Properties

Label 112.8.i.d.81.1
Level 112112
Weight 88
Character 112.81
Analytic conductor 34.98734.987
Analytic rank 00
Dimension 1010
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] = [112,8,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 112=247 112 = 2^{4} \cdot 7
Weight: k k == 8 8
Character orbit: [χ][\chi] == 112.i (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-27] 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: 34.987122854234.9871228542
Analytic rank: 00
Dimension: 1010
Relative dimension: 55 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x10)\mathbb{Q}[x]/(x^{10} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x10x9+342x8+2165x7+113605x6+319380x5+1438128x4+1705752x3++23619600 x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 2183475 2^{18}\cdot 3^{4}\cdot 7^{5}
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 81.1
Root 1.383712.39666i-1.38371 - 2.39666i of defining polynomial
Character χ\chi == 112.81
Dual form 112.8.i.d.65.1

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+(37.9099+65.6620i)q3+(274.614+475.646i)q5+(907.484+4.07000i)q7+(1780.833084.49i)q9+(2225.01+3853.83i)q113918.83q1341642.4q15+(3001.55+5198.83i)q17+(1058.25+1832.95i)q19+(34135.459741.4i)q21+(32443.3+56193.4i)q23+(111763.+193580.i)q25+104226.q27+118135.q29+(42724.474000.8i)q31+(168700.292197.i)q33+(251144.430523.i)q35+(13176.1+22821.6i)q37+(148563.257318.i)q39459420.q41+511775.q43+(978081.1.69409e6i)q45+(76217.4132012.i)q47+(823510.7386.92i)q49+(227577.394175.i)q51+(30522.552866.5i)q532.44408e6q55160473.q57+(1.03829e61.79837e6i)q59+(475849.+824194.i)q61+(1.62863e6+2.79187e6i)q63+(1.07617e61.86397e6i)q65+(558691.+967681.i)q674.91969e6q69+4.62273e6q71+(1.25326e6+2.17070e6i)q73+(8.47389e61.46772e7i)q75+(2.00348e63.50635e6i)q77+(1.81828e6+3.14935e6i)q79+(56543.1+97935.5i)q81194336.q833.29707e6q85+(4.47850e6+7.75699e6i)q87+(2.85298e64.94150e6i)q89+(3.55627e615949.7i)q91+(3.23936e6+5.61073e6i)q93+(581222.+1.00671e6i)q95+1.01861e7q97+1.58495e7q99+O(q100)q+(-37.9099 + 65.6620i) q^{3} +(274.614 + 475.646i) q^{5} +(-907.484 + 4.07000i) q^{7} +(-1780.83 - 3084.49i) q^{9} +(-2225.01 + 3853.83i) q^{11} -3918.83 q^{13} -41642.4 q^{15} +(-3001.55 + 5198.83i) q^{17} +(1058.25 + 1832.95i) q^{19} +(34135.4 - 59741.4i) q^{21} +(32443.3 + 56193.4i) q^{23} +(-111763. + 193580. i) q^{25} +104226. q^{27} +118135. q^{29} +(42724.4 - 74000.8i) q^{31} +(-168700. - 292197. i) q^{33} +(-251144. - 430523. i) q^{35} +(13176.1 + 22821.6i) q^{37} +(148563. - 257318. i) q^{39} -459420. q^{41} +511775. q^{43} +(978081. - 1.69409e6i) q^{45} +(-76217.4 - 132012. i) q^{47} +(823510. - 7386.92i) q^{49} +(-227577. - 394175. i) q^{51} +(30522.5 - 52866.5i) q^{53} -2.44408e6 q^{55} -160473. q^{57} +(1.03829e6 - 1.79837e6i) q^{59} +(475849. + 824194. i) q^{61} +(1.62863e6 + 2.79187e6i) q^{63} +(-1.07617e6 - 1.86397e6i) q^{65} +(-558691. + 967681. i) q^{67} -4.91969e6 q^{69} +4.62273e6 q^{71} +(-1.25326e6 + 2.17070e6i) q^{73} +(-8.47389e6 - 1.46772e7i) q^{75} +(2.00348e6 - 3.50635e6i) q^{77} +(1.81828e6 + 3.14935e6i) q^{79} +(-56543.1 + 97935.5i) q^{81} -194336. q^{83} -3.29707e6 q^{85} +(-4.47850e6 + 7.75699e6i) q^{87} +(-2.85298e6 - 4.94150e6i) q^{89} +(3.55627e6 - 15949.7i) q^{91} +(3.23936e6 + 5.61073e6i) q^{93} +(-581222. + 1.00671e6i) q^{95} +1.01861e7 q^{97} +1.58495e7 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q27q3+249q5332q75702q96399q1126988q1319294q15+3609q17+12403q19+16099q21+13959q23162364q25161550q27+26148q29++119277812q99+O(q100) 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/112Z)×\left(\mathbb{Z}/112\mathbb{Z}\right)^\times.

nn 1515 1717 8585
χ(n)\chi(n) 11 e(23)e\left(\frac{2}{3}\right) 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 −37.9099 + 65.6620i −0.810641 + 1.40407i 0.101774 + 0.994808i 0.467548π0.467548\pi
−0.912416 + 0.409265i 0.865785π0.865785\pi
44 0 0
55 274.614 + 475.646i 0.982489 + 1.70172i 0.652601 + 0.757702i 0.273678π0.273678\pi
0.329888 + 0.944020i 0.392989π0.392989\pi
66 0 0
77 −907.484 + 4.07000i −0.999990 + 0.00448489i
88 0 0
99 −1780.83 3084.49i −0.814279 1.41037i
1010 0 0
1111 −2225.01 + 3853.83i −0.504032 + 0.873009i 0.495957 + 0.868347i 0.334817π0.334817\pi
−0.999989 + 0.00466169i 0.998516π0.998516\pi
1212 0 0
1313 −3918.83 −0.494715 −0.247357 0.968924i 0.579562π-0.579562\pi
−0.247357 + 0.968924i 0.579562π0.579562\pi
1414 0 0
1515 −41642.4 −3.18579
1616 0 0
1717 −3001.55 + 5198.83i −0.148175 + 0.256646i −0.930553 0.366158i 0.880673π-0.880673\pi
0.782378 + 0.622804i 0.214006π0.214006\pi
1818 0 0
1919 1058.25 + 1832.95i 0.0353958 + 0.0613073i 0.883181 0.469033i 0.155397π-0.155397\pi
−0.847785 + 0.530340i 0.822064π0.822064\pi
2020 0 0
2121 34135.4 59741.4i 0.804336 1.40769i
2222 0 0
2323 32443.3 + 56193.4i 0.556003 + 0.963026i 0.997825 + 0.0659227i 0.0209991π0.0209991\pi
−0.441822 + 0.897103i 0.645668π0.645668\pi
2424 0 0
2525 −111763. + 193580.i −1.43057 + 2.47782i
2626 0 0
2727 104226. 1.01907
2828 0 0
2929 118135. 0.899470 0.449735 0.893162i 0.351518π-0.351518\pi
0.449735 + 0.893162i 0.351518π0.351518\pi
3030 0 0
3131 42724.4 74000.8i 0.257579 0.446139i −0.708014 0.706198i 0.750409π-0.750409\pi
0.965593 + 0.260059i 0.0837420π0.0837420\pi
3232 0 0
3333 −168700. 292197.i −0.817178 1.41539i
3434 0 0
3535 −251144. 430523.i −0.990112 1.69730i
3636 0 0
3737 13176.1 + 22821.6i 0.0427641 + 0.0740696i 0.886615 0.462508i 0.153050π-0.153050\pi
−0.843851 + 0.536577i 0.819717π0.819717\pi
3838 0 0
3939 148563. 257318.i 0.401036 0.694615i
4040 0 0
4141 −459420. −1.04104 −0.520519 0.853850i 0.674262π-0.674262\pi
−0.520519 + 0.853850i 0.674262π0.674262\pi
4242 0 0
4343 511775. 0.981611 0.490806 0.871269i 0.336703π-0.336703\pi
0.490806 + 0.871269i 0.336703π0.336703\pi
4444 0 0
4545 978081. 1.69409e6i 1.60004 2.77135i
4646 0 0
4747 −76217.4 132012.i −0.107081 0.185470i 0.807506 0.589860i 0.200817π-0.200817\pi
−0.914586 + 0.404390i 0.867484π0.867484\pi
4848 0 0
4949 823510. 7386.92i 0.999960 0.00896969i
5050 0 0
5151 −227577. 394175.i −0.240233 0.416096i
5252 0 0
5353 30522.5 52866.5i 0.0281614 0.0487770i −0.851601 0.524190i 0.824368π-0.824368\pi
0.879763 + 0.475413i 0.157701π0.157701\pi
5454 0 0
5555 −2.44408e6 −1.98082
5656 0 0
5757 −160473. −0.114773
5858 0 0
5959 1.03829e6 1.79837e6i 0.658167 1.13998i −0.322923 0.946425i 0.604666π-0.604666\pi
0.981090 0.193553i 0.0620011π-0.0620011\pi
6060 0 0
6161 475849. + 824194.i 0.268420 + 0.464917i 0.968454 0.249193i 0.0801652π-0.0801652\pi
−0.700034 + 0.714109i 0.746832π0.746832\pi
6262 0 0
6363 1.62863e6 + 2.79187e6i 0.820596 + 1.40671i
6464 0 0
6565 −1.07617e6 1.86397e6i −0.486052 0.841866i
6666 0 0
6767 −558691. + 967681.i −0.226939 + 0.393071i −0.956900 0.290419i 0.906205π-0.906205\pi
0.729960 + 0.683490i 0.239539π0.239539\pi
6868 0 0
6969 −4.91969e6 −1.80288
7070 0 0
7171 4.62273e6 1.53283 0.766415 0.642346i 0.222039π-0.222039\pi
0.766415 + 0.642346i 0.222039π0.222039\pi
7272 0 0
7373 −1.25326e6 + 2.17070e6i −0.377059 + 0.653086i −0.990633 0.136552i 0.956398π-0.956398\pi
0.613574 + 0.789637i 0.289731π0.289731\pi
7474 0 0
7575 −8.47389e6 1.46772e7i −2.31936 4.01725i
7676 0 0
7777 2.00348e6 3.50635e6i 0.500111 0.875260i
7878 0 0
7979 1.81828e6 + 3.14935e6i 0.414921 + 0.718664i 0.995420 0.0955966i 0.0304759π-0.0304759\pi
−0.580499 + 0.814261i 0.697143π0.697143\pi
8080 0 0
8181 −56543.1 + 97935.5i −0.0118218 + 0.0204759i
8282 0 0
8383 −194336. −0.0373060 −0.0186530 0.999826i 0.505938π-0.505938\pi
−0.0186530 + 0.999826i 0.505938π0.505938\pi
8484 0 0
8585 −3.29707e6 −0.582320
8686 0 0
8787 −4.47850e6 + 7.75699e6i −0.729147 + 1.26292i
8888 0 0
8989 −2.85298e6 4.94150e6i −0.428977 0.743009i 0.567806 0.823162i 0.307792π-0.307792\pi
−0.996783 + 0.0801532i 0.974459π0.974459\pi
9090 0 0
9191 3.55627e6 15949.7i 0.494710 0.00221874i
9292 0 0
9393 3.23936e6 + 5.61073e6i 0.417608 + 0.723318i
9494 0 0
9595 −581222. + 1.00671e6i −0.0695520 + 0.120468i
9696 0 0
9797 1.01861e7 1.13320 0.566601 0.823992i 0.308258π-0.308258\pi
0.566601 + 0.823992i 0.308258π0.308258\pi
9898 0 0
9999 1.58495e7 1.64169
100100 0 0
101101 7.53754e6 1.30554e7i 0.727956 1.26086i −0.229790 0.973240i 0.573804π-0.573804\pi
0.957746 0.287616i 0.0928627π-0.0928627\pi
102102 0 0
103103 −1.82282e6 3.15722e6i −0.164367 0.284692i 0.772063 0.635546i 0.219225π-0.219225\pi
−0.936430 + 0.350854i 0.885891π0.885891\pi
104104 0 0
105105 3.77898e7 169485.i 3.18575 0.0142879i
106106 0 0
107107 4.56669e6 + 7.90974e6i 0.360378 + 0.624193i 0.988023 0.154307i 0.0493144π-0.0493144\pi
−0.627645 + 0.778500i 0.715981π0.715981\pi
108108 0 0
109109 −1.20607e7 + 2.08898e7i −0.892033 + 1.54505i −0.0545997 + 0.998508i 0.517388π0.517388\pi
−0.837434 + 0.546539i 0.815945π0.815945\pi
110110 0 0
111111 −1.99801e6 −0.138665
112112 0 0
113113 −1.51746e7 −0.989337 −0.494668 0.869082i 0.664710π-0.664710\pi
−0.494668 + 0.869082i 0.664710π0.664710\pi
114114 0 0
115115 −1.78188e7 + 3.08630e7i −1.09253 + 1.89232i
116116 0 0
117117 6.97876e6 + 1.20876e7i 0.402836 + 0.697732i
118118 0 0
119119 2.70269e6 4.73007e6i 0.147022 0.257308i
120120 0 0
121121 −157767. 273260.i −0.00809593 0.0140226i
122122 0 0
123123 1.74166e7 3.01664e7i 0.843909 1.46169i
124124 0 0
125125 −7.98587e7 −3.65710
126126 0 0
127127 −3.12338e7 −1.35305 −0.676523 0.736422i 0.736514π-0.736514\pi
−0.676523 + 0.736422i 0.736514π0.736514\pi
128128 0 0
129129 −1.94014e7 + 3.36042e7i −0.795735 + 1.37825i
130130 0 0
131131 4.64512e6 + 8.04559e6i 0.180529 + 0.312686i 0.942061 0.335442i 0.108886π-0.108886\pi
−0.761532 + 0.648128i 0.775552π0.775552\pi
132132 0 0
133133 −967806. 1.65906e6i −0.0356704 0.0611479i
134134 0 0
135135 2.86220e7 + 4.95748e7i 1.00123 + 1.73417i
136136 0 0
137137 4.48735e6 7.77232e6i 0.149097 0.258243i −0.781797 0.623533i 0.785697π-0.785697\pi
0.930894 + 0.365290i 0.119030π0.119030\pi
138138 0 0
139139 −4.36201e7 −1.37764 −0.688819 0.724934i 0.741870π-0.741870\pi
−0.688819 + 0.724934i 0.741870π0.741870\pi
140140 0 0
141141 1.15576e7 0.347217
142142 0 0
143143 8.71944e6 1.51025e7i 0.249352 0.431890i
144144 0 0
145145 3.24416e7 + 5.61905e7i 0.883719 + 1.53065i
146146 0 0
147147 −3.07342e7 + 5.43533e7i −0.798015 + 1.41129i
148148 0 0
149149 2.62443e7 + 4.54565e7i 0.649955 + 1.12575i 0.983133 + 0.182891i 0.0585455π0.0585455\pi
−0.333178 + 0.942864i 0.608121π0.608121\pi
150150 0 0
151151 2.09853e7 3.63477e7i 0.496017 0.859127i −0.503972 0.863720i 0.668129π-0.668129\pi
0.999989 + 0.00459273i 0.00146192π0.00146192\pi
152152 0 0
153153 2.13810e7 0.482622
154154 0 0
155155 4.69309e7 1.01227
156156 0 0
157157 1.76464e7 3.05645e7i 0.363922 0.630331i −0.624681 0.780880i 0.714771π-0.714771\pi
0.988603 + 0.150549i 0.0481042π0.0481042\pi
158158 0 0
159159 2.31421e6 + 4.00833e6i 0.0456576 + 0.0790813i
160160 0 0
161161 −2.96704e7 5.08625e7i −0.560317 0.960522i
162162 0 0
163163 2.54191e7 + 4.40271e7i 0.459730 + 0.796276i 0.998946 0.0458910i 0.0146127π-0.0146127\pi
−0.539216 + 0.842167i 0.681279π0.681279\pi
164164 0 0
165165 9.26549e7 1.60483e8i 1.60574 2.78122i
166166 0 0
167167 −6.77967e7 −1.12642 −0.563211 0.826313i 0.690434π-0.690434\pi
−0.563211 + 0.826313i 0.690434π0.690434\pi
168168 0 0
169169 −4.73913e7 −0.755257
170170 0 0
171171 3.76913e6 6.52832e6i 0.0576441 0.0998425i
172172 0 0
173173 −2.54193e7 4.40275e7i −0.373252 0.646492i 0.616812 0.787111i 0.288424π-0.288424\pi
−0.990064 + 0.140619i 0.955091π0.955091\pi
174174 0 0
175175 1.00636e8 1.76125e8i 1.41944 2.48421i
176176 0 0
177177 7.87229e7 + 1.36352e8i 1.06707 + 1.84823i
178178 0 0
179179 −2.33223e7 + 4.03954e7i −0.303939 + 0.526437i −0.977024 0.213127i 0.931635π-0.931635\pi
0.673086 + 0.739565i 0.264969π0.264969\pi
180180 0 0
181181 5.38874e7 0.675480 0.337740 0.941239i 0.390338π-0.390338\pi
0.337740 + 0.941239i 0.390338π0.390338\pi
182182 0 0
183183 −7.21576e7 −0.870369
184184 0 0
185185 −7.23666e6 + 1.25343e7i −0.0840306 + 0.145545i
186186 0 0
187187 −1.33570e7 2.31349e7i −0.149369 0.258715i
188188 0 0
189189 −9.45837e7 + 424202.i −1.01906 + 0.00457042i
190190 0 0
191191 −4.24528e7 7.35305e7i −0.440849 0.763573i 0.556904 0.830577i 0.311989π-0.311989\pi
−0.997753 + 0.0670042i 0.978656π0.978656\pi
192192 0 0
193193 3.14432e7 5.44613e7i 0.314830 0.545302i −0.664571 0.747225i 0.731386π-0.731386\pi
0.979401 + 0.201923i 0.0647190π0.0647190\pi
194194 0 0
195195 1.63190e8 1.57605
196196 0 0
197197 −1.65100e8 −1.53856 −0.769281 0.638910i 0.779385π-0.779385\pi
−0.769281 + 0.638910i 0.779385π0.779385\pi
198198 0 0
199199 2.85712e6 4.94867e6i 0.0257005 0.0445146i −0.852889 0.522092i 0.825152π-0.825152\pi
0.878590 + 0.477578i 0.158485π0.158485\pi
200200 0 0
201201 −4.23599e7 7.33695e7i −0.367933 0.637279i
202202 0 0
203203 −1.07206e8 + 480811.i −0.899461 + 0.00403402i
204204 0 0
205205 −1.26163e8 2.18521e8i −1.02281 1.77156i
206206 0 0
207207 1.15552e8 2.00142e8i 0.905483 1.56834i
208208 0 0
209209 −9.41849e6 −0.0713624
210210 0 0
211211 −1.18425e8 −0.867870 −0.433935 0.900944i 0.642875π-0.642875\pi
−0.433935 + 0.900944i 0.642875π0.642875\pi
212212 0 0
213213 −1.75247e8 + 3.03537e8i −1.24258 + 2.15220i
214214 0 0
215215 1.40541e8 + 2.43424e8i 0.964423 + 1.67043i
216216 0 0
217217 −3.84705e7 + 6.73284e7i −0.255575 + 0.447290i
218218 0 0
219219 −9.50217e7 1.64582e8i −0.611320 1.05884i
220220 0 0
221221 1.17626e7 2.03733e7i 0.0733041 0.126967i
222222 0 0
223223 −2.90507e8 −1.75424 −0.877121 0.480270i 0.840539π-0.840539\pi
−0.877121 + 0.480270i 0.840539π0.840539\pi
224224 0 0
225225 7.96125e8 4.65954
226226 0 0
227227 3.42802e7 5.93750e7i 0.194515 0.336909i −0.752227 0.658905i 0.771020π-0.771020\pi
0.946741 + 0.321995i 0.104353π0.104353\pi
228228 0 0
229229 −1.27632e8 2.21064e8i −0.702318 1.21645i −0.967651 0.252294i 0.918815π-0.918815\pi
0.265332 0.964157i 0.414518π-0.414518\pi
230230 0 0
231231 1.54282e8 + 2.64478e8i 0.823518 + 1.41171i
232232 0 0
233233 −2.98377e7 5.16805e7i −0.154533 0.267658i 0.778356 0.627823i 0.216054π-0.216054\pi
−0.932889 + 0.360165i 0.882721π0.882721\pi
234234 0 0
235235 4.18608e7 7.25050e7i 0.210412 0.364444i
236236 0 0
237237 −2.75723e8 −1.34541
238238 0 0
239239 −3.11142e8 −1.47423 −0.737117 0.675765i 0.763813π-0.763813\pi
−0.737117 + 0.675765i 0.763813π0.763813\pi
240240 0 0
241241 1.80211e8 3.12135e8i 0.829319 1.43642i −0.0692534 0.997599i 0.522062π-0.522062\pi
0.898573 0.438824i 0.144605π-0.144605\pi
242242 0 0
243243 1.09684e8 + 1.89979e8i 0.490369 + 0.849344i
244244 0 0
245245 2.29661e8 + 3.89670e8i 0.997714 + 1.69284i
246246 0 0
247247 −4.14711e6 7.18300e6i −0.0175108 0.0303296i
248248 0 0
249249 7.36725e6 1.27605e7i 0.0302418 0.0523804i
250250 0 0
251251 1.56577e8 0.624985 0.312492 0.949920i 0.398836π-0.398836\pi
0.312492 + 0.949920i 0.398836π0.398836\pi
252252 0 0
253253 −2.88747e8 −1.12097
254254 0 0
255255 1.24992e8 2.16492e8i 0.472053 0.817619i
256256 0 0
257257 1.70225e8 + 2.94838e8i 0.625544 + 1.08347i 0.988435 + 0.151642i 0.0484561π0.0484561\pi
−0.362892 + 0.931831i 0.618211π0.618211\pi
258258 0 0
259259 −1.20499e7 2.06566e7i −0.0430959 0.0738771i
260260 0 0
261261 −2.10379e8 3.64386e8i −0.732419 1.26859i
262262 0 0
263263 −1.95033e8 + 3.37808e8i −0.661095 + 1.14505i 0.319233 + 0.947676i 0.396575π0.396575\pi
−0.980328 + 0.197374i 0.936759π0.936759\pi
264264 0 0
265265 3.35276e7 0.110673
266266 0 0
267267 4.32625e8 1.39098
268268 0 0
269269 1.80155e8 3.12038e8i 0.564306 0.977406i −0.432808 0.901486i 0.642477π-0.642477\pi
0.997114 0.0759200i 0.0241894π-0.0241894\pi
270270 0 0
271271 1.24337e8 + 2.15358e8i 0.379496 + 0.657307i 0.990989 0.133943i 0.0427640π-0.0427640\pi
−0.611493 + 0.791250i 0.709431π0.709431\pi
272272 0 0
273273 −1.33771e8 + 2.34117e8i −0.397917 + 0.696407i
274274 0 0
275275 −4.97349e8 8.61435e8i −1.44211 2.49780i
276276 0 0
277277 −1.93090e8 + 3.34442e8i −0.545860 + 0.945458i 0.452692 + 0.891667i 0.350464π0.350464\pi
−0.998552 + 0.0537907i 0.982870π0.982870\pi
278278 0 0
279279 −3.04339e8 −0.838964
280280 0 0
281281 −1.38669e8 −0.372826 −0.186413 0.982471i 0.559686π-0.559686\pi
−0.186413 + 0.982471i 0.559686π0.559686\pi
282282 0 0
283283 2.13509e8 3.69809e8i 0.559969 0.969894i −0.437529 0.899204i 0.644146π-0.644146\pi
0.997498 0.0706904i 0.0225202π-0.0225202\pi
284284 0 0
285285 −4.40682e7 7.63283e7i −0.112763 0.195312i
286286 0 0
287287 4.16916e8 1.86984e6i 1.04103 0.00466894i
288288 0 0
289289 1.87151e8 + 3.24155e8i 0.456089 + 0.789969i
290290 0 0
291291 −3.86155e8 + 6.68840e8i −0.918620 + 1.59110i
292292 0 0
293293 5.59769e8 1.30009 0.650043 0.759898i 0.274751π-0.274751\pi
0.650043 + 0.759898i 0.274751π0.274751\pi
294294 0 0
295295 1.14051e9 2.58657
296296 0 0
297297 −2.31905e8 + 4.01671e8i −0.513644 + 0.889657i
298298 0 0
299299 −1.27140e8 2.20212e8i −0.275063 0.476423i
300300 0 0
301301 −4.64427e8 + 2.08293e6i −0.981601 + 0.00440242i
302302 0 0
303303 5.71496e8 + 9.89860e8i 1.18022 + 2.04420i
304304 0 0
305305 −2.61350e8 + 4.52671e8i −0.527439 + 0.913552i
306306 0 0
307307 1.50479e8 0.296819 0.148409 0.988926i 0.452585π-0.452585\pi
0.148409 + 0.988926i 0.452585π0.452585\pi
308308 0 0
309309 2.76413e8 0.532971
310310 0 0
311311 −4.31084e8 + 7.46659e8i −0.812644 + 1.40754i 0.0983630 + 0.995151i 0.468639π0.468639\pi
−0.911007 + 0.412390i 0.864694π0.864694\pi
312312 0 0
313313 4.79502e8 + 8.30522e8i 0.883864 + 1.53090i 0.847010 + 0.531577i 0.178400π0.178400\pi
0.0368540 + 0.999321i 0.488266π0.488266\pi
314314 0 0
315315 −8.80698e8 + 1.54134e9i −1.58760 + 2.77850i
316316 0 0
317317 7.64346e7 + 1.32389e8i 0.134767 + 0.233423i 0.925508 0.378727i 0.123638π-0.123638\pi
−0.790742 + 0.612150i 0.790305π0.790305\pi
318318 0 0
319319 −2.62852e8 + 4.55274e8i −0.453361 + 0.785245i
320320 0 0
321321 −6.92492e8 −1.16855
322322 0 0
323323 −1.27056e7 −0.0209790
324324 0 0
325325 4.37982e8 7.58606e8i 0.707724 1.22581i
326326 0 0
327327 −9.14444e8 1.58386e9i −1.44624 2.50496i
328328 0 0
329329 6.97034e7 + 1.19489e8i 0.107912 + 0.184987i
330330 0 0
331331 −3.64647e8 6.31587e8i −0.552681 0.957271i −0.998080 0.0619390i 0.980272π-0.980272\pi
0.445399 0.895332i 0.353062π-0.353062\pi
332332 0 0
333333 4.69286e7 8.12827e7i 0.0696438 0.120627i
334334 0 0
335335 −6.13698e8 −0.891862
336336 0 0
337337 9.99520e8 1.42261 0.711306 0.702882i 0.248104π-0.248104\pi
0.711306 + 0.702882i 0.248104π0.248104\pi
338338 0 0
339339 5.75270e8 9.96397e8i 0.801997 1.38910i
340340 0 0
341341 1.90125e8 + 3.29305e8i 0.259656 + 0.449737i
342342 0 0
343343 −7.47292e8 + 1.00552e7i −0.999909 + 0.0134543i
344344 0 0
345345 −1.35102e9 2.34003e9i −1.77131 3.06799i
346346 0 0
347347 2.89931e8 5.02175e8i 0.372513 0.645211i −0.617439 0.786619i 0.711830π-0.711830\pi
0.989951 + 0.141408i 0.0451630π0.0451630\pi
348348 0 0
349349 −1.18267e9 −1.48928 −0.744638 0.667469i 0.767378π-0.767378\pi
−0.744638 + 0.667469i 0.767378π0.767378\pi
350350 0 0
351351 −4.08445e8 −0.504149
352352 0 0
353353 −2.27190e7 + 3.93505e7i −0.0274903 + 0.0476145i −0.879443 0.476004i 0.842085π-0.842085\pi
0.851953 + 0.523618i 0.175418π0.175418\pi
354354 0 0
355355 1.26947e9 + 2.19878e9i 1.50599 + 2.60845i
356356 0 0
357357 2.08127e8 + 3.56781e8i 0.242097 + 0.415014i
358358 0 0
359359 4.45017e7 + 7.70792e7i 0.0507628 + 0.0879238i 0.890290 0.455393i 0.150501π-0.150501\pi
−0.839527 + 0.543317i 0.817168π0.817168\pi
360360 0 0
361361 4.44696e8 7.70236e8i 0.497494 0.861685i
362362 0 0
363363 2.39237e7 0.0262516
364364 0 0
365365 −1.37665e9 −1.48183
366366 0 0
367367 −2.39219e8 + 4.14340e8i −0.252619 + 0.437548i −0.964246 0.265009i 0.914625π-0.914625\pi
0.711627 + 0.702557i 0.247958π0.247958\pi
368368 0 0
369369 8.18149e8 + 1.41708e9i 0.847696 + 1.46825i
370370 0 0
371371 −2.74835e7 + 4.80997e7i −0.0279424 + 0.0489028i
372372 0 0
373373 5.82022e8 + 1.00809e9i 0.580709 + 1.00582i 0.995395 + 0.0958535i 0.0305580π0.0305580\pi
−0.414686 + 0.909965i 0.636109π0.636109\pi
374374 0 0
375375 3.02744e9 5.24368e9i 2.96460 5.13484i
376376 0 0
377377 −4.62952e8 −0.444981
378378 0 0
379379 1.78772e9 1.68679 0.843395 0.537294i 0.180553π-0.180553\pi
0.843395 + 0.537294i 0.180553π0.180553\pi
380380 0 0
381381 1.18407e9 2.05088e9i 1.09683 1.89977i
382382 0 0
383383 1.98793e8 + 3.44319e8i 0.180803 + 0.313159i 0.942154 0.335180i 0.108797π-0.108797\pi
−0.761352 + 0.648339i 0.775464π0.775464\pi
384384 0 0
385385 2.21796e9 9.94741e6i 1.98080 0.00888377i
386386 0 0
387387 −9.11384e8 1.57856e9i −0.799305 1.38444i
388388 0 0
389389 2.46267e8 4.26547e8i 0.212121 0.367404i −0.740257 0.672324i 0.765296π-0.765296\pi
0.952378 + 0.304920i 0.0986297π0.0986297\pi
390390 0 0
391391 −3.89520e8 −0.329542
392392 0 0
393393 −7.04386e8 −0.585378
394394 0 0
395395 −9.98650e8 + 1.72971e9i −0.815311 + 1.41216i
396396 0 0
397397 2.85231e7 + 4.94034e7i 0.0228786 + 0.0396269i 0.877238 0.480056i 0.159384π-0.159384\pi
−0.854359 + 0.519683i 0.826050π0.826050\pi
398398 0 0
399399 1.45627e8 653126.i 0.114772 0.000514745i
400400 0 0
401401 −9.15297e8 1.58534e9i −0.708854 1.22777i −0.965283 0.261207i 0.915879π-0.915879\pi
0.256429 0.966563i 0.417454π-0.417454\pi
402402 0 0
403403 −1.67430e8 + 2.89997e8i −0.127428 + 0.220712i
404404 0 0
405405 −6.21101e7 −0.0464590
406406 0 0
407407 −1.17268e8 −0.0862179
408408 0 0
409409 −1.03052e9 + 1.78492e9i −0.744777 + 1.28999i 0.205522 + 0.978653i 0.434111π0.434111\pi
−0.950299 + 0.311339i 0.899223π0.899223\pi
410410 0 0
411411 3.40230e8 + 5.89296e8i 0.241728 + 0.418685i
412412 0 0
413413 −9.34910e8 + 1.63621e9i −0.653047 + 1.14292i
414414 0 0
415415 −5.33673e7 9.24349e7i −0.0366528 0.0634845i
416416 0 0
417417 1.65364e9 2.86418e9i 1.11677 1.93430i
418418 0 0
419419 2.50238e8 0.166190 0.0830948 0.996542i 0.473520π-0.473520\pi
0.0830948 + 0.996542i 0.473520π0.473520\pi
420420 0 0
421421 −8.67705e8 −0.566741 −0.283370 0.959011i 0.591453π-0.591453\pi
−0.283370 + 0.959011i 0.591453π0.591453\pi
422422 0 0
423423 −2.71460e8 + 4.70183e8i −0.174387 + 0.302048i
424424 0 0
425425 −6.70926e8 1.16208e9i −0.423949 0.734301i
426426 0 0
427427 −4.35180e8 7.46006e8i −0.270502 0.463708i
428428 0 0
429429 6.61107e8 + 1.14507e9i 0.404270 + 0.700216i
430430 0 0
431431 −6.03709e7 + 1.04565e8i −0.0363209 + 0.0629097i −0.883614 0.468215i 0.844897π-0.844897\pi
0.847294 + 0.531125i 0.178231π0.178231\pi
432432 0 0
433433 −8.11123e8 −0.480153 −0.240076 0.970754i 0.577172π-0.577172\pi
−0.240076 + 0.970754i 0.577172π0.577172\pi
434434 0 0
435435 −4.91944e9 −2.86552
436436 0 0
437437 −6.86663e7 + 1.18934e8i −0.0393603 + 0.0681741i
438438 0 0
439439 −1.15817e9 2.00601e9i −0.653352 1.13164i −0.982304 0.187293i 0.940029π-0.940029\pi
0.328952 0.944347i 0.393305π-0.393305\pi
440440 0 0
441441 −1.48931e9 2.52695e9i −0.826897 1.40301i
442442 0 0
443443 1.25883e9 + 2.18036e9i 0.687947 + 1.19156i 0.972501 + 0.232899i 0.0748213π0.0748213\pi
−0.284554 + 0.958660i 0.591845π0.591845\pi
444444 0 0
445445 1.56694e9 2.71401e9i 0.842930 1.46000i
446446 0 0
447447 −3.97968e9 −2.10752
448448 0 0
449449 1.98108e7 0.0103286 0.00516429 0.999987i 0.498356π-0.498356\pi
0.00516429 + 0.999987i 0.498356π0.498356\pi
450450 0 0
451451 1.02222e9 1.77053e9i 0.524716 0.908835i
452452 0 0
453453 1.59111e9 + 2.75588e9i 0.804184 + 1.39289i
454454 0 0
455455 9.84189e8 + 1.68715e9i 0.489823 + 0.839678i
456456 0 0
457457 3.90833e8 + 6.76943e8i 0.191551 + 0.331776i 0.945764 0.324853i 0.105315π-0.105315\pi
−0.754213 + 0.656630i 0.771982π0.771982\pi
458458 0 0
459459 −3.12840e8 + 5.41855e8i −0.151000 + 0.261540i
460460 0 0
461461 −6.02108e8 −0.286234 −0.143117 0.989706i 0.545713π-0.545713\pi
−0.143117 + 0.989706i 0.545713π0.545713\pi
462462 0 0
463463 1.26552e9 0.592565 0.296282 0.955100i 0.404253π-0.404253\pi
0.296282 + 0.955100i 0.404253π0.404253\pi
464464 0 0
465465 −1.77915e9 + 3.08157e9i −0.820591 + 1.42130i
466466 0 0
467467 −2.01050e8 3.48229e8i −0.0913471 0.158218i 0.816731 0.577018i 0.195784π-0.195784\pi
−0.908078 + 0.418801i 0.862451π0.862451\pi
468468 0 0
469469 5.03065e8 8.80429e8i 0.225174 0.394084i
470470 0 0
471471 1.33795e9 + 2.31740e9i 0.590020 + 1.02195i
472472 0 0
473473 −1.13871e9 + 1.97230e9i −0.494763 + 0.856955i
474474 0 0
475475 −4.73095e8 −0.202545
476476 0 0
477477 −2.17421e8 −0.0917250
478478 0 0
479479 8.08815e8 1.40091e9i 0.336260 0.582419i −0.647466 0.762094i 0.724171π-0.724171\pi
0.983726 + 0.179675i 0.0575046π0.0575046\pi
480480 0 0
481481 −5.16347e7 8.94340e7i −0.0211560 0.0366433i
482482 0 0
483483 4.46454e9 2.00232e7i 1.80286 0.00808570i
484484 0 0
485485 2.79725e9 + 4.84498e9i 1.11336 + 1.92839i
486486 0 0
487487 1.52284e8 2.63764e8i 0.0597453 0.103482i −0.834606 0.550848i 0.814304π-0.814304\pi
0.894351 + 0.447366i 0.147638π0.147638\pi
488488 0 0
489489 −3.85454e9 −1.49071
490490 0 0
491491 2.14359e9 0.817252 0.408626 0.912702i 0.366008π-0.366008\pi
0.408626 + 0.912702i 0.366008π0.366008\pi
492492 0 0
493493 −3.54588e8 + 6.14165e8i −0.133279 + 0.230845i
494494 0 0
495495 4.35248e9 + 7.53872e9i 1.61294 + 2.79370i
496496 0 0
497497 −4.19505e9 + 1.88145e7i −1.53281 + 0.00687457i
498498 0 0
499499 −1.24815e9 2.16187e9i −0.449693 0.778891i 0.548673 0.836037i 0.315133π-0.315133\pi
−0.998366 + 0.0571461i 0.981800π0.981800\pi
500500 0 0
501501 2.57017e9 4.45166e9i 0.913124 1.58158i
502502 0 0
503503 −2.89416e9 −1.01399 −0.506997 0.861948i 0.669244π-0.669244\pi
−0.506997 + 0.861948i 0.669244π0.669244\pi
504504 0 0
505505 8.27966e9 2.86084
506506 0 0
507507 1.79660e9 3.11180e9i 0.612243 1.06044i
508508 0 0
509509 −1.24625e9 2.15857e9i −0.418883 0.725526i 0.576945 0.816783i 0.304245π-0.304245\pi
−0.995827 + 0.0912573i 0.970911π0.970911\pi
510510 0 0
511511 1.12847e9 1.97498e9i 0.374126 0.654770i
512512 0 0
513513 1.10298e8 + 1.91041e8i 0.0360708 + 0.0624764i
514514 0 0
515515 1.00115e9 1.73404e9i 0.322978 0.559414i
516516 0 0
517517 6.78339e8 0.215889
518518 0 0
519519 3.85458e9 1.21029
520520 0 0
521521 1.06085e9 1.83745e9i 0.328641 0.569223i −0.653601 0.756839i 0.726743π-0.726743\pi
0.982242 + 0.187616i 0.0600760π0.0600760\pi
522522 0 0
523523 −4.49299e8 7.78208e8i −0.137334 0.237870i 0.789152 0.614197i 0.210520π-0.210520\pi
−0.926487 + 0.376327i 0.877187π0.877187\pi
524524 0 0
525525 7.74965e9 + 1.32848e10i 2.33735 + 4.00681i
526526 0 0
527527 2.56478e8 + 4.44234e8i 0.0763333 + 0.132213i
528528 0 0
529529 −4.02719e8 + 6.97530e8i −0.118279 + 0.204865i
530530 0 0
531531 −7.39605e9 −2.14372
532532 0 0
533533 1.80039e9 0.515017
534534 0 0
535535 −2.50816e9 + 4.34425e9i −0.708135 + 1.22653i
536536 0 0
537537 −1.76830e9 3.06278e9i −0.492771 0.853504i
538538 0 0
539539 −1.80385e9 + 3.19011e9i −0.496181 + 0.877494i
540540 0 0
541541 −1.77642e9 3.07685e9i −0.482342 0.835441i 0.517452 0.855712i 0.326880π-0.326880\pi
−0.999795 + 0.0202709i 0.993547π0.993547\pi
542542 0 0
543543 −2.04287e9 + 3.53836e9i −0.547572 + 0.948423i
544544 0 0
545545 −1.32482e10 −3.50565
546546 0 0
547547 1.53131e9 0.400043 0.200021 0.979792i 0.435899π-0.435899\pi
0.200021 + 0.979792i 0.435899π0.435899\pi
548548 0 0
549549 1.69481e9 2.93550e9i 0.437137 0.757144i
550550 0 0
551551 1.25017e8 + 2.16536e8i 0.0318374 + 0.0551440i
552552 0 0
553553 −1.66287e9 2.85058e9i −0.418140 0.716796i
554554 0 0
555555 −5.48683e8 9.50347e8i −0.136237 0.235970i
556556 0 0
557557 −9.29428e8 + 1.60982e9i −0.227889 + 0.394715i −0.957182 0.289486i 0.906516π-0.906516\pi
0.729294 + 0.684201i 0.239849π0.239849\pi
558558 0 0
559559 −2.00556e9 −0.485617
560560 0 0
561561 2.02545e9 0.484340
562562 0 0
563563 −3.22008e9 + 5.57735e9i −0.760480 + 1.31719i 0.182124 + 0.983276i 0.441703π0.441703\pi
−0.942604 + 0.333914i 0.891631π0.891631\pi
564564 0 0
565565 −4.16717e9 7.21775e9i −0.972013 1.68358i
566566 0 0
567567 5.09133e7 8.91050e7i 0.0117298 0.0205287i
568568 0 0
569569 2.37325e9 + 4.11058e9i 0.540070 + 0.935429i 0.998899 + 0.0469042i 0.0149355π0.0149355\pi
−0.458829 + 0.888524i 0.651731π0.651731\pi
570570 0 0
571571 2.87568e9 4.98083e9i 0.646420 1.11963i −0.337552 0.941307i 0.609599π-0.609599\pi
0.983972 0.178325i 0.0570679π-0.0570679\pi
572572 0 0
573573 6.43754e9 1.42948
574574 0 0
575575 −1.45039e10 −3.18161
576576 0 0
577577 −3.45836e9 + 5.99005e9i −0.749471 + 1.29812i 0.198606 + 0.980079i 0.436359π0.436359\pi
−0.948077 + 0.318042i 0.896975π0.896975\pi
578578 0 0
579579 2.38402e9 + 4.12925e9i 0.510429 + 0.884089i
580580 0 0
581581 1.76356e8 790947.i 0.0373056 0.000167313i
582582 0 0
583583 1.35826e8 + 2.35257e8i 0.0283885 + 0.0491703i
584584 0 0
585585 −3.83293e9 + 6.63884e9i −0.791564 + 1.37103i
586586 0 0
587587 2.21194e9 0.451378 0.225689 0.974199i 0.427537π-0.427537\pi
0.225689 + 0.974199i 0.427537π0.427537\pi
588588 0 0
589589 1.80853e8 0.0364688
590590 0 0
591591 6.25893e9 1.08408e10i 1.24722 2.16025i
592592 0 0
593593 1.71060e9 + 2.96285e9i 0.336866 + 0.583470i 0.983842 0.179041i 0.0572996π-0.0572996\pi
−0.646975 + 0.762511i 0.723966π0.723966\pi
594594 0 0
595595 2.99204e9 1.34191e7i 0.582314 0.00261164i
596596 0 0
597597 2.16626e8 + 3.75208e8i 0.0416678 + 0.0721708i
598598 0 0
599599 −4.61146e9 + 7.98728e9i −0.876687 + 1.51847i −0.0217322 + 0.999764i 0.506918π0.506918\pi
−0.854955 + 0.518703i 0.826415π0.826415\pi
600600 0 0
601601 6.81256e9 1.28012 0.640059 0.768326i 0.278910π-0.278910\pi
0.640059 + 0.768326i 0.278910π0.278910\pi
602602 0 0
603603 3.97973e9 0.739168
604604 0 0
605605 8.66499e7 1.50082e8i 0.0159083 0.0275540i
606606 0 0
607607 −2.00349e9 3.47015e9i −0.363603 0.629778i 0.624948 0.780666i 0.285120π-0.285120\pi
−0.988551 + 0.150888i 0.951787π0.951787\pi
608608 0 0
609609 4.03260e9 7.05757e9i 0.723476 1.26618i
610610 0 0
611611 2.98683e8 + 5.17335e8i 0.0529745 + 0.0917545i
612612 0 0
613613 −4.14048e9 + 7.17153e9i −0.726005 + 1.25748i 0.232555 + 0.972583i 0.425292π0.425292\pi
−0.958559 + 0.284893i 0.908042π0.908042\pi
614614 0 0
615615 1.91314e10 3.31653
616616 0 0
617617 3.35578e7 0.00575169 0.00287584 0.999996i 0.499085π-0.499085\pi
0.00287584 + 0.999996i 0.499085π0.499085\pi
618618 0 0
619619 1.45841e9 2.52604e9i 0.247151 0.428079i −0.715583 0.698528i 0.753839π-0.753839\pi
0.962734 + 0.270449i 0.0871722π0.0871722\pi
620620 0 0
621621 3.38144e9 + 5.85683e9i 0.566606 + 0.981391i
622622 0 0
623623 2.60914e9 + 4.47272e9i 0.432305 + 0.741078i
624624 0 0
625625 −1.31988e10 2.28610e10i −2.16250 3.74555i
626626 0 0
627627 3.57054e8 6.18437e8i 0.0578493 0.100198i
628628 0 0
629629 −1.58194e8 −0.0253462
630630 0 0
631631 7.72632e9 1.22425 0.612125 0.790761i 0.290315π-0.290315\pi
0.612125 + 0.790761i 0.290315π0.290315\pi
632632 0 0
633633 4.48948e9 7.77601e9i 0.703531 1.21855i
634634 0 0
635635 −8.57726e9 1.48562e10i −1.32935 2.30251i
636636 0 0
637637 −3.22720e9 + 2.89481e7i −0.494695 + 0.00443744i
638638 0 0
639639 −8.23228e9 1.42587e10i −1.24815 2.16186i
640640 0 0
641641 3.01543e9 5.22288e9i 0.452216 0.783261i −0.546307 0.837585i 0.683967π-0.683967\pi
0.998523 + 0.0543235i 0.0173002π0.0173002\pi
642642 0 0
643643 −3.27128e9 −0.485265 −0.242633 0.970118i 0.578011π-0.578011\pi
−0.242633 + 0.970118i 0.578011π0.578011\pi
644644 0 0
645645 −2.13116e10 −3.12720
646646 0 0
647647 3.36141e9 5.82214e9i 0.487929 0.845118i −0.511974 0.859001i 0.671086π-0.671086\pi
0.999904 + 0.0138825i 0.00441906π0.00441906\pi
648648 0 0
649649 4.62041e9 + 8.00278e9i 0.663474 + 1.14917i
650650 0 0
651651 −2.96250e9 5.07846e9i −0.420848 0.721438i
652652 0 0
653653 2.47194e9 + 4.28152e9i 0.347409 + 0.601730i 0.985788 0.167992i 0.0537282π-0.0537282\pi
−0.638379 + 0.769722i 0.720395π0.720395\pi
654654 0 0
655655 −2.55123e9 + 4.41887e9i −0.354736 + 0.614421i
656656 0 0
657657 8.92733e9 1.22813
658658 0 0
659659 −3.36078e9 −0.457448 −0.228724 0.973491i 0.573455π-0.573455\pi
−0.228724 + 0.973491i 0.573455π0.573455\pi
660660 0 0
661661 1.05592e9 1.82890e9i 0.142208 0.246312i −0.786120 0.618074i 0.787913π-0.787913\pi
0.928328 + 0.371762i 0.121246π0.121246\pi
662662 0 0
663663 8.91835e8 + 1.54470e9i 0.118847 + 0.205849i
664664 0 0
665665 5.23352e8 9.15935e8i 0.0690110 0.120778i
666666 0 0
667667 3.83269e9 + 6.63842e9i 0.500108 + 0.866212i
668668 0 0
669669 1.10131e10 1.90752e10i 1.42206 2.46308i
670670 0 0
671671 −4.23508e9 −0.541168
672672 0 0
673673 1.12547e10 1.42324 0.711622 0.702562i 0.247961π-0.247961\pi
0.711622 + 0.702562i 0.247961π0.247961\pi
674674 0 0
675675 −1.16487e10 + 2.01761e10i −1.45785 + 2.52507i
676676 0 0
677677 −1.57599e9 2.72970e9i −0.195206 0.338107i 0.751762 0.659435i 0.229204π-0.229204\pi
−0.946968 + 0.321327i 0.895871π0.895871\pi
678678 0 0
679679 −9.24373e9 + 4.14575e7i −1.13319 + 0.00508228i
680680 0 0
681681 2.59912e9 + 4.50181e9i 0.315363 + 0.546225i
682682 0 0
683683 4.30544e9 7.45724e9i 0.517065 0.895583i −0.482739 0.875764i 0.660358π-0.660358\pi
0.999804 0.0198183i 0.00630877π-0.00630877\pi
684684 0 0
685685 4.92916e9 0.585943
686686 0 0
687687 1.93540e10 2.27731
688688 0 0
689689 −1.19612e8 + 2.07175e8i −0.0139319 + 0.0241307i
690690 0 0
691691 −6.39246e8 1.10721e9i −0.0737046 0.127660i 0.826818 0.562470i 0.190149π-0.190149\pi
−0.900522 + 0.434810i 0.856816π0.856816\pi
692692 0 0
693693 −1.43831e10 + 6.45073e7i −1.64167 + 0.00736280i
694694 0 0
695695 −1.19787e10 2.07477e10i −1.35351 2.34436i
696696 0 0
697697 1.37897e9 2.38845e9i 0.154255 0.267178i
698698 0 0
699699 4.52459e9 0.501082
700700 0 0
701701 −7.24407e9 −0.794273 −0.397136 0.917760i 0.629996π-0.629996\pi
−0.397136 + 0.917760i 0.629996π0.629996\pi
702702 0 0
703703 −2.78872e7 + 4.83020e7i −0.00302734 + 0.00524350i
704704 0 0
705705 3.17388e9 + 5.49732e9i 0.341137 + 0.590866i
706706 0 0
707707 −6.78706e9 + 1.18782e10i −0.722294 + 1.26411i
708708 0 0
709709 −2.12572e9 3.68185e9i −0.223998 0.387976i 0.732020 0.681283i 0.238578π-0.238578\pi
−0.956018 + 0.293307i 0.905244π0.905244\pi
710710 0 0
711711 6.47608e9 1.12169e10i 0.675723 1.17039i
712712 0 0
713713 5.54448e9 0.572858
714714 0 0
715715 9.57793e9 0.979942
716716 0 0
717717 1.17954e10 2.04302e10i 1.19508 2.06993i
718718 0 0
719719 3.48927e9 + 6.04360e9i 0.350093 + 0.606379i 0.986265 0.165168i 0.0528166π-0.0528166\pi
−0.636172 + 0.771547i 0.719483π0.719483\pi
720720 0 0
721721 1.66703e9 + 2.85771e9i 0.165642 + 0.283952i
722722 0 0
723723 1.36636e10 + 2.36660e10i 1.34456 + 2.32885i
724724 0 0
725725 −1.32032e10 + 2.28686e10i −1.28676 + 2.22873i
726726 0 0
727727 1.21909e9 0.117670 0.0588349 0.998268i 0.481261π-0.481261\pi
0.0588349 + 0.998268i 0.481261π0.481261\pi
728728 0 0
729729 −1.68798e10 −1.61370
730730 0 0
731731 −1.53612e9 + 2.66063e9i −0.145450 + 0.251927i
732732 0 0
733733 8.20330e9 + 1.42085e10i 0.769351 + 1.33255i 0.937915 + 0.346864i 0.112754π0.112754\pi
−0.168564 + 0.985691i 0.553913π0.553913\pi
734734 0 0
735735 −3.42930e10 + 3.07609e8i −3.18566 + 0.0285755i
736736 0 0
737737 −2.48619e9 4.30620e9i −0.228769 0.396240i
738738 0 0
739739 −1.86533e9 + 3.23084e9i −0.170020 + 0.294483i −0.938427 0.345479i 0.887717π-0.887717\pi
0.768407 + 0.639962i 0.221050π0.221050\pi
740740 0 0
741741 6.28867e8 0.0567799
742742 0 0
743743 −1.82498e10 −1.63229 −0.816144 0.577849i 0.803892π-0.803892\pi
−0.816144 + 0.577849i 0.803892π0.803892\pi
744744 0 0
745745 −1.44141e10 + 2.49660e10i −1.27715 + 2.21208i
746746 0 0
747747 3.46078e8 + 5.99425e8i 0.0303775 + 0.0526154i
748748 0 0
749749 −4.17639e9 7.15937e9i −0.363174 0.622571i
750750 0 0
751751 5.97037e9 + 1.03410e10i 0.514353 + 0.890886i 0.999861 + 0.0166536i 0.00530125π0.00530125\pi
−0.485508 + 0.874232i 0.661365π0.661365\pi
752752 0 0
753753 −5.93582e9 + 1.02811e10i −0.506639 + 0.877524i
754754 0 0
755755 2.30515e10 1.94933
756756 0 0
757757 1.34863e10 1.12994 0.564971 0.825111i 0.308887π-0.308887\pi
0.564971 + 0.825111i 0.308887π0.308887\pi
758758 0 0
759759 1.09464e10 1.89597e10i 0.908707 1.57393i
760760 0 0
761761 5.63097e9 + 9.75313e9i 0.463166 + 0.802228i 0.999117 0.0420218i 0.0133799π-0.0133799\pi
−0.535950 + 0.844250i 0.680047π0.680047\pi
762762 0 0
763763 1.08599e10 1.90062e10i 0.885095 1.54903i
764764 0 0
765765 5.87151e9 + 1.01698e10i 0.474171 + 0.821288i
766766 0 0
767767 −4.06887e9 + 7.04750e9i −0.325605 + 0.563964i
768768 0 0
769769 −2.13869e10 −1.69592 −0.847961 0.530059i 0.822170π-0.822170\pi
−0.847961 + 0.530059i 0.822170π0.822170\pi
770770 0 0
771771 −2.58129e10 −2.02837
772772 0 0
773773 8.36012e9 1.44802e10i 0.651005 1.12757i −0.331874 0.943324i 0.607681π-0.607681\pi
0.982879 0.184250i 0.0589857π-0.0589857\pi
774774 0 0
775775 9.55004e9 + 1.65412e10i 0.736969 + 1.27647i
776776 0 0
777777 1.81317e9 8.13193e6i 0.138664 0.000621899i
778778 0 0
779779 −4.86182e8 8.42093e8i −0.0368484 0.0638232i
780780 0 0
781781 −1.02856e10 + 1.78152e10i −0.772595 + 1.33817i
782782 0 0
783783 1.23128e10 0.916623
784784 0 0
785785 1.93838e10 1.43020
786786 0 0
787787 −4.72932e9 + 8.19142e9i −0.345849 + 0.599029i −0.985508 0.169631i 0.945743π-0.945743\pi
0.639658 + 0.768659i 0.279076π0.279076\pi
788788 0 0
789789 −1.47874e10 2.56126e10i −1.07182 1.85645i
790790 0 0
791791 1.37707e10 6.17609e7i 0.989327 0.00443706i
792792 0 0
793793 −1.86477e9 3.22988e9i −0.132791 0.230001i
794794 0 0
795795 −1.27103e9 + 2.20149e9i −0.0897162 + 0.155393i
796796 0 0
797797 −4.25156e9 −0.297471 −0.148736 0.988877i 0.547520π-0.547520\pi
−0.148736 + 0.988877i 0.547520π0.547520\pi
798798 0 0
799799 9.15081e8 0.0634667
800800 0 0
801801 −1.01613e10 + 1.75999e10i −0.698613 + 1.21003i
802802 0 0
803803 −5.57701e9 9.65967e9i −0.380100 0.658352i
804804 0 0
805805 1.60446e10 2.80802e10i 1.08404 1.89721i
806806 0 0
807807 1.36594e10 + 2.36587e10i 0.914899 + 1.58465i
808808 0 0
809809 −2.76390e9 + 4.78722e9i −0.183528 + 0.317880i −0.943080 0.332567i 0.892085π-0.892085\pi
0.759551 + 0.650447i 0.225419π0.225419\pi
810810 0 0
811811 −4.99197e9 −0.328624 −0.164312 0.986408i 0.552540π-0.552540\pi
−0.164312 + 0.986408i 0.552540π0.552540\pi
812812 0 0
813813 −1.88544e10 −1.23054
814814 0 0
815815 −1.39609e10 + 2.41810e10i −0.903361 + 1.56467i
816816 0 0
817817 5.41587e8 + 9.38056e8i 0.0347449 + 0.0601799i
818818 0 0
819819 −6.38231e9 1.09409e10i −0.405961 0.695918i
820820 0 0
821821 5.29502e9 + 9.17125e9i 0.333939 + 0.578399i 0.983280 0.182098i 0.0582888π-0.0582888\pi
−0.649342 + 0.760497i 0.724955π0.724955\pi
822822 0 0
823823 −1.00055e10 + 1.73301e10i −0.625663 + 1.08368i 0.362749 + 0.931887i 0.381838π0.381838\pi
−0.988412 + 0.151793i 0.951495π0.951495\pi
824824 0 0
825825 7.54180e10 4.67612
826826 0 0
827827 −2.82024e9 −0.173387 −0.0866935 0.996235i 0.527630π-0.527630\pi
−0.0866935 + 0.996235i 0.527630π0.527630\pi
828828 0 0
829829 −2.35107e9 + 4.07217e9i −0.143326 + 0.248247i −0.928747 0.370714i 0.879113π-0.879113\pi
0.785421 + 0.618961i 0.212446π0.212446\pi
830830 0 0
831831 −1.46401e10 2.53574e10i −0.884994 1.53285i
832832 0 0
833833 −2.43340e9 + 4.30346e9i −0.145867 + 0.257965i
834834 0 0
835835 −1.86179e10 3.22472e10i −1.10670 1.91686i
836836 0 0
837837 4.45301e9 7.71283e9i 0.262491 0.454648i
838838 0 0
839839 2.73548e10 1.59907 0.799534 0.600621i 0.205080π-0.205080\pi
0.799534 + 0.600621i 0.205080π0.205080\pi
840840 0 0
841841 −3.29394e9 −0.190954
842842 0 0
843843 5.25692e9 9.10525e9i 0.302228 0.523474i
844844 0 0
845845 −1.30143e10 2.25415e10i −0.742032 1.28524i
846846 0 0
847847 1.44283e8 + 2.47337e8i 0.00815873 + 0.0139861i
848848 0 0
849849 1.61882e10 + 2.80388e10i 0.907868 + 1.57247i
850850 0 0
851851 −8.54949e8 + 1.48081e9i −0.0475540 + 0.0823659i
852852 0 0
853853 1.99998e10 1.10333 0.551665 0.834066i 0.313993π-0.313993\pi
0.551665 + 0.834066i 0.313993π0.313993\pi
854854 0 0
855855 4.14023e9 0.226539
856856 0 0
857857 1.79869e10 3.11543e10i 0.976167 1.69077i 0.300139 0.953896i 0.402967π-0.402967\pi
0.676028 0.736876i 0.263700π-0.263700\pi
858858 0 0
859859 1.05362e10 + 1.82492e10i 0.567161 + 0.982351i 0.996845 + 0.0793721i 0.0252915π0.0252915\pi
−0.429684 + 0.902979i 0.641375π0.641375\pi
860860 0 0
861861 −1.56825e10 + 2.74464e10i −0.837345 + 1.46546i
862862 0 0
863863 1.22382e10 + 2.11971e10i 0.648154 + 1.12264i 0.983563 + 0.180563i 0.0577919π0.0577919\pi
−0.335410 + 0.942072i 0.608875π0.608875\pi
864864 0 0
865865 1.39610e10 2.41812e10i 0.733433 1.27034i
866866 0 0
867867 −2.83795e10 −1.47890
868868 0 0
869869 −1.61828e10 −0.836533
870870 0 0
871871 2.18942e9 3.79218e9i 0.112270 0.194458i
872872 0 0
873873 −1.81397e10 3.14189e10i −0.922743 1.59824i
874874 0 0
875875 7.24705e10 3.25025e8i 3.65707 0.0164017i
876876 0 0
877877 1.65279e9 + 2.86271e9i 0.0827404 + 0.143311i 0.904426 0.426630i 0.140299π-0.140299\pi
−0.821686 + 0.569941i 0.806966π0.806966\pi
878878 0 0
879879 −2.12208e10 + 3.67555e10i −1.05390 + 1.82541i
880880 0 0
881881 −3.05557e10 −1.50549 −0.752743 0.658315i 0.771270π-0.771270\pi
−0.752743 + 0.658315i 0.771270π0.771270\pi
882882 0 0
883883 1.20403e10 0.588540 0.294270 0.955722i 0.404923π-0.404923\pi
0.294270 + 0.955722i 0.404923π0.404923\pi
884884 0 0
885885 −4.32368e10 + 7.48884e10i −2.09678 + 3.63173i
886886 0 0
887887 −1.16411e9 2.01629e9i −0.0560093 0.0970109i 0.836661 0.547721i 0.184504π-0.184504\pi
−0.892671 + 0.450710i 0.851171π0.851171\pi
888888 0 0
889889 2.83442e10 1.27122e8i 1.35303 0.00606826i
890890 0 0
891891 −2.51618e8 4.35815e8i −0.0119171 0.0206410i
892892 0 0
893893 1.61315e8 2.79405e8i 0.00758042 0.0131297i
894894 0 0
895895 −2.56186e10 −1.19447
896896 0 0
897897 1.92794e10 0.891909
898898 0 0
899899 5.04726e9 8.74211e9i 0.231684 0.401289i
900900 0 0
901901 1.83229e8 + 3.17362e8i 0.00834561 + 0.0144550i
902902 0 0
903903 1.74697e10 3.05742e10i 0.789545 1.38181i
904904 0 0
905905 1.47983e10 + 2.56313e10i 0.663652 + 1.14948i
906906 0 0
907907 −9.46293e9 + 1.63903e10i −0.421114 + 0.729391i −0.996049 0.0888082i 0.971694π-0.971694\pi
0.574934 + 0.818199i 0.305028π0.305028\pi
908908 0 0
909909 −5.36923e10 −2.37104
910910 0 0
911911 1.12249e10 0.491891 0.245945 0.969284i 0.420902π-0.420902\pi
0.245945 + 0.969284i 0.420902π0.420902\pi
912912 0 0
913913 4.32399e8 7.48937e8i 0.0188034 0.0325685i
914914 0 0
915915 −1.98155e10 3.43215e10i −0.855128 1.48113i
916916 0 0
917917 −4.24812e9 7.28234e9i −0.181930 0.311873i
918918 0 0
919919 −1.64634e10 2.85155e10i −0.699706 1.21193i −0.968568 0.248747i 0.919981π-0.919981\pi
0.268863 0.963179i 0.413352π-0.413352\pi
920920 0 0
921921 −5.70465e9 + 9.88074e9i −0.240614 + 0.416755i
922922 0 0
923923 −1.81157e10 −0.758313
924924 0 0
925925 −5.89040e9 −0.244708
926926 0 0
927927 −6.49227e9 + 1.12449e10i −0.267681 + 0.463637i
928928 0 0
929929 4.16396e9 + 7.21219e9i 0.170393 + 0.295129i 0.938557 0.345124i 0.112163π-0.112163\pi
−0.768164 + 0.640253i 0.778830π0.778830\pi
930930 0 0
931931 8.85021e8 + 1.50163e9i 0.0359443 + 0.0609873i
932932 0 0
933933 −3.26847e10 5.66116e10i −1.31753 2.28202i
934934 0 0
935935 7.33601e9 1.27064e10i 0.293508 0.508370i
936936 0 0
937937 −3.35209e10 −1.33115 −0.665576 0.746330i 0.731814π-0.731814\pi
−0.665576 + 0.746330i 0.731814π0.731814\pi
938938 0 0
939939 −7.27116e10 −2.86599
940940 0 0
941941 1.21961e10 2.11243e10i 0.477152 0.826452i −0.522505 0.852636i 0.675002π-0.675002\pi
0.999657 + 0.0261843i 0.00833566π0.00833566\pi
942942 0 0
943943 −1.49051e10 2.58164e10i −0.578821 1.00255i
944944 0 0
945945 −2.61758e10 4.48718e10i −1.00899 1.72967i
946946 0 0
947947 5.80141e8 + 1.00483e9i 0.0221977 + 0.0384476i 0.876911 0.480653i 0.159600π-0.159600\pi
−0.854713 + 0.519101i 0.826267π0.826267\pi
948948 0 0
949949 4.91129e9 8.50661e9i 0.186537 0.323091i
950950 0 0
951951 −1.15905e10 −0.436990
952952 0 0
953953 1.31548e10 0.492333 0.246167 0.969228i 0.420829π-0.420829\pi
0.246167 + 0.969228i 0.420829π0.420829\pi
954954 0 0
955955 2.33163e10 4.03850e10i 0.866259 1.50040i
956956 0 0
957957 −1.99294e10 3.45188e10i −0.735027 1.27310i
958958 0 0
959959 −4.04056e9 + 7.07151e9i −0.147937 + 0.258909i
960960 0 0
961961 1.01056e10 + 1.75033e10i 0.367306 + 0.636193i
962962 0 0
963963 1.62650e10 2.81718e10i 0.586897 1.01653i
964964 0 0
965965 3.45390e10 1.23727
966966 0 0
967967 1.67834e10 0.596880 0.298440 0.954428i 0.403534π-0.403534\pi
0.298440 + 0.954428i 0.403534π0.403534\pi
968968 0 0
969969 4.81667e8 8.34272e8i 0.0170065 0.0294561i
970970 0 0
971971 1.69538e10 + 2.93648e10i 0.594292 + 1.02934i 0.993646 + 0.112547i 0.0359009π0.0359009\pi
−0.399355 + 0.916796i 0.630766π0.630766\pi
972972 0 0
973973 3.95845e10 1.77534e8i 1.37762 0.00617855i
974974 0 0
975975 3.32077e10 + 5.75175e10i 1.14742 + 1.98739i
976976 0 0
977977 −1.65238e10 + 2.86200e10i −0.566863 + 0.981835i 0.430011 + 0.902824i 0.358510π0.358510\pi
−0.996874 + 0.0790114i 0.974824π0.974824\pi
978978 0 0
979979 2.53916e10 0.864871
980980 0 0
981981 8.59124e10 2.90546
982982 0 0
983983 1.60564e10 2.78105e10i 0.539151 0.933837i −0.459799 0.888023i 0.652079π-0.652079\pi
0.998950 0.0458138i 0.0145881π-0.0145881\pi
984984 0 0
985985 −4.53388e10 7.85291e10i −1.51162 2.61820i
986986 0 0
987987 −1.04883e10 + 4.70395e7i −0.347213 + 0.00155723i
988988 0 0
989989 1.66037e10 + 2.87584e10i 0.545779 + 0.945317i
990990 0 0
991991 4.68129e9 8.10824e9i 0.152795 0.264648i −0.779459 0.626453i 0.784506π-0.784506\pi
0.932254 + 0.361805i 0.117839π0.117839\pi
992992 0 0
993993 5.52949e10 1.79210
994994 0 0
995995 3.13842e9 0.101002
996996 0 0
997997 −2.62304e10 + 4.54323e10i −0.838246 + 1.45188i 0.0531148 + 0.998588i 0.483085π0.483085\pi
−0.891360 + 0.453295i 0.850248π0.850248\pi
998998 0 0
999999 1.37329e9 + 2.37861e9i 0.0435796 + 0.0754822i
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 112.8.i.d.81.1 10
4.3 odd 2 28.8.e.a.25.5 yes 10
7.2 even 3 inner 112.8.i.d.65.1 10
12.11 even 2 252.8.k.c.109.1 10
28.3 even 6 196.8.a.e.1.5 5
28.11 odd 6 196.8.a.d.1.1 5
28.19 even 6 196.8.e.f.177.1 10
28.23 odd 6 28.8.e.a.9.5 10
28.27 even 2 196.8.e.f.165.1 10
84.23 even 6 252.8.k.c.37.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.e.a.9.5 10 28.23 odd 6
28.8.e.a.25.5 yes 10 4.3 odd 2
112.8.i.d.65.1 10 7.2 even 3 inner
112.8.i.d.81.1 10 1.1 even 1 trivial
196.8.a.d.1.1 5 28.11 odd 6
196.8.a.e.1.5 5 28.3 even 6
196.8.e.f.165.1 10 28.27 even 2
196.8.e.f.177.1 10 28.19 even 6
252.8.k.c.37.1 10 84.23 even 6
252.8.k.c.109.1 10 12.11 even 2