566216 + 0.283108i
306306 0 0
307307 1.81966i 0.103853i 0.998651 + 0.0519267i 0.0165362π0.0165362\pi
−0.998651 + 0.0519267i 0.983464π0.983464\pi
308308 0 0
309309 −2.76393 1.05573i −0.157235 0.0600583i
310310 0 0
311311 −16.0000 −0.907277 −0.453638 0.891186i 0.649874π-0.649874\pi
−0.453638 + 0.891186i 0.649874π0.649874\pi
312312 0 0
313313 24.0000i 1.35656i 0.734803 + 0.678280i 0.237274π0.237274\pi
−0.734803 + 0.678280i 0.762726π0.762726\pi
314314 0 0
315315 20.1803 8.00000i 1.13703 0.450749i
316316 0 0
317317 16.9443i 0.951685i −0.879530 0.475843i 0.842143π-0.842143\pi
0.879530 0.475843i 0.157857π-0.157857\pi
318318 0 0
319319 17.8885i 1.00157i
320320 0 0
321321 −15.7082 6.00000i −0.876746 0.334887i
322322 0 0
323323 −16.0000 −0.890264
324324 0 0
325325 −19.4164 25.8885i −1.07703 1.43604i
326326 0 0
327327 8.00000 20.9443i 0.442401 1.15822i
328328 0 0
329329 29.8885i 1.64781i
330330 0 0
331331 2.47214 0.135881 0.0679404 0.997689i 0.478357π-0.478357\pi
0.0679404 + 0.997689i 0.478357π0.478357\pi
332332 0 0
333333 −3.41641 3.05573i −0.187218 0.167453i
334334 0 0
335335 23.4164 + 11.7082i 1.27938 + 0.639688i
336336 0 0
337337 17.8885i 0.974451i −0.873276 0.487226i 0.838009π-0.838009\pi
0.873276 0.487226i 0.161991π-0.161991\pi
338338 0 0
339339 23.4164 + 8.94427i 1.27180 + 0.485786i
340340 0 0
341341 −9.88854 −0.535495
342342 0 0
343343 11.4164 0.616428
344344 0 0
345345 −3.23607 3.52786i −0.174224 0.189934i
346346 0 0
347347 4.76393 0.255741 0.127871 0.991791i 0.459186π-0.459186\pi
0.127871 + 0.991791i 0.459186π0.459186\pi
348348 0 0
349349 18.8328i 1.00810i 0.863675 + 0.504049i 0.168157π0.168157\pi
−0.863675 + 0.504049i 0.831843π0.831843\pi
350350 0 0
351351 −15.4164 29.8885i −0.822867 1.59533i
352352 0 0
353353 11.4164 0.607634 0.303817 0.952730i 0.401739π-0.401739\pi
0.303817 + 0.952730i 0.401739π0.401739\pi
354354 0 0
355355 0 0
356356 0 0
357357 −33.8885 12.9443i −1.79357 0.685084i
358358 0 0
359359 −30.8328 −1.62729 −0.813647 0.581359i 0.802521π-0.802521\pi
−0.813647 + 0.581359i 0.802521π0.802521\pi
360360 0 0
361361 −12.8885 −0.678344
362362 0 0
363363 8.09017 + 3.09017i 0.424624 + 0.162192i
364364 0 0
365365 −9.88854 4.94427i −0.517590 0.258795i
366366 0 0
367367 14.6525 0.764853 0.382427 0.923986i 0.375089π-0.375089\pi
0.382427 + 0.923986i 0.375089π0.375089\pi
368368 0 0
369369 8.00000 8.94427i 0.416463 0.465620i
370370 0 0
371371 28.9443i 1.50271i
372372 0 0
373373 30.4721 1.57779 0.788894 0.614530i 0.210654π-0.210654\pi
0.788894 + 0.614530i 0.210654π0.210654\pi
374374 0 0
375375 −19.0344 3.56231i −0.982934 0.183957i
376376 0 0
377377 28.9443 1.49071
378378 0 0
379379 −8.58359 −0.440910 −0.220455 0.975397i 0.570754π-0.570754\pi
−0.220455 + 0.975397i 0.570754π0.570754\pi
380380 0 0
381381 −18.1803 6.94427i −0.931407 0.355766i
382382 0 0
383383 22.1803i 1.13336i −0.823937 0.566681i 0.808227π-0.808227\pi
0.823937 0.566681i 0.191773π-0.191773\pi
384384 0 0
385385 −25.8885 12.9443i −1.31940 0.659701i
386386 0 0
387387 −18.4721 + 20.6525i −0.938991 + 1.04982i
388388 0 0
389389 −18.0000 −0.912636 −0.456318 0.889817i 0.650832π-0.650832\pi
−0.456318 + 0.889817i 0.650832π0.650832\pi
390390 0 0
391391 8.00000i 0.404577i
392392 0 0
393393 −0.583592 + 1.52786i −0.0294383 + 0.0770705i
394394 0 0
395395 −20.9443 10.4721i −1.05382 0.526910i
396396 0 0
397397 −3.41641 −0.171465 −0.0857323 0.996318i 0.527323π-0.527323\pi
−0.0857323 + 0.996318i 0.527323π0.527323\pi
398398 0 0
399399 12.9443 + 4.94427i 0.648024 + 0.247523i
400400 0 0
401401 0 0 1.00000 00
−1.00000 π\pi
402402 0 0
403403 16.0000i 0.797017i
404404 0 0
405405 −18.8885 6.94427i −0.938579 0.345064i
406406 0 0
407407 6.11146i 0.302934i
408408 0 0
409409 −4.47214 −0.221133 −0.110566 0.993869i 0.535266π-0.535266\pi
−0.110566 + 0.993869i 0.535266π0.535266\pi
410410 0 0
411411 −2.47214 0.944272i −0.121941 0.0465775i
412412 0 0
413413 28.9443i 1.42425i
414414 0 0
415415 8.18034 16.3607i 0.401557 0.803114i
416416 0 0
417417 37.8885 + 14.4721i 1.85541 + 0.708704i
418418 0 0
419419 4.00000i 0.195413i −0.995215 0.0977064i 0.968849π-0.968849\pi
0.995215 0.0977064i 0.0311506π-0.0311506\pi
420420 0 0
421421 4.94427i 0.240969i 0.992715 + 0.120485i 0.0384448π0.0384448\pi
−0.992715 + 0.120485i 0.961555π0.961555\pi
422422 0 0
423423 −18.4721 + 20.6525i −0.898146 + 1.00416i
424424 0 0
425425 19.4164 + 25.8885i 0.941834 + 1.25578i
426426 0 0
427427 16.0000i 0.774294i
428428 0 0
429429 −16.0000 + 41.8885i −0.772487 + 2.02240i
430430 0 0
431431 35.7771 1.72332 0.861661 0.507485i 0.169425π-0.169425\pi
0.861661 + 0.507485i 0.169425π0.169425\pi
432432 0 0
433433 3.05573i 0.146849i 0.997301 + 0.0734245i 0.0233928π0.0233928\pi
−0.997301 + 0.0734245i 0.976607π0.976607\pi
434434 0 0
435435 12.7639 11.7082i 0.611984 0.561365i
436436 0 0
437437 3.05573i 0.146175i
438438 0 0
439439 7.41641i 0.353966i 0.984214 + 0.176983i 0.0566337π0.0566337\pi
−0.984214 + 0.176983i 0.943366π0.943366\pi
440440 0 0
441441 7.76393 + 6.94427i 0.369711 + 0.330680i
442442 0 0
443443 16.1803 0.768751 0.384376 0.923177i 0.374417π-0.374417\pi
0.384376 + 0.923177i 0.374417π0.374417\pi
444444 0 0
445445 −16.0000 8.00000i −0.758473 0.379236i
446446 0 0
447447 −19.2361 7.34752i −0.909835 0.347526i
448448 0 0
449449 2.11146i 0.0996458i 0.998758 + 0.0498229i 0.0158657π0.0158657\pi
−0.998758 + 0.0498229i 0.984134π0.984134\pi
450450 0 0
451451 −16.0000 −0.753411
452452 0 0
453453 6.47214 16.9443i 0.304087 0.796111i
454454 0 0
455455 20.9443 41.8885i 0.981883 1.96377i
456456 0 0
457457 41.8885i 1.95946i −0.200314 0.979732i 0.564196π-0.564196\pi
0.200314 0.979732i 0.435804π-0.435804\pi
458458 0 0
459459 15.4164 + 29.8885i 0.719576 + 1.39508i
460460 0 0
461461 20.4721 0.953482 0.476741 0.879044i 0.341818π-0.341818\pi
0.476741 + 0.879044i 0.341818π0.341818\pi
462462 0 0
463463 22.2918 1.03599 0.517994 0.855384i 0.326679π-0.326679\pi
0.517994 + 0.855384i 0.326679π0.326679\pi
464464 0 0
465465 −6.47214 7.05573i −0.300138 0.327202i
466466 0 0
467467 19.2361 0.890139 0.445070 0.895496i 0.353179π-0.353179\pi
0.445070 + 0.895496i 0.353179π0.353179\pi
468468 0 0
469469 37.8885i 1.74953i
470470 0 0
471471 2.47214 + 0.944272i 0.113910 + 0.0435098i
472472 0 0
473473 36.9443 1.69870
474474 0 0
475475 −7.41641 9.88854i −0.340288 0.453718i
476476 0 0
477477 −17.8885 + 20.0000i −0.819060 + 0.915737i
478478 0 0
479479 −14.8328 −0.677729 −0.338864 0.940835i 0.610043π-0.610043\pi
−0.338864 + 0.940835i 0.610043π0.610043\pi
480480 0 0
481481 −9.88854 −0.450879
482482 0 0
483483 2.47214 6.47214i 0.112486 0.294492i
484484 0 0
485485 −25.8885 12.9443i −1.17554 0.587769i
486486 0 0
487487 −6.65248 −0.301452 −0.150726 0.988576i 0.548161π-0.548161\pi
−0.150726 + 0.988576i 0.548161π0.548161\pi
488488 0 0
489489 9.12461 23.8885i 0.412629 1.08028i
490490 0 0
491491 0.944272i 0.0426144i −0.999773 0.0213072i 0.993217π-0.993217\pi
0.999773 0.0213072i 0.00678280π-0.00678280\pi
492492 0 0
493493 −28.9443 −1.30358
494494 0 0
495495 9.88854 + 24.9443i 0.444457 + 1.12116i
496496 0 0
497497 0 0
498498 0 0
499499 −28.3607 −1.26960 −0.634799 0.772677i 0.718917π-0.718917\pi
−0.634799 + 0.772677i 0.718917π0.718917\pi
500500 0 0
501501 0.763932 2.00000i 0.0341300 0.0893534i
502502 0 0
503503 37.0132i 1.65034i 0.564888 + 0.825168i 0.308919π0.308919\pi
−0.564888 + 0.825168i 0.691081π0.691081\pi
504504 0 0
505505 −3.52786 + 7.05573i −0.156988 + 0.313976i
506506 0 0
507507 −46.7426 17.8541i −2.07591 0.792929i
508508 0 0
509509 5.41641 0.240078 0.120039 0.992769i 0.461698π-0.461698\pi
0.120039 + 0.992769i 0.461698π0.461698\pi
510510 0 0
511511 16.0000i 0.707798i
512512 0 0
513513 −5.88854 11.4164i −0.259986 0.504047i
514514 0 0
515515 −1.70820 + 3.41641i −0.0752725 + 0.150545i
516516 0 0
517517 36.9443 1.62481
518518 0 0
519519 2.47214 6.47214i 0.108515 0.284095i
520520 0 0
521521 8.00000i 0.350486i −0.984525 0.175243i 0.943929π-0.943929\pi
0.984525 0.175243i 0.0560712π-0.0560712\pi
522522 0 0
523523 3.70820i 0.162148i 0.996708 + 0.0810742i 0.0258351π0.0258351\pi
−0.996708 + 0.0810742i 0.974165π0.974165\pi
524524 0 0
525525 −7.70820 26.9443i −0.336414 1.17594i
526526 0 0
527527 16.0000i 0.696971i
528528 0 0
529529 21.4721 0.933571
530530 0 0
531531 −17.8885 + 20.0000i −0.776297 + 0.867926i
532532 0 0
533533 25.8885i 1.12136i
534534 0 0
535535 −9.70820 + 19.4164i −0.419722 + 0.839445i
536536 0 0
537537 −0.583592 + 1.52786i −0.0251839 + 0.0659322i
538538 0 0
539539 13.8885i 0.598222i
540540 0 0
541541 7.05573i 0.303349i 0.988430 + 0.151675i 0.0484666π0.0484666\pi
−0.988430 + 0.151675i 0.951533π0.951533\pi
542542 0 0
543543 0.583592 1.52786i 0.0250443 0.0655669i
544544 0 0
545545 −25.8885 12.9443i −1.10894 0.554472i
546546 0 0
547547 14.7639i 0.631260i −0.948882 0.315630i 0.897784π-0.897784\pi
0.948882 0.315630i 0.102216π-0.102216\pi
548548 0 0
549549 9.88854 11.0557i 0.422033 0.471847i
550550 0 0
551551 11.0557 0.470990
552552 0 0
553553 33.8885i 1.44109i
554554 0 0
555555 −4.36068 + 4.00000i −0.185101 + 0.169791i
556556 0 0
557557 5.88854i 0.249506i −0.992188 0.124753i 0.960186π-0.960186\pi
0.992188 0.124753i 0.0398138π-0.0398138\pi
558558 0 0
559559 59.7771i 2.52830i
560560 0 0
561561 16.0000 41.8885i 0.675521 1.76854i
562562 0 0
563563 29.1246 1.22746 0.613728 0.789518i 0.289669π-0.289669\pi
0.613728 + 0.789518i 0.289669π0.289669\pi
564564 0 0
565565 14.4721 28.9443i 0.608847 1.21769i
566566 0 0
567567 −3.23607 28.9443i −0.135902 1.21555i
568568 0 0
569569 31.7771i 1.33216i −0.745878 0.666082i 0.767970π-0.767970\pi
0.745878 0.666082i 0.232030π-0.232030\pi
570570 0 0
571571 18.4721 0.773035 0.386517 0.922282i 0.373678π-0.373678\pi
0.386517 + 0.922282i 0.373678π0.373678\pi
572572 0 0
573573 8.00000 + 3.05573i 0.334205 + 0.127655i
574574 0 0
575575 −4.94427 + 3.70820i −0.206190 + 0.154643i
576576 0 0
577577 8.00000i 0.333044i −0.986038 0.166522i 0.946746π-0.946746\pi
0.986038 0.166522i 0.0532537π-0.0532537\pi
578578 0 0
579579 −12.9443 + 33.8885i −0.537946 + 1.40836i
580580 0 0
581581 26.4721 1.09825
582582 0 0
583583 35.7771 1.48174
584584 0 0
585585 −40.3607 + 16.0000i −1.66871 + 0.661519i
586586 0 0
587587 36.7639 1.51741 0.758705 0.651434i 0.225832π-0.225832\pi
0.758705 + 0.651434i 0.225832π0.225832\pi
588588 0 0
589589 6.11146i 0.251818i
590590 0 0
591591 −5.52786 + 14.4721i −0.227386 + 0.595304i
592592 0 0
593593 −30.4721 −1.25134 −0.625670 0.780088i 0.715174π-0.715174\pi
−0.625670 + 0.780088i 0.715174π0.715174\pi
594594 0 0
595595 −20.9443 + 41.8885i −0.858631 + 1.71726i
596596 0 0
597597 8.36068 21.8885i 0.342180 0.895838i
598598 0 0
599599 4.94427 0.202017 0.101009 0.994886i 0.467793π-0.467793\pi
0.101009 + 0.994886i 0.467793π0.467793\pi
600600 0 0
601601 4.47214 0.182422 0.0912111 0.995832i 0.470926π-0.470926\pi
0.0912111 + 0.995832i 0.470926π0.470926\pi
602602 0 0
603603 23.4164 26.1803i 0.953590 1.06615i
604604 0 0
605605 5.00000 10.0000i 0.203279 0.406558i
606606 0 0
607607 −48.1803 −1.95558 −0.977790 0.209588i 0.932788π-0.932788\pi
−0.977790 + 0.209588i 0.932788π0.932788\pi
608608 0 0
609609 23.4164 + 8.94427i 0.948881 + 0.362440i
610610 0 0
611611 59.7771i 2.41832i
612612 0 0
613613 −19.4164 −0.784221 −0.392111 0.919918i 0.628255π-0.628255\pi
−0.392111 + 0.919918i 0.628255π0.628255\pi
614614 0 0
615615 −10.4721 11.4164i −0.422277 0.460354i
616616 0 0
617617 35.4164 1.42581 0.712905 0.701260i 0.247379π-0.247379\pi
0.712905 + 0.701260i 0.247379π0.247379\pi
618618 0 0
619619 34.4721 1.38555 0.692776 0.721153i 0.256387π-0.256387\pi
0.692776 + 0.721153i 0.256387π0.256387\pi
620620 0 0
621621 −5.70820 + 2.94427i −0.229062 + 0.118150i
622622 0 0
623623 25.8885i 1.03720i
624624 0 0
625625 −7.00000 + 24.0000i −0.280000 + 0.960000i
626626 0 0
627627 −6.11146 + 16.0000i −0.244068 + 0.638978i
628628 0 0
629629 9.88854 0.394282
630630 0 0
631631 13.5279i 0.538536i 0.963065 + 0.269268i 0.0867817π0.0867817\pi
−0.963065 + 0.269268i 0.913218π0.913218\pi
632632 0 0
633633 −12.0000 4.58359i −0.476957 0.182181i
634634 0 0
635635 −11.2361 + 22.4721i −0.445890 + 0.891780i
636636 0 0
637637 22.4721 0.890378
638638 0 0
639639 0 0
640640 0 0
641641 13.8885i 0.548564i 0.961649 + 0.274282i 0.0884403π0.0884403\pi
−0.961649 + 0.274282i 0.911560π0.911560\pi
642642 0 0
643643 21.5967i 0.851693i 0.904796 + 0.425846i 0.140024π0.140024\pi
−0.904796 + 0.425846i 0.859976π0.859976\pi
644644 0 0
645645 24.1803 + 26.3607i 0.952100 + 1.03795i
646646 0 0
647647 20.2918i 0.797753i 0.917005 + 0.398876i 0.130600π0.130600\pi
−0.917005 + 0.398876i 0.869400π0.869400\pi
648648 0 0
649649 35.7771 1.40437
650650 0 0
651651 4.94427 12.9443i 0.193781 0.507326i
652652 0 0
653653 32.9443i 1.28921i 0.764516 + 0.644604i 0.222978π0.222978\pi
−0.764516 + 0.644604i 0.777022π0.777022\pi
654654 0 0
655655 1.88854 + 0.944272i 0.0737915 + 0.0368958i
656656 0 0
657657 −9.88854 + 11.0557i −0.385789 + 0.431325i
658658 0 0
659659 7.05573i 0.274852i −0.990512 0.137426i 0.956117π-0.956117\pi
0.990512 0.137426i 0.0438830π-0.0438830\pi
660660 0 0
661661 3.05573i 0.118854i 0.998233 + 0.0594270i 0.0189274π0.0189274\pi
−0.998233 + 0.0594270i 0.981073π0.981073\pi
662662 0 0
663663 67.7771 + 25.8885i 2.63224 + 1.00543i
664664 0 0
665665 8.00000 16.0000i 0.310227 0.620453i
666666 0 0
667667 5.52786i 0.214040i
668668 0 0
669669 34.1803 + 13.0557i 1.32149 + 0.504764i
670670 0 0
671671 −19.7771 −0.763486
672672 0 0
673673 28.9443i 1.11572i 0.829935 + 0.557860i 0.188377π0.188377\pi
−0.829935 + 0.557860i 0.811623π0.811623\pi
674674 0 0
675675 −11.3262 + 23.3820i −0.435947 + 0.899972i
676676 0 0
677677 24.9443i 0.958686i 0.877628 + 0.479343i 0.159125π0.159125\pi
−0.877628 + 0.479343i 0.840875π0.840875\pi
678678 0 0
679679 41.8885i 1.60753i
680680 0 0
681681 −20.6525 7.88854i −0.791405 0.302290i
682682 0 0
683683 −11.2361 −0.429936 −0.214968 0.976621i 0.568965π-0.568965\pi
−0.214968 + 0.976621i 0.568965π0.568965\pi
684684 0 0
685685 −1.52786 + 3.05573i −0.0583767 + 0.116753i
686686 0 0
687687 15.4164 40.3607i 0.588173 1.53986i
688688 0 0
689689 57.8885i 2.20538i
690690 0 0
691691 −17.3050 −0.658311 −0.329156 0.944276i 0.606764π-0.606764\pi
−0.329156 + 0.944276i 0.606764π0.606764\pi
692692 0 0
693693 −25.8885 + 28.9443i −0.983424 + 1.09950i
694694 0 0
695695 23.4164 46.8328i 0.888235 1.77647i
696696 0 0
697697 25.8885i 0.980599i
698698 0 0
699699 −15.4164 5.88854i −0.583102 0.222725i
700700 0 0
701701 −10.0000 −0.377695 −0.188847 0.982006i 0.560475π-0.560475\pi
−0.188847 + 0.982006i 0.560475π0.560475\pi
702702 0 0
703703 −3.77709 −0.142456
704704 0 0
705705 24.1803 + 26.3607i 0.910684 + 0.992801i
706706 0 0
707707 −11.4164 −0.429358
708708 0 0
709709 32.9443i 1.23725i 0.785687 + 0.618624i 0.212310π0.212310\pi
−0.785687 + 0.618624i 0.787690π0.787690\pi
710710 0 0
711711 −20.9443 + 23.4164i −0.785472 + 0.878184i
712712 0 0
713713 −3.05573 −0.114438
714714 0 0
715715 51.7771 + 25.8885i 1.93635 + 0.968177i
716716 0 0
717717 17.8885 + 6.83282i 0.668060 + 0.255176i
718718 0 0
719719 25.8885 0.965480 0.482740 0.875764i 0.339642π-0.339642\pi
0.482740 + 0.875764i 0.339642π0.339642\pi
720720 0 0
721721 −5.52786 −0.205868
722722 0 0
723723 20.1803 + 7.70820i 0.750515 + 0.286671i
724724 0 0
725725 −13.4164 17.8885i −0.498273 0.664364i
726726 0 0
727727 −12.7639 −0.473388 −0.236694 0.971584i 0.576064π-0.576064\pi
−0.236694 + 0.971584i 0.576064π0.576064\pi
728728 0 0
729729 −15.6525 + 22.0000i −0.579721 + 0.814815i
730730 0 0
731731 59.7771i 2.21094i
732732 0 0
733733 −1.52786 −0.0564329 −0.0282165 0.999602i 0.508983π-0.508983\pi
−0.0282165 + 0.999602i 0.508983π0.508983\pi
734734 0 0
735735 9.90983 9.09017i 0.365530 0.335296i
736736 0 0
737737 −46.8328 −1.72511
738738 0 0
739739 −18.4721 −0.679509 −0.339754 0.940514i 0.610344π-0.610344\pi
−0.339754 + 0.940514i 0.610344π0.610344\pi
740740 0 0
741741 −25.8885 9.88854i −0.951039 0.363265i
742742 0 0
743743 47.4853i 1.74207i 0.491224 + 0.871033i 0.336550π0.336550\pi
−0.491224 + 0.871033i 0.663450π0.663450\pi
744744 0 0
745745 −11.8885 + 23.7771i −0.435563 + 0.871125i
746746 0 0
747747 −18.2918 16.3607i −0.669262 0.598606i
748748 0 0
749749 −31.4164 −1.14793
750750 0 0
751751 17.3050i 0.631467i 0.948848 + 0.315733i 0.102250π0.102250\pi
−0.948848 + 0.315733i 0.897750π0.897750\pi
752752 0 0
753753 13.5279 35.4164i 0.492983 1.29065i
754754 0 0
755755 −20.9443 10.4721i −0.762240 0.381120i
756756 0 0
757757 −27.4164 −0.996466 −0.498233 0.867043i 0.666018π-0.666018\pi
−0.498233 + 0.867043i 0.666018π0.666018\pi
758758 0 0
759759 8.00000 + 3.05573i 0.290382 + 0.110916i
760760 0 0
761761 9.88854i 0.358459i 0.983807 + 0.179230i 0.0573605π0.0573605\pi
−0.983807 + 0.179230i 0.942639π0.942639\pi
762762 0 0
763763 41.8885i 1.51647i
764764 0 0
765765 40.3607 16.0000i 1.45924 0.578481i
766766 0 0
767767 57.8885i 2.09023i
768768 0 0
769769 27.8885 1.00569 0.502843 0.864378i 0.332287π-0.332287\pi
0.502843 + 0.864378i 0.332287π0.332287\pi
770770 0 0
771771 7.41641 + 2.83282i 0.267095 + 0.102021i
772772 0 0
773773 52.7214i 1.89626i 0.317890 + 0.948128i 0.397026π0.397026\pi
−0.317890 + 0.948128i 0.602974π0.602974\pi
774774 0 0
775775 −9.88854 + 7.41641i −0.355207 + 0.266405i
776776 0 0
777777 −8.00000 3.05573i −0.286998 0.109624i
778778 0 0
779779 9.88854i 0.354294i
780780 0 0
781781 0 0
782782 0 0
783783 −10.6525 20.6525i −0.380688 0.738059i
784784 0 0
785785 1.52786 3.05573i 0.0545318 0.109064i
786786 0 0
787787 14.1803i 0.505475i −0.967535 0.252737i 0.918669π-0.918669\pi
0.967535 0.252737i 0.0813308π-0.0813308\pi
788788 0 0
789789 8.76393 22.9443i 0.312004 0.816838i
790790 0 0
791791 46.8328 1.66518
792792 0 0
793793 32.0000i 1.13635i
794794 0 0
795795 23.4164 + 25.5279i 0.830494 + 0.905380i
796796 0 0
797797 23.0557i 0.816676i −0.912831 0.408338i 0.866109π-0.866109\pi
0.912831 0.408338i 0.133891π-0.133891\pi
798798 0 0
799799 59.7771i 2.11476i
800800 0 0
801801 −16.0000 + 17.8885i −0.565332 + 0.632061i
802802 0 0
803803 19.7771 0.697918
804804 0 0
805805 −8.00000 4.00000i −0.281963 0.140981i
806806 0 0
807807 −32.1803 12.2918i −1.13280 0.432692i
808808 0 0
809809 40.0000i 1.40633i −0.711029 0.703163i 0.751771π-0.751771\pi
0.711029 0.703163i 0.248229π-0.248229\pi
810810 0 0
811811 −8.58359 −0.301411 −0.150705 0.988579i 0.548154π-0.548154\pi
−0.150705 + 0.988579i 0.548154π0.548154\pi
812812 0 0
813813 16.3607 42.8328i 0.573794 1.50221i
814814 0 0
815815 −29.5279 14.7639i −1.03432 0.517158i
816816 0 0
817817 22.8328i 0.798819i
818818 0 0
819819 −46.8328 41.8885i −1.63647 1.46370i
820820 0 0
821821 −4.11146 −0.143491 −0.0717454 0.997423i 0.522857π-0.522857\pi
−0.0717454 + 0.997423i 0.522857π0.522857\pi
822822 0 0
823823 −14.2918 −0.498181 −0.249090 0.968480i 0.580132π-0.580132\pi
−0.249090 + 0.968480i 0.580132π0.580132\pi
824824 0 0
825825 33.3050 9.52786i 1.15953 0.331718i
826826 0 0
827827 −24.5410 −0.853375 −0.426687 0.904399i 0.640320π-0.640320\pi
−0.426687 + 0.904399i 0.640320π0.640320\pi
828828 0 0
829829 20.9443i 0.727425i −0.931511 0.363712i 0.881509π-0.881509\pi
0.931511 0.363712i 0.118491π-0.118491\pi
830830 0 0
831831 18.4721 + 7.05573i 0.640791 + 0.244760i
832832 0 0
833833 −22.4721 −0.778613
834834 0 0
835835 −2.47214 1.23607i −0.0855518 0.0427759i
836836 0 0
837837 −11.4164 + 5.88854i −0.394609 + 0.203538i
838838 0 0
839839 30.8328 1.06447 0.532234 0.846598i 0.321353π-0.321353\pi
0.532234 + 0.846598i 0.321353π0.321353\pi
840840 0 0
841841 −9.00000 −0.310345
842842 0 0
843843 −7.41641 + 19.4164i −0.255435 + 0.668737i
844844 0 0
845845 −28.8885 + 57.7771i −0.993796 + 1.98759i
846846 0 0
847847 16.1803 0.555963
848848 0 0
849849 −2.29180 + 6.00000i −0.0786542 + 0.205919i
850850 0 0
851851 1.88854i 0.0647384i
852852 0 0
853853 −41.5279 −1.42189 −0.710943 0.703249i 0.751732π-0.751732\pi
−0.710943 + 0.703249i 0.751732π0.751732\pi
854854 0 0
855855 −15.4164 + 6.11146i −0.527230 + 0.209007i
856856 0 0
857857 −40.3607 −1.37869 −0.689347 0.724431i 0.742103π-0.742103\pi
−0.689347 + 0.724431i 0.742103π0.742103\pi
858858 0 0
859859 39.4164 1.34487 0.672435 0.740156i 0.265248π-0.265248\pi
0.672435 + 0.740156i 0.265248π0.265248\pi
860860 0 0
861861 8.00000 20.9443i 0.272639 0.713779i
862862 0 0
863863 38.7639i 1.31954i −0.751468 0.659770i 0.770654π-0.770654\pi
0.751468 0.659770i 0.229346π-0.229346\pi
864864 0 0
865865 −8.00000 4.00000i −0.272008 0.136004i
866866 0 0
867867 −40.2705 15.3820i −1.36766 0.522399i
868868 0 0
869869 41.8885 1.42097
870870 0 0
871871 75.7771i 2.56761i
872872 0 0
873873 −25.8885 + 28.9443i −0.876194 + 0.979615i
874874 0 0
875875 −35.5967 + 6.47214i −1.20339 + 0.218798i
876876 0 0
877877 −33.5279 −1.13216 −0.566078 0.824352i 0.691540π-0.691540\pi
−0.566078 + 0.824352i 0.691540π0.691540\pi
878878 0 0
879879 −8.58359 + 22.4721i −0.289517 + 0.757966i
880880 0 0
881881 29.8885i 1.00697i 0.864004 + 0.503485i 0.167949π0.167949\pi
−0.864004 + 0.503485i 0.832051π0.832051\pi
882882 0 0
883883 1.23607i 0.0415970i −0.999784 0.0207985i 0.993379π-0.993379\pi
0.999784 0.0207985i 0.00662085π-0.00662085\pi
884884 0 0
885885 23.4164 + 25.5279i 0.787134 + 0.858110i
886886 0 0
887887 21.5967i 0.725148i −0.931955 0.362574i 0.881898π-0.881898\pi
0.931955 0.362574i 0.118102π-0.118102\pi
888888 0 0
889889 −36.3607 −1.21950
890890 0 0
891891 35.7771 4.00000i 1.19858 0.134005i
892892 0 0
893893 22.8328i 0.764071i
894894 0 0
895895 1.88854 + 0.944272i 0.0631271 + 0.0315635i
896896 0 0
897897 −4.94427 + 12.9443i −0.165084 + 0.432197i
898898 0 0
899899 11.0557i 0.368729i
900900 0 0
901901 57.8885i 1.92855i
902902 0 0
903903 −18.4721 + 48.3607i −0.614714 + 1.60934i
904904 0 0
905905 −1.88854 0.944272i −0.0627773 0.0313887i
906906 0 0
907907 55.4853i 1.84236i −0.389138 0.921179i 0.627227π-0.627227\pi
0.389138 0.921179i 0.372773π-0.372773\pi
908908 0 0
909909 7.88854 + 7.05573i 0.261646 + 0.234024i
910910 0 0
911911 −57.8885 −1.91793 −0.958967 0.283519i 0.908498π-0.908498\pi
−0.958967 + 0.283519i 0.908498π0.908498\pi
912912 0 0
913913 32.7214i 1.08292i
914914 0 0
915915 −12.9443 14.1115i −0.427924 0.466510i
916916 0 0
917917 3.05573i 0.100909i
918918 0 0
919919 42.4721i 1.40103i 0.713639 + 0.700513i 0.247046π0.247046\pi
−0.713639 + 0.700513i 0.752954π0.752954\pi
920920 0 0
921921 1.12461 2.94427i 0.0370572 0.0970171i
922922 0 0
923923 0 0
924924 0 0
925925 4.58359 + 6.11146i 0.150708 + 0.200943i
926926 0 0
927927 3.81966 + 3.41641i 0.125454 + 0.112210i
928928 0 0
929929 53.8885i 1.76803i −0.467463 0.884013i 0.654832π-0.654832\pi
0.467463 0.884013i 0.345168π-0.345168\pi
930930 0 0
931931 8.58359 0.281316
932932 0 0
933933 25.8885 + 9.88854i 0.847553 + 0.323736i
934934 0 0
935935 −51.7771 25.8885i −1.69329 0.846646i
936936 0 0
937937 38.8328i 1.26861i 0.773082 + 0.634306i 0.218714π0.218714\pi
−0.773082 + 0.634306i 0.781286π0.781286\pi
938938 0 0
939939 14.8328 38.8328i 0.484051 1.26726i
940940 0 0
941941 53.4164 1.74133 0.870663 0.491881i 0.163690π-0.163690\pi
0.870663 + 0.491881i 0.163690π0.163690\pi
942942 0 0
943943 −4.94427 −0.161008
944944 0 0
945945 −37.5967 + 0.472136i −1.22302 + 0.0153586i
946946 0 0
947947 50.0689 1.62702 0.813510 0.581551i 0.197554π-0.197554\pi
0.813510 + 0.581551i 0.197554π0.197554\pi
948948 0 0
949949 32.0000i 1.03876i
950950 0 0
951951 −10.4721 + 27.4164i −0.339582 + 0.889038i
952952 0 0
953953 1.52786 0.0494924 0.0247462 0.999694i 0.492122π-0.492122\pi
0.0247462 + 0.999694i 0.492122π0.492122\pi
954954 0 0
955955 4.94427 9.88854i 0.159993 0.319986i
956956 0 0
957957 −11.0557 + 28.9443i −0.357381 + 0.935635i
958958 0 0
959959 −4.94427 −0.159659
960960 0 0
961961 24.8885 0.802856
962962 0 0
963963 21.7082 + 19.4164i 0.699537 + 0.625685i
964964 0 0
965965 41.8885 + 20.9443i 1.34844 + 0.674220i
966966 0 0
967967 −43.5967 −1.40198 −0.700988 0.713173i 0.747258π-0.747258\pi
−0.700988 + 0.713173i 0.747258π0.747258\pi
968968 0 0
969969 25.8885 + 9.88854i 0.831660 + 0.317666i
970970 0 0
971971 28.0000i 0.898563i 0.893390 + 0.449281i 0.148320π0.148320\pi
−0.893390 + 0.449281i 0.851680π0.851680\pi
972972 0 0
973973 75.7771 2.42930
974974 0 0
975975 15.4164 + 53.8885i 0.493720 + 1.72581i
976976 0 0
977977 −12.5836 −0.402585 −0.201292 0.979531i 0.564514π-0.564514\pi
−0.201292 + 0.979531i 0.564514π0.564514\pi
978978 0 0
979979 32.0000 1.02272
980980 0 0
981981 −25.8885 + 28.9443i −0.826558 + 0.924119i
982982 0 0
983983 11.7082i 0.373434i 0.982414 + 0.186717i 0.0597847π0.0597847\pi
−0.982414 + 0.186717i 0.940215π0.940215\pi
984984 0 0
985985 17.8885 + 8.94427i 0.569976 + 0.284988i
986986 0 0
987987 −18.4721 + 48.3607i −0.587975 + 1.53934i
988988 0 0
989989 11.4164 0.363021
990990 0 0
991991 51.1935i 1.62621i −0.582114 0.813107i 0.697774π-0.697774\pi
0.582114 0.813107i 0.302226π-0.302226\pi
992992 0 0
993993 −4.00000 1.52786i −0.126936 0.0484853i
994994 0 0
995995 −27.0557 13.5279i −0.857724 0.428862i
996996 0 0
997997 42.2492 1.33805 0.669023 0.743242i 0.266713π-0.266713\pi
0.669023 + 0.743242i 0.266713π0.266713\pi
998998 0 0
999999 3.63932 + 7.05573i 0.115143 + 0.223233i
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 1920.2.m.d.959.1 yes 4
3.2 odd 2 1920.2.m.t.959.3 yes 4
4.3 odd 2 1920.2.m.p.959.4 yes 4
5.4 even 2 1920.2.m.o.959.4 yes 4
8.3 odd 2 1920.2.m.g.959.1 yes 4
8.5 even 2 1920.2.m.s.959.4 yes 4
12.11 even 2 1920.2.m.h.959.2 yes 4
15.14 odd 2 1920.2.m.g.959.2 yes 4
20.19 odd 2 1920.2.m.c.959.1 4
24.5 odd 2 1920.2.m.c.959.2 yes 4
24.11 even 2 1920.2.m.o.959.3 yes 4
40.19 odd 2 1920.2.m.t.959.4 yes 4
40.29 even 2 1920.2.m.h.959.1 yes 4
60.59 even 2 1920.2.m.s.959.3 yes 4
120.29 odd 2 1920.2.m.p.959.3 yes 4
120.59 even 2 inner 1920.2.m.d.959.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1920.2.m.c.959.1 4 20.19 odd 2
1920.2.m.c.959.2 yes 4 24.5 odd 2
1920.2.m.d.959.1 yes 4 1.1 even 1 trivial
1920.2.m.d.959.2 yes 4 120.59 even 2 inner
1920.2.m.g.959.1 yes 4 8.3 odd 2
1920.2.m.g.959.2 yes 4 15.14 odd 2
1920.2.m.h.959.1 yes 4 40.29 even 2
1920.2.m.h.959.2 yes 4 12.11 even 2
1920.2.m.o.959.3 yes 4 24.11 even 2
1920.2.m.o.959.4 yes 4 5.4 even 2
1920.2.m.p.959.3 yes 4 120.29 odd 2
1920.2.m.p.959.4 yes 4 4.3 odd 2
1920.2.m.s.959.3 yes 4 60.59 even 2
1920.2.m.s.959.4 yes 4 8.5 even 2
1920.2.m.t.959.3 yes 4 3.2 odd 2
1920.2.m.t.959.4 yes 4 40.19 odd 2