Properties

Label 120.2.r.a.113.1
Level 120120
Weight 22
Character 120.113
Analytic conductor 0.9580.958
Analytic rank 00
Dimension 44
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,2,Mod(17,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 120=2335 120 = 2^{3} \cdot 3 \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 120.r (of order 44, degree 22, 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: 0.9582048242550.958204824255
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

Embedding invariants

Embedding label 113.1
Root 0.7071070.707107i-0.707107 - 0.707107i of defining polynomial
Character χ\chi == 120.113
Dual form 120.2.r.a.17.1

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+(1.707110.292893i)q3+(2.12132+0.707107i)q5+(3.000003.00000i)q7+(2.82843+1.00000i)q9+1.41421iq11+(3.414211.82843i)q15+(4.242644.24264i)q17+4.00000iq19+(6.00000+4.24264i)q21+(2.82843+2.82843i)q23+(4.00000+3.00000i)q25+(4.535532.53553i)q271.41421q292.00000q31+(0.4142142.41421i)q33+(8.485284.24264i)q35+(2.00000+2.00000i)q375.65685iq41+(2.000002.00000i)q43+(5.29289+4.12132i)q45+(5.65685+5.65685i)q4711.0000iq49+(6.00000+8.48528i)q51+(8.48528+8.48528i)q53+(1.00000+3.00000i)q55+(1.171576.82843i)q571.41421q596.00000q61+(11.48535.48528i)q63+(4.000004.00000i)q67+(5.656854.00000i)q692.82843iq71+(3.00000+3.00000i)q73+(5.949756.29289i)q75+(4.24264+4.24264i)q77+10.0000iq79+(7.00000+5.65685i)q81+(2.828432.82843i)q83+(6.0000012.0000i)q85+(2.41421+0.414214i)q87+2.82843q89+(3.41421+0.585786i)q93+(2.82843+8.48528i)q95+(13.0000+13.0000i)q97+(1.41421+4.00000i)q99+O(q100)q+(-1.70711 - 0.292893i) q^{3} +(2.12132 + 0.707107i) q^{5} +(3.00000 - 3.00000i) q^{7} +(2.82843 + 1.00000i) q^{9} +1.41421i q^{11} +(-3.41421 - 1.82843i) q^{15} +(-4.24264 - 4.24264i) q^{17} +4.00000i q^{19} +(-6.00000 + 4.24264i) q^{21} +(-2.82843 + 2.82843i) q^{23} +(4.00000 + 3.00000i) q^{25} +(-4.53553 - 2.53553i) q^{27} -1.41421 q^{29} -2.00000 q^{31} +(0.414214 - 2.41421i) q^{33} +(8.48528 - 4.24264i) q^{35} +(-2.00000 + 2.00000i) q^{37} -5.65685i q^{41} +(-2.00000 - 2.00000i) q^{43} +(5.29289 + 4.12132i) q^{45} +(5.65685 + 5.65685i) q^{47} -11.0000i q^{49} +(6.00000 + 8.48528i) q^{51} +(-8.48528 + 8.48528i) q^{53} +(-1.00000 + 3.00000i) q^{55} +(1.17157 - 6.82843i) q^{57} -1.41421 q^{59} -6.00000 q^{61} +(11.4853 - 5.48528i) q^{63} +(4.00000 - 4.00000i) q^{67} +(5.65685 - 4.00000i) q^{69} -2.82843i q^{71} +(3.00000 + 3.00000i) q^{73} +(-5.94975 - 6.29289i) q^{75} +(4.24264 + 4.24264i) q^{77} +10.0000i q^{79} +(7.00000 + 5.65685i) q^{81} +(2.82843 - 2.82843i) q^{83} +(-6.00000 - 12.0000i) q^{85} +(2.41421 + 0.414214i) q^{87} +2.82843 q^{89} +(3.41421 + 0.585786i) q^{93} +(-2.82843 + 8.48528i) q^{95} +(-13.0000 + 13.0000i) q^{97} +(-1.41421 + 4.00000i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q3+12q78q1524q21+16q254q278q314q338q378q43+24q45+24q514q55+16q5724q61+12q63+16q67+12q73+52q97+O(q100) 4 q - 4 q^{3} + 12 q^{7} - 8 q^{15} - 24 q^{21} + 16 q^{25} - 4 q^{27} - 8 q^{31} - 4 q^{33} - 8 q^{37} - 8 q^{43} + 24 q^{45} + 24 q^{51} - 4 q^{55} + 16 q^{57} - 24 q^{61} + 12 q^{63} + 16 q^{67} + 12 q^{73}+ \cdots - 52 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/120Z)×\left(\mathbb{Z}/120\mathbb{Z}\right)^\times.

nn 3131 4141 6161 9797
χ(n)\chi(n) 11 1-1 11 e(34)e\left(\frac{3}{4}\right)

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 −1.70711 0.292893i −0.985599 0.169102i
44 0 0
55 2.12132 + 0.707107i 0.948683 + 0.316228i
66 0 0
77 3.00000 3.00000i 1.13389 1.13389i 0.144370 0.989524i 0.453885π-0.453885\pi
0.989524 0.144370i 0.0461154π-0.0461154\pi
88 0 0
99 2.82843 + 1.00000i 0.942809 + 0.333333i
1010 0 0
1111 1.41421i 0.426401i 0.977008 + 0.213201i 0.0683888π0.0683888\pi
−0.977008 + 0.213201i 0.931611π0.931611\pi
1212 0 0
1313 0 0 0.707107 0.707107i 0.250000π-0.250000\pi
−0.707107 + 0.707107i 0.750000π0.750000\pi
1414 0 0
1515 −3.41421 1.82843i −0.881546 0.472098i
1616 0 0
1717 −4.24264 4.24264i −1.02899 1.02899i −0.999567 0.0294245i 0.990633π-0.990633\pi
−0.0294245 0.999567i 0.509367π-0.509367\pi
1818 0 0
1919 4.00000i 0.917663i 0.888523 + 0.458831i 0.151732π0.151732\pi
−0.888523 + 0.458831i 0.848268π0.848268\pi
2020 0 0
2121 −6.00000 + 4.24264i −1.30931 + 0.925820i
2222 0 0
2323 −2.82843 + 2.82843i −0.589768 + 0.589768i −0.937568 0.347801i 0.886929π-0.886929\pi
0.347801 + 0.937568i 0.386929π0.386929\pi
2424 0 0
2525 4.00000 + 3.00000i 0.800000 + 0.600000i
2626 0 0
2727 −4.53553 2.53553i −0.872864 0.487964i
2828 0 0
2929 −1.41421 −0.262613 −0.131306 0.991342i 0.541917π-0.541917\pi
−0.131306 + 0.991342i 0.541917π0.541917\pi
3030 0 0
3131 −2.00000 −0.359211 −0.179605 0.983739i 0.557482π-0.557482\pi
−0.179605 + 0.983739i 0.557482π0.557482\pi
3232 0 0
3333 0.414214 2.41421i 0.0721053 0.420261i
3434 0 0
3535 8.48528 4.24264i 1.43427 0.717137i
3636 0 0
3737 −2.00000 + 2.00000i −0.328798 + 0.328798i −0.852129 0.523331i 0.824689π-0.824689\pi
0.523331 + 0.852129i 0.324689π0.324689\pi
3838 0 0
3939 0 0
4040 0 0
4141 5.65685i 0.883452i −0.897150 0.441726i 0.854366π-0.854366\pi
0.897150 0.441726i 0.145634π-0.145634\pi
4242 0 0
4343 −2.00000 2.00000i −0.304997 0.304997i 0.537968 0.842965i 0.319192π-0.319192\pi
−0.842965 + 0.537968i 0.819192π0.819192\pi
4444 0 0
4545 5.29289 + 4.12132i 0.789018 + 0.614370i
4646 0 0
4747 5.65685 + 5.65685i 0.825137 + 0.825137i 0.986840 0.161703i 0.0516985π-0.0516985\pi
−0.161703 + 0.986840i 0.551699π0.551699\pi
4848 0 0
4949 11.0000i 1.57143i
5050 0 0
5151 6.00000 + 8.48528i 0.840168 + 1.18818i
5252 0 0
5353 −8.48528 + 8.48528i −1.16554 + 1.16554i −0.182300 + 0.983243i 0.558354π0.558354\pi
−0.983243 + 0.182300i 0.941646π0.941646\pi
5454 0 0
5555 −1.00000 + 3.00000i −0.134840 + 0.404520i
5656 0 0
5757 1.17157 6.82843i 0.155179 0.904447i
5858 0 0
5959 −1.41421 −0.184115 −0.0920575 0.995754i 0.529344π-0.529344\pi
−0.0920575 + 0.995754i 0.529344π0.529344\pi
6060 0 0
6161 −6.00000 −0.768221 −0.384111 0.923287i 0.625492π-0.625492\pi
−0.384111 + 0.923287i 0.625492π0.625492\pi
6262 0 0
6363 11.4853 5.48528i 1.44701 0.691080i
6464 0 0
6565 0 0
6666 0 0
6767 4.00000 4.00000i 0.488678 0.488678i −0.419211 0.907889i 0.637693π-0.637693\pi
0.907889 + 0.419211i 0.137693π0.137693\pi
6868 0 0
6969 5.65685 4.00000i 0.681005 0.481543i
7070 0 0
7171 2.82843i 0.335673i −0.985815 0.167836i 0.946322π-0.946322\pi
0.985815 0.167836i 0.0536780π-0.0536780\pi
7272 0 0
7373 3.00000 + 3.00000i 0.351123 + 0.351123i 0.860527 0.509404i 0.170134π-0.170134\pi
−0.509404 + 0.860527i 0.670134π0.670134\pi
7474 0 0
7575 −5.94975 6.29289i −0.687018 0.726641i
7676 0 0
7777 4.24264 + 4.24264i 0.483494 + 0.483494i
7878 0 0
7979 10.0000i 1.12509i 0.826767 + 0.562544i 0.190177π0.190177\pi
−0.826767 + 0.562544i 0.809823π0.809823\pi
8080 0 0
8181 7.00000 + 5.65685i 0.777778 + 0.628539i
8282 0 0
8383 2.82843 2.82843i 0.310460 0.310460i −0.534628 0.845088i 0.679548π-0.679548\pi
0.845088 + 0.534628i 0.179548π0.179548\pi
8484 0 0
8585 −6.00000 12.0000i −0.650791 1.30158i
8686 0 0
8787 2.41421 + 0.414214i 0.258831 + 0.0444084i
8888 0 0
8989 2.82843 0.299813 0.149906 0.988700i 0.452103π-0.452103\pi
0.149906 + 0.988700i 0.452103π0.452103\pi
9090 0 0
9191 0 0
9292 0 0
9393 3.41421 + 0.585786i 0.354037 + 0.0607432i
9494 0 0
9595 −2.82843 + 8.48528i −0.290191 + 0.870572i
9696 0 0
9797 −13.0000 + 13.0000i −1.31995 + 1.31995i −0.406138 + 0.913812i 0.633125π0.633125\pi
−0.913812 + 0.406138i 0.866875π0.866875\pi
9898 0 0
9999 −1.41421 + 4.00000i −0.142134 + 0.402015i
100100 0 0
101101 15.5563i 1.54791i −0.633238 0.773957i 0.718274π-0.718274\pi
0.633238 0.773957i 0.281726π-0.281726\pi
102102 0 0
103103 −11.0000 11.0000i −1.08386 1.08386i −0.996145 0.0877167i 0.972043π-0.972043\pi
−0.0877167 0.996145i 0.527957π-0.527957\pi
104104 0 0
105105 −15.7279 + 4.75736i −1.53489 + 0.464271i
106106 0 0
107107 −2.82843 2.82843i −0.273434 0.273434i 0.557047 0.830481i 0.311934π-0.311934\pi
−0.830481 + 0.557047i 0.811934π0.811934\pi
108108 0 0
109109 10.0000i 0.957826i 0.877862 + 0.478913i 0.158969π0.158969\pi
−0.877862 + 0.478913i 0.841031π0.841031\pi
110110 0 0
111111 4.00000 2.82843i 0.379663 0.268462i
112112 0 0
113113 4.24264 4.24264i 0.399114 0.399114i −0.478806 0.877920i 0.658930π-0.658930\pi
0.877920 + 0.478806i 0.158930π0.158930\pi
114114 0 0
115115 −8.00000 + 4.00000i −0.746004 + 0.373002i
116116 0 0
117117 0 0
118118 0 0
119119 −25.4558 −2.33353
120120 0 0
121121 9.00000 0.818182
122122 0 0
123123 −1.65685 + 9.65685i −0.149394 + 0.870729i
124124 0 0
125125 6.36396 + 9.19239i 0.569210 + 0.822192i
126126 0 0
127127 −3.00000 + 3.00000i −0.266207 + 0.266207i −0.827570 0.561363i 0.810277π-0.810277\pi
0.561363 + 0.827570i 0.310277π0.310277\pi
128128 0 0
129129 2.82843 + 4.00000i 0.249029 + 0.352180i
130130 0 0
131131 15.5563i 1.35916i −0.733599 0.679582i 0.762161π-0.762161\pi
0.733599 0.679582i 0.237839π-0.237839\pi
132132 0 0
133133 12.0000 + 12.0000i 1.04053 + 1.04053i
134134 0 0
135135 −7.82843 8.58579i −0.673764 0.738947i
136136 0 0
137137 12.7279 + 12.7279i 1.08742 + 1.08742i 0.995793 + 0.0916263i 0.0292065π0.0292065\pi
0.0916263 + 0.995793i 0.470793π0.470793\pi
138138 0 0
139139 16.0000i 1.35710i −0.734553 0.678551i 0.762608π-0.762608\pi
0.734553 0.678551i 0.237392π-0.237392\pi
140140 0 0
141141 −8.00000 11.3137i −0.673722 0.952786i
142142 0 0
143143 0 0
144144 0 0
145145 −3.00000 1.00000i −0.249136 0.0830455i
146146 0 0
147147 −3.22183 + 18.7782i −0.265732 + 1.54880i
148148 0 0
149149 −4.24264 −0.347571 −0.173785 0.984784i 0.555600π-0.555600\pi
−0.173785 + 0.984784i 0.555600π0.555600\pi
150150 0 0
151151 8.00000 0.651031 0.325515 0.945537i 0.394462π-0.394462\pi
0.325515 + 0.945537i 0.394462π0.394462\pi
152152 0 0
153153 −7.75736 16.2426i −0.627145 1.31314i
154154 0 0
155155 −4.24264 1.41421i −0.340777 0.113592i
156156 0 0
157157 8.00000 8.00000i 0.638470 0.638470i −0.311708 0.950178i 0.600901π-0.600901\pi
0.950178 + 0.311708i 0.100901π0.100901\pi
158158 0 0
159159 16.9706 12.0000i 1.34585 0.951662i
160160 0 0
161161 16.9706i 1.33747i
162162 0 0
163163 12.0000 + 12.0000i 0.939913 + 0.939913i 0.998294 0.0583818i 0.0185941π-0.0185941\pi
−0.0583818 + 0.998294i 0.518594π0.518594\pi
164164 0 0
165165 2.58579 4.82843i 0.201303 0.375893i
166166 0 0
167167 0 0 0.707107 0.707107i 0.250000π-0.250000\pi
−0.707107 + 0.707107i 0.750000π0.750000\pi
168168 0 0
169169 13.0000i 1.00000i
170170 0 0
171171 −4.00000 + 11.3137i −0.305888 + 0.865181i
172172 0 0
173173 9.89949 9.89949i 0.752645 0.752645i −0.222327 0.974972i 0.571365π-0.571365\pi
0.974972 + 0.222327i 0.0713654π0.0713654\pi
174174 0 0
175175 21.0000 3.00000i 1.58745 0.226779i
176176 0 0
177177 2.41421 + 0.414214i 0.181463 + 0.0311342i
178178 0 0
179179 15.5563 1.16274 0.581368 0.813641i 0.302518π-0.302518\pi
0.581368 + 0.813641i 0.302518π0.302518\pi
180180 0 0
181181 10.0000 0.743294 0.371647 0.928374i 0.378793π-0.378793\pi
0.371647 + 0.928374i 0.378793π0.378793\pi
182182 0 0
183183 10.2426 + 1.75736i 0.757158 + 0.129908i
184184 0 0
185185 −5.65685 + 2.82843i −0.415900 + 0.207950i
186186 0 0
187187 6.00000 6.00000i 0.438763 0.438763i
188188 0 0
189189 −21.2132 + 6.00000i −1.54303 + 0.436436i
190190 0 0
191191 2.82843i 0.204658i 0.994751 + 0.102329i 0.0326294π0.0326294\pi
−0.994751 + 0.102329i 0.967371π0.967371\pi
192192 0 0
193193 −7.00000 7.00000i −0.503871 0.503871i 0.408768 0.912639i 0.365959π-0.365959\pi
−0.912639 + 0.408768i 0.865959π0.865959\pi
194194 0 0
195195 0 0
196196 0 0
197197 0 0 0.707107 0.707107i 0.250000π-0.250000\pi
−0.707107 + 0.707107i 0.750000π0.750000\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 0 0
201201 −8.00000 + 5.65685i −0.564276 + 0.399004i
202202 0 0
203203 −4.24264 + 4.24264i −0.297775 + 0.297775i
204204 0 0
205205 4.00000 12.0000i 0.279372 0.838116i
206206 0 0
207207 −10.8284 + 5.17157i −0.752628 + 0.359449i
208208 0 0
209209 −5.65685 −0.391293
210210 0 0
211211 −16.0000 −1.10149 −0.550743 0.834675i 0.685655π-0.685655\pi
−0.550743 + 0.834675i 0.685655π0.685655\pi
212212 0 0
213213 −0.828427 + 4.82843i −0.0567629 + 0.330838i
214214 0 0
215215 −2.82843 5.65685i −0.192897 0.385794i
216216 0 0
217217 −6.00000 + 6.00000i −0.407307 + 0.407307i
218218 0 0
219219 −4.24264 6.00000i −0.286691 0.405442i
220220 0 0
221221 0 0
222222 0 0
223223 3.00000 + 3.00000i 0.200895 + 0.200895i 0.800383 0.599489i 0.204629π-0.204629\pi
−0.599489 + 0.800383i 0.704629π0.704629\pi
224224 0 0
225225 8.31371 + 12.4853i 0.554247 + 0.832352i
226226 0 0
227227 4.24264 + 4.24264i 0.281594 + 0.281594i 0.833744 0.552151i 0.186193π-0.186193\pi
−0.552151 + 0.833744i 0.686193π0.686193\pi
228228 0 0
229229 6.00000i 0.396491i 0.980152 + 0.198246i 0.0635244π0.0635244\pi
−0.980152 + 0.198246i 0.936476π0.936476\pi
230230 0 0
231231 −6.00000 8.48528i −0.394771 0.558291i
232232 0 0
233233 −7.07107 + 7.07107i −0.463241 + 0.463241i −0.899716 0.436475i 0.856227π-0.856227\pi
0.436475 + 0.899716i 0.356227π0.356227\pi
234234 0 0
235235 8.00000 + 16.0000i 0.521862 + 1.04372i
236236 0 0
237237 2.92893 17.0711i 0.190255 1.10889i
238238 0 0
239239 25.4558 1.64660 0.823301 0.567605i 0.192130π-0.192130\pi
0.823301 + 0.567605i 0.192130π0.192130\pi
240240 0 0
241241 20.0000 1.28831 0.644157 0.764894i 0.277208π-0.277208\pi
0.644157 + 0.764894i 0.277208π0.277208\pi
242242 0 0
243243 −10.2929 11.7071i −0.660289 0.751011i
244244 0 0
245245 7.77817 23.3345i 0.496929 1.49079i
246246 0 0
247247 0 0
248248 0 0
249249 −5.65685 + 4.00000i −0.358489 + 0.253490i
250250 0 0
251251 12.7279i 0.803379i −0.915776 0.401690i 0.868423π-0.868423\pi
0.915776 0.401690i 0.131577π-0.131577\pi
252252 0 0
253253 −4.00000 4.00000i −0.251478 0.251478i
254254 0 0
255255 6.72792 + 22.2426i 0.421319 + 1.39289i
256256 0 0
257257 −15.5563 15.5563i −0.970378 0.970378i 0.0291953 0.999574i 0.490706π-0.490706\pi
−0.999574 + 0.0291953i 0.990706π0.990706\pi
258258 0 0
259259 12.0000i 0.745644i
260260 0 0
261261 −4.00000 1.41421i −0.247594 0.0875376i
262262 0 0
263263 −11.3137 + 11.3137i −0.697633 + 0.697633i −0.963899 0.266266i 0.914210π-0.914210\pi
0.266266 + 0.963899i 0.414210π0.414210\pi
264264 0 0
265265 −24.0000 + 12.0000i −1.47431 + 0.737154i
266266 0 0
267267 −4.82843 0.828427i −0.295495 0.0506989i
268268 0 0
269269 −9.89949 −0.603583 −0.301791 0.953374i 0.597585π-0.597585\pi
−0.301791 + 0.953374i 0.597585π0.597585\pi
270270 0 0
271271 −24.0000 −1.45790 −0.728948 0.684569i 0.759990π-0.759990\pi
−0.728948 + 0.684569i 0.759990π0.759990\pi
272272 0 0
273273 0 0
274274 0 0
275275 −4.24264 + 5.65685i −0.255841 + 0.341121i
276276 0 0
277277 2.00000 2.00000i 0.120168 0.120168i −0.644465 0.764634i 0.722920π-0.722920\pi
0.764634 + 0.644465i 0.222920π0.222920\pi
278278 0 0
279279 −5.65685 2.00000i −0.338667 0.119737i
280280 0 0
281281 25.4558i 1.51857i 0.650759 + 0.759284i 0.274451π0.274451\pi
−0.650759 + 0.759284i 0.725549π0.725549\pi
282282 0 0
283283 −4.00000 4.00000i −0.237775 0.237775i 0.578153 0.815928i 0.303774π-0.303774\pi
−0.815928 + 0.578153i 0.803774π0.803774\pi
284284 0 0
285285 7.31371 13.6569i 0.433227 0.808962i
286286 0 0
287287 −16.9706 16.9706i −1.00174 1.00174i
288288 0 0
289289 19.0000i 1.11765i
290290 0 0
291291 26.0000 18.3848i 1.52415 1.07773i
292292 0 0
293293 −5.65685 + 5.65685i −0.330477 + 0.330477i −0.852768 0.522291i 0.825078π-0.825078\pi
0.522291 + 0.852768i 0.325078π0.325078\pi
294294 0 0
295295 −3.00000 1.00000i −0.174667 0.0582223i
296296 0 0
297297 3.58579 6.41421i 0.208068 0.372190i
298298 0 0
299299 0 0
300300 0 0
301301 −12.0000 −0.691669
302302 0 0
303303 −4.55635 + 26.5563i −0.261755 + 1.52562i
304304 0 0
305305 −12.7279 4.24264i −0.728799 0.242933i
306306 0 0
307307 10.0000 10.0000i 0.570730 0.570730i −0.361602 0.932332i 0.617770π-0.617770\pi
0.932332 + 0.361602i 0.117770π0.117770\pi
308308 0 0
309309 15.5563 + 22.0000i 0.884970 + 1.25154i
310310 0 0
311311 31.1127i 1.76424i 0.471025 + 0.882120i 0.343884π0.343884\pi
−0.471025 + 0.882120i 0.656116π0.656116\pi
312312 0 0
313313 −15.0000 15.0000i −0.847850 0.847850i 0.142014 0.989865i 0.454642π-0.454642\pi
−0.989865 + 0.142014i 0.954642π0.954642\pi
314314 0 0
315315 28.2426 3.51472i 1.59129 0.198032i
316316 0 0
317317 −1.41421 1.41421i −0.0794301 0.0794301i 0.666276 0.745706i 0.267887π-0.267887\pi
−0.745706 + 0.666276i 0.767887π0.767887\pi
318318 0 0
319319 2.00000i 0.111979i
320320 0 0
321321 4.00000 + 5.65685i 0.223258 + 0.315735i
322322 0 0
323323 16.9706 16.9706i 0.944267 0.944267i
324324 0 0
325325 0 0
326326 0 0
327327 2.92893 17.0711i 0.161970 0.944032i
328328 0 0
329329 33.9411 1.87123
330330 0 0
331331 −16.0000 −0.879440 −0.439720 0.898135i 0.644922π-0.644922\pi
−0.439720 + 0.898135i 0.644922π0.644922\pi
332332 0 0
333333 −7.65685 + 3.65685i −0.419593 + 0.200394i
334334 0 0
335335 11.3137 5.65685i 0.618134 0.309067i
336336 0 0
337337 15.0000 15.0000i 0.817102 0.817102i −0.168585 0.985687i 0.553920π-0.553920\pi
0.985687 + 0.168585i 0.0539198π0.0539198\pi
338338 0 0
339339 −8.48528 + 6.00000i −0.460857 + 0.325875i
340340 0 0
341341 2.82843i 0.153168i
342342 0 0
343343 −12.0000 12.0000i −0.647939 0.647939i
344344 0 0
345345 14.8284 4.48528i 0.798336 0.241479i
346346 0 0
347347 −12.7279 12.7279i −0.683271 0.683271i 0.277465 0.960736i 0.410506π-0.410506\pi
−0.960736 + 0.277465i 0.910506π0.910506\pi
348348 0 0
349349 2.00000i 0.107058i 0.998566 + 0.0535288i 0.0170469π0.0170469\pi
−0.998566 + 0.0535288i 0.982953π0.982953\pi
350350 0 0
351351 0 0
352352 0 0
353353 18.3848 18.3848i 0.978523 0.978523i −0.0212513 0.999774i 0.506765π-0.506765\pi
0.999774 + 0.0212513i 0.00676499π0.00676499\pi
354354 0 0
355355 2.00000 6.00000i 0.106149 0.318447i
356356 0 0
357357 43.4558 + 7.45584i 2.29993 + 0.394605i
358358 0 0
359359 22.6274 1.19423 0.597115 0.802156i 0.296314π-0.296314\pi
0.597115 + 0.802156i 0.296314π0.296314\pi
360360 0 0
361361 3.00000 0.157895
362362 0 0
363363 −15.3640 2.63604i −0.806399 0.138356i
364364 0 0
365365 4.24264 + 8.48528i 0.222070 + 0.444140i
366366 0 0
367367 −7.00000 + 7.00000i −0.365397 + 0.365397i −0.865795 0.500398i 0.833187π-0.833187\pi
0.500398 + 0.865795i 0.333187π0.333187\pi
368368 0 0
369369 5.65685 16.0000i 0.294484 0.832927i
370370 0 0
371371 50.9117i 2.64320i
372372 0 0
373373 12.0000 + 12.0000i 0.621336 + 0.621336i 0.945873 0.324537i 0.105208π-0.105208\pi
−0.324537 + 0.945873i 0.605208π0.605208\pi
374374 0 0
375375 −8.17157 17.5563i −0.421978 0.906606i
376376 0 0
377377 0 0
378378 0 0
379379 24.0000i 1.23280i −0.787434 0.616399i 0.788591π-0.788591\pi
0.787434 0.616399i 0.211409π-0.211409\pi
380380 0 0
381381 6.00000 4.24264i 0.307389 0.217357i
382382 0 0
383383 0 0 −0.707107 0.707107i 0.750000π-0.750000\pi
0.707107 + 0.707107i 0.250000π0.250000\pi
384384 0 0
385385 6.00000 + 12.0000i 0.305788 + 0.611577i
386386 0 0
387387 −3.65685 7.65685i −0.185888 0.389220i
388388 0 0
389389 −12.7279 −0.645331 −0.322666 0.946513i 0.604579π-0.604579\pi
−0.322666 + 0.946513i 0.604579π0.604579\pi
390390 0 0
391391 24.0000 1.21373
392392 0 0
393393 −4.55635 + 26.5563i −0.229837 + 1.33959i
394394 0 0
395395 −7.07107 + 21.2132i −0.355784 + 1.06735i
396396 0 0
397397 −10.0000 + 10.0000i −0.501886 + 0.501886i −0.912024 0.410138i 0.865481π-0.865481\pi
0.410138 + 0.912024i 0.365481π0.365481\pi
398398 0 0
399399 −16.9706 24.0000i −0.849591 1.20150i
400400 0 0
401401 14.1421i 0.706225i −0.935581 0.353112i 0.885123π-0.885123\pi
0.935581 0.353112i 0.114877π-0.114877\pi
402402 0 0
403403 0 0
404404 0 0
405405 10.8492 + 16.9497i 0.539103 + 0.842240i
406406 0 0
407407 −2.82843 2.82843i −0.140200 0.140200i
408408 0 0
409409 0 0 1.00000 00
−1.00000 π\pi
410410 0 0
411411 −18.0000 25.4558i −0.887875 1.25564i
412412 0 0
413413 −4.24264 + 4.24264i −0.208767 + 0.208767i
414414 0 0
415415 8.00000 4.00000i 0.392705 0.196352i
416416 0 0
417417 −4.68629 + 27.3137i −0.229489 + 1.33756i
418418 0 0
419419 −24.0416 −1.17451 −0.587255 0.809402i 0.699792π-0.699792\pi
−0.587255 + 0.809402i 0.699792π0.699792\pi
420420 0 0
421421 −34.0000 −1.65706 −0.828529 0.559946i 0.810822π-0.810822\pi
−0.828529 + 0.559946i 0.810822π0.810822\pi
422422 0 0
423423 10.3431 + 21.6569i 0.502901 + 1.05299i
424424 0 0
425425 −4.24264 29.6985i −0.205798 1.44059i
426426 0 0
427427 −18.0000 + 18.0000i −0.871081 + 0.871081i
428428 0 0
429429 0 0
430430 0 0
431431 22.6274i 1.08992i −0.838461 0.544962i 0.816544π-0.816544\pi
0.838461 0.544962i 0.183456π-0.183456\pi
432432 0 0
433433 −1.00000 1.00000i −0.0480569 0.0480569i 0.682670 0.730727i 0.260819π-0.260819\pi
−0.730727 + 0.682670i 0.760819π0.760819\pi
434434 0 0
435435 4.82843 + 2.58579i 0.231505 + 0.123979i
436436 0 0
437437 −11.3137 11.3137i −0.541208 0.541208i
438438 0 0
439439 8.00000i 0.381819i 0.981608 + 0.190910i 0.0611437π0.0611437\pi
−0.981608 + 0.190910i 0.938856π0.938856\pi
440440 0 0
441441 11.0000 31.1127i 0.523810 1.48156i
442442 0 0
443443 7.07107 7.07107i 0.335957 0.335957i −0.518887 0.854843i 0.673653π-0.673653\pi
0.854843 + 0.518887i 0.173653π0.173653\pi
444444 0 0
445445 6.00000 + 2.00000i 0.284427 + 0.0948091i
446446 0 0
447447 7.24264 + 1.24264i 0.342565 + 0.0587749i
448448 0 0
449449 −19.7990 −0.934372 −0.467186 0.884159i 0.654732π-0.654732\pi
−0.467186 + 0.884159i 0.654732π0.654732\pi
450450 0 0
451451 8.00000 0.376705
452452 0 0
453453 −13.6569 2.34315i −0.641655 0.110091i
454454 0 0
455455 0 0
456456 0 0
457457 1.00000 1.00000i 0.0467780 0.0467780i −0.683331 0.730109i 0.739469π-0.739469\pi
0.730109 + 0.683331i 0.239469π0.239469\pi
458458 0 0
459459 8.48528 + 30.0000i 0.396059 + 1.40028i
460460 0 0
461461 32.5269i 1.51493i 0.652876 + 0.757465i 0.273562π0.273562\pi
−0.652876 + 0.757465i 0.726438π0.726438\pi
462462 0 0
463463 1.00000 + 1.00000i 0.0464739 + 0.0464739i 0.729962 0.683488i 0.239538π-0.239538\pi
−0.683488 + 0.729962i 0.739538π0.739538\pi
464464 0 0
465465 6.82843 + 3.65685i 0.316661 + 0.169583i
466466 0 0
467467 25.4558 + 25.4558i 1.17796 + 1.17796i 0.980264 + 0.197692i 0.0633445π0.0633445\pi
0.197692 + 0.980264i 0.436655π0.436655\pi
468468 0 0
469469 24.0000i 1.10822i
470470 0 0
471471 −16.0000 + 11.3137i −0.737241 + 0.521308i
472472 0 0
473473 2.82843 2.82843i 0.130051 0.130051i
474474 0 0
475475 −12.0000 + 16.0000i −0.550598 + 0.734130i
476476 0 0
477477 −32.4853 + 15.5147i −1.48740 + 0.710370i
478478 0 0
479479 −2.82843 −0.129234 −0.0646171 0.997910i 0.520583π-0.520583\pi
−0.0646171 + 0.997910i 0.520583π0.520583\pi
480480 0 0
481481 0 0
482482 0 0
483483 4.97056 28.9706i 0.226168 1.31821i
484484 0 0
485485 −36.7696 + 18.3848i −1.66962 + 0.834810i
486486 0 0
487487 −3.00000 + 3.00000i −0.135943 + 0.135943i −0.771804 0.635861i 0.780645π-0.780645\pi
0.635861 + 0.771804i 0.280645π0.280645\pi
488488 0 0
489489 −16.9706 24.0000i −0.767435 1.08532i
490490 0 0
491491 15.5563i 0.702048i −0.936366 0.351024i 0.885834π-0.885834\pi
0.936366 0.351024i 0.114166π-0.114166\pi
492492 0 0
493493 6.00000 + 6.00000i 0.270226 + 0.270226i
494494 0 0
495495 −5.82843 + 7.48528i −0.261968 + 0.336438i
496496 0 0
497497 −8.48528 8.48528i −0.380617 0.380617i
498498 0 0
499499 20.0000i 0.895323i 0.894203 + 0.447661i 0.147743π0.147743\pi
−0.894203 + 0.447661i 0.852257π0.852257\pi
500500 0 0
501501 0 0
502502 0 0
503503 25.4558 25.4558i 1.13502 1.13502i 0.145690 0.989330i 0.453460π-0.453460\pi
0.989330 0.145690i 0.0465401π-0.0465401\pi
504504 0 0
505505 11.0000 33.0000i 0.489494 1.46848i
506506 0 0
507507 −3.80761 + 22.1924i −0.169102 + 0.985599i
508508 0 0
509509 38.1838 1.69247 0.846233 0.532813i 0.178865π-0.178865\pi
0.846233 + 0.532813i 0.178865π0.178865\pi
510510 0 0
511511 18.0000 0.796273
512512 0 0
513513 10.1421 18.1421i 0.447786 0.800995i
514514 0 0
515515 −15.5563 31.1127i −0.685495 1.37099i
516516 0 0
517517 −8.00000 + 8.00000i −0.351840 + 0.351840i
518518 0 0
519519 −19.7990 + 14.0000i −0.869079 + 0.614532i
520520 0 0
521521 8.48528i 0.371747i −0.982574 0.185873i 0.940489π-0.940489\pi
0.982574 0.185873i 0.0595115π-0.0595115\pi
522522 0 0
523523 24.0000 + 24.0000i 1.04945 + 1.04945i 0.998712 + 0.0507346i 0.0161562π0.0161562\pi
0.0507346 + 0.998712i 0.483844π0.483844\pi
524524 0 0
525525 −36.7279 1.02944i −1.60294 0.0449283i
526526 0 0
527527 8.48528 + 8.48528i 0.369625 + 0.369625i
528528 0 0
529529 7.00000i 0.304348i
530530 0 0
531531 −4.00000 1.41421i −0.173585 0.0613716i
532532 0 0
533533 0 0
534534 0 0
535535 −4.00000 8.00000i −0.172935 0.345870i
536536 0 0
537537 −26.5563 4.55635i −1.14599 0.196621i
538538 0 0
539539 15.5563 0.670059
540540 0 0
541541 14.0000 0.601907 0.300954 0.953639i 0.402695π-0.402695\pi
0.300954 + 0.953639i 0.402695π0.402695\pi
542542 0 0
543543 −17.0711 2.92893i −0.732590 0.125693i
544544 0 0
545545 −7.07107 + 21.2132i −0.302891 + 0.908674i
546546 0 0
547547 30.0000 30.0000i 1.28271 1.28271i 0.343586 0.939121i 0.388358π-0.388358\pi
0.939121 0.343586i 0.111642π-0.111642\pi
548548 0 0
549549 −16.9706 6.00000i −0.724286 0.256074i
550550 0 0
551551 5.65685i 0.240990i
552552 0 0
553553 30.0000 + 30.0000i 1.27573 + 1.27573i
554554 0 0
555555 10.4853 3.17157i 0.445075 0.134626i
556556 0 0
557557 16.9706 + 16.9706i 0.719066 + 0.719066i 0.968414 0.249348i 0.0802163π-0.0802163\pi
−0.249348 + 0.968414i 0.580216π0.580216\pi
558558 0 0
559559 0 0
560560 0 0
561561 −12.0000 + 8.48528i −0.506640 + 0.358249i
562562 0 0
563563 −24.0416 + 24.0416i −1.01323 + 1.01323i −0.0133227 + 0.999911i 0.504241π0.504241\pi
−0.999911 + 0.0133227i 0.995759π0.995759\pi
564564 0 0
565565 12.0000 6.00000i 0.504844 0.252422i
566566 0 0
567567 37.9706 4.02944i 1.59461 0.169220i
568568 0 0
569569 −14.1421 −0.592869 −0.296435 0.955053i 0.595798π-0.595798\pi
−0.296435 + 0.955053i 0.595798π0.595798\pi
570570 0 0
571571 32.0000 1.33916 0.669579 0.742741i 0.266474π-0.266474\pi
0.669579 + 0.742741i 0.266474π0.266474\pi
572572 0 0
573573 0.828427 4.82843i 0.0346080 0.201710i
574574 0 0
575575 −19.7990 + 2.82843i −0.825675 + 0.117954i
576576 0 0
577577 21.0000 21.0000i 0.874241 0.874241i −0.118690 0.992931i 0.537869π-0.537869\pi
0.992931 + 0.118690i 0.0378694π0.0378694\pi
578578 0 0
579579 9.89949 + 14.0000i 0.411409 + 0.581820i
580580 0 0
581581 16.9706i 0.704058i
582582 0 0
583583 −12.0000 12.0000i −0.496989 0.496989i
584584 0 0
585585 0 0
586586 0 0
587587 19.7990 + 19.7990i 0.817192 + 0.817192i 0.985700 0.168508i 0.0538950π-0.0538950\pi
−0.168508 + 0.985700i 0.553895π0.553895\pi
588588 0 0
589589 8.00000i 0.329634i
590590 0 0
591591 0 0
592592 0 0
593593 −1.41421 + 1.41421i −0.0580748 + 0.0580748i −0.735548 0.677473i 0.763075π-0.763075\pi
0.677473 + 0.735548i 0.263075π0.263075\pi
594594 0 0
595595 −54.0000 18.0000i −2.21378 0.737928i
596596 0 0
597597 0 0
598598 0 0
599599 11.3137 0.462266 0.231133 0.972922i 0.425757π-0.425757\pi
0.231133 + 0.972922i 0.425757π0.425757\pi
600600 0 0
601601 22.0000 0.897399 0.448699 0.893683i 0.351887π-0.351887\pi
0.448699 + 0.893683i 0.351887π0.351887\pi
602602 0 0
603603 15.3137 7.31371i 0.623622 0.297837i
604604 0 0
605605 19.0919 + 6.36396i 0.776195 + 0.258732i
606606 0 0
607607 −25.0000 + 25.0000i −1.01472 + 1.01472i −0.0148286 + 0.999890i 0.504720π0.504720\pi
−0.999890 + 0.0148286i 0.995280π0.995280\pi
608608 0 0
609609 8.48528 6.00000i 0.343841 0.243132i
610610 0 0
611611 0 0
612612 0 0
613613 −30.0000 30.0000i −1.21169 1.21169i −0.970471 0.241218i 0.922453π-0.922453\pi
−0.241218 0.970471i 0.577547π-0.577547\pi
614614 0 0
615615 −10.3431 + 19.3137i −0.417076 + 0.778804i
616616 0 0
617617 4.24264 + 4.24264i 0.170802 + 0.170802i 0.787332 0.616530i 0.211462π-0.211462\pi
−0.616530 + 0.787332i 0.711462π0.711462\pi
618618 0 0
619619 48.0000i 1.92928i −0.263566 0.964641i 0.584899π-0.584899\pi
0.263566 0.964641i 0.415101π-0.415101\pi
620620 0 0
621621 20.0000 5.65685i 0.802572 0.227002i
622622 0 0
623623 8.48528 8.48528i 0.339956 0.339956i
624624 0 0
625625 7.00000 + 24.0000i 0.280000 + 0.960000i
626626 0 0
627627 9.65685 + 1.65685i 0.385658 + 0.0661684i
628628 0 0
629629 16.9706 0.676661
630630 0 0
631631 34.0000 1.35352 0.676759 0.736204i 0.263384π-0.263384\pi
0.676759 + 0.736204i 0.263384π0.263384\pi
632632 0 0
633633 27.3137 + 4.68629i 1.08562 + 0.186263i
634634 0 0
635635 −8.48528 + 4.24264i −0.336728 + 0.168364i
636636 0 0
637637 0 0
638638 0 0
639639 2.82843 8.00000i 0.111891 0.316475i
640640 0 0
641641 28.2843i 1.11716i −0.829450 0.558581i 0.811346π-0.811346\pi
0.829450 0.558581i 0.188654π-0.188654\pi
642642 0 0
643643 8.00000 + 8.00000i 0.315489 + 0.315489i 0.847032 0.531542i 0.178387π-0.178387\pi
−0.531542 + 0.847032i 0.678387π0.678387\pi
644644 0 0
645645 3.17157 + 10.4853i 0.124881 + 0.412858i
646646 0 0
647647 2.82843 + 2.82843i 0.111197 + 0.111197i 0.760516 0.649319i 0.224946π-0.224946\pi
−0.649319 + 0.760516i 0.724946π0.724946\pi
648648 0 0
649649 2.00000i 0.0785069i
650650 0 0
651651 12.0000 8.48528i 0.470317 0.332564i
652652 0 0
653653 4.24264 4.24264i 0.166027 0.166027i −0.619203 0.785231i 0.712544π-0.712544\pi
0.785231 + 0.619203i 0.212544π0.212544\pi
654654 0 0
655655 11.0000 33.0000i 0.429806 1.28942i
656656 0 0
657657 5.48528 + 11.4853i 0.214001 + 0.448084i
658658 0 0
659659 −46.6690 −1.81797 −0.908984 0.416831i 0.863141π-0.863141\pi
−0.908984 + 0.416831i 0.863141π0.863141\pi
660660 0 0
661661 −14.0000 −0.544537 −0.272268 0.962221i 0.587774π-0.587774\pi
−0.272268 + 0.962221i 0.587774π0.587774\pi
662662 0 0
663663 0 0
664664 0 0
665665 16.9706 + 33.9411i 0.658090 + 1.31618i
666666 0 0
667667 4.00000 4.00000i 0.154881 0.154881i
668668 0 0
669669 −4.24264 6.00000i −0.164030 0.231973i
670670 0 0
671671 8.48528i 0.327571i
672672 0 0
673673 −19.0000 19.0000i −0.732396 0.732396i 0.238698 0.971094i 0.423279π-0.423279\pi
−0.971094 + 0.238698i 0.923279π0.923279\pi
674674 0 0
675675 −10.5355 23.7487i −0.405513 0.914089i
676676 0 0
677677 18.3848 + 18.3848i 0.706584 + 0.706584i 0.965815 0.259231i 0.0834691π-0.0834691\pi
−0.259231 + 0.965815i 0.583469π0.583469\pi
678678 0 0
679679 78.0000i 2.99337i
680680 0 0
681681 −6.00000 8.48528i −0.229920 0.325157i
682682 0 0
683683 −25.4558 + 25.4558i −0.974041 + 0.974041i −0.999671 0.0256307i 0.991841π-0.991841\pi
0.0256307 + 0.999671i 0.491841π0.491841\pi
684684 0 0
685685 18.0000 + 36.0000i 0.687745 + 1.37549i
686686 0 0
687687 1.75736 10.2426i 0.0670474 0.390781i
688688 0 0
689689 0 0
690690 0 0
691691 −12.0000 −0.456502 −0.228251 0.973602i 0.573301π-0.573301\pi
−0.228251 + 0.973602i 0.573301π0.573301\pi
692692 0 0
693693 7.75736 + 16.2426i 0.294678 + 0.617007i
694694 0 0
695695 11.3137 33.9411i 0.429153 1.28746i
696696 0 0
697697 −24.0000 + 24.0000i −0.909065 + 0.909065i
698698 0 0
699699 14.1421 10.0000i 0.534905 0.378235i
700700 0 0
701701 1.41421i 0.0534141i −0.999643 0.0267071i 0.991498π-0.991498\pi
0.999643 0.0267071i 0.00850213π-0.00850213\pi
702702 0 0
703703 −8.00000 8.00000i −0.301726 0.301726i
704704 0 0
705705 −8.97056 29.6569i −0.337851 1.11694i
706706 0 0
707707 −46.6690 46.6690i −1.75517 1.75517i
708708 0 0
709709 6.00000i 0.225335i 0.993633 + 0.112667i 0.0359394π0.0359394\pi
−0.993633 + 0.112667i 0.964061π0.964061\pi
710710 0 0
711711 −10.0000 + 28.2843i −0.375029 + 1.06074i
712712 0 0
713713 5.65685 5.65685i 0.211851 0.211851i
714714 0 0
715715 0 0
716716 0 0
717717 −43.4558 7.45584i −1.62289 0.278444i
718718 0 0
719719 −45.2548 −1.68772 −0.843860 0.536563i 0.819722π-0.819722\pi
−0.843860 + 0.536563i 0.819722π0.819722\pi
720720 0 0
721721 −66.0000 −2.45797
722722 0 0
723723 −34.1421 5.85786i −1.26976 0.217856i
724724 0 0
725725 −5.65685 4.24264i −0.210090 0.157568i
726726 0 0
727727 −15.0000 + 15.0000i −0.556319 + 0.556319i −0.928257 0.371938i 0.878693π-0.878693\pi
0.371938 + 0.928257i 0.378693π0.378693\pi
728728 0 0
729729 14.1421 + 23.0000i 0.523783 + 0.851852i
730730 0 0
731731 16.9706i 0.627679i
732732 0 0
733733 14.0000 + 14.0000i 0.517102 + 0.517102i 0.916693 0.399592i 0.130848π-0.130848\pi
−0.399592 + 0.916693i 0.630848π0.630848\pi
734734 0 0
735735 −20.1127 + 37.5563i −0.741868 + 1.38529i
736736 0 0
737737 5.65685 + 5.65685i 0.208373 + 0.208373i
738738 0 0
739739 16.0000i 0.588570i −0.955718 0.294285i 0.904919π-0.904919\pi
0.955718 0.294285i 0.0950814π-0.0950814\pi
740740 0 0
741741 0 0
742742 0 0
743743 25.4558 25.4558i 0.933884 0.933884i −0.0640616 0.997946i 0.520405π-0.520405\pi
0.997946 + 0.0640616i 0.0204054π0.0204054\pi
744744 0 0
745745 −9.00000 3.00000i −0.329734 0.109911i
746746 0 0
747747 10.8284 5.17157i 0.396191 0.189218i
748748 0 0
749749 −16.9706 −0.620091
750750 0 0
751751 −30.0000 −1.09472 −0.547358 0.836899i 0.684366π-0.684366\pi
−0.547358 + 0.836899i 0.684366π0.684366\pi
752752 0 0
753753 −3.72792 + 21.7279i −0.135853 + 0.791809i
754754 0 0
755755 16.9706 + 5.65685i 0.617622 + 0.205874i
756756 0 0
757757 −6.00000 + 6.00000i −0.218074 + 0.218074i −0.807686 0.589613i 0.799280π-0.799280\pi
0.589613 + 0.807686i 0.299280π0.299280\pi
758758 0 0
759759 5.65685 + 8.00000i 0.205331 + 0.290382i
760760 0 0
761761 19.7990i 0.717713i 0.933393 + 0.358856i 0.116833π0.116833\pi
−0.933393 + 0.358856i 0.883167π0.883167\pi
762762 0 0
763763 30.0000 + 30.0000i 1.08607 + 1.08607i
764764 0 0
765765 −4.97056 39.9411i −0.179711 1.44407i
766766 0 0
767767 0 0
768768 0 0
769769 10.0000i 0.360609i −0.983611 0.180305i 0.942292π-0.942292\pi
0.983611 0.180305i 0.0577084π-0.0577084\pi
770770 0 0
771771 22.0000 + 31.1127i 0.792311 + 1.12050i
772772 0 0
773773 18.3848 18.3848i 0.661254 0.661254i −0.294421 0.955676i 0.595127π-0.595127\pi
0.955676 + 0.294421i 0.0951269π0.0951269\pi
774774 0 0
775775 −8.00000 6.00000i −0.287368 0.215526i
776776 0 0
777777 3.51472 20.4853i 0.126090 0.734905i
778778 0 0
779779 22.6274 0.810711
780780 0 0
781781 4.00000 0.143131
782782 0 0
783783 6.41421 + 3.58579i 0.229225 + 0.128146i
784784 0 0
785785 22.6274 11.3137i 0.807607 0.403804i
786786 0 0
787787 −12.0000 + 12.0000i −0.427754 + 0.427754i −0.887863 0.460109i 0.847810π-0.847810\pi
0.460109 + 0.887863i 0.347810π0.347810\pi
788788 0 0
789789 22.6274 16.0000i 0.805557 0.569615i
790790 0 0
791791 25.4558i 0.905106i
792792 0 0
793793 0 0
794794 0 0
795795 44.4853 13.4558i 1.57773 0.477230i
796796 0 0
797797 −21.2132 21.2132i −0.751410 0.751410i 0.223332 0.974742i 0.428307π-0.428307\pi
−0.974742 + 0.223332i 0.928307π0.928307\pi
798798 0 0
799799 48.0000i 1.69812i
800800 0 0
801801 8.00000 + 2.82843i 0.282666 + 0.0999376i
802802 0 0
803803 −4.24264 + 4.24264i −0.149720 + 0.149720i
804804 0 0
805805 −12.0000 + 36.0000i −0.422944 + 1.26883i
806806 0 0
807807 16.8995 + 2.89949i 0.594890 + 0.102067i
808808 0 0
809809 45.2548 1.59108 0.795538 0.605904i 0.207189π-0.207189\pi
0.795538 + 0.605904i 0.207189π0.207189\pi
810810 0 0
811811 −44.0000 −1.54505 −0.772524 0.634985i 0.781006π-0.781006\pi
−0.772524 + 0.634985i 0.781006π0.781006\pi
812812 0 0
813813 40.9706 + 7.02944i 1.43690 + 0.246533i
814814 0 0
815815 16.9706 + 33.9411i 0.594453 + 1.18891i
816816 0 0
817817 8.00000 8.00000i 0.279885 0.279885i
818818 0 0
819819 0 0
820820 0 0
821821 26.8701i 0.937771i 0.883259 + 0.468886i 0.155344π0.155344\pi
−0.883259 + 0.468886i 0.844656π0.844656\pi
822822 0 0
823823 −29.0000 29.0000i −1.01088 1.01088i −0.999940 0.0109363i 0.996519π-0.996519\pi
−0.0109363 0.999940i 0.503481π-0.503481\pi
824824 0 0
825825 8.89949 8.41421i 0.309841 0.292945i
826826 0 0
827827 −12.7279 12.7279i −0.442593 0.442593i 0.450289 0.892883i 0.351321π-0.351321\pi
−0.892883 + 0.450289i 0.851321π0.851321\pi
828828 0 0
829829 14.0000i 0.486240i −0.969996 0.243120i 0.921829π-0.921829\pi
0.969996 0.243120i 0.0781709π-0.0781709\pi
830830 0 0
831831 −4.00000 + 2.82843i −0.138758 + 0.0981170i
832832 0 0
833833 −46.6690 + 46.6690i −1.61699 + 1.61699i
834834 0 0
835835 0 0
836836 0 0
837837 9.07107 + 5.07107i 0.313542 + 0.175282i
838838 0 0
839839 −39.5980 −1.36707 −0.683537 0.729916i 0.739559π-0.739559\pi
−0.683537 + 0.729916i 0.739559π0.739559\pi
840840 0 0
841841 −27.0000 −0.931034
842842 0 0
843843 7.45584 43.4558i 0.256793 1.49670i
844844 0 0
845845 9.19239 27.5772i 0.316228 0.948683i
846846 0 0
847847 27.0000 27.0000i 0.927731 0.927731i
848848 0 0
849849 5.65685 + 8.00000i 0.194143 + 0.274559i
850850 0 0
851851 11.3137i 0.387829i
852852 0 0
853853 −18.0000 18.0000i −0.616308 0.616308i 0.328274 0.944582i 0.393533π-0.393533\pi
−0.944582 + 0.328274i 0.893533π0.893533\pi
854854 0 0
855855 −16.4853 + 21.1716i −0.563785 + 0.724053i
856856 0 0
857857 12.7279 + 12.7279i 0.434778 + 0.434778i 0.890250 0.455472i 0.150530π-0.150530\pi
−0.455472 + 0.890250i 0.650530π0.650530\pi
858858 0 0
859859 40.0000i 1.36478i 0.730987 + 0.682391i 0.239060π0.239060\pi
−0.730987 + 0.682391i 0.760940π0.760940\pi
860860 0 0
861861 24.0000 + 33.9411i 0.817918 + 1.15671i
862862 0 0
863863 −25.4558 + 25.4558i −0.866527 + 0.866527i −0.992086 0.125559i 0.959928π-0.959928\pi
0.125559 + 0.992086i 0.459928π0.459928\pi
864864 0 0
865865 28.0000 14.0000i 0.952029 0.476014i
866866 0 0
867867 5.56497 32.4350i 0.188996 1.10155i
868868 0 0
869869 −14.1421 −0.479739
870870 0 0
871871 0 0
872872 0 0
873873 −49.7696 + 23.7696i −1.68444 + 0.804477i
874874 0 0
875875 46.6690 + 8.48528i 1.57770 + 0.286855i
876876 0 0
877877 36.0000 36.0000i 1.21563 1.21563i 0.246488 0.969146i 0.420724π-0.420724\pi
0.969146 0.246488i 0.0792765π-0.0792765\pi
878878 0 0
879879 11.3137 8.00000i 0.381602 0.269833i
880880 0 0
881881 19.7990i 0.667045i −0.942742 0.333522i 0.891763π-0.891763\pi
0.942742 0.333522i 0.108237π-0.108237\pi
882882 0 0
883883 −12.0000 12.0000i −0.403832 0.403832i 0.475749 0.879581i 0.342177π-0.342177\pi
−0.879581 + 0.475749i 0.842177π0.842177\pi
884884 0 0
885885 4.82843 + 2.58579i 0.162306 + 0.0869203i
886886 0 0
887887 36.7696 + 36.7696i 1.23460 + 1.23460i 0.962178 + 0.272423i 0.0878251π0.0878251\pi
0.272423 + 0.962178i 0.412175π0.412175\pi
888888 0 0
889889 18.0000i 0.603701i
890890 0 0
891891 −8.00000 + 9.89949i −0.268010 + 0.331646i
892892 0 0
893893 −22.6274 + 22.6274i −0.757198 + 0.757198i
894894 0 0
895895 33.0000 + 11.0000i 1.10307 + 0.367689i
896896 0 0
897897 0 0
898898 0 0
899899 2.82843 0.0943333
900900 0 0
901901 72.0000 2.39867
902902 0 0
903903 20.4853 + 3.51472i 0.681707 + 0.116963i
904904 0 0
905905 21.2132 + 7.07107i 0.705151 + 0.235050i
906906 0 0
907907 −26.0000 + 26.0000i −0.863316 + 0.863316i −0.991722 0.128406i 0.959014π-0.959014\pi
0.128406 + 0.991722i 0.459014π0.459014\pi
908908 0 0
909909 15.5563 44.0000i 0.515972 1.45939i
910910 0 0
911911 5.65685i 0.187420i −0.995600 0.0937100i 0.970127π-0.970127\pi
0.995600 0.0937100i 0.0298726π-0.0298726\pi
912912 0 0
913913 4.00000 + 4.00000i 0.132381 + 0.132381i
914914 0 0
915915 20.4853 + 10.9706i 0.677223 + 0.362676i
916916 0 0
917917 −46.6690 46.6690i −1.54115 1.54115i
918918 0 0
919919 2.00000i 0.0659739i 0.999456 + 0.0329870i 0.0105020π0.0105020\pi
−0.999456 + 0.0329870i 0.989498π0.989498\pi
920920 0 0
921921 −20.0000 + 14.1421i −0.659022 + 0.465999i
922922 0 0
923923 0 0
924924 0 0
925925 −14.0000 + 2.00000i −0.460317 + 0.0657596i
926926 0 0
927927 −20.1127 42.1127i −0.660588 1.38316i
928928 0 0
929929 −36.7696 −1.20637 −0.603185 0.797601i 0.706102π-0.706102\pi
−0.603185 + 0.797601i 0.706102π0.706102\pi
930930 0 0
931931 44.0000 1.44204
932932 0 0
933933 9.11270 53.1127i 0.298336 1.73883i
934934 0 0
935935 16.9706 8.48528i 0.554997 0.277498i
936936 0 0
937937 11.0000 11.0000i 0.359354 0.359354i −0.504221 0.863575i 0.668220π-0.668220\pi
0.863575 + 0.504221i 0.168220π0.168220\pi
938938 0 0
939939 21.2132 + 30.0000i 0.692267 + 0.979013i
940940 0 0
941941 38.1838i 1.24476i −0.782717 0.622378i 0.786167π-0.786167\pi
0.782717 0.622378i 0.213833π-0.213833\pi
942942 0 0
943943 16.0000 + 16.0000i 0.521032 + 0.521032i
944944 0 0
945945 −49.2426 2.27208i −1.60186 0.0739107i
946946 0 0
947947 15.5563 + 15.5563i 0.505513 + 0.505513i 0.913146 0.407633i 0.133646π-0.133646\pi
−0.407633 + 0.913146i 0.633646π0.633646\pi
948948 0 0
949949 0 0
950950 0 0
951951 2.00000 + 2.82843i 0.0648544 + 0.0917180i
952952 0 0
953953 −21.2132 + 21.2132i −0.687163 + 0.687163i −0.961604 0.274441i 0.911507π-0.911507\pi
0.274441 + 0.961604i 0.411507π0.411507\pi
954954 0 0
955955 −2.00000 + 6.00000i −0.0647185 + 0.194155i
956956 0 0
957957 −0.585786 + 3.41421i −0.0189358 + 0.110366i
958958 0 0
959959 76.3675 2.46604
960960 0 0
961961 −27.0000 −0.870968
962962 0 0
963963 −5.17157 10.8284i −0.166652 0.348941i
964964 0 0
965965 −9.89949 19.7990i −0.318676 0.637352i
966966 0 0
967967 15.0000 15.0000i 0.482367 0.482367i −0.423520 0.905887i 0.639205π-0.639205\pi
0.905887 + 0.423520i 0.139205π0.139205\pi
968968 0 0
969969 −33.9411 + 24.0000i −1.09035 + 0.770991i
970970 0 0
971971 41.0122i 1.31614i 0.752955 + 0.658072i 0.228628π0.228628\pi
−0.752955 + 0.658072i 0.771372π0.771372\pi
972972 0 0
973973 −48.0000 48.0000i −1.53881 1.53881i
974974 0 0
975975 0 0
976976 0 0
977977 21.2132 + 21.2132i 0.678671 + 0.678671i 0.959699 0.281029i 0.0906757π-0.0906757\pi
−0.281029 + 0.959699i 0.590676π0.590676\pi
978978 0 0
979979 4.00000i 0.127841i
980980 0 0
981981 −10.0000 + 28.2843i −0.319275 + 0.903047i
982982 0 0
983983 8.48528 8.48528i 0.270638 0.270638i −0.558719 0.829357i 0.688707π-0.688707\pi
0.829357 + 0.558719i 0.188707π0.188707\pi
984984 0 0
985985 0 0
986986 0 0
987987 −57.9411 9.94113i −1.84429 0.316430i
988988 0 0
989989 11.3137 0.359755
990990 0 0
991991 16.0000 0.508257 0.254128 0.967170i 0.418211π-0.418211\pi
0.254128 + 0.967170i 0.418211π0.418211\pi
992992 0 0
993993 27.3137 + 4.68629i 0.866774 + 0.148715i
994994 0 0
995995 0 0
996996 0 0
997997 24.0000 24.0000i 0.760088 0.760088i −0.216250 0.976338i 0.569383π-0.569383\pi
0.976338 + 0.216250i 0.0693827π0.0693827\pi
998998 0 0
999999 14.1421 4.00000i 0.447437 0.126554i
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 120.2.r.a.113.1 yes 4
3.2 odd 2 inner 120.2.r.a.113.2 yes 4
4.3 odd 2 240.2.v.d.113.2 4
5.2 odd 4 inner 120.2.r.a.17.2 yes 4
5.3 odd 4 600.2.r.d.257.1 4
5.4 even 2 600.2.r.d.593.2 4
8.3 odd 2 960.2.v.b.833.1 4
8.5 even 2 960.2.v.l.833.2 4
12.11 even 2 240.2.v.d.113.1 4
15.2 even 4 inner 120.2.r.a.17.1 4
15.8 even 4 600.2.r.d.257.2 4
15.14 odd 2 600.2.r.d.593.1 4
20.3 even 4 1200.2.v.c.257.2 4
20.7 even 4 240.2.v.d.17.1 4
20.19 odd 2 1200.2.v.c.593.1 4
24.5 odd 2 960.2.v.l.833.1 4
24.11 even 2 960.2.v.b.833.2 4
40.27 even 4 960.2.v.b.257.2 4
40.37 odd 4 960.2.v.l.257.1 4
60.23 odd 4 1200.2.v.c.257.1 4
60.47 odd 4 240.2.v.d.17.2 4
60.59 even 2 1200.2.v.c.593.2 4
120.77 even 4 960.2.v.l.257.2 4
120.107 odd 4 960.2.v.b.257.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.r.a.17.1 4 15.2 even 4 inner
120.2.r.a.17.2 yes 4 5.2 odd 4 inner
120.2.r.a.113.1 yes 4 1.1 even 1 trivial
120.2.r.a.113.2 yes 4 3.2 odd 2 inner
240.2.v.d.17.1 4 20.7 even 4
240.2.v.d.17.2 4 60.47 odd 4
240.2.v.d.113.1 4 12.11 even 2
240.2.v.d.113.2 4 4.3 odd 2
600.2.r.d.257.1 4 5.3 odd 4
600.2.r.d.257.2 4 15.8 even 4
600.2.r.d.593.1 4 15.14 odd 2
600.2.r.d.593.2 4 5.4 even 2
960.2.v.b.257.1 4 120.107 odd 4
960.2.v.b.257.2 4 40.27 even 4
960.2.v.b.833.1 4 8.3 odd 2
960.2.v.b.833.2 4 24.11 even 2
960.2.v.l.257.1 4 40.37 odd 4
960.2.v.l.257.2 4 120.77 even 4
960.2.v.l.833.1 4 24.5 odd 2
960.2.v.l.833.2 4 8.5 even 2
1200.2.v.c.257.1 4 60.23 odd 4
1200.2.v.c.257.2 4 20.3 even 4
1200.2.v.c.593.1 4 20.19 odd 2
1200.2.v.c.593.2 4 60.59 even 2