Properties

Label 768.2.f.b.383.1
Level 768768
Weight 22
Character 768.383
Analytic conductor 6.1336.133
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 768=283 768 = 2^{8} \cdot 3
Weight: k k == 2 2
Character orbit: [χ][\chi] == 768.f (of order 22, degree 11, not 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: 6.132510875236.13251087523
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,5)\Q(i, \sqrt{5})
comment: defining polynomial
 
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: no (minimal twist has level 384)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 383.1
Root 0.618034i-0.618034i of defining polynomial
Character χ\chi == 768.383
Dual form 768.2.f.b.383.2

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) == q+(1.618030.618034i)q33.23607q5+1.23607iq7+(2.23607+2.00000i)q9+5.23607iq114.47214iq13+(5.23607+2.00000i)q152.47214iq17+0.763932q19+(0.7639322.00000i)q212.47214q23+5.47214q25+(2.381974.61803i)q27+4.76393q295.23607iq31+(3.236078.47214i)q334.00000iq358.47214iq37+(2.76393+7.23607i)q396.47214iq41+7.23607q43+(7.236076.47214i)q458.00000q47+5.47214q49+(1.52786+4.00000i)q513.23607q5316.9443iq55+(1.236070.472136i)q571.23607iq59+0.472136iq61+(2.47214+2.76393i)q63+14.4721iq65+9.70820q67+(4.00000+1.52786i)q69+15.4164q71+2.00000q73+(8.854103.38197i)q756.47214q770.291796iq79+(1.00000+8.94427i)q812.76393iq83+8.00000iq85+(7.708202.94427i)q874.00000iq89+5.52786q91+(3.23607+8.47214i)q932.47214q95+0.472136q97+(10.4721+11.7082i)q99+O(q100)q+(-1.61803 - 0.618034i) q^{3} -3.23607 q^{5} +1.23607i q^{7} +(2.23607 + 2.00000i) q^{9} +5.23607i q^{11} -4.47214i q^{13} +(5.23607 + 2.00000i) q^{15} -2.47214i q^{17} +0.763932 q^{19} +(0.763932 - 2.00000i) q^{21} -2.47214 q^{23} +5.47214 q^{25} +(-2.38197 - 4.61803i) q^{27} +4.76393 q^{29} -5.23607i q^{31} +(3.23607 - 8.47214i) q^{33} -4.00000i q^{35} -8.47214i q^{37} +(-2.76393 + 7.23607i) q^{39} -6.47214i q^{41} +7.23607 q^{43} +(-7.23607 - 6.47214i) q^{45} -8.00000 q^{47} +5.47214 q^{49} +(-1.52786 + 4.00000i) q^{51} -3.23607 q^{53} -16.9443i q^{55} +(-1.23607 - 0.472136i) q^{57} -1.23607i q^{59} +0.472136i q^{61} +(-2.47214 + 2.76393i) q^{63} +14.4721i q^{65} +9.70820 q^{67} +(4.00000 + 1.52786i) q^{69} +15.4164 q^{71} +2.00000 q^{73} +(-8.85410 - 3.38197i) q^{75} -6.47214 q^{77} -0.291796i q^{79} +(1.00000 + 8.94427i) q^{81} -2.76393i q^{83} +8.00000i q^{85} +(-7.70820 - 2.94427i) q^{87} -4.00000i q^{89} +5.52786 q^{91} +(-3.23607 + 8.47214i) q^{93} -2.47214 q^{95} +0.472136 q^{97} +(-10.4721 + 11.7082i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q2q34q5+12q15+12q19+12q21+8q23+4q2514q27+28q29+4q3320q39+20q4320q4532q47+4q4924q514q53+4q57+24q99+O(q100) 4 q - 2 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{19} + 12 q^{21} + 8 q^{23} + 4 q^{25} - 14 q^{27} + 28 q^{29} + 4 q^{33} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 32 q^{47} + 4 q^{49} - 24 q^{51} - 4 q^{53} + 4 q^{57}+ \cdots - 24 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/768Z)×\left(\mathbb{Z}/768\mathbb{Z}\right)^\times.

nn 257257 511511 517517
χ(n)\chi(n) 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 −3.23607 −1.44721 −0.723607 0.690212i 0.757517π-0.757517\pi
−0.723607 + 0.690212i 0.757517π0.757517\pi
66 0 0
77 1.23607i 0.467190i 0.972334 + 0.233595i 0.0750489π0.0750489\pi
−0.972334 + 0.233595i 0.924951π0.924951\pi
88 0 0
99 2.23607 + 2.00000i 0.745356 + 0.666667i
1010 0 0
1111 5.23607i 1.57873i 0.613922 + 0.789367i 0.289591π0.289591\pi
−0.613922 + 0.789367i 0.710409π0.710409\pi
1212 0 0
1313 4.47214i 1.24035i −0.784465 0.620174i 0.787062π-0.787062\pi
0.784465 0.620174i 0.212938π-0.212938\pi
1414 0 0
1515 5.23607 + 2.00000i 1.35195 + 0.516398i
1616 0 0
1717 2.47214i 0.599581i −0.954005 0.299791i 0.903083π-0.903083\pi
0.954005 0.299791i 0.0969168π-0.0969168\pi
1818 0 0
1919 0.763932 0.175258 0.0876290 0.996153i 0.472071π-0.472071\pi
0.0876290 + 0.996153i 0.472071π0.472071\pi
2020 0 0
2121 0.763932 2.00000i 0.166704 0.436436i
2222 0 0
2323 −2.47214 −0.515476 −0.257738 0.966215i 0.582977π-0.582977\pi
−0.257738 + 0.966215i 0.582977π0.582977\pi
2424 0 0
2525 5.47214 1.09443
2626 0 0
2727 −2.38197 4.61803i −0.458410 0.888741i
2828 0 0
2929 4.76393 0.884640 0.442320 0.896857i 0.354156π-0.354156\pi
0.442320 + 0.896857i 0.354156π0.354156\pi
3030 0 0
3131 5.23607i 0.940426i −0.882553 0.470213i 0.844177π-0.844177\pi
0.882553 0.470213i 0.155823π-0.155823\pi
3232 0 0
3333 3.23607 8.47214i 0.563327 1.47481i
3434 0 0
3535 4.00000i 0.676123i
3636 0 0
3737 8.47214i 1.39281i −0.717649 0.696405i 0.754782π-0.754782\pi
0.717649 0.696405i 0.245218π-0.245218\pi
3838 0 0
3939 −2.76393 + 7.23607i −0.442583 + 1.15870i
4040 0 0
4141 6.47214i 1.01078i −0.862892 0.505389i 0.831349π-0.831349\pi
0.862892 0.505389i 0.168651π-0.168651\pi
4242 0 0
4343 7.23607 1.10349 0.551745 0.834013i 0.313962π-0.313962\pi
0.551745 + 0.834013i 0.313962π0.313962\pi
4444 0 0
4545 −7.23607 6.47214i −1.07869 0.964809i
4646 0 0
4747 −8.00000 −1.16692 −0.583460 0.812142i 0.698301π-0.698301\pi
−0.583460 + 0.812142i 0.698301π0.698301\pi
4848 0 0
4949 5.47214 0.781734
5050 0 0
5151 −1.52786 + 4.00000i −0.213944 + 0.560112i
5252 0 0
5353 −3.23607 −0.444508 −0.222254 0.974989i 0.571341π-0.571341\pi
−0.222254 + 0.974989i 0.571341π0.571341\pi
5454 0 0
5555 16.9443i 2.28477i
5656 0 0
5757 −1.23607 0.472136i −0.163721 0.0625359i
5858 0 0
5959 1.23607i 0.160922i −0.996758 0.0804612i 0.974361π-0.974361\pi
0.996758 0.0804612i 0.0256393π-0.0256393\pi
6060 0 0
6161 0.472136i 0.0604508i 0.999543 + 0.0302254i 0.00962251π0.00962251\pi
−0.999543 + 0.0302254i 0.990377π0.990377\pi
6262 0 0
6363 −2.47214 + 2.76393i −0.311460 + 0.348223i
6464 0 0
6565 14.4721i 1.79505i
6666 0 0
6767 9.70820 1.18605 0.593023 0.805186i 0.297934π-0.297934\pi
0.593023 + 0.805186i 0.297934π0.297934\pi
6868 0 0
6969 4.00000 + 1.52786i 0.481543 + 0.183933i
7070 0 0
7171 15.4164 1.82959 0.914796 0.403917i 0.132352π-0.132352\pi
0.914796 + 0.403917i 0.132352π0.132352\pi
7272 0 0
7373 2.00000 0.234082 0.117041 0.993127i 0.462659π-0.462659\pi
0.117041 + 0.993127i 0.462659π0.462659\pi
7474 0 0
7575 −8.85410 3.38197i −1.02238 0.390516i
7676 0 0
7777 −6.47214 −0.737568
7878 0 0
7979 0.291796i 0.0328296i −0.999865 0.0164148i 0.994775π-0.994775\pi
0.999865 0.0164148i 0.00522523π-0.00522523\pi
8080 0 0
8181 1.00000 + 8.94427i 0.111111 + 0.993808i
8282 0 0
8383 2.76393i 0.303381i −0.988428 0.151690i 0.951528π-0.951528\pi
0.988428 0.151690i 0.0484717π-0.0484717\pi
8484 0 0
8585 8.00000i 0.867722i
8686 0 0
8787 −7.70820 2.94427i −0.826406 0.315659i
8888 0 0
8989 4.00000i 0.423999i −0.977270 0.212000i 0.932002π-0.932002\pi
0.977270 0.212000i 0.0679975π-0.0679975\pi
9090 0 0
9191 5.52786 0.579478
9292 0 0
9393 −3.23607 + 8.47214i −0.335565 + 0.878520i
9494 0 0
9595 −2.47214 −0.253636
9696 0 0
9797 0.472136 0.0479381 0.0239691 0.999713i 0.492370π-0.492370\pi
0.0239691 + 0.999713i 0.492370π0.492370\pi
9898 0 0
9999 −10.4721 + 11.7082i −1.05249 + 1.17672i
100100 0 0
101101 1.70820 0.169973 0.0849863 0.996382i 0.472915π-0.472915\pi
0.0849863 + 0.996382i 0.472915π0.472915\pi
102102 0 0
103103 14.1803i 1.39723i 0.715498 + 0.698615i 0.246200π0.246200\pi
−0.715498 + 0.698615i 0.753800π0.753800\pi
104104 0 0
105105 −2.47214 + 6.47214i −0.241256 + 0.631616i
106106 0 0
107107 14.1803i 1.37087i −0.728136 0.685433i 0.759613π-0.759613\pi
0.728136 0.685433i 0.240387π-0.240387\pi
108108 0 0
109109 8.47214i 0.811483i 0.913988 + 0.405742i 0.132987π0.132987\pi
−0.913988 + 0.405742i 0.867013π0.867013\pi
110110 0 0
111111 −5.23607 + 13.7082i −0.496986 + 1.30113i
112112 0 0
113113 8.00000i 0.752577i −0.926503 0.376288i 0.877200π-0.877200\pi
0.926503 0.376288i 0.122800π-0.122800\pi
114114 0 0
115115 8.00000 0.746004
116116 0 0
117117 8.94427 10.0000i 0.826898 0.924500i
118118 0 0
119119 3.05573 0.280118
120120 0 0
121121 −16.4164 −1.49240
122122 0 0
123123 −4.00000 + 10.4721i −0.360668 + 0.944241i
124124 0 0
125125 −1.52786 −0.136656
126126 0 0
127127 13.2361i 1.17451i −0.809402 0.587256i 0.800208π-0.800208\pi
0.809402 0.587256i 0.199792π-0.199792\pi
128128 0 0
129129 −11.7082 4.47214i −1.03085 0.393750i
130130 0 0
131131 6.76393i 0.590967i 0.955348 + 0.295484i 0.0954808π0.0954808\pi
−0.955348 + 0.295484i 0.904519π0.904519\pi
132132 0 0
133133 0.944272i 0.0818788i
134134 0 0
135135 7.70820 + 14.9443i 0.663417 + 1.28620i
136136 0 0
137137 1.52786i 0.130534i −0.997868 0.0652671i 0.979210π-0.979210\pi
0.997868 0.0652671i 0.0207899π-0.0207899\pi
138138 0 0
139139 −9.70820 −0.823439 −0.411720 0.911311i 0.635072π-0.635072\pi
−0.411720 + 0.911311i 0.635072π0.635072\pi
140140 0 0
141141 12.9443 + 4.94427i 1.09010 + 0.416383i
142142 0 0
143143 23.4164 1.95818
144144 0 0
145145 −15.4164 −1.28026
146146 0 0
147147 −8.85410 3.38197i −0.730274 0.278940i
148148 0 0
149149 9.70820 0.795327 0.397664 0.917531i 0.369821π-0.369821\pi
0.397664 + 0.917531i 0.369821π0.369821\pi
150150 0 0
151151 19.7082i 1.60383i −0.597438 0.801915i 0.703814π-0.703814\pi
0.597438 0.801915i 0.296186π-0.296186\pi
152152 0 0
153153 4.94427 5.52786i 0.399721 0.446901i
154154 0 0
155155 16.9443i 1.36100i
156156 0 0
157157 4.47214i 0.356915i −0.983948 0.178458i 0.942889π-0.942889\pi
0.983948 0.178458i 0.0571108π-0.0571108\pi
158158 0 0
159159 5.23607 + 2.00000i 0.415247 + 0.158610i
160160 0 0
161161 3.05573i 0.240825i
162162 0 0
163163 −4.18034 −0.327429 −0.163715 0.986508i 0.552348π-0.552348\pi
−0.163715 + 0.986508i 0.552348π0.552348\pi
164164 0 0
165165 −10.4721 + 27.4164i −0.815255 + 2.13436i
166166 0 0
167167 −13.5279 −1.04682 −0.523409 0.852082i 0.675340π-0.675340\pi
−0.523409 + 0.852082i 0.675340π0.675340\pi
168168 0 0
169169 −7.00000 −0.538462
170170 0 0
171171 1.70820 + 1.52786i 0.130630 + 0.116839i
172172 0 0
173173 22.6525 1.72224 0.861118 0.508405i 0.169765π-0.169765\pi
0.861118 + 0.508405i 0.169765π0.169765\pi
174174 0 0
175175 6.76393i 0.511305i
176176 0 0
177177 −0.763932 + 2.00000i −0.0574206 + 0.150329i
178178 0 0
179179 6.18034i 0.461940i −0.972961 0.230970i 0.925810π-0.925810\pi
0.972961 0.230970i 0.0741900π-0.0741900\pi
180180 0 0
181181 20.4721i 1.52168i 0.648938 + 0.760841i 0.275213π0.275213\pi
−0.648938 + 0.760841i 0.724787π0.724787\pi
182182 0 0
183183 0.291796 0.763932i 0.0215702 0.0564715i
184184 0 0
185185 27.4164i 2.01569i
186186 0 0
187187 12.9443 0.946579
188188 0 0
189189 5.70820 2.94427i 0.415211 0.214164i
190190 0 0
191191 −12.9443 −0.936615 −0.468307 0.883566i 0.655136π-0.655136\pi
−0.468307 + 0.883566i 0.655136π0.655136\pi
192192 0 0
193193 −11.8885 −0.855756 −0.427878 0.903836i 0.640739π-0.640739\pi
−0.427878 + 0.903836i 0.640739π0.640739\pi
194194 0 0
195195 8.94427 23.4164i 0.640513 1.67688i
196196 0 0
197197 4.76393 0.339416 0.169708 0.985494i 0.445718π-0.445718\pi
0.169708 + 0.985494i 0.445718π0.445718\pi
198198 0 0
199199 6.18034i 0.438113i 0.975712 + 0.219056i 0.0702979π0.0702979\pi
−0.975712 + 0.219056i 0.929702π0.929702\pi
200200 0 0
201201 −15.7082 6.00000i −1.10797 0.423207i
202202 0 0
203203 5.88854i 0.413295i
204204 0 0
205205 20.9443i 1.46281i
206206 0 0
207207 −5.52786 4.94427i −0.384213 0.343651i
208208 0 0
209209 4.00000i 0.276686i
210210 0 0
211211 −11.2361 −0.773523 −0.386761 0.922180i 0.626406π-0.626406\pi
−0.386761 + 0.922180i 0.626406π0.626406\pi
212212 0 0
213213 −24.9443 9.52786i −1.70915 0.652838i
214214 0 0
215215 −23.4164 −1.59699
216216 0 0
217217 6.47214 0.439357
218218 0 0
219219 −3.23607 1.23607i −0.218673 0.0835257i
220220 0 0
221221 −11.0557 −0.743689
222222 0 0
223223 10.1803i 0.681726i −0.940113 0.340863i 0.889281π-0.889281\pi
0.940113 0.340863i 0.110719π-0.110719\pi
224224 0 0
225225 12.2361 + 10.9443i 0.815738 + 0.729618i
226226 0 0
227227 2.18034i 0.144714i 0.997379 + 0.0723571i 0.0230521π0.0230521\pi
−0.997379 + 0.0723571i 0.976948π0.976948\pi
228228 0 0
229229 11.5279i 0.761783i −0.924620 0.380891i 0.875617π-0.875617\pi
0.924620 0.380891i 0.124383π-0.124383\pi
230230 0 0
231231 10.4721 + 4.00000i 0.689016 + 0.263181i
232232 0 0
233233 24.9443i 1.63415i 0.576529 + 0.817077i 0.304407π0.304407\pi
−0.576529 + 0.817077i 0.695593π0.695593\pi
234234 0 0
235235 25.8885 1.68878
236236 0 0
237237 −0.180340 + 0.472136i −0.0117143 + 0.0306685i
238238 0 0
239239 8.00000 0.517477 0.258738 0.965947i 0.416693π-0.416693\pi
0.258738 + 0.965947i 0.416693π0.416693\pi
240240 0 0
241241 −2.00000 −0.128831 −0.0644157 0.997923i 0.520518π-0.520518\pi
−0.0644157 + 0.997923i 0.520518π0.520518\pi
242242 0 0
243243 3.90983 15.0902i 0.250816 0.968035i
244244 0 0
245245 −17.7082 −1.13134
246246 0 0
247247 3.41641i 0.217381i
248248 0 0
249249 −1.70820 + 4.47214i −0.108253 + 0.283410i
250250 0 0
251251 23.7082i 1.49645i −0.663446 0.748224i 0.730907π-0.730907\pi
0.663446 0.748224i 0.269093π-0.269093\pi
252252 0 0
253253 12.9443i 0.813799i
254254 0 0
255255 4.94427 12.9443i 0.309622 0.810602i
256256 0 0
257257 20.9443i 1.30647i −0.757156 0.653234i 0.773412π-0.773412\pi
0.757156 0.653234i 0.226588π-0.226588\pi
258258 0 0
259259 10.4721 0.650707
260260 0 0
261261 10.6525 + 9.52786i 0.659372 + 0.589760i
262262 0 0
263263 −10.4721 −0.645740 −0.322870 0.946443i 0.604648π-0.604648\pi
−0.322870 + 0.946443i 0.604648π0.604648\pi
264264 0 0
265265 10.4721 0.643298
266266 0 0
267267 −2.47214 + 6.47214i −0.151292 + 0.396088i
268268 0 0
269269 −16.1803 −0.986533 −0.493266 0.869878i 0.664197π-0.664197\pi
−0.493266 + 0.869878i 0.664197π0.664197\pi
270270 0 0
271271 0.291796i 0.0177253i −0.999961 0.00886267i 0.997179π-0.997179\pi
0.999961 0.00886267i 0.00282111π-0.00282111\pi
272272 0 0
273273 −8.94427 3.41641i −0.541332 0.206770i
274274 0 0
275275 28.6525i 1.72781i
276276 0 0
277277 7.52786i 0.452306i 0.974092 + 0.226153i 0.0726149π0.0726149\pi
−0.974092 + 0.226153i 0.927385π0.927385\pi
278278 0 0
279279 10.4721 11.7082i 0.626950 0.700952i
280280 0 0
281281 24.9443i 1.48805i −0.668151 0.744025i 0.732914π-0.732914\pi
0.668151 0.744025i 0.267086π-0.267086\pi
282282 0 0
283283 −20.7639 −1.23429 −0.617144 0.786850i 0.711710π-0.711710\pi
−0.617144 + 0.786850i 0.711710π0.711710\pi
284284 0 0
285285 4.00000 + 1.52786i 0.236940 + 0.0905029i
286286 0 0
287287 8.00000 0.472225
288288 0 0
289289 10.8885 0.640503
290290 0 0
291291 −0.763932 0.291796i −0.0447825 0.0171054i
292292 0 0
293293 20.7639 1.21304 0.606521 0.795068i 0.292565π-0.292565\pi
0.606521 + 0.795068i 0.292565π0.292565\pi
294294 0 0
295295 4.00000i 0.232889i
296296 0 0
297297 24.1803 12.4721i 1.40309 0.723707i
298298 0 0
299299 11.0557i 0.639369i
300300 0 0
301301 8.94427i 0.515539i
302302 0 0
303303 −2.76393 1.05573i −0.158784 0.0606500i
304304 0 0
305305 1.52786i 0.0874852i
306306 0 0
307307 27.5967 1.57503 0.787515 0.616296i 0.211367π-0.211367\pi
0.787515 + 0.616296i 0.211367π0.211367\pi
308308 0 0
309309 8.76393 22.9443i 0.498563 1.30525i
310310 0 0
311311 −5.52786 −0.313456 −0.156728 0.987642i 0.550095π-0.550095\pi
−0.156728 + 0.987642i 0.550095π0.550095\pi
312312 0 0
313313 0.472136 0.0266867 0.0133434 0.999911i 0.495753π-0.495753\pi
0.0133434 + 0.999911i 0.495753π0.495753\pi
314314 0 0
315315 8.00000 8.94427i 0.450749 0.503953i
316316 0 0
317317 −13.1246 −0.737152 −0.368576 0.929598i 0.620155π-0.620155\pi
−0.368576 + 0.929598i 0.620155π0.620155\pi
318318 0 0
319319 24.9443i 1.39661i
320320 0 0
321321 −8.76393 + 22.9443i −0.489155 + 1.28062i
322322 0 0
323323 1.88854i 0.105081i
324324 0 0
325325 24.4721i 1.35747i
326326 0 0
327327 5.23607 13.7082i 0.289555 0.758065i
328328 0 0
329329 9.88854i 0.545173i
330330 0 0
331331 −32.5410 −1.78862 −0.894308 0.447452i 0.852332π-0.852332\pi
−0.894308 + 0.447452i 0.852332π0.852332\pi
332332 0 0
333333 16.9443 18.9443i 0.928540 1.03814i
334334 0 0
335335 −31.4164 −1.71646
336336 0 0
337337 22.3607 1.21806 0.609032 0.793146i 0.291558π-0.291558\pi
0.609032 + 0.793146i 0.291558π0.291558\pi
338338 0 0
339339 −4.94427 + 12.9443i −0.268536 + 0.703036i
340340 0 0
341341 27.4164 1.48468
342342 0 0
343343 15.4164i 0.832408i
344344 0 0
345345 −12.9443 4.94427i −0.696896 0.266191i
346346 0 0
347347 23.7082i 1.27272i −0.771391 0.636362i 0.780439π-0.780439\pi
0.771391 0.636362i 0.219561π-0.219561\pi
348348 0 0
349349 28.4721i 1.52408i −0.647531 0.762039i 0.724198π-0.724198\pi
0.647531 0.762039i 0.275802π-0.275802\pi
350350 0 0
351351 −20.6525 + 10.6525i −1.10235 + 0.568587i
352352 0 0
353353 9.88854i 0.526314i −0.964753 0.263157i 0.915236π-0.915236\pi
0.964753 0.263157i 0.0847637π-0.0847637\pi
354354 0 0
355355 −49.8885 −2.64781
356356 0 0
357357 −4.94427 1.88854i −0.261679 0.0999523i
358358 0 0
359359 31.4164 1.65809 0.829047 0.559178i 0.188883π-0.188883\pi
0.829047 + 0.559178i 0.188883π0.188883\pi
360360 0 0
361361 −18.4164 −0.969285
362362 0 0
363363 26.5623 + 10.1459i 1.39416 + 0.532522i
364364 0 0
365365 −6.47214 −0.338767
366366 0 0
367367 5.23607i 0.273321i −0.990618 0.136660i 0.956363π-0.956363\pi
0.990618 0.136660i 0.0436369π-0.0436369\pi
368368 0 0
369369 12.9443 14.4721i 0.673852 0.753389i
370370 0 0
371371 4.00000i 0.207670i
372372 0 0
373373 2.58359i 0.133773i 0.997761 + 0.0668867i 0.0213066π0.0213066\pi
−0.997761 + 0.0668867i 0.978693π0.978693\pi
374374 0 0
375375 2.47214 + 0.944272i 0.127661 + 0.0487620i
376376 0 0
377377 21.3050i 1.09726i
378378 0 0
379379 −10.6525 −0.547181 −0.273590 0.961846i 0.588211π-0.588211\pi
−0.273590 + 0.961846i 0.588211π0.588211\pi
380380 0 0
381381 −8.18034 + 21.4164i −0.419092 + 1.09720i
382382 0 0
383383 28.9443 1.47898 0.739492 0.673166i 0.235066π-0.235066\pi
0.739492 + 0.673166i 0.235066π0.235066\pi
384384 0 0
385385 20.9443 1.06742
386386 0 0
387387 16.1803 + 14.4721i 0.822493 + 0.735660i
388388 0 0
389389 −32.1803 −1.63161 −0.815804 0.578328i 0.803705π-0.803705\pi
−0.815804 + 0.578328i 0.803705π0.803705\pi
390390 0 0
391391 6.11146i 0.309070i
392392 0 0
393393 4.18034 10.9443i 0.210870 0.552065i
394394 0 0
395395 0.944272i 0.0475115i
396396 0 0
397397 17.4164i 0.874104i −0.899436 0.437052i 0.856022π-0.856022\pi
0.899436 0.437052i 0.143978π-0.143978\pi
398398 0 0
399399 0.583592 1.52786i 0.0292161 0.0764889i
400400 0 0
401401 28.3607i 1.41626i 0.706080 + 0.708132i 0.250462π0.250462\pi
−0.706080 + 0.708132i 0.749538π0.749538\pi
402402 0 0
403403 −23.4164 −1.16645
404404 0 0
405405 −3.23607 28.9443i −0.160802 1.43825i
406406 0 0
407407 44.3607 2.19888
408408 0 0
409409 31.8885 1.57679 0.788394 0.615171i 0.210913π-0.210913\pi
0.788394 + 0.615171i 0.210913π0.210913\pi
410410 0 0
411411 −0.944272 + 2.47214i −0.0465775 + 0.121941i
412412 0 0
413413 1.52786 0.0751813
414414 0 0
415415 8.94427i 0.439057i
416416 0 0
417417 15.7082 + 6.00000i 0.769234 + 0.293821i
418418 0 0
419419 10.1803i 0.497342i 0.968588 + 0.248671i 0.0799938π0.0799938\pi
−0.968588 + 0.248671i 0.920006π0.920006\pi
420420 0 0
421421 21.4164i 1.04377i −0.853015 0.521886i 0.825229π-0.825229\pi
0.853015 0.521886i 0.174771π-0.174771\pi
422422 0 0
423423 −17.8885 16.0000i −0.869771 0.777947i
424424 0 0
425425 13.5279i 0.656198i
426426 0 0
427427 −0.583592 −0.0282420
428428 0 0
429429 −37.8885 14.4721i −1.82928 0.698721i
430430 0 0
431431 8.00000 0.385346 0.192673 0.981263i 0.438284π-0.438284\pi
0.192673 + 0.981263i 0.438284π0.438284\pi
432432 0 0
433433 16.4721 0.791600 0.395800 0.918337i 0.370467π-0.370467\pi
0.395800 + 0.918337i 0.370467π0.370467\pi
434434 0 0
435435 24.9443 + 9.52786i 1.19599 + 0.456826i
436436 0 0
437437 −1.88854 −0.0903413
438438 0 0
439439 22.1803i 1.05861i 0.848432 + 0.529305i 0.177547π0.177547\pi
−0.848432 + 0.529305i 0.822453π0.822453\pi
440440 0 0
441441 12.2361 + 10.9443i 0.582670 + 0.521156i
442442 0 0
443443 16.2918i 0.774047i 0.922070 + 0.387023i 0.126497π0.126497\pi
−0.922070 + 0.387023i 0.873503π0.873503\pi
444444 0 0
445445 12.9443i 0.613617i
446446 0 0
447447 −15.7082 6.00000i −0.742973 0.283790i
448448 0 0
449449 0.583592i 0.0275414i 0.999905 + 0.0137707i 0.00438349π0.00438349\pi
−0.999905 + 0.0137707i 0.995617π0.995617\pi
450450 0 0
451451 33.8885 1.59575
452452 0 0
453453 −12.1803 + 31.8885i −0.572282 + 1.49825i
454454 0 0
455455 −17.8885 −0.838628
456456 0 0
457457 22.0000 1.02912 0.514558 0.857455i 0.327956π-0.327956\pi
0.514558 + 0.857455i 0.327956π0.327956\pi
458458 0 0
459459 −11.4164 + 5.88854i −0.532872 + 0.274854i
460460 0 0
461461 −8.18034 −0.380996 −0.190498 0.981688i 0.561010π-0.561010\pi
−0.190498 + 0.981688i 0.561010π0.561010\pi
462462 0 0
463463 34.7639i 1.61562i 0.589445 + 0.807808i 0.299346π0.299346\pi
−0.589445 + 0.807808i 0.700654π0.700654\pi
464464 0 0
465465 10.4721 27.4164i 0.485634 1.27141i
466466 0 0
467467 10.7639i 0.498095i −0.968491 0.249048i 0.919882π-0.919882\pi
0.968491 0.249048i 0.0801176π-0.0801176\pi
468468 0 0
469469 12.0000i 0.554109i
470470 0 0
471471 −2.76393 + 7.23607i −0.127355 + 0.333420i
472472 0 0
473473 37.8885i 1.74212i
474474 0 0
475475 4.18034 0.191807
476476 0 0
477477 −7.23607 6.47214i −0.331317 0.296339i
478478 0 0
479479 −6.11146 −0.279240 −0.139620 0.990205i 0.544588π-0.544588\pi
−0.139620 + 0.990205i 0.544588π0.544588\pi
480480 0 0
481481 −37.8885 −1.72757
482482 0 0
483483 −1.88854 + 4.94427i −0.0859317 + 0.224972i
484484 0 0
485485 −1.52786 −0.0693767
486486 0 0
487487 32.6525i 1.47962i −0.672813 0.739812i 0.734914π-0.734914\pi
0.672813 0.739812i 0.265086π-0.265086\pi
488488 0 0
489489 6.76393 + 2.58359i 0.305876 + 0.116834i
490490 0 0
491491 32.0689i 1.44725i −0.690194 0.723624i 0.742475π-0.742475\pi
0.690194 0.723624i 0.257525π-0.257525\pi
492492 0 0
493493 11.7771i 0.530413i
494494 0 0
495495 33.8885 37.8885i 1.52318 1.70296i
496496 0 0
497497 19.0557i 0.854766i
498498 0 0
499499 −27.2361 −1.21925 −0.609627 0.792688i 0.708681π-0.708681\pi
−0.609627 + 0.792688i 0.708681π0.708681\pi
500500 0 0
501501 21.8885 + 8.36068i 0.977908 + 0.373528i
502502 0 0
503503 −21.5279 −0.959880 −0.479940 0.877301i 0.659342π-0.659342\pi
−0.479940 + 0.877301i 0.659342π0.659342\pi
504504 0 0
505505 −5.52786 −0.245987
506506 0 0
507507 11.3262 + 4.32624i 0.503016 + 0.192135i
508508 0 0
509509 25.7082 1.13950 0.569748 0.821819i 0.307041π-0.307041\pi
0.569748 + 0.821819i 0.307041π0.307041\pi
510510 0 0
511511 2.47214i 0.109361i
512512 0 0
513513 −1.81966 3.52786i −0.0803400 0.155759i
514514 0 0
515515 45.8885i 2.02209i
516516 0 0
517517 41.8885i 1.84226i
518518 0 0
519519 −36.6525 14.0000i −1.60887 0.614532i
520520 0 0
521521 24.3607i 1.06726i 0.845718 + 0.533630i 0.179173π0.179173\pi
−0.845718 + 0.533630i 0.820827π0.820827\pi
522522 0 0
523523 −11.8197 −0.516838 −0.258419 0.966033i 0.583201π-0.583201\pi
−0.258419 + 0.966033i 0.583201π0.583201\pi
524524 0 0
525525 4.18034 10.9443i 0.182445 0.477647i
526526 0 0
527527 −12.9443 −0.563861
528528 0 0
529529 −16.8885 −0.734285
530530 0 0
531531 2.47214 2.76393i 0.107282 0.119944i
532532 0 0
533533 −28.9443 −1.25372
534534 0 0
535535 45.8885i 1.98393i
536536 0 0
537537 −3.81966 + 10.0000i −0.164831 + 0.431532i
538538 0 0
539539 28.6525i 1.23415i
540540 0 0
541541 26.3607i 1.13333i 0.823947 + 0.566667i 0.191767π0.191767\pi
−0.823947 + 0.566667i 0.808233π0.808233\pi
542542 0 0
543543 12.6525 33.1246i 0.542970 1.42151i
544544 0 0
545545 27.4164i 1.17439i
546546 0 0
547547 −7.23607 −0.309392 −0.154696 0.987962i 0.549440π-0.549440\pi
−0.154696 + 0.987962i 0.549440π0.549440\pi
548548 0 0
549549 −0.944272 + 1.05573i −0.0403005 + 0.0450574i
550550 0 0
551551 3.63932 0.155040
552552 0 0
553553 0.360680 0.0153377
554554 0 0
555555 16.9443 44.3607i 0.719244 1.88301i
556556 0 0
557557 14.6525 0.620845 0.310423 0.950599i 0.399529π-0.399529\pi
0.310423 + 0.950599i 0.399529π0.399529\pi
558558 0 0
559559 32.3607i 1.36871i
560560 0 0
561561 −20.9443 8.00000i −0.884268 0.337760i
562562 0 0
563563 30.5410i 1.28715i −0.765383 0.643575i 0.777450π-0.777450\pi
0.765383 0.643575i 0.222550π-0.222550\pi
564564 0 0
565565 25.8885i 1.08914i
566566 0 0
567567 −11.0557 + 1.23607i −0.464297 + 0.0519100i
568568 0 0
569569 46.4721i 1.94821i −0.226089 0.974107i 0.572594π-0.572594\pi
0.226089 0.974107i 0.427406π-0.427406\pi
570570 0 0
571571 5.12461 0.214458 0.107229 0.994234i 0.465802π-0.465802\pi
0.107229 + 0.994234i 0.465802π0.465802\pi
572572 0 0
573573 20.9443 + 8.00000i 0.874960 + 0.334205i
574574 0 0
575575 −13.5279 −0.564151
576576 0 0
577577 28.4721 1.18531 0.592655 0.805456i 0.298080π-0.298080\pi
0.592655 + 0.805456i 0.298080π0.298080\pi
578578 0 0
579579 19.2361 + 7.34752i 0.799424 + 0.305353i
580580 0 0
581581 3.41641 0.141736
582582 0 0
583583 16.9443i 0.701760i
584584 0 0
585585 −28.9443 + 32.3607i −1.19670 + 1.33795i
586586 0 0
587587 21.5967i 0.891393i 0.895184 + 0.445697i 0.147044π0.147044\pi
−0.895184 + 0.445697i 0.852956π0.852956\pi
588588 0 0
589589 4.00000i 0.164817i
590590 0 0
591591 −7.70820 2.94427i −0.317073 0.121111i
592592 0 0
593593 28.9443i 1.18860i −0.804244 0.594299i 0.797429π-0.797429\pi
0.804244 0.594299i 0.202571π-0.202571\pi
594594 0 0
595595 −9.88854 −0.405391
596596 0 0
597597 3.81966 10.0000i 0.156328 0.409273i
598598 0 0
599599 7.41641 0.303026 0.151513 0.988455i 0.451585π-0.451585\pi
0.151513 + 0.988455i 0.451585π0.451585\pi
600600 0 0
601601 −33.7771 −1.37780 −0.688898 0.724858i 0.741905π-0.741905\pi
−0.688898 + 0.724858i 0.741905π0.741905\pi
602602 0 0
603603 21.7082 + 19.4164i 0.884026 + 0.790697i
604604 0 0
605605 53.1246 2.15982
606606 0 0
607607 34.7639i 1.41102i 0.708698 + 0.705512i 0.249283π0.249283\pi
−0.708698 + 0.705512i 0.750717π0.750717\pi
608608 0 0
609609 3.63932 9.52786i 0.147473 0.386089i
610610 0 0
611611 35.7771i 1.44739i
612612 0 0
613613 19.3050i 0.779720i 0.920874 + 0.389860i 0.127477π0.127477\pi
−0.920874 + 0.389860i 0.872523π0.872523\pi
614614 0 0
615615 12.9443 33.8885i 0.521963 1.36652i
616616 0 0
617617 21.8885i 0.881200i −0.897704 0.440600i 0.854766π-0.854766\pi
0.897704 0.440600i 0.145234π-0.145234\pi
618618 0 0
619619 29.1246 1.17062 0.585308 0.810811i 0.300973π-0.300973\pi
0.585308 + 0.810811i 0.300973π0.300973\pi
620620 0 0
621621 5.88854 + 11.4164i 0.236299 + 0.458125i
622622 0 0
623623 4.94427 0.198088
624624 0 0
625625 −22.4164 −0.896656
626626 0 0
627627 2.47214 6.47214i 0.0987276 0.258472i
628628 0 0
629629 −20.9443 −0.835103
630630 0 0
631631 19.1246i 0.761339i 0.924711 + 0.380669i 0.124306π0.124306\pi
−0.924711 + 0.380669i 0.875694π0.875694\pi
632632 0 0
633633 18.1803 + 6.94427i 0.722604 + 0.276010i
634634 0 0
635635 42.8328i 1.69977i
636636 0 0
637637 24.4721i 0.969621i
638638 0 0
639639 34.4721 + 30.8328i 1.36370 + 1.21973i
640640 0 0
641641 25.3050i 0.999485i 0.866174 + 0.499743i 0.166572π0.166572\pi
−0.866174 + 0.499743i 0.833428π0.833428\pi
642642 0 0
643643 −13.3475 −0.526375 −0.263187 0.964745i 0.584774π-0.584774\pi
−0.263187 + 0.964745i 0.584774π0.584774\pi
644644 0 0
645645 37.8885 + 14.4721i 1.49186 + 0.569840i
646646 0 0
647647 −33.3050 −1.30935 −0.654676 0.755909i 0.727195π-0.727195\pi
−0.654676 + 0.755909i 0.727195π0.727195\pi
648648 0 0
649649 6.47214 0.254054
650650 0 0
651651 −10.4721 4.00000i −0.410435 0.156772i
652652 0 0
653653 −14.2918 −0.559281 −0.279641 0.960105i 0.590215π-0.590215\pi
−0.279641 + 0.960105i 0.590215π0.590215\pi
654654 0 0
655655 21.8885i 0.855256i
656656 0 0
657657 4.47214 + 4.00000i 0.174475 + 0.156055i
658658 0 0
659659 28.2918i 1.10209i −0.834475 0.551046i 0.814229π-0.814229\pi
0.834475 0.551046i 0.185771π-0.185771\pi
660660 0 0
661661 51.3050i 1.99553i 0.0668107 + 0.997766i 0.478718π0.478718\pi
−0.0668107 + 0.997766i 0.521282π0.521282\pi
662662 0 0
663663 17.8885 + 6.83282i 0.694733 + 0.265365i
664664 0 0
665665 3.05573i 0.118496i
666666 0 0
667667 −11.7771 −0.456011
668668 0 0
669669 −6.29180 + 16.4721i −0.243255 + 0.636850i
670670 0 0
671671 −2.47214 −0.0954358
672672 0 0
673673 −25.4164 −0.979731 −0.489865 0.871798i 0.662954π-0.662954\pi
−0.489865 + 0.871798i 0.662954π0.662954\pi
674674 0 0
675675 −13.0344 25.2705i −0.501696 0.972662i
676676 0 0
677677 −14.2918 −0.549278 −0.274639 0.961547i 0.588558π-0.588558\pi
−0.274639 + 0.961547i 0.588558π0.588558\pi
678678 0 0
679679 0.583592i 0.0223962i
680680 0 0
681681 1.34752 3.52786i 0.0516372 0.135188i
682682 0 0
683683 13.2361i 0.506464i 0.967406 + 0.253232i 0.0814936π0.0814936\pi
−0.967406 + 0.253232i 0.918506π0.918506\pi
684684 0 0
685685 4.94427i 0.188911i
686686 0 0
687687 −7.12461 + 18.6525i −0.271821 + 0.711636i
688688 0 0
689689 14.4721i 0.551344i
690690 0 0
691691 29.7082 1.13015 0.565077 0.825038i 0.308847π-0.308847\pi
0.565077 + 0.825038i 0.308847π0.308847\pi
692692 0 0
693693 −14.4721 12.9443i −0.549751 0.491712i
694694 0 0
695695 31.4164 1.19169
696696 0 0
697697 −16.0000 −0.606043
698698 0 0
699699 15.4164 40.3607i 0.583102 1.52658i
700700 0 0
701701 −14.2918 −0.539794 −0.269897 0.962889i 0.586990π-0.586990\pi
−0.269897 + 0.962889i 0.586990π0.586990\pi
702702 0 0
703703 6.47214i 0.244101i
704704 0 0
705705 −41.8885 16.0000i −1.57761 0.602595i
706706 0 0
707707 2.11146i 0.0794095i
708708 0 0
709709 7.52786i 0.282715i 0.989959 + 0.141357i 0.0451467π0.0451467\pi
−0.989959 + 0.141357i 0.954853π0.954853\pi
710710 0 0
711711 0.583592 0.652476i 0.0218864 0.0244698i
712712 0 0
713713 12.9443i 0.484767i
714714 0 0
715715 −75.7771 −2.83390
716716 0 0
717717 −12.9443 4.94427i −0.483413 0.184647i
718718 0 0
719719 −30.8328 −1.14987 −0.574935 0.818199i 0.694973π-0.694973\pi
−0.574935 + 0.818199i 0.694973π0.694973\pi
720720 0 0
721721 −17.5279 −0.652772
722722 0 0
723723 3.23607 + 1.23607i 0.120351 + 0.0459699i
724724 0 0
725725 26.0689 0.968174
726726 0 0
727727 14.1803i 0.525920i 0.964807 + 0.262960i 0.0846987π0.0846987\pi
−0.964807 + 0.262960i 0.915301π0.915301\pi
728728 0 0
729729 −15.6525 + 22.0000i −0.579721 + 0.814815i
730730 0 0
731731 17.8885i 0.661632i
732732 0 0
733733 29.4164i 1.08652i 0.839565 + 0.543260i 0.182810π0.182810\pi
−0.839565 + 0.543260i 0.817190π0.817190\pi
734734 0 0
735735 28.6525 + 10.9443i 1.05686 + 0.403686i
736736 0 0
737737 50.8328i 1.87245i
738738 0 0
739739 −13.1246 −0.482797 −0.241398 0.970426i 0.577606π-0.577606\pi
−0.241398 + 0.970426i 0.577606π0.577606\pi
740740 0 0
741741 −2.11146 + 5.52786i −0.0775663 + 0.203071i
742742 0 0
743743 −7.41641 −0.272082 −0.136041 0.990703i 0.543438π-0.543438\pi
−0.136041 + 0.990703i 0.543438π0.543438\pi
744744 0 0
745745 −31.4164 −1.15101
746746 0 0
747747 5.52786 6.18034i 0.202254 0.226127i
748748 0 0
749749 17.5279 0.640454
750750 0 0
751751 17.5967i 0.642114i 0.947060 + 0.321057i 0.104038π0.104038\pi
−0.947060 + 0.321057i 0.895962π0.895962\pi
752752 0 0
753753 −14.6525 + 38.3607i −0.533966 + 1.39794i
754754 0 0
755755 63.7771i 2.32109i
756756 0 0
757757 5.41641i 0.196863i −0.995144 0.0984313i 0.968618π-0.968618\pi
0.995144 0.0984313i 0.0313825π-0.0313825\pi
758758 0 0
759759 −8.00000 + 20.9443i −0.290382 + 0.760229i
760760 0 0
761761 42.2492i 1.53153i 0.643119 + 0.765767i 0.277640π0.277640\pi
−0.643119 + 0.765767i 0.722360π0.722360\pi
762762 0 0
763763 −10.4721 −0.379117
764764 0 0
765765 −16.0000 + 17.8885i −0.578481 + 0.646762i
766766 0 0
767767 −5.52786 −0.199600
768768 0 0
769769 49.7771 1.79501 0.897504 0.441007i 0.145378π-0.145378\pi
0.897504 + 0.441007i 0.145378π0.145378\pi
770770 0 0
771771 −12.9443 + 33.8885i −0.466177 + 1.22047i
772772 0 0
773773 7.81966 0.281254 0.140627 0.990063i 0.455088π-0.455088\pi
0.140627 + 0.990063i 0.455088π0.455088\pi
774774 0 0
775775 28.6525i 1.02923i
776776 0 0
777777 −16.9443 6.47214i −0.607872 0.232187i
778778 0 0
779779 4.94427i 0.177147i
780780 0 0
781781 80.7214i 2.88844i
782782 0 0
783783 −11.3475 22.0000i −0.405527 0.786216i
784784 0 0
785785 14.4721i 0.516533i
786786 0 0
787787 −28.1803 −1.00452 −0.502260 0.864716i 0.667498π-0.667498\pi
−0.502260 + 0.864716i 0.667498π0.667498\pi
788788 0 0
789789 16.9443 + 6.47214i 0.603232 + 0.230414i
790790 0 0
791791 9.88854 0.351596
792792 0 0
793793 2.11146 0.0749800
794794 0 0
795795 −16.9443 6.47214i −0.600951 0.229543i
796796 0 0
797797 −50.0689 −1.77353 −0.886765 0.462220i 0.847053π-0.847053\pi
−0.886765 + 0.462220i 0.847053π0.847053\pi
798798 0 0
799799 19.7771i 0.699663i
800800 0 0
801801 8.00000 8.94427i 0.282666 0.316030i
802802 0 0
803803 10.4721i 0.369554i
804804 0 0
805805 9.88854i 0.348525i
806806 0 0
807807 26.1803 + 10.0000i 0.921592 + 0.352017i
808808 0 0
809809 3.41641i 0.120115i 0.998195 + 0.0600573i 0.0191283π0.0191283\pi
−0.998195 + 0.0600573i 0.980872π0.980872\pi
810810 0 0
811811 52.1803 1.83230 0.916150 0.400836i 0.131280π-0.131280\pi
0.916150 + 0.400836i 0.131280π0.131280\pi
812812 0 0
813813 −0.180340 + 0.472136i −0.00632480 + 0.0165585i
814814 0 0
815815 13.5279 0.473860
816816 0 0
817817 5.52786 0.193395
818818 0 0
819819 12.3607 + 11.0557i 0.431917 + 0.386318i
820820 0 0
821821 50.4296 1.76000 0.880002 0.474970i 0.157541π-0.157541\pi
0.880002 + 0.474970i 0.157541π0.157541\pi
822822 0 0
823823 51.1246i 1.78209i 0.453913 + 0.891046i 0.350028π0.350028\pi
−0.453913 + 0.891046i 0.649972π0.649972\pi
824824 0 0
825825 17.7082 46.3607i 0.616521 1.61407i
826826 0 0
827827 19.1246i 0.665028i −0.943098 0.332514i 0.892103π-0.892103\pi
0.943098 0.332514i 0.107897π-0.107897\pi
828828 0 0
829829 4.47214i 0.155324i −0.996980 0.0776619i 0.975255π-0.975255\pi
0.996980 0.0776619i 0.0247455π-0.0247455\pi
830830 0 0
831831 4.65248 12.1803i 0.161393 0.422531i
832832 0 0
833833 13.5279i 0.468713i
834834 0 0
835835 43.7771 1.51497
836836 0 0
837837 −24.1803 + 12.4721i −0.835795 + 0.431100i
838838 0 0
839839 31.4164 1.08461 0.542307 0.840180i 0.317551π-0.317551\pi
0.542307 + 0.840180i 0.317551π0.317551\pi
840840 0 0
841841 −6.30495 −0.217412
842842 0 0
843843 −15.4164 + 40.3607i −0.530969 + 1.39010i
844844 0 0
845845 22.6525 0.779269
846846 0 0
847847 20.2918i 0.697234i
848848 0 0
849849 33.5967 + 12.8328i 1.15304 + 0.440421i
850850 0 0
851851 20.9443i 0.717960i
852852 0 0
853853 39.3050i 1.34578i −0.739744 0.672888i 0.765054π-0.765054\pi
0.739744 0.672888i 0.234946π-0.234946\pi
854854 0 0
855855 −5.52786 4.94427i −0.189049 0.169091i
856856 0 0
857857 7.63932i 0.260954i 0.991451 + 0.130477i 0.0416509π0.0416509\pi
−0.991451 + 0.130477i 0.958349π0.958349\pi
858858 0 0
859859 −16.7639 −0.571978 −0.285989 0.958233i 0.592322π-0.592322\pi
−0.285989 + 0.958233i 0.592322π0.592322\pi
860860 0 0
861861 −12.9443 4.94427i −0.441140 0.168500i
862862 0 0
863863 −22.8328 −0.777238 −0.388619 0.921399i 0.627048π-0.627048\pi
−0.388619 + 0.921399i 0.627048π0.627048\pi
864864 0 0
865865 −73.3050 −2.49244
866866 0 0
867867 −17.6180 6.72949i −0.598340 0.228545i
868868 0 0
869869 1.52786 0.0518292
870870 0 0
871871 43.4164i 1.47111i
872872 0 0
873873 1.05573 + 0.944272i 0.0357310 + 0.0319588i
874874 0 0
875875 1.88854i 0.0638444i
876876 0 0
877877 18.5836i 0.627523i −0.949502 0.313762i 0.898411π-0.898411\pi
0.949502 0.313762i 0.101589π-0.101589\pi
878878 0 0
879879 −33.5967 12.8328i −1.13319 0.432840i
880880 0 0
881881 24.0000i 0.808581i −0.914631 0.404290i 0.867519π-0.867519\pi
0.914631 0.404290i 0.132481π-0.132481\pi
882882 0 0
883883 0.763932 0.0257084 0.0128542 0.999917i 0.495908π-0.495908\pi
0.0128542 + 0.999917i 0.495908π0.495908\pi
884884 0 0
885885 2.47214 6.47214i 0.0830999 0.217558i
886886 0 0
887887 20.3607 0.683645 0.341822 0.939765i 0.388956π-0.388956\pi
0.341822 + 0.939765i 0.388956π0.388956\pi
888888 0 0
889889 16.3607 0.548720
890890 0 0
891891 −46.8328 + 5.23607i −1.56896 + 0.175415i
892892 0 0
893893 −6.11146 −0.204512
894894 0 0
895895 20.0000i 0.668526i
896896 0 0
897897 6.83282 17.8885i 0.228141 0.597281i
898898 0 0
899899 24.9443i 0.831938i
900900 0 0
901901 8.00000i 0.266519i
902902 0 0
903903 5.52786 14.4721i 0.183956 0.481603i
904904 0 0
905905 66.2492i 2.20220i
906906 0 0
907907 39.2361 1.30281 0.651406 0.758729i 0.274179π-0.274179\pi
0.651406 + 0.758729i 0.274179π0.274179\pi
908908 0 0
909909 3.81966 + 3.41641i 0.126690 + 0.113315i
910910 0 0
911911 33.8885 1.12278 0.561389 0.827552i 0.310267π-0.310267\pi
0.561389 + 0.827552i 0.310267π0.310267\pi
912912 0 0
913913 14.4721 0.478958
914914 0 0
915915 −0.944272 + 2.47214i −0.0312167 + 0.0817263i
916916 0 0
917917 −8.36068 −0.276094
918918 0 0
919919 58.5410i 1.93109i −0.260236 0.965545i 0.583800π-0.583800\pi
0.260236 0.965545i 0.416200π-0.416200\pi
920920 0 0
921921 −44.6525 17.0557i −1.47135 0.562005i
922922 0 0
923923 68.9443i 2.26933i
924924 0 0
925925 46.3607i 1.52433i
926926 0 0
927927 −28.3607 + 31.7082i −0.931487 + 1.04143i
928928 0 0
929929 47.4164i 1.55568i 0.628461 + 0.777841i 0.283685π0.283685\pi
−0.628461 + 0.777841i 0.716315π0.716315\pi
930930 0 0
931931 4.18034 0.137005
932932 0 0
933933 8.94427 + 3.41641i 0.292822 + 0.111848i
934934 0 0
935935 −41.8885 −1.36990
936936 0 0
937937 14.3607 0.469143 0.234571 0.972099i 0.424631π-0.424631\pi
0.234571 + 0.972099i 0.424631π0.424631\pi
938938 0 0
939939 −0.763932 0.291796i −0.0249300 0.00952240i
940940 0 0
941941 1.70820 0.0556859 0.0278429 0.999612i 0.491136π-0.491136\pi
0.0278429 + 0.999612i 0.491136π0.491136\pi
942942 0 0
943943 16.0000i 0.521032i
944944 0 0
945945 −18.4721 + 9.52786i −0.600899 + 0.309941i
946946 0 0
947947 24.0689i 0.782134i −0.920362 0.391067i 0.872106π-0.872106\pi
0.920362 0.391067i 0.127894π-0.127894\pi
948948 0 0
949949 8.94427i 0.290343i
950950 0 0
951951 21.2361 + 8.11146i 0.688627 + 0.263032i
952952 0 0
953953 24.3607i 0.789120i −0.918870 0.394560i 0.870897π-0.870897\pi
0.918870 0.394560i 0.129103π-0.129103\pi
954954 0 0
955955 41.8885 1.35548
956956 0 0
957957 15.4164 40.3607i 0.498342 1.30468i
958958 0 0
959959 1.88854 0.0609843
960960 0 0
961961 3.58359 0.115600
962962 0 0
963963 28.3607 31.7082i 0.913910 1.02178i
964964 0 0
965965 38.4721 1.23846
966966 0 0
967967 9.23607i 0.297012i 0.988912 + 0.148506i 0.0474464π0.0474464\pi
−0.988912 + 0.148506i 0.952554π0.952554\pi
968968 0 0
969969 −1.16718 + 3.05573i −0.0374954 + 0.0981641i
970970 0 0
971971 17.0132i 0.545978i 0.962017 + 0.272989i 0.0880123π0.0880123\pi
−0.962017 + 0.272989i 0.911988π0.911988\pi
972972 0 0
973973 12.0000i 0.384702i
974974 0 0
975975 −15.1246 + 39.5967i −0.484375 + 1.26811i
976976 0 0
977977 60.3607i 1.93111i −0.260200 0.965555i 0.583789π-0.583789\pi
0.260200 0.965555i 0.416211π-0.416211\pi
978978 0 0
979979 20.9443 0.669382
980980 0 0
981981 −16.9443 + 18.9443i −0.540989 + 0.604844i
982982 0 0
983983 −21.5279 −0.686632 −0.343316 0.939220i 0.611550π-0.611550\pi
−0.343316 + 0.939220i 0.611550π0.611550\pi
984984 0 0
985985 −15.4164 −0.491208
986986 0 0
987987 −6.11146 + 16.0000i −0.194530 + 0.509286i
988988 0 0
989989 −17.8885 −0.568823
990990 0 0
991991 50.1803i 1.59403i −0.603959 0.797016i 0.706411π-0.706411\pi
0.603959 0.797016i 0.293589π-0.293589\pi
992992 0 0
993993 52.6525 + 20.1115i 1.67088 + 0.638218i
994994 0 0
995995 20.0000i 0.634043i
996996 0 0
997997 18.3607i 0.581489i −0.956801 0.290744i 0.906097π-0.906097\pi
0.956801 0.290744i 0.0939029π-0.0939029\pi
998998 0 0
999999 −39.1246 + 20.1803i −1.23785 + 0.638478i
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 768.2.f.b.383.1 4
3.2 odd 2 768.2.f.c.383.2 4
4.3 odd 2 768.2.f.e.383.4 4
8.3 odd 2 768.2.f.c.383.1 4
8.5 even 2 768.2.f.f.383.4 4
12.11 even 2 768.2.f.f.383.3 4
16.3 odd 4 384.2.c.c.383.2 yes 4
16.5 even 4 384.2.c.d.383.2 yes 4
16.11 odd 4 384.2.c.a.383.3 4
16.13 even 4 384.2.c.b.383.3 yes 4
24.5 odd 2 768.2.f.e.383.3 4
24.11 even 2 inner 768.2.f.b.383.2 4
48.5 odd 4 384.2.c.a.383.4 yes 4
48.11 even 4 384.2.c.d.383.1 yes 4
48.29 odd 4 384.2.c.c.383.1 yes 4
48.35 even 4 384.2.c.b.383.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.c.a.383.3 4 16.11 odd 4
384.2.c.a.383.4 yes 4 48.5 odd 4
384.2.c.b.383.3 yes 4 16.13 even 4
384.2.c.b.383.4 yes 4 48.35 even 4
384.2.c.c.383.1 yes 4 48.29 odd 4
384.2.c.c.383.2 yes 4 16.3 odd 4
384.2.c.d.383.1 yes 4 48.11 even 4
384.2.c.d.383.2 yes 4 16.5 even 4
768.2.f.b.383.1 4 1.1 even 1 trivial
768.2.f.b.383.2 4 24.11 even 2 inner
768.2.f.c.383.1 4 8.3 odd 2
768.2.f.c.383.2 4 3.2 odd 2
768.2.f.e.383.3 4 24.5 odd 2
768.2.f.e.383.4 4 4.3 odd 2
768.2.f.f.383.3 4 12.11 even 2
768.2.f.f.383.4 4 8.5 even 2