Properties

Label 80.9.p.b.33.1
Level 8080
Weight 99
Character 80.33
Analytic conductor 32.59032.590
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] = [80,9,Mod(17,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.17"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: N N == 80=245 80 = 2^{4} \cdot 5
Weight: k k == 9 9
Character orbit: [χ][\chi] == 80.p (of order 44, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 32.590288804932.5902888049
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(i,249)\Q(i, \sqrt{249})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+125x2+3844 x^{4} + 125x^{2} + 3844 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 252 2\cdot 5^{2}
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

Embedding invariants

Embedding label 33.1
Root 8.38987i8.38987i of defining polynomial
Character χ\chi == 80.33
Dual form 80.9.p.b.17.1

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+(52.949352.9493i)q3+(490.341+387.544i)q5+(2624.012624.01i)q7953.736iq95372.73q11+(27972.7+27972.7i)q13+(46483.4+5443.05i)q15+(53444.253444.2i)q17144788.iq19277879.q21+(52148.752148.7i)q23+(90244.3380058.i)q25+(397900.+397900.i)q27+41723.4iq29244148.q31+(284482.+284482.i)q33+(269741.+2.30358e6i)q35+(2.00653e6+2.00653e6i)q372.96228e6iq394.17322e6q41+(4.01446e64.01446e6i)q43+(369615.+467656.i)q45+(2.26940e6+2.26940e6i)q478.00606e6iq495.65967e6q51+(3.72734e63.72734e6i)q53+(2.63447e62.08217e6i)q55+(7.66643e6+7.66643e6i)q57+1.12165e7iq59+2.02002e7q61+(2.50261e62.50261e6i)q63+(2.45569e72.87552e6i)q65+(1.30857e71.30857e7i)q67+5.52248e6iq692.61041e7q71+(1.85930e71.85930e7i)q73+(2.49022e7+1.53454e7i)q75+(1.40981e7+1.40981e7i)q773.07594e6iq79+3.58796e7q81+(2.41522e72.41522e7i)q83+(5.49393e6+4.69179e7i)q85+(2.20923e62.20923e6i)q875.76618e6iq89+1.46802e8q91+(1.29275e7+1.29275e7i)q93+(5.61117e7+7.09955e7i)q95+(2.86891e72.86891e7i)q97+5.12416e6iq99+O(q100)q+(-52.9493 - 52.9493i) q^{3} +(-490.341 + 387.544i) q^{5} +(2624.01 - 2624.01i) q^{7} -953.736i q^{9} -5372.73 q^{11} +(27972.7 + 27972.7i) q^{13} +(46483.4 + 5443.05i) q^{15} +(53444.2 - 53444.2i) q^{17} -144788. i q^{19} -277879. q^{21} +(-52148.7 - 52148.7i) q^{23} +(90244.3 - 380058. i) q^{25} +(-397900. + 397900. i) q^{27} +41723.4i q^{29} -244148. q^{31} +(284482. + 284482. i) q^{33} +(-269741. + 2.30358e6i) q^{35} +(-2.00653e6 + 2.00653e6i) q^{37} -2.96228e6i q^{39} -4.17322e6 q^{41} +(-4.01446e6 - 4.01446e6i) q^{43} +(369615. + 467656. i) q^{45} +(-2.26940e6 + 2.26940e6i) q^{47} -8.00606e6i q^{49} -5.65967e6 q^{51} +(-3.72734e6 - 3.72734e6i) q^{53} +(2.63447e6 - 2.08217e6i) q^{55} +(-7.66643e6 + 7.66643e6i) q^{57} +1.12165e7i q^{59} +2.02002e7 q^{61} +(-2.50261e6 - 2.50261e6i) q^{63} +(-2.45569e7 - 2.87552e6i) q^{65} +(1.30857e7 - 1.30857e7i) q^{67} +5.52248e6i q^{69} -2.61041e7 q^{71} +(-1.85930e7 - 1.85930e7i) q^{73} +(-2.49022e7 + 1.53454e7i) q^{75} +(-1.40981e7 + 1.40981e7i) q^{77} -3.07594e6i q^{79} +3.58796e7 q^{81} +(-2.41522e7 - 2.41522e7i) q^{83} +(-5.49393e6 + 4.69179e7i) q^{85} +(2.20923e6 - 2.20923e6i) q^{87} -5.76618e6i q^{89} +1.46802e8 q^{91} +(1.29275e7 + 1.29275e7i) q^{93} +(5.61117e7 + 7.09955e7i) q^{95} +(2.86891e7 - 2.86891e7i) q^{97} +5.12416e6i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q54q3+90q5+1186q7+19852q11+73704q13+137490q15+198944q17766572q21631334q23+545600q25184680q27+2329892q31+1362948q331403870q35++139631604q97+O(q100) 4 q - 54 q^{3} + 90 q^{5} + 1186 q^{7} + 19852 q^{11} + 73704 q^{13} + 137490 q^{15} + 198944 q^{17} - 766572 q^{21} - 631334 q^{23} + 545600 q^{25} - 184680 q^{27} + 2329892 q^{31} + 1362948 q^{33} - 1403870 q^{35}+ \cdots + 139631604 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/80Z)×\left(\mathbb{Z}/80\mathbb{Z}\right)^\times.

nn 1717 2121 3131
χ(n)\chi(n) e(34)e\left(\frac{3}{4}\right) 11 11

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 −52.9493 52.9493i −0.653695 0.653695i 0.300185 0.953881i 0.402951π-0.402951\pi
−0.953881 + 0.300185i 0.902951π0.902951\pi
44 0 0
55 −490.341 + 387.544i −0.784546 + 0.620070i
66 0 0
77 2624.01 2624.01i 1.09288 1.09288i 0.0976629 0.995220i 0.468863π-0.468863\pi
0.995220 0.0976629i 0.0311367π-0.0311367\pi
88 0 0
99 953.736i 0.145364i
1010 0 0
1111 −5372.73 −0.366964 −0.183482 0.983023i 0.558737π-0.558737\pi
−0.183482 + 0.983023i 0.558737π0.558737\pi
1212 0 0
1313 27972.7 + 27972.7i 0.979403 + 0.979403i 0.999792 0.0203888i 0.00649040π-0.00649040\pi
−0.0203888 + 0.999792i 0.506490π0.506490\pi
1414 0 0
1515 46483.4 + 5443.05i 0.918192 + 0.107517i
1616 0 0
1717 53444.2 53444.2i 0.639890 0.639890i −0.310638 0.950528i 0.600543π-0.600543\pi
0.950528 + 0.310638i 0.100543π0.100543\pi
1818 0 0
1919 144788.i 1.11101i −0.831513 0.555505i 0.812525π-0.812525\pi
0.831513 0.555505i 0.187475π-0.187475\pi
2020 0 0
2121 −277879. −1.42882
2222 0 0
2323 −52148.7 52148.7i −0.186351 0.186351i 0.607765 0.794117i 0.292066π-0.292066\pi
−0.794117 + 0.607765i 0.792066π0.792066\pi
2424 0 0
2525 90244.3 380058.i 0.231025 0.972948i
2626 0 0
2727 −397900. + 397900.i −0.748720 + 0.748720i
2828 0 0
2929 41723.4i 0.0589913i 0.999565 + 0.0294956i 0.00939012π0.00939012\pi
−0.999565 + 0.0294956i 0.990610π0.990610\pi
3030 0 0
3131 −244148. −0.264367 −0.132183 0.991225i 0.542199π-0.542199\pi
−0.132183 + 0.991225i 0.542199π0.542199\pi
3232 0 0
3333 284482. + 284482.i 0.239883 + 0.239883i
3434 0 0
3535 −269741. + 2.30358e6i −0.179753 + 1.53508i
3636 0 0
3737 −2.00653e6 + 2.00653e6i −1.07063 + 1.07063i −0.0733185 + 0.997309i 0.523359π0.523359\pi
−0.997309 + 0.0733185i 0.976641π0.976641\pi
3838 0 0
3939 2.96228e6i 1.28046i
4040 0 0
4141 −4.17322e6 −1.47685 −0.738424 0.674336i 0.764430π-0.764430\pi
−0.738424 + 0.674336i 0.764430π0.764430\pi
4242 0 0
4343 −4.01446e6 4.01446e6i −1.17423 1.17423i −0.981191 0.193038i 0.938166π-0.938166\pi
−0.193038 0.981191i 0.561834π-0.561834\pi
4444 0 0
4545 369615. + 467656.i 0.0901362 + 0.114045i
4646 0 0
4747 −2.26940e6 + 2.26940e6i −0.465071 + 0.465071i −0.900313 0.435242i 0.856663π-0.856663\pi
0.435242 + 0.900313i 0.356663π0.356663\pi
4848 0 0
4949 8.00606e6i 1.38878i
5050 0 0
5151 −5.65967e6 −0.836586
5252 0 0
5353 −3.72734e6 3.72734e6i −0.472384 0.472384i 0.430301 0.902685i 0.358407π-0.358407\pi
−0.902685 + 0.430301i 0.858407π0.858407\pi
5454 0 0
5555 2.63447e6 2.08217e6i 0.287901 0.227544i
5656 0 0
5757 −7.66643e6 + 7.66643e6i −0.726262 + 0.726262i
5858 0 0
5959 1.12165e7i 0.925653i 0.886449 + 0.462826i 0.153165π0.153165\pi
−0.886449 + 0.462826i 0.846835π0.846835\pi
6060 0 0
6161 2.02002e7 1.45893 0.729467 0.684016i 0.239768π-0.239768\pi
0.729467 + 0.684016i 0.239768π0.239768\pi
6262 0 0
6363 −2.50261e6 2.50261e6i −0.158866 0.158866i
6464 0 0
6565 −2.45569e7 2.87552e6i −1.37569 0.161088i
6666 0 0
6767 1.30857e7 1.30857e7i 0.649377 0.649377i −0.303465 0.952843i 0.598144π-0.598144\pi
0.952843 + 0.303465i 0.0981436π0.0981436\pi
6868 0 0
6969 5.52248e6i 0.243634i
7070 0 0
7171 −2.61041e7 −1.02725 −0.513624 0.858015i 0.671697π-0.671697\pi
−0.513624 + 0.858015i 0.671697π0.671697\pi
7272 0 0
7373 −1.85930e7 1.85930e7i −0.654722 0.654722i 0.299404 0.954126i 0.403212π-0.403212\pi
−0.954126 + 0.299404i 0.903212π0.903212\pi
7474 0 0
7575 −2.49022e7 + 1.53454e7i −0.787032 + 0.484991i
7676 0 0
7777 −1.40981e7 + 1.40981e7i −0.401049 + 0.401049i
7878 0 0
7979 3.07594e6i 0.0789714i −0.999220 0.0394857i 0.987428π-0.987428\pi
0.999220 0.0394857i 0.0125720π-0.0125720\pi
8080 0 0
8181 3.58796e7 0.833505
8282 0 0
8383 −2.41522e7 2.41522e7i −0.508914 0.508914i 0.405279 0.914193i 0.367174π-0.367174\pi
−0.914193 + 0.405279i 0.867174π0.867174\pi
8484 0 0
8585 −5.49393e6 + 4.69179e7i −0.105246 + 0.898800i
8686 0 0
8787 2.20923e6 2.20923e6i 0.0385623 0.0385623i
8888 0 0
8989 5.76618e6i 0.0919027i −0.998944 0.0459513i 0.985368π-0.985368\pi
0.998944 0.0459513i 0.0146319π-0.0146319\pi
9090 0 0
9191 1.46802e8 2.14075
9292 0 0
9393 1.29275e7 + 1.29275e7i 0.172815 + 0.172815i
9494 0 0
9595 5.61117e7 + 7.09955e7i 0.688905 + 0.871639i
9696 0 0
9797 2.86891e7 2.86891e7i 0.324063 0.324063i −0.526260 0.850324i 0.676406π-0.676406\pi
0.850324 + 0.526260i 0.176406π0.176406\pi
9898 0 0
9999 5.12416e6i 0.0533436i
100100 0 0
101101 7.95260e7 0.764229 0.382115 0.924115i 0.375196π-0.375196\pi
0.382115 + 0.924115i 0.375196π0.375196\pi
102102 0 0
103103 −4.96462e7 4.96462e7i −0.441100 0.441100i 0.451282 0.892382i 0.350967π-0.350967\pi
−0.892382 + 0.451282i 0.850967π0.850967\pi
104104 0 0
105105 1.36256e8 1.07690e8i 1.12098 0.885972i
106106 0 0
107107 −6.12534e7 + 6.12534e7i −0.467300 + 0.467300i −0.901039 0.433739i 0.857194π-0.857194\pi
0.433739 + 0.901039i 0.357194π0.357194\pi
108108 0 0
109109 9.33557e7i 0.661355i −0.943744 0.330678i 0.892723π-0.892723\pi
0.943744 0.330678i 0.107277π-0.107277\pi
110110 0 0
111111 2.12489e8 1.39973
112112 0 0
113113 1.17246e8 + 1.17246e8i 0.719093 + 0.719093i 0.968419 0.249327i 0.0802093π-0.0802093\pi
−0.249327 + 0.968419i 0.580209π0.580209\pi
114114 0 0
115115 4.57806e7 + 5.36075e6i 0.261752 + 0.0306503i
116116 0 0
117117 2.66786e7 2.66786e7i 0.142370 0.142370i
118118 0 0
119119 2.80477e8i 1.39865i
120120 0 0
121121 −1.85493e8 −0.865337
122122 0 0
123123 2.20969e8 + 2.20969e8i 0.965409 + 0.965409i
124124 0 0
125125 1.03039e8 + 2.21332e8i 0.422046 + 0.906574i
126126 0 0
127127 −1.16482e7 + 1.16482e7i −0.0447759 + 0.0447759i −0.729140 0.684364i 0.760080π-0.760080\pi
0.684364 + 0.729140i 0.260080π0.260080\pi
128128 0 0
129129 4.25126e8i 1.53518i
130130 0 0
131131 −3.83537e8 −1.30233 −0.651167 0.758934i 0.725720π-0.725720\pi
−0.651167 + 0.758934i 0.725720π0.725720\pi
132132 0 0
133133 −3.79925e8 3.79925e8i −1.21420 1.21420i
134134 0 0
135135 4.09031e7 3.49311e8i 0.123146 1.05166i
136136 0 0
137137 −1.46786e7 + 1.46786e7i −0.0416681 + 0.0416681i −0.727634 0.685966i 0.759380π-0.759380\pi
0.685966 + 0.727634i 0.259380π0.259380\pi
138138 0 0
139139 4.00314e8i 1.07236i 0.844103 + 0.536181i 0.180134π0.180134\pi
−0.844103 + 0.536181i 0.819866π0.819866\pi
140140 0 0
141141 2.40326e8 0.608030
142142 0 0
143143 −1.50290e8 1.50290e8i −0.359406 0.359406i
144144 0 0
145145 −1.61697e7 2.04587e7i −0.0365788 0.0462814i
146146 0 0
147147 −4.23916e8 + 4.23916e8i −0.907842 + 0.907842i
148148 0 0
149149 5.62457e8i 1.14115i −0.821244 0.570577i 0.806719π-0.806719\pi
0.821244 0.570577i 0.193281π-0.193281\pi
150150 0 0
151151 2.75254e7 0.0529452 0.0264726 0.999650i 0.491573π-0.491573\pi
0.0264726 + 0.999650i 0.491573π0.491573\pi
152152 0 0
153153 −5.09717e7 5.09717e7i −0.0930172 0.0930172i
154154 0 0
155155 1.19716e8 9.46182e7i 0.207408 0.163926i
156156 0 0
157157 −3.22628e8 + 3.22628e8i −0.531010 + 0.531010i −0.920873 0.389863i 0.872522π-0.872522\pi
0.389863 + 0.920873i 0.372522π0.372522\pi
158158 0 0
159159 3.94720e8i 0.617591i
160160 0 0
161161 −2.73678e8 −0.407320
162162 0 0
163163 −2.36679e8 2.36679e8i −0.335281 0.335281i 0.519307 0.854588i 0.326190π-0.326190\pi
−0.854588 + 0.519307i 0.826190π0.826190\pi
164164 0 0
165165 −2.49743e8 2.92440e7i −0.336944 0.0394549i
166166 0 0
167167 5.45436e8 5.45436e8i 0.701258 0.701258i −0.263422 0.964681i 0.584851π-0.584851\pi
0.964681 + 0.263422i 0.0848512π0.0848512\pi
168168 0 0
169169 7.49218e8i 0.918462i
170170 0 0
171171 −1.38089e8 −0.161501
172172 0 0
173173 −6.41655e8 6.41655e8i −0.716337 0.716337i 0.251516 0.967853i 0.419071π-0.419071\pi
−0.967853 + 0.251516i 0.919071π0.919071\pi
174174 0 0
175175 −7.60474e8 1.23408e9i −0.810834 1.31580i
176176 0 0
177177 5.93905e8 5.93905e8i 0.605095 0.605095i
178178 0 0
179179 9.41081e8i 0.916673i 0.888779 + 0.458337i 0.151555π0.151555\pi
−0.888779 + 0.458337i 0.848445π0.848445\pi
180180 0 0
181181 −4.09543e8 −0.381580 −0.190790 0.981631i 0.561105π-0.561105\pi
−0.190790 + 0.981631i 0.561105π0.561105\pi
182182 0 0
183183 −1.06959e9 1.06959e9i −0.953699 0.953699i
184184 0 0
185185 2.06266e8 1.76150e9i 0.176092 1.50382i
186186 0 0
187187 −2.87141e8 + 2.87141e8i −0.234817 + 0.234817i
188188 0 0
189189 2.08819e9i 1.63652i
190190 0 0
191191 −5.56054e8 −0.417814 −0.208907 0.977935i 0.566991π-0.566991\pi
−0.208907 + 0.977935i 0.566991π0.566991\pi
192192 0 0
193193 1.72776e8 + 1.72776e8i 0.124524 + 0.124524i 0.766622 0.642098i 0.221936π-0.221936\pi
−0.642098 + 0.766622i 0.721936π0.721936\pi
194194 0 0
195195 1.14801e9 + 1.45253e9i 0.793977 + 1.00458i
196196 0 0
197197 −1.39026e9 + 1.39026e9i −0.923060 + 0.923060i −0.997245 0.0741846i 0.976365π-0.976365\pi
0.0741846 + 0.997245i 0.476365π0.476365\pi
198198 0 0
199199 1.80023e8i 0.114793i −0.998351 0.0573964i 0.981720π-0.981720\pi
0.998351 0.0573964i 0.0182799π-0.0182799\pi
200200 0 0
201201 −1.38576e9 −0.848990
202202 0 0
203203 1.09483e8 + 1.09483e8i 0.0644705 + 0.0644705i
204204 0 0
205205 2.04630e9 1.61731e9i 1.15866 0.915750i
206206 0 0
207207 −4.97361e7 + 4.97361e7i −0.0270888 + 0.0270888i
208208 0 0
209209 7.77906e8i 0.407701i
210210 0 0
211211 4.17651e8 0.210709 0.105355 0.994435i 0.466402π-0.466402\pi
0.105355 + 0.994435i 0.466402π0.466402\pi
212212 0 0
213213 1.38219e9 + 1.38219e9i 0.671507 + 0.671507i
214214 0 0
215215 3.52423e9 + 4.12675e8i 1.64934 + 0.193132i
216216 0 0
217217 −6.40648e8 + 6.40648e8i −0.288922 + 0.288922i
218218 0 0
219219 1.96897e9i 0.855978i
220220 0 0
221221 2.98996e9 1.25342
222222 0 0
223223 −1.19002e9 1.19002e9i −0.481210 0.481210i 0.424308 0.905518i 0.360517π-0.360517\pi
−0.905518 + 0.424308i 0.860517π0.860517\pi
224224 0 0
225225 −3.62475e8 8.60692e7i −0.141432 0.0335829i
226226 0 0
227227 −2.69749e9 + 2.69749e9i −1.01591 + 1.01591i −0.0160418 + 0.999871i 0.505106π0.505106\pi
−0.999871 + 0.0160418i 0.994894π0.994894\pi
228228 0 0
229229 4.90322e9i 1.78295i −0.453069 0.891475i 0.649671π-0.649671\pi
0.453069 0.891475i 0.350329π-0.350329\pi
230230 0 0
231231 1.49297e9 0.524328
232232 0 0
233233 −2.60187e9 2.60187e9i −0.882798 0.882798i 0.111020 0.993818i 0.464588π-0.464588\pi
−0.993818 + 0.111020i 0.964588π0.964588\pi
234234 0 0
235235 2.33288e8 1.99227e9i 0.0764929 0.653247i
236236 0 0
237237 −1.62869e8 + 1.62869e8i −0.0516232 + 0.0516232i
238238 0 0
239239 4.57968e9i 1.40360i −0.712374 0.701800i 0.752380π-0.752380\pi
0.712374 0.701800i 0.247620π-0.247620\pi
240240 0 0
241241 5.82501e9 1.72675 0.863373 0.504566i 0.168348π-0.168348\pi
0.863373 + 0.504566i 0.168348π0.168348\pi
242242 0 0
243243 7.10820e8 + 7.10820e8i 0.203861 + 0.203861i
244244 0 0
245245 3.10270e9 + 3.92570e9i 0.861144 + 1.08957i
246246 0 0
247247 4.05012e9 4.05012e9i 1.08813 1.08813i
248248 0 0
249249 2.55769e9i 0.665350i
250250 0 0
251251 4.01845e9 1.01243 0.506214 0.862408i 0.331045π-0.331045\pi
0.506214 + 0.862408i 0.331045π0.331045\pi
252252 0 0
253253 2.80181e8 + 2.80181e8i 0.0683843 + 0.0683843i
254254 0 0
255255 2.77517e9 2.19337e9i 0.656340 0.518742i
256256 0 0
257257 −1.77795e9 + 1.77795e9i −0.407556 + 0.407556i −0.880885 0.473330i 0.843052π-0.843052\pi
0.473330 + 0.880885i 0.343052π0.343052\pi
258258 0 0
259259 1.05303e10i 2.34014i
260260 0 0
261261 3.97931e7 0.00857523
262262 0 0
263263 −1.87568e9 1.87568e9i −0.392046 0.392046i 0.483370 0.875416i 0.339412π-0.339412\pi
−0.875416 + 0.483370i 0.839412π0.839412\pi
264264 0 0
265265 3.27218e9 + 3.83161e8i 0.663519 + 0.0776958i
266266 0 0
267267 −3.05315e8 + 3.05315e8i −0.0600764 + 0.0600764i
268268 0 0
269269 5.91528e9i 1.12971i 0.825191 + 0.564854i 0.191068π0.191068\pi
−0.825191 + 0.564854i 0.808932π0.808932\pi
270270 0 0
271271 1.60432e9 0.297449 0.148725 0.988879i 0.452483π-0.452483\pi
0.148725 + 0.988879i 0.452483π0.452483\pi
272272 0 0
273273 −7.77304e9 7.77304e9i −1.39940 1.39940i
274274 0 0
275275 −4.84858e8 + 2.04195e9i −0.0847781 + 0.357037i
276276 0 0
277277 −1.67266e9 + 1.67266e9i −0.284111 + 0.284111i −0.834746 0.550635i 0.814386π-0.814386\pi
0.550635 + 0.834746i 0.314386π0.314386\pi
278278 0 0
279279 2.32853e8i 0.0384295i
280280 0 0
281281 9.29346e9 1.49057 0.745285 0.666747i 0.232314π-0.232314\pi
0.745285 + 0.666747i 0.232314π0.232314\pi
282282 0 0
283283 −4.61061e9 4.61061e9i −0.718808 0.718808i 0.249553 0.968361i 0.419716π-0.419716\pi
−0.968361 + 0.249553i 0.919716π0.919716\pi
284284 0 0
285285 7.88088e8 6.73024e9i 0.119453 1.02012i
286286 0 0
287287 −1.09506e10 + 1.09506e10i −1.61402 + 1.61402i
288288 0 0
289289 1.26318e9i 0.181082i
290290 0 0
291291 −3.03814e9 −0.423678
292292 0 0
293293 −6.99292e9 6.99292e9i −0.948829 0.948829i 0.0499241 0.998753i 0.484102π-0.484102\pi
−0.998753 + 0.0499241i 0.984102π0.984102\pi
294294 0 0
295295 −4.34688e9 5.49990e9i −0.573970 0.726217i
296296 0 0
297297 2.13781e9 2.13781e9i 0.274753 0.274753i
298298 0 0
299299 2.91749e9i 0.365026i
300300 0 0
301301 −2.10680e10 −2.56659
302302 0 0
303303 −4.21085e9 4.21085e9i −0.499573 0.499573i
304304 0 0
305305 −9.90498e9 + 7.82845e9i −1.14460 + 0.904642i
306306 0 0
307307 −2.31844e9 + 2.31844e9i −0.261001 + 0.261001i −0.825461 0.564459i 0.809085π-0.809085\pi
0.564459 + 0.825461i 0.309085π0.309085\pi
308308 0 0
309309 5.25746e9i 0.576690i
310310 0 0
311311 1.45111e10 1.55117 0.775584 0.631245i 0.217456π-0.217456\pi
0.775584 + 0.631245i 0.217456π0.217456\pi
312312 0 0
313313 2.03082e9 + 2.03082e9i 0.211590 + 0.211590i 0.804943 0.593353i 0.202196π-0.202196\pi
−0.593353 + 0.804943i 0.702196π0.702196\pi
314314 0 0
315315 2.19701e9 + 2.57262e8i 0.223146 + 0.0261296i
316316 0 0
317317 9.69018e8 9.69018e8i 0.0959611 0.0959611i −0.657497 0.753458i 0.728385π-0.728385\pi
0.753458 + 0.657497i 0.228385π0.228385\pi
318318 0 0
319319 2.24168e8i 0.0216477i
320320 0 0
321321 6.48666e9 0.610943
322322 0 0
323323 −7.73808e9 7.73808e9i −0.710924 0.710924i
324324 0 0
325325 1.31556e10 8.10688e9i 1.17918 0.726641i
326326 0 0
327327 −4.94312e9 + 4.94312e9i −0.432325 + 0.432325i
328328 0 0
329329 1.19099e10i 1.01654i
330330 0 0
331331 7.63072e9 0.635702 0.317851 0.948141i 0.397039π-0.397039\pi
0.317851 + 0.948141i 0.397039π0.397039\pi
332332 0 0
333333 1.91370e9 + 1.91370e9i 0.155631 + 0.155631i
334334 0 0
335335 −1.34517e9 + 1.14877e10i −0.106807 + 0.912126i
336336 0 0
337337 −2.47412e9 + 2.47412e9i −0.191824 + 0.191824i −0.796484 0.604660i 0.793309π-0.793309\pi
0.604660 + 0.796484i 0.293309π0.293309\pi
338338 0 0
339339 1.24162e10i 0.940135i
340340 0 0
341341 1.31174e9 0.0970132
342342 0 0
343343 −5.88110e9 5.88110e9i −0.424895 0.424895i
344344 0 0
345345 −2.14020e9 2.70790e9i −0.151070 0.191142i
346346 0 0
347347 2.63557e9 2.63557e9i 0.181785 0.181785i −0.610348 0.792133i 0.708971π-0.708971\pi
0.792133 + 0.610348i 0.208971π0.208971\pi
348348 0 0
349349 2.46965e10i 1.66469i −0.554257 0.832346i 0.686998π-0.686998\pi
0.554257 0.832346i 0.313002π-0.313002\pi
350350 0 0
351351 −2.22607e10 −1.46660
352352 0 0
353353 −1.25084e10 1.25084e10i −0.805566 0.805566i 0.178393 0.983959i 0.442910π-0.442910\pi
−0.983959 + 0.178393i 0.942910π0.942910\pi
354354 0 0
355355 1.27999e10 1.01165e10i 0.805923 0.636966i
356356 0 0
357357 −1.48510e10 + 1.48510e10i −0.914290 + 0.914290i
358358 0 0
359359 2.30279e10i 1.38636i 0.720763 + 0.693181i 0.243792π0.243792\pi
−0.720763 + 0.693181i 0.756208π0.756208\pi
360360 0 0
361361 −3.97999e9 −0.234344
362362 0 0
363363 9.82172e9 + 9.82172e9i 0.565667 + 0.565667i
364364 0 0
365365 1.63225e10 + 1.91131e9i 0.919634 + 0.107686i
366366 0 0
367367 9.47400e9 9.47400e9i 0.522239 0.522239i −0.396008 0.918247i 0.629605π-0.629605\pi
0.918247 + 0.396008i 0.129605π0.129605\pi
368368 0 0
369369 3.98015e9i 0.214681i
370370 0 0
371371 −1.95612e10 −1.03252
372372 0 0
373373 6.05089e9 + 6.05089e9i 0.312597 + 0.312597i 0.845915 0.533318i 0.179055π-0.179055\pi
−0.533318 + 0.845915i 0.679055π0.679055\pi
374374 0 0
375375 6.26354e9 1.71752e10i 0.316734 0.868513i
376376 0 0
377377 −1.16712e9 + 1.16712e9i −0.0577763 + 0.0577763i
378378 0 0
379379 4.27189e9i 0.207044i −0.994627 0.103522i 0.966989π-0.966989\pi
0.994627 0.103522i 0.0330113π-0.0330113\pi
380380 0 0
381381 1.23353e9 0.0585397
382382 0 0
383383 2.68406e10 + 2.68406e10i 1.24738 + 1.24738i 0.956874 + 0.290502i 0.0938224π0.0938224\pi
0.290502 + 0.956874i 0.406178π0.406178\pi
384384 0 0
385385 1.44925e9 1.23765e10i 0.0659628 0.563320i
386386 0 0
387387 −3.82873e9 + 3.82873e9i −0.170691 + 0.170691i
388388 0 0
389389 9.16592e8i 0.0400293i 0.999800 + 0.0200146i 0.00637128π0.00637128\pi
−0.999800 + 0.0200146i 0.993629π0.993629\pi
390390 0 0
391391 −5.57410e9 −0.238489
392392 0 0
393393 2.03080e10 + 2.03080e10i 0.851330 + 0.851330i
394394 0 0
395395 1.19206e9 + 1.50826e9i 0.0489678 + 0.0619567i
396396 0 0
397397 5.24090e9 5.24090e9i 0.210981 0.210981i −0.593703 0.804684i 0.702335π-0.702335\pi
0.804684 + 0.593703i 0.202335π0.202335\pi
398398 0 0
399399 4.02336e10i 1.58744i
400400 0 0
401401 1.76862e10 0.684000 0.342000 0.939700i 0.388896π-0.388896\pi
0.342000 + 0.939700i 0.388896π0.388896\pi
402402 0 0
403403 −6.82950e9 6.82950e9i −0.258922 0.258922i
404404 0 0
405405 −1.75933e10 + 1.39049e10i −0.653923 + 0.516832i
406406 0 0
407407 1.07805e10 1.07805e10i 0.392882 0.392882i
408408 0 0
409409 2.18253e9i 0.0779950i 0.999239 + 0.0389975i 0.0124164π0.0124164\pi
−0.999239 + 0.0389975i 0.987584π0.987584\pi
410410 0 0
411411 1.55445e9 0.0544765
412412 0 0
413413 2.94321e10 + 2.94321e10i 1.01163 + 1.01163i
414414 0 0
415415 2.12029e10 + 2.48278e9i 0.714829 + 0.0837040i
416416 0 0
417417 2.11964e10 2.11964e10i 0.700998 0.700998i
418418 0 0
419419 3.02443e10i 0.981268i −0.871366 0.490634i 0.836765π-0.836765\pi
0.871366 0.490634i 0.163235π-0.163235\pi
420420 0 0
421421 1.77840e10 0.566109 0.283055 0.959104i 0.408652π-0.408652\pi
0.283055 + 0.959104i 0.408652π0.408652\pi
422422 0 0
423423 2.16441e9 + 2.16441e9i 0.0676048 + 0.0676048i
424424 0 0
425425 −1.54889e10 2.51349e10i −0.474749 0.770410i
426426 0 0
427427 5.30055e10 5.30055e10i 1.59444 1.59444i
428428 0 0
429429 1.59155e10i 0.469884i
430430 0 0
431431 2.12717e10 0.616443 0.308221 0.951315i 0.400266π-0.400266\pi
0.308221 + 0.951315i 0.400266π0.400266\pi
432432 0 0
433433 −2.24808e10 2.24808e10i −0.639529 0.639529i 0.310910 0.950439i 0.399366π-0.399366\pi
−0.950439 + 0.310910i 0.899366π0.899366\pi
434434 0 0
435435 −2.27103e8 + 1.93945e9i −0.00634257 + 0.0541653i
436436 0 0
437437 −7.55051e9 + 7.55051e9i −0.207038 + 0.207038i
438438 0 0
439439 1.46904e10i 0.395526i −0.980250 0.197763i 0.936632π-0.936632\pi
0.980250 0.197763i 0.0633676π-0.0633676\pi
440440 0 0
441441 −7.63567e9 −0.201880
442442 0 0
443443 −1.79953e10 1.79953e10i −0.467245 0.467245i 0.433776 0.901021i 0.357181π-0.357181\pi
−0.901021 + 0.433776i 0.857181π0.857181\pi
444444 0 0
445445 2.23465e9 + 2.82740e9i 0.0569861 + 0.0721019i
446446 0 0
447447 −2.97817e10 + 2.97817e10i −0.745968 + 0.745968i
448448 0 0
449449 2.81226e10i 0.691943i 0.938245 + 0.345972i 0.112451π0.112451\pi
−0.938245 + 0.345972i 0.887549π0.887549\pi
450450 0 0
451451 2.24216e10 0.541951
452452 0 0
453453 −1.45745e9 1.45745e9i −0.0346100 0.0346100i
454454 0 0
455455 −7.19829e10 + 5.68921e10i −1.67951 + 1.32741i
456456 0 0
457457 5.32217e10 5.32217e10i 1.22018 1.22018i 0.252614 0.967567i 0.418710π-0.418710\pi
0.967567 0.252614i 0.0812901π-0.0812901\pi
458458 0 0
459459 4.25310e10i 0.958196i
460460 0 0
461461 2.80695e9 0.0621485 0.0310743 0.999517i 0.490107π-0.490107\pi
0.0310743 + 0.999517i 0.490107π0.490107\pi
462462 0 0
463463 1.84228e10 + 1.84228e10i 0.400896 + 0.400896i 0.878549 0.477653i 0.158512π-0.158512\pi
−0.477653 + 0.878549i 0.658512π0.658512\pi
464464 0 0
465465 −1.13489e10 1.32891e9i −0.242739 0.0284239i
466466 0 0
467467 1.19286e10 1.19286e10i 0.250796 0.250796i −0.570501 0.821297i 0.693251π-0.693251\pi
0.821297 + 0.570501i 0.193251π0.193251\pi
468468 0 0
469469 6.86739e10i 1.41939i
470470 0 0
471471 3.41658e10 0.694238
472472 0 0
473473 2.15686e10 + 2.15686e10i 0.430900 + 0.430900i
474474 0 0
475475 −5.50278e10 1.30663e10i −1.08095 0.256672i
476476 0 0
477477 −3.55490e9 + 3.55490e9i −0.0686679 + 0.0686679i
478478 0 0
479479 8.08280e10i 1.53539i −0.640813 0.767697i 0.721403π-0.721403\pi
0.640813 0.767697i 0.278597π-0.278597\pi
480480 0 0
481481 −1.12256e11 −2.09715
482482 0 0
483483 1.44911e10 + 1.44911e10i 0.266263 + 0.266263i
484484 0 0
485485 −2.94916e9 + 2.51857e10i −0.0533006 + 0.455185i
486486 0 0
487487 −7.93140e10 + 7.93140e10i −1.41005 + 1.41005i −0.650797 + 0.759252i 0.725565π0.725565\pi
−0.759252 + 0.650797i 0.774435π0.774435\pi
488488 0 0
489489 2.50640e10i 0.438343i
490490 0 0
491491 −6.00911e10 −1.03391 −0.516957 0.856011i 0.672935π-0.672935\pi
−0.516957 + 0.856011i 0.672935π0.672935\pi
492492 0 0
493493 2.22988e9 + 2.22988e9i 0.0377479 + 0.0377479i
494494 0 0
495495 −1.98584e9 2.51259e9i −0.0330768 0.0418505i
496496 0 0
497497 −6.84974e10 + 6.84974e10i −1.12266 + 1.12266i
498498 0 0
499499 1.08749e11i 1.75397i −0.480516 0.876986i 0.659551π-0.659551\pi
0.480516 0.876986i 0.340449π-0.340449\pi
500500 0 0
501501 −5.77610e10 −0.916819
502502 0 0
503503 4.36650e10 + 4.36650e10i 0.682122 + 0.682122i 0.960478 0.278356i 0.0897896π-0.0897896\pi
−0.278356 + 0.960478i 0.589790π0.589790\pi
504504 0 0
505505 −3.89949e10 + 3.08198e10i −0.599573 + 0.473876i
506506 0 0
507507 3.96706e10 3.96706e10i 0.600394 0.600394i
508508 0 0
509509 6.95461e10i 1.03610i −0.855350 0.518050i 0.826658π-0.826658\pi
0.855350 0.518050i 0.173342π-0.173342\pi
510510 0 0
511511 −9.75763e10 −1.43107
512512 0 0
513513 5.76112e10 + 5.76112e10i 0.831835 + 0.831835i
514514 0 0
515515 4.35836e10 + 5.10349e9i 0.619576 + 0.0725502i
516516 0 0
517517 1.21929e10 1.21929e10i 0.170665 0.170665i
518518 0 0
519519 6.79504e10i 0.936533i
520520 0 0
521521 1.18226e10 0.160459 0.0802294 0.996776i 0.474435π-0.474435\pi
0.0802294 + 0.996776i 0.474435π0.474435\pi
522522 0 0
523523 −5.44052e10 5.44052e10i −0.727166 0.727166i 0.242888 0.970054i 0.421905π-0.421905\pi
−0.970054 + 0.242888i 0.921905π0.921905\pi
524524 0 0
525525 −2.50770e10 + 1.05610e11i −0.330095 + 1.39017i
526526 0 0
527527 −1.30483e10 + 1.30483e10i −0.169166 + 0.169166i
528528 0 0
529529 7.28720e10i 0.930546i
530530 0 0
531531 1.06975e10 0.134557
532532 0 0
533533 −1.16736e11 1.16736e11i −1.44643 1.44643i
534534 0 0
535535 6.29669e9 5.37735e10i 0.0768595 0.656377i
536536 0 0
537537 4.98296e10 4.98296e10i 0.599225 0.599225i
538538 0 0
539539 4.30144e10i 0.509634i
540540 0 0
541541 1.29671e11 1.51375 0.756873 0.653562i 0.226726π-0.226726\pi
0.756873 + 0.653562i 0.226726π0.226726\pi
542542 0 0
543543 2.16850e10 + 2.16850e10i 0.249437 + 0.249437i
544544 0 0
545545 3.61794e10 + 4.57761e10i 0.410087 + 0.518864i
546546 0 0
547547 −2.70067e10 + 2.70067e10i −0.301664 + 0.301664i −0.841664 0.540001i 0.818424π-0.818424\pi
0.540001 + 0.841664i 0.318424π0.318424\pi
548548 0 0
549549 1.92656e10i 0.212077i
550550 0 0
551551 6.04105e9 0.0655399
552552 0 0
553553 −8.07130e9 8.07130e9i −0.0863064 0.0863064i
554554 0 0
555555 −1.04192e11 + 8.23487e10i −1.09815 + 0.867930i
556556 0 0
557557 1.51528e9 1.51528e9i 0.0157425 0.0157425i −0.699192 0.714934i 0.746457π-0.746457\pi
0.714934 + 0.699192i 0.246457π0.246457\pi
558558 0 0
559559 2.24591e11i 2.30009i
560560 0 0
561561 3.04079e10 0.306997
562562 0 0
563563 1.16988e11 + 1.16988e11i 1.16441 + 1.16441i 0.983500 + 0.180911i 0.0579045π0.0579045\pi
0.180911 + 0.983500i 0.442095π0.442095\pi
564564 0 0
565565 −1.02929e11 1.20526e10i −1.01005 0.118273i
566566 0 0
567567 9.41486e10 9.41486e10i 0.910923 0.910923i
568568 0 0
569569 6.87794e10i 0.656159i −0.944650 0.328080i 0.893599π-0.893599\pi
0.944650 0.328080i 0.106401π-0.106401\pi
570570 0 0
571571 1.03730e10 0.0975795 0.0487897 0.998809i 0.484464π-0.484464\pi
0.0487897 + 0.998809i 0.484464π0.484464\pi
572572 0 0
573573 2.94427e10 + 2.94427e10i 0.273123 + 0.273123i
574574 0 0
575575 −2.45257e10 + 1.51134e10i −0.224362 + 0.138258i
576576 0 0
577577 5.84784e10 5.84784e10i 0.527584 0.527584i −0.392267 0.919851i 0.628309π-0.628309\pi
0.919851 + 0.392267i 0.128309π0.128309\pi
578578 0 0
579579 1.82967e10i 0.162802i
580580 0 0
581581 −1.26751e11 −1.11237
582582 0 0
583583 2.00260e10 + 2.00260e10i 0.173348 + 0.173348i
584584 0 0
585585 −2.74249e9 + 2.34208e10i −0.0234165 + 0.199976i
586586 0 0
587587 −2.36584e10 + 2.36584e10i −0.199266 + 0.199266i −0.799685 0.600420i 0.795000π-0.795000\pi
0.600420 + 0.799685i 0.295000π0.295000\pi
588588 0 0
589589 3.53497e10i 0.293714i
590590 0 0
591591 1.47226e11 1.20680
592592 0 0
593593 1.22246e11 + 1.22246e11i 0.988589 + 0.988589i 0.999936 0.0113464i 0.00361175π-0.00361175\pi
−0.0113464 + 0.999936i 0.503612π0.503612\pi
594594 0 0
595595 1.08697e11 + 1.37529e11i 0.867261 + 1.09730i
596596 0 0
597597 −9.53208e9 + 9.53208e9i −0.0750396 + 0.0750396i
598598 0 0
599599 9.59446e10i 0.745269i 0.927978 + 0.372635i 0.121546π0.121546\pi
−0.927978 + 0.372635i 0.878454π0.878454\pi
600600 0 0
601601 1.94971e11 1.49442 0.747210 0.664588i 0.231393π-0.231393\pi
0.747210 + 0.664588i 0.231393π0.231393\pi
602602 0 0
603603 −1.24803e10 1.24803e10i −0.0943964 0.0943964i
604604 0 0
605605 9.09547e10 7.18866e10i 0.678897 0.536570i
606606 0 0
607607 1.26277e11 1.26277e11i 0.930188 0.930188i −0.0675290 0.997717i 0.521512π-0.521512\pi
0.997717 + 0.0675290i 0.0215115π0.0215115\pi
608608 0 0
609609 1.15941e10i 0.0842882i
610610 0 0
611611 −1.26963e11 −0.910984
612612 0 0
613613 1.84169e11 + 1.84169e11i 1.30429 + 1.30429i 0.925468 + 0.378826i 0.123672π0.123672\pi
0.378826 + 0.925468i 0.376328π0.376328\pi
614614 0 0
615615 −1.93986e11 2.27151e10i −1.35603 0.158786i
616616 0 0
617617 −1.20679e11 + 1.20679e11i −0.832707 + 0.832707i −0.987886 0.155180i 0.950404π-0.950404\pi
0.155180 + 0.987886i 0.450404π0.450404\pi
618618 0 0
619619 1.43750e11i 0.979143i 0.871963 + 0.489572i 0.162847π0.162847\pi
−0.871963 + 0.489572i 0.837153π0.837153\pi
620620 0 0
621621 4.15000e10 0.279050
622622 0 0
623623 −1.51305e10 1.51305e10i −0.100439 0.100439i
624624 0 0
625625 −1.36300e11 6.85961e10i −0.893255 0.449551i
626626 0 0
627627 4.11896e10 4.11896e10i 0.266512 0.266512i
628628 0 0
629629 2.14475e11i 1.37017i
630630 0 0
631631 1.08715e11 0.685760 0.342880 0.939379i 0.388598π-0.388598\pi
0.342880 + 0.939379i 0.388598π0.388598\pi
632632 0 0
633633 −2.21144e10 2.21144e10i −0.137740 0.137740i
634634 0 0
635635 1.19741e9 1.02258e10i 0.00736456 0.0628930i
636636 0 0
637637 2.23952e11 2.23952e11i 1.36018 1.36018i
638638 0 0
639639 2.48964e10i 0.149325i
640640 0 0
641641 −3.25454e11 −1.92778 −0.963890 0.266299i 0.914199π-0.914199\pi
−0.963890 + 0.266299i 0.914199π0.914199\pi
642642 0 0
643643 −4.11926e10 4.11926e10i −0.240977 0.240977i 0.576278 0.817254i 0.304505π-0.304505\pi
−0.817254 + 0.576278i 0.804505π0.804505\pi
644644 0 0
645645 −1.64755e11 2.08457e11i −0.951918 1.20442i
646646 0 0
647647 8.13749e10 8.13749e10i 0.464380 0.464380i −0.435708 0.900088i 0.643502π-0.643502\pi
0.900088 + 0.435708i 0.143502π0.143502\pi
648648 0 0
649649 6.02630e10i 0.339682i
650650 0 0
651651 6.78438e10 0.377734
652652 0 0
653653 1.29147e11 + 1.29147e11i 0.710285 + 0.710285i 0.966595 0.256310i 0.0825067π-0.0825067\pi
−0.256310 + 0.966595i 0.582507π0.582507\pi
654654 0 0
655655 1.88064e11 1.48638e11i 1.02174 0.807539i
656656 0 0
657657 −1.77328e10 + 1.77328e10i −0.0951733 + 0.0951733i
658658 0 0
659659 2.74018e10i 0.145291i 0.997358 + 0.0726453i 0.0231441π0.0231441\pi
−0.997358 + 0.0726453i 0.976856π0.976856\pi
660660 0 0
661661 −1.19547e11 −0.626228 −0.313114 0.949716i 0.601372π-0.601372\pi
−0.313114 + 0.949716i 0.601372π0.601372\pi
662662 0 0
663663 −1.58317e11 1.58317e11i −0.819355 0.819355i
664664 0 0
665665 3.33531e11 + 3.90553e10i 1.70549 + 0.199707i
666666 0 0
667667 2.17582e9 2.17582e9i 0.0109931 0.0109931i
668668 0 0
669669 1.26021e11i 0.629129i
670670 0 0
671671 −1.08530e11 −0.535377
672672 0 0
673673 −2.17592e11 2.17592e11i −1.06068 1.06068i −0.998036 0.0626416i 0.980047π-0.980047\pi
−0.0626416 0.998036i 0.519953π-0.519953\pi
674674 0 0
675675 1.15317e11 + 1.87133e11i 0.555492 + 0.901438i
676676 0 0
677677 1.34337e11 1.34337e11i 0.639503 0.639503i −0.310930 0.950433i 0.600640π-0.600640\pi
0.950433 + 0.310930i 0.100640π0.100640\pi
678678 0 0
679679 1.50561e11i 0.708327i
680680 0 0
681681 2.85661e11 1.32820
682682 0 0
683683 2.62510e11 + 2.62510e11i 1.20632 + 1.20632i 0.972210 + 0.234110i 0.0752177π0.0752177\pi
0.234110 + 0.972210i 0.424782π0.424782\pi
684684 0 0
685685 1.50893e9 1.28862e10i 0.00685339 0.0585277i
686686 0 0
687687 −2.59622e11 + 2.59622e11i −1.16551 + 1.16551i
688688 0 0
689689 2.08528e11i 0.925310i
690690 0 0
691691 −3.13746e11 −1.37615 −0.688075 0.725639i 0.741544π-0.741544\pi
−0.688075 + 0.725639i 0.741544π0.741544\pi
692692 0 0
693693 1.34459e10 + 1.34459e10i 0.0582982 + 0.0582982i
694694 0 0
695695 −1.55139e11 1.96290e11i −0.664940 0.841318i
696696 0 0
697697 −2.23035e11 + 2.23035e11i −0.945021 + 0.945021i
698698 0 0
699699 2.75534e11i 1.15416i
700700 0 0
701701 2.56197e11 1.06097 0.530483 0.847696i 0.322011π-0.322011\pi
0.530483 + 0.847696i 0.322011π0.322011\pi
702702 0 0
703703 2.90521e11 + 2.90521e11i 1.18948 + 1.18948i
704704 0 0
705705 −1.17842e11 + 9.31370e10i −0.477027 + 0.377021i
706706 0 0
707707 2.08677e11 2.08677e11i 0.835213 0.835213i
708708 0 0
709709 1.51234e10i 0.0598499i 0.999552 + 0.0299250i 0.00952683π0.00952683\pi
−0.999552 + 0.0299250i 0.990473π0.990473\pi
710710 0 0
711711 −2.93364e9 −0.0114796
712712 0 0
713713 1.27320e10 + 1.27320e10i 0.0492651 + 0.0492651i
714714 0 0
715715 1.31937e11 + 1.54494e10i 0.504828 + 0.0591136i
716716 0 0
717717 −2.42491e11 + 2.42491e11i −0.917527 + 0.917527i
718718 0 0
719719 4.20879e11i 1.57486i −0.616404 0.787430i 0.711411π-0.711411\pi
0.616404 0.787430i 0.288589π-0.288589\pi
720720 0 0
721721 −2.60544e11 −0.964140
722722 0 0
723723 −3.08430e11 3.08430e11i −1.12877 1.12877i
724724 0 0
725725 1.58573e10 + 3.76530e9i 0.0573954 + 0.0136285i
726726 0 0
727727 −2.08188e11 + 2.08188e11i −0.745277 + 0.745277i −0.973588 0.228312i 0.926680π-0.926680\pi
0.228312 + 0.973588i 0.426680π0.426680\pi
728728 0 0
729729 3.10681e11i 1.10003i
730730 0 0
731731 −4.29099e11 −1.50275
732732 0 0
733733 −2.48791e11 2.48791e11i −0.861826 0.861826i 0.129725 0.991550i 0.458591π-0.458591\pi
−0.991550 + 0.129725i 0.958591π0.958591\pi
734734 0 0
735735 4.35774e10 3.72149e11i 0.149318 1.27517i
736736 0 0
737737 −7.03058e10 + 7.03058e10i −0.238298 + 0.238298i
738738 0 0
739739 1.15406e11i 0.386946i 0.981106 + 0.193473i 0.0619752π0.0619752\pi
−0.981106 + 0.193473i 0.938025π0.938025\pi
740740 0 0
741741 −4.28902e11 −1.42261
742742 0 0
743743 −1.60556e11 1.60556e11i −0.526833 0.526833i 0.392794 0.919627i 0.371509π-0.371509\pi
−0.919627 + 0.392794i 0.871509π0.871509\pi
744744 0 0
745745 2.17977e11 + 2.75796e11i 0.707596 + 0.895289i
746746 0 0
747747 −2.30348e10 + 2.30348e10i −0.0739780 + 0.0739780i
748748 0 0
749749 3.21459e11i 1.02141i
750750 0 0
751751 5.98277e10 0.188080 0.0940400 0.995568i 0.470022π-0.470022\pi
0.0940400 + 0.995568i 0.470022π0.470022\pi
752752 0 0
753753 −2.12774e11 2.12774e11i −0.661819 0.661819i
754754 0 0
755755 −1.34969e10 + 1.06673e10i −0.0415379 + 0.0328297i
756756 0 0
757757 −2.88999e11 + 2.88999e11i −0.880060 + 0.880060i −0.993540 0.113480i 0.963800π-0.963800\pi
0.113480 + 0.993540i 0.463800π0.463800\pi
758758 0 0
759759 2.96708e10i 0.0894050i
760760 0 0
761761 −1.15311e11 −0.343819 −0.171910 0.985113i 0.554994π-0.554994\pi
−0.171910 + 0.985113i 0.554994π0.554994\pi
762762 0 0
763763 −2.44966e11 2.44966e11i −0.722783 0.722783i
764764 0 0
765765 4.47473e10 + 5.23975e9i 0.130654 + 0.0152991i
766766 0 0
767767 −3.13755e11 + 3.13755e11i −0.906587 + 0.906587i
768768 0 0
769769 1.71633e11i 0.490791i 0.969423 + 0.245395i 0.0789178π0.0789178\pi
−0.969423 + 0.245395i 0.921082π0.921082\pi
770770 0 0
771771 1.88283e11 0.532835
772772 0 0
773773 −3.60143e11 3.60143e11i −1.00869 1.00869i −0.999962 0.00872501i 0.997223π-0.997223\pi
−0.00872501 0.999962i 0.502777π-0.502777\pi
774774 0 0
775775 −2.20330e10 + 9.27905e10i −0.0610754 + 0.257215i
776776 0 0
777777 5.57572e11 5.57572e11i 1.52974 1.52974i
778778 0 0
779779 6.04232e11i 1.64079i
780780 0 0
781781 1.40250e11 0.376963
782782 0 0
783783 −1.66018e10 1.66018e10i −0.0441679 0.0441679i
784784 0 0
785785 3.31653e10 2.83230e11i 0.0873383 0.745866i
786786 0 0
787787 −3.81822e11 + 3.81822e11i −0.995318 + 0.995318i −0.999989 0.00467136i 0.998513π-0.998513\pi
0.00467136 + 0.999989i 0.498513π0.498513\pi
788788 0 0
789789 1.98632e11i 0.512557i
790790 0 0
791791 6.15310e11 1.57177
792792 0 0
793793 5.65054e11 + 5.65054e11i 1.42888 + 1.42888i
794794 0 0
795795 −1.52972e11 1.93548e11i −0.382950 0.484529i
796796 0 0
797797 4.32433e11 4.32433e11i 1.07173 1.07173i 0.0745090 0.997220i 0.476261π-0.476261\pi
0.997220 0.0745090i 0.0237389π-0.0237389\pi
798798 0 0
799799 2.42573e11i 0.595189i
800800 0 0
801801 −5.49941e9 −0.0133594
802802 0 0
803803 9.98949e10 + 9.98949e10i 0.240260 + 0.240260i
804804 0 0
805805 1.34195e11 1.06062e11i 0.319561 0.252567i
806806 0 0
807807 3.13210e11 3.13210e11i 0.738485 0.738485i
808808 0 0
809809 4.04427e11i 0.944160i 0.881556 + 0.472080i 0.156497π0.156497\pi
−0.881556 + 0.472080i 0.843503π0.843503\pi
810810 0 0
811811 6.23463e11 1.44121 0.720605 0.693346i 0.243864π-0.243864\pi
0.720605 + 0.693346i 0.243864π0.243864\pi
812812 0 0
813813 −8.49475e10 8.49475e10i −0.194441 0.194441i
814814 0 0
815815 2.07777e11 + 2.43299e10i 0.470941 + 0.0551456i
816816 0 0
817817 −5.81245e11 + 5.81245e11i −1.30458 + 1.30458i
818818 0 0
819819 1.40010e11i 0.311188i
820820 0 0
821821 3.67266e11 0.808366 0.404183 0.914678i 0.367556π-0.367556\pi
0.404183 + 0.914678i 0.367556π0.367556\pi
822822 0 0
823823 −1.17460e11 1.17460e11i −0.256030 0.256030i 0.567407 0.823437i 0.307947π-0.307947\pi
−0.823437 + 0.567407i 0.807947π0.807947\pi
824824 0 0
825825 1.33793e11 8.24468e10i 0.288813 0.177975i
826826 0 0
827827 5.29740e11 5.29740e11i 1.13251 1.13251i 0.142748 0.989759i 0.454406π-0.454406\pi
0.989759 0.142748i 0.0455939π-0.0455939\pi
828828 0 0
829829 2.39704e11i 0.507525i −0.967267 0.253762i 0.918332π-0.918332\pi
0.967267 0.253762i 0.0816681π-0.0816681\pi
830830 0 0
831831 1.77132e11 0.371444
832832 0 0
833833 −4.27878e11 4.27878e11i −0.888669 0.888669i
834834 0 0
835835 −5.60694e10 + 4.78831e11i −0.115340 + 0.984999i
836836 0 0
837837 9.71467e10 9.71467e10i 0.197937 0.197937i
838838 0 0
839839 5.22857e11i 1.05520i −0.849493 0.527601i 0.823092π-0.823092\pi
0.849493 0.527601i 0.176908π-0.176908\pi
840840 0 0
841841 4.98506e11 0.996520
842842 0 0
843843 −4.92082e11 4.92082e11i −0.974378 0.974378i
844844 0 0
845845 −2.90355e11 3.67372e11i −0.569511 0.720576i
846846 0 0
847847 −4.86735e11 + 4.86735e11i −0.945712 + 0.945712i
848848 0 0
849849 4.88258e11i 0.939764i
850850 0 0
851851 2.09276e11 0.399025
852852 0 0
853853 −2.94110e11 2.94110e11i −0.555537 0.555537i 0.372496 0.928034i 0.378502π-0.378502\pi
−0.928034 + 0.372496i 0.878502π0.878502\pi
854854 0 0
855855 6.77110e10 5.35158e10i 0.126705 0.100142i
856856 0 0
857857 −1.69558e11 + 1.69558e11i −0.314336 + 0.314336i −0.846587 0.532251i 0.821346π-0.821346\pi
0.532251 + 0.846587i 0.321346π0.321346\pi
858858 0 0
859859 3.81187e11i 0.700109i −0.936729 0.350055i 0.886163π-0.886163\pi
0.936729 0.350055i 0.113837π-0.113837\pi
860860 0 0
861861 1.15965e12 2.11016
862862 0 0
863863 −8.42734e10 8.42734e10i −0.151931 0.151931i 0.627049 0.778980i 0.284263π-0.284263\pi
−0.778980 + 0.627049i 0.784263π0.784263\pi
864864 0 0
865865 5.63300e11 + 6.59605e10i 1.00618 + 0.117820i
866866 0 0
867867 6.68848e10 6.68848e10i 0.118373 0.118373i
868868 0 0
869869 1.65262e10i 0.0289797i
870870 0 0
871871 7.32085e11 1.27200
872872 0 0
873873 −2.73618e10 2.73618e10i −0.0471073 0.0471073i
874874 0 0
875875 8.51151e11 + 3.10402e11i 1.45203 + 0.529533i
876876 0 0
877877 3.72176e10 3.72176e10i 0.0629144 0.0629144i −0.674950 0.737864i 0.735835π-0.735835\pi
0.737864 + 0.674950i 0.235835π0.235835\pi
878878 0 0
879879 7.40541e11i 1.24049i
880880 0 0
881881 −4.91908e10 −0.0816545 −0.0408273 0.999166i 0.512999π-0.512999\pi
−0.0408273 + 0.999166i 0.512999π0.512999\pi
882882 0 0
883883 −4.82360e11 4.82360e11i −0.793467 0.793467i 0.188589 0.982056i 0.439609π-0.439609\pi
−0.982056 + 0.188589i 0.939609π0.939609\pi
884884 0 0
885885 −6.10518e10 + 5.21380e11i −0.0995235 + 0.849927i
886886 0 0
887887 −7.58872e11 + 7.58872e11i −1.22595 + 1.22595i −0.260472 + 0.965481i 0.583878π0.583878\pi
−0.965481 + 0.260472i 0.916122π0.916122\pi
888888 0 0
889889 6.11301e10i 0.0978697i
890890 0 0
891891 −1.92772e11 −0.305867
892892 0 0
893893 3.28582e11 + 3.28582e11i 0.516699 + 0.516699i
894894 0 0
895895 −3.64710e11 4.61451e11i −0.568402 0.719173i
896896 0 0
897897 −1.54479e11 + 1.54479e11i −0.238616 + 0.238616i
898898 0 0
899899 1.01867e10i 0.0155953i
900900 0 0
901901 −3.98410e11 −0.604548
902902 0 0
903903 1.11553e12 + 1.11553e12i 1.67777 + 1.67777i
904904 0 0
905905 2.00816e11 1.58716e11i 0.299367 0.236606i
906906 0 0
907907 8.25250e11 8.25250e11i 1.21943 1.21943i 0.251597 0.967832i 0.419044π-0.419044\pi
0.967832 0.251597i 0.0809558π-0.0809558\pi
908908 0 0
909909 7.58468e10i 0.111092i
910910 0 0
911911 −5.86650e11 −0.851737 −0.425869 0.904785i 0.640031π-0.640031\pi
−0.425869 + 0.904785i 0.640031π0.640031\pi
912912 0 0
913913 1.29763e11 + 1.29763e11i 0.186753 + 0.186753i
914914 0 0
915915 9.38973e11 + 1.09951e11i 1.33958 + 0.156860i
916916 0 0
917917 −1.00641e12 + 1.00641e12i −1.42330 + 1.42330i
918918 0 0
919919 1.27566e12i 1.78843i 0.447640 + 0.894214i 0.352265π0.352265\pi
−0.447640 + 0.894214i 0.647735π0.647735\pi
920920 0 0
921921 2.45520e11 0.341231
922922 0 0
923923 −7.30203e11 7.30203e11i −1.00609 1.00609i
924924 0 0
925925 5.81519e11 + 9.43674e11i 0.794322 + 1.28901i
926926 0 0
927927 −4.73493e10 + 4.73493e10i −0.0641202 + 0.0641202i
928928 0 0
929929 1.11916e12i 1.50255i −0.659991 0.751274i 0.729440π-0.729440\pi
0.659991 0.751274i 0.270560π-0.270560\pi
930930 0 0
931931 −1.15918e12 −1.54295
932932 0 0
933933 −7.68353e11 7.68353e11i −1.01399 1.01399i
934934 0 0
935935 2.95174e10 2.52077e11i 0.0386217 0.329828i
936936 0 0
937937 3.48571e11 3.48571e11i 0.452202 0.452202i −0.443883 0.896085i 0.646399π-0.646399\pi
0.896085 + 0.443883i 0.146399π0.146399\pi
938938 0 0
939939 2.15061e11i 0.276631i
940940 0 0
941941 3.10729e11 0.396299 0.198150 0.980172i 0.436507π-0.436507\pi
0.198150 + 0.980172i 0.436507π0.436507\pi
942942 0 0
943943 2.17628e11 + 2.17628e11i 0.275213 + 0.275213i
944944 0 0
945945 −8.09265e11 1.02393e12i −1.01476 1.28393i
946946 0 0
947947 4.81109e11 4.81109e11i 0.598197 0.598197i −0.341636 0.939832i 0.610981π-0.610981\pi
0.939832 + 0.341636i 0.110981π0.110981\pi
948948 0 0
949949 1.04019e12i 1.28247i
950950 0 0
951951 −1.02618e11 −0.125459
952952 0 0
953953 2.19961e11 + 2.19961e11i 0.266670 + 0.266670i 0.827757 0.561087i 0.189617π-0.189617\pi
−0.561087 + 0.827757i 0.689617π0.689617\pi
954954 0 0
955955 2.72656e11 2.15495e11i 0.327794 0.259074i
956956 0 0
957957 −1.18696e10 + 1.18696e10i −0.0141510 + 0.0141510i
958958 0 0
959959 7.70338e10i 0.0910767i
960960 0 0
961961 −7.93283e11 −0.930110
962962 0 0
963963 5.84196e10 + 5.84196e10i 0.0679287 + 0.0679287i
964964 0 0
965965 −1.51677e11 1.77609e10i −0.174909 0.0204812i
966966 0 0
967967 −6.94014e11 + 6.94014e11i −0.793711 + 0.793711i −0.982095 0.188384i 0.939675π-0.939675\pi
0.188384 + 0.982095i 0.439675π0.439675\pi
968968 0 0
969969 8.19453e11i 0.929456i
970970 0 0
971971 −1.29419e12 −1.45587 −0.727935 0.685646i 0.759520π-0.759520\pi
−0.727935 + 0.685646i 0.759520π0.759520\pi
972972 0 0
973973 1.05043e12 + 1.05043e12i 1.17197 + 1.17197i
974974 0 0
975975 −1.12584e12 2.67328e11i −1.24582 0.295819i
976976 0 0
977977 2.00533e11 2.00533e11i 0.220093 0.220093i −0.588444 0.808538i 0.700259π-0.700259\pi
0.808538 + 0.588444i 0.200259π0.200259\pi
978978 0 0
979979 3.09801e10i 0.0337250i
980980 0 0
981981 −8.90367e10 −0.0961375
982982 0 0
983983 −4.14637e11 4.14637e11i −0.444072 0.444072i 0.449306 0.893378i 0.351671π-0.351671\pi
−0.893378 + 0.449306i 0.851671π0.851671\pi
984984 0 0
985985 1.42915e11 1.22049e12i 0.151821 1.29655i
986986 0 0
987987 6.30619e11 6.30619e11i 0.664505 0.664505i
988988 0 0
989989 4.18698e11i 0.437638i
990990 0 0
991991 6.51779e11 0.675780 0.337890 0.941186i 0.390287π-0.390287\pi
0.337890 + 0.941186i 0.390287π0.390287\pi
992992 0 0
993993 −4.04041e11 4.04041e11i −0.415555 0.415555i
994994 0 0
995995 6.97667e10 + 8.82725e10i 0.0711796 + 0.0900603i
996996 0 0
997997 −6.71342e11 + 6.71342e11i −0.679459 + 0.679459i −0.959878 0.280419i 0.909527π-0.909527\pi
0.280419 + 0.959878i 0.409527π0.409527\pi
998998 0 0
999999 1.59680e12i 1.60320i
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 80.9.p.b.33.1 4
4.3 odd 2 10.9.c.a.3.2 4
5.2 odd 4 inner 80.9.p.b.17.1 4
12.11 even 2 90.9.g.b.73.2 4
20.3 even 4 50.9.c.e.7.1 4
20.7 even 4 10.9.c.a.7.2 yes 4
20.19 odd 2 50.9.c.e.43.1 4
60.47 odd 4 90.9.g.b.37.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.9.c.a.3.2 4 4.3 odd 2
10.9.c.a.7.2 yes 4 20.7 even 4
50.9.c.e.7.1 4 20.3 even 4
50.9.c.e.43.1 4 20.19 odd 2
80.9.p.b.17.1 4 5.2 odd 4 inner
80.9.p.b.33.1 4 1.1 even 1 trivial
90.9.g.b.37.2 4 60.47 odd 4
90.9.g.b.73.2 4 12.11 even 2